Aggregation Operators for Various Extensions of Fuzzy Set and Its Applications in Transportation Problems [1st ed.] 9789811569975, 9789811569982

Table of contents :
Front Matter ....Pages i-xvii
Appropriate Weighted Averaging Aggregation Operator Under Some Extensions of the Fuzzy Environment (Akansha Mishra, Amit Kumar)....Pages 1-86
Mehar Method to Find a Unique Fuzzy Optimal Value of Balanced Fully Triangular Fuzzy Transportation Problems (Akansha Mishra, Amit Kumar)....Pages 87-118
Vaishnavi Approach for Solving Triangular Intuitionistic Transportation Problems of Type-2 (Akansha Mishra, Amit Kumar)....Pages 119-141
JMD Approach for Solving Unbalanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problems (Akansha Mishra, Amit Kumar)....Pages 143-203
JMD Approach for Solving Unbalanced Fully Generalized Trapezoidal Intuitionistic Fuzzy Transportation Problems (Akansha Mishra, Amit Kumar)....Pages 205-262

Citation preview

Studies in Fuzziness and Soft Computing

Akansha Mishra Amit Kumar

Aggregation Operators for Various Extensions of Fuzzy Set and Its Applications in Transportation Problems

Studies in Fuzziness and Soft Computing Volume 399

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by ISI, DBLP and Ulrichs, SCOPUS, Zentralblatt Math, GeoRef, Current Mathematical Publications, IngentaConnect, MetaPress and Springerlink. The books of the series are submitted for indexing to Web of Science.

More information about this series at http://www.springer.com/series/2941

Akansha Mishra Amit Kumar •

Aggregation Operators for Various Extensions of Fuzzy Set and Its Applications in Transportation Problems

123

Akansha Mishra School of Mathematics Thapar Institute of Engineering & Technology Patiala, Punjab, India

Amit Kumar School of Mathematics Thapar Institute of Engineering & Technology Patiala, Punjab, India

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-981-15-6997-5 ISBN 978-981-15-6998-2 (eBook) https://doi.org/10.1007/978-981-15-6998-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Dedicated to Parents and God

Preface

In the last few years, several methods have been proposed in the literature for solving transportation problems under fuzzy environment and its various extensions [1–8 and references therein]. But, all these methods [1–8 and references therein] have been proposed by considering the assumption that the aggregated values of all the parameters are known. Therefore, instead of the aggregated values of the parameters, the values of the various parameters, collected from experts, are provided. Then, the existing methods [1–8 and references therein] cannot be used to solve transportation problems under fuzzy environment and its various extensions. To overcome this limitation of the existing methods [1–8 and references therein], firstly, there is a need to use an appropriate weighted aggregation operator to aggregate the values of the parameters provided by the experts in terms of the fuzzy set and its various extensions. But, there exist several weighted aggregation operators under a fuzzy environment and its various extensions. Therefore, to overcome the limitation of the existing methods [1–8 and references therein], firstly, there is a need to choose an appropriate weighted aggregation operator under fuzzy environment and its various extensions. It is pertinent to mention that after aggregating the provided values of the parameters by the selected weighted aggregation operator, researchers may use the recently proposed methods [1–6] for solving transportation problems under fuzzy environment and its various extensions. However, after a deep study, some drawbacks have been observed in the existing methods [1–6]. Therefore, it is scientifically incorrect to use the existing methods [1–6]. Keeping all above in mind, the aim of this book is (i) To choose/propose an appropriate weighted averaging aggregation operator under some extensions of the fuzzy environment. Also, to show that it is illogical to define the weighted geometric aggregation operator under fuzzy environment and its various extensions. (ii) To make the researchers aware about the inappropriateness of the existing methods [2–4] for solving balanced fully triangular fuzzy transportation

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problems. Also, to propose a new method (named as Mehar method) for solving balanced fully triangular fuzzy transportation problems. (iii) To make the researchers aware about the inappropriateness of the existing approach [5] for solving the triangular intuitionistic fuzzy transportation problem of type-2. Also, to propose a new approach (named as Vaishnavi approach) for solving triangular intuitionistic fuzzy transportation problems of type-2. (iv) To make the researchers aware about some limitations and a drawback of the existing approach [6] for solving balanced fully trapezoidal intuitionistic fuzzy transportation problems. Also, to propose a new approach (named as JMD approach) for solving unbalanced fully trapezoidal intuitionistic fuzzy transportation problems. (v) To make the researchers aware about some limitations and a drawback of the existing approach [1] for solving balanced fully generalized trapezoidal intuitionistic fuzzy transportation problems. Also, to propose a new approach (named as JMD approach) to solve unbalanced fully generalized trapezoidal intuitionistic fuzzy transportation problems. Patiala, India

Akansha Mishra Amit Kumar

References 1. D. Chakraborty, D. K. Jana and T. K. Roy, Arithmetic operations on generalized intuitionistic fuzzy number and its applications to transportation problem, OPSEARCH 52 (2015) 431–471. 2. A. Ebrahimnejad, An improved approach for solving fuzzy transportation problem with triangular fuzzy numbers, Journal of Intelligent & Fuzzy Systems 29 (2015) 963–974. 3. A. Ebrahimnejad, New method for solving fuzzy transportation problems with LR flat fuzzy numbers, Information Sciences 357 (2016) 108-124. 4. A. Ebrahimnejad, A lexicographic ordering-based approach for solving fuzzy transportation problems with triangular fuzzy numbers, International Journal of Management and Decision Making 16 (2017) 346–374. 5. A. Ebrahimnejad and J. L. Verdegay, An efficient computational approach for solving type-2 intuitionistic fuzzy numbers based transportation problems, International Journal of Computational Intelligence Systems 9 (2016) 1154–1173. 6. A. Ebrahimnejad and J. L. Verdegay, A new approach for solving fully intuitionistic fuzzy transportation problems, Fuzzy Optimization and Decision Making 17 (2018) 447–474. 7. G. Gupta, Transportation Problems in Intuitionistic Fuzzy Environment, Ph.D Thesis, 2016, Thapar Institute of Engineering & Technology, Patiala, Punjab, India. 8. A. Kaur, J. Kacprzyk and A. Kumar. Fuzzy Transportation and Transshipment Problems, Studies in Fuzziness and Soft Computing Springer Nature, Switzerland AG, Vol. 385, 2020.

Acknowledgements

The authors would like to thank the Series Editor Prof. Janusz Kacprzyk for his valuable suggestions. Dr. Akansha Mishra would like to express her heartfelt gratitude to her mother Mrs. Uma Mishra and her father Mr. R. P. Mishra for their unconditional love, support and blessings. She gratefully acknowledges the patience and love of her sibling Vinayak Mishra who helped her to overcome the difficulties encountered during her life. Dr. Amit Kumar would like to acknowledge the adolescent inner blessings of Mehar (Daughter of his Cousin Dr. Parmpreet Kaur). He believes that Mata Vaishno Devi has appeared on the earth in the form of Mehar and without Mehar’s blessings it would not have been possible to think the ideas presented in this thesis. Last but not the least, both the authors are ever grateful to Almighty God. Thank you God for the numerous blessings bestowed upon us in every aspect of our life.

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Contents

1 Appropriate Weighted Averaging Aggregation Operator Under Some Extensions of the Fuzzy Environment . . . . . . . . . . . . . . . . . 1.1 Organization of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Need of the Fuzzy Set and Its Various Extensions in Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Need of a Weighted Aggregation Operator Under Various Extensions of the Fuzzy Environment . . . . . . . . . . . . . . . . . . 1.4 Need to Choose an Appropriate Weighted Aggregation Operator Under Various Extensions of the Fuzzy Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Some Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Drawbacks of Some Existing Weighted Geometric Aggregation Operators Under Various Extensions of the Fuzzy Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Drawbacks of Xu and Yager’s Intuitionistic Fuzzy Weighted Geometric Aggregation Operator . . . . . . . . 1.6.2 Drawbacks of Wang and Liu’s Intuitionistic Fuzzy Weighted Geometric Aggregation Operator . . . . . . . . 1.6.3 Drawbacks of Garg’s Intuitionistic Fuzzy Interactive Weighted Geometric Aggregation Operator . . . . . . . . 1.6.4 Drawbacks of Garg’s Intuitionistic Fuzzy Hamacher Interactive Weighted Geometric Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.5 Drawbacks of He et al.’s Intuitionistic Fuzzy Interaction Weighted Geometric Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.6 Drawbacks of Chen and Chang’s Intuitionistic Fuzzy Weighted Geometric Aggregation Operator . . . . . . . . 1.6.7 Drawbacks of Garg and Kumar’s Connection Number-Based Power Geometric Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Drawbacks of Zhang’s Linguistic Intuitionistic Fuzzy Weighted Geometric Aggregation Operator . . . . . . . . 1.6.9 Drawbacks of Garg and Kumar’s Linguistic Connection Number-Based Weighted Geometric Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . 1.6.10 Drawbacks of Garg and Kumar’s Linguistic Connection Number-Based Power Geometric Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . 1.6.11 Drawbacks of Garg and Kumar’s Linguistic Connection Number-Based Prioritized Geometric Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . 1.6.12 Drawbacks of Arora and Garg’s Intuitionistic Fuzzy Soft Weighted Geometric Aggregation Operator . . . . 1.6.13 Drawbacks of Arora and Garg’s Intuitionistic Fuzzy Soft Prioritized Weighted Geometric Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.14 Drawbacks of Garg and Arora’s Dual Hesitant Fuzzy Soft Weighted Geometric Aggregation Operator . . . . 1.6.15 Drawbacks of Garg’s Intuitionistic Fuzzy Multiplicative Weighted Geometric Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.16 Drawbacks of Garg’s Pythagorean Fuzzy Weighted Geometric Aggregation Operator . . . . . . . . . . . . . . . . 1.6.17 Drawbacks of Garg’s Pythagorean Fuzzy Interactive Weighted Geometric Aggregation Operator . . . . . . . . 1.6.18 Drawbacks of Garg’s Confidence Levels-Based Pythagorean Fuzzy Weighted Geometric Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.19 Drawbacks of Garg’s Interval-Valued Intuitionistic Fuzzy Weighted Geometric Aggregation Operator . . . 1.6.20 Drawbacks of Garg and Kumar’s Linguistic Interval-Valued Intuitionistic Fuzzy Weighted Geometric Aggregation Operator . . . . . . . . . . . . . . . . 1.6.21 Drawbacks of Nancy and Garg’s Single-Valued Neutrosophic Weighted Geometric Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drawbacks of Some Existing Weighted Averaging Aggregation Operators Under Various Extensions of the Fuzzy Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Drawbacks of Xu’s Intuitionistic Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . 1.7.2 Drawbacks of Wang and Liu’s Intuitionistic Fuzzy Weighted Averaging Aggregation Operator . . . . . . . .

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Drawbacks of Garg’s Intuitionistic Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . Drawbacks of Yu’s Intuitionistic Fuzzy Interaction Weighted Averaging Aggregation Operator . . . . . . . . Drawbacks of Huang’s Hamacher Intuitionistic Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . Drawbacks of Garg’s Intuitionistic Fuzzy Hamacher Weighted Averaging Aggregation Operator . . . . . . . . Drawbacks of Chen et al.’s Linguistic Intuitionistic Fuzzy Weighted Averaging Aggregation Operator . . . Drawbacks of Arora and Garg’s Linguistic Intuitionistic Fuzzy Prioritized Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . Drawbacks of Arora and Garg’s Intuitionistic Fuzzy Soft Weighted Averaging Aggregation Operator . . . . Drawbacks of Arora and Garg’s Intuitionistic Fuzzy Soft Weighted Prioritized Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drawbacks of Garg and Arora’s Intuitionistic Fuzzy Soft Weighted Interaction Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drawbacks of Garg and Arora’s Intuitionistic Fuzzy Soft Possibility Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drawbacks of Garg and Arora’s Intuitionistic Fuzzy Soft Power Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drawbacks of Garg and Arora’s Intuitionistic Fuzzy Soft Bonferroni Mean Aggregation Operator . . . . . . . Drawbacks of Garg and Arora’s Generalized Intuitionistic Fuzzy Soft Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . Drawbacks of Garg and Arora’s Dual Hesitant Fuzzy Soft Weighted Averaging Aggregation Operator . . . . Drawbacks of Garg’s Intuitionistic Fuzzy Multiplicative Averaging Aggregation Operator . . . . . Drawbacks of Ma and Xu’s Pythagorean Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . Drawbacks of Garg’s Pythagorean Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . Drawbacks of Garg’s Confidence Levels-Based Pythagorean Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.7.21 Drawbacks of Garg et al.’s Interval-Valued Intuitionistic Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.22 Drawbacks of Linguistic Interval-Valued Atanassov Intuitionistic Fuzzy Weighted Averaging Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.23 Drawbacks of Nancy and Garg’s Single-Valued Neutrosophic Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.24 Drawbacks of Liu and Luo’s Single-Valued Neutrosophic Hesitant Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Appropriate Weighted Averaging Aggregation Operators Under Some Extensions of the Fuzzy Environment . . . . . . . . . . . . . . 1.8.1 Appropriate Intuitionistic Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Appropriate Pythagorean Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Appropriate Connection Number Weighted Averaging Aggregation Operator . . . . . . . . . . . . . . . . . . . . . . . . 1.8.4 Appropriate Interval-Valued Intuitionistic Fuzzy Weighted Averaging Aggregation Operator . . . . . . . . 1.9 Limitation of the Weighted Geometric Aggregation Operators Under Various Extensions of the Fuzzy Environment . . . . . . . 1.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mehar Method to Find a Unique Fuzzy Optimal Value of Balanced Fully Triangular Fuzzy Transportation Problems . . . . . . . . . . . . . 2.1 Organization of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 An Existing Method for Comparing Triangular Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Ebrahimnejad’s Methods for Solving Balanced Fully Triangular Fuzzy Transportation Problems . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Ebrahimnejad’s First Method . . . . . . . . . . . . . . . . . . 2.4.2 Ebrahimnejad’s Second Method . . . . . . . . . . . . . . . . 2.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Fuzzy Optimal Solution by Ebrahimnejad’s First Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Fuzzy Optimal Solution by Ebrahimnejad’s Second Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Limitation of Ebrahimnejad’s Methods . . . . . . . . . . . . . . . . . . 2.7 Drawback of Ebrahimnejad’s Methods . . . . . . . . . . . . . . . . . .

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Reason for the Occurrence of the Drawback in Ebrahimnejad’s Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Rank, Mode and Divergence Based Approach for Comparing Triangular Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Validity of Rank, Mode and Divergence Based Approach for Comparing Triangular Fuzzy Numbers . . . . . . . . . . . . . . . 2.11 Proposed Mehar Method to Find a Unique Optimal Value of Balanced Fully Triangular Fuzzy Transportation Problems . 2.12 Unique Fuzzy Transportation Cost of the Considered Balanced Fully Triangular Fuzzy Transportation Problem . . . . . . . . . . . 2.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Vaishnavi Approach for Solving Triangular Intuitionistic Transportation Problems of Type-2 . . . . . . . . . . . . . . . . . . . . . . . 3.1 Organization of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Tabular Representation of a Triangular Intuitionistic Fuzzy Transportation Problem of Type-2 . . . . . . . . . . . . . . . . . . . . . 3.4 An Existing Approach for Comparing Triangular Intuitionistic Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Singh and Yadav’s Approach . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Ebrahimnejad and Verdegay’s Approach . . . . . . . . . . . . . . . . 3.7 Limitation of Ebrahimnejad and Verdegay’s Approach . . . . . . 3.8 Drawback of Ebrahimnejad and Verdegay’s Approach . . . . . . 3.9 Reasons for the Occurrence of the Drawback . . . . . . . . . . . . . 3.10 Proposed MEHAR Approach for Comparing Triangular Intuitionistic Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Proposed Vaishnavi Approach . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Crisp Optimal Solution of the Considered Triangular Intuitionistic Fuzzy Transportation Problem of Type-2 . . . . . . 3.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 JMD Approach for Solving Unbalanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problems . . . . . . . . . . . . . . . . 4.1 Organization of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Intuitionistic Fully Fuzzy Linear Programming Problem of a Balanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Existing Approach for Comparing Trapezoidal Intuitionistic Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Ebrahimnejad and Verdegay’s Approach for Solving Balanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problems .

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4.6 4.7 4.8 4.9 4.10

Limitations of Ebrahimnejad and Verdegay’s Approach . . . . . Drawback of Ebrahimnejad and Verdegay’s Approach . . . . . . Reasons for the Occurrence of the Limitations . . . . . . . . . . . . Reasons for the Occurrence of the Drawback . . . . . . . . . . . . . DAUGHTER Approach for Comparing Trapezoidal Intuitionistic Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Mehar Representation of a Trapezoidal Intuitionistic Fuzzy Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Multiplication of a Trapezoidal Intuitionistic Fuzzy Number in Its Existing Representation with a Trapezoidal Intuitionistic Fuzzy Number in Its Mehar Representation . . . . . . . . . . . . . . 4.13 Proposed JMD Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14.1 Intuitionistic Fuzzy Optimal Solution of the First Unbalanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problem . . . . . . . . . . . . . . . . . . . . . . 4.14.2 Intuitionistic Fuzzy Optimal Solution of the Second Unbalanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problem . . . . . . . . . . . . . . . . . . . . . . 4.15 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 JMD Approach for Solving Unbalanced Fully Generalized Trapezoidal Intuitionistic Fuzzy Transportation Problems . . . . . . 5.1 Organization of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Tabular Representation of a Fully Generalized Trapezoidal Intuitionistic Fuzzy Transportation Problem . . . . . . . . . . . . . . 5.4 Existing Approach for Comparing Generalized Trapezoidal Intuitionistic Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Chakraborty et al.’s Approach for Solving Balanced Fully Generalized Trapezoidal Intuitionistic Fuzzy Transportation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Origin of the Generalized Intuitionistic Fuzzy Linear Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Limitations of Chakraborty et al.’s Approach . . . . . . . . . . . . . 5.8 Invalidity of Chakraborty et al.’s Approach . . . . . . . . . . . . . . 5.9 Inappropriateness of the Existing Approach for Comparing Generalized Trapezoidal Intuitionistic Fuzzy Numbers . . . . . . 5.10 PRABHUS Approach for Comparing Generalized Trapezoidal Intuitionistic Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Proposed JMD Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

150 153 155 156

. . 156 . . 160

. . 161 . . 162 . . 180

. . 180

. . 191 . . 202 . . 203 . . 205 . . 206 . . 207 . . 208 . . 209

. . 210 . . 213 . . 218 . . 219 . . 220 . . . . .

. . . . .

220 225 243 262 262

About the Authors

Dr. Akansha Mishra received her M.Sc. degree in Mathematics from Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, India, in 2015, and her Ph.D. in Mathematics from Thapar Institute of Engineering & Technology, Patiala, Punjab, India, in 2019. Her main research interest is in fuzzy optimization. She has published six research papers in SCI/SCIE indexed journals, and presented another at the International Congress on Industrial and Applied Mathematics, organized by the International Council for Industrial and Applied Mathematics in Valencia, Spain on July 15–19, 2019. Dr. Amit Kumar is an Associate Professor at the School of Mathematics, Thapar Institute of Engineering & Technology, Patiala, Punjab, India. Holding a Ph.D. from the Indian Institute of Technology Roorkee (2008), Dr. Kumar has made significant contributions to the development of solution methods for various types of fuzzy linear programming problems, fuzzy transportation problems, fuzzy game theory and fuzzy multi-criteria decision-making problems. He has published over 60 research papers in SCI/SCIE indexed journals, and has co-authored three books in the series “Studies in Fuzziness and Soft Computing,” published by Springer, Germany.

xvii

Chapter 1

Appropriate Weighted Averaging Aggregation Operator Under Some Extensions of the Fuzzy Environment

Weighted aggregation operator plays an important role in optimization problems. In the last few years, several weighted geometric aggregation operators and several weighted averaging aggregation operators have been proposed in the literature [1–3, 6, 9–11, 13, 14, 16–39, 42–65, 67, 76–93, 95–104, 106–108, 110–131, 135–138, 140–146] under various extensions of the fuzzy environment [139]. The aim of this chapter is to make the researchers aware about: (i) Some drawbacks of the existing weighted geometric aggregation operators [1, 2, 10, 16, 17, 19, 20, 22, 23, 25, 28, 33, 42–45, 47, 60, 97, 114, 130, 142] which have been defined under various extensions of the fuzzy environment. (ii) Some drawbacks of the existing weighted averaging aggregation operators [2, 3, 13, 18, 21, 22, 24, 28, 31–37, 42, 46, 65, 90, 92, 97, 114, 127, 137] which have been defined under various extensions of the fuzzy environment. (iii) The fact that it is appropriate to use the existing intuitionistic fuzzy weighted averaging aggregation operator [6] to aggregate intuitionistic fuzzy numbers. (iv) The fact that it is appropriate to use the generalization of the existing intuitionistic fuzzy weighted averaging aggregation operator [6] to aggregate Pythagorean fuzzy numbers. (v) The fact that it is appropriate to use the existing weighted averaging aggregation operator [73, 74] to aggregate connection numbers. (vi) The fact that it is appropriate to use the existing weighted averaging aggregation operator [129] to aggregate interval-valued intuitionistic numbers. (vii) The fact that it is illogical to propose weighted geometric aggregation operators under various extensions of the fuzzy environment.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Mishra and A. Kumar, Aggregation Operators for Various Extensions of Fuzzy Set and Its Applications in Transportation Problems, Studies in Fuzziness and Soft Computing 399, https://doi.org/10.1007/978-981-15-6998-2_1

1

2

1.1

1 Appropriate Weighted Averaging Aggregation Operator …

Organization of the Chapter

This chapter is organized as follows: (i) In Sect. 1.2, a need of the fuzzy set and its various extensions in optimization problems has been discussed. (ii) In Sect. 1.3, a need to define weighted aggregation operators under various extensions of the fuzzy environment has been discussed. (iii) In Sect. 1.4, a need to choose an appropriate weighted averaging aggregation operator under various extensions of the fuzzy environment has been discussed. (iv) In Sect. 1.5, some basic definitions have been discussed. (v) In Sect. 1.6, some drawbacks of the existing weighted geometric aggregation operators [1, 2, 10, 16, 17, 19, 20, 22, 23, 25, 28, 33, 42–45, 47, 60, 97, 114, 130, 142], defined under various extensions of the fuzzy environment, have been discussed. (vi) In Sect. 1.7, some drawbacks of the existing weighted averaging aggregation operators [2, 3, 13, 18, 21, 22, 24, 28, 31–37, 42, 46, 65, 90, 92, 97, 114, 127, 137], defined under various extensions of the fuzzy environment, have been discussed. (vii) In Sect. 1.8, appropriate weighted averaging aggregation operators under some extensions of the fuzzy environment have been discussed. (viii) In Sect. 1.9, it is pointed out that it is illogical to propose weighted geometric aggregation operators under various extensions of the fuzzy environment. (ix) Section 1.10 concludes the chapter.

1.2

Need of the Fuzzy Set and Its Various Extensions in Optimization Problems

In most of the real-life optimization problems, firstly, a numerical value is assigned corresponding to each linguistic variable like poor, good, excellent etc. Then, two or more experts are asked to provide their opinion in terms of the assigned numerical values. After that, an appropriate mathematical expression (named as an aggregation operator) is used to obtain a single numerical value by aggregating all the numerical values provided by the experts. Finally, the obtained single numerical value is used to finalize the decision. The following example has been considered to explain this process. An institute may apply the following process to select the best candidate out of a finite number of shortlisted candidates (say, m).

1.2 Need of the Fuzzy Set and Its Various Extensions …

3

Step 1: Assign the real numbers 1, 2, 5, 8 and 10 corresponding to the linguistic variables poor, average, good, very good and excellent, respectively. Step 2: Constitute a committee of finite number of experts (say, n). Step 3: Ask each committee member to provide his/her opinion about each candidate in terms of only the real numbers 1, 2, 5, 8 and 10. Step 4: Use the crisp weighted averaging aggregation operator (1.1.1) or the crisp weighted geometric aggregation operator (1.1.2) to obtain a single real number Ck corresponding to the kth candidate. Pn i¼1 wi aik ; k ¼ 1; 2; . . .; m: Ck ¼ P n i¼1 wi Ck ¼

n Y ðaik Þwi ; k ¼ 1; 2; . . .; m:

ð1:1:1Þ

ð1:1:2Þ

i¼1

where, (i) aik = 1 or 2 or 5 or 8 or 10 represents the numerical value assigned to kth candidate by the by the ith expert. (ii) wi  0 represents the weightage of the ith expert’s opinion. Step 5: Find maximum1  k  m fCk g. Step 6: If maximum1  k  m fCk g ¼ Cp , then the pth candidate is the best candidate. It is obvious that in the above-discussed process; a real number has been assigned to represent a linguistic variable, while, in the literature, it is pointed out that it is not appropriate to represent a linguistic variable with a real number. The following example has been considered to explain the same. Four students appear in an examination of a subject. Out of these four students (i) The first student secures 98 marks. (ii) The second student secures 80 marks. (iii) The third student secures 79.95 marks. Finally, the following is decided for the considered subject: (i) The numerical value “10”, representing the linguistic term excellent, will be awarded to all those students who have secured 80 or more than 80 marks. (ii) The numerical value “9”, representing the linguistic term very good, will be awarded to all those students who have secured less than 80 marks. In this case, (i) The first student will always feel it is injustice with him/her as there is a huge difference between his/her marks and the second student’s marks, while, the

4

1 Appropriate Weighted Averaging Aggregation Operator …

numerical value, assigned to him/her, is same as the numerical value assigned to the second student. (ii) The third student will always feel it is injustice with him/her as there is a very small difference between his/her marks and the third student’s marks, while, the numerical value, assigned to him/her, is 9 and the numerical value assigned to the second student is 10. To resolve such type of issues, in the literature, fuzzy set and its various extensions [7] have been used to represent the linguistic terms of various optimization problems like multi-attribute decision-making problems [12], transportation problems [69], game theory [75, 109], linear programming problems [70], etc. Hence, various methods have been proposed in the literature to solve different types of optimization problems under fuzzy environment and its various extensions.

1.3

Need of a Weighted Aggregation Operator Under Various Extensions of the Fuzzy Environment

It is pertinent to mention that most of methods for solving optimization problems under fuzzy environment and its extensions have been proposed by considering the assumption that the aggregated value of the required parameters, provided by all the decision-makers in terms of fuzzy set or any specific extension of the fuzzy set, is known. Therefore, if instead of the aggregated values of required parameters, the values of the parameters, provided by each expert in terms of the fuzzy set or in terms of a specific extension of the fuzzy set, are known. Then, most of the existing methods, proposed for solving optimization problems under fuzzy environment and its extensions, cannot be used directly. For example, as the existing method [75] for solving matrix games with intuitionistic fuzzy payoffs, the existing method [75] for solving matrix games with interval-valued intuitionistic fuzzy payoffs and the existing method [72] for solving Pythagorean fuzzy transportation problems have been proposed by considering the assumption that the aggregated values of required parameters, provided by every expert in terms of an intuitionistic fuzzy set, an interval-valued intuitionistic fuzzy set and a Pythagorean fuzzy set, respectively, are known. Therefore, if instead of the aggregated values of required parameters, the values of the parameters, provided by each expert, are known. Then, these existing methods cannot be used directly. To overcome the limitation of the existing methods for solving optimization problems under a specific extension of the fuzzy environment, there is a need to use an appropriate weighted aggregation operator to aggregate the opinions of all the experts provided in terms of a specific extension of the fuzzy set.

1.4 Need to Choose an Appropriate Weighted Aggregation …

1.4

5

Need to Choose an Appropriate Weighted Aggregation Operator Under Various Extensions of the Fuzzy Environment

It is obvious from Sect. 1.3 that to overcome the limitation of the existing methods for solving optimization problems under a specific extension of the fuzzy environment, there is a need to use an appropriate weighted aggregation operator to aggregate the opinions of all the experts provided in terms of a specific extension of the fuzzy set. However, as there exist several weighted aggregation operators under each extension of the fuzzy environment. Therefore, to overcome the limitation of the existing methods for solving optimization problems under fuzzy environment and its extensions, firstly, there is a need to choose an appropriate weighted aggregation under each extension of the fuzzy environment. The reasons for the existence of several weighted aggregation operators under each extension of the fuzzy environment are as follows: It is a well-known fact that to aggregate real numbers with the classical weighted averaging aggregation operator (1.1.1), there is a need to use an expression to evaluate the addition of real numbers as well as an expression to evaluate the multiplication of a non-negative real number with a real number. Similarly, to aggregate positive real numbers with the weighted geometric aggregation operator (1.1.2), there is a need to use an expression to evaluate the multiplication of real numbers as well as an expression to evaluate the non-negative power of a positive real number. On the same direction, to generalize the crisp weighted averaging aggregation operator (1.1.1) under a specific extension of the fuzzy environment, there is a need to use (i) An expression to evaluate the addition for the considered extension of the fuzzy set. (ii) An expression to evaluate the multiplication of a non-negative real number with the considered extension of the fuzzy set. Similarly, to generalize the crisp weighted geometric aggregation operator (1.1.2) under a specific extension of the fuzzy environment, there is a need to use (i) An the (ii) An the

expression to evaluate the multiplication for the considered extension of fuzzy set. expression to evaluate the positive power of the considered extension of fuzzy set.

However, as arithmetic operations for each extension of the fuzzy set depend upon a t-norm and a t-conorm and different types of t-norm and t-conorm have been defined in the literature. Therefore, in the literature [1–3, 6, 9–11, 13, 14, 16–39, 42–68, 71, 76–93, 95–104, 106–108, 110–131, 134–137, 139–145], various

1 Appropriate Weighted Averaging Aggregation Operator …

6

arithmetic operations, based upon different types of t-norm and t-conorm, have been proposed for each extension of the fuzzy set. Hence, in the literature, various weighted aggregation operators have been proposed for each extension of the fuzzy set.

1.5

Some Basic Definitions

In this section, some basic definitions have been presented. Definition 1.5.1 [127]: The weighted averaging operator is the mapping WA: Rn ! R defined by the following formula: WAw ða1 ; a2 ; . . .; an Þ ¼

n X

wj aj ;

j¼1

where w ¼ ðw1 ; w2 ; . . .; wn ÞT is the weight vector of aj ðj ¼ 1; 2; . . .; nÞ, with wj 2 P ½0; 1 and nj¼1 wj ¼ 1. Definition 1.5.2 [130]: The weighted geometric operator is the mapping n WG: R þ ! R þ defined by the following formula: WGw ða1 ; a2 ; . . .; an Þ ¼

n Y

w

aj j

j¼1

where w ¼ ðw1 ; w2 ; . . .; wn ÞT is the weight vector of aj ðj ¼ 1; 2; . . .; nÞ, with wj 2 P ½0; 1 and nj¼1 wj ¼ 1. Definition 1.5.3 [133]: The power average operator is the mapping PA: Rn ! R defined by the following formula: Pn ð1 þ T ðai ÞÞai PAða1 ; a2 ; . . .; an Þ ¼ Pi¼1 n i¼1 ð1 þ T ðai ÞÞ where

  Pn : Sup a ; a i j j¼1 j 6¼ i 2     (ii) Sup ai ; aj is the support for ai from aj , Sup ai ; aj ¼ Keaðai aj Þ for K 2 ½0; 1; a  0. (i) T ðai Þ ¼

1.5 Some Basic Definitions

7

Definition 1.5.4 [131]: The power geometric operator is the mapping PG: R þ ! R þ defined by the following formula: n

PGða1 ; a2 ; . . .; an Þ ¼

n Y

ai 1 þ T ðai Þ=

Pn i¼1

ð1 þ T ðai ÞÞ

i¼1

Definition 1.5.5 [100]: A function T: ½0; 1  ½0; 1 ! ½0; 1 is called a t-norm if it satisfies the following conditions: (1) (2) (3) (4)

8x 2 ½0; 1; T ð1; xÞ ¼ x. 8x; y 2 ½0; 1; T ðx; yÞ ¼ T ðy; xÞ. 8x; y; z 2 ½0; 1; T ðx; T ðy; zÞÞ ¼ T ðT ðx; yÞ; zÞ. If x  x0 , y  y0 , then T ðx; yÞ  T ðx0 ; y0 Þ.

Definition 1.5.6 [100]: A function, S: ½0; 1  ½0; 1 ! ½0; 1 is called a t-conorm if it satisfies the following conditions: (1) (2) (3) (4)

8x 2 ½0; 1; Sð0; xÞ ¼ x. 8x; y 2 ½0; 1; Sðx; yÞ ¼ Sðy; xÞ. 8x; y; z 2 ½0; 1; Sðx; Sðy; zÞÞ ¼ Sððx; yÞ; zÞ. If x  x0 , y  y0 , then Sðx; yÞ  Sðx0 ; y0 Þ.

~ over X is defined as Definition 1.5.7 [139]: Let X be a universal set. A fuzzy set A ~ A ¼ fx; lA ð xÞjx 2 X g, where lA~ : X ! ½0; 1 and lA~ ð xÞ indicate the degree of ~ membership of x in A.” Definition 1.5.8 [4]: An intuitionistic fuzzy set, over the universal set X, is defined as a ¼ fx; la ð xÞ; ma ð xÞjx 2 X; 0  la ð xÞ  1; 0  ma ð xÞ  1; la ð xÞ þ ma ð xÞ  1g. The values la ð xÞ, ma ð xÞ and 1  la ð xÞ  ma ð xÞ, respectively, are called the degree of membership, the degree of non-membership and the degree of hesitation for the element x. Also, the pair hla ; ma i is called an intuitionistic fuzzy number. ~ on X defined Definition 1.5.9 [15]: Let X be an initial universe of objects. A set A   ðsÞ   ~ as A ¼ x; lA~ x jx 2 X is called a hesitant fuzzy set, where (i) lA~ is a mapping defined by lA~ : X ! ½0; 1.   (ii) lA~ xðsÞ is a set of some different values in [0,1]. (iii) s represents the number of possible membership degrees of the element ~ x 2 X to A. ~ on X defined Definition 1.5.10 [33]: Let X be an initial universe of objects. A set A   ðsÞ   ðtÞ   ~ ¼ x; l ~ x ; m ~ x jx 2 X is called a dual hesitant fuzzy set, where l ~ ; m ~ as A A A A A is a mapping defined by

8

1 Appropriate Weighted Averaging Aggregation Operator …

lA~ ; mA~ : X ! ½0; 1; where

    (i) lA~ xðsÞ ; mA~ xðtÞ is a set of some different values in [0,1]. (ii) s represents the number of possible membership degrees of the element ~ x 2 X to A. (iii) t represents the number of possible non-membership degrees of the element ~ x 2 X to A.

Definition 1.5.11 [124]: An intuitionistic fuzzy multiplicative set, over the universal set X, is defined as a ¼ fhx; la ð xÞ; ma ð xÞijx 2 X; 1q  la ð xÞ  q;  ma ð xÞ  q; la ð xÞma ð xÞ  1; q [ 1g. The values la ð xÞ, ma ð xÞ and 1  la ð xÞ  ma ð xÞ are called the degree of membership, the degree of non-membership and the degree of hesitation, respectively, for the element x. Also, the pair hla ; ma i is called an intuitionistic fuzzy multiplicative number.

1 q

Definition 1.5.12 [134]: A Pythagorean fuzzy set, over the universal set X, is defined as a ¼ fx; la ð xÞ; ma ð xÞjx 2 X; 0  la ð xÞ  1; 0  ma ð xÞ  1; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðla ð xÞÞ þ ðma ð xÞÞ  1g. The values la ð xÞ, ma ð xÞ and 1  ðla ð xÞÞ2 ðma ð xÞÞ2 are called the degree of membership, the degree of non-membership and the degree of hesitation, respectively, for the element x. Also, the pair hla ; ma i is called a Pythagorean fuzzy number. Definition 1.5.13 [73, 74]: If two sets A and B, having same number of characteristics (say N), are put together to form a pair H with respect  to the  problem W, then the pair H is called a set pair and the number lðH; W Þ ¼ NS þ NP j is called a connection number of the set pair H, where S represents the number of identity characteristics S  P  and P represents the number of contrary characteristics. Furthermore, and degree and the contrary degree of these two sets, N N are called identical     respectively. Assuming NS ¼ a and NP ¼ c, NP ¼ c the connection number     P lðH; W Þ ¼ NS þ NSP i þ N j can also be written as lðH; W Þ ¼ N a þ ð1  a  cÞi þ cj. Definition 1.5.14 [5]: An interval-valued intuitionistic fuzzy set,    L over Uthe universal  ð x Þ ; ma ð xÞ; ma ð xÞ jx 2 X; set X, is defined as a ¼ fx; lLa ð xÞ; lU a U U 0  lL ð xÞ  lU ð xÞ  1; 0  mLa ð xÞ  mU a ðxÞ  1; la ð xÞ þ ma ð xÞ  1g. The intervals  L a U a  L   U L L la ð xÞ; la ð xÞ , ma ð xÞ; mU and 1  lU are a ð xÞ a ð xÞ  ma ð xÞ; 1  la ð xÞ  ma ð xÞ called the interval of degree of membership, the interval of degree of non-membership and the interval of   degree  of hesitation, respectively, for the L U ; m ; m element x. Also, the pair h lLa ; lU a a a i is called an interval-valued intuitionistic fuzzy number.

1.5 Some Basic Definitions

9

Definition 1.5.15 [66]: Let X be an initial universe of objects and E the set of parameters in relation to objects in X and AE. Parameters are often attributes, characteristics or properties of objects. Let P ð X Þ denote the power set of X. Then,   ~ A is called a soft set over X, where F ~ is a mapping defined by the pair F; ~ : A ! P ð X Þ: F Definition 1.5.16 [66]: Let F ð X Þ be the set of all fuzzy subsets in X. Then, the pair   ~ A is called a fuzzy soft set over X, where F ~ is a mapping defined by F; ~ : A ! F ð X Þ: F Definition 1.5.17 [66]: Let Hð X Þ be the set of all hesitant fuzzy subsets in X.   ~ A is called a hesitant fuzzy soft set over X, where F ~ is a mapping Then, the pair F; defined by ~ : A ! Hð X Þ: F Definition 1.5.18 [33]: Let DHð X Þ be the set of all dual hesitant fuzzy subsets in   ~ A is called a dual hesitant fuzzy soft set over X, where F ~ is a X. Then, the pair F; mapping defined by ~ : A ! DHð X Þ: F Definition 1.5.19 [105]: A single-valued neutrosophic set, over the universal set X, is defined as a ¼ fx; la ð xÞ; ma ð xÞ; ha ð xÞjx 2 X; 0  la ð xÞ  1; 0  ma ð xÞ  1; 0  ha ð xÞ  1; la ð xÞ þ ma ð xÞ þ ha ð xÞ  3g. The values la ð xÞ, ma ð xÞ and ha ð xÞ, respectively, are called the degree of membership, the degree of non-membership and the degree of indeterminacy for the element x. Also, the triplet hla ; ma ; ha i is called a single-valued neutrosophic number. Definition 1.5.20 [90]: A single-valued neutrosophic hesitant fuzzy set, over  the universal set X, is defined as a ¼ f x; la ð xÞ; ma ð xÞ; ha ð xÞjx 2 X; la ð xÞ ¼ cj ; j ¼    1; 2; . . .; l : 0  cj  1g; ma ð xÞ ¼ dj ; j ¼ 1; 2; . . .; p : 0  dj  1 ; ha ð xÞ ¼ fj ; j ¼        1; 2; . . .; m : 0  fj  1g; max1  j  l cj þ max1  j  p dj þ max1  j  m fj  3 . The values la ð xÞ, ma ð xÞ and ha ð xÞ, respectively, are called the set of degree of membership, the set of degree of non-membership and the set of degree of indeterminacy for the element x. Also, the set fla ; ma ; ha g is called a single-valued neutrosophic hesitant fuzzy number.

10

1.6

1 Appropriate Weighted Averaging Aggregation Operator …

Drawbacks of Some Existing Weighted Geometric Aggregation Operators Under Various Extensions of the Fuzzy Environment

In this section, some drawbacks of the existing weighted geometric aggregation operators [1, 2, 10, 16, 17, 19, 20, 22, 23, 25, 28, 33, 42–45, 47, 60, 97, 114, 130, 142], defined under various extensions of the fuzzy environment, have been pointed out. It can be easily verified that the same drawbacks are also occurring in the other existing weighted geometric aggregation operators defined under various extensions of the fuzzy environment.

1.6.1

Drawbacks of Xu and Yager’s Intuitionistic Fuzzy Weighted Geometric Aggregation Operator

Using the t-norm T ðx; yÞ ¼ xy and the t-conorm Sðx; yÞ ¼ x þ y  xy ¼ 1  ð1  xÞð1  yÞ, Xu and Yager [130] proposed (i) The expression (1.6.1.1) to evaluate the multiplication of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. a1  a2 ¼ hl1 l2 ; 1  ð1  m1 Þð1  m2 Þi

ð1:6:1:1Þ

(ii) The expression (1.6.1.2) to evaluate the positive power k of an intuitionistic fuzzy number a ¼ hl; mi. D E ak ¼ lk ; 1  ð1  mÞk

ð1:6:1:2Þ

Also, using the expression (1.6.1.1) and the expression (1.6.1.2), Xu and Yager [130] proposed the intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3). * ni¼1 ðai Þwi ¼

n Y i¼1

lwi i ; 1

n Y  ð 1  m i Þ wi

+ ð1:6:1:3Þ

i¼1

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy numbers. (ii) wi is the weight assigned P to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1.

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

11

Beliakov et al. [6] pointed out that if there exist an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 0 and mi ¼ 1. Then, the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3), will always be h0; 1i. Beliakov et al. [6] further pointed out that due to this drawback, the mono w tonicity property “ai 4a0i 8 i ) ni¼1 ðai Þwi 4 ni¼1 a0i i ; where ai ¼ hli ; mi i and

a0i ¼ l0i ; m0i are intuitionistic fuzzy numbers” will not be satisfied for the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3). Hence, the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3) is not valid as an aggregation operator is said to be valid if it satisfies the monotonicity property in addition to some other necessary properties [132]. To validate this claim, Beliakov et al. [6] considered the intuitionistic fuzzy numbers a1 ¼ h0:3; 0:2i; a2 ¼ h0; 1i; a01 ¼ h0:5; 0:4i and a02 ¼ h0; 1i. Beliakov et al. [6] claimed that as a2 ¼ a02 and according to the existing approach for comparing intuitionistic fuzzy numbers [130], a1 a01 . Therefore,  w according to the monotonicity property, the relation 2i¼1 ðai Þwi 2i¼1 a0i i should hold. While, it can be easily verified that if w1 ¼ 0:4 and w2 ¼ 0:6. Then, on applying the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3),  w 2i¼1 ðai Þwi ¼ 2i¼1 a0i i ¼ a02 ¼ h0; 1i. This clearly indicates that the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3) is not valid.

1.6.2

Drawbacks of Wang and Liu’s Intuitionistic Fuzzy Weighted Geometric Aggregation Operator

Garg [19] claimed that using the t-conorm Sðx; yÞ ¼ 1xþþxyy and the t-norm xy T ðx; yÞ ¼ 1 þ ð1x Þð1yÞ, Wang and Liu [114] have proposed (i) The expression (1.6.2.1) to evaluate the multiplication of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i.  a1  a2 ¼

l1 l2 m1 þ m2 ; 1 þ ð1  l1 Þð1  l2 Þ 1 þ m1 m2

ð1:6:2:1Þ

(ii) The expression (1.6.2.2) to evaluate the positive power k of an intuitionistic fuzzy number a ¼ hl; mi. * a ¼ k

2lk

ð 1 þ mÞ k  ð 1  mÞ k

; ð2  lÞk þ lk ð1 þ mÞk þ ð1  mÞk

+ ð1:6:2:2Þ

1 Appropriate Weighted Averaging Aggregation Operator …

12

(iii) The intuitionistic fuzzy weighted geometric aggregation operator (1.6.2.3).  ni¼1 awi i

¼

 Qn Qn wi  wi Q n wi 2 i¼1 ðli Þ i¼1 ð1 þ mi Þ  Qi¼1 ð1  mi Þ Q Qn Q ; wi n wi n wi n wi i¼1 ð2  li Þ þ i¼1 ðli Þ i¼1 ð1 þ mi Þ þ i¼1 ð1  mi Þ ð1:6:2:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy numbers. (ii) wi is the weight assigned P to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. Garg [19] also claimed that the membership value of the aggregated intuitionistic fuzzy number, obtained on applying the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.2.3), is independent from the change in the membership degree and the non-membership degree of the intuitionistic fuzzy numbers. To validate the claim, Garg [19] considered the four intuitionistic fuzzy numbers a1 ¼ h0; 0:52i; a2 ¼ h0:23; 0:73i; a3 ¼ h0:50; 0:43i; a4 ¼ h0:22; 0:76i with w1 ¼ 0:1; w2 ¼ 0:4; w3 ¼ 0:2; w4 ¼ 0:1 and showed that aggregated intuitionistic fuzzy number, obtained on applying the intuitionistic fuzzy weighted geometric aggregation operator (1.6.2.3), is h0; 0:6751i. Then, Garg [19] considered the four different intuitionistic fuzzy numbers b1 ¼ h0; 0:45i; b2 ¼ h0:32; 0:63i; b3 ¼ h0:55; 0:45i, b4 ¼ h0:63; 0:36i and showed that that aggregated intuitionistic fuzzy number, obtained on applying the intuitionistic fuzzy weighted geometric aggregation operator (1.6.2.3), is h0; 0:5043i. Furthermore, Garg [19] claimed that it is obvious that in both the cases, the membership value of the aggregated intuitionistic fuzzy number is same, and hence, it may be concluded that the membership value of the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.2.3), is not depending upon the changes in the membership value and non-membership value of the considered intuitionistic fuzzy numbers.

1.6.3

Drawbacks of Garg’s Intuitionistic Fuzzy Interactive Weighted Geometric Aggregation Operator

To resolve the shortcomings of the existing intuitionistic fuzzy weighted geometric aggregation operator [114], Garg [19] proposed (i) The expression (1.6.3.1) to evaluate the multiplication of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i.

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

o Q2 * n Q2 2 ð 1  m Þ  ð 1  l  m Þ i i i i¼1 i¼1 a1  a2 ¼ ; Q2 Q2 i¼1 ð1 þ mi Þ þ i¼1 ð1  mi Þ + Q2 Q2 i¼1 ð1 þ mi Þ  i¼1 ð1  mi Þ Q2 Q2 i¼1 ð1 þ mi Þ þ i¼1 ð1  mi Þ

13

ð1:6:3:1Þ

(ii) The expression (1.6.3.2) to evaluate the positive power k of an intuitionistic fuzzy number a ¼ hl; mi. o * n + 2 ð1  mÞk ð1  l  mÞk ð1 þ mÞk ð1  mÞk k a ¼ ; ð1 þ mÞk þ ð1  mÞk ð1 þ mÞk þ ð1  mÞk

ð1:6:3:2Þ

Also, using the expression (1.6.3.1) and the expression (1.6.3.2), Garg [19] proposed the intuitionistic fuzzy interactive weighted geometric aggregation operator (1.6.3.3). ni¼1 awi i

  Qn wi Q n wi  2 i¼1 ð1  mi Þ  i¼1 ð1  li  mi Þ Qn Qn ¼ ; wi wi i¼1 ð1 þ mi Þ þ i¼1 ð1  mi Þ Q Qn ð1 þ mi Þwi  ni¼1 ð1  mi Þwi Q Qni¼1 wi n wi i¼1 ð1 þ mi Þ þ i¼1 ð1  mi Þ

ð1:6:3:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy numbers. (ii) wi is the weight assigned P to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3), is also occurring in the existing intuitionistic fuzzy interactive weighted geometric aggregation operator (1.6.3.3). Therefore, the existing intuitionistic fuzzy interactive weighted geometric aggregation operator (1.6.3.3) is not valid. “If there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 0 and mi ¼ 1. Then, the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy interactive weighted geometric aggregation operator (1.6.3.3), will always be h0; 1i. Hence, the monotonicity property ‘ai 4a0i 8 i )  w

ni¼1 ðai Þwi 4 ni¼1 a0i i , where ai ¼ hli ; mi i and a0i ¼ l0i ; m0i are intuitionistic fuzzy numbers’ will not be satisfied for the existing intuitionistic fuzzy interactive weighted geometric aggregation operator (1.6.3.3).”

1 Appropriate Weighted Averaging Aggregation Operator …

14

1.6.4

Drawbacks of Garg’s Intuitionistic Fuzzy Hamacher Interactive Weighted Geometric Aggregation Operator

Garg [17, 28] proposed (i) The expression (1.6.4.1) to evaluate the multiplication of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i.  a1  a2 ¼

cð1  m1 Þð1  m2 Þ  cð1  l1  m1 Þð1  l2  m2 Þ ; ð1 þ ðc  1Þm1 Þð1 þ ðc  1Þm2 þ ðc  1Þð1  m1 Þð1  m2 ÞÞ ð1 þ ðc  1Þm1 Þð1 þ ðc  1Þm2 Þ  ð1  m1 Þð1  m2 Þ ð1 þ ðc  1Þm1 Þð1 þ ðc  1Þm2 þ ðc  1Þð1  m1 Þð1  m2 ÞÞ ð1:6:4:1Þ

(ii) The expression (1.6.4.2) to evaluate the positive power k of an intuitionistic fuzzy number a ¼ hl; mi. * a ¼ k

cð1  mÞk cð1  l  mÞk

ð1 þ ðc  1ÞmÞk ð1  mÞk

+

; ð1 þ ðc  1ÞmÞk þ ðc  1Þð1  mÞk ð1 þ ðc  1ÞmÞk þ ðc  1Þð1  mÞk ð1:6:4:2Þ

Also, using the expression (1.6.4.1) and the expression (1.6.4.2), Garg [17, 28] proposed the intuitionistic fuzzy Hamacher interactive weighted geometric aggregation operator (1.6.4.3). Q Q c ni¼1 ð1  mi Þwi  ni¼1 ð1  li  mi Þwi Qn ¼ Qn wi wi ; i¼1 ð1 þ ðc  1Þmi Þ þ ðc  1Þ i¼1 ð1  mi Þ Q Qn ð1 þ ðc  1Þmi Þwi  ni¼1 ð1  mi Þwi i¼1 Qn Qn wi wi i¼1 ð1 þ ðc  1Þmi Þ þ ðc  1Þ i¼1 ð1  mi Þ 

ni¼1 awi i

ð1:6:4:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy numbers. (ii) wi is the weight assigned P to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3), is also occurring in the existing intuitionistic fuzzy Hamacher interactive weighted geometric aggregation operator (1.6.4.3). Therefore, the existing intuitionistic fuzzy Hamacher interactive weighted geometric aggregation operator (1.6.4.3) is not valid.

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

15

“If there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 0 and mi ¼ 1, then the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy Hamacher interactive weighted geometric aggregation operator (1.6.4.3), will always be h0; 1i. Hence, the monotonicity property  w

‘ai 4a0i 8 i ) ni¼1 ðai Þwi 4 ni¼1 a0i i , where ai ¼ hli ; mi i and a0i ¼ l0i ; m0i are intuitionistic fuzzy numbers’ will not be satisfied for the existing intuitionistic fuzzy Hamacher interactive weighted geometric aggregation operator (1.6.4.3).”

1.6.5

Drawbacks of He et al.’s Intuitionistic Fuzzy Interaction Weighted Geometric Aggregation Operator

He et al. [60] proposed (i) The expression (1.6.5.1) to evaluate the multiplication of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. a1  a2 ¼ hð1  m1 Þð1  m2 Þ  ð1  ðl1 þ m1 ÞÞð1  ðl2 þ m2 ÞÞ;

ð1:6:5:1Þ

1  ð1  m1 Þð1  m2 Þi

(ii) The expression (1.6.5.2) to evaluate the positive power k of an intuitionistic fuzzy number a ¼ hl; mi. D E ak ¼ ð1  mÞk ð1  ðl þ mÞÞk ; 1  ð1  mÞk

ð1:6:5:2Þ

Also, using the expression (1.6.5.1) and the expression (1.6.5.2), He et al. [60] proposed the intuitionistic fuzzy interaction weighted geometric aggregation operator (1.6.5.3). * ni¼1 ðai Þwi ¼

n n n Y Y Y ð1  mi Þwi  ð1  ðli þ mi ÞÞwi ; 1  ð1  mi Þwi i¼1

i¼1

+ ð1:6:5:3Þ

i¼1

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy numbers. (ii) wi is the weight assigned Pn to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and i¼1 wi ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3), is also occurring in the existing intuitionistic fuzzy interaction weighted geometric aggregation operator (1.6.5.3). Therefore, the existing

1 Appropriate Weighted Averaging Aggregation Operator …

16

intuitionistic fuzzy interaction weighted geometric aggregation operator (1.6.5.3) is not valid. “If there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 0 and mi ¼ 1, then the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy weighted interaction geometric aggregation operator (1.6.5.3), will always be h0; 1i. Hence, the monotonicity property ‘ai 4a0i 8 i )  w

ni¼1 ðai Þwi 4 ni¼1 a0i i , where ai ¼ hli ; mi i and a0i ¼ l0i ; m0i are intuitionistic fuzzy numbers’ will not be satisfied for the existing intuitionistic fuzzy interaction weighted geometric aggregation operator (1.6.5.3).”

1.6.6

Drawbacks of Chen and Chang’s Intuitionistic Fuzzy Weighted Geometric Aggregation Operator

Chen and Chang [10] pointed out the drawbacks of the existing intuitionistic fuzzy weighted geometric aggregation operator [88]. To resolve the drawbacks, Chen and Chang [10] proposed (i) The expression (1.6.6.1) to evaluate the multiplication of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. a1  a2 ¼ h1  ð1  l1 Þð1  l2 Þ; ð1  l1 Þð1  l2 Þ ð1  ðl1 þ m1 ÞÞð1  ðl2 þ m2 ÞÞi

ð1:6:6:1Þ

(ii) The expression (1.6.6.2) to evaluate the positive power k of an intuitionistic fuzzy number a ¼ hl; mi. D E ak ¼ 1  ð1  lÞk ; ð1  lÞk ð1  ðl þ mÞÞk

ð1:6:6:2Þ

Also, using the expression (1.6.6.1) and the expression (1.6.6.2), Chen and Chang [10] proposed the intuitionistic fuzzy weighted geometric aggregation operator (1.6.6.3). * ni¼1 ðai Þwi ¼

n  n  n  Y w Y w Y w 1 1  la i i ; 1  la i i  1  lai  mai i i¼1

i¼1

+

i¼1

ð1:6:6:3Þ where, (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy numbers. (ii) wi is the weight assigned P to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1.

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

17

It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3), is also occurring in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.6.3). Therefore, the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.6.3) is not valid. “If there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 0 and mi ¼ 1, then the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.6.3), will always  w be h0; 1i. Hence, the monotonicity property ‘ai 4a0i 8 i ) ni¼1 ðai Þwi 4 ni¼1 a0i i ,

where ai ¼ hli ; mi i and a0i ¼ l0i ; m0i are intuitionistic fuzzy numbers’ will not be satisfied for the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.6.3).”

