Optimization Theory Based on Neutrosophic and Plithogenic Sets 9780128196700

Table of contents :
Cover......Page 1
Optimization Theory
Based on Neutrosophic
and Plithogenic Sets
......Page 3
Copyright......Page 4
Contributors......Page 5
Introduction......Page 8
Multi-criteria decision-making (MCDM)......Page 9
The best-worst method (BWM)......Page 10
Plithogenic set......Page 12
Neutrosophic set......Page 13
Proposed model......Page 14
Case 1: Warehouse location problem......Page 15
Case 2: Plant evaluation problem......Page 18
Discussion......Page 23
Conclusion and future research directions......Page 24
References......Page 25
Microservices......Page 27
Statefulness......Page 28
Containers......Page 29
Rules engine......Page 30
Neutrosophic theory......Page 31
Problem definition......Page 32
Case study problem definition: e-Commerce......Page 36
DSL......Page 37
Neutrosophic DSL......Page 39
Neutrosophic rules engine......Page 40
Business rule example......Page 41
ANTLR......Page 42
Decentralized rules engine......Page 44
Conclusion and future work......Page 47
References......Page 48
Introduction......Page 50
Proposed sampling plan......Page 52
Application of the proposed plan......Page 61
References......Page 64
Proposed model......Page 67
Pedagogical eLearning challenges......Page 68
Adaptive eLearning......Page 69
Intelligent eLearning systems......Page 70
Neutrosophic theory......Page 71
Microservices architecture......Page 72
Model components......Page 73
Scenario 1: New student......Page 74
Scenario 4: Suspended student......Page 75
Proposed intelligent microservices......Page 76
Intelligent adaptive online lecture LOs advisor specifications......Page 78
Intelligent cheat depressor......Page 80
Intelligent study plan advisor......Page 81
Intelligent LOs recommender......Page 82
Intelligent meeting manager for suspended students......Page 83
Evaluation......Page 84
Comments on evaluation results......Page 86
Conclusion......Page 87
References......Page 88
Introduction......Page 90
Neutrosophic set (NS)......Page 93
Single valued neutrosophic set (SVNS)......Page 94
Complex fuzzy set (CFS)......Page 95
Complex intuitionistic fuzzy set (CIFS)......Page 96
Complex neutrosophic set (CNS)......Page 97
Complex neutrosophic cosine similarity measure (CNCSM)......Page 99
Weighted complex neutrosophic cosine similarity measure (WCNCSM)......Page 100
Complex neutrosophic Dice similarity measure (CNDSM)......Page 101
Weighted complex neutrosophic Dice similarity measure (WCNDSM)......Page 102
Complex neutrosophic Jaccard similarity measure (CNJSM)......Page 104
Weighted complex neutrosophic Jaccard similarity measure (WCNJSM)......Page 105
Tangent function for CNS......Page 107
Decision-making steps......Page 108
Selection of educational stream for higher secondary education......Page 109
Comparison analysis......Page 110
References......Page 113
Introduction......Page 120
Basic concepts......Page 122
K-means clustering algorithm for SVNS......Page 124
Support vector machine classifier......Page 125
Sentiment analysis using neutrosophic sets......Page 126
Historical significance of the #MeToo movement......Page 127
Description of dataset......Page 128
Analysis of tweets using neutrosophy......Page 129
K-means clustering results......Page 130
Classification of data......Page 131
k-NN classification results......Page 132
SVM classifier results......Page 134
Neutrosophic sentiment analysis......Page 135
Results and further study......Page 136
References......Page 137
Introduction......Page 139
Design of the proposed plan......Page 141
Limitations and advantages......Page 143
Comparison......Page 148
Implementation in real-life datasets......Page 149
Acknowledgments......Page 150
References......Page 151
Introduction......Page 153
Review of literature......Page 155
Markov chain [22]......Page 156
Intuitionistic fuzzy Markov chain [19]......Page 157
Single valued neutrosophic set (SVNS) [44]......Page 158
Interval neutrosophic Markov chain and long-run behavior of the neutrosophic Markov chain using interval neutrosophi .........Page 159
Experimental analysis......Page 160
Comparative analysis......Page 165
Comparative analysis with the existing methods......Page 167
References......Page 168
Recommender systems......Page 171
Neutrosophic sets and theory......Page 172
Stage 1: Synthesization......Page 173
Multicriteria recommender systems (MC-RS)......Page 174
Recommender systems in eLearning......Page 175
Proposed system......Page 176
Popularity-based recommender systems......Page 177
Item-based collaborative filtering......Page 178
Learning objects......Page 179
General learning style......Page 183
ATLAS learning style......Page 184
Phase 1: LOs finding, gathering, and analyzing......Page 185
Phase 2: Personalized supervised generated LOs......Page 187
Neutrosophic theory in the proposed recommender system......Page 189
Students manager service......Page 190
Students usage data manager......Page 194
Crawler module......Page 197
Removing stop words module......Page 199
Intelligent LOs recommender challenges......Page 200
Evaluation results......Page 203
Comments on results and optimized solution......Page 207
Information retrieval evaluation......Page 209
Intelligent LOs classifier evaluation......Page 211
References......Page 214
Preliminaries......Page 216
New types of continuity in FNTSs......Page 219
Interrelations......Page 227
Conflict of interests......Page 233
References......Page 234
Introduction......Page 235
Preliminaries......Page 236
Zimmermann´s method......Page 239
Werners method......Page 241
Guu and Wu´s method......Page 242
Skandari and Ghaznavi´s method......Page 243
Klir and Yuan´s method......Page 244
Verdegay´s method......Page 246
Chanas method......Page 247
Proposed ranking method......Page 248
Comparing with other methods......Page 251
NLPs with fuzzy relation......Page 252
Numerical example......Page 254
Empirical application......Page 255
Conclusion......Page 257
References......Page 258
Further reading......Page 259
Introduction......Page 260
Some concepts related to trapezoidal fuzzy numbers......Page 262
Some concepts related to neutrosophic sets and neutrosophic numbers......Page 263
Dice similarity measure between two vectors......Page 266
Dice similarity measure of trapezoidal neutrosophic fuzzy numbers......Page 267
Jaccard similarity measure of trapezoidal neutrosophic fuzzy numbers......Page 271
Multicriteria decision-making method......Page 275
Illustrative example......Page 277
Ranking method of alternatives based on similarity measure methods......Page 281
Conclusion......Page 283
References......Page 284
Further reading......Page 286
Introduction......Page 287
Neutrosophic set......Page 290
Bipolar neutrosophic set......Page 291
Some refinements on neutrosophic sets......Page 293
Extended neutrosophic optimization and bipolar neutrosophic optimization technique......Page 296
Computational algorithm......Page 298
Application of bipolar neutrosophic in riser design......Page 307
Conclusion......Page 308
References......Page 311
Introduction......Page 313
Neutrosophic set......Page 315
Single valued neutrosophic sets......Page 316
Existing similarity measures......Page 317
A new similarity measure of SVNSs......Page 319
A comparison approach with existing similarity measures......Page 322
Pattern recognition......Page 323
Cluster analysis......Page 326
Discussions and comparison......Page 331
Conclusions......Page 334
References......Page 338
Introduction......Page 340
Literature review......Page 342
Research contribution......Page 345
Description of CLSC network......Page 346
Multiple objective function......Page 351
Constraints related to the capacity of different echelons in the CLSC network......Page 354
Constraints related to production requirement......Page 355
Constraints related to the testing capacity at testing facility centers......Page 356
Proposed CLSC model formulation under uncertainty......Page 357
Treating fuzzy parameters and constraints......Page 360
Neutrosophic fuzzy programming approach......Page 366
Modified neutrosophic fuzzy programming with intuitionistic fuzzy preference relations......Page 371
Computational study......Page 375
Results and discussions......Page 378
Sensitivity analyses......Page 386
Sensitivity analyses of objective functions......Page 388
Sensitivity analyses of intuitionistic fuzzy linguistic preference relations......Page 395
Acknowledgments......Page 397
References......Page 398
Introduction......Page 401
Neutrosophic image......Page 404
Entropy of neutrosophic subsets......Page 405
Concept of optimization......Page 406
Particle swarm optimization......Page 407
OptNS-based CAD medical image processing applications......Page 408
CAD using neutrosophic set without optimizing NS......Page 409
CAD using neutrosophic set with optimization......Page 411
Discussion and future perceptions in OptNS-based CAD systems......Page 412
References......Page 414
Further reading......Page 417
C......Page 418
D......Page 419
F......Page 420
I......Page 421
M......Page 422
N......Page 423
P......Page 424
S......Page 425
T......Page 426
W......Page 427
Back Cover......Page 428

Citation preview

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Edited by

Florentin Smarandache Mohamed Abdel-Basset

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/ permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-12-819670-0 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

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Contributors

Mohamed Abdel-Basset Department of Operations Research, Faculty of Computers and Informatics, Zagazig University, Sharqiyah, Egypt Ahmad Yusuf Adhami Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, India Firoz Ahmad Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, India Liaquat Ahmad Department of Statistics and Computer Science, University of Veterinary and Animal Sciences, Lahore, Pakistan Wadei AL-Omeri Department of Mathematics, Al-Balqa Applied University, Salt, Jordan Amira S. Ashour Department of Electronics and Electrical Communications Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt Muhammad Aslam Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Muhammad Azam Department of Statistics and Computer Science, University of Veterinary and Animal Sciences, Lahore, Pakistan Mayank Bisht School of Computer Science and Engineering, VIT, Vellore, India Said Broumi Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, Sidi Othman, Casablanca, Morocco Majid Darehmiraki Department of Mathematics, Behbahan Khatam Alanbia University of Technology, Behbahan, Iran Haitham A. El-Ghareeb Information Systems Department, Faculty of Computers and Information Sciences, Mansoura University, Mansoura, Egypt Gul Freen International Islamic University, Islamabad, Pakistan

xii

Contributors

Abduallah Gamal Department of Operations Research, Faculty of Computers and Informatics, Zagazig University, Sharqiyah, Egypt Bibhas C. Giri Department of Mathematics, Jadavpur University, Kolkata, India Yanhui Guo Department of Computer Science, University of Illinois at Springfield, Springfield, IL, United States Chiranjibe Jana Department of Applied Mathematics With Oceanology and Computer Programming, Vidyasagar University, Midnapore, India Ilanthenral Kandasamy School of Computer Science and Engineering, VIT, Vellore, India Faruk Karaaslan Department of Mathematics, Faculty and Sciences, C¸ankırı Karatekin University, C¸ankırı, Turkey Mesut Karabacak Atat€ urk University, Faculty of Science, Department of Mathematics, Erzurum, Turkey J. Kavikumar Department of Mathematics and Statistics, Faculty of Applied Science and Technology, Universiti Tun Hussein Onn, Batu Pahat, Malaysia Sumbal Khalil International Islamic University, Islamabad, Pakistan Sajida Kousar International Islamic University, Islamabad, Pakistan M. Lathamaheswari Department of Mathematics, Hindustan Institute of Technology & Science, Chennai, India Niharika Mathur School of Computer Science and Engineering, VIT, Vellore, India Rehab Mohamed Department of Operations Research, Faculty of Computers and Informatics, Zagazig University, Sharqiyah, Egypt Fatimah M. Mohammed College of Education for Pure Sciences, Tikrit University, Tikrit, Iraq Kalyan Mondal Department of Mathematics, Jadavpur University, Kolkata, India D. Nagarajan Department of Mathematics, Hindustan Institute of Technology & Science, Chennai, India Surapati Pramanik Department of Mathematics, Nandalal Ghosh B.T. College, Kolkata, India

Contributors

xiii

Rıdvan Şahin G€ um€ u¸shane University, Faculty of Natural Sciences and Engineering, Department of Mathematical Engineering, G€ um€ u¸shane, Turkey Florentin Smarandache Department of Mathematics, University of New Mexico, Gallup, NM, United States W.B. Vasantha School of Computer Science and Engineering, VIT, Vellore, India Abd El-Nasser H. Zaied Department of Operations Research, Faculty of Computers and Informatics, Zagazig University, Sharqiyah, Egypt

Solving the supply chain problem using the best-worst method based on a novel Plithogenic model

1

Mohamed Abdel-Basseta, Rehab Mohameda, Abd El-Nasser H. Zaieda, Abduallah Gamala, Florentin Smarandacheb a Department of Operations Research, Faculty of Computers and Informatics, Zagazig University, Sharqiyah, Egypt, bDepartment of Mathematics, University of New Mexico, Gallup, NM, United States

1.1

Introduction

Decision-making is a critical process of selecting the optimal decision among a set of alternatives based on the decision-maker’s experience and preferences. Multi-criteria decision-making (MCDM) is one of most important operations research topics that deals with the decision-making process. Most MCDM models consider evaluating criteria resulting from expert opinions, which may be individual evaluations based on the expert’s preferences and experience [1]. One of the most functional MCDM methods is the best-worst method (BWM, proposed by Rezaei in 2015), which considers pairwise comparison among alternatives. In addition, the Analytic Hierarchy Process (AHP) is a pairwise comparison technique, but has some limitations that are overcome in the BWM, such as that the BWM requires fewer comparisons than AHP, and the comparison is easier and has less complexity [2]. Multi-optimality results are a necessary feature of the BWM, which means solving the problem for diverse criteria weights. This feature offers the decision-maker a large scale of information and more flexibility in decisionmaking [3]. On the other hand, in some problems DMs need a unique optimal solution, and this can be defined based on the problem’s nature. The BWM is based on pairwise comparison between the best and worst criteria with the rest of the criteria. In this method, DMs have to determine the best and worst criteria by the pairwise comparison of two vectors. The first vector is best-to-others and the second is worst-to-others. The BWM shows reliable results in many fields such as green practices and innovation [4], evaluation of the technologies for ballast water treatment [5], logistics performance measurements [6], research and development performance measurement [7], and supplier selection [8]. Plithogeny is the genesis, construction, development, and progression of new entities from syntheses of contradictory or non-contradictory multiple old entities [9]. It was introduced by Smarandache in 2017 as a generalization of neutrosophy.

Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00001-9 © 2020 Elsevier Inc. All rights reserved.

2

Optimization Theory Based on Neutrosophic and Plithogenic Sets

A plithogenic set (as a generality of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), is a set whose elements are categorized by attribute values which have corresponding contradiction degree values c(vj,vD) between each attribute value vj and the dominant (most important) attribute value vD. The contradiction degree function acquires better accuracy for plithogenic aggregation operators (intersection, union, complement, inclusion, equality) [9]. Plithogenic sets, logic, probability, and statistics, introduced by Smarandache in 2017, are derived from plithogeny, and they are generalizations of neutrosophic sets, logic, probability, and statistics, respectively. To combine several decision-makers’ opinions in order to evaluate a particular problem using the BWM, we propose a novel plithogenic model based on the BWM. Among the plithogenic set aggregation operators, we use the intersection operator to aggregate more than one DM’s opinions. Then the result is used to solve the BWM to find the weight of each criterion. Nowadays, supply chain problems are one of the most pioneer and interesting topics in decision-making studies. In this paper, we consider an important problem: inventory location. This problem, similar to many of decision-making problems, has a set of criteria that restrict the decision-maker in terms of the way of finding the optimal decision. The BWM helps the DM to find the weight of these criteria, taking into consideration the best criterion and the worst. There are other supply chain problems that are solved using the BWM, such as assessment of the exterior forces affecting the sustainability of supply chain [10], supplier segmentation [11], and third party logistics [12]. This chapter is arranged as follows: Section 1.2 is a literature review of multicriteria decision-making concepts, BWM definitions and steps, explanation of the plithogenic set definition, and the review of neutrosophic sets. Section 1.3 shows the proposed plithogenic model based on the BWM, and Section 1.4 provides a real-world case study to examine the proposed model. Section 1.5 discusses the model results, and Section 1.6 presents the conclusion and future research.

1.2

Literature review

1.2.1 Multi-criteria decision-making (MCDM) Multi-criteria decision-making (MCDM) is a common offshoot of decision-making science. There are a huge number of MCDM techniques which assist individuals in constructing and solving decision problems that concern multiple criteria. Each technique has its own physiognomies and no single one is the best [13, 14]. The proper MCDM technique should be designated consistent with the problem structure. It is recognized that without integrating preference information, no unique optimal solution to an MCDM problem can be acquired. The steps of any MCDM method can be summarized as shown in Fig. 1.1. Regardless of the chosen MCDM technique for the problem we are dealing with, the significant phase is to achieve the decision factor weights. Either the subjective or objective method can regulate the criteria weights

Solving the supply chain problem using the best-worst method based on a novel Plithogenic model

3

Fig. 1.1 MCDM process.

in MCDM techniques [15]. One of the most popular subjective methods is the analytic hierarchy process (AHP). This process first splits the desired problem into a hierarchy of sub-problems, then the decision-maker (DM) applies pairwise comparisons among elements. However, Løken [14] observed that various critiques relate to AHP, such as inefficiency when the number of decision criteria is large. Rezaei et al. [11] discovered that the origin of this problem is in the unstructured comparisons, and proposed a new subjective technique—the BWM. More details about the BWM will be given in the next section.

1.2.2 The best-worst method (BWM) The BWM technique uses pairwise comparisons of the selection criteria on the basis of a DM’s opinion. However, to handle the inconsistency of AHP comparison, the DM should identify the most preferred criterion and the least preferred criterion, and apply the pairwise comparison between these two criteria and the other criteria [11]. That is why the BWM value requires fewer comparisons than the AHP. In addition, the BWM consists of less complex comparisons, as it exploits only whole numbers. Finally, the BWM is differentiated because redundant comparisons are eradicated. The model of this method is used to find the weight of each selection criterion [16]. Some of the latest applications of the BWM include: engineering sustainability [17]; financial performance evaluation [18]; sustainable supplier selection and order allocation [19]; evaluating the community sustainability of supply chains [20]; and location selection for wind farms [21]. The steps of the method are illustrated in Fig. 1.2 and explained as follows: Step 1. The decision-maker identifies the set of decision measures (criteria) according to the problem nature N ¼ {c1, c2,…, cn}. Step 2. Determine the best (most significant) and the worst (least significant) criteria.

4

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Fig. 1.2 BWM steps.

Step 3. Establish best-to-other vector AB ¼ (aB1, aB2, …, aBn), which is the decision-maker’s preference of the best criterion in comparison with other criteria, using a (1–9) scale, where aBn designates the preference of the best criterion over criterion n. It is explicit that aBB ¼ 1. Step 4. Establish others-to-worst vector Aw ¼ (aw1, aw2, …, awn), which is the decision-maker’s preference of all criteria in comparison with the worst one, using a (1–9) scale, where awn designates the preference of criterion n over the worst criterion. It is explicit also that aww ¼ 1. Step 5. Use the BWM model [2, 11] to find the optimal criteria weights (W1∗, W2∗, …, Wn∗ ).




 


wj

wB


min max
 aBj
,
 ajW

wj wW s.t. X

wj ¼ 1

j

wj  0, for all j

(1.1)

The equivalent model is: Min ε
s.t.

wB

wj  aBj
 ε, for all j


wj

ww  ajW
 ε, for all j X wj ¼ 1

(1.2)

j

wj  0, for all j

Solving the supply chain problem using the best-worst method based on a novel Plithogenic model

5

1.2.2.1 Consistency ratio A comparison is completely consistent when aBj  ajW ¼ aBW, for all j. The consistency ratio is calculated as follows: Consistency ratio ¼

ε* Consistency index

(1.3)

The consistency ratio is 2[0, 1]; if it is close to 0, that means more consistency, while being close to 1 means inconsistent.

1.2.3 Plithogenic set Plithogeny is the initiation, construction, development, and evolution of new entities from syntheses of contradictory (dissimilar) or non-contradictory multiple old entities [22]. A plithogenic set (P, A, V, d, c) is a set that encloses several elements characterized by a number of attributes A ¼ {α1, α2,…, αm}, m  1, and each attribute has values V ¼ {v1, v2, …, vn}, for n  1. Each attribute’s value v has an appurtenance degree d(x,v) of the element x, with respect to some given criteria [23]. The contradiction (dissimilarity) degree function is defined among each attribute value and most important (dominant) attribute value (determined by the expert), in order to achieve more accurate results of the plithogenic aggregation operators (intersection, union, complement, inclusion, and equality). Let A ¼ {α1, α2, …, αm}, m  1 be a set of uni-dimensional attributes, and let α Œ A be an attribute whose attribute value spectrum is the set S, where S can be a finite discrete set, S ¼ {s1, s2, …, sl}, 1  l < ∞, or an infinitely countable set, S ¼ {s1, s2,…, s∞}, or an infinitely uncountable (continuum) set S ¼ ]a,b[, a < b, where ]…[ is any open, semi-open, or closed interval from the set of real numbers or from other general set [24]. Let V be a subset of S, where V is the range of all attribute α’s values, V ¼ {v1, v2,…, vn}. The degree of appurtenance d(x,v) may be a fuzzy, or intuitionistic fuzzy, or neutrosophic degree of appurtenance to the plithogenic set. Therefore, the appurtenance degree function of attribute value is defined as follows: 8x 2 P,d : P  V ! P ð½0, 1z Þ

(1.4)

So d(x, v) is a subset of [0, 1]z, and P ð½0, 1z Þ is the power set of [0, 1]z, where z ¼ 1, 2, 3, for fuzzy, intuitionistic fuzzy, and neutrosophic degrees of appurtenance, respectively. Let c: V  V ! [0, 1] be the attribute value contradiction degree function (it can be fuzzy, or intuitionistic fuzzy, or neutrosophic as well) between attribute values v1 and v2, represented by c(v1, v2), and satisfying the following axioms: c(v1, v1) ¼ 0, no contradiction degree between the same attribute values c(v1, v2) ¼ c(v2, v1).

6

Optimization Theory Based on Neutrosophic and Plithogenic Sets

1.2.4 Neutrosophic set Neutrosophy is a branch of philosophy identified by Florentin Smarandache in 1980. It is a generalization of dialectics and paradoxism in the literary and scientific movement. Definition 1. [25] Let the element x(T,I,F) belong to the set U in the following way: the element has truth-membership function TA, indeterminate-membership function IA, and falsity-membership function FA; these are called neutrosophic components. They are subsets of ]0, 1+[. There is no constraint on the sum of TA(x), IA(x) and FA(x), so 0  sup TA(x) + sup IA(x) + sup FA(x)  3+. Definition 2. [26] Let X be a universal set of objects, with a generic element denoted by x. A neutrosophic set N  X is considered a set such that each element x from N is characterized by TN(x), which is the truth-membership function, IN(x), which is the indeterminacy-membership function, and FN(x), which is the falsity-membership function. TN(x), IN(x), and FN(x) are subsets of [0, 0+], so the three neutrosophic components are TN(x) 2 [0, 1+], IN(x) 2 [0, 1+], and FN(x) 2 [0, 1+]. IN(x) represents the uncertainty, undefined, unknown, or error value. The sum of the three components is 0  TN(x) + IN(x) + FN(x)  3+. Definition 3. [27] Let a˜¼ h(a1, a2, a3); α, θ, βi be a single valued triangular neutrosophic set, which contains truth membership Ta(x), indeterminate membership Ia(x), and falsity membership function Fa(x) as follows: 8   x  a1 > > > α < a a  a if a1  x  a2 2 1 T a ð xÞ ¼ α if x ¼ a2 > a > > : o otherwise

(1.5)

8 ða2  x + θa ðx  a1 ÞÞ > > > if a1  x  a2 > > ð a2  a1 Þ < I a ð xÞ ¼ θa if x ¼ a2 > > > ðx  a2 + θa ða3  xÞÞ > > otherwise : ð a3  a2 Þ

(1.6)

8 ð a2  x + β a ð x  a1 Þ Þ > > if a1  x  a2 > > ð a2  a1 Þ > > > > > βa if x ¼ a2 < Fa ðxÞ ¼ ðx  a2 + βa ða3  xÞÞ if a2 < x  a3 > > > ð a3  a2 Þ > > > > 1 otherwise > > :

(1.7)

where αa, θa, βa Є [0,1] and they represent the extreme truth membership degree, least indeterminacy membership degree, and minimum falsity membership degree, respectively.

Solving the supply chain problem using the best-worst method based on a novel Plithogenic model

1.3

7

Proposed model

In this study, we propose a combination of plithogenic aggregation operations and the best-worst method to solve MCDM problems. The steps of this suggested technique will be described in detail in this section, and are summarized in Fig. 1.3. Step 1. Define the problem and criteria, and organize a group of decision-makers who will evaluate the problem. Let the problem p have a set of criteria N ¼ {C1, C2, …, Cn}, taking into consideration the group of experts’ (DM1, DM2, …, DMn) opinions. Step 2. Conduct the linguistic scale that describes the evaluation of each criterion by the DMs. In this model, the scale is signified as a triangular neutrosophic scale. Step 3. Depict the evaluation matrix of each criterion based on the decision-makers’ experience using the proposed linguistic scale in the previous step. Step 4. The DMs determine the dominant (most important) criterion. Then, define the contradiction degree c of each criterion with respect to the dominant criterion. Let c: V V ! [0, 1] if you use the fuzzy contradiction degree function and denote it by cF: intuitionistic contradiction degree function cIF ¼ V V ! [0, 1]2. Lastly, the neutrosophic contradiction degree function should be cN ¼ V  V ! [0, 1]3. This step contribute to more accurate results. Step 5. Aggregate all DMs’ opinions using plithogenic aggregation operations.

The plithogenic neutrosophic set intersection is defined as follows: ððai1,  ai2 , ai3 Þ, 1  i  nÞ ^ p ððbi1 , bi2 , bi3 Þ, 1  i  nÞ  1 1 ¼ ai1 ^F bi1 , ðai2 ^F bi2 Þ + ðai2 _F bi2 Þ, ai2 _F bi3 , 1  i  n: 2 2 where ^F and _F are the fuzzy t-norm and t-conorm, respectively.

Fig. 1.3 Proposed model phases.

(1.8)

8

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Step 6. Transform the neutrosophic number into a crisp number using the following equation:

1 Sð aÞ ¼ ð a1 + b1 + c 1 Þ  ð 2 + α  θ  β Þ 8

(1.9)

Step 7. Start applying the BWM. Define the best criterion and worst criterion based on the crisp values of the DMs’ aggregated opinions. Step 8. Establish the best-to-others vector based on the importance scale. Step 9. Establish the others-to-worst vector, again based on the importance scale. Step 10. Apply the BWM model (3) to find the optimal weight of each criterion (W1∗, W2∗,…, Wn∗).

1.4

Case studies

Supply chain problems are one of the leading issues in multi-criteria decision-making. In this section, the proposed model will be examined in two real-world case studies regarding the supply chain. One of them is about the problem of warehouse location, and the second concerns the plant evaluation problem.

1.4.1 Case 1: Warehouse location problem One of the most significant supply chain problems is inventory location. The main challenge in the inventory location problem is to decide whether to establish or not a warehouse in a specific geographical region to be able to serve customers within a planned service level and keep on minimizing logistic costs. This section presents a real-world company example of a warehouse location problem. Greenfinch Ltd. specializes in the good industry. It has been operating satisfactorily in recent years. Greenfinch has a duplicate warehouse problem. For example, there are two central warehouses in Germany (which is an essential market) which are shown to be unproductive and insufficient. That is why the company’s management has been persuaded to identigy a new central warehouse location. In order to minimize the administrative exertion and to increase profitability, one central warehouse will be more efficient to cover the demand of the German market. Knowing that such a strategic problem might be affected by operational decisions, Greenfinch Ltd. determined the significant criteria—economic factors (C1), service level and effectiveness (C2), capacity and size of facility (C3), competition (C4), population (C5), and environmental risk (C6)—to find the best location for the central warehouse, as shown in Fig. 1.4. These criteria are measured by four decision-makers who have significant experience in supply chain and inventory management. The steps of the proposed model to solve this problem are as follows: l

Step 1: Determine the goal and criteria, and organize a committee of decision-makers. Four decision-makers (DM1, DM2, DM3, DM4) have made their opinions about the central warehouse location, based on six criteria (C1, C2,…, C6), as mentioned before.

Solving the supply chain problem using the best-worst method based on a novel Plithogenic model

9

Fig. 1.4 Criteria for identifying a new central warehouse location.

l

Step 2: Evaluate each criterion through all DMs using a linguistic scale as shown in Table 1.1.

l

Step 3: Acquire the DMs’ opinions regarding each criterion as set out in Table 1.2.

According to each DM’s experience, they gave their opinion about the importance of each criterion based on their preference. Table 1.1 Linguistic scale. Linguistic variable

Score

Triangular neutrosophic scale

Very weakly important (VWI) Weakly important (WI) Fairly weakly important (FWI) Equal important (EI) Strong important (SI) Very strongly important (VSI) Absolutely important (AI)

0 1 2 3 4 5 6

((0.10, 0.30,0.35), 0.1,0.2,0.15) ((0.15.25,0.10), 0.6,0.2,0.3) ((0.40,0.35,0.50), 0.6,0.1,0.2) (0.65,0.60,0.70),0.8,0.1,0.1) ((0.70,0.65,0.80),0.9,0.2,0.1) ((0.90,0.85,0.90),0.7,0.2,0.2) ((0.95,0.90,0.95),0.9,0.10,0.10)

Table 1.2 Evaluation matrix in case 1.

DM1 DM2 DM3 DM4

C1

C2

C3

C4

C5

C6

SI AI VSI SI

VSI SI AI VSI

EI EI FWI EI

WI FWI EI EI

FWI WI VWI WI

SI EI WI FWI

10 l

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Step 4: To help the plithogenic aggregation operators to obtain more accurate result, the contradiction degree function is applied. If the contradiction degree is not used, the aggregation will be less accurate, but it still works.

It was unanimous that C1 (Economic factor) was the most important (Dominant) criterion for the company to take into consideration, so the contradiction degrees were obtained as shown in Table 1.3. l

Step 5: Apply the aggregation intersection operation to aggregate the opinions of all DMs about the six criteria. For example, to aggregate the four DMs’ opinions about criterion 1 (C1):

ai1 ^ p bi1 ¼ ½1  cðvD , v1 Þ:tnorm ðvD , v1 Þ + cðvD , v1 Þ:tconorm ðvD , v1 Þ

(1.10)

ai1 _ p bi1 ¼ ½1  cðvD , v1 Þ:tconorm ðvD , v1 Þ + cðvD , v1 Þ:tnorm ðvD , v1 Þ

(1.11)

where tnorm(a, b)¼ a ^ F b ¼ ab, and tconorm(a, b) ¼ a _ F b ¼ a + b  ab. DM1 ^ p DM2 (0.7, 0.65, 0.8) ^ p (0.95, 0.9, 0.95) ¼ (0.7 ^ p 0.95, ½ (0.65 ^F 0.9) + ½ (0.65 _F 0.9), 0.8 _p0.9) ¼ (1-0) (0.7 0.95) + 0, ½ (0.65 0.9) + ½ (0.65+ 0.9-0.65 0.9), (1-0) (0.8+ 0.95-0.8 0.95) ¼ (0.665, 0.775, 0.99)

DM3 ^ p DM4 (0.9, 0.85, 0.9) ^ p (0.7, 0.65, 0.8) ¼ (0.9 ^ p 0.7, ½ (0.85 ^F0.65) + ½ (0.85 _F 0.65), 0.9 _p 0.8) ¼ (1-0) (0.9 0.7) + 0, ½ (0.8 0.65) + ½ (0.85+ 0.65-0.850.65), (1-0) (0.9 + 0.8 - 0.9 0.8) +0 ¼ (0.63, 0.73, 0.98)

DM1 ^ p DM2 ^ p DM3 ^ p DM4 ¼ ((0.42, 0.76, 1.00), 0.05, 0.15, 0.32). l

Step 6: Transform the triangular neutrosophic numbers to crisp values:

1 S ðC1Þ ¼ ð0:42 + 0:76 + 1:00Þ  ð2 + 0:05  0:15  0:32Þ ¼ 0:43055: 8 l

Step 7: Determine the best criterion and worst criterion based on the results from plithogenic aggregation of all decision-makers in Table 1.6. As we can see, the best criterion is C2 and the worst criterion is C5. Table 1.3 Contradiction degree in case 1.

Contradiction degree

C1

C2

C3

C4

C5

C6

0

1/6

1/3

1/2

2/3

5/6

Solving the supply chain problem using the best-worst method based on a novel Plithogenic model

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Table 1.4 Importance rating scale. Numerical scale

Verbal scale

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Equally important Weakly more important Moderately more important Moderately plus more important Strongly more important Strongly plus more important Very strongly plus more important Very strongly more important Extremely more important

Table 1.5 DMs’ evaluation of each criterion represented as triangular NN in case 1. Criteria

DM1

DM2

DM3

DM4

C1

(0.70,0.65,0.80), 0.9,0.2,0.1)) ((0.90,0.85,0.90), 0.7,0.2,0.2) (0.65,0.60,0.70), 0.8,0.1,0.1) ((0.15,0.25,0.10), 0.6,0.2,0.3) ((0.40,0.35,0.50), 0.6,0.1,0.2) ((0.70,0.65,0.80), 0.9,0.2,0.1)

((0.95,0.90,0.95), 0.9,0.10,0.10) ((0.70,0.65,0.80), 0.9,0.2,0.1) (0.65,0.60,0.70), 0.8,0.1,0.1) ((0.40,0.35,0.50), 0.6,0.1,0.2) ((0.15,0.25,0.10), 0.6,0.2,0.3) (0.65,0.60,0.70), 0.8,0.1,0.1)

((0.90,0.85,0.90), 0.7,0.2,0.2) ((0.95,0.90,0.95), 0.9,0.10,0.10) ((0.40,0.35,0.50), 0.6,0.1,0.2) (0.65,0.60,0.70), 0.8,0.1,0.1) ((0.10, 0.30,0.35), 0.1,0.2,0.15) ((0.15,0.25,0.10), 0.6,0.2,0.3)

((0.70,0.65,0.80), 0.9,0.2,0.1) ((0.90,0.85,0.90), 0.7,0.2,0.2) (0.65,0.60,0.70), 0.8,0.1,0.1) (0.65,0.60,0.70), 0.8,0.1,0.1) ((0.15,0.25,0.10), 0.6,0.2,0.3) ((0.40,0.35,0.50), 0.6,0.1,0.2)

C2 C3 C4 C5 C6

l

Step 8: Establish the best-to-other vector (Table 1.7) based on the importance rating scale shown in Table 1.4 (Tables 1.5–1.7).

l

Step 9: Establish the others-to-worst vector (Table 1.8).

l

Step 10: Find the optimal weights. Using the BWM model as shown in Table 1.9, we found that W1∗¼ 0.2591, W2∗¼ 0.4126, W3∗¼ 0.0864, W4∗¼ 0.1295, W5∗¼ 0.03838, W6∗¼ 0.07402, and ε ¼ 0.1056. The results shows that the weight of C2 (the best criterion) is the highest value where C5 (the worst criterion) is the lowest, as illustrated in Fig. 1.5. Based on the CI table (Table 1.10) in Ref. [11], the consistency ratio is CR ¼ 0:1056 4:47 ¼ 0:0236.

1.4.2 Case 2: Plant evaluation problem One of the world’s leading motor companies is Toyota. In this case study, we consider a plant of Toyota located in Portugal. Its main role is to perform the soldering, painting, and final assembly of a particular automobile model. Firstly, the necessary parts

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 1.6 Aggregation operator result in case 1. Criteria

DM1 ^p DM2

DM3 ^p DM4

DM1 ^p DM2 ^p DM3 ^p DM4

Crisp value

C1

((0.665,0.775,0.99), 0.81,0.1,0.1) ((0.687,0.75,0.937), 0.06,0.2,0.24) ((0.574,0.6,0.77), 0.75,0.1,0.13) ((0.275,0.3,0.3), 0.6,0.15,0.25) ((0.35,0.3,0.216), 0.68,0.15,0.19) ((0.82,0.625,0.624), 0.94,0.15,0.04)

((0.63,0.73,0.98), 0.06,0.2,0.24) ((0.84,0.88,0.97), 0.69,0.15,0.24) ((0.17,0.48,0.45), 0.63,0.1,0.19) ((0.574,0.6,0.77), 0.75,0.1,0.13) ((0.16,0.25,0.16), 0.45,0.2,0.165) ((0.275,0.3,0.3), 0.6,0.15,0.25)

((0.42,0.76,1.00), 0.05,0.15,0.32) ((0.64,0.82,0.98), 0.15,0.18,0.24) ((0.28,0.54,0.70), 0.62,0.1,0.21) ((0.42,0.45,0.54), 0.68,0.13,0.19) ((0.29,0.28,0.14), 0.65,0.18,0.13) ((0.76,0.47,0.28), 0.91,0.15,0.06)

0.43055

C2 C3 C4 C5 C6

0.52765 0.4389 0.41595 0.20768 0.50963

Table 1.7 Best-to-others vector in case 1. Criteria

C1

C2

C3

C4

C5

C6

The best criterion C 2

0.2

0.1

0.6

0.4

0.8

0.7

Table 1.8 Others-to-worst vector in case 1. Criteria

The worst criteria C 5

C1 C2 C3 C4 C5 C6

0.7 0.8 0.5 0.2 0.1 0.3

Table 1.9 BWM results in case 1. Economic factor Weights (C1)

Service level and effectiveness (C2)

0.25908156 0.41261138

Capacity and size of warehouse (C3)

Competition (C4)

Population (C5)

Environmental risk (C6)

0.08636052

0.12954078

0.03838245

0.0740233

Solving the supply chain problem using the best-worst method based on a novel Plithogenic model

13

Weights 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 C1

C2

C3

C4

C5

C6

Fig. 1.5 BWM results in case 1. Table 1.10 Consistency index. aBW

1

2

3

4

5

6

7

8

9

Consistency index

0.00

0.44

1.00

1.63

2.30

3.00

3.73

4.47

5.23

of the vehicles are forwarded to the soldering sector based on production planning. The soldered body of the vehicle then heads to the painting phase. The main problem of this model is to optimize the painting sector, which is the goal of the operation manager. A number of meetings were convened with the operation manager and the painting sector crew to collate information about the criteria that should be considered in this phase. As mentioned above, the goal is to optimize the painting sector of this plant. Therefore, our model will assist in evaluating each criterion of this problem. After the meetings between the operation manager and the painting sector crew, four criteria were considered: quality indicator C1, the level of energy consumption C2, the amount of painting used C3, and lastly the quality of painted vehicle C4. Firstly, the quality indicator is defined by the average number of imperfections per vehicle. Secondly, the energy consumption includes the amount of electricity and gas needed in the painting sector. Thirdly, the amount of paint consumption reflects the cost of raw material needed in this phase. The last criterion measures the quality of painted vehicles per day. After defining the criteria that need to be considered in this case, three of the decision-makers were asked to evaluate the relative importance of the determined criteria. These DMs have significant experience in the motor production process. l

The evaluation matrix (Table 1.11) of each criterion based on DMs’ opinions was based on the linguistic scale shown in Table 1.1.

l

Criterion 1 is obtained to be the dominant one, thus the contradiction degree (Table 1.12) will be as follows

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 1.11 Evaluation matrix.

DM1 DM2 DM3

C1

C2

C3

C4

AI VSI AI

SI EI EI

FWI EI WI

WI VWI FWI

Table 1.12 Contradiction degree.

Contradiction degree

C1

C2

C3

C4

0

1/4

1/2

3/4

l

The result of the aggregation operation is shown in Tables 1.13 and 1.14.

l

As we can see from the aggregation results (Table 1.14), the best criterion is C1 and the worst is C3. According to this result, the best-to-others vector (Table 1.15) and others-to-worst vector (Table 1.16) should be established based on the importance scaling in Table 1.4.

l

The optimal weights is W∗.

Table 1.13 DMs’ evaluation of each criterion represented as triangular NN in case 2. Criteria

DM1

DM2

DM3

C1

((0.95,0.90,0.95), 0.9,0.10,0.10) ((0.70,0.65,0.80), 0.9,0.2,0.1) ((0.40,0.35,0.50), 0.6,0.1,0.2) ((0.15,0.25,0.10), 0.6,0.2,0.3)

((0.90,0.85,0.90), 0.7,0.2,0.2) (0.65,0.60,0.70), 0.8,0.1,0.1) (0.65,0.60,0.70), 0.8,0.1,0.1) ((0.10, 0.30,0.35), 0.1,0.2,0.15)

((0.95,0.90,0.95), 0.9,0.10,0.10) (0.65,0.60,0.70), 0.8,0.1,0.1) ((0.15,0.25,0.10), 0.6,0.2,0.3) ((0.40,0.35,0.50), 0.6,0.1,0.2)

C2 C3 C4

Table 1.14 Aggregation operator result in case 2. Criteria

DM1 ^p DM2

DM1 ^p DM2 ^p DM3

Crisp value

C1 C2 C3 C4

((0.86,0.92,0.995),0.63,0.15,0.0.28) ((0.57,0.63,0.86),0.79,0.15,0.15) ((0.53,0.48,0.6),0.7,0.1,0.15) ((0.18,0.28,0.13),0.5,0.2,0.14)

((0.82,0.91,1),0.57,0.13,0.35) ((0.49,0.62,0.87),0.7,0.13,0.18) ((0.34,0.37,0.35),0.65,0.15,0.23) ((0.4,0.28,0.47),0.83,0.15,0.3)

0.7132 0.5915 0.3008 0.3421

Solving the supply chain problem using the best-worst method based on a novel Plithogenic model

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Table 1.15 Best-to-others vector in case 2. Criteria

C1

C2

C3

C4

Best criterion C1

0.1

0.2

0.7

0.3

Table 1.16 Others-to-worst vector in case 2. Criteria

Worst criterion C3

C1 C2 C3 C4

0.7 0.5 0.1 0.4

Table 1.17 BWM results in case 2. Weights

l

l

C1

C2

C3

C4

0.48484848

0.27272727

0.06060606

0.18181818

Using the BWM model as in Table 1.17, we found that W1∗ ¼ 0.4848, W2∗ ¼ 0.2727, W3∗ ¼ 0.0606, W4∗ ¼ 0.1818, and ε ¼ 0.06060. The results show that the weight of C1 (the best criterion) is the highest value where C3 (the worst criterion) is the lowest, as illustrated in Fig. 1.6. Based on the CI table in Ref. [11], the consistency ratio is CR ¼ 0:0606 3:73 ¼ 0:0162.

The analysis of this proposed model results shows that in case 1 the maximum weight is W2 (service level and effectiveness) and the lowest is W5 (population). The sum of all criteria weights is 1, which is one of the constraints. We can conclude that the sixth Weights 0.6 0.5 0.4 0.3 0.2 0.1 0 C1

Fig. 1.6 BWM results in case 2.

C2

C3

C4

16

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Case 1

C6

Case 2

C1

C1

0.5

0.5

0.4

0.4

0.3

0.3

C2

0.2

0.2

0.1

0.1 C4

0

C5

C2

0

C3

C4

C3

Fig. 1.7 Criteria weights of cases 1 and 2.

criterion is important in different levels. The service level and the economic factors criteria are the most that must be considered. On the other hand, the case 2 results show that the maximum weight is W1∗ (quality indicator) and the worst is W3∗ (the amount of painting used). The consistency ratio is obtained in both cases, in case 1 CR ¼ 0.0236, and in case 2 CR ¼ 0.0162. The ratio in both cases was excellent, which means that the BWM model is consistent enough (Fig. 1.7).

1.5

Discussion

In this study, a model of plithogenic aggregation was proposed, based on the BWM, to solve a multi-criteria decision-making problem in the supply chain. This model is investigated by two numerical real-world case studies regarding warehouse location problem and plant evaluation. In the first case, six criteria were evaluated by the experience of four decisionmakers in the supply chain field. To combine the four DMs’ opinions, we used a plithogenic intersection aggregation operation. The evaluation of the DMs is expressed in the form of a triangular neutrosophic number. The result of the plithogenic intersection is then converted to a crisp value in order to use them in the BWM. In this phase, we consider the contradiction degree between each criterion and the dominant criterion (economic factors C1). As mentioned before, the contradiction degree function increases the accuracy of the aggregation result. After solving the BWM, the results show that the service level and effectiveness criterion is the most important one, with weight W2∗ ¼ 0.4126. The second most important criterion is the economic factor, with weight W1∗ ¼ 0.2591. These are therefore the most important criteria that must be considered. On the other side, the population criterion has the lowest weight W5∗ ¼ 0.0383, and environmental risk comes next with weight W6∗ ¼ 0.074. As noted above, the best criterion was C2 and the worst was C5, which fit with the model results. Therefore, the order of new central warehouse location criteria are as follows: C2 > C1 > C4 > C3 > C6 > C5.

Solving the supply chain problem using the best-worst method based on a novel Plithogenic model

17

In case 2 there are four criteria to evaluate the plant performance by assessments done by three experts in this field. The same model was used to evaluate these criteria, and the results show that the quality indicator C1 is the most important criterion, with weight W1∗ ¼ 0.4848. The least important factor is the amount of painting used C3, with weight W3∗ ¼ 0.0606. If the best and worst selected criteria are changed, then all the weights will be changed too. The consistency ratio is a measurement of the results that we found; the small ratio means more consistency. Moreover, we calculated the consistency ratio and it was 0.0236, which means that we obtained a high consistency of evaluations.

1.6

Conclusion and future research directions

This study proposed a combination between plithogenic aggregation operations and the best-worst method. The purpose of this combination is to aggregate the decision-maker’s opinions in order to apply the BWM to find the optimal weight of each criterion. The MCDM field provided impressive results in countless problem categories where it is needed to identify the optimal decision. In addition, the BWM was appreciated in pairwise comparison issues. In this paper, we examined the proposed model in two significant supply chain problems: the warehouse location determination of Greenfinch Ltd. and the plant evaluation of Toyota Motors. The problem of Greenfinch Ltd. was that in Germany, which was a core market for the company, it had two central warehouses, which was not cost-efficient. Therefore, the company decided to establish a single central warehouse to reduce administrative costs. After the company defined the main six criteria to be taken into consideration when selecting the central warehouse location, a group of four experts in supply chain and inventory management evaluated the criteria based on their preferences. The plithogenic aggregation operation combined these preferences of the DMs, which were represented as triangular neutrosophic numbers. The result of the aggregation was taken as input to the BWM. We found that the BWM identified the weight of each criterion based on comparison to the best and the worst one. The second case, that of Toyota Motors, was to optimize the painting sector performance in the plant based on four criteria defined by the operation manager and the sector crew. This problem was estimated by three experts in motor production, and their opinions were accumulated by plithogenic aggregation operations. It is worth mentioning that the contradiction degree function, which is an essential property of plithogenic aggregation operations, added more accuracy to the results. The consistency ratio was evaluated and its value was near to zero, which means that it was acceptable enough. In future research directions, the plithogenic aggregation operations may be combined with other MCDM techniques to solve several types of problems. Moreover, this model may examine many issues in different fields of decision-making problems.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

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Neutrosophic-based domainspecific languages and rules engine to ensure data sovereignty and consensus achievement in microservices architecture

2

Haitham A. El-Ghareeb Information Systems Department, Faculty of Computers and Information Sciences, Mansoura University, Mansoura, Egypt

2.1

Introduction

A monolithic application is an application with a single large codebase/repository that offers tens or hundreds of services using different interfaces such as HTML pages, web services, and/or REST services [1]. A monolithic application has most of its functionality within a single process that is componentized with internal layers or libraries [2]. Monolithic applications scale out by cloning the application on multiple servers or multiple virtual machines. Fig. 2.1 presents a typical monolithic application structure and scaling mechanism. Monolithic applications rely heavily on layered architectures (as depicted in Fig. 2.1 per application within a physical or virtual server). Each box in Fig. 2.1 presents a running instance of the application on a separate server. Microservices architecture has developed from a promising architecture into a de facto standard in designing and developing distributed applications. Microservices architecture is used by large companies like Amazon, Netflix, and LinkedIn to deploy large applications in the cloud as a set of small services that can be developed, tested, deployed, scaled, operated, and upgraded independently, allowing these companies to gain agility, reduce complexity, and scale their applications in the cloud in a more efficient way [1].

2.2

Microservices

Microservices can be thought of as an architectural style for developing applications as suites of small and independent (micro)services. Each Microservice is built around a business capability; it runs in its own process and it communicates with the other Microservices in an application through lightweight mechanisms (e.g., HTTP Application Programming Interfaces (APIs)) [3]. Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00002-0 © 2020 Elsevier Inc. All rights reserved.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Layer

Layer

Layer

Layer

Layer

Layer

Layer

Layer

Layer

Application A

Application A

Application A

Physical/virtual server

Physical/virtual server

Physical/virtual server

Fig. 2.1 Monolithic application structure and scaling mechanism.

A Microservice is either a service-oriented architecture (SOA), if done right, or it is a totally new architecture. Microservices have learned from SOA mistakes and failures [4–6]. One important rule for Microservices is data sovereignty.

2.2.1 Data sovereignty Each Microservice must hold and control its data. No Microservice should be able to access another Microservice’s data. Integration at the database layer is not recommended, because it introduces coupling between the data representation internally used by multiple Microservices. Instead, Microservices should interact only through well-defined APIs, to provide a clear mechanism for managing the state of the resources exposed by each Microservice.

2.2.2 Integrated data Microservices architecture-based solutions do not act in isolation. There must be relationships between different Microservices. Relationships between related entities can be implemented using hypermedia, so that representations retrieved from one Microservice API can include links to other entities found on other Microservice APIs [7]. Such a solution faces many of challenges, and can easily become unreliable. Broken links on the World Wide Web are a live example of such a challenge.

2.2.3 Statefulness A stateful system produces outputs that depend on the state generated in previous interactions. An e-Commerce application case study will be presented in Section 2.7 to depict this idea. e-Commerce is stateful if it remembers previous shopping activities and past visits to the online shop. Such activities are important for personalization, recommendation, and security.

Neutrosophic-based domain-specific languages

23

2.2.3.1 Stateless microservices In many cases, stateless Microservices are ideal because they exploit the benefits of cloud computing, such as on-demand elasticity, load balancing, high availability, and high reliability through redundancy [8].

2.2.3.2 Stateful microservices Stateful Microservices require a more complex logic for managing the state, are less scalable, and do not facilitate high availability and high reliability. Hot standby is a redundant method in which one system runs simultaneously with an identical primary system. Upon failure of the primary system, the hot standby system immediately takes over, replacing the primary system. Fail-over scenario is switching to a redundant or standby computer server, system, hardware component or network upon the failure or abnormal termination of the previously active application, server, system, hardware component, or network. State information has to be made persistent and synchronized in hot standby configurations and recreated in fail-over scenarios. Even without failure, statefulness leads to session affinity, which can decrease throughput and increase latency due to the need to wait for specific stateful instances [8]. Like any other architectural style, Microservices have easy parts and tough parts. Tough parts always define success from failure. Easy parts in Microservices architecture include communication between different Microservices. It is widely accepted that the REST protocol is the de facto one for communication. Restful APIs solved a big problem in Microservices architecture, mainly in standardizing communication. Stateless Microservices facilitate autonomy and isolation, so we guarantee that we acquire all Microservices benefits and promises. It has become widely accepted that we tend to make Microservices as stateless as possible. Stateless Microservices are perfect. However, applications are not stateless. Microservices need to reflect applications’ statefullness nature, and hence comes the challenge.

2.2.4 Containerized applications using Docker 2.2.4.1 Containers Container technology is having a profound impact on distributed systems and cloud computing. While virtual machines provide an abstraction at the hardware level, the container model virtualizes operating system calls so that, typically, many containers share the same kernel instance [9]. Containers are like a very lightweight virtual machine. Virtual machines are a fairly large-weight computer resource. An average virtual machine is a copy of an operating system running on top of a hypervisor running on top of physical hardware, which an application is then on top of. That presents some challenges for speed and performance [10].

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

2.2.4.2 Docker This is a disruptive technology which changes the way applications are being developed and distributed. With a lot of advantages, a Docker is a very good fit to implementing Microservices architecture [11]. A Docker is a container virtualization technology that attempts to overcome virtual machine’s challenges.

2.2.4.3 Docker Swarm The Docker has recently introduced its distributed system development tool called Swarm, which extends the Docker container-based software development process on multiple hosts in multiple clouds without any interoperability issues. Docker Swarm-based distributed software development is a new approach for the cloud industry; nonetheless, it has huge potential to provide a multicloud development environment without increasing the complexity of it [12].

2.3

Domain-specific languages

A domain-specific language (DSL) is a programming language that is specialized to a particular application domain [13]. Although DSL advantages have been known for a long time, they have not achieved widespread adoption yet [14]. Languages that were created for particular domains include FORTRAN used to allow direct mathematical formula, Structured Query Language for database access and manipulation, and Algol for algorithm specification [15].

2.3.1 DSL components Programs that recognize languages are called parsers or syntax analyzers [16]. Syntax refers to the rules governing language membership. Grammar is just a set of rules, each one expressing the structure of a phrase. Lexer. The program that tokenizes the input is a lexer. It is also known as a tokenizer. It is the first step toward a parser. The process of grouping characters into words or symbols (tokens) is called lexical analysis or simply tokenizing [16]. Parser. The second stage is the actual parser and feeds off of these tokens to recognize the sentence structure. Most parsers build a data structure called a parse tree or syntax tree that records how the parser recognized the structure of the input sentence and its component phrases [16].

2.4

Rules engine

Business rules are full of changes in the business world. Businesses need to adapt to changeable business rules. A business rules engine is a software system that executes one or more business rules in a runtime production environment. The rules might come from legal regulation (“An employee can be fired for any reason or no reason

Neutrosophic-based domain-specific languages

Rule i

25

Fact i

Rule i+1

Fact i+1

Inference engine

Pattern matcher Production memory

Working memory Decision

Fig. 2.2 Rules engine components.

but not for an illegal reason”), company policy (“All customers who spend more than $100 at one time will receive a 10% discount”), or other sources. Rule engine software is commonly provided as a component of a business rule management system which, among other functions, provides the ability to: register, define, classify, and manage all the rules, verify consistency of rules definitions, define the relationships between different rules, and relate some of these rules to IT applications that are affected or need to enforce one or more of the rules [17]. Fig. 2.2 depicts the most important rules engine components. Rule-based systems have become increasingly popular due to three factors [18]: 1. better separation of concerns between the knowledge and its implementation logic in contrast to a hard-coded approach; 2. rule repositories that increase the visibility and readability of the knowledge; and 3. graphical user interfaces that render rules more usable while bridging the gap between users (e.g., domain experts) and IT specialists.

Drools is a hybrid chaining engine, meaning it can react to changes in data and also provides advanced query capabilities. Drools provides built-in temporal reasoning for complex event processing and is fully integrated with the jBPM project for BPMN2based workflow [19]. The rest of this chapter is arranged as follows: Section 2.5 introduces the basic and required concepts of neutrosophic sets and theory. Section 2.6 presents the problem definition that is tackled for a solution in this chapter. Section 2.7 presents the problem definition in a concrete description. Section 2.8 presents a case study adopting DSL and highlighting how it can be utilized in solving different challenges. Section 2.9 presents the conclusion and future work. The chapter concludes with references.

2.5

Neutrosophic theory

Neutrosophic sets have been introduced to the literature by Smarandache to handle incomplete, indeterminate, and inconsistent information [20]. In neutrosophic sets, indeterminacy is quantified explicitly through a new parameter I. Truth-membership

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

(T), indeterminacy membership (I), and falsity-membership (F) are three independent parameters that are used to define a neutrosophic number.


N¼

nD

E o x;TN ðxÞ, IN ðxÞ, FN ðxÞ , x 2 X








x 2 X, TN ðxÞ, IN ðxÞ, FN ðxÞ 2 ½0, 1








(2.1) (2.2)

2.5.1 Single-valued neutrosophic number This presents the basic element and the most widely used neutrosophic number. Single-valued neutrosophic numbers’ (SVNNs) operations and their implementation are presented in this section. Deneutrosophy is the process where neutrosophic scales/numbers are converted to crisp values by applying score functions of s(aij) as illustrated in Eq. (2.3) [22].  Trij + Irij + Frij      9 s aij ¼  lr ij  mr ij  ur ij 

(2.3)

2.5.2 Single-valued neutrosophic set A single-valued neutrosophic set (SVNS) consists of multiple SVNNs. Aggregation operations are presented in this section. Implementation details simplified the calculation, and hopefully will act as an enabler for researchers in academia and developers in industry as a guidance and concrete implementation on how to adopt neutrosophic sets in real-world applications.

2.6

Problem definition

A Microservices architecture approach introduces a lot of new complexity and requires in application developers a certain level of maturity in order to apply the architectural style confidently [11]. Let us take a closer look at some Microservice design architecture alternatives. Microservices mesh architecture. Fig. 2.3 depicts a Microservices architecture, in which each Microservice is autonomous and satisfies data sovereignty rules. This figure presents the communication method between different Microservices, and between different Microservices and front-end applications. They are connected using mesh topology. Full mesh topologies have the advantages of high fault tolerance. For this Microservices setup, an advanced front-end application based on a modern technology like Angular, React, Vue, or others can make use of component-based frontend applications. Each component can get its own data from a different Microservice. However, it is clear that this architectural setup suffers in scalability. Full mesh

Neutrosophic-based domain-specific languages

27

Frontend application

API

API

Layer

Layer API Layer Stateless microservice Microservice A

Microservices Z

Physical/virtual server

Physical/virtual server

Stateful microservice Microservices B Physical/virtual server

Fig. 2.3 Microservices mesh architecture structure.

topology requires tremendous connections, and they are tough to manage, monitor, and scale. Eq. (2.4) represents the total number of connections required for full mesh topology. Besides, this architecture setup suffers from a severe disadvantage: a global application state. In this Microservices architecture setup, each Microservice has its own state. What if our application needs to take decisions based on a global state, that is, a state from different Microservices? What about integrated data from different Microservices? It is not applicable and not accepted to leave such a decision for the front-end application. Tc ¼

n∗ðn  1Þ 2

where n ¼ no: of nodes

(2.4)

Microservices mesh architecture with gateway. Fig. 2.4 presents an adaptation to Fig. 2.3 that adds an API Gateway to the system. This architecture solves the need for the polynomial number of connections between Microservices, and presents a centralized method to manage the global system state. However, presenting a central point is a clear contradiction with Microservices architectural objectives. This is a reproduction of SOA choreography and orchestration: Two services-based design

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Frontend application

API

API Gateway

API

Layer

Layer API Layer Stateless microservice Microservice A

Microservices Z Physical/virtual server

Physical/virtual server

Stateful microservice Microservices B Physical/virtual server

Fig. 2.4 Microservices mesh architecture structure with gateway.

patterns that were not successful within SOA, besides of course, the central point of failure, scalability, and availability bottleneck that this solution suffers from. Microservices mesh architecture with Swarm manager and event bus. Fig. 2.5 presents a midway solution between the previous two Microservices architectures that attempts to make use of a simple event bus that supports publish/subscribe events management, so we get rid of full mesh topology, and relies on Swarm/cluster management for monitoring, tracking, load balancing, and directing requests and messages among Microservices. Certainly there are different Microservices architectural setups, each with its own advantages and challenges. Picking the most suitable one relies mainly on the problem in hand. Attempting to present highly modular, easily scalable, enterprise-wide applications, chapter focus will be on Microservices mesh architecture structure with Swarm manager and event bus. This architecture, though it is the most sophisticated among the presented ones, is the one that makes the most benefit from different cloud computing providers’ services (benefiting from the Swarm manager). However, it suffers from different challenges. Those challenges can be generalized to different Microservice architectures. In this chapter, we attempt to solve the following challenges: 1. What happens if the service provided to the user needs integration of data between different Microservices?

Neutrosophic-based domain-specific languages

29

Frontend application

Swarm manager/ load balancer

API

Simple event bus

API

Layer

Layer API Layer Stateless microservice Microservice A

Microservices Z

Physical/virtual server

Physical/virtual server

Stateful microservice Microservices B Physical/virtual server

Fig. 2.5 Microservices mesh architecture structure with Swarm manager and event bus. 2. How can we handle the case that data will be calculated depending on values from other Microservices? 3. What happens if one of those Microservices fails? 4. What are the procedures to be followed to guarantee accepted levels of scalability and availability? 5. Applications must have the ability to present clones of the current Microservices as needed. What are the means to do so? 6. What about databases for those newly generated Microservices?

We can rephrase the previous challenges in simple words: We need to manage the global application state, on both infrastructure and database levels. However, Microservices-based applications now consist of autonomous Microservices, that are independent, and do not rely on each other. Each Microservice has its own data, and they must satisfy data sovereignty rules. We need to find a suitable solution that satisfies the following conditions: 1. The proposed solution does not rely on centralized or shared databases. 2. The proposed solution must be as flexible and adaptable as possible. 3. The Microservices must adhere to different changing business rules, especially for global software solutions.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

2.7

Case study problem definition: e-Commerce

This section presents a concrete description of the problems presented in Section 2.6. The classical attempt to design and implement a Microservices-based e-Commerce includes some activities as follows. e-Commerce is divided into basic Microservices, using Domain-Driven Design, into: 1. 2. 3. 4.

products Microservice; orders Microservice; users Microservice; and front-end Microservice.

Such a design can be containerized into different Microservices easily. Taking the easy way in designing and implementing an e-Commerce Microservices solution will guide us to the following steps: 1. identify the different Microservices; 2. design, develop, and deploy those Microservices (most probably as containers); 3. Microservices’ main communication method would be the REST Protocol through Restful APIs; 4. deploy those Microservices into Swarm to provide scalability, monitor, and manage different services to detect failures, ease scalability, and benefit from other Swarm capabilities; and 5. develop the front-end application, which communicates with different Microservices via the Swarm manager(s).

Hence come the main challenges. Those challenges are best depicted through different scenarios, as follows: 1. Scenario 1: The user is viewing the catalog items. It has become widely accepted that users do not have to know the item price displayed to other users, because our e-Commerce needs to provide different prices for different items to different users based on different criteria. Criteria might include: purchase history; user loyalty; and personalized parameters, like birthdays, geolocation, personal interests, etc. So, our e-Commerce needs certain information about the user from other Microservices (mainly user Microservice in this scenario) to be able to respond to certain queries. In addition, those rules must not be hard-coded in the Microservice, to be able to change them easily in the future, and provide high levels of adaptability and configurability. Hard-coding such rules will lead to application redeployment to change such rules. 2. Scenario 2: The user has placed a new order; however, this order is very suspicious. Order characteristics include: The order is originating from an IP address that the user has never used before. This IP address belongs to a country/region that the user was never recorded as having logged in from before. The order includes a new product/category that the user has not ordered before. The shipping address is completely a new one. The shipping address belongs to an address that is in a country that was not logged for this user before. l

l

l

l

l

l

l

l

Neutrosophic-based domain-specific languages

31

Such a scenario might or might not be natural; however, it is certainly suspicious. To resolve this issue, e-Commerce solution must be able to utilize integrated data from different Microservices, and to check certain rules. Those rules need to be changed based on current security threats, and depending on many criteria. 3. Scenario 3: It is White Friday, and suddenly we have a real traffic overload. Our e-Commerce needs to make new sets of different Microservices. As stated earlier, Microservices can be divided into stateless and stateful Microservices. Replicating stateless Microservices is easy and efficient. However, replicating stateful Microservices is a challenging task. Containerized applications rely on building and creating containers from images. Images are like blueprints in object-oriented programming, and containers are more like objects. Every newly created container is a representation of the image state. It is completely in an initial state. So, if our e-Commerce application relies on creating new containers from images, the newly created containers will fall behind current executing containers. A proposed solution might be to make clones of current executing containers. To make a clone of the current executing containers, we have to: – stop the current container, which will affect overall system availability; and – find a solution to manage the nonconsistent state duplication, for example, if the user is in the middle of placing an order, and we replicated the container, this means we have two running Microservices instances (containers) with an incomplete order, and the user will continue with one of them only. So, what about the status of the newly cloned container? l

l

l

l

l

2.8

Case study solution

2.8.1 DSL In this chapter, we present neutrosophic-based DSL and rules engine as a solution to the above-mentioned problems. DSLs have existed for a long time. It is time to re-evaluate their suitability to tackle tough challenges in Microservices. Listing 2.1 presents an event of type Purchase Operation Pending with Global ID abc123xyz that took place on Node ID 1234 with items on the cart right then. There are some notes on this grammar:

Listing 2.1 e-Commerce in JSON format {"guid" : "abc123xyz", "description" : "Purchase Operation Pending", "cartItems" : {"book":"Microservices and DSLs","chapter”: "Enterprise Integration"}, "nodeID" : "1234" }

32 l

l

Optimization Theory Based on Neutrosophic and Plithogenic Sets

The JSON format has been chosen as e-Commerce DSL representation, so it can be processed later, even without the need of the proposed parser. JSON has native support in most modern programming languages. The proposed DSL grammar is not exclusive or complete. This is a proof of work that can be easily expanded and built upon.

The main contribution of this chapter is: our e-Commerce application uses the Microservices logs in an innovative manner. Logs here are not just entries that are used to record important events, so administrators can come back later to check what went wrong, or identify certain performance issues. Logs have become a DSL that can be parsed based on a syntax tree, like the one presented in Fig. 2.6 generated from Listing 2.1. Listeners and Visitors can later be generated for each node in this parse tree. Code can be executed based on the visited node, giving large flexibility and enabling an efficient rule-based engine among Microservices. Generated Listeners and Visitors form the basis of the Rules Engine that will direct the system’s decision based on the different inputs. Treating log events as DSL helped solve e-Commerce challenges in different ways: l

l

l

DSL is programming language independent; it is more a grammar. DSL can be understood by different programming languages, so each Microservice can have its own autonomy, and can utilize different technology. DSL can specify the rules for the system to follow globally. DSL can easily act as an enabler for rules engines. Our e-Commerce Microservices can adjust their behavior based on DSL logs that are exchanged and delivered. Those rules include specifications about promotions, special discounts, special pricing, and security, among other specifications. Those rules are enough to solve the majority of issues discussed in the above-mentioned scenarios.

json value obj { “guid”

pair :

’

pair

value “description” : value

“abc123xyz”

pair

’

“cartItems” : value

“Purchase⋅Operation⋅Pending” {

pair

“book” : value

pair }

’

“nodelD” : value

obj

“1234” pair }

’

“chapter” : value

“Microservices⋅and⋅DSLs”

Fig. 2.6 Original DSL parse tree.

“Enterprise⋅Integration”

Neutrosophic-based domain-specific languages l

l

33

For newly created Microservice container instances, those logs can be used to move the newly created Microservice state forward, so it overcomes the challenge of cloning the state from similar Microservices. Those events can be easily used to build event-dependency graphs. Such graphs are very useful and helpful for identifying conflicts, defining deadlocks, helping to solve eventual consistency problems. Such challenges are very well known in distributed systems, and they need communication and collaboration between different Microservices.

2.8.2 Neutrosophic DSL Section 2.8.1 presented the simplest and most straightforward case, but it was necessary to present it this way to make the problem scenario and solution workflow clear. However, the scenario presented at Section 2.8.1 faces some challenges. Section 2.8.1 assumed that the user is dealing through the purchase process with only one Microservice, something that is not true. In distributed systems, it is well known that the system must keep a global state, so the user can continue processing from any Microservice—of course that of the same type of the user’s request. That means all running Microservices must negotiate a global state and reach consensus. Caching is a highly utilized technique to enhance distributed web applications. There are different caching algorithms and implementations depending on the case at hand; however, the simplest case involves that: l

l

if the user requests an item that is not in the cache, the Microservice reads it and keeps it in cache; and if the user updates an item, the Microservice updates its value at the cache first, then persists that update to the database later.

Caching provides many advantages to systems; however, it presents the problems of Dirty Reads and Dirty Writes. Simply, reading a value from a cache that was never persisted in the database, and executing further operations yields a wrong value. Another version of the problem is that writing a value to the cache that was never persisted leads to the loss of that value. There are different proposed solutions on infrastructure and information technology levels for this problem; however, to ensure persistent global status, we need a solution on an application level. The solution would be: l

l

l

for for for I

values persisted in the database, assign (T 1, I 0, F 0); values in cache, assign (T 0, I 1, F 0); and values that are neither in cache nor persisted in the database (Default), assign (T 0, F 1).

0,

Exchanging messages between Microservices right now would be different; that represents the Microservice state. Listing 2.2 presents the enhanced neutrosophicbased e-Commerce cart. The status represents the neutrosophic-based proposed enhancement, where book status means that this purchase has been persisted, while chapter status means that it is still in the cache. Fig. 2.7 presents the enhanced neutrosophic-based Microservice state representation.

34

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Listing 2.2 Neutrosophic-based e-Commerce in JSON format { "guid" : "abc123xyz", "description" : "Purchase Operation Pending", "cartItems" : [ { "book":"Microservices and DSLs", "status": "(1,0,0)" }, { "chapter":"Enterprise Integration", "status": "(0,1,0)" } ], "nodeID" : "1234" }

Fig. 2.7 Neutrosophic DSL parse tree.

2.8.3 Neutrosophic rules engine The main advantage of a Rules Engine is the separation of rules from the programming language, and the application. Using a Rules Engine generally gives business developers the ability to focus on technical domains, and business experts the ability to

Neutrosophic-based domain-specific languages

35

focus on the business rules. In this chapter, JSON will also be utilized to represent Rules in Rules Engines. To continue the case study, the focus will be on business rules. Unified Service-Oriented Rules Language was presented in Yin et al. [17]. In this chapter, we will utilize only the rules, and reformat them in JSON format instead of XML. This means that Microservices have to exchange two types of JSON messages: Business Rules and Microservice state.

2.8.3.1 Business rules engine language design Four types of tags are presented in Yin et al. [17]: 1. Logical tag set (a) ruleset (b) rule (c) when (d) then (e) param 2. Service tag set (a) service 3. Operator tag set (a) opt (b) and (c) or (d) not (e) equal (f ) add (g) sub (h) multiply (i) divide (j) modulo (k) addequal (l) subequal (m) multiplyequal (n) divideequal (o) moduloequal (p) gt (q) lt (r) openparenthesi (s) closeparenthesi 4. Environmental tag set (a) package

2.8.3.2 Business rule example Using the above-mentioned Rules Engine proposed modeling language, assume that business domain experts want to add the following business rule: Special Discount on Purchases if Delivery Country is Egypt, Items greater than 3, and Today is User Birthday. Listing 2.3 presents the specification of the above business rule in JSON

36

Optimization Theory Based on Neutrosophic and Plithogenic Sets

format. This specification will be the JSON message exchanged between different Microservices. Fig. 2.8 presents the parse tree of the same business rule.

2.8.3.3 ANTLR Another Tool For Language Recognition (ANTLR) is a parser generator that uses LL(*) for parsing. ANTLR takes as input a grammar that specifies a language and generates as output source code for a recognizer for that language. ANTLR supports

Listing 2.3 Example of business rule in JSON format {"guid" : "BR000001 ", "Title" : "Business Rule 001", "Date" : "23/06/2019", "description" : "Special Discount on Purchases if Delivery Country is Egypt, Items greater than 3, and Today is User Birthday", "rule" : [ { "Today" : { "equal" : "User Birthday" } }, { "and" : { "items in cart" : { "gt" : "3" } } }, { "and" : { "Delivery" : { "equal" : "Egypt" } } } ] }

37

Fig. 2.8 Business rule parse tree.

Neutrosophic-based domain-specific languages

38

Optimization Theory Based on Neutrosophic and Plithogenic Sets

generating code in Java, C#, JavaScript, Python2, and Python3. Language applications read sentences and react appropriately to the phrases and tokens (input symbols). A language is just a set of valid sentences. To react appropriately, an interpreter or translator has to recognize the input. The parser feeds off of tokens from the lexer, which feeds off of a char stream, to check syntax (language membership). ANTLR builds a parse tree while parsing. An ANTLR sequence of actions is presented in Eq. (2.5). ANTLR autogenerates listeners and visitors. For more information on ANTLR, the reader is referred to Parr [16]. The proposed Neutrosophic Rules Engine utilizes ANTLR as follows: 1. Define JSON Grammar file, as listed in Listing 2.4. 2. Parse the JSON Grammar file with ANTLR. 3. The proposed model will focus on Python3. Listing 2.5 displays the generated JSON Listener methods. 4. Taking a closer look at the generated JSON Listener, it is clear that generated methods are built based on Depth-First Search Graph Traversing of the JSON Grammar. 5. Business developers need to implement the required methods. For example, Listing 2.6 presents part of the implementation of the above-mentioned business rule. The proposed neutrosophic-based DSL and Rules Engine utilizes the novel open-source neutrosophic package presented in El-Ghareeb [22].

chars ! LEXER ! tokens ! PARSER ! parse tree

(2.5)

2.8.3.4 Decentralized rules engine Right now, proposed Microservices architecture has the capability to exchange Microservice state in JSON format, and the capability to exchange different business rules. That leaves one last important decision for the proposed architecture: Rules Engine architecture. There are two models to apply the Business Rules at Microservices architecture: Centralized and Decentralized. The Centralized Rules Engine solution is the easiest, where there will be a central Rules Engine server, where business domain experts design and apply the business rules. The Rules Engine server is responsible for distributing the business rules to the different Microservices, and it is also responsible for consensus agreements. However, any centralized solution is not accepted any more for the same reasons discussed earlier, including scalability, fault tolerance, availability, and many other benefits. Decentralization is the de facto standard in Microservices architecture. Microservices act as peers, and they must achieve consensus. This is a real challenge, as different Microservices are not the same state. This is where neutrosophy can tackle the challenge. There are different group aggregate methods and calculations based on Neutrosophic Sets and Theory. For detailed illustration, discussion, and examples of the utilized Neutrosophic Analytic Hierarchy Process, the reader is referred to Refs. [21, 23]. For example, assume that there are different Microservices state messages presenting the cart of a certain user based on the presentation at Section 2.8.2. There must be methods for aggregating different SVNSs. A single-valued neutrosophic soft

Neutrosophic-based domain-specific languages

Listing 2.4 JSON grammar [16] /** Source: "The Definitive ANTLR 4 Reference" by Terence Parr from http://json.org */ grammar JSON; json: object j array ; object : ’{’ pair (’,’ pair)* ’}’ j ’{’ ’}’ // empty object ; pair: STRING ’:’ value ; array : ’[’ value (’,’ value)* ’]’ j ’[’ ’]’ // empty array ; value : j j j j j j ;

STRING NUMBER object // recursion array // recursion ’true’ // keywords ’false’ ’null’

STRING : ’"’ (ESC j ~["\\])* ’"’ ; fragment ESC : ’\\’ (["\\/bfnrt] j UNICODE) ; fragment UNICODE : ’u’ HEX HEX HEX HEX ; fragment HEX : [0-9a-fA-F] ; NUMBER : ’-’? INT ’.’ [0-9]+ EXP? // 1.35, 1.35E-9, 0.3, -4.5 j ’-’? INT EXP // 1e10 -3e4 j ’-’? INT // -3, 45 ; fragment INT : ’0’ j [1-9] [0-9]* ; // no leading zeros fragment EXP : [Ee] [+\-]? INT ; // \- since - means "range" inside [...] WS : [ \t\n\r]+ -> skip ;

39

40

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Listing 2.5 JSON parser-Automatically generated # Generated from java–escape by ANTLR 4.4 from antlr4 import * # This class defines a complete listener for a parse tree produced by ↪ JSONParser. class JSONListener(ParseTreeListener): # Enter a parse tree produced by JSONParser#array. def enterArray(self, ctx): pass # Exit a parse tree produced by JSONParser#array. def exitArray(self, ctx): pass # Enter a parse tree produced by JSONParser#json. def enterJson(self, ctx): pass # Exit a parse tree produced by JSONParser#json. def exitJson(self, ctx): pass # Enter a parse tree produced by JSONParser#value. def enterValue(self, ctx): pass # Exit a parse tree produced by JSONParser#value. def exitValue(self, ctx): pass # Enter a parse tree produced by JSONParser#pair. def enterPair(self, ctx): pass # Exit a parse tree produced by JSONParser#pair. def exitPair(self, ctx): pass # Enter a parse tree produced by JSONParser#object. def enterObject(self, ctx): pass # Exit a parse tree produced by JSONParser#object. def exitObject(self, ctx): pass

Neutrosophic-based domain-specific languages

41

Listing 2.6 Business Rule 01-Subimplementation # Business Rule 01 Implementation import datetime from JSONListener import JSONListener class BusinessRule_One(JSONListener): self._no_of_items_in_cart = 0 self._delivery_country = "" self._today = datetime.datetime.date() # Enter a parse tree produced by JSONParser#pair. def enterPair(self, ctx): if ctx = "items-in-cart": self._no_of_items_in_cart = ↪self.enterValue(ctx) # Enter a parse tree produced by JSONParser#value. def enterValue(self, ctx): return ctx

weighted arithmetic averaging (SVNSWA) operator and single-valued neutrosophic soft weighted geometric averaging (SVNSWGA) operator have been used to compare two single-valued neutrosophic soft numbers (SVNSNs) for aggregating different single-valued neutrosophic soft input arguments in a neutrosophic soft environment [24]. Both operators are implemented in El-Ghareeb [22]. * + n n n n X Y Y Y wj wj wj SVNNWAAðz1 ,z2 ,…,zn Þ ¼ wj zj ¼ 1  ð1  Tj Þ , ðUj Þ , ðVj Þ j¼1

j¼1

j¼1

j¼1

(2.6) * + n n n n Y Y Y Y wj wj wj wj SVNNWGAðz1 ,z2 ,…,zn Þ ¼ zj ¼ ðTj Þ ,1  ð1  Uj Þ ,1  ð1  Vj Þ j¼1

j¼1

j¼1

j¼1

(2.7)

2.9

Conclusion and future work

This chapter highlighted the importance to move from monolithic architectures to Microservices ones. Different Microservices setup architectures have been presented, highlighting the pros and cons of each one. We then moved forward with the most suitable Microservices architecture: The one that provides the most advantages to fulfill and satisfy Microservices architecture objectives. However, Microservices architectures are very complex and face some significant challenges. Some of those challenges are depicted through an e-Commerce case study. Those challenges can

42

Optimization Theory Based on Neutrosophic and Plithogenic Sets

be solved utilizing DSLs. The main contribution of this chapter is utilizing Microservices logs as DSL in an innovative manner to present rules engines capabilities to Microservices architecture, and to help it overcome architectural challenges, and provide flexibility and configurability.

References [1] M. Villamizar, O. Garces, H. Castro, M. Verano, L. Salamanca, R. Casallas, S. Gil, Evaluating the monolithic and the Microservice architecture pattern to deploy web applications in the cloud, in: 2015 10th Computing Colombian Conference (10CCC), 2015, pp. 583–590. https://doi.org/10.1109/ColumbianCC.2015.7333476. [2] Cesardelatorre, .NET Microservices. Architecture for Containerized .NET Applications (Online), 2019, https://docs.microsoft.com/en-us/dotnet/standard/microservices-architecture. Accessed 14 February 2019. [3] J. Soldani, D.A. Tamburri, W.-J.V.D. Heuvel, The pains and gains of Microservices: a systematic grey literature review, J. Syst. Softw. 146 (2018) 215–232, https://doi.org/ 10.1016/j.jss.2018.09.082. [4] O. Zimmermann, Microservices tenets, Comput. Sci. Res. Dev. 32 (3) (2017) 301–310, https://doi.org/10.1007/s00450-016-0337-0. [5] T. Cerny, Aspect-oriented challenges in system integration with Microservices, SOA and IoT, Enterp. Inf. Syst. (2018) 1–23, https://doi.org/10.1080/17517575.2018.1462406. [6] T. Cerny, M.J. Donahoo, J. Pechanec, Disambiguation and comparison of SOA, Microservices and self-contained systems, in: Proceedings of the 2017 Research in Adaptive and Convergent Systems, RACS 2017, 2017, pp. 228–235. [7] G. Pardon, C. Pautasso, O. Zimmermann, Consistent disaster recovery for Microservices: the BAC theorem, IEEE Cloud Comput. 5 (1) (2018) 49–59, https://doi.org/10.1109/ MCC.2018.011791714. [8] A. Furda, C. Fidge, O. Zimmermann, W. Kelly, A. Barros, Migrating enterprise legacy source code to Microservices: on multitenancy, statefulness, and data consistency, IEEE Softw. 35 (3) (2018) 63–72, https://doi.org/10.1109/MS.2017.440134612. [9] J. Stubbs, W. Moreira, R. Dooley, Distributed systems of Microservices using Docker and Serfnode, in: 2015 7th International Workshop on Science Gateways, 2015, pp. 34–39. https://doi.org/10.1109/IWSG.2015.16. [10] C. Anderson, Docker [Software engineering], IEEE Softw. 32 (3) (2015) 102-c3, https:// doi.org/10.1109/MS.2015.62. [11] D. Jaramillo, D.V. Nguyen, R. Smart, Leveraging Microservices architecture by using Docker technology, in: SoutheastCon 2016, 2016, pp. 1–5, https://doi.org/10.1109/ SECON.2016.7506647. [12] N. Naik, Building a virtual system of systems using Docker swarm in multiple clouds, in: 2016 IEEE International Symposium on Systems Engineering (ISSE), 2016, pp. 1–3, https://doi.org/10.1109/SysEng.2016.7753148. [13] D.R. Cacciagrano, R. Culmone, IRON: reliable domain specific language for programming IoT devices, Internet Things (2018), https://doi.org/10.1016/j.iot.2018.09.006.  . Vukovic, TextX: a Python tool for domain[14] I. Dejanovic, R. Vaderna, G. Milosavljevic, Z specific languages implementation, Knowl. Based Syst. 115 (2017) 1–4, https://doi.org/ 10.1016/j.knosys.2016.10.023. [15] D. Kramer, T. Clark, S. Oussena, MobDSL: a domain specific language for multiple mobile platform deployment, in: 2010 IEEE International Conference on Networked Embedded Systems for Enterprise Applications (NESEA), IEEE, 2010, pp. 1–7.

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43

[16] T. Parr, The Definitive ANTLR 4 Reference, Pragmatic Bookshelf, USA, 2013. [17] Z. Yin, C. Wu, S. Li, X. Li, T. Liu, A service-oriented business rules designer based on rule engine, in: S. Sambath, E. Zhu (Eds.), Frontiers in Computer Education, Springer, Berlin, Heidelberg, 2012, pp. 675–682, https://doi.org/10.1007/978-3-642-27552-4_90. [18] P. de Leusse, B. Kwolek, K. Zielinski, A middleware infrastructure for multi-rule-engine distributed systems, in: E. Di Nitto, R. Yahyapour (Eds.), Towards a Service-Based Internet, Springer, Berlin, Heidelberg, 2010, pp. 231–232. [19] M. Proctor, Drools: a rule engine for complex event processing, in: A. Sch€ urr, D. Varro´, G. Varro´ (Eds.), Applications of Graph Transformations With Industrial Relevance, Springer, Berlin, Heidelberg, 2012. _ Otay, C. Kahraman, A state-of-the-art review of neutrosophic sets and theory, in: [20] I. _ Otay (Eds.), Fuzzy Multi-Criteria Decision-Making Using Neutrosophic C. Kahraman, I. Sets, Springer International Publishing, Cham, 2019, pp. 3–24. https://doi.org/ 10.1007/978-3-030-00045-5_1. [21] N.A. Nabeeh, F. Smarandache, M. Abdel-Basset, H.A. El-Ghareeb, A. Aboelfetouh, An integrated neutrosophic-TOPSIS approach and its application to personnel selection: a new trend in brain processing and analysis, IEEE Access 7 (2019) 29734–29744, https://doi.org/10.1109/ACCESS.2019.2899841. [22] H.A. El-Ghareeb, Novel open source python neutrosophic package, Neutrosophic Sets Syst. 25 (2019) 136. [23] N.A. Nabeeh, M. Abdel-Basset, H.A. El-Ghareeb, A. Aboelfetouh, Neutrosophic multicriteria decision making approach for IoT-based enterprises, IEEE Access 7 (2019) 59559–59574, https://doi.org/10.1109/ACCESS.2019.2908919. [24] C. Jana, M. Pal, A robust single-valued neutrosophic soft aggregation operators in multicriteria decision making, Symmetry 11 (1) (2019), https://doi.org/10.3390/sym11010110.

Product acceptance determination using repetitive sampling based on process loss consideration under neutrosophic numbers

3

Muhammad Aslama, Muhammad Azamb, Liaquat Ahmadb, Florentin Smarandachec a Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia, bDepartment of Statistics and Computer Science, University of Veterinary and Animal Sciences, Lahore, Pakistan, cDepartment of Mathematics, University of New Mexico, Gallup, NM, United States

3.1

Introduction

The acceptance sampling plan is an important and commonly used tool of statistical process control for deciding whether to accept or reject the submitted lot. These acceptance sampling plans are normally used in industries for inspection of the quality of products based upon different sampling schemes. Collecting few but frequent samples can effectively probe the lot for its quality. When the sentencing of the lot is based upon a single sample, this is called the single sampling scheme; when the decision is based upon two samples (a decision is made on the findings of both the samples), it is called the double sampling scheme [1]. The repetitive sampling scheme is better than the single or double sampling schemes [2]. The attribute repetitive group sampling scheme was introduced by Sherman [3], who claimed that it is intermediate in sample size efficiency between ratio to probability sampling plan and the single sampling plan. In repetitive sampling, which is different from double sampling, the sentencing of the submitted lot is made on the conditions described as “accept the lot” if v
Ka and “reject
the lot”  if v < Kr, and repeat the process if Kr  v < Ka, where v is calculated as USL  X =σ and Ka is prespecified acceptance and Kr is a prespecified rejection number. It should be noted that if Ka ¼ Kr, then the repetitive sampling tends to form the single sampling scheme. Aslam et al. [4] developed a repetitive group sampling plan for Weibull and generalized exponential distributions. A repetitive group sampling approach is used based upon the median of the product’s lifetime. The optimal values of the acceptance sampling plan parameters are determined by equating the producer’s risk to the probability of type-I errors and the consumer’s risk to the probability of type-II errors. Azam et al. [5] described a repetitive acceptance sampling plan based upon an exponentially weighted moving average regression estimator. Lee et al. [6] designed repetitive group sampling plans with Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00003-2 © 2020 Elsevier Inc. All rights reserved.

46

Optimization Theory Based on Neutrosophic and Plithogenic Sets

double specification limits. The design parameters of the variable repetitive group sampling plan for the normal distribution are obtained by satisfying the producer’s risk and the consumer’s risk. Upper and lower specification limits for the fraction defectives for the symmetric and non-symmetric cases have been targeted. Aslam et al. [7] suggested repetitive sampling plans for Burr type XII percentiles. This distribution is a generalization of many statistical distributions, including the log-logistic and Pareto Type II (Lomax) distribution. It is also a member of a system of continuous distributions. Traditionally, the acceptance sampling plans were developed for the reliability and the quality of products using the mean lifetime of items/products. An acceptance sampling plan based upon the repetitive sampling for the truncated life test using lifetime percentiles under this distribution was proposed. Aslam et al. [8] described the decision rule of repetitive acceptance sampling plans assuring percentile life. Repetitive group sampling plans for the Weibull as well as generalized exponential distribution were used for the median of a lifetime of the product, and then a decision-making frame was developed. A new approach on the median of the product lifetime was introduced using the repetitive group sampling scheme. Aslam et al. [9] developed a group acceptance sampling plan for lifetime data using the generalized Pareto distribution. When the lifetime of the product follows the generalized Pareto distribution, then the repetitive group sampling scheme performs better in terms of minimum sample size to reach any decision regarding whether to accept or reject the submitted lot. Dobbah et al. [10] proposed a new synthetic sampling plan assuming that the quality characteristic follows the normal distribution with known and unknown standard deviation. A synthetic statistic sampling plan is the integration of the customary single sampling plan and a plan based upon conforming run length. Two types of sampling plans are commonly used in the quality control literature. Attribute sampling plans are used for the count data in the form of yes or no, conforming and non-conforming items, good or defective items, etc., whereas variable plans are applied for the measurement quality characteristics such as weight, height, pressure, temperature, etc. Duarte and Saraiva [11] presented an optimization-based approach for an attribute sampling plan for lot acceptance purposes. Aslam et al. [12] proposed an attribute sampling plan using exponentially weighted moving average (EWMA) for Weibull and Burr type X distributions. More information about attribute and variable sampling plans can be seen in Pearn and Wu [13], Aslam et al. [14, 15], Wu and Pearn [16], Jun et al. [17], Seidel [18], Collani [19], Suresh and Devaarul [20], Li et al. [21], Mussidaa et al. [22], and Gregory and Resnikoff [23]. Neutrosophic statistics are used when the numbers, observations, parameters, or logic under consideration are vague, imprecise, incomplete, unclear, uncertain, vague, or indeterminate. Such a situation of determinate and/or indeterminate information is common in the real world. Classical statistics cannot be useful if the information under study is imprecise [24]. Therefore, the neutrosophic statistics introduced by Smarandache [25] are suitable for describing determinate and/or indeterminate information. Neutrosophic statistics are the generalization of classical statistics [26]. In classical statistics, only the crisp, clear, complete, certain, and precise information is considered [27]. More literature on neutrosophic statistics can be found in Wang et al. [26], Haibin et al. [27], Smarandache [28–30], Chen et al. [31], Ye et al.

Product acceptance determination using repetitive sampling

47

[24], Aslam et al. [32, 33], Aslam [34], Aslam and Al-Marshadi [35], and Aslam and Arif [36]. To the best of the author’s knowledge, the repetitive sampling plan for neutrosophic statistics using process loss consideration has not been considered in the literature of the acceptance sampling plan. Thus, in this chapter, a repetitive sampling plan under neutrosophic statistics using process loss consideration has been developed which shows an effective and flexible sampling plan. The rest of the chapter is organized as follows: Section 3.2 describes neutrosophic process loss consideration. Section 3.3 discusses the proposed sampling plan, and Section 3.4 presents the comparative study of the proposed plan with the existing plan. Section 3.5 provides concluding remarks.

3.2

Neutrosophic process loss consideration

In this section, we describe the neutrosophic process loss function and its expansion for the operating characteristic (OC) function. Assume that XNiE{XL, XU} ¼ i ¼ 1, 2, 3, …, n denotes the neutrosophic random variable with incomplete, imprecise, uncertain, and indeterminate observations. Also assume that the neutrosophic mean and standard deviation may be defined as μNE{L, μU} and σ NE{σ L, σ U}, respectively. Again we define the crisp target value T from XNiE{XL, XU} with a lower specification limit (LSL) and upper specification limit (USL). By following Johnson [37], the neutrosophic process loss index is defined as follows: LNe ¼

σ 2N ðμN  T Þ2 + ;μN EfμL , μU g, σ N Efσ L , σ U g d2 d2

(3.1)

where d ¼ (USL  LSL)/2. In general, the mean μN and the standard deviation σ N are unknown, hence we define the estimated neutrosophic process loss index as follows:
2 2   XN 2T S N ^Ne 5 + L ;XN E XL , XU , sN ¼fsL , sU g 2 2 d d where XL ¼

3.3

Pn

L i¼1 xi =nL , XU ¼

Pn

U i¼1 xi =nU ,sL ¼

(3.2)

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Pn  L Pn  U x  X =n ¼ x  X =n . and s L L U U U i¼1 i i¼1 i

Proposed sampling plan

In this section, we describe the proposed sampling plan in the form of steps that may be stated as follows: Step 1: From a lot of the submitted product, select a random sample of size nNE{nL, nU} and   ^ Ne ; XN E XL , XU , sN ¼ {sL, sU}. compute L

48

Optimization Theory Based on Neutrosophic and Plithogenic Sets

^Ne kaN ; kaNE{kaL, kaU}, accept the lot of the product where kaN is the Step 2: If L neutrosophic acceptance number and reject the lot if L^Ne
krN ; krNE{krL, krU}, where krN is the neutrosophic rejection number. Otherwise, return to step 1.

Thus, there are four parameters of the proposed sampling plan, nNE{nL, nU}, kaNE{kaL, kaU}, krNE{krL, krU}, and ASNNE{ASNL, ASNU}. It can be observed that the proposed plan approaches to the plan presented by Yen and Chang [38], when kaL ¼krU. The probability of acceptance of the proposed sampling plan can be derived as follows:     ^Ne  kaN ;nN EfnL , nU g, XN E XL , XU , sN PNa ¼ P L ¼ fsL , sU g, kaN EfkaL , kaU g

(3.3)

From Yen and Chang [38], L^Ne is distributed as LNeχ nN2/nN, where χ nN2 is a neutrosophic chi-square distribution. So, the neutrosophic operating characteristic (NOC) function is obtained by the following: n o PNa ¼ P χ 2nN  ðnN kaN =LNe Þ ;nN EfnL , nU g,kaN EfkaL , kaU g

(3.4)

Likewise, the probability that the submitted lot is rejected:   ^Ne ≥krN ;nN EfnL , nU g, krN EfkrL , krU g PNr ¼P L n o PNr ¼ 1  P χ 2nN  ðnN krN =LNe Þ ;nN EfnL , nU g,krN EfkrL , krU g

(3.5)

Then the NOC function of the proposed plan can be written as follows: n o P χ 2nN  ðnN kaN =LNe Þ o n o ;nN EfnL , nU g,krN EfkrL , krU g LðpÞN ¼ n P χ 2nN  ðnN kaN =LNe Þ + 1  P χ 2nN  ðnN krN =LNe Þ (3.6)

A neutrosophic sampling plan is considered to be efficient if it satisfies the producer’s risk α and the consumer’s risk β. Therefore, the NOC function must pass through (p1, 1  α) and (p2, β), where p1 and p2 are the acceptable quality level (AQL) and the limiting quality level (LQL), respectively. Thus, the proposed plan parameters nNE{nL, nU}, kaNE{kaL, kaU}, krNE{krL, krU}, and ASNNE{ASNL, ASNU} can be estimated by using the following neutrosophic non-linear problem: Minimize ASN N EfASN L , ASN U g Subject to Lðp1 ÞN
1  α

(3.7)

Product acceptance determination using repetitive sampling

49

and Lðp1 ÞN β:

(3.8)

The parameters of the proposed plan nNE{nL, nU}, kaNE{kaL, kaU}, krNE{krL, krU}, and ASNNE{ASNL, ASNU} are estimated and placed in Tables 3.1–3.4 using Eqs. (3.7) and (3.8) under the search grid method for various values of the AQL and LQL. To save space, we study only α ¼ 0.05, β ¼ 0.10 and 0.05. The similar tables for any other values of α and β can be made utilizing the same lines. From Table 3.1, we note that for the fixed value of AQL, nNE{nL, nU}, kaNE{kaL, kaU}, krNE{krL, krU}, and ASNNE{ASNL, ASNU} decreases as the LQL increases. For instance, when p1 ¼ 0.001 and p2 ¼ 0.002, 0.003, 0.004, 0.006 the values of nN decrease as 88, 29, 13, and 10, and the same is the case with ASNN ¼ 93.318, 33.8, 15.8819, and 10.6128. From Table 3.3, we note that for the fixed value of AQL, nNE{nL, nU}, kaNE{kaL, kaU}, krNE{krL, krU}, and ASNNE{ASNL, ASNU} decreases as the LQL increases when β ¼ 0.05. For instance, when p1 ¼ 0.001 and p2 ¼ 0.002, 0.003, 0.004, 0.006, the values of nN decrease as 82, 14, 15, and 9, and the same is the case with ASNN ¼ 85.8292, 22.2598, 16.2526, and 11.4204. Therefore, a smaller nNE{nL, nU} is needed for the lot sentencing for the larger values of the LQL, which shows that at the same time as the producer’s confidence level increases for the acceptance of the good lot, the consumer’s risk decreases, and they are both willing to select a larger sample of the submitted lot to inspect. We can also observe that the indeterminacy interval in parameters increases at the larger values of LQL. Tables 3.2 and 3.4 have been generated for the non-repetitive sampling scheme using β ¼ 0.10 and 0.05. The nN are estimated for different values of p1 and p2. The variable two-stage acceptance sampling plan using process loss consideration was probed by Aslam et al. [15] and provided the same protection with a smaller sample size than the single sampling plan when the process mean is exactly at target and when it is not. The design parameters of the proposed study were found when the process is at target for various values of AQL ¼ 0.033, 0.05, 0.067, and 0.1 for different values of lot tolerance percent defective (LTPD) ¼ 0.05, 0.06, 0.1, and 0.133. The average sample number (ASN) is also probed for various combinations of α ¼ 0.01, 0.025, 0.050, 0.075, and 0.10, and β ¼ 0.010, 0.025, 0.050, 0.075, and 0.100. From Tables 3.1 to 3.4, the trend of the ASN can be observed that when the value of β increases for a fixed value of α, the sample size decreases. This can be observed for other values of the plan parameters. For instance, if we use α ¼ 0.010, β ¼ 0.010, AQL ¼ 0.033, and LTPD ¼ 0.05, then the values of the design parameters are estimated as n ¼ 117, Ka1 ¼ 0.0345, Ka2 ¼ 0.0781, Kr ¼ 0.0782, and ASN ¼ 195.70. The effect of β can be noted when it is increased from 0.010 to 0.025 in the form of a decreasing trend in ASN as it is 160.73 when the plan parameters are estimated as n ¼98, Ka1 ¼ 0.0345,Ka2 ¼ 0.0787,and Kr ¼ 0.0788. The repetitive group sampling plans with double specification limits were developed by Lee et al. [6] by solving the nonlinear optimization problem with the sequential quadratic programming. The plan parameters n, K1a,K2a, K1, and K2 have been estimated by various fixed n values (in increasing order) and then selected n to

50

Table 3.1 Plan parameters of repetitive neutrosophic sampling plan using loss function when β ¼ 0.10. p2

nN

ASNN

kaN

krN

L(CAQL)

L(CLQL)

0.001

0.002 0.003 0.004 0.006 0.008 0.01 0.15 0.02 0.005 0.01 0.15 0.2 0.25 0.3 0.5 0.001 0.015 0.02 0.03 0.04 0.05 0.1 0.02 0.03 0.04 0.05 0.1 0.15 0.2

[27,88] [8,29] [8,13] [6,10] [5,6] [4,6] [3,3] [3,3] [24,27] [5,8] [5,8] [3,5] [3,4] [3,4] [3,3] [22,38] [8,6] [6,10] [5,6] [4,4] [3,3] [3,3] [19,38] [9,17] [5,10] [5,5] [3,3] [3,3] [3,3]

[55.810,93.3168] [13.451,33.6866] [9.7934,15.8819] [6.9951,10.6128] [5.21,6.9215] [4.8933,6.7811] [3.5914,3.7819] [3.196,4.0456] [34.908,70.6623] [8.181,12.6786] [6.1797,8.1961] [5.123,6.558] [4.734,5.6697] [3.7758,5.1697] [3.2515,3.9742] [29.389,45.995] [12.3,17.9882] [10.172,13.4843] [6.236,8.0153] [4.9875,6.4786] [4.248,4.5953] [3.3058,3.4693] [27.593,46.174] [12.212,19.4288] [7.8597,13.2284] [6.88,8.2872] [3.9943,4.881] [3.4755,4.0664] [3.2127,3.3913]

[0.002,0.0015] [0.0023,0.0021] [0.0021,0.0023] [0.0021,0.0021] [0.002,0.0025] [0.003,0.0034] [0.0028,0.0028] [0.0022,0.0041] [0.0041,0.0051] [0.006,0.0066] [0.0061,0.0053] [0.0083,0.0077] [0.0093,0.0091] [0.0061,0.0085] [0.0061,0.01] [0.0079,0.0076] [0.0105,0.0146] [0.0131,0.012] [0.0111,0.0137] [0.0116,0.0181] [0.0144,0.0159] [0.0125,0.0128] [0.0162,0.0153] [0.0192,0.0175] [0.0228,0.0239] [0.0236,0.0278] [0.0253,0.0344] [0.0239,0.0349] [0.0231,0.0244]

[0.0011,0.0007] [0.0009,0.0009] [0.0011,0.0009] [0.0008,0.001] [0.0016,0.0004] [0.001,0.0017] [0.0009,0.0004] [0.0013,0.0001] [0.0023,0.0016] [0.0015,0.0011] [0.0028,0.0047] [0.0006,0.0014] [0.0011,0.0009] [0.0012,0.0004] [0.0032,0.0005] [0.0055,0.0046] [0.0044,0.0014] [0.003,0.0026] [0.0037,0.0025] [0.003,0.0007] [0.0019,0.0011] [0.0054,0.0027] [0.0099,0.0091] [0.0098,0.0082] [0.0059,0.0089] [0.0076,0.0027] [0.0046,0.0012] [0.0079,0.0033] [0.013,0.0071]

[0.9981,0.9632] [0.9791,0.9991] [0.9751,0.9955] [0.9584,0.9842] [0.9586,0.9573] [0.9899,0.9989] [0.981,0.9681] [0.959,0.9701] [0.9603,0.9932] [0.9613,0.977] [0.9792,0.9816] [0.9784,0.9903] [0.9922,0.9929] [0.952,0.9721] [0.9714,0.9905] [0.9601,0.9675] [0.969,0.9664] [0.981,0.9761] [0.9554,0.9809] [0.9524,0.9757] [0.9705,0.9704] [0.9711,0.9632] [0.9501,0.9659] [0.9589,0.9523] [0.9507,0.992] [0.9674,0.9519] [0.9568,0.9664] [0.9618,0.9863] [0.965,0.962]

[0.0938,0] [0.0953,0.0002] [0.0706,0.0048] [0.0305,0.0052] [0.0951,0.003] [0.0762,0.0458] [0.099,0.0486] [0.0973,0.0052] [0.024,0.0017] [0.0869,0.005] [0.0967,0.0763] [0.0805,0.0174] [0.0977,0.0199] [0.0724,0.0047] [0.0986,0.0193] [0.0831,0.0036] [0.0978,0.0298] [0.0516,0.0021] [0.0476,0.0103] [0.0503,0.0077] [0.08,0.0492] [0.0858,0.0461] [0.073,0.0029] [0.0841,0.0031] [0.0834,0.017] [0.0773,0.0141] [0.0888,0.0291] [0.0881,0.0434] [0.0996,0.0585]

0.0025

0.005

0.01

Optimization Theory Based on Neutrosophic and Plithogenic Sets

p1

0.05

0.06 0.09 0.12 0.15 0.3 0.1 0.15 0.2 0.25 0.5

[17,30] [6,13] [6,6] [5,7] [3,3] [22,26] [6,12] [6,7] [4,6] [3,3]

[25.619,41.6353] [11.277,15.8231] [7.7842,11.008] [6.5197,8.4274] [4.0346,4.481] [28.962,36.1187] [11.142,15.7529] [8.2531,10.3611] [6.3881,8.0239] [3.9332,4.1469]

[0.0495,0.0485] [0.0674,0.0549] [0.0604,0.0843] [0.0667,0.063] [0.076,0.0931] [0.0788,0.0797] [0.1116,0.0971] [0.1076,0.1198] [0.124,0.1171] [0.1218,0.1341]

[0.0299,0.0247] [0.0174,0.032] [0.0277,0.0169] [0.0264,0.0215] [0.0123,0.0085] [0.056,0.0452] [0.0296,0.049] [0.0424,0.0353] [0.0221,0.0303] [0.0238,0.0188]

[0.9512,0.9588] [0.9512,0.9694] [0.9503,0.9868] [0.9624,0.9528] [0.9536,0.9733] [0.9603,0.9561] [0.9505,0.9747] [0.9608,0.9778] [0.9517,0.963] [0.9518,0.9601]

[0.0999,0.0041] [0.097,0.0372] [0.0957,0.0475] [0.0944,0.0174] [0.0803,0.0622] [0.0918,0.0159] [0.0995,0.0365] [0.0854,0.0368] [0.0806,0.0249] [0.0903,0.0756]

Product acceptance determination using repetitive sampling

0.03

51

52

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 3.2 Plan parameters of neutrosophic sampling plan using loss function when β ¼ 0.10. p1

p2

nN

kaN

L(CAQL)

L(CLQL)

0.001

0.002 0.003 0.004 0.006 0.008 0.01 0.15 0.02 0.005 0.01 0.15 0.2 0.25 0.3 0.5 0.001 0.015 0.02 0.03 0.04 0.05 0.1 0.02 0.03 0.04 0.05 0.1 0.15 0.2 0.06 0.09 0.12 0.15 0.3 0.1 0.15 0.2 0.25 0.5

[49,54] [18,38] [12,27] [7,15] [6,8] [5,10] [4,6] [4,6] [39,67] [11,16] [7,10] [6,7] [5,6] [5,8] [4,5] [39,76] [16,27] [11,19] [7,11] [6,8] [5,8] [4,4] [41,56] [16,25] [11,17] [9,14] [5,7] [4,6] [4,4] [37,52] [16,21] [11,11] [8,10] [5,6] [39,50] [16,22] [11,11] [8,11] [5,6]

[0.0014,0.0014] [0.0015,0.0018] [0.0016,0.0017] [0.0019,0.0025] [0.0021,0.002] [0.0021,0.0021] [0.002,0.0034] [0.0025,0.0048] [0.0035,0.0036] [0.0042,0.0044] [0.0046,0.0051] [0.0049,0.0062] [0.0048,0.006] [0.0062,0.0078] [0.0056,0.0083] [0.007,0.0075] [0.0079,0.0085] [0.0086,0.0105] [0.0092,0.0129] [0.0102,0.0119] [0.0102,0.0143] [0.0102,0.0111] [0.0136,0.0148] [0.0157,0.017] [0.0174,0.0208] [0.0174,0.023] [0.0202,0.0262] [0.0213,0.0376] [0.0278,0.0258] [0.0415,0.0436] [0.0474,0.0504] [0.05,0.0512] [0.0528,0.0574] [0.061,0.0621] [0.0698,0.0711] [0.0787,0.087] [0.0839,0.085] [0.0885,0.1032] [0.0986,0.1111]

[0.9596,0.9856] [0.9522,0.9982] [0.9502,0.991] [0.9576,0.9992] [0.9745,0.9721] [0.9641,0.9893] [0.9507,0.9989] [0.9814,1] [0.958,0.9913] [0.9544,0.9777] [0.9568,0.9848] [0.9635,0.9915] [0.9511,0.9869] [0.986,0.9993] [0.9709,0.9978] [0.9608,0.9975] [0.9556,0.9906] [0.9579,0.9979] [0.954,0.9984] [0.9691,0.9921] [0.9621,0.9982] [0.9571,0.9691] [0.951,0.9915] [0.9518,0.9883] [0.9611,0.9965] [0.952,0.9978] [0.961,0.9945] [0.9633,0.9996] [0.9889,0.9839] [0.9517,0.9858] [0.9539,0.9813] [0.9504,0.9567] [0.9502,0.976] [0.9622,0.9704] [0.9589,0.9786] [0.9526,0.988] [0.9522,0.9556] [0.9516,0.9881] [0.9571,0.9795]

[0.0513,0.0753] [0.068,0.0273] [0.0651,0.0064] [0.097,0.0372] [0.0984,0.0382] [0.0946,0.0111] [0.0862,0.0705] [0.081,0.0807] [0.0952,0.0487] [0.0868,0.0415] [0.0961,0.0543] [0.0851,0.0949] [0.0834,0.0805] [0.0965,0.0452] [0.0704,0.0662] [0.0996,0.0601] [0.0969,0.0483] [0.0906,0.0684] [0.0933,0.0912] [0.0913,0.0646] [0.0926,0.0579] [0.0614,0.069] [0.0752,0.0901] [0.0923,0.0563] [0.0945,0.0802] [0.0738,0.0716] [0.0915,0.0656] [0.096,0.0872] [0.0937,0.0845] [0.0985,0.085] [0.0947,0.0757] [0.0828,0.0892] [0.0985,0.0776] [0.0926,0.0591] [0.0966,0.0747] [0.0931,0.0831] [0.0846,0.0881] [0.0999,0.0803] [0.088,0.0685]

0.0025

0.005

0.01

0.03

0.05

minimize the ASN. Table 3.1 [6] was generated by when α ¼ 0.05, β ¼ 0.10 for a known sigma case and for the specified values of the AQL and LQL. The ASNs have been listed as 80.20 for the symmetric case of the variable repetitive group sampling when p1 ¼ 0.001, p2 ¼ 0.002,n ¼ 41, Ka ¼ 3.08,and Kr ¼ 2.97. Here again the values of

p1

p2

nN

ASNN

kaN

krN

L(CAQL)

L(CLQL)

0.001

0.002 0.003 0.004 0.006 0.008 0.01 0.15 0.02 0.005 0.01 0.15 0.2 0.25 0.3 0.5 0.001 0.015 0.02 0.03 0.04 0.05 0.1

[18,82] [20,14] [9,15] [6,9] [4,6] [4,6] [4,6] [3,4] [28,50] [9,10] [5,6] [5,4] [4,3] [4,4] [3,3] [34,49] [8,14] [9,11] [4,8] [4,4] [4,5] [3,3]

[36.2529,85.8292] [20.8495,22.2598] [10.5943,16.2526] [7.6599,11.4204] [5.7684,7.2503] [5.1002,6.4583] [4.2594,6.3737] [3.6978,4.539] [34.7316,61.9036] [10.4462,15.5487] [7.3775,8.3231] [6.069,6.4966] [5.1794,5.2165] [4.8763,6.4857] [3.5916,3.6367] [43.2454,56.2128] [14.7349,18.6491] [10.5001,13.5529] [6.5117,9.2861] [5.3758,6.6507] [5.1504,5.7462] [3.5897,4.1818]

[0.0019,0.0015] [0.0017,0.0023] [0.0019,0.002] [0.0027,0.0031] [0.003,0.0029] [0.0029,0.0023] [0.0022,0.0031] [0.0035,0.0033] [0.0038,0.004] [0.0047,0.0069] [0.0072,0.0072] [0.0067,0.009] [0.0077,0.0096] [0.0079,0.0135] [0.0074,0.0072] [0.0079,0.0076] [0.0115,0.01] [0.0099,0.011] [0.0142,0.0122] [0.0134,0.0188] [0.0152,0.0137] [0.0151,0.0224]

[0.0008,0.0009] [0.0015,0.0006] [0.001,0.0008] [0.0009,0.0003] [0.0002,0.0005] [0.0005,0.0006] [0.0012,0.0003] [0.0005,0.0001] [0.0026,0.0023] [0.0025,0.0008] [0.001,0.0006] [0.0021,0.0002] [0.0014,0.0002] [0.0019,0.0004] [0.0011,0.0007] [0.0053,0.0041] [0.003,0.0034] [0.0057,0.0027] [0.0014,0.002] [0.0021,0.0006] [0.0029,0.0044] [0.0026,0.0002]

[0.9752,0.9873] [0.9813,0.9907] [0.963,0.9779] [0.9895,0.9912] [0.9532,0.9892] [0.9807,0.9571] [0.9641,0.9785] [0.99,0.9524] [0.9541,0.9933] [0.9553,0.9748] [0.977,0.9534] [0.985,0.9614] [0.9863,0.974] [0.9913,0.9993] [0.976,0.9609] [0.9799,0.9634] [0.9685,0.9666] [0.9703,0.9629] [0.9591,0.9535] [0.9644,0.9744] [0.986,0.9874] [0.9803,0.9818]

[0.045,0] [0.0499,0.0032] [0.0326,0.0012] [0.0406,0.0001] [0.0124,0.0057] [0.0297,0.0035] [0.0448,0.0002] [0.0489,0.0011] [0.0315,0.0009] [0.0312,0.0003] [0.0188,0.002] [0.035,0.0046] [0.0334,0.0219] [0.0391,0.0049] [0.0398,0.0236] [0.019,0.0002] [0.0383,0.0036] [0.0487,0.0013] [0.0334,0.0008] [0.0327,0.0056] [0.0351,0.0241] [0.0457,0.0046]

0.0025

0.005

Product acceptance determination using repetitive sampling

Table 3.3 Plan parameters of repetitive neutrosophic sampling plan using loss function when β ¼ 0.05.