1.6.7

Drawbacks of Garg and Kumar’s Connection Number-Based Power Geometric Aggregation Operator

Garg and Kumar [47] claimed that if (i) The intuitionistic fuzzy numbers a11 ¼ h0:3; 0:5i, a12 ¼ h0; 1i and a13 ¼ h0:4; 0:2i represents the rating values of the alternative H1 corresponding to the attributes G1 ; G2 and G3 , respectively. (ii) The intuitionistic fuzzy numbers a21 ¼ h0:4; 0:3i, a22 ¼ h0:2; 0:3i and a23 ¼ h0; 1i represents the rating values of the alternative H2 corresponding to the attributes G1 ; G2 and G3 , respectively. Then, on applying the existing intuitionistic fuzzy weighted geometric aggregation operators [19, 60, 128, 130, 135], the same intuitionistic fuzzy number h0; 1i, representing the aggregated value, is obtained corresponding to both the alternatives H1 and H2 . To resolve this issue, Garg and Kumar [47], firstly, used the existing method [40] to transform the ith intuitionistic ai ¼ hli ; mi i into its equivalent  fuzzy  number  ith connection number di ¼ a1i þ a2i i þ a3i j, where (i) (ii) (iii) (iv) (v)

a1i ¼ li ð1  mi Þ. a2i ¼ 1  li ð1  mi Þ  mi ð1  li Þ. a3i ¼ mi ð1  li Þ. 0  a1i  a2i  a3i  1. a1i þ a2i þ a3i ¼ 1.

Then, using

1 Appropriate Weighted Averaging Aggregation Operator …

18

(i) The existing expression (1.6.7.1)  to evaluate   the multiplication   oftwo  connection numbers [8] d1 ¼ a11 þ a21 i þ a31 j and d2 ¼ a12 þ a22 i þ a32 j.           d1  d2 ¼ a11 a12 þ 1  1  a21 1  a22 i þ a11 þ a31 a12 þ a32  a11 a12 j ð1:6:7:1Þ (ii) The existing expression (1.6.7.2) to evaluate the positive power k of a connection number [8] d ¼ a1 þ ða2 Þi þ ða3 Þj.  k  k   k  k  dk ¼ a1 þ 1  1  a2 i þ a1 þ a 3  a1 j

ð1:6:7:2Þ

(iii) The existing expression (1.6.7.3) to evaluate between two oftwo   the distance  connection numbers [41] d1 ¼ a11 þ a21 i þ a31 j and d2 ¼ a12 þ a22 i þ  3 a2 j. d ð d1 ; d2 Þ ¼

     1  1 a1  a12  þ a21  a22  þ a31  a32  3

ð1:6:7:3Þ

Garg [47] proposed the connection number-based power geometric aggregation operator (1.6.7.4). nt¼1

ð dt Þ

Pn1 þ T ðdt Þ t¼1

ð1 þ T ðdt ÞÞ

! 1 þ T ðdt Þ n  Pn1 þ T ðdt Þ n  Y Y  P n ð 1 þ T ð dt Þ Þ ð 1 þ T ð dt Þ Þ 1 2 t¼1 t¼1 ¼ at þ 1 1  at i t¼1

þ

n  Y t¼1

a1t þ a3t

t¼1

! n  Pn1 þ T ðdt Þ Y ð 1 þ T ð dt Þ Þ ð 1 þ T ð d Þ Þ t t¼1 t¼1  a1t j

1 þ T ð dt Þ  Pn

t¼1

ð1:6:7:4Þ where T ð dt Þ ¼ 1 

n X

  d dp ; di :

p¼1 p 6¼ t It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.1), is also occurring in the existing connection number-based power geometric aggregation operator (1.6.7.4). Therefore, the existing connection number-based power geometric aggregation operator (1.6.7.4) is not valid. “If there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 0 and mi ¼ 0, then using the existing method [40], used by Garg and Kumar [47] to

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

19

transform an intuitionistic fuzzy number into its equivalent connection number, the intuitionistic fuzzy number ai ¼ h0; 0i will be transformed into its equivalent connection number di ¼ a1i þ a2i i þ a3i j ¼ 0 þ ð1Þi þ ð0Þj. Therefore, the aggregated connection number, obtained on applying the existing connection number-based power geometric aggregation operator (1.6.7.4), will be 0 þ ð1Þi þ ð0Þj. Hence, the monotonicity property ‘di 4d0i 8 i ) ni¼1 ðdi Þwi 4  w      0  0  0 ni¼1 d0i i , where di ¼ a1i þ a2i i þ a3i j and d0i ¼ a1i þ a2i i þ a3i j are connection numbers’ will not be satisfied for the existing connection number-based power geometric aggregation operator (1.6.7.4).”

1.6.8

Drawbacks of Zhang’s Linguistic Intuitionistic Fuzzy Weighted Geometric Aggregation Operator

Zhang [142] proposed (i) The expression (1.6.8.1) to evaluate the of Etwo linguistic intuD E multiplication D itionistic fuzzy numbers a1 ¼ sa11 ; sa21 and a2 ¼ sa12 ; sa22 . * a1  a2 ¼

+ s 2  2  ; s t

a

a

1 t

2 t

a

 1

tt 1 t1

a

 1

1 t2

ð1:6:8:1Þ

(ii) The expression (1.6.8.2) to evaluate the positive power k of a linguistic intuitionistic fuzzy number a ¼ hsa1 ; sa2 i.  a ¼ k

s

2 k

tðat

Þ

;s

ð1:6:8:2Þ

1 k

ttð1at

Þ

Also, using the expression (1.6.8.1) and the expression (1.6.8.2), Zhang [142] proposed the linguistic intuitionistic fuzzy weighted geometric aggregation operator (1.6.8.3). * ni¼1 ðai Þwi ¼

+

2 wi ; s sQ n a i t

t i¼1

where

tt

n Q

a1

1 ti

wi

ð1:6:8:3Þ

i¼1



(i) ai ¼ a1i ; a2i , i ¼ 1; 2; . . .; n are n linguistic intuitionistic fuzzy numbers. (ii) wi is the weight assigned to the ith linguistic intuitionistic fuzzy number P ai ¼ ha1i ; a2i i such that wi  0 and ni¼1 wi ¼ 1.

20

1 Appropriate Weighted Averaging Aggregation Operator …

(iii) 0  a1i  t; 0  a2i  t, (iv) t is a non-negative integer. (v) a1i þ a2i  t. Garg and Kumar [42] claimed that the non-membership value of the aggregated linguistic intuitionistic fuzzy number, obtained on applying the existing linguistic intuitionistic fuzzy weighted geometric aggregation operator (1.6.8.3), is independent from the change in the non-membership degree of the linguistic intuitionistic fuzzy numbers. To validate the claim, Garg and Kumar [42] considered the four linguistic intuitionistic fuzzy numbers a1 ¼ hs4 ; s3 i; a2 ¼ hs2 ; s4 i; a3 ¼ hs5 ; s1 i; a4 ¼ hs4 ; s2 i with w1 ¼ 0:3; w2 ¼ 0:2; w3 ¼ 0:2; w4 ¼ 0:2 and showed that aggregated linguistic intuitionistic fuzzy number, obtained on applying the linguistic intuitionistic fuzzy weighted geometric aggregation operator (1.6.8.3), is hs3:7233 ; s2:5139 i. Then, Garg and Kumar [42] obtained the linguistic intuitionistic fuzzy numbers b1 ¼ hs4 ; s1 i; b2 ¼ hs2 ; s2 i; b3 ¼ hs5 ; s3 i; b4 ¼ hs4 ; s4 i by replacing the non-membership values s3 , s4 , s1 and s2 with s1 ; s2 ; s3 and s4 , respectively, and showed that the aggregated linguistic intuitionistic fuzzy number, obtained on applying the linguistic intuitionistic fuzzy weighted geometric aggregation operator (1.6.8.3), is hs3:7233 ; s2:5139 i. Furthermore, Garg and Kumar [42] claimed that it is obvious that in both the cases, the non-membership value of the aggregated linguistic intuitionistic fuzzy number is same, and hence, it may be concluded that the non-membership value of the aggregated linguistic intuitionistic fuzzy number, obtained by the linguistic intuitionistic fuzzy weighted geometric aggregation operator (1.6.8.3), is not depending upon the changes in the non-membership values of the considered linguistic intuitionistic fuzzy numbers, which is illogical.

1.6.9

Drawbacks of Garg and Kumar’s Linguistic Connection Number-Based Weighted Geometric Aggregation Operator

To resolve the drawbacks of the existing linguistic intuitionistic fuzzy weighted geometric aggregation operator [142], Garg and Kumar [42], firstly, proposed the a1 ðta2 Þ a1 ðta2 Þ a2 ðta1 Þ a2 ðta1 Þ expressions a1i ¼ i t i , a2i ¼ t  i t i  i t i and a3i ¼ i t i to transform the ith linguistic intuitionistic fuzzy number ai ¼ hsa1i ; sa2i i into its equivalent ith

  linguistic connection number di ¼ sa1i þ sa2i i þ sa3i j, where, 0  a1i  a2i  a3i  t and a1i þ a2i þ a3i ¼ t. Then, Garg and Kumar [42] proposed

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

21

(i) The expression (1.6.9.1) to evaluate multiplication of two  the 

linguistic  connection numbers d1 ¼ sa11 þ sa21 i þ sa31 j and d2 ¼ sa12 þ sa22 i þ sa32 j. 0

1

d1  d2 ¼ s a1 a1  þ @s t

1 2 t2

a2



t 1 1 t1

a2

1 t2

0

1

 Ai þ @s

1 a

1 t

t

þ

a3 1 t

 1 a

2 t

þ

a3 2 t

 1  1   A j 

a

1 t

a

2 t

ð1:6:9:1Þ (ii) The expression (1.6.9.2) to evaluate the positive power k of a linguistic connection number d ¼ sa1 þ ðsa2 Þi þ ðsa3 Þj.    d ¼ s a1 k þ s  k

t

t 1ð1at

t2

2

Þ

   iþ s k t

k

ðat1 þ at3 Þ ðat1 Þ

  j k

ð1:6:9:2Þ

Also, using the expression (1.6.9.1) and the expression (1.6.9.2), Garg and Kumar [42] proposed the linguistic connection number-based weighted geometric aggregation operator (1.6.9.3). 0

1

ni¼1 ðdi Þwi ¼ s Qn a1 wi þ @s Qn i t

i¼1

0

t 1

t2

þ @ s Q n a1 t

i¼1

i t

þ

a3 i t

i¼1

a2

1 ti

wi  Ai 1

ð1:6:9:3Þ

 wi Q 1  wi  A j a n i 

i¼1

t

It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3), is also occurring in the existing linguistic connection number-based weighted geometric aggregation operator (1.6.9.3). Therefore, the existing linguistic connection number-based weighted geometric aggregation operator (1.6.9.3) is not valid. D E “If there exists a linguistic intuitionistic fuzzy number ai ¼ sa1i ; sa2i

such that

¼ 0 and ¼ 0, then using the method, proposed by Garg and Kumar [42] to transform a linguistic intuitionistic fuzzy number into its equivalent linguistic connection number, the linguistic intuitionistic fuzzy number ai ¼ hs0 ; s0 i will be transformed into its equivalent linguistic connection number di ¼ di ¼ sa1i þ

  sa2i i þ sa3i j ¼ s0 þ ðst Þi þ ðs0 Þj. Therefore, the aggregated linguistic connection a1i

a2i

number, obtained on applying the existing linguistic connection number-based weighted geometric aggregation operator (1.6.9.3), will be s0 þ ðst Þi þ ðs0 Þj. Hence,  w the monotonicity property ‘di 4d0i 8 i ) ni¼1 ðdi Þwi 4 ni¼1 d0i i , where di ¼

1 Appropriate Weighted Averaging Aggregation Operator …

22

 

  sa1i þ sa2i i þ sa3i j and d0i ¼ sða1 Þ þ sða2 Þ i þ sða3 Þ j are linguistic connection i i i numbers’ will not be satisfied for the existing linguistic connection number-based weighted geometric aggregation operator (1.6.9.3).”

1.6.10 Drawbacks of Garg and Kumar’s Linguistic Connection Number-Based Power Geometric Aggregation Operator Garg and Kumar [43] pointed out the drawbacks of the existing intuitionistic fuzzy weighted geometric aggregation operators [13, 91]. To resolve the drawbacks, Garg and Kumar [43], firstly, used the existing method [42] to transform the ith linguistic intuitionistic fuzzy number ai ¼ D E sa1i ; sa2i into its equivalent ith linguistic connection number

  di ¼ sa1i þ sa2i i þ sa3i j: where a1i ðta2i Þ . t 1 2 a ta ð Þ a2 ðta1 Þ a2i ¼ t  i t i  i t i . a2 ðta1 Þ a3i ¼ i t i . 0  a1i  a2i  a3i  t. a1i þ a2i þ a3i ¼ t.

(i) a1i ¼ (ii) (iii) (iv) (v)

Then, using (i) The existing expression (1.6.10.1) [42] to evaluate the

multiplication   of two and linguistic connection numbers d1 ¼ sa11 þ sa21 i þ sa31 j

  d2 ¼ sa12 þ sa22 i þ sa32 j. 0

1

d1  d2 ¼ s a1 a1  þ @s t

1 2 t2

a2

t 1 1 t1



a2

1 t2

0

1

 Ai þ @s

1 t

a

1 t

þ

a3 1 t

 1 a

2 t

þ

a3 2 t

 1  1   A j 

a

1 t

a

2 t

ð1:6:10:1Þ (ii) The existing expression (1.6.10.2) [42] to evaluate the positive power k of a linguistic connection number d ¼ sa1 þ ðsa2 Þi þ ðsa3 Þj.

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

   d ¼ s a1 k þ s  k

t

t 1ð1at

t2

2

Þ

   iþ s k t

k

ðat1 þ at3 Þ ðat1 Þ

  j k

23

ð1:6:10:2Þ

(iii) The existing expression (1.6.10.3) [42] to evaluate the distance

 between

 two

of two linguistic connection numbers d1 ¼ sa11 þ sa21 i þ sa31 j and

  d2 ¼ sa12 þ sa22 i þ sa32 j. d ðd1 ; d2 Þ ¼

     1  1 a1  a12  þ a21  a22  þ a31  a32  3t

ð1:6:10:3Þ

Garg and Kumar [43] proposed the linguistic connection number-based power geometric aggregation operator (1.6.10.4). 0 nh¼1

ðdh Þ

Pn1 þ T ðdh Þ

ð 1 þ T ðd h Þ Þ h¼1

1

B C B C B 0 C 1 ¼s þ s B Ci 1 þ T ðd h Þ

2 Pn1 þ T ðdh Þ C Qn a1h Pn ð1 þ T ðdh ÞÞ B Q a n 1 þ T d ð ð Þ Þ @ h AA h¼1 h¼1 t t@1 1 th 2 i¼1

t

i¼1

1

0

C B C B C B 0 1 þ Bs Cj 1 þ T ðd h Þ C B Q a1 a3 wi Q a1 Pn 1 þ T ðd h Þ Þ A A @ t@ n h þ h  n h ð h¼1 t t t i¼1

i¼1

ð1:6:10:4Þ where T ð dh Þ ¼ 1 

n X

  d dh ; dp :

p¼1 p 6¼ h It is pertinent to mention that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.1), is also occurring in the existing linguistic connection number-based power geometric aggregation operator (1.6.10.4). Therefore, the existing linguistic connection number-based power geometric aggregation operator (1.6.10.4) is not valid. D E “If there exists a linguistic intuitionistic fuzzy number ai ¼ sa1i ; sa2i

such that

¼ 0 and ¼ 0, then using the existing method [42], used by Garg and Kumar [43] to transform a linguistic intuitionistic fuzzy number into its equivalent a1i

a2i

1 Appropriate Weighted Averaging Aggregation Operator …

24

linguistic connection number, the linguistic intuitionistic fuzzy number ai ¼ D E sa1i ; sa2i will be transformed into its equivalent linguistic connection number

  di ¼ sa1i þ sa2i i þ sa3i j ¼ s0 þ ðst Þi þ ðs0 Þj. Therefore, the aggregated linguistic connection number, obtained on applying the existing linguistic connection number-based power geometric aggregation operator (1.6.10.4), will be Hence, the monotonicity s0 þ ðst Þi þ ðs0 Þj.

  property  0  wi wi 0 n n ‘di 4di 8 i ) i¼1 ðdi Þ 4 i¼1 di , where di ¼ sa1i þ sa2i i þ sa3i j and d0i ¼

  sða1 Þ þ sða2 Þ i þ sða3 Þ j are connection numbers’ will not be satisfied for the i

i

i

existing linguistic connection number-based power geometric aggregation operator (1.6.10.4).”

1.6.11 Drawbacks of Garg and Kumar’s Linguistic Connection Number-Based Prioritized Geometric Aggregation Operator Garg and Kumar [44], firstly, used the existing method [42] to transform the ith linguistic intuitionistic fuzzy number ai ¼ hsa1i ; sa2i i into its equivalent ith linguistic

  connection number di ¼ sa1i þ sa2i i þ sa3i j: where a1i ðta2i Þ . t a1i ðta2i Þ a2 ðta1 Þ 2 ai ¼ t   i t i . t a2 ðta1 Þ a3i ¼ i t i : 0  a1i  a2i  a3i  t. a1i þ a2i þ a3i ¼ t.

(i) a1i ¼ (ii) (iii) (iv) (v)

Then, using (i) The existing expression (1.6.11.1) [42] to evaluate the

multiplication   of two and linguistic connection numbers d1 ¼ sa11 þ sa21 i þ sa31 j

  d2 ¼ sa12 þ sa22 i þ sa32 j. 0

1

d1  d2 ¼ s a1 a1  þ @s t

1 2 t2

a2

t 1 1 t1



a2

1 t2

0

1

 Ai þ @s

1 t

a

1 t

þ

a3 1 t

 1 a

2 t

þ

a3 2 t

 1  1   A j 

a

1 t

a

2 t

ð1:6:11:1Þ

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

25

(ii) The existing expression (1.6.11.2) [42] to evaluate the positive power k of a linguistic connection number d ¼ sa1 þ ðsa2 Þi þ ðsa3 Þj. 

d ¼ s  a1  k þ s  k

t

t2

t 1ð1at

2

Þ

   iþ s k t

k

ðat1 þ at3 Þ ðat1 Þ

  j k

ð1:6:11:2Þ

Garg and Kumar [44] proposed the linguistic connection number-based prioritized geometric aggregation operator (1.6.11.3). 1

0 PnT ðdh Þ

nh¼1 ðdh Þ

h¼1

ðT ðdh ÞÞ

C B C B C B 0 1 ¼s þ s Ci B T ð dh Þ T ðdh Þ

2  Pn C Qn a1h Pn ðT ðdh ÞÞ B Q a n T d @ ð ð Þ Þ h AA h¼1 h¼1 t t@1 1 h i¼1

t2

i¼1

t

1

0

C B C B B 0 1C þ Bs Cj T ðdh Þ C B Q a1 a3 wi Q a1 Pn T ðdh ÞÞ A A @ t@ n h þ h  n h ð h¼1 i¼1

t

t

i¼1

t

ð1:6:11:3Þ where, Qn Sðdh Þ p ¼ 2 t , h ¼ 2; 3; . . .; n with T ðd1 Þ ¼ 1. p 6¼ h  ða1 a3 Þðt þ a2h Þ 1 (ii) Sðdh Þ ¼ 2 t þ h h t . (i) T ðdh Þ ¼

It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3), is also occurring in the existing linguistic connection number-based prioritized geometric aggregation operator (1.6.11.3). Therefore, the existing linguistic connection number-based prioritized geometric aggregation operator (1.6.11.3) is not valid. D E “If there exists a linguistic intuitionistic fuzzy number ai ¼ sa1i ; sa2i

such that

a1i ¼ a2i ¼ 0, then using the existing expression [42], used by Garg and Kumar [44] to transform a linguistic intuitionistic fuzzy number into a linguistic connection number, the linguistic intuitionistic fuzzy number ai ¼ hs0 ; s0 i will be transformed into its equivalent linguistic connection number di ¼ s0 þ ðst Þi þ ðs0 Þj, i.e., a1i ¼ 0; a2i ¼ t and a3i ¼ 0. Therefore, the aggregated linguistic connection number, obtained on applying the existing linguistic connection number-based prioritized geometric aggregation operator (1.6.11.3), will be s0 þ ðst Þi þ ðs0 Þj. Hence, the

1 Appropriate Weighted Averaging Aggregation Operator …

26

 w monotonicity property ‘di 4d0i 8 i ) ni¼1 ðdi Þwi 4 ni¼1 d0i i , where di ¼      0  0  0 a1i þ a2i i þ a3i j and d0i ¼ a1i þ a2i i þ a3i j are linguistic connection numbers’ will not be satisfied for the existing linguistic connection number-based prioritized geometric aggregation operator (1.6.11.3).”

1.6.12 Drawbacks of Arora and Garg’s Intuitionistic Fuzzy Soft Weighted Geometric Aggregation Operator Arora and Garg [2] proposed (i) The expression (1.6.12.1) to evaluate the multiplication of two intuitionistic fuzzy soft numbers a11 ¼ hl11 ; m11 i, a12 ¼ hl12 ; m12 i. a11  a12 ¼ hl11 l12 ; 1  ð1  m11 Þð1  m12 Þi

ð1:6:12:1Þ

(ii) The expression (1.6.12.2) to evaluate the positive power k of an intuitionistic fuzzy soft number a ¼ hl; mi. D E ak ¼ lk ; 1  ð1  mÞk

ð1:6:12:2Þ

. Using the expression (1.6.12.1) and the expression (1.6.12.2), Arora and Garg [2] proposed the intuitionistic fuzzy soft weighted geometric aggregation operator (1.6.12.3). m j¼1



ni¼1

 gi nj aij ¼

*

m n Y Y j¼1

i¼1

!nj g liji

;1 

m n  Y Y j¼1

1  mij

gi

!nj + ð1:6:12:3Þ

i¼1

where (a) gi [ 0 8 i ¼ 1; 2; . . .; n: (b) nj [ 0 8 j ¼ 1; 2; . . .; m: Pn (i) gi ¼ 1. Pi¼1 m (ii) j¼1 nj ¼ 1. Garg and Arora [32] pointed out the following drawbacks in the existing intuitionistic fuzzy soft weighed geometric aggregation operator (1.6.12.3).

(i) If there exists an intuitionistic fuzzy soft number aij ¼ lij ; mij such that lij ¼ 0, then the membership value of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted geometric aggregation operator (1.6.12.3),

will be 0, i.e., if there exists an intuitionistic fuzzy soft number aij ¼ lij ; mij such that lij ¼ 0, then the membership value of the

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

27

aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted geometric aggregation operator (1.6.12.3), will be independent from the membership values of remaining intuitionistic fuzzy soft numbers, which is illogical. To validate this claim, Garg and Arora [32] considered the intuitionistic fuzzy soft numbers a11 ¼ h0; 0:5i, a12 ¼ h0:3; 0:6i, a21 ¼ h0:5; 0:4i, a22 ¼ h0:2; 0:7i with g1 ¼ 0:7, g2 ¼ 0:3; n1 ¼ 0:4; n2 ¼ 0:6 and claimed that the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted geometric aggregation operator (1.6.12.3), is h0; 0:5126i. It is obvious that the membership value of the aggregated intuitionistic fuzzy soft number is 0. (ii) The membership value of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted geometric aggregation operator (1.6.12.3), is independent from the changes in the non-membership values of the considered intuitionistic fuzzy soft numbers. To validate this claim, Garg and Arora [32], firstly, considered the intuitionistic fuzzy soft numbers a11 ¼ h0:6; 0:3i, a12 ¼ h0:3; 0:4i, a21 ¼ h0:2; 0:6i, a22 ¼ h0:1; 0:7i with g1 ¼ 0:6, g2 ¼ 0:4; n1 ¼ 0:8; n2 ¼ 0:2 and claimed that the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted geometric aggregation operator aggregation operator (1.6.12.3), is h0:2730; 0:5268i. Then, Garg and Arora [32] replaced the intuitionistic fuzzy soft numbers a12 ¼ h0:3; 0:4i and a22 ¼ h0:1; 0:7i with the intuitionistic fuzzy soft numbers b12 ¼ h0:3; 0:6i and b22 ¼ h0:1; 0:8i. Garg and Arora [32] claimed that the new aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted geometric aggregation operator (1.6.12.3), is h0:2730; 0:55565i. It is obvious that the membership value of the new aggregated intuitionistic fuzzy soft number is also 0:2730.

1.6.13 Drawbacks of Arora and Garg’s Intuitionistic Fuzzy Soft Prioritized Weighted Geometric Aggregation Operator Arora and Garg [1] proposed (i) The expression (1.6.13.1) to evaluate the multiplication of two intuitionistic fuzzy soft numbers a11 ¼ hl11 ; m11 i, a12 ¼ hl12 ; m12 i. a11  a12 ¼ hl11 l12 ; 1  ð1  m11 Þð1  m12 Þi

ð1:6:13:1Þ

(ii) The expression (1.6.13.2) to evaluate the positive power k of an intuitionistic fuzzy soft number a ¼ hl; mi.

1 Appropriate Weighted Averaging Aggregation Operator …

28

D E ak ¼ lk ; 1  ð1  mÞk

ð1:6:13:2Þ

Using the expression (1.6.13.1) and the expression (1.6.13.2), Arora and Garg [1] proposed the intuitionistic fuzzy soft prioritized weighted geometric aggregation operator (1.6.13.3).

m i¼1

1PTmi !PTmi *0 gj Pgnj Ti P n T n Y i i¼1 gj   i¼1 gj j¼1 A n @ j¼1 j¼1 aij ¼ lij ; j¼1 m Y i¼1

1

m n  Y Y i¼1

 Pn

!PTmi

gj

1  mij

j¼1

gj

i¼1

Ti

+

j¼1

ð1:6:13:3Þ where (i) g1 ¼ T1 ¼ 1. Q Sðail Þ; j ¼ 2; 3; . . .; n. (ii) gj ¼ j1 Ql¼1 i1 (iii) Ti ¼ k¼1 Sðaik Þ; i ¼ 2; 3; . . .; m.   1 þ lij mij . (iv) S aij ¼ 2 Garg and Arora [32] pointed out the following drawbacks in the existing intuitionistic fuzzy soft prioritized weighted geometric aggregation operator (1.6.13.3).

(i) If there exists an intuitionistic fuzzy soft numbers aij ¼ lij ; mij such that lij ¼ 0, then the membership values of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft prioritized weighted geometric aggregation operator (1.6.13.3), will be

0, i.e., if there exists an intuitionistic fuzzy soft number aij ¼ lij ; mij such that lij ¼ 0, then the membership value of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft prioritized weighted geometric aggregation operator (1.6.13.3), will be independent from the membership values of remaining intuitionistic fuzzy soft numbers, which is illogical. To validate this claim, Garg and Arora [32] considered the intuitionistic fuzzy soft numbers a11 ¼ h0; 0:5i, a12 ¼ h0:3; 0:6i, a21 ¼ h0:5; 0:4i, a22 ¼ h0:2; 0:7i with g1 ¼ 0:7, g2 ¼ 0:3; n1 ¼ 0:4; n2 ¼ 0:6 and claimed that the aggregated intuitionistic fuzzy soft number, obtained by the expression (1.6.13.3), is h0; 0:5144i. It is obvious that the membership value of the aggregated intuitionistic fuzzy soft number is 0. (ii) The membership value of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft prioritized weighted geometric aggregation operator (1.6.13.3), is independent from the changes in the degree of non-membership values of the considered intuitionistic fuzzy soft numbers.

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

29

To validate this claim, Garg and Arora [32], firstly, considered the intuitionistic fuzzy soft numbers a11 ¼ h0:6; 0:3i, a12 ¼ h0:3; 0:4i, a21 ¼ h0:2; 0:6i, a22 ¼ h0:1; 0:7i with g1 ¼ 0:6, g2 ¼ 0:4; n1 ¼ 0:8; n2 ¼ 0:2 and claimed that the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft prioritized weighted geometric aggregation operator (1.6.13.3), is h0:4423; 0:3966i. Then, Garg and Arora [32] replaced the intuitionistic fuzzy soft numbers a12 ¼ h0:3; 0:4i and a22 ¼ h0:1; 0:7i with the intuitionistic fuzzy soft numbers b12 ¼ h0:3; 0:6i and b22 ¼ h0:1; 0:8i. Garg and Arora [32] claimed that the new aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft prioritized weighted geometric aggregation operator (1.6.13.3), is h0:4423; 0:4140i. It is obvious that the membership value of the new aggregated intuitionistic fuzzy soft number is also 0:4423.

1.6.14 Drawbacks of Garg and Arora’s Dual Hesitant Fuzzy Soft Weighted Geometric Aggregation Operator Garg and Arora [33] proposed (i) The expression (1.6.14.1) to evaluate the multiplication of two dual hesitant fuzzy soft sets a11 ¼ hfh11 g; fg11 gi and a12 ¼ hfh12 g; fg12 gi. a11  a12 ¼

[



  

k1 ðkðc11 Þ þ kðc12 ÞÞ ; l1 ðlðd11 Þ þ lðd12 ÞÞ

c11 2 h11 ; c12 2 h12 ; d11 2 g11 ; d12 2 g12 ð1:6:14:1Þ (ii) The expression (1.6.14.2) to evaluate positive power k of a dual hesitant fuzzy soft set a ¼ hfhg; fggi. ak ¼

[ 

  

k1 ðkkðcÞÞ ; l1 ðklðdÞÞ

ð1:6:14:2Þ

c2h d2g where (i) l : ½0; 1 ! ½0; 1 is a continuous increasing function with lð0Þ ¼ 0. (ii) k : ½0; 1 ! ½0; 1 is a continuous decreasing function with k ð1Þ ¼ 0. (iii) lð xÞ ¼ k ð1  xÞ.

1 Appropriate Weighted Averaging Aggregation Operator …

30

Using the expressions (1.6.14.1) and (1.6.14.2), Garg and Arora [33] proposed the dual hesitant fuzzy soft weighted geometric aggregation operator (1.6.14.3). m j¼1



ni¼1

*(

[

 gi nj aij ¼ c

(

k

1

m X j¼1

ij 2 hij

nj

n X    gi k cij

m X j¼1

nj

;

i¼1

dij 2 hij l1

!!)

n X    gi l dij

!!)+

ð1:6:14:3Þ

i¼1

It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.1), is also occurring in the existing dual hesitant fuzzy soft weighted geometric aggregation operator (1.6.14.3). Therefore, the existing dual hesitant fuzzy soft weighted geometric aggregation operator (1.6.14.3)  isnot  valid. 

“If there exists one dual hesitant fuzzy soft number aij ¼ hij ; gij such that     hij ¼ f0g and gij ¼ f1g, then the aggregated dual hesitant fuzzy soft number, obtained on applying the dual hesitant fuzzy soft weighted geometric aggregation operator (1.6.14.3), will be h0; 1i, i.e., if one intuitionistic fuzzy number is  n  gi nj ai ¼ h0; 1i. Hence, the monotonicity property ‘aij 4a0ij 8 i ) m 4 j¼1 i¼1 aij

gi nj n 0 m , where aij and a0ij are dual hesitant fuzzy soft numbers’ will j¼1 i¼1 aij not be satisfied for the existing dual hesitant fuzzy soft weighted geometric aggregation operator (1.6.14.3).”

1.6.15 Drawbacks of Garg’s Intuitionistic Fuzzy Multiplicative Weighted Geometric Aggregation Operator Garg [16] pointed out the drawbacks of the existing intuitionistic fuzzy multiplicative weighted geometric aggregation operator [124]. To resolve the drawbacks, Garg [16] proposed (i) The expression (1.6.15.1) to evaluate the multiplication of two intuitionistic fuzzy multiplicative numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i.  a1  a2 ¼

2f1  ð1  l1 m1 Þð1  l2 m2 Þg ð1 þ 2m1 Þð1 þ 2m2 Þ  1 ; ð1:6:15:1Þ ; 2 ð1 þ 2m1 Þð1 þ 2m1 Þ  1

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

31

(ii) The expression (1.6.15.2) to evaluate the positive power k of an intuitionistic fuzzy multiplicative number a ¼ hl; mi. o * n + 2 1  ð1  lmÞk ð1 þ 2mÞk 1 ak ¼ ; : 2 ð1 þ 2mÞk 1

ð1:6:15:2Þ

Also, using the expression (1.6.15.1) and the expression (1.6.15.2), Garg [16] proposed the intuitionistic fuzzy multiplicative weighted geometric aggregation operator (1.6.15.3). ni¼1 ðai Þwi ¼

 Qn Q   wi 2 1  ni¼1 ð1  li mi Þwi i¼1 ð1 þ 2mi Þ 1 Qn ; wi 2 i¼1 ð1 þ 2mi Þ 1

ð1:6:15:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy multiplicative numbers. (ii) wi is the weight assigned to the ith fuzzy multiplicative number Pintuitionistic n ai ¼ hli ; mi i such that wi  0 and i¼1 wi ¼ 1. Mishra [94] considered the following example to show that the expression (1.6.15.1) is not valid, and hence, the expression (1.6.15.2) and the intuitionistic fuzzy multiplicative weighted geometric aggregation operator (1.6.15.3) are also not valid.



Let a1 ¼ hl1 ; m1 i ¼ 14 ; 3 and a2 ¼ hl2 ; m2 i ¼ 16 ; 4 be two intuitionistic fuzzy multiplicative numbers. Then, on applying the existing expression (1.6.15.1), 

2f1  ð1  l1 m1 Þð1  l2 m2 Þg ð1 þ 2m1 Þð1 þ 2m2 Þ  1 a1  a2 ¼ ; 2m1 Þð1 þ 2m2 Þ  1 2  ð1 þ 11 ; 31 : ¼ 372



It is a well-known fact that if a ¼ hl; mi is an intuitionistic fuzzy multiplicative number, then for l and m, the conditions 19  l; m  9 and lm  1 should necessarily 11

11 ; 31 , the value of l is 372 be satisfied. However, it is obvious that in a1  a2 ¼ 372 which is less than 19 . Also, the value of m is 31, which is greater than 9. Therefore, 11

; 31 is not an intuitionistic fuzzy multiplicative number. Hence, the a1  a2 ¼ 372 expression (1.6.14.1) to evaluate the multiplication of two intuitionistic fuzzy multiplicative numbers is not valid. Furthermore, it is pertinent to mention that the existing expression (1.6.15.2) i.e.,  k 2 1 ð 1lm Þ f g ð1 þ 2mÞk 1 ak ¼ ; is obtained by considering a1 ¼ a2 in the existing 2 ð1 þ 2mÞk 1 D E ð1l1 m1 Þð1l2 m2 Þg ð1 þ 2m1 Þð1 þ 2m2 Þ1 expression (1.6.15.1), i.e., a1  a2 ¼ 2fð1 . 1 þ 2m1 Þð1 þ 2m2 Þ1 ; 2 However, as discussed above, the existing expression (1.6.15.1) is not valid. Therefore, the existing expression (1.6.15.2) is also not valid.

1 Appropriate Weighted Averaging Aggregation Operator …

32

1.6.16 Drawbacks of Garg’s Pythagorean Fuzzy Weighted Geometric Aggregation Operator Garg [20] proposed (i) The expression (1.6.16.1) to evaluate the multiplication of two Pythagorean fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. * a1  a2 ¼

l1 ; l2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ; 1 þ 1  l21 1  l22

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi+ m21 þ m22 1 þ m21 m22

ð1:6:16:1Þ

(ii) The expression (1.6.16.2) to evaluate the positive power k of a Pythagorean fuzzy number a ¼ hl; mi. * a ¼ k

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi+ pffiffiffi k 2l ð1 þ m2 Þk ð1  m2 Þk qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð1 þ m2 Þk þ ð1  m2 Þk ð2  l2 Þk þ ðl2 Þk

ð1:6:16:2Þ

Also, using the expression (1.6.16.1) and the expression (1.6.16.2), Garg [20] proposed the Pythagorean fuzzy weighted geometric aggregation operator (1.6.16.3). *

ni¼1 awi i

pffiffiffi Qn wi 2 i¼1 li ffi; ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qn Qn 2 wi 2 wi i¼1 ð2  li Þ þ i¼1 ðli Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi+ Qn Qn 2 Þ wi  2 Þwi ð 1 þ m ð 1  m i i Qi¼1 Qni¼1 n 2 wi 2 wi i¼1 ð1 þ mi Þ þ i¼1 ð1  mi Þ

ð1:6:16:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n Pythagorean fuzzy numbers. (ii) wi is the weight assigned P to the ith Pythagorean fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3), is also occurring in the existing Pythagorean fuzzy weighted geometric aggregation operator (1.6.16.3). Therefore, the existing Pythagorean fuzzy weighted geometric aggregation operator (1.6.16.3) is not valid. “If there exists a Pythagorean fuzzy number ai ¼ hli ;mi i such that li ¼ 0 and mi ¼ 1, then the aggregated Pythagorean fuzzy number, obtained by the existing Pythagorean fuzzy weighted geometric aggregation operator (1.6.16.3), will always  w be h0;1i, and hence, the monotonicity property ‘ai 4a0i 8 i ) ni¼1 ðai Þwi 4ni¼1 a0i i ,

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

33

where ai ¼ hli ;mi i and a0i ¼ hl0i ;m0i i are Pythagorean fuzzy numbers’ will not be satisfied for the existing Pythagorean fuzzy weighted geometric aggregation _operator (1.6.16.3).”

1.6.17 Drawbacks of Garg’s Pythagorean Fuzzy Interactive Weighted Geometric Aggregation Operator Garg [25] pointed out the drawbacks of the existing Pythagorean fuzzy weighted geometric aggregation operators [18, 20]. To resolve the drawbacks, Garg [25] proposed (i) The expression (1.6.17.1) to evaluate the multiplication of two Pythagorean fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nQ o Q2 *u 2 2Þ  2  m2 Þ u2 ð 1  m ð 1  l i i i i¼1 i¼1 t a1  a2 ¼ ; Q2 Q2 2Þ þ 2 ð 1 þ m i i¼1 i¼1 ð1  mi Þ ffi+ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q2 Q2 2 2 i¼1 ð1 þ mi Þ  i¼1 ð1  mi Þ Q2 Q2 2 2 i¼1 ð1 þ mi Þ þ i¼1 ð1  mi Þ

ð1:6:17:1Þ

(ii) The expression (1.6.17.2) to evaluate the positive power k of a Pythagorean fuzzy number a ¼ hl; mi. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi+ *u u2 ð1  m2 Þk ð1  l2  m2 Þk ð 1 þ m2 Þ k  ð 1  m2 Þ k t ak ¼ ; ð 1 þ m2 Þ k þ ð 1  m2 Þ k ð 1 þ m2 Þ k þ ð 1  m2 Þ k

ð1:6:17:2Þ

Also, using the expressions (1.6.17.1) and the expression (1.6.17.2), Garg [25] proposed the Pythagorean fuzzy interactive weighted geometric aggregation operator (1.6.17.3). *sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qn ffi Qn 2 Þ wi  2  m 2 Þ wi 2 ð 1  m ð 1  l i i i i¼1 i¼1 Qn Q ni¼1 awi i ¼ ; n 2 wi 2 wi i¼1 ð1 þ mi Þ þ i¼1 ð1  mi Þ ffi+ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qn Qn 2 Þ wi  2 Þ wi ð 1 þ m ð 1  m i i Qi¼1 Qni¼1 n 2 wi 2 wi i¼1 ð1 þ mi Þ þ i¼1 ð1  mi Þ

ð1:6:17:3Þ

1 Appropriate Weighted Averaging Aggregation Operator …

34

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n Pythagorean fuzzy numbers. (ii) wi is the weight assigned P to the ith Pythagorean fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.1), is also occurring in the existing Pythagorean fuzzy weighted interactive geometric aggregation operator (1.6.17.3). Therefore, the existing Pythagorean fuzzy weighted interactive geometric aggregation operator (1.6.17.3) is not valid. “If there exists a Pythagorean fuzzy number ai ¼ hli ; mi i such that li ¼ 0 and mi ¼ 1, then the aggregated Pythagorean fuzzy number, obtained by the existing Pythagorean fuzzy interactive weighted geometric aggregation operator (1.6.17.3), will always be h0; 1i. Hence, the monotonicity property ‘ai 4a0i 8 i )  w ni¼1 ðai Þwi 4 ni¼1 a0i i , where ai ¼ hli ; mi i and a0i ¼ hl0i ; m0i i are Pythagorean fuzzy numbers’ will not be satisfied for the existing Pythagorean fuzzy interactive weighted geometric aggregation operator (1.6.17.3).”

1.6.18 Drawbacks of Garg’s Confidence Levels-Based Pythagorean Fuzzy Weighted Geometric Aggregation Operator Garg [22] pointed out that all the aggregation operators have been proposed by considering the assumption that decision-makers are surely familiar with the evaluated objects. But, it is not a realistic assumption. Therefore, to handle this situation, Garg [22] proposed the confidence levels-based Pythagorean fuzzy weighted geometric aggregation operator (1.6.18.1). nj¼1

 wj g aj j ¼

*

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi+ u n   n gj wj Y Y gj wj u lj ; t1  1  m2j j¼1

ð1:6:18:1Þ

j¼1

where (i) 0\gj  1 is confidence level of aj . P (ii) wj is the weight vector associated with aj such that wj  0 and nj¼1 wj ¼ 1. P (iii) gj is the confidence level of aj such that gj [ 0 and nj¼1 gj ¼ 1.

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

35

It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.1), is also occurring in the existing confidence levels-based Pythagorean fuzzy weighted geometric aggregation operator (1.6.18.1). Therefore, the existing confidence levels-based Pythagorean fuzzy weighted geometric aggregation operator (1.6.18.1) is not valid. “If there exists a Pythagorean fuzzy number ai ¼ hli ; mi i such that li ¼ 0 and mi ¼ 1, then the aggregated Pythagorean fuzzy number, obtained by the existing confidence levels-based Pythagorean fuzzy weighted geometric aggregation operator (1.6.18.1), will always be h0; 1i. Hence, the monotonicity property  w

‘ai 4a0i 8 i ) ni¼1 ðai Þwi 4 ni¼1 a0i i , where ai ¼ hli ; mi i and a0i ¼ l0i ; m0i are Pythagorean fuzzy numbers’ will not be satisfied for the existing confidence levels-based Pythagorean fuzzy weighted geometric aggregation operator (1.6.18.1).”

1.6.19 Drawbacks of Garg’s Interval-Valued Intuitionistic Fuzzy Weighted Geometric Aggregation Operator Garg [23] pointed out the drawbacks of the existing interval-valued intuitionistic fuzzy weighted geometric aggregation operator [85]. To resolve the drawbacks, Garg [23] proposed (i) The expression (1.6.19.1) to evaluate the multiplication of two interval-valued intuitionistic fuzzy numbers a1 ¼ h½a1 ; b1 ; ½c1 ; d1 i and a2 ¼ h½a2 ; b2 ; ½c2 ; d2 i. *"

Q2

Q2

i¼1 ð1

 ci Þ  c

Q2

 ai  c i  ; Q2 i¼1 ½1 þ ðc  1Þci  þ ðc  1Þ i¼1 ð1  ci Þ # Q Q c 2i¼1 ð1  di Þ  c 2i¼1 ½1  bi  di  Q2 Q2 i¼1 ½1 þ ðc  1Þdi  þ ðc  1Þ i¼1 ð1  di Þ i 2 Q2 h Q2 i¼1 1 þ ðc  1Þci  i¼1 ð1  ci Þ 4Q ; Q2 2 i¼1 ½1 þ ðc  1Þci  þ ðc  1Þ i¼1 ð1  ci Þ i 3+ Q2 Q2 h i¼1 1 þ ðc  1Þdi  i¼1 ð1  di Þ 5 Q2 Q2 i¼1 ½1 þ ðc  1Þbi  þ ðc  1Þ i¼1 ð1  di Þ

a1  a2 ¼

c

i¼1 ½1

ð1:6:19:1Þ

(ii) The expression (1.6.19.2) to evaluate the positive power k of an interval-valued intuitionistic fuzzy number a ¼ h½a; b; ½c; d i.

1 Appropriate Weighted Averaging Aggregation Operator …

36

*" a ¼

c½1  ck c½1  a  ck

c½1  d k c½1  b  d k

#

; ; ½1 þ ðc  1Þck þ ðc  1Þ½1  ck ½1 þ ðc  1Þd k þ ðc  1Þ½1  d k " #+ : ½1 þ ðc  1Þck ½1  ck ½1 þ ðc  1Þd k ½1  d k ; ½1 þ ðc  1Þck þ ðc  1Þ½1  ck ½1 þ ðc  1Þd k þ ðc  1Þ½1  d k k

ð1:6:19:2Þ Also, using the expression (1.6.19.1) and the expression (1.6.19.2), Garg [23] proposed the interval-valued intuitionistic fuzzy weighted geometric aggregation operator (1.6.19.3). Q Q c ni¼1 ½1  ci wi c ni¼1 ð1  ai  ci Þwi Qn Qn wi wi ; i¼1 ½1 þ ðc  1Þci  þ ðc  1Þ i¼1 ð1  ci Þ Q Q  c ni¼1 ½1  di wi c ni¼1 ð1  bi  d Þwi Q Qn wi n wi ; i¼1 ½1 þ ðc  1Þdi  þ ðc  1Þ i¼1 ð1  di Þ Qn Q  ð1 þ ðc  1Þci Þwi  ni¼1 ½1  ci wi Qn Qn i¼1 wi wi ; i¼1 ½1 þ ðc  1Þci  þ ðc  1Þ i¼1 ð1  ci Þ Q Qn  ð1 þ ðc  1Þdi Þwi  ni¼1 ½1  di wi Q Qn i¼1 wi n wi i¼1 ½1 þ ðc  1Þdi  þ ðc  1Þ i¼1 ð1  di Þ 

ni¼1 awi i ¼

ð1:6:19:3Þ

where (i) ai ¼ h½ai ; bi ; ½ci ; di i i ¼ 1; 2; . . .; n are n interval-valued intuitionistic fuzzy numbers. (ii) wi is the weight assigned to the ith interval-valued intuitionistic fuzzy number P ai ¼ h½ai ; bi ; ½ci ; di i such that wi  0 and ni¼1 wi ¼ 1. Mishra [94] considered the following example to show that the existing expression (1.6.19.1) is not valid, and hence, the existing interval-valued intuitionistic fuzzy weighted geometric aggregation operator (1.6.19.3) is also not valid. If c ¼ 1, k ¼ 2 and n ¼ 2, then the existing expression (1.6.19.1) and (1.6.19.2) will be transformed into the expression (1.6.19.4) and the expression (1.6.19.5), respectively. *" a1  a 2 ¼ "

2 Y

# 2 2 2 Y Y Y ð1  ai Þ  ½1  ai  ci ; ð1  bi Þ  ½1  bi  di  ;

i¼1

i¼1

i¼1

2 2 Y Y 1  ð1  ai Þ; 1  ð1  bi Þ i¼1

#+

i¼1

i¼1

ð1:6:19:4Þ

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

Dh i a2 ¼ ½1  a2 ½1  a  c2 ; ½1  b2 ½1  b  d 2 ; h iE 1  ½1  a2 ; 1  ½1  b2

37

ð1:6:19:5Þ

Now, let a1 ¼ h½0:65; 0:73; ½0:17; 0:21i and a2 ¼ h½0:50; 0:60; ½0:35; 0:40i be two interval-valued intuitionistic fuzzy numbers. Then, a1  a2 ¼ h½0:1480; 0:1080; ½0:8250; 0:8920i a21 ¼ h½0:0901; 0:0693; ½0:8775; 0:9271i It is a well-known fact that for an interval-valued intuitionistic fuzzy number, a ¼ h½a; b; ½c; d i, the conditions a  b, c  d and b þ d  1 should always be satisfied. However, it is obvious that for a1  a2 ¼ h½0:1480; 0:1080; ½0:8250; 0:8920i and a21 ¼ h½0:0901; 0:0693; ½0:8775; 0:9271i, the condition a  b is not satisfying. Therefore, a1  a2 , obtained by the expression (1.6.19.1), and a21 ¼ h½0:0901; 0:0693; ½0:8775; 0:9271i, obtained by the expression (1.6.19.2), are not interval-valued intuitionistic fuzzy numbers. Hence, the expression (1.6.19.1) and the expression (1.6.19.2) are not valid.

1.6.20 Drawbacks of Garg and Kumar’s Linguistic Interval-Valued Intuitionistic Fuzzy Weighted Geometric Aggregation Operator Garg and Kumar [45] proposed (i) The expression (1.6.20.1) to evaluate the multiplication of two linguistic interval-valued intuitionistic fuzzy numbers a1 ¼ h½sa1 ; sb1 ; ½sc1 ; sd1 i and a2 ¼ h½sa2 ; sb2 ; ½sc2 ; sd2 i. a1  a2 ¼

Dh i h iE sa1 a2 ; sb1 b2 ; sc1 þ c2 c1 c2 ; sd1 þ d2 d1 d2 t

t

t

ð1:6:20:1Þ

t

(ii) The expression (1.6.20.2) to evaluate the positive power k of a linguistic interval-valued intuitionistic fuzzy number a ¼ h½sa ; sb ; ½sc ; sd i.  ak ¼

st

ðÞ

a k t

; st

ðÞ

b k t

  ; s

t 1ð

Þ

k 1ct

; s 

t 1ð

Þ

k 1dt



 ð1:6:20:2Þ

1 Appropriate Weighted Averaging Aggregation Operator …

38

where (i) (ii) (iii) (iv) (v) (vi)

½sai ; sbi  represents the linguistic membership degree. ½sci ; sdi  represents the linguistic non-membership degree. ai ; bi ; ci ; di 2 ½0; t. bi þ di  t;i ¼ 1; 2: t is a positive integer. k is a positive real number.

Using the expression (1.6.20.1) and the expression (1.6.20.2), Garg and Kumar [45, 46] proposed the linguistic interval-valued intuitionistic fuzzy weighted geometric aggregation operator (1.6.20.3). * ni¼1 ðawi i Þ

¼

2

3 2

6 6s  n 4 Q t

i¼1

ðati Þ

wi

; s  t

n Q i¼1

ðbti Þ

3

7 6  7; 6s  n 5 4 Q wi t

1

i¼1

ð1cti Þ

wi

; s  t

1

n Q

ð1dti Þ

7 7 5 wi

+

i¼1

ð1:6:20:3Þ It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted geometric aggregation operator (1.6.1.3), is also occurring in the existing linguistic interval-valued intuitionistic fuzzy weighted geometric aggregation operator (1.6.20.3). Therefore, the existing linguistic interval-valued intuitionistic fuzzy weighted geometric aggregation operator (1.6.20.3) is not valid. “If there exists a linguistic interval-valued intuitionistic fuzzy number ai ¼ h½sai ; sbi ; ½sci ; sdi i such that ai ¼ bi ¼ 0 and ci ¼ di ¼ t, then the aggregated linguistic interval-valued intuitionistic fuzzy number, obtained by the existing linguistic interval-valued intuitionistic fuzzy weighted geometric aggregation operator (1.6.20.3), will always be h½s0 ; s0 ; ½sh ; sh i, and hence, the monotonicity property  w ‘ai 4a0i 8 i ) ni¼1 ðai Þwi 4 ni¼1 a0i i , where ai ¼ h½sai ; sbi ; ½sci ; sdi i and a0i ¼ h½sa0i ; sb0i ; ½sc0i ; sdi0 i are linguistic interval-valued intuitionistic fuzzy numbers’ will not be satisfied for the existing linguistic interval-valued intuitionistic fuzzy weighted geometric aggregation operator (1.6.20.3).”

1.6 Drawbacks of Some Existing Weighted Geometric Aggregation …

39

1.6.21 Drawbacks of Nancy and Garg’s Single-Valued Neutrosophic Weighted Geometric Aggregation Operator Nancy and Garg [97] proposed (i) The expression (1.6.21.1) to evaluate the multiplication of two single-valued neutrosophic numbers a ¼ ha; b; ci, a1 ¼ ha1 ; b1 ; c1 i and a2 ¼ ha2 ; b2 ; c2 i. *

 1b1  !  k  1 k1b2  1 ðka1  1Þðka2  1Þ a1  a2 ¼ logk 1 þ ; 1  logk 1 þ ; k1 k1  1c1  !+ k  1 k1c2  1 1  logk 1 þ ;k[1 k1 

ð1:6:21:1Þ (ii) The expression (1.6.21.2) to evaluate the positive power n of a single-valued neutrosophic number a ¼ ha; b; ci. * a ¼ n

logk 1 þ

1  logk

ðka  1Þn

!



k1b  1

n !

; 1  logk 1 þ ðk  1Þn1 ðk  1Þn1 !+  1c n k 1 1þ ðk  1Þn1

; ð1:6:21:2Þ

Also, using the expression (1.6.21.1) and the expression (1.6.21.2), Nancy and Garg [97] proposed the single-valued neutrosophic weighted geometric aggregation operator (1.6.21.3). * ni¼1 awi i

¼

logk

! ! n n  Y Y wi wi ai 1bi ; 1þ ðk  1Þ ; 1  logk 1 þ k 1 i¼1

1  logk 1 þ

n  Y

k1ci  1

 wi

!+

i¼1

i¼1

ð1:6:21:3Þ where (i) ai ¼ hai ; bi ; ci i, i ¼ 1; 2; . . .; n are n single-valued neutrosophic numbers. (ii) wi is the weight assigned to thePith single-valued neutrosophic number ai ¼ hai ; bi ; ci i such that wi  0 and ni¼1 wi ¼ 1.

1 Appropriate Weighted Averaging Aggregation Operator …

40

Mishra [93] considered a numerical example to show that the existing single-valued neutrosophic weighted geometric aggregation operator (1.6.21.3) is not valid as the monotonicity property is not satisfying for it.

1.7

Drawbacks of Some Existing Weighted Averaging Aggregation Operators Under Various Extensions of the Fuzzy Environment

In this section, the drawbacks of the existing weighted averaging aggregation operators [2, 3, 13, 18, 21, 22, 24, 28, 31–37, 42, 46, 65, 90, 92, 97, 114, 127, 137], defined under various extensions of the fuzzy environment, have been pointed out. It can be easily verified that the same drawbacks are also occurring in the other existing weighted averaging aggregation operators defined under various extensions of fuzzy environment.

1.7.1

Drawbacks of Xu’s Intuitionistic Fuzzy Weighted Averaging Aggregation Operator

Using the t-norm T ðx; yÞ ¼ xy and the Sðx; yÞ ¼ x þ y  xy ¼ 1  ð1  xÞð1  yÞ, Xu [127] proposed

t-conorm

(i) The expression (1.7.1.1) to evaluate the sum of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. ð1:7:1:1Þ

a1 a2 ¼ h1  ð1  l1 Þð1  l2 Þ; m1 m2 i

(ii) The expression (1.7.1.2) to evaluate the multiplication of a positive real number k with an intuitionistic fuzzy number a ¼ hl; mi. D E k  a ¼ 1  ð 1  l Þ k ; mk

ð1:7:1:2Þ

Also, using the expression (1.7.1.1) and the expression (1.7.1.2), Xu [127] proposed the intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.3). *

ni¼1 ðwi

 ai Þ ¼

n n Y Y 1  ð 1  li Þ w i ; mwi i i¼1

i¼1

+ ð1:7:1:3Þ

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

41

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy numbers. (ii) wi is the weight assigned P to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. Beliakov et al. [6] pointed out that if there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0, then the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy weighted averaging operator (1.7.1.3), will always be h1; 0i. Beliakov et al. [6] further pointed out that due to the mono this drawback,  tonicity property “ai 4a0i 8 i ) ni¼1 ðwi  ai Þ4 ni¼1 wi  a0i where ai ¼ hli ; mi i

and a0i ¼ l0i ; m0i are intuitionistic fuzzy numbers” will not be satisfied for the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.3). Hence, the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.3) is not valid as an aggregation operator is said to be valid if it satisfies the monotonicity property in addition to some other necessary properties [132]. To validate this claim, Beliakov et al. [6] considered the intuitionistic fuzzy numbers a1 ¼ h0:3; 0:2i; a2 ¼ h1; 0i; a01 ¼ h0:5; 0:4i and a02 ¼ h1; 0i. Beliakov et al. [6] claimed that as a2 ¼ a02 and according to the existing approach [127] for comparing intuitionistic fuzzy numbers, a1 a01 . Therefore,   according to the monotonicity property, the relation 2i¼1 ðwi  ai Þ 2i¼1 wi  a0i should hold. While, it can be easily verified that if with w1 ¼ 0:4 and w2 ¼ 0:6. Then, on applying the existing intuitionistic  fuzzy weighted averaging aggregation operator (1.7.1.3), 2i¼1 ðwi  ai Þ 2i¼1 wi  a0i ¼ h1; 0i. This clearly indicates that the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.3) is not valid.

1.7.2

Drawbacks of Wang and Liu’s Intuitionistic Fuzzy Weighted Averaging Aggregation Operator

xy Using the t-conorm Sðx; yÞ ¼ 1xþþxyy and the t-norm T ðx; yÞ ¼ 1 þ ð1x Þð1yÞ, Wang and Liu [114] proposed

(i) The expression (1.7.2.1) to evaluate the addition of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i.  a1 a2 ¼

ð1 þ l1 Þð1 þ l2 Þ  ð1  l1 Þð1  l2 Þ 2m1 m2 ; ð1 þ l1 Þð1 þ l2 Þ þ ð1  l1 Þð1  l2 Þ ð2  m1 Þð2  m2 Þ þ m1 m2



ð1:7:2:1Þ

1 Appropriate Weighted Averaging Aggregation Operator …

42

(ii) The expression (1.7.2.2) to evaluate the multiplication of a positive real number with an intuitionistic fuzzy number a ¼ hl; mi. *

ð1 þ lÞk ð1  lÞk

2mk

+

; ð1 þ lÞk þ ð1  lÞk ð2  mÞk þ mk

ka¼

ð1:7:2:2Þ

Also, using the expression (1.7.2.1) and the expression (1.7.2.2), Wang and Liu [114] proposed the intuitionistic fuzzy weighted averaging aggregation operator (1.7.2.3). Q  Qn ð1 þ li Þwi  ni¼1 ð1  li Þwi Qn Qni¼1 wi wi ; i¼1 ð1 þ li Þ þ i¼1 ð1  li Þ Qn wi  2 i¼1 ðmi Þ Qn Q wi n wi i¼1 ð2  mi Þ þ i¼1 ðmi Þ

ni¼1 ðwi  ai Þ ¼

ð1:7:2:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy numbers. (ii) wi is the weight assigned Pn to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and i¼1 wi ¼ 1. Garg [21] pointed out that the following drawbacks in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.2.3). (i) If there exists one intuitionistic fuzzy number ai ¼ hli ; mi i such that mi ¼ 0, then the degree of non-membership of the aggregated intuitionistic fuzzy number, obtained on applying the intuitionistic fuzzy weighted averaging aggregation operator (1.7.2.3), will always be 0, i.e., the degree of non-membership of the aggregated intuitionistic fuzzy number ni¼1 ðwi  ai Þ is independent from the non-membership degree of the remaining ðn  1Þ intuitionistic fuzzy numbers. To validate the claim, Garg [21] considered the four intuitionistic fuzzy numbers a1 ¼ h0:72; 0i; a2 ¼ h0:63; 0:36i; a3 ¼ h0:31; 0:62i; a4 ¼ h0:71; 0:22i with w1 ¼ 0:2; w2 ¼ 0:3; w3 ¼ 0:4; w4 ¼ 0:1 and showed that aggregated intuitionistic fuzzy number, obtained on applying the intuitionistic fuzzy weighted averaging aggregation operator (1.7.2.3), is h0:5517; 0i. Furthermore, Garg [21] claimed that it is obvious that degree of the non-membership of the aggregated intuitionistic fuzzy number is 0, and hence, it may be concluded that the degree of non-membership of the aggregated intuitionistic fuzzy number, obtained by the intuitionistic fuzzy weighted averaging aggregation operator (1.7.2.3), is not depending upon the degree of non-membership of the remaining three intuitionistic fuzzy numbers. (ii) The membership value of the aggregated intuitionistic fuzzy number, obtained on applying the intuitionistic fuzzy weighted averaging operator (1.7.2.3), is

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

43

independent from the change in the membership degree and the non-membership degree of the intuitionistic fuzzy numbers. To validate the claim, Garg [21] considered the four intuitionistic fuzzy numbers a1 ¼ h0:23; 0:35i; a2 ¼ h0:45; 0:23i; a3 ¼ h0:65; 0:17i; a4 ¼ h0:50; 0:20i with w1 ¼ 0:2; w2 ¼ 0:3; w3 ¼ 0:4; w4 ¼ 0:1 and showed that the aggregated intuitionistic fuzzy number, obtained on applying the expression (1.7.2.3), is h0:5060; 0:2196i. Then, Garg [21] replaced the intuitionistic fuzzy numbers a2 ¼ h0:45; 0:23i and a3 ¼ h0:65; 0:17i with the intuitionistic fuzzy numbers a2 ¼ h0:33; 0:23i and a3 ¼ h0:37; 0:17i and showed that the aggregated intuitionistic fuzzy number, obtained on applying the intuitionistic fuzzy weighted averaging operator (1.7.2.3), is h0:3422; 0:2196i. Furthermore, Garg [21] claimed that it is obvious that in both the cases, the non-membership value of the aggregated intuitionistic fuzzy number is same. Hence, it may be concluded that the non-membership value of the aggregated intuitionistic fuzzy number, obtained by the intuitionistic fuzzy weighted averaging aggregation operator (1.7.2.3), is not depending upon the changes in the membership values of the considered intuitionistic fuzzy numbers.

1.7.3

Drawbacks of Garg’s Intuitionistic Fuzzy Weighted Averaging Aggregation Operator

To resolve the drawbacks of the existing intuitionistic fuzzy weighted averaging operators [114], Garg [21] proposed (i) The expression (1.7.3.1) to evaluate the addition of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. *Q

2 ð 1 þ li Þ  Q2i¼1 i¼1 ð1 þ li Þ þ

a1 a2 ¼ 2

nQ 2

Q2 Qi¼1 2

ð 1  li Þ

i¼1 ð1

 li Þ

;

o+ Q2 ð 1  l Þ  ð 1  l  m Þ i i i i¼1 i¼1 Q2 Q2 i¼1 ð1 þ li Þ þ i¼1 ð1  li Þ

ð1:7:3:1Þ

(ii) The expression (1.7.3.2) to evaluate the multiplication of a positive real number k with an intuitionistic fuzzy number a ¼ hl; mi . * ka¼

k

ð1 þ lÞ ð1  lÞ

k

ð1 þ lÞk þ ð1  lÞk

;

n o+ 2 ð1  lÞk ð1  l  mÞk ð 1 þ l Þ k þ ð 1  lÞ k

ð1:7:3:2Þ

1 Appropriate Weighted Averaging Aggregation Operator …

44

Also, using the expression (1.7.3.1) and the expression (1.7.3.2), Garg [21] proposed the intuitionistic fuzzy weighted averaging aggregation operator (1.7.3.2).

ni¼1 ðwi

 Qn wi Q n wi i¼1 ð1 þ li Þ  Qi¼1 ð1  li Þ Q  ai Þ ¼ ; n wi n ð1 þ li Þ þ i¼1 ð1  li Þwi Qni¼1 Q wi n wi  2 i¼1 ð1  li Þ  i¼1 ð1  li  mi Þ Qn Qn wi wi i¼1 ð1 þ li Þ þ i¼1 ð1  li Þ

ð1:7:3:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy number s. (ii) wi is the weight assigned P to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.3.1), is also occurring in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.3.3). Therefore, the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.3.3) is not valid. “If there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0, then the aggregated intuitionistic fuzzy number, obtained on applying the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.3.3), will always be h1; 0i. Hence, monotonicity property   the ‘ai 4a0i 8 i ) ni¼1 ðwi  ai Þ4 ni¼1 wi  a0i , where ai ¼ hli ; mi i and a0i ¼ hl0i ; m0i i are intuitionistic fuzzy numbers’ will not be satisfied for the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.3.3).”

1.7.4

Drawbacks of Yu’s Intuitionistic Fuzzy Interaction Weighted Averaging Aggregation Operator

Yu [137] pointed out the drawbacks of the existing intuitionistic fuzzy interaction weighted averaging aggregation operator [127]. To resolve the drawbacks, Yu [137] proposed (i) The expression (1.7.4.1) to evaluate the sum of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. a1 a2 ¼ h1  ð1  l1 Þð1  l2 Þ; ð1  l1 Þð1  l2 Þ ð1  ðl1 þ m1 ÞÞð1  ðl2 þ m2 ÞÞi

ð1:7:4:1Þ

(ii) The expression (1.7.4.1) to evaluate the multiplication of a positive real number k with an intuitionistic fuzzy number a ¼ hl; mi.

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

45

D E k  a ¼ 1  ð1  lÞk ; ð1  lÞk ð1  ðl þ mÞÞk

ð1:7:4:2Þ

Also, using the expression (1.7.4.1) and the expression (1.7.4.2), Yu [137] proposed the intuitionistic fuzzy interaction weighted averaging aggregation operator (1.7.4.3). *

ni¼1 ðwi  ai Þ ¼

n n n Y Y Y 1  ð1  li Þwi ; ð1  li Þwi  ð1  ðli þ mi ÞÞwi i¼1

i¼1

+

i¼1

ð1:7:4:3Þ where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy numbers. (ii) wi is the weight assigned P to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.3), is also occurring in the existing intuitionistic fuzzy interaction weighted averaging aggregation operator (1.7.4.3). Therefore, the existing intuitionistic fuzzy interaction weighted averaging aggregation operator (1.7.4.3) is not valid. “If there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0, then the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy weighted interaction averaging aggregation operator (1.7.4.3), will always be h1; 0i. Hence, the monotonicity property ‘ai 4a0i 8 i )  

ni¼1 ðwi  ai Þ4 ni¼1 wi  a0i , where ai ¼ hli ; mi i and a0i ¼ hl0i ; m0i i are intuitionistic fuzzy numbers’ will not be satisfied for the existing intuitionistic fuzzy weighted interaction averaging aggregation operator (1.7.4.3).”

1.7.5

Drawbacks of Huang’s Hamacher Intuitionistic Fuzzy Weighted Averaging Aggregation Operator

Using

the

t-norm

Sðx; yÞ ¼

ð1xÞð1yÞð1cÞxy , 1ð1cÞxy

T ðx; yÞ ¼ c þ ð1cÞðxy1xÞð1yÞ

and

the

t-conorm

Huang [65] proposed

(i) The expression (1.7.5.1) to evaluate the addition of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i.

1 Appropriate Weighted Averaging Aggregation Operator …

46

 a1 a2 ¼

ð1  l1 Þð1  l2 Þ  ð1  cÞl1 l2 ð1  m1 Þð1  m2 Þ  ð1  cÞm1 m2 ; 1  ð1  cÞl1 l2 1  ð1  cÞm1 m2



ð1:7:5:1Þ (ii) The expression (1.7.5.2) to evaluate the multiplication of a positive real number with an intuitionistic fuzzy number a ¼ hl; mi. * ka¼

ð1 þ ðc  1ÞlÞk ð1  lÞk

cmk

+

; ð1 þ ðc  1ÞlÞk þ ðc  1Þð1  lÞk ð1 þ ðc  1Þð1  mÞÞk þ ðc  1Þmk ð1:7:5:2Þ

Also, using the expression (1.7.5.1) and the expression (1.7.5.2), Huang [65] proposed the intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.5.3). Q  1Þli Þwi  ni¼1 ð1  li Þwi Qn wi wi ; i¼1 ð1 þ ðc  1Þli Þ þ ðc  1Þ i¼1 ð1  li Þ Q c ni¼1 ðmi Þwi Q Qn wi n wi i¼1 ð1 þ ðc  1Þð1  mi ÞÞ þ ðc  1Þ i¼1 ðmi Þ 

ni¼1 ðwi  ai Þ ¼

Qn

Qn

i¼1 ð1 þ ðc

ð1:7:5:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy numbers. (ii) wi is the weight assigned P to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. Garg [28] pointed out that the following drawbacks in the existing intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.5.3). (i) If there exists one ai ¼ hli ; mi i such that mi ¼ 0, then the degree of non-membership of the aggregated intuitionistic fuzzy number, obtained on applying the existing intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.5.3), will always be 0, i.e., the degree of non-membership of the aggregated intuitionistic fuzzy number ni¼1 ðwi  ai Þ is independent from the non-membership degree of the remaining ðn  1Þ intuitionistic fuzzy numbers. To validate the claim, Garg [28] considered the four intuitionistic fuzzy numbers a1 ¼ h0:85; 0i; a2 ¼ h0:35; 0:40i; a3 ¼ h0:55; 0:35i; a4 ¼ h0:76; 0:17i with w1 ¼ 0:2; w2 ¼ 0:3; w3 ¼ 0:4; w4 ¼ 0:1, c ¼ 1 and showed that aggregated intuitionistic fuzzy number, obtained on applying the existing intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.5.3), is h0:6212; 0i. Furthermore, Garg [28] claimed that it is obvious that degree of the non-membership of the aggregated intuitionistic fuzzy number is 0, and hence,

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

47

it may be concluded that the degree of non-membership of the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.5.3), is not depending upon the degree of non-membership of the remaining three intuitionistic fuzzy numbers. (ii) The membership value of the aggregated intuitionistic fuzzy number, obtained on applying the existing intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.5.3), is independent from the change in the membership degree and the non-membership degree of the intuitionistic fuzzy numbers. To validate the claim, Garg [28] considered the four intuitionistic fuzzy number s a1 ¼ h0:85; 0i; a2 ¼ h0:35; 0:40i; a3 ¼ h0:55; 0:35i; a4 ¼ h0:76; 0:17i with w1 ¼ 0:2; w2 ¼ 0:3; w3 ¼ 0:4; w4 ¼ 0:1, c ¼ 1 and showed that aggregated intuitionistic fuzzy number, obtained on applying the existing intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.5.3), is h0:6212; 0i. Then, Garg [28] replaced the intuitionistic fuzzy numbers a2 ¼ h0:35; 0:40i and a3 ¼ h0:55; 0:35i with the intuitionistic fuzzy numbers a2 ¼ h0:45; 0:40i and a3 ¼ h0:15; 0:35i; and showed that the aggregated intuitionistic fuzzy number, obtained on applying the intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.5.3), is h0:5354; 0i. Furthermore, Garg [28] claimed that it is obvious that in both the cases, the non-membership value of the aggregated intuitionistic fuzzy number is same, and hence, it may be concluded that the non-membership value of the aggregated intuitionistic fuzzy number, obtained by the intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.5.3), is not depending upon the changes in the membership values of the considered intuitionistic fuzzy numbers.

1.7.6

Drawbacks of Garg’s Intuitionistic Fuzzy Hamacher Weighted Averaging Aggregation Operator

To resolve the drawbacks of the existing intuitionistic fuzzy weighted averaging aggregation operators [65], Garg [28] proposed (i) The expression (1.7.6.1) to evaluate the addition of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. 

ð1 þ ðc  1Þl1 Þð1 þ ðc  1Þl2 Þ  ð1  l1 Þð1  l2 Þ ; ð1 þ ðc  1Þl1 Þð1 þ ðc  1Þl2 þ ðc  1Þð1  l1 Þð1  l2 ÞÞ cð1  l1 Þð1  l2 Þ  cð1  l1  m1 Þð1  l2  m2 Þ ð1 þ ðc  1Þl1 Þð1 þ ðc  1Þl2 þ ðc  1Þð1  l1 Þð1  l2 ÞÞ

a1 a2 ¼

ð1:7:6:1Þ

1 Appropriate Weighted Averaging Aggregation Operator …

48

(ii) The expression (1.7.6.2) to evaluate the positive power k of an intuitionistic fuzzy number a ¼ hl; mi. * ka¼

ð1 þ ðc  1ÞlÞk ð1  lÞk ð1 þ ðc  1ÞlÞk þ ðc  1Þð1  lÞk + cð1  lÞk cð1  l  mÞk

; ð1:7:6:2Þ

ð1 þ ðc  1ÞlÞk þ ðc  1Þð1  lÞk Also, using the expression (1.7.6.1) and the expression (1.7.6.2), Garg [28] proposed the intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.6.3). Q  1Þli Þwi  ni¼1 ð1  li Þwi Qn  ai Þ ¼ wi wi ; i¼1 ð1 þ ðc  1Þli Þ þ ðc  1Þ i¼1 ð1  li Þ Qn Q c i¼1 ð1  li Þwi  ni¼1 ð1  li  mi Þwi Qn Qn wi wi i¼1 ð1 þ ðc  1Þli Þ þ ðc  1Þ i¼1 ð1  li Þ Qn



ni¼1 ðwi

Qn

i¼1 ð1 þ ðc

ð1:7:6:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy number s. (ii) wi is the weight assigned P to the ith intuitionistic fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.3), is also occurring in the existing intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.6.3). Therefore, the existing intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.6.3) is not valid. “If there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0, then the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.6.3), will always be 1; 0. Hence, the monotonicity property ‘ai 4a0i 8 i )  

ni¼1 ðwi  ai Þ4 ni¼1 wi  a0i , where ai ¼ hli ; mi i and a0i ¼ l0i ; m0i are intuitionistic fuzzy numbers’ will not be satisfied for the existing intuitionistic fuzzy Hamacher weighted averaging aggregation operator (1.7.6.3).”

1.7.7

Drawbacks of Chen et al.’s Linguistic Intuitionistic Fuzzy Weighted Averaging Aggregation Operator

Chen et al. [13] proposed

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

49

(i) The expression (1.7.7.1) the addition of D to evaluate E D E two linguistic intuitionistic fuzzy numbers a1 ¼ sa11 ; sa21

and a2 ¼ sa12 ; sa22 .

* a1 a2 ¼

+ s

a1



tt 1 t1

a1

1 t2

 ; s 2  2  a

1 t

t

ð1:7:7:1Þ

a

2 t

(ii) The expression (1.7.7.2) to evaluate the multiplication of a non-negative real number k with a linguistic intuitionistic fuzzy number a ¼ hsa1 ; sa2 i.  ka¼

s

1 k

ttð1at

Þ

;s

ð1:7:7:2Þ

2 k

tðat

Þ

Also, using the expression (1.7.7.1) and the expression (1.7.7.2), Chen et al. [13] proposed the linguistic intuitionistic fuzzy weighted averaging aggregation operator (1.7.7.3) and its extensions. *

ni¼1 ðwi

 ai Þ ¼

+ s tt

n Q i¼1

 wi ; s Q

2  wi n a1 a i i

1 t

t

ð1:7:7:3Þ

t i¼1

where (i) ai ¼ ha1i ; a2i i, i ¼ 1; 2; . . .; n are n linguistic intuitionistic fuzzy numbers. assigned to the ith linguistic intuitionistic fuzzy number (ii) wi is the weight

P ai ¼ a1i ; a2i such that wi [ 0 and ni¼1 wi ¼ 1. (iii) 0  a1i  t; 0  a2i  t, (iv) t is a non-negative integer. (v) a1i þ a2i  t. Garg and Kumar [42] pointed out that the following drawbacks in the existing linguistic intuitionistic fuzzy weighted averaging aggregation operator (1.7.7.3). The non-membership value of the aggregated linguistic intuitionistic fuzzy number, obtained on applying the existing linguistic intuitionistic fuzzy weighted averaging aggregation operator (1.7.7.3), is independent from the change in the non-membership degree of the linguistic intuitionistic fuzzy numbers. To validate the claim, Garg and Kumar [42] considered the four linguistic intuitionistic fuzzy numbers a1 ¼ hs4 ; s3 i; a2 ¼ hs2 ; s4 i; a3 ¼ hs5 ; s1 i; a4 ¼ hs4 ; s2 i with w1 ¼ 0:3; w2 ¼ 0:2; w3 ¼ 0:2; w4 ¼ 0:2 and showed that aggregated linguistic intuitionistic fuzzy number, obtained on applying the linguistic intuitionistic fuzzy weighted averaging aggregation operator (1.7.7.3), is hs3:7233 ; s2:5139 i. Then, Garg and Kumar [42] obtained the linguistic intuitionistic fuzzy numbers b1 ¼ by replacing the hs4 ; s1 i; b2 ¼ hs2 ; s2 i; b3 ¼ hs5 ; s3 i; b4 ¼ hs4 ; s4 i non-membership values s3 , s4 , s1 and s2 with s1 ; s2 ; s3 and s4 , respectively, and

1 Appropriate Weighted Averaging Aggregation Operator …

50

showed that the aggregated linguistic intuitionistic fuzzy number, obtained on applying the linguistic intuitionistic fuzzy weighted averaging aggregation operator (1.7.7.3), is hs3:7233 ; s2:5139 i. Furthermore, Garg and Kumar [42] claimed that it is obvious that in both the cases, the non-membership value of the aggregated linguistic intuitionistic fuzzy number is same, and hence, it may be concluded that the non-membership value of the aggregated linguistic intuitionistic fuzzy number, obtained by the linguistic intuitionistic fuzzy weighted averaging aggregation operator (1.7.7.3), is not depending upon the changes in the non-membership values of the considered linguistic intuitionistic fuzzy numbers, which is illogical.

1.7.8

Drawbacks of Arora and Garg’s Linguistic Intuitionistic Fuzzy Prioritized Weighted Averaging Aggregation Operator

Arora and Garg [3] proposed (i) The expression (1.7.8.1) the addition ofE two linguistic intuitionistic D evaluates E D fuzzy numbers a1 ¼ sa11 ; sa21 and a2 ¼ sa12 ; sa22 . * a1 a2 ¼

+ s

1 t

a

1 t

þ

a

 1 3

1 t

a

2 t

þ

a

 1  1  ; s 1 1  3

2 t



a

1 t

a

2 t

t

a a 1 2 t2

ð1:7:8:1Þ

(ii) The expression (1.7.8.2) to evaluate the multiplication of a non-negative real number k with a linguistic intuitionistic fuzzy number a ¼ hsa1 ; sa2 i.  ka¼

s  a1 a2 k a1 k  ; s  a2  k t ð t þ t Þ ð t Þ t 2 t

ð1:7:8:2Þ

(iii) The expression (1.7.8.3) to evaluate the score of a linguistic intuitionistic fuzzy number a ¼ hsa1 ; sa2 i. Sð aÞ ¼

 1 t þ a1  a2 2

(v) The expression (1.7.8.4) to evaluate the total score value.

ð1:7:8:3Þ

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

T ð ah Þ ¼

51

n Y Sð ah Þ ; h ¼ 2; 3; . . .; n with T ða1 Þ ¼ 1 t p¼2 p 6¼ h

ð1:7:8:4Þ

Also, using the expressions (1.7.8.1)–(1.7.8.4), Arora and Garg [3] proposed the linguistic intuitionistic fuzzy prioritized averaging aggregation operator (1.7.8.5). 

nh¼1

 *  T ð ah h Þ  ah ¼ s Rnh¼1 ðT ðah ÞÞ

+ t

1

Qn i¼1

a1 1 th



T ðah Þ Rn ðT ðah ÞÞ h¼1

!; s t

T ðah Þ Qn a2h Rnh¼1 ðT ðah ÞÞ i¼1

t

ð1:7:8:5Þ It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.3), is also occurring in the linguistic intuitionistic fuzzy prioritized averaging aggregation operator (1.7.8.5). Therefore, the linguistic intuitionistic fuzzy prioritized averaging aggregation operator (1.7.8.5) is not valid. D E “If there exists a linguistic intuitionistic fuzzy number ai ¼ sa1i ; sa2i

such that

¼ t and ¼ 0, then the aggregated linguistic intuitionistic fuzzy number, obtained by the existing linguistic intuitionistic fuzzy prioritized averaging aggregation operator (1.7.8.5),   will always  behst ; s0 i. Hence,  the monotonicity  property 0 T a ð Þ Pn h 0 PnT ðah Þ  a0h ; where ‘ai 4a0i 8 i ) nh¼1  ah 4 nh¼1 ðT ðah ÞÞ ðT ðah ÞÞ h¼1 h¼1

ai ¼ hli ; mi i and a0i ¼ l0i ; m0i are linguistic intuitionistic fuzzy numbers will not be satisfied for the existing linguistic intuitionistic fuzzy prioritized averaging aggregation operator (1.7.8.5).” a1i

1.7.9

a2i

Drawbacks of Arora and Garg’s Intuitionistic Fuzzy Soft Weighted Averaging Aggregation Operator

Arora and Garg [2] proposed (i) The expression (1.7.9.1) to evaluate the addition of two intuitionistic fuzzy soft numbers a11 ¼ hl11 ; m11 i, a12 ¼ hl12 ; m12 i. a11 a12 ¼ h1  ð1  l11 Þð1  l12 Þ; m11 m12 i

ð1:7:9:1Þ

(ii) The expression (1.7.9.2) to evaluate the multiplication of a positive real number k with the intuitionistic fuzzy soft number a ¼ hl; mi.

1 Appropriate Weighted Averaging Aggregation Operator …

52

D E k  a ¼ 1  ð 1  l Þ k ; mk

ð1:7:9:2Þ

Using the expression (1.7.9.1) and the expression (1.7.9.2), Arora and Garg [2] proposed the intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.9.3).

m j¼1



nj 



ni¼1

  ¼ gi  aij

* 1

m n  Y Y j¼1

i¼1

1  lij

gi

! nj ;

m n Y Y j¼1

! nj + g miji

i¼1

ð1:7:9:3Þ where (i) (ii) (iii) (c)

gi [ 0. P n i¼1 gi ¼ 1. nj [ 0. Pm j¼1 nj ¼ 1:

Garg and Arora [32] pointed out the following drawbacks in the existing intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.9.3)

(i) If there exists an intuitionistic fuzzy soft number aij ¼ lij ; mij such that mij ¼ 0, then the non-membership values of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.9.3), will

be 0, i.e., if there exists an intuitionistic fuzzy soft number aij ¼ lij ; mij such that mij ¼ 0, then the non-membership value of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.9.3), will be independent from the non-membership values of remaining intuitionistic fuzzy soft numbers, which is illogical. To validate this claim, Garg and Arora [32] considered the intuitionistic fuzzy soft numbers a11 ¼ h0:5; 0i, a12 ¼ h0:6; 0:3i, a21 ¼ h0:4; 0:5i, a22 ¼ h0:7; 0:2i with g1 ¼ 0:7, g2 ¼ 0:3; n1 ¼ 0:4; n2 ¼ 0:6 and claimed that the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.9.3), is h0:5126; 0i. It is obvious that the non-membership value of the aggregated intuitionistic fuzzy soft number is 0. (ii) The non-membership value of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft prioritized averaging aggregation operator (1.7.9.3), is independent from the changes in the degree of membership values of the considered intuitionistic fuzzy soft numbers. To validate this claim, Garg and Arora [32], firstly, considered the intuitionistic fuzzy soft numbers a11 ¼ h0:3; 0:6i, a12 ¼ h0:4; 0:3i, a21 ¼ h0:6; 0:2i,

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

53

a22 ¼ h0:7; 0:1i with g1 ¼ 0:6, g2 ¼ 0:4; n1 ¼ 0:8; n2 ¼ 0:2 and claimed that the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.9.3), is h0:5268; 0:2730i. Then, Garg and Arora [32] replaced the intuitionistic fuzzy soft numbers a12 ¼ h0:4; 0:3i and a22 ¼ h0:7; 0:1i with the intuitionistic fuzzy soft numbers b12 ¼ h0:6; 0:3i and b22 ¼ h0:8; 0:1i. Garg and Arora [32] claimed that the new aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.9.3), is h0:55565; 0:2730i.

1.7.10 Drawbacks of Arora and Garg’s Intuitionistic Fuzzy Soft Weighted Prioritized Averaging Aggregation Operator Arora and Garg [2] proposed (i) The expression (1.7.10.1) to evaluate the addition of two intuitionistic fuzzy soft numbers a11 ¼ hl11 ; m11 i, a12 ¼ hl12 ; m12 i. ð1:7:10:1Þ

a11 a12 ¼ h1  ð1  l11 Þð1  l12 Þ; m11 m12 i

(ii) The expression (1.7.10.2) to evaluate the multiplication of a positive real number k with the intuitionistic fuzzy soft number a ¼ hl; mi. D E k  a ¼ 1  ð 1  l Þ k ; mk

ð1:7:10:2Þ

Using the expression (1.7.10.1) and the expression (1.7.10.2), Arora and Garg [2] proposed the intuitionistic fuzzy soft weighted prioritized averaging aggregation operator aggregation operator (1.7.10.3).

m j¼1





  ¼ nj  ni¼1 gi  aij

* 1

m n  Y Y j¼1

i¼1

1  lij

gi

! nj ;

m n Y Y j¼1

! nj + g miji

i¼1

ð1:7:10:3Þ where (i) gi ¼ PRn i i¼1

Ri

  S aip , i ¼ 2; 3; . . .; n:   Q with T1 ¼ 1 and Tj ¼ np¼2 S ajp , j ¼ 2; 3; . . .; m: with R1 ¼ 1 and Ri ¼

T (ii) nj ¼ Pmj Tj   j¼1 (iii) S aij ¼ lij  mij .

Qn

p¼2

54

1 Appropriate Weighted Averaging Aggregation Operator …

Garg and Arora [32] pointed out the following drawbacks in the existing intuitionistic fuzzy soft weighted prioritized averaging aggregation operator (1.7.10.3)

(i) If there exists an intuitionistic fuzzy soft number aij ¼ lij ; mij such that mij ¼ 0, then the non-membership values of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted prioritized averaging aggregation operator (1.7.10.3), will be 0, i.e., if there exists an intuitionistic fuzzy soft number aij ¼ lij ; mij such that mij ¼ 0, then the non-membership value of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted prioritized averaging aggregation operator (1.7.10.3), will be independent from the non-membership values of remaining intuitionistic fuzzy soft numbers, which is illogical. To validate this claim, Garg and Arora [32] considered the intuitionistic fuzzy soft numbers a11 ¼ h0:5; 0i, a12 ¼ h0:6; 0:3i, a21 ¼ h0:4; 0:5i, a22 ¼ h0:7; 0:2i with g1 ¼ 0:7, g2 ¼ 0:3; n1 ¼ 0:4; n2 ¼ 0:6 and claimed that the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted prioritized averaging aggregation operator (1.7.10.3), is h0:5144; 0i. It is obvious that the non-membership value of the aggregated intuitionistic fuzzy soft number is 0. (ii) The non-membership value of the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted prioritized averaging aggregation operator aggregation operator (1.7.10.3), is independent from the changes in the degree of membership values of the considered intuitionistic fuzzy soft numbers. To validate this claim, Garg and Arora [32], firstly, considered the intuitionistic fuzzy soft numbers a11 ¼ h0:3; 0:6i, a12 ¼ h0:4; 0:3i, a21 ¼ h0:6; 0:2i, a22 ¼ h0:7; 0:1i with g1 ¼ 0:6, g2 ¼ 0:4; n1 ¼ 0:8; n2 ¼ 0:2 and claimed that the aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted prioritized averaging aggregation operator (1.7.10.3), is h0:3966; 0:4423i. Then, Garg and Arora [32] replaced the intuitionistic fuzzy soft numbers a12 ¼ h0:4; 0:3i and a22 ¼ h0:7; 0:1i with the intuitionistic fuzzy soft numbers b12 ¼ h0:6; 0:3i and b22 ¼ h0:8; 0:1i. Garg and Arora [32] claimed that the new aggregated intuitionistic fuzzy soft number, obtained by the intuitionistic fuzzy soft weighted prioritized averaging aggregation operator (1.7.10.3), is h0:4140; 0:4423i. It is obvious that the non-membership value of the new aggregated intuitionistic fuzzy soft number is also 0:4423.

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

55

1.7.11 Drawbacks of Garg and Arora’s Intuitionistic Fuzzy Soft Weighted Interaction Averaging Aggregation Operator Garg and Arora [32] proposed (i) The expression (1.7.11.1) to evaluate the addition of two intuitionistic fuzzy soft numbers a11 ¼ hl11 ; m11 i, a12 ¼ hl12 ; m12 i. a11 a12 ¼ h1  ð1  l11 Þð1  l12 Þ; ð1  l11 Þð1  l12 Þ ð1  l11  m11 Þð1  l12  m12 Þi

ð1:7:11:1Þ

(ii) The expression (1.7.11.2) to evaluate the multiplication of a positive real number k with the intuitionistic fuzzy soft number a ¼ hl; mi. D E k  a ¼ 1  ð1  lÞk ; ð1  lÞk ð1  l  mÞk

ð1:7:11:2Þ

Using the expression (1.7.11.1) and the expression (1.7.11.2), Garg and Arora [32] proposed the intuitionistic fuzzy soft weighted interaction averaging aggregation operator (1.7.11.3).

m j¼1



nj 



ni¼1



gi  aij



* ¼

m n  Y Y g 1 1  lij i j¼1



m Y

n  Y

j¼1

i¼1

!nj

i¼1

1  lij  mij

m n  Y Y g ; 1  lij i j¼1

gi

!nj +

!nj

i¼1

ð1:7:11:3Þ Mishra et al. [96] pointed out that an aggregation operator is said to be valid if and only if it satisfies some necessary properties.

The monotonicity

 property   n   0 m m n 0 “aij 4aij ) j¼1 nj  i¼1 gi  aij 4 j¼1 nj  i¼1 gi  aij 8 i; j D E

where aij ¼ lij ; mij and a0ij ¼ l0ij ; m0ij are intuitionistic fuzzy soft numbers” is one of these necessary properties [93, 95]. Mishra et al. [96] further pointed out that although Garg and Arora [32] have claimed that for the intuitionistic fuzzy soft weighted interaction averaging aggregation operator (1.7.11.3), the monotonicity property is not satisfying. But, in actual case, for the intuitionistic fuzzy soft weighted interaction averaging aggregation operator (1.7.11.3), the monotonicity property is not satisfying. Therefore, the intuitionistic fuzzy soft weighted interaction averaging aggregation operator (1.7.11.3) is not valid.

56

1 Appropriate Weighted Averaging Aggregation Operator …

To validate this claim, Mishra et al. [96] pointed out that according to the   n   monotonicity property, the relation

m ¼ j¼1 nj  i¼1 gi  aij

 n 0

m should hold only if lij ¼ l0ij ; mij ¼ m0ij for all i; j i.e., j¼1 nj  i¼1 gi  aij aij ¼ a0ij for all i; j, i.e., if there exist the intuitionistic fuzzy soft numbers aij ¼ D E

lij ; mij and a0ij ¼ l0ij ; m0ij for which the condition lij ¼ l0ij ; mij ¼ m0ij 8 i; j   n   ¼ does not hold. But, the relation

m j¼1 nj  i¼1 gi  aij

 n 0

m hold. Then the intuitionistic fuzzy soft weighted j¼1 nj  i¼1 gi  aij interaction averaging aggregation operator (1.7.11.3) will not be valid. Mishra et al. [96] considered the following example to show that the monotonicity property is not satisfying for the intuitionistic fuzzy soft weighted interaction averaging aggregation operator (1.7.11.3). Example 1.7.11.1 Let us consider preference of the ith expert for the alternatives A D E

and B with respect to the jth attribute be represented by the element aAij ¼ lAij ; mAij D E

 ¼ and the element aBij ¼ lBij ; mBij , respectively, of the matrix A ¼ aAij 22  



 h0:6; 0:3i h1:0; 0:0i ¼ and the matrix B ¼ aBij ¼ lAij ; mAij h0:7; 0:2i h0:4; 0:5i 22 22 22  

 h0:5; 0:4i h1:0; 0:0i lBij ; mBij ¼ : h0:6; 0:3i h0:3; 0:6i 22 It is obvious from the matrices A and B that



(i) aA11 ¼ lA11 ; mA11 ¼ h0:6; 0:3i is greater than aB11 ¼ lB11 ; mB11 ¼ h0:5; 0:4i lA11 ¼ 0:6 is greater than lB11 ¼ 0:5 and mA11 ¼ 0:3 is less than mB11 ¼ 0:4.



(ii) aA12 ¼ lA12 ; mA12 ¼ h1:0; 0:0i is equal to aB12 ¼ lB12 ; mB12 ¼ h1:0; 0:0i lA12 ¼ lB12 ¼ 1:0 and mA12 ¼ mB12 ¼ 0:0.



(iii) aA21 ¼ lA21 ; mA21 ¼ h0:7; 0:2i is greater than aB21 ¼ lB21 ; mB21 ¼ h0:6; 0:3i lA21 ¼ 0:7 is greater than lB21 ¼ 0:6 and mA21 ¼ 0:2 is less than mB21 ¼ 0:3.



(iv) aA22 ¼ lA22 ; mA22 ¼ h0:4; 0:5i is greater than aB22 ¼ lB22 ; mB22 ¼ h0:3; 0:6i lA22 ¼ 0:4 is greater than lB22 ¼ 0:3 and mA22 ¼ 0:5 is less than mB22 ¼ 0:6.

as as as as

 aB11 ; aA12 ¼ aB12 ; aA21  aB21 and aA22  aB22 , i.e.,  B B B B  a11 ; a12 ; a21 ; a22 . So, according to the monotonicity property, the existing intuitionistic fuzzy soft interaction averaging operator

aggregation





Since, aA11  A A A A a11 ; a12 ; a21 ; a22

(1.7.11.3) will be valid only if the relation 2j¼1 nj  2i¼1 gi  aAij



2j¼1 nj  2i¼1 gi  aBij holds.



1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

57

While, using the existing intuitionistic fuzzy soft weighted interaction averaging aggregation operator (1.7.11.3),





2j¼1 nj  2i¼1 gi  aAij  2j¼1 nj  2i¼1 gi  aBij ¼ h1; 0i: This clearly indicates that the monotonicity property is not satisfying for the existing intuitionistic fuzzy soft weighted interaction averaging aggregation operator (1.7.11.3). Hence, the existing intuitionistic fuzzy soft weighted interaction averaging aggregation operator (1.7.11.3) is not valid. Mishra et al. [96] used the following procedure to point out that the expression (1.7.11.1), proposed by Garg and Arora [32], is not valid. If ai ¼ hli ; mi i; i ¼ 1; 2; . . .; n are n intuitionistic fuzzy soft numbers, then the expression (1.7.11.1) can be generalized as the expression (1.7.11.4). *

ni¼1 ai

¼

1

n Y

ð1  li Þ;

i¼1

n Y

ð1  li Þ 

i¼1

n Y

+ ð1  li  mi Þ

ð1:7:11:4Þ

i¼1

If there exists an intuitionistic fuzzy soft number ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0, then *

ni¼1 ai ¼ * ¼

n n n Y Y Y 1  ð1  li Þ; ð1  li Þ  ð1  li  mi Þ i¼1

1  ð1  li Þ

i¼1

i¼1

n1 Y

n1 Y

i¼1

i¼1

ð1  li Þ; ð1  li Þ

n1 Y ð1  li  mi Þ ð1  li  mi Þ

+

ð1  li Þ

i¼1

* ¼

+

1  ð1  1ÞÞ

n1 Y

ð1  li Þ; ð1  1Þ

i¼1

ð1  1  0Þ

n1 Y

+ ð1  li  mi Þ

n1 Y ð1  li Þ i¼1

¼ h1; 0i

i¼1

This indicates that if there exists a ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0. Then, ni¼1 ai is independent from the remaining ai ¼ hli ; mi i; i ¼ 1; 2; . . .; n  1; which is mathematically incorrect.

1 Appropriate Weighted Averaging Aggregation Operator …

58

1.7.12 Drawbacks of Garg and Arora’s Intuitionistic Fuzzy Soft Possibility Weighted Averaging Aggregation Operator Garg and Arora [37] proposed (i) The expression (1.7.12.1) to evaluate the addition of two possibility intuitionistic fuzzy soft numbers a11 ¼ hl11 ; m11 ; p11 i, a12 ¼ hl12 ; m12 ; p12 i. a11 a12 ¼ h1  ð1  l11 Þð1  l12 Þ; m11 m12 ; 1  ð1  p11 Þð1  p12 Þi ð1:7:12:1Þ (ii) The expression (1.7.12.2) to evaluate the multiplication of a positive real number k with the possibility intuitionistic fuzzy soft number a ¼ hl; m; pi. D E k  a ¼ 1  ð 1  lÞ k ; m k ; 1  ð 1  p Þ k

ð1:7:12:2Þ

Using the expression (1.7.12.1) and the expression (1.7.12.2), Garg and Arora [37] proposed the possibility intuitionistic fuzzy soft possibility weighted averaging aggregation operator (1.7.12.3). 





n

m j¼1 nj  i¼1 gi  aij



* ¼

1

m n  Y Y j¼1

1

m n  Y Y j¼1

1  lij

i¼1

1  pij

gi

gi

!nj

!nj +

;

m n Y Y j¼1

!nj g miji

;

i¼1

i¼1

ð1:7:12:3Þ It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.1), is also occurring in the intuitionistic fuzzy soft possibility weighted averaging aggregation operator (1.7.12.3). Therefore, the existing intuitionistic fuzzy soft possibility weighted averaging aggregation operator (1.7.12.3) is not valid.

“If there exists a possibility intuitionistic fuzzy soft number aij ¼ lij ; mij ; pij such that lij ¼ 1, mij ¼ 0 and pij ¼ 0, then the aggregated possibility intuitionistic fuzzy soft number, obtained by the existing intuitionistic fuzzy soft possibility weighted averaging aggregation operator (1.7.12.3), will always be h1; 0; 0i. Hence,

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

59

  n   monotonicity property ‘aij 4a0ij 8 i ) m 4 j¼1 nj  i¼1 gi  aij

 D E

n 0 , where aij ¼ lij ; mij ; pij and a0ij ¼ l0ij ; m0ij ; p0ij are

m j¼1 nj  i¼1 gi  aij

the

possibility intuitionistic fuzzy numbers’ will not be satisfied for the existing intuitionistic fuzzy soft possibility weighted averaging aggregation operator (1.7.12.3).”

1.7.13 Drawbacks of Garg and Arora’s Intuitionistic Fuzzy Soft Power Weighted Averaging Aggregation Operator Garg and Arora [36] proposed (i) The expression (1.7.13.1) to evaluate the addition of two intuitionistic fuzzy soft numbers a11 ¼ hl11 ; m11 i, a12 ¼ hl12 ; m12 i.

a11 a12 ¼ p1 ðpðl11 Þ þ pðl12 ÞÞ; q1 ðqðm11 Þ þ qðm12 ÞÞ

ð1:7:13:1Þ

(ii) The expression (1.7.13.2) to evaluate the multiplication of a positive realumber k with the possibility intuitionistic fuzzy soft number a ¼ hl; mi.

k  a ¼ p1 ðpðkl11 Þ þ pðkl12 ÞÞ; q1 ðqðkm11 Þ þ qðkm12 ÞÞ

ð1:7:13:2Þ

where (i) p : ½0; 1 ! ½0; 1 is a continuous increasing function with pð0Þ ¼ 0. (ii) q : ½0; 1 ! ½0; 1 is a continuous decreasing function with qð1Þ ¼ 0. (iii) qð xÞ ¼ pð1  xÞ. Using the expression (1.7.13.1) and the expression (1.7.13.2), Garg and Arora [36] proposed the intuitionistic fuzzy soft power weighted averaging aggregation operator (1.7.13.3).

m j¼1



nj 



ni¼1



 gi  aij



* ¼

p

1

m X

nj

n X

j¼1

q

1

m X j¼1

nj

  gi p lij

i¼1 n X

  gi q mij

!!

!!+

; ð1:7:13:3Þ

i¼1

It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.1), is also occurring in the existing intuitionistic fuzzy soft power weighted averaging aggregation operator (1.7.13.3). Therefore, the existing intuitionistic fuzzy soft power weighted averaging aggregation operator (1.7.13.3) is not valid.

60

1 Appropriate Weighted Averaging Aggregation Operator …



“If there exists an intuitionistic fuzzy soft number aij ¼ lij ; mij such that li ¼ 1 and mi ¼ 0, then the aggregated intuitionistic fuzzy soft number, obtained by the existing intuitionistic fuzzy soft power weighted averaging aggregation operator (1.7.13.3), will always be h1; 0i. Hence, the monotonicity

property    n   0 m m n 0 ‘aij 4aij 8 i ) j¼1 nj  i¼1 gi  aij 4 j¼1 nj  i¼1 gi  aij , D E

where aij ¼ lij ; mij and a0ij ¼ l0ij ; m0ij are intuitionistic fuzzy soft numbers’ will not be satisfied for the existing intuitionistic fuzzy soft power weighted averaging aggregation operator (1.7.13.3).”

1.7.14 Drawbacks of Garg and Arora’s Intuitionistic Fuzzy Soft Bonferroni Mean Aggregation Operator Garg and Arora [34] proposed (i) The expression (1.7.14.1) to evaluate the addition of two intuitionistic fuzzy soft numbers a11 ¼ hl11 ; m11 i, a12 ¼ hl12 ; m12 i. a11 a12 ¼ h1  ð1  l11 Þð1  l12 Þ; m11 m12 i

ð1:7:14:1Þ

(ii) The expression (1.7.14.2) to evaluate the multiplication of two intuitionistic fuzzy soft numbers a11 ¼ hl11 ; m11 i, a12 ¼ hl12 ; m12 i. a11  a12 ¼ hl11 l12 ; 1  ð1  m11 Þð1  m12 Þi

ð1:7:14:2Þ

(iii) The expression (1.7.14.3) to evaluate the multiplication of a positive real number k with the intuitionistic fuzzy soft number a ¼ hl; mi. D E k  a ¼ 1  ð 1  l Þ k ; mk

ð1:7:14:3Þ

(iv) The expression (1.7.14.4) to evaluate the positive power k of an intuitionistic fuzzy soft number a ¼ hl; mi. D E ak ¼ lk ; 1  ð1  mÞk

ð1:7:14:4Þ

Using the expressions (1.7.14.1)–(1.7.14.4), Garg and Arora [34] proposed the intuitionistic fuzzy soft Bonferroni mean aggregation operator (1.7.14.5).

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

0

61

1p þ1 q

 B 1 p q C m n B C @mnðm  1Þðn  1Þ k; l ¼ 1 i; j ¼ 1 aik  ajl A 0

0

k 6¼ l

i 6¼ j

11p þ1 q

CC B *B m n B CC B Y 1

 Y B B p q mnðm1Þðn1Þ CC ¼ B1  B 1  lik ljl CC B CC B @ AA @ k; l ¼ 1 i; j ¼ 1 k 6¼ l i 6¼ j 0

;

ð1:7:14:5Þ 1p þ1 q

C B m n B Y Y  q mnðm11 Þðn1Þ C C B p 1  B1  1  ð1  mik Þ 1  mjl C C B A @ k; l ¼ 1 i; j ¼ 1 k 6¼ l i ¼ 6 j

+

It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.1), is also occurring in the intuitionistic fuzzy soft Bonferroni mean aggregation operator (1.7.14.5). Therefore, the existing intuitionistic fuzzy soft Bonferroni mean aggregation operator (1.7.14.5) is not valid.

“If there exists an intuitionistic fuzzy soft number aij ¼ lij ; mij such that lij ¼ 0 and mij ¼ 1, then the aggregated intuitionistic fuzzy soft number, obtained by the existing intuitionistic fuzzy soft Bonferroni mean aggregation operator (1.7.14.5), will always be h1; 0i. Hence, the monotonicity property ‘aij 4a0ij 8 i; j ) 0 1p þ1 q

 B p q C @mnðm11 Þðn1Þ mk; l ¼ 1 ni; j ¼ 1 aik  ajl A 4 k 6¼ l i 6¼ j 0 1p þ1 q

p q 

B C @mnðm11 Þðn1Þ mk; l ¼ 1 ni; j ¼ 1 a0ij  a0jl A , where aij ¼ lij ; mij i 6¼ j D Ek 6¼ l and a0ij ¼ l0ij ; m0ij are intuitionistic fuzzy soft numbers’ will not be satisfied for the

existing intuitionistic fuzzy soft Bonferroni mean aggregation operator (1.7.14.5).”

1 Appropriate Weighted Averaging Aggregation Operator …

62

1.7.15 Drawbacks of Garg and Arora’s Generalized Intuitionistic Fuzzy Soft Weighted Averaging Aggregation Operator Garg and Arora [35] proposed the generalized intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.15.1). *

ni¼1 ðwi

 ai Þ ¼

l bi 1 

n Y i¼1

! ð1  l i Þ

wi

; m bi þ

n Y i¼1

mwi i

 m bi

n Y

+ mwi i

i¼1

ð1:7:15:1Þ It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.1.3), is also occurring in the existing generalized intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.15.1). Therefore, the existing generalized intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.15.1) is not valid.

“If there exists an intuitionistic fuzzy soft number ai ¼ lij ; mij such that li ¼ 1 and mi ¼ 0, then the aggregated intuitionistic fuzzy soft number, obtained by the existing generalized intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.15.1), will always be h1; 0i. Hence, the monotonicity property

‘ai 4a0i 8 i; j ) ni¼1 ðwi  ai Þ4 ni¼1 ðwi  ai Þ, where ai ¼ lij ; mij and a0ij ¼ D E l0ij ; m0ij are intuitionistic fuzzy soft numbers’ will not be satisfied for the existing generalized intuitionistic fuzzy soft weighted averaging aggregation operator (1.7.15.1).”

1.7.16 Drawbacks of Garg and Arora’s Dual Hesitant Fuzzy Soft Weighted Averaging Aggregation Operator Garg and Arora [33] proposed (i) The expression (1.7.16.1) to evaluate the addition of two dual hesitant fuzzy soft sets a11 ¼ hfh11 g; fg11 gi and a12 ¼ hfh12 g; fg12 gi.

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

[

a11 a12 ¼

63

 1   

l ðlðc11 Þ þ lðc12 ÞÞ ; k1 ðk ðd11 Þ þ kðd12 ÞÞ

c11 2 h11 ; c12 2 h12 ; d11 2 g11 ; d12 2 g12 ð1:7:16:1Þ (ii) The expression (1.7.16.2) to evaluate the multiplication of a positive real number k with a dual hesitant fuzzy soft set a ¼ hfhg; fggi. ka¼

[    

l1 ðklðcÞÞ ; k 1 ðkkðdÞÞ c2h d2g

ð1:7:16:2Þ

where (i) l : ½0; 1 ! ½0; 1 is a continuous increasing function with lð0Þ ¼ 0. (ii) k : ½0; 1 ! ½0; 1 is a continuous decreasing function with k ð1Þ ¼ 0. (iii) lð xÞ ¼ k ð1  xÞ. Using the expressions (1.7.16.1) and the expression (1.7.16.2), Garg and Arora [33] proposed the dual hesitant fuzzy soft weighted averaging aggregation operator (1.7.16.3).

m j¼1



nj 



ni¼1



gi  aij



*

[

¼ c

(

1

l

m X j¼1

ij 2 hij

nj

n X    gi l cij

k

m X j¼1

nj

;

i¼1

dij 2 hij 1

!!

n X    gi k dij

!!)+

i¼1

ð1:7:16:3Þ It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.3), is also occurring in the existing dual hesitant fuzzy soft weighted averaging aggregation operator (1.7.16.3). Therefore, the existing dual hesitant fuzzy soft weighted averaging aggregation operator (1.7.16.3) is not  valid.   

“If there exists one dual hesitant fuzzy soft number aij ¼ hij ; gij such that     hij ¼ f1g and gij ¼ f0g, then the aggregated dual hesitant fuzzy number, obtained on applying the existing dual hesitant fuzzy soft weighted averaging aggregation operator (1.7.16.3), will be h1; 0i. Hence,

the monotonicity

property    n   0 m m n 0 ‘aij 4aij 8 i; j ) j¼1 nj  i¼1 gi  aij 4 j¼1 nj  i¼1 gi  aij ,

1 Appropriate Weighted Averaging Aggregation Operator …

64

D E

where aij ¼ lij ; mij and a0ij ¼ l0ij ; m0ij are dual hesitant fuzzy soft numbers’ will not be satisfied.”

1.7.17 Drawbacks of Garg’s Intuitionistic Fuzzy Multiplicative Averaging Aggregation Operator Garg [24] pointed out the drawbacks of the existing intuitionistic fuzzy multiplicative averaging aggregation operator [124]. To resolve the drawbacks, Garg [24] proposed (i) The expression (1.7.17.1) to evaluate the sum of two intuitionistic fuzzy multiplicative numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i.  a1 a2 ¼

ð1 þ 2l1 Þð1 þ 2l2 Þ  1 2f1  ð1  l1 m1 Þð1  l2 m2 Þg ; 2 ð1 þ 2l1 Þð1 þ 2l2 Þ  1

ð1:7:17:1Þ

(ii) The expression (1.7.17.2) to evaluate the multiplication of a positive real number k with an intuitionistic fuzzy multiplicative number a ¼ hl; mi. * ka¼

n o k + 2 1  ð 1  lm Þ ð1 þ 2lÞ 1 ; 2 ð1 þ 2lÞk 1 k

ð1:7:17:2Þ

Also, using the expression (1.7.17.1) and the expression (1.7.17.2), Garg [24] proposed the intuitionistic fuzzy multiplicative averaging aggregation operator (1.7.17.3). Q n

ni¼1 ðwi

 ai Þ ¼

i¼1 ð1 þ 2li Þ

2

wi

 Qn w  1 2 1  i¼1 ð1  li mi Þ i ; Qn wi i¼1 ð1 þ 2li Þ 1

ð1:7:17:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n intuitionistic fuzzy multiplicative numbers. (ii) wi is the weight assigned to the ith Pintuitionistic fuzzy multiplicative number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. Mishra [94] considered the following example to show that the expression (1.7.17.1) is not valid, and hence, the expression (1.7.17.2) and the existing intuitionistic fuzzy multiplicative averaging aggregation operator (1.7.17.3) are also not valid.