Continued

53

54

Table 3.3 Continued p2

nN

ASNN

kaN

krN

L(CAQL)

L(CLQL)

0.01

0.02 0.03 0.04 0.05 0.1 0.15 0.2 0.06 0.09 0.12 0.15 0.3 0.1 0.15 0.2 0.25 0.5

[19,33] [11,16] [7,6] [7,4] [4,4] [3,3] [3,3] [16,39] [11,14] [6,8] [5,9] [4,4] [18,35] [7,15] [7,9] [5,7] [3,3]

[33.8181,50.2974] [14.2608,20.0446] [9.099,13.8833] [8.0778,9.6243] [4.6872,5.2333] [3.782,3.9528] [3.4863,3.6914] [31.2697,48.6093] [14.7702,19.2167] [9.0296,10.8401] [7.4566,9.8891] [4.7041,5.1007] [31.4967,49.8426] [13.1447,18.6673] [9.2821,12.5568] [7.1693,9.4086] [4.3562,5.1404]

[0.0176,0.0171] [0.0185,0.0194] [0.0204,0.0325] [0.0199,0.036] [0.0238,0.0302] [0.0282,0.0315] [0.0249,0.0301] [0.0545,0.0469] [0.0572,0.0616] [0.0686,0.0663] [0.0758,0.0578] [0.0717,0.0882] [0.0873,0.0835] [0.1134,0.0965] [0.1046,0.1198] [0.1183,0.1258] [0.1413,0.1848]

[0.008,0.0072] [0.009,0.0075] [0.0081,0.0017] [0.0097,0.0009] [0.0076,0.0034] [0.0039,0.003] [0.0042,0.002] [0.0235,0.0239] [0.0244,0.018] [0.0182,0.0168] [0.0163,0.026] [0.0219,0.0165] [0.0429,0.0433] [0.0267,0.0513] [0.0389,0.0216] [0.0261,0.0205] [0.0116,0.0031]

[0.9592,0.9681] [0.9548,0.9742] [0.9532,0.962] [0.9561,0.9584] [0.9637,0.9757] [0.9682,0.9758] [0.9518,0.9599] [0.9581,0.9553] [0.9559,0.9628] [0.9578,0.9531] [0.9677,0.9533] [0.9633,0.9827] [0.9611,0.9842] [0.9521,0.9838] [0.9567,0.9583] [0.9525,0.96] [0.9532,0.9578]

[0.029,0.0007] [0.0341,0.0028] [0.0459,0.0037] [0.0366,0.013] [0.0475,0.017] [0.0487,0.0384] [0.0362,0.0184] [0.0493,0.0006] [0.0244,0.002] [0.0458,0.0099] [0.0462,0.0092] [0.0452,0.0326] [0.0486,0.0031] [0.0477,0.0199] [0.0422,0.0022] [0.041,0.0043] [0.0497,0.0159]

0.03

0.05

Optimization Theory Based on Neutrosophic and Plithogenic Sets

p1

Product acceptance determination using repetitive sampling

55

Table 3.4 Plan parameters of neutrosophic sampling plan using loss function when β ¼ 0.05. p1

p2

nN

kaN

L(CAQL)

L(CLQL)

0.001

0.002 0.003 0.004 0.006 0.008 0.01 0.15 0.02 0.005 0.01 0.15 0.2 0.25 0.3 0.5 0.001 0.015 0.02 0.03 0.04 0.05 0.1 0.02 0.03 0.04 0.05 0.1 0.15 0.2 0.06 0.09 0.12 0.15 0.3 0.1 0.15 0.2 0.25 0.5

[49,102] [25,35] [14,29] [9,17] [7,11] [7,11] [5,7] [5,7] [49,68] [15,25] [9,17] [7,12] [6,8] [6,8] [5,6] [52,82] [21,32] [14,22] [9,13] [7,11] [6,8] [5,6] [49,61] [21,31] [14,19] [11,14] [6,8] [5,6] [5,5] [48,64] [20,27] [14,18] [11,14] [6,8] [48,64] [20,24] [14,16] [11,12] [6,7]

[0.0013,0.0015] [0.0016,0.0016] [0.0016,0.002] [0.0018,0.0024] [0.0018,0.0027] [0.0018,0.0031] [0.0021,0.0028] [0.0019,0.003] [0.0033,0.0036] [0.0043,0.0051] [0.0044,0.0049] [0.0046,0.0064] [0.0047,0.0051] [0.0049,0.0072] [0.0061,0.0064] [0.0068,0.0073] [0.0076,0.0081] [0.0081,0.01] [0.009,0.011] [0.0093,0.012] [0.0093,0.01] [0.0104,0.0109] [0.0134,0.0139] [0.0152,0.015] [0.0168,0.017] [0.0179,0.021] [0.0187,0.0254] [0.0201,0.0277] [0.0191,0.0268] [0.0402,0.0405] [0.0454,0.0454] [0.0493,0.0539] [0.0506,0.0578] [0.0558,0.0704] [0.0668,0.0705] [0.0756,0.0768] [0.0802,0.0836] [0.0875,0.0902] [0.094,0.1155]

[0.9512,0.9991] [0.9757,0.9871] [0.9533,0.9991] [0.9558,0.9993] [0.9508,0.9989] [0.9516,0.9998] [0.9683,0.9967] [0.9546,0.9979] [0.951,0.9922] [0.9741,0.999] [0.9539,0.9932] [0.9559,0.9988] [0.9559,0.9782] [0.9607,0.9984] [0.9838,0.9913] [0.967,0.997] [0.9539,0.9884] [0.9537,0.9976] [0.9594,0.9954] [0.9565,0.9967] [0.9511,0.9749] [0.9655,0.9776] [0.9548,0.9809] [0.9563,0.9719] [0.964,0.9798] [0.9676,0.9943] [0.9533,0.995] [0.9606,0.9947] [0.9517,0.9905] [0.952,0.9727] [0.9512,0.9679] [0.9586,0.9864] [0.9539,0.9875] [0.9516,0.9911] [0.9512,0.9861] [0.9513,0.9664] [0.9513,0.9693] [0.9627,0.9727] [0.954,0.9871]

[0.0443,0.0274] [0.0366,0.0129] [0.0421,0.0152] [0.0454,0.0216] [0.046,0.0392] [0.0266,0.0301] [0.0493,0.0287] [0.0252,0.0157] [0.0442,0.0493] [0.0478,0.0308] [0.0444,0.0077] [0.0486,0.0257] [0.0495,0.0229] [0.0354,0.0362] [0.0379,0.0211] [0.0492,0.04] [0.0436,0.0214] [0.0423,0.0368] [0.0476,0.0346] [0.049,0.0265] [0.047,0.0214] [0.0284,0.0147] [0.047,0.0416] [0.0453,0.0133] [0.0495,0.0226] [0.0498,0.0498] [0.0481,0.0418] [0.0451,0.0467] [0.0244,0.045] [0.048,0.0264] [0.0491,0.0223] [0.0454,0.0353] [0.0406,0.0347] [0.0473,0.0337] [0.0474,0.0431] [0.0491,0.0342] [0.0409,0.0343] [0.0461,0.0407] [0.0485,0.0486]

0.0025

0.005

0.01

0.03

0.05

56

Optimization Theory Based on Neutrosophic and Plithogenic Sets

ASN decreases from 80.20 to 3.50 as p2 values increases from 0.002 to 0.020. Tables 3.3 and 3.4 have included the other ASN values under different plan settings.

3.4

Comparative study

In this section, a comparative study of the proposed repetitive neutrosophic sampling plan with the existing neutrosophic sampling plan given by Aslam [34] is discussed. As described by Chen et al. [39], under the situation of uncertainty and an unclear environment, a procedure that provides the plan parameters in indeterminacy interval rather than the determined values is known as the most adequate and effective procedure. Thus, the neutrosophic plan parameters under the process loss function have been placed in Tables 3.1–3.4 under the indeterminacy interval. Table 3.5 has been constructed for the comparison purpose of the proposed and the existing sampling plans for ready reference. Using the same p1 and p2 values, the ASNs of the proposed plan and existing plan have been calculated using the R-language code program. The same settings of parameters for the proposed plan and the existing plan have been ensured so that the real and exact difference and comparison be seen. For this purpose, the calculations of both plans have been determined and cross-checked for similarity of conditions. It can be easily observed that under the same values of α ¼0.05 and β ¼ 0.10, the proposed plan is effective for sentencing the submitted lots with smaller sample sizes than the plan proposed by Aslam [34]. For example, when p1 ¼ 0.001 and p2 ¼ 0.003, the proposed plan declares the submitted lot need a sample between 8 and 29, whereas the existing sampling plan provides this sample size between 18 and 38. Fig. 3.1 has been generated for the comparison of the proposed and existing sampling plans. The lower and upper ASN are shown for both proposed and existing sampling plans. From Fig. 3.1 it can be observed very easily that the proposed repetitive neutrosophic sampling plan under process loss consideration is better in the smaller ASN. From this comparison, it can be concluded that the proposed procedure of the repetitive neutrosophic sampling plan is effective and flexible compared to the existing procedure of the neutrosophic sampling plan under an uncertain environment.

3.5

Application of the proposed plan

The application of the proposed sampling plan is given using the amplified pressure sensor selected from a company located in Taiwan. The company manufactured the amplified pressure sensor product with target T ¼ 10 V, upper specification limit (USL) ¼12 V, and lower specification limit (LSL) ¼10 V. Yen and Chang [38] used the amplified pressure sensor for the single sampling plan under classical statistics. As the amplified pressure sensor is recorded with the help of instruments, not all observations in the recorded data are precise, determined, or clear. There is a chance that observations are in an interval rather than the exact values. Therefore, when observations are uncertain in the data, the sampling plan under classical statistics cannot be applied for the inspection of the lot of the product. Suppose that the experimenter is interested to use the proposed plan for the inspection of the amplified pressure sensor

Product acceptance determination using repetitive sampling

57

Table 3.5 Comparison of the proposed and the existing plans. p1

p2

Proposed

Existing

0.001

0.002 0.003 0.004 0.006 0.008 0.01 0.15 0.02 0.005 0.01 0.15 0.2 0.25 0.3 0.5 0.001 0.015 0.02 0.03 0.04 0.05 0.1 0.02 0.03 0.04 0.05 0.1 0.15 0.2 0.06 0.09 0.12 0.15 0.3 0.1 0.15 0.2 0.25 0.5

[27,88] [8,29] [8,13] [6,10] [5,6] [4,6] [3,3] [3,3] [24,27] [5,8] [5,8] [3,5] [3,4] [3,4] [3,3] [22,38] [8,6] [6,10] [5,6] [4,4] [3,3] [3,3] [19,38] [9,17] [5,10] [5,5] [3,3] [3,3] [3,3] [17,30] [6,13] [6,6] [5,7] [3,3] [22,26] [6,12] [6,7] [4,6] [3,3]

[49,54] [18,38] [12,27] [7,15] [6,8] [5,10] [4,6] [4,6] [39,67] [11,16] [7,10] [6,7] [5,6] [5,8] [4,5] [39,76] [16,27] [11,19] [7,11] [6,8] [5,8] [4,4] [41,56] [16,25] [11,17] [9,14] [5,7] [4,6] [4,4] [37,52] [16,21] [11,11] [8,10] [5,6] [39,50] [16,22] [11,11] [8,11] [5,6]

0.0025

0.005

0.01

0.03

0.05

product when AQL ¼ 0.001, LQL ¼ 0.002, α ¼ 5%, and β ¼ 10%. From Table 3.1, we note that nE{27, 83}, krNE{0.002,0.0015}, and kaNE{0.0011,0.0007} at these specified parameters. This means that the experimenter can select any sample from 27 to 83. Suppose that the experimenter decided to select a random sample of size n ¼ 50. The data having neutrosophic observations have been given in Table 3.6.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Lower ASN of Upper ASN of Lower ASN of Upper ASN of

100

proposed sampling plan proposed sampling plan existing sampling plan existing sampling plan

ASN’s

80

60 40

20

0 0.000

0.010

0.005

0.015

0.020

p2

Fig. 3.1 Graphical comparison of the proposed and the existing plans.

Table 3.6 Data for example. XL

XU

XL

XU

10.3556 10.1252 10.3158 10.0274 10.6130 10.0777 10.0900 10.2715 10.3930 10.0457 10.2255 10.3361 10.3715 10.3187 10.2158 10.7686 10.0387 10.0502 10.1261 10.4034 10.2279 10.0122 10.5616 10.1222 10.2138

10.4250 10.2132 10.8973 10.0631 10.7452 10.6895 10.1129 10.4185 10.9496 10.2681 10.4711 10.6050 10.6311 10.5770 10.9043 10.9396 10.1056 10.1265 10.6947 10.5254 10.2324 10.0166 10.7732 10.2772 10.9967

10.5379 10.1857 10.4178 10.3705 10.2038 10.0225 10.0341 10.5563 10.0482 10.0788 10.1768 10.3196 10.2496 10.1129 10.0075 10.7149 10.0978 10.2696 10.2302 10.2601 10.0170 10.3632 10.0235 10.2570 10.4246

10.8046 10.2160 10.6802 10.5650 10.3446 10.0990 10.1974 10.5819 10.6652 10.8806 10.8255 10.5323 10.3244 10.5233 10.0116 10.7417 10.4074 10.4539 10.8078 10.9192 10.0426 10.5428 10.3450 10.8058 10.9417

0.025

Product acceptance determination using repetitive sampling

59

The neutrosophic descriptive statistics for the data given in Table 3.6 are XN Ef10:24574, 10:51836 g and s2 N ¼f0:03659062 0:08643689g. The neutrosophic 2 2 Þ ^ ^ Ne ¼ SN2 + ðXN 2T statistic with d¼ 1 is calculated as follows: L ¼LNe E 2 d d f0:09697863, 0:3551381g. We note that the product of the amplified pressure sensor ^Ne Ef0:09697863, 0:3551381g >krN Ef0:002;0:0015g. As the lot should be rejected as L of the product is rejected on the basis of the first sample, we do not need to repeat the sampling plan. The existing sampling plan proposed by Aslam [34] gives the same decision when nE{49, 94}. By comparing both sampling plans, we conclude that the proposed plan needs a smaller sample size than the existing plan to reach the same decision.

3.6

Concluding remarks

In this chapter, a new sampling plan for the repetitive neutrosophic sampling under process loss consideration was proposed. The plan parameters were estimated for different AQL and LQLs. The plan was studied for the producer’s and the consumer’s risks by developing the inequalities to calculate the probabilities. The comparison of the proposed and the existing plans was discussed. It was observed that the proposed plan is a useful addition to quality control personnel as it provides the smaller sample size for the sentencing of the submitted lots compared to the existing procedure of the neutrosophic sampling plan. Numerical tables were provided for different practical conditions for the implementation of the proposed plan under neutrosophic statistics. Thus a wide variety of situations can be dealt for lot sentencing efficiently and accurately under the repetitive sampling scheme. It can be observed from the proposed plan that it utilizes smaller ASNs compared to traditional and existing plans. The application of the proposed plan can reduce the time, inspection cost, and efforts in an environment involving uncertainty. The proposed sampling plan can further be extended to some other sampling schemes.

References [1] C.D. Montgomery, Introduction to Statistical Quality Control, John Wiley & Sons, Inc., New York, 2009. [2] S. Balamurali, H. Park, C.H. Jun, K.J. Kim, J. Lee, Designing of variables repetitive group sampling plan involving minimum average sample number, Commun. Stat.—Simul. Comput. 34 (2005) 799–809. [3] R.E. Sherman, Design and evaluation of a repetitive group sampling plan, Technometrics 7 (1965) 11–21. [4] M. Aslam, S. Niaki, M. Rasool, M. Fallahnezhad, Decision rule of repetitive acceptance sampling plans assuring percentile life, Scient. Iran. 19 (2012) 879–884. [5] M. Azam, O.H. Arif, M. Aslam, W. Ejaz, Repetitive acceptance sampling plan based on exponentially weighted moving average regression estimator, J. Comput. Theor. Nanosci. 13 (2016) 4413–4426.

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[6] S.-H. Lee, M. Aslam, C.-H. Jun, Repetitive group sampling plans with double specification limits, Pak. J. Statist. 28 (2012) 41–57. [7] M. Aslam, Y.L. Lio, C.-H. Jun, Repetitive acceptance sampling plans for burr type XII percentiles, Int. J. Adv. Manuf. Technol. 68 (2013) 495–507. [8] M. Aslam, S.T.A. Niaki, M. Rasool, M.S. Fallahnezhad, Decision rule of repetitive acceptance sampling plans assuring percentile life, Scient. Iran. Trans. E, Ind. Eng. 19 (2012) 879. [9] M. Aslam, M. Ahmad, A.R. Mughal, Group acceptance sampling plan for lifetime data using generalized pareto distribution, Pak. J. Commer. Soc. Sci. 4 (2010) 185–193. [10] S.A. Dobbah, M. Aslam, K. Khan, Design of a new synthetic acceptance sampling plan, Symmetry 10 (2018) 653. [11] B.P. Duarte, P.M. Saraiva, An optimization-based approach for designing attribute acceptance sampling plans, Int. J. Qual. Reliab. Manage. 25 (2008) 824–841. [12] M. Aslam, S. Balamurali, C.-H. Jun, A. Meer, Time-truncated attribute sampling plans using EWMA for Weibull and Burr type X distributions, Commun. Stat.-Simulat. Comput. (2016) 1–12. [13] W. Pearn, C.-W. Wu, Critical acceptance values and sample sizes of a variables sampling plan for very low fraction of defectives, Omega 34 (2006) 90–101. [14] M. Aslam, M. Azam, S. Balamurali, C.-H. Jun, A new mixed acceptance sampling plan based on sudden death testing under the Weibull distribution, J. Chin. Inst. Ind. Eng. 29 (2012) 427–433. [15] M. Aslam, C.-H. Yen, C.-H. Chang, C.-H. Jun, M. Ahmad, M. Rasool, Two-stage variables acceptance sampling plans using process loss functions, Commun. Stat.-Theory Methods 41 (2012) 3633–3647. [16] C.-W. Wu, W.L. Pearn, A variables sampling plan based on C pmk for product acceptance determination, Eur. J. Oper. Res. 184 (2008) 549–560. [17] C.-H. Jun, S. Balamurali, S.-H. Lee, Variables sampling plans for Weibull distributed lifetimes under sudden death testing, IEEE Trans. Reliab. 55 (2006) 53–58. [18] W. Seidel, A possible way out of the pitfall of acceptance sampling by variables: treating variances as unknown, Comput. Stat. Data Anal. 25 (1997) 207–216. [19] E.V. Collani, A note on acceptance sampling for variables, Metrika 38 (1991) 19–36. [20] K. Suresh, S. Devaarul, Designing and selection of mixed sampling plan with chain sampling as attribute plan, Qual. Eng. 15 (2002) 155–160. [21] Y. Li, X. Pu, D. Xiang, Mixed variables-attributes test plans for single and double acceptance sampling under exponential distribution, Math. Probl. Eng. 2011 (2011). [22] A. Mussidaa, U. Gonzales-Barron, F. Butler, Operating characteristic curves for single, double and multiple fraction defective sampling plans developed for Cronobacter in powder infant formula, Proc. Food Sci. 1 (2011) 979–986. [23] G. Gregory, G. Resnikoff, Some notes on mixed variables and attributes sampling plans, Technical Report No. 10, Applied Mathematics and Statistics Laboratory, Stanford University, CA, 1955. [24] J. Ye, J. Chen, R. Yong, S. Du, Expression and analysis of joint roughness coefficient using neutrosophic number functions, Information 8 (2017) 69. [25] F. Smarandache, Neutrosophy: neutrosophic probability, set, and logic: analytic synthesis & synthetic analysis, (1998). https://arxiv.org/ftp/math/papers/0101/0101228.pdf. [26] H. Wang, F. Smarandache, R. Sunderraman, Y.-Q. Zhang, Interval Neutrosophic Sets and Logic: Theory and Applications in Computing: Theory and Applications in Computing, Infinite Study, 2005.

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[27] W. Haibin, F. Smarandache, Y. Zhang, R. Sunderraman, Single Valued Neutrosophic Sets, Infinite Study, 2010. [28] F. Smarandache, A geometric interpretation of the neutrosophic set—A generalization of the intuitionistic fuzzy set, (2004). arXiv preprint math/0404520. [29] F. Smarandache, Neutrosophic set—a generalization of the intuitionistic fuzzy set, J. Defense Resour. Manage. 1 (2010) 107. [30] F. Smarandache, n-Valued Refined Neutrosophic Logic and Its Applications to Physics, Infinite Study, 2013. https://arxiv.org/ftp/arxiv/papers/1407/1407.1041.pdf. [31] J. Chen, J. Ye, S. Du, Scale effect and anisotropy analyzed for neutrosophic numbers of rock joint roughness coefficient based on neutrosophic statistics, Symmetry 9 (2017) 208. [32] M. Aslam, N. Khan, M. Albassam, Control chart for failure-censored reliability tests under uncertainty environment, Symmetry 10 (2018) 690. [33] M. Aslam, N. Khan, M. Khan, Monitoring the variability in the process using neutrosophic statistical interval method, Symmetry 10 (2018) 562. [34] M. Aslam, A new sampling plan using neutrosophic process loss consideration, Symmetry 10 (2018) 132. [35] M. Aslam, A. Al-Marshadi, Design of sampling plan using regression estimator under indeterminacy, Symmetry 10 (2018) 754. [36] M. Aslam, O. Arif, Testing of grouped product for the weibull distribution using neutrosophic statistics, Symmetry 10 (2018) 403. [37] T. Johnson, The relationship of Cpm to squared error loss, J. Qual. Technol. 24 (1992) 211–215. [38] C.-H. Yen, C.-H. Chang, Designing variables sampling plans with process loss consideration, Commun. Stat.-Simulat. Comput. 38 (2009) 1579–1591. [39] J. Chen, J. Ye, S. Du, R. Yong, Expressions of rock joint roughness coefficient using neutrosophic interval statistical numbers, Symmetry 9 (2017) 123.

Intelligent and adaptive microservices and neutrosophic-based learning management systems

4

Haitham A. El-Ghareeb Information Systems Department, Faculty of Computers and Information Sciences, Mansoura University, Mansoura, Egypt

4.1

Introduction

Adaptive eLearning that is supported by intelligent techniques and methods is one of the ways to support personalized eLearning. Adaptive eLearning overcomes many eLearning limitations and solves some of its challenges. This chapter reviewed different adaptive and intelligent eLearning systems that were proposed over a long period of eLearning research. This chapter introduces novel Neutrosophic and Microservicesbased eLearning system to overcome many of the technical and pedagogical eLearning challenges. Technical challenges includes reusability, scalability, integration, and interoperability. New learning model that attempts to solve different challenges is presented. Though there is no single unified learning model that can be the only right model, this chapter is a step toward a better learning model supported by appropriate adaptive and intelligent features. Most of the artificial intelligence (AI) applications have not yet been expanded to or adopted in widely used eLearning systems, especially open-source systems such as Moodle and Sakai. Current intelligent learning management systems (LMSs) are still in their early stages. AI applications need to handle some problems or to be modified before applying them into LMSs, and AI technology also needs to be brought to opensource communities. The presented adaptive eLearning model integrates different intelligent features, mainly Neutrosophic theory within the system to empower the presented model.

4.1.1 Proposed model This chapter presents innovative intelligent features to improve students’ and instructors’ performance and enhance the presented adaptive eLearning model. Neutrosophic theory utilization in different Microservices to present intelligent features is highlighted in different aspects. Presenting adaptive and intelligent features as Microservices

Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00004-4 © 2020 Elsevier Inc. All rights reserved.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

with standard interfaces will allow different eLearning systems to adopt them, so they will be reusable. This chapter goes as follows: Section 4.2 discusses different eLearning aspects, mainly pedagogical, technical, adaptive, and intelligent. Section 4.3 presents the basic ideas of Neutrosophic Sets and Theory, focusing on what will be utilized in this chapter. Section 4.4 illustrates the importance of moving eLearning systems from monolithic systems to Microservices-based ones. Section 4.5 discusses the proposed adaptive eLearning model that is enabled via the presented intelligent Microservices. Section 4.6 presents novel neutrosophic utilization in empowering adaptive eLearning with intelligent services. Section 4.7 presents the evaluation of the proposed novel adaptive eLearning model and neutrosophic-based intelligent microservices. The chapter ends with a conclusion and references.

4.2

eLearning

eLearning can be thought of as the learning process created by interaction with digitally delivered content, services, and support. eLearning involves intensive usage of Information and Communication Technology to serve, facilitate, and revolutionize the learning process [1–3]. Learning methods include traditional learning (face-to-face), distance learning (complete asynchronous; both time and place independent learning delivery; mainly online), and blended learning that combines instruction-led learning with online learning activities leading to reduced classroom contact hours. Blended learning has the potential to increase student learning while lowering attendance rates compared to equivalent fully online courses [1]. Blended learning is the learning paradigm that attempts to optimize both traditional learning and distance learning advantages while eliminating learning paradigm shortages and challenges. When compared to traditional learning paradigms, blended learning is found to be consistent with the values of traditional learning paradigms adopted in almost all higher education learning institutions for decades, and has the proven potential to enhance both the effectiveness and efficiency of meaningful learning experiences [2].

4.2.1 Learning management systems An LMS is the software that automates the administration of education. An LMS registers students, tracks courses in a catalog, records data from learners, and provides reports to management. An LMS is typically designed to handle courses by multiple publishers and providers. It usually does not include its own authoring capabilities; instead it focuses on managing courses created by a variety of other learning resources. A prototypical LMS is presented in Riad [3]. Technology and the great advancement in recent web technologies and informal learning methods allowed complete education programs and courses to be presented online.

4.2.2 Pedagogical eLearning challenges Pedagogically, educational psychologists agree that students differ in the ways they learn and very few teachers can adapt learning to each student in typical large

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classes. Computer-based learning systems are criticized by many researchers for their limited adaptability of teaching actions compared to rich tactics and strategies employed by human expert teachers [4]. Many universities in developing countries started to adopt eLearning by modifying network infrastructure, establishing new labs, providing an internet connection, and purchasing different tools for creating eLearning courses and using different LMSs. However, these modifications and supplements were not enough to ensure successful eLearning outcomes because other important elements for eLearning success were missing such as flexibility of the system, adaptability toward students’ needs, reusability of learning objects (LOs), interoperability between LMSs, and effective and official design of e-Content [5]. Pedagogically, most training methods and technologies produce, at best, trained novices. That is, they introduce facts and concepts to students, present them with relatively simple questions to test this new knowledge, and provide them with a few opportunities to practice using this knowledge in exercises or scenarios. However, becoming proficient requires extensive proactive solving of realistically complex problems in a wide range of situations, combined with coaching and feedback from managers, more experienced peers, or other types of experts [6].

4.2.3 Adaptive eLearning Adaptive learning for students with many different backgrounds, learning styles, and interests is virtually a must. Educational psychologists by and large agree that students differ greatly in the ways they learn and very few teachers or professors can adapt learning to each student in typical large classes, as the costs associated with delivering different instruction for varied learning styles are prohibitive [7]. Benjamin Bloom [7a] showed 35 years ago, as reported in his 2 sigma paper, that almost all students can learn to a level of mastery, given the right learning environment [8]. In Bloom’s experiments, the most successful learning strategy was tutoring. Adaptive eLearning that is supported with intelligent techniques and methods is one way to support tutoring in eLearning, and thus may be the way to solve many of the limitations and challenges of eLearning today. Adaptive eLearning systems would be a good solution for better eLearning. The vast majority of web-enhanced courses rely on LMSs because they are powerful integrated systems that support a number of teachers’ and students’ needs. Though LMSs are doing great job, indeed for every function that a typical LMS performs there is an Adaptive Web-Based Educational System that can do it much better [9]. Adaptivity is the ability to modify eLearning lessons using different parameters and a set of predefined rules. Researchers differentiate slightly between adaptivity and adaptability by thinking about adaptability as the possibility for learners to personalize an eLearning lesson by themselves. These two approaches go from machine centered (adaptivity) to learner centered (adaptability). In practice, it is quite difficult to isolate one from the other due to their close relationship [10, 11]. Adaptive eLearning is often meant to be new or in an early development stage [4]. An adaptive eLearning system is the environment of software modules, which comprises a set of features for adaptivity and adaptability [12].

66

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Important factors for adapting to student needs and desires include the following [7]: l

l

l

l

l

Each student should move at a unique pace: Given all the variations between students backgrounds, interests, and abilities, it is highly desirable to allow each student to move at a unique pace in the learning units. Adaptation should be very frequent: Changes based on occasional exams are inadequate. Learning activities should adapt to each student on a moment-by-moment basis. Students should feel that the adaptive program is responding to them as individuals. Each student should be successful in learning: A major advantage of adaptive variable placing is that the students can continue to learn in a given area until they have learned the material. Almost all learners can succeed and achieve mastery, but some learners need more time and more practice than others. When something is successfully learned, the learner should move on: Often in classroom learning, after a student has learned something, the class continues working on the topic, boring the student. This will not occur in a fully adaptive learning environment. No one should be taught something he or she already knows: By assuring learner competencies, avoiding unneeded instruction, and moving each student forward when ready, students are expected to achieve a major reduction of learning time, but this cannot be verified empirically until there is a full range of computer-based adaptive learning units.

The provision of static learning material will not meet the requirements of students. Adaptive eLearning enables personalizing the learning process to individual learners via adapting some parameters, like identifying, analyzing, and monitoring relevant aspects of instructions, such as different velocities, paths, or strategies of learning. Performance improvements within the learning process can be gained via adaptive eLearning systems [12]. Adaptation and personalization will improve the learning process; therefore, a paradigm shift from the consumption of static learning contents to well-tailored and highly personalized learning sessions is needed. Over recent decades, various types of adaptation systems and possible areas for their applicability have been identified, leading to the emergence of specialized research fields, like adaptive hypermedia systems, computer-aided instruction, computer managed instruction, recommender systems, intelligent tutoring systems, personalized systems of instruction, and many others. Adaptive multimedia systems as an improved learning environment are well documented in the research work of G€ utl et al. [12].

4.2.4 Intelligent eLearning systems AI utilizes programming algorithms to simulate thought processes and reasoning that produce behavior similar to humans. The applications of AI within eLearning can produce the potential of creating realistic environments with which students can interact. The student essentially would interact with the intelligent agents, which in turn perceive changes in the simulated environment. The intelligent agents would then communicate perceived changes in the environment back to the student who in turn makes decisions based upon their own perceptions of the environment. For modern eLearning systems, we would refer to intelligent agents as intelligent services.

Intelligent and adaptive microservices and neutrosophic-based learning management systems

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Current learning technologies can help create trained novices more efficiently, but they are really not up to the job of creating true experts. For example, multimedia computer-based training (CBT) systems are good at presenting information and then testing factual recall using multiple choice or fill-in-the-blank questions. However, traditional CBT systems are incapable of providing the intelligent, individualized coaching, performance assessment, and feedback that students need to acquire deep expertise [6]. Employing state-of-the-art AI technology in current eLearning systems can bring personalized, adaptive, and intelligent services to both students and educators. Most AI applications have not yet been expanded to or adopted in widely used eLearning systems, especially open-source systems such as Moodle and Sakai. Current intelligent LMSs are still in their early stages, while AI applications need to handle some problems or to be modified before applying them into LMSs, and AI technology needs to be brought to open-source communities [13]. For detailed study on adaptive and intelligent eLearning systems, and how to integrate Web 3.0 and social networks to present personalized and adaptive eLearning systems, the reader is highly encouraged to review [14].

4.3

Neutrosophic theory

Neutrosophic sets have been introduced to the literature by Smarandache to handle incomplete, indeterminate, and inconsistent information [10]. Neutrosophic theory helps in addressing vagueness, inconsistencies, and missing information. These three challenges face intelligent Microservices, and need to be addressed carefully. In neutrosophic sets, indeterminacy is quantified explicitly through a new parameter I. Truth-membership (T), indeterminacy membership (I), and falsity-membership (F) are three independent parameters that are used to define a Neutrosophic Number. For detailed illustration, discussion, and examples of the utilized Neutrosophic Analytic Hierarchy Process that is utilized in the proposed eLearning model, the reader is referred to Nabeeh et al. [11, 15]. nD E o
(4.1) N ¼ x;TN ðxÞ,IN ðxÞ, FN ðxÞ ,x 2 X








x 2 X, TN ðxÞ, IN ðxÞ,FN ðxÞ 2 ½0,1








(4.2)

Deneutrosophy is the process where Neutrosophic scales/numbers are converted to crisp values by applying score functions of s(aij) as illustrated in Eq. (4.3) [16].   s aij ¼ 1 

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  Tx Þ2 + ðIx Þ2 + ðFx Þ2 3

(4.3)

Different Neutrosophic Numbers and Sets are available. They include the following, among others: l

l

single-valued neutrosophic number (SVNN); interval valued neutrosophic number (IVNN);

68 l

l

Optimization Theory Based on Neutrosophic and Plithogenic Sets

single-valued neutrosophic sets (SVNSs); and interval valued neutrosophic sets (IVNSs).

Proposed eLearning model utilizes the novel open-source Neutrosophic package presented in El-Ghareeb [16]. Proposed model utilizes SVNN and SVNS.

4.4

Services-based eLearning systems

4.4.1 Monolithic architecture A monolithic application is an application with a single large codebase/repository that offers tens or hundreds of services using different interfaces such as HTML pages, web services, and/or REST services [17]. A monolithic application has most of its functionality within a single process that is componentized with internal layers or libraries [18]. Monolithic applications scale out by cloning the application on multiple servers or multiple virtual machines. Monolithic applications rely heavily on layered architectures. Monolithic applications face many challenges. An intensive review study of eLearning systems convergence from traditional monolithic systems to services-based adaptive and intelligent systems is presented in Riad [3]. This study highlights the reasons and discusses the necessity of discarding monolithic design when dealing with eLearning systems.

4.4.2 Service-oriented architecture Service-oriented architecture (SOA) is a design pattern that presents IT infrastructure and information systems architecture as loosely coupled, fine granular services that can address system requirements once they are presented by either adding new services or modifying existing ones. SOA also addresses enterprises information systems inefficiency by enhancing reusability, thus theoretically shortening information systems development time and effort required. Besides reusability, interoperability and integration are the other main driving forces for adopting SOA in eLearning systems. W3C defines a Service as A Component capable of performing a task. Consumers need or want is satisfied via a service according to a negotiated contract (implicit or explicit) which includes service agreement, function offered, and so on. SOA is the design pattern that utilizes a services concept to achieve architectural advantages. W3C defines SOA as A set of components which can be invoked, and whose interface descriptions can be published and discovered. This definition can be expanded to include the science, art, and practice of building applications, so SOA can be defined as the policies, practices, and frameworks that enable application functionality to be provided and consumed as sets of services published at a granularity relevant to the service consumer. Services can be invoked, published, and discovered, and are abstracted away from the implementation using a single, standards-based form of interface [19].

4.4.3 Microservices architecture Either Microservices is an SOA done right, or it is a totally new architecture. Microservices has learned from SOA mistakes and failures [20–22]. One important rule for

Intelligent and adaptive microservices and neutrosophic-based learning management systems

69

Microservices is data sovereignty. Microservices is considered the next wave of services-based architecture that has moved SOA from promising architecture to a real success. Microservices architecture has moved from being a promising architecture to become a de facto standard in designing and developing distributed applications. Microservices architecture is used by large companies like Amazon, Netflix, and LinkedIn to deploy large applications in the cloud as a set of small services that can be developed, tested, deployed, scaled, operated, and upgraded independently, allowing these companies to gain agility, reduce complexity, and scale their applications in the cloud in a more efficient way [17]. Microservices can be thought of as an architectural style for developing applications as suites of small and independent (micro)services. Each Microservice is built around a business capability, it runs in its own process, and it communicates with the other Microservices in an application through lightweight mechanisms (e.g., HTTP APIs) [23].

4.5

Proposed adaptive eLearning model

This section presents an adaptive eLearning model as a solution to pedagogical eLearning challenges facing students, and an Adaptive Online Lecture as an enabler to instructors to address adaptivity features in eLearning in a new and innovative form.

4.5.1 Model components Proposed adaptive eLearning model components include: adaptive LMS, quality assurance and accreditation project (QAAP) management system, exam management system, and learning content management system (LCMS) as depicted in Fig. 4.1.

Adaptive LMS

QAAP

Exam management

Learning content management

Student learning subsystem

Course specification module

Exam data

Questions

Student learning profile subsystem

Instructor timetable module

Exam application

Learning objects

Fig. 4.1 Adaptive eLearning model components.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

The proposed adaptive eLearning model requires the integration of different systems to achieve the required goal. There are four subsystems: l

l

l

l

Adaptive LMS: Responsible for providing the adaptation features to each student via determining the learning road, topics and time required for each student based on performance, learning profile, and learning preferences. Adaptive LMS provides the basic functionalities provided by different LMSs in an adaptive manner. Adaptive LMS contains two subsystems: student learning and student learning profile. Quality assurance and accreditation project (QAAP) management system: An Egyptian National Initiative and Project that is maintained by the Egyptian Ministry of Higher Education, QAAP includes a course specification module and an instructor module. The course specification module focuses on defining and determining course content, learning objectives, and other course resources. The instructor module contains the instructor timetable that will be used to define suitable times for meetings between students and instructors. Exam management system: A blended model of online questions repository and desktop application delivery exam is used to overcome web-based exam systems’ vulnerabilities to cheating. Students will run the desktop application at exam times. The application will retrieve questions from online repositories. Those repositories are maintained by an LCMS. LCMS: This is critical and vital to the success of the proposed model implementation. The LCMS holds questions items, and LOs. The proposed adaptive eLearning model addresses extra needed metadata for questions and LOs to support needed adaptivity features. The LCMS focuses on providing a stand-alone LOs management that can be utilized by different LMSs. Though the LCMS is thought to be part of the LMS, it is best practice to provide it as a stand-alone system for two reasons: support different LMSs and isolate LOs metadata management from LMS.

4.5.2 Model scenarios To make the proposed model clearer, it will be illustrated in words describing what takes place with students in four different scenarios. Scenario 1 presents Student (A) who uses the system for the first time and has not built the learning profile yet. Scenario 2 continues with Student (A) in the learning phase. Scenario 3 presents Student (B) who is currently doing well through learning, and now has an exam due. Scenario 4 presents Student (C) who failed twice before in the exam, and is taking the exam for the third time.

4.5.2.1 Scenario 1: New student Student (A) attempts to log in but, as it is the first time, finds himself forced to register. During registration, the student completes the forms needed to identify students’ learning profiles and preferences. The second time the student logs in; the student learning subsystem tends to retrieve the student learning preferences from the student learning profile subsystem. If it is not complete, the system forces the student to complete it before starting to learn. Otherwise, the student learning system extracts the student information and registered courses, then checks if this student has an exam. Student (A) does not have an exam, so again, the student learning

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system checks if the student has a meeting with an instructor. Student (A) does not have a meeting; otherwise the system would have displayed the calendar.

4.5.2.2 Scenario 2: Studying student The student learning system calls up the QAAP course specification module, and acquires the course specifications and list of topics. The student learning system displays to the student the list of topics, which are available or unavailable as a result of the requirements’ prerequisites, so the student can identify his position on the roadmap. The student selects the topics to learn within the rules. The student learning profile is then updated with the topics selected and not selected. The student learning system displays the suitable learning material and a list of recommended learning materials and what previous students learning the same topics have seen learning materials and what other students currently learning the same topic have seen learning materials list. The student learning system qualifies the student to make sure he understands the learning objectives of the topic. If the student is not qualified, then he goes back to the study plan. The student can quit learning at any time and continue later. Then the student learning system checks if the topics that the student has learned form an exam; if yes, the student becomes eligible for an exam. The student can go through the learn via questions (LVQ) experiment. LVQ is a learning method that simulates the exam environment by presenting questions with feedback, so students can measure their readiness for an exam. The main objective of LVQ is helping students define their readiness level, not testing them. Then the student receives an exam date, and the learning process moves to Student (B).

4.5.2.3 Scenario 3: Due exam student Student (B) logs into the system and has a due exam, so the student learning system attempts to initialize the desktop application responsible for the examination process. The exam application retrieves questions from the LCMS questions repository based on the exam ID submitted by the student learning system. The exam application ranks the exam, and updates the student profile with this rank. The next time the student logs in, the student continues learning new topics.

4.5.2.4 Scenario 4: Suspended student Student (C) faces troubles with some topics. The student took the exam twice but did not pass. So, the third time the student logs in, attends the exam, and does not pass, an automatic initiation of the Intelligent Meeting Manager for Suspended Students occurs to arrange a meeting for this student with one of the instructors to help the student. The meeting details with the student’s detailed profile are mailed to the instructor. The next time the student logs in, he finds the system paused and the calendar is displayed, directing the student to a meeting with the instructor. Only the instructor can solve the situation after meeting the student by reactivating the student account after the proper action has been taken and recorded in the study profile. The instructor can illustrate the topic more than once to the student, examines the student orally, in written form, or by whatever method the instructor finds appropriate.

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4.5.3 Adaptive features in proposed model Adaptive features in the proposed adaptive eLearning model include the following: l

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Building learning profile and identifying learning preferences for each student using different methodologies. Checking student profile and learning preferences before recommending LOs. Allowing the students to choose among the topics to learn within the constraints of the prerequisites (partial control). Capability to arrange meetings for suspended students. Students are given the chance to selfstudy the subjects and attend the exams three times. If the student fails to pass the exam three times, a meeting must be arranged between the instructor and the student to submit a report by the instructor to the student profile, so the student can continue the learning process again in the adaptive way. This sort of blended learning gives strength to the model. Providing the capability to calculate the required time to study a topic. Tracking student’s behavior in the exams and attempting to identify cheating incidences. Integration with different online forum, wiki, and blog services is available to enhance collaboration between students and encourage them to help each other. Facilities to enable online study groups—like chatting applications—are available.

For detailed study on empowering adaptive lectures through activation of intelligent and Web 2.0 technologies related to the proposed model, the reader is referred to El-Ghareeb and Riad [24]. Supporting online lectures with adaptive and intelligent features related to the proposed model is presented in details in Riad et al. [25].

4.6

Proposed intelligent features

The presented adaptive eLearning model sheds light on supporting eLearning with intelligent features. Intelligence can be addressed in different aspects of the proposed model. This section presents detailed design of the intelligent features to improve student’s performance and help instructors through the eLearning process. Those goals can be achieved by applying different technologies available to educational institutions, instructors, and students in an innovative way. Different intelligent services are presented to enable the intelligent features. Generally, an intelligent service is presented for each intelligent feature. Presented intelligent services can be grouped into two categories based on their users. Fig. 4.2 presents the two proposed intelligent services categories: instructor intelligent services and student intelligent services.

4.6.1 Proposed intelligent microservices From the adaptive eLearning model, intelligent features are as follows: l

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Students are grouped. Each group is delimited by the same start date. Students who do not catch this start date are delayed to the next group, which is 15 days later. Crawlers keep searching the Internet for newly and updated LOs; in addition, instructors add different resources to the LOs repository. Intelligent learning objects classifier is used to classify found LOs.

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Intelligent LO classifier

Intelligent online lecture LOs advisor Instructor intelligent services

Proposed intelligent eLearning services

Intelligent student tracker

Intelligent cheat depressor

Intelligent study plan advisor

Intelligent time-to-learn topic calculation Student intelligent services

Intelligent LOs recommender

Intelligent agenda study time planner Intelligent meeting manager for suspended students

Fig. 4.2 Proposed intelligent eLearning features. l

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Learning goals are identified by instructors. Based on these learning goals, instructors define learning paths. Instructors can create some branching at certain points to give students the flexibility to customize their learning paths. The Intelligent study plan advisor helps students at those points. The intelligent time-to-learn topic calculation is the service that is used to advise students about the time needed to study a certain topic. Based on students’ study time of previous topics and the available LOs for this topic, this service can intelligently advise students about the study time issue. Students attend one or more adaptive online lectures within the same learning goal. Adaptive online lectures make use of the intelligent online lecture LOs advisor to recommend LOs for the instructor to use during the lecture, based on the students’ learning profiles. The intelligent LOs recommender is the intelligent service that will recommend LOs for students based on their learning profiles. The recommended LOs list is approved by the instructor and reordered based on the students’ preferences.

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The intelligent agenda study time planner is used to organize the students’ timetables and organize their different activities by connecting them to different activities available in the university based on their preferences through announcements. The intelligent student tracker service is used to track the students’ performances during the online learning journey, and to verify the completeness of students’ learning profiles. Peaks and performance degradation in students’ performances need to be recorded and studied. The learning path is marked by different learning checkpoints. At each checkpoint, students attend an online exam. Those who pass will continue the learning path. Those who fail will have to re-attend the exam within 4 days. If they fail again, they will have to re-attend the exam within 2 days. If they do not pass the third time, they are suspended. The intelligent meeting manager for suspended students service is responsible for managing a meeting between an instructor and the suspended student to handle the learning issues that prevent a student from keeping up with their peer group. Suspended students drop behind their group. The intelligent cheat depressor service focuses on utilizing intelligent techniques in prohibiting students from cheating. When combining the intelligent student tracking service and the intelligent cheat depressor service, cheating instances might be identified.

4.6.2 Instructor intelligent microservices 4.6.2.1 Intelligent LOs classifier Different intelligent techniques can be utilized in classifying LOs based on LO type. Classifying multimedia-based LOs can be via metadata, tags, and annotations, while classifying text-based LOs can be done through accessing and analyzing content. Text classification or categorization is the process of organizing information logically. It can be used in many fields such as document retrieval, routing, and clustering. Document classification tasks can be divided into two sorts: supervised document classification where some external mechanism—such as human feedback—provides information on the correct classification for documents. The second sort is unsupervised document classification, where the classification must be done entirely without reference to external information. The presented intelligent LOs classifier utilizes two of the supervised document classification algorithms: the Naive Bayes Classifier and Term Frequency-Inverse Document Frequency (TF-IDF) algorithms. Both belong to probabilistic classifiers. Intelligent LOs classifier microservice will be covered in a later chapter due to their extensive details.

4.6.2.2 Intelligent adaptive online lecture LOs advisor specifications The intelligent online lecture LOs advisor accesses students’ profiles and learning preferences side by side with data from previous online lectures and course specification data. This service provides the instructor with a recommended list of LOs based on the attending students. This list can be used during the lecture. Table 4.1 presents intelligent online lecture LOs advisor specifications. Algorithm 4.1 presents a detailed algorithm specification highlighting Neutrosophic utilization for this intelligent Microservice.

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Table 4.1 Intelligent adaptive online lecture LOs advisor specifications.

Input Student preferences

Related LOs specifications Learning topics data Previous online lectures data

Different learning preferences that identify the student learning behavior are stored. Those preferences are considered for identifying different study plans LOs satisfy students’ classes by percentage. The more available LOs that match students’ preferences, the more this topic is recommended for teaching Data about courses and topics to be pedagogically used in learning scenarios LOs that were used by previous instructors during online lectures for the same topic and students’ feedback for those LOs are important data for this recommendation process

Processing By assigning different weights to the different inputs, Neutrosophic sets are used to generate a weighted list summary report. The intelligent advisor does the following: Identify LOs presented at previous lectures Classify attending students to one of the learning styles Check the LOs specifications and metadata Identify the most suitable LOs to use with attending students l

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Output Recommendation report

The instructor can use this report to identify LOs suitability

Algorithm 4.1 Intelligent online LOs advisor algorithm 1: Instructor initiates adaptive online lecture preparation 2: Extract LOs related to the learning topic 3: Extract LOs utilized by previous instructors and students’ feedback 4: Extract instructor recommended LOs for topic 5: Extract student preferences and learning class 6: Extract related LOs specs 7: for all LOs do 8: if instructor previously assigned T, I, F weights to LO then 9: assign T, I, F to instructor recommended LOs for topic 10: end if 11: if LO was utilized by previous instructor then 12: assign T, I, F to LO based on previous instructor evaluation 13: end if 14: if LO was utilized in previous lectures then 15: assign T, I, F to LO based on previous students’ feedback (cumulative)

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16: 17: 18: 19: 20: 21: 22: 23: 24:

end if assign T, I, F to LO based on LO matching with student preference end for for all LOs do Deneutrosophy LO and calculate its relevancy to the learning topic Deneutrosophy LO and calculate its similarity to student learning preferences end for Sort different LOs based on their similarity to students Return sorted list of recommended LOs

4.6.2.3 Intelligent student tracker This service tracks student behavior and ensures that there is a complete learning profile that helps the system to identify the students learning preferences and styles all over the system. The learning style is the individual’s characteristic ways of processing information and behaving in learning situations. Knowledge of learning styles can help instructors better understand learners and has important implications for program planning, teaching, and learning.

4.6.2.4 Intelligent cheat depressor The intelligent cheat depressor service tracks students’ behavior in exams and records both: students’ marks, and exam times, trying to identify peaks in marks. Though this service does not detect incidents of cheating for certain; it is used as an indicator to the

Table 4.2 Intelligent cheat depressor Microservice specifications.

Input Student’s previous exam data

Student’s latest exam data

Data include time consumed by student at each exam, type of exam, and mark scored at previous exams, besides different other exam metadata, like number of exam questions, each question difficulty level, etc. Data include time consumed by student at each exam, type of exam, and mark scored at this exam, besides different other exam metadata, like number of exam questions, each question difficulty, etc.

Processing Utilizing Neutrosophic theory to calculate cheating susceptibility. Membership values are adjusted to track and allow the increase of a student’s performance when getting better; however, peak changes are definitely identified

Output Informing instructor to investigate the case when needed

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instructor to track certain students. Table 4.2 presents the details of this intelligent Microservice.

4.6.3 Student intelligent microservices 4.6.3.1 Intelligent study plan advisor The intelligent study plan advisor is an intelligent advisory service used to help students identify the appropriate study plans by: l

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identifying older study plans; and identifying study plans of colleges.

Students differ in their learning behavior and learning preferences. The intelligent study plan advisor service considers different students as classes based on their learning preferences. Table 4.3 presents the intelligent study plan advisor specifications. In this Microservice, instructors identify branching capabilities in the learning path where students can have the opportunity to study a learning topic. Algorithm 4.2 Table 4.3 Intelligent study plan advisor Microservice specifications.

Input Student preferences

Learning class

Study plans for previous students Study plan for colleges

The proposed model stores different learning preferences that identify student learning behavior. Those preferences are considered for identifying different study plans. Students are grouped into classes to ease educational tasks. Classes include auditory, visual, and other classes that are discussed in detail in the learning profile section. Students need to take a closer look of previous instructor plans, grades that students scored by following certain plans, and other data What students in the same groups are studying now

Processing To generate the recommended study plan, the system utilizes Neutrosophic theory and sets as follows: Identify the class to which the students belong Check the branching decision that is assigned by instructor for that class, and double the weights of this decision Check the average of branching decisions taken by students in the same class Ranks recommendations from top-down based on the generated weights l

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Output Recommended study plan Study plan for colleges as information

Recommended choice to take in the study plan Display hints on what colleges allow students to study, so student is free to follow their path

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Algorithm 4.2 Intelligent study plan advisor algorithm 1: 2: 3: 4: 5: 6: 7: 8:

Student initiates intelligent study plan advisor microservice Extract study plans for previous students Extract instructor recommendations for student class Extract student preferences and learning class Identify student class for all Topics to learn do if instructor previously assigned T, I, F weights to topics to learn then assign T, I, F to instructor recommended next topic to learn for student class 9: end if 10: if Topic was studied before by previous students then 11: assign T, I, F to topic learned by elder students in the same student class 12: end if 13: end for 14: Utilize Neutrosophic rules engine to calculate top-three recommendations of next topics to learn 15: Return three recommendation lists: instructor, previous students, and colleges

presents detailed intelligent study plan advisor implementation specifications utilizing Neutrosophic Sets and Theory.

4.6.3.2 Intelligent time-to-learn topic calculation The intelligent time-to-learn topic calculation is an intelligent Microservice that helps students identify time needed to learn a certain topic. From a study time point of view, the time needed to study a topic is the summation of the time needed to study LOs composing this topic. The instructors define the learning time for each LO as one of the LOs educational metadata attributes. The system can identify learning time variances between instructors’ identified learning time and the students’ actual consumed learning time through tracking students. This time can help students estimate the time needed to finish studying. Table 4.4 presents the intelligent time-to-learn topic calculation specifications.

4.6.3.3 Intelligent LOs recommender The intelligent LOs recommender is the Microservice that aims to find the most pedagogically suitable LO for helping students learn a topic, then personalizing the recommended list based on students’ preferences. Thus, the intelligent LOs recommender must analyze newly introduced LOs efficiently, then store information about them for further processing and ordering to each student. From high-level view, the intelligent LOs recommender executes through two phases: the LOs finding, gathering, and analyzing phase, and the intelligent personalized supervised LOs recommendation phase. The

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Table 4.4 Intelligent time-to-learn topic calculation Microservice specifications.

Input Instructor defined learning time Student learning time shift

LOs author defines learning time for each LO. Later, different instructors can identify learning times for the same LO to match students’ skills Tracking the students’ learning progress helped the system to calculate the time-to-learn shift between the defined time and the student actual time to learn. Average time-to-learn calculation will be presented

Processing To calculate total time-to-learn topic for student, system utilizes Neutrosophic Sets and Theory in the following process: System identifies the LOs list the student must learn to finish the topic based on instructors’ directions System identifies the time shift between instructor’s identified learning time and the actual time taken by student. Such an entry is identified over time through tracking student System estimates time needed for learning each LO, and for all LOs forming topic, system calculates timed needed to learn that topic l

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Output Time-to-learn topic

Time estimated for student to learn certain topic

intelligent LOs recommender Microservice will be covered in Chapter 9 due to its extensive details.

4.6.3.4 Intelligent agenda study time planner This helps students identify study times, and integrate activities with their agenda to improve performance. It uses the study time shift estimated via Neutrosophic theory between the instructor LO study time and student actual study time, and can intelligently suggest time needed for students to finish their studies. In addition, it integrates different activities in the university within students’ timetable based on students’ preferences. It presents students’ timetables that combine lecture times, study times, and activity times, so they are personalized for each student. Table 4.5 presents the intelligent agenda study time planner specifications.

4.6.3.5 Intelligent meeting manager for suspended students Students who fail three times when taking an exam are suspended from accessing the system. Not being able to pass the exam after three attempts indicates that there are some pedagogical issues that need taking care of. Suspended students must meet one of the instructors to help them identify and work on solving challenges. Identifying the

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Table 4.5 Intelligent agenda study time planner Microservice specifications.

Input Student preferences Related LOs specifications Study time shift

Model stores different learning preferences that identify student learning behavior. In addition, students register in their preferred activities Specifications of LOs those students will study, including instructoridentified study time Neutrosophic-based estimate of the time shift between actual study time identified for each LO and estimation of the actual time the student needs to study this LO

Processing Intelligence in processing takes place in different activities, mainly when conflicts, vagueness, data incompleteness, or inconsistencies occur. The system can resolve conflicts using Neutrosophic Sets and Theory as follows: The system identifies LOs list that student has to study, assigning them the highest weight value The system identifies activities available this week that match student’s interests The system identifies lecture times, and assigns them the highest weight value The system identifies student free time The system attempts to suggest a weekly agenda for student to satisfy all of the above. When conflicts, vagueness, data incompleteness, or inconsistencies occur, Neutrosophic theory is used to assign different T, I, F weights for different activities, and to calculate the importance of each entry. Highest priorities are identified and override low-priority activities l

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Output Personalized agenda

Personalized agenda for each student that combines lecture times, suggested study times for LOs, and activity times

time for suspended students to meet instructors is an intelligent process that utilizes Neutrosophic Sets and Theory to reach the most suitable time for both students and instructors. Table 4.6 presents specifications of the intelligent meeting manager for suspended students. Suspended students cannot access the system until they are reactivated by the instructor after the meeting.

4.7

Evaluation

Preparing a computer networks course and presenting it to students in the form of the presented adaptive eLearning models and experiencing it resulted in the following satisfaction measures. Table 4.7 presents a summary of students’ opinions about

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Table 4.6 Intelligent meeting manager for suspended students Microservice specifications.

Input Student timetable Instructors to meet

Proposed model extracts suitable student meeting times Different instructors can support the same course. Students are able to give priorities for different instructors

Processing By finding matches between students’ available time and instructors’ available time, proposed meeting times are presented. Three different proposed meeting times are presented, and wait for instructors’ approval in order. Arranging meetings faces challenges, especially when there are no free times available. When this is the case, the system needs to break some time constraints using Neutrosophic Sets and Theory to identify what time constraint to break. Instructors must approve meetings before they are sent to the student. Neutrosophic theory is perfect here for calculating and formulating indeterminacy for different schedules that might take place because of unforeseen problems

Output Proposed meeting time

Proposed meeting is approved by instructor. If approved, student is informed of this meeting and now the instructor has full control on the student’s status. The instructor can reactivate the student, make him access LOs, and attend exams if needed

presented features and how they evaluate the need for these and their performance. The presented adaptive eLearning model was tested on a sample of 10 students. Table 4.8 presents a summary of instructors’ thoughts about presented features and how they evaluate the need for these and their performance and behavior. Table 4.7 Summary of students’ evaluation of presented eLearning model features.

Feature Learning preferences Learning profile Customizing course Separate groups Exams check points

Strongly agree (%)

Agree (%)

Neutral (%)

Disagree (%)

Strongly disagree (%)

90

10

–

–

–

85 80

15 10

– 10

– –

– –

30 30

10 10

10 10

25 30

25 20 Continued

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Table 4.7 Continued

Feature

Strongly agree (%)

Agree (%)

Neutral (%)

Disagree (%)

Strongly disagree (%)

LVQ Video LOs LOs recommender Agenda study plan Study plan advisor Meeting manager

90 90 70 63 65 40

5 5 20 17 30 10

5 5 – 5 – 8

– – 5 8 5 2

– – 5 7 – 40

Table 4.8 Summary of instructors’ evaluation of presented eLearning model features.

Feature Learning preferences Learning profile Customizing course Separate groups Exams check points LVQ Video LOs LOs recommender LOs advisor Cheat depressor Student tracker

Strongly agree (%)

Agree (%)

Neutral (%)

Disagree (%)

Strongly disagree (%)

90

10

–

–

–

90 70

10 20

– 10

– –

– –

30 70

10 15

10 15

25 –

25 –

90 90 70 73 50 80

5 5 20 17 10 20

5 5 10 10 10 –

– – – – 20 –

– – – – 10 –

4.7.1 Comments on evaluation results To evaluate the adaptive eLearning model, the two target categories of the model were students and instructors. Both of them showed interest in the presented adaptive eLearning model and felt that it could enhance the eLearning experience greatly. Students have issues with the repeated exams process, and grouping students in smaller groups. However, they liked the presented adaptive features. Instructors suspected the applicability of the intelligent cheat depressor service; however, they agreed to use it as an indicator, and the final decision remains their decision.

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Conclusion

eLearning is an important part of the future; however, it is facing challenges. One of the eLearning challenges is the absence of current eLearning systems that adaptively and intelligently invoke students’ capabilities. Adaptive eLearning is the solution to exploit unlimited eLearning advantages. Adaptive eLearning that is supported by intelligent methods and techniques, such as Neutrosophic Sets and Theory is a need. Adaptive eLearning that is supported by intelligent techniques is the solution to present efficient and effective learning. This chapter presented an adaptive eLearning model that blends instructor-led education with eLearning capabilities to provide an enhanced eLearning environment as the solution to current eLearning challenges. Presenting adaptive and intelligent features in the form of Microservices with standard interfaces allows different eLearning systems to adopt them, so they will be reusable, and the newly introduced information systems will not have to reinvent the wheel. In addition, wrapping adaptive and intelligent features with standard interfaces will present a separation of interests that help adaptive and intelligent features researchers and developers to focus more on their target, and transfer the responsibility of utilizing these features in different information systems to information systems specialists. Intelligent Microservices were categorized into two categories based on the user of those Microservices: instructor and student Microservices. The instructor intelligent Microservices are: intelligent LO classifier, intelligent online lecture LOs advisor, intelligent student performance tracker, and intelligent cheating depressor. The student intelligent Microservices are: intelligent time-to-learn topic calculation, intelligent study plan advisor, intelligent agenda study time planner, and intelligent meeting manager for suspended students and intelligent LOs recommender. All those intelligent Microservices utilized Neutrosophic Sets and Theory to overcome difficult situations, mainly incomplete, inconsistent, and missing data. Future work includes the focus on what eLearning would look like in a Web 3.0 world, and how it might differ from current eLearning. eLearning 3.0 is the eLearning empowered by Web 3.0 technologies. eLearning 3.0 will have four key drivers: distributed computing, extended smart mobile technology, collaborative intelligent filtering, and 3D visualization and interaction. eLearning 3.0 will cross the boundaries of traditional learning institutions, and there will be an increase in self-organized learning. With cloud computing and increased reliability of data storage and retrieval, the mashup is a viable replacement for the portal. Both mashups and portals are supporting integration of content provided by other websites and presentation. This will lead to less reliance on centralized provision. Mobiles will play a big part in the eLearning 3.0. There will be a need of ubiquitous access to tools, services, and learning resources, including people—peer learning group, subject specialists, and expert support. Collaborative eLearning will be possible in all contexts. eLearning 3.0 will make collaborating across distance much easier. Three-dimensional visualization will become more readily available. Neutrosophic Sets and Theory will take place in all those scenarios.

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[15] N.A. Nabeeh, M. Abdel-Basset, H.A. El-Ghareeb, A. Aboelfetouh, Neutrosophic multicriteria decision making approach for IoT-based enterprises, IEEE Access 7 (2019) 59559–59574, https://doi.org/10.1109/ACCESS.2019.2908919. [16] H.A. El-Ghareeb, Novel open source Python neutrosophic package, Neutrosophic Sets Syst. 25 (2019) 136. [17] M. Villamizar, O. Garces, H. Castro, M.V. Merino, L. Salamanca, R. Casallas, S. Gil, Evaluating the monolithic and the microservice architecture pattern to deploy web applications in the cloud, in: 2015 10th Computing Colombian Conference (10CCC), 2015, pp. 583–590. https://doi.org/10.1109/ColumbianCC.2015.7333476. [18] Cesardelatorre, .NET Microservices. Architecture for Containerized .NET Applications (Online), 2019. Available from: https://docs.microsoft.com/en-us/dotnet/standard/ microservices-architecture. (Accessed 14 February 2019). [19] C.C. Chang, K. Hsiao, A SOA-based E-learning system for teaching fundamental information management courses, J. Converg. Inf. Technol. 6 (4) (2011) 298–305. [20] O. Zimmermann, Microservices tenets, Comput. Sci. Res. Dev. 32 (3) (2017) 301–310, https://doi.org/10.1007/s00450-016-0337-0. [21] T. Cerny, Aspect-oriented challenges in system integration with microservices, SOA and IoT, Enterp. Inf. Syst. (2018) 1–23, https://doi.org/10.1080/17517575.2018.1462406. [22] T. Cerny, M.J. Donahoo, J. Pechanec, Disambiguation and comparison of SOA, Microservices and self-contained systems, in: Proceedings of the 2017 Research in Adaptive and Convergent Systems, RACS 2017, 2017, pp. 228–235. [23] J. Soldani, D.A. Tamburri, W.-J.V.D. Heuvel, The pains and gains of Microservices: a systematic grey literature review, J. Syst. Softw. 146 (2018) 215–232, https://doi.org/ 10.1016/j.jss.2018.09.082. [24] H. El-Ghareeb, A. Riad, Empowering adaptive lectures through activation of intelligent and Web 2.0 technologies, Int. J. E-Learning 10 (4) (2011) 365–391. [25] A.M. Riad, H.K. El-Minir, H.M. El-Bakry, H.A. El-Ghareeb, Supporting online lectures with adaptive and intelligent features, Adv. Inf. Sci. Serv. Sci. 3 (1) (2011) 47.

Some similarity measures for MADM under a complex neutrosophic set environment

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Kalyan Mondala, Surapati Pramanikb, Bibhas C. Giric a Department of Mathematics, Jadavpur University, Kolkata, India, bDepartment of Mathematics, Nandalal Ghosh B.T. College, Kolkata, India, cDepartment of Mathematics, Jadavpur University, Kolkata, India

Abbreviations NS SVNS MADM CFS CIFS CNS FS IFS INS RNS CNCSM CNDSM CNJSM Re Im

5.1

neutrosophic set single valued neutrosophic set multiple attribute decision making complex fuzzy set complex intuitionistic fuzzy set complex neutrosophic set fuzzy set intuitionistic fuzzy set interval neutrosophic set rough neutrosophic set complex neutrosophic cosine similarity measure complex neutrosophic Dice similarity measure complex neutrosophic Jaccard similarity measure real part imaginary part

Introduction

Uncertainty plays a vital role in modeling of real world problems. So, it is necessary to bridge the gap between mathematical models and uncertainty, and their explorative explanations. This gap can be found in problems of mathematics, operations research, social sciences, biological sciences, modern technology, and other applied sciences. Zadeh [1] introduced the mathematical concept of fuzzy set (FS), incorporating membership degree of an element. Fuzzy sets are widely used by researchers in decisionmaking [2–5] to deal with uncertain situations. To tackle uncertain situations more precisely, Atanassov [6] introduced intuitionistic fuzzy set (IFS), incorporating non-membership degree with membership degree of an element. The concept of IFS has been studied and applied in various areas such as decision-making problems Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00005-6 © 2020 Elsevier Inc. All rights reserved.

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[7–9], selection problems [10, 11], educational problems [12], medical diagnosis [13–15], etc. Smarandache [16] introduced indeterminacy degree as an independent part and proposed the neutrosophic set (NS) to tackle uncertainty, inconsistency, and indeterminacy. To use the concept in a real scientific field, Wang et al. [17] restricted the notion of NS to single valued neutrosophic set (SVNS). SVNS is a significant mathematical tool in the research of medical analysis [18], engineering problems [19], decision-making [20–64], clustering study [65], social problems [66, 67] and neutrosophic applications [68, 69] in indeterminate, uncertain, and inconsistent environments. Several similarity measures under SVNSs have been proposed to tackle decision-making problems. Majumdar and Samanta [70] proposed the similarity measures under SVNSs based on a matching function, distances, and membership grades, and developed an entropy measure for SVNS. Ye [71] studied several vector similarity measures under SNS environment. Ye [72] presented cosine similarity measure (improved form) for SVNSs. Ye [73] also presented the similarity measures for SVNS to solve MADM problems with totally unknown weights. Ye and Zhang [74] studied the similarity measures for SVNS to tackle MADM problems. Ye and Fu [75] proposed medical diagnosis strategy under SVNS using tangent function based similarity measure. Pramanik and Mondal [76] studied cosine function based similarity measure and presented its application for medical diagnosis in RNS environment. Mondal and Pramanik [77] studied tangent function based similarity measure for neutrosophic refined set and applied it to MADM problem. Mondal and Pramanik [78] also studied cotangent function based similarity measure in refined neutrosophic environment. Mondal and Pramanik [79] further studied cotangent function based similarity measure under RNS environment. Biswas et al. [80] proposed MADM for trapezoidal fuzzy neutrosophic environment based on cosine similarity measure. Wang et al. [81] extended TODIM strategy by employing Choquet integral and distance measure for linguistic Z-numbers. In the same study, Wang et al. [81] provided a numerical example and conducted comparative analysis. Hu et al. [82] presented a VIKOR strategy based on vector projection to deal with selection of doctors using PAGD App. in INS environment. In the same work, Hu et al. [82] proposed transition functions using maximizing deviation strategy. Thanh et al. [83] developed a new clustering strategy for medical diagnosis in neutrosophic environment and presented a comparison with the existing algorithms in the literature. To deal with planning methodology selection, Abdel-Basset et al. [84] integrated AHP and SWOT strategy in neutrosophic environment. Abdel-Basset et al [85] proposed a novel group decision-making model based on triangular neutrosophic numbers. Chang et al. [86] developed a reuse strategic decision pattern framework from theories to practices. Abdel-Basset et al. [52] proposed a group decision-making framework based on neutrosophic VIKOR approach for e-government website evaluation. Abdel-Basset and Mohamed [87] showed the role of SVNS and rough sets in a smart city under imperfect and incomplete information. Mondal et al. [88] proposed tangent similarity measure based MADM model in interval neutrosophic set environment. Pramanik et al. [89] developed NC-VIKOR based MAGDM strategy under neutrosophic cubic set environment. Dalapati et al. [90] proposed IN-cross entropy

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based MAGDM strategy under interval neutrosophic set environment. Mondal et al. [36] introduced hyperbolic sine similarity measure based MADM strategy in SVNS environment. Several researchers [20, 80, 91–105] present decision-making models in neutrosophic hybrid environment. Ramot et al. [106] introduced an innovative concept of complex fuzzy set (CFS); it is an extension of FS. Here, the value of the membership function of CFS ranges in a circle of radius one. Ramot et al. [107] also proposed union, intersection, complement of CFS. Chen et al. [108] proposed a neuro-fuzzy system architecture rule as a realistic application of CFS logic. Alkouri and Salleh [109] introduced complex intuitionistic fuzzy sets (CIFS), which generalize CFS. CFS is transformed to CIFS by adding nonmembership grade (complex-valued). The CIFS can deal the situations involving uncertainty and periodicity simultaneously. Uncertainty, inconsistency, and indeterminacy in information may be in periodic form. To tackle these problems, Ali and Smarandache [110] proposed the concept of complex neutrosophic set (CNS). A CNS is structured by a truth membership degree (complex-valued), indeterminate membership degree (complex-valued), and false membership degree (complex-valued), whose range is extended from [0, 1] to the unit circle in the complex space. The CNS can handle a situation that is indeterminate, uncertain, incomplete, inconsistent, and ambiguous, because the truth amplitude value with phase term, the indeterminate amplitude value with phase term, and the false amplitude value with phase term can tackle the indeterminacy with periodicity [110]. CNS is an extension of NS. Thus, the CNS deals with information that has indeterminacy, uncertainty, and falsity that is in periodicity while both the CFSs and CIFS cannot deal with indeterminacy in periodicity. In this study, we propose cosine, Dice, and Jaccard similarity-based measures as three strategies for CNS and applied these strategies in decision-making. A decision-making strategy based on similarity measure is yet to appear in complex neutrosophic set (CNS) environment. To fill the research gap, we propose an MADM strategy based on cosine, Dice, and Jaccard similarity measures in CNS environment. Contributions of the chapter are stated as follows: l

l

l

l

l

l

We define cosine, Dice and Jaccard similarity measures in complex neutrosophic set environment. We define weighted cosine, weighted Dice, and weighted Jaccard similarity measures in complex neutrosophic set environment. We define a tangent function to determine unknown weights of attributes in complex neutrosophic set environment. We present multi-attribute decision-making steps based on proposed similarity measures. We provide a numerical example to show the effectiveness of the proposed approach. We conduct a comparison analysis between existing strategy and the proposed strategy.

The chapter is structured as follows. Section 5.2 presents several basic concepts of FSs, IFSs, NSs, SVNSs, CFSs, CIFSs, and CNSs. Section 5.3 defines cosine and weighted cosine similarity measures in CNSs environment. Section 5.4 defines Dice and weighted Dice similarity measures in CNSs environment. Section 5.5 defines Jaccard and weighted Jaccard similarity measures in CNSs environment. Section 5.6 proposes

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a tangent function to determine unknown attributes weights. Section 5.7 presents the steps of decision-making based on weighted complex neutrosophic cosine similarity measure, weighted complex neutrosophic Dice similarity measure, and weighted complex neutrosophic Jaccard similarity measure. Section 5.8 provides a numerical problem to demonstrate the effectiveness and applicability of the proposed strategy. Section 5.9 conducts a comparison analysis between the existing strategy [111] and the proposed strategy. Section 5.10 concludes the chapter and discusses future scopes of research.

5.2

Preliminaries

In this section, the concepts of FSs, IFSs, NSs, SVNS, CFSs, CIFSs, and CNSs are outlined.

5.2.1 Fuzzy set A fuzzy set (FS) [1] X in a universe of discourse U is defined as the following set of pairs X ¼ {x, μX(x) : x 2 X}. Here, μX(x) : x ! [0, 1] is a mapping called the membership value of x 2 U in a fuzzy set X. The value of ηX(x) ¼ 1  μX(x) is called the degree of non-membership of the element x 2 U to the fuzzy set. Example 5.1. Let S ¼ {S1, S2, S3, S4} be the reference set of students. Let A be the fuzzy set of “smart” students where “smart” is fuzzy word then, A ¼ {(S1, 0.7), (S2, 0.6), (S3, 0.8), (S4, 0.4)}. Here A indicates that the smartness of S1 is 0.7 and so on.

5.2.2 Intuitionistic fuzzy set An intuitionistic fuzzy set (IFS) [6] A ¼ {(x, μA(x), νA(x) ) : x 2 X}, where the functions μA(x) : x ! [0, 1] and νA(x) : x ! [0, 1] define the degree of membership and degree of non-membership, respectively, of the element x 2 X to the set A that is a subset of X, and for every x 2 X, 0 < μA(x) + νA(x) < 1. The value of ξA(x) ¼ 1  (μA(x) + νA(x)) is called the degree of non-determinacy (or uncertainty or hesitancy) of the element x 2 X to the intuitionistic fuzzy set. Example 5.2. Let S ¼ {S1, S2, S3, S4} be the reference set of students. Let A be the intuitionistic fuzzy set of “smart” students where “smart” is intuitionistic fuzzy word then, A ¼ {(S1, 0.7, 0.2), (S2, 0.6, 0.1), (S3, 0.8, 0.1), (S4, 0.4, 0.3)}. Here A indicates that the smartness of S1 is (0.7, 0.2) and so on.

5.2.3 Neutrosophic set (NS) Let G be a space of objects/points with generic element in E denoted by y. Then an NS [16] P in G is characterized by a truth membership term TP, an indeterminacy

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membership term IP, and a falsity membership term FP. The functions TP, IP and FP are real non-standard or standard subsets of ] 0, 1+[, that is, TP: G ! 0, 1 + ½; IP: G ! 0, 1 + ½; FP: G ! 0, 1 + ½ and 

0  sup TP ðyÞ + sup IP ðyÞ + sup FP ðyÞ  3 + :

Example 5.3. Let S ¼ {S1, S2, S3, S4} be the reference set of students. Let A be the neutrosophic set of “smart” students where “smart” is a neutrosophic word then, A ¼ {(S1, 0.7, 0.2, 0.4), (S2, 0.6, 0.1, 0.2), (S3, 0.8, 0.1, 0.5), (S4, 0.4, 0.3, 0.5)}. Here A indicates that the smartness of S1 is (0.7, 0.2, 0.4) and so on. Definition 5.1. [16] The complement of an NS P is denoted by Pc and is defined as follows: TPc ðyÞ ¼ f1 + g  TP ðyÞ; IPc ðyÞ ¼ f1 + g  IP ðyÞ; FPc ðyÞ ¼ f1 + g  FP ðyÞ: Definition 5.2. [16] An NS P is contained in the other NS Q, P  Q if and only if the following results hold: inf TP ðyÞ  inf TQ ðyÞ, sup TP ðyÞ  sup TQ ðyÞ; inf IP ðyÞ  inf IQ ðyÞ, sup IP ðyÞ  sup IQ ðyÞ; inf FP ðyÞ  inf FQ ðyÞ, sup FP ðyÞ  sup FQ ðyÞ, 8y 2 G:

5.2.4 Single valued neutrosophic set (SVNS) An SVNS [17] S is characterized by a truth membership term TS(y), a falsity membership term FS(y), and indeterminacy term IS(y) with TS(y), FS(y), IS(y) 2 [0, 1] for all y in G. When G is continuous, an SNVS S can be written as follows: Z hTS ðyÞ, FS ðyÞ, IS ðyÞ i=y, 8y 2 G

S¼ y

and when G is discrete, an SVNS S can be written as follows: S¼

X hTS ðyÞ, FS ðyÞ, IS ðyÞ i=y, 8y 2 G

It should be observed that for an SVNS S: 0  sup TS ðyÞ + sup FS ðyÞ + sup IS ðyÞ  3, 8y 2 G

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Example 5.4. Let S ¼ {S1, S2, S3, S4} be the reference set of students. Let A be the single valued neutrosophic set of “smart” students where “smart” is a single valued neutrosophic word then, A ¼ {(S1, 0.7, 0.4, 0.3), (S2, 0.6, 0.1, 0.3), (S3, 0.8, 0.1, 0.1), (S4, 0.4, 0.3, 0.1)}. Here A indicates that the smartness of S1 is (0.7, 0.2, 0.3) and so on. Definition 5.3. [17] The complement of an SVNS S is denoted by Sc and is defined as follows: TS c ðyÞ ¼ FS ðyÞ; IS c ðyÞ ¼ 1  IS ðyÞ; FS c ðyÞ ¼ TS ðyÞ: Definition 5.4. [17] An SVNS S1 is contained in the other SVNS S2, denoted by S1  S2 if and only if: TS1 ðyÞ  TS2 ðyÞ; IS1 ðyÞ  IS2 ðyÞ; FS1 ðyÞ  FS2 ðyÞ, 8y 2 G: Definition 5.5. [17] Two SVNSs S1 and S2 are equal, i.e., S1 ¼ S2, if and only if S1  S2 and S1  S2. Definition 5.6. [17] The union of two SVNSs S1 and S2 is an SVNS S3, written as S3 ¼ S1 [ S2. Its truth membership, indeterminacy-membership, and falsity membership terms are related to S1 and S2 by the following relations: TS3 ðyÞ ¼ max ðTS1 ðyÞ, TS2 ðyÞÞ; IS3 ðyÞ ¼ min ðIS1 ðyÞ, IS2 ðyÞÞ; FS3 ðyÞ ¼ min ðFS1 ðyÞ, FS2 ðyÞÞ 8y 2 G: Definition 5.7. [17] The intersection of two SVNSs S1 and S2 is an SVNS S2, written as S3 ¼ S1 \ S2. Its truth membership, indeterminacy membership, and falsity membership terms are related to S1 an S2 by the following relations: TS3(y) ¼ min(TS1(y), TS2(y) ); IS3(y) ¼ max(IS1(y), IS2(y)); FS3(y) ¼ max(FS1(y), FS2(y)), 8 y 2 G.

5.2.5 Complex fuzzy set (CFS) A CFS [106] M, defined on a universal set X, is structured by a membership function ηM(x) that assigns any element x 2 X. The function ηM(x) ¼ pM(x).ei.μM(x) lies within the unit circle in the complex space, where, pM(x) and μM(x) are both real valued funcpffiffiffiffiffiffiffi tions, i ¼ 1 and pM(x) 2 [0, 1]. Here, pM(x) is the amplitude term and ei.μM(x) is the phase term. The CFS is presented as follows: M ¼ { x, ηM(x)} : x 2 X.

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Example 5.5. Let S ¼ {S1, S2, S3, S4} be the reference set of students. Let A be the complex fuzzy set of “smart” students where “smart” is a complex fuzzy word then, A ¼ {(S1, 0.6e1.0i), (S2, 0.7e1.2i), (S3, 0.3e0.8i), (S4, 0.4e1.5i)}. Here A indicates that the smartness of S1 is 0.6e1.0i and so on. Definition 5.8. [106] Let M be a CFS on X and ηM(x)¼ pM(x).ei.μM(x) its complex-valued membership term. The complement of M, denoted by c(M) is presented as follows:   ηcðMÞ ðxÞ ¼ pcðMÞ ðxÞ:ei:μc ðMÞðxÞ ¼ 1  pcðMÞ ðxÞ:ei:ð2πμc ðMÞðxÞÞ Definition 5.9. [106] Let M and N be any two CFSs on X, and ηM(x) ¼ pM(x).ei.μM(x) and ηN(x)¼ pN(x). i.μN(x) e be their membership terms, respectively. The union of M and N is denoted by M [ N, which is expressed as follows: ηM[N ðxÞ ¼ pM[N ðxÞ:ei:μM[N ðxÞ ¼ maxð pM ðxÞ, pN ðxÞ Þ:ei:½maxðμM ðxÞ,μN ðxÞÞ Definition 5.10. [106] Let M and N be any two CFSs on X, and ηM(x)¼ pM(x).ei.μM(x) and ηN(x)¼ pN(x). i.μN(x) e be their membership terms, respectively. The intersection of M and N is denoted by M \ N, which is expressed as follows: ηM\N ðxÞ ¼ pM\N ðxÞ:ei:μM\N ðxÞ ¼ minð pM ðxÞ, pN ðxÞ Þ:ei:½minðμM ðxÞ,μN ðxÞÞ: Definition 5.11. [106] Let M and N be any two CFSs on X, and ηM(x)¼ pM(x).ei.μM(x) and ηN(x)¼ pN(x). i.μN(x) e be their membership terms, respectively. The product of CFSs M and N is denoted by M ∘ N, which is expressed as follows: μMðxÞμN ðxÞ 2π

ηM∘N ðxÞ ¼ pM∘N ðxÞ:ei:μM∘N ðxÞ ¼ ð pM ðxÞ:pN ðxÞ Þ:ei

5.2.6 Complex intuitionistic fuzzy set (CIFS) A CIFS [109] M, defined on a universal set X, is characterized by a membership term ηM(x) and a non-membership term ψ M(x) that assigns any element x 2 X. The functions ηM(x) ¼ pM(x).ei.μM(x) and ψ M(x) ¼ qM(x).ei.ϑM(x) lie within the unit circle in the complex pffiffiffiffiffiffiffi space, where pM(x), μM(x), qM(x) and ϑM(x) are real valued terms, i ¼ 1 and qM(x), pM(x) 2 [0, 1]. Here pM(x) and qM(x) are amplitude terms and ei.μM(x) and ei.ϑM(x) are phase terms. The CIFS be presented as follows: M ¼ f x, ηM ðxÞ, ψ M ðxÞg : x 2 X:

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Example 5.6. Let S ¼ {S1, S2, S3, S4} be the reference set of students. Let A be the complex intuitionistic fuzzy set of “smart” students where “smart” is a complex intuitionistic fuzzy word, then: A ¼ {(S1, 0.6e1.0i, 0.2e0.8i), (S2, 0.5e1.2i, 0.2e0.7i), (S3, 0.7e1.0i, 0.1e0.8i), (S4, 0.3e1.0i 0.4e1.5i)}. Here A indicates that the smartness of S1 is (0.6e1.0i, 0.2e1.0i) and so on. Definition 5.12. [109] Complement of CIFS: Let M be a CIFS on X. Let ηM(x) ¼ pM(x).ei.μM(x) and ψ M(x) ¼ qM(x).ei.ϑM(x) be its complex-valued membership term and non-membership term, respectively. The complement of M, denoted by c(M), is expressed as follows:   ηcðMÞ ðxÞ ¼ pcðMÞ ðxÞ:ei:μcðMÞ ðxÞ ¼ pcðMÞ ðxÞ:ei:ð2πμcðMÞ ðxÞÞ and   ψ cðMÞ ðxÞ ¼ qcðMÞ ðxÞ:ei:ϑcðMÞ ðxÞ ¼ qcðMÞ ðxÞ:ei:ð2πϑcðMÞ ðxÞÞ Definition 5.13. [109] Union of CIFS: Let M and N be any two CIFSs on X and ηM(x) ¼ pM(x).ei.μM(x), ψ M(x) ¼ qM(x). i.ζ M(x) e and ηN(x) ¼ pN(x).ei.μN(x), ψ N(x) ¼ qN(x).ei.ζN(x) represent the membership term and non-membership terms of M and N, respectively. The union of M and N is denoted by M [ N, which is expressed as follows: ηM[N ðxÞ ¼ pM[N ðxÞ:ei:μM[N ðxÞ ¼ maxð pM ðxÞ, pN ðxÞ Þ:ei:½maxðμM ðxÞ,μN ðxÞÞ ψ M[N(x) ¼ pM[N(x).ei.ζM[N(x) ¼ min( qM(x), qN(x) ).ei.[min(ζM(x),ζN(x)). Definition 5.14. [109] Intersection of CIFS The intersection of two CIFSs M and N is denoted by M \ N, which is defined as follows: ηM\N ðxÞ ¼ pM\N ðxÞ:ei:μM\N ðxÞ ¼ minð pM ðxÞ, pN ðxÞ Þ:ei:½minðμM ðxÞ,μN ðxÞÞ ψ M\N(x) ¼ pM\N(x).ei.ζM\N(x) ¼ max( qM(x), qN(x) ).ei.[max(μM(x),μN(x)).

5.2.7 Complex neutrosophic set (CNS) A CNS [110] N, defined on a universal set X, which is structured by a truth membership term TN(x), an indeterminacy membership term IN(x), and a falsity membership term FN(x) that assigns a complex-valued grade of TN(x), IN(x), FN(x) in N for all x 2 X. The values TN(x), IN(x) and FN(x) lie in the unit circle in the complex plane. So, it is of the following form: T N ðxÞ ¼ pN ðxÞeiμN ðxÞ , I N ðxÞ ¼ qN ðxÞeiϑN ðxÞ , FN ðxÞ ¼ r N ðxÞeiωN ðxÞ :

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where, pN(x), qN(x), rN(x) and μN(x), ϑN(x), ωN(x) are real valued, and pN(x), qN(x), rN(x) 2[0,1] such that 0  pN(x) + qN(x) + rN(x)  3. Example 5.7. Let S ¼ {S1, S2, S3, S4} be the reference set of students. Let A be the complex neutrosophic set of “smart” students where “smart” is a complex neutrosophic word, then: A ¼ {(S1, 0.6e1.0i, 0.2e0.8i, 0.5e0.8i), (S2, 0.5e1.2i, 0.4e0.7i, 0.2e0.8i), (S3, 0.7e1.0i, 0.5e0.8i, 0.4e0.8i), (S4, 0.3e1.0i 0.4e1.5i, 0.2e0.8i)}. Here A indicates that the smartness of S1 is (0.6e1.0i, 0.2e0.8i, 0.5e0.8i), and so on. Definition 5.15. [110] A CNS CN1 is contained in the other CNS CN2 denoted by CN1  CN2 if and only if pCN1(x)  pCN2(x), qCN1(x)  qCN2(x), rCN1(x)  rCN2(x), and μCN1(x)  μCN2(x), ϑCN1(x)  ϑCN2(x), ωCN1(x)  ωCN2(x). Definition 5.16. [110] Two CNSs CN1 and CN2 are equal, i.e., CN1 ¼ CN2 if and only if pCN1(x) ¼ pCN2(x), qCN1(x)¼ qCN2(x), rCN1(x)¼ rCN2(x), μCN1(x)¼ μCN2(x), ϑCN1(x)¼ ϑCN2(x), and ωCN1(x)¼ ωCN2(x). Definition 5.17. [110] Complement of CNS: Let N be a CNS on X. Let TN(x) ¼ pN(x)eiμN(x), IN(x) ¼ qN(x)eiϑN(x), FN(x) ¼ rN(x) iωN(x) e be its complex-valued membership term, an indeterminacy membership term, and non-membership term, respectively. The complement of N, denoted by c(N), is expressed as follows: T cðN Þ ðxÞ ¼ r cðNÞ ðxÞ:ei:μcðNÞ ðxÞ where, μcðNÞ ðxÞ ¼ μN ðxÞ or 2π  μN ðxÞ or μN ðxÞ + π I cðNÞ ðxÞ ¼ ð1 I N ðxÞÞ:ei:ϑcðNÞ ðxÞ where, ϑcðNÞ ðxÞ¼ϑN ðxÞ or 2π  ϑN ðxÞ or ϑN ðxÞ + π FcðNÞ ðxÞ ¼ pN ðxÞ:ei:ωcðNÞ ðxÞ where, ωcðNÞ ðxÞ ¼ ωN ðxÞ or 2π  ωN ðxÞ or ωN ðxÞ + π Definition 5.18. [110] Union of CNSs: Let M and N be any two CNSs on X, and TM(x) ¼ pM(x)eiμM(x), IM(x) ¼ qM(x)eiϑM(x), FM(x) ¼ rM(x)eiωM(x), and TN(x) ¼ pN(x)eiμN(x), IN(x) ¼ qN(x)eiϑN(x), FN(x) ¼ rN(x)eiωN(x) be their membership term, an indeterminacy membership term, and non-membership term, respectively. The union of M and N is denoted by M [ N, which is expressed as follows: T M[N ðxÞ ¼ pM[N ðxÞ:ei:μM[N ðxÞ ¼ maxð pM ðxÞ, pN ðxÞ Þ:ei:½maxðμM ðxÞ,μN ðxÞÞ I M[N ðxÞ ¼ pM[N ðxÞ:ei:ζM[N ðxÞ ¼ minð qM ðxÞ, qN ðxÞ Þ:ei:½minðϑM ðxÞ,ϑN ðxÞÞ FM[N ðxÞ ¼ r M[N ðxÞ:ei:ωM[N ðxÞ ¼ minð r M ðxÞ, r N ðxÞ Þ:ei:½minðωM ðxÞ,ωN ðxÞÞ:

96

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Definition 5.19. [110] Intersection of CNS: Let M and N be any two CNSs on X, and TM(x) ¼ pM(x)eiμM(x), IM(x) ¼ qM(x)eiϑM(x), FM(x) ¼ rM(x)eiωM(x) and TN(x) ¼ pN(x)eiμN(x), IN(x) ¼ qN(x)eiϑN(x), FN(x) ¼ rN(x)eiωN(x) be their membership term, an indeterminacy membership term, and non-membership term, respectively. The intersection of M and N is denoted by M \ N, which is expressed as follows: T M\N ðxÞ ¼ pM\N ðxÞ:ei:μM\N ðxÞ ¼ minð pM ðxÞ, pN ðxÞ Þ:ei:½minðμM ðxÞ,μN ðxÞÞ I M\N ðxÞ ¼ pM\N ðxÞ:ei:ζM\N ðxÞ ¼ maxð qM ðxÞ, qN ðxÞ Þ:ei:½maxðϑM ðxÞ,ϑN ðxÞÞ FM\N ðxÞ ¼ r M\N ðxÞ:ei:ωM\N ðxÞ ¼ maxð r M ðxÞ, r N ðxÞ Þ:ei:½maxðωM ðxÞ,ωN ðxÞÞ

5.3

Complex neutrosophic cosine similarity measure (CNCSM)

Similarity measure based on cosine function is determined as the ratio of scalar product of two vectors and the product of the lengths. Cosine similarity measure for CNS environment is defined as follows: Definition 5.20. Let two CNSs CN1 ¼ hpCN1(xi)eiμCN1(xi), qCN1(xi)eiϑCN1(xi), rCN1(xi)eiωCN1(xi)i and CN2 ¼ hpCN2(xi)eiμCN2(xi), qCN2(xi)eiϑCN2(xi), rCN2(xi)eiωCN2(xi)i for all xi (i ¼ 1, 2, …, n) belong to X. A complex cosine similarity measure between CNSs CN1 and CN2 is proposed as follows:   CCNS CN 1 , CN 2      a1 ðxi Þ + b1 ðxi Þ  a2 ðxi Þ + b2 ðxi Þ + c1 ðxi Þ + d 1 ðxi Þ  n c2 ðxi Þ + d2 ðxi Þ + ðe1 ðxi Þ + f1 ðxi ÞÞ e2 ðxi Þ + f 2ðxi Þ 1X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2  2  2ffi n i¼1  a1 ðxi Þ + b1 ðxi Þ + c1 ðxi Þ + d1 ðxi Þ + e1 ðxi Þ + f1 ðxi Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5.1) ða2 ðxi Þ + b2 ðxi ÞÞ2 + ðc2 ðxi Þ + d2 ðxi ÞÞ2 + ðe2 ðxi Þ + f2 ðxi ÞÞ2 Here, h i h i h i iμ ðxi Þ iμ ðxi Þ iμ ðxi Þ , b1 ðxi Þ ¼ Im pCN ðxi Þe CN1 , a2 ðxi Þ ¼ Re pCN ðxi Þe CN2 , a1 ðxi Þ ¼ Re pCN ðxi Þe CN1 1

1

2

h i h i h i iμ ðxi Þ iϑ ðxi Þ iϑ ðxi Þ b2 ðxi Þ ¼ Im pCN ðxi Þe CN2 , c1 ðxi Þ ¼ Re qCN ðxi Þe CN1 , d1 ðxi Þ ¼ Im qCN ðxi Þe CN1 , 2

h

1

c2 ðxi Þ ¼ Re qCN ðxi Þe

iϑCN

2

i ðx Þ i

2

h

f 1 ðxi Þ ¼ Im rCN ðxi Þe 1

iωCN

1

h i h i iϑ ðxi Þ iω ðxi Þ , d 2 ðxi Þ ¼ Im qCN ðxi Þe CN2 , e1 ðxi Þ ¼ Re r CN ðxi Þe CN1 , 2

1

i ðx Þ i

1

h i h i iω ðxi Þ iω ðxi Þ , e2 ðxi Þ ¼ Re r CN ðxi Þe CN2 , f 2 ðxi Þ ¼ Im rCN ðxi Þe CN2 2

2

Some similarity measures for MADM under a complex neutrosophic set

97

Theorem 5.1. Let CN1 and CN2 be two CNSs then, I. II. III. IV.

0  CCNS(CN1, CN2)  1 CCNS(CN1, CN2) ¼ CCNS(CN2, CN1) CCNS(CN1, CN2) ¼ 1, if and only if CN1 ¼ CN2 If CN is a CNS in X and CN1  CN2  CN then, CCNS(CN1, CN) CCNS(CN1, CN2), and CCNS(CN1, CN)  CCNS(CN2, CN).

Proof I. It is obvious because all positive values of cosine function lie within 0 and 1. II. It is obvious that the theorem is true. III. When CN1 ¼ CN2, then CCNS(CN1, CN2) ¼ 1. On the other hand, if CCNS(CN1, CN2) ¼ 1 then a1(xi) ¼ a2(xi), b1(xi) ¼ b2(xi), c1(xi) ¼ c2(xi), d1(xi) ¼ d2(xi), e1(xi) ¼ e2(xi), f1(xi) ¼ f2(xi). This implies that CN1 ¼ CN2. IV. Let CN ¼ hpCN(xi)eiμCN(xi), qCN(xi)eiϑCN(xi), rS(xi)eiωCN(xi)i be a CNS. Let l1(xi) ¼ Re[pCN(xi)eiμCN(xi)], m1(xi) ¼ Re[qCN(xi)eiϑCN(xi)], n1(xi) ¼ Re[rCN(xi)eiωCN(xi)], and l2(xi) ¼ Im[pCN(xi)eiμCN(xi)], m2(xi) ¼ Im[qCN(xi)eiϑCN(xi)], n2(xi) ¼ Im[rCN(xi)eiωCN(xi)] represent the real and imaginary components of truth, indeterminacy and falsity terms of the CN. If CN1  CN2  CN then we can write: a1(xi) + b1(xi)  a2(xi) + b2(xi)  l1(xi) + l2(xi), c1(xi) + d1(xi)  c2(xi) + d2(xi)  m1(xi) + m2(xi), e1(xi) + f1(xi)  e2(xi) + f2(xi)  n1(xi) + n2(xi). The cosine function is a decreasing function in the interval [0, π/2]. Hence we can write: CCNS(CN1, CN)  CCNS(CN, CN2), and CCNS(CN1, CN)  CCNS(CN2, CN).

5.3.1 Weighted complex neutrosophic cosine similarity measure (WCNCSM) Definition 5.21. Weighted cosine similarity measure between two CNSs: D E iμ ðxi Þ iϑ ðxi Þ iω ðxi Þ CN 1 ¼ pCN ðxi Þe CN1 , qCN ðxi Þe CN1 , r CN ðxi Þe CN1 1

and

1

1

D E iμ ðxi Þ iϑ ðxi Þ iω ðxi Þ CN 2 ¼ pCN ðxi Þe CN2 , qCN ðxi Þe CN2 , r CN ðxi Þe CN2 2

2

2

for all xi (i ¼ 1, 2, …, n) belong to X is defined as follows:   CWCNS CN 1 , CN 2

     a1 ðxi Þ + b1 ðxi Þ  a2 ðxi Þ + b2 ðxi Þ + c1 ðxi Þ + d 1 ðxi Þ  n X c2 ðxi Þ + d 2 ðxi Þ + ðe1 ðxi Þ + f1 ðxi ÞÞ e2 ðxi Þ + f 2ðxi Þ wi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼  2  2  2 i¼1 a1 ðxi Þ + b1 ðxi Þ + c1 ðxi Þ + d 1 ðxi Þ + e1 ðxi Þ + f1 ðxi Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða2 ðxi Þ + b2 ðxi ÞÞ2 + ðc2 ðxi Þ + d2 ðxi ÞÞ2 + ðe2 ðxi Þ + f2 ðxi ÞÞ2

where

Pn

i¼1wi ¼ 1.

(5.2)

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Theorem 5.2. Let CN1 and CN2 be two CNSs, then: I. II. III. IV.

0  CWCNS(CN1, CN2)  1 CWCNS(CN1, CN2) ¼ CWCNS(CN2, CN1) CWCNS(CN1, CN2) ¼ 1, if and only if CN1 ¼ CN2 If CN be a CNS in X and CN1  CN2  CN then, CWCNS(CN1, CN)  CWCNS(CN1, CN2), and CWCNS(CN1, CN)  CWCNS(CN2, CN).

Proof I. It is obvious because all positive values of cosine function lie within 0 and 1. II. It is obvious that the theorem is true. III. When CN1 ¼ CN2, then CWCNS(CN1, CN2) ¼ 1. On the other hand, if CWCNS(CN1, CN2) ¼ 1 then a1(xi) ¼ a2(xi), b1(xi) ¼ b2(xi), c1(xi) ¼ c2(xi), d1(xi) ¼ d2(xi), e1(xi) ¼ e2(xi), f1(xi) ¼ f2(xi). This implies that CN1 ¼ CN2. IV. Let CN ¼ hpCN(xi)eiμCN(xi), qCN(xi)eiϑCN(xi), rS(xi)eiωCN(xi)i be a CNS. Let l1(xi) ¼ Re[pCN(xi)eiμCN(xi)], m1(xi) ¼ Re[qCN(xi)eiϑCN(xi)], n1(xi) ¼ Re[rCN(xi)eiωCN(xi)], and l2(xi) ¼ Im[pCN(xi)eiμCN(xi)], m2(xi) ¼ Im[qCN(xi)eiϑCN(xi)], n2(xi) ¼ Im[rCN(xi)eiωCN(xi)] represent the real and imaginary components of truth, indeterminacy, and falsity terms of the CN. If CN1  CN2  CN then we can write: a1(xi) + b1(xi) a2(xi) + b2(xi)  l1(xi) + l2(xi), c1(xi) + d1(xi)  c2(xi) + d2(xi)  m1(xi)+ m2(xi), e1(xi) + f1(xi)  e2(xi) + f2(xi)  n1(xi) + n2(xi). Pn The cosine function is a decreasing function in the interval [0, π/2] and i¼1wi ¼ 1. Hence we can write, CWCNS(CN1, CN)  CWCNS(CN, CN2), and CWCNS(CN1, CN)  CWCNS(CN2, CN).

5.4

Complex neutrosophic Dice similarity measure (CNDSM)

Definition 5.22. Let two CNSs, namely, CN1 ¼ hpCN1(xi)eiμCN1(xi), qCN1(xi)eiϑCN1(xi), rCN1(xi)eiωCN1(xi)i and CN2 ¼ hpCN2(xi)eiμCN2(xi), qCN2(xi)eiϑCN2(xi), rCN2(xi)eiωCN2(xi)i for all xi (i ¼ 1, 2, …, n) belong to X. A complex Dice similarity measure between CNSs CN1 and CN2 is defined as follows:      2 a1 ðxi Þ + b1 ðxi Þ a2 ðxi Þ + b2 ðxi Þ + c1 ðxi Þ + d 1 ðxiÞ n   1X c2 ðxi Þ + d 2 ðxi Þ + ðe1 ðxi Þ + f1 ðxi ÞÞ e2 ðxi Þ + f 2ðxi Þ DCNS CN 1 , CN 2 ¼       n i¼1 a ðxi Þ + b ðxi Þ 2 + c ðxi Þ + d ðxi Þ 2 + e ðxi Þ + f1 ðxi Þ 2 1 1 1 1 1 ða2 ðxi Þ + b2 ðxi ÞÞ2 + ðc2 ðxi Þ + d2 ðxi ÞÞ2 + ðe2 ðxi Þ + f2 ðxi ÞÞ2

Theorem 5.3. Let CN1 and CN2 be two CNSs, then: I. 0  DCNS(CN1, CN2)  1 II. DCNS(CN1, CN2) ¼ DCNS(CN2, CN1)

(5.3)

Some similarity measures for MADM under a complex neutrosophic set

99

III. DCNS(CN1, CN2) ¼ 1, if and only if CN1 ¼ CN2 IV. If CN be a CNS in S and CN1  CN2  CN, then DCNS(CN1, CN)  DCNS(CN1, CN2), and DCNS(CN1, CN)  DCNS(CN2, CN).

Proof I. For Dice similarity measure,       a1 ðxi Þ + b1 ðxi Þ a2 ðxi Þ + b2 ðxi Þ c1 ðxi Þ + d 1 ðxi Þ c2 ðxi Þ + d 2 ðxi Þ 2 + 4 4    e 1 ðx i Þ + f 1 ðx i Þ e 2 ðx i Þ + f 2 ðx i Þ + 4       ! a1 ðxi Þ + b1 ðxi Þ 2 c 1 ðx i Þ + d 1 ðx i Þ 2 e 1 ðx i Þ + f 1 ðx i Þ 2  + + 4 4 4  2  2   ! a2 ðxi Þ + b2 ðxi Þ c2 ðxi Þ + d 2 ðxi Þ e2 ðxi Þ + f 2 ðxi Þ 2 + + + 4 4 4 Hence, 0  DCNS(CN1, CN2)  1. II. It is obvious that the theorem is true. III. When CN1 ¼ CN2, then obviously DCNS(CN1, CN2) ¼ 1. On the other hand, if DCNS(CN1, CN2) ¼ 1 then:

a1(xi) ¼ a2(xi), f1(xi) ¼ f2(xi).

b1(xi) ¼ b2(xi),

c1(xi) ¼ c2(xi),

d1(xi) ¼ d2(xi),

e1(xi) ¼ e2(xi),

This implies that CN1 ¼ CN2. IV. Let l1(xi) ¼ Re[pCN(xi)eiμCN(xi)], m1(xi) ¼ Re[qCN(xi)eiϑCN(xi)], n1(xi) ¼ Re[rCN(xi)eiωCN(xi)], and l2(xi) ¼ Im[pCN(xi)eiμCN(xi)], m2(xi) ¼ Im[qCN(xi)eiϑCN(xi)], n2(xi) ¼ Im[rCN(xi)eiωCN(xi)] represent the real and imaginary components of truth, indeterminacy and falsity terms of the CN. If CN1  CN2  CN then, we can write a1(xi) + b1(xi)  a2(xi )+b2(xi)  l1(xi) + l2(xi), c1(xi) + d1(xi)  c2(xi) + d2(xi)  m1(xi) + m2(xi), e1(xi) + f1(xi)  e2(xi) + f2(xi)  n1(xi) + n2(xi). So, DCNS(CN1, CN)  DCNS(CN, CN2), and DCNS(CN1, CN)  DCNS(CN2, CN).

5.4.1 Weighted complex neutrosophic Dice similarity measure (WCNDSM) Definition 5.23. Weighted Dice similarity measure between two CNSs: CN1 ¼ hpCN1(xi)eiμCN1(xi), qCN1(xi)eiϑCN1(xi), rCN1(xi)eiωCN1(xi)i and CN2 ¼ hpCN2(xi)eiμCN2(xi), qCN2(xi)eiϑCN2(xi), rCN2(xi)eiωCN2(xi)i

100

Optimization Theory Based on Neutrosophic and Plithogenic Sets

for all xi (i ¼ 1, 2, …, n) belong to X is defined as follows: DWCNS(CN1, CN2)¼

     2 a1 ðxi Þ + b1 ðxiÞ a2 ðxi Þ + b2 ðxi Þ + c1 ðxi Þ + d1 ðx iÞ n X c ðxi Þ + d2 ðxi Þ + ðe1 ðxi Þ + f1 ðxi ÞÞ e2 ðxi Þ + f 2ðxi Þ w i D 2 2  2  2 a1 ðxi Þ + b1 ðxi Þ + c1 ðxi Þ + d 1 ðxi Þ + e1 ðxi Þ + f1 ðxi Þ i¼1 E ða2 ðxi Þ + b2 ðxi ÞÞ2 + ðc2 ðxi Þ + d2 ðxi ÞÞ2 + ðe2 ðxi Þ + f2 ðxi ÞÞ2

where

(5.4)

Pn

i¼1wi ¼ 1.

Theorem 5.4. Let CN1 and CN2 be two CNSs, then: I. II. III. IV.

0  DWCNS(CN1, CN2)  1 DWCNS(CN1, CN2) ¼ DWCNS(CN2, CN1) DWCNS(CN1, CN2) ¼ 1, if and only if CN1 ¼ CN2 If CN be a CNS in X and CN1  CN2  CN then DWCNS(CN1, CN)  DWCNS(CN1, CN2), and DWCNS(CN1, CN)  DWCNS(CN2, CN).

Proof I. For Dice similarity measure,       a1 ðxi Þ + b1 ðxi Þ a2 ðxi Þ + b2 ðxi Þ c ðxi Þ + d 1 ðxi Þ c2 ðxi Þ + d 2 ðxi Þ + 1 2 4 4    e 1 ðx i Þ + f 1 ðx i Þ e 2 ðx i Þ + f 2 ðx i Þ + 4       ! a1 ðxi Þ + b1 ðxi Þ 2 c1 ðxi Þ + d 1 ðxi Þ 2 e1 ðxi Þ + f 1 ðxi Þ 2 + +  4 4 4  2  2   ! a2 ðxi Þ + b2 ðxi Þ c 2 ðx i Þ + d 2 ðx i Þ e 2 ðx i Þ + f 2 ðx i Þ 2 + + + 4 4 4 and

Pn

i¼1wi ¼ 1.

Hence, 0  DWCNS(CN1, CN2)  1.

II. It is obvious that the theorem Pn is true. III. When CN1 ¼ CN2 and i¼1wi ¼ 1, then obviously DWCNS(CN1, CN2) ¼ 1. On the other hand, if DWCNS(CN1, CN2) ¼ 1 then, a1(xi) ¼ a2(xi), b1(xi) ¼ b2(xi), c1(xi) ¼ c2(xi), d1(xi) ¼ d2(xi), e1(xi) ¼ e2(xi), f1(xi) ¼ f2(xi). This implies that CN1 ¼ CN2. IV. Let l1(xi) ¼ Re[pCN(xi)eiμCN(xi)], m1(xi) ¼ Re[qCN(xi)eiϑCN(xi)], n1(xi) ¼ Re[rCN(xi)eiωCN(xi)], and l2(xi) ¼ Im[pCN(xi)eiμCN(xi)], m2(xi) ¼ Im[qCN(xi)eiϑCN(xi)], n2(xi) ¼ Im[rCN(xi)eiωCN(xi)]

Some similarity measures for MADM under a complex neutrosophic set

101

represent the real and imaginary components of truth, indeterminacy, and falsity terms of the CN. If CN1  CN2  CN, then we can write a1(xi)+ b1(xi)  a2(xi)+ b2(xi)  l1(xi) + l2(xi), c1(xi) + d1(xi)  c2(xi) + d2(xi)  m1(xi) + m2(xi), e1(xi) + fP 1(xi)  e2(xi) + f2(xi)  n1(xi) + n2(xi). Again, ni¼1wi ¼ 1. So, DWCNS(CN1, CN)  DWCNS(CN, CN2) and DWCNS(CN1, CN)  DWCNS(CN2, CN).

5.5

Complex neutrosophic Jaccard similarity measure (CNJSM)

Definition 5.24. Let two CNSs: D E CN 1 ¼ pCN1 ðxi ÞeiμCN1 ðxi Þ , qCN1 ðxi ÞeiϑCN1 ðxi Þ , r CN1 ðxi ÞeiωCN1 ðxi Þ and D E CN 2 ¼ pCN2 ðxi ÞeiμCN2 ðxi Þ , qCN2 ðxi ÞeiϑCN2 ðxi Þ , r CN 2 ðxi ÞeiωCN2 ðxi Þ for all xi’s (i ¼ 1, 2, …, n) belong to X. A complex Jaccard similarity measure between CNSs CN1 and CN2 is defined as follows: JCNS(CN1, CN2)      a1 ðxi Þ + b1 ðxi Þ a2 ðxi Þ + b2 ðxi Þ + c1 ðxi Þ + d1 ðx iÞ  n c2 ðxi Þ + d 2 ðxi Þ + ðe1 ðxi Þ + f1 ðxi ÞÞ e2 ðxi Þ + f 2ðxi Þ 1X ¼       n i¼1 a ðxi Þ + b ðxi Þ 2 + c ðxi Þ + d ðxi Þ 2 + e ðxi Þ + f1 ðxi Þ 2 1

1

1

2

1

1

2

+ða2 ðxi Þ + b2 ðxi ÞÞ + ðc2 ðxi Þ + d2 ðxi ÞÞ + ðe2 ðxi Þ + f2 ðxi ÞÞ2       a1 ð x i Þ + b1 ð x i Þ a2 ð x i Þ + b2 ð x i Þ + c 1 ð x i Þ + d 1 ð x i Þ     c2 ðxi Þ + d2 ðxi Þ + ðe1 ðxi Þ + f1 ðxi ÞÞ e2 ðxi Þ + f 2ðxi Þ

(5.5)

Theorem 5.5. Let CN1 and CN2 be two CNSs, then: I. II. III. IV.