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

65





Let a1 ¼ hl1 ; m1 i ¼ 3; 14 ; and a2 ¼ hl2 ; m2 i ¼ 4; 16 be two intuitionistic fuzzy multiplicative numbers. Then on applying the existing expression (1.7.17.1), 

ð1 þ 2l1 Þð1 þ 2l2 Þ  1 2f1  ð1  l1 m1 Þð1  l2 m2 Þg ; a1 a2 ¼ ð1 þ 2l1 Þð1 þ 2l2 Þ  1  2 11 ¼ 31; : 372



It is a well-known fact that if a ¼ hl; mi is an intuitionistic fuzzy multiplicative number, then for l and m, the conditions 19  l; m  9 and lm  1 should necessarily

11 , the value be satisfied. However, it is obvious that in, a1 a2 ¼ hl; mi ¼ 31; 372 11 of l is 31 which is greater than 9. Also, the value of m is 372, which is less than 19.

11 is not an intuitionistic fuzzy multiplicative number. Therefore, a1 a2 ¼ 31; 372 Hence, D

the

existing

expression (1.7.17.1), E ð1 þ 2l1 Þð1 þ 2l2 Þ1 2f1ð1l1 m1 Þð1l2 m2 Þg ; ð1 þ 2l Þð1 þ 2l Þ1 is not valid. 2 1

Furthermore, 

as

ð1 þ 2lÞk 1 2f1ð1lmÞ g ; ð1 þ 2lÞk 1 2 k

the

a1 a2 ¼

i.e.,

2

existing

expression

(1.7.17.2),

i.e.,

ka¼

is obtained by considering a1 ¼ a2 in the existing D E ð1l1 m1 Þð1l2 m2 Þg expression (1.7.17.1), i.e., a1 a2 ¼ ð1 þ 2l1 Þð21 þ 2l2 Þ1 ; 2fð1 : 1 þ 2l Þð1 þ 2l Þ1 1

2

However, as discussed, the existing expression (1.7.17.1) is not valid. Therefore, k ð1 þ 2lÞk 1 2f1ð1lmÞ g ; ð1 þ 2lÞk 1 is also the existing expression (1.7.17.2), i.e., k  a ¼ 2 not valid.

1.7.18 Drawbacks of Ma and Xu’s Pythagorean Fuzzy Weighted Averaging Aggregation Operator Ma and Xu [92] proposed (i) The expression (1.7.18.1) to evaluate the addition of two Pythagorean fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. * a1 a2 ¼



l1 l2

m1 m2

+

  1 ;    1 : 1  l21 1  l22 þ l21 l22 2 1  m21 1  m22 þ m21 m22 2 ð1:7:18:1Þ

1 Appropriate Weighted Averaging Aggregation Operator …

66

(ii) The expression (1.7.18.2) to evaluate the multiplication of a positive real number k with a Pythagorean fuzzy number a ¼ hl; mi. * ka¼

h

lk ð1  l2 Þk þ l2k

i12 ; h

+

lk ð1  l2 Þk þ l2k

i12

ð1:7:18:2Þ

Also, using the expression (1.7.18.1) and the expression (1.7.18.2), Ma and Xu [92] proposed the Pythagorean fuzzy weighted averaging aggregation operator (1.7.18.3) and its extensions. *

ni¼1 ðwi

 ai Þ ¼

Qn  Qn

i¼1 ð1

Qn

i¼1 ð1

 Qn



wi i¼1 li Q w l2i Þ i þ ni¼1

1 ; i 2 l2w i +

wi i¼1 mi 1 Q w i 2 m2i Þ i þ ni¼1 m2w i

ð1:7:18:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n Pythagorean fuzzy numbers. (ii) wi is the weight assigned P to the ith Pythagorean fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.1), is also occurring in the existing Pythagorean fuzzy weighted averaging aggregation operator (1.7.18.3). Therefore, the existing Pythagorean fuzzy weighted averaging aggregation operator (1.7.18.3) is not valid. “If there exists a Pythagorean fuzzy number ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0, then the aggregated Pythagorean fuzzy number, obtained by the existing Pythagorean fuzzy weighted averaging aggregation operator (1.7.18.3), will always be h1; 0i. Hence, the monotonicity property ‘ai 4a0i 8 i ) ni¼1 ðwi  ai Þ4  

ni¼1 wi  a0i , where ai ¼ hli ; mi i and a0i ¼ l0i ; m0i are Pythagorean fuzzy numbers’ will not be satisfied for the existing Pythagorean fuzzy weighted averaging aggregation operator (1.7.18.3).”

1.7.19 Drawbacks of Garg’s Pythagorean Fuzzy Weighted Averaging Aggregation Operator Garg [18] proposed (i) The expression (1.7.19.1) to evaluate the sum of two Pythagorean fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i.

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

67

*sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + l21 þ l22 m1 ; m2 a1 a2 ¼ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ l21 :l22 1 þ 1  m2 :1  m2 

ð1:7:19:1Þ

1

2

(ii) The expression (1.7.19.2) to evaluate the multiplication of a positive real number k with a Pythagorean fuzzy number a ¼ hl; mi. ka¼

*sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 þ l2 Þ k  ð 1  l2 Þ k

+ pffiffiffi k 2m ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 þ l2 Þ k þ ð 1  l 2 Þ k ð 2  m 2 Þ k þ ð m 2 Þ k

ð1:7:19:2Þ

Also, using the expression (1.7.19.1) and the expression (1.7.19.2), Garg [18] proposed the Pythagorean fuzzy weighted averaging aggregation operator (1.7.19.3).

ni¼1 ðwi

ffi *sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qn Qn 2 wi 2 wi i¼1 ð1 þ li Þ  i¼1 ð1  li Þ Qn Qn ;  ai Þ ¼ 2 wi 2 wi i¼1 ð1 þ li Þ þ i¼1 ð1  li Þ + pffiffiffi Qn wi 2 i¼1 mi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Qn Qn 2 wi 2 wi i¼1 ð2  mi Þ þ i¼1 ðmi Þ

ð1:7:19:3Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n Pythagorean fuzzy numbers. (ii) wi is the weight assigned P to the ith Pythagorean fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.1), is also occurring in the existing Pythagorean fuzzy weighted averaging aggregation operator (1.7.19.3). Therefore, the existing Pythagorean fuzzy weighted averaging aggregation operator (1.7.19.3) is not valid. “If there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0, then the aggregated Pythagorean fuzzy number, obtained by the existing Pythagorean fuzzy weighted averaging aggregation operator (1.7.19.3), will always be h1; 0i. Hence, the monotonicity property ‘ai 4a0i 8 i ) ni¼1 ðwi  ai Þ4  

ni¼1 wi  a0i , where ai ¼ hli ; mi i and a0i ¼ hl0i ; m0i i are Pythagorean fuzzy numbers’ will not be satisfied for the existing Pythagorean fuzzy weighted averaging aggregation operator (1.7.19.3).”

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1.7.20 Drawbacks of Garg’s Confidence Levels-Based Pythagorean Fuzzy Weighted Averaging Aggregation Operator Garg [22] pointed out that all the aggregation operators have been proposed by considering the assumption that decision-makers are surely familiar with the evaluated objects. But, it is not a realistic assumption. Therefore, to handle this situation, Garg [22] proposed the confidence levels-based Pythagorean fuzzy weighted averaging aggregation operator (1.7.20.1). 



nj¼1 wj  gj  aj



ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *v + u n n   gj wj Y Y u gj wj 2 ; 1  lj mj ¼ t1  j¼1

ð1:7:20:1Þ

j¼1

where (i) 0  gj  1 is confidence level of aj . P (ii) wj is the weight vector associated with aj such that wj  0 and nj¼1 wj ¼ 1. P (i) gj is the confidence level of aj such that gj 2 ½0; 1 and nj¼1 gj ¼ 1. It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.1), is also occurring in the existing confidence levels-based Pythagorean fuzzy weighted averaging aggregation operator (1.7.20.1). Therefore, the existing confidence levels-based Pythagorean fuzzy weighted averaging aggregation operator (1.7.20.1) is not valid. “If there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0, then the aggregated Pythagorean fuzzy number, obtained by the existing Pythagorean fuzzy weighted averaging aggregation operator (1.7.20.1),  will always  be h1; 0i. Hence, the monotonicity property ‘aj 4a0j 8 j ) nj¼1 wj  gj  aj 4



nj¼1 wj  gj  a0j , where ai ¼ hli ; mi i and a0i ¼ hl0i ; m0i i are Pythagorean fuzzy numbers’ will not be satisfied for the existing Pythagorean fuzzy weighted averaging aggregation operator (1.7.20.1).”

1.7.21 Drawbacks of Garg et al.’s Interval-Valued Intuitionistic Fuzzy Weighted Averaging Aggregation Operator Garg et al. [31] pointed out the drawbacks of the existing interval-valued intuitionistic fuzzy weighted averaging aggregation operator [85]. To resolve the drawbacks, Garg et al. [31] proposed

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

69

(i) The expression (1.7.21.1) to evaluate the sum of two interval-valued intuitionistic fuzzy numbers a1 ¼ h½a1 ; b1 ; ½c1 ; d1 i and a2 ¼ h½a2 ; b2 ; ½c2 ; d2 i. *" a1 a2 ¼

Q2 Q2

i¼1 ½1 þ ðc

i¼1 ½1 þ ðc

 1Þai  

Q2

i¼1 ð1  ai Þ Q 1Þ 2i¼1 ð1 

 1Þai  þ ðc  ai Þ # Q2 Q2 ½1 þ ðc  1Þbi   i¼1 ð1  bi Þ ; Q2 Q2 i¼1 i¼1 ½1 þ ðc  1Þbi  þ ðc  1Þ i¼1 ð1  bi Þ " Q Q c 2i¼1 ð1  ai Þ  c 2i¼1 ½1  ai  ci  ; Q2 Q2 i¼1 ½1 þ ðc  1Þai  þ ðc  1Þ i¼1 ð1  ai Þ #+ Q Q c 2i¼1 ð1  bi Þ  c 2i¼1 ½1  bi  di  Q2 Q2 i¼1 ½1 þ ðc  1Þbi  þ ðc  1Þ i¼1 ð1  bi Þ

;

ð1:7:21:1Þ

(ii) The expression (1.7.21.2) to evaluate the multiplication of a positive real number k with an interval-valued intuitionistic fuzzy number a ¼ h½a; b; ½c; d i. *" ka¼

½1 þ ðc  1Þak ½1  ak

½1 þ ðc  1Þak þ ðc  1Þ½1  ak # ½1 þ ðc  1Þbk ½1  bk ; ½1 þ ðc  1Þbk þ ðc  1Þ½1  bk " c½1  ak c½1  a  ck ; ½1 þ ðc  1Þak þ ðc  1Þ½1  ak #+ c½1  bk c½1  b  d k

;

ð1:7:21:2Þ

½1 þ ðc  1Þbk þ ðc  1Þ½1  bk Also, using the expression (1.7.21.1) and the expression (1.7.21.2), Garg et al. [31] proposed the interval-valued intuitionistic fuzzy weighted averaging aggregation operator (1.7.21.3).

1 Appropriate Weighted Averaging Aggregation Operator …

70

Q  1Þai wi  ni¼1 ð1  ai Þwi Qn Q  ai Þ ¼ ; ½1 þ ðc  1Þai wi þ ðc  1Þ ni¼1 ð1  ai Þwi Q Qi¼1  n ½1 þ ðc  1Þbi wi  ni¼1 ð1  bi Þwi Q Qn i¼1 ; ½1 þ ðc  1Þbi wi þ ðc  1Þ ni¼1 ð1  bi Þwi Q ð1:7:21:3Þ  i¼1 Qn c i¼1 ð1  ai Þwi c ni¼1 ½1  ai  ci wi Qn Qn wi wi ; i¼1 ½1 þ ðc  1Þai  þ ðc  1Þ i¼1 ð1  ai Þ Qn Q  c i¼1 ð1  bi Þwi c ni¼1 ½1  bi  di wi Qn Qn wi wi i¼1 ½1 þ ðc  1Þbi  þ ðc  1Þ i¼1 ð1  bi Þ 

ni¼1 ðwi

Qn

i¼1 ½1 þ ðc

where (i) ai ¼ h½ai ; bi ; ½ci ; di i, i ¼ 1; 2; . . .; n are n interval-valued intuitionistic fuzzy numbers. (ii) wi is the weight assigned to the ith interval-valued intuitionistic fuzzy number P ai ¼ h½ai ; bi ; ½ci ; di i such that wi  0 and ni¼1 wi ¼ 1. Mishra [94] considered the following example to show that the expression (1.7.21.1) is not valid, and hence, the interval-valued intuitionistic fuzzy weighted averaging aggregation operator (1.7.21.3) is not valid. If c ¼ 1, k ¼ 2 and n ¼ 2, then the existing expression (1.7.21.1) will be transformed into the expression (1.7.21.4). "

# 2 2 Y Y a1 a2 ¼ 1  ð1  ai Þ; 1  ð1  bi Þ ; "

i¼1

i¼1

2 2 2 2 Y Y Y Y ð1  ai Þ  ½1  ai  ci ; ð1  bi Þ  ½1  bi  di  i¼1

i¼1

i¼1

#

ð1:7:21:4Þ

i¼1

Now, let a1 ¼ h½0:65; 0:73; ½0:17; 0:21i and a2 ¼ h½0:50; 0:60; ½0:30; 0:40i be two interval-valued intuitionistic fuzzy numbers. Then, using the existing interval-valued intuitionistic fuzzy weighted averaging aggregation operator (1.7.21.4), a1 a2 ¼ h½0:8250; 0:8920; ½0:1480; 0:1080i It is a well-known fact that for an interval-valued intuitionistic fuzzy number, a ¼ h½a; b; ½c; d i, the conditions a  b, c  d and b þ d  1 should always be satisfied. However, it can be easily verified that for a1 a2 ¼ h½0:8250; 0:8920; ½0:1480; 0:1080i the condition c  d is not satisfying. Therefore, a1 a2 , obtained by the existing interval-valued intuitionistic fuzzy weighted averaging aggregation operator (1.7.21.4), is not an interval-valued intuitionistic fuzzy number. Hence, the existing interval-valued intuitionistic fuzzy weighted averaging aggregation operator (1.7.21.4) is not valid.

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

71

1.7.22 Drawbacks of Linguistic Interval-Valued Atanassov Intuitionistic Fuzzy Weighted Averaging Aggregation Operators Garg and Kumar [46] proposed (i) The expression (1.7.22.1) to evaluate the addition of two linguistic intervalvalued Atanassov intuitionistic fuzzy numbers a1 ¼ h½sa1 ; sb1 ; ½sc1 ; sd1 i and a2 ¼ h½sa2 ; sb2 ; ½sc2 ; sd 2 i.

a1 a2 ¼

Dh

i h iE sa1 þ a2 a1 a2 ; sb1 þ b2 b1 b2 ; sc1 þ c2 c1 c2 ; sd1 þ d2 d1 d2 t

t

t

ð1:7:22:1Þ

t

(ii) The expression (1.7.22.2) to evaluate the multiplication of a positive real number k with a linguistic interval-valued Atanassov intuitionistic fuzzy number a ¼ h½sa ; sb ; ½sc ; sd i.  ka¼ s

t 1ð1at Þ

    ; st c k ; st d k t 1ð1btÞ ðt Þ ð t Þ

  t ;s t

ð1:7:22:2Þ

where (i) (ii) (iii) (iv) (v)

½sai ; sbi  represents the linguistic membership degree. ½sci ; sdi  represents the linguistic non-membership degree. ai ; bi ; ci ; di 2 ½0; t. bi þ di  t. t is a positive integer.

Using the expression (1.7.22.1) and the expression (1.7.22.2), Garg and Kumar [46] proposed the linguistic interval-valued Atanassov intuitionistic fuzzy weighted averaging aggregation operator (1.7.22.3).

ni¼1 ðwi  ai Þ ¼

Dh iE i h stð1Qn ð1aÞwi Þa ; stð1Qn ð1bÞwi Þ ; st Qn ðcÞwi ; st Qn ðdÞwi t t i¼1 i¼1 i¼1 t i¼1 t ð1:7:22:3Þ

It can be easily verified that the following drawback, pointed out by Beliakov et al. [6] in the existing intuitionistic fuzzy weighted averaging aggregation operator (1.7.1.1), is also occurring in the existing linguistic interval-valued Atanassov intuitionistic fuzzy weighted averaging aggregation operator (1.7.22.3). Therefore, the existing linguistic interval-valued Atanassov intuitionistic fuzzy weighted averaging aggregation operator (1.7.22.3) is not valid.

1 Appropriate Weighted Averaging Aggregation Operator …

72

“If there exists a linguistic interval-valued Atanassov intuitionistic fuzzy number ai ¼ h½sai ; sbi ; ½sci ; sdi i such that sai ¼ sbi ¼ h and sci ¼ sdi ¼ 0, then the aggregated linguistic interval-valued Atanassov intuitionistic fuzzy number, obtained by the existing linguistic interval-valued Atanassov intuitionistic fuzzy weighted averaging aggregation operator (1.7.22.3), will always be ai ¼ h½1;1; ½0; 0i. Hence, the monotonicity property ‘ai 4a0i 8 i ) ni¼1 ðwi  ai Þ4 ni¼1 wi  a0i , h i h i where ai ¼ h½sai ; sbi ; ½sci ; sdi i and a0i ¼ h sa0i ; sb0i ; sc0i ; sdi0 i are linguistic interval-valued Atanassov intuitionistic fuzzy numbers’ will not be satisfied for the existing linguistic interval-valued Atanassov intuitionistic fuzzy weighted averaging aggregation operator (1.7.22.3).”

1.7.23 Drawbacks of Nancy and Garg’s Single-Valued Neutrosophic Weighted Averaging Aggregation Operator Nancy and Garg [97] proposed (i) The expression (1.7.23.1) to evaluate the sum of two single-valued neutrosophic numbers a1 ¼ ha* i. 1 ; b1 ; c1 i and a2 ¼ ha2 ; b2 ;c 2 !  b1  ! k1a1  1 k1a2  1 k  1 kb2  1 ; logk 1 þ ; a1 a2 ¼ 1  logk 1 þ k1 k1   ðkc1  1Þðkc2  1Þ logk 1 þ ;k[1 k1 ð1:7:23:1Þ (ii) The expression (1.7.23.2) to evaluate the multiplication of a positive real number n with a single-valued neutrosophic number a ¼ ha; b; ci. * na¼

 1  logk 1 þ

logk

k1a  1

n !



kb  1

n !

; logk 1 þ ; ðk  1Þn1 ðk  1Þn1 !+ ð kc  1Þ n 1þ ; k[0 ðk  1Þn1

ð1:7:23:2Þ

1.7 Drawbacks of Some Existing Weighted Averaging Aggregation …

73

Also, using the expression (1.7.23.1) and the expression (1.7.23.2), Nancy and Garg [97] proposed the single-valued neutrosophic weighted averaging aggregation operator (1.7.23.3). *

ni¼1 ðwi

 ai Þ ¼

1  logk 1 þ

n  Y

k

1ai

i¼1

logk

n Y 1þ ð k c i  1 Þ wi

1

 wi

! ; logk 1 þ

n  Y

k 1 bi

 wi

! ;

i¼1

!+

i¼1

ð1:7:23:3Þ where (i) ai ¼ hai ; bi ; ci i, i ¼ 1; 2; . . .; n are n single-valued neutrosophic numbers. (ii) wi is the weight assigned to thePith single-valued neutrosophic number ai ¼ hai ; bi ; ci i such that wi  0 and ni¼1 wi ¼ 1. Mishra [93] considered a numerical example to show that the existing single-valued neutrosophic weighted averaging aggregation operator (1.7.23.3) is not valid as it is not satisfying the monotonicity property.

1.7.24 Drawbacks of Liu and Luo’s Single-Valued Neutrosophic Hesitant Fuzzy Weighted Averaging Aggregation Operator Liu and Luo [90] proposed the single-valued neutrosophic hesitant fuzzy weighted averaging aggregation operator (1.7.24.1). [

ki¼1 ðwi  ai Þ ¼

c1 2l1 ;...;ck 2lk ;d1 2m1 ;...;dk 2mk ;g1 2h1 ;...;gk 2hk

(

) ( )) k k ki ki Y Y crðiÞ grðiÞ ; i¼1

(( 1

k Y

1  crðiÞ

ki

) ;

i¼1

i¼1

ð1:7:24:1Þ where (i) ai ¼ fli ; mi ; hi gði ¼ 1; 2; . . .; kÞ is a collection of single-valued neutrosophic hesitant fuzzy elements, where li , mi and hi are three sets of some values in ½0; 1. (ii) wi is the weight associated with ai ði ¼ 1; 2; . . .; k Þ satisfying wi  0, Pk i¼1 wi ¼ 1.

1 Appropriate Weighted Averaging Aggregation Operator …

74

(iii) r: f1; 2; . . .; k g ! f1; 2; . . .; kg is a permutation such that arðiÞ is the largest number in ða1 ; a2 ; . . .; ak Þ. Mishra and Kumar [95] considered the following example to show the monotonicity property is not satisfying for the existing single-valued neutrosophic hesitant fuzzy weighted averaging aggregation operator (1.7.24.1). Hence, the existing single-valued neutrosophic hesitant fuzzy weighted averaging aggregation operator (1.7.24.1) is not valid. Let n1 ¼ ff1g; f0g; f1gg, n2 ¼ ff0:1g; f0:2g; f0gg and m1 ¼ ff1g; f0g; f0gg, m2 ¼ ff0:3g; f0:2g; f0gg be four single-valued neutrosophic hesitant fuzzy numbers. Then, using the existing comparing method [90], (i) n1 ¼ ff1g; f0g; f1gg is less than m1 ¼ ff1g; f0g; f0gg as sðn1 Þ ¼ 0:6667 is less than sðm1 Þ ¼ 1. (ii) n2 ¼ ff0:1g; f0:2g; f0gg is less than m2 ¼ ff0:3g; f0:2g; f0gg as sðn2 Þ ¼ 0:6333 is less than sðm2 Þ ¼ 0:7. Since, n1 m1 and n2 m2 . So, according to the monotonicity property, the relation w1  n1 w2  n2 w1  m1 w2  m2 should hold. While, using the existing single-valued neutrosophic hesitant fuzzy weighted averaging aggregation S operator (1.7.24.1), i.e., ki¼1 ðwi  ai Þ = c1 2l1 ;...;ck 2lk ;d1 2m1 ;...;dk 2mk ;g1 2h1 ;...;gk 2hk  ki   Q ki   Q ki  k k k Q 1 1  crðiÞ crðiÞ grðiÞ ; ; , i¼1

i¼1

i¼1

w1  n1 w2  n2 ¼ w1  m1 w2  m2 ¼ ff1g; f0g; f0gg This clearly indicates that the monotonicity property is not satisfying for the existing single-valued neutrosophic hesitant fuzzy weighted averaging aggregation operator (1.7.24.1). Hence, the existing single-valued neutrosophic hesitant fuzzy weighted averaging aggregation operator (1.7.24.1) is not valid.

1.8

Appropriate Weighted Averaging Aggregation Operators Under Some Extensions of the Fuzzy Environment

In this section, the appropriate weighted averaging aggregation operators under some extensions of the fuzzy environment have been discussed. The appropriate weighted averaging aggregation operators under the remaining extensions of the fuzzy environment can be defined in the same manner.

1.8 Appropriate Weighted Averaging Aggregation …

1.8.1

75

Appropriate Intuitionistic Fuzzy Weighted Averaging Aggregation Operator

Using the Lukasiewicz t-conorm Sðx; yÞ ¼ minð1; x þ yÞ and the Lukasiewicz t-norm T ðx; yÞ ¼ maxð0; 1  ðð1  xÞ þ ð1  yÞÞÞ, Beliakov et al. [6] proposed (i) The expression (1.8.1.1) to evaluate the addition of two intuitionistic fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i. a1 a2 ¼ hminð1; l1 þ l2 Þ; maxð0; 1  ðð1  m1 Þ þ ð1  m2 ÞÞÞi

ð1:8:1:1Þ

(ii) The expression (1.8.1.2) to evaluate the multiplication of a positive real number k with an intuitionistic fuzzy number a ¼ hl; mi, and hence, the expression (1.8.1.3) to evaluate the multiplication of a real number k 2 ð0; 1 with an intuitionistic fuzzy number a ¼ hl; mi. k  a ¼ hminð1; klÞ; maxð0; 1  kð1  mÞÞi; k [ 0 k  a ¼ hkl; 1  kð1  mÞi; 0\k  1

ð1:8:1:2Þ ð1:8:1:3Þ

Also, using the expression (1.8.1.1) and the expression (1.8.1.3), Beliakov et al. [6] proposed the intuitionistic fuzzy weighted averaging aggregation operator (1.8.1.4). *

ni¼1 ðwi

 ai Þ ¼

min 1; *

¼

n X

! wi li

i¼1 n X i¼1

wi li ;

n X

n X ðwi ð1  mi ÞÞ ; max 0; 1 

+

!+

i¼1

wi mi

i¼1

ð1:8:1:4Þ In Sect. 1.7, it is pointed out that if there exists an intuitionistic fuzzy number ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0, then the aggregated intuitionistic fuzzy number, obtained by the existing intuitionistic fuzzy weighted averaging operators [2, 3, 13, 21, 24, 32–37, 65, 114, 127, 137], is h1; 0i. While, it can be easily verified that the aggregated intuitionistic fuzzy number, obtained by the intuitionistic fuzzy weighted averaging aggregation operator (1.8.1.4), is not h1; 0i. Hence, it is appropriate to use the intuitionistic fuzzy weighted averaging aggregation operator (1.8.1.4) for aggregating intuitionistic fuzzy numbers as compared to the existing intuitionistic fuzzy weighted averaging operators [2, 3, 13, 21, 24, 32–37, 65, 114, 127, 137].

1 Appropriate Weighted Averaging Aggregation Operator …

76

1.8.2

Appropriate Pythagorean Fuzzy Weighted Averaging Aggregation Operator

Using the generalized form of the Lukasiewicz t-conorm Sðx; yÞ ¼ minð1; x2 þ y2 Þ and the generalized form of the Lukasiewicz t-norm T ðx; yÞ ¼ maxð0; 1  ðð1  x2 Þ þ ð1  y2 ÞÞÞ, (i) The expression (1.8.2.1) has proposed to evaluate the addition of two Pythagorean fuzzy numbers a1 ¼ hl1 ; m1 i and a2 ¼ hl2 ; m2 i.       

a1 a2 ¼ min 1; l21 þ l22 ; max 0; 1  1  m21 þ 1  m22

ð1:8:2:1Þ

(ii) The expression (1.8.2.2) has been proposed to evaluate the multiplication of a positive real number k with a Pythagorean fuzzy number a ¼ hl; mi, and hence, the expression (1.8.2.3) has been proposed to evaluate the multiplication of a real number k 2 ð0; 1 with an intuitionistic fuzzy number a ¼ hl; mi.     

k  a ¼ min 1; kl2 ; max 0; 1  k 1  m2 ; k [ 0  

k  a ¼ kl2 ; 1  k 1  m2 ; 0\k  1

ð1:8:2:2Þ ð1:8:2:3Þ

Also, using the expression (1.8.2.2) and the expression (1.8.2.3), the Pythagorean fuzzy weighted averaging aggregation operator (1.8.2.4) has been proposed. *

ni¼1 ðwi

 ai Þ ¼

min 1; *

¼

n X

! wi l2i

i¼1 n X i¼1

wi l2i ;

n X

n   X  wi 1  m2i ; max 0; 1 

+

!+

i¼1

wi m2i

i¼1

ð1:8:2:4Þ In Sect. 1.7, it is pointed out that if there exists a Pythagorean fuzzy number ai ¼ hli ; mi i such that li ¼ 1 and mi ¼ 0, then the aggregated Pythagorean fuzzy number, obtained by the existing Pythagorean fuzzy weighted averaging operators [18, 22, 92], is h1; 0i. While, it can be easily verified that the aggregated Pythagorean fuzzy number, obtained by the Pythagorean fuzzy weighted averaging aggregation operator (1.8.2.4), is not h1; 0i. Hence, it is appropriate to use the intuitionistic fuzzy

1.8 Appropriate Weighted Averaging Aggregation …

77

weighted averaging aggregation operator (1.8.2.4) for aggregating Pythagorean fuzzy numbers as compared to the existing Pythagorean fuzzy weighted averaging operators [18, 22, 92].

1.8.3

Appropriate Connection Number Weighted Averaging Aggregation Operator

Kumar and Garg [73, 74] proposed the connection number weighted averaging aggregation  operator (1.8.3.1) to aggregate connection numbers   dh ¼ a1h þ a2h i þ a3h j, h ¼ 1; 2; . . .; n:

nh¼1 ðwh

 dh Þ ¼

n  X

wh a1h

h¼1



! ! n  n  X X   2 3 w h ah i þ wh ah j þ h¼1

ð1:8:3:1Þ

h¼1

It can be easily verified that if there exists a connection number dh ¼     þ a2h i þ a3h j such that a1h ¼ a3h ¼ 0 and a2h ¼ 1, then the aggregated connection number, obtained by the existing connection number weighted averaging aggregation operator (1.8.3.1), is not 0 þ ð1Þi þ ð0Þj. Hence, it is appropriate to use the connection number weighted averaging aggregation operator (1.8.3.1) for aggregating connection numbers. a1h

1.8.4

Appropriate Interval-Valued Intuitionistic Fuzzy Weighted Averaging Aggregation Operator

Xu and Yager [129] proposed the interval-valued intuitionistic fuzzy weighted averaging aggregation  operator  (1.8.4.1)  to aggregate interval-valued intuitionistic L U L U fuzzy numbers ai ¼ li ; li ; mi ; mi ; i ¼ 1; 2; . . .; n: *

ni¼1 ðwi

 ai Þ ¼

n  n  X  X  ; wi lLi ; wi lU wi mLi ; wi mU i i i¼1

+ ð1:8:4:1Þ

i¼1

In Sect. 1.7, it is pointed if there intuitionistic  out

that

exists anL interval-valued L U U L U ; m such that l ; m ¼ l ¼ 1 and m fuzzy number ai ¼ lLi ; lU i i i i i i ¼ mi ¼ 0, then the aggregated interval-valued intuitionistic fuzzy number, obtained by the existing interval-valued intuitionistic fuzzy weighted averaging aggregation operator [46], is h½1; 1; ½0; 0i.” While, it can be easily verified that the aggregated interval-valued intuitionistic fuzzy number, obtained by the existing interval-valued intuitionistic fuzzy weighted

1 Appropriate Weighted Averaging Aggregation Operator …

78

averaging operator (1.8.4.1), is not h½1; 1; ½0; 0i. Hence, it is appropriate to use the existing interval-valued intuitionistic fuzzy weighted averaging aggregation operator (1.8.4.1) for aggregating interval-valued intuitionistic fuzzy numbers as compared to the existing interval-valued intuitionistic fuzzy weighted averaging aggregation operator [46].

1.9

Limitation of the Weighted Geometric Aggregation Operators Under Various Extensions of the Fuzzy Environment

It is a well-known fact that if ai ; i ¼ 1; 2; . . .; n are n real numbers, then the weighted geometric mean of these real numbers can be defined only if ai [ 0 8 i ¼ 1; 2; . . .; n: Since, this condition will not be satisfied for the numbers under various extensions of the fuzzy environment. Therefore, it is illogical to define a weighted geometric aggregation operator under various extensions of the fuzzy environment. The following clearly indicates that this claim is valid: Garg [29] proposed the weighted neutral geometric aggregation operator (1.9.1) to aggregate ‘n’ Pythagorean fuzzy numbers ai ¼ hli ; mi i and claimed that it can be used only if neither li ¼ 0 nor mi ¼ 0 for any i. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *v !wiffi u Qn n 2 Þ wi u Y ð l i¼1 i Qn ni¼1 ðai Þwi ¼ tQn ; 1 1  l2i  m2i 2 Þwi þ 2 Þ wi ð l ð m i¼1 i i¼1 i i¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !wiffi+ u Qn n 2 Þ wi u Y ð m i¼1 i tQ Qn 1 1  l2i  m2i n 2 Þ wi þ 2 Þ wi ð l ð m i¼1 i i¼1 i i¼1

ð1:9:1Þ

where (i) ai ¼ hli ; mi i, i ¼ 1; 2; . . .; n are n Pythagorean fuzzy numbers. (ii) wi is the weight assigned P to the ith Pythagorean fuzzy number ai ¼ hli ; mi i such that wi  0 and ni¼1 wi ¼ 1.

1.10

Conclusions

On the basis of the present study, the following can be concluded (i) It is inappropriate to use the existing weighted aggregation operators [1–3, 6, 9–11, 13, 14, 16–39, 42–65, 67, 76–93, 95–104, 106–108, 110–131, 135– 138, 140–146] under various extensions of the fuzzy environment.

1.10

Conclusions

79

(ii) It is appropriate to use the existing intuitionistic fuzzy weighted averaging aggregation operator [6] to aggregate intuitionistic fuzzy numbers. (iii) It is appropriate to use the generalization of existing intuitionistic fuzzy weighted averaging aggregation operator [6] to aggregate Pythagorean fuzzy numbers. (iv) It is appropriate to use the existing connection number weighted averaging aggregation operator [73, 74] to aggregate connection numbers. (v) It is appropriate to use the existing interval-valued intuitionistic fuzzy weighted averaging aggregation operator [129] to aggregate interval-valued intuitionistic fuzzy numbers. (vi) It is illogical to propose weighted geometric aggregation operators under various extensions of the fuzzy environment.

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Chapter 2

Mehar Method to Find a Unique Fuzzy Optimal Value of Balanced Fully Triangular Fuzzy Transportation Problems

In the last few years, several methods have been proposed to solve transportation problems under fuzzy environment [11 and the references therein]. Ebrahimnejad [1, 3] pointed out some drawbacks of the existing methods [4–8, 10, 12–18] for solving fully triangular fuzzy transportation problems (transportation problems in which each parameter is represented by a triangular fuzzy number). Also, to resolve the drawbacks, (i) Ebrahimnejad [1] proposed a method for solving balanced fully triangular fuzzy transportation problems. (ii) Ebrahimnejad [3] proposed a method for solving balanced fully triangular fuzzy transportation problems. One may claim that Ebrahimnejad’s methods [1, 3] can be used only to solve such balanced fully triangular fuzzy transportation problems for which the aggregated value of the fuzzy transportation cost, fuzzy availability and fuzzy demand, provided by all the decision-makers, is available. While, Ebrahimnejad’s methods [1, 3] cannot be used to solve such balanced fully triangular fuzzy transportation problems for which, instead of the aggregated data, the data of each decision-maker is provided separately. To overcome this limitation, one may modify Ebrahimnejad’s methods [1, 3] with the help of the existing triangular fuzzy weighted aggregation operators [19, 20]. Also, one may use Ebrahimnejad’s methods [1, 3] in its present form or its modified version to solve real-life balanced fully triangular fuzzy transportation problems. However, after a deep study, it is observed that on applying Ebrahimnejad’s methods [1, 3] more than one triangular fuzzy numbers, representing the total minimum fuzzy transportation cost, are obtained, which is mathematically incorrect as the physical meaning of all the obtained triangular fuzzy numbers will be different. Hence, it is inappropriate to use Ebrahimnejad’s methods [1, 3] in their present form or the modified version of Ebrahimnejad’s methods [1, 3] to solve real-life balanced fully triangular fuzzy transportation problems. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Mishra and A. Kumar, Aggregation Operators for Various Extensions of Fuzzy Set and Its Applications in Transportation Problems, Studies in Fuzziness and Soft Computing 399, https://doi.org/10.1007/978-981-15-6998-2_2

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2 Mehar Method to Find a Unique Fuzzy Optimal Value …

88

The aim of this chapter is (i) To make the researchers aware about the inappropriateness of Ebrahimnejad’s methods [1, 3]. (ii) To propose a new method (named as Mehar method) for solving balanced fully triangular fuzzy transportation problems to overcome a limitation and to resolve a drawback of Ebrahimnejad’s methods [1, 3].

2.1

Organization of the Chapter

This chapter is organized as follows: (i) In Sect. 2.2, some basic definitions have been presented. (ii) In Sect. 2.3, an existing method for comparing triangular fuzzy numbers, used in Ebrahimnejad’s methods [1, 3], has been discussed. (iii) In Sect. 2.4, Ebrahimnejad’s methods [1, 3] for solving balanced fully triangular fuzzy transportation problems have been discussed. (iv) In Sect. 2.5, to illustrate Ebrahimnejad’s methods [1, 3], a balanced fully triangular fuzzy transportation problem has been solved by Ebrahimnejad’s methods [1, 3]. (v) In Sect. 2.6, a limitation of Ebrahimnejad’s methods [1, 3] has been discussed. (vi) In Sect. 2.7, a drawback of Ebrahimnejad’s methods [1, 3] has been discussed. (vii) In Sect. 2.8, the reason for the occurrence of the drawback in Ebrahimnejad’s methods [1, 3] has been discussed. (viii) In Sect. 2.9, the existing Rank, Mode and Divergence based approach for comparing triangular fuzzy numbers [9] has been discussed. (ix) In Sect. 2.10, it is proved that the existing Rank, Mode and Divergence based approach for comparing triangular fuzzy numbers [9] is valid. (x) In Sect. 2.11, a new method (named as Mehar method) has been proposed to solve balanced fully triangular fuzzy transportation problems. (xi) In Sect. 2.12, a unique fuzzy optimal value of the balanced fully triangular fuzzy transportation problem, considered in Sect. 2.5, has been obtained by the proposed Mehar method. (xii) Sect. 2.13 concludes the chapter.

2.2 Preliminaries

2.2

89

Preliminaries

In this section, some basic definitions, used in the further sections, have been presented [9]. Definition 2.2.1 A convex normalized fuzzy set having piecewise continuous membership function is called a fuzzy number. e ¼ ða; b; cÞ is said to be a triangular fuzzy Definition 2.2.2 A fuzzy number A number if its membership function le ð xÞ is defined as A

8 xa ; > > < ba 1; le ð xÞ ¼ cx A > cb ; > : 0;

a  x\b; x ¼ b; b\x  c; elsewhere:

e 1 ¼ ða1 ; b1 ; c1 Þ and A e2 ¼ Definition 2.2.3 Two triangular fuzzy numbers A e e ða2 ; b2 ; c2 Þ are said to be equal, i.e., A 1 ¼ A 2 if and only if a1 ¼ a2 , b1 ¼ b2 , c1 ¼ c2 . e ¼ ða; b; cÞ is said to be Definition 2.2.4 A triangular fuzzy number A non-negative triangular fuzzy number if and only if a  0. e 2 ¼ ða2 ; b2 ; c2 Þ are two triangular fuzzy e 1 ¼ ða1 ; b1 ; c1 Þ and A Definition 2.2.5 If A e e numbers. Then, A 1  A 2 ¼ ða1 þ a2 ; b1 þ b2 ; c1 þ c2 Þ. e 2 ¼ ða2 ; b2 ; c2 Þ are two non-fsubjects. e 1 ¼ ða1 ; b1 ; c1 Þ and A Definition 2.2.6 If A e 2 ¼ ða1 a2 ; b1 b2 ; c1 c2 Þ. e1  A Then, A

2.3

An Existing Method for Comparing Triangular Fuzzy Numbers

If A1 and A2 are two distinct real numbers, i.e., A1 6¼ A2 , then it can be easily e 1 ¼ ða1 ; b1 ; c1 Þ and A e2 ¼ concluded that A1 \A2 or A1 [ A2 . However, if A e e ða2 ; b2 ; c2 Þ are two triangular fuzzy numbers such that A 1 6¼ A 2 . Then, it cannot e 2 or A e1  A e 2 . Different methods have been proposed e1  A easily concluded that A in the literature for comparing triangular fuzzy numbers. Ebrahimnejad [1, 3] has used the following method for comparing two distinct e 2 ¼ ða2 ; b2 ; c2 Þ. e 1 ¼ ða1 ; b1 ; c1 Þ and A triangular fuzzy numbers A e 2 if a1 \a2 , b1  b2 , c1  c2 or a1  a2 , b1 \b2 , c1  c2 or a1  a2 , e1  A (i) A b1  b2 , c1 \c2 .

2 Mehar Method to Find a Unique Fuzzy Optimal Value …

90

e1  A e 2 if a1 [ a2 , b1  b2 , c1  c2 or a1  a2 , b1 [ b2 , c1  c2 or a1  a2 , (ii) A b1  b2 , c 1 [ c 2 .

2.4

Ebrahimnejad’s Methods for Solving Balanced Fully Triangular Fuzzy Transportation Problems

Ebrahimnejad [1, 3] claimed that if it is assumed that the triangular fuzzy number  ~xij ¼ xij;1 ; xij;2 ; xij;3 represents the fuzzy quantity of the product to be supplied   from the ith source ðSi Þ to the jth destination Dj . Then, to solve a balanced fully triangular fuzzy transportation problem, represented by Table 2.1, is equivalent to solve the fully fuzzy linear programming problem (2.4.1). Hence, Ebrahimnejad [1, 3] proposed the following two methods to solve the fully fuzzy linear programming problem (2.4.1). where   (i) The triangular fuzzy number ~cij ¼ cij;1 ; cij;2 ; cij;3 represents the fuzzy transportation cost for supplying the one  unit  quantity of the product from the ith source ðSi Þ to the jth destination Dj .   (ii) The triangular fuzzy number ~ai ¼ ai;1 ; ai;2 ; ai;3 represents the fuzzy availability of the product at the ith source ðSi Þ.   (iii) The triangular fuzzy number ~bj ¼ bj;1 ; bj;2 ; bj;3 represents the fuzzy   demand of the product at the jth destination Dj . (iv) m represents the number of sources. (v) n represents the number of destinations. Fully fuzzy linear programming problem (2.4.1) h    i n Minimize m  x  c ; c ; c ; x ; x ij;1 ij;2 ij;3 ij;1 ij;2 ij;3 i¼1 j¼1

Table 2.1 Tabular representation of a balanced fully fuzzy transportation problem Destinations Sources

D1

D2



Dj



Dn

Fuzzy availability

S1 .. . Si .. . Sm Fuzzy demand

~c11 .. . ~ci1 .. . ~cm1 ~b1

~c12 .. . ~ci2 .. . ~cm2 ~b2

.. . .. .

~c1j .. . ~cij .. . ~cmj ~bj

.. . .. .

~c1n .. . ~cin .. . ~cmn ~ bn

~ a1 .. . ~ ai .. . ~ am Pm i¼1

~ ai ¼

Pn j¼1

~ bj

2.4 Ebrahimnejad’s Methods for Solving Balanced Fully …

91

Subject to     nj¼1 xij;1 ; xij;2 ; xij;3 ¼ ai;1 ; ai;2 ; ai;3 ; i ¼ 1; 2; . . .; m;     m i¼1 xij;1 ; xij;2 ; xij;3 ¼ bj;1 ; bj;2 ; bj;3 ; j ¼ 1; 2; . . .; n; 

 xij;1 ; xij;2 ; xij;3 is a non-negative 8 i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n.

2.4.1

triangular

fuzzy

number

Ebrahimnejad’s First Method

Ebrahimnejad [1] proposed the following method to solve the fully fuzzy linear programming problem (2.4.1). Step 1: Using the multiplication of two non-negative triangular fuzzy numbers ða1 ; b1 ; c1 Þ and ða2 ; b2 ; c2 Þ, i.e., ða1 ; b1 ; c1 Þ  ða2 ; b2 ; c2 Þ = ða1 a2 ; b1 b2 ; c1 c2 Þ, transform the fully fuzzy linear programming problem (2.4.1) into its equivalent fully fuzzy linear programming problem (2.4.1.1). Fully fuzzy linear programming problem (2.4.1.1) h  i n Minimize m  c x ; c x ; c x ij;1 ij;1 ij;2 ij;2 ij;3 ij;3 i¼1 j¼1 Subject to Constraints of the fully fuzzy linear programming problem (2.4.1).  Pm Pm Pm  Step 2: Using the relation, m i¼1 ðai ; bi ; ci Þ ¼ i¼1 ai ; i¼1 bi ; i¼1 ci , transform the fully fuzzy linear programming problem (2.4.1.1) into its equivalent fully fuzzy linear programming problem (2.4.1.2). Fully fuzzy linear programming problem (2.4.1.2) " Minimize

m X n X i¼1 j¼1

Subject to

cij;1 xij;1 ;

m X n X i¼1 j¼1

cij;2 xij;2 ;

m X n X i¼1 j¼1

!# cij;3 xij;3

2 Mehar Method to Find a Unique Fuzzy Optimal Value …

92 n X

xij;1 ;

n X

xij;2 ;

n X

j¼1

j¼1

j¼1

n X

n X

n X

xij;1 ;

j¼1

xij;2 ;

j¼1

! xij;3 ! xij;3

  ¼ ai;1 ; ai;2 ; ai;3 ;

i ¼ 1; 2; . . .; m;

  ¼ bj;1 ; bj;2 ; bj;3 ;

j ¼ 1; 2; . . .; n;

j¼1



 is a non-negative xij;1 ; xij;2 ; xij;3 8 i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n.

triangular

fuzzy

number

Step 3: Using the relation, ða1 ; b1 ; c1 Þ ¼ ða2 ; b2 ; c2 Þ ) a1 ¼ a2 ; b1 ¼ b2 ; c1 ¼ c2 and the relation ða; b; cÞ is a non-negative triangular fuzzy number ) a  0, b  a  0; c  b  0; transform the fully fuzzy linear programming problem (2.4.1.2) into its equivalent fuzzy linear programming problem (2.4.1.3). Fuzzy linear programming problem (2.4.1.3) " Minimize

m X n X

cij;1 xij;1 ;

m X n X

i¼1 j¼1

i¼1 j¼1

n X

n X

cij;2 xij;2 ;

m X n X

!# cij;3 xij;3

i¼1 j¼1

Subject to n X

xij;1 ¼ ai;1 ;

j¼1 m X i¼1

xij;2 ¼ ai;2 ;

j¼1

xij;1 ¼ bj;1 ;

m X

xij;3 ¼ ai;3 ;

i ¼ 1; 2; . . .; m;

j¼1

xij;2 ¼ bj;2 ;

i¼1

m X

xij;3 ¼ bj;3 ; j ¼ 1; 2; . . .; n;

i¼1

xij;1  0; xij;2  xij;1  0; xij;3  xij;2  0:

Step 4: Using the existing method for comparing triangular fuzzy numbers, discussed in Sect. 2.3, transform the fuzzy linear programming problem (2.4.1.3) into its equivalent crisp multi-objective linear programming problem (2.4.1.4). Crisp multi-objective linear programming problem (2.4.1.4) " Minimize

m X n X i¼1 j¼1

# cij;1 xij;1

2.4 Ebrahimnejad’s Methods for Solving Balanced Fully …

" Minimize

m X n X

93

# cij;2 xij;2

i¼1 j¼1

" Minimize

m X n X

# cij;3 xij;3

i¼1 j¼1

Subject to Constraints of the fuzzy linear programming problem (2.4.1.3).   Step 5: Find a crisp optimal solution xij;1 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n and the P Pn crisp optimal value m i¼1 j¼1 cij;1 xij;1 of the crisp linear programming problem (2.4.1.5). Crisp linear programming problem (2.4.1.5) " Minimize

m X n X

# cij;1 xij;1

i¼1 j¼1

Subject to

n X

xij;1 ¼ ai;1 ;

i ¼ 1; 2; . . .; m;

j¼1 m X

xij;1 ¼ bj;1 ; j ¼ 1; 2; . . .; n;

i¼1

xij;1  0

8 i ¼ 1; 2; . . .; m;

j ¼ 1; 2; . . .; n:

  Step 6: Find a crisp optimal solution xij;2 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n and the Pm Pn crisp optimal value i¼1 j¼1 cij;2 xij;2 of the crisp linear programming problem (2.4.1.6). Crisp linear programming problem (2.4.1.6) " Minimize

m X n X

# cij;2 xij;2

i¼1 j¼1

Subject to n X j¼1

xij;2 ¼ ai;2 ;

i ¼ 1; 2; . . .; m;

2 Mehar Method to Find a Unique Fuzzy Optimal Value …

94 m X

xij;2 ¼ bj;2 ;

j ¼ 1; 2; . . .; n;

i¼1

x ij;1  xij;2 8 i ¼ 1; 2; . . .; m;

j ¼ 1; 2; . . .; n:

where x ij;1 is the crisp optimal value of the variable xij;1 in the crisp optimal solution of the crisp linear programming problem (2.4.1.5).   Step 7: Find a crisp optimal solution xij;3 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n and the Pm Pn crisp optimal value i¼1 j¼1 cij;3 xij;3 of the crisp linear programming problem (2.4.1.7). Crisp linear programming problem (2.4.1.7) " Minimize

m X n X

# cij;3 xij;3

i¼1 j¼1

Subject to

n X

xij;3 ¼ ai;3 ;

i ¼ 1; 2; . . .; m;

j¼1 m X

xij;3 ¼ bj;3 ; j ¼ 1; 2; . . .; n;

i¼1

x ij;2  xij;3 8 i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n: where x ij;2 is the crisp optimal value of the variable xij;2 in the crisp optimal solution of the crisp linear programming problem (2.4.1.6).   Step 8: Using the crisp optimal solution xij;1 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n , the   crisp optimal solution xij;2 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n and the crisp optimal   solution xij;3 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n of the crisp linear programming problem (2.4.1.5), the crisp linear programming problem (2.4.1.6) and the crisp linear programming    problem (2.4.1.7), respectively, find a fuzzy optimal solution xij;1 ; xij;2 ; xij;3 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; ng of the fully fuzzy linear programming problem (2.4.1). P Pn Step 9: Using the crisp optimal value m i¼1 j¼1 cij;1 xij;1 , the crisp optimal value Pm Pn Pm Pn i¼1 j¼1 cij;2 xij;2 and the crisp optimal value i¼1 j¼1 cij;3 xij;3 of the crisp linear programming problem (2.4.1.5), the crisp linear programming problem (2.4.1.6) and the crisp linear programming problem (2.4.1.7), respectively, find the

2.4 Ebrahimnejad’s Methods for Solving Balanced Fully …

fuzzy optimal value

P

m i¼1

Pn

j¼1 cij;1 xij;1 ;

95

Pm Pn i¼1

j¼1 cij;2 xij;2 ;,

Pm Pn



j¼1 cij;3 xij;3

i¼1

of the fully fuzzy linear programming problem (2.4.1).

2.4.2

Ebrahimnejad’s Second Method

Ebrahimnejad [3] proposed the following method to solve the fully fuzzy linear programming problem (2.4.1). Step 1: Using the multiplication of two non-negative triangular fuzzy numbers ða1 ; b1 ; c1 Þ and ða2 ; b2 ; c2 Þ, i.e., ða1 ; b1 ; c1 Þ  ða2 ; b2 ; c2 Þ = ða1 a2 ; b1 b2 ; c1 c2 Þ, transform the fully fuzzy linear programming problem (2.4.1) into its equivalent fully fuzzy linear programming problem (2.4.2.1). Fully fuzzy linear programming problem (2.4.2.1) h  i n Minimize m i¼1 j¼1 cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 Subject to Constraints of the fully fuzzy linear programming problem (2.4.1).  Pm Pm Pm  Step 2: Using the relation, m i¼1 ðai ; bi ; ci Þ ¼ i¼1 ai ; i¼1 bi ; i¼1 ci , transform the fully fuzzy linear programming problem (2.4.2.1) into its equivalent fully fuzzy linear programming problem (2.4.2.2). Fully fuzzy linear programming problem (2.4.2.2) " Minimize

m X n X

cij;1 xij;1 ;

i¼1 j¼1

m X n X

cij;2 xij;2 ;

i¼1 j¼1

m X n X

!# cij;3 xij;3

i¼1 j¼1

Subject to n X

xij;2 ;

n X

j¼1

j¼1

j¼1

n X

n X

n X

j¼1



xij;1 ;

n X

xij;1 ;

j¼1

xij;2 ;

! xij;3 ! xij;3

  ¼ ai;1 ; ai;2 ; ai;3 ;

i ¼ 1; 2; . . .; m;

  ¼ bj;1 ; bj;2 ; bj;3 ;

j ¼ 1; 2; . . .; n;

j¼1

 is a non-negative xij;1 ; xij;2 ; xij;3 8 i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n.

triangular

fuzzy

number

2 Mehar Method to Find a Unique Fuzzy Optimal Value …

96

Step 3: Using the relation, ða1 ; b1 ; c1 Þ ¼ ða2 ; b2 ; c2 Þ ) a1 ¼ a2 ; b1 ¼ b2 ; c1 ¼ c2 and the relation ða; b; cÞ is a non-negative triangular fuzzy number ) a  0, b  a  0; c  b  0; transform the fully fuzzy linear programming problem (2.4.2.2) into its equivalent fuzzy linear programming problem (2.4.2.3). Fuzzy linear programming problem (2.4.2.3) " Minimize

m X n X

cij;1 xij;1 ;

m X n X

i¼1 j¼1

i¼1 j¼1

n X

n X

cij;2 xij;2 ;

m X n X

!# cij;3 xij;3

i¼1 j¼1

Subject to n X

xij;1 ¼ ai;1 ;

j¼1 m X i¼1

xij;2 ¼ ai;2 ;

j¼1

xij;1 ¼ bj;1 ;

m X

xij;3 ¼ ai;3 ;

i ¼ 1; 2; . . .; m;

xij;3 ¼ bj;3 ;

j ¼ 1; 2; . . .; n;

j¼1

xij;2 ¼ bj;2 ;

i¼1

m X i¼1

xij;1  0; xij;2  xij;1  0; xij;3  xij;2  0:

Step 4: Using the existing method for comparing triangular fuzzy numbers, discussed in Sect. 2.3, transform the fuzzy linear programming problem (2.4.2.3) into its equivalent crisp multi-objective linear programming problem (2.4.2.4). Crisp multi-objective linear programming problem (2.4.2.4) " Minimize

m X n X

# cij;1 xij;1

i¼1 j¼1

" Minimize

m X n X

# cij;2 xij;2

i¼1 j¼1

" Minimize

m X n X

# cij;3 xij;3

i¼1 j¼1

Subject to Constraints of the crisp linear programming problem (2.4.2.3).