0  JCNS(CN1, CN2)  1 JCNS(CN1, CN2) ¼ JCNS(CN2, CN1) CCNS(CN1, CN2) ¼ 1, if and only if CN1 ¼ CN2 If CN is a CNS in S and CN1  CN2  CN then, JCNS(CN1, CN)  JCNS(CN1, CN2), and JCNS(CN1, CN)  JCNS(CN2, CN).

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Proof I. Since: 

     a1 ðxi Þ + b1 ðxi Þ a2 ðxi Þ + b2 ðxi Þ c1 ðxi Þ + d 1 ðxi Þ c2 ðxi Þ + d2 ðxi Þ + 4 4   

e ðxi Þ + f 1 ðxi Þ e2 ðxi Þ + f 2 ðxi Þ + 1 4 * 2  2 a1 ðxi Þ + b1 ðxi Þ c1 ðxi Þ + d 1 ðxi Þ ðe ðx Þ + f 1 ðx i ÞÞ2  + + 1 i 4 4 4

 2  2 c ðx i Þ + d 2 ðx i Þ e ðxi Þ + f 2 ðxi Þ ða2 ðxi Þ + b2 ðxi ÞÞ2 + 2 + 2 4 4 4       a1 ðxi Þ + b1 ðxi Þ a2 ðxi Þ + b2 ðxi Þ c1 ðxi Þ + d 1 ðxi Þ c2 ðxi Þ + d 2 ðxi Þ  + 4 4    

e ðxi Þ + f 1 ðxi Þ e2 ðxi Þ + f 2 ðxi Þ + 1 4 +

Then, we can write 0  JCNS(CN1, CN2)  1. II. It is obvious that the theorem is true. III. When CN1 ¼ CN2, then JCNS(CN1, CN2) ¼ 1. On the other hand, if JCNS(CN1, CN2) ¼ 1, then a1(xi) ¼ a2(xi), b1(xi) ¼ b2(xi), c1(xi) ¼ c2(xi), d1(xi) ¼ d2(xi), e1(xi) ¼ e2(xi), f1(xi) ¼ f2(xi). This implies that CN1 ¼ CN2. IV. Let l1(xi) ¼ Re[pCN(xi)eiμCN(xi)], m1(xi) ¼ Re[qCN(xi)eiϑCN(xi)], n1(xi) ¼ Re[rCN(xi)eiωCN(xi)], and l2(xi) ¼ Im[pCN(xi)eiμCN(xi)], m2(xi) ¼ Im[qCN(xi)eiϑCN(xi)], n2(xi) ¼ Im[rCN(xi)eiωCN(xi)] represent the real and imaginary components of truth, indeterminacy, and falsity terms of the CN. If CN1  CN2  CN, then we can write: a1(xi) + b1(xi)  a2(xi) + b2(xi)  l1(xi) + l2(xi), c1(xi) + d1(xi)  c2(xi) + d2(xi)  m1(xi) + m2(xi), e1(xi) + f1(xi)  e2(xi) + f2(xi)  n1(xi) + n2(xi). So, JCNS(CN1, CN)  JCNS(CN, CN2), and JCNS(CN1, CN)  JCNS(CN2, CN).

5.5.1 Weighted complex neutrosophic Jaccard similarity measure (WCNJSM) Definition 5.25. Weighted Jaccard similarity measure between two CNSs: CN1 ¼ hpCN1(xi)eiμCN1(xi), qCN1(xi)eiϑCN1(xi), rCN1(xi)eiωCN1(xi)i and CN2 ¼ hpCN2(xi)eiμCN2(xi), qCN2(xi)eiϑCN2(xi), rCN2(xi)eiωCN2(xi)i for all xi’s (i ¼ 1, 2, …, n) belong to X is defined as follows:

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JWCNS(CN1, CN2)      a1 ðxi Þ + b1 ðxi Þ a2 ðxi Þ + b2 ðxi Þ + c1 ðxi Þ + d 1 ðx iÞ  n X c ðxi Þ + d 2 ðxi Þ + ðe1 ðxi Þ + f1 ðxi ÞÞ e2 ðxi Þ + f 2ðxi Þ wi  2 ¼ 2  2  2 a1 ðxi Þ + b1 ðxi Þ + c1 ðxi Þ + d1 ðxi Þ + e1 ðxi Þ + f1 ðxi Þ i¼1 +ða2 ðxi Þ + b2 ðxi ÞÞ2 + ðc2 ðxi Þ + d2 ðxi ÞÞ2 + ðe2 ðxi Þ + f2 ðxi ÞÞ2       a1 ðxi Þ + b1 ðxi Þ a2 ðxi Þ + b2 ðxi Þ + c1 ðxi Þ + d1 ðxi Þ     c2 ðxi Þ + d2 ðxi Þ + ðe1 ðxi Þ + f1 ðxi ÞÞ e2 ðxi Þ + f 2ðxi Þ Here,

(5.6)

Pn

i¼1wi ¼ 1.

Theorem 5.6. Let CN1 and CN2 be two CNSs then, I. II. III. IV.

0  JWCNS(CN1, CN2)  1 JWCNS(CN1, CN2) ¼ JWCNS(CN2, CN1) CWCNS(CN1, CN2) ¼ 1, if and only if CN1 ¼ CN2 If CN is a CNS in S and CN1  CN  CN then, JWCNS(CN1, CN)  JWCNS(CN1, CN2), and JWCNS(CN1, CN)  JWCNS(CN2, CN).

Proof I. Since:       a1 ðxi Þ + b1 ðxi Þ a2 ðxi Þ + b2 ðxi Þ c ðxi Þ + d 1 ðxi Þ c2 ðxi Þ + d 2 ðxi Þ + 1 4 4   

e ðx i Þ + f 1 ðx i Þ e 2 ðx i Þ + f 2 ðx i Þ + 1 4 * 2  2 a1 ðxi Þ + b1 ðxi Þ c 1 ðx i Þ + d 1 ðx i Þ ðe ðx Þ + f 1 ðx i ÞÞ2 + + 1 i  4 4 4  2  2 c ðxi Þ + d2 ðxi Þ e ðx i Þ + f 2 ðx i Þ ða2 ðxi Þ + b2 ðxi ÞÞ2 + 2 + 2 4 4 4       a1 ðxi Þ + b1 ðxi Þ a2 ðxi Þ + b2 ðxi Þ c1 ðxi Þ + d 1 ðxi Þ c2 ðxi Þ + d 2 ðxi Þ  + 4 4    

e ðxi Þ + f 1 ðxi Þ e2 ðxi Þ + f 2 ðxi Þ + 1 4 +

and

Pn

i¼1wi ¼ 1;

so, we can write 0  JCNS(CN1, CN2)  1.

II. It is obvious that the theorem P is true. III. When CN1 ¼ CN2, and ni¼1wi ¼ 1 then JWCNS(CN1, CN2) ¼ 1. On the other hand, if JWCNS(CN1, CN2) ¼ 1 then: a1(xi) ¼ a2(xi), b1(xi) ¼ b2(xi), c1(xi) ¼ c2(xi), d1(xi) ¼ d2(xi), e1(xi) ¼ e2(xi), f1(xi )¼f2(xi). This implies that CN1 ¼ CN2.

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IV. Let l1(xi) ¼ Re[pCN(xi)eiμCN(xi)], m1(xi) ¼ Re[qCN(xi)eiϑCN(xi)], n1(xi) ¼ Re[rCN(xi)eiωCN(xi)], and l2(xi) ¼ Im[pCN(xi)eiμCN(xi)], m2(xi) ¼ Im[qCN(xi)eiϑCN(xi)], n2(xi) ¼ Im[rCN(xi)eiωCN(xi)] represent the real and imaginary components of truth, indeterminacy, and falsity terms of the CN. If CN1  CN2  CN, then we can write a1(xi) + b1(xi) a2(xi) + b2(xi)  l1(xi) + l2(xi), c1(xi) + d1(xi)  c2(xi) + d2(xi)  m1(xi) + m2(xi), e1(xi) + fP 1(xi)  e2(xi) + f2(xi)  n1(xi) + n2(xi). Again, ni¼1wi ¼ 1. So, JWCNS(CN1, CN)  JWCNS(CN, CN2), and JWCNS(CN1, CN)  JWCNS(CN2, CN).

5.6

Tangent function for CNS

In this section, we define a tangent function to determine partially or completely unknown attributes weights. Definition 5.26. The tangent function of a CNS CN ¼ hpCN(xj)eiμCN(xi), qCN(xj)eiϑCN(xj), pffiffiffiffiffiffiffi rCN(xj)eiωCN(xj)i, ( j ¼ 1, 2,…, n) and i ¼ 1 is defined as follows: Tj ðCN Þ ¼

n       1X π   2 + pCN xj  qCN xj  r CN xj  tan n j¼1 12

(5.7)

The weight structure is defined as: Tj ðCN Þ wj ¼ Xn ; j ¼ 1, 2,…, n T ðCN Þ j¼1 j Here,

n P

(5.8)

wj ¼ 1.

j¼1

Theorem 5.7. For any CNS CN ¼ hpCN(xj)eiμCN(xi), qCN(xj)eiϑCN(xj), rCN(xj)eiωCN(xj)i, 0  Tj(CN)  1. Proof       0  pCN xj  1, 0  qCN xj  1, 0  r CN xj  1       ) 0  2 + pCN xj  qCN xj  r CN xj  3       π 2 + pCN xj  qCN xj  r CN xj  1 12        1 Xn π  2 + p tan x x  q  r 1 )0  j j CN xj  CN CN j¼1 n 12

) 0  tan

) 0  Tj ðCN Þ  1

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105

Theorem 5.8. Tj(CN) ¼ 1 if pCN(xj) ¼ 1, qCN(xj) ¼ 0, rCN(xj) ¼ 0. Proof       pCN xj ¼ 1, qCN xj ¼ 0, r CN xj ¼ 0 ) Tj ðCN Þ ¼

n n 1X π 1X π 1 tan ð j2 + 1  0  0jÞ ¼ tan ¼ :n ¼ 1: n j¼1 12 n j¼1 4 n

Theorem 5.9. Tj(CN) ¼ 0 if pCN(xj) ¼ 0, qCN(xj) ¼ 1, rCN(xj) ¼ 1. Proof       pCN xj ¼ 0, qCN xj ¼ 1, r CN xj ¼ 1 ) Tj ðCN Þ ¼

n n 1X π 1X 1 tan ð j2 + 0  1  1jÞ ¼ tan 0 ¼ :0 ¼ 0: n j¼1 12 n j¼1 n

  π π  π π  Example 5.8. Let CN 1 ¼ 0:7eπ , 0:4e2i , 0:4e4i , and CN 2 ¼ 0:5e2i , 0:5eπi , 0:5e2i : Then, we obtain T(CN1) ¼ 0.3689 and T(CN2) ¼ 0.1989.

5.7

Decision-making steps

In this section, the steps of decision-making based on WCNCSM, WCNDSM, and WCNJSM in CNS environment are presented (see Fig. 5.1).

Start

Set the candidates, their attributes, and possible selection options

Calculate the complex neutrosophic cosine/Dice/Jaccard similarity measure values

Rank the priority

Determine the relation between candidates and their attributes in terms of CNSs

Determine the relation between candidates attributes and selection options in terms of CNSs

Determine the candidates selection option

End

Fig. 5.1 Decision-making based on cosine/Dice/Jaccard similarity measure in CNS environment.

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5.7.1 Decision-making based on WCNCSM Step 1: Determine the relation (R-1) between candidates and their attributes in terms of CNS information. Step 2: Determine the relation (R-2) between attributes and selection options in CNS information. Step 3: Using Eq. (5.8) and the relation (R-1), determine attribute weight structure. Step 4: Determine complex neutrosophic cosine similarity measures (WCNCSM) between R-1 and R-2. All the probable selection options of a candidate have a WCNCSM value. The selection option with the highest WCNCSM value reflects the best choice. Step 5: End.

5.7.2 Decision-making based on WCNDSM Step 1: Determine the relation (R-1) between candidates and their attributes in terms of CNS information. Step 2: Determine the relation (R-2) between attributes and selection options in CNS information. Step 3: Using Eq. (5.8) and the relation (R-1), determine attribute weight structure. Step 4: Determine complex neutrosophic Dice similarity measures (WCNDSM) between R-1 and R-2. All the probable selection options of a candidate have a WCNDSM value. The selection option with highest WCNDSM value reflects the best choice. Step 5: End.

5.7.3 Decision-making based on WCNJSM Step 1: Determine the relation (R-1) between candidates and their attributes in terms of CNS information. Step 2: Determine the relation (R-2) between attributes and selection options in CNS information. Step 3: Using Eq. (5.8) and the relation (R-1), determine attribute weight structure. Step 4: Determine complex neutrosophic Jaccard similarity measures (CNJSM) between R-1 and R-2. All the probable selection options of a candidate have a WCNJSM value. The selection option with the highest WCNJSM value reflects the best choice. Step 5: End.

5.8

Selection of educational stream for higher secondary education

In this section, we present a numerical example (which is adopted from [112]) to demonstrate the proposed MADM strategies. It is very important for students after secondary examination to select an appropriate educational stream for higher secondary education. After secondary examination, the student takes up subjects of his/her choice and focuses on hard work for better career prospects in the future. Students often have a problem to decide which stream they should choose to follow.

Some similarity measures for MADM under a complex neutrosophic set

107

Selecting a career in a particular branch right at this point of time has a long lasting impact on a student’s future. If the chosen stream is inappropriate, the student may encounter a negative impact to his/her career. It is very important for a student to choose carefully from different options available to him/her in which he/she is interested. So, it is necessary to use a suitable mathematical strategy for decision-making. The proposed similarity measure among the students versus attributes and attributes versus educational streams will give the proper selection of educational stream of students. The feature of the proposed strategy is that it includes periodic truth membership, periodic indeterminate, and periodic falsity membership function simultaneously. Let S ¼ {S1, S2, S3} be a set of students, E ¼ {science (E1), humanities/arts (E2), commerce (E3), vocational course (E4)} be a set of educational streams and C ¼ {depth in basic science and mathematics (C1), depth in language (C2), good grade point in secondary examination (C3), concentration (C4), and laborious (C5)} be a set of attributes. Our solution is to examine the students and make the decision to choose a proper educational stream for them (see Tables 5.1, 5.2, and 5.3). The decision-making procedure is presented using the following steps. Step 1: The relation between students and their attributes in CNS form is presented in Table 5.1. Step 2: The relation between student’s attributes and educational streams in the form CNSs is presented in Table 5.2. Step 3: Using Eq. (5.8) and the relation (R-1), we determine attributes’ weight structure as follows: w1 ¼ 0.2055, w2 ¼ 0.1873, w3 ¼ 0.1919, w4 ¼ 0.2046, w5 ¼ 0.2107. Step 4: Determine the weighted similarity measure values between Tables 5.1 and 5.2 using WCNCSM, WCNDSM, and WCNJSM. The obtained measure values are presented in Tables 5.3, 5.4, and 5.5. Table 5.3 shows that the highest similarity measure values of S1, S2, and S3 are 0.9318, 0.8694, and 0.9282, respectively. Table 5.4 shows that the highest similarity measure values of S1, S2, and S3 are 0.8636, 0.8320. and 0.9020, respectively. Table 5.5 shows that the highest similarity measure values of S1, S2, and S3 are 0.8608, 0.8515, and 0.8721, respectively. Therefore, student S1 should select the arts stream, student S2 should select the science stream, and student S3 should select the science stream for higher secondary education purpose. Step 5: End.

5.9

Comparison analysis

Only one decision-making strategy [111] exists in the literature in complex neutrosophic environments. The ranking results obtained from proposed strategies and the existing strategy [111] are furnished in Table 5.6.

108

Table 5.1 Relation (R-1) between students and attributes in CNS form.

S1 S2 S3

C2

0:4e1:2i , 0:4e1:1i , 0:3e0:7i

0:4e1:5i , 0:6e1:5i , 0:5i 0:3e

0:5e1:3i , 0:4e1:2i , 0:4e0:4i

C1

0:6e1:0i ,0:4e1:2i , 0:2e0:8i

0:7e1:3i ,0:4e1:2i , 0:9i 0:5e

0:5e0:6i ,0:5e1:2i , 0:5e0:9i

C3

0:3e1:0i ,0:4e1:0i , 0:4e0:6i

0:5e1:4i ,0:4e1:2i , 1:0i 0:4e

0:4e1:0i ,0:4e1:0i , 0:2e0:6i

C4

0:6e1:0i ,0:5e1:2i , 0:3e0:8i

0:6e1:0i ,0:4e1:0i , 0:6i 0:4e

0:4e1:0i ,0:5e1:1i , 0:2e1:2i

C5

0:4e1:0i , 0:3e1:0i , 0:2e0:5i

0:3e1:5i , 0:4e1:0i , 1:0i 0:5e

0:5e1:2i , 0:2e1:2i , 0:2e1:4i

E1 C1 C2 C3 C4 C5



E2

0:4e1:2i ,0:4e1:4i , 0:3e0:6i

0:5e0:6i ,0:4e0:7i , 0:2e0:8i

0:4e1:0i ,0:4e1:1i , 0:4e1:2i

0:3e1:4i ,0:4e1:5i , 0:5e0:6i

0:4e0:8i ,0:4e0:9i , 1:0i 0:5e



E3

0:6e1:3i ,0:4e1:4i , 0:2e1:5i

0:4e0:7i ,0:4e0:8i , 0:3e0:9i

0:5e1:1i ,0:2e1:2i , 0:2e1:3i

0:4e1:5i ,0:5e0:6i , 0:3e0:7i

0:6e1:0i ,0:4e1:2i , 1:4i 0:3e



E4

0:5e1:4i , 0:5e1:5i , 0:2e0:6i

0:5e0:8i , 0:4e0:9i , 0:2e1:0i

0:4e1:2i , 0:4e1:3i , 0:5e1:4i

0:5e0:6i , 0:4e0:7i , 0:3e0:8i

0:4e1:2i , 0:4e1:4i , 0:6i 0:5e



0:6e1:5i ,0:4e0:6i , 0:5e0:7i

0:5e0:9i ,0:4e1:0i , 0:5e0:8i

0:4e1:3i ,0:4e1:4i , 0:3e1:5i

0:3e0:7i ,0:4e0:8i , 0:4e0:9i

0:4e1:4i ,0:3e0:6i , 0:8i 0:2e

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 5.2 Relation (R-2) between attributes and educational streams in CNS form.

Some similarity measures for MADM under a complex neutrosophic set

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Table 5.3 The complex cosine neutrosophic measures between R-1 and R-2. CNCSM

E1

E2

E3

E4

S1 S2 S3

0.9318 0.8594 0.9282

0.9287 0.7523 0.8615

0.8675 0.8161 0.8422

0.8455 0.8694 0.7875

Table 5.4 The complex Dice neutrosophic measures between R-1 and R-2. CNDSM

E1

E2

E3

E4

S1 S2 S3

0.8636 0.8037 0.9020

0.8294 0.7331 0.8486

0.8609 0.7946 0.8200

0.8464 0.8320 0.7683

Table 5.5 The complex Jaccard neutrosophic measures between R-1 and R-2. CNJSM

E1

E2

E3

E4

S1 S2 S3

0.8608 0.8213 0.8721

0.8126 0.8032 0.8159

0.8510 0.7922 0.8481

0.8456 0.8515 0.7438

Table 5.6 Comparison analysis. Strategies

Students

Ranking order

Proposed strategy (based on CNCSM)

S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3

E1 E2 E3 E4 E4 E1 E3 E2 E1 E2 E3 E4 E1 E3 E4 E2 E4 E1 E3 E2 E1 E2 E3 E4 E1 E3 E4 E2 E4 E1 E2 E3 E1 E3 E2 E4 E1 E4 E3 E2 E4 E1 E2 E3 E1 E3 E4 E2

Proposed strategy (based on CNDSM) Proposed strategy (based on CNJSM) Al-Quran and Hassan strategy [111]

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

It is observed that ranking results of the proposed strategies is similar to the ranking result of the existing strategy [111].

5.10

Conclusion

In this chapter, we have proposed three similarity measures, viz., cosine, Dice, and Jaccard in CNS environment. We have also proved some of their basic properties. We have developed three new strategies for MADM based on proposed similarity measures in CNS environment. We have defined a tangent function for determining unknown attributes weights. We have demonstrated the proposed strategies with a numerical example. We have conducted a comparison analysis between the results of the proposed strategies and the results of the existing strategy. The proposed strategies simply and reliably represent human cognition by considering the interactivity of attributes and the cognition towards indeterminacy involves in the problem. In future research, it will be interesting to develop new aggregation operators of CNSs and their applications in MADM problems such as logistics center selection [113], conflict resolution [114], personnel selection [115], image processing [116], medical diagnosis [117], and so on.

Conflict of interests The authors declare that there is no conflict of interests regarding the publication of the chapter.

Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.

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[27] P. Biswas, S. Pramanik, B.C. Giri, GRA method of multiple attribute decision making with single valued neutrosophic hesitant fuzzy set information, in: F. Smarandache, S. Pramanik (Eds.), New Trends in Neutrosophic Theory and Applications, Vol. I, Pons Editions, Brussels, 2016, pp. 55–63. [28] P. Biswas, S. Pramanik, B.C. Giri, Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making, Neutrosophic Sets Syst. 12 (2016) 20–40. [29] P. Biswas, S. Pramanik, B.C. Giri, Value and ambiguity index based ranking method of single-valued trapezoidal neutrosophic numbers and its application to multi-attribute decision making, Neutrosophic Sets Syst. 12 (2016) 127–138. [30] P. Biswas, S. Pramanik, B.C. Giri, Non-linear programming approach for single-valued neutrosophic TOPSIS method, New Math. Natural Comput. (2019), https://doi.org/ 10.1142/S1793005719500169. [31] P. Biswas, S. Pramanik, B.C. Giri, NH-MADM strategy in neutrosophic hesitant fuzzy set environment based on extended GRA, Informatica 30 (2) (2019) 213–242. [32] K. Mondal, S. Pramanik, F. Smarandache, NN-harmonic mean aggregation operatorsbased MCGDM strategy in a neutrosophic number environment, Axioms 7 (2018) 12, https://doi.org/10.3390/axioms7010012. [33] K. Mondal, S. Pramanik, Neutrosophic decision making model for clay-brick selection in construction field based on grey relational analysis, Neutrosophic Sets Syst. 9 (2015) 64–71. [34] K. Mondal, S. Pramanik, Neutrosophic decision making model of school choice, Neutrosophic Sets Syst. 7 (2015) 62–68. [35] K. Mondal, S. Pramanik, F. Smarandache, Role of neutrosophic logic in data mining, in: F. Smarandache, S. Pramanik (Eds.), New Trends in Neutrosophic Theory and Application, Vol. I, Pons Editions, Brussels, Belgium, 2016, pp. 15–23. [36] K. Mondal, S. Pramanik, B.C. Giri, Single valued neutrosophic hyperbolic sine similarity measure based MADM strategy, Neutrosophic Sets Syst. 20 (2018) 3–11. [37] J.J. Peng, J.Q. Wang, H.Y. Zhang, X.H. Chen, An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets, Appl. Soft Comput. 25 (2014) 336–346. [38] S. Pramanik, P.P. Dey, B.C. Giri, Hybrid vector similarity measure of single valued refined neutrosophic sets to multi-attribute decision making problems, in: F. Smarandache, S. Pramanik (Eds.), New Trends in Neutrosophic Theory and Applications, Vol. II, Pons Editions, Brussels, 2018, pp. 156–174. [39] S. Pramanik, S. Dalapati, T.K. Roy, Neutrosophic multi-attribute group decision making strategy for logistics center location selection, in: F. Smarandache, M.A. Basset, V. Chang (Eds.), Neutrosophic Operational Research, Vol. III, Pons asbl, Brussels, 2016, pp. 13–32. [40] S. Pramanik, P.P. Dey, B.C. Giri, TOPSIS for single valued neutrosophic soft expert set based multi-attribute decision making problems, Neutrosophic Sets Syst. 10 (2015) 88–95. [41] S. Pramanik, R. Roy, T.K. Roy, Teacher selection strategy based on bidirectional projection measure in neutrosophic number environment, in: F. Smarandache, M.A. Basset, V. Chang (Eds.), Neutrosophic Operational Research, Vol. II, Pons asbl, Brussels, 2016, pp. 29–53. [42] S. Pramanik, S. Dalapati, T.K. Roy, Logistics center location selection approach based on neutrosophic multi-criteria decision making, in: F. Smarandache, S. Pramanik (Eds.), New Trends in Neutrosophic Theory and Applications, Vol. I, Pons Editions, Brussels, 2016, pp. 161–174.

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Sentiment analysis of the #MeToo movement using neutrosophy: Application of single-valued neutrosophic sets

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Ilanthenral Kandasamya, W.B. Vasanthaa, Niharika Mathura, Mayank Bishta, Florentin Smarandacheb a School of Computer Science and Engineering, VIT, Vellore, India, bDepartment of Mathematics, University of New Mexico, Gallup, NM, United States

6.1

Introduction

The #MeToo movement is a highly publicized online movement that started on the social media platform Twitter. It dealt with the ideology of equal rights and that harassment, in any form, directed toward any individual, is a punishable offense. It originated with several women across the world coming out with their stories of sexual harassment at work and in their personal lives. Advocating the way that the overall #MeToo movement developed has forced a renewed consideration regarding the issue of lewd behavior in the workplace, which has led to appropriate discussions about workplace ethics [1]. It has been argued that the #MeToo movement is yet to take off extensively [2]. The establishment may overlook the complaint of a single woman, but they are less inclined to disregard aggregate voices. The implications of hashtagbased movements and their reach in terms of what they mean for the internet generation and what their reach is has been researched [3]. Sentiment analysis, also known as opinion mining, is a process of analyzing the polarity of the emotion behind a line of text. It is helpful in understanding social media on specific subjects. The existing conventional sentiment analysis tools that work on various platforms aim at categorizing the target subject into negative or positive polarity. The recent emergence of opinion-based sentiments rather than the conventional fact-based analysis for information-seeking systems has been studied extensively in Pang et al. [4]. The difference in the linguistic approach in a micro-blogging site such as Twitter has been highlighted in Kouloumpis et al. [5] with emphasis on the lexiconbased approach to the way people express themselves on social media, which is generally quite informal. The seemingly nonexistent relation between sentiment analysis and social network analysis based on assigning sentiments to nodes which build social connections is explored in Fornacciari et al. [6]. The conception of Valence Aware Dictionary and Sentiment Reasoner (VADER) as a simple rule-based model for general sentiment analysis as well as the comparison of its effectiveness with other Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00006-8 © 2020 Elsevier Inc. All rights reserved.

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sentiment analysis tools has been set out in Hutto and Gilbert [7]. Fuzzy logic was used for sentiment analysis in Jefferson et al. [8]; even while using fuzzy logic the indeterminate part was ignored. The polarity projected by the tool is not accurate since sometimes it is neither positive nor negative, that is, it is indeterminate. Essentially to capture this indeterminate nature of sentiment analysis, for the first time Neutrosophy is used, which is a branch of philosophy that deals with indeterminacy. Indeterminacy supports the fact that a proposition lies between two extremes, supporting neither of the two. In the context of social media text, Neutrosophy is a very powerful tool in analyzing the popular opinion around a trending topic or movement. This is because many opinions on social media platforms such as Twitter or Facebook are based on a user’s opinion and perception of the concerned topic. A user can take a variety of stances on an issue, and these can be strongly positive to strongly negative. However, in many cases, the user might not have a strong opinion. This could be because the user requires more information to make an informed choice of stance, or the user simply does not wish to actively take any side. In other words, the sentiment of the user falls in the indeterminate category. This is what Neutrosophy bases its predictions on: the indeterminacies of a sentiment presented in the form of a tweet or a post on social media. The concept of a neutrosophic set was proposed by Smarandache [9], which was later developed as single-valued neutrosophic sets (SVNSs) by Wang et al. [10]. New similarity measures for link prediction dependent on fuzzy and Neutrosophic situations has been derived by using predictive measures [11]. SVNSs are essentially a generalization of fuzzy sets [12]. The notion of distance between two SVNSs and their properties has been defined clearly in terms of distance measures. Several similarly measures between two SVNSs and their properties have been investigated in Refs. [13–15]. To make more refined and accurate representation of the indeterminacy present in the real-world data, double-valued neutrosophic sets (DVNSs) were defined [16, 17] with two indeterminate memberships. The usage of neutrosophic sets for social network analysis in proposing a learning that consolidates the underlying social networks helps in molding their learning, takes into account the Neutrosophic value of e-learning systems to present user generated opinions in a sensitive way. [18]. Triple refined indeterminate neutrosophic sets (TRINSs) have been used for personality classification and Likert scaling [19]. The algebraic concept of Neutrosophic duplets and triplets have been linked to SVNSs recently [20–22]. The existing conventional sentiment analysis or fuzzy sentiment analysis does not capture the indeterminacy present in the tweets/opinion. However, our final analysis showed that many of the tweets are indeterminate in nature. Our analysis yields better accuracy in capturing the indeterminacy present. Recently refined neutrosophic sets have been used in Likert scaling [23], the indeterminate scaling can be extended to sentiment analysis. Individual analysis of each tweet is not carried out and importance is not given to the indeterminacy represent in each tweet. This chapter provides a remarkable application of SVNSs to sentiment analysis, where each tweet is analyzed separately and represented as SVNSs with negative,

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positive, and indeterminate memberships. It is an interesting and innovative study that gives an accurate evaluation of the tweets. We also dealt with a huge dataset of more than 400,000 tweets. Our analysis shows that the actual trend of the tweets was indeterminate in nature and not positive as suggested by conventional sentiment analysis. Clustering and classification of SVNSs is carried out. The chapter is organized as follows: Section 6.1 is introductory in nature. Section 6.2 gives theoretical and mathematical foundations of neutrosophy and SVNSs along with their distance measures in terms of dealing with opinions. Section 6.3 describes the algorithms used for the clustering and classification of SVNSs. Section 6.4 uses sentiment analysis in a neutrosophic environment using the concept of SVNS. Section 6.5 deals a the case study of the #MeToo movement and analyzes the tweets using SVNSs. Clustering and classification are carried out on this to validate the predominance of indeterminacy in the opinion. Section 6.6 compares neutrosophy sentiment analysis with conventional and fuzzy sentiment analysis; comparisons between the various classification algorithms used on SVNSs are also done. The results and possible future studies in these directions are discussed in Section 6.7.

6.2

Basic concepts

The concept of neutrosophy and related definitions are rewritten here with relation to opinion mining. Neutrosophy is analysis of opinion “O” and its relation to the opposite opinion, “Anti-O” and not O, “Non-O,” and as neither “O” nor “Anti-O” as indeterminate opinion. Definition 6.1 Wang et al. [10]. Let X be a space of points (opinions) with elements in X denoted by x. An SVNS R in X is characterized by negative (false) NR(x), indeterminacy IR(x), and positive (truth) PR(x) membership functions. 8x 2 X, there are NR(x), IR(x), PR(x) 2 [0, 1], and 0  NR(x) + IR(x) + PR(x)  3. Therefore, an SVNS of an opinion O can be represented by R ¼ fhx, NR ðxÞ, IR ðxÞ,PR ðxÞi|x 2 Xg: The distance measures over two opinions represented as SVNSs R and Q in a universe of discourse, X ¼ xl, x2, …, xn, is given in the following paragraph. R and Q are denoted by R ¼ fhxi , NR ðxi Þ,IR ðxi Þ, PR ðxi Þi|xi 2 Xg, and Q ¼ fhxi , NQ ðxi Þ, IQ ðxi Þ, PQ ðxi Þi|xi 2 Xg, where NR(xi), IR(xi), PR(xi), NQ(xi), IQ(xi), PQ(xi) 2 P [0, 1];8xi 2 X. Let wi be the weight of an element xi with wi  0(i ¼ 1, 2, …, n) and ni ¼ 1 wi ¼ 1:

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Then, the neutrosophic weighted distance of R and Q is defined as follows: ( dλ ðR, QÞ ¼

n 1X wi ½|NR ðxi Þ  NQ ðxi Þ|λ + |IR ðxi Þ  IQ ðxi Þ|λ 3 i¼1 )1=λ

(6.1)

λ

+|PR ðxi Þ  PQ ðxi Þ|  where λ > 0. Eq. (6.1) will reduce to Hamming distance and Euclidean distance, when λ ¼ 1, 2, respectively. The Euclidean distance of two SVNSs P and Q is given as ( dλ ðR, QÞ ¼

n 1X wi ½|NR ðxi Þ  NQ ðxi Þ|2 + |IR ðxi Þ  IQ ðxi Þ|2 3 i¼1 )1=2

+|PR ðxi Þ  PQ ðxi Þ|2  where λ ¼ 2 in Eq. (6.1). The algorithm for finding dλ(R, Q) is given by Algorithm 6.1.

Algorithm 6.1 Neutrosophic weighted distance dλ(R, Q)

(6.2)

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Clustering is grouping of data points to classify them into specific classes. Neutrosophic C-means clustering was defined in Guo and Sengur [24] and has been extensively used in image processing [25, 26]. In the case of neutrosophic refined sets like SVNSs, DVNSs, etc., the clustering method is the minimum spanning tree (MST) [13, 16, 19]. An MST is a spanning tree with the minimum length. Algorithms to obtain the MST of a graph operate iteratively. At any stage, the edge belongs to one of two sets, that is, set A, which contains those edges that belong to the MST and set B, which contains those that do not belong to the MST. Another algorithm that works in assigning iteratively to set A the shortest edge in the set B is Kruskal’s algorithm. The edge in set B which does not form a closed loop with any of the edges in A is assigned to A. Initially, A is empty, and the iteration stops when A contains (n  1) edges. Another algorithm, called Prim’s algorithm, begins with any one of the given nodes and initially assigns to A the shortest edge. After that it traverses the nodes and, starting from the first assigned node, continues to add the shortest edge from B, which connects at least one edge from A without forming a closed loop with the edge in A. The iteration stops when A has (n  1) edges.

6.3

Clustering and classification algorithms of SVNSs

To the best of our knowledge, K-means clustering has not been used along with SVNSs for opinion mining, which is an innovative approach of our study. Since real-world data like tweets can also be indeterminate in nature, we have taken up the sentiment analysis using neutrosophy. Furthermore, to substantiate the presence of indeterminate class other than positive and negative classes, clustering of the tweets represented as data points is carried out, which is innovative.

6.3.1 K-means clustering algorithm for SVNS The clustering technique with neutrosophy has been applied in various domains. But the use of clustering for sentiment analysis is fairly rare, and presents a novel edge to the work. The approach is to transform the data points into neutrosophic sets by calculating positive, negative, and indeterminacy memberships and to create clusters. The iterative process converges when the required number of clusters are formed. When using the K-means algorithm, it is essential to determine the number of clusters that the dataset gets clustered into. The optimal number of clusters for a dataset is found through the elbow method. It specifies what can be an appropriate K (number of clusters) which is based on the sum of squared distance between data points and their assigned clusters’ centroids. The algorithm for clustering SVNS values using K-means clustering is given in Algorithm 6.2; it takes in all the SVNS values and returns the K clusters.

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Algorithm 6.2 K-means algorithm for clustering SVNS values

6.3.2 Classification algorithms for SVNS 6.3.2.1 k-Nearest neighbors algorithm k-Nearest neighbors (k-NN) is an algorithm that saves all available instances and classifies new instances on the basis of a similarity measure, in this case, the Euclidean distance. The k-NN algorithm is defined by feature similarity: how accurately the outliers look alike to the training set, and that similarity determines how the test data point gets classified. An SVNS (data point) is represented by a large number of upvotes from its neighboring sets, with the most common neighbor becoming the recipient of the incoming set. The k-NN algorithm for classifying SVNS values is given in Algorithm 6.3.

6.3.2.2 Support vector machine classifier The SVM is a machine learning algorithm that predominantly deals with supervised learning algorithms. It is used for classification and sometimes regression problems. The basis of the algorithm is defined by the decision planes that exist in space and those

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Algorithm 6.3 k-NN algorithm for classifying SVNS values

which establish decision boundaries. Members having different classes of membership are divided by decision planes. In the algorithm, each data point is plotted as a point in an n-dimensional space, and a decision boundary, also called a hyperplane, separates each of the similar points (similarity in terms of feature). In a two-dimensional world, hyperplanes look simply like lines. The SVM performs excellently with a restricted quantity of data. Here the SVM is used to separate SVNS values.

6.4

Sentiment analysis using neutrosophic sets

Sentiment analysis is an area where abundant research has taken place. Today, sentiment analysis is used by companies, corporations, and institutions to understand and gauge their consumers. Social media platforms such as Twitter are invaluable for opinion mining since people tend to review and express their emotions on their profiles as the outreach is high. Conventional sentiment analysis has usually dealt with categorizing emotions into negative and positive. However, the concept of neutrosophy gave rise to a new dimension in the research field of sentiment analysis. In the literature review phase, extensive research was done among the existing works regarding sentiment analysis and neutrosophy. As a result of the literature review, it was found that there has not been any significant research in the field of neutrosophic sentiment analysis. Sentiment analysis is a general natural language processing (NLP) task that can be performed on various platforms using in-built or trained libraries. Python presents a lot of flexibility and modularity when it comes to feeding data and using packages designed specifically for sentiment analysis. Further, the natural language

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toolkit (NLTK) is a top platform for creating Python programs to work with humanbased language data. It is useful for statistical analysis of NLP-based tasks that rely on extracting sentimental information from texts. NLTK has libraries for tokenization, stemming, parsing, classification, tagging, and semantic reasoning. VADER is a Python package that is based on the lexical approach to sentiment analysis. It has been developed with a human-centered approach, in the sense that the lexicon is based on words and expressions that are most frequently used on social networks and microblogging websites like Twitter. VADER generates three tuples for every tweet that gives the proportion of the negative, positive, and neutral (indeterminate) polarity in it. It assigns polarity scores based on the predefined library of gold standard lexicons used effectively in social media texts. Through experimental results, VADER has been observed to match the accuracy of human rates with a very high similar correlation coefficient between the two. It is also conceptualized to understand the tonality differences that uses of punctuation can create. VADER performs as well as or better across domains than the machine learning approaches do in the same domain for which they were trained. It is a dictionary and lexicon-based assumption investigation instrument that is explicitly receptive to social media text, and functions well on writings from different spaces. It is based upon the sentiment-related lexicons. The classification of positive and negative, or in some cases, the extent of positivity or negativity, is provided on a lexicon basis. Every word in the lexicon is analyzed and corroborated with whether the text is present in the built-in dictionary. VADER is used to map each tweet to an SVNS, with three memberships: positive, negative, and indeterminate.

6.5

Case study: Sentiment analysis of the #MeToo movement

6.5.1 Historical significance of the #MeToo movement Tarana Burke, a social activist and community organizer, sowed the initial seeds of the #MeToo movement in 2006 by using the phrase “Me Too” on the Myspace social network as part of a campaign to promote “empowerment through empathy” among women of color who have experienced sexual abuse, particularly within underprivileged communities. She stated that in her work, when a young woman confided in her about her experience with a certain kind of abuse, she felt like comforting her by letting her know that she was not alone and that she has also gone through something similar. She mentions that she felt like telling her “me too.” Burke conceptualized the idea of MeToo at that moment. Later, in October 2017, the idea for the hashtag came from American actress/singer Alyssa Milano, who prompted women to retweet her post with the hashtag #MeToo if they have ever experienced sexual violence of any kind. The screenshot of the first tweet on #MeToo is given in Fig. 6.1.

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Fig. 6.1 First Tweet on #MeToo.

Her idea behind this was to give people a sense of the magnitude of this problem. It was also to have several positive social and political ramifications and aimed at leading to a better understanding of how the world sees such allegations against perpetrators and also the lawmakers’ need to formulate policies and laws that protect as well as empower the survivors. Eventually, the hashtag spread to various industries, starting with Hollywood in regard to the Harvey Weinstein case and eventually trickling down to the corporate space as well. The movement eventually broke out through a single hashtag, with several variations. Our dataset is not from this time period, but right after this month from November 2017 to December 2017. The analysis of the movement forms an interesting basis for opinion mining since it produced highly diverse reactions online. There are some reactions that are strongly positive and others that are strongly negative, and some indeterminate. But there are various sentiments that are indeterminately positive and others that are indeterminately negative.

6.5.2 Description of dataset The collection of dataset is an important task since it forms the basis of the research. The language of all the tweets under consideration was established to be English and the number of tweets is approximately 400,000. The dataset has been collected via an open-source availability platform Data World [27]. The dataset is dated between November 29 and December 25, 2017. The tweets come from several demographic regions with the language restricted to English and a live feed streaming. The hashtag #MeToo and its variations, such as #metoo, Metoo, meToo, were used for the dataset from the open source. The size of the dataset stands at approximately 400,000 tweets, with line breaks differentiating every separate tweet as well as a retweet. The presence of stopwords, hashtags, links, and other jargon decreases the efficiency of the model and hence the tweets are cleaned using Python libraries. A cleaned dataset provides more lucid conclusion and also helps with increasing the accuracy of the polarity of the sentences.

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6.5.3 Conventional and fuzzy sentiment analysis Conventional sentiment analysis was carried out on the same dataset. The overall trend of the tweets was classified as positive, since the tool such as the TextBlob Python package considered neutral and positive tweets as positive sentiment. Similarly, TextBlob was used for fuzzy sentiment analysis on the same dataset. In TextBlob, the result is represented as a polarity score, which is a float number which represents whether the sentence is positive or negative and assigns a positive or negative emotion to the sentence. Analysis using TextBlob for the same dataset resulted in 75.3% of the tweets being classified as positive and the rest as negative. The indeterminate tweets are not represented separately in fuzzy sentiment analysis: They are not captured at all, and instead are clubbed with positive.

6.5.4 Analysis of tweets using neutrosophy The dataset was decreased to nearly 161,000 tweets after the removal of all retweets so as to increase the accuracy of clustering and classification results. The cleaned dataset is presented in the form of a newly created .txt file that goes for processing into the VADER tool for analysis, which then creates the tuple, hNx, Ix, Pxi. The collected polarities of the individual tweets are transferred from the stored .txt file to a .csv file for further processing. The first column represents the negative polarity, the second indeterminate polarity, while the third represents positive polarity in the sentence. The final column groups the sentence into a particular class—-negative, positive, or neutral (indeterminate), based on the compound score generated by the VADER tool. Out of the 161,000 tweets, nearly 64,225 tweets were classified as negative, 41,864 tweets as indeterminate, and 55,082 tweets as positive. This classification of tweets is not done in conventional or fuzzy sentiment analysis, while using the VADER sentiment tool, the overall sentiment turns out to be negative. The working of the VADER sentiment tool analysis for three tweets is shown on three sample tweets. Tweet 1: “we can not keep turning blind eye and pretend this is not real MeToo.” The SVNS generated from the polarity scores given by VADER: h0.089, 0.766, 0.144i. Overall polarity given by VADER: Positive. Tweet 2: “what will come out is eventually judges by what the story represents.” The SVNS generated from the polarity scores given by VADER: h0,1, 0i. Overall polarity given: Indeterminate. Tweet 3: “And yet fakemedia Hollywood tries to make it all political again by using some thought up movement getalife.” The SVNS generated from the polarity scores given by VADER h0.181, 0.819, 0i. Overall polarity given: Negative.

Tweet 2 is correctly classified as indeterminate by the tool. In case of Tweet 1, the compound polarity is tagged as positive under the VADER tool, but the indeterminate nature of the tweet is ignored. Similarly in Tweet 3, it is tagged as negative, but the tweet has an amount of indeterminate involved which is not represented in the classification as negative. The clustering process accounts for such anomalies and

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aims to classify tweets as accurately as possible. Also the classification of data into eight classes which are described later in this chapter helps in increasing the accuracy in such cases.

6.5.5 K-means clustering results The K-means clustering algorithm given in Algorithm 6.2 is used to cluster the tweets represented as SVNS values. Clustering is carried out to show the dominant presence of a different polarity other than just positive and negative polarities. Python programming language was used for clustering the SVNS values. The elbow curve method is used to find the optimal K value. The graph is plotted between the Within Cluster Sum of Squares (WCSS) and the number of clusters. The result is given in Fig. 6.2. From the figure it can be clearly seen that there is a sharp bend when K ¼ 3. So, for the working of the K-means algorithm, the value of K has been fixed as 3. It shows that there should be three clusters, which highlights the presence of a polarity other than the regular positive and negative. The results of K-means clustering carried out on 1000 tweets is shown in Fig. 6.3 and the result obtained in case of the entire dataset is shown in Fig. 6.4. The results show three clusters: positive, negative, and indeterminate. The largest cluster is categorized as indeterminate, and neither positive nor negative clusters are as huge as the indeterminate cluster. This shows that the actual polarity of the #MeToo movement was indeterminate in nature. The same K-means clustering was also carried out with Weka software with nearly 161,000 tweets. Missing values were globally replaced with mean or mode. Clusters were attained after 16 iterations; initial cluster points were randomly selected. The largest cluster was cluster 2 with nearly 48% of data points: it is the indeterminate cluster. Over 22% of the tweets which are actually indeterminate in nature The elbow method 12,000 10,000

WCSS

8000 6000 4000 2000

2

Fig. 6.2 Elbow curve for all tweets.

4 6 Number of clusters

8

10

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were wrongly classified as positive or negative by the VADER tool. The clusters are as follows: 47.4% belongs to the indeterminate cluster, 29.2% belongs to the negative cluster, and 23.2% belongs to the positive cluster. Our study accurately captures the indeterminacy present in the tweets, while neither conventional and fuzzy sentiment analysis considered the indeterminacy. Even when the VADER sentiment tool is used the overall sentiment is projected as negative (39%) in nature, whereas with our analysis we are clearly able to show that indeterminate is the overall sentiment.

6.5.6 Classification of data To increase the accuracy of the indeterminacy involved, the tweets were classified as eight classes: 1. 2. 3. 4. 5. 6. 7. 8.

strong indeterminate; positive indeterminate; negative indeterminate; weak indeterminate; strong positive; weak positive; strong negative; and weak negative.

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Fig. 6.4 Three-dimensional K-means for entire dataset.

The classification was done on the basis of the SVNS values given by hNx, Ix, Pxi. An SVNS of the tweet is classified as strong positive if the value of Px 2 [0.3, 1] and Nx < Px; it is classified as strong negative if the value of Nx 2 [0.3, 1] and Px < Nx. Since most of the tweets have a high indeterminate membership value, we have classified only tweets where Ix 2 [0.7, 1] as strong indeterminate. In case Ix 2 [0.7, 1] and Px 2 [0.2, 0.3), it is classified as positive indeterminate; similarly Ix 2 [0.7, 1] and Nx 2 [0.2, 0.3), it is classified as negative indeterminate. If Px 2 (0, 0.2), Ix 2 (0, 0.7), and Nx < Px, it is classified as weak positive. If Nx 2 (0, 0.2), Ix 2 (0, 0.7), and Px < Nx, it is classified as weak negative. The others are classified as weak indeterminate. All re-tweets were removed since it would affect the results. The overall dataset decreased to nearly 161,000 tweets. The k-NN classifier and SVM classifier were implemented in Weka Software.

6.5.7 k-NN classification results The dataset was then divided into 80% training data and the remaining 20% was taken as test data. The model was trained using the training data and later validated by both conventional validation and 10-fold cross-validation using the test data. The value of k was taken from 1 to 10. The results are tabulated in Table 6.1; in each case the test run

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Table 6.1 Performance outcome of the k-NN classifier for SVNS. k-Value 1 2 3 4 5 6 7 8 9 10

Validation method

Classification accuracy

Validation method

Classification accuracy

Conventional validation

99.8666

10-fold crossvalidation

99.9007

99.8449 99.8201 99.817 99.817 99.8139 99.8201 99.7859 99.7921 99.7797

99.8654 99.8734 99.8492 99.8498 99.8455 99.8455 99.8461 99.8461 99.8356

was performed three times and the best result was considered, since there was little variation in the accuracy achieved. An accuracy of 99.9007% was achieved when k’s value was taken as 1 and a 10-fold cross-validation method was used for validation. The result of the same is given in Fig. 6.5. As the k-value was increased, the accuracy achieved by the classifier decreased. It is noted that better accuracy was achieved in cases of 10-fold crossvalidation being used instead of the conventional validation method. Further detailed and comparative discussions about the results are carried out in the next section.

Fig. 6.5 k-NN results for k ¼ 1 and 10-fold cross-validation method.

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6.5.8 SVM classifier results The eight classes are further studied with the supervised learning model, the SVM. The SVM classifier was modeled with a linear kernel and radial basis function (RBF) kernel, and validated with conventional validation and 10-fold cross-validation methods. The same split of 80% as training data and 20% as test data was taken. The results of the SVM classifier are tabulated in Table 6.2. The best accuracy of 99.1407% was achieved with an RBF kernel and 10-fold cross-validation method. The results of the SVM classifier for the same are given in Fig. 6.6. Similar to the k-NN classifier, a better accuracy was achieved when the 10-fold cross-validation method was used as the validation method instead of the conventional validation method.

Table 6.2 Performance outcome of the SVM classifier. Kernel

Validation method

Classification accuracy

Linear RBF

Conventional validation

Linear RBF

10-fold cross-validation

99.0197 99.0073 99.0364 99.1407

Fig. 6.6 SVM classifier results for RBF kernel and 10-fold cross-validation method.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Comparison and discussion

6.6.1 Neutrosophic sentiment analysis The concept of indeterminacy has not been dealt with in normal/conventional and fuzzy sentiment analysis. SVNSs take care of the indeterminate aspect of a sentence or emotion present in a tweet. In a conventional sentiment analysis tool such as the TextBlob Python package, the result is represented as a polarity score, which is a float number which represents whether the sentence is positive or negative and assigns a positive or negative emotion to the sentence. TextBlob was used for fuzzy sentiment analysis, for the same dataset, 75.3% of the tweets were classified as positive and the rest as negative. The indeterminate membership is totally ignored in the case of conventional sentiment analysis or fuzzy sentiment analysis. While TextBlob is used for sentiment analysis, tweets with no polarity are considered as positive in nature, hence the overall polarity of the movement is projected as positive. Sentiment analysis can be carried out using neutrosophy, but the analysis will be done as a whole and not on individual tweets (Table 6.3). In our neutrosophic sentiment analysis, each tweet is represented as an SVNS, with three memberships, namely positive, indeterminate, and negative. If neutrosophic analysis is carried out on the dataset, using the different neutrosophic refined set, like SVNS, DVNS, TRINS, and MRNS, the overall indeterminacy will be represented individually. For example, the results from the VADER sentiment tool will be represented as a whole as an SVNS Swhole ¼ h0.398, 0.26, 0.342i. It shows that the movement is negative in nature. The concept behind neutrosophy is that every element needs to be mapped with three memberships, so in our analysis to capture the true essence of neutrosophy we have taken each tweet individually. Each tweet has been mapped to negative, indeterminate, and positive membership in our analysis.

Table 6.3 Comparison of different sentiment analysis approaches. Conventional sentiment analysis

Fuzzy sentiment analysis

Neutrosophic sentiment analysis

Gives the overall polarity. Major overall sentiment as positive

Gives the percentage of positive sentiment as 75.3%

Gives the proportion of positive sentiment of a particular tweet Gives the proportion of indeterminate sentiment of a particular tweet Gives the proportion of negative sentiment of a particular tweet

Gives the percentage of negative sentiment as 24.7%

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6.6.2 Clustering and classification algorithm of SVNS values In common sentiment analysis, results are not clustered or classified. In this chapter, the notable research is that each tweet is represented as SVNS to portray accurately the indeterminacy existing and both clustering using the K-means and classification using k-NN classifier and SVM classifier is carried out. Each tweet available in the dataset of 400,000 tweets of the #MeToo movement was evaluated using VADER and represented as SVNS. K-means clustering was carried out on the 400,000 SVNS values. From K-means clustering, it was evident that the genuine polarity of the movement was indeterminate in nature, unlike the prediction from normal sentiment analysis or fuzzy sentiment analysis. Majority of the tweets fell into the indeterminate cluster, undoubtedly showing that the opinions of people were indeterminate. To increase the accuracy of the indeterminacy involved, the tweets were classified as eight classes. The dataset was then divided into training and test data. For the k-NN classifier, k’s value was taken from 1 to 10 and both conventional and 10-fold crossvalidation was used. The best accuracy of 99.9% was achieved when k ¼ 1. Similarly, for the SVM classifier both linear and RBF kernels were used and both conventional and 10-fold cross-validation methods were used. The best accuracy achieved was 99.1% in the case of using an RBF Kernel and 10-fold cross-validation. It is clearly seen that the k-NN classifier achieved a better accuracy. It is acknowledged that the computational complexity of the k-NN classifier is higher when compared with the SVM classifier [28]. From the tabulated results of the k-NN classifier, it is seen that as k’s value increases the classifier’s accuracy decreases. When k ¼ 1, the nearest SVNS only is selected. As the value of k, that is, the number of neighbors, grows, it may lead to a decline in accuracy, causing the chances of counting an SVNS from a different class to become higher with the increase of nearest neighbors [29]. The incremental property of the k-NN algorithm is better than the SVM classifier. This property allows the k-NN classifier to perform better than the SVM classifier in classifying SVNS values.

6.7

Results and further study

The #MeToo movement was a heavily publicized online crusade that began on the social media platform, Twitter. The study carried out is novel, since neutrosophy has not been used in sentiment analysis. To explore sentiment analysis using neutrosophy, a dataset of 400,000 tweets of the #MeToo movement was used. Each tweet was represented as an SVNS (hNx, Ix, Pxi), using the VADER Python package. Clustering of the 400,000 SVNS representation of the tweets using K-means shows that sentiment/polarity of the tweets was frequently indeterminate, clearly indicating that people did not have a fixed positive opinion or a fixed negative opinion about the movement. Refined neutrosophic sets like DVNS, TRINS, or MRNS can also be used to study the same movement; it will thus be taken up for future study. Both conventional and fuzzy sentiment analysis results project the movement to be positive in nature. When the VADER sentiment tool was used, the analysis projected the movement to be negative. Our analysis of the on the other hand individual tweets using SVNS clearly shows that the movement was actually indeterminate in nature.

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To increase the accuracy of the classification, eight different classes were used, and the dataset was classified. Both the k-NN classifier and SVM classifier were implemented for the SVNS #MeToo dataset. Classification of the SVNS representation of the tweets shows that the sentiment/polarity of the tweets was mostly strong indeterminate, as opposed to positive, shown by conventional sentiment analysis. The best accuracy of 99.9001% was achieved with the k-NN classifier when k ¼ 1 and the 10-fold cross-validation method was used on the SVNS tweeter dataset. In the SVM classifier, an accuracy of 99.14% was achieved, when the RBF kernel was used along with the 10-fold cross-validation method. This study compared the performance of the k-NN classifier and SVM classifier, and the findings demonstrate that the k-NN classifier performs better than the SVM classifier in classifying the SVNS values.

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A new sudden death testing using repetitive sampling under a neutrosophic statistical interval system

7

Muhammad Aslama, Muhammad Azamb, Florentin Smarandachec a Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia, bDepartment of Statistics and Computer Science, University of Veterinary and Animal Sciences, Lahore, Pakistan, cDepartment of Mathematics, University of New Mexico, Gallup, NM, United States

7.1

Introduction

For sudden death testing, the group-sampling plan is usually adopted to minimize the cost, time, and resources (i.e., labor force, machinery, raw material, other efforts) for the life testing experiments of the products under investigation. As mentioned by Meeker and Hamada [1]; rapid advances in technology, development of highly sophisticated products, intense global competition, and increasing customer expectations have combined to put new pressures on manufacturers to produce high-quality products. Customers expect purchased products to be reliable and safe (expected to be at least after the warranty time). Systems, vehicles, machines, devices and so on should, with high probability, be able to perform their intended function under encountered operating conditions, for some specified period.

The group testing procedure to test the life period/failure time of a product is performed by installing the final product or the product under investigation in groups of testers. The testers are expensive and increase the cost of the whole procedure. In practice, a single item is placed on each tester; this is why testers must be equal in number to the required sample size. However, group testing procedures have cut down the cost of the procedure by planting more than one item to be investigated on a single tester and allowing both producer and consumer to implement group testing methods in sentencing. Therefore, group testing-based acceptance sampling plans are preferred to obtain economical inspection procedures. According to Jun et al. [2]:

Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00007-X © 2020 Elsevier Inc. All rights reserved.

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The specimens in each group are tested identically & simultaneously on different testers. The first group of specimens is run until the first failure occurs. At this point, the remaining survived specimens are suspended and removed from the testers. An equal set of new specimens numbering is next placed on testers for testing until the first failure. This process is repeated until one failure is generated from each of the group.

Due to market pressure, competition, and safety, at least until the product warranty time, testing of the products is preferred, using failure censoring and time-censoring experiments, rather than waiting for the failure of all items inserted on the testers. Nowadays, these failure time testing schemes are widely used in industry to generate profit. More discussion and applications of sudden death tests under classical statistics can be found in Refs. [2–4]. Jun et al. [2] assessed that the products passing through the sudden death testing procedure follow the Weibull distribution. Further reading about distribution behavior can be found in Refs. [5–11]. Usually, sudden death test plans are designed under the assumption that all observations and specified parameters are exact and crisp, in order to give precise, reliable, and accurate results. According to Shafiq et al. [12], “in real life practices, the lifetime measurements recorded are not precise quantities but more or less fuzzy in nature. Hence, in addition to standard statistical tools, fuzzy model based NSIS approaches are also essential.” To deal with such matters, many authors have proposed various sampling plans using the NSIS fuzzy logic: Kanagawa and Ohta [13], Tamaki et al. [14], Cheng et al. [15], Zarandi et al. [16], Alaeddini et al. [17], Sadeghpour Gildeh et al. [18], Jamkhaneh et al. [19], Turanog˘lu et al. [20], Divya [21], Jamkhaneh and Gildeh [22,23], Venkateh and Elango [24], Afshari and Gildeh [25], Afshari et al. [26], Elango et al. [27], and Afshari et al. [28]. A generalization of the fuzzy logic that deals with indeterminacy measures is known as the neutrosophic logic [29]. Smarandache [30] proposed the neutrosophic statistics (NS) based on the neutrosophic logic. The NS is used to analyze the data with imprecise or uncertain observations. Classical statistics is the special case of the NS when the recorded observations are crisp, exact, determined, and free from doubts. Researchers have employed fuzzy NS approaches in some other fields and obtained efficient results. Chen et al. [31,32] used NS in a joint rock study. Aslam [33,34] introduced the NS in the area of inspection schemes by designing variable sampling plan for the exponential distribution and process loss consideration. Aslam and Arif [35] proposed an NS sudden death testing using single sampling. Aslam and Raza [36] proposed an NS plan for multiple manufacturing lines. Usually, life test sampling plans are developed using single sampling schemes. However, sometimes, the variables deviate from the parametric values. Still, this deviation remains less than the rejection number; hence, Sherman [37] presented the repetitive sampling scheme with two control constraints as acceptance and rejection numbers, and found it to be more efficient than the single constraint-based sampling plans, outperforming previous attempts in terms of least average sample. Due to these advantages, several authors adopted the repetitive sampling scheme to deal with a variety of datasets, see for example, Refs. [38–43].

A new sudden death testing using repetitive sampling

139

This study is an extension of the grouped sudden death testing plan proposed by Aslam and Arif [35] with an adaptation of the repetitive sampling scheme under NSIMs. The contribution of the author’s development of the neutrosophic operating characteristic function under NSIMs, along with mathematical derivations and construction of the algorithm in R-software to simulate the optimization process for obtaining the minimized NASN among the various parametric combinations. The methodology of the proposed plan is extensively discussed in Section 7.2. This is followed by the discussion of the parametric results in Section 3. The limitations of the study with respect to the advantages are discussed are discussed in Section 4. The comparison with existing single sampling-based plans is explained in Section 5 and proved to outperform the previously existing plans [35]. The implementation of the concept of real-life dataset is presented in Section 6. The study concludes in the last section and some future research ideas are suggested.

7.2

Design of the proposed plan

Let a neutrosophic interval variable of interest TNiE{TL, TU} ¼ i ¼ 1, 2, 3, …, nN follow the neutrosophic Weibull distribution with the subsequent neutrosophic cumulative distribution function (Ncdf ): FN ðtN ; mN , λN Þ ¼ 1  exp ððtN =λN ÞmN Þ,tN  0,mN E½mL , mU , λN E½λL , λU 

(7.1)

It should be noted that mNE[mL, mU] is the neutrosophic fuzzy shape parameter and λNE[λL, λU] is the neutrosophic fuzzy scale parameter of the neutrosophic fuzzy Weibull distribution Ncdf given in Eq. (7.1). More details about the neutrosophic distributions can be seen in Smarandache [30] and Aslam and Arif [35]. As mentioned earlier, sudden death testing is performed in each group identically and at once. Suppose that L shows the lower specification limit, and the quality characteristic away from L is known as the defective item. The neutrosophic probability of the defective item is defined as follows: pN ¼ prN fTN < Lg ¼ FN ðLÞ; where TN EfTL , TU g

(7.2)

If pNE{pL, pU} is to be known in advance as prior information, the expression for λNL is given as follows (see [2]): wN ¼  ln ð1  pN Þ ¼ ðλN LÞmN

(7.3)

Based on the above information, the following sampling plan is being proposed using repetitive sampling under the NSIS. Step 1. Pick a random sample nNE{nL, nU} from a lot and allocate r items in gNiE{gL, gU} groups. Step 2. Every first failure from the ith group (i ¼ 1, 2, …gN) is noted and the neutrosophic stagN P m tistic is calculated as vN ¼ YiN ;gN E½gL , gU ,vN E½vL , vU . i¼1

140

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Step 3. The lot will be accepted if vN  kaNLm, where kaNE[kaL, kaU] is a neutrosophic fuzzy acceptance number and the lot will be rejected if vN  krNLm, where krNE[krL, krU], is a neutrosophic fuzzy rejection number. Otherwise, the procedure will move to Step 1.

To implement the proposed plan, the values of neutrosophic fuzzy parameters gNE[gL, gU], kaNE[kaL, kaU] and krNE[krL, krU] should be known. The proposed plan reduces to the existing plan given by Aslam and Arif [35] under the NSIS if both the acceptance, rejection numbers, and the respective groups coincide, such as kaN ¼ krN ¼ kN and gL ¼ gU ¼ gN; this means that both plans work in a similar way for a lot sentencing procedure. The proposed plan is reduced to that provided by Jun et al. [2] under classical statistics if no uncertainty is found in observations or the parameters are exact and crisp in nature in every attempt. By following Jun et al. [2], the neutrosophic statistics Ym iN is modeled by the neutrosophic exponential N distribution with the neutrosophic parameter λm N rN; λNE[λL, λU], mNE[mL, mU], g N P m rNE[rL, rU]. Based on this assumption, vN ¼ YiN ;vN E½vL , vU  has the neutrosophic i¼1

N fuzzy gamma distribution approach with parameter (gN, λm N rN); gNE[gL, gU]. The neutrosophic fuzzy operating characteristic (NOC) function for the single sampling plan under the NSIS is derived by Aslam and Arif [35] and given by

Pa ðpN Þ ¼ 1  G2gN ð2rN kaN wN Þ; kaN E½kaL , kaU 

(7.4)

where GδN denotes the Ncdf of the neutrosophic fuzzy gamma distribution with neutrosophic degree of freedom δN. The probability of rejection of a lot under the NSIS is given by Pr ðpN Þ ¼ G2gN ð2rN krN wN Þ; krN E½krL , krU :

(7.5)

By following Sherman [37], the NOC under the repetitive sampling scheme is derived for the proposed sampling plan and given as follows: LðpN Þ ¼

Pa ð p N Þ ; LðpN ÞE½LðpL Þ, LðpU Þ Pa ðpN Þ + Pr ðpN Þ

(7.6)

For the proposed plan, Eq. (7.6) can be written as follows:   1  G2gN ð2rN kaN wN Þ    ; LðpN ÞE½LðpL Þ, LðpU Þ: LðpN Þ ¼  1  G2gN ð2rN kaN wN Þ + G2gN ð2rN krN wN Þ (7.7) In the proposed plan, the neutrosophic average sample number (NASN) is derived by ASN N ðpN Þ ¼ 

nN   : 1  G2gN ð2rN kaN wN Þ + G2gN ð2rN krN wN Þ

(7.8)

A new sudden death testing using repetitive sampling

141

The parameters of the study are α, β, AQL and LQL, which are the producer’s risk, consumer’s risk, the acceptable quality level at pL and the limiting quality level at pU, respectively. The proposed sudden death testing plan parameters under NSIS repetitive sampling scheme will be determined through the following neutrosophic fuzzy non-linear optimization grid search method under the NSIS: After minimizing ASN N |LQLE½ASN L , ASN U 

(7.9a)

Subjected to   L CAQL ¼  

L CLQL





 1  G2gN ð2rN kaN wN Þ   1α 1  G2gN ð2rN kaN wN Þ + G2gN ð2rN krN wN Þ

 1  G2gN ð2rN kaN wN Þ     β: ¼ 1  G2gN ð2rN kaN wN Þ + G2gN ð2rN krN wN Þ

(7.9b)



(7.9c)

The proposed plan parameters krNE[krL, krU], kaNE[kaL, kaU], & gNE[gL, gU] are determined through the above-mentioned non-inear optimization procedure under NSIS. During the simulation process, it is also obvious that there exists more than one combination satisfying the provided constraints. From these combinations, the neutrosophic plan parameters’ combinations having the smaller NASN are chosen (see Tables 7.1 and 7.2). Moreover, Tables 7.1 and 7.2 are constructed for the various pre-specified neutrosophic parametric values when rNE[5, 5] and rNE[10, 10], respectively.

7.3

Results and discussion

After completion of the optimization grid search method, Tables 7.1 and 7.2 are constructed as mentioned and the following trend has been observed: 1. A decreasing trend is noted in the indeterminacy interval of NASN as ASNN, i.e., krNE[krL ¼ 65.52, krU ¼ 270.25], kaNE[kaL ¼ 148.36, kaU ¼ 273.57], ASNN  [ASNL ¼ 15.4, ASNU ¼ 20.05] and gNE[gL ¼ 2, gU ¼ 4] at LQL ¼ 0.006 and krNE[krL ¼ 8.7, krU ¼ 69.87], kaNE[kaL ¼ 38.97, kaU72.79], ASNN  [ASNL ¼ 9.36, ASNU ¼ 10.05] and gNE[gL ¼ 1, gU ¼ 2] at LQL ¼ 0.15 when rNE[5, 5] from Table 7.1 and AQL ¼ 0.001 is fixed. 2. An increasing trend is seen in the indeterminacy interval of NASN as ASNN when rNE[5, 5] changes to rNE[10, 10], i.e., krNE[krL ¼ 32.71, krU ¼ 129.4], kaNE[kaL ¼ 75.19, kaU ¼ 132.52], ASNN  [ASNL ¼ 30.99, ASNU ¼ 40.23] and gNE[gL ¼ 2, gU ¼ 4] at LQL ¼ 0.006 and krNE[krL ¼ 4.4, krU ¼ 32.85], kaNE[kaL ¼ 19.47, kaU ¼ 56.85], ASNN  [ASNL ¼ 18.58, ASNU ¼ 20.83] and gNE[gL ¼ 1, gU ¼ 2] at LQL ¼ 0.15 when rNE[10, 10] from Table 7.2 and AQL ¼ 0.001 is fixed.

7.4

Limitations and advantages

Firstly, the sampling schemes are adopted in lot sentencing procedures to reduce cost, time, and resources; these factors are directly proportional to the sample size ASNN

142

Table 7.1 Plan parameters of repetitive neutrosophic sampling plan using Weibull distribution when r ¼ 5. pU

gN

ASNN

kaN

krN

L(CAQL)

L(CLQL)

0.001

0.002 0.003 0.004 0.006 0.008 0.01 0.15 0.02 0.005 0.01 0.15 0.2 0.25 0.3 0.5 0.001 0.015 0.02 0.03 0.04 0.05 0.1

[7,9] [5,7] [3,5] [2,4] [2,3] [2,3] [1,2] [1,2] [9,15] [3,5] [2,4] [2,3] [2,3] [2,3] [1,2] [12,15] [5,6] [3,5] [2,4] [2,3] [2,3] [1,2]

[46.47,51.46] [33.93,37.28] [23.3,25.5] [15.4,20.05] [11.87,15.16] [10.39,15.06] [9.36,10.05] [7.43,10.03] [76.24,86.58] [23.12,26.91] [15.39,20.02] [11.77,15.1] [10.38,15.01] [10,15.01] [7.31,10.04] [77.54,86.89] [33.88,40.2] [22.82,25.19] [15.2,20.21] [11.76,15.45] [10.31,15.04] [7.18,10.04]

[895.93,1076.5] [572.45,714.97] [303.32,404.36] [148.36,273.57] [104.35,134.31] [79.02,170.66] [38.97,72.79] [27.62,64.32] [563.83,842.5] [118.84,179.28] [59.69,107.72] [40.69,65.84] [31.51,61.12] [26.36,58.93] [10.58,28.73] [344.45,411.48] [112.98,135.36] [59.34,79.46] [28.97,53.75] [20.19,35.33] [15.41,33.59] [5.11,16.61]

[636.73,903.8] [376.76,644.95] [151.55,387.79] [65.52,270.25] [69,130.52] [70.5,160.79] [8.7,69.87] [9.35,62.02] [355.39,721.85] [60.47,151.07] [26.24,107.15] [27.52,63.68] [28.19,60.75] [26.35,58.51] [3.75,26.02] [269.5,355.24] [74.76,91.41] [30.52,78.17] [13.09,51.33] [13.68,30.3] [14.05,32.37] [1.88,13.87]

[0.95,0.96] [0.95,0.95] [0.95,0.95] [0.95,0.95] [0.95,0.97] [0.95,0.95] [0.95,0.95] [0.95,0.96] [0.95,0.95] [0.95,0.96] [0.95,0.95] [0.95,0.95] [0.95,0.96] [0.96,0.96] [0.95,0.96] [0.95,0.96] [0.95,0.97] [0.95,0.95] [0.95,0.96] [0.95,0.96] [0.95,0.95] [0.95,0.95]

[0.09,0.09] [0.1,0.1] [0.09,0.1] [0.1,0.04] [0.09,0.1] [0.1,0.01] [0.1,0.03] [0.09,0.01] [0.1,0.08] [0.1,0.06] [0.09,0.04] [0.1,0.04] [0.1,0.02] [0.09,0.01] [0.1,0.01] [0.1,0.09] [0.1,0.08] [0.09,0.1] [0.1,0.04] [0.1,0.03] [0.1,0.01] [0.1,0]

0.0025

0.005

Optimization Theory Based on Neutrosophic and Plithogenic Sets

pL

0.03

0.05

0.02 0.03 0.04 0.05 0.1 0.15 0.2 0.06 0.09 0.12 0.15 0.3 0.1 0.15 0.2 0.25 0.5

[11,12] [5,8] [3,5] [3,4] [2,3] [1,2] [1,2] [10,14] [5,8] [3,6] [3,5] [2,3] [12,17] [5,9] [4,5] [3,4] [2,3]

[75.65,91.39] [32.99,44.97] [22.68,27.37] [17.78,21.97] [10.22,15.06] [8.74,10.45] [7.08,10.01] [73.67,82.33] [32.42,45.01] [21.94,30.95] [17.11,25.23] [10.04,15.13] [73.47,89.74] [32.03,46.69] [21.79,26.58] [16.57,21.69] [10.01,15.05]

[160.86,201.91] [55.78,115.74] [29.14,47.51] [22.03,33.64] [7.5,15.1] [3.54,7.98] [2.56,7.13] [48.74,64.95] [18.15,38.46] [9.43,18.75] [6.99,10.75] [2.29,5.42] [32.55,45.05] [10.71,21.5] [6.29,8.34] [3.82,5.69] [1.17,3.39]

[119.26,127.94] [37.9,75.45] [15.15,37.44] [15.66,25.05] [7.02,14.52] [0.89,5.72] [0.94,6.83] [34.17,54.35] [12.48,24.38] [5.03,16.38] [5.25,10.48] [2.25,4.59] [26.51,41.85] [7.38,17.97] [5.26,7.09] [3.07,4.48] [1.16,2.76]

[0.95,0.96] [0.95,0.95] [0.95,0.96] [0.95,0.96] [0.95,0.96] [0.95,0.96] [0.95,0.95] [0.95,0.95] [0.95,0.96] [0.95,0.96] [0.95,0.98] [0.95,0.97] [0.95,0.95] [0.95,0.95] [0.95,0.96] [0.95,0.97] [0.96,0.96]

[0.1,0.03] [0.1,0] [0.1,0.04] [0.09,0.03] [0.1,0.01] [0.1,0.01] [0.08,0] [0.1,0.07] [0.09,0] [0.09,0.02] [0.09,0.07] [0.09,0] [0.1,0.07] [0.08,0.01] [0.09,0.05] [0.1,0.04] [0.09,0]

A new sudden death testing using repetitive sampling

0.01

143

144

Table 7.2 Plan parameters of repetitive neutrosophic sampling plan using Weibull distribution when r ¼ 10. pU

gN

ASNN

kaN

krN

L(CAQL)

L(CLQL)

0.001

0.002 0.003 0.004 0.006 0.008 0.01 0.15 0.02 0.005 0.01 0.15 0.2 0.25 0.3 0.5 0.001 0.015 0.02 0.03 0.04 0.05 0.1

[9,11] [5,7] [3,5] [2,4] [2,3] [2,3] [1,2] [1,2] [11,10] [3,5] [2,4] [2,3] [2,3] [2,3] [1,2] [11,14] [5,8] [4,5] [2,4] [2,3] [2,3] [1,2]

[154.83,159.77] [68.12,84.56] [46.6,54.88] [30.99,40.23] [23.59,31.25] [20.95,30.02] [18.58,20.83] [14.88,20] [152.39,174.86] [45.91,53.38] [31.39,40.29] [23.68,31.21] [20.73,30.32] [20,30.02] [14.65,20.09] [153.54,181.43] [67.49,91.82] [45.84,56.02] [30.84,40.42] [23.42,30.08] [20.75,30.07] [14.62,20.15]

[710.48,840.67] [284.08,418.99] [148.8,250.15] [75.19,132.52] [51.18,73.81] [40.24,70.77] [19.47,56.85] [13.39,24.52] [323.3,338.1] [59.45,89.72] [30.63,47.48] [20.65,33.23] [15.72,37.39] [13.96,32.13] [5.29,14.71] [162.03,195.32] [57.43,110.51] [34.22,43.77] [14.93,24.87] [10.06,13.09] [7.71,15.4] [2.59,3.72]

[442.54,591.19] [187.13,300.17] [75.23,192.29] [32.71,129.4] [34.52,65.33] [35.03,70.49] [4.4,32.85] [4.6,24.51] [239.09,199.97] [30.45,76.69] [12.95,46.5] [13.74,28.33] [14.11,31.2] [13.96,31.34] [1.87,13.23] [118.97,155.58] [37.77,73.61] [26.71,34.92] [6.51,24.02] [6.87,12.99] [6.92,15.02] [0.91,3.64]

[0.95,0.95] [0.95,0.96] [0.95,0.95] [0.95,0.96] [0.95,0.97] [0.95,0.97] [0.95,0.95] [0.95,0.97] [0.95,0.95] [0.95,0.95] [0.95,0.97] [0.95,0.96] [0.95,0.95] [0.95,0.95] [0.95,0.96] [0.95,0.97] [0.95,0.96] [0.95,0.97] [0.95,0.97] [0.95,0.97] [0.95,0.96] [0.95,0.99]

[0.1,0.08] [0.1,0.04] [0.1,0.03] [0.09,0.04] [0.1,0.07] [0.09,0.03] [0.1,0] [0.1,0.04] [0.1,0.05] [0.1,0.06] [0.09,0.07] [0.09,0.04] [0.1,0] [0.07,0] [0.1,0] [0.1,0.1] [0.09,0.01] [0.1,0.07] [0.09,0.06] [0.1,0.1] [0.1,0.01] [0.1,0.1]

0.0025

0.005

Optimization Theory Based on Neutrosophic and Plithogenic Sets

pL

0.03

0.05

0.02 0.03 0.04 0.05 0.1 0.15 0.2 0.06 0.09 0.12 0.15 0.3 0.1 0.15 0.2 0.25 0.5

[10,14] [4,8] [3,6] [3,4] [2,3] [1,2] [1,2] [10,19] [4,7] [3,6] [3,5] [2,3] [10,17] [4,9] [3,6] [3,4] [2,3]

[153.78,169.75] [68.85,94.61] [45.7,61.4] [35.14,44.94] [20.61,30.04] [18.05,20.11] [13.79,20.08] [147.89,191.27] [65.13,82.54] [44.07,63.44] [33.8,51.95] [20.3,30.04] [146.65,202.2] [63.08,91.63] [41.79,62.93] [33.53,46.66] [20.09,30.13]

[76.11,97.72] [25.28,40.85] [14.81,26.35] [10.94,22.72] [3.77,8.14] [1.85,2.87] [1.19,3.27] [24.54,40.77] [7.98,14.37] [4.73,12.4] [3.39,8.3] [1.11,2.7] [14.4,29.32] [4.61,9.59] [2.64,6.86] [1.99,3.28] [0.56,4.09]

[51.51,80.82] [12.52,32.61] [7.57,24.55] [7.93,12.64] [3.44,7.98] [0.43,2.8] [0.48,2.99] [17.1,40.42] [4.18,10] [2.51,8.08] [2.63,5.85] [1.06,2.59] [10.08,19.95] [2.49,9] [1.5,4.73] [1.54,2.06] [0.56,1.36]

[0.95,0.96] [0.95,0.98] [0.95,0.96] [0.95,0.95] [0.95,0.95] [0.95,0.97] [0.95,0.96] [0.95,0.95] [0.95,0.96] [0.95,0.95] [0.95,0.96] [0.96,0.95] [0.95,0.95] [0.95,0.95] [0.95,0.96] [0.95,0.98] [0.97,0.95]

[0.09,0.09] [0.09,0.08] [0.09,0.04] [0.1,0] [0.1,0.01] [0.09,0.05] [0.1,0.01] [0.09,0.09] [0.09,0.02] [0.09,0] [0.1,0] [0.1,0] [0.09,0] [0.09,0.03] [0.09,0] [0.08,0.02] [0.1,0]

A new sudden death testing using repetitive sampling

0.01

145

146

Optimization Theory Based on Neutrosophic and Plithogenic Sets

required for the inspection of a lot. Therefore, a plan having smaller values of sample size ASNN or NASN is preferred. Secondly, the concept dealing with uncertainty in the provided parameters as indeterminacy intervals rather than determined, crisp, and exact values is adequate and effective—see Refs. [31, 32]. Hence, a plan with a smaller indeterminacy interval in sample size or NASN is considered an efficient sampling plan. The parameters can be decided by consensus between the consumer and the producer.

7.5

Comparison

The efficiency of the proposed plan is compared with that of the plan proposed by Aslam and Arif [35] at the same level of AQL and LQL discussed by them, to enable ease of understanding and comparison. The neutrosophic plan parameters of both sampling plans are shown in Table 7.3 when rNE[5, 5]. From this table, it is clear that the proposed sampling plan has a smaller indeterminacy interval in gNE[gL, gU] compared to Aslam and Arif’s plan. Similarly, the proposed control chart has smaller values of NASN ASNN compared to Aslam and Arif’s plan for every combination of AQL and Table 7.3 Comparison of performance w.r.t. group size with repetitive NSIS and single sampling plan using Weibull distribution when r ¼ 5. Proposed

Single

pL

pU

gN

gN

0.001

0.002 0.003 0.004 0.006 0.008 0.01 0.15 0.02 0.005 0.01 0.15 0.2 0.25 0.3 0.5 0.001 0.015 0.02 0.03 0.04 0.05 0.1

[7,9] [5,7] [3,5] [2,4] [2,3] [2,3] [1,2] [1,2] [9,15] [3,5] [2,4] [2,3] [2,3] [2,3] [1,2] [12,15] [5,6] [3,5] [2,4] [2,3] [2,3] [1,2]

[19,21] [8,10] [6,8] [4,6] [3,5] [3,5] [2,4] [2,4] [19,21] [6,8] [4,6] [3,5] [3,5] [2,4] [2,4] [19,21] [8,10] [6,8] [4,6] [3,5] [3,5] [2,4]

0.0025

0.005

A new sudden death testing using repetitive sampling

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Table 7.3 Continued Proposed

Single

pL

pU

gN

gN

0.01

0.02 0.03 0.04 0.05 0.1 0.15 0.2 0.06 0.09 0.12 0.15 0.3 0.1 0.15 0.2 0.25 0.5

[11,12] [5,8] [3,5] [3,4] [2,3] [1,2] [1,2] [10,14] [5,8] [3,6] [3,5] [2,3] [12,17] [5,9] [4,5] [3,4] [2,3]

[19,21] [8,10] [5,7] [4,6] [3,5] [2,4] [2,4] [19,21] [8,10] [5,7] [4,6] [2,5] [18,20] [8,10] [5,7] [4,6] [2,6]

0.03

0.05

LQL. For example, when AQL ¼ 0.001 and LQL ¼ 0.02, the number of groups required for the inspection of the product are smaller for the proposed plan than Aslam and Arif [35]. Similarly, the ASNN for the proposed sampling plan from Table 7.1 is ASNN  [ASNL ¼ 46.47, ASNU ¼ 51.46], and for Aslam and Arif’s plan, the ASNN is ASNN  [ASNL ¼ 5  19 ¼ 98, ASNU ¼ 21  5 ¼ 105]. Therefore, it can be seen that the proposed sampling plan has smaller values of gNE[gL, gU] and ASNN than the existing sampling plan. The Friedman test has been used to test the significance of the difference between the values of gN of the current plan with those of Aslam and Arif [35]; the result revealed a significant difference with P  value ¼ 0.004678 for AQL as pL ¼ 0.001 and with P  value ¼ 0.01431 for AQL as pL ¼ 0.0025 in favor of the proposed method results with a repetitive sampling scheme. The test is applied with α ¼ 0.05 and the gLof both plans are observed. Hence, it is concluded that use of the proposed sampling plan for the inspection of a lot will minimize the inspection cost.

7.6

Implementation in real-life datasets

In this section, a practical application taken from Aslam and Arif [35] is discussed. The example is treated with a repetitive sampling scheme under two inspection criteria, i.e., rejection and acceptance numbers for a lot of inspection. Moreover, the respective dataset follows the neutrosophic Weibull distribution with mNE{2, 2}. The failure time measurement may not be crisp or exact. In this scenario, most

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

observations will remain determined at a specific inspection. Hence, the proposed plan might be appropriate to solve this problem. The values will be converted in an interval dataset, provided that data naturally follows the Weibull distribution for the failure time of the inspection units of ball bearings. For the inspection of this product, let AQL ¼ 0.001, LQL ¼ 0.04, α ¼ 0.05, rNE[5, 5], L ¼ 200, and β ¼ 0.10. From Table 7.1, the neutrosophic plan parameters are stated as: gNE[3, 5], krNE[151.55,387.79], and kaNE[303.32,404.36]. It is also feasible to get the dataset with gN ¼ 5. The cost and time remain manageable to run this study. However, the industrial engineers decided to select gN ¼ 5 rather than gN ¼ 3, to make decisions more reliable. The proposed plan will be implemented using the following steps: Step 1. Pick a random sample nNE{25, 25} from a lot and allocate 5 items gNiE{5, 5} groups. Step 2. Note every first failure from Y1 ¼ [220,230], Y2 ¼ [300,320], Y3 ¼ 285, Y4 ¼ [155,165], and Y5 ¼ [365,375] ith group (i ¼ 1, 2, …5), and calculate neutrosophic statistic gN P m vN ¼ YiN ¼ ½2202, 2302 + ½3002, 3202 + ½2852, 2852 + ½1552, 1652 + ½3652, 3752 ¼ i¼1

vN E ½376875, 404375, kaNLmE [12132800, 16174400], and krNLmE [6062000, 15515600]. Step 3. As the values of vNE [376875, 404375] are smaller than kaNLmE [12132800, 16174400], we will therefore accept a lot of ball bearing product. On the basis of the abovementioned results, it can be concluded that in spite of the significance level α at 5% and β at 10%, the lot is good enough to be accepted after passing the pre-specified quality measures.

7.7

Conclusion and directions for further research

In this study, the basic problem of the data collection process has been identified as not getting crisp, exact, and determined values—not only for the samples, but also for the parameters, which are the backbone of any statistical or mathematical procedure to be implanted for obtaining results. The proposed design is based on a sudden death testing procedure using a repetitive sampling scheme under the neutrosophic statistical interval system (NSIS), and it is presented with extensive tables, constructed through a nonlinear optimization grid search. The proposed plan is an extension of Aslam and Arif’s [35] sampling plan and a generalization of Jun et al. [2] under classical statistics in terms of the crisp dataset. The comparative study shows the efficiency of the proposed plan in terms of ASNN, as it provides a minimum average sample size to implement the concept in real-life datasets; it is economical for both producer and consumer, and is also easy to apply and simple to understand. This concept can be extended with other distributions along with other sampling schemes. The usage of the big dataset and an increased number of testers to obtain reliable and economical results in similar patterns might be of further research interest. The proposed plan using a cost model can be considered in future research.

Acknowledgments The authors are deeply grateful to the editor and reviewers for their valuable suggestions to improve the quality and presentation of this chapter.

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D. Nagarajana, M. Lathamaheswaria, Said Broumib, J. Kavikumarc, Florentin Smarandached a Department of Mathematics, Hindustan Institute of Technology & Science, Chennai, India, b Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, Sidi Othman, Casablanca, Morocco, cDepartment of Mathematics and Statistics, Faculty of Applied Science and Technology, Universiti Tun Hussein Onn, Batu Pahat, Malaysia, d Department of Mathematics, University of New Mexico, Gallup, NM, United States

8.1

Introduction

As the world is a competitive one, prediction of the future trend is an important task for the survival of any organization. There are many statistical and technical methods available for doing this task and this can be done in an optimized way using Markov chain with time series where random changes are allowed. Markov chains are an essential technique in random process underlying the Markov property [1]. Longrun behavior is the behavior of the system where each and every input can be different and the free entry is unconditional. In addition, the cost of this behavior is the minimum of short run behavior. Markov chain (MC) is a meticulous mathematical model to study the behavior of the process, derived non-deterministically, where the future events depend only on the present event. In addition, MC is a random process without memory where the probable values from a random variable are called states. A sequence of possible events can be obtained using MC. An MC on finite state space is called a discrete time MC. If a system changes over time, then the changes should be tracked and this can be done using MC according to the given probabilities. MC may be applied to any system where a certain number of states is available and inclined probabilities that the system transforms from any state to another state. MC helps to construct the data using a state diagram. If an MC has 2 states, then the transition matrix would be a 2 by 2 matrix whose elements are the evidence of the probability of moving from every state to another state. It can be extended to any finite number of states. In a transition probability matrix (TPM), the rows represent the current state and the columns represent the next state. The position of the states can be tracked using state vector [2]. MC is a discrete stochastic process with Markov property, where the distribution of the probability for the system at the next and all future steps depends only on the current step not on any previous steps. It is commonly used to predict the future trend. In addition, it is a convenient methodology when the random changes are difficult to Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00008-1 © 2020 Elsevier Inc. All rights reserved.

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express. Since it has descriptive power, it will point out the direction and consequence of changes in the future. One might select biased possibilities over biased probabilities for uncertain information and, hence, MC can be applied using fuzzy environment where impreciseness can be handled [3]. Markov chains have been applied in many fields; namely, statistical models, control system in motor vehicles, currency exchange rate, and queuing systems. However, the world has indeterminacy (true and false at the same time) in nature, which is different from randomness. Due to construction type, items involved in the space and materials indeterminacy may occur, and neutrosophic probability (NP) was introduced as an extension of conventional and fuzzy probabilities. Various rules of conventional probability can be adjusted by NP. In the experiment of rolling a die, the die and the surface on which it rolls are considered ideal in the conventional probability and hence only randomness occurs, not indeterminacy incurred by the materials. In this experiment, due to damaged die, cracked space indeterminacy may occur. If the die has incompatible density, then the outcome would be influenced by that factor. Hence NP analyzes both randomness and indeterminacy, and deals with their variables and processes as well. A conventional stochastic (random) variable is accountably changed due to only randomness, but neutrosophic variable (NV) is subject to change due to randomness and indeterminacy. The values of the NV represent the possible outcomes and indeterminacies that are impartial or partial. NV can be classified as discrete (takes values from particular exact values and limited number of indeterminacies), continuous (values taken from an interval or a set of intervals), and mixt (takes values as for the discrete and continuous cases). NP estimates the occurrence and non-occurrence of the event together with indeterminacy estimation. A neutrosophic random variable (NRV) is a variable that has vague and ambiguous outcome (indeterminate). The neutrosophic random process (NRP) performs the change over time of some neutrosophic random values, a collection of NRVS. The conventional probability deals with dice, coins, decks of cards, and random walks, whereas NP deals with improper, incomplete objects, variables, and processes. If the chance of getting indeterminacy of a random process is zero then the conventional probability and neutrosophic probability will coincide. The probability for a system that moves from one state to another in a required number of steps can be calculated by matrix multiplication [4]. If there is finite or countably infinite number of states then the Markov process is called MC. Developing a model to study the movements of the events is possible using MC. The traditional MC requires only actual and crisp data for any analysis of the system. It is thus unable to cope with an imprecise state, called a fuzzy state, and its transition probability (TP). To handle uncertainty, we need to change TP in the MC, and fuzzy set represents the fuzzy transition probability (FTP). Generally, in the applications of MC, data will be collected by assessment or experience and this makes the data incomplete. This kind of problem needs to be handled by fuzzy MC (FMC) where FTP is the base of the FMC, whereas traditional MC is unable to deal with and analyze impreciseness in the decision-making problem. In FMC, TP will be a fuzzy number [5]. A fuzzy Markov system is introduced to portray determined and random functioning for the complex dynamical systems. Fuzzy logic has been applied in most of the

Long-run behavior of interval neutrosophic Markov chain

153

areas for analyzing and controlling purposes. Random processes can be expressed in terms of the approach of the Markov model. Since conventional fuzzy systems are unable to cope with randomness, fuzzy Markov systems have been introduced for stable non-linear approximation of a probability density function with multidimensional Fuzzy Markov system [6]. The concept of neutrosophic set contributes a new base for handling issues related to indeterminate data, which may be numbers or neutrosophic numbers. The clarity of the study can be obtained by neutrosophic probability distributions. Due to the controversy of the crisp and fuzzy numbers, interval numbers can be used to deal with randomness, fuzziness, and indeterminacy of the data using the optimized method. To date, neutrosophic Markov chain has been studied and applied in the decisionmaking problem using single valued neutrosophic numbers, but not by using interval neutrosophic numbers. Hence, in this chapter, neutrosophic probabilities have been taken as interval neutrosophic numbers and the long-run behavior of the Markov chain analyzed under interval neutrosophic environment. The rest of the chapter is organized as follows. In Section 8.2, a review of literature related to the present study is given. In Section 8.3, some of the basic concepts are given. In Section 8.4, interval neutrosophic Markov chain is introduced, long-run behavior of the Markov chain for the economical year is studied under interval neutrosophic environment, comparative analysis is performed to study the behavior of the MC under crisp, fuzzy, intuitionistic, and neutrosophic environments, and comparative analysis is done with the existing method. In Section 8.5, conclusion of the present work is given with the future direction.

8.2

Review of literature

Hunter [7] applied time to stationarity in the Markov chain (MC), which is perturbed. Garcia et al. [8] made a simulation study on MC under fuzzy environment. Ponomarev et al. [9] designed residence tine distribution based on the concept of MC in a single screw extruder. Fort et al. [2] extended fluid limit methodology to a general class of skip free MCS using Monto Carlo technique. Hunter [10] applied coupling and mixing time in an MC. Jung et al. [11] proposed an algorithm to find the fastest possible rate of convergence using a non-reversible MC for a given network. Garcia [12] applied the concept of interval type-2 fuzzy set in MC. Kalenatic et al. [13] proposed a methodology to convert fuzzy MC into crisp MC using equivalence matrix from scalar cardinality. Kazemi et al. [1] applied a gray forecasting model for energy demand of industry in Iran. Bai and Wang [14] analyzed economic time series using conditional MC. Mallak et al. [15] studied ergodicity of the fuzzy MC using max-min composition. Gerencs [16] proposed a methodology to find the lower bound on the mixing time of MCs. Xin et al. [3] introduced spatiotemporal method for changes in the usage of land. Masoumi and Vajargah [17] examined the behavior of fuzzy MC. Kou et al. [18] proposed a novel methodology to study incidence of disease of insured persons. Sujatha [19] introduced intuitionistic Markov chain and its path transition and future behavior.

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Smarandache [4] introduced the concepts of measurement, integral, and probability under neutrosophic environment. Chan et al. [20] explained the concept of MC with examples. Adeleke et al. [5] applied the concept of MC in the assessment of admission of students and academic performance score. Vajargah and Gharehdaghi [21] applied Faure and Kronecker sequences to generate the membership values of fuzzy MC and found the number of ergodic MCs. Li and Xiu [22] introduced fuzzy MC based on fuzzy transition probability. Kanyinda et al. [23] introduced a method to calculate fuzzy Eigen values and Eigen vectors of a FMC. Lei et al. [24] proposed a prediction algorithm for multi aggregation and occasional demand forecasting with fuzzy MC. Smarandache et al. [25] applied PCR5 and probability under neutrosophic environment to identify the target. Piriyakumar and Sreevinotha studied the ergodic behavior of the FMCs. Garcia et al. [26] proposed quasi MCs under type-2 fuzzy environment. Zhu et al. [27] provided the sufficient conditions for the ergodicity of FMCs. Liu and Liang [28] applied MCMC method in geochemical inverse problems. Awiszus and Rosenhahn [29] applied the concept of MC in neural networks. Alhabib et al. [30] proposed some of the concepts of neutrosophic probability distributions. Petrov [31] described aggregated MC and applied it in rule-based designing. Broumi et al. [32] studied the shortest path problem under crisp, fuzzy, intuitionistic, and neutrosophic environments as an overview. Broumi et al. [33] solved a shortest path problem using interval triangular and trapezoidal neutrosophic environments. Broumi et al. [34] extended Bellman algorithm under interval neutrosophic environment. Nagarajan et al. [35] introduced Blockchain single and interval valued neutrosophic graph and applied it in Blockchain technology. Nagarajan et al. [36] proposed a Dombi interval valued neutrosophic graph and its operational laws. Nagarajan et al. [37] studied traffic control management under interval type-2 fuzzy and interval neutrosophic environments. Abdel-Basset et al. [38–42] applied the concept of neutrosophic logic to several fields. From this literature study, Markov chain concept has not been studied under interval neutrosophic environment, which is the motivation of the present studied.

8.3

Basic concepts

In this section, some of the basic concepts required for the present study have been given.

8.3.1 Markov chain [22] A Markov chain is a sequence of random variables X ¼ {X0, X1, X3…} with the following properties. For n 2 {0, 1, 2, …}, Xn is defined on the sample space ℧ and takes values from the finite set S. Thus Xn : ℧ ! S. Also for n 2 {0, 1, 2, …} and {i, j, in1, in2, …, i0}  S PfXn + 1 ¼ j=Xn ¼ i, Xn1 ¼ i  1, Xn2 ¼ i  2, …, Xo ¼ i0 g ¼ PfXn + 1 ¼ j=Xn ¼ ig

(8.1)

Long-run behavior of interval neutrosophic Markov chain

155

And the transition probabilities P{Xn+1 ¼ j/Xn ¼ i} ¼ pij are independent of n.

8.3.2 Fuzzy set (FS) A fuzzy set in a universe of discourse X is a set: Ae ¼

n

 o μeðxÞ, x =x 2 X

(8.2)

A

where μeð Þ : X ! ½0, 1 is called the membership function of the FS Ae and μeðxÞ A A e denotes the membership degrees of the element x in A. l

8.3.3 Fuzzy Markov chain [22] A fuzzy stochastic process {X(n) : n 2 ℕ} is said to be a fuzzy Markov chain if it satisfies the Markov property: ϑðXn + 1 ¼ j=Xn1 ¼ i, Xn ¼ k, …, X0 ¼ mÞ ¼ ϑðXn + 1 ¼ j=Xn1 ¼ iÞ

(8.3)

where i, j, k establish the state space S of the process. fij ¼ ϑðXn + 1 ¼ j=Xn ¼ iÞ are called the fuzzy probabilities of moving from Here P   fij ¼ μ state i to state j in one step. Hence P eij , where μPeij is the membership of P   fij is called the fuzzy transition the transition from state i to state j. The matrix P ¼ P probability matrix.

8.3.4 Intuitionistic fuzzy set (IFS) Let X be a universal set. An intuitionistic fuzzy set B assigns to every element x 2 X, a membership degree μB(x) 2 [0, 1] and a non-membership degree νB(x) 2 [0, 1], such that 0  μB(x) + νB(x)  1. For all the values of x 2 X, the number π B(x) ¼ 1  μB(x)  νB(x) is the degree of hesitation of the element. An IFS on X is an object of the form: B ¼ fðx, μB ðxÞ, νB ðxÞÞ=x 2 Xg

(8.4)

8.3.5 Intuitionistic fuzzy Markov chain [19] An intuitionistic stochastic process {X(n) : n 2 ℕ} is said to be an intuitionistic Markov chain if it satisfies the Markov property: βðXn + 1 ¼ j=Xn1 ¼ i, Xn ¼ k, …, X0 ¼ mÞ ¼ βðXn + 1 ¼ j=Xn1 ¼ iÞ

(8.5)

where i, j, k establish the state space S of the process. fij ¼ βðXn + 1 ¼ j=Xn ¼ iÞ are called the intuitionistic probabilities of moving Here P   f from state i to state j in one step. Hence Pij ¼ μe , νe , where μe is the membership Pij

Pij

Pij

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of the transition from state i to state j and νe is the non-membership of the transition  Pij fij is called the intuitionistic transition probfrom state i to state j. The matrix P ¼ P ability matrix.

8.3.6 Neutrosophic set [43] Consider the space X consists of universal elements characterized by x. The NS C is a phenomenon that has the structure N ¼ {(TN(x), IN(x), FN(x))/x 2 X} where the three grades of memberships are defined from X to ] 0,1+[, degrees of the element x 2 X with respect to the set X, satisfying the criterion: 

0  TN ðxÞ + IN ðxÞ + FN ðxÞ  3 +

The functions TN(x), IN(x) and FN(x) are the truth, indeterminate, and falsity grades lying in real standard/non-standard subsets of ] 0, 1+[.

8.3.7 Neutrosophic Markov chain [4] A neutrosophic stochastic process {X(n) : n 2 ℕ} is said to be a neutrosophic Markov chain if it satisfies the Markov property: βðXn + 1 ¼ j=Xn1 ¼ i, Xn ¼ k, …, X0 ¼ mÞ ¼ βðXn + 1 ¼ j=Xn1 ¼ iÞ

(8.6)

where i, j, k establish the state space S of the process. fij ¼ βðXn + 1 ¼ j=Xn ¼ iÞ are called the neutrosophic probabilities of moving Here P   f from state i to state j in one step. Hence Pij ¼ Te , Ie , Fe , where Te is the truth Pij

Pij

Pij

Pij

membership of the transition from state i to state j and Ie is the indeterminate memPij

bership of the transition from state i to state j and Fe is the falsity membership of the  Pij fij is called the neutrosophic trantransition from state i to state j. The matrix P ¼ P sition probability matrix.

8.3.8 Single valued neutrosophic set (SVNS) [44] Because the space X of objects contains global elements x, an SVNS is represented by degrees of membership grades truth, indeterminacy, and falsity. For all x in X, TN(x), IN(x), FN(x) 2 [0, 1], and the SVNS is defined as: N ¼ fhx : TN ðxÞ, IN ðxÞ, FN ðxÞi=x 2 Xg

(8.7)

Long-run behavior of interval neutrosophic Markov chain

157

8.3.9 Interval neutrosophic set [45] Let X be the space of objects with generic elements in X is denoted by x. An interval valued neutrosophic set (IVNS) N in X is characterized by truth-membership function, TN(x), indeterminacy-membership function IN(x), and falsity membership function FN(x). For each point x in X, TN(x), IN(x), FN(x) 2 [0, 1], and an IVNS N is defined by: N¼



   

 TNL ðxÞ, TNU ðxÞ , INL ðxÞ, INU ðxÞ , FLN ðxÞ, FU N ðxÞ j x 2 X

(8.8)

L U L U where TN(x) ¼ [TLN(x), TU N (x)], IN(x) ¼ [IN(x), IN (x)], and FN(x) ¼ [FN(x), FN (x)].

8.3.10 Operations on neutrosophic probabilities [4] Consider two single valued neutrosophic probabilities NP1 ¼ (l1, m1, n1) and NP2 ¼ (l2, m2, n2) then: NP1  NP2 ¼ ðl1  l2 , max fm1 , m2 g, max fn1 , n2 gÞ ðMultiplicationÞ

(8.9)

and NP1 + NP2 ¼ ðl1 + l2 , min fm1 , m2 g, min fn1 , n2 gÞ ðAdditionÞ

(8.10)

8.3.11 Operations on interval neutrosophic numbers [32] L U L U L U L U L U Let N1 ¼ h[TL1 , TU 1 ], [I1 , I1 ], [F1 , F1 ]i and N2 ¼ h[T2 , T2 ], [I2 , I2 ], [F2 , F2 ]i be two interval neutrosophic numbers then,

N1  N2 ¼



    

U T1L + T2L  T1L T2L , T1U + T2U  T1U T2U , I1L I2L , I1U I2U , FL1 FL2 , FU 1 F2

ðAdditionÞ N1  N2 ¼

(8.11) 

    T1L T2L , T1U T2U , I1L + I2L  I1L I2L , I1U + I2U  I1U I2U , FL1 + FL2

U U U FL1 FL2 , FU 1 + F2  F1 F2 i ðMultiplicationÞ

8.4

(8.12)

Interval neutrosophic Markov chain and long-run behavior of the neutrosophic Markov chain using interval neutrosophic probabilities

Here we define interval neutrosophic Markov chain, and its long-run behavior of the system has been studied using interval neutrosophic probabilities.

158

Optimization Theory Based on Neutrosophic and Plithogenic Sets

8.4.1 Interval neutrosophic Markov chain An interval neutrosophic stochastic process {X(n) : n 2 N} is said to be an interval neutrosophic Markov chain if it satisfies the Markov property: βðXn + 1 ¼ j=Xn1 ¼ i, Xn ¼ k, …, X0 ¼ mÞ ¼ βðXn + 1 ¼ j=Xn1 ¼ iÞ

(8.13)

where i, j, k establish the state space S of the process. fij ¼ βðXn + 1 ¼ j=Xn ¼ iÞ are called the interval-valued neutrosophic probabilHere P ities of moving

from state i  to state j in one step. Hence

fij ¼ P

T L , T U , I L , I U , F L , FU , where T L , T U are the lower and upper eij Peij eij Peij eij Peij eij Peij P P P P truth membership of the transition from state i to state j, respectively, I L , I U are eij Peij P the lower and upper indeterminate membership of the transition from state i to state j respectively and FL , FU are the lower and upper falsity membership of the transition eij Peij P   fij is called the interval-valued neutrosophic from state i to state j. The matrix P ¼ P

transition probability matrix.

8.4.2 Long-run behavior of the Markov chain using interval neutrosophic probabilities Here, the long-run behavior of the interval neutrosophic Markov chain has been studied for the world economic year problem by considering the transition probabilities are the interval neutrosophic numbers. In addition, stability of the long-run behavior has been studied.

8.4.3 Experimental analysis The problem is taken from the book entitled Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability by F. Smarandache [4] and the data taken from Broumi et al. [34]. Consider the world economy and its states: economic benefit (B), economic decline (D), and economic abjection (A). Consider an economic benefit year (EBY) is followed by another EBY 40% of the time while 10% of the time is unknown, an economic decline year (EDY) 20% of the time, while 10% of the time is not followed by an economic decline, and an economic abjection year (EAY) 10% of the time and 5% of the time is unknown while 5% of the time is not followed by an EDY. Fig. 8.1 represents the interval neutrosophic graph during a year. Here we studied the economical trend after two years and after four years. The interval neutrosophic transition matrix of the graph is represented in Fig. 8.1.

Long-run behavior of interval neutrosophic Markov chain

159

[0.1,0.2] , [0.2,0.3] , [0.2,0.5]

B

[0.2,0.4] , [0.3,0.5] , [0.1,0.2]

[0.3,0.4] , [0.1,0.2] , [0.3,0.5]

[0.1,0.3] , [0.3,0.4] , [0.2,0.3]

[0.2,0.3] , [0.2,0.5] , [0.4,0.5] D

[0.4,0.6] , [0.2,0.4] , [0.1,0.3] [0.2,0.3] , [0.3,0.4] , [0.1,0.3] A

[0.3,0.6] , [0.1,0.2] , [0.1,0.4] [0.3,0.4] , [0.1,0.3] , [0.2,0.4]

Fig. 8.1 Interval-valued neutrosophic transition probability graph during a year.

Interval neutrosophic transition probability matrix (INTPM) during a year is given by B D A 3 h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i 6 7 INTP ¼ D 4 h½0:1, 0:3, ½0:3, 0:4, ½0:2, 0:3i h½0:2, 0:3, ½0:2, 0:5, ½0:4, 0:5i h½0:3, 0:6, ½0:1, 0:2, ½0:1, 0:4i 5 B A

2

h½0:4, 0:6, ½0:2, 0:4, ½0:1, 0:3i h½0:2, 0:3, ½0:3, 0:4, ½0:1, 0:3i h½0:3, 0:4, ½0:1, 0:3, ½0:2, 0:4i (8.14)

Here the state space is {B, D, A}. The random row vectors are: B ¼ ½1 0 0, D ¼ ½ 0 1 0 , A ¼ ½ 0 0 1 : Let X be any of these random vectors with the neutrosophic relation: Xðn + 1Þ ¼ Xn INTP

(8.15)

Xðn + 2Þ ¼ Xn + 1 INTP ¼ ½Xn INTPNP ¼ Xn ðINTPÞ2

(8.16)

And in general: Xðn + mÞ ¼ Xn ðINTPÞm for any time n

(8.17)

where (INTP)2 ¼ {cij}i.j, the behavior of the interval neutrosophic Markov chain in two steps (after two years) using Eqs. (8.11) and (8.12), and the elements of the transition are:

160

Optimization Theory Based on Neutrosophic and Plithogenic Sets

c11 ¼ ½h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i  2 3 h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i 6 7 6 h½0:1, 0:3, ½0:3, 0:4, ½0:2, 0:3i 7 4 5 h½0:4, 0:6, ½0:2, 0:4, ½0:1, 0:3i ¼ h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i  h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i +h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i  h½0:1, 0:3, ½0:3, 0:4, ½0:2, 0:3i +h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i  h½0:4, 0:6, ½0:2, 0:4, ½0:1, 0:3i ¼ h½0:146, 0:297, ½0:774, 0:929, ½0:71, 0:951i c12 ¼ ½h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i 2 3 h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i 6 7 6 h½0:2, 0:3, ½0:2, 0:5, ½0:4, 0:5i 7 4 5 0:2, 0:3 ½ , ½ 0:3, 0:4 , ½ 0:1, 0:3  h i ¼ h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i  h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i +h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i  h½0:2, 0:3, ½0:2, 0:5, ½0:4, 0:5i +h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i  h½0:2, 0:3, ½0:3, 0:4, ½0:1, 0:3i ¼ h½0:116, 0:297, ½0:802, 0:958, ½0:755, 0:944i c13 ¼ ½h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i 2 3 h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i 6 7 6 h½0:3, 0:6, ½0:1, 0:2, ½0:1, 0:4i 7 4 5 h½0:3, 0:4, ½0:1, 0:3, ½0:2, 0:4i ¼ h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i  h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i +h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i  h½0:3, 0:6, ½0:1, 0:2, ½0:1, 0:4i +h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i  h½0:3, 0:4, ½0:1, 0:3, ½0:2, 0:4i ¼ h½0:17, 0:433, ½0:633, 0:875, ½0:746, 0:964i

Similarly,   c21 ¼ h½0:1, 0:3, ½0:3, 0:4, ½0:2, 0:3i h½0:2, 0:3, ½0:2, 0:5, ½0:4, 0:5i h½0:3, 0:6, ½0:1, 0:2, ½0:1, 0:4i  2 3 h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i 6 7 6 h½0:1, 0:3, ½0:3, 0:4, ½0:2, 0:3i 7 ¼ h½0:146, 0:312, ½0:774, 0:94, ½0:751, 0:949i 4 5 h½0:4, 0:6, ½0:2, 0:4, ½0:1, 0:3i   c22 ¼ h½0:1, 0:3, ½0:3, 0:4, ½0:2, 0:3i h½0:2, 0:3, ½0:2, 0:5, ½0:4, 0:5i h½0:3, 0:6, ½0:1, 0:2, ½0:1, 0:4i  2 3 h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i 6 7 6 h½0:2, 0:3, ½0:2, 0:5, ½0:4, 0:5i 7 ¼ h½0:116, 0:329, ½0:802, 0:964, ½0:79, 0:941i 4 5 h½0:2, 0:3, ½0:3, 0:4, ½0:1, 0:3i

Long-run behavior of interval neutrosophic Markov chain

161

  c23 ¼ h½0:1, 0:3, ½0:3, 0:4, ½0:2, 0:3i h½0:2, 0:3, ½0:2, 0:5, ½0:4, 0:5i h½0:3, 0:6, ½0:1, 0:2, ½0:1, 0:4i  2 3 h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i 6 7 6 h½0:3, 0:6, ½0:1, 0:2, ½0:1, 0:4i 7 ¼ h½0:17, 0:459, ½0:633, 0:892, ½0:782, 0:962i 4 5 h½0:3, 0:4, ½0:1, 0:3, ½0:2, 0:4i   c31 ¼ h½0:4, 0:6, ½0:2, 0:4, ½0:1, 0:3i h½0:2, 0:3, ½0:3, 0:4, ½0:1, 0:3i h½0:3, 0:4, ½0:1, 0:3, ½0:2, 0:4i  2 3 h½0:1, 0:2, ½0:2, 0:3, ½0:2, 0:5i 6 7 6 h½0:1, 0:3, ½0:3, 0:4, ½0:2, 0:3i 7 ¼ h½0:172, 0:336, ½0:774, 0:936, ½0:627, 0:928i 4 5 h½0:4, 0:6, ½0:2, 0:4, ½0:1, 0:3i   c32 ¼ h½0:4, 0:6, ½0:2, 0:4, ½0:1, 0:3i h½0:2, 0:3, ½0:3, 0:4, ½0:1, 0:3i h½0:3, 0:4, ½0:1, 0:3, ½0:2, 0:4i  2 3 h½0:2, 0:4, ½0:3, 0:5, ½0:1, 0:2i 6 7 6 h½0:2, 0:3, ½0:2, 0:5, ½0:4, 0:5i 7 ¼ h½0:17, 0:327, ½0:802, 0:962, ½0:685, 0:918i 4 5 h½0:2, 0:3, ½0:3, 0:4, ½0:1, 0:3i   c33 ¼ h½0:4, 0:6, ½0:2, 0:4, ½0:1, 0:3i h½0:2, 0:3, ½0:3, 0:4, ½0:1, 0:3i h½0:3, 0:4, ½0:1, 0:3, ½0:2, 0:4i  2 3 h½0:3, 0:4, ½0:1, 0:2, ½0:3, 0:5i 6 7 6 h½0:3, 0:6, ½0:1, 0:2, ½0:1, 0:4i 7 ¼ h½0:247, 0:438, ½0:633, 0:887, ½0:673, 0:947i 4 5 h½0:3, 0:4, ½0:1, 0:3, ½0:2, 0:4i

Therefore, 2

h½0:146, 0:297, ½0:774, 0:929, ½0:71, 0:951i

h½0:116, 0:297, ½0:802, 0:958, ½0:755, 0:944i

ðINTPÞ2 ¼ 6 4 h½0:146, 0:312, ½0:774, 0:94, ½0:751, 0:949i

h½0:116, 0:329, ½0:802, 0:964, ½0:79, 0:941i

6

h½0:172, 0:336, ½0:774, 0:936, ½0:627, 0:928i

h½0:17, 0:327, ½0:802, 0:962, ½0:685, 0:918i 3 h½0:17, 0:433, ½0:633, 0:875, ½0:746, 0:964i 7 h½0:17, 0:459, ½0:633, 0:892, ½0:782, 0:962i 7 5

h½0:311, 0:619, ½0:927, 0:995, ½0:92, 0:997i (8.18)

Now, (INTP)4 ¼ (INTP)2  (INTP)2 ¼ {cij}i.j, the behavior of the interval neutrosophic Markov chain in four steps (after four years) using Eqs. (8.11) and (8.12) and the elements of the transition are: "

c11 ¼

# h½0:146, 0:297, ½0:774, 0:929, ½0:71, 0:951i h½0:116, 0:297, ½0:802, 0:958, ½0:755, 0:944i  h½0:17, 0:433, ½0:633, 0:875, ½0:746, 0:964i 2 3 h½0:146, 0:297, ½0:774, 0:929, ½0:71, 0:951i 6 7 4 h½0:146, 0:312, ½0:774, 0:94, ½0:751, 0:949i 5 h½0:172, 0:336, ½0:774, 0:936, ½0:627, 0:928i

¼ h½0:146, 0:297, ½0:774, 0:929, ½0:71, 0:951i  h½0:146, 0:297, ½0:774, 0:929, ½0:71, 0:951i + h½0:116, 0:297, ½0:802, 0:958, ½0:755, 0:944i  h½0:146, 0:312, ½0:774, 0:94, ½0:751, 0:949i + h½0:17, 0:433, ½0:633, 0:875, ½0:746, 0:964i  h½0:172, 0:336, ½0:774, 0:936, ½0:627, 0:928i ¼ h½0:066, 0:642, ½1, 1, ½1, 1i

162

Optimization Theory Based on Neutrosophic and Plithogenic Sets

" c12 ¼

h½0:146, 0:297, ½0:774, 0:929, ½0:71, 0:951i h½0:116, 0:297, ½0:802, 0:958, ½0:755, 0:944i

# 

h½0:17, 0:433, ½0:633, 0:875, ½0:746, 0:964i 2 3 h½0:116, 0:297, ½0:802, 0:958, ½0:755, 0:944i 6 7 6 h½0:116, 0:329, ½0:802, 0:964, ½0:79, 0:941i 7 4 5 h½0:17, 0:327, ½0:802, 0:962, ½0:685, 0:918i

¼ h½0:058, 0:646, ½1, 1, ½1, 1i " c13 ¼

h½0:146, 0:297, ½0:774, 0:929, ½0:71, 0:951i h½0:116, 0:297, ½0:802, 0:958, ½0:755, 0:944i

# 

h½0:17, 0:433, ½0:633, 0:875, ½0:746, 0:964i 3 h½0:17, 0:433, ½0:633, 0:875, ½0:746, 0:964i 6 7 6 h½0:17, 0:459, ½0:633, 0:892, ½0:782, 0:962i 7 4 5 h½0:311, 0:619, ½0:927, 0:995, ½0:92, 0:997i 2

¼ h½0:095, 0:864, ½1, 1, ½1, 1i " c21 ¼

h½0:146, 0:312, ½0:774, 0:94, ½0:751, 0:949i h½0:116, 0:329, ½0:802, 0:964, ½0:79, 0:941i

# 

h½0:17, 0:459, ½0:633, 0:892, ½0:782, 0:962i 3 h½0:146, 0:297, ½0:774, 0:929, ½0:71, 0:951i 6 7 6 h½0:146, 0:312, ½0:774, 0:94, ½0:751, 0:949i 7 4 5 h½0:172, 0:336, ½0:774, 0:936, ½0:627, 0:928i 2

¼ h½0:066, 0:647, ½1, 1, ½1, 1i " c22 ¼

h½0:146, 0:312, ½0:774, 0:94, ½0:751, 0:949i h½0:116, 0:329, ½0:802, 0:964, ½0:79, 0:941i

# 

h½0:17, 0:459, ½0:633, 0:892, ½0:782, 0:962i 2 3 h½0:116, 0:297, ½0:802, 0:958, ½0:755, 0:944i 6 7 6 h½0:116, 0:329, ½0:802, 0:964, ½0:79, 0:941i 7 4 5 h½0:17, 0:327, ½0:802, 0:962, ½0:685, 0:918i

¼ h½0:058, 0:651, ½1, 1, ½1, 1i " c23 ¼

h½0:146, 0:312, ½0:774, 0:94, ½0:751, 0:949i h½0:116, 0:329, ½0:802, 0:964, ½0:79, 0:941i

h½0:17, 0:459, ½0:633, 0:892, ½0:782, 0:962i 2 3 ½ 0:17, 0:433 , ½ 0:633, 0:875, ½0:746, 0:964i h 6 7 6 h½0:17, 0:459, ½0:633, 0:892, ½0:782, 0:962i 7 4 5 h½0:311, 0:619, ½0:927, 0:995, ½0:92, 0:997i

¼ h½0:095, 0:874, ½1, 1, ½1, 1i

# 

Long-run behavior of interval neutrosophic Markov chain

" c31 ¼

163

h½0:172, 0:336, ½0:774, 0:936, ½0:627, 0:928i h½0:17, 0:327, ½0:802, 0:962, ½0:685, 0:918i

# 

h½0:311, 0:619, ½0:927, 0:995, ½0:92, 0:997i 2 3 h½0:146, 0:297, ½0:774, 0:929, ½0:71, 0:951i 6 7 6 h½0:146, 0:312, ½0:774, 0:94, ½0:751, 0:949i 7 4 5 h½0:172, 0:336, ½0:774, 0:936, ½0:627, 0:928i

¼ h½0:1, 0:664, ½1, 1, ½1, 1i " c32 ¼

h½0:172, 0:336, ½0:774, 0:936, ½0:627, 0:928i h½0:17, 0:327, ½0:802, 0:962, ½0:685, 0:918i

# 

h½0:311, 0:619, ½0:927, 0:995, ½0:92, 0:997i 3 h½0:116, 0:297, ½0:802, 0:958, ½0:755, 0:944i 6 7 6 h½0:116, 0:329, ½0:802, 0:964, ½0:79, 0:941i 7 4 5 h½0:17, 0:327, ½0:802, 0:962, ½0:685, 0:918i 2

¼ h½0:09, 0:667, ½1, 1, ½1, 1i " c33 ¼

h½0:172, 0:336, ½0:774, 0:936, ½0:627, 0:928i h½0:17, 0:327, ½0:802, 0:962, ½0:685, 0:918i

h½0:311, 0:619, ½0:927, 0:995, ½0:92, 0:997i 3 h½0:17, 0:433, ½0:633, 0:875, ½0:746, 0:964i 6 7 6 h½0:17, 0:459, ½0:633, 0:892, ½0:782, 0:962i 7 4 5 h½0:311, 0:619, ½0:927, 0:995, ½0:92, 0:997i

# 

2

¼ h½0:183, 0:927, ½1, 1, ½1, 1i

Therefore, 3 h½0:066, 0:642, ½1, 1, ½1, 1i h½0:058, 0:646, ½1, 1, ½1, 1i h½0:095, 0:864, ½1, 1, ½1, 1i 7 6 ðINTPÞ4 ¼ 4 h½0:066, 0:647, ½1, 1, ½1, 1i h½0:058, 0:651, ½1, 1, ½1, 1i h½0:095, 0:874, ½1, 1, ½1, 1i 5 h½0:1, 0:664, ½1, 1, ½1, 1i h½0:09, 0:667, ½1, 1, ½1, 1i h½0:176, 0:955, ½1, 1, ½1, 1i 2

(8.19)

Hence, according to this interval-valued neutrosophic transition probability matrix, after four years, economic stability can be obtained and the largest chance of economy to be in the state of abjection (A).

8.4.4 Comparative analysis Here the comparative analysis has been done in Table 8.1 for Markov chain, fuzzy MC, intuitionistic MC, neutrosophic MC, and interval-valued neutrosophic MC and, hence, one can understand the behavior of all the types and the possible step for obtaining steady state.

164

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 8.1 Advantages and limitations of the Markov chain under different environments. Type

Advantage

Limitations

Markov chain

▪ Simple Markov model. ▪ Used to determine behavior of the system. ▪ Very traditional probabilistic models of an immense variety of real systems that develop along with time. ▪ Used to model profile or outline. ▪ Lends itself to dynamic programming solutions. ▪ Used to predict future trend; depends only on the present state. ▪ Parameters will be calculated with certainty. ▪ Use linguistic labels directly. ▪ States are connected to other states with crisp transition rate. ▪ Able to cope where uncertainties exist in the system. ▪ Estimation of the parameters is possible even for uncertain states. ▪ Membership functions replaced by related probability functions. ▪ States are connected to other states with fuzzy transition rate using fuzzy numbers. ▪ Able to handle interval data.

▪ Unable to cope where uncertainties exist in the system.

▪ Able to handle nonmembership and hesitation degree of the states along with membership degree. ▪ Transition rate is intuitionistic fuzzy.

▪ Unable to cope with indeterminacy of the state.

Fuzzy MC

Interval fuzzy MC

Intuitionistic MC

▪ Unable to cope with nonmembership and hesitation degree of the states. ▪ Unable to handle interval-based data.

▪ Incapable of handling non-membership of the state.

Long-run behavior of interval neutrosophic Markov chain

165

Table 8.1 Continued Type

Advantage

Limitations

Interval valued intuitionistic MC

▪ Able to deal with interval data.

▪ Does not have the capability of dealing with indeterminacy of the interval data.

Neutrosophic MC

▪ Able to deal with indeterminacy of the state. ▪ Transition rate is neutrosophic. ▪ Able to deal with interval data.

▪ Unable to deal with interval data of the system.

Interval neutrosophic MC

▪ Steady state behavior can be obtained in lower number of steps.

▪ Unable to handle incomplete weight information.

8.4.5 Comparative analysis with the existing methods In this section, the proposed work is compared with the existing methods in Table 8.2., which shows the novelty of the proposed work. From this comparative analysis it is observed that, when we use the interval neutrosophic Markov chain, stability (robustness) can be obtained with a lower number of steps, whereas the single valued neutrosophic Markov chain takes more steps to reach the stability of the system; hence, the effectiveness of the proposed work is proved.

Table 8.2 Comparative analysis of the proposed work with the existing method. Existing method

Proposed method

In [4], neutrosophic Markov chain has been defined and studied regarding the economical changes after two years using neutrosophic transition matrix under single valued neutrosophic environment and it is concluded that the economical state will be in the state of recession. In this proposed work, long-run behavior of the interval neutrosophic Markov chain has been studied to predict the economical changes and the robustness of the behavior under interval neutrosophic environment by considering the neutrosophic probabilities as interval neutrosophic numbers. It is found that, after four years, economic stability can be obtained and the chance of economic state to be is in the abjection state.

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Conclusion

Future trend prediction is an important task in the decision-making environment in all industries and companies, and all real-world problems have indeterminacy in their nature. Hence, these two things have been taken into account and the interval neutrosophic Markov chain introduced to study the long-run behavior of the system. In this chapter, the long-run behavior of the economic year was studied using interval neutrosophic Markov chain by considering the neutrosophic transition probabilities as the interval neutrosophic numbers. It was concluded that after four years, stability can be obtained in the economic position and will be in the state of abjection using a lower number of steps (quickly). Comparative study was done for analyzing the behavior of the Markov chain under different environments. Further proposed work was compared with the existing method, and it was found that the interval neutrosophic Markov chain reaches stability faster than the single valued neutrosophic Markov chain, indicating the novelty and effectiveness of the present work. In future, this study may be done using special types of neutrosophic environments.

References [1] A. Kazemi, M. Modarres, M.R. Mehregan, N. Neshat, A. Foroughi, A Markov chain grey forecasting model: a case study of energy demand of industry sector in Iran, in: International Conference on Information and Financial Engineering IPEDR Vol. 12 (2011) 13–18, IACSIT Press, Singapore, 2011, p. 2011. [2] G. Fort, S. Meyn, E. Moulines, P. Priouret, The ode method for stability of skip-free Markov chains with applications to MCMC, Ann. Appl. Probab. 18 (2) (2008) 664–707. [3] Y. Xin, Z.X. Qi, L.L. Na, A spatiotemporal model of land use change based on ant colony optimization, Markov chain and cellular automata, Ecol. Model. 233 (2012) 11–19. [4] F. Smarandache, Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability, Sitech & Educational, Craiova, Columbus, 2013. 140 p, https://arxiv.org/ftp/arxiv/papers/1311/1311.7139.pdf. [5] R.A. Adeleke, K.A. Oguntuase, R.E. Ogunsakin, Application of Markov chain to the assessment of students’ admission and academic performance in Ekiti state university, Int. J. Sci. Technol. Res. 3 (7) (2014) 349–357. [6] J.A.L. PiriyaKumar, V. Sreevinotha, On the ergodic behaviour of fuzzy Markov chains, IOSR J. Math. (IOSR-JM) 12 (5) (2016) 28–34. [7] J.J. Hunter, Mixing times with applications to perturbed Markov chains, Linear Algebra Appl. 417 (2006) 108–123. [8] J.C.F. Garcı´a, D. Kalenatic, C.A.L. Bello (2008) A simulation study on fuzzy Markov chains. In Conference: International Conference On Intelligent Computing, 2008. [9] D. Ponomarev, M. Sauceau, C. Nikitine, E. Rodier, J. Fages, Application of the Markov chain theory for modelling residence time distribution in a single screw extruder, in: 11th European Meeting on Supercritical Fluids: Reactions, Materials and Natural Products Processing, May 2008, Barcelona, Spain, 2008. ISBN 2-905267-58-5, 8. [10] J.J. Hunter, Coupling and mixing times in a Markov chain, Linear Algebra Appl. 430 (2009) 2607–2621. [11] K. Jung, D. Shah, J. Shin, Distributed averaging via lifted Markov chains, IEEE Trans. Inform. Theory 56 (1) (2010) 634–647.

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[12] J.C.F. Garcia, Interval type-2 fuzzy Markov chains, Stud. Fuzziness Soft Comput. 301 (2010) 1–6. [13] D. Kalenatic, J.C. Figueroa-Garcı´a, C.A. Lopez, Scalarization of Type-1 fuzzy Markov chains, in: Conference: Advanced Intelligent Computing Theories and Applications, 6th International Conference on Intelligent Computing, ICIC 2010, Changsha, China, 2010. https://doi.org/10.1007/978-3-642-14922-1_15. [14] J. Bai, P. Wang, Conditional Markov chain and its application in economic time series analysis, Munich Personal RePEc Archive, MPRA Paper No. 33369 (2011) pp. 1–26. http://mpra.ub.uni-muenchen.de/33369/. [15] S.F. Mallak, M.M. Beh, A. Zaiqan, A particular class of ergodic finite fuzzy Markov chains, Adv. Fuzzy Math. 6 (2) (2011) 253–268. [16] B.A. GerencsEr, Markov chain mixing time on cycles, Stoch. Process. Appl. 121 (2011) 2553–2570. [17] H. Masoumi, B.F. Vajargah, Study on behavior of fuzzy Markov chains, Adv. Comput. Sci. Appl. (ACSA) 2 (2) (2012) 373–376. [18] Y. Kou, L. Jia, Y. Wang, The study on the incidence of disease based on fuzzy Markov chain, 2012, https://doi.org/10.1007/978-3-642-25766-7_6. [19] R. Sujatha, An Introduction to intuitionistic Markov chain, Int. Math. Forum 7 (50) (2012) 2449–2456. [20] K.C. Chan, C.T. Lenard, T.M. Mills, On Markov chains, Math. Gazette 97 (2013) 515–520, https://doi.org/10.2307/3616733. [21] B.F. Vajargah, M. Gharehdaghi, Ergodicity of fuzzy Markov chains based on simulation using sequences, J. Math. Comput. Sci. 11 (2014) 159–165. [22] Li G, Xiu B (2014) Fuzzy Markov chains based on the fuzzy transition probability. In: 26th Chinese Control and Decision Conference (CCDC), 978-1-4799-3708-0/14/$31.00_c 2014 IEEE 4351-4356. [23] J.P.M. Kanyinda, R.M.M. Matendo, B.U.E. Lukata, D.N. Ibula, Fuzzy eigen values and fuzzy eigenvectors of fuzzy Markov chain transition matrix under max-min composition, J. Fuzzy Set Valued Anal. 1 (2015) 25–35. [24] M. Lei, S. Li, Q. Tan, Intermittent demand forecasting with fuzzy Markov chain and multi aggregation prediction algorithm, J. Intell. Fuzzy Syst. 31 (6) (2016) 2911–2918. [25] F. Smarandache, N. Abbas, Y. Chibani, B. Hadjadji, Z.A. Omar, PCR5 and neutrosophic probability in target identification, Progr. Nonlinear Dyn. Chaos 4 (2) (2016) 45–50. [26] J.C.F. Garcia, L.C.G. Arcos, S.K.L. Rivera, Quasi type-2 fuzzy Markov chains: an approach, in: Conference: 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2016. https://doi.org/10.1109/FUZZ- IEEE.2016.7737734. [27] D.M. Zhu, W.K. Ching, S.M. Guu, Sufficient conditions for the ergodicity of fuzzy Markov chains, Fuzzy Set. Syst. 304 (2017), https://doi.org/10.1016/j.fss.2016.01.005. [28] B. Liu, Y. Liang, An introduction of Markov chain Monte Carlo method to geochemical inverse problems: Reading melting parameters from REE abundances in abyssal peridotites, Geochim. Cosmochim. Acta 203 (2017) 216–234. [29] M. Awiszus, B. Rosenhahn, Markov chain neural networks, in: Conference: 2018 IEEE/ CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), 2018. https://doi.org/10.1109/CVPRW.2018.00293. [30] R. Alhabib, M.M. Ranna, H. Farah, A.A. Salama, Some neutrosophic probability distributions, Neutrosophic Sets Syst. 22 (2018) 30–38. [31] Petrov T (2019) Markov chain aggregation and its application to rule-based modelling. In book: Model. Biomol. Site Dyn., DOI: https://doi.org/10.1007/978-1-4939-9102-0_14.

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[32] S. Broumi, M. Talea, A. Bakali, F. Smarandache, D. Nagarajan, M. Lathamaheswari, M. Parimala, Shortest path problem in fuzzy, intuitionistic fuzzy and neutrosophic environment: an overview, Complex Intell. Syst. (2019), https://doi.org/10.1007/ s40747-019-0098-z. [33] S. Broumi, D. Nagarajan, A. Bakali, M. Talea, F. Smarandache, M. Lathamaheswari, The shortest path problem in interval valued trapezoidal and triangular neutrosophic environment, Complex Intell. Syst. (2019), https://doi.org/10.1007/s40747-019-0092-5. [34] S. Broumi, A. Dey, M. Talea, A. Bakali, F. Smarandache, D. Nagarajan, M. Lathamaheswari, R. Kumar, Shortest path problem using Bellman algorithm under neutrosophic environment, Complex Intell. Syst. (2019), https://doi.org/10.1007/ s40747-019-0101-8. [35] D. Nagarajan, M. Lathamaheswari, S. Broumi, J. Kavikumar, Blockchain single and interval valued neutrosophic graphs, Neutrosophic Sets Syst. 24 (2019) 23–35. [36] D. Nagarajan, M. Lathamaheswari, S. Broumi, J. Kavikumar, Dombi interval valued neutrosophic graph and its role in traffic control management, Neutrosophic Sets Syst. 24 (2019) 114–133. [37] D. Nagarajan, M. Lathamaheswari, S. Broumi, J. Kavikumar, A new perspective on traffic control management using triangular interval type-2 fuzzy sets and interval neutrosophic sets, Oper. Res. Perspect. (2019), https://doi.org/10.1016/j.orp.2019.100099. [38] M. Abdel-Basset, M. Saleh, A. Gamal, F. Smarandache, An approach of TOPSIS technique for developing supplier selection with group decision making under type-2 neutrosophic number, Appl. Soft Comput. 77 (2019) 438–452. [39] M. Abdel-Baset, V. Chang, A. Gamal, F. Smarandache, An integrated neutrosophic ANP and VIKOR method for achieving sustainable supplier selection: a case study in importing field, Comput. Ind. 106 (2019) 94–110. [40] M. Abdel-Basset, G. Manogaran, A. Gamal, F. Smarandache, A group decision making framework based on neutrosophic TOPSIS approach for smart medical device selection, J. Med. Syst. 43 (2) (2019) 38. [41] M. Abdel-Baset, V. Chang, A. Gamal, Evaluation of the green supply chain management practices: A novel neutrosophic approach, Comput. Ind. 108 (2019) 210–220. [42] M. Abdel-Basset, A. Atef, F. Smarandache, A hybrid Neutrosophic multiple criteria group decision making approach for project selection, Cognit. Syst. Res. (2019) 216–227. [43] Smarandache F (2006) Neutrosophic set—a generalization of the intuitionistic fuzzy set. Granular Computing, 2006 IEEE International Conference, 38–42, DOI: https://doi.org/ 10.1109/GRC.2006.1635754. [44] H. Wang, F. Smarandache, Y. Zhang, R. Sunderraman, Single valued neutrosophic sets, Multisspace Multistruct. 4 (2010) 410–413. [45] H. Wang, Y. Zhang, R. Sunderraman, Truth-value based interval neutrosophic sets, in: Granular Computing, IEEE International Conference, 1 2005, pp. 274–277, https:// doi.org/10.1109/GRC.2005.1547284.

Neutrosophic-based recommender for eLearning systems

9

Haitham A. El-Ghareeb Information Systems Department, Faculty of Computers and Information Sciences, Mansoura University, Mansoura, Egypt

9.1 9.1.1

Introduction Recommender systems

Recommendations drive user engagement. Relevance is at the heart of modern information systems. User-level personalization is an important goal to achieve, and recommendation engines are one of the best early examples of how this goal can be achieved. The main goal of recommender system is to identify the items most relevant to the user (e.g., top n items). A recommender system creates a matching between users and items and exploits the similarity between users/items to make recommendations. Recommender systems are everywhere. Among many of the available examples are the following: l

l

l

Netflix displays trending now movies that users can choose from. This list is generated by a Popularity-Based Recommender System, which will be discussed in Section 9.3.1. Amazon displays product recommendations in the form of: customers who bought this item also bought. This type of recommendation is classified under Collaborative-Based Recommender Systems, specifically item-based filtering, which will be discussed in Section 9.3.3. Facebook displays friend recommendations in the form of: you have a friend suggestion. Digging deeper into Facebook’s friend recommendations, we obtain the following information: “Your friend suggestions are generated when one of your friends select you as someone who knows someone else on Facebook. If you add your suggested friends as friends, a normal friend request will be sent. If you do not, no one will be notified that you ignored a suggestion.” This type of recommendation is classified under Collaborative-Based Recommender Systems, specifically user-based filtering, which will be discussed in Section 9.3.3.

Recommender systems success stories are many. Thirty-five percent of what consumers purchase on Amazon and 75% of what they watch on Netflix comes from product recommendations [1]. In 1988, a British mountain climber named Joe Simpson wrote a book called “Touching the Void,” a harrowing account of a near-death experience in the Peruvian Andes. It got good reviews, but only a modest success, and was soon forgotten. Then, a decade later, a strange thing happened. Jon Krakauer wrote “Into Thin Air,” another book about a mountain-climbing tragedy, which became a publishing sensation. Suddenly, “Touching the Void” started to sell again [2].

Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00009-3 © 2020 Elsevier Inc. All rights reserved.

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People who bought side table also bought:

Similar product in from home office series:

Coffee table

Low book shelf

Bed side table

Wood side table

Green side table

Walnut side table

$235

$150

$90

$110

$135

$120

Personalized real-time recommendations

Personalized promotions

Personalized real-time recommendations

Fig. 9.1 Online store recommender system example.

Fig. 9.1 presents an example of recommender systems utilization in online stores. There are two types of recommendations that are applied in this figure: personalized promotions (upper part of the page) and personalized real-time recommendations (lower part of the page). For the personalized real-time recommendations, there are two types: l

l

Collaborative filtering (bottom left-hand side): An algorithm that considers users’ interaction with products, with the assumption that other users will behave in similar ways. Content based (bottom right-hand side): An algorithm that considers similarities between products and categories of products.

9.1.2

Neutrosophic sets and theory

Neutrosophic sets have been introduced to the literature by Smarandache to handle incomplete, indeterminate, and inconsistent information [3]. Neutrosophic theory helps in addressing vagueness, inconsistencies, and missing information, three challenges that face intelligent Microservices, and need to be addressed carefully. In neutrosophic sets, indeterminacy is quantified explicitly through a new parameter I. Truth membership (T), indeterminacy membership (I), and falsity membership (F) are three independent parameters that are used to define a Neutrosophic Number. For detailed illustration, discussion, and examples of Neutrosophic Set and Theory in decision making, the reader is referred to Refs. [4–6].


N¼

nD E o x;TN ðxÞ, IN ðxÞ,FN ðxÞ , x 2 X








x 2 X, TN ðxÞ, IN ðxÞ,FN ðxÞ 2 ½0, 1








(9.1) (9.2)

Neutrosophic-based recommender for eLearning systems

171

9.1.2.1 Neutrosophication The main purpose of Neutrosophication is to map input variables to Neutrosophic sets [7]. If x is a crisp input, then 8 b x 8 c2  x 8 x  a1 2 > b1  x < b2 > > > a  x < a 1 2 > > > > > > b  b c2  c1 2 1 > > > a2  a1 > > > > > > > > > a  x x x  b > > > 3 2 > > > a  x < a3 > > > > > > < c3 < b3  b2 b2  x < b3 < a3  a2 2 FðxÞ ¼ c4  x I ðxÞ ¼ b  x T ðxÞ ¼ x  a3 4 > > > a3  x < a4 > > > b3  x < b4 > > > a  a c4  c3 > > > 4 3 > > > b4  b3 > > > > > > > > > 0 otherwise 1 > > > 1 otherwise > > > > > > : > : :

c1  x < c2 c2  x < c3 c3  x < c4 otherwise

(9.3)

are the truth, indeterminacy, and falsehood memberships for the crisp input x 2 X and aj  x  ak for truth membership, bj  x  bk for indeterminacy membership, and cj  x  ck for falsehood membership, respectively, and j, k ¼ 1, 2, 3, 4.

9.1.2.2 Deneutrosophication Deneutrosophy is the process where Neutrosophic scales/numbers are converted to crisp values by applying score functions of s(aij) as illustrated in Eq. (9.4) presented in Nabeeh et al. [4].    Trij + Irij + Frij   sðaij Þ ¼ ðlrij  mrij  urij Þ   9

(9.4)

Another deneutrosophication function is presented in Ali et al. [7]. This step involves the following two stages.

Stage 1: Synthesization In this stage, neutrosophic set A is transformed into a fuzzy set B by the following function: f ðTA ðyÞ, IA ðyÞ, FA ðyÞÞ : ½0, 1  ½0,1  ½0, 1 ! ½0, 1 TB ðyÞ ¼ αTA ðyÞ + β

FA ðyÞ IA ðyÞ +γ 4 2

where 0  α, β, γ  1 such that α + β + γ ¼ 1.

(9.5) (9.6)

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Stage 2: Typical neutrosophic value In this stage, a typical deneutrosophicated value denðTB ðyÞÞ can be calculated by the centroid or center of gravity method, which is given as follows: Z

b

denðTB ðyÞÞ ¼ Za

TB ðyÞydy

b

(9.7) TB ðyÞdy

a

where the integrand is a continuous function.

9.1.2.3

Neutrosophic numbers family

Different neutrosophic numbers and sets are available. The most relevant neutrosophic numbers and sets for our Proposed Neutrosophic-Based Recommender System are: l

l

l

l

single-valued neutrosophic number (SVNN); interval valued neutrosophic number; single-valued neutrosophic sets (SVNSs); and interval valued neutrosophic sets.

For further details on the above-mentioned neutrosophic numbers and sets, the reader is referred to El-Ghareeb [8]. For further details on neutrosophic operators and the different neutrosophic numbers, the reader is referred to Refs. [9–13].

9.1.3 Neutrosophic-based recommender systems 9.1.3.1

Single-criterion recommender systems (SC-RS)

Suppose U is a set of all users and ω is the set of items in the system. The utility function R is a mapping specified on U1  U and ω1  ω follows: R : U1  Ω1 ! ℘ ðu1 ;ω1 Þ 7! Rðu1 ;ω1 Þ

(9.8)

where Rðu1 ;ω1 Þ is a nonnegative integer or a real number within a certain range (set of real numbers R). R is a set of available ratings in the system. Thus, SC-RS is the system that provides two basic functions. 1. Prediction: Determine Rðu*; w*Þ for any (u*, w*) 2 (U, ω)n(U1;ω1). 2. Recommendation: Choose w*2 ω satisfying w* ¼ argmaxi2I Rðu, wÞ,u 2 U.

9.1.3.2

Multicriteria recommender systems (MC-RS)

MC-RS provides similar basic functions with SC-RS but follows multiple criteria.

Neutrosophic-based recommender for eLearning systems

R : U1  Ω1 ! ℘1  ℘2  ⋯  ℘k ðu1 ;ω1 Þ 7! ðR1 , R2 , …,Rk Þ

173

(9.9)

where Ri(i ¼ 1, 2, …, k) is the rating of user u1 2 U1 for item w1 2 ω1 followed by criterion i in this case, the recommendation is performed according to a given criterion.

9.1.3.3 Medical recommender systems As one of the most common decision-making problems, medical diagnosis has received full attention from both the computer science and computer applications and mathematics research communities since the introduction of artificial intelligence. Medical diagnosis is the process of analyzing the relationship between symptoms and diseases according to uncertain and inconsistent information, a process that leads to a decision-making problem [14]. Indeed, a medical diagnosis problem usually involves a large amount of uncertain, inconsistent, incomplete, and indeterminate data that are notably difficult to retrieve, handle, and process. The neutrosophic set proposed by Smarandache displays advantages in handling this type of information and is characterized independently by a truth membership function, indeterminate membership function, and false membership function [15]. Recommender systems have been commonly used in medical diagnosis to make recommendations based on patient preferences [16]. Further details on Neutrosophic Recommender Systems in medical diagnosis can be found at Refs. [7, 15, 17, 18].

9.1.3.4 Hotel recommender system in tourism The hotel recommendation approach based on the online consumer reviews using interval neutrosophic linguistic numbers is presented in Wang et al. [19].

9.1.3.5 Stock trending The neutrosophic soft set decision making for stock trending analysis is presented in Jha et al. [20].

9.1.4

Recommender systems in eLearning

Recommender systems have been widely utilized in eLearning systems. Recommender systems can help students and enhance the eLearning process in many aspects, including recommending new learning activities, learning objects (LOs) to study, study groups to join, academic supervisor to contact, lectures to attend, among many other things that recommender systems can achieve and recommend. The main benefit of recommender systems in eLearning is the support of personalized and adaptive eLearning experience.

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An attempt to support personalized and adaptive lectures, either online or in lecture halls via recommender systems, is presented in Refs. [21, 22]. The rest of the chapter is organized as follows. Section 9.2 presents an overview of the Proposed Neutrosophic-Based Recommender System. Section 9.3 sheds light on the recommender systems algorithms, some of which will be utilized in the proposed model. Section 9.4 presents the characteristics and the main data attributes of the objects to be recommended that are the focus of this chapter. Different LOs’ attributes and metadata are available. Here, the most important and relevant LOs’ attributes and metadata are discussed and presented to be further utilized in the proposed model. Section 9.5 presents the characteristics considered for students in the eLearning system. Different learning styles are considered to enhance the eLearning process. Attempting the recommendation process between different LOs with different learning styles presents vague, inconsistent, and indeterminate values. Section 9.6 presents the different required Microservices to enable and support the proposed Intelligent Recommender System. Different Microservices and components are discussed. Section 9.7 discusses how Neutrosophic Sets and Theory are utilized in the proposed Intelligent Neutrosophic-Based Recommender System. Section 9.8 discusses some technical and implementation aspects, to illustrate further the implementation details, act as a reference for regenerating the proposed model, and shed light on some challenges that faced the implementation of the proposed model. Section 9.9 presents a case study that utilized the proposed model, highlighting its pros and cons. Section 9.10 presents the evaluation of the proposed model in the case study, discusses some comments on the results, presents some optimization techniques for the proposed model, and compares the results before and after the optimization. The chapter concludes with references.

9.2

Proposed system

The main aim of this chapter is to present a new hybrid method between Recommender Systems and Neutrosophic Sets and Theory for eLearning systems. The Proposed Neutrosophic-Based Recommender System was used in an eLearning system to overcome the traditional recommender systems shortage of dealing with the huge amount of uncertain, inconsistent, incomplete, and indeterminate data which are very difficult to deal with. Neutrosophic Sets and Theory can handle those challenges accurately. The substantial difference of Neutrosophic-Based Recommender Systems compared to the traditional recommender systems is as follows. First, the new approach handles the difficulties of vague data, and the impossibilities to generate rigid boundaries between topics, courses, and LOs. Second, Neutrosophic Sets and Theory was shown to be better at modeling real-life problems than other sets and theories used in uncertainty modeling, like fuzzy logic sets and intuitionistic fuzzy logic sets. Third, the Proposed Neutrosophic Recommender System is constructed based on the neutrosophic measures. Fourth, the proposed system is a generalization of the other existing recommender systems which could improve the accuracy of the neutrosophic set as well as the recommender system. Lastly, the Proposed Neutrosophic-Based Recommender System is designed based on a strong

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175

mathematical foundation. It is indeed recognizable that the significance and importance of the proposed work can be seen from both theoretical and practical aspects. This chapter presents an Intelligent LOs Classifier: a Microservice that utilizes Neutrosophic Sets and Theory to classify LOs. Such a Microservice is very useful, and can be used to enable the Intelligent Online Lecture LOs Advisor: a Microservice that enables the Adaptive Online Lecture Model to present different pedagogical aspects via: l

l

l

recommending LOs based on students’ learning preferences; involving students in the learning process from the very beginning of the lecture; and preparing for the next lecture, so students feel the lecture’s adaptivity.

Neutrosophic Sets and Theory is utilized in the two Intelligent LOs Recommender phases: classification and personalization. For the system to generate recommendations, it needs to classify a certain LO to an objective in a certain course lecture. However, LOs may be related to different objectives, which is why the system needs to be flexible enough so that the same LOs may belong to different objectives, lectures, or courses. That is why Neutrosophic Set and Theory is a suitable solution.

9.3 9.3.1

Recommender systems algorithms Popularity-based recommender systems

Popularity-based recommender systems recommend items viewed/purchased by most people. Recommendations are presented as a list of items ranked by their purchase count. Usually in most stores, the bestsellers or the most popular items will be listed first. This means in this recommendation engine that the number of sales governs the selling choices. However, there are differences between physical and online stores. The main difference can be explained by the long tail phenomenon depicted in Fig. 9.2. The

Fig. 9.2 Long tail phenomenon representation.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

physical stores only provide the popular items, while the online world provides all items. Therefore, in the online world, there is a need for recommendations. The main strengths of the Popularity-Based Recommender Systems are that it: l

l

l

l

l

can use context (e.g., time of day); can make use of user features; can make use of item features; can use purchase history; and is scalable.

However, the main limitation of Popularity-Based Recommender Systems is the lack of personalized recommendations.

9.3.2 Classification-based recommender systems Classification-Based Recommender Systems use features of both items as well as users in order to predict whether a user will like an item or not. It is difficult to collect high-quality information about items and users. The main Classification-Based Recommender System strengths are that it can: l

l

l

l

l

generate personalized recommendations; use context (e.g., time of day); use user features; use item features; and use purchase history.

However, the main drawback of Classification-Based Recommender Systems is that they are not scalable.

9.3.3 9.3.3.1

Collaborative filtering-based recommender systems Nearest neighbor

User-based collaborative filtering This finds users who have a similar taste of products to the current user. It is based upon similarity in users’ purchasing behavior. User-Based Collaborative Filtering can be expressed with the following statement: User x is similar to user y because both purchased items A, B, C.

Item-based collaborative filtering This recommends items that are similar to other items that the user bought. It is based upon cooccurrence of purchases. Item-Based Collaborative Filtering can be expressed with the following statement: Items A and B were purchased by both users x and y, so they are similar. The simplest way to apply Item-Based Collaborative Filtering is the Cooccurrence matrix, which is a matrix of items x items. The history matrix is reviewed, where the users’ purchases are stored, and every time there is a purchase that holds two items, we increment the cooccurrence matrix between both of them. It is worth noting that in the

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177

cooccurrence matrix, the diameter is kept empty. Though Item-Based Collaborative Filtering took a step further in personalizing the recommendations, it lacks the correct recommendation with very popular items. If everybody has watched the movie Titanic, for example, then it is not a very good indicator of what to recommend next, so the recommender would become similar to a popularity-based recommender engine. One of the solutions to this problem is to normalize the cooccurrence matrix. There are different normalization functions. One of the most popular methods is normalize by popularity (Jaccard similarity), where the numbers presented in the cooccurrence matrix is represented by Eq. (9.10) rather than cumulative occurrences. In simple words, Jaccard similarity for items can be represented as: JðA,BÞ ¼

j A \ Bj j A \ Bj ¼ jA [ Bj jAj + jBj  jA \ Bj

(9.10)

All previous solutions face great challenges when the matrix is very sparse. Sparse matrices represent a great challenge on memory and for computation. An example of a sparse matrix is presented at Netflix, with the prize of one million dollars. It is known as the Million Dollar Matrix. It is a matrix of dimensions 480, 189  17, 770. Only 100 million out of possible 8.5 billion ratings are nonzero. In such sparse matrices, it is almost useless to use the algorithms previous discussed.

9.3.3.2 Matrix factorization Model-Based Collaborative Filtering, also known as Matrix Factorization, identifies latent (hidden) features from the input user x item ratings matrix to represent users and items as vectors in an N-dimensional space. The probability that two users who have bought 100 books each have a common book, assuming that there is a catalog of one million books, is 0.01. Netflix announced a competition which was completed in 2009 to solve the Netflix matrix sparsity challenge. Netflix competition showed that many matrix factorizationbased methods could be used for solving different problems of recommender systems.

9.4

Learning objects

The LO is the basic building block of a learning resource; it is the electronic representation of media, such as text, images, sounds, assessment objects, or any other piece of data that can be rendered by a web client and presented to a learner. LOs play an important role in the proposed system to present different adaptivity features. LOs metadata needed to support the adaptivity features are presented in detail below, highlighting the focus area of each category and presenting the needed attributes to be stored for each category. LOs’ metadata are grouped as follows: 1. General: Groups the general information that describes the LO as a whole. (a) Identifier: Represents a mechanism for assigning a globally unique label that identifies the LO.

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(b) Catalog: Represents the name or designator of the identification or cataloging scheme for the LO. There are a variety of cataloging systems available. Some types of cataloging systems: Universal Resource Identifier (URI) Universal Resource Name (URN) Digital Object Identifier (DOI) International Standard Book Numbers (ISBN) International Standard Serial Numbers (ISSNs) (c) Title: Name given to the LO. (d) Language: Primary language or languages used in the LO. (e) Description: Textual description of the LO. (f ) Keyword: Defines common keywords that describe the LO. (g) Coverage: Describes the time, culture, geography, or region to which the LO applies. (h) Structure: Describes the underlying organizational structure of the LO. Values are as follows: Atomic: Object that is indivisible. Collection: Set of objects with no specified relationship between them. Networked: Set of objects with relationships that are unspecified. Hierarchical: Set of objects whose relationships can be represented by a tree structure. Linear: Set of objects that are fully ordered. (i) Aggregation level: Describes the functional granularity of the LO. Values are as follows: The smallest level of aggregations, for example, raw media data or fragments. A collection of level 1 LOs, for example, a lesson. A collection of level 2 LOs, for example, a course. The largest level of granularity, for example, a set of courses that lead to a certificate. 2. Life cycle: Groups the features related to the history and current state of the LO and those who have affected the component during its evolution. (a) Version: Describes the edition of the LO. (b) Status: Describes the completion status or condition of LO. Values are as follows: Draft: The component is in a draft state (as determined by the developer). Final: The component is in a final state (as determined by the developer). Revised: The component has been revised since the last version. Unavailable: The status information is unavailable. (c) Contribute: Describes those entities (i.e., people, organizations) that have contributed to the state of the LO during its lifecycle. (d) Role: Defines the kind or type of contribution made by the contributor (identified by the Entity element). Values are: author publisher unknown initiator terminator validator editor graphical designer technical implementer content provider l

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technical validator educational validator script writer instructional designer subject matter expert (e) Date: Identifies the date of the contribution. 3. Technical: Describes all of the technical characteristics and requirements of LOs. (a) Format: Represents the technical datatype(s) of all of the components used in the makeup of the LO. (b) Size: Represents the size of the LO in bytes. (c) Location: Specifies the location of the LO. (d) Requirement: Expresses the technical capabilities necessary for using the LO. (e) Type: Represents the technology required to use the LO (e.g., hardware, software, network). Vocabulary tokens include: operating system browser (f ) Name: Represents the required technology to use the LO. (g) Minimum version: Represents the lowest possible version of the required technology to use the LO. (h) Maximum version: Represents the highest possible version of the required technology to the LO. (i) Installation remarks: Used to represent any specific instructions on how to install the LO. (j) Other platform requirements: Used to represent information about other software and hardware requirements. (k) Duration: Represents the time a continuous LO takes when played at the intended speed. This element is useful for sounds, movies, simulations, and the like. 4. Educational: Describes the key educational or pedagogic characteristics of the LO. This category is typically used by teachers, managers, authors, and learners. (a) Interactivity type: Represents the dominant mode of learning supported by the LO. Vocabulary tokens include: Active: Active learning (e.g., learning by doing) is supported by content that directly induces productive action by the learner. Expositive: Expositive learning (e.g., passive learning) occurs when the learner’s job mainly consists of absorbing the content exposed to them. Mixed: A blend of active and expositive interactivity types. (b) Learning resource type: Represents the specific kind of the LO. Vocabulary tokens include: exercise simulation questionnaire diagram figure graph index slide table narrative text exam experiment l

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problem statement self-assessment lecture (c) Interactivity level: Represents the degree of interactivity characterizing the LO. Interactivity refers to the degree to which the learner can influence the aspect or behavior of the LO. Vocabulary tokens include: very low low medium high very high (d) Intended end-user role: Represents the principal user(s) for which the LO was designed. Vocabulary tokens include: teacher author learner manager (e) Context: Represents the principal environment within which the learning and use of the LO is intended to take place. Vocabulary tokens include: school higher education training other (f ) Typical age range: Represents the age of the typical end user. Value should be formatted as minimum age–maximum age. (g) Difficulty: Represents how hard it is to work with or through the LO for the typical intended target audience. Vocabulary elements include: very easy easy medium difficult very difficult (h) Typical learning time: Represents the approximate of typical time it takes to work with or through the LO. (i) Description: Used to comment on how the LO will be used in the learning process. (j) Language: Represents the human language used by the typical intended user of the LO. 5. Rights: Describes the intellectual property rights and conditions of use for the LO. Cost: Represents whether the LO requires some sort of payment. Vocabulary tokens include: – yes – no Description: Allows comments on conditions of use of the LO. 6. Annotation: Provides comments on the educational use of the LO and information on when and by whom the comments were created. This category enables educators to share their assessments of the LO. Entity: Identifies the entity created the annotation. Date: Identifies the date the annotation was created. Description: Used to represent contents of the annotation. l

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7. Classification: Describes where the LO falls within a particular classification system. Multiple classification categories may be used. Purpose defines the purpose for classifying the LO. Vocabulary tokens include: – discipline – idea – prerequisite – educational objective – accessibility restrictions – educational level – skill level – security level – competency Description: Represents the content of classification. l

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9.5

Learning styles

There are different learning styles for people. The Proposed LOs Recommender System makes use of four of them, namely: 1. 2. 3. 4.

General Felder ATLAS BrainWorks

9.5.1

General learning style

During registration, students are asked to complete general learning style preferences. In case a student selects more than one style, they are asked to rank choices, so later recommendations can define to what extent the style is fulfilling the student’s requirements. General learning profile preferences are as follows: 1. Visual: Individuals who learn best when ideas or subjects are presented in a visual format, whether that is written language, pictures, diagrams, or videos, are visual learners. 2. Auditory: Individuals who learn best by participating in class discussion, or by listening to teacher lecture, audio tapes, or other language formats, are auditory learners. 3. Tactile: Tactile learners are hand-on learners. They learn best when they are able to participate physically directly in what they are required to learn or understand. 4. Logical: Logical learners like using their brain for logical and mathematical reasoning. They can recognize patterns easily and are good at making logical connections between what would appear to most people to be meaningless content. 5. Social: Social learners communicate well with others. They are good listeners and are able to understand others’ views. 6. Solitary: Solitary learners tend to be private, introspective, and independent. They are able to concentrate and focus on a specific subject, topic, or concept without outside help.

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Felder learning style

The Felder learning model can be identified by encouraging the student to answer questions that help identify their learning preferences. Though Felder identifies that the student’s learning model is middle between different models, the Felder model can help the system to identify the student learning features, and prepare the most appropriate learning environment. The Felder learning model categories are: 1. Active and Reflective (a) Active: Tend to retain and understand information best by doing something active with it, like discussing or applying it or explaining it to others. Tend to work in a group. (b) Reflective: Prefer to think about things first. Prefer working alone. 2. Sensing and Intuitive (a) Sensing: Like learning facts, solving problems by well-established methods, dislike complications and surprises. Tend to be patient with details and good at memorizing facts and doing hands-on (laboratory) work. (b) Intuitive: Prefer discovering possibilities and relationships. Like innovation and dislike repetition. Better at grasping new concepts and are often more comfortable with abstractions and mathematical formulations. 3. Visual and Verbal (a) Visual: Remember best what they see: pictures, diagrams, flow charts, timelines, films, and demonstrations. Most people are visual learners. (b) Verbal: Get more out of words—written and spoken explanations. 4. Sequential and Global (a) Sequential: Tend to gain understanding in linear steps, with each step following logically from the previous one. Tend to follow logical stepwise paths in finding solutions. (b) Global: Global learners tend to learn in large jumps, absorbing material almost randomly without seeing connections, and then suddenly “getting it.” Able to solve complex problems quickly or put things together in novel ways, but they have difficulty explaining how they did so.

9.5.3

ATLAS learning style

The ATLAS learning model can be identified by encouraging the student to answer questions that help identify the student’s learning preferences. ATLAS learning model categories are as follows: 1. Navigator: Focused learner who charts a course for learning and follows it. Initiates a learning activity by looking externally at the utilization of resources that will help them accomplish the learning task and by immediately beginning to narrow and focus these resources. Relies heavily on planning their learning. 2. Problem solver: Relies on critical thinking skills. Like the navigator, the problem solver initiates a learning activity by looking externally at available resources; however, instead of narrowing the options available, they immediately begin to generate alternatives. They do not do well on multiple-choice tests because these limit divergent thinking. 3. Engager: Passionate learner who loves to learn, learns with feeling, and learns best when they are actively engaged in a meaningful manner with the learning task. The key to learning is engagement.

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BrainWorks learning style

The BrainWorks learning style can be identified by encouraging the student to answer questions. BrainWorks tries to determine which hemisphere of the brain is dominant. It also determines whether the learner reacts in a more auditory or visual manner. Each hemisphere of the brain has prescribed functions or specialities. In this manner, the brain avoids duplication of function. Hemispheres always work together so that a combination of the right and left hemisphere in everything is achieved. There is, however, a tendency for one hemisphere to be dominant. BrainWorks learning style categories are: 1. visual vs. auditory; and 2. left vs. right brain hemisphere.

9.6

Proposed recommender system components

The Intelligent LOs Recommender aims to find the most pedagogically suitable LOs for helping students learning a topic, then personalizing the recommended list based on students’ preferences. Thus, the Intelligent LOs Recommender must efficiently analyze newly introduced LOs, then store information about them for further processing and ordering to each student. From a high-level view, the Intelligent LOs Recommender executes through two phases: 1. LOs finding, gathering, and analyzing phase: In this phase, the system completes different data input resources, mainly crawler for supporting open learning environment, digital library data, and students’ learning preferences. Web content can be of different types. Audio and video types are identified and handled via annotations that are managed by instructors and learning specialists. Information extraction techniques are employed in this process to provide further processing of textual LOs. 2. Intelligent personalized supervised LOs recommendation phase: In this phase, the Intelligent LOs Classifier utilizes Neutrosophic Sets and Theory to discover intelligently the degree of relevancy between LOs and certain course specifications. LOs that satisfy a specific item of the course with a certain threshold are then recommended. Those recommendations are not considered valid unless they were approved by the instructor. After improvement, LOs are then ranked based on a final score which is a combination of relevancy degrees and the user’s preferences. Thus, our system guarantees a minimum level of pedagogical and learning quality with a personalization spirit.

Intelligent LOs Recommender phases are interdisciplinary, overlapping and it is not easy to set boundaries between them because they build over each other.

9.6.1

Phase 1: LOs finding, gathering, and analyzing

Extensive amount of data is needed to enable model processing. Different input sources means responsibility in following different input resources and extra effort in completing all input forms. Fig. 9.3 presents the steps taken in this phase to ensure

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Fig. 9.3 LOs finding, gathering, and analyzing (Phase 1).

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optimal data collection. Four external main data sources for our proposed system must be handled carefully: 1. Internet: This provides an open learning environment for LOs via Crawler. 2. Digital library: This presents different representations of different types of LOs. Digital libraries provide exclusive information resources that are higher in quality and effectiveness when compared to LOs coming from the Internet. 3. Student preferences: In order to be able to personalize the system for each student based on her/his preferences, there should be a mechanism to inform students with the importance of building their profiles, and to track this process. 4. Courses’ specifications: The Quality Assurance and Accreditation Project (QAAP) server holds course specifications to be utilized to classify LOs.

The process of document processing (divided into two steps) is intended to make clear the border of each language structure and to eliminate as much as possible the language-dependent factors, tokenization, stop words removal, and stemming. Removing stop words and stemming words are the preprocessing tasks. The documents in text classification are represented by a significant number of features and most of them could be irrelevant, vague, or ambiguous. The steps taken for the feature extractions are as follows: 1. Tokenization: A document is treated as a string, and then partitioned into a list of tokens. 2. Removing stop words: Stop words, such as the, a, and, etc., frequently occur, so the insignificant words need to be removed. 3. Stemming words: Applying the stemming algorithm that converts a different word form into a similar canonical form. This step is the process of conflating tokens to their root form, for example, connection to connect, computing to compute.

Next comes the process of text classification. In this process, a text document may partially match many categories. We need to find the best matching category for the text document. The term frequency-inverse document frequency (TF-IDF) approach is commonly used to weight each word in the text document according to how unique it is. TF/IDF weights are then fed to the Neutrosophic classifier, which specifies to what degree textual content is relevant to a certain category making use of a threshold for more focused and relevant content. Using Neutrosophic Sets and Theory allows the participation of a single LO in different items in course specifications with different membership degrees.

9.6.2

Phase 2: Personalized supervised generated LOs

Fig. 9.4 presents detailed phase activities. By identifying different information resources, and getting them integrated, the Intelligent LOs Recommender can generate a list of LOs that match course specifications based on the former Neutrosophic classifier. The generated list is not submitted directly to students to guarantee minimum levels of quality and accuracy. This list is appended for the instructor to approve/ modify it before submitting it. Once the instructor approves the list, it becomes available to the student. The second phase allows ranking those approved LOs based on their relevancy together with their suitability to students’ preferences.

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Fig. 9.4 Personalized supervised automatically generated LOs list (Phase 2).

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9.7

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Neutrosophic theory in the proposed recommender system

Let ℘ be a list of students such that n 2 N+ is the number of LOs. ℘ ¼ fs1 ,s2 , …, sn g

(9.11)

Let Γ be a list of topics to learn such that m 2 N+ is the topics to learn. Γ ¼ ft1 ,t2 , …,tm g

(9.12)

Let D be a list of LOs such that k 2 N+ is the number of LOs. D ¼ fd1 , d2 , …,dk g

(9.13)

Let R℘Γ be the set of relations between students and topics to learn where   R℘Γ si , tj is the evaluation of the student si who studied the topic tj. The value of   R℘Γ si , tj is either a numeric number or a neutrosophic number which depends on the proposed domain of the problem.     R℘Γ ¼ R℘Γ si , tj : 8i ¼ 1,2,…, n; j ¼ 1,2, …, m

(9.14)

Similarly, let     RΓD ¼ RΓD ti , dj : 8i ¼ 1, 2,…, m; j ¼ 1, 2, …,k

(9.15)

be the set which represents the relationship between the topic to learn and LO   RΓD ti , dj reveals the probability that topic to learn ti is satisfied by LO dj. The purpose of the Intelligent Recommender System is to determine the relationship between the students and the LOs described as     R℘D ¼ R℘D pi , dj : 8i ¼ 1, 2,…, m; j ¼ 1, 2, …,k

(9.16)

  where the value of R℘D pi , dj is either 0 or 1, which indicates that the student si shall study LO loj to learn the required topic or to gain the required skill(s). Mathematically, the problem of the Intelligent Recommender System in eLearning can be thought of as an implication operator given by the mapping 

 R℘Γ ,RΓD ! R℘D

(9.17)

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Technical details and implementation

This section presents deeper technical details and implementation specifications of aspects of the proposed Intelligent LOs Recommender System, and highlights some of the intelligent Microservices. The proposed eLearning model utilizes the novel open source Neutrosophic package presented in El-Ghareeb [8] for Neutrosophic Sets and Theory functionalities. The proposed model utilizes the SVNN and SVNS. Upon starting to build the proposed model, some challenges became clear. Challenges include: 1. The lack of ability to access the Internet in certain occasions. The Internet is not available to all students all the time. 2. The large size of some learning resources, mainly video lectures. 3. LOs’ copyrights preventing them from being uploaded online. 4. A need to provide meanings of learning to students all the time.

Technical implementation consists of three complementary parts: 1. Student desktop application: Available to registered students to download, including instructor-recommended LOs for course topics based on student registration information. This will be the focus of this part. 2. Student eLearning environment and adaptive features: The adaptive learning management system (LMS) website where the student registers, download the application, update profile and learning preferences, and connects the application so the student’s usage and learning data is synchronized automatically, and the desktop application itself where the student attends the Adaptive eLearning Model and use the different services and features. 3. Instructor portal: The adaptive LMS administration where instructors manage students, courses, topics, LOs, instructors’ data, and other system configurations that affect the learning process.

9.8.1

Students’ manager service

The students’ manager service enables different systems to manage students’ data. Students’ data include learning preferences, learning profiles, timetables, and students’ usage data. The students’ manager service includes three inner services: students’ general data manager, students’ learning profile manager, and students’ usage data manager. While the students’ general data manager handles basic create, retrieve, update, and delete (CRUD) operations for basic information like username, password, email, and other general data; the students’ learning profile manager and students’ usage data manager are of more pedagogical importance to the proposed model. Fig. 9.5 presents the required database tables to support the students’ learning profile manager service. Four different learning profiles are available for each student: General, Felder, ATLAS, and BrainWorks. Fig. 9.6 presents the general profile manager screen. Fig. 9.7 presents the Felder learning profile manager and displays a sample of Felder questions.

StudentBrainWorksLearningModelView

Student PK

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StudentFelderLearningModelView STDID FullName ActiveVSReflective SensingVSIntuitive VisualVSVerbal SequentialVSGlobal

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STDID FLMQID

FLMQID Category Question OptionA OptionB Category

TotalActive TotalReflective TotalSensing TotalIntuitive TotalVisual TotalVerbal TotalSequential TotalGlobal

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StudentBrainWorksLearningModelQuestions ATLASLearningModelQuestions PK

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STDID FLMQID

PK

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Question OptionA OptionB OptionC OptionACategory OptionBCategory OptionCCategory

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Choice Choice

Fig. 9.5 Students’ learning profile manager database tables.

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Fig. 9.6 Welcome message indicating the need to build a learning profile.

Fig. 9.7 Student learning profile manager screen: general learning profile.

Fig. 9.8 presents the Felder learning style report. Felder proposes a calculation method to identify to which category each student belongs. Fig. 9.9 presents an instance of the Felder report. Fig. 9.10 presents the ATLAS learning style profile manager. The ATLAS learning style does not rely heavily on questions as Felder does. However, it is a step-by-step series of questions that are capable of identifying

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Fig. 9.8 Student learning profile manager screen: Felder learning profile. 191

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Fig. 9.9 Student learning profile manager screen: Felder report.

students’ learning styles at the end. Fig. 9.11 displays a BrainWorks questions sample, and Fig. 9.12 displays the BrainWorks report. The BrainWorks learning style identifies two aspects for each student: auditory or visual, and left or right brain hemisphere directed. The proposed eLearning system utilizes two phases for the LOs recommendation: Intelligent LOs Classification and Intelligent LOs Recommendation. Both phases are provided by stand-alone Microservices that can be integrated in different LMSs.

9.8.2

Students’ usage data manager

The students’ usage data manager keeps track of three students’ behavior that is used to adjust the overall recommender system performance and behavior. Fig. 9.13 presents database tables to support this service. Usage data include the following: 1. Browsing behavior: The Intelligent LOs Recommender depends on students browsing through different LOs for initiating relations between different LOs. The tracking referrer and target URL help the Intelligent LOs Recommender based on students’ browsing to relate LOs together.

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Fig. 9.10 Student learning profile manager screen: ATLAS learning profile.

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Fig. 9.11 Student learning profile manager screen: BrainWorks learning profile. Fig. 9.12 Student learning profile manager screen: BrainWorks report.

2. LOs study time: Among specifications that instructors associate with LOs, they identify the time needed to study it. However, that time might differ from one student to another based on their personal differences. Recording the time taken by a student to study a certain LO is very important for the system. It helps both the student and the system. 3. LOs ranking: Students can express their thoughts about each LO in any of three ways: like/ not like, starring the LO from 1 to 5, or leaving a comment/feedback about it.

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Student

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Fig. 9.13 Students’ usage manager database tables.

9.9

Case study

With the availability of 221 unique LOs varying from word documents (.txt, .doc, .docx, .pdf ) to presentations (.ppt, .pptx), a service read the contents of digital libraries in the previous formats and extracted the contents to be processed further. Processing those 221 files yielded 1623 high-quality keywords when compared to those generated from the web pages. The Intelligent LOs Recommender includes different modules and services, which are Pending LOs for Recommendation Manager Module, Crawler Module, and Document Processor Service.

9.9.1

Pending LOs for recommendation manager module

Pending LOs represent LOs that a crawler has collected information about through querying online search engines, but have not been processed yet. They are marked as pending, waiting for processing to be either stored or deleted. Fig. 9.14 presents the Pending LOs for Recommendation Manager database tables that store collected information temporarily about the two LO types: files and online resources.

9.9.2

Crawler module

A crawler is a program that searches for information on the web, and is widely used by web search engines to index all the pages on a site by following the links from page to page. Based on the crawler model, we developed a web-based java crawler that reads Google Search results via an open source library, that is, Selenium. Working on the knowledge domain of information systems analysis and design, we ran the search for queries including different combinations of the keywords: System + Analysis + Design + Tutorial. The crawler extracted three main pieces of information for

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Fig. 9.14 Learning objects recommendation process database tables.

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com.application.Crawler Main +main(args : String [ ]) : void -getDateTime() : String

MySQLConnector +main(args: String [ ]) : void LoadDriver

~conn : Connection = null ~stmt : Statement = null ~rs : ResultSet = null +main(args : String []) : void +insertData(Title : String, Description : String, URL : String, Date : String) : void

Fig. 9.15 Crawler package diagram.

each search result: Title, URL, and Description. The database includes the summary of 2236 records, yielding 161 unique stemmed keywords. Fig. 9.15 presents a crawler module package diagram. Challenges faced in this process included the following: l

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9.9.3

Document processor service

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9.9.3.1 Tokenizer module The document processor service tokenizes sentences into words based on any of the following symbols: dot, comma, space, semicolon, and colon. It takes the String to tokenize as an input and returns a list of tokenized words. Those words will be the input to the next module: removing stop words.

9.9.3.2 Removing stop words module Stop words are words which are filtered out after Tokenizer module processing of text. There is no one definite list of stop words which all tools use. Observations lead to a customized list of stop words that includes the following list as a sample. Course Objectives usually include words like those in the following list. Listing 9.1 presents a sample of stop words encountered in the proposed Recommender system.

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Listing 9.1 Stop words sample stopwords = {"details", "course", "lecture", "explains", "discusses", "outside", "inside", "curriculum", "syllabus", "explain", "discuss"}

9.9.3.3

Stemmer module

Fig. 9.16 illustrates the Stemmer module pseudo code and presents the conducted activities. Stemmer module activities include the following: l

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9.9.3.4

Query expansion module

WordNet is a large lexical database of English used to find synonyms of stemmed keywords generated by the Stemmer module to be further used in the search and recommendation process in later services/modules. Fig. 9.17 presents WordNet relational database tables implementation to provide Query Expansion capabilities to the Adaptive eLearning Model. Each set of synonyms (synset) has a unique index and shares its properties, such as a dictionary definition (lemma). Query expansion takes place as follows: 1. Take input (the term to expand) 2. Search WordNet synsets for the term. When Found, Return Synonyms to Expand Query

9.10

Evaluation and optimization

Applying the presented solution yielded some challenges and bottlenecks that need optimization. Optimization in this phase focuses on the system performance. The Intelligent LOs Recommender is evaluated from a performance perspective in order to accelerate overall system performance.

9.10.1 Intelligent LOs recommender challenges The Intelligent LOs Recommender was tested on different files (221) and online LOs (342). File LOs generated 7388 tokenized term frequencies (TFs) and online LOs generated 169,876 TFs. Challenges in designing and implementing the Intelligent LOs Recommender include the following: 1. Identifying seeds: With the increasing quantity of online content, defining seeds for the crawler is an important task.

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Fig. 9.16 Stemmer module pseudo code. 2. Identifying online and offline phases: Identifying bottlenecks in the Intelligent LOs Recommender performance is an important issue to avoid dead ends and long times of processing that affects systems. 3. Evaluating accuracy of generated terms: Generated terms should be evaluated to avoid nonrelated terms.

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Fig. 9.17 WordNet relational database tables.

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9.10.2 Evaluation results Unleashing an online crawler to collect different LOs available online, and storing metadata about them in an offline database, with URLs made available for later visits of the Intelligent LOs Recommender service was the first task achieved in building this service. Other tasks include: visiting those URLs later, retrieving the LOs, tokenizing and stemming, calculating TFs, and storing calculated TFs in the database for later matching with course objectives. Random LO groups of the crawler’s results are used as the test set, with a capacity of 254 LOs. Reading times in seconds for each of those LOs are presented in Fig. 9.18. Table 9.1 presents a summary of the main statistical measures of LOs’ reading times in seconds. Optimizing LOs’ retrieval can be done through increasing network bandwidth and the server’s memory that affects window sizing. Tokenization duration for retrieved LOs is presented in Fig. 9.19, followed by Table 9.2 summarizing the main statistical measures of tokenizing LOs’ times in seconds. Tokenization duration falls below half a second at its worst case, and it is believed that tokenization is in an optimized form. TFs calculation time for each LO is presented in Fig. 9.20, followed by Table 9.3, which highlights a summary of the main statistical measures of LOs’ TFs calculation time in seconds. TF processing range from fractions of milliseconds to almost 3 seconds for the LOs. TF processing is efficient enough to be utilized in the recommendation process. The first performance bottleneck appears in extracted terms database insertion times. Fig. 9.21 presents LOs’ keywords insertion time in seconds. As summarized

Fig. 9.18 Learning objects’ reading time in seconds.

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Table 9.1 Summary of the main statistical measures of learning objects’ reading times in seconds. Min. Max. Range Mean Mode Median

0.819568872 38.5457058 37.72613692 5.108815395 N/A 3.159333944

Fig. 9.19 Learning objects’ tokenization duration in seconds.

Table 9.2 Summary of the main statistical measures of learning objects’ tokenization times in seconds. Min. Max. Range Mean Mode Median

0.000496149 0.461540937 0.461044788 0.051718733 N/A 0.026743531

in Table 9.4, the average keywords insertion time for LOs is 373 seconds, with the worst cases exceeding 2536 seconds. Such a performance issue is not accepted and leads to dead ends in the system. It takes a lot of time to insert keywords in the database for later recommendation processing.

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Fig. 9.20 Learning objects’ term frequencies calculation times in seconds. Table 9.3 Summary of the main statistical measures of TF calculation time in seconds. Min. Max. Range Mean Mode Median

6.48499E05 2.913298845 2.913233995 0.095572824 0.001230955 0.015192032

Fig. 9.21 Learning objects’ keywords insertion times in seconds.

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Table 9.4 Summary of the main statistical measures of learning objects’ keywords insertion time in seconds. Min. Max. Range Mean Mode Median

4.576031208 2536.919661 2532.34363 373.0938357 N/A 260.3440971

Fig. 9.22 Learning objects’ tokenized no. of words versus total no. of words.

Fig. 9.22 compares the percentages that the tokenized number of words and the total number of words contribute to the total. The Tokenized number of words when compared to total number of words does not exceed 15%. One challenge with online LOs is the tremendous amount of Hyper Text Markup Language (HTML) used for web-based user interface. The tokenization process is responsible for handling this challenge. Regular expressions (RE) are used to extract text from online LOs. Tables 9.5 and 9.6 present statistical measures about the total and tokenized number of words, respectively. Though the tokenized number of words percentage when compared to total number of words does not exceed 15%, there is still a challenge facing our proposed Intelligent LOs Recommender which is the tremendous amount of extracted keywords that are not related to course objectives. Processing books, for example, is a significant challenge. One of the processed books was read in less than 0.25 seconds with 12,522,837 original words number, generated 258,457 tokenized words (1.6 MB of data) in

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Table 9.5 Summary of the main statistical measures of learning objects’ original no. of words count. Min. Max. Range Mean Mode Median

5 66,491 66,486 7238.443137 1085 3803

Table 9.6 Summary of the main statistical measures of learning objects’ tokenized no. of words count. Min. Max. Range Mean Mode Median

0 9737 9737 912.6745098 121 469

2188 seconds for tokenization processing, and had a TF calculation of 521 seconds. Such processing never finished uploading extracted keywords into the database. The situation changes a lot when adding the course objectives into inputs. Course objectives’ extracted keywords and expanded by WordNet to increase the system’s efficiency percentage when compared to the number of tokenized words and total number of words are presented in Fig. 9.23. Handling more than the keywords really needed yields a significant performance degradation that we can remove by presenting a design solution and taking the decision of including course objectives as an input parameter, thus overcoming the performance bottleneck. Fig. 9.24, followed by Table 9.7, presents the optimization gained in insertion.

9.10.3 Comments on results and optimized solution Optimizing and enhancing the Intelligent LOs Recommender can be done through different design decisions: 1. Taking course objectives into consideration while determining crawler seeds: A preprocessing step that processes course objectives and generates search keywords that are used to find related websites using one of the many internet search engines helps in finding more near and accurate seeds for the crawler. In our proposed model, we developed a Java-based crawler that takes course objectives keywords as input, uses the Google Search engine to invoke the search query, loops through search results, and stores metadata about found URLs in the database.

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80% 60% 40% 20% 0% 1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241 251

Objectives no. of words vs. tokenized no. of words vs. total no. of words

100%

Learning object ID Course objectives no. of words

Tokenized no. of words

Total no. of words

Fig. 9.23 Course objectives’ extracted and expanded keywords vs. tokenized no. of words vs. total no. of words.

Fig. 9.24 Learning objects’ optimized keywords insertion times in seconds.

Table 9.7 Summary of the main statistical measures of learning objects’ keywords insertion time in seconds. Min. Max. Range Mean Mode Median

0 132 132 124 N/A 74

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2. Query expansion: To increase the accuracy of search terms resulting from processing course objectives and specifications, query expansion methods can be utilized. WordNet is a lexical database for the English language that groups English words into sets of synonyms to provide short, general definitions, and records the various semantic relations between these synonym sets. Extracted keywords are expanded by WordNet synonyms and then passed to the crawler when seeking seeds. 3. Taking course objectives into consideration while calculating TFs for LOs: The main objective of the Intelligent LOs Recommender is to contribute intelligently in personalizing the learning experience for the student by recommending the most accurate LOs, not indexing the online LOs. There is thus no need to get bogged down in analyzing what does not matter for the recommendation process. An enhanced solution will not calculate frequencies for terms that do not exist in course objectives and will not store TFs at all. The processing time and cost are much lower when compared to the storage cost.

9.10.4 Evaluation of the optimized solution The Presented Intelligent Microservices will be evaluated from different perspectives. Evaluation perspectives include: 1. Information Retrieval (IR) evaluation of Intelligent LOs Recommender; and 2. Intelligent LOs Classifier evaluation.

The following sections present these evaluation aspects and results.

9.10.4.1 Information retrieval evaluation One of the fundamental problems in IR is the ranking problem, ordering the results of a query such that the most relevant results show up first. Ranking algorithms employ scoring functions that assign scores to each result of the query at hand. Therefore, ranking the results of a query consists of assigning a score to each result and then sorting the results by score, from highest to lowest. Many performance measures are used by the IR community to evaluate the effectiveness of ranking functions. In order to measure the relevance of the Ranked Recommended LOs list, three performance measures are used: Precision, Recall, and F-measure. Precision measures the ratio of relevant documents within a given number of documents returned to the number of returned documents. The second performance measure is Recall. Recall is defined as the number of relevant documents retrieved by a search divided by the total number of existing relevant documents (which should have been retrieved). The Recall measure quantifies what fraction of all the relevant results was ranked to fall within the first k documents. Precision and Recall scores are not discussed in isolation. Instead, both may be combined into a single measure, such as the F-measure. This is the weighted harmonic mean of Precision and Recall. Fig. 9.25 shows the Precision measures of the proposed Intelligent LOs Recommender followed by Table 9.8 highlighting a summary of the main statistical measures. Fig. 9.26 shows the Recall measures of the proposed Intelligent LOs Recommender followed by Table 9.9 highlighting a summary of the main statistical measures. Fig. 9.27 shows the F-measure of the proposed Intelligent LOs Recommender followed by Table 9.10 highlighting a summary of the main statistical measures.

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Fig. 9.25 Precision evaluation of proposed intelligent LOs recommender.

Table 9.8 Summary of the main statistical measures of Precision. Min. Max. Variance Mean Mode Median

0.333333333 0.909090909 0.008883086 0.700674926 0.727272727 0.727272727

Fig. 9.26 Recall evaluation of proposed intelligent LOs recommender.

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Table 9.9 Summary of the main statistical measures of Recall. Min. Max. Variance Mean Mode Median

0.434782609 0.434782609 0.005640442 0.31440623 0.347826087 0.347826087

Fig. 9.27 F-measure evaluation of proposed intelligent LOs recommender. Table 9.10 Summary of the main statistical measures of F-measure. Min. Max. Variance Mean Mode Median

0 0.588235294 0.04558944 0.308731541 0.470588235 0.441176471

9.10.4.2 Intelligent LOs classifier evaluation The presented Intelligent LOs Classifier is evaluated to check its accuracy and capability to classify unclassified documents. The Intelligent LOs Classifier implements the Naive Bayes classifier algorithm. The following training set was given to the classifier. The following items present categories and documents for each document that the classifier was trained on.

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1. Bioinformatics (a) biology-inspired computation (b) dynamical systems in neuroscience (c) molecular analysis of cancer (d) probabilistic models of the brain perception and neural function 2. BPM (a) BPM success: How a travel giant turned its ship around CIO 3. Computer networks (a) Interconnecting CISCO network devices Part 1 (b) Interconnecting CISCO network devices Part 2 (c) Portable command guide 4. eLearning (a) 101 free eLearning tools (b) Adaptive and personal LMS (c) eLearning technologies (d) Questionmark-tools-effective-assessments (e) SCROM-v1 5. English (a) New interchange 1-key (b) New interchange 1-student book (c) New interchange 1-workbook (d) Intro Workbook 3rd edition 6. Programming (a) ASPNet in VB (b) ASPnet MVC (c) Applied numeric methods using Matlab (d) Essential Matlab for engineers and scientists (e) Visual Basic 2008 7. SOA (a) SOA description (b) SOA lab setup guide (c) SOA release notes (d) SOA design patterns (e) SOA instructor exercises guide (f ) SOA lab setup guide classroom (g) SOA lab setup guide online (h) SOA student exercises (i) SOA student book

The following list presents the results of testing. The testing set is presented to the trained classifier. It displays the document under test, and the class percentage for the given document, followed by the final result. The Intelligent LOs Classifier resulted in 100% accuracy. 1. Document: eLearning by design (a) English: NaN (b) eLearning: 17,510.210584564997 (c) Bioinformatics: 20,216.649226425725 (d) BPM: 17,698.0930699629 (e) Computer Networks: 20,366.694928274974

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2.

3.

4.

5.

6.

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(f ) SOA: 19,893.456729741945 (g) Programming: 22,055.679402138958 (h) Decision eLearning Document: effective eLearning environment personalization using web usage mining technology (a) English: NaN (b) eLearning: 21,879.011486226136 (c) Bioinformatics: 25,730.60082626081 (d) BPM: 25,055.906020316877 (e) Computer Networks: 25,432.70422899807 (f ) SOA: 25,199.419281366278 (g) Programming: 28,851.363753332593 (h) Decision eLearning Document: CCNP2 (a) English: NaN (b) eLearning: 769,103.2978993196 (c) Bioinformatics: 839,262.6497608843 (d) BPM: 695,368.4959748124 (e) Computer Networks: 610,877.2165053524 (f ) SOA: 784,774.4069853874 (g) Programming: 855,032.0508602582 (h) Decision Computer Networks Document: Forrester BPM Wave (a) English: NaN (b) eLearning: 83,003.91116782578 (c) Bioinformatics: 94,739.36297244373 (d) BPM: 78,479.22990639707 (e) Computer Networks: 89,490.77817552155 (f ) SOA: 84,576.42321484128 (g) Programming: 101,919.05505022638 (h) Decision BPM Document: Network Fundamentals (a) English: NaN (b) eLearning: 31,300.31339492937 (c) Bioinformatics: 33,352.432161580524 (d) BPM: 29,577.875485405537 (e) Computer Networks: 29,199.402824383295 (f ) SOA: 32,654.278796223974 (g) Programming: 36,143.44494075163 (h) Decision Computer Networks Document: Prentice.Hall.SOA.Principles.of.Service.Design.Jul.2007 (a) English: NaN (b) eLearning: 810,529.5972492553 (c) Bioinformatics: 991,026.2503207972 (d) BPM: 962,086.7346663276 (e) Computer Networks: 933,969.3054615034 (f ) SOA: 730,561.9506475903 (g) Programming: 1,049,997.9439706628 (h) Decision SOA

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Conclusion

This chapter presented an Intelligent LOs Recommender that uses Neutrosophic Sets and Theory to overcome intelligently the vagueness, indeterminacy, and missing data of the LO, and of the topic to learn, and the course specification to determine the suitability of an LO with course objectives and topic to learn. Components of the presented Intelligent LOs Recommender system are: crawler, intelligent recommender microservice, data repositories that hold course specifications, student learning preferences, and approved LOs lists. The proposed Intelligent LOs Recommender initially faced some performance challenges due to the tremendous amount of generated keywords that affect processing and insertion times. Including course objectives as an input parameter in the keywords term frequency processing optimizes the performance by focusing on the important and needed keywords instead of wasting the processing time and storage spaces for nonimportant keywords. The presented Intelligent LOs Recommender System is evaluated from different perspectives: performance and IR. Performance measurements were addressed on a Microservice level focusing on the Intelligent LOs Recommender as it has to deal with tremendous amounts of data, showing enhancements in Intelligent LOs Recommender performance measures after adopting the presented optimization techniques. Intelligent LOs Recommender performance increased almost 10 times and is capable of analyzing the presented LOs, extracting the relevant keywords, and expanding the queries based on course specifications. IR measures of proposed Intelligent LOs Recommender show an achievement in Precision measure, with challenges at Recall and F-measure due to the increased number of relevant LOs as a result of including the course objectives at the crawling phase. That means that almost all of the LOs stored in the database are already relevant. The Intelligent LOs Classifier is evaluated showing high accuracy levels in classifying the test set.

References [1] McKinsey & Company, How Retailers Can Keep Up With Consumers (Online), 2019. Available from: https://www.mckinsey.com/industries/retail/our-insights/how-retailerscan-keep-up-with-consumers (Accessed 7 July 2019). [2] C. Anderson, M.P. Andersson, Long Tail, Bonnier Fakta, 2004. http://arlt-lectures.com/ The-Long-Tail.pdf. _ Otay, C. Kahraman, A State-of-the-Art Review of Neutrosophic Sets and Theory, [3] I. Springer International Publishing, Cham, 2019, pp. 3–24. https://doi.org/10.1007/ 978-3-030-00045-5_1. [4] N.A. Nabeeh, F. Smarandache, M. Abdel-Basset, H.A. El-Ghareeb, A. Aboelfetouh, An integrated neutrosophic-TOPSIS approach and its application to personnel selection: a new trend in brain processing and analysis, IEEE Access 7 (2019) 29734–29744, https://doi.org/10.1109/ACCESS.2019.2899841. [5] N.A. Nabeeh, M. Abdel-Basset, H.A. El-Ghareeb, A. Aboelfetouh, Neutrosophic multicriteria decision making approach for IoT-based enterprises, IEEE Access (2019) 59559–59574, https://doi.org/10.1109/ACCESS.2019.2908919.

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[6] M. Abdel-Basset, N.A. Nabeeh, H.A. El-Ghareeb, A. Aboelfetouh, Utilising neutrosophic theory to solve transition difficulties of IoT-based enterprises, Enterp. Inf. Syst. (2019), https://doi.org/10.1080/17517575.2019.1633690. [7] M. Ali, L.H. Son, N.D. Thanh, N.V. Minh, A neutrosophic recommender system for medical diagnosis based on algebraic neutrosophic measures, Appl. Soft Comput. 71 (2018) 1054–1071, https://doi.org/10.1016/j.asoc.2017.10.012. [8] H.A. El-Ghareeb, Novel open source python neutrosophic package, Neutrosophic Sets Syst. 25 (2019) 136–160, https://doi.org/10.5281/zenodo.2631514. [9] J. Chen, J. Ye, Some single-valued neutrosophic Dombi weighted aggregation operators for multiple attribute decision-making, Symmetry 9 (6) (2017) 82. [10] P. Liu, X. Zhang, Some Maclaurin symmetric mean operators for single-valued trapezoidal neutrosophic numbers and their applications to group decision making, Int. J. Fuzzy Syst. 20 (1) (2018) 45–61. [11] J. Ye, Multiple-attribute decision-making method under a single-valued neutrosophic hesitant fuzzy environment, J. Intell. Syst. 24 (1) (2015) 23–36. [12] J.-Q. Wang, Y. Yang, L. Li, Multi-criteria decision-making method based on singlevalued neutrosophic linguistic Maclaurin symmetric mean operators, Neural Comput. Appl. 30 (5) (2018) 1529–1547. [13] X. Zhang, C. Bo, F. Smarandache, C. Park, New operations of totally dependentneutrosophic sets and totally dependent-neutrosophic soft sets, Symmetry 10 (6) (2018) 187. [14] E. Szmidt, J. Kacprzyk, Intuitionistic fuzzy sets in some medical applications, in: Fuzzy Days, 2001. [15] N.D. Thanh, M. Ali, A novel clustering algorithm in a neutrosophic recommender system for medical diagnosis, Cogn. Comput. 9 (4) (2017) 526–544. [16] L.H. Son, N.T. Thong, Intuitionistic fuzzy recommender systems: an effective tool for medical diagnosis, Knowl. Based Syst. 74 (2015) 133–150, https://doi.org/10.1016/j. knosys.2014.11.012. [17] M. Ali, N.D. Thanh, N. Van Minh, A neutrosophic recommender system for medical diagnosis based on algebraic neutrosophic measures, Appl. Soft Comput. 71 (2018) 1054–1071. [18] N.D. Thanh, M. Ali, Neutrosophic recommender system for medical diagnosis based on algebraic similarity measure and clustering, in: 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), IEEE, 2017, pp. 1–6. [19] J.-Q. Wang, X. Zhang, H.-Y. Zhang, Hotel recommendation approach based on the online consumer reviews using interval neutrosophic linguistic numbers, J. Intell. Fuzzy Syst. 34 (1) (2018) 381–394. [20] S. Jha, R. Kumar, J.M. Chatterjee, M. Khari, N. Yadav, F. Smarandache, Neutrosophic soft set decision making for stock trending analysis, Evol. Syst. (2018) 1–7. [21] A. Riad, H.K. El-Minir, H.M. El-Bakry, H.A. El-Ghareeb, Supporting online lectures with adaptive and intelligent features, Adv. Inf. Sci. Serv. Sci. 3 (1) (2011) 47. [22] H. El-Ghareeb, A. Riad, Empowering adaptive lectures through activation of intelligent and Web 2.0 technologies, Int. J. E-Learn. 10 (4) (2011) 365–391.

Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology

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Fatimah M. Mohammed a, Wadei AL-Omeri b a College of Education for Pure Sciences, Tikrit University, Tikrit, Iraq, bDepartment of Mathematics, Al-Balqa Applied University, Salt, Jordan

10.1

Introduction

The term fuzzy sets (FS, for short) was introduced in the classic paper of Zadeh in 1965 [1]. The fuzzy set theory was subsequently investigated by many researchers. Intuitionistic fuzzy sets (IFS) one of the extension sets was defined by K. Atanassov in 1983 [2–4], when the fuzzy set gave the degree of membership function of an element in the sets. Then, the IFSs gave a degree of membership function and a degree of nonmembership function. After that several researches were conducted on the generalizations of the notion of IFSs. The concept of neutrosophy, neutrosophic sets (NSs), and neutrosophic components were studied by Smarandache in 1999 [5]. The concept of a NS and neutrosophic topological space (NTS) was defined by Salama and Alblowi [6]. Recently, Al-Omeri, Smarandache, and Jafari [7–9] introduced and studied the notions of openness, continuity, semiopenness, precontinuity, and irresoluteness and preirresoluteness degree of functions in an NTS. In 2013, Arockiarani et al. [10] defined the fuzzy NS. In 2014, Arockiarani and Martina Jency [11] studied the fuzzy neutrosophic topological space (FNTS). The fuzzy NSs are defined with membership, nonmembership, and indeterminacy degrees. In 2017, Arockiarani and Martina Jency [11] introduced the fuzzy neutrosophic continuous function. In this work, we generalize the studying of the last year (2018) when Mohammed and Matar [12, 13] introduced fuzzy neutrosophic αm-closed sets and new types of continuity in FNTSs. Finally, some new relationships between the defined functions with comparative among them have been identified.

10.2

Preliminaries

Throughout this chapter, X and Y denote nonempty sets and FNS is the fuzzy NS on the closed unit interval I ¼ [0, 1]. The smallest and greatest members in FNS are denoted by 0N and 1N, respectively. In this section, we present some basic concepts of Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00010-X © 2020 Elsevier Inc. All rights reserved.

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intutionistic fuzzy sets, fuzzy NSs, and some operators on fuzzy NSs with the definitions of FNτ0,1 and FNτ0,2 spaces has been reviewed. Definition 10.1 (see Atanassov [3]). An intutionistic fuzzy set λ on a nonempty set X is an object of the form λ ¼ {hx, μλ(x), νλ(x)i : x 2 X}, where μλ(x) 2 [0, 1] is called the “degree of membership of x in λ,” and νλ(x) 2 [0, 1] is called the “degree of nonmembership of x in λ,” where μλ(x) and νλ(x) satisfy the following condition: For all x 2 X, we have 0  μλ ðxÞ + νλ ðxÞ  1: The fuzzy NS was introduced by Arockiarani et al. [10] as follows. Definition 10.2 (see Arockiarani et al. [10]). Let X be a nonempty fixed set. A fuzzy NS (FNS), λfn, is an object having the form λfn ¼ {hx, αλfn(x), βλfn(x), γ λfn(x)i : x 2 X} where the functions αλfn, βλfn, γ λfn : X ! [0, 1] denote the degree of membership function (namely αλfn(x)), the degree of indeterminacy function (namely βλfn(x)), and the degree of nonmembership (namely γ λfn(x)), respectively, of each set λfn we have 0  αλfn(x) + βλfn(x) + γ λfn(x)  3, for each x 2 X. The FNS λfn ¼ {hx, αλfn(x), βλfn(x), γ λfn(x)i : x 2 X} can be identified to an ordered triple αλ, βλ, γ λ in [0, 3] on X. In this section, we devoted what to review briefly the notions and the properties of an implication operations on FNS, which are essential for establishing the main point of the article which was listed in the following definition. Definition 10.3 (see Veereswari [14]). Let X be a nonempty set and the FNS’s λfn and μfn be in the form λfn ¼ {hx, αλ(x), βλ(x), γ λ(x)i : x 2 X} and μfn ¼ {hx, αμ(x), βμ(x), γ μ(x)i : x 2 X} on X. Then: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

0fn ¼ hx, 0, 0, 1i and 1fn ¼ hx, 1, 0, 0i; λ  μ iff (for all x 2 X, αλ(x)  αμ(x), βλ(x)  βμ(x), and γ λ(x)  γ μ(x)); λ ¼ μ iff λ  μ and μ  λ; Coλ ¼ λc ¼ 1fn  λ ¼ {hx, γ λ(x), 1  βλ(x), αλ(x)i : x 2 X}; λ [ β ¼ fhx, max ðαλ ðxÞ, αμ ðxÞÞ, min ðβλ ðxÞ,βμ ðxÞÞ, min ðγ λ ðxÞ, γ μ ðxÞÞi : x 2 Xg; λ \ β ¼ fhx, minðαλ ðxÞ,αμ ðxÞÞ, min ðβλ ðxÞ, βμ ðxÞÞ, max ðγ λ ðxÞ, γ μ ðxÞÞi : x 2 Xg; Coλ ¼ λc ¼ {hx, νλ(x), ηλ(x), μλ(x)i : x 2 X}; []λfn ¼ {hx, αλ(x), βλ(x), 1  αλ(x)i : x 2 X}; and hiλfn ¼ {hx, 1  γ λ(x), βλ(x), γ λ(x)i : x 2 X}-1

Definition 10.4 (see Veereswari [14]). A fuzzy neutrosophic topology (FNT) on a nonempty set X is a family τfn of fuzzy neutrosophic subsets in X satisfying the following axioms: (T1) 0fn, 1fn 2 τfn; (T2) λ1 \ λ2 2 τfn for any λ1, λ2 2 τfn; and S (T3) i2Iλi 2 τfn for a family {λiji 2 I} τfn, where I is an index set.

In this case, the pair (X,τfn) is called FNTS. The elements of τfn are called fuzzy neutrosophic open sets (fn-open sets). The complement of an fn-open set in the FNTS (X,τfn) is called a fuzzy neutrosophic closed set (fn-closed set).

Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 217

Definition 10.5 (see Arockiarani and Martina Jency [11]). Let (X, τfn) be FNTS and λfn ¼ hx, αλfn(x), βλfn(x), γ λfn(x)i is FNS in X. Then, the fuzzy neutrosophic-closure (FNCl) and fuzzy neutrosophic-Interior of λfn (FNInt) are defined by: T FNClðλfn Þ ¼ fμfn : μfn is an fn  closed set in X and λfn  μfn g, S FNIntðλfn Þ ¼ fμfn : μfn is an fn  open set in X and μfn  λfn g: Note that FNCl(λfn) is an fn-closed set and FNInt(λfn) is an fn-open set in X. Further: (i) λfn is an fn-closed set in X iff FNCl(λfn) ¼ λfn; and (ii) λfn is an fn-open set in X iff FNInt(λfn) ¼ λfn.

Definition 10.6 (see Iswarya and Bageerathi [15]). A fuzzy neutrosophic subset λfn of FNTS (X, τfn) is called: (i) a fuzzy neutrosophic-clopen set (fn-clopen) if λfn is an fn-closed set and an fn-open set at the same time; (ii) a fuzzy neutrosophic preopen set (fnp-open) if λfn  FNInt(FNCl(λfn)); (iii) a fuzzy neutrosophic preclosed set (fnp-closed) if FNCl(FNInt(λfn))  λfn; (iv) a fuzzy neutrosophic preclopen set (fnp-clopen) if λfn is an fnp-closed set and an fnp-open set at the same time; (v) a fuzzy neutrosophic α-open set (fnα-open) if λfn  FNInt(FNCl(FNInt(λfn))); (vi) a fuzzy neutrosophic α-closed set (fnα-closed) if FNCl(FNInt(FNCl(λfn)))  λfn; T (vii) FNPClðλfn Þ ¼ fμfn : μfn is an fnp  closed set in X and λfn  μfn g; and S (viii) FNPIntðλfn Þ ¼ fμfn : μfn is an fnp  open set in X and μfn  λfn g.

Note: Every fuzzy neutrosophic open set is a fuzzy neutrosophic preopen set. Definition 10.7 (see Veereswari [14]). Let (X, τfn) be an FNTS on X. We can then also construct several FNTS on X in the following way: (i) FNτ0, 1 ¼ {[]ψ fn : ψ fn 2 τfn}, where []ψ fn ¼ hx, αψ , βψ , 1  αψ i; and (ii) FNτ0, 2 ¼ {hiψ fn : ψ fn 2 τfn}, where hiψ fn ¼ hx, 1  γ ψ , βψ , γ ψ i

are FNT on X. Definition 10.8 (see Veereswari [14]). If μfn ¼ {hy, αμ(y), βμ(y), γ μ(y)i : y 2 Y} is FNS in Y, then the inverse image of μfn under ϕ, (ϕ1(μfn)) is FNS in X defined by ϕ1 ðμfn Þ ¼ fhx,ϕ1 ðαμ ÞðxÞ, ϕ1 ðβμ ÞðxÞ, ϕ1 ðγ μ ÞðxÞi : x 2 Xg, where ϕ1(αμ)(x) ¼ αμϕ(x), ϕ1(βμ)(x) ¼ βμϕ(x), and ϕ1(γ μ)(x) ¼ γ μϕ(x). Definition 10.9 (see Salama and Alblowi [6] and Veereswari [14]). Let (X, τX) and (Y, τY) are two FNTS. Then a function ϕ : (X, τX) ! (Y, τY) is called: (i) a fuzzy neutrosophic-continuous (fn-con.) if the inverse image of every fn-open (fn-closed) set in (Y, τY) is an fn-open (fn-closed) set in (X, τX); and (ii) a fuzzy neutrosophic-contra continuous (fn-ccon.) if the inverse image of every fn-open (fn-closed) set in (Y, τY) is an fn-closed (fn-open) set in (X, τX).

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Definition 10.10 (see Mohammed and Matar [13]). (1) Let ðX,τx0, 1 Þ and (Y, τy,0,1) be two constructions of FNTS (X, τX) and (Y, τY), respectively. Then a function ϕ : ðX,τx0, 1 Þ ! ðY,τY0, 1 Þ is called: (i) a fuzzy neutrosophic-τ0, 1 continuous (τ0, 1 con.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 1 Þ is an fn-open (fn-closed) set in ðX,τx0, 1 Þ; and (ii) a fuzzy neutrosophic-τ0, 1 contra continuous (τ0, 1 ccon.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 1 Þ is an fn-closed (fn-open) set in ðX,τx0, 1 Þ. (2) Let ðX,τx0, 2 Þ and (Y, τy,0,2) be two construction of FNTS (X, τX) and (Y, τY), respectively. Then a function ϕ : ðX,τx0, 2 Þ ! ðY,τY0, 2 Þ is called: (i) fuzzy neutrosophic-τ0, 2 continuous (τ0,2 con.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 2 Þ is an fn-open (fn-closed) set in ðX,τx0, 2 Þ; and (ii) fuzzy neutrosophic-τ0, 2 contra continuous (τ0,2 ccon.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 2 Þ is an fn-closed (fn-open) set in ðX,τx0, 2 Þ.

10.3

New types of continuity in FNTSs

We will now introduce a new concept in FNTSs called fuzzy neutrosophic pre-τ0,1 continuous, fuzzy neutrosophic pre-τ0,2 continuous, fuzzy neutrosophic pre-τ0,1 contra continuous, and fuzzy neutrosophic pre-τ0, 2 contra continuous functions when each of τ0,1 and τ0,2 is construction of fuzzy neutrosophic topology. Definition 10.11 Let ðX, τx0, 1 Þ, ðY, τy0, 1 Þ, ðX, τx0, 2 Þ, and ðY, τy0, 2 Þ be PFTSs. Then: (i) a function ϕ : ðX,τx0, 1 Þ ! ðY,τy0, 1 Þ is called a fuzzy neutrosophic pre-τ0,1 continuous function (FNP-τ0, 1-con.) if the inverse image of every fn-open (fn-closed) set in ðY,τy0, 1 Þ is an fnp-open (fnp-closed) set in ðX,τx0, 1 Þ; (ii) a function ϕ : ðX,τx0, 2 Þ ! ðY,τy0, 2 Þ is called a fuzzy neutrosophic pre-τ0,2 continuous function (FNP-τ0,2 con.) if the inverse image of every fn-open (fn-closed) set in (Y, τy0,2) is an fnp-open (fnp-closed) set in ðX,τx0, 2 ); (iii) a function ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is called a fuzzy neutrosophic pre-τ0,1 contra continuous function (FNP-τ0,1 ccon.) if the inverse image of every fn-open (fn-closed) set in (Y, τY0,1) is an fnp-closed set in ðX, τx0, 1 ); and (iv) a function ϕ : ðX,τx0, 2 Þ ! ðY, τy0, 2 Þ is called a fuzzy neutrosophic pre-τ0, 2 contra continuous function (FNP-τ0, 1 ccon.) if the inverse image of every fn-open (fn-closed) set in (Y, τY0,2) is an fnp-closed set in ðX, τx0, 2 ).

Example 10.1 (i) Let X ¼ Y ¼ {x, y} and define FNSs λfn in X and μfn in Y as follows: λfn ¼ hx,ð0:4,0:5Þ,ð0:5, 0:5Þ,ð0:9,0:6Þi: And the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. In addition: μfn ¼ hy,ð0:5, 0:4Þ,ð0:5,0:5Þ,ð0:6, 0:9Þi: The family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y.

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Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðxÞ ¼ y and ϕðyÞ ¼ x: So, μfn is an fn-open set in τY. Then, ϕ1 ðμfn Þ ¼ λfn 2 τX : That is, ϕ1(μfn) is an fnp-open set. Hence, ϕ is FNP-con. function. (ii) Take (i) so from τX we get: The family, τx0, 1 ¼ f0fn , 1fn ,hx, ð0:4, 0:5Þ,ð0:5,0:5Þ, ð0:6, 0:5Þig, is FNT on X. And from τY we get: The family, τy0, 1 ¼ f0fn , 1fn ,hy, ð0:5, 0:4Þ,ð0:5,0:5Þ, ð0:5, 0:6Þig, is FNT on Y. Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as in (i). Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.5, 0.6)i is an fn-open set in τy0, 1 . Then ϕ1 ðμfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 2 FNP  τx0, 1 : So, ϕ1(μfn) is an fnp-open set in τx0, 1 . Hence, ϕ is FNP-τ0,1 con. function. (iii) Again take (i) so from τX we get The family, τx0, 2 ¼ f0fn , 1fn ,hx,ð0:1,0:4Þ, ð0:5, 0:5Þ,ð0:9,0:6Þig to be FNT on X. And from τY we get The family, τy0, 2 ¼ f0fn , 1fn ,hy,ð0:4,0:1Þ, ð0:5, 0:5Þ,ð0:6,0:9Þig to be FNT on Y. And define ϕ : ðX, τx0, 2 Þ ! ðY,τy0, 2 Þ as in (i). So if μfn ¼ hy, (0.4, 0.1), (0.5, q/0.5), (0.6, 0.9)i is an fn-open set in FN-τy0, 2 . Then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þi 2 FNP  τ0, 2 : Hence, ϕ is an FNP-τ0,2 con. function.

Proposition 10.1 Let (X, τX) and (Y, τY) be two FNTSs. If ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is a function, then the following statements are equivalent: (i) ϕ is FNP-τ0,1 con; (ii) f1(μfn) is an fnp-closed set in X for each fn-closed set μfn in Y; and (iii) FNPCl(f1(λfn))  f1(Ufn), wherever λfn  Ufn for each fnα-open set Ufn and fnp-closed set λfn in Y.

Proof. (i) ) (ii) The proof is straightforward. (ii) ) (iii) Let λfn be an fnp-closed set in ðY,τy0, 1 Þ. Then FNPCl(λfn) ¼ λfn is a fuzzy neutrosophic closed set. It follows from (ii) that λfn is an fn-closed set so it is an fnp-closed set in ðX, τx0, 1 Þ. Therefore, ϕ1 ðFNPClðλfn ÞÞ ¼ FNPClðϕ1 ðλfnÞÞ  ϕ1 ðUfn Þ,

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since λfn  Ufn for each fnα-open set Ufn and fnp-closed set λfn in ðY,τy0, 1 Þ. (iii) ) (i) Let λfn be an fn-open set so λfn is an fnp-open set in ðY, τy0, 1 Þ. Then (1fn  λfn) is a fuzzy fnp-closed set in ðY, τy0, 1 Þ, and so by (iii) there exists an fnα-open set Ufn such that FNPClðϕ1 ð1fn  λfn ÞÞ  ϕ1 ðUfn Þ, wherever λfn  Ufn and (1fn  λfn) is an fnp-closed set in ðY,τy0, 1 Þ. However, FNPCl (ϕ1(1fn  λfn)) is an fnp-closed set in ðX,τx0, 1 Þ; in addition, FNPClðϕ1 ð1fn  λfn ÞÞ ¼ 1fn  FNPIntðϕ1 ðλfn ÞÞ and FNPInt(ϕ1(λfn)) is an fnp-open set in ðX, τx0, 1 Þ, and ϕ is an FNP-τ0,1 con. function.□

Theorem 10.1 Let (X, τX) and (Y, τY) be two FNTSs and ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ be a function, so: (i) if ϕ is an FNτ0,1-con., then ϕ is an FNP-τ0,1 con. function for each ϕ1(μfn) is an fnp-clopen set in ðX,τx0, 1 Þ, whenever μfn is an fn-open set in ðY, τy0, 1 Þ; and (ii) if ϕ is an FNτ0,2-con., then ϕ is an FNP-τ0,2 con. function for each ϕ1(μfn) is an fnp-clopen set in ðX,τx0, 2 Þ, whenever μfn is an fn-open set in ðY, τy0, 2 Þ.

Proof. (i) Let ϕ be an FNτ0, 1-con. function and μfn ¼ {hy, αμ(y), βμ(y), γ μ(y)i : y 2 Y}2 τY, so ϕ1 ðμfn Þ ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg, where ϕ1 ðαμ ÞðxÞ ¼ ðαμ ÞϕðxÞ, ϕ1 ðβμ ÞðxÞ ¼ ðβμ ÞϕðxÞ and ϕ1 ðγ μ ÞðxÞ ¼ ðγ μ ÞϕðxÞ are an fn-open set in τX. However, μfn 2 τy0, 1 , so μfn ¼ fhy,αμ ðyÞ, βμ ðyÞ, 1fn  γ μ ðyÞi : y 2 Yg 2 τy0, 1 : Then, by definition of the inverse image, we get ϕ1 ðμfn Þ ¼ fhx,ϕ1 ðαμ ÞðxÞ, ϕ1 ðβμ ÞðxÞ,ϕ1 ð1fn  γ μ ÞðxÞi : x 2 Xg, ¼ fhx,ϕ1 ðαμ ÞðxÞ, ϕ1 ðβμ ÞðxÞ, 1fn  ϕ1 ðγ μ ÞðxÞi : x 2 Xg, which is an fnp-τx0, 1 (by Definition 10.11 (i)).Hence, ϕ is an FNP-τ0,1 con. function. (ii) The proof is the same of proof (i) by taking Let ϕ be an FNτ0,1-con. function and μfn ¼ {hy, αμ(y), βμ(y), γ μ(y)i : y 2 Y}2 τY, so ϕ1 ðμfn Þ ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg,

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where ϕ1 ðαμ ÞðxÞ ¼ ðαμ ÞϕðxÞ,ϕ1 ðβμ ÞðxÞ ¼ ðβμ ÞϕðxÞ and ϕ1 ðγ μ ÞðxÞ ¼ ðγ μ ÞϕðxÞ are an fn-open set in τX. However, μfn 2 τy0, 1 , so μfn ¼ fhy,αμ ðyÞ,βμ ðyÞ, 1fn  γ μ ðyÞi : y 2 Yg 2 τy0, 1 : Then, by definition of the inverse image, we get ϕ1 ðμfn Þ ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ð1fn  γ μ ÞðxÞi : x 2 Xg, ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ, 1fn  ϕ1 ðγ μ ÞðxÞi : x 2 Xg, which is an fnp-τx0, 1 (by Definition 10.11 (i)). Hence, ϕ is an FNP-τ0,1 con. function. (ii) The proof is the same of proof (i) by taking ϕ1 ðμfn Þ ¼ fhx, ϕ1 ð1fn  ðγ μ ÞÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg, ¼ fhx, ð1fn  ϕ1 ðγ μ ÞÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg: □

Remark 10.1 The converse of Theorem 10.1 is not true in general, and we can show this by the following examples. Example 10.2 (i) Let X ¼ Y ¼ {p, q} and define FNSs λ in X and μ in Y as follows: λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi, and the family, τX ¼ {0fn, 1fn, λfn}, is an FNT on X. Now, let μfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:4,0:7Þi with the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:4,0:7Þi 2 τY ,

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then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:7, 0:4Þi 2 τX , which is not in an fnp-open set. Hence, ϕ is not an FNP-τ0,2-con. function. However, from τX we get The family; τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þig, is FNT on X, and from τY we get The family; τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5, 0:4Þ,ð0:5,0:5Þ,ð0:5, 0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðpÞ ¼ q: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 2 τx0, 1 : Hence, ϕ is an FNP-τ0,1 con. function. (ii) Let we put λfn ¼ hx,ð0:2,0:5Þ,ð0:5, 0:5Þ,ð0:8,0:6Þi, where the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X and μfn ¼ hy,ð0:1, 0:6Þ,ð0:5,0:5Þ,ð0:5, 0:9Þi, where the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as in (i) so, if μfn ¼ hy, ð0:2, 0:6Þ, ð0:5,q=0:5Þ, ð0:6, 0:9Þi 2 τY , then ϕ1 ðμfn Þ ¼ hx,ð0:6,0:2Þ,ð0:5, 0:5Þ, ð0:9, 0:6Þi 2 τX : Hence, ϕ is not an FNPp-con. function. However, from τX we get The family; τx0, 2 ¼ f0fn , 1fn ,hx, ð0:1, 0:4Þ, ð0:5,0:5Þ,ð0:9,0:6Þig, is FNT on X, and from τY we get The family; τy0, 2 ¼ f0fn , 1fn ,hy, ð0:4, 0:1Þ, ð0:5,0:5Þ,ð0:6,0:9Þig, is FNT on Y:

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Define ϕ : ðX, τx0, 2 Þ ! ðY,τy0, 2 Þ as in (i). So if, μfn ¼ hy, ð0:4,0:1Þ,ð0:5,0:5Þ,ð0:6, 0:9Þi 2 τy0, 2 , then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ,ð0:5, 0:5Þ, ð0:9, 0:6Þi 2 fnp  τx0, 2 : Therefore, ϕ is an FNP-τ0,2 con. function.

Theorem 10.2 (i) Let (X, τX) and (Y, τY) be two FNTSs; the function ϕ : (X, τX) ! (Y, τY) is an FNP-con. function iff ϕ is FNP-ccon., where ϕ1(η) is an fnp-clopen set in τX for each η 2 τY. (ii) Let ðX,τx0, 1 Þ and ðY,τy0, 1 Þ be two FNTSs; the function ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is an FNP-τ0,1 con. function iff ϕ is FNP-τ0,1 ccon., where ϕ1 ðηÞ 2 τy0, 1 is an fnp-clopen set in τX for each η 2 τy0, 1 in Y. (iii) Let ðX,τx0, 2 Þ and ðY, τy0, 2 Þ be two FNTSs; the function ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is an FNP-τ0, 2 con. function iff ϕ is FNP-τ0,2 ccon., where ϕ1 ðηÞ 2 τy0, 2 is an fnp-clopen set in τX for each η 2 τy0, 2 in Y.

Proof. (i) Let ϕ be an FNP-con. function and ηfn 2 τY. Then, by Definition 10.8, we have ϕ1(ηfn) ¼ ωfn 2 fnp-open set in X. However, ωfn is an fnp-clopen set in X. Therefore, ϕ1(ηfn) ¼ ωfn 2 (1fn- fnp-closed set). Hence, by Definition 10.9, ϕ is an FNP-ccon. function. Conversely, the proof is direct. (ii) Let ϕ be an fnp-τ0,1 con. function. If, ηfn 2 τy0, 1 . Then, by Definition 10.11 (i), ϕ1(ηfn) ¼ ωfn 2 fnp-τx0, 1 space. However, ωfn is an fnp-clopen set in τx0, 1 . So, ϕ1(ηfn) ¼ ωfn 2 (1fn  fnp) set in τx0, 1 . Hence, ϕ is an FNP-τ0,1 ccon. function. Conversely, the proof is direct. (iii) The proof is the same as for (i) and (ii). □

Example 10.3 (i) Let X ¼ Y ¼ {p, q} and define FNSs λfn in X and μ in Y as follows: λfn ¼ fhx, ð0:9,0:6Þ,ð0:5, 0:5Þ, ð0:4,0:5Þig: The family,τx0, 1 ¼ f0fn , 1fn , λfn g, is FNT on X, such that 1  τx ¼ f0fn , 1fn , hx, ð0:4,0:5Þ,ð0:5, 0:5Þ, ð0:9,0:6Þig,

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and βfn ¼ {hy, (0.5, 0.4), (0.5, 0.5), (0.6, 0.9)i}. The family,τy ¼ f0fn , 1fn , βfn g, is FNT on Y: Define ϕ : (X, τx) ! (Y, τy) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If βfn ¼ {hy, (0.5, 0.4), (0.5, 0.5), (0.6, 0.9)i} is an FN-open set in τX, then ϕ1 ðβfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:9, 0:6Þi 2 1  τx : So, ϕ1(βfn) is an FNP-closed set in τx. Hence, ϕ is an (FNP-ccon.) function. (ii) Let X ¼ Y ¼ {p, q} and define FNSs λfn in X and μ in Y as follows: λfn ¼ fhx,ð0:4,0:2Þ,ð0:6, 0:5Þ,ð0:5,0:7Þig: The family,τx0, 1 ¼ f0fn , 1fn , λfn g, is FNT on X: From τX we get The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4, 0:2Þ,ð0:6,0:5Þ,ð0:6, 0:8Þig, is FNT on X, such that 1  τx0, 1 ¼ f0fn , 1fn ,hx,ð0:6, 0:8Þ,ð0:4,0:5Þ,ð0:4, 0:2Þig is FNPT on X, and βfn ¼ fhy,ð0:8, 0:6Þ,ð0:5,0:4Þ,ð0:4, 0:3Þig: The family,τy ¼ f0fn , 1fn , βfn g, is FNT on Y: From τY we get: The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:8, 0:6Þ,ð0:5,0:4Þ,ð0:2, 0:4Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:8,0:6Þ, ð0:5,0:4Þ,ð0:2,0:4Þi is FN  open set on τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:6, 0:8Þ,ð0:4,0:5Þ,ð0:4, 0:2Þi 2 1  τx0, 1 :

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So, ϕ1(ηfn) is FN-closed and FNP-closed sets in τx0, 1 . Hence, ϕ is an (FNP-τ0,1 ccon.) function.