2.4 Ebrahimnejad’s Methods for Solving Balanced Fully …

97

  Step 5: Find a crisp optimal solution xij;2 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n and the P Pn crisp optimal value m i¼1 j¼1 cij;2 xij;2 of the crisp linear programming problem (2.4.2.5). Crisp linear programming problem (2.4.2.5) " Minimize

m X n X

# cij;2 xij;2

i¼1 j¼1

Subject to n X

xij;2 ¼ ai;2 ;

i ¼ 1; 2; . . .; m;

xij;2 ¼ bj;2 ;

j ¼ 1; 2; . . .; n;

j¼1 m X i¼1

xij;2  0 8 i ¼ 1; 2; . . .; m;

j ¼ 1; 2; . . .; n:

 Step 6: Find a crisp optimal solution xij;1 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; ng and the P Pn crisp optimal value m i¼1 j¼1 cij;1 xij;1 of the crisp linear programming problem (2.4.2.6). Crisp linear programming problem (2.4.2.6) " Minimize

m X n X

# cij;1 xij;1

i¼1 j¼1

Subject to n X

xij;1 ¼ ai;1 ;

i ¼ 1; 2; . . .; m;

xij;1 ¼ bj;1 ;

j ¼ 1; 2; . . .; n;

j¼1 m X i¼1

0  xij;1  x ij;2 8 i ¼ 1; 2; . . .; m;

j ¼ 1; 2; . . .; n:

where x ij;2 is the crisp optimal value of the variable xij;2 in the crisp optimal solution of the crisp linear programming problem (2.4.2.5).

2 Mehar Method to Find a Unique Fuzzy Optimal Value …

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 Step 7: Find a crisp optimal solution xij;3 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; ng and the P Pn crisp optimal value m i¼1 j¼1 cij;3 xij;3 of the crisp linear programming problem (2.4.2.7). Crisp linear programming problem (2.4.2.7) " Minimize

m X n X

# cij;3 xij;3

i¼1 j¼1

Subject to n X

xij;3 ¼ ai;3 ;

i ¼ 1; 2; . . .; m;

j¼1 m X

xij;3 ¼ bj;3 ; j ¼ 1; 2; . . .; n;

i¼1

x ij;2  xij;3 8 i ¼ 1; 2; . . .; m;

j ¼ 1; 2; . . .; n:

where x ij;2 is the crisp optimal value of the variable xij;2 in the crisp optimal solution of the crisp linear programming problem (2.4.2.5).  Step 8: Using the crisp optimal solution xij;2 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; ng, the  crisp optimal solution xij;1 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; ng and the crisp optimal  solution xij;3 ; i ¼ 1; 2; . . .; m, j ¼ 1; 2; . . .; ng of the crisp linear programming problem (2.4.2.5), the crisp linear programming problem (2.4.2.6) and the crisp linear programming problem (2.4.2.7), respectively,     find a fuzzy optimal solution xij;1 ; xij;2 ; xij;3 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n of the fully fuzzy linear programming problem (2.4.1). P Pn Step 9: Using the crisp optimal value m i¼1 j¼1 cij;2 xij;2 , the crisp optimal value Pm Pn Pm Pn i¼1 j¼1 cij;1 xij;1 and the crisp optimal value i¼1 j¼1 cij;3 xij;3 of the crisp linear programming problem (2.4.2.5), the crisp linear programming problem (2.4.2.6) and the crisp linear programming problem (2.4.2.7), respectively, find the  Pm Pn Pm Pn Pm Pn c x c x , c x fuzzy optimal value , i¼1 j¼1 ij;1 ij;1 i¼1 j¼1 ij;2 ij;2 i¼1 j¼1 ij;3 ij;3 of the fully fuzzy linear programming problem (2.4.1).

2.5 Illustrative Example

2.5

99

Illustrative Example

In this section, Ebrahimnejad’s methods [1, 3] have been illustrated with the help of the balanced fully fuzzy transportation problem represented by Table 2.2. It is pertinent to mention that as in Ebrahimnejad methods [1, 3], the fully fuzzy linear programming problem (2.4.1) has been solved to find a fuzzy optimal solution of the balanced fully triangular fuzzy transportation problem represented by Table 2.1. Therefore, in this section, the fully fuzzy linear programming problem (2.5.1) has been solved to find a fuzzy optimal solution of the balanced fully triangular fuzzy transportation problem represented by Table 2.2. Fully fuzzy linear programming problem (2.5.1)      Minimize ð20; 40; 60Þ  x11;1 ; x11;2 ; x11;3  ð25; 50; 50Þ  x12;1 ; x12;2 ; x12;3      ð25; 50; 50Þ  x21;1 ; x21;2 ; x21;3  ð20; 40; 60Þ  x22;1 ; x22;2 ; x22;3 Subject to 

   x11;1 ; x11;2 ; x11;3  x12;1 ; x12;2 ; x12;3 ¼ ð40; 60; 80Þ;



   x21;1 ; x21;2 ; x21;3  x22;1 ; x22;2 ; x22;3 ¼ ð40; 60; 80Þ;



   x11;1 ; x11;2 ; x11;3  x21;1 ; x21;2 ; x21;3 ¼ ð40; 60; 80Þ;



   x12;1 ; x12;2 ; x12;3  x22;1 ; x22;2 ; x22;3 ¼ ð40; 60; 80Þ;



       x11;1 ; x11;2 ; x11;3 , x12;1 ; x12;2 ; x12;3 , x21;1 ; x21;2 ; x21;3 and x22;1 ; x22;2 ; x22;3 are non-negative triangular fuzzy numbers.

Table 2.2 Balanced fully triangular fuzzy transportation problem

Destinations Sources S1 S2 Fuzzy demand

D1

D2

Fuzzy availability

ð20; 40; 60Þ ð25; 50; 50Þ ð40; 60; 80Þ

ð25; 50; 50Þ ð20; 40; 60Þ ð40; 60; 80Þ

ð40; 60; 80Þ ð40; 60; 80Þ

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2.5.1

Fuzzy Optimal Solution by Ebrahimnejad’s First Method

Using Ebrahimnejad’s first method [1], the fuzzy optimal solution of the fully fuzzy linear programming problem (2.5.1) can be obtained as follows: Step 1: Using the multiplication of two non-negative triangular fuzzy numbers ða1 ; b1 ; c1 Þ and ða2 ; b2 ; c2 Þ, i.e., ða1 ; b1 ; c1 Þ  ða2 ; b2 ; c2 Þ = ða1 a2 ; b1 b2 ; c1 c2 Þ, the fully fuzzy linear programming problem (2.5.1) can be transformed into its equivalent fully fuzzy linear programming problem (2.5.1.1). Fully fuzzy linear programming problem (2.5.1.1)     Minimize 20x11;1 ; 40x11;2 ; 60x11;3  25x12;1 ; 50x12;2 ; 50x12;3      25x21;1 ; 50x21;2 ; 50x21;3  20x22;1 ; 40x22;2 ; 60x22;3 Subject to Constraints of the fully fuzzy linear programming problem (2.5.1).  Pm Pm Pm  Step 2: Using the relation, m i¼1 ðai ; bi ; ci Þ ¼ i¼1 ai ; i¼1 bi ; i¼1 ci , the fully fuzzy linear programming problem (2.5.1.1) can be transformed into its equivalent fully fuzzy linear programming problem (2.5.1.2). Fully fuzzy linear programming problem (2.5.1.2)  Minimize 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1 ; 40x11;2 þ 50x12;2  þ 50x21;2 þ 40x22;2 ; 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3 Subject to





 x11;1 þ x12;1 ; x11;2 þ x12;2 ; x11;3 þ x12;3 ¼ ð40; 60; 80Þ;



 x21;1 þ x22;1 ; x21;2 þ x22;2 ; x21;3 þ x22;3 ¼ ð40; 60; 80Þ;



 x11;1 þ x21;1 ; x11;2 þ x21;2 ; x11;3 þ x21;3 ¼ ð40; 60; 80Þ;



 x12;1 þ x22;1 ; x12;2 þ x22;2 ; x12;3 þ x22;3 ¼ ð40; 60; 80Þ;

       x11;1 ; x11;2 ; x11;3 , x12;1 ; x12;2 ; x12;3 , x21;1 ; x21;2 ; x21;3 and x22;1 ; x22;2 ; x22;3 are non-negative triangular fuzzy numbers.

2.5 Illustrative Example

101

Step 3: Using the relation, ða1 ; b1 ; c1 Þ ¼ ða2 ; b2 ; c2 Þ ) a1 ¼ a2 ; b1 ¼ b2 ; c1 ¼ c2 and the relation ða; b; cÞ is a non-negative triangular fuzzy number ) a  0, b  a  0; c  b  0;, the fully fuzzy linear programming problem (2.5.1.2) can be transformed into its equivalent fuzzy linear programming problem (2.5.1.3). Fully fuzzy linear programming problem (2.5.1.3)  Minimize 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1 ; 40x11;2 þ 50x12;2  þ 50x21;2 þ 40x22;2 ; 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3 Subject to x11;1 þ x12;1 ¼ 40; x11;2 þ x12;2 ¼ 60; x11;3 þ x12;3 ¼ 80; x21;1 þ x22;1 ¼ 40; x21;2 þ x22;2 ¼ 60; x21;3 þ x22;3 ¼ 80; x11;1 þ x21;1 ¼ 40; x11;2 þ x21;2 ¼ 60; x11;3 þ x21;3 ¼ 80; x12;1 þ x22;1 ¼ 40; x12;2 þ x22;2 ¼ 60; x12;3 þ x22;3 ¼ 80; x11;2  x11;1  0; x12;2  x12;1  0; x21;2  x21;1  0; x22;2  x22;1  0; x11;3  x11;2  0; x12;3  x12;2  0; x21;3  x21;2  0; x22;3  x22;2  0; x11;1 ; x12;1 ; x21;1 ; x22;1  0:

Step 4: Using the existing method for comparing triangular fuzzy numbers, discussed in Sect. 2.3, the fuzzy linear programming problem (2.5.1.3) can be transformed into its equivalent crisp multi-objective linear programming problem (2.5.1.4). Crisp multi-objective linear programming problem (2.5.1.4)  Minimize 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1  Minimize 40x11;2 þ 50x12;2 þ 50x21;2 þ 40x22;2  Minimize 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3 Subject to Constraints of the fuzzy linear programming problem (2.5.1.3). Step 5: According to Step 5 of Ebrahimnejad’s first method [1], discussed in Sect. 2.4.1, there is a need to find a crisp optimal solution and the crisp optimal value of the crisp linear programming problem (2.5.1.5).

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Crisp linear programming problem (2.5.1.5)  Minimize 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1 Subject to x11;1 þ x12;1 ¼ 40; x21;1 þ x22;1 ¼ 40; x11;1 þ x21;1 ¼ 40; x12;1 þ x22;1 ¼ 40; x11;1 ; x12;1 ; x21;1 ; x22;1  0: It can be easily verified that on solving the crisp linear programming problem (2.5.1.5), the obtained crisp optimal solution is x11;1 ¼ 40; x12;1 ¼ 0, x21;1 ¼ 0; x22;1 ¼ 40 and the corresponding crisp optimal value is 1600. Step 6: According to Step 6 of Ebrahimnejad’s first method [1], discussed in Sect. 2.4.1, there is a need to find a crisp optimal solution and the crisp optimal value of the crisp linear programming problem (2.5.1.6). Crisp linear programming problem (2.5.1.6)  Minimize 40x11;2 þ 50x12;2 þ 50x21;2 þ 40x22;2 Subject to x11;2 þ x12;2 ¼ 60; x21;2 þ x22;2 ¼ 60; x11;2 þ x21;2 ¼ 60; x12;2 þ x22;2 ¼ 60; x11;2  40; x12;2  0; x21;2  0; x22;2  40: It can be easily verified that on solving the crisp linear programming problem (2.5.1.6), the obtained crisp optimal solution is x11;2 ¼ 60; x12;2 ¼ 0, x21;2 ¼ 0; x22;2 ¼ 60 and the corresponding crisp optimal value is 4800. Step 7: According to Step 7 of Ebrahimnejad’s first method [1], discussed in Sect. 2.4.1, there is a need to find a crisp optimal solution and the crisp optimal value of the crisp linear programming problem (2.5.1.7). Crisp linear programming problem (2.5.1.7)  Minimize 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3 Subject to x11;3 þ x12;3 ¼ 80; x21;3 þ x22;3 ¼ 80; x11;3 þ x21;3 ¼ 80; x12;3 þ x22;3 ¼ 80; x11;3  60; x12;3  0; x21;3  0; x22;3  60: It can be easily verified that on solving the crisp linear programming problem (2.5.1.7), the obtained crisp optimal solution is x11;3 ¼ 60; x12;3 ¼ 20, x21;3 ¼ 20; x22;3 ¼ 60 and the corresponding crisp optimal value is 9200.

2.5 Illustrative Example

103

Step 8: Using Step 8 of Ebrahimnejad’s first method [1], the fuzzy optimal solution of the fully fuzzy linear programming problem (2.5.1) is ~x11 ¼ ð40; 60; 60Þ, ~x12 ¼ ð0; 0; 20Þ, ~x21 ¼ ð0; 0; 20Þ, ~x22 ¼ ð40; 60; 60Þ. Step 9: Using Step 9 of Ebrahimnejad’s first method [1], the fuzzy optimal value of the fully fuzzy linear programming problem (2.5.1) is ð1600; 4800; 9200Þ:

2.5.2

Fuzzy Optimal Solution by Ebrahimnejad’s Second Method

Using Ebrahimnejad’s second method [3], a fuzzy optimal solution of the fully fuzzy linear programming problem (2.5.1) can be obtained as follows: Step 1: Using the multiplication of two non-negative triangular fuzzy numbers ða1 ; b1 ; c1 Þ and ða2 ; b2 ; c2 Þ, i.e., ða1 ; b1 ; c1 Þ  ða2 ; b2 ; c2 Þ = ða1 a2 ; b1 b2 ; c1 c2 Þ, the fully fuzzy linear programming problem (2.5.1) can be transformed into its equivalent fully fuzzy linear programming problem (2.5.2.1). Fully fuzzy linear programming problem (2.5.2.1)     Minimize 20x11;1 ; 40x11;2 ; 60x11;3  25x12;1 ; 50x12;2 ; 50x12;3  

   25x21;1 ; 50x21;2 ; 50x21;3  20x22;1 ; 40x22;2 ; 60x22;3

Subject to Constraints of the fully fuzzy linear programming problem (2.5.1).  Pm Pm Pm  Step 2: Using the relation, m i¼1 ðai ; bi ; ci Þ ¼ i¼1 ai ; i¼1 bi ; i¼1 ci , the fully fuzzy linear programming problem (2.5.2.1) can be transformed into its equivalent fully fuzzy linear programming problem (2.5.2.2). Fully fuzzy linear programming problem (2.5.2.2)  Minimize 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1 ; 40x11;2 þ 50x12;2 þ 50x21;2 þ 40x22;2 ; 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3 Subject to 

 x11;1 þ x12;1 ; x11;2 þ x12;2 ; x11;3 þ x12;3 ¼ ð40; 60; 80Þ;



 x21;1 þ x22;1 ; x21;2 þ x22;2 ; x21;3 þ x22;3 ¼ ð40; 60; 80Þ;



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 x11;1 þ x21;1 ; x11;2 þ x21;2 ; x11;3 þ x21;3 ¼ ð40; 60; 80Þ;



 x12;1 þ x22;1 ; x12;2 þ x22;2 ; x12;3 þ x22;3 ¼ ð40; 60; 80Þ;



       x11;1 ; x11;2 ; x11;3 , x12;1 ; x12;2 ; x12;3 , x21;1 ; x21;2 ; x21;3 and x22;1 ; x22;2 ; x22;3 are non-negative triangular fuzzy numbers. Step 3: Using the relation, ða1 ; b1 ; c1 Þ ¼ ða2 ; b2 ; c2 Þ ) a1 ¼ a2 ; b1 ¼ b2 ; c1 ¼ c2 and the relation ða; b; cÞ is a non-negative triangular fuzzy number ) a  0, b  a  0; c  b  0, the fully fuzzy linear programming problem (2.5.2.2) can be transformed into its equivalent fuzzy linear programming problem (2.5.2.3). Fully fuzzy linear programming problem (2.5.2.3)  Minimize 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1 ; 40x11;2 þ 50x12;2 þ 50x21;2 þ 40x22;2 ; 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3



Subject to x11;1 þ x12;1 ¼ 40; x11;2 þ x12;2 ¼ 60; x11;3 þ x12;3 ¼ 80; x21;1 þ x22;1 ¼ 40; x21;2 þ x22;2 ¼ 60; x21;3 þ x22;3 ¼ 80; x11;1 þ x21;1 ¼ 40; x11;2 þ x21;2 ¼ 60; x11;3 þ x21;3 ¼ 80; x12;1 þ x22;1 ¼ 40; x12;2 þ x22;2 ¼ 60; x12;3 þ x22;3 ¼ 80; x11;2  x11;1  0; x12;2  x12;1  0; x21;2  x21;1  0; x22;2  x22;1  0; x11;3  x11;2  0; x12;3  x12;2  0; x21;3  x21;2  0; x22;3  x22;2  0; x11;1 ; x12;1 ; x21;1 ; x22;1  0: Step 4: Using the existing method for comparing triangular fuzzy numbers, discussed in Sect. 2.3, the fuzzy linear programming problem (2.5.2.3) can be transformed into its equivalent crisp multi-objective linear programming problem (2.5.2.4). Crisp multi-objective linear programming problem (2.5.2.4)  Minimize 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1  Minimize 40x11;2 þ 50x12;2 þ 50x21;2 þ 40x22;2

2.5 Illustrative Example

105

 Minimize 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3 Subject to Constraints of the fuzzy linear programming problem (2.5.2.3). Step 5: According to Step 5 of Ebrahimnejad’s second method [3], discussed in Sect. 2.4.2, there is a need to find a crisp optimal solution and the crisp optimal value of the crisp linear programming problem (2.5.2.5). Crisp linear programming problem (2.5.2.5)  Minimize 40x11;2 þ 50x12;2 þ 50x21;2 þ 40x22;2 Subject to x11;2 þ x12;2 ¼ 60; x21;2 þ x22;2 ¼ 60; x11;2 þ x21;2 ¼ 60; x12;2 þ x22;2 ¼ 60; x11;2  0; x12;2  0; x21;2  0; x22;2  0: It can be easily verified that on solving the crisp linear programming problem (2.5.2.5), the obtained crisp optimal solution is x11;2 ¼ 60; x12;2 ¼ 0, x21;2 ¼ 0; x22;2 ¼ 60 and the corresponding crisp optimal value is 4800. Step 6: According to Step 6 of Ebrahimnejad’s second method [3], discussed in Sect. 2.4.2, there is a need to find a crisp optimal solution and the crisp optimal value of the crisp linear programming problem (2.5.2.6). Crisp linear programming problem (2.5.2.6)  Minimize 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1 Subject to x11;1 þ x12;1 ¼ 40; x21;1 þ x22;1 ¼ 40; x11;1 þ x21;1 ¼ 40; x12;1 þ x22;1 ¼ 40; 0  x11;1  60; 0  x12;1  0; 0  x21;1  0; 0  x22;1  60: It can be easily verified that on solving the crisp linear programming problem (2.5.2.6), the obtained crisp optimal solution is x11;1 ¼ 40; x12;1 ¼ 0, x21;1 ¼ 0; x22;1 ¼ 40 and the corresponding crisp optimal value is 1600. Step 7: According to Step 7 of Ebrahimnejad’s second method [3], discussed in Sect. 2.4.2, there is a need to find a crisp optimal solution and the crisp optimal value of the crisp linear programming problem (2.5.2.7).

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Crisp linear programming problem (2.5.2.7)  Minimize 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3 Subject to x11;3 þ x12;3 ¼ 80; x21;3 þ x22;3 ¼ 80; x11;3 þ x21;3 ¼ 80; x12;3 þ x22;3 ¼ 80; x11;3  60; x12;3  0; x21;3  0; x22;3  60: It can be easily verified that on solving the crisp linear programming problem (2.5.2.7), the obtained crisp optimal solution is x11;3 ¼ 60; x12;3 ¼ 20, x21;3 ¼ 20; x22;3 ¼ 60 and the corresponding crisp optimal value is 9200. Step 8: Using Step 8 of Ebrahimnejad’s second method [3], the fuzzy optimal solution of the fully fuzzy linear programming problem (2.5.1) is ~x11 ¼ ð40; 60; 60Þ, ~x12 ¼ ð0; 0; 20Þ, ~x21 ¼ ð0; 0; 20Þ, ~x22 ¼ ð40; 60; 60Þ. Step 9: Using Step 9 of Ebrahimnejad’s second method [3], the fuzzy optimal solution of the fully fuzzy linear programming problem (2.5.1) is ð1600; 4800; 9200Þ.

2.6

Limitation of Ebrahimnejad’s Methods

In general, to solve a real-life fully fuzzy transportation problem, the opinion of two or more experts about the parameters is collected. Then, all the collected information is aggregated to obtain a single value of each parameter. Since, Ebrahimnejad’s methods [1, 3] have been proposed by considering the assumption that the aggregated value of each parameter is available. Therefore, Ebrahimnejad’s methods [1, 3] can be used to solve such balanced fully triangular fuzzy transportation problems for which the aggregated value of the fuzzy transportation cost, the fuzzy availability and the fuzzy demand are provided. But, Ebrahimnejad’s methods [1, 3] cannot be used to solve such balanced fully triangular fuzzy transportation problems for which instead of providing the aggregated fuzzy data, the fuzzy data of each decision-maker is provided separately. For example, Ebrahimnejad’s methods [1, 3] cannot be used to solve the balanced fully triangular fuzzy transportation problem considered in Example 2.6.1. Example 2.6.1 Let us consider a product needs to be supplied from two sources to two destinations. For the same purpose, the information about each parameter is collected from two experts. If (i) Table 2.3 represents the fuzzy transportation cost, the fuzzy availability and the fuzzy demand provided by the first decision-maker.

2.6 Limitation of Ebrahimnejad’s Methods Table 2.3 Fuzzy data provided by the first decision-maker

Destinations Sources S1 S2 Fuzzy demand

Table 2.4 Fuzzy data provided by the second decision-maker

Destinations Sources S1 S2 Fuzzy demand

107

D1

D2

Fuzzy availability

ð10; 30; 40Þ ð25; 30; 50Þ ð40; 60; 80Þ

ð25; 50; 50Þ ð20; 40; 60Þ ð40; 45; 60Þ

ð20; 60; 70Þ ð25; 45; 60Þ

D1

D2

Fuzzy availability

ð10; 20; 25Þ ð20; 35; 40Þ ð20; 55; 60Þ

ð25; 30; 60Þ ð20; 25; 40Þ ð40; 45; 50Þ

ð30; 50; 70Þ ð20; 40; 80Þ

(ii) Table 2.4 represents the fuzzy transportation cost, the fuzzy availability and the fuzzy demand provided by the second decision-maker. Then, this balanced fully triangular fuzzy transportation problem cannot be solved by Ebrahimnejad’s methods [1, 3].

2.7

Drawback of Ebrahimnejad’s Methods

It is obvious from Sect. 2.5 that according to Ebrahimnejad’s methods [1, 3], the triangular fuzzy number ð1600; 4800; 9200Þ represents the total minimum fuzzy transportation cost for the balanced fully triangular fuzzy transportation problem represented by Table 2.2. While, in actual case, the triangular fuzzy number ð1600; 4800; 9200Þ does not represents the total minimum fuzzy transportation cost for the balanced fully triangular fuzzy transportation problem, represented by Table 2.2, due to the following reason: (i) It is obvious that ~x11 ¼ ð40; 40; 40Þ, ~x12 ¼ ð0; 20; 40Þ, ~x21 ¼ ð0; 20; 40Þ, ~x22 ¼ ð40; 40; 40Þ is a feasible solution of the considered balanced fully triangular fuzzy transportation problem represented by Table 2.2. (ii) The total fuzzy transportation cost, i.e., ð20; 40; 60Þ  ~x11  ð25; 50; 50Þ ~x12  ð25; 50; 50Þ ~x21  ð20; 40; 60Þ  ~x22 corresponding to this fuzzy optimal solution is ð1600; 5200; 8800Þ. (iii) According to the existing method for comparing triangular fuzzy numbers, discussed in Sect. 2.3 and used in Ebrahimnejad’s methods [1, 3], the fuzzy

108

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transportation cost ð1600; 4800; 9200Þ is not less than the fuzzy transportation cost ð1600; 5200; 8800Þ. Remark 2.1 The limitation and drawback, pointed out in Ebrahimnejad’s methods [1, 3], are also occurring in Ebrahimnejad’s method [2].

2.8

Reason for the Occurrence of the Drawback in Ebrahimnejad’s Methods

In this section, the reason for the occurrence of drawback in Ebrahimnejad’s methods [1, 3] has been discussed. Ebrahimnejad [1, 3] has used the method, discussed in Sect. 2.3, for comparing triangular fuzzy numbers. However, it is not appropriate to use this method due to the following reason: It is well-known fact that if A1 and A2 are two real numbers such that A1 6¼ A2 . Then, either the relation A1 [ A2 or the relation A1 \A2 will be satisfied. On the e 2 ¼ ða2 ; b2 ; c2 Þ are two triangular fuzzy e 1 ¼ ða1 ; b1 ; c1 Þ and A same direction, if A e 2 . Then, either the relation A e1  A e 2 or the relation e 1 6¼ A numbers such that A e e A 1  A 2 should be satisfied. While, if the method, used by Ebrahimnejad [1, 3] for comparing triangular e1 ¼ fuzzy numbers, is applied for comparing two triangular fuzzy numbers A e e e ða1 ; b1 ; c1 Þ and A 2 ¼ ða2 ; b2 ; c2 Þ such that A 1 6¼ A 2 . Then, it may happen that e 2 nor the relation A e1  A e 2 will be satisfied, which is e1  A neither the relation A mathematically incorrect. The following example has been considered to validate this claim. e 1 ¼ ð1; 2; 5Þ and A e 2 ¼ ð1; 3; 4Þ be two triangular fuzzy numbers. Then, as Let A for these triangular fuzzy numbers, none of the conditions a1 \a2 , b1  b2 , c1  c2 , or a1  a2 , b1 \b2 , c1  c2 , a1  a2 , b1  b2 , c1 \c2 , a1 [ a2 , b1  b2 , c1  c2 , a1  a2 , b1 [ b2 , c1  c2 , a1  a2 , b1  b2 , c1 [ c2 is satisfying. Therefore, according to the method, discussed in Sect. 2.3 and used by e 2 nor e1  A Ebrahimnejad [1, 3] for comparing triangular fuzzy numbers, neither A e e A1  A2.

2.9

Rank, Mode and Divergence Based Approach for Comparing Triangular Fuzzy Numbers

It is obvious from Sect. 2.8 that the drawback in Ebrahimnejad’s methods [1, 3] is occurring due to the drawback of the method for comparing triangular fuzzy numbers.

2.9 Rank, Mode and Divergence Based Approach …

109

It is pertinent to mention that this drawback is not occurring in the existing Rank, Mode and Divergence based approach for comparing triangular fuzzy numbers [9], i.e., on applying the existing Rank, Mode and Divergence based approach for e 1 ¼ ða1 ; b1 ; c1 Þ and comparing two distinct triangular fuzzy numbers [9], A e 2 ¼ ða2 ; b2 ; c2 Þ, either the relation A e1  A e 2 or the relation A e1  A e 2 will be A e e e 2 , i.e., e obtained. The relation A 1 ¼ A 2 will be obtained only if in actual case A 1 ¼ A a1 ¼ a2 , b1 ¼ b2 , c 1 ¼ c 2 . Hence, it is better to use the existing Rank-, Mode- and Divergence-based approach for comparing triangular fuzzy numbers [9] instead of the existing method discussed in Sect. 2.3. Using the existing Rank-, Mode- and Divergence-based approach for comparing e 1 ¼ ða1 ; b1 ; c1 Þ and triangular fuzzy numbers [9], two triangular fuzzy numbers A e 2 ¼ ða2 ; b2 ; c2 Þ can be compared as follows: A     e 2 ¼ a2 þ 2b2 þ c2 and check that e 1 ¼ a1 þ 2b1 þ c1 and Rank A Step 1: Find Rank A 4 4             e e e e e 1 ¼ Rank A e2 . Rank A 1 [ Rank A 2 or Rank A 1 \Rank A 2 or Rank A     e 2. e 1 [ Rank A e 2 , then A e1  A Case (i): If Rank A     e 1 \Rank A e 2. e 2 , then A e1  A Case (ii): If Rank A     e 1 ¼ Rank A e 2 , then go to Step 2. Case (iii): If Rank A       e 2 ¼ b2 and check that Mode A e1 [ e 1 ¼ b1 , Mode A Step 2: Find A           e 2 or Mode A e 1 \Mode A e 2 or Mode A e 1 ¼ Mode A e2 . Mode A     e 2. e 1 [ Mode A e 2 , then A e1  A Case (i): If Mode A     e 1 \Mode A e 2. e 2 , then A e1  A Case (ii): If Mode A     e 1 ¼ Mode A e 2 , then go to Step 3. Case (iii): If Mode A     e 2 ¼ c2  a2 and check e 1 ¼ c1  a1 , Divergence A Step 3: Find Divergence A         e 1 [ Divergence A e 2 or Divergence A e 1 \Divergence A e2 that Divergence A     e 1 ¼ Divergence A e2 . or Divergence A     e 2. e 1 [ Divergence A e 2 , then A e1  A Case (i): If Divergence A     e 1 \Divergence A e 2. e 2 , then A e1  A Case (ii): If Divergence A     e 1 ¼ Divergence A e 2. e 2 , then A e1 ¼ A Case (iii): If Divergence A

110

2.10

2 Mehar Method to Find a Unique Fuzzy Optimal Value …

Validity of Rank, Mode and Divergence Based Approach for Comparing Triangular Fuzzy Numbers

In Case (iii) of Step 3 of the existing Rank, Mode and Divergence based approach for comparing triangular fuzzy numbers [9], it is claimed that if             e 1 ¼ Rank A e 2 , Mode A e 1 ¼ Mode A e 2 and Div A e 1 ¼ Div A e2 Rank A e 2 , i.e., a1 ¼ a2 , b1 ¼ b2 , c1 ¼ c2 . e1 ¼ A then A The following clearly indicates that this claim is valid.     e 1 ¼ Rank A e 2 ) a1 þ 2b1 þ c1 ¼ a2 þ 2b2 þ c2 Rank A 4 4     e 1 ¼ Mode A e 2 ) b1 ¼ b2 Mode A     e 1 ¼ Div A e 2 ) c 1  a1 ¼ c 2  a2 Div A

ð2:10:1Þ ð2:10:2Þ ð2:10:3Þ

It can be easily verified that on solving Eqs. (2.10.1), (2.10.2) and (2.10.3), the e1 ¼ A e 2. obtained solution is a1 ¼ a2 , b1 ¼ b2 , c1 ¼ c2 , d1 ¼ d2 , i.e., A

2.11

Proposed Mehar Method to Find a Unique Optimal Value of Balanced Fully Triangular Fuzzy Transportation Problems

In this section, using the existing Rank, Mode and Divergence based approach for comparing triangular fuzzy numbers [9], a method (named as Mehar method) has been proposed to find a unique triangular fuzzy number, representing the total minimum fuzzy transportation cost of the fully triangular fuzzy transportation problem represented by Table 2.1. The steps of the proposed Mehar method are as follows: Step 1: Check that the aggregated value of the fuzzy transportation cost, the fuzzy availability and the fuzzy demand, provided by all the decision-makers, is available or not. Case (i): If it is available, then go to Step 2. Case (ii): If it is not available, then using the existing triangular fuzzy weighted averaging aggregation operator [19], find,

2.11

Proposed Mehar Method to Find a Unique Optimal Value …

(i) The

p P k¼1

triangular wk ckij;1 ;

p P k¼1

fuzzy

wk ckij;2 ;

p P k¼1

wk ckij;3



number

111

  ~cij ¼ pk¼1 wk  ~ckij

=

representing the aggregated value of the

fuzzy transportation cost for supplying the one unit quantity of the product from the ith source to the jth destination.   ~ aki = (ii) The triangular fuzzy number ai ¼ pk¼1 wk  ~

p  p p P P P wk aki;1 ; wk aki;2 ; wk aki;3 representing the aggregated value of the k¼1

k¼1

k¼1

fuzzy availability of the product at the ith source.   ~ bkj = (iii) The triangular fuzzy number bj ¼ pk¼1 wk  ~

p  p p P P P wk bkj;1 ; wk bkj;2 ; wk bkj;3 representing the aggregated value of the k¼1

k¼1

k¼1

fuzzy demand the product at the jth destination. where weight of the kth decision-maker. (i) wk 2 ½0; 1 represents the normalized  k (ii) The triangular fuzzy number ~cij ¼ ckij;1 ; ckij;2 ; ckij;3 represents the fuzzy transportation cost for supplying the one unit quantity of the product from the ith source to the jth destination according  to the kth decision-maker.

(iii) The triangular fuzzy number a~ki ¼ aki;1 ; aki;2 ; aki;3

represents the fuzzy

availability of the product at the ith source according to the kth decision-maker.   (iv) The triangular fuzzy number ~bkj ¼ bkj;1 ; bkj;2 ; bkj;3 represents the fuzzy demand of the product at the jth destination according to the kth decision-maker. and go to Step 2. For example, if in Example 2.6.1, the normalized weights of the first and second decision-makers are 0.4 and 0.6, repsectively. Then, the triangular fuzzy number

Table 2.5 Aggregated fuzzy data of decision-makers

Destinations Sources S1 S2 Fuzzy demand

D1

D2

Fuzzy availability

ð10; 24; 31Þ ð22; 33; 44Þ ð28; 57; 68Þ

ð25; 38; 56Þ ð20; 31; 48Þ ð40; 45; 53Þ

ð26; 54; 70Þ ð22; 42; 76Þ

2 Mehar Method to Find a Unique Fuzzy Optimal Value …

112

~cij ; i ¼ 1; 2; j ¼ 1; 2; ~ai ; i ¼ 1; 2 and ~bj ; j ¼ 1; 2, presented in Table 2.5, represents the aggregated fuzzy cost for supplying the one unit quantity of the product from the ith source to the jth destination, the aggregated fuzzy availability of the product at the ith source and the aggregated fuzzy demand of the product at the jth destination, respectively. Step 2: Write the fully fuzzy linear programming problem (2.5.1) with the help of the provided/aggregated fuzzy data. Step 3: Using Step 1 to Step 3 of Ebrahimnejad’s method [1], transform the fully fuzzy linear programming problem (2.5.1) into its equivalent fuzzy linear programming problem (2.5.1.3). Step 4: Using the existing Rank, Mode and Divergence based approach for comparing triangular fuzzy numbers [9], transform the fuzzy linear programming problem (2.5.1.3) into its equivalent crisp linear programming problem (2.11.1). Crisp linear programming problem (2.11.1) " Minimize Rank

m X n X

cij;1 xij;1 ;

i¼1 j¼1

m X n X

cij;2 xij;2 ;

i¼1 j¼1

m X n X

!# cij;3 xij;3

i¼1 j¼1

Subject to Constraints of the fuzzy linear programming problem (2.5.1.3). þc Step 5: Using the expression, Rankða; b; cÞ ¼ a þ 2b , transform the crisp linear 4 programming problem (2.11.1) into its equivalent crisp linear programming problem (2.11.2).

Crisp linear programming problem (2.11.2) 2P m P n 6i¼1 j¼1 Minimize6 4

cij;1 xij;1 þ 2

m P n P

cij;2 xij;2 þ

i¼1 j¼1

m P n P

3 cij;3 xij;3

i¼1 j¼1

4

7 7 5

Subject to Constraints of the fuzzy linear programming problem (2.5.1.3). Step 6: Solve the crisp linear programming problem (2.11.2) and check that a unique crisp optimal solution exists for the crisp linear programming problem (2.11.2) or not. Case (i) If a unique crisp optimal solution exists for the crisp linear programming problem (2.11.2), then go to Step 9.

2.11

Proposed Mehar Method to Find a Unique Optimal Value …

113

Case (ii) If more than one crisp optimal solution exists for the crisp linear programming problem (2.11.2), then go to Step 7. Step 7: Solve the crisp linear programming problem (2.11.3) and check that a unique crisp optimal solution exists for the crisp linear programming problem (2.11.3) or not. Crisp linear programming problem (2.11.3) " Minimize

m X n X

# cij;2 xij;2

i¼1 j¼1

Subject to Constraints of the crisp linear programming problem (2.11.2) with the additional constraint. P P P P m

n

cij;1 xij;1 þ 2

m

n

i¼1 “ i¼1 j¼1 4 gramming problem (2.11.2).”

c x j¼1 ij;2 ij;2

Crisp optimal value of the crisp linear pro-

Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (2.11.3), then go to Step 9. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (2.11.3), then go to Step 8. Step 8: Solve the crisp linear programming problem (2.11.4). Crisp linear programming problem (2.11.4) Minimize

hXm Xn i¼1

c x  j¼1 ij;3 ij;3

Xm Xn i¼1

i

c x j¼1 ij;1 ij;1

Subject to Constraints the crisp linear programming problem (2.11.3) with the additional P of P n constraint “ m i¼1 j¼1 cij;2 xij;2 ¼ Crisp optimal value of the crisp linear programming problem (2.11.3).” Step 9: Using the obtained i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; ng,

crisp

optimal

solution



xij;1 ; xij;2 ; xij;3 ;

  (i) Find the fuzzy optimal solution xij;1 ; xij;2 ; xij;3 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; ng of the fully fuzzy linear programming problem (2.5.1).   n of the fully  c x ; c x (ii) The fuzzy optimal value m ij;1 ij;1 ij;2 ij;2 ; cij;3 xij;3 i¼1 j¼1 fuzzy linear programming problem (2.5.1).

2 Mehar Method to Find a Unique Fuzzy Optimal Value …

114

2.12

Unique Fuzzy Transportation Cost of the Considered Balanced Fully Triangular Fuzzy Transportation Problem

In Sect. 2.5, a balanced fully triangular fuzzy transportation problem, represented by Table 2.2, is solved by Ebrahimnejad’s methods [1, 3]. Also, in Sect. 2.7, it is showed that on applying Ebrahimnejad’s methods [1, 3] more than one triangular fuzzy numbers, representing the optimal fuzzy transportation cost, are obtained, which is mathematically incorrect. In this section, the same balanced fully triangular fuzzy transportation problem is solved by proposed Mehar method and showed that a unique triangular fuzzy number, representing the optimal fuzzy transportation cost, is obtained. Using the Mehar method, proposed in Sect. 2.11, a unique fuzzy optimal value, representing the optimal fuzzy transportation cost of the balanced fully triangular fuzzy transportation problem represented by Table 2.2, can be obtained as follows. Step 1: Since, in the balanced fully triangular fuzzy transportation problem, represented by Table 2.2, the aggregated values of fuzzy transportation cost, fuzzy availability and fuzzy demand are provided. So, there is no need to apply Step 1 of the proposed Mehar method. Step 2: The balanced fully triangular fuzzy transportation problem, represented by Table 2.2, can also be represented by the fully fuzzy linear programming problem (2.12.1). Fully fuzzy linear programming problem (2.12.1)     ! ð20; 40; 60Þ  x11;1 ; x11;2 ; x11;3  ð25; 50; 50Þ  x12;1 ; x12;2 ; x12;3     Minimize  ð25; 50; 50Þ  x21;1 ; x21;2 ; x21;3  ð20; 40; 60Þ  x22;1 ; x22;2 ; x22;3

Subject to





   x11;1 ; x11;2 ; x11;3  x12;1 ; x12;2 ; x12;3 ¼ ð40; 60; 80Þ;



   x21;1 ; x21;2 ; x21;3  x22;1 ; x22;2 ; x22;3 ¼ ð40; 60; 80Þ;



   x11;1 ; x11;2 ; x11;3  x21;1 ; x21;2 ; x21;3 ¼ ð40; 60; 80Þ;



   x12;1 ; x12;2 ; x12;3  x22;1 ; x22;2 ; x22;3 ¼ ð40; 60; 80Þ;

       x11;1 ; x11;2 ; x11;3 , x12;1 ; x12;2 ; x12;3 , x21;1 ; x21;2 ; x21;3 and x22;1 ; x22;2 ; x22;3 are non-negative triangular fuzzy numbers.

2.12

Unique Fuzzy Transportation Cost of the Considered Balanced …

115

Step 2: Using the multiplication of non-negative triangular fuzzy numbers ða1 ; b1 ; c1 Þ and ða2 ; b2 ; c2 Þ, i.e., ða1 ; b1 ; c1 Þ  ða2 ; b2 ; c2 Þ = ða1 a2 ; b1 b2 ; c1 c2 Þ, the fully fuzzy linear programming problem (2.12.1) can be transformed into its equivalent fully fuzzy linear programming problem (2.12.2). Fully fuzzy linear programming problem (2.12.2)     Minimize 20x11;1 ; 40x11;2 ; 60x11;3  25x12;1 ; 50x12;2 ; 50x12;3      25x21;1 ; 50x21;2 ; 50x21;3  20x22;1 ; 40x22;2 ; 60x22;3 Subject to Constraints of the fully fuzzy linear programming problem (2.12.1).  Pm Pm Pm  Step 3: Using the relation, m i¼1 ðai ; bi ; ci Þ ¼ i¼1 ai ; i¼1 bi ; i¼1 ci , the fully fuzzy linear programming problem (2.12.2) can be transformed into its equivalent fully fuzzy linear programming problem (2.12.3). Fully fuzzy linear programming problem (2.12.3)  Minimize 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1 ; 40x11;2 þ 50x12;2 þ 50x21;2 þ 40x22;2 ; 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3



Subject to 

 x11;1 þ x12;1 ; x11;2 þ x12;2 ; x11;3 þ x12;3 ¼ ð40; 60; 80Þ;



 x21;1 þ x22;1 ; x21;2 þ x22;2 ; x21;3 þ x22;3 ¼ ð40; 60; 80Þ;



 x11;1 þ x21;1 ; x11;2 þ x21;2 ; x11;3 þ x21;3 ¼ ð40; 60; 80Þ;



 x12;1 þ x22;1 ; x12;2 þ x22;2 ; x12;3 þ x22;3 ¼ ð40; 60; 80Þ;



       x11;1 ; x11;2 ; x11;3 , x12;1 ; x12;2 ; x12;3 , x21;1 ; x21;2 ; x21;3 and x22;1 ; x22;2 ; x22;3 are non-negative triangular fuzzy numbers. Step 4: Using the relation, ða1 ; b1 ; c1 Þ ¼ ða2 ; b2 ; c2 Þ ) a1 ¼ a2 ; b1 ¼ b2 ; c1 ¼ c2 and the relation ða; b; cÞ is a non-negative triangular fuzzy number ) a  0, b  a  0; c  b  0, the fully fuzzy linear programming problem (2.12.3) can be transformed into its equivalent fuzzy linear programming problem (2.12.4).

2 Mehar Method to Find a Unique Fuzzy Optimal Value …

116

Fully fuzzy linear programming problem (2.12.4)  Minimize 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1 ; 40x11;2  þ 50x12;2 þ 50x21;2 þ 40x22;2 ; 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3 Subject to x11;1 þ x12;1 ¼ 40; x11;2 þ x12;2 ¼ 60; x11;3 þ x12;3 ¼ 80; x21;1 þ x22;1 ¼ 40; x21;2 þ x22;2 ¼ 60; x21;3 þ x22;3 ¼ 80; x11;1 þ x21;1 ¼ 40; x11;2 þ x21;2 ¼ 60; x11;3 þ x21;3 ¼ 80; x12;1 þ x22;1 ¼ 40; x12;2 þ x22;2 ¼ 60; x12;3 þ x22;3 ¼ 80; x11;2  x11;1  0; x12;2  x12;1  0; x21;2  x21;1  0; x22;2  x22;1  0; x11;3  x11;2  0; x12;3  x12;2  0; x21;3  x21;2  0; x22;3  x22;2  0; ; x11;1 ; x12;1 ; x21;1 ; x22;1  0: Step 5: Using the existing Rank, Mode and Divergence based approach for comparing triangular fuzzy numbers [9], the fuzzy linear programming problem (2.12.4) can be transformed into its equivalent crisp linear programming problem (2.12.5). Crisp linear programming problem (2.12.5)   Minimize Rank 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1 ; 40x11;2 þ 50x12;2  þ 50x21;2 þ 40x22;2 ; 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3 Subject to Constraints of the fuzzy linear programming problem (2.12.4). þc Step 6: Using the expression Rankða; b; cÞ ¼ a þ 2b , the crisp linear programming 4 problem (2.12.5) can be transformed into its equivalent crisp linear programming problem (2.12.6).

2.12

Unique Fuzzy Transportation Cost of the Considered Balanced …

117

Crisp linear programming problem (2.12.6) 20x11;1 þ 25x12;1 þ 25x21;1 þ 20x22;1 Minimize 4   þ 2 40x11;2 þ 50x12;2 þ 50 x21;2 þ 40x22;2 4

þ 60x11;3 þ 50x12;3 þ 50x21;3 þ 60x22;3 4 Subject to Constraints of the fuzzy linear programming problem (2.12.4). Step 7: It can be easily verified that the following unique crisp optimal solution is obtained on solving the crisp linear programming problem (2.12.6). x11;1 ¼ 40; x11;2 ¼ 40; x11;3 ¼ 40; x12;1 ¼ 0; x12;2 ¼ 20; x12;3 ¼ 40; x21;1 ¼ 0; x21;2 ¼ 20; x21;3 ¼ 40; x22;1 ¼ 40; x22;2 ¼ 40; x22;3 ¼ 40:

Therefore, according to Case (i) of Step 6 of the proposed Mehar method, there is a need to go to Step 9 of the proposed Mehar method. Step 8: Using the crisp optimal solution of the crisp linear programming problem (2.12.6), obtained in Step 7, the fuzzy optimal solution for the balanced fully trian-  gular fuzzy transportation problem, represented by Table 2.2, is x11;1 ; x11;2 ; x11;3     ¼ ð40; 40; 40Þ, x12;1 ; x12;2 ; x12;3 ¼ ð0; 20; 40Þ; x21;1 ; x21;2 ; x21;3 ¼ ð0; 20; 40Þ,   x22;1 ; x22;2 ; x22;3 ¼ ð40; 40; 40Þ and the total minimum fuzzy transportation cost is ð1600; 5200; 8800Þ.

2.13

Conclusions

It is showed that on solving the balanced fully triangular fuzzy transportation problem, represented by Table 2.2, by Ebrahimnejad’s methods [1, 3] more than one triangular fuzzy numbers, representing the optimal fuzzy transportation cost, are obtained, which is mathematically incorrect. Therefore, it is inappropriate to use Ebrahimnejad’s methods [1, 3] for solving balanced fully triangular fuzzy transportation problems. Also, a new method (named as Mehar method) has been proposed for solving balanced fully triangular fuzzy transportation problems. Furthermore, it is showed that on solving a balanced fully triangular fuzzy transportation problem by the proposed Mehar method, always a unique triangular fuzzy number, representing the total minimum fuzzy transportation cost, will be obtained. So, it is appropriate to use the proposed Mehar method for solving a balanced fully triangular fuzzy transportation problem.

118

2 Mehar Method to Find a Unique Fuzzy Optimal Value …

References 1. A. Ebrahimnejad, An improved approach for solving fuzzy transportation problem with triangular fuzzy numbers. J. Intell. Fuzzy Syst. 29, 963–974 (2015) 2. A. Ebrahimnejad, New method for solving fuzzy transportation problems with LR flat fuzzy numbers. Inf. Sci. 357, 108–124 (2016) 3. A. Ebrahimnejad, A lexicographic ordering-based approach for solving fuzzy transportation problems with triangular fuzzy numbers. Int. J. Manag. Decis. Making 16, 346–374 (2017) 4. H. Basirzadeh, An approach for solving fuzzy transportation problem. Appl. Math. Sci. 5, 1549–1566 (2011) 5. S. Chanas, W. Kolodziejczyk, A. Machaj, A fuzzy approach to the transportation problem. Fuzzy Sets Syst. 13, 211–221 (1984) 6. S. Chanas, D. Kuchta, A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets Syst. 82, 299–305 (1996) 7. S. Chanas, D. Kuchta, Fuzzy integer transportation. Fuzzy Sets Syst. 98, 291–298 (1998) 8. D.S. Dinagar, K. Palanivel, The transportation problem in fuzzy environment. Int. J. Algorithm Comput. Math. 2, 65–71 (2009) 9. A. Gani, K.A. Razak, Two stage fuzzy transportation problem. J. Phys. Sci. 10, 63–69 (2006) 10. A. Kaufmann, M.M. Gupta, Fuzzy Mathematical Models in Engineering and Management science (Elsevier, Amsterdam, Netherland, 1988) 11. A. Kaur, J. Kacprzyk, A. Kumar, Fuzzy Transportation and Transshipment Problems, Studies in Fuzziness and Soft Computing, vol. 385 (Springer Nature, Switzerland AG, 2020) 12. A. Kaur, A. Kumar, A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers. Appl. Soft Comput. 12, 1201–1213 (2012) 13. A. Kumar, A. Kaur, A new method for solving fuzzy transportation problems using ranking function. Appl. Math. Model. 35, 5652–5661 (2011) 14. A. Kumar, A. Kaur, Application of classical transportation methods for solving fuzzy transportation problems. J. Transp. Syst. Eng. Inform. Technol. 11, 68–80 (2011) 15. S.T. Liu, C. Kao, Solving fuzzy transportation problems based on extension principle. Eur. J. Oper. Res. 153, 661–674 (2004) 16. M. Oheigeartaigh, A fuzzy transportation algorithm. Fuzzy Sets Syst. 8, 235–243 (1982) 17. P. Pandian, G. Natarajan, A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Appl. Math. Sci. 4, 79–90 (2010) 18. M. Shanmugasundari, K. Ganesan, A novel approach for the fuzzy optimal solution of fuzzy transportation problem. Int. J. Eng. Res. Appl. 3, 1416–1421 (2013) 19. X.R. Wang, Z.P. Fan, Fuzzy ordered weighted averaging (FOWA) operator and its application. Fuzzy Syst. Math. 17, 67–72 (2003) 20. Z.S. Xu, A priority method for triangular fuzzy number complementary judgement matrix. Syst. Eng. Theor. Pract. 23, 86–89 (2003)

Chapter 3

Vaishnavi Approach for Solving Triangular Intuitionistic Transportation Problems of Type-2

In the last few years, several methods have been proposed to solve transportation problems under intuitionistic fuzzy environment [1 and references therein]. Singh and Yadav [2] proposed an approach for solving triangular intuitionistic fuzzy transportation problems of type-2 (transportation problems in which the cost for supplying the one unit quantity of the product from each source to each destination is represented by a triangular intuitionistic fuzzy number. While, all the other parameters are represented by non-negative real numbers). In this approach, firstly, one of the well-known methods (North–west corner method or least cost method or Vogel’s approximation method, etc.) is used to find an initial basic feasible solution of the considered triangular intuitionistic fuzzy transportation problem of type-2. Then, the well-known MODI approach (or u
v approach) is used to find a crisp optimal solution with the help of the obtained initial basic feasible solution. Ebrahimnejad and Verdegay [3] pointed out that for solving a triangular intuitionistic fuzzy transportation problem of type-2 by Singh and Yadav’s approach [2], there is a need to use arithmetic operations of triangular intuitionistic fuzzy numbers. Due to which, more computational efforts are required for solving a triangular intuitionistic fuzzy transportation problem of type-2 by Singh and Yadav’s approach [2]. To reduce the computational efforts, Ebrahimnejad and Verdegay [3] proposed an alternative approach for solving triangular intuitionistic fuzzy transportation problems of type-2. One may claim that Ebrahimnejad and Verdegay’s approach [3] can be used to solve such balanced triangular intuitionistic fuzzy transportation problems of type-2 for which the aggregated value of the intuitionistic fuzzy transportation cost, the crisp availability and the crisp demand, provided by all the decision-makers, is available. While, Ebrahimnejad and Verdegay’s approach [3] cannot be used to solve such triangular intuitionistic fuzzy transportation problems of type-2 for which, instead of the aggregated data, the data of each decision-maker is provided

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Mishra and A. Kumar, Aggregation Operators for Various Extensions of Fuzzy Set and Its Applications in Transportation Problems, Studies in Fuzziness and Soft Computing 399, https://doi.org/10.1007/978-981-15-6998-2_3

119

120

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

separately. To overcome this limitation, one may modify Ebrahimnejad and Verdegay’s approach [3] with the help of the existing triangular intuitionistic fuzzy weighted averaging aggregation operator [4]. Also, one may use Ebrahimnejad and Verdegay’s approach [3] to solve real-life triangular intuitionistic fuzzy transportation problems of type-2. However, after a deep study, it is observed that on applying Ebrahimnejad and Verdegay’s approach [3] more than one triangular intuitionistic fuzzy numbers, representing the total minimum intuitionistic fuzzy transportation cost, are obtained, which is mathematically incorrect as the physical meaning of all the obtained triangular intuitionistic fuzzy numbers will be different. Hence, it is inappropriate to use Ebrahimnejad and Verdegay’s approah [3] to solve real-life triangular intuitionistic fuzzy transportation problems of type-2. The aim of this chapter is (i) To make the researchers aware about the inappropriateness of Ebrahimnejad and Verdegay’s approach [3]. (ii) To propose a new approach (named as MEHAR approach) for comparing triangular intuitionistic fuzzy numbers. (iii) To propose a new approach (named as Vaishnavi approach) for solving triangular intuitionistic fuzzy transportation problems of type-2.

3.1

Organization of the Chapter

This chapter is organized as follows: (i) In Sect. 3.2, some basic definitions have been presented. (ii) In Sect. 3.3, an existing approach for comparing triangular fuzzy numbers, used in Ebrahimnejad and Verdegay’s approach [3], has been discussed. (iii) In Sect. 3.4, Ebrahimnejad and Verdegay’s approach [3] for solving triangular intuitionistic fuzzy transportation problems of type-2 has been discussed. (iv) In Sect. 3.5, a limitation of Ebrahimnejad and Verdegays’s approach [3] has been discussed. (v) In Sect. 3.6, a drawback of Ebrahimnejad and Verdegay’s approach [3] has been discussed. (vi) In Sect. 3.7, the reason for the occurrence of the drawback in Ebrahimnejad and Verdegay’s approach [3] has been discussed. (vii) In Sect. 3.8, a new approach (named as MEHAR approach) has been proposed for comparing triangular intuitionistic fuzzy numbers. (viii) In Sect. 3.9, a new approach (named as Vaishnavi approach) has been proposed for solving triangular intuitionistic fuzzy transportation problems of type-2.