Now, before we ended this section we gave the following Theorem which get the FNP-τ0,1 con. function with FNP-τ0,2 con. functions and FNP-τ0,1 ccon. function with FNP-τ0,2 ccon. functions to be equivalent. Definition 10.12 Let (X, τX) and (Y, τY) be two FNTSs. Then a function ϕ : (X, τX) ! (Y, τY) is called fuzzy neutrosophic perfectly continuous function (FNPT-con.) if the inverse image of every fn-open set in (Y, τY) is an fn-clopen set in (X, τX). Theorem 10.3 Let (X, τX) and (Y, τY) be two FNTSs and the function ϕ : (X, τX) ! (Y, τY) is an FNPT-con. function so: (i) FNPτ0,1-con. and FNPτ0,2-con. functions are equivalent; and (ii) FNPτ0,1-ccon. and FNPτ0,1-ccon. functions are equivalent.

Proof. (i) Necessity: ϕ : (X, τX) ! (Y, τY) is an FNPT-con. function where ϕ is a function between two FNTSs (X, τX) and (Y, τY), and put λfn ¼ {hy, αλ(y), βλ(y), γ λ(y)i : y 2 Y}2 τY. So ϕ1 ðλfn Þ ¼ fhx,ϕ1 ðαλ ÞðxÞ, ϕ1 ðβλ ÞðxÞ, ϕ1 ðγ λ ÞðxÞi : x 2 Xg ¼ fhx,ðαλ ÞϕðxÞ,ðβλ ÞϕðxÞ, ðγ λ ÞϕðxÞi : x 2 Xg: Since, ϕ is an FNPT-con. function, then ϕ1 ðλfn Þ ¼ fhx,ðγ λ ÞϕðxÞ,ðβλ ÞϕðxÞ, ðαλ ÞϕðxÞi : x 2 Xg ¼ fhx,ϕ1 ðγ λ ÞðxÞ, ϕ1 ðβλ ÞðxÞ, ϕ1 ðαλ ÞðxÞi : x 2 Xg: That is ϕ1 ðαλ ÞðxÞ ¼ 1  ϕ1 ðαλ ÞðxÞ 2 FNP  τ0, 1 which means that ϕ is an FNP-τ0, 1 con. function. Also, if we put 1  ϕ1 ðαλ ÞðxÞ ¼ ϕ1 ðγ λ ÞðxÞ, then ψ 1 ðγ λ ÞðxÞ ¼ 1  ϕ1 ðγ λ ÞðxÞ 2 FNP  τ0, 2 , which means that ϕ is an FNP-τ0,2-con. function. That is, FNPτ0,1-con. and FNPτ0,2-con. functions are equivalent. Sufficiency: Let ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ be an FNPτ0,1-con. function and λfn ¼ fhy, αλ ðyÞ,βλ ðyÞ, ð1  αλ ÞðyÞi : y 2 Yg 2 τy0, 1 :

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So, ϕ1 ðλfn Þ ¼ fhx,ϕ1 ðμλ ÞðxÞ,ϕ1 ðβλ ÞðxÞ,ϕ1 ð1  μλ ÞðxÞi : x 2 Xg 2 fnp  τx0, 1 : But ϕ if also an FNPτ0,2-con. function, then ϕ1 ðμλ ÞðxÞ ¼ 1  ϕ1 ðμλ ÞðxÞ ¼ 1  ϕ1 ðγ λ ÞðxÞ ¼ ϕ1 ðγ λ ÞðxÞ: That is, ϕ1(λfn) is an fnp-clopen set in X, which implies that ϕ is an FNPT-con. function. (ii) Similar to proof (i). □

10.4

Interrelations

Fig. 10.1 shows the relationships among different fuzzy neutrosophic functions. None of these implications is reversible where P ) Q represents X implies Y and P
Q represents the negation, as shown by the following examples. Remark 10.2. (i) (ii) (iii) (iv)

Every FNP-con. function is FNP-τ0,1 con. functions. Every FNP-con. function is FNP-τ0,2 con. functions. Every FNP-ccon. function is FNP-τ0,1 con. functions. Every FNP-ccon. function is FNP-τ0,2 con. functions.

The converse of the implications is not true in general and we can show that by the following examples.

Fig. 10.1 Solution 1.

Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 227

Example 10.4 Let X ¼ Y ¼ {p, q} and define FNSs λ in X and μ in Y as follows: (i)

λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi: And the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.4, 0.7)i, with the family, τY ¼ {0fn, 1fn, μfn}, being FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:4,0:7Þi 2 τY , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:7,0:4Þi 2 τX , which is not in an FNP-open set. Hence, ϕ is not an FNP-con. function. However, from τX we get: The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þig, is FNT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY,τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þi 2 τx0, 1 :

Hence, ϕ is an FNP-τ0,1 con. function. (ii) Let λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi: And the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.4, 0.7)i, with the family, τY ¼ {0fn, 1fn, μfn}, being FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p:

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If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:4,0:7Þi 2 τY , then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:7, 0:4Þi 2 τX , which is not in an FNP-open set. Hence, ϕ is not an FNP-con. function. However, from τX we get: The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þig, is FNT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5, 0:4Þ,ð0:5,0:5Þ,ð0:5, 0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 2 τx0, 1 : Hence, ϕ is an FNP-τ0,1 con. function. (ii)

λfn ¼ hx,ð0:2,0:5Þ,ð0:5, 0:5Þ,ð0:8,0:6Þi, where the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X and μfn ¼ hy,ð0:1, 0:6Þ,ð0:5,0:5Þ,ð0:5, 0:9Þi, where the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as in (i) so, if μfn ¼ hy, (0.2, 0.6), (0.5, q/0.5), (0.6, 0.9)i2 τY, then ϕ1 ðμfn Þ ¼ hx, ð0:6, 0:2Þ,ð0:5,0:5Þ, ð0:9, 0:6Þi 2 τX : Hence ϕ is not an FNP-con. function. However, from τX we get: The family,τx0, 2 ¼ f0fn , 1fn ,hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þig, is FNT on X, and from τY we get The family,τy0, 2 ¼ f0fn , 1fn ,hy,ð0:4, 0:1Þ,ð0:5,0:5Þ,ð0:6, 0:9Þig, is FNT on Y:

Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 229

Define ϕ : ðX,τx0, 2 Þ ! ðY, τy0, 2 Þ as in (i). So if μfn ¼ hy,ð0:4,0:1Þ, ð0:5,0:5Þ,ð0:6,0:9Þi 2 τy0, 2 , then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þi 2 fnp  τx0, 2 : Therefore, ϕ is an FNP-τ0,2 con. function. (iii) Let X ¼ Y ¼ {p, q} and define FNSs λ in X and μ in Y as follows: λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi: The family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. 1  τx ¼ f0fn , 1fn , hx, ð0:3,0:6Þ,ð0:5, 0:5Þ, ð0:4,0:5Þig: Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.4, 0.7)i, with the family, τY ¼ {0fn, 1fn, μfn}, being FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:4,0:7Þi 2 τY , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:7,0:4Þi 2 τX , which is not in an FNP-open set. Hence, ϕ is not an FNP-con. function. However, from τX we get The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þig, is FNT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY,τy0, 1 Þas follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þi 2 τx0, 1 ,

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but ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 62 1  τx : Hence, ϕ is an FNP-τ0,1 con but not an FNP-ccon. function. (iv) Let λfn ¼ hx,ð0:2,0:5Þ,ð0:5, 0:5Þ,ð0:8,0:6Þi, where the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X and μfn ¼ hy,ð0:1, 0:6Þ,ð0:5,0:5Þ,ð0:5, 0:9Þi, where the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. 1  τx ¼ f0fn , 1fn , hx,ð0:8,0:6Þ,ð0:5, 0:5Þ,ð0:2,0:5Þig: Define ϕ : (X, τX) ! (Y, τY) as in (i) so, if μfn ¼ hy, (0.2, 0.6), (0.5, q/0.5), (0.6, 0.9)i2 τY. Then, ϕ1 ðμfn Þ ¼ hx, ð0:6, 0:2Þ,ð0:5,0:5Þ, ð0:9, 0:6Þi 2 τX : Hence, ϕ is not an FNP-con. function. However, from τX we get The family,τx0, 2 ¼ f0fn , 1fn ,hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þig, is FNT on X, and from τY we get The family,τy0, 2 ¼ f0fn , 1fn ,hy,ð0:4, 0:1Þ,ð0:5,0:5Þ,ð0:6, 0:9Þig, is FNT on Y:

Define ϕ : ðX, τx0, 2 Þ ! ðY, τy0, 2 Þ as in (i). So if μfn ¼ hy,ð0:4,0:1Þ, ð0:5, 0:5Þ, ð0:6,0:9Þi 2 τy0, 2 , then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ, ð0:5, 0:5Þ,ð0:9,0:6Þi 2 fnp  τx0, 2 , but ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ, ð0:5, 0:5Þ,ð0:9,0:6Þi 62 1  τx : So, ϕ is an FNP-τ0,2 con. but not an FNP-ccon. function. Remark 10.3 The following functions have independent relations: (i) The relations between FNP-τ0,1 con. and FNP-τ0,2 con. functions. (ii) The relations between FNP-ccon. and FNP-τ0,1 ccon. functions.

Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 231

(iii) (iv) (v) (vi) (vii)

The The The The The

relations relations relations relations relations

between between between between between

FNP-τ0,1 ccon. and FNP-τ0,2 ccon. functions. FNP-ccon. and FNP-τ0,2 ccon. functions. FNP-τ0,1 ccon. and FNP-τ0,2 ccon. functions. FNP-τ0,1 ccon. and FNP-τ0,1 con. functions. FNP-τ0,2 ccon. and FNP-τ0,2 con. functions.

We can show these cases by the following examples: Example 10.5 (i), (iii), (v), and (vii) follow from Examples 10.4 and 10.3. (ii) (1) Take Example 10.3 (i). Then, ϕ is an (FNP-ccon.) function. However, ϕ is not an (FNP-τ0,1 ccon.) function. From τX we get The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:9,0:6Þ, ð0:5,0:5Þ,ð0:1,0:4Þig, is FNT on X, such that 1  τx0, 1 ¼ f0fn , 1fn ,hx,ð0:1,0:4Þ, ð0:5,0:5Þ,ð0:9,0:6Þig is FNPT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þi 62 1  τx0, 1 : Hence, ϕ is an (FNP-ccon.) function. However, ϕ is not an (FNP-τ0,1 ccon.) function. (2) Take Example 10.3 (ii). Then, ϕ is an (FNP-τ0,1 ccon.) function. However, ϕ is not an (FNP-ccon.) function. Since 1  τx ¼ f0fn , 1fn , hx, ð0:5,0:7Þ,ð0:4, 0:5Þ, ð0:4,0:2Þig, define ϕ : (X, τx) ! (Y, τy) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:8,0:6Þ, ð0:5, 0:4Þ,ð0:4,0:3Þi 2 τy ,

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then ϕ1 ðηfn Þ ¼ hx,ð0:6, 0:8Þ,ð0:4,0:5Þ,ð0:3, 0:4Þi 62 1  τx0, 1 :

(vi) (1) Take Example 10.3 (2). Then, ϕ is an (FNP-τ0,1 ccon.) function. However, ϕ is not an (FNP-τ0,1 con.) function, since, ϕ1 ðηfn Þ 62 τx0, 1 . (2) Take Example 10.2. Then, ϕ is an (FNP-τ0,1 con.) function. However, ϕ is not an (FNP-τ0,1 ccon.) function, since, ϕ1 ðηfn Þ 62 1  τx0, 1 .

10.5

Conclusion

In this work, we represented many useful basic characterizations and properties of new sets and functions in FNTSs where research is identifying a new class of sets in FNTSs, called fuzzy neutrosophic preopen sets. In addition, some new relationships between the definding sets with comparative studies among them have been established. We then discussed the relationship between different types of continuous functions in FNTSs via the new construction fuzzy neutrosophic pre-τ0,1 and fuzzy neutrosophic pre-τ0,2 spaces. Finally, we think our results can be considered as a generalization of the same results in other kinds of topological spaces. In addition, it is possible to study this topic for a completely distributive De Morgan algebra where the laws of excluded middle in neutrosophic sets and noncontradiction hold with indeterminacy between them.

Future work In this section, the following subjects are suggested for future work as follows: (i) The fuzzy set theory is a main theory in neutrosophic fuzzy topology. Thus one can continue this work by investigating the properties of the fuzzy field based on neutrosophic fuzzy space. The notions such as connectedness and compactness define via fuzzy preopen sets on the new construction fuzzy neutrosophic pre-τ0,1 and fuzzy neutrosophic pre-τ0,2 spaces. (ii) Studying the separation axioms in FNTSs via the new defined concepts. (iii) Investigate the extremally disconnectedness in FNTSs via the new construction, fuzzy neutrosophic pre-τ0,1 and fuzzy neutrosophic pre-τ0,2 spaces. (iv) Finding the undefined relation between fuzzy precontra continuous functions based on τ0,1 and τ0,2 spaces in FNTSs with other fuzzy neutrosophic continuous functions. (v) Use some programs to propose applications on computer sciences by using fuzzy NSs.

Conflict of interests The authors declare that there is no conflict of interests regarding this manuscript.

Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 233

Acknowledgments The authors wish to gratefully acknowledge all those who have generously given their time to referee our chapter.

References [1] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 1822-1190. [2] K. Atanassov, S. Stoeva, Intuitionistic fuzzy sets, in: Polish Syrup. on Interval & Fuzzy Mathematics, Poznan, August 1983, pp. 23–26 [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96. [4] K. Atanassov, Review and new results on intuitionistic fuzzy sets, in: Mathematical Foundations of Artificial Intelligence Seminar, Sofia, 1988, pp. 1–88. [5] F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Research Press, Rehoboth, NM, 1999. [6] A.A. Salama, S.A. Alblowi, Neutrosophic set and neutrosophic topological spaces, IOSR J. Math. 3 (2012) 31–35. [7] W.F. Al-Omeri, S. Jafari, On generalized closed sets and generalized pre-closed sets in neutrosophic topological spaces, Mathematics 1 (2019) 1–12. [8] W.F. Al-Omeri, Neutrosophic crisp sets via neutrosophic crisp topological spaces, Neutrosophic Sets Syst. 13 (2016) 96–104. [9] W.F. Al-Omeri, F. Smarandache, New neutrosophic sets via neutrosophic topological spaces, in: F. Smarandache, S. Pramanik (Eds.), Neutrosophic Operational Research, vol. I, Pons Editions, Brussels, Belgium, 2017, pp. 189–209. [10] I. Arockiarani, I.R. Sumathi, J. Martina Jency, Fuzzy neutrosophic soft topological spaces, IJMA 10 (4) (2013) 225–238. [11] I. Arockiarani, J. Martina Jency, More on fuzzy neutrosophic sets and fuzzy neutrosophic topological spaces, Int. J. Innov. Res. Stud. 3 (2014) 642–652. [12] F.M. Mohammed, S.F. Matar, Fuzzy neutrosophic αm-closed sets in fuzzy neutrosophic topological spaces, Neutrosophic Sets Syst. 21 (2018) 56–65. [13] F.M. Mohammed, S.F. Matar, Some new kinds of continuous functions via fuzzy neutrosophic topological spaces, Tikrit J. Pure Sci. 24 (2019) 118–124. [14] Y. Veereswari, An introduction to fuzzy neutrosophic topological spaces, IJMA 8 (3) (2017) 144–149. [15] P. Iswarya, K. Bageerathi, On neutrosophic semi-open sets in neutrosophic topological spaces, Int. J. Math. Trends Technol. 37 (3) (2016) 214–223.

A solution for the neutrosophic linear programming problem with a new ranking function

11

Majid Darehmiraki Department of Mathematics, Behbahan Khatam Alanbia University of Technology, Behbahan, Iran

11.1

Introduction

Decision-making, artificial intelligence, control theory, management sciences, and job placement interventions are the fields where linear programming problems (LPP) have effective applications. In many practical applications, the available information in the system under consideration is not precise. Therefore, we would like to use concepts to reflect uncertainty that exist in data. In such cases it is better to use LPP with imprecise data. From the time that Zadeh proposed fuzzy theory [1], researchers used it when confronted with uncertainty. Later, when intuitionistic fuzzy theory was introduced by Atanassov [2], some scholars also used it to model uncertainty in optimization problems. However, these two concepts can only investigate incomplete information, not the peddling information and incompatible information which exist generally in belief systems. However, Smarandache [3, 4] solved this problem by adding an independent indeterminacy membership to intuitionistic fuzzy sets and calling it a neutrosophic set. The use of neutrosophic sets is growing rapidly due to their outstanding features. It can therefore be helpful to investigate the knowledge of experts about the parameters of neutrosophic data. A fuzzy linear programming (FLP) model is formulated in one of the following ways: (i) The decision-maker tries to achieve an ideal level for the objective function, which is not directly possible through a crisp linear programming problem. (ii) The constraints of the problem can be ambiguous. In order to overcome this ambiguity, one can consider a tolerance for the violation of the constraints. (iii) The technological coefficients and the right-hand side values can also be fuzzy. In this case, ranking functions can be used to encounter the constraints.

In this chapter, we investigate the models related to the second and third models with neutrosophic data instead of fuzzy data.

Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00011-1 © 2020 Elsevier Inc. All rights reserved.

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11.1.1 Literature review So far, various models of fuzzy linear programming have been proposed and solved. In some of them only the coefficients are fuzzy; in others, the inequality relations are fuzzy; and in some cases they are both fuzzy. The fuzzy linear programming problems in which all the parameters as well as the variables are represented by fuzzy numbers are known as FFLP problems. These issues were initially formulated and solved by Zimmerman. Mahdavi-Amiri and Nasseri employed a dual simplex algorithm for solving FLPs [5, 6]. Considering the acceptance degree of the decision-maker that the fuzzy constraints may violate, Dong and Wang proposed a new method to solve trapezoidal FLPs [7]. Javanmard and Nehi used the nearest interval approximation to solve interval type-2 FLPs [8]. In addition, the authors of [9] proposed a novel method for solving FLPs. Ye first proposed a neutrosophic function involving neutrosophic numbers, then used it for solving neutrosophic linear programming problems (NLPPs) [10, 11]. Abdel-Basset et al. presented a novel technique for neutrosophic LP models, using the trapezoidal neutrosophic numbers as a parameter [12]. Das et al. proposed a method based on the multi-objective linear programming problem and lexicographic ordering method [13]. Khan et al. employed a modified version of the simplex method [14].

11.1.2 The main aim In this chapter, a new linear ranking function is proposed to rank neutrosophic numbers (NNs). This method is then used to solve NLPPs. Because this method leads to ranking NNs in different levels, the decision-maker can therefore make decisions in different levels. For example, decision-making in top levels is done by using ∝ > 0.5, and correspondingly decision-making in lower levels is done by using ∝
0.5. The method presented in this chapter is an extension of the existing procedure in [15].

11.2

Preliminaries

Neutrosophic sets were presented to address the weaknesses of the fuzzy sets. Smarandache [3, 4] defined neutrosophic sets as follows: Definition 11.1. Let X be a universe of discourse, with a class of elements in X denoted by x. A neutrosophic set B in X is summarized by a truth membership function μB(x), an indeterminacy membership function νB(x), and a falsity membership function ηB(x). The functions μB(x), νB(x), and ηB(x) are real standard or nonstandard subsets of ]0, 1+[. That is μB(x) : X ! ]0, 1+[, νB(x) : X ! ]0, 1+[, and ηB(x) : X ! ]0, 1+[. Definition 11.2. Let X be a universe of discourse. A single valued neutrosophic set A over X is an object having the form A ¼ {(x, μA(x), νA(x), ηA(x)) : x 2 X}, where μA(x) : X ! [0, 1], νA(x) : X ! [0, 1], and ηA(x) : X ! [0, 1], with 0
μA(x) + νA(x) + ηA(x)
3 for all x 2 X. The functions μA(x), νA(x), and ηA(x) denote the truth membership degree, the indeterminacy membership degree, and the falsity membership degree of x to A, respectively. For convenience, an SVN number is denoted by A ¼ (a, b, c), where a, b, c 2 [0, 1] and a + b + c
3.

A solution for the neutrosophic linear programming problem with a new ranking function

237

Definition 11.3. A fuzzy number u in parametric form is a pair ðuðr Þ, uðr ÞÞ of functions uðr Þ,uðr Þ, 0
r
ω, which satisfy the following requirements: 1. u is a bounded monotonic increasing left continuous function; 2. u is a bounded monotonic decreasing left continuous function; and 3. u
u,0
r
ω.

ω is an arbitrary constant between zero and one (0 < ω
1). Similar to Definition 11.3, we can imagine a parametric form to SVNN (single valued neutrosophic number). An SVNN y in parametric form is a (uðr Þ,uðr Þ, vðr Þ,vðr Þ,wðr Þ,wðr ÞÞ of functions uðr Þ,uðr Þ, vðr Þ, vðr Þ,wðr Þ,wðr Þ,0
r
1, which satisfy the above requirements. In Fig. 11.1, a parametric fuzzy number can be observed. In special cases, a single valued triangular neutrosophic number is defined as follows. Definition 11.4. Let va ,wa , and ta and a100 , a1, a10 , a2, a3, a40 , a4, a400 2 ℝ such that a100
a1
a10
a2
a3
a40
a4
a400 . Then a single valued trapezoidal neutrosophic 
   number (SVTNN), a ¼ a001 , a1 , a01 , a2 , a3 , a04 , a4 , a004 , va , wa , ta , is a special neutrosophic set on the real line set ℝ, whose truth membership, indeterminacy membership, and falsity membership functions are given as follows:

μa ðxÞ ¼

8 ð x  a1 Þ > > > v  a1
x
a2 > > a2  a1 Þ a ð > > > > < v a2
x
a3 a > ð a4  x Þ > > > v  a3
x
a4 > > ða4  a3 Þ a > > > : 0 otherwise

Fig. 11.1 A parametric fuzzy number.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

8
 a2  x + wa ðx  a1 Þ > > > > > ða2  a1 Þ > > > < wa
 νa ðxÞ ¼
> x  a3 + wa a04  x > >
0  > > > a4  a3 > > : 1 8
 a2  x + ta ðx  a1 Þ > > > > > ð a2  a1 Þ > > > < t a
 ηa ðxÞ ¼
> x  a3 + ta a004  x > >
 > > > a004  a3 > > : 1

a01
x
a2 a2
x
a3 a3
x
a04 otherwise a001
x
a2 a2
x
a3

,

a3
x
a004 otherwise

where va ,wa , and ta denote the maximum truth membership degree, minimum indeterminacy membership degree, and minimum falsity membership degree, respectively.    Definition 11.5. Let a ¼ ða1 , a2 , a3 , a4 Þ, va , wa , ta and D E  b ¼ ðb1 , b2 , b3 , b4 Þ, vb , wb , tb be two SVTNNs and λ  0 be any real number. Then: (1) Addition of two SVTNNs:

D E   a + b ¼ ða1 + b1 , a2 + b2 , a3 + b3 , a4 + b4 Þ, va ^ vb , wa _ wb , ta _ tb : (2) Subtraction of two SVTNNs:

D E   a  b ¼ ða1  b4 , a2  b3 , a3  b2 , a4  b1 Þ, va ^ vb , wa _ wb , ta _ tb : (3) Inverse of an SVTNN: 1

a

 ¼

 1 1 1 1  , , , , va , wa , ta , where a 6¼ 0: a4 a3 a2 a1

(4) Multiplication of an SVTNN by constant value: 

λa ¼



 ðλa1 , λa2 , λa3 , λa4 Þ, va , wa , ta  λ > 0 , ðλa4 , λa3 , λa2 , λa1 Þ, va , wa , ta λ < 0: 



Definition 11.6. A (α, β, γ)-cut set of an SVNN a denoted by a ðα, β, γÞ is defined as 

a ðα, β, γÞ ¼ fxj μðxÞ  α, νðxÞ
β,ηðxÞ
γ, x 2 ℝg,

A solution for the neutrosophic linear programming problem with a new ranking function

239

which satisfies the conditions as follows: 0
α
va , wa
β
1,ta
γ
1, 0
α + β + γ
1: 



Definition 11.7. A ∝-cut set of an SVNN a denoted by a α is defined as 

a α ¼ fxj μðxÞ  α,x 2 ℝg,  where α 2 0, va . 







Definition 11.8. A β-cut set of an SVNN a denoted by a β is defined as 

a β ¼ fxj νðxÞ
β,x 2 ℝg,  where 2 wa , 1 . Definition 11.9. A γ-cut set of an SVNN a denoted by a γ is defined as 

a γ ¼ fxj ηðxÞ
γ, x 2 ℝg,  where 2 ta , 1 .

11.3

An overview of existing methods

This section contains a review of the methods proposed for FLPs by Zimmermann [1, 16], Werners [15], Guu and Wu [17], Skandari and Ghaznavi [18], Wu et al. [10], Klir and Yuan [14], Das et al. [13], Verdegay [19], and Chanas [20].

11.3.1 Zimmermann’s method Consider the following FLP: 

ma x z ¼ cw S:t:  Ai w
bi , i ¼ 1, …, m w 2 Κ:

(11.1)

where w ¼ (w1, w2, …, wn)T, c ¼ (c1, c2, …, cn) 2 ℝn, Ai ¼ (ai1, ai2, …, ain)T 2 ℝn, bi 2 ℝ, i ¼ 1, 2, …, m and Κ ¼ {w 2 ℝn : Bw
d} with B 2 ℝln, and d 2 ℝn. The maximum fuzzy here seeks to reach a level of optimality (aspiration level) such as b0 2 ℝ. In addition, the fuzzy inequality refers to the fuzzy version of inequality, which means “essentially less than or equal to.”

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

According to the given description, problem (11.1) can be rewritten as follows: Find w S:t:  cw  b0 , 

(11.2)

Ai w
bi , i ¼ 1,…, m w 2 Κ: Now, in order to solve problem (11.2), we must propose tolerances to overcome the fuzzy constraints of the problem as well as an appropriate membership function for them. To this end, the membership function of the objective function and the constraints of the problem are defined as follows: 8 1, cw > b0 , > > < b0  cw , b0  p0 < cw < b0 , ν0 ðcwÞ ¼ 1  > p0 > : 0, cw < b0  p0 ,

(11.3)

8 > < 1, A w  b Ai w < bi , i i , bi < Ai w < b i + pi , νi ðAi wÞ ¼ 1  > p i : 0, A i w > bi + pi ,

(11.4)

By assuming β ¼ min {ν0(cw), ν1(A1w), ν2(A2w), …, νm(Amw)} and using the membership functions given in Eqs. (11.3) and (11.4), FLP (1) can be converted to the following crisp nonlinear programming problem: max β S:t: ν0 ðcwÞ  β, νi ðAi wÞ  β, i ¼ 1, 2,…,m, 0
β
1, w 2 Κ: cw Let’s assume, for simplicity, however, by reducing the generality, ν0 ðcwÞ ¼ 1  b0  p0

and υi ðAi wÞ ¼ 1  Ai wpi bi , so the above problem will be simplified as follows: max β S:t: cw  b0  ð1  βÞp0 , Ai w
bi + ð1  βÞpi , i ¼ 1, 2,…, m, 0
β
1, w 2 Κ:

A solution for the neutrosophic linear programming problem with a new ranking function

241

11.3.2 Werners’ method Werners examined the following FLP: max z ¼ cw S:t:  Ai w
bi , i ¼ 1,…, m w 2 Κ,

(11.5)

where in which only constraints are fuzzy. Taking into account the maximum tolerance pi for the right-hand side value bi of the equal constraints of problem (11.5), with respect to two following problems, the membership function (11.8) for this problem is defined. Consider the two following crisp LPs: z0 ¼ max z ¼ cw S:t: Ai w
bi , i ¼ 1, …, m w 2 Κ,

(11.6)

z1 ¼ max z ¼ cw S:t: Ai w
bi + pi , i ¼ 1, …, m w 2 Κ:

(11.7)

and

The resulting membership function for a crisp version of the objective function of FLP (11.5) is 8 > < 1, z  cw cw > z1 , 1 , z0
cw
z1 , ν0 ðcwÞ ¼ 1  z > 1  z0 : 0, cw < z0 : In addition, the membership function associated with the constraints of this problem is similar to Equation (11.4). Now, assuming β ¼ min {ν0(cw), ν1(A1w), ν2(A2w), …, νm(Amw)} and using the principle of Bellman and Zadeh, the main problem becomes the nonlinear planning problem below. max β S:t: ν0 ðcwÞ  β, νi ðAi wÞ  β, i ¼ 1, 2,…, m, 0
β
1, w 2 Κ:

(11.8)

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Again, similar to the previous case, if we set ν0 ðcwÞ ¼ 1  zz11  cw and z0

νi ðAi wÞ ¼ 1  Ai wpi bi , i ¼ 1, 2, …,m. In this case, the nonlinear problem becomes the following linear problem: max β S:t: cw  z1  ð1  βÞðz1  z0 Þ, Ai w
bi + ð1  βÞpi , i ¼ 1, 2, …,m, 0
β
1, w 2 Κ:

11.3.3 Guu and Wu’s method In this subsection, the two-phase method proposed by Guu and Wu is presented. These two phases are as follows. Phase 1. Solve the crisp problem below and display the optimal solution to (w∗, β∗). maxβ S:t: 0
β
υi ðwÞ
1, i ¼ 0,1, …,m w 2 Κ: Phase 2. In this phase, by solving the following problem, the optimal solution of the main problem is obtained. max

m X βi i¼0

S:t: ν0 ðcw∗ Þ
β0
ν0 ðcwÞ
1, ν0 ðAi w∗ Þ
β0
ν0 ðAi wÞ
1,i ¼ 1, 2,…,m 0
β
1, i ¼ 0,1,…,m w 2 Κ: Ai wbi At this point, for comfort put υ0 ðcwÞ ¼ 1  zz11cw z0 and υi ðAi wÞ ¼ 1  pi ,i ¼ 1, 2, …,m. In this case, the nonlinear problem becomes the following linear problem:

max

m X βi i¼0

S:t: z0 + βðz1  z0 Þ
cw
z1 , bi
Ai w
bi + ð1  βi Þpi , i ¼ 1, 2,…, m, υ0 ðcw∗ Þ
β0 , υi ðAi w∗ Þ
βi , 0
β
1, i ¼ 0, 1,…, m, w 2 Κ:

A solution for the neutrosophic linear programming problem with a new ranking function

243

11.3.4 Wu et al.’s method To illustrate this method, consider FLP (11.5) again. In this approach, the membership functions are proposed as follows: 8 < cw  b0 , z < cw, 0 ν0 ðcwÞ ¼ z1  z0 : 0, cw
z0 , 8 A i w  bi < 1 , Ai w
b i + p i : νi ðAi wÞ ¼ pi : 0 Ai w > b i + pi

(11.9)

(11.10)

In Phase 1 of this method, the following crisp LP is solved: max τ S:t: υ0 ðcwÞ  τ, υi ðAi wÞ  τ, w 2 Κ:

(11.11)

Assuming that τ∗ is the optimal solution of problem (11.11), in order to obtain the solution to the main problem, we solve the following problem in phase 2: max

m X

hi

i¼0

S:t: υ0 ðcwÞ  h0  τ∗ , νi ðAi wÞ  hi  τ∗ , w 2 Κ, hi  0,i ¼ 1, 2,…, m:

11.3.5 Skandari and Ghaznavi’s method Skandari and Ghaznavi proposed the following membership function for constraints of problems (11.1) and (11.5):



A i w  bi υi ðAi wÞ ¼ min 1, 1  ,w 2 Κi i ¼ 1, 2,…, m, pi

(11.12)

where Κi ¼ {w  0 : Aiw
bi + pi}. In addition, they suggested the following membership functions for the objective functions of problems (11.1) and (11.5), respectively:



b0  cw υ0 ðcwÞ ¼ min 1, 1  ,w 2 Κ0 p0

(11.13)

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Optimization Theory Based on Neutrosophic and Plithogenic Sets



z1  cw υ0 ðcwÞ ¼ min 1, 1  ,w 2 Κ0 , z1  z0

(11.14)

where Κ0 ¼ {w 2 Κ : cw  b0  p0} and Κ0 ¼ fw 2 Κ : cw  z0 g. Now, using the membership functions introduced, problems (11.1) and (11.5) can be written in the form of crisp LPs: max τ +

m 1 X τi m + 1 i¼0

S:t:
 τ
τ0
υ0 ðcwÞ, w 2 Κ0 or Κ0 τ
τi
υi ðAi wÞ, w 2 Κi , i ¼ 1,2, …, m, τi  0, i ¼ 0, 1, …,m, w 2 Κ:

11.3.6 Klir and Yuan’s method Klir and Yuan investigated the FLP with the fuzzy technological coefficients as follows: max

n X cj w j j¼1

S:t: n X  a ij wj
bi ,

(11.15) 1
i
m

j¼1

wj  0 1
j
n: Assume that the membership functions of the technological coefficients are as follows: 8 1 > >  >

pij > > : 0

w < aij aij < w < aij + pij : w  aij + pij

In the process of solving the problem, we solve the following two crisp LPs: z1 ¼ max

n X

cj wj

j¼1

S:t: n X aij wj
bi , j¼1

(11.16) 1
i
m

wj  0 1
j
n ,

A solution for the neutrosophic linear programming problem with a new ranking function

245

and z2 ¼ max

n X cj wj j¼1

S:t: n
X

 aij + pij wj
bi ,

(11.17) 1
i
m

j¼1

wj  0 1
j
n: Now, assuming that z1 and z2 are finite, the membership function of the objective function is defined as follows:

υ0 ðwÞ ¼

8 > > 0 > > > > > ! > > n > X > > < c j w j  z1 > > > > > > > > > > > > :1

j¼1

z2  z1

n X cj wj < z1 , j¼1

z1
n X

n X

cj w j < z2 ,

j¼1

cj wj  z2 :

j¼1

In addition, the membership functions of the constraints are written as follows:

υi ðwÞ ¼

8 > > 0 > > > > > ! > > n > X > > > aij wj > < bi  > > > > > > > > > > > > > > :1

j¼1 n X

pij wj ,

b1
z0 , > < ∗ z  z ð δ Þ 0 υ 0 ðz∗ ðδ ÞÞ ¼ 1  z 0  p0
z ∗ ð δ Þ
z 0 , > p0 : 0 z∗ ðδÞ < z0  p0 :

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

11.4

Proposed ranking method

Various ranking functions have been proposed for solving FLP problems [5, 18, 21, 22]. In this section, a new ranking function to solve NLP problems is introduced. Defining a ranking function R : NF ðℝÞ!ℝ which maps each NN into the real line is a useful approach for ordering the elements of NN(ℝ). Orders on NN(ℝ) are defined as follows: 











a
R b a > t0 : 0 otherwise 8 1 > >
1 > > >
1 
Q Q b a x
Q b i + di > i i α , β , γ α , β , γ α , β , γ > d > i : 0 otherwise

By considering the linear ranking function Qα,β,γ and using Bellman and Zadeh’s fuzzy decision, FLP (11.31) is transformed to the following model: 

 

 

  max min μ0 Qα,β,γ c x , μ1 Qα,β, γ a 1 x , …, μm Qα,β,γ a m x

  S:t: Qα , β,γ z  t0
Qα,β,γ c x  
   Qα,β,γ a i x
Qα,β,γ b i + di i ¼ 1,…, m x0

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

This is equivalent to max λ

  S:t: λ
μ0 Qα,β, γ c x

  λ
μi Qα,β,γ a i x i ¼ 1, …,m

  Qα,β,γ z  t0
Qα,β,γ c x  
  Qα, β,γ a i x
Qα,β,γ b i + di i ¼ 1,…, m x  0, 0
λ
1:

11.6.1 Numerical example The proposed method in this chapter is explained with the help of the following example: 





max 4 x1 + 5 x2 + 6 x3 







2 x1 + 15 x2 + 6 x3 ≲ 12 







4 x1 + 6 x2 + 11 x3 ≲ 16 







10 x1 + 9 x2 + 5 x3 ≲ 18 x1 ,x2 ,x3  0,0
λ
1: The uncertain parameters are estimated by the following (triangular or trapezoidal) NNs: 





2 ¼ ð2, 2, 0:2;0:3;1Þ, 15 ¼ ð15;15;0:4;0:1;1Þ, 6 ¼ ð6;6:5;0:2;0:2;1Þ,



4 ¼ ð4, 5, 0:3;0:4;1Þ, 





11 ¼ ð11;11; 0:5; 0:2; 1Þ, 10 ¼ ð10; 10;0:1;0:3;1Þ, 9 ¼ ð8:5;9, 0:2;0:1;1Þ, 

5 ¼ ð5, 5, 0:3;0:3;1Þ 



12 ¼ ð12;12;0:5;0:2;1Þ, 16 ¼ ð15;75; 16:25; 0:2; 0:2; 1Þ,  18 ¼ ð18; 18;0:5;0:5;1Þ: The approval degree of the above information is (1, 0, 0). According to the proposed ranking function and the above methods, we have the following classical linear programming problems for α ¼ 0.2,0.8, respectively. The first LP corresponds to a low-level decision and the second LP corresponds to a top-level decision (z0 ¼ 6, t0 ¼ 2, d1 ¼ 2, d2 ¼ 3, d3 ¼ 4). α ¼ 0.2

A solution for the neutrosophic linear programming problem with a new ranking function

255

max λ 1 S:t: λ
ð7:232x1 + 8x2 + 10x3  4Þ 2 1 λ
ð3:232x1  23:904x2  10x3 + 21:104Þ 2 1 λ
ð7:232x1  10x2  17:504x3 + 28:6Þ 3 1 λ
ð16:064x1  13:968x2  8x3 + 32:8Þ 4 4
7:232x1 + 8x2 + 10x3 3:232x1 + 23:904x2 + 10x3
21:104 7:232x1 + 10x2 + 17:504x3
28:6 16:064x1 + 13:968x2 + 8x3
32:8 x1 , x2 , x3  0, 0
λ
1: The optimal solutions of this problem are obtained as x1 ¼ 0.5539, x2 ¼ 0.0780, x3 ¼ 0.5532, λ ¼ 1. α ¼ 0.8 max λ 1 S:t: λ
ð1:802x1 + 2x2 + 2:5x3  4Þ 2 1 λ
ð0:802x1 + 5:994x2 + 2:5x3 + 6:704Þ 2 1 λ
ð1:802x1 + 2:5x2 + 4:394x3 + 9:4Þ 3 1 λ
ð4:004x1 + 3:498x2 + 2x3 + 11:2Þ 4 4
1:802x1 + 2x2 + 2:5x3 0:802x1 + 5:994x2 + 2:5x3
6:704 1:802x1 + 2:5x2 + 4:394x3
9:4 4:004x1 + 3:498x2 + 2x3
11:2 x1 ,x2 , x3  0,0
λ
1: The optimal solutions of this problem are obtained as x1 ¼ 1.7036, x2 ¼ 0.3646, x3 ¼ 1.0629, λ ¼ 0.2385.

11.7

Empirical application

A company produces four technical products for aerospace companies [12]. The outputs should be obtained through four sections before they are shipped, and these sections include: wiring, drilling, assembling, and finally inspection. The profit and the time required to produce each unit item are set out in Table 11.1.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 11.1 Sections. Products

Wiring

Drilling

Assembly

Inspection

Unit profit

P1 P2 P3 P4

0.5 1.5 1 1

3 1 2 3

2 4 1 2

0.5 1 0.5 0.5

9$ f 12$ f 15$ f 11$



Table 11.2 Time capacity and minimum production level. Sections

Capacity (h)

Products

Minimum production level

Wiring Drilling Assembly

g 1500 g 2350 g 2600

P1 P2 P3

g 150 g 100 g 300

Inspection

g 1200

P4

g 400

The minimum production amounts to do and time capacities are presented in Table 11.2. The goal is to provide a program to supply products for the company in order to maximize profits. The degree of approval for the NNs in this example is (0.9,0.1,0.1). Here, the decision variables are defined as follows: xi : the number of Pi,

1
i
4:

The NLP associated with this problem is formulated as follows: 12x2 + f 15x3 + f 11x4 Max Ze ¼ e 9x1 + f S:t: g 0:5x1 + 1:5x2 + 1:5x3 + x4
1500, g 3x1 + x2 + 2x3 + 3x4
2350, g 2x1 + 4x2 + x3 + 2x4
2600, g 0:5x1 + x2 + 0:5x3 + 0:5x4
1200, g 2  100,x g 3  300,x g 4  400, g x1  150,x x1 , x2 , x3 ,x4  0: The NNs used in the previous model are described as follows: e 9 ¼ ð8, 9, 2, 3Þ, f 12 ¼ ð10;12;1, 2Þ, f 15 ¼ ð10;12;1, 2Þ, f 11 ¼ ð9;11; 1, 2Þ, g ¼ ð130;150;10;20Þ, 100 g ¼ ð80;100; 10;20Þ, 300 g ¼ ð280;300;10; 20Þ, 150

A solution for the neutrosophic linear programming problem with a new ranking function

257

g ¼ ð1300;1500;100;200Þ, g ¼ ð380; 400;10;20Þ, 1500 400 g 2350 ¼ ð2250; 2350;50;50Þ, g ¼ ð2400; 2600;200;200Þ, 1200 g ¼ ð1100;1200;100;200Þ: 2600 Using the introduced ranking function, the above model is modified to the following crisp model: Max Z ¼ 1:57x1 + 2:14x2 + 2:74 x3 + 1:94x4 S:t: 0:5x1 + 1:5x2 + 1:5x3 + x4
274:5, 3x1 + x2 + 2x3 + 3x4
456:5, 2x1 + 4x2 + x3 + 2x4
486,

(11.32)

0:5x1 + x2 + 0:5x3 + 0:5x4
224:5, x1  27:45, x2  17:45, x3  57:45, x4  77:45, x1 ,x2 ,x3 ,x4  0, x1 ¼ 27:45, x2 ¼ 26:9, x3 ¼ 57:45, x4 ¼ 77:45, Z ¼ 408:6157 Max Ze ¼ 12:07x1 + 15:64x2 + 19:84x3 + 14:24x4 S:t: 0:5x1 + 1:5x2 + 1:5x3 + x4
1984:5, 3x1 + x2 + 2x3 + 3x4
3216:5, 2x1 + 4x2 + x3 + 2x4
3486, 0:5x1 + x2 + 0:5x3 + 0:5x4
1634:5, x1  198:45, x2  128:45, x3  408:45, x4  548:45,

(11.33)

x1 , x2 ,x3 ,x4  0: x1 ¼198:45, x2 ¼ 158:9, x3 ¼ 408:45, x4 ¼ 548:45, Z ¼ 20794: Models (11.32) and (11.33) correspond to decision-making at the upper and lower level, respectively.

11.8

Conclusion

In this chapter, we tried to provide a decision-based parametric method to solve neutrosophic linear programming problems. In order to achieve this goal, we defined a new parametric ranking function. The proposed method is easily understood and applicable in real-life situations, and gives the decision-maker the authority to solve the problem at various levels of decision-making. Several examples have been used to demonstrate the capability and efficiency of the method. We also showed that the proposed method resolves some of the disadvantages of existing methods.

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References [1] L.A. Zadeh, Fuzzy sets, Inform. Contr. 8 (1965) 338–356. [2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst. 20 (1) (1986) 87–96. [3] F. Smarandache, Neutrosophic set, a generalisation of the intuitionistic fuzzy sets, Int. J. Pure Appl. Math. 24 (2005) 287–297. [4] F. Smarandache, A Unifying Field in Logics: Neutrsophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Probability, Infinite Study, Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Research Press, 2005 ISBN 978-1-59973-080-6. [5] N. Mahdavi-Amiri, S.H. Nasseri, Duality in fuzzy number linear programming by use of a certain linear ranking function, Appl. Math Comput. 180 (1) (2006) 206–216. [6] N. Mahdavi-Amiri, S.H. Nasseri, Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables, Fuzzy Set. Syst. 158 (17) (2007) 1961–1978. [7] J.Y. Dong, S.P. Wan, A new trapezoidal fuzzy linear programming method considering the acceptance degree of fuzzy constraints violated, Knowledge-Based Syst. 148 (2018) 100–114. [8] M. Javanmard, H.M. Nehi, Solving the interval type-2 fuzzy linear programming problem by the nearest interval approximation, in: Fuzzy and Intelligent Systems (CFIS), 2018 6th Iranian Joint Congress on, IEEE, 2018, , pp. 4–6. [9] M.N. Skandari, M. Ghaznavi, An efficient algorithm for solving fuzzy linear programming problems, Neural Process. Lett. (2018) 1–20. [10] J. Ye, Neutrosophic number linear programming method and its application under neutrosophic number environments, Soft Comput. 22 (14) (2018) 4639–4646. [11] J. Ye, Prioritized aggregation operators of trapezoidal intuitionistic fuzzy sets and their application to multicriteria decision making, Neural Comput. Applic. 25 (6) (2014) 1447–1454. [12] M. Abdel-Basset, M. Gunasekaran, M. Mohamed, F. Smarandache, A novel method for solving the fully neutrosophic linear programming problems, Neural Comput. Applic. (2018) 1–11. [13] S.K. Das, T. Mandal, S.A. Edalatpanah, A mathematical model for solving fully fuzzy linear programming problem with trapezoidal fuzzy numbers, Appl. Intell. 46 (3) (2017) 509–519. [14] I.U. Khan, T. Ahmad, N. Maan, A simplified novel technique for solving fully fuzzy linear programming problems, J. Optim. Theory Appl. 159 (2) (2013) 536–546. [15] R.A. Shureshjani, M. Darehmiraki, A new parametric method for ranking fuzzy numbers, Indagat. Math. 24 (3) (2013) 518–529. [16] H.J. Zimmermann, Fuzzy Set Theory and Its Applications, third ed., Kluwer Academic Publishers, Nowell, 1996. [17] S.M. Guu, Y.K. Wu, Two phase approach for solving the fuzzy linear programming problems, Fuzzy Set. Syst. 107 (1999) 191–195. [18] H.M. Nehi, H. Hajmohamadi, A ranking function method for solving fuzzy multiobjective linear programming problem, Ann. Fuzzy Math. Inform. 3 (1) (2012) 31–38. [19] X. Wang, E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy Set. Syst. 118 (3) (2001) 375–385. [20] S. Chanas, The use of parametric programming in fuzzy linear programming problems, Fuzzy Set. Syst. 11 (1983) 243–251.

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[21] T. Allahviranloo, F.H. Lotfi, M.K. Kiasary, N.A. Kiani, L. Alizadeh, Solving fully fuzzy linear programming problem by the ranking function, Appl. Math. Sci. 2 (1) (2008) 19–32. [22] H.R. Maleki, Ranking functions and their applications to fuzzy linear programming, Far East J. Appl. Math. 4 (2002) 283–301. [23] A. Ebrahimnejad, M. Tavana, A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers, App. Math. Model. 38 (2014) 4388–4395. [24] K. Ganesan, P. Veeramani, Fuzzy linear programs with trapezoidal fuzzy numbers, Ann. Operat. Res. 143 (1) (2006) 305–315. [25] A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzy linear programming problems, App. Math. Model. 35 (2011) 817–823.

Further reading [26] M. Darehmiraki, A novel parametric ranking method for intuitionistic fuzzy numbers, Iranian J. Fuzzy Syst. (2018). [27] G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic-Theory and Applications, Prentice-Hall Inc., 1995. 574p. [28] J.L. Verdegay, Fuzzy mathematical programming, in: M.M. Gupta, E. Sanchez (Eds.), Fuzzy Information and Decision Processes, North Holland, Amsterdam, 1982. [29] B. Werners, Interactive multiple objective programming subject to flexible constraints, Eur. J. Oper. Res. 31 (1987) 342–349. [30] Y.K. Wu, C.C. Liu, Y.Y. Lur, Pareto optimal solution for multiobjective linear programming problems with fuzzy goals, Fuzzy Optim. Decis. Making 14 (2015) 43–55.

Dice and Jaccard similarity measures based on expected intervals of trapezoidal neutrosophic fuzzy numbers and their applications in multicriteria decision making

12

Chiranjibe Janaa, Faruk Karaaslanb a Department of Applied Mathematics With Oceanology and Computer Programming, Vidyasagar University, Midnapore, India, bDepartment of Mathematics, Faculty and Sciences, C¸ankırı Karatekin University, C¸ankırı, Turkey

12.1

Introduction

In real life, humankind encounters many situations involving uncertainty, inconsistency, and incomplete information. Many theories have been proposed by researchers in order to overcome these situations. Some of the well-known theories are the fuzzy set theory proposed by Zadeh [1], the intuitionistic fuzzy set theory suggested by Atanassov [2], and the interval-valued intuitionistic fuzzy set theory suggested by Atanassov and Gargov [3]. A fuzzy set is characterized by a membership function. The intuitionistic fuzzy set, which is a generalization of the fuzzy sets, is characterized by the membership and nonmembership function. Dubois and Prade [4, 5] introduced the notion of fuzzy numbers and operations between fuzzy numbers. Intuitionistic trapezoidal fuzzy numbers are the extension of intuitionistic triangular fuzzy numbers. Some operators of intuitionistic trapezoidal fuzzy numbers were presented by Nehi and Maleki [6] based on the intuitionistic fuzzy numbers defined by Grzegrorzewski [7]. Even though the fuzzy sets and the intuitionistic fuzzy set theories are very successful in order to model some decision-making problems containing uncertainty and incomplete information, sometimes these theories may not suffice for modeling of indeterminate and inconsistent information encountered in the real world. Therefore, Smarandache [8, 9] introduced the concept of neutrosophic set. A neutrosophic set is characterized independently by three functions called truth (T), indeterminacy (I), and falsity (F)-membership functions, defined from a nonempty set to real standard or nonstandard subsets of ]
0, 1+[. Although the concept of the neutrosophic set is a useful tool to model indeterminate and inconsistent information, modeling of some

Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00012-3 © 2020 Elsevier Inc. All rights reserved.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

problems in engineering and other scientific fields may not be easy by using real standard or nonstandard subsets of ]
0, 1+[. To overcome these difficulties, Wang et al. [10] introduced the concept of the single-valued neutrosophic set (SVNS) by using the truth, indeterminacy, and falsity-membership functions defined from a nonempty set to the interval [0, 1]. SVNSs can be seen as a subclass of the neutrosophic set. The theories mentioned earlier have an important role in solving problems containing multicriteria decision making (MCDM) and multicriteria group decision making. Many researchers have studied MCDM methods under neutrosophic, single-valued neutrosophic, and interval neutrosophic information. For example, the neutrosophic analytic hierarchy process [11, 12], cross-entropy under neutrosophic environment [13–16], gray relational analysis [17, 18], similarity measures [19–27], correlation coefficient [16, 28], the TOPSIS method [29], the VIKOR method [30], aggregating operators [31–34], the outranking method [35], and the decision-making method based on neutrosophic graphs and matrices [36–39] were studied by researchers. Ye [40] defined the concept of trapezoidal neutrosophic sets by combining trapezoidal fuzzy numbers and the SVNS, and proposed some operational rules such as score and accuracy functions of trapezoidal neutrosophic sets. He also suggested a trapezoidal neutrosophic number weighted arithmetic averaging operator and a trapezoidal neutrosophic number weighted geometric averaging operator. Based on these operators, he developed a multiattribute decision-making method. Deli and Subas [31] introduced the concept of single-valued triangular neutrosophic numbers (SVTrNNs), which can be regarded as special cases of single-valued trapezoidal neutrosophic numbers. Biswas et al. [41] proposed the concepts of expected interval (EI) and expected value (EV), and based on these concepts they developed the Cosine similarity measure under a trapezoidal fuzzy neutrosophic environment. They also developed an MCDM method based on this similarity measure method. Thamaraiselvi and Santhi [42] used SVTrNNs to make a mathematical representation of a transportation problem. Liang et al. [43] defined the concept of single-valued trapezoidal neutrosophic preference relations (SVTNPRs) as a strategy for tackling MCDM problems, and gave two aggregation operators called the single-valued trapezoidal neutrosophic weighted arithmetic average operator and the single-valued trapezoidal neutrosophic weighted geometric average operator. They also developed a decision-making method based on SVTNPRs to address green supplier selection problems. Ye [27] proposed another form of the Dice measures of simplified neutrosophic sets. He also developed the generalized Dice measures-based multiple-attribute decision-making (MADM) methods with simplified neutrosophic information and applied this to a real example on the selection of manufacturing schemes. In 2018, Biswas et al. [44] introduced the concept of interval trapezoidal neutrosophic numbers and presented some of the arithmetic operations of them. They also proposed an MADM method under an interval trapezoidal neutrosophic environment. In [45], Biswas et al. presented an EV-based method for multiple-attribute group decision making (MAGDM) with neutrosophic trapezoidal numbers. In this study, we introduce Dice and Jaccard similarity measures and weighted Dice and Jaccard similarity measures between two trapezoidal fuzzy

Dice and Jaccard similarity measures

263

neutrosophic numbers based on the EI and EVs of the trapezoidal neutrosophic fuzzy numbers defined by Biswas et al. [41]. Later, we define Dice and Jaccard similarity measures and weighted Dice and Jaccard similarity measures between trapezoidal fuzzy neutrosophic sets. We also propose an MCDM method based on these similarity measures. Furthermore, we give an application of the proposed MCDM method and compare the proposed MCDM method with the other methods used in the ranking of alternatives under a trapezoidal fuzzy neutrosophic environment.

12.2

Preliminaries

12.2.1 Some concepts related to trapezoidal fuzzy numbers Definition 12.1. Dubois and Prade [4]). A trapezoidal fuzzy number A is a fuzzy set on  (set of real numbers) in which its membership function is defined as follows: 8 0, > > > > F > A ðxÞ, > < 1, μA ðxÞ ¼ > GA ðxÞ, > > > 0, > > :

if if if if if

x < a1 , a1  x  a2 , a2  x  a3 , a3  x  a4 , x > a4 ,

(12.1)

where a1 , a2 , a3 , a4 2  and a1  a2  a3  a4. F A : ½a1 , a2  ! ½0, 1 is the continuous increasing function and GA : ½a3 ,a4  ! ½0,1 is the continuous decreasing function. Here, the left side of fuzzy number A is F A ða1 Þ ¼ 0, F A ða2 Þ ¼ 1 and the right side of fuzzy number A is GA ða3 Þ ¼ 1, GA ða4 Þ ¼ 0. Definition 12.2. Dubois and Prade [46]). Let A ¼ (a1, a2, a3, a4) be a trapezoidal fuzzy number over . Then, the membership value of A is defined by 8 0, > > x
a1 > > > , > > a > < 2
a1 1, μA ¼ x
a4 > , > > a3
a4 > > > 0, > > :

if x < a1 , if a1  x  a2 , if a2  x  a3 , if a3  x  a4 ,

(12.2)

if a4 > x:

If a2 ¼ a3, then a trapezoidal fuzzy number is called a triangular fuzzy number. In other words, a triangular fuzzy number is a special case of a trapezoidal fuzzy number.

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

In [47], Heilpern defined the concepts of (EI) and (EV ) of a fuzzy number A as follows: EIðAÞ ¼ ½EL ðAÞ, EU ðAÞ, EVðAÞ ¼

ðEL ðAÞ + EU ðAÞÞ , 2

(12.3)

respectively. Here, Z

a2

EL ðAÞ ¼ a2


Z F A ðxÞdx ¼ a2


a1

Z EU ðAÞ ¼ a2


a2

a1 a4

Z GA ðxÞdx ¼ a3


a3

a4 a3

x
a1 a1 + a2 , dx ¼ a2
a1 2

(12.4)

x
a4 a3 + a4 : dx ¼ a3
a4 2

(12.5)

Thus, the EI for a trapezoidal fuzzy number A ¼ (a1, a2, a3, a4) can be written as EIðAÞ ¼

ha + a a + a i 1 2 3 4 , : 2 2

(12.6)

The EV of the center of the EI can be written as follows: EVðAÞ ¼

EL ðAÞ + EU ðAÞ a1 + a2 + a3 + a4 ¼ : 2 4

(12.7)

12.2.2 Some concepts related to neutrosophic sets and neutrosophic numbers In this section, we present some basic concepts in the theory of neutrosophic sets [8] to be required in the next sections. Definition 12.3. Smarandache [8]). Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set a in X is defined by a ¼ fhta ðxÞ,ia ðxÞ, fa ðxÞijx 2 Xg, where ta(x) is a truth-membership function, ia(x) is an indeterminacy-membership function, and fa(x) is a falsity-membership function. ta(x), ia(x), and fa(x) are real standard or nonstandard subsets of ]
0, 1+[ such that ta : X !]
0, 1+[, ia : X !]
0, 1+[, and fa : X !]
0, 1+[. There is no restriction on the sum of ta(x), ia(x), and fa(x), and so 0
 ta(x) + ia(x) + fa(x)  3+. The concept of the SVNS, which is more useful in real applications, was defined by Wang et al. [10].

Dice and Jaccard similarity measures

265

Definition 12.4. Wang et al. [10]). Let X 6¼ Ø, with a generic element in X denoted by x. A single-valued neutrosophic set (SVN-set) a~ is characterized by three functions called truth-membership function ta(x), indeterminacy-membership function ia(x), and falsity-membership function fa(x) such that ta(x), ia(x), fa(x) 2 [0, 1] for all x 2 X. If X is continuous, an SV N-set a~ can be written as follows: Z a~ ¼

hta ðxÞ, ia ðxÞ, fa ðxÞi=x, for all x 2 X: X

If X is a crisp set, an SV N-set a~ can be written as follows: a~ ¼

X

hta ðxÞ,ia ðxÞ, fa ðxÞi=x, for all x 2 X:

x

Here, 0  ta(x) + ia(x) + fa(x)  3 for all x 2 X. From now on, we say that for x 2 X a~ðxÞ ¼ ðta ðxÞ, ia ðxÞ, fa ðxÞÞ is single-valued neutrosophic number (SVN-number). Some operations between two SVN-numbers are given in Wang et al. [10], Ye [48], and Liu et al. [34] as follows. Let a~ðxÞ ¼ ðta ðxÞ,ia ðxÞ, fa ðxÞÞ and b~ðxÞ ¼ ðtb ðxÞ, ib ðxÞ, fb ðxÞÞ be two SVN-numbers. Then, 1. a~ðxÞ + b~ðxÞ ¼ ðta ðxÞ + tb ðxÞ
ta ðxÞtb ðxÞ,ia ðxÞib ðxÞ, fa ðxÞfb ðxÞÞ. 2. a~ b~ðxÞ ¼ ðta ðxÞtb ðxÞ,ia ðxÞ + ib ðxÞ
ia ðxÞib ðxÞ, fa ðxÞ + fb ðxÞ
fa ðxÞfb ðxÞÞ. 3. The SVN-number a~ðxÞ is smaller than the SVN-number b~ðxÞ, that is, a~  b~ if and only if the following conditions hold: ta(x)  tb(x); ia(x)  ib(x); fa(x)  fb(x). 4. The complement of a~, denoted by a~c , is defined as: tca ðxÞ ¼ fa ðxÞ; ica ðxÞ ¼ 1
ia ðxÞ; fac ðxÞ ¼ ta ðxÞ. 5. The intersection of a~ and b~, denoted by a~ \ b~, is defined as ð~ a \ b~ÞðxÞ ¼ ð minfta ðxÞ,tb ðxÞg;maxfia ðxÞ, ib ðxÞg;max ffa ðxÞ, fb ðxÞgÞ: 6. The union of a~ and b~, denoted by a~ [ b~, is defined as ð~ a [ b~ÞðxÞ ¼ ð maxfta ðxÞ, tb ðxÞg;min fia ðxÞ, ib ðxÞg;min ffa ðxÞ, fb ðxÞgÞ:

Definition 12.5. Biswas et al. [41]). Let a~ðxÞ ¼ ðta ðxÞ,ia ðxÞ, fa ðxÞÞ be a neutrosophic fuzzy number in the set of real numbers . Then, its truth-membership, indeterminacy-membership, and falsity-membership functions can be defined as follows:

266

Optimization Theory Based on Neutrosophic and Plithogenic Sets

8L t ðxÞ, > > ta ðxÞ, > : 8 0,L > ia ðxÞ, > < 0, ia ðxÞ ¼ U > ia ðxÞ, > : 8 1,L f ðxÞ, > > faU ðxÞ, > : 1,

if a11  x  a21 , if a21  x  a31 , if a31  x  a41 , otherwise, if b11  x  b21 , if b21  x  b31 , if b31  x  b41 , otherwise, if c11  x  c21 , if c21  x  c31 , if c31  x  c41 , otherwise,

,

,

respectively. Here, 0  sup ta ðxÞ + sup ia ðxÞ + sup fa ðxÞ  3, for all x 2 X and a11, a21, a31, a41 ,b11 , b21 ,b31 ,b41 , c11 ,c21 , c31 ,c41 2  such that a11  a21  a31  a41, b11  b21  b31  b41, and c11  c21  c31  c41. Here, tLa ðxÞ 2 ½0, 1, iU a ðxÞ 2 ½0,1, and faU ðxÞ 2 ½0,1 are continuous increasing functions and tU ðxÞ 2 ½0, 1, iLa ðxÞ 2 ½0, 1, a L and fa ðxÞ 2 ½0, 1 are continuous decreasing functions for all x 2 X. Definition 12.6. Biswas et al. [41]). A trapezoidal neutrosophic fuzzy number (TrNFN) n~ with parameters a1  a2  a3  a4, b1  b2  b3  b4, and c1  c2  c3  c4 is denoted as n~ ¼ fða1 , a2 , a3 , a4 Þ, ðb1 , b2 , b3 , b4 Þ, (c1, c2, c3, c4)} in the set of real numbers . In this case, its truth-membership function (tn(x)), indeterminacy-membership function (in(x)), and falsity-membership function (fn(x)) are defined as follows: 8 x
a 1 > , > > > a
a 2 1 < 1, tn ðxÞ ¼ a4
x > , > > > : a4
a3 8 0, b2
x > > , > > > < b2
b1 0, in ðxÞ ¼ x
b 3 > > , > > b
b > 4 3 : 8 1, > c2
x , > > > < c2
c1 0, fn ðxÞ ¼ x
c3 > , > > > : c4
c3 1,

if a1  x  a2 , if a2  x  a3 , if a3  x  a4 , otherwise, if b1  x  b2 , if b2  x  b3 , if b3  x  b4 , otherwise, if c1  x  c2 , if c2  x  c3 , if c3  x  c4 , otherwise:

Dice and Jaccard similarity measures

267

Biswas et al. [41] define EI and EVs of a TrNFN as follows: Definition 12.7. Biswas et al. [41]). Let n~ ¼ fða1 ,a2 ,a3 ,a4 Þ, ðb1 ,b2 , b3 ,b4 Þ,ðc1 ,c2 , c3 ,c4 Þg be a TrNFN. Then, 1. the EI and EV of the truth-membership function tn of TrNFN are defined as follows: ha + a a + a i 1 2 3 4 EI T ð~ , ¼ ½ETL ð~ nÞ ¼ nÞ,ETU ð~ nÞ, (12.8) 2 2 EV T ð~ nÞ ¼

a1 + a2 + a3 + a4 , 4

(12.9)

respectively. 2. The EI and EV of indeterminacy-membership function in are defined as follows: 

 b1 + b2 b3 + b4 nÞ ¼ nÞ,EIU ð~ nÞ , ¼ ½EIL ð~ EI ð~ 2 2 I

(12.10)

and nÞ ¼ EV I ð~

b1 + b2 + b3 + b4 , 4

(12.11)

respectively. 3. The EI and EV of indeterminacy-membership function fn are defined as follows: nÞ ¼ EI F ð~

hc + c c + c i 1 2 3 4 , ¼ ½EFL ð~ nÞ,EFU ð~ nÞ 2 2

(12.12)

and EV F ð~ nÞ ¼

c1 + c2 + c3 + c4 , 4

(12.13)

respectively.

12.2.3 Dice similarity measure between two vectors In 1945, Dice’s similarity measure between two vectors defined in this following way: Let X ¼ (x1, x2, …, xn) and Y ¼ (y1, y2, …, yn) be two vectors of length n where all coordinates are positive. Then, the Dice similarity measure is defined as follows:

DðX,YÞ ¼

2X  Y jjXjj22 + jjYjj22

2 ¼

n X xi yi i¼1

n n X X x2i + y2i i¼1

i¼1

:

(12.14)

P Here, X  Y ¼ ni¼1 xi yi is the inner product of the vectors X and Y, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 Pn 2 jjXjj2 ¼ i¼1 xi and jjYjj2 ¼ i¼1 yi are Euclidean norms of X and Y. When

268

Optimization Theory Based on Neutrosophic and Plithogenic Sets

xi ¼ yi, the Dice similarity measure is not definable. In this case, it is accepted that the Dice similarity measure is zero. The Dice similarity measure satisfied the following properties: 1. 0  D(X, Y )  1; 2. D(X, Y ) ¼ D(Y, X); and 3. D(X, Y ) ¼ 1 if and only if X ¼ Y, that is, xi ¼ yi for i ¼ 1, 2, …, n.

12.2.4 Dice similarity measure of trapezoidal fuzzy numbers The Dice similarity measure of trapezoidal fuzzy numbers was proposed by Ye [24] as follows. Let a1 ¼ (a11, a12, a13, a14) and a2 ¼ (a21, a22, a23, a24) be two trapezoidal fuzzy numbers of the set of real numbers . If EVs of a1 and a2 are considered as two vector representations with the elements, then the Dice similarity measure of a1 and a2 is defined as follows: Sða1 , a2 Þ ¼ ¼

2ðEL ða1 ÞEL ða2 Þ + EU ða1 ÞEU ða2 ÞÞ

, ðEL ða1 ÞÞ2 + ðEU ða1 ÞÞ2 + ðEL ða2 ÞÞ2 + ðEU ða2 ÞÞ2 2½ða11 + a12 Þða21 + a22 Þ + ða13 + a14 Þða23 + a24 Þ 2

2

2

(12.15) 2

ða11 + a12 Þ + ða13 + a14 Þ + ða21 + a22 Þ + ða23 + a24 Þ

:

Based on the properties of the Dice similarity measure between two vectors, the Dice similarity measure of trapezoidal fuzzy numbers between a1 and a2 satisfies the following properties: 1. 0  S(a1, a2)  1; 2. S(a1, a2) ¼ S(a2, a1); and 3. S(a1, a2) ¼ 1 if and only if a1 ¼ a2, that is, a1j ¼ a2j for j ¼ 1, 2, 3, 4.

12.3

Dice similarity measure of trapezoidal neutrosophic fuzzy numbers

In this section, we define the Dice similarity measure between two TrNFNs based on the EIs and EVs of TrNFNs. Let n~1 ¼ ðtn1 , in1 , fn1 Þ and n~2 ¼ ðtn2 , in2 , fn2 Þ be two TrNFNs, where tn1 ¼ ða11 , a12 , a13 ,a14 Þ, in1 ¼ ðb11 , b12 ,b13 , b14 Þ, and fn1 ¼ ðc11 , c12 , c13 , c14 Þ. tn2 ¼ ða21 , a22 , a23 ,a24 Þ, in2 ¼ ðb21 ,b22 ,b23 , b24 Þ, and fn2 ¼ ðc21 , c22 , c23 , c24 Þ are truthmembership function, indeterminacy-membership function, and falsity-membership function in the set of real numbers , respectively. If EIs of n~1 and n~2 are considered as the two vector presentations with three elements, the Dice similarity measure between TrNFNs n~1 and n~2 can be defined as follows:

Dice and Jaccard similarity measures

269

0

SD ð~ n1 , n~2 Þ ¼ 

¼0 @

1 ETL ð~ n1 ÞETL ð~ n2 Þ + ETU ð~ n1 ÞETU ð~ n2 Þ + EIL ð~ n1 ÞEIL ð~ n2 Þ @ A 2 F ð~ F ð~ F ð~ n1 ÞEIU ð~ n2 Þ + EF ð~ n ÞE n Þ + E n ÞE n Þ +EIU ð~ 1 2 1 2 L L U U

 : ETL ð~ n1 Þ2 + ðETU ð~ n1 ÞÞ2 + ðEIL ð~ n1 ÞÞ2 + ðEIU ð~ n1 ÞÞ2 + ðEF n1 ÞÞ2 + ðEF n1 ÞÞ2 L ð~ U ð~   n2 Þ2 + ðETU ð~ n2 ÞÞ2 + ðEIL ð~ n2 ÞÞ2 + ðEIU ð~ n2 ÞÞ2 + ðEF n2 ÞÞ2 + ðEF n2 ÞÞ2 + ETL ð~ L ð~ U ð~ 0 1 ða11 + a12 Þða21 + a22 Þ + ða13 + a14 Þða23 + a24 Þ B C B C 2B +ðb11 + b12 Þðb21 + b22 Þ + ðb13 + b14 Þðb23 + b24 Þ C @ A +ðc11 + c12 Þðc21 + c22 Þ + ðc13 + c14 Þðc23 + c24 Þ

ða11 + a12 Þ2 + ða13 + a14 Þ2 + ða21 + a22 Þ2 + ða23 + a24 Þ2 + ðb11 + b12 Þ2 + ðb13 + b14 Þ2

1: A

+ðb21 + b22 Þ2 + ðb23 + b24 Þ2 + ðc11 + c12 Þ2 + ðc13 + c14 Þ2 + ðc21 + c22 Þ2 + ðc23 + c24 Þ2 (12.16)

Proposition 12.1. The Dice similarity measure between two TrNFNs Sð~ n1 , n~2 Þ satisfies the following properties: 1. 0  SD ð~ n1 , n~2 Þ  1; 2. SD ð~ n1 , n~2 Þ ¼ SD ð~ n2 , n~1 Þ; and 3. SD ð~ n1 , n~2 Þ ¼ 1, if n~1 ¼ n~2 , that is, a1j ¼ a2j, b1j ¼ b2j and c1j ¼ c2j, where j ¼ 1, 2, 3, 4.

Proof 1. It is clear that 0  SD ð~ n1 , n~2 Þ. We must prove that SD ð~ n1 , n~2 Þ  1. If it is proved that the

n1 , n~2 Þ is greater than or equal to dividend of SD ð~ n1 , n~2 Þ, the denominator of SD ð~ proof is completed. We know that (x
y)2  0, for all x 2 . Since (x
y)2 ¼ x2 + y2
2xy  0, 2xy  x2 + y2. By using this inequality, we obtain the following equations: 2ða11 + a12 Þða21 + a22 Þ 2ða13 + a14 Þða23 + a24 Þ 2ðb11 + b12 Þðb21 + b22 Þ 2ðb13 + b14 Þðb23 + b24 Þ 2ðc11 + c12 Þðc21 + c22 Þ 2ðc13 + c14 Þðc23 + c24 Þ

 ða11 + a12 Þ2 + ða21 + a22 Þ2 ,  ða13 + a14 Þ2 + ða23 + a24 Þ2 ,  ðb11 + b12 Þ2 + ðb21 + b22 Þ2 ,  ðb13 + b14 Þ2 + ðb23 + b24 Þ2 ,  ðc11 + c12 Þ2 + ðc21 + c22 Þ2 ,  ðc13 + c14 Þ2 + ðc23 + c24 Þ2 :

Then, 2ðða11 + a12 Þða21 + a22 Þ + ða13 + a14 Þða23 + a24 Þ + ðb11 + b12 Þðb21 + b22 Þ + ðb13 + b14 Þ ðb23 + b24 Þ + ðc11 + c12 Þðc21 + c22 Þ + ðc13 + c14 Þðc23 + c24 ÞÞ  ðða11 + a12 Þ2 + ða21 + a22 Þ2 + ða13 + a14 Þ2 + ða23 + a24 Þ2 + ðb11 + b12 Þ2 + ðb21 + b22 Þ2 + ðb13 + b14 Þ2 + ðb23 + b24 Þ2 + ðc11 + c12 Þ2 + ðc21 + c22 Þ2 + ðc13 + c14 Þ2 + ðc23 + c24 Þ2 Þ:

270

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Thus, we have 0

1 ða11 + a12 Þða21 + a22 Þ + ða13 + a14 Þða23 + a24 Þ 2@ +ðb11 + b12 Þðb21 + b22 Þ + ðb13 + b14 Þðb23 + b24 Þ A +ðc11 + c12 Þðc21 + c22 Þ + ðc13 + c14 Þðc23 + c24 Þ

SD ð~ n1 , n~2 Þ ¼ 0

1 ða11 + a12 Þ2 + ða13 + a14 Þ2 + ða21 + a22 Þ2 + ða23 + a24 Þ2 @ +ðb11 + b12 Þ2 + ðb13 + b14 Þ2 + ðb21 + b22 Þ2 + ðb23 + b24 Þ2 A

 1:

+ðc11 + c12 Þ2 + ðc13 + c14 Þ2 + ðc21 + c22 Þ2 + ðc23 + c24 Þ2

2.

0

SD ð~ n1 , n~2 Þ ¼ 0

1 ða11 + a12 Þða21 + a22 Þ + ða13 + a14 Þða23 + a24 Þ 2@ +ðb11 + b12 Þðb21 + b22 Þ + ðb13 + b14 Þðb23 + b24 Þ A +ðc11 + c12 Þðc21 + c22 Þ + ðc13 + c14 Þðc23 + c24 Þ

1 ða11 + a12 Þ2 + ða13 + a14 Þ2 + ða21 + a22 Þ2 + ða23 + a24 Þ2 2 2 2 2 @ +ðb11 + b12 Þ + ðb13 + b14 Þ + ðb21 + b22 Þ + ðb23 + b24 Þ A 2 2 2 +ðc Þ2 011 + c12 Þ + ðc13 + c14 Þ + ðc21 + c22 Þ + ðc23 + c241 ða21 + a22 Þða11 + a12 Þ + ða23 + a24 Þða13 + a14 Þ 2@ +ðb21 + b22 Þðb11 + b12 Þ + ðb23 + b24 Þðb13 + b14 Þ A +ðc21 + c22 Þðc11 + c12 Þ + ðc23 + c24 Þðc13 + c14 Þ 1 ¼0 ða21 + a22 Þ2 + ða23 + a24 Þ2 + ða11 + a12 Þ2 + ða13 + a14 Þ2 2 2 2 2 @ +ðb21 + b22 Þ + ðb23 + b24 Þ + ðb11 + b12 Þ + ðb13 + b14 Þ A +ðc21 + c22 Þ2 + ðc23 + c24 Þ2 + ðc11 + c12 Þ2 + ðc13 + c14 Þ2 ¼ SD ð~ n2 , n~1 Þ:

3. Let n~1 ¼ n~2 . Then, 2ðða11 + a12 Þða21 + a22 Þ + ða13 + a14 Þða23 + a24 Þ + ðb11 + b12 Þðb21 + b22 Þ + ðb13 + b14 Þðb23 + b24 Þ + ðc11 + c12 Þðc21 + c22 Þ + ðc13 + c14 Þðc23 + c24 ÞÞ ¼ 2ðða11 + a12 Þ2 + ða13 + a14 Þ2 + ðb11 + b12 Þ2 + ðb13 + b14 Þ2 + ðc11 + c12 Þ2 + ðc13 + c14 Þ2 Þ and  ða11 + a12 Þ2 + ða13 + a14 Þ2 + ða21 + a22 Þ2 + ða23 + a24 Þ2 + ðb11 + b12 Þ2 + ðb13 + b14 Þ2

 + ðb21 + b22 Þ2 + ðb23 + b24 Þ2 + ðc11 + c12 Þ2 + ðc13 + c14 Þ2 + ðc21 + c22 Þ2 + ðc23 + c24 Þ2   ¼ ða11 + a12 Þ2 + ða13 + a14 Þ2 + ða11 + a12 Þ2 + ða13 + a14 Þ2 + ðb11 + b12 Þ2 + ðb13 + b14 Þ2 + ðb11 + b12 Þ2 + ðb13 + b14 Þ2 + ðc11 + c12 Þ2 + ðc13 + c14 Þ2 + ðc11 + c12 Þ2 + ðc13 + c14 Þ2  ¼ 2 ða11 + a12 Þ2 + ða13 + a14 Þ2 + ðb11 + b12 Þ2 + ðb13 + b14 Þ2 + ðc11 + c12 Þ2 + ðc13 + c14 Þ2 :

n1 , n~2 Þ ¼ 1: Thus, SD ð~

□

Dice and Jaccard similarity measures

271 



Definition 12.8. Let X ¼ {x1, x2, …, xn}, αi , and β i (i ¼ 1, 2, …, n) be TrNFNs and 















α ¼ fα1 , α2 , …, αn g and β ¼ fβ 1 , β 2 ,…,β n g be two TrNF-sets on X. Then, the Dice sim



ilarity measure between TrNF-sets α and β is defined as follows: SeD ðe α, e βÞ ¼

α i ÞETL ðe β i Þ + ETU ðe α i ÞETU ðe β i Þ + EIL ðe α i ÞEIL ðe βiÞ ETL ðe 2 I I F F F e α i ÞEU ðe β i Þ + EL ðe α i ÞEL ðe β i Þ + EU ðe α i ÞEF +EU ðe U ðβ i Þ

!

n 1X   : n ðe α Þ2 + ðETU ðe α i ÞÞ2 + ðEIL ðe α i ÞÞ2 + ðEIU ðe α i ÞÞ2 + ðEF α i ÞÞ2 + ðEF α i ÞÞ2 i¼1 ET L ðe U ðe L i  2 + ðEF ðe 2 e + ETL ðe β i Þ2 + ðETU ðe β i ÞÞ2 + ðEIL ðe β i ÞÞ2 + ðEIU ðe β i ÞÞ2 + ðEF ð β ÞÞ β ÞÞ i i L U

Proposition 12.2. The Dice similarity measure between two TrNF-sets e α and e β  e α , β Þ satisfies the following properties: denoted by S D ðe 





1. 0  SD ðα , β Þ  1; 

















2. SD ðα , β Þ ¼ SD ðβ , α Þ; and 



3. SD ðα , β Þ ¼ 1 for α ¼β , that is, αi ¼ βi, i ¼ 1, 2, …, n.

Proof The proof is obvious from Proposition 12.1.

□

In some applications, xi 2 X may have different weights. If we take the weight of i i i i , k12 ,k13 , k14 Þ, ðli11 , li12 ,li13 ,li14 Þ,ðmi11 ,mi12 ,mi13 , mi14 ÞÞ, then the xi 2 X as TrNFN wi ¼ ððk11 



weighted Dice similarity measures of TrNF sets α and β is defined as follows:    S WD ðα , β Þ

0

n 

1     ETL ð~aα i ÞETL ðe β i Þ + ETU ðα i ÞETU ðβ i Þ + EIL ðα i ÞEIL ðβ i Þ @ A 2    F   F  F ðα +EIU ðα i ÞEIU ðβ i Þ + EF ðα ÞE ðβ Þ + E ÞE ðβ Þ i i i i L L U U

¼  wi  i¼1

 :      2 F  2 ETL ðα i Þ2 + ðETU ðα i ÞÞ2 + ðEIL ðα i ÞÞ2 + ðEIU ðα i ÞÞ2 + ðEF L ðα i ÞÞ + ðEU ðα i ÞÞ        2 F  2 + ETL ðβ i Þ2 + ðETU ðβ i ÞÞ2 + ðEIL ðβ i ÞÞ2 + ðEIU ðβ i ÞÞ2 + ðEF L ðβ i ÞÞ + ðEU ðβ i ÞÞ

(12.17)



Here, wi values are obtained by using Eq. (30) in Biswas et al. [41] as follows: EV T ðwi Þ  wi ¼ X , n EV T ðwi Þ i¼1

where

(12.18)

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

3

kij

j¼1

EV ðwi Þ ¼ T

4 4 4 X X X kij + lij + mij j¼1

Pn

4 X

j¼1

! for i ¼ 1, 2, …,n:

j¼1





Note that i¼1 wi ¼ 1. If for i ¼ 1, 2, …, n, wi ¼ 1n, then Eq. (12.17) is reduced to Eq. (12.16).  Proposition 12.3. The weighted Dice similarity measure between two TrNF sets α and 







β denoted by SWD ðα , β Þ satisfies the following properties: 





1. 0  SWD ðα , β Þ  1; 











2. SWD ðα , β Þ ¼ SWD ðβ , α Þ; and 3.

   SWD ðα , β Þ ¼ 1,





if α ¼β , that is, αi ¼ βi, where i ¼ 1, 2, …, n.

Proof 











1. It is clear that 0  SWD ðα , β Þ . We must prove that SWD ðα , β Þ  1. From Proposition 

 





 

12.1, we have SD ðαi ,β i Þ  1 (i ¼ 1, 2, …, n). Since wi 2 ½0, 1, wi SD ðαi ,β i Þ  1 (i ¼ Pn  Pn      1 1, 2, …, n). Then, i¼1 w i SD ðα i , β i Þ  n. Thus, n i¼1 wi SD ðαi ,β i Þ  1 and 



SWD ðα,βÞ  1. It is concluded that 0  SWD ðα,βÞ  1:     2. By Proposition 12.1, we know that SD ðαi , β i Þ ¼ SD ðβ i ,αi Þ (i ¼ 1, 2, …, n). Then, it is obvi











ous that SWD ðα , β Þ ¼ SWD ðβ , α Þ.

3.

 

  Let αi ¼ β i

(i ¼ 1, 2, …, n). Then, from Proposition 12.1 we have that SD ðαi , β i Þ ¼ 1.  If it is multiplied by wi both sides of the equality, it can be obtained that  P  P  P       wi SD ðαi , β i Þ ¼ wi . Then, ni¼1 wi SD ðαi , β i Þ ¼ ni¼1 wi . Since ni¼1 wi ¼ 1, we have  P      SWD ðα , β Þ ¼ ni¼1 wi SD ðαi ,β i Þ ¼ 1:

□

12.4

Jaccard similarity measure of trapezoidal neutrosophic fuzzy numbers 

Definition 12.9. Ye [49]). Let a~ and b be two SVN-sets over universe



X ¼ fx1 ,x2 , …, xn g. Then the Jaccard similarity measure between SVN-sets a~ and b in the vector space is defined as follows: 

ð~ a, bÞJ ¼

n 1X ta ðxi Þtb ðxi Þ + ia ðxi Þib ðxi Þ + fa ðxi Þfb ðxi Þ  : n i¼1 ðt2a ðxi Þ + i2a ðxi Þ + fa2 ðxi ÞÞ + ðt2b ðxi Þ + i2b ðxi Þ + fb2 ðxi ÞÞ
ðta ðxi Þtb ðxi Þ + ia ðxi Þib ðxi Þ + fa ðxi Þfb ðxi ÞÞ

(12.19)

Dice and Jaccard similarity measures

273

We define the Jaccard similarity measure between two TrNFNs by a similar way to the previous definition. Let n~1 ¼ ðtn1 ,in1 , fn1 Þ and n~2 ¼ ðtn2 , in2 , fn2 Þ be two TrNFNs, where tn1 ¼ ða11 , a12 ,a13 , a14 Þ, in1 ¼ ðb11 ,b12 ,b13 , b14 Þ, and fn1 ¼ ðc11 , c12 ,c13 , c14 Þ. tn2 ¼ ða21 , a22 ,a23 , a24 Þ, in2 ¼ ðb21 , b22 ,b23 , b24 Þ, and fn2 ¼ ðc21 , c22 , c23 , c24 Þ are the truth-membership function, intermediate-membership function, and falsitymembership function in the set of real numbers , respectively. If the EIs of n~1 and n~2 are considered as the two vector presentations with three elements, the Jaccard similarity measure between n~1 and n~2 is defined as follows:  SJ ð~ n1 , n~2 Þ ¼ 0

ðETL ð~ n1 ÞETL ð~ n2 Þ + ETU ð~ n1 ÞETU ð~ n2 Þ + EIL ð~ n1 ÞEIL ð~ n2 Þ I I F F F n1 ÞEU ð~ n2 Þ + EL ð~ n1 ÞEL ð~ n2 Þ + EU ð~ n1 ÞEFU ð~ n2 ÞÞ +EU ð~



1 ðETL ð~ n1 ÞÞ2 + ðETU ð~ n1 ÞÞ2 + ðEIL ð~ n1 ÞÞ2 + ðEIU ð~ n1 ÞÞ2 + ðEFL ð~ n1 ÞÞ2 + ðEFU ð~ n1 ÞÞ2 2 2 2 2 2 2 T n ÞÞ + ðEI ð~ I n ÞÞ + ðEF ð~ F n ÞÞ C B ðET ð~ 2 2 2 L n2 ÞÞ + ðEU ð~ L n2 ÞÞ + ðEU ð~ B L n2 ÞÞ + ðEU ð~ C T T T I I @
ET ð~ A n2 Þ + EU ð~ n1 ÞEU ð~ n2 Þ + EL ð~ n1 ÞEL ð~ n2 Þ

L n1 ÞEL ð~ I ð~ I ð~ F ð~ F ð~ F ð~ F ð~ n ÞE n Þ + E n ÞE n Þ + E n ÞE n Þ +E L 1 L 2 U 1 U 2  U 1 U 2  ða11 + a12 Þða21 + a22 Þ + ða13 + a14 Þða23 + a24 Þ + ðb11 + b12 Þðb21 + b22 Þ +ðb13 + b14 Þðb23 + b24 Þ + ðc11 + c12 Þðc21 + c22 Þ + ðc13 + c14 Þðc23 + c24 Þ 1: ¼0 ða11 + a12 Þ2 + ða13 + a14 Þ2 + ða21 + a22 Þ2 + ða23 + a24 Þ2 B C 2 2 2 2 B +ðb11 + b12 Þ + ðb13 + b14 Þ + ðb21 + b22 Þ + ðb23 + b24 Þ C B C 2 2 2 2 B +ðc11 + c12 Þ + ðc13 + c14 Þ + ðc21 + c22 Þ + ðc23 + c24 Þ C B C @
ðða11 + a12 Þða21 + a22 Þ + ða13 + a14 Þða23 + a24 Þ + ðb11 + b12 Þðb21 + b22 Þ A +ðb13 + b14 Þðb23 + b24 Þ + ðc11 + c12 Þðc21 + c22 Þ + ðc13 + c14 Þðc23 + c24 ÞÞ (12.20)

Proposition 12.4. The Jaccard similarity measure between two TrNFNs n~1 and n~2 satisfies the following properties: 1. 0  SJ ð~ n1 , n~2 Þ  1; 2. SJ ð~ n1 , n~2 Þ ¼ SJ ð~ n2 , n~1 Þ; and 3. SJ ð~ n1 , n~2 Þ ¼ 1, if n~1 ¼ n~2 , that is, a1j ¼ a2j, b1j ¼ b2j, and c1j ¼ c2j, where j ¼ 1, 2, 3, 4.

Proof. 1. It is clear that the dividend of SJ ð~ n1 , n~2 Þ is greater than or equal to 0.

Also for all x, y 2  x2 + y2
xy  0. Then (a11 + a12)2 + (a21 + a22)2
(a11 + a12)(a21 + a22)  0, (a13 + a14)2 + (a23 + a24)2
(a13 + a14)(a23 + a24)  0, (b11 + b12)2 + (b21 + b22)2
(b11 + b12)(b21 + b22)  0, (b13 + b14)2 + (b23+b24)2
(b13 + b14) (b23 + b24)  0. Therefore, the denominator of SJ ð~ n1 , n~2 Þ is positive. Thus 0  SJ ð~ n1 , n~2 Þ. Since 2xy  x2 + y2, xy  x2 + y2
xy. Then, it is clear that the denominator of SJ ð~ n1 , n~2 Þ is n1 , n~2 Þ. Hence, it is concluded that greater than or equal to the dividend of SJ ð~ SJ ð~ n1 , n~2 Þ  1.

274

Optimization Theory Based on Neutrosophic and Plithogenic Sets

2. The symmetry of Eq. (12.20) validates the proof. 3. Let n~1 ¼ n~2 . Then Eq. (12.20) is as follows: 

 ða11 + a12 Þ2 + ða13 + a14 Þ2 + ðb11 + b12 Þ2 2 2 2 +ðb13 + b14 Þ + ðc11 + c12 Þ + ðc13 + c14 Þ 1 SJ ð~ n1 , n~2 Þ ¼ 0 2ða11 + a12 Þ2 + 2ða13 + a14 Þ2 2 2 C B +2ðb11 + b12 Þ + 2ðb13 + b14 Þ C B C B +2ðc11 + c12 Þ2 + 2ðc13 + c14 Þ2 C B @
ðða11 + a12 Þ2 + ða13 + a14 Þ2 + ðb11 + b12 Þ2 A +ðb13 + b14 Þ2 + ðc11 + c12 Þ2 + ðc13 + c14 Þ2 Þ ¼ 1:

Definition 12.10. Let X ¼ {x1, x2, …, xn},    α ¼ fα1 ,α2 , …,αn g

and

   β ¼ fβ 1 , β 2 ,…, β n g 

 αi ,

and

 βi

□

(i ¼ 1, 2, …, n) be TrNFNs and

be two TrNF sets on X. Then, the Jaccard 

similarity measure between TrNF sets α and β is defined as follows: 0 @







1





T T T I I ðET L ðα i ÞEL ðβi Þ + EU ðα i ÞEU ðβ i Þ + EL ðα i ÞEL ðβ i Þ

A

   F  F  F  n +EIU ðα i ÞEIU ðβ i Þ + EF    1X L ðα i ÞEL ðβ i Þ + EU ðα i ÞEU ðβ i ÞÞ : S J ðα , β Þ ¼ 0 T  2 T  2 I  2 I  2 F  2 F  21 n i¼1 ðEL ðα i ÞÞ + ðEU ðα i ÞÞ + ðEL ðα i ÞÞ + ðEU ðα i ÞÞ + ðEL ðα i ÞÞ + ðEU ðα i ÞÞ C B T  2      B ðE ðβ i ÞÞ + ðET ðβ i ÞÞ2 + ðEI ðβ i ÞÞ2 + ðEI ðβ i ÞÞ2 + ðEF ðβ i ÞÞ2 + ðEF ðβ i ÞÞ2 C C B L U L U L U C B     T  T  I T ðα I ðα C B
ðET ðα ÞE ðβ Þ + E ÞE ðβ Þ + E ÞE ðβ Þ i i i i i i A @ L L U U L L  I   F   F  I F F +EU ðα i ÞEU ðβ i Þ + EL ðα i ÞEL ðβ i Þ + EU ðα i ÞEU ðβ i ÞÞ

(12.21)





Proposition 12.5. The Jaccard similarity measure between two TrNF sets α and β satisfies the following properties: 





1. 0  SJ ðα , β Þ  1; 

















2. SJ ðα , β Þ ¼ SJ ðβ , α Þ; and 







3. SJ ðα , β Þ ¼ 1, if α ¼β , that is, αi ¼ β i , i ¼ 1, 2, …, n.

In some applications, xi 2 X (i ¼ 1, 2, …, n) may have different weights. If we take the i i i i , k12 , k13 ,k14 Þ, ðli11 ,li12 , li13 , li14 Þ,ðmi11 ,mi12 , mi13 ,mi14 ÞÞ, weight of xi 2 X as TrNFN wi ¼ ððk11 



then the weighted Jaccard similarity measures of TrNF sets α and β is defined as follows: ðETL ðn1 ÞETL ðn2 Þ + ETU ðn1 ÞETU ðn2 Þ + EIL ðn1 ÞEIL ðn2 Þ

!

F F F n X +EIU ðn1 ÞEIU ðn2 Þ + EF     L ðn1 ÞEL ðn2 Þ + EU ðn1 ÞEU ðn2 ÞÞ 1: S WJ ðα , β Þ ¼ wi 0  ðETL ð~ n1 ÞÞ2 + ðETU ð~ n1 ÞÞ2 + ðEIL ð~ n1 ÞÞ2 + ðEIU ð~ n1 ÞÞ2 + ðEF n1 ÞÞ2 + ðEF n1 ÞÞ2 i¼1 L ð~ U ð~

C B B C n ÞÞ2 + ðETU ð~ n2 ÞÞ2 + ðEIL ð~ n2 ÞÞ2 + ðEIU ð~ n2 ÞÞ2 + ðEF n2 ÞÞ2 + ðEF n2 ÞÞ2 C B ðETL ð~ L ð~ U ð~ B  2 C B C B
ET ðn1 ÞET ðn2 Þ + ET ðn1 ÞET ðn2 Þ + EI ðn1 ÞEI ðn2 Þ C L L U U L L @ A  F F F +EIU ðn1 ÞEIU ðn2 Þ + EF L ðn1 ÞEL ðn2 Þ + EU ðn1 ÞEU ðn2 Þ (12.22)

Dice and Jaccard similarity measures

275



Here, wi is calculated by using Eq. (12.18). P   Note that ni¼1 wi ¼ 1. If for i ¼ 1, 2, …, n, wi ¼ 1n, then Eq. (12.22) is reduced to Eq. (12.20).  Proposition 12.6. The weighted Jaccard similarity measure between two TrNF sets α 

and β satisfies the following properties: 





1. 0  SWJ ðα , β Þ  1; 











2. SWJ ðα , β Þ ¼ SWJ ðβ , α Þ; and 3.

   SWJ ðα , β Þ ¼ 1,









if α ¼β , that is, αi ¼ β i , i ¼ 1, 2, …, n.

Proof 











1. It is clear that 0  SWJ ðα , β Þ. We must prove that SWJ ðα , β Þ  1. From Proposition  





 

12.5, we have SJ ðαi , β i Þ  1 (i ¼ 1, 2, …, n). Since wi 2 ½0,1, wi SJ ðαi ,β i Þ  1 Pn  Pn      1 (i ¼ 1, 2, …, n). Then i¼1 w i SJ ðαi ,β i Þ  n. Thus, n i¼1 wi SJ ðαi , β i Þ  1 and 



SWJ ðα, βÞ  1. It is concluded that 0  SJW ðα, βÞ  1:     2. By Proposition 12.1, SJ ðαi ,β i Þ ¼ SJ ðβ i ,αi Þ (i ¼ 1, 2, …, n). Then it is obvious that 











SWJ ðα , β Þ ¼ SJW ðβ , α Þ.

 

  αi ¼ β i

(i ¼ 1, 2, …, n). Then, from Proposition 12.5, we have that SJ ðαi , β i Þ ¼ 1  (i ¼ 1, 2, …, n). If it is multiplied by wi both sides of the equality, it can be  P P  P        obtained wi SJ ðαi , β i Þ ¼ wi . Then ni¼1 wi SJ ðαi ,β i Þ ¼ ni¼1 wi . Since ni¼1 wi ¼ 1,    P    we get S ðα , β Þ ¼ ni¼1 wi SJ ðαi , β i Þ ¼ 1:

3. Let

□

Example 12.1. Let us consider TrNFNs n~1 ¼ hð0:1, 0:2, 0:4,0:5Þð0:2,0:4, 0:6, 0:7Þ ð0:1,0:4, 0:5,0:8Þi and n~2 ¼ hð0:2,0:4, 0:7,0:9Þð0:3, 0:5,0:6, 0:7Þð0:1,0:2, 0:3,0:5Þi. Let w1 ¼ h(0.3, 0.5, 0.8, 0.9)(0.1, 0.3, 0.6, 0.7)(0.2, 0.3, 0.6, 0.6)i and w2 ¼ h(0.5, 0.6, 0.7, 0.9)(0.3, 0.5, 0.6, 0.8)(0.2, 0.4, 0.7, 0.8)i. Then similarity measures are obtained as in Table 12.1:

Table 12.1 Similarity measures under a TrNF environment. Similarity measures

Values

n1 , n~2 Þ SD ð~ SWD ð~ n1 , n~2 Þ SJ ð~ n1 , n~2 Þ SWJ ð~ n1 , n~2 Þ

0.916 0.921 0.846 0.853

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

12.5

Multicriteria decision-making method

In this section, applications of weighted Dice and weighted Jaccard similarity measures in multicriteria decision-making problems under a TrNFN environment are given. Let us consider an MCDM problem with k alternatives and r criteria. Let A ¼ {A1, A2, …, Ak} be a set of alternatives and C ¼ {C1, C2, …, Cr} be a set of criteria. Alternatives Ai(i ¼ 1, 2, …, k) are characterized by TrNFNs for each Cj(j ¼ 1, 2, …, r) as follows: n o n~Ai ¼ h~ nj ¼ ðtinj , iinj , fnij Þi : j ¼ 1, 2, …,r , where tinj ¼ ðaij1 , aij2 , aij3 , aij4 Þ, iinj ¼ ðbij1 , bij2 , bij3 , bij4 Þ, and fnij ¼ ðcij1 , cij2 , cij3 , cij4 Þ. Step 1: Construction of the decision matrix with TrNFNs. The evaluation of the alternative Ai with respect to the criterion Cj made by an expert or decision maker can be briefly written as γij(i ¼ 1, 2, …, k; j ¼ 1, 2, …, r). Hence, the TrNFdecision matrix D ¼ [γij]k r can be constructed as follows: 0

γ11 γ12 B γ21 γ22 B D ¼ ½γij k r ¼ B ⋮ B⋮ @ γk1 γk2

⋯ ⋯ ⋯ ⋯

1 γ1r γ2r C C ⋮ C C: γkr A

Here, γij ¼ hðaij1 , aij2 , aij3 , aij4 Þ,ðbij1 , bij2 , bij3 , bij4 Þ,ðcij1 , cij2 , cij3 , cij4 Þi. Step 2: Determination of the weight of criteria with TrNFNs. Assume that the weight of each criterion determined by n decision makers is denoted by i i i i , kj2 , kj3 , kj4 Þ,ðlij1 , lij2 , lij3 , lij4 Þ,ðmij1 , mij2 , mij3 , mij4 Þiði ¼ 1,2, …,nÞ, ðj ¼ 1,2,…,rÞ. wj ¼ hðkj1 Then by using Eq. (12.18), the weight of a criterion Cj is calculated as follows: EV T ðwij Þ  wj ¼ X : n EV T ðwip Þ

(12.23)

p¼1

Here, 4 X 3 kij

EV T ðwij Þ ¼

i¼1

4 4 4 X X X kij + lij + mij i¼1

i¼1

:

i¼1

Step 3: Determination of benefit criteria (BC) and cost criteria (CC). In the MCDM environment, to characterize the best alternative properly in the decision set, the notion of the ideal point is used. In order to evaluate the criteria, two type modifiers called BC and CC are used generally.

Dice and Jaccard similarity measures

277

In this study, BC and CC ideal TrNF-values are defined by a similar way in Biswas et al. [41] as follows: l

γ∗j + ¼ hða∗j1 , a∗j2 , a∗j3 , a∗j4 Þ,ðb∗j1 , b∗j2 , b∗j3 , b∗j4 Þ,ðc∗j1 , c∗j2 , c∗j3 , c∗j4 Þi * ð max i ðaij1 Þ, max i ðaij2 Þ, max i ðaij3 Þ, max i ðaij4 ÞÞ, + ¼

ð min i ðbij1 Þ, min i ðbij2 Þ, min i ðbij3 Þ, min i ðbij4 ÞÞ,

, for j ¼ 1, 2,…, r, and

ð min i ðcij1 Þ, min i ðcij2 Þ, min i ðcij3 Þ, min i ðcij4 ÞÞ (12.24) l

γ∗
¼ hða∗j1 , a∗j2 , a∗j3 , a∗j4 Þ, ðb∗j1 , b∗j2 , b∗j3 , b∗j4 Þ, ðc∗j1 , c∗j2 , c∗j3 , c∗j4 Þi j * ð min i ðaij1 Þ, min i ðaij2 Þ, min i ðaij3 Þ, min i ðaij4 ÞÞ, + ¼

ð max i ðbij1 Þ, max i ðbij2 Þ, max i ðbij3 Þ, max i ðbij4 ÞÞ,

, for j ¼ 1, 2,…, r,

(12.25)

ð max i ðcij1 Þ, max i ðcij2 Þ, max i ðcij3 Þ, max i ðcij4 ÞÞ respectively. Here equations are called the positive ideal solution and negative ideal solution, respectively. Also a set of ideal TrNF values will be denoted by  ∗+ γ , if Cj is BC , n~* ¼ j∗
γj , if Cj is CC : Step 4: Determination of the weighted Dice and the weighted Jaccard similarity measures. For k alternatives and r criteria, by using Eqs. (12.26), (12.27), which are open expressions of Eqs. (12.17), (12.22), weighted Dice and weighted Jaccard similarity measures are obtained as follows: 0 1 ðaij1 + aij2 Þða∗j1 + a∗j2 Þ + ðaij3 + aij4 Þða∗j3 + a∗j4 Þ B C 2@ +ðbij1 + bij2 Þðb∗j1 + b∗j2 Þ + ðbij3 + bij4 Þðb∗j3 + b∗j4 Þ A i i ∗ ∗ i i ∗ ∗ r X +ðcj1 + cj2 Þðcj1 + cj2 Þ + ðcj3 + cj4 Þðcj3 + cj4 Þ  ∗  SWD ð~ nAi , n~ Þ ¼ wj 0  1 ðaij1 + aij2 Þ2 + ðaij3 + aij4 Þ2 + ðbij1 + bij2 Þ2 + ðbij3 + bij4 Þ2 j¼1 B C   B 2 2 2 2C B +ðcij1 + cij2 Þ + ðcij3 + cij4 Þ + ða*j1 + a*j2 Þ + ða*j3 + a*j4 Þ C @  A +ðb*j1 + b*j2 Þ2 + ðb*j3 + b*j4 Þ2 + ðc*j1 + c*j2 Þ2 + ðc*j3 + c*j4 Þ2 (12.26) and ! ðaij1 + aij2 Þða∗j1 + a∗j2 Þ + ðaij3 + aij4 Þða∗j3 + a∗j4 Þ + ðbij1 + bij2 Þðb∗j1 + b∗j2 Þ r +ðbij3 + bij4 Þðb∗j3 + b∗j4 Þ + ðcij1 + cij2 Þðc∗j1 + c∗j2 Þ + ðcij3 + cij4 Þðc∗j3 + c∗j4 Þ X  ∗  1, S WD ð~ nAi , n~ Þ ¼ wj 0  j¼1 ðaij1 + aij2 Þ2 + ðaij3 + aij4 Þ2 + ðbij1 + bij2 Þ2 + ðbij3 + bij4 Þ2 B C   B C B +ðcij1 + cij2 Þ2 + ðcij3 + cij4 Þ2 + ða*j1 + a*j2 Þ2 + ða*j3 + a*j4 Þ2 C B C  B C B +ðb* + b* Þ2 + ðb* + b* Þ2 + ðc* + c* Þ2 + ðc* + c* Þ2 C j3 j4 j1 j2 j3 j4 B  j1 j2 C B C B
ðai + ai Þða∗ + a∗ Þ + ðai + ai Þða∗ + a∗ Þ + ðbi + bi Þðb∗ + b∗ Þ C B j1 j2 j3 j4 j1 j2 C j1 j2 j3 j4 j1 j2  A @ +ðbij3 + bij4 Þðb∗j3 + b∗j4 Þ + ðcij1 + cij2 Þðc∗j1 + c∗j2 Þ + ðcij3 + cij4 Þðc∗j3 + c∗j4 Þ (12.27)

respectively.

278

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Step 1: Construction of the decision matrix with TrNFNs:

w~j =

...

...

g11 g12 ... g1r g21 g22 ... g2r ... gk1 gk2 ... gkr

...

D = [gij]k´r =

Step 2: Determination of the weight of criteria with TrNFNs:

Step 4: Determination of weighted Dice and Jaccard similarity measures: By using defined similarity measure methods, weighted Dice and Jaccard similarity measures are obtained for each alternative

Step 5: Ranking of alternatives:

Similarity measure values of alternatives are ordered from large to small

EVT (wij) n ∑ p =1 EVT (wip)

Step 3: Determination of benefit criteria (BC) and cost criteria (CC) (maxi (a ij1 ), maxi (a ij2), maxi (a ij3), maxi (a ij4)), (mini (b ij1 ), mini (b ij2), mini (a ij3), mini (bij4)), (mini (cij1 ), mini (cij2), mini (cij3), mini (cij4)) , (mini (a ij1 ), mini (a ij2), mini (a ij3), mini (a ij4)), (maxi (bij1 ), maxi (bij2), maxi (bij3), maxi (bij4)), (maxi (cij1 ), maxi (cij2), maxi (cij3), maxi (cij4))

Step 6: Choosing of optimum alternative alternatives:

Alternative of which similarity measures bigger that others is selected as optimum alternative

Fig. 12.1 Flowchart of the proposed method.

Step 5: Ranking of alternatives. Step 6: Choosing of optimum alternative.

A flowchart of the proposed method is depicted in Fig. 12.1.

12.6

Illustrative example

Let us consider the decision-making problem given in Biswas et al. [41]. We adapt this decision-making problem to a TrNFN environment. “Let us consider the decisionmaking problem in which a customer intends to buy a tablet from the set of primarily

Dice and Jaccard similarity measures

279

chosen five alternatives A ¼ {A1, A2, A3, A4, A5}. The customer takes into account the following four attributes: (1) (2) (3) (4)

features (C1); hardware (C2); affordable price (C3); and customer care (C4)” [41].