3.1 Organization of the Chapter

121

(ix) In Sect. 3.10, a unique intuitionistic fuzzy optimal value of the triangular intuitionistic fuzzy transportation problem of type-2, considered in Sect. 3.5, has been obtained by the proposed Vaishnavi’s approach. (x) Sect. 3.11 concludes the chapter.

3.2

Preliminaries

In this section, some basic definitions have been presented [2, 3]. Definition 3.2.1 An intuitionistic fuzzy set, over the universal set X, is defined as a ¼ fx; la ð xÞ; ma ð xÞjx 2 X, 0  la ð xÞ  1; 0  ma ð xÞ  1, la ð xÞ þ ma ð xÞ  1g. The values la ð xÞ, ma ð xÞ and 1
la ð xÞ
ma ð xÞ, respectively, are called the degree of membership, the degree of non-membership and the degree of hesitation for the element x.
 ~ I ¼ x; l ~ I ð xÞ; m ~ I ð xÞ : x 2 R , Definition 3.2.2 An intuitionistic fuzzy set, A A A defined over the set of real numbers ℝ, is said to be an intuitionistic fuzzy number if the following conditions holds: (i) There exist m 2 R such that lA~ I ðmÞ ¼ 1 and mA~ I ðmÞ ¼ 0, (m is called the ~ I ). mean value of A (ii) lA~ I ð xÞ and mA~ I ð xÞ are piecewise continuous mapping from ℝ to the closed interval ½0; 1. (iii) 0  lA~ I ð xÞ; mA~ I ð xÞ  1 8x 2 R. (iv) 0  lA~ I ð xÞ þ mA~ I ð xÞ  1 8x 2 R. ~ I is said to be a triangular Definition 3.2.3 An intuitionistic fuzzy number A intuitionistic fuzzy number if its membership function lA~ I ð xÞ and non-membership function mA~ I ð xÞ are defined as 8 x
a 1 ; > > < a2
a1 1; lA~ I ðxÞ ¼ a3
x ; > > : a3
a2 0

a1  x\a2 a2 a2 \x  a3 otherwise

8 a0
x 2 > 0 0 ; > > < a2
a1 0; and mA~ I ðxÞ ¼ a02
x > > 0 0 ; > : a2
a3 1;

~I A triangular intuitionistic fuzzy number A   I 0 0 ~ A ¼ a1 ; a2 ; a3 ; a1 ; a2 ; a3 .

a01  x\a2 a2 a2 \x  a03 otherwise

may be represented as

Definition 3.2.4 A triangular intuitionistic fuzzy number is said to be a non-negative triangular intuitionistic fuzzy number if a01  0.

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

122

    ~ I ¼ a1 ; a2 ; a3 ; a0 ; a2 ; a0 and B ~ I ¼ b1 ; b2 ; b3 ; b01 ; b2 ; b03 be Definition 3.2.5 Let A 1 3 ~I ~I two triangular intuitionistic fuzzy  numbers. Then, A  B ¼ ða1 þ b1 ; a2 þ b2 ; 0 0 a3 þ b3 :; a1 þ b1 ; a2 þ b2 ; a3 þ b3 .     ~ I ¼ a1 ; a2 ; a3 ; a0 ; a2 ; a0 and B ~ I ¼ b1 ; b2 ; b3 ; b01 ; b2 ; b03 be Definition 3.2.6 Let A 1 3 two triangular intuitionistic fuzzy numbers. Then,   ~ I B ~ I ¼ a1
b3 ; a2
b2 ; a3
b1 ; a01
b03 ; a2
b2 ; a03
b01 : A     ~ I ¼ a1 ; a2 ; a3 ; a0 ; a2 ; a0 and B ~ I ¼ b1 ; b2 ; b3 ; b01 ; b2 ; b03 be Definition 3.2.7 Let A 1 3 ~I  B ~I ¼ intuitionistic fuzzy numbers. Then, A two non-negative 0 triangular  a1 b1 ; a2 b2 ; a3 b3 ; a1 b01 ; a2 b2 ; a03 b03 .   ~ I ¼ a1 ; a2 ; a3 ; a0 ; a2 ; a0 be a triangular intuitionistic fuzzy Definition 3.2.8 Let A 1 3 number and k be a real number. Then, ~I ¼ kA

3.3

  0 0 ka1 ; ka2 ; ka3 ; ka10 ; ka2 ; ka30 ; k  0; ka3 ; ka2 ; ka1 ; ka3 ; ka2 ; ka1 ; k\0:

Tabular Representation of a Triangular Intuitionistic Fuzzy Transportation Problem of Type-2

A triangular intuitionistic fuzzy transportation problem of type-2 having m sources and n destinations can be represented by Table 3.1 [2, 3]. where

Table 3.1 Tabular representation of triangular intuitionistic fuzzy transportation problem of type-2 Destinations Sources

D1

D2



Dj



Dn

S1

~cI11

~cI12



~cI1j



~cI1n

a1

.. . Si

.. .

.. .

.. .

~cIi2

.. .

.. .

~cIi1

.. .

.. . ai

.. . Sm

.. .

.. . ~cIm2

.. .

~cIm1

Crisp demand

b1

b2



~cIij .. .

~cIin

~cImj

.. .

.. . ~cImn

bj



bn

Crisp availability

.. . am

3.3 Tabular Representation of a Triangular Intuitionistic Fuzzy …

123

(i) The triangular intuitionistic fuzzy number ~cIij ¼ ðcij;1 ; cij;2 ; cij;3 ; c0ij;1 ; cij;2 ; c0ij;3 Þ represents the intuitionistic fuzzy transportation cost for supplying the one unit quantity of the product from the ith source ðSi Þ to the jth destination Dj , (ii) The non-negative real number ai represents the crisp availability of the product at the ith source ðSi Þ, (iii) The non-negative real number   (iv) bj represents the crisp demand of the product at the jth destination Dj ,

3.4

An Existing Approach for Comparing Triangular Intuitionistic Fuzzy Numbers

If A and B are two distinct real numbers, i.e., A 6¼ B, then it can be easily concluded   ~ I ¼ a1 ; a2 ; a3 ; a0 ; a2 ; a0 ~I ¼ and B that A\B or A [ B. However, if A 1 3   ~ I 6¼ B ~ I . Then, it b1 ; b2 ; b3 ; b01 ; b2 ; b03 are triangular fuzzy numbers such that A I I I I ~ ~ ~ or A  B ~ . Different approaches have been cannot easily conclude that A B proposed in the literature for comparing triangular intuitionistic fuzzy numbers. Singh and Yadav [2] as well as Ebrahimnejad and Vedegay [3] have used the following approach for comparing triangular intuitionistic fuzzy numbers     ~ I ¼ a1 ; a2 ; a3 ; a0 ; a2 ; a0 and B ~ I ¼ b1 ; b2 ; b3 ; b01 ; b2 ; b03 : A 1 3     Check that f a1 ; a2 ; a3 ; a01 ; a2 ; a03 > f b1 ; b2 ; b3 ; b01 ; b2 ; b03 or       f a1 ; a2 ; a3 ; a01 ; a2 ; a03 < f b1 ; b2 ; b3 ; b01 ; b2 ; b03 or f a1 ; a2 ; a3 ; a01 ; a2 ; a03 =   f b1 ; b2 ; b3 ; b01 ; b2 ; b03 , where,   a1 þ 2a2 þ a3 þ a01 þ 2a2 þ a03 f a1 ; a2 ; a3 ; a01 ; a2 ; a03 ¼ 8 and   b1 þ 2b2 þ b3 þ b01 þ 2b2 þ b03 f b1 ; b2 ; b3 ; b01 ; b2 ; b03 ¼ : 8

124

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

    Case (i): If f a1 ; a2 ; a3 ; a01 ; a2 ; a03 > f b1 ; b2 ; b3 ; b01 ; b2 ; b03 then ða1 ; a2 ; a3 ;  
  a0 ; a ; a0 Þ ≻ b1 ; b2 ; b3 ; b01 ; b2 ; b03 . Hence, maximum a1 ; a2 ; a3 ; a01 ; a2 ; a03 , 1 2 3 0   
 = a1 ; a2 ; a3 ; a01 ; a2 ; a03 and minimum a1 ; a2 ; a3 ; a01 ; b1 ; b2 ; b3 ; b1 ; b2 ; b03     a2 ; a03 Þ:, b1 ; b2 ; b3 ; b01 ; b2 ; b03 = b1 ; b2 ; b3 ; b01 ; b2 ; b03 .     Case (ii): If f a1 ; a2 ; a3 ; a01 ; a2 ; a03 < f b1 ; b2 ; b3 ; b01 ; b2 ; b03 then ða1 ; a2 ; a3 ;  
  a0 ; a ; a0 Þ ≺ b1 ; b2 ; b3 ; b01 ; b2 ; b03 . Hence, maximum a1 ; a2 ; a3 ; a01 ; a2 ; a03 , 1 2 3 0   
 = b1 ; b2 ; b3 ; b01 ; b2 ; b03 and minimum a1 ; a2 ; a3 ; a01 ; b1 ; b2 ; b3 ; b1 ; b2 ; b03     a2 ; a03 Þ:, b1 ; b2 ; b3 ; b01 ; b2 ; b03 = a1 ; a2 ; a3 ; a01 ; a2 ; a03 .     Case (iii): If f a1 ; a2 ; a3 ; a01 ; a2 ; a03 = f b1 ; b2 ; b3 ; b01 ; b2 ; b03 then ða1 ; a2 ; a3 ;  
  a01 ; a2 ; a03 Þ b1 ; b2 ; b3 ; b01 ; b2 ; b03 . Hence, maximum a1 ; a2 ; a3 ; a01 ; a2 ; a03 ,  
    = minimum a1 ; a2 ; a3 ; a01 ; a2 ; a03 , b1 ; b2 ; b3 ; b01 ; b2 ; b03 b1 ; b2 ; b3 ; b01 ; b2 ; b03     = a1 ; a2 ; a3 ; a01 ; a2 ; a03 and b1 ; b2 ; b3 ; b01 ; b2 ; b03 .

3.5

Singh and Yadav’s Approach

Singh and Yadav [2] proposed following approach for solving triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.1, by considering (i) The arithmetic operations of triangular intuitionistic fuzzy numbers instead of the arithmetic operations of real numbers. (ii) The approach, discussed in Sect. 3.4, for comparing triangular intuitionistic fuzzy numbers. Step 1: Find an initial basic feasible solution of the triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.1, by applying one of the well-known methods (North–west corner method or least cost method or Vogel’s approximation method, etc.). Step 2: Apply MODI method (or u
v method) to find a crisp optimal solution with the help of the obtained initial basic feasible solution.

3.6

Ebrahimnejad and Verdegay’s Approach

Ebrahimnejad and Verdegay [3] pointed out that for solving a triangular intuitionistic fuzzy transportation problem of type-2 by Singh and Yadav’s approach [2], there is a need to use arithmetic operations of triangular intuitionistic fuzzy numbers. Due to which, more computational efforts are required for solving a triangular intuitionistic fuzzy transportation problem of type-2 by Singh and Yadav’s approach [2].

3.6 Ebrahimnejad and Verdegay’s Approach

125

Table 3.2 Transformed crisp transportation problem Destinations Sources

D1   f ~cI11

D2   f ~cI12



.. . Si

.. .

.. .

.. . Sm

.. .

Crisp demand

b1

S1

  f ~cIi1   f ~cIm1

Dj   f ~cI1j



.. .

.. .

.. .

  f ~cIm2

.. .

b2



  f ~cIi2



Dn   f ~cI1n

Crisp availability

.. .

.. .

.. . ai

.. .

.. .

.. .

.. . am

bj



bn

  f ~cIij   f ~cImj



  f ~cIin   f ~cImn

a1

To reduce the computational efforts, Ebrahimnejad and Verdegay [3] proposed the following approach for solving the triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.1. Step 1: Transform the triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.1, into its equivalent crisp transportation problem represented by Table 3.2. where       cij;1 þ 2cij;2 þ cij;3 þ c0ij;1 þ 2cij;2 þ c0ij;3 f ~cIij ¼ : 8
 Step 2: Find a crisp optimal solution xij ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n of the transformed crisp transportation problem represented by Table 3.2.
 Step 3: Using the crisp optimal solution xij ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n , n obtained in Step 2, find the intuitionistic fuzzy transportation cost m i¼1 j¼1   cij;1 xij ; cij;2 xij ; cij;3 xij ; c0ij;1 xij ; c0ij;2 xij ; c0ij;3 xij .

3.7

Limitation of Ebrahimnejad and Verdegay’s Approach

In general, to solve a real-life transportation problem, the opinion of two or more experts about the parameters is collected. Then, all the collected information is aggregated to obtain a single value of each parameter. Since, Ebrahimnejad and Verdegay’s approach [3] has been proposed by considering the assumption that the aggregated value of each parameter is available. Therefore, Ebrahimnejad and Verdegay’s approach [3] cannot be used to solve several real-life triangular intuitionistic fuzzy transportation problems of type-2. For example, Ebrahimnejad and

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

126

Table 3.3 Data provided by the first decision-maker Destinations Sources

D1

D2

Crisp availability

S1 S2 Crisp demand

ð10; 30; 40; 5; 30; 45Þ ð15; 30; 50; 10; 30; 70Þ 60

ð25; 50; 60; 10; 50; 70Þ ð20; 40; 60; 15; 40; 70Þ 50

50 60

Table 3.4 Data provided by the second decision-maker Destinations Sources

D1

D2

Crisp availability

S1 S2 Crisp demand

ð15; 20; 45; 10; 20; 50Þ ð20; 35; 55; 15; 35; 75Þ 50

ð30; 55; 65; 15; 55; 75Þ ð25; 45; 65; 20; 45; 75Þ 60

60 50

Verdegay’s approach [3] cannot be used to solve the triangular intuitionistic fuzzy transportation problem of type-2 considered in Example 3.7.1. Example 3.7.1 Let us consider a product needs to be supplied from two sources to two destinations. For the same purpose, the information about each parameter is collected from two experts. If

(i) Table 3.3 represents the intuitionistic fuzzy availability and the crisp demand provided by (ii) Table 3.4 represents the intuitionistic fuzzy availability and the crisp demand provided by

transportation cost, the crisp the first decision-maker transportation cost, the crisp the second decision-maker.

Then, this triangular intuitionistic fuzzy transportation problem of type-2 cannot be solved by Ebrahimnejad and Verdegay’s approach [3].

3.8

Drawback of Ebrahimnejad and Verdegay’s Approach

In this section, the triangular intuitionistic fuzzy transportation problems of type-2, represented by Table 3.5, have been solved by Ebrahimnejad and Verdegay’s approach [3] and showed that two distinct triangular intuitionistic fuzzy numbers (875, 3125, 3875; 375, 3125, 4375) and (1375, 2625, 4375; 875, 2625, 4875), representing the intuitionistic fuzzy optimal transportation cost, are obtained, which are mathematically incorrect as the physical meaning of these triangular intuitionistic fuzzy numbers is different.

3.8 Drawback of Ebrahimnejad and Verdegay’s Approach

127

Table 3.5 Triangular intuitionistic fuzzy transportation problem of type-2 Destinations Sources

D1

D2

Crisp availability

S1 S2 Crisp demand

ð20; 60; 80; 10; 60; 90Þ ð30; 50; 90; 20; 50; 100Þ 25

ð25; 55; 85; 15; 55; 95Þ ð15; 65; 75; 5; 65; 85Þ 25

25 25 50

Using Ebrahimnejad and Verdegay’s approach [3], discussed in Sect. 3.6, a crisp optimal solution and the intuitionistic fuzzy optimal transportation cost of the triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.5, can be obtained as follows. Step 1: Using Step 1 of Ebrahimnejad and Verdegay’s approach [3], the triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.5, can be transformed into a crisp transportation problem represented by Table 3.6. Step 2: On solving the crisp transportation problem, represented by Table 3.6, the following two crisp optimal basic feasible solutions are obtained (i) x11 ¼ 25, x12 ¼ 0, x21 ¼ 0, x22 ¼ 25, (ii) x11 ¼ 0, x12 ¼ 25, x21 ¼ 25, x22 ¼ 0. Step 3: Using Step 3 of Ebrahimnejad and Verdegay’s approach [3] (i) The triangular intuitionistic fuzzy number, representing the intuitionistic fuzzy optimal transportation cost, corresponding to the first crisp optimal basic feasible solution is (875, 3125, 3875; 375, 3125, 4375). (ii) The triangular intuitionistic fuzzy number, representing the intuitionistic fuzzy optimal transportation cost, corresponding to the second crisp optimal basic feasible solution is (1375, 2625, 4375; 875, 2625, 4875) . Hence, both the distinct triangular intuitionistic fuzzy numbers (875, 3125, 3875; 375, 3125, 4375) and (1375, 2625, 4375; 875, 2625, 4875) represent the intuitionistic fuzzy optimal transportation cost of the triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.5, which is mathematically incorrect as the physical meaning of these triangular intuitionistic fuzzy numbers is different. Table 3.6 Transformed crisp transportation problem Destinations Sources S1 S2 Crisp demand

D1

D2

Crisp availability

f ð20; 60; 80; 10; 60; 90Þ ¼ 55 f ð30; 50; 90; 20; 50; 100Þ ¼ 55 25

f ð25; 55; 85; 15; 55; 95Þ ¼ 55 f ð15; 65; 75; 5; 65; 85Þ ¼ 55 25

25 25 50

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

128

3.9

Reasons for the Occurrence of the Drawback

The drawback, pointed out in Sect. 3.8, is occurring due to the following reason. It is obvious from Step 2 of Ebrahimnejad and Verdegay’s approach [3] that   Ebrahimnejad and Verdegay [3] have used the expression f ~cij ¼ ðcij;1 þ 2cij;2 þ cij;3 Þ þ ðc0ij;1 þ 2cij;2 þ c0ij;3 Þ for transforming a triangular intuitionistic fuzzy 8  I 0 number ~cij ¼ cij;1 ; cij;2 ; cij;3 ; cij;1 ; cij;2 ; c0ij;3 into a real number, i.e., Ebrahimnejad and Verdegay [3] have used the approach, discussed in Sect. 3.3, to find minimum of triangular intuitionistic fuzzy numbers. However, there may exist such distinct triangular intuitionistic fuzzy numbers for which the real number, obtained by the   ðcij;1 þ 2cij;2 þ cij;3 Þ þ ðc0ij;1 þ 2cij;2 þ c0ij;3 Þ , will be equal and hence, the expression ~cij ¼ 8 obtained minimum will not be a unique triangular intuitionistic fuzzy number. For example, (20, 60, 80; 10, 60, 90) and (30, 50, 90; 20, 50, 100) are two different triangular intuitionistic fuzzy numbers. While, f ð20; 60; 80; 10; 60; 90Þ = f ð30; 50; 90; 20; 50; 100Þ ¼ 55. Hence, minimum (20, 60, 80; 10, 60, 90), (30, 50, 90; 20, 50, 100) = (20, 60, 80; 10, 60, 90) and (30, 50, 90; 20, 50, 100) .

3.10

Proposed MEHAR Approach for Comparing Triangular Intuitionistic Fuzzy Numbers

It is obvious from Sect. 3.9 that it is not appropriate to use the existing approach, discussed in Sect. 3.3, for comparing triangular intuitionistic fuzzy numbers. In this section, a new approach (named as MEHAR) has been proposed for comparing triangular intuitionistic fuzzy numbers. Using the  proposed MEHAR approach,  the triangular  intuitionistic fuzzy num I 0 0 I ~ ~ bers A ¼ a1;1 ; a1;2 ; a1;3 ; a ; a1;2 ; a and A ¼ a2;1 ; a2;2 ; a2;3 ; a0 ; a2;2 ; a0 1

1;1

1;3

2

2;1

2;3

can be compared as follows:    Step 1: Check that M a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 > M a2;1 ; a2;2 ; a2;3 ;    a02;1 ; a2;2 ; a02;3 Þ or M a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 < M a2;1 ; a2;2 ; a2;3 ; a02;1 ;     a2;2 ; a02;3 Þ or M a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = M a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , where   M a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ¼ a1;1 þ a1;3 þ 2a1;2 þ a01;1 þ a01;3   M a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ¼ a2;1 þ a2;3 þ 2a2;2 þ a02;1 þ a02;3

and

:

3.10

Proposed MEHAR Approach for Comparing Triangular Intuitionistic …

129

    Case (i): If M a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 > M a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ,     then a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ≻ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .     Case (ii): If M a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 < M a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ,     then a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ≺ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .     Case (iii): If M a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = M a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , then go to Step 2.

   Step 2: Check that E a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 > E a2;1 ; a2;2 ; a2;3 ;     a02;1 ; a2;2 ; a02;3 Þ or E a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 < E a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3     or E a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = E a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , where   E a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ¼ a1;1 þ a1;3 þ 2a1;2 þ a01;3 and   E a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ¼ a2;1 þ a2;3 þ 2a2;2 þ a02;3 :     Case (i): If E a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 > E a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , then     a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ≻ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .     Case (ii): If E a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 < E a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ,     then a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ≺ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .     Case (iii): If E a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = E a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , then go to Step 3.

   > H a2;1 ; a2;2 ; a2;3 ; Step 3: Check that H a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3    < H a2;1 ; a2;2 ; a2;3 ; a02;1 ; a02;1 ; a2;2 ; a02;3 Þ or H a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3     a2;2 ; a02;3 Þ or H a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = H a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , where   H a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ¼ a1;3 þ 2a1;2 þ a01;3 and   H a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ¼ a2;3 þ 2a2;2 þ a02;3 :

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

130

    Case (i): If H a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 > H a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ,     then a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ≻ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .     Case (ii): If H a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 < H a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ,     then a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ≺ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .     Case (iii): If H a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = H a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , then go to Step 2.

   Step 4: Check that A a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 > A a2;1 ; a2;2 ; a2;3 ;     a02;1 ; a2;2 ; a02;3 Þ or A a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 < A a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3     or A a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = A a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , where   A a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ¼ a1;3 þ a01;3 and   A a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ¼ a2;3 þ a02;3 :     Case (i): If A a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 > A a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , then     a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ≻ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .     Case (ii): If A a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 < A a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ,     then a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ≺ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .     Case (iii): If A a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = A a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , then go to Step 2.

   > R a2;1 ; a2;2 ; a2;3 ; Step 5: Check that R a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3     a02;1 ; a2;2 ; a02;3 Þ or R a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 < R a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3     or R a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = R a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , where   R a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ¼ a01;3 and   R a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ¼ a02;3 :

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    Case (i): If R a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 > R a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 , then     a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ≻ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .     Case (ii): If R a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 < R a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ,     then a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 ≺ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .     Case (iii): If R a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = R a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ,     then a1;1 ; a1;2 ; a1;3 ; a01;1 ; a1;2 ; a01;3 = a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 .

3.11

Proposed Vaishnavi Approach

In this section, to overcome the limitation as well as to resolve the drawback of Ebrahimnejad and Verdegay’s approach [3], a new approach (named as Vaishnavi approach) has been proposed to solve the triangular intuitionistic fuzzy transportation problems of type-2, represented by Table 3.1. Step 1: Check that the aggregated value of the intuitionistic fuzzy transportation cost, the crisp availability and the crisp demand, provided by all the decision-makers, is available or not. Case (i): If it is available, then go to Step 2. Case (ii): If it is not available, then use the existing triangular intuitionistic fuzzy weighted averaging aggregation operator [4]. Find,

  (i) The triangular intuitionistic fuzzy number ~cij ¼ pk¼1 wk  ~ckij = ! Pp Pp Pp k k k wk cij;1 ; k¼1 wk cij;2 ; k¼1 wk cij;3 ; Pp Pp Pk¼1 representing the aggregated p k k k k¼1 wk cij;5 ; k¼1 wk cij;2 ; k¼1 wk cij;7 value of the intuitionistic fuzzy transportation cost for supplying the one unit quantity of the product from the ith source to the jth destination. P (ii) The non-negative real number ai ¼ pk¼1 wk aki representing the aggregated value of the crisp availability of the product at the ith source. P (iii) The non-negative real number bj ¼ pk¼1 wk bkj representing the aggregated value of the crisp demand of the product at the jth destination. where, (i) wk 2 ½0; 1 represents the normalized weight of the kth decision-maker.  (ii) The triangular intuitionistic fuzzy number ~ckij ¼ ckij;1 ; ckij;2 ; ckij;3 ; ckij;5 ; ckij;2 ; ckij;7 Þ represents the intuitionistic fuzzy cost for supplying the one

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

132

unit quantity of the product from the ith source to the jth destination according to the kth decision-maker. (iii) The non-negative real number aki represents the crisp availability of the product at the ith source according to the kth decision-maker. (iv) The non-negative real number bkj represents the crisp demand of the product at the jth destination according to the kth decision-maker. and go to Step 2. For example, if in Example 3.7.1, the normalized weights of first and second decision-makers are 0:4 and 0:6, repsectively. Then, Table 3.7 will represent the aggregated intuitionistic fuzzy cost for supplying one unit quantity of the product, the crisp availability of the product and the crisp demand of the product. Step 2: Check that the considered/transformed triangular intuitionistic fuzzy transportation problem of type-2 problem P Pn is a balanced Pm Pn or an unbalanced problem, i.e., check that m a ¼ b or a ¼ 6 i j i i¼1 j¼1 i¼1 j¼1 bj . Case (i): If the considered/transformed triangular Pn fuzzy transportation P intuitionistic a ¼ problem of type-2 is a balanced problem, i.e., m i i¼1 j¼1 bj , then go to Step 4. Case (ii): If the considered/transformed triangular intuitionistic fuzzy transportation P P n problem of type-2 is an unbalanced problem, i.e., m a ¼ 6 i¼1 i j¼1 bj , then go to Step 3. P  Pm n Step 3: Add a dummy source having availability and j¼1 bj
i¼1 ai consider the transportation cost for supplying the one unit quantity of the product from each source to the dummy destination as the triangular intuitionistic fuzzy number ð0; 0; 0; 0; 0; 0Þ. P  Pn m Also, add a dummy destination having demand a
b i¼1 i j¼1 j and consider the transportation cost for supplying the one unit quantity of the product from the dummy source to each destination as the triangular intuitionistic fuzzy number ð0; 0; 0; 0; 0; 0Þ. Step 4: Write the intuitionistic fuzzy linear programming problem (3.11.1) corresponding to the considered/transformed balanced triangular intuitionistic fuzzy transportation problem of type-2.

Table 3.7 Aggregated data Destinations Sources

D1

D2

Crisp availability

S1 S2 Crisp demand

ð13; 24; 43; 8; 24; 48Þ ð18; 33; 53; 13; 33; 73Þ 54

ð28; 53; 63; 13; 53; 73Þ ð23; 43; 63; 18; 43; 73Þ 56

56 54

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Intuitionistic fuzzy linear programming problem (3.11.1) h   i n Minimize m cij;1 ; cij;2 ; cij;3 ; c0ij;1 ; cij;2 ; c0ij;3  xij i¼1 j¼1 Subject to nX þ1

xij ¼ ai ; i ¼ 1; 2; . . .; m;

j¼1 nX þ1

xij ¼

j¼1

n X

bj


j¼1 m þ1 X

m X

ai ; i ¼ m þ 1;

i¼1

xij ¼ bj ; j ¼ 1; 2; . . .; n;

i¼1 m þ1 X i¼1

xij ¼

m X i¼1

ai


n X

bj ; j ¼ n þ 1;

j¼1

xij is a non-negative real number.

   Step 5: Using the relation a  cij;1 ; cij;2 ; cij;3 ; c0ij;1 ; cij;2 ; c0ij;3 = acij;1 ; acij;2 ; acij;3 ; ac0ij;1 ; acij;2 ; ac0ij;3 Þ, a  0, transform the intuitionistic fuzzy linear programming problem (3.11.1) into its equivalent intuitionistic fuzzy linear programming problem (3.11.2). Intuitionistic fuzzy linear programming problem (3.11.2) h  i n 0 0 Minimize m  c x ; c x ; c x ; c x ; c x ; c x ij;1 ij ij;2 ij ij;3 ij ij ij;2 ij ij i¼1 j¼1 ij;1 ij;3 Subject to Constraints of the intuitionistic fuzzy linear programming problem (3.11.1). P   n Step 6: Using the relation nj¼1 aj;1 ; aj;2 ; aj;1 ; a0j;1 ; aj;2 ; a0j;3 = j¼1 aj;1 ;  Pn Pn Pn 0 Pn Pn 0 j¼1 aj;2 ; j¼1 aj;3 :; j¼1 aj;1 ; j¼1 aj;2 ; j¼1 aj;3 , transform the intuitionistic fuzzy linear programming problem (3.11.2) into its equivalent intuitionistic fuzzy linear programming problem (3.11.3).

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

134

Intuitionistic fuzzy linear programming problem (3.11.3) m X n X

Minimize

cij;1 xij

m X n X

i¼1 j¼1

cij;2 xij ;

m X n X

i¼1 j¼1 m X n X

cij;3 xij ;

i¼1 j¼1

cij;2 xij ;

m X n X

i¼1 j¼1

m X n X

c0ij;1 xij ;

i¼1 j¼1

! c0ij;3 xij

i¼1 j¼1

Subject to Constraints of the intuitionistic fuzzy linear programming problem (3.11.1). Step 7: Using the proposed MEHAR approach for comparing triangular intuitionistic fuzzy numbers, transform the intuitionistic fuzzy linear programming problem (3.11.3) into its equivalent crisp linear programming problem (3.11.4) and hence, in the crisp linear programming problem (3.11.5). Crisp linear programming problem (3.11.4) " Minimize M

m X n X

cij;1 xij

i¼1 j¼1

m X n X

cij;2 xij ;

i¼1 j¼1

m X n X

cij;2 xij ;

i¼1 j¼1

m X n X

m X n X

cij;3 xij ;

i¼1 j¼1

m X n X

c0ij;1 xij ;

i¼1 j¼1

!# c0ij;3 xij

i¼1 j¼1

Subject to Constraints of the intuitionistic fuzzy linear programming problem (3.11.1). Crisp linear programming problem (3.11.5) " Minimize

m X n X

cij;1 xij þ

i¼1 j¼1

þ

m X n X

#

m X n X i¼1 j¼1

cij;3 xij þ 2

m X n X i¼1 j¼1

cij;2 xij þ

m X n X

c0ij;1 xij

i¼1 j¼1

c0ij;3 xij

i¼1 j¼1

Subject to Constraints of the intuitionistic fuzzy linear programming problem (3.11.1). Step 8: Check that a unique crisp optimal solution exists for the crisp linear programming problem (3.115) or not.

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135

Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (3.11.5), then go to Step 18. Case (ii): If more than one crisp optimal solutions exist for the crisp linear programming problem (3.11.5), then go to Step 9. Step 9: Using the proposed MEHAR approach for comparing triangular intuitionistic fuzzy numbers, transform the intuitionistic fuzzy linear programming problem (3.11.3) into its equivalent crisp linear programming problem (3.11.6) and hence, in the crisp linear programming problem (3.11.7). Crisp linear programming problem (3.11.6) " Minimize E

m X n X

cij;1 xij

i¼1 j¼1 m X n X

c0ij;1 xij ;

m X n X

i¼1 j¼1

m X n X

cij;2 xij ;

m X n X

i¼1 j¼1

cij;2 xij ;

i¼1 j¼1

cij;3 xij ;

i¼1 j¼1

m X n X

!#

c0ij;3 xij

i¼1 j¼1

Subject to m X n X

cij;1 xij þ

i¼1 j¼1 m X n X

þ

m X n X

cij;3 xij þ 2

i¼1 j¼1

m X n X

cij;2 xij þ

i¼1 j¼1

m X n X

c0ij;1 xij

i¼1 j¼1

c0ij;3 xij ¼ Crisp optimal value of the crisp linear programming problem ð3:11:5Þ:

i¼1 j¼1

and Constraints of the crisp linear programming problem (3.11.5). Crisp linear programming problem (3.11.7)

Minimize

" m X n X i¼1 j¼1

cij;1 xij þ

m X n X i¼1 j¼1

cij;3 xij þ 2

m X n X

cij;2 xij þ

m X n X

i¼1 j¼1

# c0ij;3 xij

i¼1 j¼1

Subject to Constraints of the crisp linear programming problem (3.11.6). Step 10: Check that a unique crisp optimal solution exists for the crisp linear programming problem (3.11.7) or not. Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (3.11.7), then go to Step 18. Case (ii): If more than one crisp optimal solutions exist for the crisp linear programming problem (3.11.7), then go to Step 11.

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

136

Step 11: Using the proposed MEHAR approach for comparing triangular intuitionistic fuzzy numbers, transform the intuitionistic fuzzy linear programming problem (3.11.3) into its equivalent crisp linear programming problem (3.11.8) and hence, in the crisp linear programming problem (3.11.9). Crisp linear programming problem (3.11.8) " Minimize H

m X n X

cij;1 xij

i¼1 j¼1 m X n X i¼1 j¼1

c0ij;1 xij ;

m X n X

m X n X

cij;2 xij ;

i¼1 j¼1

cij;2 xij ;

i¼1 j¼1

m X n X

cij;3 xij ;

i¼1 j¼1

m X n X

!#

c0ij;3 xij

i¼1 j¼1

Subject Pm Pto n

P Pn Pm Pn Pm Pn þ m i¼1 j¼1 cij;3 xij + 2 i¼1 j¼1 cij;2 xij þ i¼1 j¼1 c0ij;3 xij ¼ Crisp optimal value of the crisp linear programming problem (3.11.7). and Constraints of the crisp linear programming problem (3.11.7). i¼1

j¼1 cij;1 xij

Crisp linear programming problem (3.11.9) " Minimize

2

m X n X i¼1 j¼1

cij;2 xij þ

m X n X i¼1 j¼1

cij;3 xij þ

m X n X

! c0ij;3 xij

i¼1 j¼1

Subject to Constraints of the crisp linear programming problem (3.11.8). Step 12: Check that a unique crisp optimal solution exists for the crisp linear programming problem (3.11.9) or not. Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (3.11.9), then go to Step 18. Case (ii): If more than one crisp optimal solutions exist for the crisp linear programming problem (3.11.9), then go to Step 13. Step 13: Using the proposed MEHAR approach for comparing triangular intuitionistic fuzzy numbers, transform the intuitionistic fuzzy linear programming problem (3.11.3) into its equivalent crisp linear programming problem (3.11.10) and hence, in the crisp linear programming problem (3.11.11).

3.11

Proposed Vaishnavi Approach

137

Crisp linear programming problem (3.11.10) " Minimize A

m X n X

cij;1 xij

i¼1 j¼1 m X n X

c0ij;1 xij ;

i¼1 j¼1

m X n X

m X n X

cij;2 xij ;

i¼1 j¼1

cij;2 xij ;

i¼1 j¼1

m X n X

m X n X

cij;3 xij ;

i¼1 j¼1

!#

c0ij;3 xij

i¼1 j¼1

Subject to Pm Pn Pm Pn 0 P P n 2 m i¼1 j¼1 cij;2 xij þ i¼1 j¼1 cij;3 xij + i¼1 j¼1 cij;3 xij ¼ Crisp optimal value of the crisp linear programming problem (3.11.9) and Constraints of the crisp linear programming problem (3.11.9). Crisp linear programming problem (3.11.11) " Minimize

m X n X i¼1 j¼1

cij;3 xij þ

m X n X

# c0ij;3 xij

i¼1 j¼1

Subject to Constraints of the crisp linear programming problem (3.11.10). Step 14: Check that a unique crisp optimal solution exists for the crisp linear programming problem (3.11.11) or not. Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (3.11.11), then go to Step 18. Case (ii): If more than one crisp optimal solutions exist for the crisp linear programming problem (3.11.11), then go to Step 15. Step 15: Using the proposed MEHAR approach for comparing triangular intuitionistic fuzzy numbers, transform the intuitionistic fuzzy linear programming problem (3.11.3) into its equivalent crisp linear programming problem (3.11.12) and hence, in the crisp linear programming problem (3.11.13).

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

138

Crisp linear programming problem (3.11.13) " Minimize R

m X n X

cij;1 xij

i¼1 j¼1 m X n X i¼1 j¼1

c0ij;1 xij ;

m X n X

cij;2 xij ;

i¼1 j¼1

m X n X

cij;2 xij ;

i¼1 j¼1

m X n X

m X n X

cij;3 xij ;

i¼1 j¼1

!#

c0ij;3 xij

i¼1 j¼1

Subject Pm Pto n

P Pn 0 þ m i¼1 j¼1 cij;3 xij ¼ Crisp optimal value of the crisp linear programming problem (3.11.12). i¼1

j¼1 cij;3 xij

and Constraints of the crisp linear programming problem (3.11.12). Crisp linear programming problem (3.11.14) " Minimize

m X n X

! c0ij;3 xij

i¼1 j¼1

Subject to Constraints of the crisp linear programming problem (3.11.13). Step 17: Find a crisp optimal solution for the crisp linear programming problem (3.11.14) and go to Step 18. Step 18: Find the intuitionistic fuzzy optimal value of the intuitionistic fuzzy linear programming problem (3.11.1) corresponding to the obtained crisp optimal solution.

3.12

Crisp Optimal Solution of the Considered Triangular Intuitionistic Fuzzy Transportation Problem of Type-2

In Sect. 3.9, the triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.5, has been solved by Ebrahimnejad and Verdegay’s approach [3] to point out that more than one triangular intuitionistic fuzzy numbers, representing the optimal intuitionistic fuzzy transportation cost, are obtained. In this section, a unique triangular intuitionistic fuzzy, representing the optimal intuitionistic fuzzy transportation cost, for the same triangular intuitionistic fuzzy transportation problem of type-2 has been obtained by the proposed Vaishnavi approach.

3.12

Crisp Optimal Solution of the Considered Triangular Intuitionistic …

139

Using the proposed Vaishnavi approach, a unique triangular intuitionistic fuzzy number, representing the intuitionistic fuzzy optimal transportation cost, can be obtained as follows: Step 1: Since, in the considered triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.5, the aggregated values of fuzzy transportation cost, the crisp availability and the crisp demand are provided. So, there is no need to apply Step 1 of the proposed Vaishnavi approach. P P Step 2: Since, 2i¼1 ai ¼ 2j¼1 bj ¼ 50. So, the triangular intuitionistic fuzzy transportation problem, represented by Table 3.5, is a balanced triangular intuitionistic fuzzy transportation problem. Step 3: Since, the triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.5, is a balanced triangular intuitionistic fuzzy transportation problem of type-2. So, there is no need to add a dummy source and a dummy destination. Step 4: The balanced triangular intuitionistic fuzzy transportation problem of type-2, represented by Table 3.5, can also be represented by the intuitionistic fuzzy linear programming problem (3.12.1). Intuitionistic fuzzy linear programming problem (3.12.1) Minimize½ð20; 60; 80; 10; 60; 90Þ  x11  ð25; 55; 85; 15; 55; 95Þ x12  ð30; 50; 90; 20; 50; 100Þ  x21  ð15; 65; 75; 5; 65; 85Þ  x22  Subject to x11 þ x12 ¼ 25; x21 þ x22 ¼ 25; x11 þ x12 ¼ 25; x12 þ x22 ¼ 25; x11  0; x12  0; x21  0; x22  0.

   Step 5: Using the relation a  cij;1 ; cij;2 ; cij;3 ; c0ij;1 ; cij;2 ; c0ij;3 = acij;1 ; acij;2 ; acij;3 ; ac0ij;1 ; acij;2 ; ac0ij;3 Þ, a  0, the intuitionistic fuzzy linear programming problem (3.12.1) can be transformed into its equivalent intuitionistic fuzzy linear programming problem (3.12.2). Intuitionistic fuzzy linear programming problem (3.12.2) Minimize½ð20x11 ; 60x11 ; 80x11 ; 10x11 ; 60x11 ; 90x11 Þ  ð25x12 ; 55x12 ; 85x12 ; 15x12 ; 55x12 ; 95x12 Þ  ð30x21 ; 50x21 ; 90x21 ; 20x21 ; 50x21 ; 100x21 Þ ð15x22 ; 65x22 ; 75x22 ; 5x22 ; 65x22 ; 85x22 Þ

140

3 Vaishnavi Approach for Solving Triangular Intuitionistic …

Subject to Constraints of the intuitionistic fuzzy linear programming problem (3.12.1).   P n Step 6: Using the relation nj¼1 aj;1 ; aj;2 ; aj;1 ; a0j;1 ; aj;2 ; a0j;3 = j¼1 aj;1 ; Pn Pn 0 Pn Pn 0  Pn j¼1 aj;2 ; j¼1 aj;3 :; j¼1 aj;1 ; j¼1 aj;2 ; j¼1 aj;3 , the intuitionistic fuzzy linear programming problem (3.12.2) can be transformed into its equivalent intuitionistic fuzzy linear programming problem (3.12.3). Intuitionistic fuzzy linear programming problem (3.12.3) Minimize½ð20x11 þ 25x12 þ 30x21 þ 15x22 ; 60x11 þ 55x12 þ 50x21 þ 65x22 ; 80x11 þ 85x12 þ 90x21 þ 75x22 ; 10x11 þ 15x12 þ 20x21 þ 5x22 ; 60x11 þ 55x12 þ 50x21 þ 65x22 ; 90x11 þ 95x12 þ 100x21 þ 85x22 Þ Subject to Constraints of the intuitionistic fuzzy linear programming problem (3.12.1). Step 7: Using the proposed MEHAR approach for comparing triangular intuitionistic fuzzy numbers, the intuitionistic fuzzy linear programming problem (3.12.3) can be transformed into its equivalent crisp linear programming problem (3.12.4) and hence, in the crisp linear programming problem (3.12.5). Crisp linear programming problem (3.12.4) Minimize½M ð20x11 þ 25x12 þ 30x21 þ 15x22 ; 60x11 þ 55x12 þ 50x21 þ 65x22 ; 80x11 þ 85x12 þ 90x21 þ 75x22 ; 10x11 þ 15x12 þ 20x21 þ 5x22 ; 60x11 þ 55x12 þ 50x21 þ 65x22 ; 90x11 þ 95x12 þ 100x21 þ 85x22 Þ Subject to Constraints of the intuitionistic fuzzy linear programming problem (3.12.1). Crisp linear programming problem (3.12.5) Minimize½320x11 þ 330x12 þ 340x21 þ 310x22  Subject to Constraints of the intuitionistic fuzzy linear programming problem (3.12.1).

3.12

Crisp Optimal Solution of the Considered Triangular Intuitionistic …

141

Step 8: It can be easily verified that on solving the crisp linear programming problem (3.12.5), the following unique crisp optimal solution is obtained. x11 ¼ 0;

x12 ¼ 25;

x21 ¼ 25;

x22 ¼ 0:

Step 9: Using the crisp optimal solution of the crisp linear programming problem (3.12.5), the intuitionistic fuzzy optimal value of the intuitionistic fuzzy linear programming problem (3.12.1) is. ð20; 60; 80; 10; 60; 90Þ  x11  ð25; 55; 85; 15; 55; 95Þ  x12  ð30; 50; 90; 20; 50; 100Þ  x21  ð15; 65; 75; 5; 65; 85Þ  x22 ¼ ð20; 60; 80; 10; 60; 90Þ  0  ð25; 55; 85; 15; 55; 95Þ  25  ð30; 50; 90; 20; 50; 100Þ  25  ð15; 65; 75; 5; 65; 85Þ  0 ¼ ð1375; 2625; 4375; 875; 2625; 4875Þ

3.13

Conclusions

A limitation and a drawback of the existing approach [3] have been pointed out. Also, it is pointed out that the drawback is occurring due to using an inappropriate existing approach for comparing triangular intuitionistic fuzzy numbers. Furthermore, a new approach (named as MEHAR approach) for comparing triangular intuitionistic fuzzy numbers as well as a new approach (named as Vaishnavi approach) for solving triangular intuitionistic fuzzy transportation problems of type-2 have been proposed.

References 1. G. Gupta, Transportation Problems in Intuitionistic Fuzzy Environment, Ph. D. Thesis, Thapar Institute of Engineering & Technology, Patiala, Punjab, India 2016 2. S.K. Singh, S.P. Yadav, A new approach for solving intuitionistic fuzzy transportation problem of type 2. Ann. Oper. Res. 43, 349–363 (2016) 3. A. Ebrahimnejad, J.L. Verdegay, An efficient computational approach for solving type-2 intuitionistic fuzzy numbers based transportation problems. Int. J. Comput. Intell. Syst. 9, 1154–1173 (2016) 4. S.-P. Wan, F. Wang, L.-L. Lin, J.-Y. Dong, Some new generalized aggregation operators for triangular intuitionistic fuzzy numbers and application to multi-attribute group decision making. Comput. Ind. Eng. 93, 286–301 (2016)

Chapter 4

JMD Approach for Solving Unbalanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problems

Ebrahimnejad and Verdegay [1] proposed an approach for solving fully trapezoidal intuitionistic fuzzy transportation problems (transportation problems in which each parameter is represented by a trapezoidal intuitionistic fuzzy number). One may claim that Ebrahimnejad and Verdegay’s approach [1] can be used to solve such fully trapezoidal intuitionistic fuzzy transportation problems for which the aggregated value of the intuitionistic fuzzy transportation cost, the intuitionistic fuzzy availability and the intuitionistic fuzzy demand, provided by all the decision-makers, is available. While Ebrahimnejad and Verdegay’s approach [1] cannot be used to solve such fully trapezoidal intuitionistic fuzzy transportation problems for which, instead of the aggregated data, the data of each decision-maker is provided separately. To overcome this limitation, one may modify Ebrahimnejad and Verdegay’s approach [1] with the help of the existing trapezoidal intuitionistic fuzzy aggregation operator [2]. Also, one may use Ebrahimnejad and Verdegay’s approach [1] to solve real-life fully trapezoidal intuitionistic fuzzy transportation problems. However, after a deep study, some limitations and a drawback have been observed in Ebrahimnejad and Verdegay’s approach [1]. The aim of this chapter is (i) To make the researchers aware about the observed limitations and the drawback of Ebrahimnejad and Verdegay’s approach [1]. (ii) To make the researchers aware about a drawback of the approach, used in Ebrahimnejad and Verdegay’s approach [1], for comparing trapezoidal intuitionistic fuzzy numbers. (iii) To propose a valid approach (named as DAUGHTER approach) for comparing trapezoidal intuitionistic fuzzy numbers. (iv) To propose a valid approach [named as JAI MATA DI (JMD)] approach for solving unbalanced fully trapezoidal intuitionistic fuzzy transportation problems. (v) To illustrate the proposed JMD approach with the help of numerical examples.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Mishra and A. Kumar, Aggregation Operators for Various Extensions of Fuzzy Set and Its Applications in Transportation Problems, Studies in Fuzziness and Soft Computing 399, https://doi.org/10.1007/978-981-15-6998-2_4

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4.1

Organization of the Chapter

This chapter has been organized as follows: (i) In Sect. 4.2, some basic definitions have been presented. (ii) In Sect. 4.3, an existing approach for comparing trapezoidal intuitionistic fuzzy numbers, used in Ebrahimnejad and Verdegay’s approach [1], has been discussed. (iii) In Sect. 4.4, the intuitionistic full fuzzy linear programming problem of a balanced fully trapezoidal intuitionistic fuzzy transportation problem has been presented. (iv) In Sect. 4.5, Ebrahimnejad and Verdegay’s approach [1] has been discussed in a brief manner. (v) In Sect. 4.6, some limitations of Ebrahimnejad and Verdegay’s approach [1] have been pointed out. (vi) In Sect. 4.7, a drawback of Ebrahimnejad and Verdegay’s approach [1] has been pointed out. (vii) In Sect. 4.8, a reason for the occurrence of the limitations has been discussed. (viii) In Sect. 4.9, the reasons for the occurrence of the drawback have been discussed. (ix) In Sect. 4.10, a new approach (named as DAUGHTER approach) has been proposed for comparing trapezoidal intuitionistic fuzzy numbers (x) In Sect. 4.11, a new representation of a trapezoidal intuitionistic fuzzy number (named as Mehar representation) has been proposed. (xi) In Sect. 4.12, an expression has been proposed to evaluate the multiplication of a trapezoidal intuitionistic fuzzy number in its existing representation with a trapezoidal intuitionistic fuzzy number in its Mehar representation. (xii) In Sect. 4.13, a new approach (named as JMD approach) has been proposed to solve unbalanced fully trapezoidal intuitionistic fuzzy transportation problems. (xiii) In Sect. 4.14, the proposed JMD approach has been illustrated with the help of two numerical problems. (xiv) Sect. 4.15 concludes the chapter.

4.2

Preliminaries

In this section, some basic definitions have been presented [1]. e I is said to be a trapezoidal Definition 4.2.1 An intuitionistic fuzzy number A intuitionistic fuzzy number if its membership function leI ðxÞ and non-membership A function meI ðxÞ is defined as A

4.2 Preliminaries

8 xa 1 ; > > < a2 a1 1; leI ðxÞ ¼ a4 x A ; > > : a4 a3 0;

145

8 a0 x > a0 2a0 ; a1  x\a2 > > < 2 1 a2  x  a3 0; and meI ðxÞ ¼ a03 x A a3 \x  a4 > ; > > a : 03 a04 otherwise 1;

a01  x\a02 a02  x  a03 a03 \x  a04 otherwise

e I may be denoted as A e I ¼ ð a1 ; a 2 ; a3 ; A trapezoidal intuitionistic fuzzy number A 0 0 0 0 a4 ; a1 ; a2 ; a3 ; a4 Þ: e I ¼ ð a1 ; a2 ; a3 ; a4 ; Definition 4.2.3 A trapezoidal intuitionistic fuzzy number A 0 0 a1 ; a2 ; a3 ; a4 Þ is said to be a non-negative trapezoidal intuitionistic fuzzy number if a01  0: e I ¼ ð a1 ; a2 ; a 3 ; a4 ; Definition 4.2.4 Two trapezoidal intuitionistic fuzzy numbers A   eI ¼ e I ¼ b1 ; b2 ; b3 ; b4 ; b01 ; b02 ; b03 ; b04 are said to be equal, i.e., A a01 ; a02 ; a03 ; a04 Þ and B I 0 0 0 0 0 0 0 e if a1 ¼ b1 ; a2 ¼ b2 ; a3 ¼ b3 ; a4 ¼ b4 ; a1 ¼ b1 ; a2 ¼ b2 ; a3 ¼ b3 and a4 ¼ b04 . B   e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a0 ; a0 ; a0 and B e I ¼ ð b1 ; b2 ; b 3 ; b4 ; Definition 4.2.5 Let A 1 2 3 4 0 0 0 0 eI  B eI ¼ b ; b ; b ; b Þ be two trapezoidal intuitionistic fuzzy numbers. Then, A 1 2 3 4  a1 þ b1 ; a2 þ b2 ; a3 þ b3 ; a4 þ b4 ; a01 þ b01 ; a02 þ b02 ; a03 þ b03 ; a04 þ b04 :   e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a0 ; a0 ; a0 and B e I ¼ ð b1 ; b2 ; b 3 ; b4 ; Definition 4.2.6 Let A 1 2 3 4 e I B eI ¼ b0 ; b0 ; b0 ; b0 Þ be two trapezoidal intuitionistic fuzzy numbers. Then, A 1 2 3 4  0 0 0 0 0 0 0 0 a1  b 4 ; a2  b 4 ; a3  b 2 ; a4  b 1 ; a1  b 4 ; a2  b 3 ; a3  b 3 ; a4  b 1 :   e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a0 ; a0 ; a0 and B e I ¼ ð b1 ; b2 ; b 3 ; b4 ; Definition 4.2.7 Let A 1 2 3 4 0 0 0 0 b1 ; b2 ; b3 ; b4 Þ be two non-negative trapezoidal intuitionistic fuzzy numbers. Then,   eI  B e I ¼ a1 b1 ; a2 b2 ; a3 b3 ; a4 b4 ; a01 b01 ; a02 b02 ; a03 b03 ; a04 b04 . A   e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a0 ; a0 ; a0 Definition 4.2.8 Let A be a trapezoidal 1 2 3 4 eI ¼ intuitionistic fuzzy number and k be a real number. Then, k  A   0 0 0 0 ka1 ; ka2 ; ka3 ; ka4 ; ka10 ; ka20 ; ka30 ; ka30 ; k  0; ka4 ; ka3 ; ka2 ; ka1 ; ka4 ; ka3 ; ka2 ; ka1 ; k\0:

4.3

Intuitionistic Fully Fuzzy Linear Programming Problem of a Balanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problem

Ebrahimnejad and Verdegay [1] have solved the intuitionistic fully fuzzy linear programming problem (4.3.1) to find an intuitionistic fuzzy optimal solution of the balanced fully trapezoidal intuitionistic fuzzy transportation problem represented by Table 4.1.

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Intuitionistic fully fuzzy linear programming problem (4.3.1) h  i n Minimize m cIij  ~xIij i¼1 j¼1 ~ Subject to nj¼1~xIij ¼ ~aIi ; i ¼ 1; 2; . . .; m; m xIij ¼ ~bIj ; j ¼ 1; 2; . . .; n; i¼1~ ~xIij is a non-negative trapezoidal intuitionistic fuzzy number. where (i) The trapezoidal intuitionistic fuzzy number ~cIij represents the intuitionistic fuzzy transportation cost for supplying the one unit  quantity of the product from the ith source ðSi Þ to the jth destination Dj . (ii) The trapezoidal intuitionistic fuzzy number ~xIij represents the quantity of the   product to be supplied from the ith source ðSi Þ to the jth destination Dj . (iii) The trapezoidal intuitionistic fuzzy number ~ aIi represents the intuitionistic fuzzy availability of the product at the ith source ðSi Þ. (iv) The trapezoidal intuitionistic fuzzy number ~ bIj represents the intuitionistic   fuzzy demand of the product at the jth destination Dj . Pm I Pn I ai ¼ j¼1 ~bj represents that the total intuitionistic fuzzy availability of (v) i¼1 ~ the product at all the sources is equal to the total intuitionistic fuzzy demand of the product at all the destinations.

Table 4.1 Tabular representation of balanced fully trapezoidal intuitionistic fuzzy transportation problem Sources

Destinations D2 D1



Dj



Dn

Intuitionistic fuzzy availability

S1

~cI11

~cI12



~cI1j



~cI1n

~ aI1

⋮ Si

⋮

⋮

⋮

~cIij

⋮ 

⋮

~cIi2

⋮ 

⋮

~cIi1

~cIin

~ aIi

⋮ Sm

⋮

⋮ ~cIm1

⋮ 

~cImj

⋮ 

⋮ ~cmn

⋮

~cIm1 ~bI 1

~bI 2



~bI j



~ bIm

Intuitionistic fuzzy demand

⋮

~ aIm Pm i¼1

~ aIi ¼

Pn j¼1

~ bIj

4.4 Existing Approach for Comparing Trapezoidal Intuitionistic Fuzzy Numbers

4.4

147

Existing Approach for Comparing Trapezoidal Intuitionistic Fuzzy Numbers

It is well-known fact that the optimal solution of a crisp linear programming problem will be that feasible solution corresponding to which the value of the objective function will be minimum. On the same direction, an intuitionistic fuzzy optimal solution of the intuitionistic fully fuzzy linear programming problem (4.3.1) will be that intuitionistic fuzzy feasible solution corresponding to which the value of the objective function of the intuitionistic fully fuzzy linear programming problem (4.3.1) will be minimum. Since, in case of intuitionistic fully fuzzy linear programming problem (4.3.1), the value of the objective function, corresponding to an intuitionistic fuzzy feasible solution, will be a trapezoidal intuitionistic fuzzy number. Therefore, to find the intuitionistic fuzzy optimal solution of the intuitionistic fully fuzzy linear programming problem (4.3.1), there is a need to find the minimum of trapezoidal intuitionistic fuzzy numbers, i.e., there is a need to compare the trapezoidal intuitionistic fuzzy numbers. In this section, the approach for comparing the trapezoidal intuitionistic fuzzy numbers, used in Ebrahimnejad and Verdegay’s approach [1], has been discussed.     e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a0 ; a0 ; a0 and B e I ¼ b1 ; b2 ; b3 ; b4 ; b01 ; b02 ; b03 ; b04 be Let A 1 2 3 4 two trapezoidal intuitionistic fuzzy numbers. Then,    I eI [ H B eI B e I if H A e , (i) A    I e I \H B eI B e I if H A e , (ii) A    I eI  B eI ¼ H B e I if H A e , (iii) A where   ða þ a þ a þ a Þ þ a0 þ a0 þ a0 þ a0  I  ðb1 þ b2 þ b3 þ b4 Þ þ ðb01 þ b02 þ b03 þ b04 Þ e I ¼ 1 2 3 4 ð 1 2 3 4 Þ and H B e ¼ H A . 8 8

4.5

Ebrahimnejad and Verdegay’s Approach for Solving Balanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problems

The aim of this chapter is to point out some limitations and a drawback of Ebrahimnejad and Verdegay’s approach [1]. Since, to achieve this aim, there is a need to discuss Ebrahimnejad and Verdegay’s approach [1]. Therefore, a brief review of Ebrahimnejad and Verdegay’s approach [1] has been presented in this section. Ebrahimnejad and Verdegay [1] proposed the following approach for solving balanced fully trapezoidal intuitionistic fuzzy transportation problems.

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Step 1: Transform the intuitionistic fully fuzzy linear programming problem (4.3.1) into its equivalent intuitionistic fully fuzzy linear programming problem (4.5.1) by replacing the parameters ~cIij , ~xIij , ~aIi and ~bIj with the trapezoidal intuitionistic fuzzy 







numbers cij;1 ; cij;2 ; cij;3 ; cij;4 ; c0ij;1 ; c0ij;2 ; c0ij;3 ; c0ij;4 , xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;2 ; x0ij;3 ; x0ij;4 ,     ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; a0i;2 ; a0i;3 ; a0i;4 and bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; b0j;2 ; b0j;3 ; b0j;4 , respectively, Intuitionistic fully fuzzy linear programming problem (4.5.1) h   n cij;1 ; cij;2 ; cij;3 ; cij;4 ; c0ij;1 ; c0ij;2 ; c0ij;3 ; c0ij;4  Minimize m i¼1 j¼1  i xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;2 ; x0ij;3 ; x0ij;4 Subject to   nj¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;2 ; x0ij;3 ; x0ij;4   ¼ ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; a0i;2 ; a0i;3 ; a0i;4 ; i ¼ 1; 2; . . .; m;   0 0 0 0 m x ; x ; x ; x ; x ; x ; x ; x ij;1 ij;2 ij;3 ij;4 i¼1 ij;1 ij;2 ij;3 ij;4   0 ¼ bj;1 ; bj;2 ; bj;3 ; bi;4 ; bj;1 ; b0j;2 ; b0j;3 ; b0j;4 ; j ¼ 1; 2; . . .; n;   xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;2 ; x0ij;3 ; x0ij;4 is a non-negative trapezoidal intuitionistic fuzzy number, i ¼ 1; 2; . . .; m, j ¼ 1; 2; . . .; n.

    Step 2: Using the relation, a1 ; a2 ; a3 ; a4 ; a01 ; a02 ; a03 ; a04 ¼ b1 ; b2 ; b3 ; b4 ; b01 ; b02 ; b03 ; b04 ) a1 ¼ b1 ; a2 ¼ b2 ; a3 ¼ b3 ; a4 ¼ b4 ; a01 ¼ b01 ; a02 ¼ b02 ; a03 ¼ b03 ; a04 ¼ b04 and   using the relation a1 ; a2 ; a3 ; a4 ; a01 ; a02 ; a03 ; a04 is a non-negative trapezoidal intuitionistic fuzzy number ) a01  0, a1  a01  0, a02  a1  0, a2  a02  0, a3  a2  0, a03  a3  0, a4  a03  0, a04  a4  0, transform the intuitionistic fully fuzzy linear programming problem (4.5.1) into its equivalent intuitionistic fuzzy linear programming problem (4.5.2). Intuitionistic fuzzy linear programming problem (4.5.2) h   n Minimize m cij;1 ; cij;2 ; cij;3 ; cij;4 ; c0ij;1 ; c0ij;2 ; c0ij;3 ; c0ij;4  i¼1 j¼1  i xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;2 ; x0ij;3 ; x0ij;4

4.5 Ebrahimnejad and Verdegay’s Approach for Solving …

149

Subject to n X j¼1 n X

xij;1 ¼ ai;1 ; x0ij;1 ¼ a0i;1 ;

j¼1 m X i¼1 m X i¼1

n X j¼1 n X

xij;2 ¼ ai;2 ; x0ij;2 ¼ a0i;2 ;

j¼1

xij;1 ¼ bj;1 ; x0ij;1 ¼ b0j;1 ;

m X i¼1 m X i¼1

n X j¼1 n X

xij;3 ¼ ai;3 ; x0ij;3 ¼ a0i;3 ;

j¼1

xij;2 ¼ bj;2 ; x0ij;2 ¼ b0j;2 ;

m X i¼1 m X

n X j¼1 n X

xij;4 ¼ ai;4 ; x0ij;4 ¼ a0i;4 ;

i ¼ 1; 2; . . .; m;

j¼1

xij;3 ¼ bj;3 ; x0ij;3 ¼ b0j;3 ;

i¼1

m X i¼1 m X

xij;4 ¼ bj;4 ; x0ij;4 ¼ b0j;4 ;

j ¼ 1; 2; . . .; n;

i¼1

x0ij;1  0, xij;1  x0ij;1  0, x0ij;2  xij;1  0, xij;2  x0ij;2  0, xij;3  xij;2  0,  xij;3  0, xij;4  x0ij;3  0, x0ij;4  xij;4  0, i ¼ 1; 2; . . .; m, j ¼ 1; 2; . . .; n:

x0ij;3

Step 3: Using of two non-negative  the multiplication  trapezoidal 0intuitionistic  fuzzy and b1 ; b2 ; b3 ; b4 ; b1 ; b02 ; b03 ; b04 i.e., numbers a1 ; a2 ; a3 ; a4 ; a01 ; a02 ; a03 ; a04     0 0 0 0 0 0 0 0 a1 ; a2 ; a3 ; a4 ; a1 ; a2 ; a3 ; a4  b1 ; b2 ; b3 ; b4 ; b1 ; b2 ; b3 ; b4 ¼ ða1 b1 ; a2 b2 ; a3 b3 ; a4 b4 ; a01 b01 ; a02 b02 ; a03 b03 ; a04 b04 Þ, transform the intuitionistic fuzzy linear programming problem (4.5.2) into its equivalent intuitionistic fuzzy linear programming problem (4.5.3). Intuitionistic fuzzy linear programming problem (4.5.3) h  i n 0 0 0 0 0 0 0 0 Minimize m  c x ; c x ; c x ; c x ; c x ; c x ; c x ; c x ij;1 ij;1 ij;2 ij;2 ij;3 ij;3 ij;4 ij;4 i¼1 j¼1 ij;1 ij;1 ij;2 ij;2 ij;3 ij;3 ij;4 ij;4

Subject to Constraints of the intuitionistic fuzzy linear programming problem (4.5.2). Step 4: Using the approach for comparing trapezoidal intuitionistic fuzzy numbers, discussed in Sect. 4.3, transform the intuitionistic fuzzy linear programming problem (4.5.3) into its equivalent crisp linear programming problem (4.5.4) and hence into its equivalent crisp linear programming problem (4.5.5). Crisp linear programming problem (4.5.4) Minimize h  i n 0 0 0 0 0 0 0 0 m  H c x ; c x ; c x ; c x ; c x ; c x ; c x ; c x ij;1 ij;1 ij;2 ij;2 ij;3 ij;3 ij;4 ij;4 ij;1 ij;1 ij;2 ij;2 ij;3 ij;3 ij;4 ij;4 i¼1 j¼1 Subject to Constraints of the intuitionistic fuzzy linear programming problem (4.5.2).

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Crisp linear programming problem (4.5.5) "

m X n  1X cij;1 xij;1 þ cij;2 xij;2 þ cij;3 xij;3 þ cij;4 xij;4 þ c0ij;1 x0ij;1 8 i¼1 j¼1 i þ c0ij;2 x0ij;2 þ c0ij;3 x0ij;3 þ c0ij;4 x0ij;4

Minimize

Subject to Constraints of the intuitionistic fuzzy linear programming problem (4.5.2). n o Step 5: Find a crisp optimal solution x0ij;1 ; x0ij;2 ; x0ij;3 ; x0ij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 of the crisp linear programming problem (4.5.5). Step 6: Using the crisp optimal solution, obtained in Step 5, find n othe intuitionistic 0 0 0 0 fuzzy optimal solution xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 and the intuitionistic fuzzy optimal value   n 0 0 0 0 0 0 0 0 m i¼1 j¼1 cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 :

4.6

Limitations of Ebrahimnejad and Verdegay’s Approach

In this section, some limitations of Ebrahimnejad and Verdegay’s approach [1] have been pointed out. 1. To solve a real-life transportation problem, the opinion of two or more experts about the parameters is collected. Then, all the collected information is aggregated to obtain a single value of each parameter. Since, Ebrahimnejad and Verdegay’s approach [1] has been proposed by considering the assumption that the aggregated value of each parameter is available. Therefore, Ebrahimnejad and Verdegay’s approach [1] cannot be used to solve several real-life balanced fully trapezoidal intuitionistic fuzzy transportation problems. For example, Ebrahimnejad and Verdegay’s approach [1] cannot be used to solve the balanced fully trapezoidal intuitionistic fuzzy transportation problem considered in Example 4.6.1. Example 4.6.1 Let us consider a product needs to be supplied from two sources to two destinations. For the same purpose, the information about each parameter is collected from two experts. If

4.6 Limitations of Ebrahimnejad and Verdegay’s Approach

151

(i) Table 4.2 represents the intuitionistic fuzzy transportation cost, the intuitionistic fuzzy availability and the intuitionistic fuzzy demand provided by the first decision-maker. (ii) Table 4.3 represents the intuitionistic fuzzy transportation cost, the intuitionistic fuzzy availability and the intuitionistic fuzzy demand provided by the second decision-maker. Then, this balanced fully trapezoidal intuitionistic fuzzy transportation problem cannot be solved by Ebrahimnejad and Verdegay’s approach [1]. 2. Ebrahimnejad and Verdegay [1] have proposed their proposed approach for solving balanced fully trapezoidal intuitionistic fuzzy transportation problems. Therefore, Ebrahimnejad and Verdegay’s approach [1] cannot be used to solve unbalanced fully trapezoidal intuitionistic fuzzy transportation problems. One may think that in actual case, it is not a limitation as an unbalanced fully trapezoidal intuitionistic fuzzy transportation problem can be easily transformed into a balanced fully trapezoidal intuitionistic fuzzy transportation problem by introducing a dummy source and a dummy destination as in the case of an unbalanced crisp transportation problem. However, the following clearly indicates that the approach to transform an unbalanced crisp transportation problem into its equivalent balanced crisp transportation problem cannot be used to transform an unbalanced fully trapezoidal Table 4.2 Intuitionistic fuzzy data provided by the first decision-maker Sources S1 S2 Intuitionistic fuzzy demand

Destinations D1   10; 30; 40; 50; 5; 15; 45; 55   15; 30; 50; 80; 10; 20; 70; 90   40; 60; 70; 90; 30; 50; 80; 95

D2   25; 50; 60; 80; 10; 30; 70; 90   20; 40; 60; 80; 15; 35; 70; 85   10; 45; 55; 70; 5; 30; 60; 80

Intuitionistic fuzzy availability   20; 60; 70; 80; 15; 50; 75; 85   25; 45; 60; 70; 20; 40; 65; 80

Table 4.3 Intuitionistic fuzzy data provided by the second decision-maker Sources S1 S2 Intuitionistic fuzzy demand

Destinations D1   15; 35; 45; 55; 10; 20; 50; 60   20; 35; 55; 85; 15; 25; 75; 95   45; 65; 75; 95; 35; 55; 85; 100

D2   30; 55; 65; 85; 15; 35; 75; 95   25; 45; 65; 85; 20; 40; 75; 90   15; 50; 60; 75; 10; 35; 65; 85

Intuitionistic fuzzy availability   25; 65; 75; 85; 20; 55; 80; 90   30; 50; 65; 75; 25; 45; 70; 85

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intuitionistic fuzzy transportation problem into its equivalent balanced fully trapezoidal intuitionistic fuzzy transportation problem. Ebrahimnejad and Verdegay [1] have solved the balanced fully trapezoidal intuitionistic fuzzy transportation problem having two sources S1 , S2 and three destinations D1 , D2 and D3 by considering ~cI11 ¼ ð10; 20; 30; 40; 5; 15; 35; 45Þ; ~cI12 ¼ ð50; 60; 70; 90; 45; 55; 75; 95Þ; ~cI13 ¼ ð80; 90; 110; 120; 75; 85; 115; 125Þ; ~cI21 ¼ ð60; 70; 80; 90; 55; 65; 85; 95Þ; ~cI22 ¼ ð70; 80; 100; 120; 65; 75; 115; 125Þ; ~cI23 ¼ ð20; 30; 50; 60; 15; 25; 35; 65Þ; ~aI1 ¼ ð60; 80; 100; 120; 50; 70; 110; 130Þ; ~aI2 ¼ ð40; 60; 80; 100; 30; 50; 90; 110Þ; ~bI ¼ ð30; 50; 70; 90; 20; 40; 80; 100Þ; ~bI ¼ ð20; 30; 40; 50; 15; 25; 45; 55Þ; 1 2 ~bI ¼ ð50; 60; 70; 80; 45; 55; 75; 85Þ: 3 If it is assumed that there are only two destinations instead of three destinations. Then, the modified fully trapezoidal intuitionistic fuzzy transportation problem can be represented by Table 4.4. It is obvious that for the modified fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.4, 2i¼1 ~ ai ¼ ð100; 140; 180; 220; 2 ~ 80; 120; 200; 240Þ and j¼1 bj ¼ ð50; 80; 110; 140; 35; 65; 125; 155Þ. Since 2 ~ai 6¼ 2 ~bj . So, the modified fully trapezoidal intuitionistic fuzzy transi¼1

j¼1

portation problem is an unbalanced fully trapezoidal intuitionistic fuzzy transportation problem. Now, according to the existing approach for transforming an unbalanced crisp transportation problem into its equivalent balanced crisp transportation problem, there is a need to check that 2i¼1 ~ai 2j¼1 ~bj or 2i¼1 ~ bj . ai 2j¼1 ~

Table 4.4 Fully trapezoidal intuitionistic fuzzy transportation problem Sources S1 S2 Intuitionistic fuzzy demand

Destinations D1   10; 20; 30; 40; 5; 15; 35; 45   60; 70; 80; 90; 55; 65; 85; 95   30; 50; 70; 90; 20; 40; 80; 100

D2   50; 60; 70; 90; 45; 55; 75; 95   70; 80; 100; 120; 65; 75; 115; 125   20; 30; 40; 50; 15; 25; 45; 55

Intuitionistic fuzzy availability   60; 80; 100; 120; 50; 70; 110; 130   40; 60; 80; 100; 30; 50; 90; 110

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153

    It can be easily verified that H 2i¼1 ~ai ¼ 160 and H 2j¼1 ~ bj ¼ 95. Since     H 2i¼1 ~ai [ H 2j¼1 ~bj . Therefore, according to the existing approach for comparing trapezoidal intuitionistic fuzzy numbers, discussed in Sect. 4.3, 2i¼1 ~ai 2j¼1 ~bj . Hence, according to the existing approach for transforming an unbalanced crisp transportation problem into a balanced crisp transportation problem, there is a  need to add a dummy destination having intuitionistic fuzzy  2  demand  ~ai  2 ~bj ¼ ð100; 140; 180; 220; 80; 120; 200; 240Þ ð50; 80; i¼1

j¼1

110; 140; 35; 65; 125; 155Þ ¼ ð40; 30; 100; 170; 75; 5; 135; 205Þ, i.e., according to the existing approach for transforming an unbalanced crisp transportation problem into its equivalent crisp transportation problem, the transformed fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.5, will be a balanced fully trapezoidal intuitionistic fuzzy transportation problem. While, in actual case, the transformed fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.5, is not a balanced fully trapezoidal intuitionistic fuzzy transportation problem as the total intuitionistic fuzzy availability 2i¼1 ~ai ¼ ð60; 80; 100; 120; 50; 70; 110; 130Þ ð40; 60; 80; 100; 30; 50; 90; 110Þ ¼ ð100; 140; 180; 220; 80; 120; 200; 240Þ is not equal to the total intuitionistic fuzzy demand 3j¼1 ~bj ¼ ð30; 50; 70; 90; 20; 40; 80; 100Þ  ð20; 30; 40; 50; 15; 25; 45; 55Þ  ð40; 30; 100; 170; 75; 5; 135; 205Þ ¼ ð10; 110; 210; 310; 40; 60; 257; 360Þ Furthermore, as the intuitionistic fuzzy demand of the dummy destination D3 does not have any physical meaning as it is not a non-negative trapezoidal intuitionistic fuzzy number.

4.7

Drawback of Ebrahimnejad and Verdegay’s Approach

Ebrahimnejad and Verdegay [1] have used an existing approach for comparing trapezoidal intuitionistic fuzzy numbers, discussed in Sect. 4.3, in their proposed approach. However, the following example clearly indicates that it is inappropriate to use this approach. It is obvious that for the fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.6, 2i¼1 ~ai ¼ ð100; 140; 180; 220; 80; 120; 200; 240Þ and 2j¼1 ~bj ¼ ð110; 155; 165; 210; 80; 105; 205; 250Þ. Since 2 ~ai 6¼ 2 b~j . So, the considered fully trapezoidal intuitionistic fuzzy transi¼1

j¼1

portation problem is an unbalanced fully trapezoidal intuitionistic fuzzy transportation problem.

Intuitionistic fuzzy demand

S2

S1

Sources

Destinations D1   10; 20; 30; 40; 5; 15; 35; 45   60; 70; 80; 90; 55; 65; 85; 95   30; 50; 70; 90; 20; 40; 80; 100 D2   50; 60; 70; 90; 45; 55; 75; 95   70; 80; 100; 120; 65; 75; 115; 125   20; 30; 40; 50; 15; 25; 45; 55

Table 4.5 Transformed fully trapezoidal intuitionistic fuzzy transportation problem D3   0; 0; 0; 0; 0; 0; 0; 0   0; 0; 0; 0; 0; 0; 0; 0   40; 30; 100; 170; 75; 5; 135; 205

 60; 80; 100; 120; 50; 70; 110; 130   40; 60; 80; 100; 30; 50; 90; 110



Intuitionistic fuzzy availability

154 4 JMD Approach for Solving Unbalanced Fully …

4.7 Drawback of Ebrahimnejad and Verdegay’s Approach

155

Table 4.6 Unbalanced fully trapezoidal intuitionistic fuzzy transportation problem Sources S1 S2 Intuitionistic fuzzy demand

Destinations D1   10; 20; 30; 40; 5; 15; 35; 45   60; 70; 80; 90; 55; 65; 85; 95   55; 85; 95; 130; 45; 75; 105; 135

D2   50; 60; 70; 90; 45; 55; 75; 95   70; 80; 100; 120; 65; 75; 115; 125   30; 70; 90; 100; 15; 65; 95; 135

Intuitionistic fuzzy availability   60; 80; 100; 125; 50; 70; 110; 130   40; 60; 80; 110; 30; 50; 90; 140

Now, according to the existing approach for transforming an unbalanced crisp transportation problem into its equivalent balanced crisp transportation problem, there is a need to check that 2i¼1 ~ai 2j¼1 ~bj or 2i¼1 ~ bj . ai 2j¼1 ~    2  1325 bj ¼ 1325 It can be easily verified that H i¼1 ~ai ¼ 8 and H 2j¼1 ~ 8 . Since    2  H i¼1 ~ai ¼ H 2j¼1 ~bj ¼ 1325 8 . Therefore, according to the existing approach for comparing trapezoidal intuitionistic fuzzy numbers, discussed in Sect. 4.3, 2i¼1 ~ai ¼ 2j¼1 ~bj . Hence, according to the existing approach for comparing trapezoidal intuitionistic fuzzy numbers, discussed in Sect. 4.3, the fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.6, is a balanced fully trapezoidal intuitionistic fuzzy transportation problem. While, in actual case, it is an unbalanced fully trapezoidal intuitionistic fuzzy transportation problem as 2i¼1 ~ai 6¼ 2j¼1 ~bj .

4.8

Reasons for the Occurrence of the Limitations

  e 1 ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a0 ; a0 ; a0 and In Sect. 4.6, it has been assumed that if A 1 2 3 4   e 2 ¼ b1 ; b2 ; b3 ; b4 ; b0 ; b0 ; b0 ; b0 are two distinct trapezoidal intuitionistic fuzzy A 1 2  3 4   e1 [ H A e 2 . Then, on adding trapezoidal intuitionistic numbers such that H A     e 2 in A e 1 , i.e., A e2  A e1  A e2 ¼ A e1  A e 2 , it will be equal to A e 1. fuzzy number A     e 1 \H A e 2 then on adding trapezoidal intuitionistic fuzzy Similarly, if H A     e 1 in A e 2 , i.e., A e1  A e2  A e1 ¼ A e2  A e 1 , it will be equal to A e 2. number A However, the following example clearly indicates that in actual case this result will not hold.

156

4 JMD Approach for Solving Unbalanced Fully …

e 1 ¼ ð10; 20; 30; 40; For the trapezoidal intuitionistic fuzzy numbers A e 2 ¼ ð15; 25; 35; 45; 10; 20; 40; 50Þ, the condition 5;15; 35; A  45Þ and e 2 is satisfying. So, according to condition (ii), on adding e 1 \H A H A     e 1 ¼ b1  a4 ; b2  a3 ; b3  a2 ; b4  a1 ; b01  a04 ; b02  a03 ; b03  a02 ; b04  a01 e2  A A e 1 , the trapezoidal intuitionistic fuzzy ¼ ð35; 15; 5; 25; 45; 25; 15; 35Þ in A   e 2 should be obtained. While A e1  A e2  A e 1 ¼ ð10; 20; 30; 40; 5; number A 15; 35; 45Þ ð35; 15; 5; 25; 45; 25; 15; 35Þ ¼ ð25; 5; 35; 65; 40; 10; e 2. 50; 80Þ 6¼ A

4.9

Reasons for the Occurrence of the Drawback

The drawback in Ebrahimnejad and Verdegay’s approach [1], discussed in Sect. 4.7, is occurring due to the following reason. It is obvious from Sect. 4.3 that Ebrahimnejad and Verdegay [1] have assumed     e 2 ¼ b1 ; b2 ; b3 ; b4 ; b0 ; b0 ; b0 ; b0 are e 1 ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a0 ; a0 ; a0 and A that if A 1 1 3 4 1 2 3 4 e e two trapezoidal   intuitionistic   fuzzy numberssuch  that A 1 6¼ A 2 , then either the e 2 or the relation H A e 1 \H A e 2 will hold. While, in e1 [ H A relation H A e actual case, theremay  existdistinct  trapezoidal intuitionistic fuzzy numbers A 1 and e1 ¼ H A e 2 , e.g., the trapezoidal intuitionistic e 2 such that H A e 2 but A e 1 6¼ A A e 1 ¼ ð10; 20; 30; 40; 5; 15; 35; 45Þ and A e 1 ¼ ð8; 22; 25; 35; 5; 15; fuzzy numbers A e e 40; 50Þ are two trapezoidal intuitionistic    fuzzy  numbers such that A 1 6¼ A 2 but it e1 ¼ H A e 2 ¼ 25. can be easily verified that H A

4.10

DAUGHTER Approach for Comparing Trapezoidal Intuitionistic Fuzzy Numbers

It is obvious from Sect. 4.9 that it is not appropriate to use the existing approach discussed in Sect. 4.3, for comparing trapezoidal intuitionistic fuzzy numbers. In this section, a new approach (named as DAUGHTER approach) has been proposed for comparing trapezoidal intuitionistic fuzzy numbers. Using the proposed DAUGHTER approach,   the trapezoidal intuitionistic fuzzy numbers a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 and a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ can be compared as follows:

4.10

DAUGHTER Approach for Comparing Trapezoidal Intuitionistic …

157

  Step 1: Check that D a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ Dða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or D a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \Dða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or D a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ Dða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ, where   D a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ a11 þ a12 þ a13 þ a14 þ a011 þ a012 þ a013   þ a014 and D a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 ¼ a21 þ a22 þ a23 þ a24 þ a021 þ a022 þ a023 þ a024 .    Case (i): If D a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ D a21 ; a22 ; a23 ; a24 ; a021 ; a022 ;    a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ    Case (ii): If D a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \ D a21 ; a22 ; a23 ; a24 ; a021 ; a022 ;    a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ.    Case (iii): If D a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ D a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ. Then, go to Step 2.    Step 2: Check that A a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ A a21 ; a22 ; a23 ; a24 ; a021 ;    a022 ; a023 ; a024 Þ or A a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \ A a21 ; a22 ; a23 ; a24 ; a021 ; a022 ;    a023 ; a024 Þ or A a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ A a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ, where   A a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ a11 þ a12 þ a13 þ a14 þ a012 þ a013 þ a014   and A a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 ¼ a21 þ a22 þ a23 þ a24 þ a022 þ a023 þ a024 .    Case (i): If A a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ A a21 ; a22 ; a23 ; a24 ; a021 ; a022 ;    a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ    Case (ii): If A a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \ A a21 ; a22 ; a23 ; a24 ; a021 ; a022 ;    a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ.    Case (iii): If A a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ A a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ. Then, go to Step 3.   Step 3: Check that U a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ U ða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or U a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \ U ða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or U a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ U ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ,

158

4 JMD Approach for Solving Unbalanced Fully …

where   U a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ a12 þ a13 þ a14 þ a012 þ a013 þ a014 and   U a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 ¼ a22 þ a23 þ a24 þ a022 þ a023 þ a024 .   [ U ða21 ; a22 ; a23 ; a24 ; Case (i): If U a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014   0 0 0 0 0 0 0 0 a21 ; a22 ; a23 ; a24 Þ. Then, a11 ; a12 ; a13 ; a14 ; a11 ; a12 ; a13 ; a14 ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ   \U ða21 ; a22 ; a23 ; a24 ; Case (ii): If U a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014   0 0 0 0 0 0 0 0 a21 ; a22 ; a23 ; a24 Þ. Then, a11 ; a12 ; a13 ; a14 ; a11 ; a12 ; a13 ; a14 ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ.   ¼ U ða21 ; a22 ; a23 ; a24 ; Case (iii): If U a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 a021 ; a022 ; a023 ; a024 Þ. Then, go to Step 4.   Step 4: Check that G a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ Gða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or G a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \ Gða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or G a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ Gða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ, where   G a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ a12 þ a13 þ a14 þ a013 þ a014 and Gða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ ¼ a22 þ a23 þ a24 þ a023 þ a024 .   [ Gða21 ; a22 ; a23 ; a24 ; Case (i) If G a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014   a021 ; a022 ; a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ   Case (ii) If G a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \Gða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ.   ¼ Gða21 ; a22 ; a23 ; a24 ; Case (iii) If G a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 a021 ; a022 ; a023 ; a024 Þ. Then, go to Step 5.   Step 5: Check that H a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ H ða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or H a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \H ða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or H a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ H ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ, where   H a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ a13 þ a14 þ a013 þ a014 and H ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ ¼ a23 þ a24 þ a023 þ a024 .   Case (i): If H a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ H ða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ

4.10

DAUGHTER Approach for Comparing Trapezoidal Intuitionistic …

159

  Case (ii): If H a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \H ða21 ; a22 ; a23 ; a24 ;   0 0 0 0 0 0 0 0 a21 ; a22 ; a23 ; a24 Þ. Then, a11 ; a12 ; a13 ; a14 ; a11 ; a12 ; a13 ; a14 ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ.   Case (iii): If H a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ H ða21 ; a22 ; a23 ; a24 ; 0 0 0 0 a21 ; a22 ; a23 ; a24 Þ. Then, go to Step 6.    Step 6: Check that T a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ T a21 ; a22 ; a23 ; a24 ; a021 ;   a022 ; a023 ; a024 Þ or T a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \T ða21 ; a22 ; a23 ; a24 ; a021 ;   a022 ; a023 ; a024 Þ or T a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ T ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ, where   T a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ a14 þ a013 þ a014 and T ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ ¼ a24 þ a023 þ a024 .   Case (i): If T a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ T ða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ   Case (ii): If T a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \T ða21 ; a22 ; a23 ; a24 ; a021 ;   a022 ; a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ.   Case (iii): If T a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ T ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ. Then, go to Step 7.   Step 7: Check that E a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ Eða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or E a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \Eða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or E a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ Eða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ, where    E a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ a14 þ a014 and E a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ ¼ a24 þ a024 .   Case (i) If E a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ Eða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ   \Eða21 ; a22 ; a23 ; a24 ; Case (ii) If E a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014   0 0 0 0 0 0 0 0 a21 ; a22 ; a23 ; a24 Þ. Then, a11 ; a12 ; a13 ; a14 ; a11 ; a12 ; a13 ; a14 ða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ.    Case (iii) If E a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ E a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ. Then, go to Step 8.

4 JMD Approach for Solving Unbalanced Fully …

160

  Step 8: Check that R a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ Rða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or R a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \Rða21 ; a22 ; a23 ; a24 ;   a021 ; a022 ; a023 ; a024 Þ or R a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ Rða21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ, where    R a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ a014 and R a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ ¼ a024 .    Case (i): If R a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 [ R a21 ; a22 ; a23 ; a24 ; a021 ;    a022 ; a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ    Case (ii): If R a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 \R a21 ; a22 ; a23 ; a24 ; a021 ;    a022 ; a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ.    Case (iii): If R a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ R a21 ; a22 ; a23 ; a24 ; a021 ;    a022 ; a023 ; a024 Þ. Then, a11 ; a12 ; a13 ; a14 ; a011 ; a012 ; a013 ; a014 ¼ a21 ; a22 ; a23 ; a24 ; a021 ; a022 ; a023 ; a024 Þ.

4.11

Mehar Representation of a Trapezoidal Intuitionistic Fuzzy Number

To overcome the second limitation of Ebrahimnejad and Verdegay’s approach [1], there is need to propose an approach for transforming an unbalanced fully trapezoidal intuitionistic fuzzy transportation problem into a balanced fully trapezoidal intuitionistic fuzzy transportation problem. It has been observed that to achieve this objective, there is a need to propose a new representation of a trapezoidal intuitionistic fuzzy number. Keeping the same in mind, in this section, a new representation of a trapezoidal intuitionistic fuzzy number (named as Mehar representation of a trapezoidal intuitionistic fuzzy number) has been proposed.   e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a0 ; a0 ; a0 be a trapezoidal intuitionistic fuzzy Let A 1 2 3 4 number, where a01  a1  a02  a2  a3  a03  a4  a04 . Then, Mehar representation  e I ¼ a0 ; b1 ; b2 ; of this trapezoidal intuitionistic fuzzy number will be A 1 b3 ; b4 ; b5 ; b6 ; b7 ÞM . where b1 ¼ a1  a01 , b2 ¼ a02  a1 , b6 ¼ a4  a03 , b7 ¼ a04  a4 .

b3 ¼ a2  a02 ,

b 4 ¼ a3  a2 ,

b5 ¼ a03  a3 ,

4.11

Mehar Representation of a Trapezoidal Intuitionistic Fuzzy Number

161

For example, the Mehar representation of the trapezoidal intuitionistic fuzzy e I ¼ ð15; 20  15; 25  e I ¼ ð20; 30; 40; 50; 15; 25; 45; 55Þ will be A number A 20; 30  25; 40  30; 45  40; 50  45; 55  50ÞM ¼ ð15; 5; 5; 5; 10; 5; 5; 5ÞM .

4.12

Multiplication of a Trapezoidal Intuitionistic Fuzzy Number in Its Existing Representation with a Trapezoidal Intuitionistic Fuzzy Number in Its Mehar Representation

In the next section, a new approach (named as JMD approach) has been proposed to solve unbalanced fully trapezoidal intuitionistic fuzzy transportation problems. In one of the steps of the proposed JMD approach, there is a need to find the multiplication of a trapezoidal intuitionistic fuzzy number in its existing form with a trapezoidal intuitionistic fuzzy number in its Mehar representation. Therefore, the same is  discussed in this section.   Let

cij;1 ; cij;2 ; cij;3 ; cij;4 ; c0ij;1 ; c0ij;2 ; c0ij;3 ; c0ij;4

and

xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;2 ;

x0ij;3 ; x0ij;4 Þ be two non-negative trapezoidal intuitionistic fuzzy numbers in its existing representation [1]. Then, using Definition 4.2.7,     cij;1 ; cij;2 ; cij;3 ; cij;4 ; c0ij;1 ; c0ij;2 ; c0ij;3 ; c0ij;4  xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;2 ; x0ij;3 ; x0ij;4   ¼ cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;2 ; c0ij;3 x0ij;3 ; c0ij;4 x0ij;4 : ð4:12:1Þ Furthermore, using Sect. 4.11, the trapezoidal intuitionistic fuzzy number, ~xij ¼  xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;2 ; x0ij;3 ; x0ij;4 in its Mehar representation can be written as   x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7 , 

M

where ¼ xij;1  x0ij;1 ) xij;1 ¼ x0ij;1 þ aij;1 . ¼ x0ij;2  xij;1 ) x0ij;2 ¼ xij;1 þ aij;2 ¼ x0ij;1 þ aij;1 þ aij;2 . ¼ xij;2  x0ij;2 ) xij;2 ¼ x0ij;2 þ aij;3 ¼ x0ij;1 þ aij;1 þ aij;2 þ aij;3 . ¼ xij;3  xij;2 ) xij;3 ¼ xij;2 þ aij;4 ¼ x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 . aij;5 ¼ x0ij;3  xij;3 ) x0ij;3 ¼ xij;3 þ aij;5 ¼ x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 . aij;6 ¼ xij;4  x0ij;3 ) xij;4 ¼ x0ij;3 þ aij;6 = x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 . (vii) aij;7 ¼ x0ij;4  xij;4 ) x0ij;4 ¼ xij;4 þ aij;7 ¼ x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 . (i) (ii) (iii) (iv) (v) (vi)

aij;1 aij;2 aij;3 aij;4

162

4 JMD Approach for Solving Unbalanced Fully …

the trapezoidal intuitionistic fuzzy number  Replacing  0 0 0 0 xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 , present in left-hand side of multiplication   (4.12.1), with its Mehar representation x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7 M

and the values of xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;2 ; x0ij;3 ; x0ij;4 , present in right-hand side of multiplication (4.12.1), with x0ij;1 þ aij;1 ; x0ij;1 þ aij;1 þ aij;2 þ aij;3 ; x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; x0ij;1 ; x0ij;1 þ aij;1 þ aij;2 ; x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 , respectively, the multiplication (4.12.1) is transformed into its equivalent multiplication (4.12.2). 

   cij;1 ; cij;2 ; cij;3 ; cij;4 ; c0ij;1 ; c0ij;2 ; c0ij;3 ; c0ij;4  x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7 M        0 0 0 ¼ cij;1 xij;1 þ aij;1 ; cij;2 xij;1 þ aij;1 þ aij;2 þ aij;3 ; cij;3 xij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ;     cij;4 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ aij;2 ;   c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ;   c0ij;4 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7

ð4:12:2Þ

4.13

Proposed JMD Approach

Mishra and Kumar [4] pointed out that the existing approach [3] cannot be used to solve unbalanced fully triangular intuitionistic fuzzy transportation problems (unbalanced transportation problems in which each parameter is represented by a triangular fuzzy number). To overcome this limitation of the existing approach [3], Mishra and Kumar [4] proposed a new approach (named as JMD approach) for transforming an unbalanced fully triangular intuitionistic fuzzy transportation problem into a balanced fully triangular intuitionistic fuzzy transportation problem. On the same direction, to overcome the limitations and to resolve the drawback of Ebrhimnejad and Verdegay’s approach [1], a new approach (named as JMD approach) has been proposed to solve unbalanced fully trapezoidal intuitionistic fuzzy transportation problems. The steps of the proposed JMD approach are as follows: Step 1: Check that the aggregated value of the intuitionistic fuzzy transportation cost, the intuitionistic fuzzy availability and the intuitionistic fuzzy demand, provided by all the decision-makers, is available or not. Case (i): If it is available then go to Step 2.

4.13

Proposed JMD Approach

163

Case (ii): If it is not available then using the existing trapezoidal intuitionistic fuzzy weighted averaging aggregation operator [2], find,   (i) The trapezoidal intuitionistic fuzzy number ~cij ¼ pk¼1 wk  ~ckij ¼ ! P P P Pp wk ckij;1 ; pk¼1 wk ckij;2 ; pk¼1 wk ckij;3 ; pk¼1 wk ckij;4 ; k¼1 Pp Pp Pp Pp representing the k k k k k¼1 wk cij;5 ; k¼1 wk cij;6 ; k¼1 wk cij;7 ; k¼1 wk cij;8 aggregated value of the intuitionistic fuzzy transportation cost for supplying the one unit quantity of the product from ith source to jth destination.   (ii) The trapezoidal intuitionistic fuzzy number ~ ai ¼ pk¼1 wk  ~ aki ¼ ! P P P Pp wk aki;1 ; pk¼1 wk aki;2 ; pk¼1 wk aki;3 ; pk¼1 wk aki;4 ; P P Pp Pk¼1 representing the p p p k k k k k¼1 wk ai;5 ; k¼1 wk ai;6 ; k¼1 wk ai;7 ; k¼1 wk ai;8 aggregated value of the intuitionistic fuzzy availability of the product at the ith source.   bkj ¼ (iii) The trapezoidal intuitionistic fuzzy number ~ bj ¼ pk¼1 wk  ~ ! P P P Pp wk bkj;1 ; pk¼1 wk bkj;2 ; pk¼1 wk bkj;3 ; pk¼1 wk bkj;4 ; k¼1 Pp Pp Pp Pp representing the k k k k k¼1 wk bj;5 ; k¼1 wk bj;6 ; k¼1 wk bj;7 ; k¼1 wk bj;8 aggregated value of the intuitionistic fuzzy demand the product at the jth destination. where (i) wk 2 ½0; 1 represents the normalized weight of the kth  decision-maker. k (ii) The trapezoidal intuitionistic fuzzy number ~cij ¼ ckij;1 ; ckij;2 ; ckij;3 ; ckij;4 ; ckij;5 ; ckij;6 ; ckij;7 ; ckij;8 Þ represents the intuitionistic fuzzy transportation cost for supplying the one unit quantity of the product from the ith source to the jth destination according to the kth decision-maker. 

~ki ¼ aki;1 ; aki;2 ; aki;3 ; aki;4 ; aki;5 ; (iii) The trapezoidal intuitionistic fuzzy number a aki;6 ; aki;7 ; aki;8 Þ represents the intuitionistic availability of the product at the ith source according to the kth decision-maker.  (iv) The trapezoidal intuitionistic fuzzy number ~ bkj ¼ bkj;1 ; bkj;2 ; bkj;3 ; bkj;4 ; bkj5; ; bkj;6 ; bkj;7 ; bkj;8 Þ represents the intuitionistic demand of the product at the jth destination according to the kth decision-maker. and go to Step 2. For example, if in Example 4.6.1, the normalized weights of the first and second decision-makers are 0.4 and 0.6, respectively. Then, the trapezoidal intuitionistic fuzzy numbers ~cij ; i ¼ 1; 2; j ¼ 1; 2; ~ai ; i ¼ 1; 2 and ~ bj ; j ¼ 1; 2, presented in Table 4.7, represent the aggregated intuitionistic fuzzy cost for supplying the one unit quantity of the product from the ith source to the jth destination, the aggregated

164

4 JMD Approach for Solving Unbalanced Fully …

Table 4.7 Aggregated intuitionistic data of decision-makers Source S1 S2 Intuitionistic fuzzy demand

Destination D1   13; 33; 43; 53; 8; 18; 48; 58   18; 33; 53; 83; 13; 23; 73; 93   43; 63; 73; 93; 33; 53; 83; 98

D2   28; 53; 63; 83; 13; 33; 73; 93   23; 43; 63; 83; 18; 38; 73; 88   15; 50; 65; 75; 8; 33; 58; 83

Intuitionistic fuzzy availability   23; 63; 73; 83; 18; 53; 78; 88   28; 48; 63; 73; 23; 43; 68; 83

intuitionistic fuzzy availability the product at the ith source and the aggregated intuitionistic fuzzy demand of the product at the jth destination, respectively.  Step 2: Transform the trapezoidal intuitionistic fuzzy number ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; a0i;2 ; a0i;3 ; a0i;4 Þ, representing the intuitionistic fuzzy availability of the product at  the ith source, and the trapezoidal intuitionistic fuzzy number bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; b0j;2 ; b0j;3 ; b0j;4 Þ, representing the intuitionistic fuzzy demand of the product at the  jth destination, into their Mehar representations a0i;1 ; bi;1 ; bi;2 ; bi;3 ; bi;4 ; bi;5 ; bi;6 ;   bi;7 ÞM and b0j;1 ; cj;1 ; cj;2 ; cj;3 ; cj;4 ; cj;5 ; cj;6 ; cj;7 , respectively, M

Step  3: Find the total intuitionistic fuzzy availability of the product, i.e., P Pm Pm m m 0 0 i¼1 ai;1 ; bi;1 ; bi;2 ; bi;3 ; bi;4 ; bi;5 ; bi;6 ; bi;7 ¼ i¼1 ai;1 ; i¼1 bi;1 ; i¼1 bi;2 ; M Pm Pm Pm Pm P m i¼1 bi;3 ; i¼1 bi;4 ; i¼1 bi;5 ; i¼1 bi;6 ; i¼1 bi;7 ÞM and the total intuitionistic P   n 0 ¼ demand of the product, i.e., nj¼1 b0j;1 ; cj;1 ; cj;2 ; cj;3 ; cj;4 ; cj;5 ; cj;6 ; cj;7 j¼1 bj;1 ; M Pn Pn Pn Pn Pn Pn Pn j¼1 cj;1 ; j¼1 cj;2 ; j¼1 cj;3 ; j¼1 cj;4 ; j¼1 cj;5 ; j¼1 cj;6 ; j¼1 cj;6 ÞM : P Pm Pm Pm Pm Pm m 0 Step 4: Check that i¼1 ai;1 ; i¼1 bi;1 ; i¼1 bi;2 ; i¼1 bi;3 ; i¼1 bi;4 ; i¼1 bi;5 ; P Pm P P P P Pm n n n n n 0 ¼ i¼1 bi;6 ; i¼1 bi;7 ÞM j¼1 bj;1 ; j¼1 cj;1 ; j¼1 cj;2 ; j¼1 cj;3 ; j¼1 cj;4 ; Pn Pn Pn Pm 0 Pm Pn 0 Pm ai;1 ¼ i¼1 ai;1 ¼ j¼1 bj;1 , b j¼1 cj;5 ; j¼1 cj;6 ; j¼1 cj;6 ÞM ; i.e., P Pm Pn Pi¼1 Pn Pm Pni¼1 i;1 m ¼ nj¼1 cj;1 , b ¼ c , b ¼ c , b ¼ c i;4 j¼1 j;4 , Pn i¼1 i;2Pm j¼1 j;2Pn i¼1 i;3 Pm j¼1 j;3 Pi¼1 Pm n i¼1 bi;5 ¼ j¼1 cj;5 , i¼1 bi;6 ¼ j¼1 cj;6 and i¼1 bi;7 ¼ j¼1 cj;7 or not. P Pm Pm Pm Pm Pm m 0 Case (i): If i¼1 ai;1 ; i¼1 bi;1 ; i¼1 bi;2 ; i¼1 bi;3 ; i¼1 bi;4 ; i¼1 bi;5 ; P Pm Pm P P P P n n n n n 0 ¼ i¼1 bi;6 ; i¼1 bi;7 ÞM j¼1 bj;1 ; j¼1 cj;1 ; j¼1 cj;2 ; j¼1 cj;3 ; j¼1 cj;4 ; Pn Pn Pn j¼1 cj;5 ; j¼1 cj;6 ; j¼1 cj;6 ÞM . Then, the considered fully trapezoidal intuitionistic fuzzy transportation problem is a balanced fully trapezoidal intuitionistic fuzzy transportation problem. Go to Step 6.

4.13

Proposed JMD Approach

165

P Pm Pm Pm Pm Pm m 0 Case (ii): If i¼1 ai;1 ; i¼1 bi;1 ; i¼1 bi;2 ; i¼1 bi;3 ; i¼1 bi;4 ; i¼1 bi;5 ; P Pm Pm P P P P n n n n n 0 6¼ i¼1 bi;6 ; i¼1 bi;7 ÞM j¼1 bj;1 ; j¼1 cj;1 ; j¼1 cj;2 ; j¼1 cj;3 ; j¼1 cj;4 ; Pn Pn Pn j¼1 cj;5 ; j¼1 cj;6 ; j¼1 cj;6 ÞM . Then, the considered fully trapezoidal intuitionistic fuzzy transportation problem is an unbalanced fully trapezoidal intuitionistic fuzzy transportation problem. Go to Step 5. Step 5: Add a dummy source Sm þ 1 having intuitionistic fuzzy availability   and qðm þ 1Þ;1 ; qðm þ 1Þ;2 ; qðm þ 1Þ3 ; qðm þ 1Þ4 ; qðm þ 1Þ5 ; qðm þ 1Þ6 ; qðm þ 1Þ7 ; qðm þ 1Þ8 M

consider the cost for supplying the one unit quantity of the product from the dummy source Sm þ 1 to all the destinations as a trapezoidal intuitionistic fuzzy number e 0 ¼ ð0; 0; 0; 0; 0; 0ÞM . where ( qðm þ 1Þ;1 ¼ max 0; ( qðm þ 1Þ;2 ¼ max 0; ( qðm þ 1Þ;3 ¼ max 0; ( qðm þ 1Þ;4 ¼ max 0; ( qðm þ 1Þ;5 ¼ max 0; ( qðm þ 1Þ;6 ¼ max 0; ( qðm þ 1Þ;7 ¼ max 0;

n X

b0j;1



j¼1

i¼1

n X

m X

cj;1 

j¼1

i¼1

n X

m X

cj;2 

j¼1

i¼1

n X

m X

cj;3 

j¼1

i¼1

n X

m X

cj;4 

j¼1

i¼1

n X

m X

cj;5 

j¼1

i¼1

n X

m X

cj;6 

j¼1

( qðm þ 1Þ;8 ¼ max 0;

m X

n X j¼1

cj;7 

) a0i;1

; )

bi;1 ; ) bi;2 ; ) bi;3 ; ) bi;4 ; ) bi;5 ; ) bi;6 ;

i¼1

Xm i¼1

) bi;7 :

166

4 JMD Approach for Solving Unbalanced Fully …

Also, add a dummy destination Dn þ 1 having intuitionistic fuzzy dummy demand   sðn þ 1Þ1 ; sðn þ 1Þ2 ; sðn þ 1Þ3 ; sðn þ 1Þ4 ; sðn þ 1Þ5 ; sðn þ 1Þ6 ; sðm þ 1Þ7 ; sðm þ 1Þ8 M and consider the cost for supplying the one unit quantity of the product from all the sources to the dummy destination Dn þ 1 as a trapezoidal intuitionistic fuzzy number e 0 ¼ ð0; 0; 0; 0; 0; 0; 0; 0Þ. where ( sðn þ 1Þ;1 ¼ max 0; ( sðn þ 1Þ;2 ¼ max 0; ( sðn þ 1Þ;3 ¼ max 0; ( sðn þ 1Þ;4 ¼ max 0; ( sðn þ 1Þ;5 ¼ max 0;

m X

a0i;1 

i¼1

j¼1

m X

n X

bi;1 

i¼1

j¼1

m X

n X

bi;2 

i¼1

j¼1

m X

n X

bi;3 

i¼1

j¼1

m X

n X

bi;4 

i¼1

( sðn þ 1Þ;6 ¼ max

m X

sðn þ 1Þ;7 ¼ max 0; ( sðn þ 1Þ;8 ¼ max 0;

m X

n X

b0j;1 ; ) cj;1 ; ) cj;2 ; ) cj;3 ; ) cj;4 ; )

cj;5 ;

j¼1

bi;6 

n X

i¼1

j¼1

m X

n X

i¼1

)

j¼1

bi;5 

i¼1

(

n X

bi;7 

) cj;6 ; ) cj;7 :

j¼1

Step 6: Write the intuitionistic fully fuzzy linear programming problem (4.13.1) of the transformed balanced fully trapezoidal intuitionistic fuzzy transportation problem.

4.13

Proposed JMD Approach

167

Intuitionistic fully fuzzy linear programming problem (4.13.1) h   n Minimize m cij;1 ; cij;2 ; cij;3 ; cij;4 ; c0ij;1 ; c0ij;2 ; c0ij;3 ; c0ij;4  i¼1 j¼1   i x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7 M

Subject to   þ1 nj¼1 x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7 M   0 ¼ ai;1 ; bi;1 ; bi;2 ; bi;3 ; bi;4 ; bi;5 ; bi;6 ; bi;7 ; i ¼ 1; 2; . . .; m; M

  þ1 nj¼1 x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7 M   ¼ qi;1 ; qi;2 ; qi;3 ; qi;4 ; qi;5 ; qi;6 ; qi;7 :qi;8 M ; i ¼ m þ 1;   þ1 0 x ; a ; a ; a ; a ; a ; a ; a m ij;1 ij;2 ij;3 ij;4 ij;5 ij;6 ij;7 i¼1 ij;1 M   0 ¼ bj;1 ; cj;1 ; cj;2 ; cj;3 ; cj;4 ; cj;5 ; cj;6 ; cj;7 ; j ¼ 1; 2; . . .; n; M



 þ1 0 m x ; a ; a ; a ; a ; a ; a ; a i¼1 ij;1 ij;1 ij;2 ij;3 ij;4 ij;5 ij;6 ij;7 M   ¼ sj;1 ; sj;2 ; sj;3 ; sj;4 ; sj;5 ; sj;6 ; sj;7 ; sj;8 M ; j ¼ n þ 1;   x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7

M

is a non-negative trapezoidal intu-

itionistic fuzzy number. where (i) (ii) (iii)

aij;1 ¼ xij;1  x0ij;1 ; aij;2 ¼ x0ij;2  xij;1 ; aij;3 ¼ xij;2  x0ij;2 ; ai;4 ¼ xij;3

.  xij;2 ; aij;5 ¼ x0ij;3  xij;3 ; aij;6 ¼ xij;4  x0ij;3 ; aij;7 ¼ x0ij;4  xij;4 : bi;1 ¼ ai;1  a0i;1 ; bi;2 ¼ a0i;2  ai;1 ; bi;3 ¼ ai;2  a0i;2 ; bi;4 ¼ ai;3  ai;2 ; bi;5 ¼ a0i;3  ai;3 ; bi;6 ¼ ai4  a0i;3 ; bi;7 ¼ a0i;4  ai;4 cj;1 ¼ bj;1  b0j;1 ; cj;2 ¼ b0j;2  bj;1 ; cj;3 ¼ bj;2  b0;2 ; cj;4 ¼ bj;3  bj;2 ; cj;5 ¼ b0j;3  bj;3 ; cj;6 ¼ bj;4  b0j;3 ; cj;7 ¼ b0j;4  bj;4

.

.

Step 7: Using the multiplication (4.12.2), proposed is Sect. 4.12, transform the intuitionistic fully fuzzy linear programming problem (4.13.1) into its equivalent intuitionistic fully fuzzy linear programming problem (4.13.2).