Assume that the weight vectors of four criteria provided by experts are expressed by the TrNFNs given as follows: w1 ¼ hð0:3, 0:5,0:8, 0:9Þ, ð0:1,0:3, 0:6, 0:7Þ, ð0:2,0:3, 0:6, 0:6Þi, w2 ¼ hð0:5, 0:6,0:7, 0:9Þ, ð0:3,0:5, 0:6, 0:8Þ, ð0, 2,0:4, 0:7, 0:8Þi, w3 ¼ hð0:6, 0:7,0:8, 0:9Þ, ð0:0,0:1:0,2, 0:3Þ, ð0:1,0:1, 0:2, 0:3Þi, w4 ¼ hð0:4, 0:6,0:7, 0:7Þ, ð0:2,0:3, 0:4, 0:5Þ, ð0:1,0:2, 0:3, 0:4Þi: The experts’ assessment of the five alternatives with respect to the four attributes is as follows: nA1 ¼ fhð0:1,0:2, 0:3,0:3Þ, ð0:0, 0:3,0:4, 0:4Þ, ð0:2, 0:5, 0:6,0:7Þi, hð0:4, 0:5, 0:6, 0:6Þ, ð0:1, 0:1, 0:4, 0:6Þ, ð0:3, 0:4, 0:4, 0:5Þi, hð0:2,0:2, 0:3,0:4Þ, ð0:5, 0:6,0:6, 0:8Þ, ð0:0, 0:2, 0:2, 0:5Þi, hð0:5, 0:5, 0:6, 0:6Þ, ð0:2,0:7, 0:7,0:7Þ, ð0:2, 0:3,0:3, 0:3Þig, nA2 ¼ fhð0:2,0:2, 0:4,0:4Þ, ð0:3, 0:3,0:5, 0:6Þ, ð0:1, 0:2, 0:2,0:5Þi, hð0:3, 0:5, 0:6, 0:7Þ, ð0:2, 0:2, 0:3, 0:4Þ, ð0:4, 0:5, 0:8, 0:9Þi, hð0:4,0:5, 0:5,0:7Þ, ð0:3, 0:3,0:4, 0:6Þ, ð0:2, 0:3, 0:4, 0:5Þi, hð0:1, 0:1, 0:2, 0:8Þ, ð0:6,0:6, 0:7,0:8Þ, ð0:0, 0:1,0:2, 0:4Þig, nA3 ¼ fhð0:5,0:7, 0:8,0:9Þ, ð0:2, 0:4,0:5, 0:8Þ, ð0:3, 0:3, 0:5,0:5Þi, hð0:1, 0:2, 0:2, 0:3Þ, ð0:2, 0:5, 0:6, 0:6Þ, ð0:1, 0:2, 0:3, 0:4Þi, hð0:3,0:3, 0:4,0:5Þ, ð0:1, 0:4,0:4, 0:6Þ, ð0:2, 0:2, 0:3, 0:7Þi, hð0:0, 0:2, 0:3, 0:9Þ, ð0:1,0:7, 0:7,0:8Þ, ð0:6, 0:7,0:7, 0:8Þig, nA4 ¼ fhð0:0,0:2, 0:3,0:7Þ, ð0:4, 0:5,0:6, 0:8Þ, ð0:4, 0:5, 0:5,0:9Þi, hð0:5, 0:5, 0:7, 0:8Þ, ð0:4, 0:5, 0:6, 0:6Þ, ð0:5, 0:6, 0:7, 0:8Þi, hð0:5,0:6, 0:6,0:9Þ, ð0:3, 0:5,0:5, 0:6Þ, ð0:1, 0:5, 0:5, 0:6Þi, hð0:5, 0:7, 0:8, 0:9Þ, ð0:5,0:6, 0:6,0:6Þ, ð0:2, 0:3,0:3, 0:3Þig, nA5 ¼ fhð0:2,0:4, 0:4,0:5Þ, ð0:3, 0:6,0:6, 0:9Þ, ð0:0, 0:2, 0:3,0:5Þi, hð0:1, 0:5, 0:7, 0:9Þ, ð0:2, 0:3, 0:3, 0:6Þ, ð0:6, 0:7, 0:7, 0:9Þi, hð0:4,0:4, 0:7,0:7Þ, ð0:1, 0:4,0:4, 0:7Þ, ð0:2, 0:4, 0:4, 0:6Þi, hð0:0, 0:1, 0:2, 0:3Þ, ð0:2,0:2, 0:4,0:5Þ, ð0:1, 0:1,0:3, 0:4Þig:

280

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Step 1: Construction of the decision matrix. By using the assessment of experts, the decision matrix is constructed as follows: 0

*

+ *

B ð0:1,0:2, 0:3,0:3Þ, B B ð0:0,0:3, 0:4,0:4Þ, B B ð0:2,0:5, 0:6,0:7Þ B B B B + B* B ð0:2,0:2, 0:4,0:4Þ, B B ð0:3,0:3, 0:5,0:6Þ, B B B ð0:1,0:2, 0:2,0:5Þ B B B B* + B B ð0:5,0:7, 0:8,0:9Þ, B D ¼ ½γij 5 4 ¼ B B ð0:2,0:4, 0:5,0:8Þ, B B ð0:3,0:3, 0:5,0:5Þ B B B* + B ð0:0,0:2, 0:3,0:7Þ, B B ð0:4,0:5, 0:6,0:8Þ, B B B ð0:4,0:5, 0:5,0:9Þ B B B B B* B ð0:2,0:4, 0:4,0:5Þ, + B B ð0:3,0:6, 0:6,0:9Þ, B B @ ð0:0,0:2, 0:3,0:5Þ

*

*

*

*

ð0:4,0:5,0:6,0:6Þ, ð0:1,0:1,0:4,0:6Þ, ð0:3,0:4,0:4,0:5Þ ð0:3,0:5,0:6,0:7Þ, ð0:2,0:2,0:3,0:4Þ, ð0:4,0:5,0:8,0:9Þ ð0:1,0:2,0:2,0:3Þ, ð0:2,0:5,0:6,0:6Þ, ð0:1,0:2,0:3,0:4Þ ð0:5,0:5,0:7,0:8Þ, ð0:4,0:5,0:6,0:6Þ, ð0:5,0:6,0:7,0:8Þ ð0:1,0:5,0:7,0:9Þ, ð0:2,0:3,0:3,0:6Þ, ð0:6,0:7,0:7,0:9Þ

+ *

+ *

+ *

+ *

+ *

ð0:2, 0:2,0:3,0:4Þ, ð0:5, 0:6,0:6,0:8Þ, ð0:0, 0:2,0:2,0:5Þ ð0:4, 0:5,0:5,0:7Þ, ð0:3, 0:3,0:4,0:6Þ, ð0:2, 0:3,0:4,0:5Þ ð0:3, 0:3,0:4,0:5Þ, ð0:1, 0:4,0:4,0:6Þ, ð0:2, 0:2,0:3,0:7Þ ð0:5, 0:6,0:6,0:9Þ, ð0:3, 0:5,0:5,0:6Þ, ð0:1, 0:5,0:5,0:6Þ ð0:4, 0:4,0:7,0:7Þ, ð0:1, 0:4,0:4,0:7Þ, ð0:2, 0:4,0:4,0:6Þ

+ *

+

+

+

+

1

ð0:5, 0:5,0:6,0:6Þ, ð0:2, 0:7,0:7,0:7Þ, ð0:2, 0:3,0:3,0:3Þ

C C C C C C C * +C C ð0:1, 0:1,0:2,0:8Þ, C C ð0:6, 0:6,0:7,0:8Þ, C C C ð0:0, 0:1,0:2,0:4Þ C C C C * +C C ð0:0, 0:2,0:3,0:9Þ, C C ð0:1, 0:7,0:7,0:8Þ, C C C ð0:6, 0:7,0:7,0:8Þ C C C * +C ð0:5, 0:7,0:8,0:9Þ, C C C ð0:5, 0:6,0:6,0:6Þ, C C ð0:2, 0:3,0:3,0:3Þ C C C C C * +C ð0:0, 0:1,0:2,0:3Þ, C C C ð0:2, 0:2,0:4,0:5Þ, C C ð0:1, 0:1,0:3,0:4Þ A

Step 2: Determination of the weight of criteria. By using Eq. (12.18), the weights of criteria are calculated as follows: EV T ðw1 Þ ¼ ¼ EV T ðw2 Þ ¼ ¼ EV T ðw3 Þ ¼ ¼ EV T ðw4 Þ ¼ ¼

3ð0:3 + 0:5 + 0:8 + 0:9Þ 0:3 + 0:5 + 0:8 + 0:9 + 0:1 + 0:3 + 0:6 + 0:7 + 0:2 + 0:3 + 0:6 + 0:6 7:5 ¼ 1:271, 5:9 3ð0:5 + 0:6 + 0:7 + 0:9Þ 0:5 + 0:6 + 0:7 + 0:9 + 0:3 + 0:5 + 0:6 + 0:8 + 0:3 + 0:4 + 0:7 + 0:8 8:1 ¼ 1:157, 7 3ð0:6 + 0:7 + 0:8 + 0:9Þ 0:6 + 0:7 + 0:8 + 0:9 + 0:0 + 0:1 + 0:2 + 0:3 + 0:1 + 0:1 + 0:2 + 0:3 9 ¼ 2:093, 4:2 3ð0:4 + 0:6 + 0:7 + 0:7Þ 0:4 + 0:6 + 0:7 + 0:7 + 0:2 + 0:3 + 0:4 + 0:5 + 0:1 + 0:2 + 0:3 + 0:4 7:2 ¼ 1:5: 4:8

Thus, 







w1 ¼ 0:211, w2 ¼ 0:192, w3 ¼ 0:348, w4 ¼ 0:249:

+C

Dice and Jaccard similarity measures

281

Step 3: Determination of BC and CC. When we consider all of the criteria as BC, by using Eqs. (12.24), (12.25), a set of positive ideal points n~* is obtained as follows: ∗

n~ ¼ fhð0:5, 0:7, 0:8,0:9Þ,ð0:0, 0:3,0:4,0:4Þ, ð0:0, 0:2,0:2,0:5Þi, hð0:5, 0:5,0:7,0:9Þ,ð0:1, 0:1,0:3,0:4Þ, ð0:1, 0:2,0:3,0:4Þi, hð0:5, 0:6,0:70:9Þ,ð0:1,0:3, 0,4,0, 6Þ,ð0:0,0:2, 0:2,0:5Þi, hð0:5, 0:7,0:8,0:9Þ,ð0:1, 0:2,0:4,0:5Þ, ð0:0, 0:1,0:2,0:3Þig: Step 4: Determination of the weighted Dice and the weighted Jaccard similarity measures.  ∗ S WD ð~ nA1 , n~ Þ ¼ 0:211

2ð0:3 1:2 + 0:6 1:7 + 0:3 0:3 + 0:8 0:8 + 0:7 0:2 + 1:3 0:7Þ 0:09 + 0:36 + 0:09 + 0:64 + 0:49 + 1:69 + 1:44 + 2:89 + 0:09 + 0:64 + 0:04 + 0:49

+ 0:192

2ð0:9 1:0 + 1:2 1:6 + 0:2 0:2 + 1:0 1:0 + 0:7 0:3 + 0:9 0:7Þ 0:81 + 1:44 + 0:04 + 1:00 + 0:49 + 0:81 + 1:00 + 2:56 + 0:04 + 0:49 + 0:09 + 0:49

+ 0:348

2ð0:4 1:1 + 0:7 1:6 + 1:1 0:4 + 1:4 1:0 + 0:2 0:2 + 0:7 0:7Þ 0:16 + 0:49 + 1:21 + 1:96 + 0:04 + 0:49 + 1:21 + 2:56 + 0:16 + 1:00 + 0:04 + 0:49

+ 0:249

2ð1:0 1:2 + 1:2 1:7 + 0:9 0:3 + 1:4 0:9 + 0:5 0:1 + 0:6 0:5Þ 1:00 + 1:44 + 0:81 + 1:96 + 0:25 + 0:36 + 1:44 + 2:89 + 0:09 + 0:81 + 0:01 + 0:25

¼ 0:211 0:706 + 0:192 0:950 + 0:348 0:801 + 0:249 0:905¼ 0:876,

and by using Eq. (12.26), the weighted Dice similarity measures of alternatives are obtained as follows: 

∗



∗



∗



∗

SWD ð~ nA2 , n~ Þ ¼ 0:904, SWD ð~ nA3 , n~ Þ ¼ 0:881, SWD ð~ nA4 , n~ Þ ¼ 0:927, SWD ð~ nA5 , n~ Þ ¼ 0:873: By using Eq. (12.27), the weighted Jaccard similarity measures of alternatives are obtained as follows:  ∗ S WJ ð~ nA1 , n~ Þ ¼ 0:211

+ 0:192

+ 0:348

+ 0:249

ð0:3 1:2 + 0:6 1:7 + 0:3 0:3 + 0:8 0:8 + 0:7 0:2 + 1:3 0:7Þ ð0:09 + 0:36 + 0:09 + 0:64 + 0:49 + 1:69 + 1:44 + 2:89 + 0:09 + 0:64 + 0:04 + 0:49Þ
ð0:3 1:2 + 0:6 1:7 + 0:3 0:3 + 0:8 0:8 + 0:7 0:2 + 1:3 0:7Þ ð0:9 1:0 + 1:2 1:6 + 0:2 0:2 + 1:0 1:0 + 0:7 0:3 + 0:9 0:7Þ

ð0:81 + 1:44 + 0:04 + 1:00 + 0:49 + 0:81 + 1:00 + 2:56 + 0:04 + 0:49 + 0:09 + 0:49Þ
ð0:9 1:0 + 1:2 1:6 + 0:2 0:2 + 1:0 1:0 + 0:7 0:3 + 0:9 0:7Þ ð0:4 1:1 + 0:7 1:6 + 1:1 0:4 + 1:4 1:0 + 0:2 0:2 + 0:7 0:7Þ ð0:16 + 0:49 + 1:21 + 1:96 + 0:04 + 0:49 + 1:21 + 2:56 + 0:16 + 1:00 + 0:04 + 0:49Þ
ð0:4 1:1 + 0:7 1:6 + 1:1 0:4 + 1:4 1:0 + 0:2 0:2 + 0:7 0:7Þ ð1:0 1:2 + 1:2 1:7 + 0:9 0:3 + 1:4 0:9 + 0:5 0:1 + 0:6 0:5Þ ð1:00 + 1:44 + 0:81 + 1:96 + 0:25 + 0:36 + 1:44 + 2:89 + 0:09 + 0:81 + 0:01 + 0:25Þ
ð1:0 1:2 + 1:2 1:7 + 0:9 0:3 + 1:4 0:9 + 0:5 0:1 + 0:6 0:5Þ

¼ 0:211 0:546 + 0:192 0905 + 0:348 0:668 + 0:249 0:827 ¼ 0:728,

!

!

!

!

282

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Table 12.2 Ranking of alternatives. Similarity measures

Values

Weighted Dice similarity measure Weighted Jaccard similarity measure

A4 A2 A3 A1 A5 A4 A2 A1 A5 A3

and in a similar way, the weighted Jaccard similarity measures of other alternatives are obtained as follows: 

∗



∗



∗



∗

SWJ ð~ nA2 , n~ Þ ¼ 0:762, SWJ ð~ nA3 , n~ Þ ¼ 0:719, SWJ ð~ nA4 , n~ Þ ¼ 0:803, SWJ ð~ nA5 , n~ Þ ¼ 0:722: Step 5: Ranking of alternatives. According to the results obtained from Step 4, the rankings of the alternatives are in Table 12.2. Step 6: Choosing of optimum alternative. It is obvious from Table 12.1 that A4 is the optimum alternative.

12.7

Comparative analysis of various similarity measures under a TrNF environment

In a TrNF environment, works on the MCDM method based on the similarity measure and TOPSIS method are new. Two of the basic works are Biswas et al. [41, 50]. Therefore, in this chapter, the proposed similarity measure methods are compared with Cosine similarity measures methods for TrNFNs given in Biswas et al. [41] and ranking-order method based on the relative closeness coefficient (RC) given in Biswas et al. [50]. Results of similarity measure in Section 12.6 and results obtained by using the relative closeness coefficient formula in Biswas et al. [50] and rankings of alternatives according to these methods are shown in Table 12.3. Note that it is shown that similarity measure methods proposed in the chapter are generally coherent by the Cosine similarity measure method and TOPSIS method for TrNF sets proposed by Biswas et al. [41, 50]. It is clear that the optimum alternative is A4 in all of the methods. Note that rankings of other alternatives are different from each other for different methods. Now we will give a new ranking method by using results in Table 12.3.

12.7.1 Ranking method of alternatives based on similarity measure methods In the literature, there are many ranking methods to select the optimum alternative based on similarity measures and some other methods under a neutrosophic environment. Sometimes, in different methods we encounter the different rankings of alternatives. In Table 12.3, we see that the optimum element is A4, the second element is A2, but in some rankings A1 is the third element, and in some rankings it is the fifth.

Dice and Jaccard similarity measures

283

Table 12.3 Rankings of alternatives according to similarity measures and relative closeness coefficients. Methods 

∗

SD ð~ nAi , n~ Þ 

∗

SWD ð~ nAi , n~ Þ 

∗

SJ ð~ nAi , n~ Þ 

∗

SWJ ð~ nAi , n~ Þ 

∗

SC ð~ nAi , n~ Þ [41] 

∗

SWC ð~ nAi , n~ Þ [41] 

R Cð~ nAi Þ [50] 

R CW ð~ nAi Þ [50]

i51

i52

i53

i54

i55

Rankings

0.841

0.846

0.816

0.870

0.812

0.876

0.904

0.881

0.927

0.873

0.737

0.742

0.708

0.780

0.701

0.728

0.762

0.719

0.803

0.722

0.846

0.852

0.823

0.884

0.844

0.837

0.863

0.828

0.896

0.857

0.385

0.423

0.423

0.756

0.410

0.349

0.433

0.407

0.784

0.417

A4 A2 A1

A3 A5 A4 A2 A3

A1 A5 A4 A2 A1

A3 A5 A4 A2 A1

A5 A3 A4 A2 A1

A5 A3 A4 A2 A5

A1 A3 A4 A2 ¼ A3

A5 A1 A4 A2 A3

A5 A1

The same situation is available for A3. In this part, we present a method to determine the order of an alternative. The following definition is given to explain some expressions and formulae required in the proposed ranking method. Definition 12.11. Let X ¼ {A1, A2, …, An} be a set and R ¼ {r1, r2, …, rm} be a set of rankings obtained by m different methods. Let eij denote the sum of the edge numbers in each of the orderings between Ai and Aj such that Ai Aj. Then the score matrix, denoted by SM, is defined as follows: 0

A1 B A1 e11 B B A2 e21 SM ¼ ½eij n n ¼ B B⋮ ⋮ B @ An en1

A2 e12 e22 ⋯ en2

⋯ ⋯ ⋯ ⋮ ⋯

1 An e1n C C e2n C C: C C enn A

Based on the score matrix, the score of each alternative, denoted by SAi (i ¼ 1, 2, …, n), is calculated by the following formula: SA i ¼

n X

eij :

j¼1

If SAi > SAj , it is said that Ai Aj.

(12.28)

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The procedures of the proposed method are shown as an algorithm as follows: Algorithm Algorithmic 1. Construct score matrix based on ranking orders. 2. Calculate score value of each alternative. 3. Rank alternatives according to score values.

12.7.2 Application Let us consider methods and orderings in Table 12.3. 

∗

Step 1. Since A4 A2 A1 A3 A5, for SD ð~ nAi , n~ Þ the edge number from A2 to A3 is 2. In a similar way,

 ∗ nAi , n~ Þ, SC ð~

  ∗ ∗ for SWD ð~ nAi , n~ Þ it is 1, for SJ ð~ nAi , n~ Þ,   ∗ is 3, SWC ð~ nAi , n~ Þ, it is 3, for RCð~ nAi Þ, it



∗

it is 2, for SWJ ð~ nAi , n~ Þ, it is 3, for 

it is 0, and for RCW ð~ nAi Þ, it is 1. Then e23 ¼ 2 + 1 + 2 + 3 + 3 + 3 + 0 + 1 ¼ 15. In a similar way, all of eij 8i, j 2{1, 2, 3, 4, 5} can be obtained. Thus, the score matrix is constructed as follows: 0 B A1 B B A2 B SM ¼ ½eij n n ¼ B B A3 B A4 B @ A5

A1 0 13 5 21 3

A2 0 0 0 8 0

A3 7 15 0 23 4

A4 0 0 0 0 0

1 A5 7C C 17 C C 6C C: 25 C C 0A

Step 2. From SM, by using formula (12.28), the score of each of the alternatives is obtained as follows: SA1 ¼ 14, SA2 ¼ 45, SA3 ¼ 11, SA4 ¼ 59, SA5 ¼ 7: Step 3. According to the score values of the alternatives, ranking is obtained as A4 A2 A1

A3 A5. 

∗



∗

Note that this ranking order matches with SD ð~ nAi , n~ Þ and SJ ð~ nAi , n~ Þ.

12.8

Conclusion

In this chapter, we proposed Dice, Jaccard, weighted Dice, and weighted Jaccard similarity measures based on the idea of the EV and EI of two TrNFNs and two TrFNSs. We developed an MCDM method and gave its application under a trapezoidal neutrosophic fuzzy environment. Finally, we gave an illustrative example to show the effectiveness of the proposed methods. There are many similarity measures and ranking methods of alternatives used in decision-making problems, but sometimes rankings obtained by using different methods may be different. By using rankings

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obtained by different methods, we proposed a method to rank alternatives precisely. We applied this method for rankings in Table 12.3. Researchers may extend our methods for hesitant neutrosophic sets of which elements are TrNFNs and trapezoidal neutrosophic soft sets.

Conflict of interests The authors declare that there is no conflict of interests regarding the publication of this chapter.

Ethical approval This chapter does not contain any studies with human participants or animals performed by any of the authors.

References [1] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353. [2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set Syst. 20 (1986) 87–96. [3] K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Set Syst. 31 (3) (1989) 343–349. [4] D. Dubois, H. Prade, Operation on fuzzy number, Int. J. Syst. Sci. 9 (1978) 613–626. [5] D. Dubois, H. Prade, The mean value of a fuzzy number, Fuzzy Sets Syst. 24 (1987) 279–300. [6] H.M. Nehi, H.R. Maleki, Intuitionistic fuzzy numbers and its applications in fuzzy optimization problem, in: Proceedings of the 9th WSEAS International Conference on Systems, Athens, Greece, 2005, pp. 1–5. [7] P. Grzegrorzewski, The hamming distance between intuitionistic fuzzy sets, in: Proceedings of the 10th IFSA World Congress, Istanbul, Turkey, 2013, pp. 35–38. [8] F. Smarandache, A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press, Rehoboth, DE, 1999. [9] F. Smarandache, Neutrosophic set—a generalization of the intuitionistic fuzzy set, Int. J. Pure Appl. Math. 24 (3) (2005) 287–297. [10] H. Wang, F. Smarandache, Y.Q. Zhang, R. Sunderraman, Single valued neutrosophic sets, Multispace Multistruct. 4 (2010) 410–413. [11] M. Abdel-Basset, M. Mohamed, Y. Zhou, I. Hezam, Multi-criteria group decision making based on neutrosophic analytic hierarchy process, J. Intell. Fuzzy Syst. 33 (6) (2017) 4055–4066. [12] M. Abdel-Basset, M. Mohamed, F. Smarandache, An extension of neutrosophic AHPSWOT analysis for strategic planning and decision-making, Symmetry 10 (2018) 116, https://doi.org/10.3390/sym10040116. [13] S. Pramanik, S. Dalapati, S. Alam, S. Smarandache, T.K. Roy, NS-cross entropy based MAGDM under single valued neutrosophic set environment, Information 9 (2) (2018) 37. [14] Z. Tian, H. Zhang, J. Wang, J. Wang, X. Chen, Multi-criteria decision-making method based on a cross-entropy with interval neutrosophic sets, Int. J. Syst. Sci. 47 (15) (2016) 3598–3608.

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[15] J. Ye, Single valued neutrosophic cross entropy for multicriteria decision making problems, Appl. Math. Model. 38 (2013) 1170–1175. [16] P. Biswas, S. Pramanik, B.C. Giri, Entropy based grey relational analysis method for multi-attribute decision making under single valued neutrosophic assessments, Neutrosophic Sets Syst. 2 (2014) 102–110. [17] P. Biswas, S. Pramanik, B.C. Giri, A new methodology for neutrosophic multi-attribute decision making with unknown weight information, Neutrosophic Sets Syst. 3 (2014) 42–52. [18] K. Mondal, S. Pramanik, Neutrosophic tangent similarity measure and its application to multiple attribute decision making, Neutrosophic Sets Syst. 9 (2015) 80–87. [19] K. Mondal, S. Pramanik, B.C. Giri, Interval neutrosophic tangent similarity measure based MADM strategy and its application to MADM problems, Neutrosophic Sets Syst. 19 (2018) 47–56, https://doi.org/10.5281/zenodo.1235201. [20] K. Mondal, S. Pramanik, B.C. Giri, Hybrid binary logarithm similarity measure for MAGDM problems under SVNS assessments. Neutrosophic Sets Syst. 20 (2018) 12–25, https://doi.org/10.5281/zenodo.1235365. [21] K. Mondal, S. Pramanik, B.C. Giri, Single valued neutrosophic hyperbolic sine similarity measure based MADM strategy. Neutrosophic Sets Syst. 20 (2018) 3–11, https://doi.org/ 10.5281/zenodo.1235383. [22] S. Pramanik, P. Biswas, B.C. Giri, Hybrid vector similarity measures and their applications to multi-attribute decision making under neutrosophic environment, Neural Comput. Appl. 28 (5) (2017) 1163–1176. [23] J. Ye, The Dice similarity measure between generalized trapezoidal fuzzy numbers based on the expected interval and its multicriteria group decision-making method, J. Chin. Inst. Ind. Eng. 29 (6) (2012) 375–382. [24] J. Ye, Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making, J. Intell. Fuzzy Syst. 26 (2014) 165–172. [25] J. Ye, A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets, J. Intell. Fuzzy Syst. 26 (2014) 2459–2466. [26] J. Ye, The generalized Dice measures for multiple attribute decision making under simplified neutrosophic environments, J. Intell. Fuzzy Syst. 31 (2016) 663–671. [27] R. Şahin, P.D. Liu, Correlation coefficients of single valued neutrosophic hesitant fuzzy sets and their applications in decision making, Neural Comput. Appl. 28 (6) (2017) 1387–1395. [28] P. Biswas, S. Pramanik, B.C. Giri, TOPSIS method for multi-attribute group decisionmaking under single valued neutrosophic environment, Neural Comput. Appl. 27 (3) (2016) 727–737. [29] R. Bausys, E.K. Zavadskas, Multicriteria decision making approach by VIKOR under interval neutrosophic set environment, Econom. Comput. Econom. Cybernet. Stud. Res. 4 (2015) 33–48. [30] I. Deli, Y. Subas, Some weighted geometric operators with SVTrN-numbers and their application to multi-criteria decision making problems, J. Intell. Fuzzy Syst. 32 (1) (2017) 291–301. [31] I. Deli, Y. Subas, A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems, Int. J. Mach. Learn. Cybernet. 8 (4) (2017) 1309–1322. [32] P. Ji, J.-Q. Wang, H.-Y. Zhang, Frank prioritized Bonferroni mean operator with singlevalued neutrosophic sets and its application in selecting third-party logistics providers, Neural Comput. Appl. (2016), https://doi.org/10.1007/s00521-016-2660-6.

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[33] P. Liu, Y. Chu, Y. Li, Y. Chen, Some generalized neutrosophic number Hamacher aggregation operators and their application to group decision making, Int. J. Fuzzy Syst. 16 (2) (2014) 242–255. [34] H. Zhan, J. Wang, X. Chen, An outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets, Neural Comput. Appl. 27 (3) (2016) 615–627. [35] S. Broumi, A. Bakali, M. Talea, F. Smarandache, A Matlab toolbox for interval valued neutrosophic matrices for computer applications, Uluslararas Ynetim Biliim Sistemleri ve Bilgisayar Bilimleri Dergisi 1 (1) (2017) 1–21. [36] S. Broumi, A. Bakali, M. Talea, F. Smarandache, R. Verma, Computing minimum spanning tree in interval valued bipolar neutrosophic environment, Int. J. Model. Optim. 7 (5) (2017) 300–304. [37] S. Broumi, A. Dey, A. Bakali, M. Talea, F. Smarandache, L.H. Son, D. Koley, Uniform single valued neutrosophic graphs, Neutrosophic Sets Syst. 17 (2017) 42–49. [38] A. Hassan, M.A. Malik, S. Broumi, A. Bakali, M. Talea, F. Smarandache, Special types of bipolar single valued neutrosophic graphs, Ann. Fuzzy Math. Inf. 14 (1) (2017) 55–73. [39] J. Ye, Trapezoidal fuzzy neutrosophic set and its application to multiple attribute decision making, Neural Comput. Appl. 26 (5) (2015) 1157–1166. [40] P. Biswas, S. Pramanik, B.C. Giri, Cosine similarity measure base multi-attribute decision making with trapezoidal fuzzy neutrosophic numbers, Neutrosophic Sets Syst. 8 (2014) 46–56. [41] A. Thamaraiselvi, R. Santhi, A new approach for optimization of real life transportation problem in neutrosophic environment, Math. Prob. Eng. (2016), https://doi.org/ 10.1155/2016/5950747. [42] R.-X. Liang, J.-Q. Wang, H.-Y. Zhang, A multi-criteria decision-making method based on single-valued trapezoidal neutrosophic preference relations with complete weight information, Neural Comput. Appl. 30 (2018) 3383–3398, https://doi.org/10.1007/s00521017-2925-8. [43] P. Biswas, S. Pramanik, B.C. Giri, Distance measure based MADM strategy with interval trapezoidal neutrosophic numbers, Neutrosophic Sets Syst. 19 (2018) 40–46. [44] P. Biswas, S. Pramanik, B.C. Giri, Multi-attribute group decision making based on expected value of neutrosophic trapezoidal numbers, in: New Trends in Neutrosophic Theory and Applications—vol. II, Pons Editions, Brussels, 2018, pp. 103–124. [45] D. Dubois, H. Prade, Ranking fuzzy number in the setting of possibility theory, Inf. Sci. 30 (1983) 183–224. [46] S. Heilpern, The expected value of fuzzy number, Fuzzy Sets Syst. 47 (1992) 81–86. [47] J. Ye, Single valued neutrosophic cross-entropy for multicriteria decision making problems, Appl. Math. Model. 38 (2014) 1170–1175. [48] J. Ye, Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making, Int. J. Fuzzy Syst. 16 (2) (2014) 204–211. [49] P. Biswas, S. Pramanik, B.C. Giri, TOPSIS strategy for multi-attribute decision making with trapezoidal neutrosophic numbers, Neutrosophic Sets Syst. 19 (2018) 29–39.

Further reading [50] J. Ye, Multicriteria decision-making method using the correlation coefficient under singlevalued neutrosophic environment, Int. J. General Syst. 42 (4) (2013) 386–394.

Multiobjective nonlinear bipolar neutrosophic optimization and its comparison with existing techniques

13

Sumbal Khalila, Florentin Smarandacheb, Sajida Kousara, Gul Freena a International Islamic University, Islamabad, Pakistan, bDepartment of Mathematics, University of New Mexico, Gallup, NM, United States

13.1

Introduction

Undoubtedly there are some situations for which high precision is required, but not many human problems require precision. Even to understand most physical processes we merely have to rely on imprecise human reasoning. This imprecision mostly carries very useful information. Until the early 19th century, uncertainty within the scientific community was considered an undesirable state that must be avoided at all costs. But gradually it was realized by physicists that Newtonian mechanics and its underlying calculus did not address the problems at the molecular level. It was a time when researchers started looking for new methods based on statistical mechanics so that particular demonstrations of microscopic entities could be replaced by statistical averages. These factual quantities, which outlined the activities of a large number of microscopic elements, could then be associated in a model with suitable macroscopic factors [1]. This statistical mechanics is based mostly on probability theory, which is capable of handling various uncertainties (random uncertainty). It was then, after the development of statistical mechanics, that uncertainties were taken into account. With the help of these, reliable solutions have been achieved as well as the amount of uncertainty being reckoned. It is a fact that from the late 19th century to the late 20th century, probability theory was the leading theory in quantifying uncertainty in scientific models. Black [2] was the first to challenge probability theory and introduces vagueness. Dempster [3], in the 1960s, for the first time included the concept of absence of information in his famous theory of evidence. In 1965, Zadeh [4] introduced his pivotal idea in logic that he named fuzzy set theory. Zadeh’s work not only influenced the concept of uncertainty but also challenged the claim of probability theory as a solitary representation for uncertainty. Zadeh questioned the binary (two-valued) logic of probability theory, and also illustrated special cases of fuzzy sets, which he called possibility theory. The 20th century witnessed the paradigm shift to address more kinds of uncertainties.

Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00013-5 © 2020 Elsevier Inc. All rights reserved.

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Glenn Shafer, in the 1970s, broadened Dempster’s work and developed a complete theory of evidence. His theory deals with information from more than one source. Later, in the 1980s, many researchers also integrated the concepts of evidence theory, possibility theory, and probability theory with the use of fuzzy measures [5]. Uncertainty in the sense of a philosophical theory of knowledge can be considered as the inverse of information. Information or data about a particular scientific or engineering problem might be inadequate, uncertain, fragmentary, questionable, dubious, conflicting, or lacking in some other way [1]. We can overcome uncertainty associated with any situation by acquiring more and more knowledge about the problem. Hence we become less uncertain about its formulation and solution. Uncertainty penetrates highly in ill-posed or complex problems. We can witness many forms of uncertainty around us: The most common of them are vague, fuzzy, ambiguous, and probability. In addition, we convert many physical problems to mathematical models after some experimentation. And wherever experimentation is involved, we have uncertainties there and need fuzzy concepts to cope with them. Continuously changing events or processes cannot always be defined as true or false, as in usual Boolean logic where we have only two values, true or false, 1 or 0. In fuzzy logic, we handle this problem by using a membership function assigned on the basis of the degree of truth or the degree to which a certain value is true or false. Zadeh [4] introduced the concept of a fuzzy set in which each object in a set assigns a grade of membership whose value lies between zero and one. He defined various properties in the context of fuzzy sets such as union, intersection, inclusion, relation, complement, convexity, etc. But not all scientific and physical models rely only on true grades of membership. Situations may arise where the degree of nonmembership is also required with that of the membership degree. Furthermore, from a designer or decision-maker’s perspective, input parameters of many problems may end up yielding imprecise parameters with more than one type of uncertainty and many key governing factors. These factors may include the degree of acceptance for any parameter, rejection degree [2], or hesitancy [6], indeterminacy, neutrality, falsity [7], etc. A wide assortment of human decision making, particularly multiagent choice and decision examination, depends on bipolar or twofold judgmental reasoning on a negative side and a positive side. The concept of bipolarity along with fuzziness may also arise in many physical applications and structural design problems, such as impact and symptom, collaboration and competition, feedback and feedforward, companionship and enmity, mutual benefits and conflict benefits, and many more [8]. Hence one can apply the fuzzy and generalized fuzzy system to the situations where uncertainty in the information lies either due to vagueness, or fuzziness, or imprecision, or due to indeterminacy, hesitancy, neutrality, bipolarity, or even when no information, as well as model, exists at all. This system also works very well in situations with continuously varying input labels. The system is rich enough to tackle uncertainties contained not only in inputs but also in the output of the system. That distinction is achieved by inculcating automation in the circumstances. Hence self-referent adjustments determined by specific guidelines make the fuzzy and generalized fuzzy system very successful. Optimization methods and theory play an important role in the variety of fields to deal with different real-life as well as decision-making problems. Many different

Multiobjective nonlinear bipolar neutrosophic optimization

291

techniques have been introduced to deal with engineering optimization problems. In the last few years, optimization methodology has repeatedly been given attention because of the advancements in computing technologies and the growing dependency on optimization-based problems in real life. Many remarkable proposals have been introduced by many researchers. However, it has been observed that decision-making processes involve the collection of information from different sources, at any rate to a limited extent, fuzzy in nature. Therefore, a wide assortment of design and decisionmaking processes often use fuzzy set-theoretic concepts, and fuzzy techniques are utilized as a helpful instrument to handle those circumstances which cannot be taken care of with established methods. Many concrete issues that might be thrown into an optimization setting are prevalent with imprecision sources. Most of the time, it is not useful to choose exact conditions, as many of these are picked up by estimation, or by analysts’ perceptions. This demonstrates the persistent need to improve optimization models’ authenticity by making it possible to merge uncertainty into mathematical programming systems and ascend to the domain of optimization under uncertainty. We could use already established optimization methods with definite and welldefined values and constraints. Linear programming is the most abundant technique for applied optimization in real-life problems. Yet if at any point we go over such a system where there is imprecision present in it, connecting the expression “imprecision” with “Fuzzy,” we think of fuzzy optimization as a final resort for our problem. Fuzzy optimization is a more recent approach to optimization under uncertainty in which imprecision is modeled by fuzzy relations and/or fuzzy parameters evolved from fuzzy sets. Fuzzy optimization techniques solve the systems involving ambiguity better than any stochastic optimization based on probability. Such systems prove fuzzy optimizations as an efficient and superior tool. Bellman and Zadeh [8a] introduce the concept of decision making in fuzzy environment. Atanassov [9] conflated the concepts of fuzziness and programming. They have proposed the basic concepts of fuzzy goals, fuzzy constraints, and fuzzy decisions. In previous optimization methods, all parameters are assumed to be exactly known and fixed. But these types of assumptions are not adequate to tackle various problems of real life where most parameters are inexact and inexplicit. The hypothesis of fuzzy linear programming was additionally improved by Tanaka and Asai [10]. They have highlighted the idea of a level set to outspread some of the standard results to problems comprising fuzzy objective functions and constraints. Zimmermann [11] modeled such situations and acquainted the concept of the fuzzy linear programming problem with numerous objective functions. One can consider Zimmerman’s work as an extension of Bellman’s. Tanaka and Asai [12] considered fuzzy numbers in the models of decision-making problems. Werners [13] in his work gave the idea of the formation of the fuzzy models and suggested the construction of membership functions under fuzzy constraints. Xu [14] has looked into the two-phase method for fuzzy programming of structures. Shih et al. [15] have created and proposed a three alphalevel cuts approach for solving structural engineering problems with fuzzy properties. In 25-bar and 72-bar truss design problems, Shih and Lee [16] have exhibited an altered double cuts approach for extensive scale fuzzy optimization. It has been established that the methodology is reliable for basic ideal structural design with fuzzy

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properties. Asimakopoulou et al. [17] have displayed another system for foreseeing the basic flashover voltage of contaminated covers in view of fuzzy logic. Uncertainty with a grade of rejection of objects as well as constraints are considered together by Angelov [18]. He formulated the intuitionistic fuzzy optimization problem and explained the construction of membership function in the intuitionistic environment. It is remarkable to note that in many optimization problems, experts’ estimations regarding parameters of optimization are different, and instead of a single grade of membership, a collection of membership grades are required. To deal with the hesitancy of information in decision-making and approximate reasoning, Pei and Yi [19] introduced some new operations and also investigated algebraic structures of hesitant fuzzy sets. A computational algorithm employing hesitant fuzzy optimization of multiobjective models was introduced by Bharati and Hesitant [20] and Bharati [21]. He also presented an effective solution procedure for hesitant optimization problems. Roy and Dey [21a,21b] used the concept of multiobjective neutrosophic optimization to structure design as well as welded beam optimization problem. Garg [21c] solved nonlinear programming problems under interval neutrosophic environment. Roy and Das [21d] used multiobjective nonlinear neutrosophic optimization technique to solve riser design problem. New and improved optimization models are required with each advancement and extension of the fuzzy system. Our motivation is to develop refined neutrosophic and bipolar neutrosophic models that could be adequately applied in the decision-making process when it is required to express an ill-known quantity with some uncertain numerical value or the situations where decision makers have to abstain from expressing their assessments. Neutrosophic sets (NS) can be refined in a number of ways [22]; in our present work, we only focus on refining NS by splitting indeterminacy in two ways. Different operators of extended concepts in neutrosophy have been defined and verified. Development of the computational algorithm for solving the nonlinear optimization problem in a refined neutrosophic environment as well as a bipolar neutrosophic one is the objective of the present work. We also aim to check the impact of bipolar truth membership, bipolar indeterminacy membership, bipolar falsity membership, and refined indeterminacies in such optimization processes. Further, we have made a comparative study in intuitionistic fuzzy, neutrosophic fuzzy, refined neutrosophic fuzzy, and bipolar neutrosophic fuzzy optimization techniques. Conclusions have been drawn on the basis of the present work and the future scope has been presented.

13.2

Preliminaries

13.2.1 Neutrosophic set N A neutrosophic set A^ in a universe of discourse X is of the form N A^ ¼ hx, TA^N ðxÞ, IA^N ðxÞ, FA^N ðxÞi,

Multiobjective nonlinear bipolar neutrosophic optimization

293

where TA^N ðxÞ, IA^N ðxÞ, and FA^N ðxÞ represent degree of truth membership, degree of falsity membership, and degree of indeterminacy membership, respectively, whose sum is less than or equal to 3 and TA^N , IA^N , FA^N : X ! ½0,1:

13.2.2 Bipolar fuzzy set A bipolar fuzzy set A^

BF

in a space of points X has a positive membership degree μ +^BF A

and negative membership degree μ^BF , where μ+^BF : X ! ½0, 1 and μ^BF : X ! ½1, 0. A

A

A

13.2.3 Bipolar neutrosophic set BN For a nonempty set X, a bipolar neutrosophic set A^ is defined as

D E BN A^ ¼ x, T +^BN ðxÞ, I +^BN ðxÞ,F+^BN ðxÞ,T ^BN ðxÞ,I ^BN ðxÞ, F^BN ðxÞ , A

A

A

A

A

A

where the positive membership degree T +^BN ðxÞ,I +^BN ðxÞ, F+^BN ðxÞ represents the truth, A

A

A

BN indeterminacy, and falsity membership of a generic element analogous to set A^ and T ^BN ðxÞ,I ^BN ðxÞ, F^BN ðxÞ represents the truth, indeterminacy, and falsity memberA

A

A

BN ship of a generic element to some implicit counter-property analogous to a set A^ , where

T +^BN ,I +^BN ,F+^BN : X ! ½0, 1, A A A T ^BN ,I ^BN ,F^BN : X ! ½1, 0: A

A

A

BN BN BN Theorem 13.1. Let A^ and B^BN be two bipolar neutrosophic sets. Then A^  A^ if and only if

T +^BN ðxÞ  TB+^BN ðxÞ,I +^BN ðxÞ  IB+^BN ðxÞ, F+^BN ðxÞ  FB+^BN ðxÞ, A A A T ^BN ðxÞ  TB^BN ðxÞ,I ^BN ðxÞ  IB^BN ðxÞ, F^BN ðxÞ  F ðxÞ, B^BN A

A

A

for all x 2 X. BN BN Theorem 13.2. Let A^ and B^BN be two bipolar neutrosophic sets. Then A^ ¼ B^BN if and only if

T +^BN ðxÞ ¼ TB+^BN ðxÞ,I +^BN ðxÞ ¼ IB+^BN ðxÞ, F+^BN ðxÞ ¼ FB+^BN ðxÞ, A A A T ^BN ðxÞ ¼ TB^BN ðxÞ,I ^BN ðxÞ ¼ IB^BN ðxÞ, F^BN ðxÞ ¼ F ðxÞ, B^BN A

for all x 2 X.

A

A

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Union, intersection, and complement of bipolar neutrosophic sets are already available in the literature [23]. Since the focus of the current study is to develop an algorithm for the optimization problem, according to requirement union, intersection, and complement of bipolar neutrosophic sets are therefore redefined here. BN The union of two bipolar neutrosophic sets A^ ¼ hx,T +BN ðxÞ,I +BN ðxÞ, A^

A^

F+^BN ðxÞ,T ^BN ðxÞ, I ^BN ðxÞ,F^BN ðxÞi and B^BN ¼ hx, TB+^BN ðxÞ, IB+^BN ðxÞ,FB+^BN ðxÞ, TB^BN ðxÞ, A

A

A

A

BN BN BN ðxÞi is a bipolar neutrosophic set U^ , where U^ ¼ A^ [ B^BN : IB^BN ðxÞ, F B^BN

  T +^BN ðxÞ ¼ max T +^BN ðxÞ,TB+^BN ðxÞ ; U A   T ^BN ðxÞ ¼ min T ^BN ðxÞ, TB^BN ðxÞ ; U A   + + I ^BN ðxÞ ¼ max I ^BN ðxÞ,IB+^BN ðxÞ ; U A     I ^BN ðxÞ ¼ min I ^BN ðxÞ, IB^BN ðxÞ ; U A   + F ^BN ðxÞ ¼ min F+^BN ðxÞ, FB+^BN ðxÞ ; U A   F^BN ðxÞ ¼ max F^BN ðxÞ,F ðxÞ : BN ^ B U

A

BN The intersection of two bipolar neutrosophic sets A^ ¼ hx,T +^BN ðxÞ,I +^BN ðxÞ, F+^BN ðxÞ, A

A

A

ðxÞi T ^BN ðxÞ,I ^BN ðxÞ, F^BN ðxÞi and B^BN ¼ hx, TB+^BN ðxÞ, IB+^BN ðxÞ, FB+^BN ðxÞ, TB^BN ðxÞ, IB^BN ðxÞ, F B^BN A

A

A

BN is a bipolar neutrosophic set V^BN , where V^BN ¼ A^ \ B^BN :

  TV+^BN ðxÞ ¼ min T +^BN ðxÞ, TB+^BN ðxÞ ; A    TV^BN ðxÞ ¼ max T ^BN ðxÞ, TB^BN ðxÞ ; A   IV+^BN ðxÞ ¼ min I +^BN ðxÞ,IB+^BN ðxÞ ; A    IV^BN ðxÞ ¼ max I ^BN ðxÞ, IB^BN ðxÞ ; A   + FV^BN ðxÞ ¼ max F+^BN ðxÞ, FB+^BN ðxÞ ; A     F ðxÞ ¼ min F BN ðxÞ, F ^BN ðxÞ : BN ^ ^ V B A

BN The Complement of a bipolar neutrosophic set A^ ¼ hx, T +^BN ðxÞ, I +^BN ðxÞ, F+^BN ðxÞ, BN T ^BN ðxÞ,I ^BN ðxÞ, F^BN ðxÞi is denoted by C^ , where A

A

A

A

A

A

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T +^BN ðxÞ ¼ F+^BN ðxÞ; C

A

I +^BN ðxÞ ¼ 1+  I +^BN ðxÞ; A C F+^BN ðxÞ ¼ T +^BN ðxÞ; C

A

T ^BN ðxÞ ¼ F^BN ðxÞ; A C I ^BN ðxÞ ¼ 1  I ^BN ðxÞ; C

A

F^BN ðxÞ ¼ T ^BN ðxÞ: A C Example 13.1. Let X ¼ {x1, x2}. Then BN A^ ¼ fhx1 ,0:8, 0:3, 0:6,  0:3,  0:2,  0:02i, hx2 , 0:4, 0:5,0:1,  0:1,  0:7,  0:03ig

and B^BN ¼ fhx1 ,0:2,0:3, 0:4,  0:5,  0:033,  0:99i,hx2 , 0:7,0:8,0:3,  0:08,  0:97,  0:5ig:

It can be easily verified that bipolar neutrosophic sets obeys many of the properties like distributivity, associativity, absorption, idempotency, involution, and De Morgan’s laws, whereas same like fuzzy sets it does not satisfy the principle of the excluded middle.

13.3

Some refinements on neutrosophic sets

Neutrosophic sets can be extended and refined in a number of ways [22]. In our present work, we only focus on refining neutrosophic sets by splitting indeterminacy in two ways and bipolar neutrosophic sets. The first type of refinement is the case when the indeterminate membership grade is split into U ¼ uncertainty and G ¼ ignorance ¼ T _ F. Hence, the resultant extended neutrosophic membership grades are comprised of T, U, G, and F. Throughout this chapter, indeterminacy of this type is taken as case i for convenience. The second type of refinement is the case when the indeterminate membership grade is split into C ¼ contradiction ¼ T ^ F and G ¼ ignorance ¼ T _ F. Hence the resultant extended neutrosophic membership grades comprises of T, C, G, and F. For convenience, this type of extended neutrosophic sets is taken as case ii and bipolar neutrosophic sets as case iii in this chapter. RNi Case i: Let A^ represent the case i of a four-valued refined neutrosophic set. Then RNi A^ ¼ fðx, TA^RNi ðxÞ,UA^RNi ðxÞ, GA^RNi ðxÞ,FA^RNi ðxÞÞ : x 2 Xg:

296

Optimization Theory Based on Neutrosophic and Plithogenic Sets

When X is continuous, then RNi A^ ¼

Z X

fx,TA^RNi ðxÞ, UA^RNi ðxÞ,GA^RNi ðxÞ,FA^RNi ðxÞ=dx : x 2 Xg,

and when X is discrete, its representation will be RNi A^ ¼

n X fTA^RNi ðxi Þ,UA^RNi ðxi Þ,GA^RNi ðxi Þ,FA^RNi ðxi Þ=xi : xi 2 Xg: i¼1

The Complement of this refinement on the neutrosophic set is denoted by C^RNi and is defined as TC^RN ðxÞ ¼ FA^RNi ðxÞ; i

UC^RN ðxÞ ¼ 1  UA^RNi ðxÞ; i

GC^RN ðxÞ ¼ 1  GA^RNi ðxÞ; i

FC^RN ðxÞ ¼ TA^RNi ðxÞ, i

for all x 2 X. RNi The union of two extended refined neutrosophic sets A^ and B^RNi is again an extended RNi RNi RNi refined neutrosophic set C^ , indicated as C^ ¼ A^ [ B^RNi , whose membership functions are defined as   TC^RNi ðxÞ ¼ max TA^RNi ðxÞ, TB^RNi ðxÞ ;   UC^RNi ðxÞ ¼ max UA^RNi ðxÞ,UB^RNi ðxÞ ;   GC^RNi ðxÞ ¼ max GA^RNi ðxÞ,GB^RNi ðxÞ ;   FC^RNi ðxÞ ¼ min FA^RNi ðxÞ,FB^RNi ðxÞ : RNi and B^RNi is again an The intersection of two extended refined neutrosophic sets A^ RNi RNi RNi RN ^ ^ extended refined neutrosophic set C , indicated as C ¼ A^ \ B^ i , whose membership functions is defined as   TC^RNi ðxÞ ¼ min TA^RNi ðxÞ,TB^RNi ðxÞ ;   UC^RNi ðxÞ ¼ min UA^RNi ðxÞ, UB^RNi ðxÞ ;   GC^RNi ðxÞ ¼ min GA^RNi ðxÞ, GB^RNi ðxÞ ;   FC^RNi ðxÞ ¼ max FA^RNi ðxÞ,FB^RNi ðxÞ : RNii represent the case ii of a refined neutrosophic set. Then Case ii: Let A^ RNii A^ ¼ fðx,TA^RNii ðxÞ, CA^RNii ðxÞ,GA^RNii ðxÞ,FA^RNii ðxÞÞ : x 2 Xg:

Multiobjective nonlinear bipolar neutrosophic optimization

297

When X is continuous, then RNii A^ ¼

Z X

fx, TA^RNii ðxÞ,CA^RNii ðxÞ, GA^RNii ðxÞ, FA^RNii ðxÞ=dx : x 2 Xg,

and when X is discrete, its representation will be RNii A^ ¼

n X fTA^RNii ðxi Þ,CA^RNii ðxi Þ, GA^RNii ðxi Þ,FA^RNii ðxi Þ=xi : xi 2 Xg: i¼1

The Complement of this refinement on the neutrosophic set is denoted by C^RNii and components of its membership grades are defined as TC^RN ðxÞ ¼ FA^RNii ðxÞ; ii

CC^RN ðxÞ ¼ 1  CA^RNii ðxÞ; ii

GC^RN ðxÞ ¼ 1  GA^RNii ðxÞ; ii

FC^RN ðxÞ ¼ TA^RNii ðxÞ, ii

for all x 2 X. The union of two extended refined neutrosophic sets A^ and B^RNii is again an extended RNii RNii RNii refined neutrosophic set C^ , indicated as C^ ¼ A^ [ B^RNii , whose membership functions is defined as   TC^RNii ðxÞ ¼ max TA^RNii ðxÞ,TB^RNii ðxÞ ;   CC^RNii ðxÞ ¼ max CA^RNii ðxÞ,CB^RNii ðxÞ ;   GC^RNii ðxÞ ¼ max GA^RNii ðxÞ,GB^RNii ðxÞ ;   FC^RNii ðxÞ ¼ min FA^RNii ðxÞ,FB^RNii ðxÞ : RNii

RNii and B^RNii is again an The intersection of two extended refined neutrosophic sets A^ RNii RNii RNii extended refined neutrosophic set C^ , indicated as C^ ¼ A^ \ B^RNii , whose membership functions are defined as   TC^RNii ðxÞ ¼ min TA^RNii ðxÞ, TB^RNii ðxÞ ;   CC^RNii ðxÞ ¼ min CA^RNii ðxÞ, CB^RNii ðxÞ ;   GC^RNii ðxÞ ¼ min GA^RNii ðxÞ,GB^RNii ðxÞ ;   FC^RNii ðxÞ ¼ max FA^RNii ðxÞ, FB^RNii ðxÞ :

It can be easily verified that extended refined neutrosophic sets (both case i and case ii) obey many of the properties like distributivity, associativity, absorption, idempotency, involution, and De Morgan’s laws, whereas same like fuzzy sets it does not satisfy the principle of the excluded middle.

298

Optimization Theory Based on Neutrosophic and Plithogenic Sets

13.4

Extended neutrosophic optimization and bipolar neutrosophic optimization technique

Consider a nonlinear multiobjective optimization problem Minimize ff^i ðxÞg i ¼ 1,2, …,p such that g^j ðxÞ  b^j j ¼ 1,2, …, q, where x are decision variables, f^i ðxÞ represents objective functions, g^j ðxÞ represents the constraint functions, and p and q represent the number of objective functions and constraints, respectively. ^ a confluence of extended neutrosophic objectives and conCase i: Now the decision set D, straints, is defined as p D^ ¼ ð\k¼1 O^k Þ \ ð\qj¼1 L^j Þ ¼ fðx,TD^ ,GD^ ,UD^ ,FD^ Þg,

where

  TD^ ðxÞ ¼ min TO^1 ðxÞ,TO^2 ðxÞ,…,TO^p ðxÞ;TL^1 ðxÞ,TL^2 ðxÞ, …,TL^q ðxÞ ;   GD^ ðxÞ ¼ min GO^1 ðxÞ, GO^2 ðxÞ,…, GO^p ðxÞ;GL^1 ðxÞ, GL^2 ðxÞ, …,GL^q ðxÞ ;   UD^ ðxÞ ¼ min UO^1 ðxÞ, UO^2 ðxÞ,…, UO^p ðxÞ;UL^1 ðxÞ, UL^2 ðxÞ, …,UL^q ðxÞ ;   FD^ ðxÞ ¼ max FO^1 ðxÞ, FO~2 ðxÞ, …,FO^p ðxÞ;FL^1 ðxÞ,FL^2 ðxÞ, …,FL^q ðxÞ ,

for all x 2 X, where TD^ ,GD^ , UD^ , and FD^ represent truth, ignorance, uncertainty, and falsity grade of an extended refined neutrosophic decision set, respectively. Now extended refined neutrosophic optimization will remodel the above problem into nonlinear optimization as Max α, Max γ, Min β, Max δ, such that TO^k ðxÞ  α, TL^j ðxÞ  α, GO^k ðxÞ  γ, GL^j ðxÞ  γ, FO^k ðxÞ  β, FL^j ðxÞ  β, UO^k ðxÞ  δ, UL^j ðxÞ  δ, α  β, α  γ α  δ, α+ β+ γ+ δ  4, α, β, γ, δ 2 ½0,1, g^j ðxÞ  b^j , x  0, j ¼ 1, 2,…, q:

Multiobjective nonlinear bipolar neutrosophic optimization

299

Case ii: Now the decision set D^ of case ii is defined as p D^ ¼ ð\k¼1 O^k Þ \ ð\qj¼1 L^j Þ ¼ fðx,TD^ ,GD^ ,CD^ ,FD^ Þg,

where

  TD^ ðxÞ ¼ min TO^1 ðxÞ,TO^2 ðxÞ,…, TO^p ðxÞ;TL^1 ðxÞ,TL^2 ðxÞ, …,TL^q ðxÞ ;   GD^ ðxÞ ¼ min GO^1 ðxÞ, GO^2 ðxÞ,…,GO^p ðxÞ;GL^1 ðxÞ, GL^2 ðxÞ,…, GL^q ðxÞ ;   CD^ ðxÞ ¼ min CO^1 ðxÞ,CO^2 ðxÞ, …,CO^p ðxÞ;CL^1 ðxÞ, CL^2 ðxÞ,…, CL^q ðxÞ ;   FD^ ðxÞ ¼ max FO^1 ðxÞ, FO~2 ðxÞ,…,FO^p ðxÞ;FL^1 ðxÞ,FL^2 ðxÞ,…, FL^q ðxÞ ,

for all x 2 X, where TD^ , GD^ , CD^ , and FD^ represent truth, ignorance, contradictory, and falsity grade of an extended refined neutrosophic decision set, respectively. Now extended refined neutrosophic optimization will remodel the earlier problem into nonlinear optimization as Max α, Max γ, Min β, Max δ, such that TO^k ðxÞ  α, TL^j ðxÞ  α, GO^k ðxÞ  γ, GL^j ðxÞ  γ, FO^k ðxÞ  β, FL^j ðxÞ  β, CO^k ðxÞ  δ, CL^j ðxÞ  δ, α  β, α  γ α  δ, α+ β+ γ+ δ  4, α, β, γ, δ 2 ½0,1, g^j ðxÞ  b^j , x  0, j ¼ 1, 2,…,q: Case iii (Bipolar neutrosophic optimization): Now the decision set D^ for the bipolar neutrosophic confluence of bipolar neutrosophic objectives and constraints is defined as n o D^ ¼ ð\pk¼1 O^k Þ \ ð\qj¼1 L^j Þ ¼ ðx,TD+^ , TD^ , ID+^ , ID^ , FD+^ ,F Þ , ^ D where

  TD+^ ðxÞ ¼ min TO+^ ðxÞ, TO+^ ðxÞ, …,TO+^ ðxÞ;TL+^1 ðxÞ, TL+^2 ðxÞ,…, TL+^q ðxÞ ; 1 2 p   TD^ ðxÞ ¼ max TO^ ðxÞ,TO^ ðxÞ,…, TO^ ðxÞ;TL^1 ðxÞ,TL^2 ðxÞ,…,TL^q ðxÞ ; 1 2 p   ID+^ ðxÞ ¼ min IO+^ ðxÞ,IO+^ ðxÞ, …,IO+^ ðxÞ;IL+^1 ðxÞ,IL+^2 ðxÞ,…, IL+^q ðxÞ ; 1 2 p       ID^ ðxÞ ¼ max IO^ ðxÞ, IO^ ðxÞ,…,IO^ ðxÞ;IL^1 ðxÞ, IL^2 ðxÞ, …,IL^q ðxÞ ; 1 2 p   + + + FD^ ðxÞ ¼ max FO^ ðxÞ,FO~ ðxÞ,…,FO+^ ðxÞ;FL+^1 ðxÞ, FL+^2 ðxÞ,…,FL+^q ðxÞ ; 2 1 p       FD^ ðxÞ ¼ min FO^ ðxÞ, FO~ ðxÞ, …,FO^ ðxÞ;F ðxÞ,F ðxÞ, …,F ðxÞ , L^1 L^2 L^q 1

2

p

300

Optimization Theory Based on Neutrosophic and Plithogenic Sets

for all x 2 X, where TD+^ , TD^ , ID+^ , ID^ , FD+^ , and F represent true positive, true negative, positive D^ indeterminacy, negative indeterminacy, positive falsity, and negative falsity grade of a bipolar neutrosophic decision set, respectively. Now bipolar neutrosophic optimization will remodel the earlier problem into nonlinear optimization as Max α+ , Max γ + , Min β + , Min α , Min γ  , Max β , such that TO+^ ðxÞ  α+ , TL+^ ðxÞ  α+ , j k TO^ ðxÞ  α , TL^ ðxÞ  α , j k IO+^ ðxÞ  δ+ , IL+^ ðxÞ  δ+ , j k IO^ ðxÞ  δ , IL^ ðxÞ  δ , j

k

FO+^ ðxÞ  β + , FL+^ ðxÞ  β + , j k   ðxÞ  β , F ðxÞ  β , F ^ ^ L O j

k

α+  β + , α+  γ + α  β , α  γ  , α+ + β + + γ + + α + β + γ   3, α+ , β + , γ + , 2 ½0,1, α , β , γ  , 2 ½1,0, g^j ðxÞ  b^j , x  0, j ¼ 1, 2,…, q:

13.4.1 Computational algorithm Step 1: Take the first objective function as a single objective function w.r.t. the given constraints and solve it to find values of decision variables. Step 2: Compute values of remaining objective function by using the values of decision variables. Step 3: Repeat Steps 1 and 2 for remaining objective functions. 2

∗ f^ ðx1 Þ 6 ^1 2 6 f 1 ðx Þ 6 4 ⋮ f^1 ðxr Þ

f^2 ðx1 Þ ∗ f^2 ðx2 Þ ⋮ f^2 ðxr Þ

⋯ ⋯ ⋱ ⋯

3 f^p ðx1 Þ 7 f^1 ðx2 Þ 7 7: ⋮ 5 ∗ r f^p ðx Þ

Step 4: Case i: Find the lower bound L^p and the upper bound U^p corresponding to each objective f^k ðxÞ. Lower and upper bounds for truth membership of objective functions are given below. T

T

T T U^ p ¼ max ff^p ðxr Þg and L^p ¼ min ff^p ðxr Þg,

where r ¼ 1, 2, …, p. F F Upper U^ p and lower L^p bounds for falsity membership of objectives are F T F T T T U^ p ¼ U^ p and L^p ¼ L^p + tðU^ p  L^p Þ:

Multiobjective nonlinear bipolar neutrosophic optimization

301

U U Upper U^p and lower L^p bounds for uncertainty membership of objectives are U T L^p ¼ L^p

U T T T U^p ¼ L^p + sðU^p  L^p Þ,

and

and upper and lower bounds for ignorance membership of objectives are G T F L^p ¼ L^p _ L^p

G T F T F T F U^p ¼ L^p _ L^p + lðU^p _ U^p  L^p _ L^p Þ,

and

where t, s, l 2 (0, 1). Case ii: Find the lower bound L^p and the upper bound U^p corresponding to each objective fk(x). Lower and upper bounds for truth membership of objective functions are given below. T

T U^p ¼ max ffp ðxr Þg

T

T L^p ¼ minffp ðxr Þg,

and

where r ¼ 1, 2, …, p. F F Upper U^p and lower L^p bounds for falsity membership of objectives are F T U^p ¼ U^p

F T T T L^p ¼ L^p + tðU^p  L^p Þ:

and

G G Upper U^p and lower L^p bounds for ignorance membership of objectives are G T F L^p ¼ L^p _ L^p

G T F T F T F U^p ¼ L^p _ L^p + sðU^p _ U^p  L^p _ L^p Þ,

and

and upper and lower bounds for contradictory membership of objectives are C T F L^p ¼ L^p ^ L^p

C T F T F T F U^p ¼ L^p ^ L^p + lðU^p ^ U^p  L^p ^ L^p Þ,

and

where t, s, l 2 (0, 1).

T

T+

T+

T

Case iii: Find the lower bound L^p , L^p and the upper bound U^p , U^p corresponding to each objective f^k ðxÞ. Lower and upper bounds for truth membership of objective function are given below. T U^p T U^p

+



+

¼ maxff^p ðxr Þg

and

T L^p ¼ min ff^p ðxr Þg,

¼ min ff^p ðxr Þg

and

T L^p ¼ max ff^p ðxr Þg,

+

where r ¼ 1, 2, …, p. F+ F+ F F Upper U^p , U^p and lower L^p , L^p bounds for falsity membership of objectives are F U^p F L^p

+



+

T F T ¼ U^p , U^p ¼ U^p







+

and





+

+

+

F T T T L^p ¼ L^p + tðU^p  L^p Þ

T T T ¼ L^p + tðU^p  L^p Þ: +





+

I I I I Upper U^p , U^p and lower L^p , L^p bounds for indeterminacy membership of objectives are +

+





+

I L^p

T I T ¼ L^p , L^p ¼ L^p

I U^p

T T T ¼ L^p + sðU^p  L^p Þ,





where t, s, 2 (0, 1).



and 

+

+

+

I T T T U^p ¼ L^p + sðU^p  L^p Þ

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Step 5: Case i: In this step, we will define truth, uncertainty, ignorance, and falsity-membership functions as follows:

Tp ð f^p ðxÞÞ ¼

8 > 1 > > > > < U^T  f^ ðxÞ p p

T T > > U^p  L^p > > > : 0 8 > 1 > > > > < U^U  f^ ðxÞ p p Up ð f^p ðxÞÞ ¼ > U^U  L^U > > p p > > : 0 8 > 1 > > > > < U^G  f^ ðxÞ p p Gp ð f^p ðxÞÞ ¼ > U^G  L^G > > p p > > : 0 8 > 0 > > > > < f^ ðxÞ  L^F p p Fp ð f^p ðxÞÞ ¼ > ^F  L^F > U > p p > > : 1

T f^p ðxÞ  L^p T T L^p  f^p ðxÞ  U^p , T f^p ðxÞ  U^p U f^p ðxÞ  L^p U U L^p  f^p ðxÞ  U^p , U f^p ðxÞ  U^p G f^p ðxÞ  L^ p

G L^p  f^p ðxÞ  U^p , G

G f^p ðxÞ  U^p F f^p ðxÞ  L^ p

F F L^p  f^p ðxÞ  U^p : F f^p ðxÞ  U^p

Case ii: In this step, we will define truth, ignorance, contradictory, and falsitymembership functions as follows:

Tp ð f^p ðxÞÞ

Gp ð f^p ðxÞÞ

Cp ð f^p ðxÞÞ

Fp ð f^p ðxÞÞ

8 > 1 > > > > > ^T ^ > < U p  f p ðxÞ T T ¼ U^p  L^p > > > > > > 0 > : 8 > 1 > > > > > ^G > < U p  fp ðxÞ G G ¼ U^p  L^p > > > > > > 0 > : 8 > 1 > > > > > ^C ^ > < U p  f p ðxÞ C C ¼ U^p  L^p > > > > > > 0 > : 8 > 0 > > > > > ^ ^F > < f p ðxÞ  L p F F ¼ U^p  L^p > > > > > > 1 > :

T f^p ðxÞ  L^p T T L^p  f^p ðxÞ  U^p

f^p ðxÞ

,

T  U^p

G f^p ðxÞ  L^p G G L^p  f^p ðxÞ  U^p

fp ðxÞ

,

G  U^p

C f^p ðxÞ  L^p C C L^p  f^p ðxÞ  U^p

f^p ðxÞ

,

C  U^p

F f^p ðxÞ  L^p F F L^p  f^p ðxÞ  U^p

f^p ðxÞ

F  U^p

:

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303

Case iii: In this step, we will define membership functions for a bipolar neutrosophic set as follows:

T + p ðf^p ðxÞÞ ¼

T  p ðf^p ðxÞÞ ¼

I + p ðf^p ðxÞÞ ¼

I p ðf^p ðxÞÞ ¼

F+ p ðf^p ðxÞÞ ¼

F p ðf^p ðxÞÞ ¼

8 > 1 > > > > T+ > > < U^p  fp ðxÞ +

+

T T U^p  L^p > > > > > >0 > : 8 > 1 > > > > T > > < U^p  fp ðxÞ T U^p

T  L^p

> > > > > 0 > > : 8 > 1 > > > > I+ > > < U^p  f^p ðxÞ +

+

I I U^p  L^p > > > > > > 0 > : 8 > 1 > > > > I > > U < ^  f^p ðxÞ p

I U^p

I  L^p

> > > > > 0 > > : 8 > 0 > > > > F+ > > < f^p ðxÞ  L^p +

+

F F U^p  L^p > > > > > > 1 > : 8 > 0 > > > > F > > < f^p ðxÞ  L^ F U^p

> > > > > 1 > > :

p F  L^p

T f^p ðxÞ  L^p

+

+

+

T T L^p  f^p ðxÞ  U^p , T f^p ðxÞ  U^p T f^p ðxÞ  L^p

+





T T L^p  f^p ðxÞ  U^p T f^p ðxÞ  U^p



,



+

I f^p ðxÞ  L^p +

+

I I L^p  f^p ðxÞ  U^p , +

I f^p ðxÞ  U^p



I f^p ðxÞ  L^p 



I I L^p  f^p ðxÞ  U^p

f^p ðxÞ

,

I  U^p

F f^p ðxÞ  L^p

+

+

+

F F L^p  f^p ðxÞ  U^p , F f^p ðxÞ  U^p F f^p ðxÞ  L^p

+





F F L^p  f^p ðxÞ  U^p

f^p ðxÞ



:

F  U^p

Step 6: Now the extended refined neutrosophic optimization and bipolar optimization method for the multiobjective nonlinear programming problem gives a corresponding nonlinear problem as follows: Case i: Max α  β+ γ+ δ,

304

Optimization Theory Based on Neutrosophic and Plithogenic Sets

such that Tp ðf^p ðxÞÞ  α, Up ðf^p ðxÞÞ  γ, Fp ðf^p ðxÞÞ  β, Gp ðf^p ðxÞÞ  δ, with α+ β+ γ+ δ  4, and α  β, α  γ, α  δ, where α, β, γ, δ 2 ½0,1, g^j ðxÞ  b^j , x  0, j ¼ 1,2, …,q, which corresponds to nonlinear programming as Max

α  β+ γ+ δ,

such that T T T f^p ðxÞ+ ðU^p  L^p Þ  α  U^p , U U U f^p ðxÞ+ ðU^  L^ Þ  γ  U^ , p

p

p

p

p

p

F F F f^p ðxÞ  ðU^p  L^p Þ  β  L^p , G G G f^p ðxÞ+ ðU^  L^ Þ  δ  U^ ,

for p ¼ 1, 2, …, k with α+ β+ γ+ δ  4 and α  β, α  γ, α  δ, where α, β, γ, δ 2 ½0,1 g^j ðxÞ  b^j , x  0, j ¼ 1, 2,…, q: Case ii: Max α  β+ γ+ δ,

Multiobjective nonlinear bipolar neutrosophic optimization

such that Tp ðf^p ðxÞÞ  α, Gp ðf^p ðxÞÞ  γ, Fp ðf^p ðxÞÞ  β, Cp ðf^p ðxÞÞ  δ, with α+ β+ γ+ δ  4, and α  β, α  γ, α  δ, where α, β, γ, δ 2 ½0,1, g^j ðxÞ  b^j , x  0, j ¼ 1,2, …,q, which corresponds to nonlinear programming as Max

α  β+ γ+ δ,

such that T T T f^p ðxÞ+ ðU^p  L^p Þ  α  U^p , G G G f^p ðxÞ+ ðU^  L^ Þ  γ  U^ , p

p

p

p

p

p

F F F f^p ðxÞ  ðU^p  L^p Þ  β  L^p , C C C f^p ðxÞ+ ðU^  L^ Þ  δ  U^ ,

for p ¼ 1, 2, …, k with α+ β+ γ+ δ  4 and α  β, α  γ, α  δ, where α, β, γ, δ 2 ½0,1 g^j ðxÞ  b^j , x  0, j ¼ 1,2, …,q: Case iii: Max α+  α  β + + β + γ +  γ  ,

305

306

Optimization Theory Based on Neutrosophic and Plithogenic Sets

such that T + p ðf^p ðxÞÞ  α+ , T  p ðf^p ðxÞÞ  α , I + p ðf^p ðxÞÞ  γ + , I  p ðf^p ðxÞÞ  γ  , F+ p ðf^p ðxÞÞ  β + , F p ðf^p ðxÞÞ  β , with α+  β + , α+  γ + α  β , α  γ  , α+ + β + + γ + + α + β + γ   3, α+ , β + , γ + , 2 ½0,1, α , β , γ  , 2 ½1,0, g^j ðxÞ  b^j , x  0, j ¼ 1, 2,…,q, which corresponds to nonlinear programming as Max α+  α  β + + β + γ +  γ  , such that +

+

+

T T T f^p ðxÞ+ ðU^p  L^p Þ  α+  U^p , T T T f^p ðxÞ+ ðU^p  L^p Þ  α  U^p , +

+

+

I I I f^p ðxÞ+ ðU^p  L^p Þ  γ +  U^p ,   I I I f^p ðxÞ+ ðU^  L^ Þ  γ   U^ , p

+

p

+

p

+

F F F f^p ðxÞ  ðU^p  L^p Þ  β +  L^p , F F F f^p ðxÞ  ðU^p  L^p Þ  β +  L^p ,

for p ¼ 1, 2, …, k. α+  β + , α+  γ + α  β , α  γ  , α+ + β + + γ + + α + β + γ   3, α+ , β + , γ + , 2 ½0,1, α , β , γ  , 2 ½1,0, g^j ðxÞ  b^j , x  0, j ¼ 1, 2,…,q:

Example 13.2. Min f1 ðx1 , x2 Þ ¼ x1 1 x2 2 , Min f2 ðx1 , x2 Þ ¼ 2x1 2 x2 3 , such that x1 + x2  1:

Multiobjective nonlinear bipolar neutrosophic optimization

307

Step 1: Solve the first objective function as a single objective nonlinear programming problem subject to given constraints, we get the value of x1 ¼ 0.33, x2 ¼ 0.67, ( f1)1 ¼ 6.75. Step 2: By using these decision variables, we compute the other remaining objective functions whose values are (f2)1 ¼ 60.78. Step 3: Now, repeat Steps 1 and 2 for other remaining objective functions. The ideal solution is given as  ¼

 6:75 60:78 : 6:94 57:87

Step 4: Find the lower and upper bounds corresponding to each objective fp(x). +





+

T T U^1 ¼ L^1 ¼ 6:94, T T U^1 ¼ L^1 ¼ 6:75, 

+

I I L^1 ¼ 6:75, L^1 ¼ 6:94, +

I U^1 ¼ 6:75+ sð0:19Þ ¼ 6:826, 

I U^1 ¼ 6:94+ sð0:19Þ ¼ 6:864, +

F L^1 ¼ 6:75+ tð0:19Þ ¼ 6:807, 

F L^1 ¼ 6:94+ tð0:19Þ ¼ 6:883, 

+

F F U^1 ¼ 6:94, U^1 ¼ 6:75, +





+

T T U^2 ¼ L^2 ¼ 60:78, T T U^2 ¼ L^2 ¼ 57:87, +



I I L^2 ¼ 57:87, L^2 ¼ 60:78, +

I U^2 ¼ 57:87+ sð2:91Þ ¼ 59:034, 

I U^2 ¼ 60:78+ sð2:91Þ ¼ 59:616, +

F L^2 ¼ 57:87+ tð2:91Þ ¼ 58:743, 

F L^2 ¼ 60:78+ tð2:91Þ ¼ 59:907, +



F F U^2 ¼ 60:78, U^2 ¼ 57:87,

where s, t ¼ (0, 1), take t ¼ 0.3 and s ¼ 0.4. Step 5: 8 1 > > > > 1 2 > < 6:94  x1 x2 + 1 2 T 1 ðx1 x2 Þ ¼ 6:94  6:75 > > > 0 > > :

T + 2 ð2x1 2 x2 3 ¼

x1 1 x2 2  6:75 6:75  x1 1 x2 2  6:94

,

x1 1 x2 2  6:94

8 1 > > > 60:78  2x1 2 x2 3
> > :0

x1 1 x2 2  60:78

57:87  2x1 2 x2 3  60:78 ,

308

Optimization Theory Based on Neutrosophic and Plithogenic Sets

8 1 x1 1 x2 2  6:94 > > 1 2 > < 6:75  x1 x2 6:94  x1 1 x2 2  6:75 , T  1 ðx1 1 x2 2 Þ ¼ 6:75  6:94 > > x1 1 x2 2  6:75 > :0 8 1 2x1 2 x2 3  60:78 > > > < 57:87  2x1 2 x2 3 57:87  2x1 2 x2 3  57:87 , T  2 ð2x1 2 x2 3 Þ ¼ 57:87  60:78 > >0 x1 1 x2 2  57:87 > : 8 1 x1 1 x2 2  6:75 > > > 6:826  x1 1 x2 2 < 6:75  x1 1 x2 2  6:826 , I + 1 ðx1 1 x2 2 Þ ¼ 6:826  6:75 > >0 x1 1 x2 2  6:826 > : 8 1 2x1 2 x2 3  57:87 > > > 59:034  2x1 2 x2 3 < 57:87  2x1 2 x2 3  59:034 , I + 2 ð2x1 2 x2 3 Þ ¼ 59:034  57:87 > > 2x1 2 x2 3  59:034 >0 : 8 1 x1 1 x2 2  6:94 > > 1 2 > < 6:864  ;2x1 x2 6:94  x1 1 x2 2  6:864 , I 1 ðx1 1 x2 2 Þ ¼ 6:826  6:75 > > x1 1 x2 2  6:864 > :0 8 1 2x1 2 x2 3  60:78 > > > < 59:616  2x1 2 x2 3 60:78  2x1 2 x2 3  59:616 , I  2 ð2x1 2 x2 3 Þ ¼ 59:616  60:78 > >0 2x1 2 x2 3  59:616 > : 8 0 x1 1 x2 2  6:807 > > > x1 1 x2 2  6:807 < 6:807  x1 1 x2 2  6:94 , F+ 1 ðx1 1 x2 2 Þ ¼ 6:94  6:807 > > x1 1 x2 2  6:94 >1 : 8 0 2x1 2 x2 3  58:743 > > > 2x1 2 x2 3  58:743 < 58:743  2x1 2 x2 3  60:78 , F+ 2 ð2x1 2 x2 3 Þ ¼ 60:78  58:743 > > 2x1 2 x2 3  60:78 >1 : 8 0 x1 1 x2 2  6:883 > > 1 2 > < x1 x2  6:883 6:883  x1 1 x2 2  6:75 , F 1 ðx1 1 x2 2 Þ ¼ 6:75  6:883 > > x1 1 x2 2  6:75 > : 1

Multiobjective nonlinear bipolar neutrosophic optimization

309

8 0 2x1 2 x2 3  59:907 > > 2 3 > < 2x1 x2  59:907 59:886  2x1 2 x2 3  57:87 : F 2 ð2x1 2 x2 3 Þ ¼ 57:87  59:907 > > 2x1 2 x2 3  57:87 > : 1 Step 6: The nonlinear programming problem in bipolar neutrosophic environment is Max α+  α  β + + β + γ +  γ  , such that x1 1 x2 2 + ð0:19Þα+  6:94, 2x1 2 x2 3 + ð2:91Þα+  60:78, x1 1 x2 2 + ð0:19Þα  6:75, 2x1 2 x2 3 + ð2:91Þα  57:87, x1 1 x2 2 + ð0:076Þγ +  6:826, 2x1 2 x2 3 + ð1:164Þγ +  59:034, x1 1 x2 2 + ð0:076Þγ   6:864, 2x1 2 x2 3 + ð1:164Þγ   59:616, x1 1 x2 2  ð0:133Þβ +  6:807, 2x1 2 x2 3  ð2:037Þβ +  58:743, x1 1 x2 2  ð0:133Þβ  6:883, 2x1 2 x2 3  ð2:037Þβ  59:907, x1 + x2  1, 0  α+  1, 0  β +  1, 0  γ +  1, 1  α  0, 1  β  0, 1  γ   0, α+  β + , α+  γ + α  β , α  γ  , α+ + β + + γ + + α + β + γ   3:

Numerical computation is done by using the Matlab optimization tool box. Comparison of proposed optimal solution with existing techniques are given in Table 13.1.

13.5

Application of bipolar neutrosophic in riser design

Multiobjective cylindrical riser design problem can be stated as follows ^ hÞ ^ ¼ πd^2 h^; Minimize vr ðd, 4

^ hÞ ^ ¼ d^h^ , Minimize st ðd, ^ 2d^ 4h+

such that 48 ^1 24 ^1 d + h  1: 19 19

310

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 13.1 Comparison of optimal solutions. Optimization technique Intuitionistic fuzzy optimization (IFO) Neutrosophic optimization (NO) Four-valued refined neutrosophic optimization (T, U, C, F) Four-valued refined neutrosophic optimization (FVRNO) (case i) Four-valued refined neutrosophic optimization (FVRNO) (case ii) Bipolar neutrosophic optimization (BNO)

Optimal decision variables x∗1 ,x∗2

Optimal objective functions f1∗ , f2∗

Sum of optimal objective values f1∗ + f2∗

0.3659009, 0.6356811 0.3635224, 0.6364776 0.365902, 0.634098

6.797078, 58.79110 6.790513, 58.68732 6.797081071, 58.59104971

65.588178

0.365902, 0.634098

6.797081071, 58.59104971

65.38813078

0.365902, 0.634098

6.797081071, 58.59104971

65.38813078

0.373367, 0.638284

6.574097696, 55.17168667

61.7457843

65.487833 65.38813078

Here the pay-off matrix is  ¼

13.6

 42:75642 0:631579 : 12:510209 0:6315786

Conclusion

In this chapter, two extensions of neutrosophic sets on the basis of indeterminacy were first defined along with their set-theoretic operators. Second, some discussion on bipolar neutrosophic sets and their operators has been carried out. Computational algorithms based on refined neutrosophic and bipolar neutrosophic sets have been developed to find the optimal solution of multiobjective nonlinear optimization models. The present technique utilizes a calculation of the upper bound of the falsity grade in such a way that the upper bound of this grade is always less than the upper bound of the truth grade of membership. Membership functions are constructed in a way that the formulation maximizes the membership and minimizes nonmembership grades while, subject to the given problem, indeterminacy grades could be maximized or minimized. From the results of the optimization algorithm of cases i and ii, discussed in Table 13.1, we can draw a very interesting conclusion. For every refinement or each extension of neutrosophic sets, whether we split indeterminacy into

Multiobjective nonlinear bipolar neutrosophic optimization

311

components C and G, or U and G, or U and C, our proposed algorithm provides us with exactly the same results with the same level of optimality. Table 13.1 also clearly shows that refined neutrosophic optimization techniques (both cases i and ii) provide improved results with respect to intuitionistic and neutrosophic optimization techniques, but the bipolar neutrosophic optimization technique provides the most optimum results. A percentage gap has been used to analyze the performance of the proposed methods, whereas Tables 13.2–13.4 and Figs. 13.1–13.3 show that the optimal solutions are attained by using a bipolar neutrosophic optimization algorithm. Thus the bipolar neutrosophic optimization technique is a more efficient, reliable, and generalized optimization technique for solving the nonlinear multiobjective optimization problems in a less certain environment. In future, we may look for those refinements in neutrosophic sets which we could use to develop new optimization algorithms with effective, improved, efficient, and reliable results while dealing with uncertainty. The present work can also be used to establish improved techniques in sampling survey analysis, cost function estimation analysis, and engineering optimization problems.

Table 13.2 Percentage gap with respect to refined neutrosophic optimization.

Optimization technique Intuitionistic fuzzy optimization (IFO) Neutrosophic optimization (NO) Four-valued refined neutrosophic optimization

Objective function f1∗ (%)

Objective function f2∗ (%)

Cumulative percentage gap (%)

0.096585621

0.321672345

0.418257966

0

0.162417483

0.162417483

0.086026083

0

0.86026083

Table 13.3 Percentage gap with respect to bipolar neutrosophic optimization.

Optimization technique Intuitionistic fuzzy optimization (IFO) Neutrosophic optimization (NO) Four-valued refined neutrosophic optimization Bipolar neutrosophic optimization

Objective function f1∗ (%)

Objective function f2∗ (%)

Cumulative percentage gap (%)

3.280531781

6.156396682

9.436928464

3.187024368

6.00646389

9.193488257

3.270308779

5.853553601

9.123862379

0

0

0

312

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 13.4 Comparison of optimal solutions.

Optimization technique

Optimal objective functions v∗r ,s∗r

Cumulative percentage gap (%)

3.152158, 3.152158 2.979, 3.50964

24.60870, 0.6315787 24.44968309, 0.5228508083

82.73991921

2.51653, 2.51541

12.5163487817, 0.419359426

0

20.28175084

Percentage GAP

IFO

0.086026083

0.162417483

0.418257966

Neutrosophic optimization (NO) Four-valued refined neutrosophic optimization (FVRNO) Bipolar neutrosophic optimization (BNO)

Optimal decision variables d*, h*

NFO

FVRNO

0

9.123862379

9.193488257

9.436928464

Fig. 13.1 Percentage gap with respect to refined neutrosophic optimization.

IFO

NFO

FVRNO

BNO

Fig. 13.2 Percentage gap with respect to bipolar neutrosophic optimization.

Multiobjective nonlinear bipolar neutrosophic optimization

313

0

20.28175084

82.73991921

GAP analysis

NFO

FVRNO

BNFO

Fig. 13.3 Comparison of proposed methodology with percentage gap.

References [1] G. Klir, B. Yuan, Fuzzy sets and fuzzy logic: theory and applications, Upper Saddle River (1995) 563. [2] M.V. Black, An exercise in logical analysis, Philos. Sci. 4 (1937) 427–455. [3] A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Stat. 38 (1967) 325–339. [4] L. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353. [5] G. Klir, M. Wierman, Uncertainty-Based Information, Physica: Verlag, Heidelberg, 2013. [6] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010) 529–539. [7] F. Smarandache, Neutrosophic set—a generalization of the intuitionistic fuzzy set, J. Defense Resour. Manag. 1 (2010) 107. [8] W.R. Zhang, NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, IEEE Service Center, 1994, pp. 305–309. [8a] R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Manag. Sci. 17 (1970) B-141. [9] K.T. Atanassov, Intuitionistic fuzzy set, Fuzzy Set. Syst. 20 (1986) 87–96. [10] H.K. Tanaka, K. Asai, On fuzzy mathematical programming, J. Cybernet. 3 (1973) 37–46. [11] H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Set. Syst. 1 (1978) 45–55. [12] H.K. Tanaka, K. Asai, Fuzzy linear programming problems with fuzzy numbers, Fuzzy Set. Syst. 139 (1984) 1–10. [13] B. Werners, An interactive fuzzy programming system, Fuzzy Set. Syst. 23 (1987) 131–147. [14] C. Xu, Fuzzy optimization of structures by the two phase method, Comput. Struct. 31 (1989) 575–580. [15] C.J. Shih, C.C. Chi, J.H. Hsiao, Alternative level cuts methods for optimum structural design with fuzzy resources, Comput. Struct. 81 (2003) 2579–2587. [16] C.J. Shih, H.W. Lee, Modified doublecuts approach in 25-bar and 72-bar fuzzy truss optimization, Comput. Struct. 84 (2006) 2100–2104.

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[17] G.E. Asimakopoulou, V.T. Kontargyri, G.J. Tsekouras, C.N. Elias, F.E. Asimakopoulou, I.A. Stathopulos, A fuzzy logic optimization methodology for the estimation of the critical flashover voltage on insulators, Electr. Power Syst. Res. 81 (2011) 580–588. [18] P.P. Angelov, Optimization in intuitionistic fuzzy environment, Fuzzy Sets Syst. 86 (1997) 299–306. [19] Z. Pei, L. Yi, A note on operations of hesitant fuzzy sets, Int. J. Comput. Intell. Syst. 8 (2015) 226–239. [20] S. Bharati, K. Hesitant, Fuzzy computational algorithm for multiobjective optimization problems, Int. J. Dyn. Control 1 (2018) 42–56. [21] S.K. Bharati, Solving optimization problems under hesitant fuzzy environment, Life Cycle Reliab. Safety Eng. 7 (2018) 165–179. [21a] M. Sarkar, S. Dey, T.K. Roy, Multi-objective neutrosophic optimization technique and its application to structural design, Int. J. Comput. Appl. 148 (12) (2016) 31–37. [21b] M. Sarkar, T. Roy, Optimization of welded beam structure using neutrosophic optimization technique: a comparative study, Int. J. Fuzzy Syst. 20 (2018) 847–860. [21c] H. Garg, Non-linear programming method for multi-criteria decision-making problems under interval neutrosophic set environment, Appl. Intell. 2 (2017) 6–15. [21d] P. Das, T.K. Roy, Multi-objective non-linear programming problem based on neutrosophic optimization technique and its application in riser design problem, Neutrosophic Sets Syst. 9 (2015) 88–95. [22] F. Smarandache, n-Valued refined neutrosophic logic and its applications to physics, Prog. Phys. 4 (2013) 143–146. [23] I. Deli, M. Ali, F. Smarandache, Bipolar Neutrosophic Sets and Their Application Based on Multi-Criteria Decision Making Problems, International Conference on Advanced Mechatronic Systems (ICAMechS), IEEE, 2015, pp. 249–254.

A novel similarity measure for single-valued neutrosophic sets and their applications in medical diagnosis, taxonomy, and clustering analysis

14

Rıdvan S¸ahina, Mesut Karabacakb G€um€u¸shane University, Faculty of Natural Sciences and Engineering, Department of Mathematical Engineering, G€um€u¸shane, Turkey, bAtat€urk University, Faculty of Science, Department of Mathematics, Erzurum, Turkey a

14.1

Introduction

Clustering analysis and medical diagnosis are among the most important research topics in a universe full of uncertainties. Due to the complexity of information or data related to the problem area, conventional tools are inadequate for processing uncertain information. Therefore, the need to develop new modeling tools is inevitable. Zadeh [1] first defined the theory of fuzzy sets (FSs) in 1965. Its apparent characteristic is that each of its members has only one membership degree. Since the definition of FSs, many researchers aimed to derive various extensions of this concept. Atanassov [2] extended the FSs to intuitionistic fuzzy sets (IFSs). Its characteristic is that each of its members has a membership and a non-membership degree. IFSs have been applied to medical diagnosis and clustering analysis by many researchers. Then distance measure and its dual, called the similarity measure, are two of the most important approaches. In order to handle the fuzzy environment, Balopoulos et al. [3] defined a new class of normalized distance measures between binary fuzzy operators based on a matrix norm and fuzzy implications (see Table 14.1). Hatzimichailidis [4] applied this idea to calculate the distance between IFSs and introduced a distance measure to determine the different between two IFSs. Then, Luo and Zhao [5] proved that it causes absurd results in practical applications, because it does not satisfy the condition of distance metric: d(A, B) ¼ 0 iff A ¼ B, and defined a new distance measure overcoming the above shortcomings. Recently, Ye [6] defined the cosine similarity measures for IFSs and applied them to solve a medical diagnosis problem. Tian [7] proposed the cotangent similarity measure of IFSs and applied it to medical diagnosis. Zhang et al. [8] proposed clustering algorithms for IFSs based on similarity measures of IFSs. Wang et al. [9] defined an approach to determine the

Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00014-7 © 2020 Elsevier Inc. All rights reserved.

316

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 14.1 Some of fuzzy implications [4, 5]. Fuzzy implications Reichenbach G€odel Lukasiewicz Kleene-Dienes Mamdani Larsen Luo-Zhau

fR(a, b) ¼ 1  a + ab
1, if a  b fG ða, bÞ ¼ b, if a > b fLU(a, b) ¼ min {1, 1  a + b} fKD(a, b) ¼ max {1  a, b} fM(a, b) ¼ min {a, b} fLA(a, b) ¼ ab fLZ(a, b) ¼ 1  a  b

similarity degree between two IFSs, based on an intuitionistic fuzzy similarity matrix and applied it to present the cluster analysis of IFSs. Khatibi and Montazer [10] proposed some similarity measures of IFSs and applied them to solve a bacteria classification problem in the medical pattern recognition. In the above medical diagnosis approaches, clustering methods, FSs, and IFSs can handle only uncertainty and incomplete information but not inconsistent and indeterminate information exists in the real world. In order to more accurately deal with inconsistent and indeterminate information, Smarandache [11] proposed philosophically the theory of neutrosophic sets (NSs), which is a generalization of FSs [1] and IFSs [2]. Its apparent characteristic is that each of its members has, independently, a truthmembership degree, an indeterminate-membership degree, and a truth-membership degree. However, it is difficult to use the NSs in solving real decision-making problems. To overcome this situation, Wang et al. [12, 13] derived the notions of a singlevalued neutrosophic set (SVNS) and an interval neutrosophic set (INS), which are the subclasses of a neutrosophic set. Since the developing technology requires more and more information processing process, it is reasonable to use SVNS and INSs for representing indeterminate and inconsistent information, which are involved in decision-making, clustering analysis, medical diagnosis, machine learning, and pattern recognition [14–20]. Similarity/distance measure are two important tools used to determine the similarity/different relationship between objects in an uncertainty environment. Therefore, more and more researchers have started to study the SVNSs and SNSs because of their capability of handling uncertainty and they have gained many studies based on similarity/distance measures in neutrosophic literature. For example, Ye [21] defined a generalized distance measure between SVNSs and proposed a single-valued neutrosophic minimum spanning tree (SVNMST) clustering algorithm based on the distance measures. Majumdar and Samanta [22] introduced several similarity measures of SVNSs based on distances, a matching function, and membership grades, and then Ye [23] applied them to develop the clustering analysis of single-valued neutrosophic data. Additionally, Ye [24] defined some approaches to determine the similarity measure between SNSs, including the Jaccard, Dice, and cosine similarity

A novel similarity measure for single-valued neutrosophic sets

317

measures for SVNSs. In medical diagnosis, recently, Ye [25] developed an improved cosine similarity measure of SVNSs. Ye [26] and Ye and Fu [27] proposed several similarity measures based on cotangent and tangent functions with applications, respectively. Moreover, Şahin and K€ uc¸€ uk [28] introduced a subsethood measure for SVNSs and applied it to decision-making. Other studies with neutrosophic information are presented in [29–35]. From the above surveys, the process of establishing a robust similarity measure is still a clear topic because of the inability to meet the axiomatic definition for existing similarity measures, the details of which are shown in Table 14.3. Considering the fact that SVNSs are a generalization of IFSs and they carry more and more uncertain information than IFSs, this chapter extends the similarity measure presented by Luo and Zhao [5] to single-valued neutrosophic environment and proposes a practical similarity measure between SVNSs. The developed similarity measure is based on both a matrix norm and fuzzy implication, and has some effective and superior properties over the other existing similarity measures. The advantage of using fuzzy implications offers not only the different final options to decision-makers but also gives a parameterized class of similarity measures of SVNSs. More generally, this chapter proposes an extended class of similarity measures for SNSs and applies them to decision-making problems. By doing so, the remaining study is constructed as follows. In Section 14.2, we summarize some basic concepts of SNSs. In Section 14.3, we review the existing similarity measures of SNSs and their drawbacks. Moreover, we define a class of similarity measures of SNSs based on fuzzy implications and discuss some of them properties. In Section 14.4, in order to demonstrate the effectiveness and validity of the developed similarity measures, the developed similarity measures are applied to solve the medical diagnosis, pattern recognition, and clustering problems under a single-valued neutrosophic environment. In Section 14.5, we discuss the effect of fuzzy implications over the developed similarity measures. Conclusions are given in Section 14.6.

14.2

Preliminaries

In this section, we concisely summarize some basic definitions related to singlevalued neutrosophic sets and their distance measures with the axiomatic properties to reach the intended goals.

14.2.1 Neutrosophic set Definition 14.1. [11] Let U be a space of points (objects) and u 2 U. A neutrosophic set A in U is defined by a truth-membership function TA(u), an indeterminacy-membership function IA(u), and a falsity-membership function FA(u). TA(u), IA(u), and FA(u) are real standard or real nonstandard subsets of ]0, 1+[. That is TA(u) : U ! ]0, 1+[, TA(u) : U ! ]0, 1+[ and TA(u) : u ! ]0, 1+[. There is no restriction on the sum of TA(u), IA(u) and FA(u), so 0  sup TA(u)  sup IA(u)  sup FA(u)  3+.

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14.2.2 Single valued neutrosophic sets A single valued neutrosophic set (SVNS) has been defined in [12] as follows: Definition 14.2. [12] Let U be a finite universe. A single-valued neutrosophic set A over U is defined by the form: A ¼ fhu, TA ðuÞ, IA ðuÞ, FA ðuÞi : u 2 Ug where TA(u) : U ! [0, 1], IA(u) : U ! [0, 1] and FA(u) : U ! [0, 1] with 0  TA(u) + IA(u) + FA(u)  3 for all u 2 U. The intervals TA(u), IA(u) and FA(u) denote the truth-membership degree, the indeterminacy-membership degree, and the falsity membership degree of u to A, respectively. If TA(u) + FA(u)  1 and IA(u) ¼ ∅, for u 2 U, then it is noted that the SVNS A is reduced an IFS. Definition 14.3. [12] The complement of single valued neutrosophic set A is denoted by Ac and is defined as TAc(u) ¼ FA(u), IAc(u) ¼ 1  IA(u), and FcA(u) ¼ TA(u) for all u 2 U. That is: Ac ¼ fhu, FA ðuÞ, 1  IA ðuÞ, FA ðuÞi : u 2 U g Definition 14.4. [12] A single valued neutrosophic set A is contained in the other single valued neutrosophic set B, A  B, iff TA (u)  TB(u), IA(u)  IB(u), and FA(u)  FB(u) for all u 2 u. Two single valued neutrosophic sets A and B are equal, written as A ¼ B, iff A  B and B  A. We will denote the set of all the SVNSs in U by P(U). A single-valued neutrosophic number (SVNN) is denoted by u ¼ (Tu, Iu, Fu) for convenience. Definition 14.5. [22] A mapping s : P(U)  P(U) ! [0, 1] is called a neutrosophic similarity measure, if s satisfies properties below (for all A, B, C 2 P(U)): ðSM1Þ: 0  sðA, BÞ  1; ðSM2Þ: ðA, BÞ ¼ 1 , A ¼ B; ðSM3Þ: sðA, BÞ ¼ sðB, AÞ; ðSM4Þ: If A  B  C ) sðA, BÞ  sðA, CÞ and d ðB, CÞ  sðA, CÞ

(14.1)

Definition 14.6. [21] A mapping d : P(U)  P(U) ! [0, 1] is called a neutrosophic distance measure, if d satisfies properties below (for all A, B, C 2 P(U)): ðDM1Þ: 0  dðA, BÞ  1; ðDM2Þ: dðA, BÞ ¼ 0 , A ¼ B; ðDM3Þ: dðA, BÞ ¼ dðB, AÞ; ðDM4Þ: If A  B  C ) d ðA, BÞ  d ðA, CÞ and dðB, CÞ  SðA, CÞ

(14.2)

A novel similarity measure for single-valued neutrosophic sets

319

Definition 14.7. [36] Let Mnn be a matrix space, kUk is a function: Mnn ! ℝ. If kUk satisfies the following conditions: ð1Þ kUk  0, kU k ¼ 0 if and only if U ¼ 0, for all U 2 Mnn , ð2Þ kaU k ¼ jaj kU k, a 2 M, for all U 2 Mnn , ð3Þ kU + V k  kU k + kV k, for all U, V 2 Mnn , ð4Þ kUV k  kU k kV k, for all U, V 2 Mnn ,then kU k is a norm of matrix U:

(14.3)

It is noted that U is a matrix and kUk is a matrix norm, then k–Uk ¼ kUk. Definition 14.8. [37] A distance metric d in a nonempty set U is a real function d : U  U ! [0, +∞), which satisfies conditions below: ð1Þ dðu, vÞ  0, for all u,v 2 U, ð2Þ dðu, vÞ ¼ 0 , u ¼ v, for all u, v 2 U, ð3Þ dðu, vÞ ¼ d ðv, uÞ, for all u,v 2 U, ð4Þ d ðu, wÞ + d ðw, vÞ  dðu, vÞ, for all u, v,w 2 U:

(14.4)

Considering the above definitions of distance and similarity measures, it is noted that the distance measure can be easily identified by the definition of the similarity measure, and vice versa. That is, for A, B 2 P(U), we can easily see d(A, B) ¼ 1  s(A, B).

14.2.3 Existing similarity measures In this part, we briefly list the existing distance measures between single-valued neutrosophic sets. Suppose that U ¼ {u1, u2, …un} is a finite universe, and A and B are two singlevalued neutrosophic sets in U. Then we can give the existing distance measures between single-valued neutrosophic sets A and B as follows: (1) Ye’s similarity measures [21,24–26,30] s1(A, B) s2(A, B) s3(A, B)

s4(A, B)

Pn 1 1  3n i¼1 ðjTA ðui Þ  TB ðui Þj + jIA ðui Þ  IB ðui Þj + jFA ðui Þ  FB ðui ÞjÞ
X  1 1 n 2 2 2 2 1 j T ð u Þ  T ð u Þ j + j I ð u Þ  I ð u Þ j + j F ð u Þ  F ð u Þ j A i B i A i B i A i B i i¼1 3n 1 Xn TA ðui ÞTB ðui Þ + IA ðui ÞIB ðui Þ + FA ðui ÞFB ðui Þ i¼1 △A ðui Þ + △B ðui Þ  ðTA ðui ÞTB ðui Þ  IA ðui ÞIB ðui Þ  FA ðui ÞFB ðui ÞÞ n 2 2 2 2 2 2 △A(ui) ¼ TA(ui) + IA(ui) + FA(ui), △B(ui) ¼ TB(ui) + IB(ui) + FB(ui)

1 Xn 2ðTA ðui ÞTB ðui Þ + IA ðui ÞIB ðui Þ + FA ðui ÞFB ðui ÞÞ   i¼1 T 2 ðu Þ + I 2 ðu Þ + F2 ðu Þ + ðT 2 ðu Þ + I 2 ðu Þ + F2 ðu ÞÞ n B i B i B i A i A i A i

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

Xn

TA ðui ÞTB ðui Þ + IA ðui ÞIB ðui Þ + FA ðui ÞFB ðui Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 TA ðui Þ + IA2 ðui Þ + F2A ðui Þ TB2 ðui Þ + IB2 ðui Þ + F2B ðui Þ
 1 Xn π ðjTA ðui Þ  TB ðui Þj + jIA ðui Þ  IB ðui Þj + jFA ðui Þ  FB ðui ÞjÞ cos i¼1 n 6
 1 Xn max ðjTA ðui Þ  TB ðui Þj, jIA ðui Þ  IB ðui Þj, jFA ðui Þ  FB ðui ÞjÞ cos π i¼1 n 2 n o X 1 π π n cot + ðjTA ðui Þ  TB ðui Þj + jIA ðui Þ  IB ðui Þj + jFA ðui Þ  FB ðui ÞjÞ i¼1 n 4 12

s5(A, B)

i¼1

s6(A, B) s7(A, B) s8(A, B)

nπ π o 1 Xn + max T cot ð ð j ð u Þ  T ð u Þ j, j I ð u Þ  I ð u Þ j, j F ð u Þ  F ð u Þ j Þ Þ A i B i A i B i A i B i i¼1 n 4 4

s9(A, B) s10(A, B) s11(A, B)


1 Pn 1  2 2 2 3 ð u Þ  T ð u Þ + I ð u Þ  I ð u Þ + F ð u Þ  F ð u Þ T j j j j j j A i B i A i B i A i B i i¼1 3 1 Xn d ðA, Bc Þ i¼1 n d ðA, Bc Þ + d ðA, BÞ where P d ðA, Bc Þ ¼ 13 ni¼1 ðjTA ðui Þ  FB ðui Þj + jIA ðui Þ  ð1  IB ðui ÞÞj + jFA ðui Þ  TB ðui ÞjÞ n 1X ðjTA ðui Þ  TB ðui Þj + jIA ðui Þ  IB ðui Þj + jFA ðui Þ  FB ðui ÞjÞ d ðA, BÞ ¼ 3 i¼1

1  1n

(2) Ye and Fu’s similarity measures [27] s12(A, B)

1  1n

s13(A, B)

1  1n

Pn

n

Pn

n

i¼1 tan i¼1 tan

π ðjTA ðui ÞTB ðui Þj + jIA ðui ÞIB ðui Þj + jFA ðui ÞFB ðui ÞjÞ 12

o

π max ðjTA ðui ÞTB ðui Þj, jIA ðu4i ÞIB ðui Þj, jFA ðui ÞFB ðui ÞjÞ

o

(3) Şahin and K€uc¸€uk’s subsethood measure [28] s14(A, B)

1  1n

Pn

i¼1 max ðjTA ðui Þ  TB ðui Þj, jIA ðui Þ  IB ðui Þj, jFA ðui Þ  FB ðui ÞjÞ

(4) Majumdar and Samanta’s similarity measures [22] s15(A, B) s16(A, B)

Pn Pni¼1 fmin fTA ðui Þ, TB ðui Þg + min fIA ðui Þ, IB ðui Þg + min fFA ðui Þ, FB ðui Þgg fmax fTA ðui Þ, TB ðui Þg + max fIA ðui Þ, IB ðui Þg + max fFA ðui Þ, FB ðui Þgg Pi¼1n fmin fTA ðui Þ, TB ðui Þg + min fIA ðui Þ, IB ðui Þg + min fFA ðui Þ, FB ðui Þgg 1 i¼1 fmax fTA ðui Þ, TB ðui Þg + max fIA ðui Þ, IB ðui Þg + max fFA ðui Þ, FB ðui Þgg n

A novel similarity measure for single-valued neutrosophic sets

321

(5) Shahzadi et al.’s similarity measure [31]  P 

1  exp  13 ni¼1 ðjTA ðuÞ  TB ðuÞj + jIA ðuÞ  IB ðuÞj + jFA ðuÞ  FB ðuÞjÞ  Pn  1 1 i¼1 1  3 ðjTA ðui Þ  TB ðui Þj + jIA ðui Þ  IB ðui Þj + jFA ðui Þ  FB ðui ÞjÞ n 1 1 exp ðnÞ

s17(A, B) s18(A, B)



(6) Mondal et al.’s similarity measures [29] s19(A, B) s20(A, B)

 

2  13 ðjTA ðui Þ  TB ðui Þj + jIA ðui Þ  IB ðui Þj + jFA ðui Þ  FB ðui ÞjÞ  Pn 1  1 λ i¼1 log 2 2  3 ðjTA ðui Þ  TB ðui Þj + jIA ðui Þ  IB ðui Þj + jFA ðui Þ  FB ðui ÞjÞ n 1 n

Pn

i¼1 log 2

+ ð1  λÞ

n X

log 2 f2  max ðjTA ðui Þ  TB ðui Þj, jIA ðui Þ  IB ðui Þj, jFA ðui Þ  FB ðui ÞjÞgÞ

i¼1

14.3

Basic results

Motivated by Luo and Zhao’s distance measure [5], we define a family of similarity measures between SVNSs based on a norm matrix and fuzzy implications. There exists a basic advantage over Luo and Zhao’s measure, which is that the developed similarity measure incorporates indeterminate and inconsistent information into the decision process independently.

14.3.1 A new similarity measure of SVNSs Definition 14.9. Let U be a finite universe of discourse, let A, B be two SVNSs in P(U), f : [0, 1]  [0, 1] ! [0, 1] a strictly increasing (or decreasing) binary function for each argument. A similarity measure is a function σ T: P(U)  P(U) ! [0, 1] defined for A, B 2 P(U) by: sf ðA, B; f Þ ¼ 1 

kΩðTA Þ  ΩðTB Þk + kΩðIA Þ  ΩðIB Þk + kΩðFA Þ  ΩðFB Þk 3n (14.5)

where     ΩðFA Þ ¼ f FA ðui Þ, FA uj nn 2 3 f ðTA ðu1 Þ, TA ðu1 ÞÞ ⋯ f ðTA ðu1 Þ, TA ðun ÞÞ 6 7 7 ⋮ ⋱ ⋮ ¼6 4 5 f ðTA ðun Þ, TA ðu1 ÞÞ ⋯ f ðTA ðun Þ, TA ðun ÞÞ nn

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

2 3 f ðIA ðu1 Þ, IA ðu1 ÞÞ ⋯ f ðIA ðu1 Þ, IA ðun ÞÞ     5 ΩðIA Þ ¼ f IA ðui Þ, IA uj nn ¼ 4 ⋮ ⋱ ⋮ f ðIA ðun Þ, IA ðu1 ÞÞ ⋯ f ðIA ðun Þ, IA ðun ÞÞ nn     ΩðFA Þ ¼ f FA ðui Þ, FA uj nn 2 3 f ðFA ðu1 Þ, FA ðu1 ÞÞ ⋯ f ðFA ðu1 Þ, FA ðun ÞÞ 6 7 7 ⋮ ⋱ ⋮ ¼6 4 5 f ðFA ðun Þ, FA ðu1 ÞÞ ⋯ f ðFA ðun Þ, FA ðun ÞÞ nn pffiffiffiffiffiffiffiffiffi f(a, b) denotes the different fuzzy implications given in Table 14.1, and kΩk ¼ λmax , λ is the largest non-negative eigenvalue of the positive definite Hermitian matrix ΩTΩ. It can easily be proved that the weighted similarity measure sf (A, B; f ) satisfies the properties given in Definition 14.5. Theorem 14.1. sf (A, B; f ) is a similarity measure between two single valued neutrosophic A and B in U. Proof For simplicity, the single-valued neutrosophic sets A and B are denoted by A ¼ {hui, TA(ui), IA(ui), FA(ui)i : ui 2 U} and B ¼ {hui, TB(ui), IB(ui), FB(ui)i : ui 2 U}, respectively. (SM1) It is noted that df  0 is obvious. Since 0  f(FA(ui), FA(uj))  1 and 0  f(FB(ui), FB(uj))  1, we have 0  kΩ(TA)  Ω(TB)k  n. Similarly, 0  kΩ(IA)  Ω(IB)k  n and 0  kΩ(FA)  Ω(FB)k  n, thus 1

kΩðTA Þ  ΩðTB Þk + kΩðIA Þ  ΩðIB Þk + kΩðFA Þ  ΩðFB Þk 1 3n

Hence, 0  sf (A, B; f )  1. (SM2) Suppose that sf (A, B; f ) ¼ 0 and f is a definitely increasing (or decreasing) binary function, then we can obtain: kΩðTA Þ  ΩðTB Þk + kΩðIA Þ  ΩðIB Þk + kΩðFA Þ  ΩðFB Þk ¼ 0, , kΩðTA Þ  ΩðTB Þk ¼ 0, kΩðIA Þ  ΩðIB Þk ¼ 0, kΩðFA Þ  ΩðFB Þk ¼ 0 , ΩðTA Þ ¼ ΩðTB Þ, ΩðIA Þ ¼ ΩðIB Þ, ΩðFA Þ ¼ ΩðFB Þ , f ðTA ðui Þ, TA ðui ÞÞ ¼ f ðTB ðui Þ, TB ðui ÞÞ, f ðIA ðui Þ, IA ðui ÞÞ ¼ f ðIB ðui Þ, IB ðui ÞÞ, f ðFA ðui Þ, FA ðui ÞÞ ¼ f ðFB ðui Þ, FB ðui ÞÞ:

A novel similarity measure for single-valued neutrosophic sets

323

So, we have TA(ui) ¼ TB(ui), IA(ui) ¼ IB(ui) and FA(ui) ¼ FB(ui), i.e. A ¼ B. Further, suppose that A ¼ B, then we have: TA ðui Þ ¼ TB ðui Þ, IA ðui Þ ¼ IB ðui Þ,FA ðui Þ ¼ FB ðui Þ , f ðTA ðui Þ, TA ðui ÞÞ ¼ f ðTB ðui Þ, TB ðui ÞÞ, f ðIA ðui Þ, IA ðui ÞÞ ¼ f ðIB ðui Þ, IB ðui ÞÞ, f ðFA ðui Þ, FA ðui ÞÞ ¼ f ðFB ðui Þ, FB ðui ÞÞ , ΩðTA Þ ¼ ΩðTB Þ,ΩðIA Þ ¼ ΩðIB Þ, ΩðFA Þ ¼ ΩðFB Þ , kΩðTA Þ  ΩðTB Þ k ¼ 0, kΩðIA Þ  ΩðIB Þ k ¼ 0, kΩðFA Þ  ΩðFB Þk ¼ 0 , df ðA, B; f Þ ¼ 0: Obviously, sf (A, B; f ) ¼ 0 ⟺ A ¼ B. (SM3) It is easy to note that the sf (A, B; f ) is commutative; that is, sf (A, B; f ) ¼ sf (B, A; f ). Really, kΩðTA Þ  ΩðTB Þk + kΩðIA Þ  ΩðIB Þk + kΩðFA Þ  ΩðFB Þk 3n kðΩðTB Þ  ΩðTA ÞÞk + kðΩðIB Þ  ΩðIA ÞÞk + kðΩðIB Þ  ΩðIA ÞÞk ¼1 3n kΩðTB Þ  ΩðTA Þk + kΩðIB Þ  ΩðIA Þk + kΩðFB Þ  ΩðFA Þk ¼ Sf ðB, A; f Þ ¼1 3n sf ðA, B; f Þ ¼ 1 

Considering the fact kUk ¼ kUk, we have sf (A, B; f ) ¼ sf (B, A; f ). (SM4) Let C be another single-valued neutrosophic set in U satisfying A  B  C. Then we have: 0  TA ðui Þ  TB ðui Þ  TC ðui Þ  1, 0  IC ðui Þ  IB ðui Þ  IA ðui Þ  1, 0  FC ðui Þ  FB ðui Þ  FA ðui Þ  1 for each ui 2 U, and so: kΩðTA Þ  ΩðTB Þk kΩðTA Þ  ΩðTC Þk kΩðTA Þ  ΩðTC Þk kΩðTA Þ  ΩðTB Þk  ; 1 1 3n 3n 3n 3n kΩðIA Þ  ΩðIB Þk kΩðIA Þ  ΩðIC Þk kΩðIA Þ  ΩðIC Þk kΩðIA Þ  ΩðIB Þk  ; 1 1 3n 3n 3n 3n kΩðFA Þ  ΩðFB Þk kΩðFA Þ  ΩðFC Þk kΩðFA Þ  ΩðFC Þk kΩðFA Þ  ΩðFB Þk  ; 1 1 3n 3n 3n 3n

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Considering the Eq. (14.5), we present the similarity measure sf (A, C; f ) as follows: kΩðTA Þ  ΩðTC Þk + kΩðIA Þ  ΩðIC Þk + kΩðFA Þ  ΩðFC Þk 3n kΩðTA Þ  ΩðTB Þk + kΩðIA Þ  ΩðIB Þk + kΩðFA Þ  ΩðFB Þk 1 ¼ sf ðA, B; f Þ: 3n sf ðA, C; f Þ ¼ 1 

Therefore, sf (A, C; f )  sf (A, B; f ). In a similar way, we can easily obtain the result sf (A, C; f )  sf (B, C; f ). Consequently, the sf (A, B; f ) is a similarity measure between single-valued neutrosophic sets.

14.3.2 A comparison approach with existing similarity measures In this subsection, we give comparison analyses to present the advantages of developed similarity measure over all the existing similarity measures, which are based on several counter-intuitive examples. Suppose that f is Luo-Zhau’s fuzzy implication given by f(a, b) ¼ 1  a  b. Then the results are provided as Table 14.2. Table 14.2 The comparison of the existing similarity measures.

Ai Bi s1(A, B) s2(A, B) s3(A, B) s4(A, B) s5(A, B) s6(A, B) s7(A, B) s8(A, B) s9(A, B) s10(A, B) s11(A, B) s12(A, B) s13(A, B) s14(A, B) s15(A, B) s16(A, B) s17(A, B) s18(A, B) s19(A, B), λ ¼ 0.5 sf (A, B; fLZ)

1

2

3

{h0.4,0.4,0.4i, h0.5,0.5,0.5i} {h0.5,0.5,0.5i, h0.6,0.6,0.6i} 0,900 0.900 0.960 0.980 1.000 0.988 0.988 0.854 0.854 0.785 0.921 0.882 0.900 0.818 0.817 0.790 0.900 0.926 0.926

{h0.4,0.4,0.4i, h0.5,0.5,0.5i} {h0.3,0.3,0.3i, h0.6,0.6,0.6i} 0.900 0.900 0.945 0.972 1.000 0.988 0.988 0.854 0.854 0.785 0.921 0.882 0.900 0.800 0.792 0.790 0.900 0.926 0.926

{h0.6,0.0,0.0i, h0.3,0.0,0.4i} {h0.0,0.5,0.0i, h0.0,0.0,0.0i} 0.700 0.621 0.000 0.000 Undefined 0.886 0.698 0.615 0.768 0.488 0.816 0.583 0.500 0.000 0.000 0.479 0.700 0.764 0.700

0.900

0.800

0.670

A novel similarity measure for single-valued neutrosophic sets

325

Looking at the first column and the second column in Table 14.2, we obtain sk(A, B) ¼ sk(A, B), for k ¼ 1, 2, 5  13, 14  15, 16  19, when A1 ¼ A2 and B1 6¼ B2. Then we say that these similarity measures are not reliable and applicable for real decision situations. Similarly, by comparing the third column, although the falsitymembership degrees in the first components of A and B sets and the indeterminacy-membership degrees in the second components of them are the same, we conclude that there is not a similarity between A and B. Thus, we find that the similarity measures sk(A, B)(k ¼ 3, 4, 14, 15) are invalid in this case. Above all, we can see that the developed similarity measure can solve the counter-intuitive cases of the existing distance measures. Thus, the developed similarity measure is the most reasonable similarity measure.

14.4

Applications

In this section, we present several practical applications to show validity and effectiveness of the developed similarity measure, which are the medical diagnosis, taxonomy classification, and clustering analysis.

14.4.1 Pattern recognition Medical diagnosis is a well-known practice area in pattern recognition, and its many applications are provided based on neutrosophic information [25, 27, 31]. Considering the newly developed similarity measure, we present another approach to solve the medical diagnosis problem. Then we can develop the following algorithm of pattern recognition for SVNSs. Algorithm I Let us consider a finite universe of discourse U ¼ {u1, u2, …, un}. Assume that there exist m patterns, which are represented by the form of SVNSs, denoted by Dj ¼ {hui, TDj(ui), IDj(ui), FDj(ui)i : ui 2 U} (j ¼ 1, 2, …, m) in U and there is a sample pattern, which is characterized by an SVNS S ¼ {hui, TS(ui), IS(ui), FS(ui)i : ui 2 U}. The recognition process is as follows: Step 1. Compute the similarity measure sf (Dj, P; f ) between Dj (j ¼ 1, 2, …, m) and S. Step 2. Select the maximum one   sf (Dj0,P; f ) from sf (Dj, P; f ) (j ¼ 1, 2, …, m), that is, sf Dj0 , P; f ¼ max 1jm sf Dj , P; f . Thus, the sample pattern P is classified to the pattern Dj0.

Example 14.1. (Medical diagnosis) Let us consider the medical diagnosis problem adapted from [25] and Reichenbach’s fuzzy implication given by fR(a, b) ¼ 1  a + ab. Assume that a set of diagnoses is: D ¼ fD1 ðviral feverÞ, D2 ðmalariaÞ, D3 ðtyphoidÞ, D4 ðgastritisÞ, D5 ðstenocardiaÞg

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Optimization Theory Based on Neutrosophic and Plithogenic Sets

and a set of symptoms is: S ¼ fs1 ðfeverÞ, s2 ðheadacheÞ, s3 ðstomach painÞ, s4 ðcoughÞ, s5 ðchest painÞg:

Table 14.3 presents the characteristic values of the considered diseases and their neutrosophic symptom values represented by the form of SVNSs. The neutrosophic symptom values are characterized with a triple of numbers corresponding to the truth-membership, indeterminacy-membership, and falsity-membership values, respectively. In the field of medical diagnosis, assume that we take a sample from a patient P with all the symptoms, which is represented by the following SVNS information: PðpatientÞ ¼ fhs1 , 0:8; 0:2; 0:1i, hs2 , 0:6;0:3; 0:1i, hs3 , 0:2; 0:1; 0:8i, hs4 , 0:6; 0:5; 0:1i, hs5 , 0:1; 0:4; 0:6ig:

Now, we apply Algorithm I in solving the mentioned diagnosis problem. Using Eq. (14.5), we compute all of the similarity measure sf (Dj, P; f ) between Dj (j ¼ 1, 2, …, m) and P. These results are presented in Table 14.4. Considering the similarity measure given in Eq. (14.5), we can determine a proper diagnosis for the patient P. According to the approach of maximum similarity degree, the largest similarity measure indicates the proper diagnosis. The obtained results are given in Table 14.4. By analyzing the data presented in the Table 14.4, we can say that the patient P suffers from D1 (viral fever), according to Reichenbach’s fuzzy implication. The final result differs from the result obtained in the method of Ye [25]. There exist some reasons for the difference, which are analyzed extensively in Section 14.5. Example 14.2. (Bacteria detection taxonomy) The second application is the detection of bacteria [10], which is an important issue in microbiology. Considering the fact that each bacterium produces specific diseases, microbiologists spend intense effort on detection of intestinal bacteria such as Shigella, Salmonella, Klebsiella, and Bacillus coli. According to microbiologists, bacteria can be broadly classified according to their shape. The characteristics used for bacterial classification in the example presented in [10] are the domical shape, the single microscopic shape, the double microscopic shape, and existence of the flagellum, denoted, respectively. Suppose that f is Luo-Zhau’s fuzzy implication, defined by fLZ(a, b) ¼ 1  a  b. First, microbiologists have gathered information about the properties of the intestinal bacteria studied. Then they provide average truth-membership, indeterminatemembership, and falsity-membership information for the characteristics of known bacteria (see Table 14.5). For each property, truth-membership, indeterminate-membership, and falsitymembership values indicate the feature’s presence, feature’s obscurity, and feature’s absence in a specific class, respectively. In addition, the experts provide the values of the properties for different undiagnosed samples (see Table 14.6).

D1 D2 D3 D4 D5

(viral fever) (malaria) (typhoid) (gastritis) (stenocardia)

s1 (fever)

s2 (headache)

s3 (stomach p.)

s4 (cough)

s5 (chest pain)

hs1, 0.4,0.6,0.0i hs1, 0.7,0.3,0.0i hs1, 0.3,0.4,0.3i hs1, 0.1,0.2,0.7i hs1, 0.1,0.1,0.8i

hs2, 0.3,0.5,0.5i hs2, 0.2,0.2,0.6i hs2, 0.6,0.3,0.1i hs2, 0.2,0.4,0.4i hs2, 0.0,0.2,0.8i

hs3, 0.1,0.3,0.7i hs3, 0.0,0.1,0.9i hs3, 0.2,0.1,0.7i hs3, 0.8,0.2,0.0i hs3, 0.2,0.0,0.8i

hs4, 0.4,0.3,0.3i hs4, 0.7,0.3,0.0i hs4, 0.2,0.2,0.6i hs4, 0.2,0.1,0.7i hs4, 0.2,0.0,0.8i

hs5, 0.1,0.2,0.7i hs5, 0.1,0.1,0.8i hs5, 0.0,0.0,0.9i hs5, 0.2,0.1,0.7i hs5, 0.8,0.1,0.1i

A novel similarity measure for single-valued neutrosophic sets

Table 14.3 Characteristic values of the diseases represented by SVNSs.

327

328

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 14.4 Similarity measure values for SVNS information.

sf (Di, P; fR) s6(Di, P)[25] s7(Di, P)[25]

D1 (viral fever)

D2 (malaria)

D3 (typhoid)

D4 (gastritis)

D5 (stenocardia)

0.855 0.313 0.298

0.840 0.319 0.299

0.792 0.308 0.281

0.620 0.274 0.204

0.593 0.255 0.187

Using Eq. (14.5), we compute all of the similarity measure sf (Bj, Ki; f ) between Bj (j ¼ 1, 2, …, 4) and Ki (i ¼ 1, 2, …, 6). These results are presented in Table 14.7. From the results of Table 14.7, since the largest similarity measure values expresses the proper diagnosis, two undiagnosed samples K1 and K3 are classified as Shigella bacteria, the three undiagnosed samples K2, K4 and K6 are classified as Bacillus coli bacteria, and the undiagnosed sample K5 is classified as Klebsiella bacteria. Obviously, these results are different from ones based on intuitionistic fuzzy information and obtained by Khatibi and Montazer [10]. There are some reasons for this difference. The IFS can only handle incomplete information but not the indeterminate information and inconsistent information. In an SVNS, its truth-membership, indeterminacymembership, and falsity-membership are characterized independently without any restriction from them, and they can offer more information to the decision process. So the notion of SVNS is more general. Therefore, the taxonomy approach under single-valued neutrosophic environment is more suitable and more effective for capturing inconsistent, imprecise, and uncertain information in handling the data.

14.4.2 Cluster analysis It is clear that the goal of cluster analysis classifies the set of objects into appropriate groups. In order to apply the developed similarity measure to a clustering problem, we present another algorithm as follows. Algorithm II Firstly, we give a clustering algorithm based on the proposed similarity measure between SVNSs and apply it to a clustering problem under a neutrosophic environment. Therefore, we basically generalize the intuitionistic fuzzy clustering algorithm proposed by Zhang et al. [8] and Xu et al. [9] to SVNSs. Let U ¼ {u1, u2, …, um} be a discrete universe of discourse, and let W ¼ (w(u1), w(u2), …, w(um)) P be the weight vector of the ui, w(ui) 2 [0, 1] and ni¼1 w ðui Þ ¼ 1. Assume that Ai (i ¼ 1, 2, …, m) is a set of m SVNSs representing different objects.

Zhang et al. [8] and Xu et al. [9] presented the following definitions to develop the clustering algorithm: Definition 14.10. A matrix composed of similarity measure values, denoted by C ¼ (cij)mm, is called a similarity matrix, where cij ¼ sfw(Aj, Ai; f )(i, j ¼ 1, 2, …, m) and cij 2 [0, 1] for i, j ¼ 1, 2, …, m, with cii ¼ 1 for i ¼ 1, 2, …, m, and cij ¼ cji for i, j ¼ 1, 2, …, m.

B1 B2 B3 B4

(Bacillus coli) (Shigella) (Salmonella) (Klebsiella)

s1 (Domical shape)

s2 (Single microscope)

s3 (Double microscope)

s4 (Flagellum)

hs1, 0.85,0.10,0.05i hs1, 0.83,0.15,0.08i hs1, 0.79,0.25,0.12i hs1, 0.82,0.15,0.15i

hs2, 0.87,0.05,0.01i hs2, 0.92,0.05,0.05i hs2, 0.78,0.10,0.11i hs2, 0.72,0.10,0.15i

hs3, 0.02,0.25,0.97i hs3, 0.05,0.15,0.92i hs3, 0.11,0.15,0.85i hs3, 0.22,0.25,0.75i

hs4, 0.92,0.10,0.06i hs4, 0.08,0.10,0.91i hs4, 0.87,0.05,0.01i hs4, 0.12,0.15,0.85i

A novel similarity measure for single-valued neutrosophic sets

Table 14.5 The single-valued neutrosophic values of the features in the classes of bacteria.

329

330

Table 14.6 The single-valued neutrosophic values of the features in the undiagnosed samples. s2 (Single microscope)

s3 (Double microscope)

s4 (Flagellum)

hs1, 0.837,0.020,0.133i hs1, 0.911,0.055,0.029i hs1, 0.929,0.125,0.037i hs1, 0.815,0.046,0.091i hs1, 0.864,0.023,0.020i hs1, 0.905,0.015,0.016i

hs2, 0.718,0.125,0.159i hs2, 0.831,0.035,0.031i hs2, 0.812,0.042,0.033i hs2, 0.949,0.055,0.048i hs2, 0.610,0.250,0.230i hs2, 0.878,0.022,0.015i

hs3, 0.064,0.015,0.897i hs3, 0.028,0.028,0.894i hs3, 0.021,0.046,0.926i hs3, 0.020,0.025,0.880i hs3, 0.243,0.152,0.624i hs3, 0.072,0.025,0.917i

hs4, 0.021,0.010,0.806i hs4, 0.952,0.252,0.036i hs4, 0.054,0.025,0.922i hs4, 0.833,0.016,0.042i hs4, 0.004,0.033,0.964i hs4, 0.789,0.150,0.114i

Optimization Theory Based on Neutrosophic and Plithogenic Sets

K1 K2 K3 K4 K5 K6

s1 (Domical shape)

A novel similarity measure for single-valued neutrosophic sets

331

Table 14.7 The obtained similarity measures.

K1 K2 K3 K4 K5 K6

B1 (Bacillus coli)

B2 (Shigella)

B3 (Salmonella)

B4 (Klebsiella)

0.486 0.902 0.513 0.884 0.486 0.883

0.854 0.515 0.918 0.536 0.813 0.548

0.496 0.859 0.522 0.870 0.500 0.846

0.824 0.500 0.833 0.511 0.858 0.525

The values in boldface indicate pairs with the largest first component.

  Definition 14.11. Let C ¼ (cij)mm be a similarity matrix. If C2  C ¼ cij mm , then C2

is called a composition matrix of C, where cij ¼ max k min cik , ckj , i, j ¼ 1, 2,…, m. Definition 14.12. Let C ¼ (cij)mm be a similarity matrix. If C2  C, i.e., cij  cij for i, j ¼ 1, 2, …, m, then C is called an equivalent similarity matrix. Definition 14.13. Let C ¼ (cij)mm be a similarity matrix. Then, after finite time compositions of C: k

C ! C 2 ! C4 ! ⋯ ! C 2 ! ⋯

(14.6) k

there must exist a positive integer k such that C2 ¼ C2 association matrix.

(k+1)

k

, and C2 is also an equivalent

Definition 14.14. Let C ¼ (cij)mm be an equivalent similarity matrix. Then, Cλ ¼ (cλij)mm is called the λ  cutting matrix of C, where: cλij ¼




0, if cij < λ ði, j ¼ 1, 2, …, mÞ 1, if cij  λ

(14.7)

and λ is the confidence level with λ 2 [0, 1]. Then the clustering process of SVNSs is as follows: Step 1. Use Eq. (14.5) to compute the association coefficients of SVNSs Ai(i ¼ 1, 2, …, m), and construct an association matrix C ¼ (cij)mm, where cij ¼ sfw(Aj, Ai; f )(i, j ¼ 1, 2, …, m). Step 2. Establish the composition matrices by the finite time compositions of C: k

ð k + 1Þ

C ! C 2 ! C4 ! ⋯ ! C 2 ¼ C 2

(14.8)

  k which implies that C2 is an equivalent similarity matrix denoted by C ¼ cij mm

  Step 3. Construct a λ-cutting matrix Cλ ¼ cλij of the equivalent similarity matrix mm   C ¼ cij mm (by Eq. (14.7)) when λ has different confidence levels, if all the elements of the ith row or column in Cλ are the same as the corresponding elements of the jth row or column, we obtain the object sets Ai and Aj, which are the same class.

332

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Example 14.3. (Clustering approach) In this section, a real example adapted from Ye [23] is utilized to demonstrate the application and effectiveness of the proposed clustering algorithms based on similarity matrices under single-valued neutrosophic data environment. A car market is going to classify five different cars of Ai (i ¼ 1, 2, …, 5). Every car has six evaluation attributes: (1) u1: fuel consumption; (2) u2: coefficient of friction; (3) u3: price; (4) u4: comfortable degree; (5) u5: design; (6) u6: security coefficient. The characteristics of each car under the six attributes are represented by the form of SVNSs, and then the single-valued neutrosophic data are as follows:


 hu1 , 0:3, 0:2, 0:5i, hu2 , 0:6, 0:3, 0:1i, hu3 , 0:4, 0:3, 0:3i, A1 ¼ , hu4 , 0:8, 0:1, 0:1i, hu5 , 0:1, 0:3, 0:6i, hu6 , 0:5, 0:2, 0:4i
 hu1 , 0:6, 0:3, 0:3i, hu2 , 0:5, 0:4, 0:2i, hu3 , 0:6, 0:2, 0:1i A2 ¼ , hu4 , 0:7, 0:2, 0:1 i, hu5 , 0:3, 0:1, 0:6i, hu6 , 0:4, 0:3, 0:3i
 hu1 , 0:4, 0:2, 0:4i, hu2 , 0:8, 0:2, 0:1i, hu3 , 0:5, 0:3, 0:1i, A3 ¼ , hu4 , 0:6, 0:1, 0:2i, hu5 , 0:4, 0:1, 0:5i, hu6 , 0:3, 0:2, 0:2i
 hu1 , 0:2, 0:4, 0:4i, hu2 , 0:4, 0:5, 0:1i, hu3 , 0:9, 0:2, 0:0i, , A4 ¼ hu4 , 0:8, 0:2, 0:1i, hu5 , 0:2, 0:3, 0:5i, hu6 , 0:7, 0:3, 0:1i
 hu1 , 0:5, 0:3, 0:2i, hu2 , 0:3, 0:2, 0:6i, hu3 , 0:6, 0:1, 0:3i, A5 ¼ , hu4 , 0:7, 0:1, 0:1i, hu5 , 0:6, 0:2, 0:2i, hu6 , 0:5, 0:2, 0:3i Then, the developed algorithm is used to classify the five different cars of Ai (i ¼ 1, 2, …, 5) under the single valued neutrosophic similarity measure, which is given respectively in following subsections in detail.


1, if a  b , we b, if a > b determine the similarity measures between each pair of SVNSs Ai and Aj(i, j ¼ 1, 2, …, 5) and construct the following similarity matrix: 3 2 1 0:7065 0:7802 0:6809 0:6260 7 6 1 0,7158 0:6364 0:7045 7 6 0:7065 7 6 C¼6 1 0:7066 0:6236 7 7 6 0:7802 0:7158 7 6 1 0:6409 5 4 0:6809 0:6364 0:6077 0:6260 0:7045 0:6236 0:6409 1

Step 1. Using Eq. (14.5) under G€odel’s fuzzy implication given by fG ða, bÞ ¼

Step 2. Using Eq. (14.6), we obtain an equivalent similarity matrix by limited time compositions of C: 3 2 1 0:7158 0:7802 0:7066 0:7045 7 6 1 0:7158 0:7066 0:7045 7 6 0:7158 7 6 C2 ¼ 6 1 0:7066 0:7045 7 7 6 0:7802 0:7158 7 6 1 0:6409 5 4 0:7066 0:7066 0:7066 0:7045 0:7045 0:7045 0:6409 1

A novel similarity measure for single-valued neutrosophic sets

2

3 0:7158 0:7802 0:7066 0:7045 7 1 0, 7158 0:7066 0:7045 7 7 0:7158 1 0:7066 0:7045 7 7 7 0:7066 0:7066 1 0:7045 5 0:7045 0:7045 0:7045 1

2

3 0:7158 0:7802 0:7066 0:7045 7 1 0, 7158 0:7066 0:7045 7 7 0:7158 1 0:7066 0:7045 7 7 7 0:7066 0:7066 1 0:7045 5 0:7045 0:7045 0:7045 1

1 6 6 0:7158 6 C4 ¼ 6 6 0:7802 6 4 0:7066 0:7045 1 6 0:7158 6 6 C8 ¼ 6 6 0:7802 6 4 0:7066 0:7045

333

Obviously, we can say that C4 ¼ C8, which indicates that C4 is an equivalent similarity matrix, denoted by C.

  Step 3. We obtained λ-cutting matrix Cλ ¼ cλij

mm

of C by Eq. (14.7). Here, we get different

categories for different values λ, and present these conclusions as follows: (1) If 0  λ  0.7045, all cars Ai(i ¼ 1, 2, …, 5) are of the same type: {A1, A2, A3, A4, A5}. (2) If 0.7045 < λ  0.7066, all cars Ai(i ¼ 1, 2, …, 5) are classified into the following two types {A1, A2, A3, A4}, {A5}. (3) If 0.7066 < λ  0.7158, all cars Ai(i ¼ 1, 2, …, 5) are classified into the following three types {A1, A2, A3}{A4}, {A5}. (4) If 0.7158 < λ  0.7802, all cars Ai(i ¼ 1, 2, …, 5) are classified into the following four types {A1, A3}{A2}, {A4}, {A5}. (5) If 0.7802 < λ  1, all cars Ai(i ¼ 1, 2, …, 5) are classified into the following five types, i.e., none of them are of the same class:{A1}, {A2}, {A3}, {A4}, {A5}.

By the motivation of the above clustering results, we can find that the clustering results based on the proposed similarity measure are slightly different from the clustering results obtained by using the distance-based similarity measures given by Ye [23]. By Step 3, we achieve the classification as {A1, A3}, {A2}, {A4}, {A5}, while it is {A1}, {A2, A3}, {A4}, {A5} according to Ye’s approach. The reason for this difference is that Ye’s method [23] is only based on the max-min operators that cause a great loss of information, and ignores the effect of similarity relationships of each truth-membership function, indeterminacy-membership function, and falsitymembership function. Others reasons are discussed extensively in Section 14.5.

14.5

Discussions and comparison

In order to illustrate the advantages of the developed similarity measure over the existing similarity measures, we prove a general comparison based on the presented examples.

334

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 14.8 Other results based on the fuzzy implications.

sf (Di, P; fR) sf (Di, P; fG) sf (Di, P; fLU) sf (Di, P; fKD) sf (Di, P; fM) sf (Di, P; fLA) sf (Di, P; fLZ)

D1 (viral fever)

D2 (malaria)

D3 (typhoid)

D4 (gastritis)

D5 (stenocardia)

0.855 0.697 0.855 0.806 0.835 0.862 0.659

0.840 0.715 0.844 0.834 0.817 0.863 0.674

0.792 0.688 0.804 0.793 0.790 0.863 0.577

0.620 0.532 0.665 0.686 0.723 0.801 0.440

0.593 0.532 0.636 0.679 0.636 0.778 0.284

By comparing with Table 14.8, the diagnosis result of the patient P is viral fever under the fuzzy implications of Reichenbach, Lukasiewicz, and Mamdani while the diagnosis result of the patient P given in this chapter is malaria under the fuzzy implications of G€ odel, Kleene-Dienes, Larsen, and Luo-Zhau. It is obviously seen that the paint P suffers from malaria according to four of the seven fuzzy implications, which are different from results obtained in [25], while the paint P suffers from viral fever according to three of the seven fuzzy implications, which are the same results obtained in [25]. Consequently, the medical diagnoses using the different fuzzy implications indicate different diagnosis results. Hence, we can obtain different diagnosis results under the different fuzzy implications. Therefore, the developed method using the similarity measures of SVNSs proposed in this chapter is more comprehensive, flexible, and effective in pattern recognition algorithms. All results obtained with existing similarity measures are given in Table 14.9. In this table, the results obtained show that P is malaria. This coincides with our ones obtained using the fuzzy implications of G€ odel, Kleene-Dienes, Larsen, and LuoZhau. Thus, it can be said that our proposed similarity measure selects the proper diagnosis appropriately for patients. By comparing with Table 14.10, it is obviously seen that the clustering results obtained by using the fuzzy implication and matrix form, and the max and min operations-based similarity measure [23], have major differences under single-valued neutrosophic environments. The basic reason is that the developed method to measure the similarity between SVNSs presents an approach based on matrix to minimize information loss, while Ye’s similarity measure [23] adopts the max-min operations that the loss of information is inevitable. Moreover, the effect of the fuzzy implications on the decision process of presented examples is clearly demonstrated in Figs. 14.1–14.3, respectively. As mentioned above, there are some drawbacks of the existing similarity measures in some cases and they cannot handle undefined (unspecified) or absurd phenomena in some real cases. Thus, the developed similarity measure of SVNSs can not only solve the counter-intuitive cases of the existing similarity measures under the single-valued neutrosophic information, but also can provide a parameterized family of similarity measures of SVNSs that provides the decision-makers with different choices, which is superior to the existing similarity measures of SVNSs.

A novel similarity measure for single-valued neutrosophic sets

335

Table 14.9 The results obtained from other existing similarity measures for the pattern recognition problem discussed in Example 14.1. Similarity Measure

D1 (viral fever)

D2 (malaria)

D3 (typhoid)

D4 (gastritis)

D5 ((stenocardia)

Result

s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15 s16 s17 s18 s19

0.793 0.762 0.249 0.282 0.286 0.313 0.298 0.723 0.640 0.869 0.455 0.575 0.720 0.557 0.570 0.351 0.793 0.841 0.641

0.833 0.786 0.264 0.288 0.289 0.319 0.299 0.773 0.667 0.877 0.544 0.670 0.740 0.615 0.628 0.431 0.833 0.873 0.716

0.800 0.729 0.236 0.267 0.272 0.308 0.281 0.744 0.638 0.857 0.354 0.282 0.700 0.552 0.606 0.364 0.800 0.843 0.622

0.633 0.560 0.127 0.170 0.170 0.274 0.204 0.551 0.369 0.802 0.100 0.161 0.440 0.304 0.324 0.154 0.633 0.703 0.211

0.593 0.511 0.109 0.141 0.141 0.255 0.187 0.521 0.363 0.212 0.112 0.139 0.420 0.265 0.315 0.125 0.593 0.663 0.118

Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria Malaria

Table 14.10 Other results based on the fuzzy implications. Fuzzy implications

λ-Cutting value

Classifications

Reichenbach

0  λ  0.8151 0.8151 < λ  0.8357 0.8357 < λ  0.8560 0.8360 < λ  0.9015 0.9015 < λ  1 0  λ  0.7045 0.7045 < λ  0.7066 0.7066 < λ  0.7158 0.7158 < λ  0.7802 0.7802 < λ  1 0  λ  0.8283 0.8283  λ  0.8376 0.8376 < λ  0.8430 0.8430 < λ  0.9032 0.9032 < λ  1

{A1, A2, A3, A4, A5} {A1, A2, A3, A5}, {A4} {A1, A3}{A2, A5}, {A4} {A1, A3}{A2}, {A4}, {A5} {A1}, {A2}, {A3}, {A4}, {A5} {A1, A2, A3, A4, A5} {A1, A2, A3, A4}, {A5} {A1, A2, A3}, {A4}, {A5} {A1, A3}{A2}, {A4}, {A5} {A1}, {A2}, {A3}, {A4}, {A5} {A1, A2, A3, A4, A5} {A1, A2, A3, A5}, {A4} {A1, A3}, {A2, A5}, {A4} {A1, A3}, {A2}, {A4}, {A5} {A1}, {A2}, {A3}, {A4}, {A5}

G€odel

Lukasiewicz

Continued

336

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 14.10 Continued Fuzzy implications

λ-Cutting value

Classifications

Kleene-Dienes

0 < λ  0.8433 0.8433 < λ  0.8468 0.8468 < λ  0.8531 0.8531 < λ  0.9095 0.9095 < λ  1 0  λ  0.8700 0.8700 < λ  0.8928 0.8928 < λ  0.8981 0.8981 < λ  0.9231 0.9231 < λ  1 0  λ  0.9244 0.9244 < λ  0.9368 0.9368 < λ  0.9383 0.9383 < λ  0.9578 0.9578 < λ  1 0  λ  0.7560 0.7560 < λ  0.7747 0.7747 < λ  0.7753 0.7753 < λ  0.8507 0.8507 < λ  1

{A1, A2, A3, A4, A5} {A1, A2, A3, A4}, {A5} {A1, A2, A3}, {A4}, {A5} {A1, A3}{A2}, {A4}, {A5} {A1}, {A2}, {A3}, {A4}, {A5} {A1, A2, A3, A4, A5} {A1, A2, A3, A5}{A4} {A1, A2, A3}, {A4}, {A5} {A1, A3}, {A2}, {A4}, {A5} {A1}, {A2}, {A3}, {A4}, {A5} {A1, A2, A3, A4, A5} {A1, A2, A3, A5}, {A4} {A1, A3}, {A2, A5}, {A4} {A1, A3}{A2}, {A4}, {A5} {A1}, {A2}, {A3}, {A4}, {A5} {A1, A2, A3, A4, A5} {A1, A2, A3, A5}, {A4} {A1, A3}, {A2, A5}, {A4} {A1, A3}, {A2}, {A4}, {A5} {A1}, {A2}, {A3}, {A4}, {A5}

Mamdani

Larsen

Luo-Zhau

14.6

Conclusions

Similarity measure plays a key role to determine the relationship between objects. The neutrosophic literature has a lot of similarity measures proposed according to different perspectives. However, most of them have some counter-intuitive cases. In this chapter, we developed a novel similarity measure between SVNSs, which handles the counter-intuitive cases. Then, since the developed similarity measure is based on matrix norms and binary functions, we presented a series of similarity measures between SVNs to minimize the loss of information. In order to check the effectiveness of our proposed similarity measure and see its advantages, it was compared with other known similarity measures. Then we applied it to real-world applications such as clustering analysis, medical diagnosis, and bacteria detection, and solved three decision-making problems under simplified neutrosophic environment, respectively. By analyzing the results obtained, we can note that not only are the proposed similarity measures able to handle the single valued neutrosophic information, but also they offer different final options to decision-makers by different fuzzy implications. In further work, it is necessary to apply the new similarity measures of SNSs to other areas such as image processing, decision-making, and clustering analysis.

A novel similarity measure for single-valued neutrosophic sets

1

0.9

0.8

Similarity

0.7

0.6 s(P, Q1)

0.5

s(P, Q2) s(P, Q3) s(P, Q4) s(P, Q5)

0.4

0.3

0.2 Reichenbach

Gödel

Lukasiewicz

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Fig. 14.1 Similarity measures with respect to different fuzzy implications for Example 14.1. 337

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s(A3, A5;f) s(A4, A5;f)

Reichenbach

Gödel

Lukasiewicz

Kleene-Dienes Function

Mamdani

Fig. 14.2 Similarity measures with respect to different fuzzy implications for Example 14.2.

Larsen

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Lukasiewicz

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Fig. 14.3 Similarity measures with respect to different fuzzy implications for Example 14.3. 339

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References [1] L.A. Zadeh, Fuzzy sets, Infect. Control 8 (1965) 338–353. [2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst. 20 (1986) 87–96. [3] V. Balopoulos, A.G. Hatzimichailidis, B.K. Papadopoulos, Distance and similarity measures for fuzzy operators, Inform. Sci. 177 (11) (2007) 2336–2348. [4] A.G. Hatzimichailidis, G.A. Papakostas, V.G. Kaburlasos, A novel distance measure of intuitionistic fuzzy sets and its application to pattern recognition problems, Int. J. Intell. Syst. 27 (4) (2012) 396–409. [5] M. Luo, R.A. Zhao, Distance measure between intuitionistic fuzzy sets and its application in medical diagnosis, Artif. Intell. Med. 89 (2018) (2018) 34–39. [6] J. Ye, Cosine similarity measures for intuitionistic fuzzy sets and their applications, Math. Comput. Model. 53 (1–2) (2011) 91–97. [7] M.Y. Tian, A new fuzzy similarity based on cotangent function for medical diagnosis, Adv. Model. Optim. 15 (2) (2013) 151–156. [8] H. Zhang, Z. Xu, Q. Chen, Clustering method of intuitionistic fuzzy sets, Contr. Decision 22 (8) (2007) 882–888. [9] Z. Wang, Z. Xu, S. Liu, J. Tang, A netting clustering analysis method under intuitionistic fuzzy environment, Appl. Soft Comput. 11 (2011) 5558–5564. [10] V. Khatibi, G.A. Montazer, Intuitionistic fuzzy set vs. fuzzy set application in medical pattern recognition, Artif. Intell. Med. 47 (2009) 43–52. [11] F. Smarandache, A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press, Rehoboth, 1999. [12] H. Wang, F. Smarandache, Y.Q. Zhang, R. Sunderraman, Single valued neutrosophic sets, Multispace Multistruct. 4 (2010) 410–413. [13] H. Wang, F. Smarandache, Y.Q. Zhang, R. Sunderraman, Interval Neutrosophic Sets and Logic: Theory and Applications in Computing, Hexis, Phoenix, AZ, 2005. [14] P. Biswas, S. Pramanik, B.C. Giri, TOPSIS method for multi-criteria group decisionmaking under simplified neutrosophic environment, Neural Comput. Applic. 27 (3) (2016) 727–737. [15] S. Pramanik, P. Biswas, B.C. Giri, Hybrid vector similarity measures and their applications to multi-attribute decision making under neutrosophic environment, Neural Comput. Applic. 28 (5) (2017) 1163–1176. [16] J. Wang, Y. Yang, L. Li, Multi-criteria decision-making method based on single-valued neutrosophic linguistic Maclaurin symmetric mean operators, Neural Comput. Applic. 30 (2018) 1529–1547. [17] J.J. Peng, J.Q. Wang, J. Wang, H.Y. Zhang, X.H. Chen, Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems, Int. J. Syst. Sci. (2015) 201, https://doi.org/10.1080/00207721.2014.994050. [18] R. Şahin, P.D. Liu, Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information, Neural Comput. Applic. (2015) https:// doi.org/10.1007/s00521-015-1995-8. [19] P.D. Liu, Y.C. Chu, Y.W. Li, Y.B. Chen, Some generalized neutrosophic number Hamacher aggregation operators and their application to Group Decision Making, Int. J. Fuzzy Syst. 16 (2) (2014) 242–255. [20] P.D. Liu, Y. Wang, Multiple attribute decision-making method based on single-valued neutrosophic normalized weighted Bonferroni mean, Neural Comput. Applic. 25 (7–8) (2014) 2001–2010.

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[21] J. Ye, Single valued neutrosophic minimum spanning tree and its clustering method, J. Intell. Syst. 23 (3) (2014) 311–324. [22] P. Majumdar, S.K. Samanta, On similarity and entropy of neutrosophic sets, J. Intell. Fuzzy Syst. 26 (3) (2014) 1245–1252. [23] J. Ye, Single-valued neutrosophic clustering algorithms based on similarity measures, J. Classif. 34 (1) (2017) 148–162. [24] J. Ye, Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making, Int. J. Fuzzy Syst. 16 (2014) 204–211. [25] J. Ye, Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses, Artif. Intell. Med. 63 (3) (2015) 171–179. [26] J. Ye, Single-valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine, Soft Comput. 21 (3) (2017) 817–825. [27] J. Ye, J. Fu, Multi-period medical diagnosis method using a single valued neutrosophic similarity measure based on tangent function, Comput. Methods Programs Biomed. 123 (2016) 142–149. [28] R. Sahin, A. Kucuk, Subsethood measure for single valued neutrosophic sets, J. Intell. Fuzzy Syst. 29 (2) (2015) 525–530. [29] K. Mondal, S. Pramanik, B.C. Giri, Hybrid binary logarithm similarity measure for MAGDM problems under SVNS assessments, Neutrosoph. Sets Syst. 20 (2018) 12–25. [30] J. Ye, Fault diagnoses of hydraulic turbine using the dimension root similarity measure of single-valued neutrosophic sets, Intell. Autom. Soft Comput. (2016), https://doi.org/ 10.1080/10798587.2016.1261955. [31] G. Shahzadi, M. Akram, A. Borumand, A.B. Saeid, An application of single-valued neutrosophic sets in medical diagnosis, Neutrosoph. Sets Syst. 18 (2017). [32] S. Broumi, F. Smarandache, Correlation coefficient of interval neutrosophic set, Appl. Mech. Mater. 436 (2013) 511–517. [33] E. Bolturk, C. Kahraman, A novel interval-valued neutrosophic AHP with cosine similarity measure, Soft Comput. 22 (15) (2018) 4941–4958. [34] A. Karas¸ an, C. Kahraman, A novel interval-valued neutrosophic EDAS method: prioritization of the United Nations national sustainable development goals, Soft Comput. 22 (15) (2018) 4891–4906. [35] R. Şahin, An approach to neutrosophic graph theory with applications, Soft Comput. 23 (2) (2019) 569–581. [36] X.Z. Liu, M. Yang, Matrix Theory, Huazhong University of Science and Technology, 2005. [37] Q.X. Cheng, D.Z. Zhang, Basis of Real Variable Function and Functional Analysis, Higher Education Press, 1983.

Modified neutrosophic fuzzy optimization model for optimal closed-loop supply chain management under uncertainty

15

Firoz Ahmada, Ahmad Yusuf Adhamia, Florentin Smarandacheb a Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, India, bDepartment of Mathematics, University of New Mexico, Gallup, NM, United States

15.1

Introduction

Needs for commodities or products for human life explored and enhanced a revolution in industrial sectors. An initially integrated mechanism for production to consumption of commodities was only the goals of any firm. Decision policies concerning the production and use of products were the prime concern. Therefore, systematic and business-oriented managerial practices were designed for the flow of products, termed as supply chain management (SCM). SCM is the procedure of procurement, processing, distribution, and consumption of finished products in a clear planning timescale. The general structural domain of SCM includes a raw material supplier point, a manufacturing plant, a distribution center, and the end-users or customers. These echelons are interconnected or interlocked to each other for the movement of different materials and products. The organizational and managerial perspective of SCM terminates at the end-users of finished products and terminates from ultimately the next stages related to the three R’s (reduce, reuse, and recycle). End-ofuse and end-of-life products create various environmental issues due to improper management of used products. Consequently, harmful impacts due to landfills, contamination of freshwater resources, and toxic air pollution generated on a large scale influenced human life drastically. These issues could not be compensated at any cost. To ensure that environmental questions and social concerns arise during supply chain design, a government has taken the initiative and established laws that include wholesome supply chain practices, termed as the closed loop supply chain (CLSC) network. The CLSC design helps in strengthening the ecofriendly practices with end-of-use products and reduces environmental impacts. Therefore, to reveal pervasiveness in SCM, extension of echelons has been located. Hence the concept of the reverse chain has been identified to execute backward processes for used products. Generally, the reverse chain consists of different echelons, such as the collection center, recycling

Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00015-9 © 2020 Elsevier Inc. All rights reserved.

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point, and disposal sites. The CLSC design contemplates the flow of different materials, products, and parts of the used commodity in a well-defined interconnected path. Various facility centers in the reverse chain reduce the environmental impacts and substantiates ecofriendly production-consumption planning scenarios. For successful completion of sustainable trade practices, the significant role of the CLSC design network would be crucial or at least prominent. Ultimate destinations for end-of-use and end-of-life products would be more rigorously depicted in CLSC design. Refurbishing and recycling centers inevitably provide services to the used products and parts to transform into their useful life. The marginal reduction in different kinds of costs and a significant increase in revenues are the counterpart for enhancement in net profit throughout the CLSC planning network. Consumerism has been a considerable part of the sustainability problem for years by imposing a burden with harmful waste through flooding and landfill issues. The CLSC business model implements highly efficient management of materials and waste minimization strategies that lead to zero-waste generation. The CLSC management network includes either putting all outputs back into the system or incineration. A combination of forwarding and reverse material flows to reuse and recycle all metals and transform waste into energy. The CLSC can enable manufacturers to take a proactive stance toward and ensure easy compliance with electronic waste regulations. Environmental value is the ease of agreement to be more conscious about the environment. A CLSC can allow the business to respond to ecological concerns by saving energy and decreasing the input of new materials. Consumer value can be achieved by a well-organized customer product returns system that can help ensure hassle-free warranties and improve customer loyalty. Improved parts management helps the business deliver extended warranties and service agreements that can boost customer satisfaction. The acquisition process in CLSC management provides valuable data on common production issues, supply defects, failure rates, product lifecycle, consumer complaints, and consumer usage patterns. This information can be used to improve product design and development. Minimize wastewater and industrial sludge production by reducing the amount of water needed for the manufacturing process. Procure raw material in bulk (where possible) to reduce the amount of packaging material that enters the waste stream. Assure precautions to avoid the process that causes hazardous waste to be mixed with nonhazardous waste, minimizing the amount of dangerous waste that must be stored, treated, and disposed of. Practice quality control strategies like ISI 14001 and Six Sigma to help minimize product defects. The implicitness of uncertainty is trivial in real-life scenarios. Inconsistent, incomplete, inappropriate, inexact, and improper information about various input parameters such as costs, capacity, and demand in the CLSC design network lead to the existence of uncertainty theory. Several aspects inherently affect the modeling and optimizing procedure of real-life optimization problem. Abrupt changes in the prices of raw materials, hike in fuel rates, increases in required facility locations, behavior of fluctuating markets, competition among different companies’ policies for customer satisfaction, environmental conditions, failing in timely shipment of ordered products,

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political and governmental policies regarding various taxes over procurement, production, distribution, and management of end-of-use products are the most dominating factors for causing uncertainty in modeling approaches. Impreciseness may be represented in different forms. The difficulty involved in parameters due to vague information can be dealt with by different fuzzy techniques. Fuzziness among parameters most frequently encounters and results in uncertainty modeling. To reflect the most common aspect of uncertainty, we have assumed that all the input parameters are a triangular and trapezoidal fuzzy number rather than stochastic random variables. Defuzzification or the ranking function executes the process of obtaining the crisp or deterministic version of a fuzzy number. A robust technique has been used, which covers an extensive range of feasibility degrees. Most of the conventional methods are limited to fuzzy-based solution schemes by defining the marginal evaluation of each objective using the membership function. Apart from metaheuristic techniques, a tremendous number of research papers have investigated and implemented the different fuzzy optimization techniques to obtain the global compromise solution of the CLSC planning problems. A detailed list of such fuzzy approaches can be found in Govindan et al. [1] and Govindan and Bouzon [2]. Here in this study, a neutrosophic fuzzy programming approach (NFPA) based on the neutrosophic decision has been suggested to solve the proposed CLSC design problem. Intuitionistic fuzzy imprecise preference relations among different objectives have also been investigated and successfully incorporated with an NFPA which is together termed as modified NFPA with intuitionistic fuzzy importance relations. The rest part of this chapter is as follows: In Section 15.2, a literature review related to the CLSC network is presented whereas Section 15.3 highlights the significant research contribution. Section 15.4 discusses the modeling CLSC design network under uncertainty while Section 15.5 represents the solution methodology to solve the final model. A real-life case study based on a laptop manufacturing firm is examined in Section 15.6, which shows the applicability and validity of the proposed approach efficiently. Finally, conclusions are highlighted based on the present work in Section 15.7.

15.2

Literature review

The CLSC planning problem has rapidly gained popularity among many researchers. The complex and challenging situation during the flow of goods and products from different sources to destination points has immensely attracted attention toward emerging research scope for the optimal policy implementation or decision-making processes to CLSC planning problems. Consequently, different approaches to solve the CLSC planning model have been introduced, along with their promising features in the context of optimality and applicability under different environments. Thus, here we review some existing CLSC models under different uncertainty and discuss the approaches adopted to solve them.

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A well-defined set of the interconnected network for the flow of multiple products has also created a very complex configuration of multiechelon CLSC design. Most of the existing studies have been presented on multiproduct and multiechelon CLSC planning problems. Gupta and Evans [3] have addressed multiple-echelon CLSC frameworks for electrical and electronic gadget scrap products. They designed a weighted nonprimitive goal programming model for the CLSC model and solved the proposed model with the aid of a discrete weighting scheme to the corresponding goal preference. Pishvaee et al. [4] designed a robust optimization model for CLSC configuration under randomly distributed parameters. The developed modeling approach then turned into the deterministic mixed-integer linear programming model € and they solved this using a robust optimization technique. Ozceylan and Paksoy [5] also presented a mixed-integer fuzzy mathematical model for CLSC under uncertainty with multiparts and multiperiods. The fuzzy solution approach has been applied for both fuzzy objectives and parameters with the help of a linear membership function. € Ozkır and Başlıgil [6] developed a multiobjective CLSC model with particular emphasis on the satisfaction level of trade, customer, and net profit incurred over the current product’s lifetime in the supply chain network. They adopted a fuzzy set (FS) theorybased solution method to deal with the proposed CLSC model. Yin and Nishi [7] also discussed an SCM problem with a quantity discount and uncertain demand at each echelon. The constructed SCM model resulted in the form of a mixed-integer nonlinear programming problem (MINLPP) with integral functions. An outerapproximation method has been suggested to solve the MINLPP. An improvement in efficiency performance has been achieved by reconstructing the MINLPP model into a stochastic programming model with the replacement of integral functions by € incorporating the normalization method. Ozceylan and Paksoy [8] addressed the CLSC planning model under tactical and strategic decision scenarios. The developed CLSC planning model has emerged as an MINLPP. They applied a fuzzy interactive solution approach to solve the propounded CLSC network design. Garg et al. [9] also designed a sustainable CLSC network with the core emphasis on environmental issues raised after the end of use and end of life of the used products. They formulated a biobjective integer nonlinear programming problem for the proposed CLSC network. The solution scheme has been adopted and applied by balancing the trade-off between socioeconomic and environmental aspects. The interactive multiobjective programming approach has been used to obtain the optimal allocation of different products. Alshamsi and Diabat [10] presented the reverse logistic (RL) system in the CLSC design network. The proposed RL texture initiates at the customer level and terminates at remanufacturing facilities in the reverse supply chain. The presented study was found to be limited to the RLs system. They modeled the deterministic mixed-integer linear programming problem with a single objective. A sustainable supply chain network has been designed by Arampantzi and Minis [11] and incorporates various factors, such as social, capital investment, environmental, political, etc., that affect the supply chain network directly and indirectly. They formulated a multiobjective mixed-integer linear programming problem and solved it by using two different conventional techniques: the goal programming method and the E-constrained method.

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Ma and Li [12] discussed a CLSC model for hazardous products under different uncertain parameters. The motive was to determine the optimal quantity of a shipment under a probabilistic environment. To address the scenario efficiently, the proposed model has been reformulated as a two-stage stochastic programming model along with risk and reward constraints. The two solution approaches, the Parallel Enumeration Method and the Genetic Algorithm (GA), have been applied to solve the designed CLSC model. Fard and Hajaghaei-Keshteli [13] also addressed a tri-level location-allocation planning problem for a CLSC network. The modeling study undertaken comprises three echelons: distribution center, customer zone, and recovery facility. The propounded tri-level CLSC planning model has been solved by using a Variable Neighborhood Search, Tabu Search (TS), and Particle Swarm Optimization in addition to these approaches; Fard and Hajaghaei-Keshteli further applied two recent metaheuristic algorithms, the Keshtel Algorithm and Water Wave Optimization, to obtain a feasible solution to the location-allocation problem. Zhen et al. [14] also designed a CLSC model with the capacitated allocation of products under uncertain demand for new and returned merchandise. The proposed decision-making model turned into a two-stage stochastic mixed-integer nonlinear programming problem (SMINLPP). Thus, the transformed model resulted in the deterministic demand and capacities parameters involved in the designed CLSC model. They also implemented the TS algorithm to solve the SMINLPP. Tsao et al. [15] formulated a sustainable supply chain design under economic and environmental objectives. The proposed supply chain model has taken the form of a multiobjective mathematical programming problem under stochastic demand and fuzzy costs. An interactive two-phase fuzzy probabilistic multiobjective programming problem has been introduced to deal with both sorts of uncertainty. Hasanov et al. [16] addressed the optimal quantity of products under four-level CLSC with a hybrid remanufacturing facility. The reverse chain includes the recovered process, which ensures the reuse of used products at a different level. The mathematical modeling framework has been carried out with a particular emphasis on remanufactured or returned products, or both. The developed modeling approach is aiming to minimize the overall cost incurred over the policies implemented during a single time horizon. Fakhrzad et al. [17] presented multiple products, periods, levels, and indices in the green CLSC planning model under uncertainty. The propounded model was then transformed into the multiobjective mixedinteger linear programming problem. Since the proposed model was NP-hard, to deal with it Nondominated Sorting Genetic Algorithm-II (NSGA-II) has been adopted to solve the proposed green CLSC network. Singh and Goh [18] also discussed the multiobjective mixed-integer linear programming problem under intuitionistic fuzzy parameters. Further, they transformed the multiobjective optimization problem into a single objective to solve the model. To achieve an acceptable satisfaction degree, different scalarization techniques such as the γ-connective approach and minimum sum bounded operator have been used. The proposed solution scheme has also been implemented to solve the pharmaceutical SCM model. Fathollahi-Fard et al. [19] designed a multiobjective stochastic CLSC model with an exclusive focus on the

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social issues associated with individual requirement and responsibility (such as job opportunity). The addressed stochastic CLSC model has been solved by using a couple of different nature-inspired algorithms and hybridized into the benefits of both, that is, social and environmental domains. Liao [20] presented a reverse logistics network design (RLND) for product recovery and remanufacturing processes. The proposed model emerged into a conventional mixed-integer nonlinear programming model for RLND under multiple echelons. The GA has been adopted as the solution method of the proposed RLND model. The formulated modeling structure has been validated and implemented with the help of the recycling bulk waste example in Taiwan. Zarbakhshnia et al. [21] have also discussed the green closed loop logistics network model as the mixed-integer linear programming problem. The undertaken study has mainly been concerned with the multiple stages, products, and objectives in the proposed model. A solution scheme, the E-constraint method, has been chosen to solve numerous targets. Dominguez et al. [22] also investigated the role of manufactured and remanufactured products in the CLSC with capacitated constraints. The research background explicitly reveals the four relevant uncertain factors to determine the efficiency of executed policies in the system. A managerial insight has been propounded that could contribute to understanding decision-making processes. Eskandarpour et al. [23] presented a study on the literature review of approximately 80 research papers in the field of CLSC planning problems. The chosen study area has been classified based on four questions: (i) What kind of socioeconomic and environmental issues have been included? (ii) How the problems related to the matters discussed have been unified or integrated in the supply chain model? (iii) What sort of solution schemes have been applied to solve the modeling problem? and (iv) Which numerical illustrations or computational studies have been taken from real-life applications? Furthermore, the shortcomings and drawbacks of different models have been pointed out, and consequently, the scope for future research has also been intimated. The interested reader may refer to the recent publications by Govindan et al. [1] and Govindan and Bouzon [2], based on reviewed work in the reverse logistic barriers and drivers.

15.3

Research contribution

A tremendous amount of work has been developed and applied successfully on the CLSC network in the last few decades. Only a few research works are available that have included the testing center as a facility location for the dissembled parts/components [3, 9]. Therefore, this chapter has put more emphasis on the reverse chain and is mainly concerned with end-of-use products and end-of-life. The modified neutrosophic fuzzy optimization techniques have been used for the first time in the field of SCM. The following are the significant and remarkable contributions to this presented research work. l

The proposed CLSC planning model has been designed for multidimensional echelons, in which five multiple echelons have been included in the forwarding chain, whereas six

Closed-loop supply chain management

l

l

l

l

l

l

l

349

various echelons have been integrated into the reverse chain which shows the great concern or influence regarding the end-of-use and end-of-life products. The different facility centers in the reverse chain ensure that the CLSC planning model is socioeconomic and environmentally friendly. The different objective functions have been presented to analyze the shares in total capital investment over the raw materials and products in the forward and reverse chain individually. A new preference scheme has been investigated to achieve better outcomes for the preferred objective functions. The uncertainty among parameters has been represented with fuzzy numbers and dealt with the expected interval and expected values of the involved parameters. Three constraints have been depicted with fuzzy equality in restrictions, which reveals the reality more closely. The fuzzy equality constraints are then efficiently transformed into two subconstraints. The NFPA has been developed to solve the proposed CLSC designed model. The proposed solution approach has been inspired by the indeterminacy degree that emerged in decisionmaking processes. Indeterminacy/neutral thoughts are the region of negligence for propositions’ values, between the degree of acceptance and rejection. It is the first time that the NFPA has been applied to solve the CLSC planning model. A novel intuitionistic fuzzy linguistic preference scheme has been investigated to assign weight/preference to the most preferred objective functions. The intuitionistic fuzzy linguistic preference relations have been efficiently integrated with an NFPA and termed as a modified NFPA. The proposed CLSC designed model has been implemented on real case study data to show the validity and applicability of the proposed solution methods. A variety of different solutions sets has been generated and summarized under the optimal choices of quantity allocation. The sensitivity analysis has also been performed on the obtained solution results based on the feasibility degree β and crisp weight parameter α by tuning them at different values between 0 and 1. The significance of the obtained results has been analyzed along with the remarkable findings. Conclusions and future research scope have been set out based on the present study.

15.4

Description of CLSC network

A well-organized systematic and interconnected network for the flow of materials, products, and parts is much needed to survive in the competitive market. Production processes explicitly adhere to the different perspectives of the finished products. The conventional supply chain design initiates with the availability of raw resources to finished goods and terminates at the consumption points. The globalization of markets, governmental legislation, and environmental practices creates many concerns for the used products and leads to the existence of a CLSC that inherently ensures the best management of end-of-use and end-of-life products. The efficiently expanded texture of the supply chain network designed has been widely adopted by the decision maker(s) with the inclusion of the reverse chain. Therefore, the CLSC network consists of two phases: forward chain and reverse chain for the flow of material, products, and used parts. In this study, a CLSC design is presented, which consists of five echelons in the forward chain and six echelons in the reverse chain, which is shown in Fig. 15.1.

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Fig. 15.1 Proposed closed loop supply chain network design.

The initialization of production processes starts with the procurement of raw materials from the storage center to the supplier point, which in turn supplies the relevant raw materials to the manufacturing plant for the production of new products. Afterward, the finished products are delivered to the distribution center to fulfill the demand of customers or markets. Unlike the forward logistics flow, the reverse logistics flow consists of a few more steps. The first step involves collecting defective products from customers at the collection center. The end-of-use products are outsourced from different customers, either directly or via markets. Collection of used products initiates the sustainable reverse chain, and collecting the used products maintains the flow cycle of products into a different facility phase. The collection center is responsible for the optimal distribution of used products for further required services. In most cases, returns processors collect fewer defective products, undertake repairs at refurbishing centers, and return them to the buyers. At this point, it is worth noting that returns processors may remanufacture defective products and ship them back to retailers and distributors, who in turn sell them to end users. Alternatively, returns processors may recycle defective products to extract materials and parts that can be reused in the production process by sending them to a disassembling center. Further, the parts and materials are taken to a testing point for the inspection of their further utility, and from there the elements that can be used to make new goods are sent back to the hybrid manufacturing plant. On the other hand, only the parts that are recyclable are shipped to the recycling point, and move forward through the supply chain until they reach the end users. The final step involves any materials or parts that are not utilized throughout the steps discussed earlier, which reach the disposal center for incineration or dumping purposes.

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To and fro movement of materials, products, and parts throughout the CLSC network contemplates over multifarious objectives associated with the entire phenomenon. Procurement, processing, distribution, and transportation processes turn into a significant investment in costs which should be optimized under optimal allocation of the commodity. The cost of purchasing raw materials and used products is also a measure of great concern. Delivery time of the finished products to the customers must be reduced to overcome cancelations of ordered products. Revenues from sales of new products and recyclable parts encourage the enhancement of shares in the net profit. Hence the proposed CLSC model comprises multiple conflicting objectives such as minimization of processing, purchasing, and transportation costs, minimization of expected product delivery time, and maximization revenue from the selling of the products. The propounded CLSC planning network configuration is based on the following postulated assumptions: The propounded CLSC network has been designed for multiple raw materials/parts multiproducts, and multiechelons along with single time horizons. Each facility location is well established and functional for the associated services over the stipulated period. Movement of new products initiates from manufacturing plants to customers, and the flow of used products starts from customers to the disassembling center. Meanwhile, the recovered products are also shipped after renovation to the distribution center. Therefore, the demand for new and refurbished products is met through the distribution center only. Set-up costs associated with different echelons are assumed to be included in the processing costs. Revenues are only derived from the selling prices of new products and recyclable products, which turn into a contribution to the net profit. The disposal facility is the only route to remove the scrap parts/components from the proposed CLSC planning model. The rest of the quantity is assumed to remain in its useful life. Uncertainty among different parameters has been considered as fuzzy numbers. The fuzzy linguistic term has assigned the preference among different objective functions.

l

l

l

l

l

Indices

Descriptions

a

The number of raw materials/parts/components storage points (a ¼ 1, 2, …, A) The number of supplier points (b ¼ 1, 2, …, B) The number of manufacturing/remanufacturing plants (c ¼ 1, 2, …, C) The number of distribution center (d ¼ 1, 2, …, D) The number of customers/markets (e ¼ 1, 2, …, E) The number of collection center ( f ¼ 1, 2, …, F) The number of refurbishing/repairing center (g ¼ 1, 2, …, G) The number of disassembling center (h ¼ 1, 2, …, H) The number of raw materials/parts/components testing points (i ¼ 1, 2, …, I) The number of recycling points ( j ¼ 1, 2, …, J) The number of disposal centers (k ¼ 1, 2, …, K) The number of different products (l ¼ 1, 2, …, L)

b c d e f g h i j k l

352

m Decision variables X1m,a,b X2m,b,c X3l,c,d X4l,d,e X5l,e, f X6l, f,g X7l,g,d X8l, f,h X9m,h,i X10m,i,c X11m,i, j X12m,i,k X13m, j,a Parameters rfl rcl,e rtm rmm rrm rdm

Optimization Theory Based on Neutrosophic and Plithogenic Sets

The number of raw materials/parts/components storage points (m ¼ 1, 2, …, M) Descriptions The quantity of raw material m shipped from raw material storage point a to supplier point b The quantity of raw material m shipped from supplier point b to manufacturing plant c The quantity of different products l shipped from manufacturing plant c to different distribution center d The quantity of different products l shipped from different distribution center d to different customers/markets e The quantity of different used products l shipped from different customers/ markets e to collection center f The quantity of different repairable products l shipped from collection center f to refurbishing center g The quantity of different recovered products l shipped from refurbishing center g to different distribution center d The quantity of different unrepairable products l shipped from collection center f to disassembling center h The quantity of parts/components m shipped from disassembling center h to testing point i The quantity of raw material m shipped from testing point i to manufacturing plant c The quantity of recyclable parts/components m shipped from testing point i to recycling point j The quantity of scrap parts/components m shipped from testing point i to disposal center k The quantity of recovered parts/components m shipped from recycling point j to raw materials storage point a Descriptions Recovery rate of used products l at refurbishing center Collection rate of used products l from customer or market e Testing rate of different parts/components m at testing center Reuse rate of different tested parts/components m at manufacturing plant Recycling rate of different recyclable parts/components m at recycling center Disposal rate of raw materials/parts/components m at disposal center

Parameters

Descriptions

PC1m,a PC2m,b PC3l,c PC4l,d PC5l, f PC6l,g PC7l,h PC8m,i

Unit storage cost incurred over raw material m at raw material storage center a Unit safety cost incurred over raw material m at supplier point b Unit production cost levied over product l at manufacturing plant c Unit inventory holding cost levied over product l at distribution center d Unit collection facility cost levied over product l at collection center f Unit refurbishing cost levied over product l at refurbishing center g Unit disassembling cost levied over product l at disassembling center h Unit testing cost levied over each component m at testing center i

Closed-loop supply chain management

PC9m, j PC10m,k TC1m,a,b TC2m,b,c TC3l,c,d TC4l,d,e TC5l,e, f TC6l, f,g TC7l,g,d TC8l, f,h TC9m,h,i TC10m,i,c TC11m,i, j TC12m,i,k TC13m, j,a Tl,d,e PU1m PU2l SP1m SP2l MC1m,a MC2m,b MC3m,c MC4l,d MC5l,e MC6l, f MC7l,g MC8m,h

353

Unit recycling cost levied over raw material m at recycling point j Unit disposal cost levied over each component m at disposal center k Unit transportation cost of raw material m shipped from raw material storage point a to supplier point b Unit transportation cost of raw material m shipped from supplier point b to manufacturing plant c Unit transportation cost of different products l shipped from manufacturing plant c to different distribution center d Unit transportation cost of different products l shipped from different distribution center d to different customer/market e Unit transportation cost of different used products l shipped from different customers/markets e to collection center f Unit transportation cost of different repairable products l shipped from collection center f to refurbishing center g Unit transportation cost of different recovered products l shipped from refurbishing center g to different distribution center d Unit transportation cost of different unrepairable products l shipped from collection center f to disassembling center h Unit transportation cost of parts/components m shipped from disassembling center h to testing point i Unit transportation cost of parts/components m shipped from testing point i to manufacturing plant c Unit transportation cost of different recyclable parts/components m shipped from testing point i to recycling point j Unit transportation cost of disposable parts/components m shipped from testing point i to disposal center k Unit transportation cost of recovered parts/components m shipped from recycling point j to raw materials storage point a Unit transportation time required to ship different products l from distribution center d to different customers/markets e Unit purchasing cost of raw materials/parts/components m Unit purchasing cost of different used products l Unit selling price of different recyclable parts/components m Unit selling price of different new products l Maximum available quantity of raw material m at raw material storage center a Maximum available quantity of raw material m at supplier b Minimum required quantity of raw material m at manufacturing plant c Maximum available quantity of new products l at distribution center d Minimum demand quantity of different new products l by customers or at markets e Maximum collection capacity of different used products l at collection center f Maximum refurbishing capacity of different repairable products l at refurbishing center g Maximum disassembling capacity of different parts/components m at disassembling center h

354

Optimization Theory Based on Neutrosophic and Plithogenic Sets

MC9m,i MC10m, j MC11m,k

Maximum testing capacity different scrap parts/components m at testing point i Maximum capacity of recyclable parts/components m at recycling point j Maximum disposal capacity of disposable parts/components m at disposal center k

15.4.1 Multiple objective function The typical and efficient CLSC model always comprises multiple conflicting objectives for both forward and reverse chains, which are to be attained simultaneously. Here, we highlight the different costs associated with ahead and change strings separately to analyze the echelon-wise effects in terms of expenditure on the overall CLSC planning problem. Objective 1: Total processing costs. Initially, the raw materials have been stored at a raw material storage center to ensure the smooth running of the CLSC design. The processing cost indicates the different sort of value at each echelon such as storage cost at the raw material storage center, safety cost at the supplier point, production cost at the manufacturing center and inventory or distribution cost at the distribution center, levied on the unit raw material or new products. The significant reduction in these processing costs automatically results in the maximum margin of profit. The reverse chain also contains multiple echelons with different processing costs associated with them. Here, the processing cost refers to the value of the collection at the collection center, the cost of disassembly at the disassembling center, the refurbishing cost at the refurbishing center, the cost of testing at the testing center, the cost of recycling at the recycling center, and the disposal cost at the disposal point, respectively. The designed network facility executed by each echelon ensures that the commonly used products in the reverse supply chain survive at their end-of-life use or disposable condition. Thus the first objective function ensures the minimization of the processing costs at different echelon in the forward chain under the optimal quantity allocation. Minimize Z1 ¼

M X A X

PC1m, a X1m, a, b +

m¼1a¼1

+

M X B X

PC2m, b X2m, b, c

m¼1b¼1

L X C L X D X X PC3l, c X3l, c, d + PC4l, d X4l, d, e l¼1 c¼1

l¼1 d¼1

L X F L X G X X + PC5l, f X5l, e, f + PC6l, g X6l, f , g l¼1 f ¼1

+

L X H X

l¼1 g¼1

PC7l, h X8l, f , h +

l¼1 h¼1

+

M X J X m¼1 j¼1

M X I X

PC8m, i X10m, i, c

i¼1 i¼1

PC9m, j X11m, i, j +

M X K X PC10m, k X12m, i, k 8 c, f ,g, h, a,b,c,d,e,i: m¼1k¼1

Closed-loop supply chain management

355

Objective 2: Total transportation costs. The transportation cost is one of the well-known objective functions under CLSC design. Typical and interconnected transportation networks within each echelon in CLSC design yield high transportation costs. In the forward chain, the shipment of raw material from the raw material storage point to the supplier point and from the supplier point to the manufacturing plant integrates the marginal shares in the total transportation cost. The delivery of new products from the manufacturing plant to the distribution center and from the distribution center to customers also has a significant role in attaining the gross profit in the proposed CLSC network. The propounded CLSC network has put more emphasis on the reverse chain by including more facility locations compared to the forward chain. The to and fro shipment of used products and raw parts/components results in high transportation costs. The reverse chain network allows the recovered products and tested parts/components to enter into the forward chain directly from the refurbishing center to the distribution center and from the testing point to the manufacturing plant without touching the recycling facility. Hence to and fro shipment of products and parts/components from multiple different echelons is turned into high transportation costs. Therefore, the second objective function results in the minimization of to and fro transportation costs to varying echelons in the forward chain for the maximum shipment quantity of products under the optimal allocation policy. Minimize Z2 ¼

M X A X B X

TC1m,a,b X1m, a,b +

m¼1 a¼1 b¼1

+

L X C X D X

L X E X F X

TC3l,c, d X3l,c,d +

L X G X D X

L X D X E X

TC4l,d, e X4l, d, e

l¼1 d¼1 e¼1

TC5l,e, f X5l, e, f +

l¼1 e¼1 f ¼1

+

TC2m,b,c X2m, b, c

m¼1 b¼1 c¼1

l¼1 c¼1 d¼1

+

M X B X C X

L X F X G X

TC6l, f ,g X6l, f , g

l¼1 f ¼1 g¼1

TC7l,g,d X7l,g,d

l¼1 g¼1 d¼1

+

L X F X H X l¼1 f ¼1 h¼1

+

M X I X C X

TC8l, f ,h X8l, f ,h +

M X I X H X

TC9m,h, i X9m, h, i

m¼1 i¼1 h¼1

TC10m,i,c X10m,i,c

m¼1 i¼1 c¼1

+

M X I X J X

TC11m,i, j X11m,i, j +

m¼1 i¼1 j¼1

+

M X J X A X m¼1 j¼1 a¼1

M X I X K X m¼1 i¼1 k¼1

TC13m, j,a X13m, j,a :

TC12m, i, k X12m, i, k

356

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Objective 3: Total purchasing cost of used products and raw materials. In this proposed CLSC design, the purchasing of raw material and used products at two echelons has been allowed. The purchasing cost of raw materials from the supplier point and the purchasing cost of used products from customers yields the total purchasing cost. However, these costs leave a significant margin among the new outsourced products by contributing less operational costs on the recovered products. Therefore, the third objective function ensures the minimization of the total purchasing cost of raw materials and different used products from suppliers and customers to maintain the efficiency of the manufacturing plant. Minimize Z3 ¼

M X

PU1m X2m,b,c +

L X PU2l X5l,e, f 8 b, c,e, f :

m¼1

l¼1

Objective 4: Products delivery time. The most critical issue in CLSC design is to determine the optimal time policy during the whole process. Notably, the shipment time of new products from the distribution center to different customers must be attained under the stipulated delivery period at the time of the ordered quantity. The goodwill and reputation of the company are strongly connected with delivery time. The latter also reduces the loss of any perishable products that happens due to delay. Moreover, cancelation from the customers’ side would be almost negligible with the timely transshipment of the products. Henceforth, the fourth objective dynamically ensures the minimization of the total shipment time of different new products from the distributor to customers to maintain the reputation and reliability of the company. Minimize Z4 ¼

L X D X E X

Tl,d,e X4l,d,e 8 d, e:

l¼1 d¼1 e¼1

Objective 5: Revenues from the sale of new products and recyclable parts/ components. By the significant increase in the sales ratio of new products and recyclable parts/ components, a marginal profit could be extracted. Selling of new products at higher quantities covers the maximum part of the capital investment during the production and distribution processes—recyclable parts/components are also a reliable source of profit from its sales. The selling price of the new products has a significant contribution toward the net profit and simultaneously yields in the contribution to gross profit. Thus the fifth or last objective function ensures the maximization of new products selling to survive in the competitive market with the maximum turnover of the new products under the optimal production policy. Maximize Z5 ¼

M X m¼1

SP1m X11m,i, j +

L X l¼1

SP2l X4l,d,e 8 i, j, d, e:

Closed-loop supply chain management

357

15.4.2 Constraints The following are the relevant constraints or restrictions under which the objective functions are to be optimized by yielding the most promising and systematic strategies for allocating different raw materials or parts/components and various products among multiple echelons in the proposed CLSC designed model. For the sake of convenience, we have categorized all the constraints under six different groups, and these can be summarized as follows.

15.4.2.1 Constraints related to the capacity of different echelons in the CLSC network The procurement of raw materials initiates from the raw material storage center where the abundance or stock of raw materials has been kept to fulfill the demand from suppliers. Therefore, the total shipment quantity of different raw materials from the raw materials storage center to the supplier must not exceed its capacities and can be represented by Eq. (15.1). Supplier points also have a limited ability for the flow of different raw materials to maintain the intake and outsourced ratio. It is essential for the supplier to hold some raw materials for distribution at times of scarcity, when raw material storage functioning is interrupted over a stipulated time. Hence the constraints imposed over the number of raw materials shipped from the supplier point to a different manufacturing plant must less than or equal to the capacity of suppliers and can be presented by Eq. (15.2). The collection of used products from different customers starts the key functioning role of the reverse chain. It is the very first stage at which the end-of-use products are collected by the collection center. It must be assured that the accumulation quantity of used products from different customers must be less than or equal to the capacity of various collection centers and can be represented by Eq. (15.3). A well-organized system of collection centers provides frequent services to the used products so that all the end-of-use products are refurbished and can be used further without significantly affecting the demand. After ensuring the required services for used products, it has been allowed to ship the used merchandise from the collection center to the refurbishing center for renovating processes. Hence the total quantity of used products must not exceed the capacity of the refurbishing center and can be given in Eq. (15.4). The number of used products that need testing services for their further utilization has been shipped to the disassembling facility to disassemble the used products into different components or parts. To ensure that the number of used products which have been sent for dismantling purpose must be less than or equal to its capacity and can be represented by Eq. (15.5). After completing the required test for the parts/components, the recyclable quantity of parts/components has been sent to the recycling facility, which denotes the last echelon of the CLSC network. To ensure the number of recyclable parts/components received from different testing points must not exceed the maximum capacity of the recycling center and can be stated in Eq. (15.6). After testing procedures, end-of-life parts or components are declared as disposable parts/components and shipped to the disposal center in good time to reduce environmental issues. Hence to avoid the burden on landfills and underground

358

Optimization Theory Based on Neutrosophic and Plithogenic Sets

disposal, the number of disposable parts/components must not exceed the maximum disposal capacity at the disposal center and this can be represented by Eq. (15.7). After recycling processes, the parts/components are transformed into new raw materials and ready the shipment to the raw material storage center. To fulfill the stock capacity of a natural material storage center, the number of raw materials must be greater than or equal to its minimum storage capacity for the smooth running of the production system, and this can be represented by Eq. (15.8). B X X1m,a,b  MC1m,a 8 m,a,

(15.1)

b¼1 C X X2m,b,c  MC2m, b 8 m, b,

(15.2)

c¼1 E X rcl,e X5l,e, f  MC6l, f 8 l, f ,

(15.3)

e¼1 G X X6l, f ,g  MC7l,g 8 l, g,

(15.4)

g¼1 H X X8l, f ,h  MC8l,h 8 l, h,

(15.5)

h¼1 I X rrm X11m,i, j  MC10m, j 8 m, j,

(15.6)

i¼1 I X rdm X12m,i,k  MC11m,k 8 m, k,

(15.7)

i¼1 J X X13m, j,a  MC1m,a 8 m, a:

(15.8)

j¼1

15.4.2.2 Constraints related to production requirement An efficient production system is an integral part of the CLSC network. Hybrid manufacturing/remanufacturing plants play a vital role in the optimal production of new products. Therefore, particular minimum requirements must be met to start the production processes. To ascertain the minimum condition of raw materials from two sources, the supplier point and testing center, the number of raw materials from two references must be greater than or equal to their production capacity at different manufacturing plants, and this can represented by Eq. (15.9). B I X X X2m,b,c + rmm X10m,i, c  MC3m,c 8 m, c: b¼1

i¼1

(15.9)

Closed-loop supply chain management

359

15.4.2.3 Constraints related to maximum inventory at the distribution center The distribution center is responsible for the shipment of products to different customers/markets. The demand for a new product is uncertain and only can be predicted based on previous information. Thus, to avoid the inventory cost and ascertain the maximum capacity restriction at the distribution center, the incoming products from manufacturing plants as well as refurbishing centers must be less than or equal to the maximum capacity of inventory at the distribution center, and this can be achieved by Eq. (15.10). C G X X X3l,c,d + rfl X7l, g,d  MC4l, d 8 l, d: c¼1

(15.10)

g¼1

15.4.2.4 Constraints related to demand of new and refurbished products The most important and critical aspect of integrated CLSC is to fulfill the demand of customers or markets. The need for products is seldom stable. However, it can be predicted through prior information from the demand pattern. The only distribution center is responsible for the delivery of new products to the customers in this proposed CLSC network. To ensure this, the number of shipped products from the distribution center to different markets must be higher than its tentative demand over the stipulated ordered period, and this can be represented by Eq. (15.11). E X X4l,d,e  MC5l,e 8 l,e:

(15.11)

e¼1

15.4.2.5 Constraints related to the testing capacity at testing facility centers The testing facility has been designed for taking the final decision over the parts or components regarding at which echelon they are to be transported. From a testing point, there are three facility options for the processing of tested parts/components. The manufacturing plant, recycling center, and disposal center have been structured for the final termination of the reverse supply chain. Hence the total sum of the number of parts/components that are transported from the testing plant to different facility locations must be less than or equal to the maximum capacity of the testing point, and this can be represented by Eq. (15.12). C J K X X X rtm X10m,i,c + rtm X11m,i, j + rtm X12m,i,k  MC9m, i 8 m, i: c¼1

j¼1

k¼1

(15.12)

360

Optimization Theory Based on Neutrosophic and Plithogenic Sets

15.4.3 Proposed CLSC model formulation under uncertainty The formulation of different conflicting objective functions and with some dynamic constraints under the proposed CLSC network has been presented in previous sections. Usually, the modeling texture of the CLSC network has been regarded as deterministic, which means that all the introduced parameters and constraints are known and predetermined well in advance. However, it is often observed that a deterministic modeling approach under CLSC design may not be an appropriate framework in decision-making processes. The typical multiechelon interconnected CLSC design model inherently yields some uncertainty. Impreciseness, vagueness, ambiguousness, randomness, incompleteness, etc., are the most common and frequent issues in the CLSC model. Different factors are responsible for the creation of uncertainty in the modeling of the CLSC network. Random fluctuation in the demand quantity, competitive market scenario, natural tragedy, variation in different kinds of costs, etc., laid down the base of uncertainty. In various adverse circumstances, the complete information about different parameters is not predetermined, but some inconsistent, improper, and incomplete information may be available to determine the deterministic value of the parameters. Uncertainty may exist in different forms, such as fuzzy, stochastic, and other types of risk. Vagueness or ambiguousness is responsible for fuzzy parameters which can be dealt with using the fuzzy techniques, whereas randomness gives birth to the stochastic parameters and can be quickly sorted out by using stochastic programming techniques with known means and variances of the parameters. Therefore, to highlight the most critical insight of the uncertainty, we have incorporated fuzzy parameters and few fuzzy equality constraints in the proposed CLSC designed network. Various cost parameters, such as processing costs, transportation costs, purchasing cost, selling prices, and time, have been taken as fuzzy parameters. The capacities or volumes of different echelons are also considered as fuzzy numbers. Inequality restrictions imposed over different constraints may avoid some aspects of getting better results from the CLSC planning model. Flexibility, among some preferred limitations, has been postulated to reveal reality more clearly. Hence we have developed a couple of fuzzy equality constraints (¼) e which means “essentially equal to” which signifies that the restrictions should more or less be satisfied and are more flexible than inequality constraints (Eqs. 15.22–15.24). The customer demand constraint has been assured with fuzzy equality constraints due to the change in utility or satisfaction behavior of the customers. The disposal facility is a single way for the removal of scrap parts/components out of the CLSC network. The testing facility plays a vital role in inspecting different parts/components. The optimum allocation of used parts/products has been decided at the testing facility point. Three various service destinations have been designed for the parts/components according to their potential utility after inspection. Therefore, more or less shipment quantity of parts/components is justifiable to ensure the optimum allocation to different facility centers. Hence, the proposed model with multiple objective functions and various constraints under uncertainty has been presented in model M1.

Closed-loop supply chain management

M1 : Minimize Z1 ¼

361

M X A M X B X X g m,a X1m,a,b + g m,b X2m,b, c PC1 PC2 m¼1 a¼1

+

L X C X

m¼1 b¼1

g l,c X3l,c,d + PC3

l¼1 c¼1

+

L X F X

l¼1 d¼1

g l, f X5l,e, f + PC5

l¼1 f ¼1

+

L X H X

+

g l,h X8l, f ,h + PC7

M X I X

g m, j X11m,i, j + PC9

g m,i X10m,i, c PC8

M X K X g m,k X12m,i,k PC10 m¼1 k¼1

M X A X B X

g m,a,b X1m,a,b + TC1

m¼1 a¼1 b¼1

+

g l,g X6l, f , g PC6

i¼1 i¼1

m¼1 j¼1

Minimize Z2 ¼

L X G X l¼1 g¼1

l¼1 h¼1 M X J X

L X D X g l,d X4l,d,e PC4

M X B X C X

g m,b, c X2m, b, c TC2

m¼1 b¼1 c¼1

L X C X D L X D X E X X g l,c,d X3l,c,d + g l,d, e X4l, d, e TC3 TC4 l¼1 c¼1 d¼1

l¼1 d¼1 e¼1

L X E X F L X F X G X X g l,e, f X5l,e, f + g l, f ,g X6l, f , g TC5 TC6 + l¼1 e¼1 f ¼1

+

l¼1 f ¼1 g¼1

L X G X D X

g l,g,d X7l,g,d TC7

l¼1 g¼1 d¼1

+

L X F X H M X I X H X X g l, f ,h X8l, f ,h + g m,h,i X9m, h, i TC8 TC9 l¼1 f ¼1 h¼1

+

m¼1 i¼1 h¼1

M X I X C X

M X I X J X

m¼1 i¼1 c¼1

m¼1 i¼1 j¼1

g m, i,c X10m,i,c + TC10

g m, i, j X11m, i, j TC11

M X I X K M X J X A X X g m, i,k X12m,i,k + g m, j, a X13m, j, a + TC12 TC13 m¼1 i¼1 k¼1

Minimize Z3 ¼

M L X X g m X2m,b, c + g l X5l,e, f PU1 PU2 m¼1

Minimize Z4 ¼

m¼1 j¼1 a¼1

L X D X E X

l¼1

e l, d,e X4l,d,e T

l¼1 d¼1 e¼1 M L X X g m X2m,b,c + g l X5l,e, f Maximize Z5 ¼ SP1 SP2 m¼1

l¼1

362

Optimization Theory Based on Neutrosophic and Plithogenic Sets

subject to B X g m,a , X1m,a,b  MC1

(15.13)

b¼1 C X g m,b , X2m,b,c  MC2

(15.14)

c¼1 B I X X g m,c , X2m,b,c + rmm X10m,i, c  MC3 b¼1

C G X X g l, d , X3l,c,d + rfl X7l,g,d  MC4 c¼1

(15.15)

i¼1

(15.16)

g¼1

E X g l, f , rcl,e X5l,e, f  MC6

(15.17)

e¼1 G X g l,g , X6l, f ,g  MC7

(15.18)

g¼1 H X g l,h , X8l, f ,h  MC8

(15.19)

h¼1 I X g m, j , rrm X11m,i, j  MC10

(15.20)

i¼1 J X g m,a , X13m, j,a  MC1

(15.21)

j¼1 E X g l,e , X4l,d,e ¼ e MC5

(15.22)

e¼1 I X g m,k , rdm X12m,i,k ¼ e MC11

(15.23)

i¼1 C J K X X X g m,i : rtm X10m, i,c + rtm X11m, i, j + rtm X12m,i,k ¼ e MC9 c¼1

j¼1

(15.24)

k¼1

Where notations (e:) over different parameters represent the triangular/trapezoidal fuzzy number for all indices’ sets, the fuzzy crisp inequality constraint has been described by (, ). The fuzzy equality constraints indicate that more or less attainment has been represented by (¼) e for the given indices’ sets.

Closed-loop supply chain management

15.5

363

Solution methodology

15.5.1 Treating fuzzy parameters and constraints The addressed CLSC mathematical model inherently involves some vagueness and ambiguousness in the value of different parameters such as costs, capacity, revenues, etc. Defuzzification and the ranking function are the processes to obtain crisp versions of the fuzzified parameters based on the upper and lower magnitude of the vague parameters. On the other hand, the vagueness or uncertainty present in the equality or inequality constraints also needs to be defuzzified, and then converted into the strict crisp equality or inequality form of the constraints. To deal with vague or fuzzy parameters and constraints, different defuzzification techniques have been used in the literature. Among all the defuzzification approaches for uncertain parameters and constraints, Jimenez [24] and Jimenez et al. [25] discussed the combo defuzzification or ranking approach, which deals efficiently with the vague parameters as well as vague constraints. They also elaborately discussed the strong justification for ranking approaches with the help of different properties such as robustness, distinguishability, fuzzy or linguistic notations, and rationality. Later on, it has been extensively used by many researchers (see [25–27]). Without more justification on the ranking function, this chapter has adopted the defuzzification or ranking function for both vague parameters and constraints based on the Jimenez [24] approaches. Definition 15.1. Jimenez et al. [25] An FS defined over any universe of discourse is said to be a fuzzy number if the membership function is increasing semicontinuously in the upper interval and decreasing semicontinuously in the lower range, respectively. Therefore, the membership function of a fuzzy number along with fϕ(x) and gϕ(x), which are the left- and right-hand sides of the membership function, can be given as follows: 8 0 > > > > f < ϕ ðxÞ μϕ ðxÞ ¼ gϕ ðxÞ > > > 1 > :

if if if if

x  ϕ1 or x  ϕ4 ϕ1  x  ϕ2 ϕ3  x  ϕ4 , ϕ2  x  ϕ3

(15.25)

e ¼ ðϕ1 ,ϕ2 ,ϕ3 ,ϕ4 ;1Þ represents a fuzzy number. A fuzzy number where ϕ e ¼ ðϕ1 ,ϕ2 ,ϕ3 ,ϕ4 Þ is said to be trapezoidal if fϕ(x) and gϕ(x) exist. Also, if ϕ2 ¼ ϕ ϕ3, then one can obtain a triangular fuzzy number. Definition 15.2. Jimenez et al. [25] e can be provided as The representation of an expected interval for the fuzzy number ϕ follows: e ¼ ½Eϕ , Eϕ  ¼ EIðϕÞ 1 2

Z

1 0

fϕ1 ðxÞdx,

Z 0

1

fϕ1 ðxÞdx

 :

(15.26)

364

Optimization Theory Based on Neutrosophic and Plithogenic Sets

e is termed as its expected The half point of the expected interval of the fuzzy number ϕ value and can be shown as follows: " # Eϕ1 + Eϕ2 e EVðϕÞ ¼ : 2

(15.27)

Hence the expected interval and expected value for a trapezoidal fuzzy number e ¼ ðϕ1 ,ϕ2 ,ϕ3 ,ϕ4 Þ can be obtained as follows: ϕ  EIðϕÞ ¼

 ϕ1 + ϕ2 ϕ3 + ϕ4 , , 2 2

 EVðϕÞ ¼

(15.28)

 ϕ1 + ϕ2 + ϕ3 + ϕ4 : 4

(15.29)

e ¼ ðϕ1 ,ϕ2 ,ϕ3 ,ϕ4 Þ, if ϕ2 ¼ ϕ3 (say ϕ) then it For any trapezoidal fuzzy number ϕ e ¼ ðϕ1 , ϕ, ϕ4 Þ and; its expected interval reduces into a triangular fuzzy number ϕ and expected value can be derived as follows: 

 ϕ1 + ϕ ϕ + ϕ4 , EIðϕÞ ¼ , 2 2

(15.30)



 ϕ1 + 2ϕ + ϕ4 : EVðϕÞ ¼ 4

(15.31)

Definition 15.3. Jimenez et al. [25] e and ψ e such that both have semicontinuous increasSuppose that there are two fuzzy ϕ ing and decreasing membership functions for upper and lower intervals, then the e is greater than ψ e can be easily pointed out by constructing the degree in which ϕ following membership function:

e ψ eÞ ¼ δV ðϕ,

8 > 0 > >
> > E2  E1  ðE1  E2 Þ :

1

if Eϕ2  Eψ1 < 0 if 0 2 ½Eϕ1  Eψ2 , Eϕ2  Eψ1  ,

(15.32)

if Eϕ2  Eψ1 > 0

e and ψ e . If where [Eϕ1 , Eϕ2 ] and [Eψ1 , Eψ2 ] represent the expected intervals of ϕ e e e Þ ¼ 0:5, then one can say that both ϕ and ψ e are indifferent. δV ðϕ, ψ e ψ e is greater than or equal to ψ e Þ  β, then one can say that ϕ e, Consequently, if δV ðϕ, e e i. at least in a degree β, and can be mathematically represented as ϕ i β ψ

Closed-loop supply chain management

365

Definition 15.4. Jimenez et al. [25] Introducing a decision vector X such that x 2Rn, then we can assign a feasibility degree β if for at least e i X, ψ e i Þ ¼ β, min ½δV ðϕ

(15.33)

i2V

e i ¼ ðϕ e i1 , ϕ e i2 ,…, ϕ e iv Þ. where ϕ Intuitionally, in another sense, it can be written as e i Xβ ψ e i 8 i ¼ 1, 2,…, v: ϕ

(15.34)

Incorporating the concept of (Jimenez et al. [25]) in the above inequality, equivalently we have

ϕX

ϕX

ψ

ψ

ϕX

E2 i  E1 i

ψ

E2 i  E1 i  ðE1 i  E2 i Þ

 β 8 i ¼ 1,2, …, v:

(15.35)

On simplifying the above inequality equation, the equivalent inequality relations with feasibility degree β have been derived as follows: ϕ

ϕ

ψ

ψ

ðð1  βÞE2 i + βE1 i Þ X  ðβE2 i + ð1  βÞE1 i Þ:

(15.36)

Furthermore, it can be concluded that the β-feasible fuzzy equalities, such as e iX ¼ e i 8 i ¼ v + 1,v + 2,…, V, eβ ψ ϕ

(15.37)

can also be defuzzified in a similar fashion to the ranking function approach for fuzzy inequalities and can be given as follows:        β ϕi β ϕi β ψi β ψi 1  E2 + E1 X  E + 1  E1 , (15.38) 2 2 2 2 2        β ϕi β ϕ β ψ β ψ E2 + 1  E1 i X  1  E2 i + E1 i : 2 2 2 2

(15.39)

Therefore, the fuzzy equality constraints result in the doubly crisp auxiliary inequality constraints for representing the restrictions with half of the β-feasibility degree by balancing an equilibrium state for the fuzzy equality constraints. In order to obtain the crisp version of the proposed CLSC model, we have used the expected values [25] of the triangular fuzzy parameters present in the objective functions such as transportation cost, processing cost, purchasing cost, time, and revenues, whereas the trapezoidal fuzzy parameters such as different capacities involved in the constraints have been defuzzified by using the concept of the expected interval [25] of the parameters. Based on the above-discussed defuzzification approaches, the fuzzy

366

Optimization Theory Based on Neutrosophic and Plithogenic Sets

parameters and constraints have been converted into their crisp versions, which has been also shown in Table 15.1. M2 : Minimize Z1 ¼

M X A X

g EVðPC1Þ m, a X1m, a, b +

m¼1a¼1

+

+

L X C X

g EVðPC3Þ l, c X3l, c, d +

L X F X

L X G X

g EVðPC5Þ l, f X5l, e, f +

L X H X

g EVðPC7Þ l, h X8l, f , h +

M X I X g EVðPC8Þ

M X J M X K X X g g EVðPC9Þ EVðPC10Þ m, j X11m, i, j + m, k X12m, i, k m¼1k¼1

g EVðTC1Þ m, a, b X1m, a, b +

L X D X E X

g EVðTC3Þ l, c, d X3l, c, d +

l¼1 c¼1d¼1

l¼1 d¼1e¼1

L X E X F X

L X F X G X

g EVðTC5Þ l, e, f X5l, e, f +

L X G X D X

l¼1 f ¼1g¼1

g EVðTC7Þ l, g, d X7l, g, d +

l¼1 g¼1d¼1

+

M X B X C X

g EVðTC2Þ m, b, c X2m, b, c

m¼1b¼1c¼1

L X C X D X

l¼1 e¼1f ¼1

+

m, i X10m, i, c

i¼1 i¼1

m¼1a¼1b¼1

+

g EVðPC6Þ l, g X6l, f , g

l¼1 g¼1

M X A X B X

+

g EVðPC4Þ l, d X4l, d , e

l¼1 d¼1

m¼1 j¼1

Minimize Z2 ¼

L X D X

l¼1 c¼1

l¼1 h¼1

+

g EVðPC2Þ m, b X2m, b, c

m¼1b¼1

l¼1 f ¼1

+

M X B X

M X I X H X

g EVðTC6Þ l, f , g X6l, f , g

L X F X H X l¼1 f ¼1h¼1

g EVðTC9Þ m, h, i X9m, h, i +

m¼1 i¼1 h¼1

g EVðTC4Þ l, d, e X4l, d, e

g EVðTC8Þ l, f , h X8l, f , h

M X I X C X

g EVðTC10Þ m, i, c X10m, i, c

m¼1 i¼1 c¼1

M X I X J M X I X K X X g g + EVðTC11Þ EVðTC12Þ m, i, j X11m, i, j + m, i, k X12m, i, k m¼1 i¼1 j¼1

+

M X J X A X

m¼1 i¼1 k¼1

g EVðTC13Þ m, j, a X13m, j, a

m¼1 j¼1 a¼1

Minimize Z3 ¼

M X

g X2 EVðPU1Þ m m, b, c +

m¼1

Minimize Z4 ¼

L X g X5 EVðPU2Þ l l , e, f l¼1

L X D X E X

EVðTe Þl, d, e X4l, d, e

l¼1 d¼1e¼1

Maximize Z5 ¼

M X m¼1

g X2 EVðSP1Þ m m , b, c +

L X g X5 EVðSP2Þ l l, e, f l¼1

Closed-loop supply chain management

Table 15.1 Information regarding triangular/trapezoidal fuzzy parameters. Fuzzy parameter

Triangular/trapezoidal fuzzy number

g ,  PC

ðPC ,  ,PC ,  ,PC,  Þ

g , ,  TC

ðTC , ,  , TC , ,  ,TC, ,  Þ

e , ,  T

ðT, ,  ,T, ,  ,T, ,  Þ

g ,  PU

ðPU ,  ,PU ,  ,PU,  Þ

g ,  SP

ðSP ,  ,SP ,  ,SP,  Þ

g ,  MC

ðMC ,  ,MC ,  ,MC ,  ,MC,  Þ

ð1Þ

ð2Þ

ð1Þ

ð1Þ

ð2Þ

ð1Þ

ð1Þ

ð1Þ



ð3Þ

ð2Þ

ð2Þ

ð2Þ

ð1Þ

ð2Þ

EV(.)

ð2Þ

ð3Þ





ð3Þ

ð1Þ

ð2Þ

ð2Þ

ð3Þ

TC, ,  + TC, ,  TC, ,  + TC, ,  , 2 2



ð1Þ

ð2Þ

ð2Þ

ð3Þ





ð3Þ

ð1Þ

ð2Þ

ð2Þ

ð3Þ



PU,  + PU,  PU,  + PU,  , 2 2



ð3Þ

ð1Þ

ð2Þ

ð2Þ

ð3Þ





ð1Þ

ð2Þ

ð3Þ

ð4Þ

MC,  + MC,  MC,  + MC,  , 2 2

ð3Þ

ð1Þ

ð2Þ

ð3Þ

TC, ,  + 2TC, ,  + TC, ,  4 ð2Þ

ð3Þ

T, ,  + 2T, ,  + T, ,  4 ð1Þ

ð2Þ

ð3Þ

PU,  + 2PU,  + PU,  4

ð1Þ

ð2Þ

ð3Þ

SP,  + 2SP,  + SP,  4

SP,  + SP,  SP,  + SP,  , 2 2

ð4Þ



ð2Þ

PC,  + 2PC,  + PC,  4

ð1Þ

T, ,  + T, ,  T, ,  + T, ,  , 2 2

ð3Þ

ð1Þ

PC,  + PC,  PC,  + PC,  , 2 2

ð3Þ

ð2Þ

(:) (:) EI(:) ¼ [E1 ,E2 ]



ð1Þ

ð2Þ

ð3Þ

ð4Þ

MC,  + MC,  + MC,  + MC,  4

Notes: ∗ represents the different numbers 1, 2, 3, … used in parameters. (∗, ∗) and (∗, ∗, ∗) in suffixes represent the different indices set.

367

368

Optimization Theory Based on Neutrosophic and Plithogenic Sets

subject to B X g gm, a MC1 X1m,a,b  ð1  βÞEMC1 + βE1 m, a , 2 b¼1 C X

g f MC2 MC2 X2m,b,c  ð1  βÞE2 m, b + βE1 m, b ,

c¼1 B X

I X

b¼1 C X

i¼1 G X

c¼1

g¼1

X2m,b,c + X3l,c,d +

(15.40) (15.41)

g g MC3 MC3 rmm X10m,i, c  βE2 m, c + ð1  βÞE1 m, c ,

(15.42)

g g MC4 MC4 rfl X7l,g,d  ð1  βÞE2 l, d + βE1 l, d ,

(15.43)

E X g g MC6 MC6 rcl,e X5l,e, f  ð1  βÞE2 l, f + βE1 l, f , e¼1 G X

(15.44)

g g MC7 MC7 X6l, f ,g  ð1  βÞE2 l, g + βE1 l, g ,

(15.45)

H X g g MC8 MC8 X8l, f ,h  ð1  βÞE2 l, h + βE1 l, h ,

(15.46)

g¼1

h¼1 I X

g g MC10 MC10 rrm X11m,i, j  ð1  βÞE2 m, j + βE1 m, j ,

i¼1 J X

gm, a gm, a X13m, j,a  βEMC1 + ð1  βÞEMC1 , 2 1

(15.47) (15.48)

j¼1

  I X g g β MC11 β MC11 rdm X12m,i,k  E2 m, k + 1  E1 m, k , 2 2 i¼1   I X g g β MC11 β MC11 rdm X12m,i,k  1  E2 m, k + E1 m, k , 2 2 i¼1   E X g g β MC5 β MC5 X4l,d,e  E2 l, e + 1  E1 l, e , 2 2 e¼1   E X g g β MC5 β MC5 X4l,d,e  1  E2 l, e + E1 l, e , 2 2 e¼1

(15.49) (15.50) (15.51) (15.52)

  C J K X X X g g β MC9 β MC9 rtm X10m, i,c + rtm X11m, i, j + rtm X12m,i,k  E2 m, i + 1  E1 m, i , 2 2 c¼1 j¼1 k¼1 (15.53)   C J K X X X gm, i gm, i β MC9 β MC9 rtm X10m, i,c + rtm X11m, i, j + rtm X12m,i,k  1  E2 + E1 : 2 2 c¼1 j¼1 k¼1 (15.54)

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369

15.5.2 Neutrosophic fuzzy programming approach The multiobjective optimization problems are prevalent in real-life scenarios. Due to the existence of complex and conflicting multiple goals or objectives, the task of obtaining optimal solutions is a vital issue. The different conventional optimization techniques for obtaining the compromise solution of multiobjective programming problems are based on the marginal evaluation (degree of validity) for each objective (say Zo) in the feasible solution set. By marginal evaluation, we mean a transformation function (say μ(Zo) ! [0, 1]jα 2[0, 1]) that assigned the values between 0 and 1 to each objective function which shows that the decision makers’ preferences have been fulfilled up to α level of satisfaction. Therefore, the quantification of marginal evaluation is based on the different decision set theory. Initially, Zadeh [28] proposed the FS theory, which explicitly contains the membership function (degree of belongingness) of the element into the feasible solution set. Later on, Zimmermann [29] introduced the fuzzy programming approach to solve multiobjective optimization problems. In a fuzzy programming approach, the quantification of marginal evaluation is represented by a membership function, which only maximizes the degree of belongingness under the fuzzy decision set. The extended version of the fuzzy optimization technique has been applied in a wide range of real-life applications. Furthermore, the generalizations or extensions of the FS were initially proposed by Atanassov [30] and named the intuitionistic fuzzy set (IFS). The analytical coverage spectrum of IFS is versatile and flexible compared to FS as it deals with the membership (degree of belongingness) as well as nonmembership (degree of nonbelongingness) functions of the element into the feasible set. Based on IFS, first Angelov [31] proposed the intuitionistic fuzzy programming approach to obtain the compromise solution of the multiobjective optimization problems. The quantification of marginal evaluation of each objective function under the IF decision set depends on the membership and nonmembership functions, which are to be achieved by maximizing the membership function and minimizing the nonmembership functions simultaneously. The intuitionistic fuzzy programming approach has been extensively studied with various real-life problems. In the past few decades, it has been observed that the situation may arise in real-life decision-making problems where the indeterminacy or neutral thoughts about an element into the feasible set exists. Indeterminacy/neutral is the region of the negligence of a proposition’s value and lies between a truth and falsity degree. Therefore, the further generalization of FS and IFS has been presented by introducing a new member into the feasible decision set. First, Smarandache [32] investigated the neutrosophic set (NS) which comprises three membership functions: truth (degree of belongingness), indeterminacy (degree of belongingness up to some extent), and falsity (degree of nonbelongingness) functions of the element into the NS. The word neutrosophic is the hybrid mixture of two different words, neutre, taken from the French, meaning neutral, and sophia, derived from the Greek, meaning skill/wisdom, which literally gives the meaning knowledge of neutral thoughts (see [32]). The independent indeterminacy degree is sufficient to differentiate itself from FS and IFS. Recent literature on the NS reveals that many researchers have taken an interest in the neutrosophic

370

Optimization Theory Based on Neutrosophic and Plithogenic Sets

domain (see [33–36]) and this is likely to be a prominent emerging research area in the future. This study has also taken advantage of the versatile and effective texture of a neutrosophic fuzzy decision set to develop the NFPA. The NFPA has been designed to solve the proposed CLSC model with multiple objectives under the set of constraints. The NFPA quantifies the marginal evaluation of each objective function under three different membership functions: truth, indeterminacy, and falsity membership functions. Thus the NFPA optimization techniques for the multiobjective optimization problem has a significant role in the implementation and execution of the neutral thoughts in decision-making processes. Definition 15.5. Neutrosophic set [32] Let there be a universe discourse Y such that y 2Y, then an NS W in Y is defined by three membership functions, truth pW(y), indeterminacy qW(y), and falsity rW(y), and denoted by the following form: W ¼ fhy, pW ðyÞ, qW ðyÞ, rW ðyÞijy 2 Yg, where pW(y), qW(y), and rW(y) are real standard or nonstandard subsets belonging to ]0, 1+[, also given as pW(y) : y ! ]0, q+[, rW(y) : Y ! ]0, 1+[, and rW(y) : Y ! ]0, 1+[. There is no restriction on the sum of pW(y), qW(y), and rW(y), so we have 0  sup pW ðyÞ + qW ðyÞ + sup rW ðyÞ  3 + : Definition 15.6. Smarandache [32] Let there be two single-valued NSs A and B, then C ¼ ðA [ BÞ with truth pC(y), indeterminacy qC(y), and falsity rC(y) membership functions are given by pC ðyÞ ¼ max ðpA ðyÞ, pB ðyÞÞ, qC ðyÞ ¼ min ðqA ðyÞ,qB ðyÞÞ, and rC ðyÞ ¼ min ðrA ðyÞ,rB ðyÞÞ for each y 2Y.

Definition 15.7. Smarandache [32] Let there be two single-valued NSs A and B, then C ¼ ðA \ BÞ with truth pC(y), indeterminacy qC(y), and falsity rC(y) membership functions are given by pC ðyÞ ¼ min ðpA ðyÞ,pB ðyÞÞ, qC ðyÞ ¼ max ðqA ðyÞ, qB ðyÞÞ, and rC ðyÞ ¼ max ðrA ðyÞ,rB ðyÞÞ for each y 2Y.

First, Bellman and Zadeh [37] introduced the idea of the fuzzy decision set (D) which contains a set of fuzzy goals (G) and fuzzy constraints (C). Later on, it was widely used in many real-life decision-making problems. Thus, a fuzzy decision set (D) can be stated as follows: D ¼ G \ C:

Closed-loop supply chain management

371

Equivalently, the neutrosophic decision set DNeutrosophic, with a set of neutrosophic goals and constraints, can be given as follows:   N  DNeutrosophic ¼ \O o¼1 Go \n¼1 Cn ¼ ðy, pD ðyÞ,qD ðyÞ, rD ðyÞÞ , where 9 8 p ðyÞ,pG2 ðyÞ, …, pGO ðyÞ > > = < G1 8 y 2 Y, pD ðyÞ ¼ min pC1 ðyÞ,pC2 ðyÞ,…, pCN ðyÞ > > ; : 9 8 q ðyÞ, qG2 ðyÞ,…, qGO ðyÞ > > = < G1 qD ðyÞ ¼ max qC1 ðyÞ, qC2 ðyÞ, …,qCN ðyÞ 8 y 2 Y, > > ; : 9 8 r ðyÞ, rG2 ðyÞ, …,rGO ðyÞ > > = < G1 rD ðyÞ ¼ max rC1 ðyÞ,rC2 ðyÞ, …, rCN ðyÞ 8 y 2 Y, > > ; : where the truth, indeterminacy, and falsity membership functions have been represented by pW(y), qW(y), and rW(y) under neutrosophic decision set DNeutrosophic, respectively. The marginal evaluation for each objective function by using the transformation functions of truth pW(y), indeterminacy qW(y), and falsity rW(y) membership functions can be derived with the help of the upper and lower bounds of each objective function. The solution of each single objective under the given set of constraints provides the upper and lower bounds for each objective function and can be denoted as Uo and Lo with a set of decision variables X1, X2, …, Xo, respectively. Mathematically, it can be shown as follows: Uo ¼ max ½Zo ðXo Þ and Lo ¼ min ½Zo ðXo Þ 8 o ¼ 1, 2,3, …, O:

(15.55)

The upper and lower bounds for o objective function under the neutrosophic environment can be obtained as follows: Uop ¼ Uo , Lpo ¼ Lo for truth membership, Uoq ¼ Lpo + so , Lqo ¼ Lpo for indeterminacy membership, Uor ¼ Uop , Lro ¼ Lpo + to for falsity membership, where so and to 2 (0, 1) are predetermined real numbers assigned by the decision maker(s). With the help of upper and lower bounds for each of the three membership

372

Optimization Theory Based on Neutrosophic and Plithogenic Sets

functions, we have presented the linear membership function under a neutrosophic decision-making framework. 8 1 > < p Uo  Zo ðxÞ po ðZo ðxÞÞ ¼ p p > : Uo  Lo 0 8 1 > < q Uo  Zo ðxÞ qo ðZo ðxÞÞ ¼ q q > : Uo  Lo 0 8 1 > > < Z ðxÞ  Lr o o ro ðZo ðxÞÞ ¼ r  Lr > U > o o : 0

if Zo ðxÞ < Lpo if Lpo  Zo ðxÞ  Uop ,

(15.56)

if Zo ðxÞ > Uop if Zo ðxÞ < Lqo if Lqo  Zo ðxÞ  Uoq , if

(15.57)

Zo ðxÞ > Uoq

if Zo ðxÞ > Uor if Lro  Zo ðxÞ  Uor :

(15.58)

if Zo ðxÞ < Lro ð:Þ

ð:Þ

In the above-discussed membership functions, Lo 6¼ Uo for all o objective functions. The value of these membership will be equal to 1, if for any membership ð:Þ ð:Þ Lo ¼ Uo . The diagrammatic representation of the objective function with different components of membership functions under a neutrosophic decision set is shown in Fig. 15.2.

mo 1 po (x)

Membership degree

Fig. 15.2 Diagrammatic representation of truth, indeterminacy, and falsity membership degrees for the objective function.

qo (x)

ro (x)

0

Lo

Uo

Objective function

Closed-loop supply chain management

373

Logically, the aim of developing the different achievement function is to achieve the maximum satisfaction degree or level according to the preference of the decision maker(s). Therefore, here also we have defined the individual achievement variables for each membership function, such as by maximization of truth membership, maximization of indeterminacy degree, and minimization of a falsity degree of each objective function efficiently. With the aid of linear truth, indeterminacy, and falsity membership functions under a neutrosophic environment, the neutrosophic fuzzy mathematical programming model can be presented as follows: M3 : Max mino¼1,2,3,…,O po ðZo ðxÞÞ Max mino¼1,2,3,…,O qo ðZo ðxÞÞ Min maxo¼1,2,3,…,O ro ðZo ðxÞÞ subject to po ðZo ðxÞÞ  qo ðZo ðxÞÞ, po ðZo ðxÞÞ  ro ðZo ðxÞÞ 0  po ðZo ðxÞÞ + qo ðZo ðxÞÞ + ro ðZo ðxÞÞ  3: Eqs: ð15:40Þ  ð15:54Þ With the help of auxiliary parameters, model M3 can be transformed into the following form M4. M4 : Max λo Max θo Min ηo subject to po ðZo ðxÞÞ  λo qo ðZo ðxÞÞ  θo ro ðZo ðxÞÞ  ηo λo  θo , λo  ηo , 0  λo + θo + ηo  3 λo , θo , ηo 2 ð0,1Þ: Eqs: ð15:40Þ  ð15:54Þ Without loss of generality, the model M4 can be rewritten as in M5. M5 : Max

O X ðλo + θo  ηo Þ o¼1

subject to Zo ðxÞ + ðUop  Lpo Þλo  Uop Zo ðxÞ + ðUoq  Lqo Þθo  Uoq Zo ðxÞ  ðUor  Lro Þηo  Lro λo  θo , λo  ηo , 0  λo + θo + ηo  3 λo , θo , ηo 2 ð0,1Þ, Eqs: ð15:40Þ  ð15:54Þ

374

Optimization Theory Based on Neutrosophic and Plithogenic Sets

where λo, θo, and ηo are auxiliary achievement variables for truth, indeterminacy, and falsity membership functions, respectively. Therefore, the proposed NFPA is a convenient conventional optimization technique that is only preferred over others due to the existence of its independent indeterminacy degree.

15.5.3 Modified neutrosophic fuzzy programming with intuitionistic fuzzy preference relations The effective modeling and optimization framework of multiobjective optimization problems explicitly results in the best possible compromise solution under adverse circumstances, since, while dealing with multiple objectives or goals, most often, DM(s) intends to provide priorities among the different objectives over each other. Generally, the preferences among the objective function have been defined by assigP ning the maximum crisp weight parameter (say wo ¼ 0:1, 0:2, …, 1j O o wo ¼ 1) to the preferred objective function. In the past few decades, Ak€oz and Petrovic [38] proposed a new methodology to assign the preference among different objectives or goals based on the linguistic importance relation and investigated three different fuzzy linguistic importance relationship such as slightly more important than, moderately more important than, and significantly more important than for different conflicting objectives. These linguistic terms have taken the advantages of membership functions associated with corresponding objectives or goals between which the important relation has been defined. Later on, this linguistic preference scheme was adopted by several researchers (see [27, 39–46]) in various real-life applications and decision-making processes. The appropriate selection of membership functions is always a crucial task for decision makers. Since the quantification of preference, the membership function has been done for the three linguistic fuzzy preference relations, but it would be more convenient and realistic to consider the nonmembership function as well as the similar linguistic fuzzy preference relations. Therefore, to incorporate the membership and nonmembership function for linguistic preference relations among the objective, we have designed the structure of our proposed linguistic preference relations among different objectives or goals. Again, we have developed the linear membership and nonmembership function for each linguistic preference relation among the different objectives in the intuitionistic fuzzy environment. The transformation function has been defined with the help of truth membership functions of each objective. The information regarding linguistic preference relations under the intuitionistic fuzzy environment is shown in Table 15.2. The membership and nonmembership function for intuitionistic fuzzy linguistic preference relations is shown in Fig. 15.3. The linear membership function for each linguistic preference relation can be defined as follows and achieved by maximizing it [38].  μe

R 1ðo, uÞ

¼

ðpo  pu + 1Þ if  1  po  pu  0 , 1 if 0  po  pu  1

(15.59)

Closed-loop supply chain management

375

Table 15.2 Linguistic relative preferences of objective o over u.

Linguistic term Slightly more important than Moderately more important than Significantly more important than

μ Re2 ðo, uÞ ¼ μ Re3 ðo, uÞ ¼

 

Intuitionistic fuzzy relation

Membership and nonmembership functions

Re1

μ Re1 and ν Re1

Re2

μ Re2 and ν Re2

Re3

μ Re3 and ν Re3

po  pu + 1 2

Transform function

po(X)pu(X) 8 o, u 2 (1… O)

 if  1  po  pu  1 ,

0 if  1  po  pu  0 : ðpo  pu Þ if 0  po  pu  1

(15.60)

(15.61)

The linear nonmembership function for the linguistic preference relations can be given as follows and achieved by minimizing it. ν Re1 ðo, uÞ ¼ ν Re2 ðo, uÞ ¼ ν Re3 ðo,uÞ ¼

  

ðpo  pu Þ if  1  po  pu  0 : 0 if 0  po  pu  1

(15.62)

1  ðpo  pu Þ if  1  po  pu  1 , 2

(15.63)

1 if  1  po  pu  0 , 1  ðpo  pu Þ if 0  po  pu  1

(15.64)

where Re1 , Re2 , and Re3 are the importance relations defined by the linguistic term slightly more important than, moderately more important than, and significantly more important than, respectively. The new achievement function for satisfaction degrees of the imprecise linguistic importance relations can be defined with the aid of the membership and nonmembership function for intuitionistic fuzzy linguistic preference relations. We have defined a score function S Rðo e , uÞ ¼ðμ Rðo e , uÞ  ν Rðo e , uÞÞ, which has been used to express the satisfactory degree of decision makers’ linguistic importance relations. Let us define a binary variable BI(o, u); o, u ¼ 1, 2, …, O, where o6¼u such that  BIo, u ¼

1 if a linguistic preference relation is defined between the objective Zo and Zu : 0 otherwise

376

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Membership Nonmembership

mR~

1

1.2 1 0.8 0.6 0.4 0.2 0 –1

0

1

(A) mR~2

Membership Nonmembership

1.2 1 0.8 0.6 0.4 0.2 0 –1

0

1

(B) Membership

mR~3

Nonmembership

1.2 1 0.8 0.6 0.4 0.2 0 –1

0

1

(C) Fig. 15.3 Linear membership and nonmembership functions for intuitionistic fuzzy linguistic e 2 . (C) R3 ðo,uÞ ¼ R e3. preference relations. (A) R1 ðo,uÞ ¼ Re1 . (B) R2 ðo,uÞ ¼ R

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The modified NFPA with intuitionistic fuzzy linguistic preference relations has been designed with the hybrid integration of the achievement function under the NFPA model and score functions for the satisfaction degree of decision makers. The achievement function for the modified NFPA can be defined as the convex combination of the sum of individual truth membership, indeterminacy function, and falsity membership function of each objective or goals and the sum of score functions of the imprecise linguistic importance relations. Thus the proposed modified NFPA can be given as follows: M6 : Max α

O O X O X X ðλo + θo  ηo Þ + ð1  αÞ BIo, u Se o¼1

subject to Zo ðxÞ + ðUop  Lpo Þλo  Uop Zo ðxÞ + ðUoq  Lqo Þθo  Uoq Zo ðxÞ  ðUor  Lro Þηo  Lro ðpo  pu + 1Þ  μe   R 1 ðo,uÞ po  pu + 1  μe R 2 ðo,uÞ 2 ðpo  pu Þ  μe R 3 ðo,uÞ ðpo  pu Þ  νe

o¼1 u¼1

R ðo,uÞ

R 1 ðo, uÞ

1  ðpo  pu Þ  νe R 2 ðo,uÞ 2 1  ðpo  pu Þ  νe R 3 ðo,uÞ Se ¼ ðμe  νe Þ R ðo,uÞ R ðo,uÞ R ðo,uÞ μe  νe R ðo, uÞ R ðo,uÞ 0  μe + νe 1 R ðo,uÞ R ðo,uÞ 0  μe , ν e  1 8 BIo,u ¼ 1 R ðo,uÞ R ðo,uÞ λo  θ o , λo  η o , 0  λo + θ o + η o  3 λo , θo , ηo 2 ð0, 1Þ, Eqs: ð15:40Þ  ð15:54Þ

where α is a nonzero parameter taking values between 0 and 1 and can be assigned by tuning it for either the membership function of objectives or linguistic preference relations. The proposed modified NFPA modeling approach considers the degree of belongingness and nonbelongingness simultaneously, which is a better representation of uncertain importance relations among objectives because it enhances the membership degree as well as efficiently reducing the nonmembership degree. In spite of all this, while dealing with a large number of goals at a time, assigning the different crisp weight to all objectives according to the decision-makers’ priority level is not feasible, because it may be time-consuming. To avoid the weight assignment complexity, it would be the best technique to assign linguistic priorities among different objectives.

378

Optimization Theory Based on Neutrosophic and Plithogenic Sets

15.5.3.1 Stepwise solution algorithm The stepwise solution procedures for the proposed modified NFPA with intuitionistic fuzzy preference relations can be represented as follows: Step 1. Design the proposed CLSC planning problem under uncertainty as given in model M1. Step 2. Convert each fuzzy parameter involved in model M1 into its crisp form by using the expected intervals and values method as given in Eqs. (15.28)–(15.31) or presented in Table 15.1. Transform fuzzy constraints into their crisp versions by using Eqs. (15.38)– (15.39). Step 3. Modify model M1 into M2 and solve for each objective function individually in order to obtain the best and worst solution set. Step 4. Determine the upper and lower bounds for each objective function by using Eq. (15.55). With the aid of Uo and Lo, define the upper and lower bounds for truth, indeterminacy, and falsity memberships as given in Eqs. (15.56)–(15.58). Step 5. Develop the neutrosophic optimization model M5 with the aid of auxiliary variables. Step 6. Assign linguistic importance relations among different objectives under an intuitionistic fuzzy environment (see Eqs. 15.59–15.64). Integrate the preference relation into model M5 and transform into model M6, which includes constraints of CLSC given in Eqs. (15.40)–(15.54). Step 7. Model M6 represents the modified neutrosophic fuzzy optimization model with intuitionistic fuzzy importance relations. Solve the model in order to obtain the compromise solution using suitable techniques or some optimizing software packages.

15.6

Computational study

The city of Nizam (Deccan), currently known as Hyderabad, is one of the leading IT hubs of India. It is well known for its IT hub service-oriented firms. A Hyderabadbased ABC (name changed) reputed multinational laptop manufacturing company has intended to model the production, transportation, distribution, and collection problems, due to the existence of a testing center facility in the proposed CLSC designed network. The prominent features of the CLSC design made it possible for the modeling and optimization approach under uncertainty. Regardless, unique, potentially functional components of the proposed CLSC design model have attracted the attention of decision makers. Less opportunity for the disposal of scrap parts/components is also a leading factor to adopt the model which ensures less accountability toward governmental managerial laws. The ecofriendly environmental nature of the modeling approach is a beneficial factor and guarantees freedom from the different governmental legislative traps. The interference of uncertainty among the various parameters reveals the realistic modeling approach. Ample scope for generating different solutions set by tuning the weight parameter and feasibility degree is the crucial promising factor for modeling choice by decision makers. To maintain sustainability in the competitive market, it would be more effective and efficient to develop the proposed CLSC design network. The company has a fully functional multiechelon facility location and a wellorganized decision policy scheme. In the forward chain, five multiechelon facilities

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are the main constituent part of the forward process. Three raw material storage centers, three supplier points, three hybrid manufacturing/remanufacturing plants, three distribution centers, and six customer/market zones explicitly represent the forward flow chain. In the reverse flow chain, six multiechelon facilities are taken into consideration, which signifies more emphasis on the opposite chain. The reverse flow chain consists of three collection centers of used laptops, three refurbishing or repair centers, three disassembling centers, three testing points, three recycling centers, and three disposal sites at which the end-of-life parts/components are removed from the designed CLSC network. Every new and refurbished laptop is a hybrid combination of three different types of raw materials and parts/components. Refurbished laptops are also usable and acceptable in the market. Manufacturing plants provide a new laptop whereas the refurbishing center is responsible for renovated or refurbished laptops. The forward chain starts from the shipment of raw parts from the raw materials storage center to three supplier points. All three suppliers are responsible for the delivery of raw materials to hybrid manufacturing plants. Afterward, the newly manufactured laptops are shipped to three distribution centers. The demand quantity of the laptops must be fulfilled by the distribution center only. There is no scope for direct shipment from the manufacturing plant to the hybrid facility center. The collection center is accountable for the accumulation of end-of-use products from customers/market zones. The used products are disassembled into three parts or components. The testing facility carefully inspects the various parts/components and decides to implement a particular service to make it usable. From the testing center, three different destinations—the manufacturing plant, recycling point, and disposal center—have been postulated. Recyclable products are sent to the recycling center, whereas scrap or end-of-life parts/components are dumped at the disposal center. Parts/components that can constitute raw materials are entered into the forward chain through manufacturing plants. The recycling process turns the pieces into new raw materials, which ensures the procurement of raw materials and initiates the forward chain. Hence to implement the proposed CLSC model efficiently, the triangular fuzzy input data for transportation cost, purchasing cost, revenues, and time have been summarized in Table 15.3. Various capacities at each echelon in the CLSC chain network have been represented by trapezoidal fuzzy data, whereas processing cost parameters have been considered as triangular fuzzy input data. Since numerous objective functions have been developed in the proposed CLSC model, the following preference relations have been decided among different objective functions. However, the preference scheme has been randomly assigned, and there are no hard and fast rules. It solely depends upon the decision maker’s choices. The type of preference relations between the objectives have been defined as follows: l

l

l

l

Objective Objective Objective Objective

Z2 Z4 Z3 Z4

is is is is

moderately more important than objective Z1 (i.e., Re2 ð2,1Þ). slightly more important than objective Z3 (i.e., Re1 ð4,3Þ). significantly more important than objective Z5 (i.e., Re3 ð3,5Þ). slightly more important than objective Z5 (i.e., Re1 ð4,5Þ).

380

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 15.3 Input fuzzy data for the parameters. Types of raw materials (m) or products (l) Transportation cost from sources to destinations (*, *)

1

2

3

(14, 24, 34)

(22, 32, 44)

(34, 36, 38)

(30, 32, 34)

(34, 36, 38)

(38, 40, 42)

(52, 56, 60)

(60, 63, 66)

(66, 67, 68)

(60, 65, 70)

(66, 69, 72)

(71, 74, 77)

(28, 29, 30)

(35, 37, 39)

(42, 44, 46)

(32, 34, 36)

(35, 39, 43)

(41, 42, 43)

(40, 42, 44)

(45, 48, 51)

(50, 55, 60)

(44, 48, 52)

(50, 53, 56)

(55, 59, 63)

(50, 51, 52)

(50, 55, 60)

(60, 63, 66)

(25, 27, 29)

(30, 32, 34)

(35, 39, 41)

(55, 60, 65)

(65, 67, 69)

(71, 73, 75)

(33, 36, 39)

(40, 43, 46)

(44, 49, 54)

(68, 71, 74)

(73, 75, 77)

(60, 62, 64)

(05, 07, 09)

(04, 06, 08)

(01, 03, 06)

(36, 38, 40)

(45, 47, 49)

(24, 26, 28)

(25, 27, 29)

(15, 17, 19)

(15, 17, 19)

gm SP1 g SP2 l

(42, 46, 50)

(40, 45, 50)

(21, 23, 26)

(36, 38, 40)

(42, 45, 48)

(24, 26, 28)

rfl rcl,e rtm rmm rrm rdm

0.71 0.82 0.23 0.81 0.32 0.12

0.53 0.76 0.49 0.67 0.43 0.19

0.58 0.38 0.73 0.35 0.61 0.23

(14, 24, 34)

(22, 32, 44)

(34, 36, 38)

(30, 32, 34)

(34, 36, 38)

(38, 40, 42)

(52, 56, 60)

(60, 63, 66)

(66, 67, 68)

(60, 65, 70)

(66, 69, 72)

(71, 74, 77)

(28, 29, 30)

(35, 37, 39)

(42, 44, 46)

(32, 34, 36)

(35, 39, 43)

(41, 42, 43)

g m, a, b TC1 g m, b, c TC2 g l, c, d TC3 g l, d, e TC4

g l, e, f TC5 g l, f , g TC6 g l, g, d TC7 g l, f , h TC8

g m, h, i TC9 g m, i, c TC10 g m, i, j TC11 g m, i, k TC12

g m, j, a TC13

Time e l, d, e T

Purchasing cost gm PU1 gl PU2

Selling price

Processing cost at each echelon g m, a PC1 g m, b PC2 g l, c PC3 g l, d PC4 g l, f PC5 g l, g PC6

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Table 15.3 Continued Types of raw materials (m) or products (l) Transportation cost from sources to destinations (*, *) g l, h PC7 g PC8 m, i

g m, j PC9 g m, k PC10

1

2

3

(40, 42, 44)

(45, 48, 51)

(50, 55, 60)

(44, 48, 52)

(50, 53, 56)

(55, 59, 63)

(50, 51, 52)

(50, 55, 60)

(60, 63, 66)

(25, 27, 29)

(30, 32, 34)

(35, 39, 41)

(512, 514, 516, 518) (613, 614, 615, 616) (724, 725, 726, 727) (212, 214, 216, 218) (314, 318, 322, 326) (115, 116, 117, 118) (124, 125, 126, 127) (110, 111, 112, 113) (224, 225, 226, 227) (324, 325, 326, 327) (212, 214, 216, 218)

(622, 624, 626, 628) (514, 516, 518, 520) (812, 813, 814, 815) (221, 222, 223, 224) (312, 314, 316, 318) (119, 120, 121, 122) (113, 114, 115, 116) (114, 116, 118, 120) (212, 214, 216, 218) (212, 213, 214, 215) (221, 222, 223, 224)

(718, 724, 726, 728) (512, 514, 516, 518) (914, 916, 918, 920) (217, 218, 219, 220) (329, 339, 349, 359) (114, 116, 118, 120) (117, 119, 121, 123) (119, 120, 121, 122) (314, 316, 318, 320) (214, 216, 218, 220) (317, 318, 319, 320)

Capacity/demand at each echelon g m, a MC1 g m, b MC2 g m, c MC3 g l, d MC4 g l, e MC5 g l, f MC6 g l, g MC7 g m, h MC8 g m, i MC9 g m, j MC10 g m, k MC11

15.6.1 Results and discussions The modified neutrosophic fuzzy optimization model for the proposed CLSC network has been written in AMPL language and solved using the solver Kintro 10.3.0 through the NEOS server version 5.0 online facility provided by Wisconsin Institutes for Discovery at the University of Wisconsin in Madison for solving optimization problems; see Refs. [47,48]. The characteristic description of the problem is presented as follows: The final multiobjective optimization model along with a set of well-defined multiple objectives comprises 459 variables including 42 binary variables and 417 linear variables, 530 constraints including 498 linear one-sided inequalities constraints and 32 linear equality constraints, respectively. The total computational time for

382

Optimization Theory Based on Neutrosophic and Plithogenic Sets

obtaining the final solution was 0.113 seconds (CPU time). Due to space limitations, only the final solution results of all decision variables obtained at a feasibility degree (β ¼ 0.5) with weight parameter (α ¼ 0.5) have been discussed in detail. The optimum allocation of raw materials, new products, and used parts/components among different echelons has been depicted in Tables 15.4 and 15.5. In the forward chain, procurement of raw materials initiates from a raw material storage center (RMS) to a supplier point (SP). The total allocation of raw materials from RMS 1 to all three SPs is found to be 504.17, 592.38, and 681.51, whereas from RMS 2 and 3 to all three SPs have been obtained as 572.14, 553.84, and 703.15, and 497.57, 457.32, and 646.87, respectively. The maximum shipment quantity has been observed from RMS 2 to SP 3 due to the lowest transportation and processing cost incurred over the raw materials. Suppliers are responsible for fulfilling the requirement for starting the manufacturing processes at the hybrid manufacturing plant (MP). The optimum shipment quantity from SP 1 to all three MPs is 706.25, 630.15, and 625.18, respectively. Similarly, from SP 2 and 3 to all three MPs have been obtained as 630.25, 630.21, and 656.51, and 563.70, 498.34, and 533.18, respectively. The highest shipment amount of raw materials has been allocated to MP 1 whereas the least amount of raw materials has been delivered to MP 2 bearing in mind the fact that the outbound capacity of manufacturing plant receives the maximum raw materials and parts from the SPs and testing points (TPs). SP 3 also provides the maximum amount of raw materials to all three MPs and are obtained as 563.7, 488.34, and 533.18 bearing in mind the fact that outbound restrictions on manufacturing plants have been satisfied, and tested and approved parts/components are sent back to the manufacturing plant for further utilization. Newly built products are transferred to the distribution center (DC) so that the demand from customers (Cs) could be met. The optimal distribution scheme among different customers has been obtained. From DC 1 to all six Cs, the total shipment of products is found to be 332.23, 400.85, 350.61, 297.21, 274.95, and 266.61, respectively. However, DC 1 has a negligible contribution to meet the demand of C 2, 3, 4, and 5, with other types of products to avoid the maximum transportation cost and late expected delivery time. Similarly, the total quantities of each product distributed from DC 2 and 3 to all six Cs have been depicted, which ensures the minimum transportation costs along with the timely shipment of products. It has been observed that no product has been shipped from DC 2 to C 1, 2, and 3 due to the maximum chances for late delivery of the products. Hence a minimum transportation cost and shipment time have been achieved without significantly affecting the demand constraint. Overall, DC 2 outsourced the maximum shipment of products to all six Cs and revealed a significant contribution to fulfilling the demand. Since refurbished products are also acceptable in the market, approximately 13.32% of total used products are renovated and shipped to DCs for the fulfillment of further needs. The significant role of the collection center (CC) starts when end-of-use and endof-life products come into existence. The potential accumulation framework for used products from the customer zone is much needed. The designed CLSC model inherently involves the CC, which is the first echelon of the reverse supply chain network. The exclusive collection of the end-of-use product from customers is found to be a significant percentage, that is, approximately 91.34% of the total fulfilled demand,

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383

Table 15.4 Optimal quantities of raw materials and products shipped from different sources to various destinations. Types of raw material (m) Raw material storage center (a)

Supplier point (b)

1

2

3

Storage center Storage center Storage center Storage center Storage center Storage center Storage center Storage center Storage center

1 2 3 1 2 3 1 2 3

127.83 248.23 201.32 164.21 321.34 221.63 116.94 213.52 329.53

232.78 151.62 312.28 264.67 109.23 368.83 219.43 142.20 127.64

143.56 192.53 167.91 143.26 123.27 112.69 161.20 101.60 189.70

1 1 1 2 2 2 3 3 3

Types of raw materials (m) Supplier point (b)

Manufacturing plant (c)

1

2

3

Supplier point Supplier point Supplier point Supplier point Supplier point Supplier point Supplier point Supplier point Supplier point

1 2 3 1 2 3 1 2 3

261.24 291.64 236.39 218.95 253.68 287.25 212.54 202.35 298.34

243.12 124.15 213.56 189.67 128.63 112.46 187.62 142.37 116.52

201.89 214.36 175.23 221.63 247.90 256.80 163.54 143.62 118.32

1 1 1 2 2 2 3 3 3

Types of products (l) Manufacturing plant (c)

Distribution center (d)

1

2

3

Manufacturing plant 1 Manufacturing plant 1 Manufacturing plant 1 Manufacturing plant 2 Manufacturing plant 2 Manufacturing plant 2 Manufacturing plant 3 Manufacturing plant 3 Manufacturing plant 3

1 2 3 1 2 3 1 2 3

127.83 148.23 121.32 164.21 171.34 181.63 196.94 113.52 129.53

132.78 151.62 112.28 64.67 119.23 68.83 89.43 42.20 127.64

143.56 192.53 67.91 163.26 143.27 152.69 61.20 101.60 189.70

Types of products (l) Distribution center (d)

Customers (e)

1

2

3

Distribution center Distribution center Distribution center Distribution center

1 2 3 4

112.34 85.26 152.36 163.98

145.26 163.23 – –

74.63 152.36 198.35 115.23

1 1 1 1

Continued

384

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 15.4 Continued Types of products (l) Distribution center (d)

Customers (e)

1

2

3

Distribution center Distribution center Distribution center Distribution center Distribution center Distribution center Distribution center Distribution center Distribution center Distribution center Distribution center Distribution center Distribution center Distribution center

5 6 1 2 3 4 5 6 1 2 3 4 5 6

165.32 154.23 198.43 165.24 180.50 155.96 169.58 187.65 169.75 – – – 184.26 179.35

– – 167.23 144.23 143.20 124.27 153.65 84.59 – 173.89 196.43 142.35 73.68 97.36

109.63 112.38 – – – 127.52 65.87 154.23 159.86 168.27 149.26 149.37 163.87 135.98

1 1 2 2 2 2 2 2 3 3 3 3 3 3

Types of products (l) Customers (e)

Collection center (f)

1

2

3

Customer 1 Customer 1 Customer 1 Customer 2 Customer 2 Customer 2 Customer 3 Customer 3 Customer 3 Customer 4 Customer 4 Customer 4 Customer 5 Customer 5 Customer 5 Customer 6 Customer 6 Customer 6

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

61.32 85.23 84.32 52.31 47.50 79.68 52.63 74.96 47.89 89.56 76.34 58.35 61.23 63.85 86.34 74.68 85.90 84.32

53.68 78.56 58.50 16.78 51.32 45.23 89.45 98.74 114.90 89.45 94.68 78.89 106.83 117.40 127.63 121.69 153.45 173.65

94.38 145.80 145.23 61.83 134.62 84.23 79.56 112.34 78.46 74.68 52.60 53.46 45.3 47.6 76.85 44.62 57.67 79.85

Collection center (f)

Refurbishing center ( g)

1

2

3

Collection center Collection center Collection center Collection center

1 2 3 1

36.24 41.58 16.23 14.23

45.32 31.25 74.32 61.32

32.65 21.32 24.12 42.37

Types of products (l)

1 1 1 2

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385

Table 15.4 Continued Types of products (l) Collection center (f)

Refurbishing center ( g)

1

2

3

Collection center Collection center Collection center Collection center Collection center

2 3 1 2 3

71.20 24.53 34.53 74.30 25.90

41.23 85.93 22.38 72.30 33.56

54.64 27.65 68.53 23.60 67.84

2 2 3 3 3

Table 15.5 Optimal quantities of used products and parts shipped from different sources to various destinations. Types of products (l) Refurbishing plant (g)

Distribution center ( d)

1

2

3

Refurbishing Refurbishing Refurbishing Refurbishing Refurbishing Refurbishing Refurbishing Refurbishing Refurbishing

1 2 3 1 2 3 1 2 3

34.28 25.36 27.85 23.89 31.45 43.56 44.87 38.45 37.84

24.89 41.98 39.38 54.23 47.68 42.89 57.98 47.56 49.63

52.37 56.35 49.35 63.45 54.38 47.86 53.78 63.45 57.68

plant 1 plant 1 plant 1 plant 2 plant 2 plant 2 plant 3 plant 3 plant 3

Types of products (l) Collection center (f )

Disassembling center (h)

1

2

3

Collection center Collection center Collection center Collection center Collection center Collection center Collection center Collection center Collection center

1 2 3 1 2 3 1 2 3

146.23 131.26 157.89 98.46 87.60 89.68 107.35 118.35 112.57

98.29 157.23 158.96 143.69 89.63 63.84 84.96 97.63 98.68

154.78 74.39 84.97 87.56 178.87 187.20 172.86 166.34 136.94

1 1 1 2 2 2 3 3 3

Types of products (m) Disassembling center (h)

Testing center (i)

1

2

3

Disassembling center 1 Disassembling center 1 Disassembling center 1

1 2 3

98.86 187.34 85.32

47.52 145.26 143.26

112.36 75.40 146.37 Continued

386

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Table 15.5 Continued Types of products (m) Disassembling center (h) Disassembling center Disassembling center Disassembling center Disassembling center Disassembling center Disassembling center

2 2 2 3 3 3

Testing center (i)

1

2

3

1 2 3 1 2 3

55.85 65.36 80.45 60.85 75.03 86.08

121.35 185.98 178.90 42.38 63.57 53.76

141.23 124.36 142.58 173.45 156.89 154.36

Types of parts (m) Recycling center ( j)

Raw material storage facility (a)

1

2

3

Recycling Recycling Recycling Recycling Recycling Recycling Recycling Recycling Recycling

1 2 3 1 2 3 1 2 3

24.23 12.54 28.34 21.50 17.35 18.53 24.37 17.98 19.63

227.35 14.80 14.25 22.24 12.36 11.98 22.35 27.85 14.32

16.35 22.35 25.36 16.80 13.52 16.39 14.35 17.68 13.84

center center center center center center center center center

1 1 1 2 2 2 3 3 3

Types of tested parts (m) Testing center (i)

Manufacturing plant (c)

1

2

3

Testing center Testing center Testing center Testing center Testing center Testing center Testing center Testing center Testing center

1 2 3 1 2 3 1 2 3

42.35 13.40 17.31 08.32 21.43 17.35 38.96 17.51 21.27

17.43 16.98 12.37 29.56 39.75 18.56 21.29 10.37 03.78

11.75 27.06 18.08 21.07 19.01 12.89 28.34 12.34 22.48

1 1 1 2 2 2 3 3 3

Types of recyclable parts (m) Testing center (i)

Recycling facility ( j)

1

2

3

Testing center Testing center Testing center Testing center Testing center Testing center

1 2 3 1 2 3

34.53 54.27 58.34 62.78 28.34 25.32

45.05 35.64 56.34 49.64 51.46 47.86

48.35 53.42 24.35 31.70 41.32 105.06

1 1 1 2 2 2

Closed-loop supply chain management

387

Table 15.5 Continued Types of recyclable parts (m) Testing center (i)

Recycling facility ( j)

1

2

3

Testing center 3 Testing center 3 Testing center 3

1 2 3

78.35 48.32 51.43

48.36 42.36 118.36

62.37 56.28 29.23

Types of scrap parts (m) Testing center (i)

Disposal facility (k)

Testing center Testing center Testing center Testing center Testing center Testing center Testing center Testing center Testing center

1 2 3 1 2 3 1 2 3

1 1 1 2 2 2 3 3 3

1

2

3

14.25 17.24 – – – 16.35 12.89 15.45 17.40

– – – 21.85 19.65 12.35 14.22 – –

16.52 13.25 11.24 18.54 – 19.32 – 19.34 21.30

which indicates the vast need for the reverse supply chain to tackle used products. The required service at different echelons in the reverse chain has been designed especially for socioenvironmental concerns. An optimal amount of used products has been collected by all three CCs from all six customer zones. The maximum amount of used products has been received by CC 3 from C 6 which is 337.82, and the least quantity 190.70 by CC 3 from C 4 to ensure the least collection and transportation costs levied over each type of product. At CCs, complete inspection of the collected, used products has been performed and a decision taken to ship either to the disassembling center (DS) or refurbishing center (RC) to initiate the required services. The total amounts of used products transported from CC 1 to all three RCs are obtained as 114.21, 94.15, and 114.67, whereas the total shipment quantities from CC 2 and 3 to all three RCs have been allocated as 117.92, 167.07, and 138.11, and 125.44, 170.20, and 127.30, respectively. The maximum quantity of used products has been transported from CC 3 to RC 2 whereas the minimum shipment quantity is found to be shipped from CC 1 to RC 2 because of the lowest transportation cost and availability of the required service for particular types of products. The quantity of used products is approximately 88.21% of the total capacity of the RC, which ensures the significant need for such a functional echelon in CLSC. The disassembling center (DS) only receives those end-of-use products that require reliability tests of each part/component. At the

388

Optimization Theory Based on Neutrosophic and Plithogenic Sets

DS, used products are disassembled into different parts/components for the testing process where all necessary measures would be taken regarding the useful life of parts. From CC 1 to all three DSs, the total shipment amounts of used products have been obtained as 399.30, 362.88, and 401.82, which shows approximately 31.98% of the entire collection of used products. Likewise, the net amount of used products transported from CC 2 and 3 to all three DSs are 329.71, 356.10, and 340.72, and 365.17, 382.32, and 348.19, respectively. The shipment of end-of-use products at DS 2 and 3 are found to be 14.29% and 39.47% of the net used products collected at all three CCs, which shows that approximately 94% of the total raised used products have been completely dealt with at the CC and signify that the design of the reverse chain is much needed to avoid environmental issues. The total disassembled parts that have been shipped from DS 1 to all three TPs are found to be 258.74, 408, and 374.95, which comprise 31.35% of the disassembled parts/components and ensures that transportation and inspection costs incurred over these parts would be minimal. Similarly, from DS 2 and 3 to all TPs the optimal amounts of pieces have been shipped, which are found to be 33.84%, and 29.27% of the total disassembled parts at DSs to minimize the total cost of inspection by ensuring the capacity restrictions at TPs, respectively. The testing point (TP) inevitably inspects the reliability or usefulness of parts/ components and provides the best decision to deal with tested parts. TPs are interconnected with three echelons: manufacturing plants, recycling centers, and disposal sites. The TP is also a promising source for the procurement of raw materials to hybrid manufacturing/remanufacturing plants. Approximately 9.84% of the total requirement for raw materials has been met by different TPs with the aid of dissembled parts of used products. However, the recycling point (RP) receives a significant amount of tested parts that ensures green practice with recyclable components. The net quantity of recyclable parts that have been transported from all three TPs to RP 1 is found to be 127.93, 143.33, and 139.03, which is 93.66% of the total recyclable capacity of tested parts at RP 1. The maximum quantity of recyclable parts has been received by RP 3 whereas the least amount of certified parts has been shipped to RP 2 bearing in mind the fact that transportation and recycling costs levied over each component are minimal at these facilities. Finished recycled products have been sent back to raw material storage centers and recycled for the smooth running of the production processes. After the inspection procedure, the declared disposable parts have been shipped for disposal. The optimal quantity of disposable parts has been obtained with the satisfaction of disposal capacity constraints of each disposal facility center (DF). The obtained results showing that at some DFs, there is no amount of tested parts for disposal. However, the total shipped amounts from all three TPs to DF 1 are found to be 30.77, 40.39, and 27.11, which is 47.32% of the full capacity of DF 1. Moreover, from all three TPs to DF 2 and 3, the net amounts of disposable parts that have been transported are found to be 30.49, 19.65, and 34.79, and 11.24, 48.02, and 38.70, respectively. At these DFs, approximately 53.57% and 69.38% shares of the total disposal capacity have been disposed of, which strictly ensures that there is still an abundant opportunity for incineration. Multiechelon CLSC design networks require potential capital investment to the flow of products throughout the supply chain processes. Processing cost,

Closed-loop supply chain management

389

transportation cost, and purchasing cost have been depicted as different objectives that inherently require capital. The obtained results of these three objectives show a remarkable contribution to the total capital investment. At each feasibility degree β and weight parameter α, the average share of processing cost has been obtained as approximately 83.39%, the total ordinary dividends of the transportation cost is found to be approximately 14.78% and that of the average purchasing cost is approximately 1.83% of the total investment in the proposed CLSC network. The maximum shares have been exhausted by the processing charge with the fact that multiple different echelons have been associated with specific functional services to raw materials, new products, and used parts in the proposed CLSC network. Transportation costs hold a slightly smaller portion of the total investment, which shows the reduced to and fro movement of products among different echelons. Due to the interconnected systematic facility centers, the optimal shipment strategy turns into fewer transportation costs. The purchasing of raw materials in bulk from raw material storage centers and used products from customers has comparatively very low in the total capital investment. The expected whole delivery time and revenues from sales have also been included as conflicting potential objectives in the proposed CLSC model, which sufficiently reflects the effective exogenous solution results. The flow of new products in the forward chain and end-of-use products in reverse chain much depends on keen managerial insight and decision-making strategy. The potential performances of each echelon would be recognized in the context of allocation and required service to the different products and parts. The solution results have been presented only for α ¼ 0.5 and β ¼ 0.5, but more information could be extracted by obtaining the solution results at different values of α and β regarding the optimum allocation of products and parts, respectively.

15.6.2 Sensitivity analyses Sensitivity analyses have been performed for all the objective functions by tuning the feasibility degree (β) and weight parameter (α) simultaneously. The feasibility degree (β) referred to the preference or acceptance level of decision makers. The higher value of (β) ensures the maximum satisfaction level of decision makers. The feasibility degree among parameters reflects the satisfaction level by offering different choices. Hence more substantial feasibility degree generally gives the worse solution of objectives. The weight parameter (α) provides the weight to either the membership function of all the objectives or the score functions of the intuitionistic fuzzy preference relations among different objectives. Therefore, a higher value of (α) signifies a higher weight to either the corresponding membership functions or the score function of linguistic preference relations. The priority structure has been designed as the convex combination between the membership function of the objectives and the score function of the linguistic preference relations. The weight parameter (α) is directly assigned to the membership functions of each goal whereas (1 α) has been assigned to the score function of linguistic preference relations. The solution results of all the objective functions and preference relations are shown in Fig. 15.4.

0.2

(A)

Total transportation cost

0.4

0.6

396,750

Cost (in $)

0.2

0.8

(B)

Weight parameter (alpha) Total purchasing cost

0.6

Time (in hours)

0.8

13,610,100 13,610,200 13,610,300 13,610,400

6,887,950 6,887,850

Cost (in $)

6,887,750

0.4

(D)

Weight parameter (alpha)

0.2

0.4

0.6

Membership function

0.8

(beta) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.2

0.4

0.6

0.8

Weight parameter (alpha)

R2 (2,1)

R3 (3,5)

R1 (4,3)

R1 (4,5)

0.7

R2 (2,1)

R3 (3,5)

R1 (4,3)

R1 (4,5)

0.3

0.4

0.30 0.25

0.5

0.6

R1 (4,5)

Feasibility degree & weight parameter Score function achieved for intuitionistic fuzzy preference relations

Score function

0.45

R3 (3,5)

0.35

0.40

R2 (2,1) R1 (4,3)

0.2

0.20

Nonmembership function

0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(F)

Weight parameter (alpha) Intuitionistic fuzzy preference relations

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(G)

0.6

Intuitionistic fuzzy preference relations 0.55 0.60 0.65 0.70 0.75 0.80 0.85

11,179,200

(beta) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

11,179,300 11,179,400

Revenue (in $)

Total revenue

(E)

0.4

Weight parameter (alpha) Total expected product delivery time

(beta) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.2

(C)

(beta) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

396,650

(beta) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

396,850

Total processing cost

396,550

Cost (in $)

Optimization Theory Based on Neutrosophic and Plithogenic Sets

39,387,700 39,387,900 39,388,100 39,388,300

390

Feasibility degree & weight parameter

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(H)

Feasibility degree & weight parameter

Fig. 15.4 Graphical representation of obtained results. (A) First objective (Z1). (B) Second objective (Z2). (C) Third objective (Z3). (D) Fourth objective (Z4). (E) Fifth objective (Z5). (F) Membership degree. (G) Nonmembership degree. (H) Score function.

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391

15.6.2.1 Sensitivity analyses of objective functions The first objective (Z1) is to minimize the total processing cost (TPC) supply chain. At β ¼ 0.1 and α ¼ 0.9, the minimum (best) value of (Z1) has been attained, which is $39,387,662. As (α) decreases, the values of (Z1) either increase or remain the same for some (α). With an increase in the feasibility degree (β), a significant increment in the objective function (Z1) has been observed. The maximum (worst) value of objective (Z1) has been obtained as $39,388,338 at β ¼ 0.9 and α ¼ 0.1. Hence it has been concluded that with the increase in feasibility degree (β) and the decrease in weight parameter (α), the value of objective (Z1) reaches its worst values. The different solution results of (Z1) ranging between $39,387,662 and $39,388,338 are summarized in Table 15.6, and Fig. 15.4A shows the trending behavior of (Z1) at different feasibility degree (β) and weight parameters (α), respectively. Furthermore, the effects of feasibility degrees (β) are severe, as the marginal increment in the value of (Z1) rapidly approaches the worst solutions, whereas the effect of the weight parameter (α) on the objective (Z1) is almost negligible. The TPC has been obtained that solely occurred over four echelons in the forwarding chain. Hence, the obtained results for TPC are due to the high processing cost at the raw material storage center, supplier point, and manufacturing plants. Inbound capacity restrictions at these echelons are also a key factor for increment in TPC. The maximum numbers of raw materials and new products require different processing costs, which turn into more capital investment in the material and product processing purposes. The minimization of total transportation costs in the CLSC has been represented by the second objective (Z2). At β ¼ 0.1 and α ¼ 0.9, the minimum (best) value of transportation cost is $396,534. As a feasibility degree (β) increases, there is a significant marginal increment in the objective (Z2) that has been found. The values of (Z2) either increase or remain stable for different values of (α) with the decrease in the weight parameter (α). The maximum (worst) value of (Z2) has been attained as $396,879 at β ¼ 0.9 and α ¼ 0.1, respectively. Thus it has emerged that with the increase in the feasibility degree (β) and the decrease in the weight parameter (α), the value of the objective (Z2) approaches its worst outcomes. The different solution results of (Z2) have been generated, which lie between $396,534 and $396,879, and are presented in Table 15.7. The fluctuating behavior of (Z2) has also been shown in Fig. 15.4B at a different feasibility degree (β) and weight parameter (α). The utmost influencing capability of the feasibility degree (β) has been reflected by the significant increase in the objective (Z2) and which lead (Z2) toward its worst values. The weight parameter (α) has fewer effects on the objective (Z2) compared to the feasibility degree (β) among all the solution choices. Due to the low processing charges at each echelon in the reverse chain, the total TPC has been obtained much less compared to TPC in the forwarding chain. Each echelon in the reverse chain dealt with either end-of-use products or end-of-life products. To perform the different required services on such products would not necessarily result in higher costs, because of less complexity in dealing with used and returned products compared to the manufacturing of new parts and products.

392

Table 15.6 Total processing costs (Z1) at different feasibility degrees (β) and weight parameters (α). Weight parameter (α)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

39,387,662 39,387,768 39,387,834 39,387,914 39,387,994 39,388,194 39,388,234 39,388,282 39,388,331

39,387,664 39,387,769 39,387,834 39,387,916 39,387,996 39,388,194 39,388,234 39,388,283 39,388,331

39,387,666 39,387,771 39,387,835 39,387,916 39,387,996 39,388,194 39,388,236 39,388,285 39,388,334

39,387,668 39,387,773 39,387,836 39,387,916 39,387,996 39,388,196 39,388,238 39,388,285 39,388,336

39,387,670 39,387,774 39,387,836 39,387,918 39,387,997 39,388,196 39,388,239 39,388,285 39,388,336

39,387,672 39,387,774 39,387,836 39,387,918 39,387,997 39,388,197 39,388,241 39,388,286 39,388,336

39,387,674 39,387,776 39,387,838 39,387,921 39,387,998 39,388,197 39,388,241 39,388,286 39,388,337

39,387,676 39,387,778 39,387,838 39,387,921 39,387,998 39,388,197 39,388,243 39,388,288 39,388,338

39,387,678 39,387,779 39,387,839 39,387,922 39,387,999 39,388,198 39,388,244 39,388,288 39,388,338

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Feasibility degree (β)

Closed-loop supply chain management

Table 15.7 Total transportation costs (Z2) at different feasibility degrees (β) and weight parameters (α). Weight parameter (α) Feasibility degree (β)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

396,534 396,593 396,642 396,686 396,723 396,767 396,798 396,837 396,873

396,535 396,593 396,642 396,686 396,724 396,767 396,798 396,837 396,874

396,535 396,593 396,642 396,687 396,724 396,768 396,798 396,838 396,874

396,534 396,595 396,643 396,687 396,724 396,768 396,799 396,838 396,874

396,535 396,595 396,644 396,687 396,725 396,768 396,799 396,838 396,875

396,537 396,596 396,644 396,688 396,725 396,769 396,802 396,839 396,876

396,538 396,598 396,646 396,688 396,725 396,769 396,802 396,839 396,876

396,538 396,599 396,648 396,689 396,726 396,769 396,805 396,839 396,877

396,539 396,599 396,649 396,689 396,728 396,770 396,805 396,840 396,879

393

394

Optimization Theory Based on Neutrosophic and Plithogenic Sets

The minimization of the total purchasing cost has been represented by the third objective (Z3). The minimum (best) value of the objective (Z3) has been obtained as $6,887,719 at β ¼ 0.1 and α ¼ 0.9. At β ¼ 0.9 and α ¼ 0.1, the maximum (worst) value of the total purchasing cost has been obtained, which is $6,887,980. As (α) decreases, the values of (Z3) either increase or remain inert for some (α). With an increase in the feasibility degree (β), the significant increment in the objective function (Z3) has been noticed. Thus it has emerged that with the increase in the feasibility degree (β) and the decrease in the weight parameter (α), the value of the objective (Z3) approaches its worst outcomes. The different solution results of (Z3) have been generated, which lie between $6,887,719 and $6,887,980 and are represented in Table 15.8. The declining performance of (Z3) has also been shown in Fig. 15.4C at different feasibility degrees (β) and weight parameters (α). Furthermore, the effect of the feasibility degree (β) is more influential, as the significant increase in the value of (Z3) rapidly approaches the worst solutions whereas the effect of the weight parameter (α) on the objective (Z3) is almost negligible. The fourth objective (Z4) is the minimization of total product delivery time to different customers/market zones. At β ¼ 0.1 and α ¼ 0.9, the minimum (best) value of the total products delivery time is 13,610,103 hours. As the feasibility degree (β) increases, a significant marginal increment in the objective (Z4) is observed. The values of (Z4) either increase or remain inactive for different values of (α) with the decrease in the weight parameter (α). The maximum (worst) value of (Z4) has been attained as 13,610,429 hours at β ¼ 0.9 and α ¼ 0.1, respectively. Thus it has been concluded that with the increase in the feasibility degree (β) and the decrease in the relative weight parameter (α), the value of the objective (Z4) approaches its worst results. The various solution outcomes of (Z4) have been generated, which lie between 13,610,103 and 13,610,429 hours, and are presented in Table 15.9. The trending feature of (Z4) is also shown in Fig. 15.4D at different feasibility degrees (β) and weight parameters (α). The powerful performance of the feasibility degree (β) has been observed by the significant increase in the objective (Z4) and which leads (Z4) toward its worst values. The weight parameter (α) has fewer effects on the objective (Z4) compared to the feasibility degree (β) among all the solution sets. The maximization of revenues earned from the selling of new products has been represented by the fifth objective (Z5). The maximum (best) value of the objective (Z5) has been obtained as $11,179,402 at β ¼ 0.1 and α ¼ 0.9. At β ¼ 0.9 and α ¼ 0.1, the minimum (worst) value of revenues has been obtained, which is $11,179,140. As (α) decreases, the values of (Z5) either decrease or remain stable for some (α). With an increase in the feasibility degree (β), the significant decrease in the objective function (Z5) has been found. Thus it has been elicited that with the increase in the feasibility degree (β) and the decrease in the weight parameter (α), the value of the objective (Z5) approaches its worst outcomes. The different solution results of (Z5) ranging between $6,887,719 and $6,887,980, and are summarized

Closed-loop supply chain management

Table 15.8 Total purchasing costs (Z3) at different feasibility degrees (β) and weight parameters (α). Weight parameter (α) Feasibility degree (β)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

6,887,719 6,887,743 6,887,795 6,887,823 6,887,847 6,887,881 6,887,916 6,887,949 6,887,976

6,887,721 6,887,743 6,887,796 6,887,823 6,887,847 6,887,881 6,887,917 6,887,951 6,887,976

6,887,723 6,887,743 6,887,797 6,887,823 6,887,848 6,887,883 6,887,917 6,887,951 6,887,976

6,887,723 6,887,744 6,887,797 6,887,825 6,887,848 6,887,883 6,887,918 6,887,952 6,887,977

6,887,724 6,887,744 6,887,797 6,887,826 6,887,848 6,887,884 6,887,918 6,887,954 6,887,978

6,887,724 6,887,745 6,887,798 6,887,827 6,887,848 6,887,884 6,887,919 6,887,954 6,887,978

6,887,725 6,887,745 6,887,798 6,887,827 6,887,849 6,887,885 6,887,921 6,887,955 6,887,978

6,887,725 6,887,747 6,887,799 6,887,828 6,887,849 6,887,886 6,887,921 6,887,955 6,887,979

6,887,727 6,887,748 6,887,799 6,887,829 6,887,851 6,887,887 6,887,923 6,887,956 6,887,980

395

Table 15.9 Total expected product delivery times (Z4) at different feasibility degrees (β) and weight parameters (α). Weight parameter (α) Feasibility degree (β)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

13,610,103 13,610,162 13,610,213 13,610,237 13,610,279 13,610,308 13,610,342 13,610,381 13,610,422

13,610,103 13,610,163 13,610,213 13,610,237 13,610,279 13,610,308 13,610,342 13,610,383 13,610,422

13,610,103 13,610,163 13,610,213 13,610,238 13,610,279 13,610,309 13,610,342 13,610,385 13,610,423

13,610,106 13,610,163 13,610,215 13,610,238 13,610,279 13,610,311 13,610,342 13,610,385 13,610,424

13,610,106 13,610,165 13,610,215 13,610,238 13,610,280 13,610,311 13,610,345 13,610,385 13,610,424

13,610,106 13,610,166 13,610,216 13,610,239 13,610,281 13,610,311 13,610,345 13,610,386 13,610,425

13,610,107 13,610,166 13,610,216 13,610,239 13,610,283 13,610,312 13,610,346 13,610,386 13,610,425

13,610,108 13,610,167 13,610,216 13,610,239 13,610,283 13,610,313 13,610,346 13,610,388 13,610,428

13,610,108 13,610,168 13,610,218 13,610,241 13,610,285 13,610,314 13,610,348 13,610,388 13,610,429

Table 15.10 Total revenues (Z5) at different feasibility degrees (β) and weight parameters (α). Weight parameter (α) Feasibility degree (β)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

11,179,402 11,179,383 11,179,351 11,179,316 11,179,281 11,179,243 11,179,209 11,179,168 11,179,143

11,179,402 11,179,383 11,179,351 11,179,316 11,179,281 11,179,243 11,179,209 11,179,168 11,179,143

11,179,401 11,179,383 11,179,349 11,179,316 11,179,281 11,179,243 11,179,209 11,179,167 11,179,143

11,179,400 11,179,383 11,179,348 11,179,314 11,179,280 11,179,241 11,179,208 11,179,165 11,179,142

11,179,398 11,179,381 11,179,347 11,179,313 11,179,278 11,179,241 11,179,206 11,179,165 11,179,142

11,179,398 11,179,381 11,179,346 11,179,313 11,179,278 11,179,240 11,179,206 11,179,165 11,179,141

11,179,397 11,179,380 11,179,346 11,179,311 11,179,275 11,179,239 11,179,206 11,179,163 11,179,140

11,179,396 11,179,380 11,179,346 11,179,310 11,179,275 11,179,237 11,179,205 11,179,162 11,179,140

11,179,395 11,179,379 11,179,345 11,179,309 11,179,274 11,179,236 11,179,203 11,179,161 11,179,140

398

Optimization Theory Based on Neutrosophic and Plithogenic Sets

in Table 15.10, and Fig. 15.4E shows the inclining behavior of (Z5) at different feasibility degrees (β) and weight parameters (α), respectively. Furthermore, the effect of the feasibility degree (β) is more influential, as the significant increase in the value of (Z5) rapidly approaches the worst solutions whereas the weight parameter (α) affects the objective (Z5) almost trivially.

15.6.2.2 Sensitivity analyses of intuitionistic fuzzy linguistic preference relations Imprecise importance relations have been represented by an intuitionistic fuzzy preference hierarchy for three different linguistic terms. The membership functions for importance relations Re2 ð2,1Þ, Re1 ð4,3Þ, Re3 ð3, 5Þ, and Re1 ð4, 5Þ have been obtained and shown in Table 15.11 and Fig. 15.4F. With the increase in the feasibility degree (β) and the weight parameter (α), the preference membership function for Re2 ð2, 1Þ also increases and reaches its maximum, that is, 0.71 at β ¼ 0.9 and α ¼ 0.9. Similarly, the preference membership functions for Re1 ð4, 3Þ, Re3 ð3, 5Þ, and Re1 ð4, 5Þ also reveal increasing behavior with the increase in the feasibility degree (β) and the weight parameter (α), and reaches their maximum attainment, that is, 0.64, 0.68, and 0.77 at β ¼ 0.9 and α ¼ 0.9, respectively. Moreover, the nonmembership functions for different linguistic preferences are summarized in Table 15.11 and are shown in Fig. 15.4G. The motive is to minimize the nonmembership functions of each linguistic preference relation. Hence the minimum attainment degrees of nonmembership functions for Re2 ð2, 1Þ, Re1 ð4,3Þ, Re3 ð3, 5Þ, and Re1 ð4,5Þ have been obtained as 0.29, 0.34, 0.26, and 0.21 at β ¼ 0.9 and α ¼ 0.9, respectively. The overall satisfaction degree of linguistic preference relations has been represented by the score function. The maximization of the score function ensures the maximum satisfaction degree for the intended preferences among different objectives and is shown in Fig. 15.4H. In Table 15.11, with the increase in value of β and α, the score function shows the enhancing trend. At β ¼ 0.9 and α ¼ 0.9, it approaches the maximum satisfactory degree, that is, 0.4256, 0.3557, 0.4253, and 0.5637 for Re2 ð2, 1Þ, Re1 ð4,3Þ, Re3 ð3,5Þ, and Re1 ð4, 5Þ, respectively. By tuning the parameters β and α, various sets of score functions for satisfaction level could be obtained effectively. Hence, intuitionistic fuzzy linguistic preference relations would be a good representative of priority structure among objectives according to the interest of decision maker(s). They would also be an effective and promising tool for assigning the preference when large numbers of objectives and goals have been dealt with simultaneously. The assignment of P crisp weight (such as wo ¼ 0:1,0:2, …,1j O o wo ¼ 1) to significant number objectives might be time-consuming and would involve more complexity to search for the best combination of crisp weight among different objectives or goals. Hence it would be tricky to assign the linguistic preferences among different objectives, which reduced the time and exempted from the best combination of crisp weight.

Table 15.11 Achievement degree of intuitionistic fuzzy linguistic preference relations at different feasibility degrees (β) and weight parameters (α). Intuitionistic fuzzy preference relations Feasibility degree

Weight parameter

(β)

(1 2 α)

μe R 2 ð2, 1Þ

μe R 1 ð4, 3Þ

μe R 3 ð3, 5Þ

μe R 1 ð4, 5Þ

νe R 2 ð2, 1Þ

νe R 1 ð4, 3Þ

νe R 3 ð3, 5Þ

νe R 1 ð4, 5Þ

Se R 2 ð2, 1Þ

Se R 1 ð4, 3Þ

Se R 3 ð3, 5Þ

Se R 1 ð4, 5Þ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.63 0.63 0.64 0.64 0.64 0.65 0.65 0.67 0.71

0.56 0.56 0.56 0.56 0.56 0.58 0.58 0.61 0.64

0.63 0.63 0.63 0.63 0.63 0.65 0.65 0.66 0.68

0.71 0.71 0.72 0.72 0.72 0.74 0.74 0.74 0.77

0.37 0.37 0.37 0.35 0.35 0.32 0.32 0.32 0.29

0.37 0.37 0.37 0.36 0.36 0.35 0.35 0.35 0.34

0.33 0.33 0.33 0.32 0.32 0.32 0.29 0.29 0.26

0.27 0.27 0.27 0.27 0.27 0.23 0.23 0.21 0.21

0.2634 0.2637 0.2721 0.2932 0.2981 0.3381 0.3393 0.3547 0.4256

0.1924 0.1938 0.1957 0.2122 0.2143 0.2311 0.2341 0.2934 0.3557

0.3091 0.3037 0.3052 0.3127 0.3149 0.3351 0.3691 0.3712 0.4253

0.4453 0.4467 0.4531 0.4567 0.4579 0.5133 0.5148 0.5321 0.5637

Membership functions

Score function achieved for intuitionistic fuzzy preference relations

Nonmembership functions

400

15.7

Optimization Theory Based on Neutrosophic and Plithogenic Sets

Conclusions

In this study, an effective modeling and optimization framework for the CLSC design has been formulated as a mixed-integer neutrosophic fuzzy programming problem under uncertainty. The proposed CLSC designed model comprises multiproduct, multiechelon, and multiobjective scenarios for the optimum allocation of new and end-of-use products. In the forward chain, five functional echelons have been designed, whereas the reverse chain consists of six potential echelons to deal with end-of-use and end-of-life products. The testing center has been depicted in the CLSC model, which ensures the promising useful life of the product. Multiple-conflicting objectives with a well-defined set of constraints reveal typical complexity under a fuzzy environment. To deal with fuzzy parameters and constraints, a fuzzy robust ranking function technique depending on a feasibility degree has been suggested. Fuzzy inequality constraints have been converted into their crisp forms by using the ranking function, whereas fuzzy equality constraints have been transformed into two equivalent auxiliary crisp inequalities. Then the obtained fresh model has been solved by using a modified NFPA which consists of independent indeterminacy thoughts in decision-making processes. A novel linguistic importance scheme named intuitionistic fuzzy preference relations among different objectives has been investigated. With the aid of the linear preference membership and nonmembership function, the marginal achievement of each linguistic preference has been attained. The overall satisfaction level has been represented by the convex combination of membership functions of each objective and score function of intuitionistic fuzzy preference relations. By tuning the feasibility degree and weight parameter, a different set of optimal solution results has been generated. A sensitivity analysis of the obtained results has been performed. Therefore, the presented CLSC modeling study under uncertainty may be helpful for practitioners and decision makers who are actively dedicated in the decision-making process of procurement, production, distribution, transportation, and management of end-of-use and end-of-life products in the CLSC network. The propounded CLSC study has some limitations that can be addressed in future research. The CLSC network has been designed for a single period, but modeling with multiple periods is much needed in real-life scenarios. Incorporation of the triple bottom lines concept, which means sustainable development of the CLSC model comprising economic policies, environmental issues, and social concerns, would be a remarkable extension of the proposed model. Uncertainty among parameters due to randomness or other uncertain forms would be a significant enhancement of the discussed CLSC model. Various metaheuristic approaches may be applied to solve the proposed model as a future research scope.

Acknowledgments All authors are very thankful to the potential reviewers and editors for improving the readability and clarity of the manuscript. The first author is grateful to Mr. Abdul Nasir Khan and Ms. Shazia Farhin for providing in-depth suggestions and continuous support.

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Optimization-based neutrosophic set in computer-aided diagnosis

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Amira S. Ashoura, Yanhui Guob a Department of Electronics and Electrical Communications Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt, bDepartment of Computer Science, University of Illinois at Springfield, Springfield, IL, United States

16.1

Introduction

Recently, advancement in the biomedical imaging domain and its association with the clinical applications have become a must with the development in the medical imaging modalities. The foremost aim is to improve computer-based techniques for designing competent computer-aided diagnosis (CAD) systems using medical images attained through numerous imaging modalities. Computer-aided detection (CADe) and diagnosis (CADx) are evolving technologies to support physicians in interpreting the medical images from different modalities. Such systems are implemented to reduce the false negative diagnostic decision of the interpreted medical images by the physicians for prompt detection of life-threatening diseases. In clinical practice, several automated diagnostic systems are ratified to aid physicians in detecting abnormalities for further diagnosis and decision-making with minimized error. CADe assists in screening the different organs by ensuring cancer and abnormalities are not overlooked, whereas CADx is used in decision-making [1]. Nevertheless, detecting small aneurysms, vessels, and lesions is challenging owing to the overlapping with neighboring normal anatomic structures and background due to their fuzziness and locations. Furthermore, subtle interval changes detection in successive abnormal scans is a challenging task due to the variants in the patient’s conditions, and the changes in the image quality, which increases the uncertainty and unpredictability in routine clinical work. The development of CAD systems for accurate and time-efficient diagnosis depends mainly on medical image analysis and processing. The CAD systems include several techniques, such as image enhancement, segmentation, feature extraction, and classification [2]. For decision-making, the classification approaches are essential, and depend predominantly on the segmentation results. However, accurate segmentation remains an open and active problem due to the inherent limitations in medical images, such as low contrast, vague edges, illumination problems, low signal-to-noise ratio, existence of artifacts, and irregular shape of abnormalities. However, the performance of traditional CADe and CADx systems is inadequate. Several requirements including high specificity, sensitivity, and accuracy are essential for instigating effective automated computer-aided diagnosis systems. Recently, to achieve these requirements in medical applications, and Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00016-0 © 2020 Elsevier Inc. All rights reserved.

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handle the incompleteness, vagueness, and fuzziness in the medical images, fuzzy approaches, including para-consistent logic theory, intuitionistic fuzzy set theory, and fuzzy set theory [3, 4], have been designed to cope with such characteristics in medical information. Nevertheless, the fuzzy-based approaches can only manage one vague problem instead of solving the limitations in the medical information in one framework. Since diagnosis is considered a challenging task owing to the existence of fuzziness in the medical images, integrating neutrosophic methods in the CAD stages verifies its dominance to resolve such fuzziness and other uncertainty limitations in medical data in one framework. Generally, neutrosophic theory was presented by Smarandache, who stated neutrosophy as a new philosophy branch in 2002 [5]. Neutrosophic theory can be involved efficiently to improve the performance of these different approaches in CAD systems. Neutrosophic set (NS) is a powerful neutrosophy method, which concerns the nature, scope, and origin of neutralities. Unlike fuzzy logic, which has true and false states, where the truth degree has values between 0 and 1, neutrosophic logic presents an indeterminacy percentage that represents the uncertainty. Accordingly, neutrosophy is more than just a form of logic as it is considered the underpinning of: (i) Neutrosophic logic, which is a general framework for many logics unification, such as paraconsistent logic, fuzzy logic, and intuitionistic logic [3, 4, 6, 7]; (ii) Neutrosophic probability [8], which gives a broad view of the imprecise and classical probabilities, where their rules are tuned in the form of neutrosophic probability rules [9]; (iii) Neutrosophic statistics [10], where the data has some indeterminacy unlike the case of classical statistics, where the data is known and designed by crisp numbers afterward; (iv) Neutrosophic sets [3, 4], which have the potential to be a broad outline for uncertainty analysis in data sets as they study the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra [11].

The preceding studies proved that neutrosophy has an important role in denoising, segmentation, clustering, and fusion of images in several applications. The NS was defined by Smarandache [3, 4] as a new extension of intuitionistic fuzzy sets to measure the indeterminacy as a new independent subset for to solving problems that fuzzy logic failed to resolve [12, 13]. Any event in NS has specific independent degrees of truth, indeterminacy, and falsity. Consequently, contradictions, fuzziness, indeterminacy, and inconsistency are solved using NS, where the performance of a neutrosophic system is similar to the medical reality compared to the fuzzy counterpart system. Since medical information is imprecise, vague, and imperfect, the diagnostic processes have low intra- and inter-reliability. Recently, NS has been adapted to solve several applications in different domains, mostly in developing CAD systems by resolving their limitations, including uncertainty and fuzziness. It provides an innovative scheme to support the design of innovative CAD systems. Several attempts have been conducted on NS in the medical applications; a single valued NS (SVNS) was developed for real-time applications [14]. In addition, for simplified NS (SNSs), a cosine similarity measures was developed for medical diagnosis by Ye [15]. Neutrosophy-based CAD systems have significant roles for breast cancer, skin cancer, lung cancer, and the detection and classification of abnormalities. Such NS-based

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systems employed the NS to deduce the intrinsic uncertainty, vagueness, and ambiguity through the different CAD stages, namely enhancement, denoising, segmentation, and classification. The main role of using the NS is to remove the uncertainty and fuzziness from medical images to guarantee efficient CAD system. Along with the NS, optimization algorithms are considered an intelligent well-established paradigm for solving computational problems in several domains. Such algorithms include simulated annealing (SA), genetic algorithm (GA), bee algorithm (BA), cuckoo search (CS), firefly algorithm (FA), particle swarm optimization (PSO), moth flame optimization (MFO), and grey wolf optimizer (GWO) [16]. Their effectiveness depends on the balance between diversification (exploration) and intensification (exploitation). Due to their powerful effect on the final optimization problem’s solution, such optimization algorithms have a great impact on the performance of the CAD in its different stages/processes, where they are involved to find the optimal solution from all probable ones based on objective function. Different optimization algorithms were applied in different CAD applications to optimize parameters in the image processing method used; for example, a coactive neuro-fuzzy inference system-based GA was implemented for heart disease prediction [17], multilevel thresholding was optimized using the ant colony algorithm (ACO) and expectation maximization (EM) algorithms [18], magnetic resonance imaging (MRI) image segmentation was performed using the Artificial Fish Swarm optimization-based fuzzy approach [19], a nested PSObased segmentation was applied to determine the optimal number of clusters [20], glowworm swarm optimization-based segmentation was used to determine the initial seed [21], hybrid firefly search algorithm-based was used for MRI image segmentation [22], and ACO-based segmentation of computed tomography (CT) images was carried out [23]. Due to the effectiveness of both the NS and optimization algorithms in the CAD applications, several researchers are interested in integrating both these powerful methods to gain their advantages. Accordingly, this chapter categorizes the incorporation between NS and the different optimization algorithms into a main framework called OptNS which uses any of the optimization algorithms to optimize the parameters in the NS parameters (in the NS subsets) themselves during medical image processing, which was first performed by Ashour et al. [24]. In addition, in the CAD systems, another use of the optimization can be conducted to optimize the image processing process, while using the NS without optimization to generate the neutrosophic image for further processing based on the target of the proposed system (i.e., for further image enhancement, segmentation, or classification). Thus, during the targeted process, an optimization or hybrid of the optimization algorithms is used to find the required optimal value of a specific parameter or set of parameters. This chapter reports the NS-based scheme and the transformation of the medical image to the NS domain with defining the three main NS subsets to handle the uncertainty in medical images. In addition, the concept of optimization algorithms is introduced. Afterwards, prior studies that involved the neutrosophic sets in CAD systems for medical image denoising, clustering, or segmentation are investigated and described without and with the use of optimization algorithms. The integrated NS

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and optimization algorithms-based diagnosis are also emphasized. Finally, the challenges with future suggestions are introduced, to enable new insights. The structure of the following sections is as follows: Section 16.2 includes the NS concept, neutrosophic image generation, and optimization algorithms in medical image analysis as well as the integrated NS procedure with different optimization algorithms in diagnosis. Section 16.3 reports the preceding studies on NS in CAD systems without and with the use of optimization algorithms. Section 16.4 sets out challenges and recommendations, and Section 16.5 provides the conclusion.

16.2

Neutrosophic set-based medical image analysis

Smarandache proposed neutrosophic theory with settling the concept of neutrosophic probability and set in 1998 [25]. The motivation to introduce neutrosophic theory is to overcome the limitations of using a single-valued membership in a fuzzy set which includes uncertainty in the attributes. One of the powerful approaches is the neutrosophic set framework, which solved the limitations of different approaches, such as the fuzzy set, intuitionistic fuzzy set [25a], classic set, paraconsistent set, interval valued fuzzy set [26], and interval valued intuitionistic fuzzy set [27]. Accordingly, neutrosophic theory including NS was introduced to include three independent membership (subset) values, namely truth “T,” indeterminacy “I,” and falsity “F” for any attribute. Moreover, these subsets are symmetric, where I represents a symmetry axis between T and F, which are opposite to each other. Accordingly, there are several types of the neutrosophic set based on the definition and range of these three subsets, such as neutrosophic faillibilist sets, neutrosophic paraconsistent sets, neutrosophic intuitionistic sets, neutrosophic pseudo-paradoxist sets, neutrosophic paradoxist sets, neutrosophic nihilist sets, neutrosophic tautological sets, neutrosophic dialetheist sets, neutrosophic cubic sets, neutrosophic trivialist sets, neutrosophic soft rough sets, neutrosophic tripolar sets and neutrosophic multipolar sets, and single valued neutrosophic sets.

16.2.1 Neutrosophic set Neutrosophic set (NS) and its variants provide frameworks to cope with indeterminacy in image processing, medical diagnoses, computational intelligence, and decision-making. In the neutrosophy theory, each entity is considered with its association to its opposite, < Anti-W > and < Non-W >, where the neutralities < Neut-W > are denoted as < Non-W>, which is neither < W > nor < Anti-W >. Hence, in medical image processing, to determine the abnormality (region of interest or ROI), the ROI, boundaries, and background are represented as < W >, < Neut-W >, and < Anti-W >, respectively.

16.2.1.1 Neutrosophic image In the neutrosophic set, three independent neutrosophic subsets (called memberships or components) are defined to estimate the truth degree, indeterminacy degree, and falsity degree, which are denoted as T, I, and F, respectively.

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Accordingly, the NA can be represented as < T, I, F >. In the medical image processing example, the neutrosophic subsets, T, I, and F, represent < W >, < Neut-W >, and < Anti-W >, where the I and F include the region information containing the ROI and its opposite “background,” while I contains the boundary information. The indeterminacy degree I refers to neither true nor false states in < W >, which handles higher degrees of uncertainty. Thus, in medical image processing, the original image is transformed to a neutrosophic image by representing T, I, and F subsets, where any pixel A(x, y) in the neutrosophic image is expressed as follows [24]: Aðx, yÞ ¼ ðT ðx, yÞ, I ðx, yÞ, Fðx, yÞÞ,

(16.1)

where in a neutrosophic image, these pixels are defined as A{t, i, f}, t 2 T, i 2 T, and f 2 F, and t, i and f are real numbers without restriction on their sum. Subsequently, every neutrosophic pixel can be represented as follows: Aim ¼ fTim , Iim , Fim g,

(16.2)

where Tim, Iim, and Fim represent the pixels of ROI (true), boundaries (indeterminate), and background (false), respectively. Hence, the T subset and its opposite F can be expressed as follows [28]: Tim ðx, yÞ ¼

Cðx, yÞ  Cmin Cmax  Cmin

Fim ðx, yÞ ¼ 1  Tim ðx, yÞ,

(16.3) (16.4)

where Cðx, yÞ is the local mean of intensity value C(x, y) of the pixel (x, y). The I subset can be expressed in terms of τ(x, y), which is the absolute value of the difference between C(x, y) and Cðx, yÞ as follows: Iim ðx, yÞ ¼

τðx, yÞ  τmin : τmax  τmin

(16.5)

16.2.1.2 Entropy of neutrosophic subsets To quantify the uncertainty amount which expressed by the NS, the fuzziness is measured in terms of the entropy. The entropy is calculated to estimate the distribution of gray levels in an image. It has minimum or maximum values in the case of nonuniform intensity, or equal-uniform distribution of the intensities, respectively. For a neutrosophic image, the entropy equals the summation of entropies of T, I, and F, which is calculated as follows [29]: Eim ¼ ETim + EIim + EFim ,

(16.6)

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ETim ¼ 

max fTim g X

pTim ðiÞ ln ðpTim ðiÞÞ,

(16.7)

pFim ðiÞ ln ðpFim ðiÞÞ,

(16.8)

i¼min fTim g

EFim ¼ 

max fFim g X i¼min fFim g

EIim ¼ 

max fIim g X

pIim ðiÞ ln ðpIim ðiÞÞ,

(16.9)

i¼min fIim g

where ET, EI and EF are the entropies of sets T, I, and F, respectively. In addition, HT(i), HI(i), and HF(i) are the probabilities in T, I, and F, respectively, of the elements whose values are equal to i.

16.2.2 Optimization algorithms in medical image analysis The optimization of CAD parameters can be defined as an efficient procedure to find the optimal values of a specific factor or parameter in the problem.

16.2.2.1 Concept of optimization Optimization can be defined as the process of modifying the characteristics of a system using mathematical processes to find the maximum or minimum output. The input to the optimizer entails the variables to be optimized and the objective function, cost function, or fitness function along with the desired fitness or cost as an output. A typical optimization problem consists of an objective function or multi-objective function followed by the associated constraints to maximize or minimize the objectives. Different procedures can be followed to solve any problem of optimization [30]. The most common methods are called meta-heuristic optimization techniques, which also include optimization techniques inspired by natural practices, where they start from initial variables set then evolved to find the global maximum or minimum of objective function. Typically, meta-heuristics is considered a stochastic optimization process which applies a random search to determine the problem’s optimal solutions under specific constraints. Such optimization procedures include PSO, GA, bacteria optimization (BO), ant colony optimization (ACO), bee colony optimization (BCO), simulated annealing (SA), and hill-climbing (HC). The primary goal of these optimization methods is to grasp the trade-off between diversification (exploration) and intensification (exploitation), which are defined as discovering the search space by producing different solutions on a global scale, and directing the search to a good solution at a local region, respectively. Generally, the associated constraints with the variable are used as the solution’s boundary. For a minimization optimization problem, assume v variables with O  1 objectives, thus, the following definitions are used [31]: min yðsÞ for single objective function

(16.10)

Optimization-based neutrosophic set in computer-aided diagnosis

411

min fy1 ðsÞ, y2 ðsÞ, y3 ðsÞ, …, yO ðsÞg for multi-objective function

(16.11)

Subject to ei  0, i ¼ 1,2, …, Oi , qj  0, j ¼ 1,2, …, Oj

(16.12)

where s is the search (decision) space, and y(s) is the objective space. In addition, ei and qj are the associated constraints, respectively. Based on the equations derivatives and variables properties in the optimization problems framework, different optimization problems can be given for single/multi-objective problems. The optimization algorithms include an enormous number of approaches; accordingly we follow two of the most dominant algorithms, the GA and PSO, which are commonly involved in different CAD systems.

16.2.2.2 Genetic algorithm The GA sets randomly a population of chromosomes for further identification of the chromosome denoting the best solution to the problem. Then, more chances to reproduce to these chromosomes (better solutions) are compared to the poorer solutions. Accordingly, the superiority of a solution is stated based on the current population. At each generation, three operators, namely reproduction, crossover, and mutation, are used with the entire population using the following procedure in Algorithm 16.1.

16.2.2.3 Particle swarm optimization PSO behavior is inspired from the strategy that bird swarms follow while searching for optimal food sources. During the search process, the direction of a bird’s movement is inclined by its current movement toward finding the best source of food, where this

Algorithm 16.1 Genetic algorithm Begin Assume random population of chromosomes (solutions) and determine the fitness function Evaluate the fitness function in the population for each solution Produce a new population by performing the following stages till ending the new population Select (Selection) two parent chromosomes according to their fitness Crossover (Crossover) the parents to produce a new offspring Mutate (Mutation) new offspring at each position Determine a new population Use new created population Assess the stopping condition Stopover to determine the finest solution Repeat preceding steps till the best solution is reached End

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Algorithm 16.2 Particle swarm optimization Begin Set initial position and velocity for each particle Define stopping criteria Evaluate fitness of the particles based on the objective function Modify individual- and global-best positions and fitness values Modify position and velocity of the particles Satisfy the stopping criteria End

movement is inspired by the bird’s knowledge, swarm knowledge, and inertia. This procedure is simulated in the PSO algorithm by representing personal-/global-best position, and inertia for each particle which represents a solution. Each particle has a specific position, velocity, and objective, and tries to reserve the global-best value based on achieving the best objective value, as well as the best global position. Algorithm 16.2 represents the PSO’s repeated steps until the stopping criterion is reached.

16.2.3 Framework for integrating NS with optimization algorithms (OptNS) In the optimization of the medical image processing phases in the NS-based CAD or the optimization of the NS itself using any optimization technique in the (OptNS), the parameters/variables to be optimized are determined initially, then their ranges are determined. In addition, the fitness function to be defined based on the problem under concern and the optimizer setting parameters are determined. Then, the iterative process of the searching process during the optimization is performed until the stopping criterion is reached or the best fitness is achieved. At this point, the values of the parameters/variables to be optimized are considered the best solution and used as the optimal values. Accordingly, in medical image processes in CAD systems, where the NS is used as a prevailing tool in numerous medical applications, two frameworks are proposed in the present chapter to integrate the optimization algorithm with the NS-based CAD, which can be formulated in one generic algorithm (Algorithm 16.3) as follows. The preceding proposed framework of the optimization-based NS-CAD system (OptNS) (Algorithm 16.3) is then used based on the optimization problem concerned.

16.3

OptNS-based CAD medical image processing applications

Accurate CAD system design attracts researchers to develop new procedures for managing huge amounts of medical information from different modalities. Currently, a novel trend to apply NS in medical image processing phases for precise diagnoses

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Algorithm 16.3 Proposed framework of optimization-based NS-CAD system (OptNS) Read original medical image Convert to gray-scale image Transform NS domain by using the following procedure: Define the NS filters Map the medical image to the NS domain by computing the three subsets (T, I, and F) Measure the entropy of I Obtain the neutrosophic image T Start 1: Training phase using optimizer to find the optimal value(s) Define the initial values (default) of the parameters to be optimized in the NS operators or filters State the fitness function Transform the medical image to the NS domain by computing T, I, and F Adapt the medical image using I to generate the updated T Repeat the previous three steps with running the optimizer using the pre-defined fitness function Save obtained optimal parameter values(s) Stop 1 Start 2: Testing phase the optimal value(s) without further optimization Transform the testing medical image to the NS domain Apply the required medical image processing phase using the optimized parameter value(s) of the used process or of the NS itself based on the optimization problem Stop 2 End

has become essential due to the important role of NS compared to the fuzzy logic approaches, where the NS includes an indeterminate value that addresses the uncertainties as an independent subset. Accordingly, the NS becomes a powerful tool in different medical image processing stages. In addition, due to the imperative role of the optimization algorithms in different applications, this chapter highlights the importance of combining the advantages of the optimization with those of the NS to design integrated NS and optimization techniques in the CAD systems’ design.

16.3.1 CAD using neutrosophic set without optimizing NS Vagueness, fuzziness, indeterminacy, incompleteness, imperfection, and imprecision are common characteristics of any medical data. This requires professional CAD systems to handle such characteristics and have the ability to resolve these limitations for superior interpretation and diagnosis of medical data. On the other side, neutrosophic theory, mainly the NS, proved its superiority to manage fuzziness and indeterminacy

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better than its fuzzy counterpart and deal with medical information in the several phases of the CADe and CDAx systems [32]. For image denoising, numerous denoising approaches based on neutrosophic theory have been implemented to remove different types of noise, such as the Rician, and Speckle noise. The NS was applied with new γ-median-filtering operators to transform the image into the NS domain for noise removal [33]. Mohan et al. [34] implemented a neutrosophic denoising method using the NS to remove the Rician noise from the MRI brain images. The results depicted that this proposed method produces superior denoised image performance compared to the median filter and the nonlocal mean approach in terms of the peak signal-to-noise ratio as well as the visual perception. Mohan et al. applied the ω-wiener filtering on the T and F operations of the NS for denoising and decreasing the set indeterminacy of the MRI images [35]. Afterward, Mohan et al. [36] transformed the nonlocal means filtered MRI images into the NS space, where entropy has been measured to determine the indeterminacy. The results revealed that the nonlocal neutrosophic set (NLNS) Wiener filter provided superior denoising results compared to the traditional Wiener filter, nonlocal means filter total variation minimization, and anisotropic diffusion filter. Furthermore, in ultrasound imaging, the dominant noise source is the Speckle, which is a type of multiplicative noise; thus, Koundal et al. used the neutrosophic fuzzy set (NFS) theory to remove such noise type [37]. Typically, the NFS handles the uncertainty in terms of T, I, and F membership sets. The results tested on Speckle synthetic images established the superiority of the proposed method with a signal-to-noise ratio (SNR) gain of 1.65 dB compared to 1.55 dB using the Lee filter and 19.2 dB using the Kuan filter. Due to the efficient performance and the ability to handle the indeterminacy of the neutrosophic-based techniques, several researchers included these techniques to solve the segmentation and clustering problems in several domains. In medical applications, Guo et al. implemented an iterative neutrosophic lung segmentation technique based on morphological operations and expectation-maximization analysis for lungs/ribs segmentation [38]. Koundal et al. [39] proposed a spatial neutrosophic distance regularizer level set method for a neutrosophic-based segmentation method for nodules in thyroid ultrasound images. Guo et al. [40] integrated the NS and Shearlet transform followed by K-means clustering to develop an automated CAD for glomerular basement membrane (GBM) image segmentation. Devi et al. proposed a neutrosophic graph-cut system with an indeterminacy filter for cervical cancer detection [41]. For dermoscopic images segmentation, a neutrosophic C-means (NCM) clustering technique was realized as a prevailing skin lesion detection approach [42]. Additionally, Ashour et al. [29] implemented a histogram-based clustering estimation neutrosophic C-means procedure for dermoscopy image segmentation. Further, for advanced CAD systems, several researchers implemented neutrosophic-based classification systems; for example, Kraipeerapun et al. [43] presented a medical binary classification using ensemble neural networks based on bagging method and interval NS (INS), while Ju [44] proposed an integrated NS and reformulated support vector machine (SVM) classifier. The results proved the efficiency of the proposed NS, and Gaber et al. [45] designed a CAD system for breast cancer classification, where a NS-based fast-FCM (F-FCM) was applied for

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segmentation, then a support vector machine classifier with diverse kernel functions was used for final classification and decision-making. However, due to the effectiveness of both the NS for removing uncertainty/ fussiness, and the optimization algorithms for optimal values determination during different medical image processing, several researchers integrated both concepts for developing efficient CAD systems and other applications. For breast cancer diagnosis using ultrasound images, Zhang et al. [12, 13] implemented a robust neutrosophybased automated CAD for breast ultrasound images segmentation. Shan et al. [46] then applied a new neutrosophic-based clustering scheme, called neutrosophic l-means (NLM), for detecting lesion boundaries accurately. Recently, for integrated NS-optimization-based CAD development, Sayed and Hassanien [47] applied the NS and moth-flame optimization (MFO)-based feature selection for the detection and classification of mitosis in histopathology images. First a Gaussian filter was used, followed by mapping the original images to the NS domain for image enhancement. Afterward, morphological operations were applied to the T component. Then, a classification phase was conducted using the extracted texture, shape, energy, and statistical features, where the MFO algorithm was used to select the optimal significant mitosis cells’ features. These optimally selected features were fed into a classification and regression tree (CART) classifier. The results established the efficiency of integrating the NS for image enhancement with the MFO optimization for optimal feature selection which maximized the classification performance with 92.99% accuracy.

16.3.2 CAD using neutrosophic set with optimization The most recent framework for the integration between the NS and the optimization has been initiated and stated by Ashour et al. [24], who optimized NS (OptNS) by using the image processing method. Ashour et al. were the first to state the concept of optimizing the parameters of the NS itself followed by any image processing process in the CADe system. Their concept can be generalized in different image processing applications. For skin lesion detection in a CAD, Ashour et al. [24] adopted a new CADe system, called optimized neutrosophic k-means (ONKM), where the genetic algorithm (GA) optimizes the NS operation while decreasing the vagueness in the dermoscopy images. Afterward, k-means clustering was implemented for lesion regions segmentation. Accordingly, in the training phase to optimize the α-mean operator of NS, an initial value for α was assumed, then the GA was applied to find an optimal threshold value in α-mean operation, where during the optimization process, the Jaccard index was considered as the fitness function. The GA determined the optimal α-mean operation of value αoptimal ¼ 0.0014 for further use in the NS directly in the test phase, while its default value was 0.01. This optimal value was used directly in the NS during the test phase to generate the neutrosophic image with the three subsets, namely true, indeterminate, and false. This novel ONKM CADe proved its efficiency in terms of different evaluation metrics, namely the accuracy, sensitivity, specificity, and Dice coefficient with achieved superior average accuracy 99.29 1.61%

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compared to using k-means with neither NS nor optimization as well as using the default NS default α-mean operator without optimization. Moreover, Ashour and Guo [47a] proposed a novel OptNS system called optimized indeterminacy filter (OIF) for adaptive white Gaussian noise (AWGN) reduction in dermoscopy images, where such noise modeled the noise caused by the dermoscopic digital camera. In this method, the kernel size of the indeterminacy filter in the NS was optimized to guarantee the effective transformation of the dermoscopy image to the NS domain as well as efficient denoising. The results established the superiority of the proposed OIF filter with proposed optimal kernel size of ho ¼ 7 compared to median, Wiener, and indeterminacy filter with kernel size ho ¼ 5, which is the default value without optimization. The achieved results were superior peak signal-to-noise ratio and maximum signal-to-noise ratio of 31.47 and 27.75, respectively, as well as superior root of MSE (RMSE), and minimum mean squared error (MSE) of 7.18 and 57.86, respectively.

16.4

Discussion and future perceptions in OptNS-based CAD systems

Different medical modalities advancements increase medical information/images resources, which are inconsistent, imprecise, and incomplete. However, the acquisition system and the physical phenomena are considered the main causes of noise during the imaging process. This leads to imperfect information as well as imprecision and uncertainty in the acquired image, which have a deleterious influence on medical image processing. In addition, the light scattering on the physical surfaces during the imaging process causes inherent ambiguities in an image’s content. Consequently, in medical image processing, the uncertainty principle is believed to be one of the major issues that should be considered during the medical image processing phases, such as enhancement, segmentation, clustering, classification, and fusion. For encoding uncertainty and imprecision, soft computing is considered an inspirational domain. Integrating new neutrosophic theories in image processing proves their superiority. By generalizing both the classical and fuzzy counterparts, neutrosophy is considered the foundation for new mathematical theories, which is used for efficient image and video processing applications. The preceding reported literatures demonstrated the impact of the different neutrosophic techniques to support the different image processing phases in several applications. The contribution of the present survey is to explain the concept of neutrosophic theory, and the neutrosophic image formation along with the role of optimization algorithms to improve the overall CAD systems. In addition, a detailed explanation of the neutrosophic techniques provides a way to establish a framework of the neutrosophic theory in image processing applications. Furthermore, some highlights are directed to link the optimization methods and the neutrosophic techniques for outstanding image processing tasks. The NS and neutrosophic logic have attracted researchers’ attention to solve several real-world decision-making problems that include indeterminacy, inconsistency,

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uncertainty, vagueness, and incompleteness. Numerous denoising methods based on NS theory have been proposed for noise removal by decreasing the image indeterminacy. However, it is recommended to solve the blurring effect that appears in some cases while using the NS due to its iterative process. The high performance with handling the indeterminacy can be attained using neutrosophic-based methods to deal with image segmentation and classification problems. Several instances of neutrosophic-based image segmentation established their effectiveness with nonmedical images. Accordingly, it is recommended to apply these methods with medical images while using optimizing algorithms. Such methods include proposed watershed segmentation in the neutrosophic logic domain with noisy as well as non-uniform images [12, 13], the implemented region growing segmentation in neutrosophic logic [48], the level set and NS-based segmentation technique [49], and neutrosophic edge detection using a new directional α-mean operation [50]. Moreover, for colored image segmentation, several studies proposed other neutrosophic-based techniques, such as the NS and wavelet transformation-based color texture image segmentation method using γ-K-means clustering [51], and α-mean and β-enhancement operations in the NS method to reduce the indeterminacy for image segmentation. For uncertain data clustering, Guo and Şeng€ ur developed several approaches, including an integrated NS and clustering algorithm [38], a neutrosophic c-means clustering (NCM) algorithm [52], and a neutrosophic evidential c-means method with Dezert–Smarandache theory for natural image clustering and segmentation [53]. Moreover, in terms of optimization with NS, an integrated NS and optimization linear programming-based method for unsupervised images by Amin [54] can be used in medical applications. It is recommended to compare different optimization methods to optimize the NS operators during the neutrosophic image generation as well as the different stages of the CAD system including denoising, clustering, segmentation, and classification. In addition, the framework of using the optimization algorithm to optimize the NS itself can be applied to medical image restoration, registration, fusion, and compression. An OptNS-based medical recommender system can be implemented for decision-making and to support the CAD, where the diagnostic systems are founded on the association between diseases and patients. Consequently, studies based on distance measure, similarity measures, and correlation coefficients can be considered to solve medical diagnostic problems. Additionally, the same OptNs framework can be applied with other neutrosophic theory models, including the similarity score, neutrosophic faillibilist sets, neutrosophic intuitionistic sets, neutrosophic tautological sets, neutrosophic dialetheist sets, neutrosophic cubic sets, neutrosophic trivialist sets, neutrosophic tripolar sets and neutrosophic multipolar sets, neutrosophic soft rough sets, single valued neutrosophic sets, and similarity score. In addition, cross entropy measures can be used with the NS instead of using the entropy while solving decision-making problems having medical information with inconsistent, incomplete, and indeterminate characteristics. Along with the preceding points, for the CAD system, two imperative principles should be considered accompanied by applying the OptNS framework, where the existing CAD systems produce more false results than true, and do not label/detect all findings. In addition, despite the commonality between the CADe and CADx systems,

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there are some differences—where the output of CADe is the location of abnormality, the CADx system outputs the probability of such an abnormality being benign or malignant, for example. Accordingly, the OptNS used should consider such differences and evaluate them using different metrics with different meta-heuristics including bio-inspired swarm algorithms which have a well-established paradigm that can be tried and integrated to support the OptNS framework. The proposed OptNS framework can be used in several applications based on system requirements. Finally, the characteristics of the NS and its ability to handle uncertainties can be considered a milestone to use the NS in the CAD decision paradigm for medical diagnosis and to handle information fusion combined from several sensors.

16.5

Conclusion

In the medical discipline, prognosis and diagnosis are two of the most challenging tasks owing to the limited subjectivity of the physicians and the existence of uncertainty, fuzziness, and incompleteness in medical images. While perceiving the diseases’ severity, subjective diagnosis may lead to a wrong diagnosis due to the vagueness in the medical information. Accordingly, implementing an intuitive diagnosis from the medical images from different modalities requires different image processing approaches, which were explored in terms of neutrosophic theory for interpreting the inherent ambiguity, vagueness and uncertainty. Typically, the NS is a prevailing generalized framework of classic-, fuzzy, interval valued fuzzy, and intuitionistic fuzzy set to reduce uncertainty, fuzziness, and ambiguity. This inspired researchers to apply the NS to the medical image processing stages of the CAD systems. The NS entails three independent neutrosophic components, true, indeterminate, and false degrees, which are used mainly to reduce the indeterminacy in an image. Recently, the NS and its varieties, such as fuzzy NS, single valued NS, interval valued NS, simplified NS, rough NS, and neutrosophic hesitant fuzzy sets, have been used in several applications including medical diagnosis and CAD design. According to [24], a new framework has been founded to optimize the NS parameters using optimization algorithms, resulting in an accurate CADe which is superior to the traditional NS without optimization. Subsequently, this chapter highlighted the superiority of using the NS, which was introduced by [25], approaches for image processing stages, including denoising based on neutrosophic filters, thresholding, and segmentation which was originated by [33]. Furthermore, this chapter spotted the prominence of the integrated NS with the optimization algorithms in several medical diagnosis applications to support scholars with the applications of NS in this comprehensive review. Finally, extensive study including future directions was introduced.

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Index Note: Page numbers followed by f indicate figures, t indicate tables, and b indicate boxes.

A Acceptance sampling plan, 45–46 Adaptive eLearning, 65–66 adaptive features, 72 components, 69–70, 69f intelligent features, 72–74, 73f model scenarios, 70–71 Adaptive white Gaussian noise (AWGN), 416 Algorithms intelligent online learning objects advisor, 75–76b intelligent study plan advisor, 78b neutrosophic weighted distance, 120, 120b recommender system, 175–177 single-valued neutrosophic sets (SVNSs) K-means clustering for, 121, 122b k-nearest neighbors (k-NN) for, 122 Amazon, 169 Amplified pressure sensor, 56–57 Analytic hierarchy process (AHP), 1–3 Another Tool For Language Recognition (ANTLR), 36–38 Appurtenance degree function, 5 Artificial intelligence (AI), 63 ATLAS learning model, 190–192, 193f engager, 182 navigator, 182 problem solver, 182 Attribute sampling plans, 46 B Bacteria classification problem, 316, 329t Best-worst method (BWM), 1 applications, 3–4 consistency ratio, 5 pairwise comparisons, 3 steps, 3–4, 4f Bipolar fuzzy set, 293

Bipolar neutrosophic optimization nonlinear multiobjective optimization problem, 299, 311t, 312f in riser design, 309–310 Bipolar neutrosophic set, 293–295 Blended learning, 64 Brain Works learning style, 183, 190–192, 194f Business rules, 24–25 implementation of, 41b in JSON format, 35–36, 36b parse tree, 35–36, 37f C Caching, 33, 36 Centralized rules engine, 38 Classification-based recommender systems, 176 Closed loop supply chain (CLSC) network, 343–344 computational study, 378–399, 383–387t constraints capacity of echelons, 357–358 demand of new and refurbished products, 359 maximum inventory at distribution center, 359 production requirement, 358 testing capacity at testing facility centers, 359 description, 349–362 design, 349, 350f forward chain, 349 literature review, 345–348 model formulation under uncertainty, 360–362 modified neutrosophic fuzzy programming with intuitionistic fuzzy preference relations, 374–378

424

Closed loop supply chain (CLSC) network (Continued) stepwise solution algorithm, 378 multiple-echelon, 346–347 multiple objective function, 354–356 processing cost, 354–355 products delivery time, 356 purchasing cost of used products and raw materials, 356 revenues from sale of new products and recyclable parts/components, 356 transportation cost, 355–356 neutrosophic fuzzy programming approach, 369–374, 372f research contribution, 348–349 reverse chain, 349 sensitivity analyses, 389–399 of intuitionistic fuzzy linguistic preference relations, 375t, 376f, 398–399 of objective functions, 390f, 391–398, 392–393t, 395t treating fuzzy parameters and constraints, 363–368 CLSC. See Closed loop supply chain (CLSC) network Cluster analysis, 316–317, 328–333 CNS. See Complex neutrosophic set (CNS) Collaborative filtering-based recommender systems item-based, 176–177 model-based, 177 user-based, 176 Collection center (CC), 382–388 Complex fuzzy set (CFS), 92–93, 294 Complex intuitionistic fuzzy sets (CIFS), 93–94, 294 Complex neutrosophic cosine similarity measure (CNCSM), 96–98 Complex neutrosophic Dice similarity measure (CNDSM), 98–101 Complex neutrosophic Jaccard similarity measure (CNJSM), 101–104 Complex neutrosophic set (CNS), 89, 94–96 comparison analysis, 107–110, 109t complex neutrosophic cosine similarity measure (CNCSM) for, 96–98 complex neutrosophic Dice similarity measure (CNDSM) for, 98–101

Index

complex neutrosophic Jaccard similarity measure (CNJSM) for, 101–104 decision-making based on weighted complex neutrosophic cosine similarity measure (WCNCSM), 106 weighted complex neutrosophic Dice similarity measure (WCNDSM), 106 weighted complex neutrosophic Jaccard similarity measure (WCNJSM), 106 educational stream selection, for higher secondary education, 106–107, 108–109t tangent function for, 104–105 Computer-aided detection (CADe), 405 Computer-aided diagnosis (CAD), 405 neutrosophic set with optimization, 415–416 without optimizing neutrosophic set, 413–415 optimization algorithms (OptNS)-based, 412–416 Computer-based training (CBT) systems, 67 Containers, 23 Continuity, in fuzzy neutrosophic topological space (FNTS), 218–226 Crawler module, 195–197, 197f D Data sovereignty, 22 Decentralized rules engine, 38–41 Decision-making, 1 in complex neutrosophic set (CNS) weighted complex neutrosophic cosine similarity measure (WCNCSM), 106 weighted complex neutrosophic Dice similarity measure (WCNDSM), 106 weighted complex neutrosophic Jaccard similarity measure (WCNJSM), 106 Defuzzification, 344–345, 363 Deneutrosophication, 171–172 neutrosophic numbers, 172 synthesization, 171 typical neutrosophic value, 172 Deneutrosophy, 26, 67, 171 Dice similarity measure of trapezoidal fuzzy numbers, 268 of trapezoidal neutrosophic fuzzy numbers, 268–272 between two vectors, 267–268

Index

Digital library, 185 Disposal facility center (DF), 388 Distance measure, 316–317 Distribution center (DC), 381–382 Docker, 24 Docker Swarm, 24 Document processor service, 197–198 query expansion module, 198, 200f removing stop words module, 197, 198b Stemmer module, 198, 199f tokenizer module, 197 Domain-specific language (DSL), 24, 31–33 components, 24 neutrosophic, 33, 34f parse tree, 32, 32f Double sampling scheme, 45–46 Double-valued neutrosophic sets (DVNSs), 118 Drools, 25 E e-Commerce, 22, 30–31 domain-specific language (DSL), 31–33, 32f in JSON format, 31b neutrosophic-based, 33, 34b eLearning, 64–67 adaptive, 65–66 evaluation of, 80–82, 81–82t intelligent learning objects recommenders, 183 challenges, 198–200 classifier evaluation, 209–211 crawler module, 195–197, 197f document processor service, 197–198 evaluation and optimization, 198–211, 201–204f, 202–206t, 206f, 208–209t, 208–209f finding, gathering, and analyzing, 183–185, 184f information retrieval evaluation, 207–208 pending learning objects for recommendation manager module, 195, 196f personalized supervised generated learning objects, 185–186, 186f technical details and implementation, 188–194 intelligent systems, 66–67, 72–80

425

learning management systems (LMS), 64 learning objects (LOs), 177–181 learning styles ATLAS, 182, 190–192, 193f Brain Works, 183, 190–192, 194f Felder, 182, 190–192, 191–192f general, 181 neutrosophic-based recommender systems for hotel recommender system in tourism, 173 medical recommender systems, 173 multicriteria recommender systems (MC-RS), 172–173 single-criterion recommender systems (SC-RS), 172 stock trending, 173 pedagogical challenges, 64–65 recommender systems in, 173–174 services-based systems microservices architecture, 68–69 monolithic architecture, 68 service-oriented architecture (SOA), 68 eLearning 3.0, 83 ε-constrained method, 346–347 Exam management system, 70 Extended refined neutrosophic optimization, 298–299, 311t, 312f F Facebook, 169 Falsity-membership function, 265–266, 273 Felder learning model, 190–192, 191–192f active and reflective, 182 sensing and intuitive, 182 sequential and global, 182 visual and verbal, 182 FS. See Fuzzy set (FS) Fuzzy linear programming (FLP), 235–236 Chanas’ method, 247 Das et al.’s method, 246 Guu and Wu’s method, 242 Klir and Yuan’s method, 244–245 Skandari and Ghaznavi’s method, 243–244 Verdegay’s method, 246–247 Werners’ method, 241–242 Wu et al.’s method, 243 Zimmermann’s method, 239–240

426

Fuzzy Fuzzy Fuzzy Fuzzy Fuzzy

logic, 117–118, 138 Markov chain (FMC), 152, 155 neutrosophic-closure (FNCl), 217 neutrosophic-Interior (FNInt), 217 neutrosophic perfectly continuous function (FNPT-con.), 225 Fuzzy neutrosophic pre-τ0,1 continuous function (FNP-τ0, 1-con.), 218 Fuzzy neutrosophic pre-τ0,2 continuous function (FNP-τ0, 2-con.), 218 Fuzzy neutrosophic pre-τ0,1 contra continuous function (FNP-τ0, 1-con.), 218 Fuzzy neutrosophic pre-τ0,2 contra continuous function (FNP-τ0, 2-con.), 218 Fuzzy neutrosophic sets (FNS), 215–216 Fuzzy neutrosophic topological space (FNTS), 215–216 continuity types in, 218–226 interrelations, 226–232 Fuzzy neutrosophic topology (FNT), 216 Fuzzy optimization, 291 Fuzzy set (FS), 90, 155, 261–262, 290, 293, 315 bipolar, 293 Fuzzy transition probability (FTP), 152 G Generalized Pareto distribution, 45–46 General learning style auditory, 181 logical, 181 social, 181 solitary, 181 tactile, 181 visual, 181 Genetic algorithm (GA), 346–348, 411, 411b Glomerular basement membrane (GBM), 414 Goal programming method, 346–347 H Hotel recommender system, in tourism, 173 Hyper Text Markup Language (HTML), 204 I IFS. See Intuitionistic fuzzy set (IFS) Indeterminacy-membership function, 265–266, 273 INS. See Interval neutrosophic set (INS)

Index

Integrated data, 22 Intelligent learning objects recommenders, 183 challenges, 198–200 classifier evaluation, 209–211 crawler module, 195–197, 197f document processor service, 197–198 query expansion module, 198, 200f removing stop words module, 197, 198b Stemmer module, 198, 199f tokenizer module, 197 evaluation and optimization, 198–211, 201–204f, 202–206t, 206f, 208–209t, 208–209f finding, gathering, and analyzing, 183–185, 184f information retrieval evaluation, 207–208 pending learning objects for recommendation manager module, 195, 196f personalized supervised generated LOs, 185–186, 186f technical details and implementation, 188–194 students’ manager service, 188–192, 189–194f students’ usage data manager, 192–194, 195f Intelligent systems, 66–67 agenda study time planner, 79, 80t cheat depressor, 76–77, 76t eLearning, 66–67 learning objects classifier, 74 learning objects recommender, 78–79 meeting manager for suspended students, 79–80, 81t online lecture learning objects advisor, 74–75, 75t, 75–76b student tracker, 76 study plan advisor, 77–78, 77t, 78b time-to-learn topic calculation, 78, 79t Internet, 185 Interrelations, 226–232 Interval neutrosophic Markov chain, 158 comparative analysis, 163–164, 164–165t with existing methods, 165, 165t experimental analysis, 158–163, 159f long-run behavior of, 158 Interval neutrosophic numbers, 157 Interval neutrosophic set (INS), 157, 316

Index

Interval neutrosophic transition probability matrix (INTPM), 159 Interval valued neutrosophic number (IVNN), 67, 172 Interval valued neutrosophic sets (IVNSs), 68, 172 Intuitionistic fuzzy Markov chain, 155–156 Intuitionistic fuzzy preference relations, modified neutrosophic fuzzy programming with, 374–378, 375t, 376f Intuitionistic fuzzy set (IFS), 90, 155, 215–216, 315, 369–370 similarity measures, 315–316 Intuitionistic fuzzy set theory, 261–262 Item-based collaborative filtering, 176–177 J Jaccard similarity measure, of trapezoidal neutrosophic fuzzy numbers, 272–275 JSON Grammar file, 38, 39b JSON listener methods, 38, 40b K Keshtel Algorithm, 347–348 K-means clustering algorithm #MeToo movement, 127–128, 127–129f for single-valued neutrosophic sets (SVNSs), 121, 122b k-Nearest neighbors (k-NN) algorithm for single-valued neutrosophic sets (SVNSs), 122, 123b, 129–130 Kruskal’s algorithm, 121 L Learning content management system (LCMS), 69–70 Learning management systems (LMSs), 63–65 adaptive, 70 Learning objects (LOs), 64–65, 70, 177–181 annotation, 180 classification, 181 educational, 179 general, 177 intelligent LOs recommender, 78–79 intelligent techniques, 74 keywords insertion times in seconds, 201–202, 203f, 204t optimized, 204–205, 206t, 206f

427

life cycle, 178 reading time in seconds, 201, 201f, 202t rights, 180 technical, 179 term frequencies calculation times in seconds, 201, 203t, 203f tokenization times in seconds, 201, 202t, 202f tokenized no. of words vs. total no. of words, 204, 204f, 205t Learning styles ATLAS, 182, 190–192, 193f Brain Works, 183, 190–192, 194f Felder, 182, 190–192, 191–192f general, 181 Learn via questions (LVQ), 71 Lexer, 24 Linear programming, 291 Linear programming problems (LPP), 235 LMSs. See Learning management systems (LMSs) Logical learners, 181 Long-run behavior, of interval neutrosophic Markov chain, 158 Long tail phenomenon, 175–176, 175f M Markov chain (MC), 151–155 fuzzy, 152, 155 intuitionistic fuzzy, 155–156 neutrosophic, 156 Matlab optimization tool box, 309 Matrix factorization, 177 MC. See Markov chain (MC) MCDM. See Multicriteria decision-making method (MCDM) Medical diagnosis, 173 bacteria classification problem, 316 fuzzy implications, 317, 334t Medical image analysis neutrosophic set entropy of neutrosophic subsets, 409–410 neutrosophic image, 408–409 optimization algorithms (OptNS) based CAD, 412–416 concept, 410–411 genetic algorithm (GA), 411, 411b neutrosophic set with, 411–412, 412b particle swarm optimization (PSO), 411–412, 412b

428

Medical pattern recognition, 316, 325–328, 334, 335t Medical recommender systems, 173 Meta-heuristic optimization techniques, 410 #MeToo movement, 117 data classification, 128–129 dataset, description of, 125 historical significance of, 124–125 K-means clustering, 127–128, 127–129f k-NN classification, 129–130, 130f, 130t sentiment analysis, conventional and fuzzy, 126 SVM classifier, 131, 131t tweets analysis, using neutrosophy, 126–127 Microservices architecture, 68–69 containerized applications using docker, 23–24 data sovereignty, 22 domain-specific language (DSL), 24, 31–33 components, 24 neutrosophic, 33, 34f parse tree, 32, 32f integrated data, 22 intelligent, 72–74 mesh architecture, 26–27, 27f with gateway, 27–28, 28f with Swarm manager and event bus, 28, 29f neutrosophic theory, 25–26 single-valued neutrosophic numbers (SVNNs), 26 single-valued neutrosophic set (SVNS), 26 rules engine, 24–25 business, language design, 35 components, 24–25, 25f decentralized, 38–41 neutrosophic, 34–41 stateful, 23 stateless, 23 Million Dollar Matrix, 177 Minimum spanning tree (MST), 121 Mixed-integer fuzzy mathematical model, 346–347

Index

Mixed-integer nonlinear programming problem (MINLPP), 346–347 Model-based collaborative filtering. See Matrix factorization Monolithic application, 21, 22f Monolithic architecture, 68 Multicriteria decision-making method (MCDM), 1–3, 3f, 261–262, 276–278, 278f Multicriteria recommender systems (MC-RS), 172–173 Multiple-attribute decision making (MADM) methods, 261–262 Multiple-attribute group decision making (MAGDM), 261–262 Multirefined neutrosophic sets (MRNSs), 118 N Natural language toolkit (NLTK), 123–124 Netflix, 169 Neutrosophic analytic hierarchy process, 38 Neutrosophication, 171 Neutrosophic average sample number (NASN), 140–141 Neutrosophic-based recommender systems hotel recommender system in tourism, 173 medical recommender systems, 173 multicriteria recommender systems (MC-RS), 172–173 single-criterion recommender systems (SC-RS), 172 stock trending, 173 Neutrosophic C-means (NCM) clustering technique, 414 Neutrosophic components, 6 Neutrosophic cumulative distribution function (Ncdf ), 139 Neutrosophic domain-specific language (DSL), 33 Neutrosophic fuzzy programming approach (NFPA), 344–345, 369–374, 372f Neutrosophic linear programming problems (NLPPs), 236 with fuzzy relation, 252–255

Index

ranking functions, 248–251 application, 255–257, 256t comparative analysis, 251–252 high-level decision, 249 linear, 248, 253 low-level decision, 249 neutrosophic number (NN), 248 numerical example, 254–255 truth membership, indeterminacy, and falsity membership functions of TNN, 249f Neutrosophic Markov chain, 156 interval, 158 Neutrosophic numbers, 25–26, 67 concepts, 264–267 Neutrosophic operating characteristic (NOC) function, 48 Neutrosophic optimization (NO) bipolar, 298–309, 311t, 312f extended refined, 298–309, 311t, 312f Neutrosophic probability (NP), 152, 157 Neutrosophic process loss index, 47 Neutrosophic random process (NRP), 152 Neutrosophic random variable (NRV), 152 Neutrosophic sampling plan, 47–56 comparative study, proposed and existing plans, 56, 57t, 58f plan parameters, 50–55t Neutrosophic sentiment analysis, 132 Neutrosophic set (NS), 6, 25–26, 67, 90–91, 118, 156, 170–172, 236–237, 261–262, 292–293, 317, 369–370, 406, 408–410 bipolar, 293–295 computer-aided diagnosis (CAD) with optimization, 415–416 without optimizing NS, 413–415 concepts, 264–267 entropy of neutrosophic subsets, 409–410 fuzzy, 215 interval, 157 neutrosophic image, 408–409 refinements on, 295–297 Neutrosophic statistical interval system (NSIS), repetitive sampling sudden death testing, 139–141 plan parameters, 141, 142–145t Weibull distribution, 139, 142–147t Neutrosophic statistics (NS), 46–47, 138

429

Neutrosophic subsets, entropy of, 409–410 Neutrosophic symptom values, diseases, 326, 327t Neutrosophic theory, 25–26, 67–68, 170–172 in computer-aided diagnosis (CAD) systems, 406 in recommender system, 187 single-valued neutrosophic numbers (SVNNs), 26 single-valued neutrosophic set (SVNS), 26 Neutrosophic topological space (NTS), 215 Neutrosophic variable (NV), 152 Neutrosophic Weibull distribution, 139 Neutrosophic weighted distance, 120, 120b Neutrosophy, 6, 118–119 tweets, analysis of, 126–127 NFPA. See Neutrosophic fuzzy programming approach (NFPA) Nondominated Sorting Genetic Algorithm-II (NSGA-II), 347–348 Nonlinear multiobjective optimization problem, 298 bipolar neutrosophic optimization, 299, 311t, 312f computational algorithm, 300–309 extended refined neutrosophic optimization, 298–299, 311t, 312f optimal solutions, 310t, 312t NS. See Neutrosophic set (NS) O Optimization defined, 410 fuzzy, 291 Optimization algorithms (OptNS) based CAD, 412–416 concept, 410–411 genetic algorithm (GA), 411, 411b neutrosophic set with, 411–412, 412b particle swarm optimization (PSO), 411–412, 412b Optimized neutrosophic k-means (ONKM), 415–416 OptNS. See Optimization algorithms (OptNS) P Parametric fuzzy number, 237, 237f Parsers, 24

430

Particle swarm optimization (PSO), 347–348, 411–412, 412b Plithogenic aggregation operations, 7, 14, 17 Plithogenic set, 2, 5 Plithogeny, 1, 5 Popularity-based recommender systems, 175–176, 175f Prim’s algorithm, 121 Process loss, neutrosophic, 47 Python, 123–124, 127 Q Quality Assurance and Accreditation Project (QAAP), 185 management system, 70–71 Query expansion module, 198, 200f R Ranking functions, neutrosophic linear programming problems (NLPPs), 248–251 application, 255–257, 256t comparative analysis, 251–252 high-level decision, 249 linear, 248, 253 low-level decision, 249 neutrosophic number (NN), 248 numerical example, 254–255 truth membership, indeterminacy, and falsity membership functions of TNN, 249f Raw material storage center (RMS), 381–382 Recommender systems, 169–170 classification-based, 176 collaborative filtering-based matrix factorization, 177 nearest neighbor, 176–177 in eLearning, 173–174 neutrosophic-based hotel recommender system in tourism, 173 medical recommender systems, 173 multicriteria recommender systems (MC-RS), 172–173 single-criterion recommender systems (SC-RS), 172 stock trending, 173

Index

neutrosophic theory in, 187 popularity-based, 175–176, 175f utilization in online stores, 170, 170f Recycling point (RP), 388 Refinements, on neutrosophic sets, 295–297 Regular expressions (RE), 204 Removing stop words module, 197, 198b Repetitive sampling, 45–46 Reverse logistics network design (RLND), 347–348 Reverse logistic (RL) system, 346–347 Riser design problem, 309–310 Robust optimization model, for closed loop supply chain (CLSC) network, 346–347 Rules engine, 24–25 business, language design, 35 components, 24–25, 25f decentralized, 38–41 neutrosophic, 34–41 S Sensitivity analyses, closed loop supply chain (CLSC) network, 389–399 of intuitionistic fuzzy linguistic preference relations, 398–399 of objective functions, 391–398 Sentiment analysis, 117 #MeToo movement, 124–131 natural language processing (NLP), 123–124 using neutrosophic sets, 123–124 Service-oriented architecture (SOA), 22, 68 Services-based eLearning systems microservices architecture, 68–69 monolithic architecture, 68 service-oriented architecture (SOA), 68 Similarity measures, 316–317, 336 clustering problem, 328 comparison approach with, 324–325 intuitionistic fuzzy sets, 315–316 neutrosophic, 318 for pattern recognition problem, 335t single-valued neutrosophic sets, 319–324, 328t Single-criterion recommender systems (SC-RS), 172 Single sampling scheme, 45–46

Index

Single-valued neutrosophic numbers (SVNNs), 26, 67, 172 Single-valued neutrosophic sets (SVNS), 26, 68, 91–92, 118, 156, 172, 261–262, 293, 318–319 applications cluster analysis, 328–333 pattern recognition, 325–328 bacteria classification problem, 329t clustering analysis, 316–317 existing similarity measures, 319–321 comparison approach with, 324–325 fuzzy implications, 316t, 317 K-means clustering algorithm for, 121, 122b k-nearest neighbors (k-NN) algorithm for, 122, 123b medical diagnosis, 315–316 new similarity measure of, 321–324 support vector machine classifier, 122–123 taxonomy approach, 328 undiagnosed samples, properties values of, 326, 330t Single-valued neutrosophic soft numbers (SVNSNs), 38–41 Single-valued neutrosophic soft weighted arithmetic averaging (SVNSWA), 38–41 Single-valued neutrosophic soft weighted geometric averaging (SVNSWGA), 38–41 Single valued trapezoidal neutrosophic number (SVTNN), 237–239 Single-valued trapezoidal neutrosophic preference relations (SVTNPRs), 261–262 Single-valued triangular neutrosophic numbers (SVTrNNs), 237–239, 261–262 Social learners, 181 Solitary learners, 181 Stateful microservices, 23 Statefulness, 22–23 Stateless microservices, 23 Stemmer module, 198, 199f Stochastic mixed-integer nonlinear programming problem (SMINLPP), 347–348 Stock trending analysis, 173

431

Stop words, 197, 198b Students’ manager service, 188–192, 189–194f Students’ usage data manager, 192–194, 195f Sudden death testing repetitive sampling under neutrosophic statistical interval system, 139–141 plan parameters, 141, 142–145t Weibull distribution, 139, 142–147t Supply chain management (SCM), 343–344 Supply chain problems, 2 plant evaluation problem, 11–16, 14–15t, 15–16f warehouse location problem, 8–11, 9f, 9–13t, 13f Support vector machine (SVM), 131, 131t SVNS. See Single valued neutrosophic sets (SVNS) Syntax, 24 T Tabu Search (TS), 347–348 Tactile learners, 181 Tangent function, for complex neutrosophic set (CNS), 104–105 Term frequency-inverse document frequency (TF-IDF), 185 Testing point (TP), 388 TextBlob, 126, 132 Tokenizer module, 197 Tourism, hotel recommender system in, 173 Transition probability (TP), 152 Transition probability matrix (TPM), 151 Trapezoidal fuzzy number, 363 concepts, 263–264 Dice similarity measure of, 268 Trapezoidal fuzzy parameters, 365–369, 367t Trapezoidal neutrosophic fuzzy number (TrNFNs), 266 Dice similarity measure of, 268–272 Jaccard similarity measure of, 272–275, 275t multicriteria decision-making method (MCDM), 261–262, 276–278, 278f ranking method of alternatives based on similarity measure methods, 282–284, 283t application, 284

432

Trapezoidal neutrosophic number, 261–262 Triangular fuzzy number, 263 Tri-level location-allocation planning problem for, 347–348 Triple refined indeterminate neutrosophic sets (TRINSs), 118 Truth-membership function, 265–266, 273 Twitter, 117–118, 123 U Uncertainty, 293 closed loop supply chain (CLSC) model formulation under, 360–362 User-based collaborative filtering, 176 V Valence Aware Dictionary and Sentiment Reasoner (VADER), 117–118, 123–124, 126

Index

Variable Neighborhood Search, 347–348 Variable sampling plans, 46 W Water Wave Optimization, 347–348 Weibull distribution, 139, 142–147t Weighted complex neutrosophic cosine similarity measure (WCNCSM), 97–98 Weighted complex neutrosophic Dice similarity measure (WCNDSM), 99–101 Weighted complex neutrosophic Jaccard similarity measure (WCNJSM), 102–104 Weighted nonprimitive goal programming model, 346–347 WordNet, 198, 200f, 204–205