4 JMD Approach for Solving Unbalanced Fully …

168

Intuitionistic fully fuzzy linear programming problem (4.13.2) h      n 0 0 Minimize m i¼1 j¼1 cij;1 xij;1 þ aij;1 ; cij;2 xij;1 þ aij;1 þ aij;2 þ aij;3 ;     cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ;     c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ aij;2 ; c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ;   c0ij;4 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7

Subject to Constraints of the intuitionistic fully fuzzy linear programming problem (4.13.1).    Pn Step 8: Using relation ni¼1 Ai1 ; Ai2 ; Ai3 ; Ai4 ; A0i;1 ; A0i;2 ; A0i;3 ; A0i;4 ¼ i¼1 Ai1 ; Pn Pn Pn Pn Pn Pn Pn 0 0 0 0 i¼1 Ai2 ; i¼1 Ai3 ; i¼1 Ai4 ; i¼1 Ai;1 ; i¼1 Ai;2 ; i¼1 Ai;3 ; i¼1 Ai;4 Þ, transform the intuitionistic fully fuzzy linear programming problem (4.13.2) into its equivalent intuitionistic fully fuzzy linear programming problem (4.13.3). Intuitionistic fully fuzzy linear programming problem (4.13.3) " Minimize

m þ 1 nX þ1 X

þ 1 nX þ1    mX  cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ aij;3 ;

i¼1 j¼1 m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX  cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ;



i¼1 j¼1 m þ 1 nX þ1 X

i¼1 j¼1

c0ij;1 x0ij;1 ;

m þ 1 nX þ1 X

i¼1 j¼1 m þ 1 nX þ1 X



 c0ij;2 x0ij;1 þ aij;1 þ aij;2 ;

i¼1 j¼1



 c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ;

i¼1 j¼1 m þ 1 nX þ1 X

  c0ij;4 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7

i¼1 j¼1

Subject to nX þ1

x0ij;1 ;

j¼1

¼



j¼1

¼

aij;1

j¼1

nX þ1

aij;2 ;

j¼1

nX þ1

aij;3 ;

nX þ1

j¼1

x0ij;1 ;

nX þ1 j¼1

aij;1

nX þ1 j¼1

aij;2 ;

nX þ1

aij;4 ;

j¼1

a0i;1 ; bi;1 ; bi;2 ; bi;3 ; bi;4 ; bi;5 ; bi;6 ; bi;7

nX þ1



nX þ1

aij;3 ;

j¼1

b0j;1 ; cj;1 ; cj;2 ; cj;3 ; cj;4 ; cj;5 ; cj;6 ; cj;7

j¼1



M

aij;5 ;

j¼1

 M

nX þ1

nX þ1

nX þ1

aij;6 ;

j¼1

nX þ1

! aij;7

j¼1

M

; i ¼ 1; 2; . . .; m;

aij;4 ;

nX þ1 j¼1

aij;5 ;

nX þ1 j¼1

; j ¼ 1; 2; . . .; n;

aij;6 ;

nX þ1 j¼1

! aij;7 M

4.13

Proposed JMD Approach

169

  x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7

is a non-negative trapezoidal intu-

M

itionistic fuzzy number. Step 9: Using the relation ða0 ; a1 ; a2 ; a3 ; a4 ; a5 ; a6 ; a7 ÞM ¼ ðb0 ; b1 ; b2 ; b3 ; b4 ; b5 ; b6 ; b7 ÞM ) a0 ¼ b0 , a1 ¼ b1 , a2 ¼ b2 , a3 ¼ b3 , a4 ¼ b4 , a5 ¼ b5 , a6 ¼ b6 , a7 ¼ b7 and the relation ða0 ; a1 ; a2 ; a3 ; a4 ; a5 ; a6 ; a7 ÞM is a non-negative trapezoidal intuitionistic fuzzy number ) a0  0, a1  0, a2  0; a3  0; a4  0; a5  0; a6  0; a7  0, transform the intuitionistic fully fuzzy linear programming problem (4.13.3) into its equivalent intuitionistic fuzzy linear programming problem (4.13.4). Intuitionistic fuzzy linear programming problem (4.13.4) " Minimize

m þ 1 nX þ1 X

þ 1 nX þ1    mX cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

i¼1 j¼1

 aij;3 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ



i¼1 j¼1

i¼1 j¼1

þ 1 nX þ1 m þ 1 nX þ1  X  mX aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

 aij;2 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ



i¼1 j¼1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7

i¼1 j¼1



Subject to nX þ1

x0ij;1 ¼ a0i;1 ;

j¼1

nX þ1 j¼1

bi;4 ;

nX þ1

nX þ1

aij;5 ¼ bi;5 ;

x0ij;1 ¼ qi;1 ;

nX þ1

nX þ1

aij;5 ¼ qi;6 ;

j¼1

cj;4 ;

nX þ1

m þ1 X

aij;1 ¼ qi;2

m þ1 X i¼1

aij;5 ¼ cj;5 ;

nX þ1

nX þ1

nX þ1

aij;2 ¼ qi;3 ; n X

nX þ1

aij;3 ¼ qi;4 ;

nX þ1

aij;4 ¼

j¼1

aij;7 ¼ qi;8 ; i ¼ m þ 1;

j¼1 m þ1 X

aij;2 ¼ cj;2 ;

i¼1

i¼1

aij;4 ¼

j¼1

j¼1

aij;6 ¼ qi;7 ;

aij;1 ¼ cj;1 ;

m þ1 X

nX þ1

aij;7 ¼ bi;7 ; i ¼ 1; 2; . . .; m;

j¼1

i¼1

aij;3 ¼ bi;3 ;

j¼1

j¼1

x0ij;1 ¼ b0j;1 ;

nX þ1 j¼1

aij;6 ¼ bi;6 ;

j¼1

i¼1

aij;2 ¼ bi;2 ;

j¼1

j¼1

m þ1 X

nX þ1 j¼1

j¼1

qi;5 ;

aij;1 ¼ bi;1

aij;6 ¼ cj;6 ;

m þ1 X i¼1

m þ1 X i¼1

aij;3 ¼ cj;3 ;

m þ1 X i¼1

aij;7 ¼ cj;7 ; j ¼ 1; 2; . . .; n;

aij;4 ¼

4 JMD Approach for Solving Unbalanced Fully …

170 m þ1 X

x0ij;1 ¼ sj;1 ;

i¼1

sj;5 ;

m þ1 X

aij;1 ¼ sj;2 ;

i¼1 m þ1 X

aij;5 ¼ sj;6 ;

m þ! X

aij;2 ¼ sj;3 ;

i¼1 m þ1 X

i¼1

aij;6 ¼ sj;7 ;

m þ1 X

aij;3 ¼ sj;4 ;

i¼1 m þ1 X

i¼1

m þ! X

aij;4 ¼

i¼1

aij;7 ¼ sj;8 ; j ¼ n þ 1;

i¼1

x0ij;1  0, aij;1  0, aij;2  0, aij;3  0, aij;4  0, aij;5  0, aij;6  0, aij;7  0, i ¼ 1; 2; . . .; m, j ¼ 1; 2; . . .; n. Step 10: Using the proposed DAUGHTER approach for comparing trapezoidal intuitionistic fuzzy numbers, transform the intuitionistic fuzzy linear programming problem (4.13.4) into its equivalent crisp multi-objective linear programming problem (4.13.5). Crisp multi-objective linear programming problem (4.13.5) "

m þ 1 nX þ1 X

Minimize D

þ 1 nX þ1    mX cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

i¼1 j¼1

 aij;3 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ



i¼1 j¼1

i¼1 j¼1

þ 1 nX þ1 m þ 1 nX þ1  X  mX aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

 aij;2 ;

m þ 1 nX þ1 X

i¼1 j¼1



þ 1 nX þ1  mX  c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ

i¼1 j¼1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 " Minimize A

m þ 1 nX þ1 X

i¼1 j¼1



þ 1 nX þ1    mX cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

i¼1 j¼1

i¼1 j¼1

þ 1 nX þ1 þ1 X    mX  mX nþ1 cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; c x0ij;1 þ aij;1 þ aij;3 ; ij;4 j¼1 i¼1 j¼1

 aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ;

i¼1

m þ 1 nX þ1 X

c0ij;1 x0ij;1 ;

i¼1 j¼1

m þ 1 nX þ1 X

 c0ij;2 x0ij;1 þ aij;1 þ

i¼1 j¼1

þ 1 nX þ1 þ 1 nX þ1    mX  mX c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ aij;2 ; i¼1 j¼1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7



i¼1 j¼1

4.13

Proposed JMD Approach

" Minimize U

m þ 1 nX þ1 X

171

þ 1 nX þ1    mX cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

i¼1 j¼1

 aij;3 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ



i¼1 j¼1

i¼1 j¼1

 aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ;

m þ 1 nX þ1 X

c0ij;1 x0ij;1 ;

m þ 1 nX þ1 X

i¼1 j¼1

 aij;2 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ



i¼1 j¼1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7

" Minimize G

m þ 1 nX þ1 X

 aij;3 ;

i¼1 j¼1



þ 1 nX þ1    mX cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

i¼1 j¼1 m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ



i¼1 j¼1

i¼1 j¼1

 aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ;

m þ 1 nX þ1 X

c0ij;1 x0ij;1 ;

i¼1 j¼1

 aij;2 ;

m þ 1 nX þ1 X

þ 1 nX þ1   mX c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ

m þ 1 nX þ1 X

 aij;3 ;

i¼1 j¼1



þ 1 nX þ1    mX cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

i¼1 j¼1 m þ 1 nX þ1 X

 c0ij;2 x0ij;1 þ aij;1 þ

i¼1 j¼1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 Minimize H

m þ 1 nX þ1 X



i¼1 j¼1

"

 c0ij;2 x0ij;1 þ aij;1 þ

i¼1 j¼1

þ 1 nX þ1   mX cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ



i¼1 j¼1

i¼1 j¼1

þ 1 nX þ1 m þ 1 nX þ1  X  mX aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

 aij;2 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ



i¼1 j¼1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7



i¼1 j¼1

4 JMD Approach for Solving Unbalanced Fully …

172

"

m þ 1 nX þ1 X

Minimize T

þ 1 nX þ1    mX cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

i¼1 j¼1

 aij;3 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ



i¼1 j¼1

i¼1 j¼1

 aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ;

m þ 1 nX þ1 X

c0ij;1 x0ij;1 ;

i¼1 j¼1

 aij;2 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 Minimize E

 c0ij;2 x0ij;1 þ aij;1 þ



i¼1 j¼1

"

m þ 1 nX þ1 X

mP þ 1 nP þ1 i¼1 j¼1

i¼1 j¼1



   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;2 ; c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 " Minimize R

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

  mP   þ 1 nP þ1 þ 1 nP þ1  mP aij;2 ; c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ þ aij;ij i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7

Subject to Constraints of the intuitionistic fuzzy linear programming problem (4.13.4). Step 11: Solve the crisp linear programming problem (4.13.6) and check that a unique crisp optimal solution exists for the crisp linear programming problem (4.13.6) or not.

4.13

Proposed JMD Approach

173

Crisp linear programming problem (4.13.6) " Minimize D

m þ 1 nX þ1 X

þ 1 nX þ1    mX cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

i¼1 j¼1

 aij;3 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ



i¼1 j¼1

i¼1 j¼1

 aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ;

m þ 1 nX þ1 X

c0ij;1 x0ij;1 ;

i¼1 j¼1

 aij;2 ;

m þ 1 nX þ1 X

m þ 1 nX þ1 X

 c0ij;2 x0ij;1 þ aij;1 þ

i¼1 j¼1

þ 1 nX þ1   mX c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ



i¼1 j¼1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7

i¼1 j¼1



Subject to Constraints of the intuitionistic fuzzy linear programming problem (4.13.4). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (4.13.6) then go to Step 19. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (4.13.6) then go to Step 12. Step 12: Solve the crisp linear programming problem (4.13.7) and check that a unique crisp optimal solution exists for the crisp linear programming problem (4.13.7) or not. Crisp linear programming problem (4.13.7) " Minimize A

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;2 ; c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 Subject to D

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

þ 1 nP þ1  mP aij;3 ; i¼1 j¼1

   mP þ 1 nP þ1 cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ i¼1 j¼1

4 JMD Approach for Solving Unbalanced Fully …

174

aij;5 þ aij;6 Þ;

mP þ 1 nP þ1 i¼1 j¼1

c0ij;1 x0ij;1 ;

mP þ 1 nP þ1

 c0ij;2 x0ij;1 þ aij;1 þ

i¼1 j¼1 mP þ 1 nP þ1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 Þ;

i¼1 j¼1

 þ 1 nP þ1  mP aij;2 ; c0ij;3 x0ij;1 þ i¼1 j¼1



c0ij;4 x0ij;1 þ : aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5

þ aij;6 þ aij;7 ÞÞ ¼ Crisp optimal value of the crisp linear programming problem (4.13.6) and Constraints of the intuitionistic fuzzy linear programming problem (4.13.6). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (4.13.7) then go to Step 19. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (4.13.7) then go to Step 13. Step 13: Solve the crisp linear programming problem (4.13.8) and check that a unique crisp optimal solution exists for the crisp linear programming problem (4.13.8) or not. Crisp linear programming problem (4.13.8) " Minimize U

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;2 ; c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 Subject to A

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

þ 1 nP þ1  mP aij;3 ;

i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ i¼1 j¼1

aij;5 þ aij;6 Þ;

mP þ 1 nP þ1 i¼1 j¼1

aij;1 þ

c0ij;1 x0ij;1 ;

mP þ 1 nP þ1

  þ 1 nP þ1  mP c0ij;2 x0ij;1 þ aij;1 þ aij;2 ; c0ij;3 x0ij;1 þ

i¼1 j¼1 mP þ 1 nP þ1

aij;2 þ aij;3 þ aij;4 þ aij;5 Þ;

i¼1 j¼1



c0ij;4 x0ij;1 þ

i¼1 j¼1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ

aij;5 þ aij;6 þ aij;7 ÞÞ ¼ Crisp optimal value of the crisp linear programming problem (4.13.7) and

4.13

Proposed JMD Approach

175

Constraints of the intuitionistic fuzzy linear programming problem (4.13.7). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (4.13.8) then go to Step 19. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (4.13.8) then go to Step 14. Step 14: Solve the crisp linear programming problem (4.13.9) and check that a unique crisp optimal solution exists for the crisp linear programming problem (4.13.9) or not. Crisp linear programming problem (4.13.9) " Minimize G

m þ 1 nX þ1 X

þ 1 nX þ1    mX cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

i¼1 j¼1

 aij;3 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ



i¼1 j¼1

i¼1 j¼1

 aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ;

m þ 1 nX þ1 X

c0ij;1 x0ij;1 ;

i¼1 j¼1

 aij;2 ;

m þ 1 nX þ1 X

m þ 1 nX þ1 X

 c0ij;2 x0ij;1 þ aij;1 þ

i¼1 j¼1

þ 1 nX þ1   mX c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ



i¼1 j¼1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7

i¼1 j¼1



Subject to U

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;2 ; c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 ¼ Crisp optimal value of the crisp linear programming problem (4.13.8) and Constraints of the intuitionistic fuzzy linear programming problem (4.13.8). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (4.13.9) then go to Step 19. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (4.13.9) then go to Step 15.

4 JMD Approach for Solving Unbalanced Fully …

176

Step 15: Solve the crisp linear programming problem (4.13.10) and check that a unique crisp optimal solution exists for the crisp linear programming problem (4.13.10) or not. Crisp linear programming problem (4.13.10) " Minimize H

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;2 ; c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 Subject to G

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;2 ; c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 ¼ Crisp optimal value of the crisp linear programming problem (4.13.9) and Constraints of the intuitionistic fuzzy linear programming problem (4.13.9). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (4.13.10) then go to Step 19. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (4.13.10) then go to Step 16. Step 16: Solve the crisp linear programming problem (4.13.11) and check that a unique crisp optimal solution exists for the crisp linear programming problem (4.13.11) or not.

4.13

Proposed JMD Approach

177

Crisp linear programming problem (4.13.11) " Minimize T

m þ 1 nX þ1 X

þ 1 nX þ1    mX cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

i¼1 j¼1

 aij;3 ;

m þ 1 nX þ1 X

i¼1 j¼1

þ 1 nX þ1   mX cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ



i¼1 j¼1

i¼1 j¼1

 aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ;

m þ 1 nX þ1 X

c0ij;1 x0ij;1 ;

i¼1 j¼1

 aij;2 ;

m þ 1 nX þ1 X

m þ 1 nX þ1 X

 c0ij;2 x0ij;1 þ aij;1 þ

i¼1 j¼1

þ 1 nX þ1   mX c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ



i¼1 j¼1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7

i¼1 j¼1



Subject to H

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ aij;2 ; i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 ¼ Crisp optimal value of the crisp linear programming problem (4.13.10) and and Constraints of the intuitionistic fuzzy linear programming problem (4.13.10). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (4.13.11) then go to Step 19. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (4.13.11) then go to Step 17. Step 17: Solve the crisp linear programming problem (4.13.12) and check that a unique crisp optimal solution exists for the crisp linear programming problem (4.13.12) or not.

4 JMD Approach for Solving Unbalanced Fully …

178

Crisp linear programming problem (4.13.12) " Minimize E

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ aij;2 ; i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 Subject to T

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ aij;2 ; i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 ¼ Crisp optimal value of the crisp linear programming problem (4.13.11) and and Constraints of the intuitionistic fuzzy linear programming problem (4.13.11). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (4.13.12) then go to Step 19. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (4.13.12) then go to Step 18. Step 18: Solve the crisp linear programming problem (4.13.13) and go to Step 19.

4.13

Proposed JMD Approach

179

Crisp linear programming problem (4.13.13) " Minimize R

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP aij;3 ; cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

 þ 1 nP þ1 mP þ 1 nP þ1  mP aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;1 þ aij;1 þ i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 þ 1 nP þ1  mP c0ij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 ; c0ij;4 x0ij;1 þ aij;2 ; i¼1 j¼1 i¼1 j¼1  aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 Subject to E

mP þ 1 nP þ1 i¼1 j¼1

   mP þ 1 nP þ1 cij;1 x0ij;1 þ aij;1 ; cij;2 x0ij;1 þ aij;1 þ aij;2 þ

þ 1 nP þ1  mP aij;3 ;

i¼1 j¼1

i¼1 j¼1

   mP þ 1 nP þ1 cij;3 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 ; cij;4 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ i¼1 j¼1

aij;5 þ aij;6 Þ;

mP þ 1 nP þ1 i¼1 j¼1

c0ij;1 x0ij;1 ;

mP þ 1 nP þ1

 c0ij;2 x0ij;1 þ aij;1 þ

i¼1 j¼1 mP þ 1 nP þ1

aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 Þ;

i¼1 j¼1



 þ 1 nP þ1  mP aij;2 ; c0ij;3 x0ij;1 þ i¼1 j¼1

c0ij;4 x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5

þ aij;6 þ aij;7 ÞÞ ¼ Crisp optimal value of the crisp linear programming problem (4.13.12) and and Constraints of the intuitionistic fuzzy linear programming problem (4.13.12). Step 19: Using the obtained crisp optimal solution fx0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7 ; i ¼ 1; 2; . . .m; j ¼ 1; 2; . . .; ng, find xij;1 ¼ x0ij;1 þ aij;1 , i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n: xij;2 ¼ x0ij;1 þ aij;1 þ aij;2 þ aij;3 , i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n: xij;3 ¼ x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 , i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n: xij;4 ¼ x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 , i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n: (v) x0ij;2 ¼ x0ij;1 þ aij;1 þ aij;2 , i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n: (vi) x0ij;3 ¼ x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 , i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n: (vii) x0ij;4 ¼ x0ij;1 þ aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;5 þ aij;6 þ aij;7 . i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n: (i) (ii) (iii) (iv)

4 JMD Approach for Solving Unbalanced Fully …

180

Step 20: Using the optimal values of xij;k and x0ij;k ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n; k ¼ 1; n2; 3; 4, obtained in Step 19, find the intuitionistic fuzzy xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;2 ; x0ij;3 ; x0ij;4

; i ¼ 1; 2; . . .; m; j ¼ m P n  P 1; 2; . . .; ng and the intuitionistic fuzzy optimal value cij;1 xij;1 ; cij;2 xij;2 ; i¼1 j¼1  cij;3 xij;3 ; cij;4 xij;4 : ; c0ij;1 x0ij;1 ; c0ij;2 x0ij;2 ; c0ij;3 x0ij;3 ; c0ij;4 x0ij;4 of the intuitionistic fully fuzzy optimal solution

linear programming problem (4.5.1).

4.14

Illustrative Examples

In Sect. 4.6, the unbalanced fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.4, has been considered to point out a limitation of Ebrahimnejad and Verdegay’s approach [1]. Also, in Sect. 4.7, the unbalanced fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.6, has been considered to point out a drawback of Ebrahimnejad and Verdegay’s approach [1]. In this section, both these unbalanced fully trapezoidal intuitionistic fuzzy transportation problems have been solved by the proposed JMD approach.

4.14.1 Intuitionistic Fuzzy Optimal Solution of the First Unbalanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problem Using the proposed JMD approach, the intuitionistic fuzzy optimal solution of the first unbalanced fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.4, can be obtained as follows: Step 1: Using Sect. 4.11, (i) The Mehar representation of the trapezoidal intuitionistic fuzzy numbers ~ aI1 ¼   a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a01;2 ; a01;3 ; a01;4 ¼ ð60; 80; 100; 120; 50; 70; 110; 130Þ   and ~aI2 ¼ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a02;2 ; a02;3 ; a02;4 ¼ ð40; 60; 80; 100; 30; 50; 90; 110Þ, representing the intuitionistic fuzzy availability of the product at first and second source, respectively, are ð50; 10; 10; 10; 20; 10; 10; 10ÞM and ð30; 10; 10; 10; 20; 10; 10; 10ÞM , respectively.

4.14

Illustrative Examples

181

(ii) The Mehar representation of the trapezoidal intuitionistic fuzzy numbers ~ bI1 ¼   b1;1 ; b1;2 ; b1;3 ; b1;4 ; b01;1 ; b01;2 ; b01;3 ; b01;4 ¼ ð30; 50; 70; 90; 20; 40; 80; 100Þ and   ~ ¼ ð20; 30; 40; 50; 15; 25; 45; 55Þ, bI2 ¼ b2;1 ; b2;2 ; b2;3 ; b2;4 ; b02;1 ; b02;2 ; b02;3 ; b02;4 representing the intuitionistic fuzzy demand of the product at first and second destination, respectively, are ð20; 10; 10; 10; 20; 10; 10; 10ÞM and ð15; 5; 5; 5; 10; 5; 5; 5ÞM , respectively. Step 2: It is obvious that 2i¼1 ~ai ¼ ð80; 20; 20; 20; 40; 20; 20; 20ÞM 15; 15; 15ÞM .

and

2i¼1 ~ bj ¼ ð35; 15; 15; 15; 30;

Step 3: Since 2i¼1 ~ai 6¼ 2i¼1 ~bj So, the fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.4, is an unbalanced fully trapezoidal intuitionistic fuzzy transportation problem. So, there is a need to add  (i) A dummy source S3 having intuitionistic fuzzy availability q3;1 ; q3;2 ; q3;3 ; q3;3 ; q3;4 ; q3;5 ; q3;6 ; q3;7 ; q3;8 ÞM with the consideration of the cost for supplying the one unit quantity of the product from the dummy source S3 to both the destinations as a trapezoidal intuitionistic fuzzy number e 0 ¼ ð0; 0; 0; 0; 0; 0; 0; 0ÞM . where q3;1 ¼ maxf0; 35  80g ¼ 0; q3;2 ¼ maxf0; 15  20g ¼ 0; q3;3 ¼ maxf0; 15  20g ¼ 0; q3;4 ¼ maxf0; 15  20g ¼ 0; q3;5 ¼ maxf0; 30  40g ¼ 0; q3;6 ¼ maxf0; 15  20g ¼ 0; q3;7 ¼ maxf0; 15  20g ¼ 0; q3;8 ¼ maxf0; 15  20g ¼ 0:  (ii) A dummy destination D3 having intuitionistic fuzzy demand s3;1 ; s3;2 ; s3;3 ; s3;4 ; s3;5 ; s3;6 ; s3;7 ; s3;8 ÞM with the consideration of the cost for supplying the one unit quantity of the product from all the sources to the dummy destination D3 as a trapezoidal intuitionistic fuzzy number e 0 ¼ ð0; 0; 0; 0; 0; 0; 0; 0Þ.

4 JMD Approach for Solving Unbalanced Fully …

182

where s3;1 ¼ maxf0; 80  35g ¼ 45; s3;2 ¼ maxf0; 20  15g ¼ 5; s3;3 ¼ maxf0; 20  15g ¼ 5; s3;4 ¼ maxf0; 20  15g ¼ 5; s3;5 ¼ maxf0; 40  30g ¼ 10; s3;6 ¼ maxf0; 20  15g ¼ 5; s3;7 ¼ maxf0; 20  15g ¼ 5; s3;8 ¼ maxf0; 20  15g ¼ 5:

Step 4: To find an intuitionistic optimal solution of the transformed fully balanced trapezoidal intuitionistic fuzzy transportation problem, there is a need to solve the intuitionistic fully fuzzy linear programming problem (4.14.1.1). Intuitionistic fully fuzzy linear programming problem (4.14.1.1) Minimize½ð10; 20; 30; 40; 5; 15; 35; 45Þ   x011;1 ; a11;1 ; a11;2 ; a11;3 ; a11;4 ; a11;5 ; a11;6 ; a11;7 ð50; 60; 70; 90; 45; 55; 75; 95Þ  M 0 x12;1 ; a12;1 ; a12;2 ; a12;3 ; a12;4 ; a125 ; a12;6 ; a12;7 ð60; 70; 80; 90; 55; 65; 85; 95Þ  M 0 x21;1 ; a21;1 ; a21;2 ; a21;3 ; a21;4 ; a21;5 ; a216 ; a21;7 ;  M   i ð70; 80; 100; 120; 65; 75; 115; 125Þ  x022;1 ; a22;1 ; a22;2 ; a22;3 ; a22;4 ; a22;5 ; a226 ; a22;7 ; M

Subject to  

x011;1 ; a11;1 ; a11;2 ; a11;3 ; a11;4 ; a11;5 ; a11;6 ; a11;7 x012;1 ; a12;1 ; a12;2 ; a12;3 ; a12;4 ; a12;5 ; a12;6 ; a12;7

 M M

 

   x013;1 ; a13;1 ; a13;2 ; a13;3 ; a13;4 ; a13;5 ; a13;6 ; a13;7 ¼ ð50; 10; 10; 10; 20; 10; 10; 10ÞM ;

M

4.14

Illustrative Examples

 

183

x021;1 ; a21;1 ; a21;2 ; a21;3 ; a21;4 ; a21;5 ; a21;6 ; a21;7 x022;1 ; a22;1 ; a22;2 ; a22;3 ; a22;4 ; a22;5 ; a22;6 ; a22;7

 M M

 

   x023;1 ; a23;1 ; a23;2 ; a23;3 ; a23;4 ; a23;5 ; a23;6 ; a23;7

M

¼ ð30; 10; 10; 10; 20; 10; 10; 10ÞM ;  

x031;1 ; a31;1 ; a31;2 ; a31;3 ; a31;4 ; a31;5 ; a31;6 ; a31;7 x032;1 ; a32;1 ; a32;2 ; a32;3 ; a32;4 ; a32;5 ; a32;6 ; a32;7

 M M

 

   x033;1 ; a33;1 ; a33;2 ; a33;3 ; a33;4 ; a33;5 ; a33;6 ; a33;7 ¼ ð0; 0; 0; 0; 0; 0; 0; 0ÞM ; M

 

x011;1 ; a11;1 ; a11;2 ; a11;3 ; a11;4 ; a11;5 ; a11;6 ; a11;7 x021;1 ; a21;1 ; a21;2 ; a21;3 ; a21;4 ; a21;5 ; a21;6 ; a21;7

 M M

 

¼ ð20; 10; 10; 10; 20; 10; 10; 10ÞM ;  

x012;1 ; a12;1 ; a12;2 ; a12;3 ; a12;4 ; a12;5 ; a12;6 ; a12;7 x022;1 ; a22;1 ; a22;2 ; a22;3 ; a22;4 ; a22;5 ; a22;6 ; a22;7

 M M

 

¼ ð15; 5; 5; 5; 10; 5; 5; 5ÞM ; 

 x013;1 ; a13;1 ; a13;2 ; a13;3 ; a13;4 ; a13;5 ; a13;6 ; a13;7   M 0 x23;1 ; a23;1 ; a23;2 ; a23;3 ; a234 ; a23;5 ; a23;6 ; a23;7  M

¼ ð45; 5; 5; 5; 10; 5; 5; 5ÞM ;   x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7

M

is a non-negative trapezoidal intu-

itionistic fuzzy number. Step 5: Using the multiplication (4.12.2), proposed in Sect. 4.12, the intuitionistic fuzzy linear programming problem (4.14.1.1) can be transformed into its equivalent intuitionistic fully fuzzy linear programming problem (4.14.1.2).

184

4 JMD Approach for Solving Unbalanced Fully …

Intuitionistic fuzzy linear programming problem (4.14.1.2) h  Minimize A11;1 ; A11;2 ; A11;3 ; A11;4 ; A011;1 ; A011;2 ; A011;3 ; A011;4    A12;1 ; A12;2 ; A12;3 ; A12;4 ; A012;1 ; A012;2 ; A012;3 ; A012;4    A21;1 ; A21;2 ; A21;3 ; A12;4 ; A021;1 ; A021;2 ; A021;3 ; A021;4   i A22;1 ; A22;2 ; A22;3 ; A22;4 ; A022;1 ; A022;2 ; A022;3 ; A022;4 Constraints of the intuitionistic fully fuzzy linear programming problem (4.14.1.1). where      A11;1 ¼ 10 x011;1 þ a11;1 ; A11;2 ¼ 20 x011;1 þ a11;1 þ a11;2 þ a11;3 ; A11;3 ¼ 30 x011;1 þ   a11;1 þ a11;2 þ a11;3 þ a11;4 ; A11;4 ¼ 40 x011;1 þ a11;1 þ a11;2 þ a11;3 þ a11;4 þ a11;5 þ     a11;6 ; A011;1 ¼ 5x011;1 ; A011;2 ¼ 15 x011;1 þ a11;1 þ a11;2 ; A011;3 ¼ 35 x011;1 þ a11;1 þ a11;2 þ   a11;3 þ a11;4 þ a11;5 ; A011;4 ¼ 45 x011;1 þ a11;1 þ a11;2 þ a11;3 þ a11;4 þ a11;5 þ a11;6 þ  a11;7 ;      A12;1 ¼ 50 x012;1 þ a12;1 ; A12;2 ¼ 60 x012;1 þ a12;1 þ a12;2 þ a12;3 ; A12;3 ¼ 70 x012;1 þ   a12;1 þ a12;2 þ a12;3 þ a12;4 ; A12;4 ¼ 90 x012;1 þ a12;1 þ a12;2 þ a12;3 þ a12;4 þ a12;5 þ     a12;6 ; A012;1 ¼ 45x012;1 ; A012;2 ¼ 55 x012;1 þ a12;1 þ a12;2 ; A012;3 ¼ 75 x012;1 þ a12;1 þ   a12;2 þ a12;3 þ a12;4 þ a12;5 ; A012;4 ¼ 95 x012;1 þ a12;1 þ a12;2 þ a12;3 þ a12;4 þ a12;5 þ  a12;6 þ a12;7 ;      A21;1 ¼ 60 x021;1 þ a21;1 ; A21;2 ¼ 70 x021;1 þ a21;1 þ a21;2 þ a21;3 ; A21;3 ¼ 80 x021;1 þ   a21;1 þ a21;2 þ a21;3 þ a21;4 ; A21;4 ¼ 90 x021;1 þ a21;1 þ a21;2 þ a21;3 þ a21;4 þ a21;5 þ     a21;6 ; A021;1 ¼ 55x021;1 ; A021;2 ¼ 65 x021;1 þ a21;1 þ a21;2 ; A021;3 ¼ 85 x021;1 þ a21;1 þ   a21;2 þ a21;3 þ a21;4 þ a21;5 ; A021;4 ¼ 95 x021;1 þ a21;1 þ a21;2 þ a21;3 þ a21;4 þ a21;5 þ  a21;6 þ a21;7 ;      A22;1 ¼ 70 x022;1 þ a22;1 ; A22;2 ¼ 80 x022;1 þ a22;1 þ a22;2 þ a22;3 ; A22;3 ¼ 100 x022;1 þ   a22;1 þ a22;2 þ a22;3 þ a22;4 ; A22;4 ¼ 120 x022;1 þ a22;1 þ a22;2 þ a22;3 þ a22;4 þ a22;5 þ     a22;6 ; A022;1 ¼ 65x022;1 ; A022;2 ¼ 75 x022;1 þ a22;1 þ a22;2 ; A022;3 ¼ 115 x022;1 þ a22;1 þ   a22;2 þ a22;3 þ a22;4 þ a22;5 ; A022;4 ¼ 125 x022;1 þ a22;1 þ a22;2 þ a22;3 þ a22;4 þ a22;5 þ  a22;6 þ a22;7 

4.14

Step n P i¼1

Illustrative Examples

185

 Ai;1 ; Ai;2 ; Ai;3 ; Ai;4 ; A0i;1 ; A0i;2 ; A0i;3 ; A0i;4 ¼ i¼1  n n n n n n n P P P P P P P 0 Ai;1 ; Ai;2 ; Ai;3 ; Ai;4 ; Ai;1 ; A0i;2 ; A0i;3 ; A0i;4 , the intuitionistic 6:

i¼1

Using

i¼1

relation

i¼1

i¼1

n  P

i¼1

i¼1

i¼1

fully fuzzy linear programming problem (4.14.1.2) can be transformed into its equivalent intuitionistic fully fuzzy linear programming problem (4.14.1.3). Intuitionistic fully fuzzy linear programming problem (4.14.1.3)

 Minimize A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ i A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4 Subject to  x011;1 þ x012;1 þ x013;1 ; a11;1 þ a12;1 þ a13;1 ; a11;2 þ a12;2 þ a13;2 ; a11;3 þ a12;3 þ a13;3 ; a11;4 þ a12;4 þ a13;4 ; a11;5 þ a12;5 þ a13;5 ; a11;6 þ a12;6 þ a13;6 ; a11;7 þ a12;7  þ a13;7 M ¼ ð50; 10; 10; 10; 20; 10; 10; 10ÞM ;  x021;1 þ x022;1 þ x023;1 ; a21;1 þ a22;1 þ a23;1 ; a21;2 þ a22;2 þ a23;2 ; a21;3 þ a22;3 þ a23;3 ; a21;4 þ a22;4 þ a23;4 ; a21;5 þ a22;5 þ a23;5 ; a21;6 þ a22;6 þ a23;6 ; a21;7 þ a22;7 þ  a23;7 M ¼ ð30; 10; 10; 10; 20; 10; 10; 10ÞM ;  x031;1 þ x032;1 þ x033;1 ; a31;1 þ a32;1 þ a33;1 ; a31;2 þ a32;2 þ a33;2 ; a31;3 þ a32;3 þ a33;3 ; a31;4 þ a32;4 þ a33;4 ; a31;5 þ a32;5 þ a33;5 ; a31;6 þ a32;6 þ a33;6 ; a31;7 þ a32;7 þ  a33;7 M ¼ ð0; 0; 0; 0; 0; 0; 0; 0ÞM ;  x011;1 þ x021;1 þ x031;1 ; a11;1 þ a21;1 þ a31;1 ; a11;2 þ a21;2 þ a31;2 ; a11;3 þ a21;3 þ a31;3 ; a11;4 þ a21;4 þ a31;4 ; a11;5 þ a21;5 þ a31;5 ; a11;6 þ a21;6 þ a31;6 ; a11;7 þ a21;7 þ  a31;7 M ¼ ð20; 10; 10; 10; 20; 10; 10; 10ÞM ;  x012;1 þ x022;1 þ x032;1 ; a12;1 þ a22;1 þ a32;1 ; a12;2 þ a22;2 þ a32;2 ; a12;3 þ a22;3 þ a32;3 ; a12;4 þ a22;4 þ a32;4 ; a12;5 þ a22;5 þ a32;5 ; a12;6 þ a22;6 þ a32;6 ; a12;7 þ a22;7 þ  a32;7 M ¼ ð15; 5; 5; 5; 10; 5; 5; 5ÞM

186

4 JMD Approach for Solving Unbalanced Fully …

 x013;1 þ x023;1 þ x033;1 ; a13;1 þ a23;1 þ a33;1 ; a13;2 þ a23;2 þ a33;2 ; a13;3 þ a23;3 þ

a33;3 ; a13;4 þ a23;4 þ a33;4 ; a13;5 þ a23;5 þ a33;5 ; a13;6 þ a23;6 þ a33;6 ; a13;7 þ a23;7 þ  a33;7 M ¼ ð45; 5; 5; 5; 10; 5; 5; 5ÞM   x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7

M

is a non-negative trapezoidal intu-

itionistic fuzzy number. Step 7: Using the relation ð a0 ; a1 ; a2 ; a3 ; a4 ; a5 ; a6 ; a7 Þ M ¼ 0 0 0 ðb ; b1 ; b2 ; b3 ; b4 ; b5 ; b6 ; b7 ÞM ) a ¼ b , a1 ¼ b1 , a2 ¼ b2 , a3 ¼ b3 , a4 ¼ b4 , a5 ¼ b5 , a6 ¼ b6 , a7 ¼ b7 and the relation ða0 ; a1 ; a2 ; a3 ; a4 ; a5 ; a6 ; a7 ÞM is a non-negative trapezoidal intuitionistic fuzzy number ) a0  0, a1  0, a2  0; a3  0; a4  0; a5  0; a6  0; a7  0, the intuitionistic fully fuzzy linear programming problem (4.14.1.3) can be transformed into its equivalent intuitionistic fuzzy linear programming problem (4.14.1.4). Intuitionistic fuzzy linear programming problem (4.14.1.4)

 Minimize A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ i A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4 Subject to x011;1 þ x012;1 þ x013;1 ¼ 50; a11;1 þ a12;1 þ a13;1 ¼ 10; a11;2 þ a12;2 þ a13;2 ¼ 10 a11;3 þ a12;3 þ a13;3 ¼ 10; a11;4 þ a12;4 þ a13;4 ¼ 20; a11;5 þ a12;5 þ a13;5 ¼ 10; a11;6 þ a12;6 þ a13;6 ¼ 10; a11;7 þ a12;7 þ a13;7 ¼ 10; x021;1 þ x022;1 þ x023;1 ¼ 30;

4.14

Illustrative Examples

187

a21;1 þ a22;1 þ a23;1 ¼ 10; a21;2 þ a22;2 þ a23;3 ¼ 10; a21;3 þ a22;3 þ a23;3 ¼ 10; a21;4 þ a22;4 þ a23;4 ¼ 20; a21;5 þ a22;5 þ a23;5 ¼ 10; a21;6 þ a22;6 þ a23;6 ¼ 10; a21;7 þ a22;7 þ a23;7 ¼ 10; x031;1 þ x032;1 þ x033;1 ¼ 0; a31;1 þ a32;1 þ a33;1 ¼ 0; a31;2 þ a32;2 þ a33;3 ¼ 0; a31;3 þ a32;3 þ a33;3 ¼ 0; a31;4 þ a32;4 þ a33;4 ¼ 0; a31;5 þ a32;5 þ a33;5 ¼ 0; a31;6 þ a32;6 þ a33;6 ¼ 0; a31;7 þ a32;7 þ a33;7 ¼ 0; x011;1 þ x021;1 þ x031;1 ¼ 20; a11;1 þ a12;1 þ a13;1 ¼ 10; a11;2 þ a12;2 þ a13;2 ¼ 10; a11;3 þ a12;3 þ a13;3 ¼ 10; a11;4 þ a12;4 þ a13;4 ¼ 20; a11;5 þ a12;5 þ a13;5 ¼ 10; a11;6 þ a12;6 þ a13;6 ¼ 10;

188

4 JMD Approach for Solving Unbalanced Fully …

a11;7 þ a12;7 þ a13;7 ¼ 10; x012;1 þ x022;1 þ x032;1 ¼ 15; a12;1 þ a22;1 þ a32;1 ¼ 5; a12;2 þ a22;2 þ a32;2 ¼ 5; a12;3 þ a22;3 þ a32;3 ¼ 5; a12;4 þ a22;4 þ a32;4 ¼ 10; a12;5 þ a32;5 þ a32;5 ¼ 5; a12;6 þ a22;6 þ a32;6 ¼ 5; a12;7 þ a22;7 þ a32;7 ¼ 5; x013;1 þ x023;1 þ x033;1 ¼ 45; a13;1 þ a23;1 þ a33;1 ¼ 5; a13;2 þ a23;2 þ a33;2 ¼ 5; a13;3 þ a23;3 þ a33;3 ¼ 5; a13;4 þ a23;4 þ a33;4 ¼ 10; a13;5 þ a33;5 þ a33;5 ¼ 5; a13;6 þ a23;6 þ a33;6 ¼ 5; a13;7 þ a23;7 þ a33;7 ¼ 5; x0ij;k ; aij;k  0:

Step 8: Using the proposed DAUGHTER approach for comparing trapezoidal intuitionistic fuzzy numbers, the intuitionistic fuzzy linear programming problem (4.14.1.4) can be transformed into its equivalent crisp multi-objective linear programming problem (4.14.1.5).

4.14

Illustrative Examples

189

Crisp multi-objective linear programming problem (4.14.1.5)

 Minimize D A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 i þ A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize A A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 i þ A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize U A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 i þ A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize G A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 i þ A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize H A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 i þ A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize T A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 i þ A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize E A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 i þ A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

Minimize RðA11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 i þ A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

Subject to

190

4 JMD Approach for Solving Unbalanced Fully …

Constraints of the intuitionistic fuzzy linear programming problem (4.14.1.4). Step 9: According to the proposed DAUGHTER approach, there is a need to check that a unique crisp optimal solution exist for the crisp linear programming problem (4.14.1.6) or not and hence, there is a need to check that a unique crisp optimal solution exist for the crisp linear programming problem (4.14.1.7) or not. Crisp linear programming problem (4.14.1.7)

 Minimize D A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2 i þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

Subject to Constraints of the intuitionistic fuzzy linear programming problem (4.14.1.4). where  D A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2   þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4 ¼ 200x011;1 þ   540x012;1 þ 0x013;1 þ 600x021;1 þ 750x022;1 þ 0x023;1 þ 195a11;1 þ 495a12;1 þ 0a13;1  þ 545a21;1 þ 685a22;1 þ 0a23;1 þ 185a11;2 þ 445a12;2 þ 0a13;2 þ 485a21;2   þ 615a22;2 þ 0a23;2 þ 170a11;3 þ 390a12;3 þ 0a13;3 þ 420a21;3 þ 540a22;3    þ 0a23;3 þ 150a11;4 þ 330a12;4 þ 0a13;4 þ 350a21;4 þ 460a22;4 þ 0a23;4    þ 120a11;5 þ 260a12;5 þ 0a13;5 þ 270a21;5 þ 360a22;5 þ 0a23;5 þ 85a11;6   þ 185a12;6 þ 0a13;6 þ 185a21;6 þ 245a22;6 þ 0a23;6 þ 45a11;7 þ 95a12;7 þ 0a13;7  þ 95a21;7 þ 125a22;7 þ 0a23;7

Step 10: It can be easily verified that on solving the crisp linear programming problem (4.14.1.7), the following unique crisp optimal solution is obtained. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

x011;1 ¼ 20, x012;1 ¼ 15, x013;1 ¼ 15, x021;1 ¼ 0, x022;1 ¼ 0, x023;1 ¼ 30. a11;1 ¼ 10, a12;1 ¼ 0, a13;1 ¼ 0, a21;1 ¼ 0, a22;1 ¼ 5, a23;1 ¼ 5. a11;2 ¼ 10, a12;2 ¼ 0, a13;2 ¼ 0, a21;2 ¼ 0, a22;2 ¼ 5, a23;2 ¼ 5. a11;3 ¼ 10, a12;3 ¼ 0, a13;3 ¼ 0, a21;3 ¼ 0, a22;3 ¼ 5, a23;3 ¼ 5. a11;4 ¼ 20, a12;4 ¼ 0, a13;4 ¼ 0, a21;4 ¼ 0, a22;4 ¼ 10, a23;4 ¼ 10. a11;5 ¼ 10, a12;5 ¼ 0, a13;5 ¼ 0, a21;5 ¼ 0, a22;5 ¼ 5; a23;5 ¼ 5. a11;6 ¼ 10; a12;6 ¼ 0; a13;6 ¼ 0; a21;6 ¼ 0; a22;6 ¼ 5; a23;6 ¼ 5. a11;7 ¼ 10; a12;7 ¼ 0; a13;7 ¼ 0; a21;7 ¼ 0; a22;7 ¼ 5; a23;7 ¼ 5.

4.14

Illustrative Examples

191

Step 11: Using Step 19 of the proposed JMD approach, (i) x11;1 ¼ 30; x11;2 ¼ 50; x11;3 ¼ 70; x11;4 ¼ 90; x011;1 ¼ 20; x011;2 ¼ 40; x011;3 ¼ 80; x011;4 ¼ 100: (ii) x12;2 ¼ 15; x12;2 ¼ 15; x12;3 ¼ 15; x12;4 ¼ 15; x012;1 ¼ 15; x012;2 ¼ 15; x012;3 ¼ 15; x012;4 ¼ 15: (iii) x13;1 ¼ 15; x13;2 ¼ 15; x13;3 ¼ 15; x13;4 ¼ 15; x013;1 ¼ 15; x013;2 ¼ 15; x013;3 ¼ 15; x013;4 ¼ 15: (iv) x21;1 ¼ 0; x21;2 ¼ 0; x21;3 ¼ 0; x21;4 ¼ 0; x021;1 ¼ 0; x021;2 ¼ 0; x021;3 ¼ 0; x021;4 ¼ 0: (v) x22;1 ¼ 5; x22;2 ¼ 15; x22;3 ¼ 25; x22;4 ¼ 35; x022;1 ¼ 0; x022;2 ¼ 10; x022;3 ¼ 30; x022;4 ¼ 0: (vi) x23;1 ¼ 0; x23;2 ¼ 45; x23;3 ¼ 55; x23;4 ¼ 65; x023;1 ¼ 30; x023;2 ¼ 40; x023;3 ¼ 60; x023;4 ¼ 70. Step 12: Using Step 20 of the proposed JMD approach, the obtained intuitionistic fuzzy optimal solution of the unbalanced fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.2, is ~xI11 ¼ ð30; 50; 70; 90; 20; 40; 80; 100Þ, ~xI12 ¼ ð15; 15; 15; 15; 15; 15; 15; 15Þ, ~xI21 ¼ ð0; 0; 0; 0; 0; 0; 0; 0Þ; ~xI22 ¼ ~xI13 ¼ ð15; 15; 15; 15; 15; 15; 15; 15Þ, I ð5; 15; 25; 35; 0; 10; 30; 40Þ, ~x23 ¼ ð35; 45; 55; 65; 30; 40; 60; 70Þ and the obtained intuitionistic fuzzy optimal value is ð3300; 5800; 9100; 13; 200; 2350; 4450; 11; 050; 15; 550Þ.

4.14.2 Intuitionistic Fuzzy Optimal Solution of the Second Unbalanced Fully Trapezoidal Intuitionistic Fuzzy Transportation Problem Using the proposed JMD approach, the intuitionistic fuzzy optimal solution of the second unbalanced fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.6, can be obtained as follows: Step 1: Using Sect. 4.11, (i) The Mehar representation of the trapezoidal intuitionistic fuzzy numbers ~ aI1 ¼   a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a01;2 ; a01;3 ; a01;4 ¼ ð60; 80; 100; 125; 50; 70; 110; 130Þ  and ~aI2 ¼ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a02;2 ; a02;3 ; a02;4 Þ¼ ð40; 60; 80; 110; 30; 50; 90; 140Þ, representing the intuitionistic fuzzy availability of the product at first

4 JMD Approach for Solving Unbalanced Fully …

192

and second source, respectively, are ð50; 10; 10; 10; 20; 10; 15; 5ÞM and ð30; 10; 10; 10; 20; 10; 20; 30ÞM , respectively. ~I ¼ (ii) The Mehar representation of the trapezoidal intuitionistic fuzzy numbers b 1   0 0 0 0 b1;1 ; b1;2 ; b1;3 ; b1;4 ; b1;1 ; b1;2 ; b1;3 ; b1;4 ¼ ð55; 85; 95; 130; 45; 75; 105; 135Þ   and ~bI2 ¼ b2;1 ; b2;2 ; b2;3 ; b2;4 ; b02;1 ; b02;2 ; b02;3 ; b02;4 ¼ ð30; 70; 90; 100; 15; 65; ; 95; 135Þ, representing the intuitionistic fuzzy demand of the product at first and second destination, respectively, are ð45; 10; 20; 10; 10; 10; 25; 5ÞM and ð15; 15; 35; 5; 20; 5; 25; 15ÞM , respectively. Step 2: It is obvious that 2i¼1 a~i ¼ ð80; 20; 20; 20; 40; 20; 35; 35ÞM and 2i¼1 ~bj ¼ ð60; 25; 55; 15; 30; 15; 50; 20ÞM . Step 3: Since 2i¼1 ~ai 6¼ 2i¼1 ~bj . So, the fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.6, is an unbalanced fully trapezoidal intuitionistic fuzzy transportation problem. So, there  is a need to add a dummy source S3 having intuitionistic fuzzy supply q3;1 ; q3;2 ; q3;3 ; q3;3 ; q3;4 ; q3;5 ; q3;6 ; q3;7 ; q3;8 ÞM with the consideration of the cost for supplying the one unit quantity of the product from the dummy source S3 to both the destinations as a trapezoidal intuitionistic fuzzy number e 0 ¼ ð0; 0; 0; 0; 0; 0; 0; 0ÞM . where q3;1 ¼ maxf0; 60  80g ¼ 0; q3;2 ¼ maxf0; 25  20g ¼ 5; q3;3 ¼ maxf0; 55  20g ¼ 35; q3;4 ¼ maxf0; 15  20g ¼ 0; q3;5 ¼ maxf0; 30  40g ¼ 0; q3;6 ¼ maxf0; 15  20g ¼ 0; q3;7 ¼ maxf0; 50  35g ¼ 15; q3;8 ¼ maxf0; 20  35g ¼ 0: Also, there is need to add a dummydestination D3 having dummy demand s3;1 ; s3;2 ; s3;3 ; s3;4 ; s3;5 ; s3;6 ; s3;7 ; s3;8 M with the consideration of the cost for supplying the one unit quantity of the product from all the sources to the dummy 0 ¼ ð0; 0; 0; 0; 0; 0; 0; 0Þ. destination D3 as a trapezoidal intuitionistic fuzzy number e

4.14

Illustrative Examples

193

where s3;1 ¼ maxf0; 80  60g ¼ 20; s3;2 ¼ maxf0; 20  25g ¼ 0; s3;3 ¼ maxf0; 20  55g ¼ 0; s3;4 ¼ maxf0; 20  15g ¼ 5; s3;5 ¼ maxf0; 40  30g ¼ 10; s3;6 ¼ maxf0; 20  15g ¼ 5; s3;7 ¼ maxf0; 35  50g ¼ 0; s3;8 ¼ maxf0; 35  20g ¼ 15:

Step 4: To find an intuitionistic fuzzy optimal solution of the transformed balanced fully trapezoidal intuitionistic fuzzy transportation problem, there is a need to solve the intuitionistic fully fuzzy linear programming problem (4.14.2.1), Intuitionistic fully fuzzy linear programming problem (4.14.2.1) Minimize½ð10; 20; 30; 40; 5; 15; 35; 45Þ   x011;1 ; a11;1 ; a11;2 ; a11;3 ; a11;4 ; a11;5 ; a11;6 ; a11;7 ð50; 60; 70; 90; 45; 55; 75; 95Þ  M 0 x12;1 ; a12;1 ; a12;2 ; a12;3 ; a12;4 ; a125 ; a12;6 ; a12;7 ð60; 70; 80; 90; 55; 65; 85; 95Þ  M 0 x21;1 ; a21;1 ; a21;2 ; a21;3 ; a21;4 ; a21;5 ; a216 ; a21;7 ;  M   i ð70; 80; 100; 120; 65; 75; 115; 125Þ  x022;1 ; a22;1 ; a22;2 ; a22;3 ; a22;4 ; a22;5 ; a226 ; a22;7 M

Subject to  

x011;1 ; a11;1 ; a11;2 ; a11;3 ; a11;4 ; a11;5 ; a11;6 ; a11;7 x012;1 ; a12;1 ; a12;2 ; a12;3 ; a12;4 ; a12;5 ; a12;6 ; a12;7

 M M

 

   x013;1 ; a13;1 ; a13;2 ; a13;3 ; a13;4 ; a13;5 ; a13;6 ; a13;7 ¼ ð50; 10; 10; 10; 20; 10; 15; 5ÞM ;

M

4 JMD Approach for Solving Unbalanced Fully …

194

 

x021;1 ; a21;1 ; a21;2 ; a21;3 ; a21;4 ; a21;5 ; a21;6 ; a21;7 x022;1 ; a22;1 ; a22;2 ; a22;3 ; a22;4 ; a22;5 ; a22;6 ; a22;7

 M M

 

   x023;1 ; a23;1 ; a23;2 ; a23;3 ; a23;4 ; a23;5 ; a23;6 ; a23;7

M

¼ ð30; 10; 10; 10; 20; 10; 20; 30ÞM ;  

x031;1 ; a31;1 ; a31;2 ; a31;3 ; a31;4 ; a31;5 ; a31;6 ; a31;7 x032;1 ; a32;1 ; a32;2 ; a32;3 ; a32;4 ; a32;5 ; a32;6 ; a32;7

 M M

 

   x033;1 ; a33;1 ; a33;2 ; a33;3 ; a33;4 ; a33;5 ; a33;6 ; a33;7 ¼ ð0; 5; 35; 0; 0; 0; 15; 0ÞM ; M

  

x011;1 ; a11;1 ; a11;2 ; a11;3 ; a11;4 ; a11;5 ; a11;6 ; a11;7 x021;1 ; a21;1 ; a21;2 ; a21;3 ; a21;4 ; a21;5 ; a21;6 ; a21;7

x031;1 ; a31;1 ; a31;2 ; a31;3 ; a31;4 ; a31;5 ; a31;6 ; a31;7  



 M

M M

 

¼ ð45; 10; 20; 10; 10; 10; 25; 5ÞM ;

x012;1 ; a12;1 ; a12;2 ; a12;3 ; a12;4 ; a12;5 ; a12;6 ; a12;7 x022;1 ; a22;1 ; a22;2 ; a22;3 ; a22;4 ; a22;5 ; a22;6 ; a22;7

x032;1 ; a32;1 ; a32;2 ; a32;3 ; a31;4 ; a32;5 ; a32;6 ; a32;7



 M

 M M

 

¼ ð15; 15; 35; 5; 20; 5; 25; 15ÞM ;



 x013;1 ; a13;1 ; a13;2 ; a13;3 ; a13;4 ; a13;5 ; a13;6 ; a13;7   M 0 x23;1 ; a23;1 ; a23;2 ; a23;3 ; a234 ; a23;5 ; a23;6 ; a23;7  M



x033;1 ; a33;1 ; a33;2 ; a33;3 ; a33;4 ; a33;5 ; a33;6 ; a33;7

  x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7

M

 M

¼ ð20; 0; 0; 5; 10; 5; 0; 15ÞM ;

is a non-negative trapezoidal intu-

itionistic fuzzy number. Step 5: Using the multiplication (4.12.2), proposed in Sect. 4.12, the intuitionistic fully fuzzy linear programming problem (4.14.2.1) can be transformed into its equivalent intuitionistic fully fuzzy linear programming problem (4.14.2.2).

4.14

Illustrative Examples

195

Intuitionistic fully fuzzy linear programming problem (4.14.2.2) hh  Minimize A11;1 ; A11;2 ; A11;3 ; A11;4 ; A011;1 ; A011;2 ; A011;3 ; A011;4    A12;1 ; A12;2 ; A12;3 ; A12;4 ; A012;1 ; A012;2 ; A012;3 ; A012;4    A21;1 ; A21;2 ; A21;3 ; A12;4 ; A021;1 ; A021;2 ; A021;3 ; A021;4   i A22;1 ; A22;2 ; A22;3 ; A22;4 ; A022;1 ; A022;2 ; A022;3 ; A022;4 Constraints of the intuitionistic fully fuzzy linear programming problem (4.14.2.1). where      A11;1 ¼ 10 x011;1 þ a11;1 ; A11;2 ¼ 20 x011;1 þ a11;1 þ a11;2 þ a11;3 ; A11;3 ¼ 30 x011;1 þ   a11;1 þ a11;2 þ a11;3 þ a11;4 ; A11;4 ¼ 40 x011;1 þ a11;1 þ a11;2 þ a11;3 þ a11;4 þ a11;5 þ     a11;6 ; A011;1 ¼ 5x011;1 ; A011;2 ¼ 15 x011;1 þ a11;1 þ a11;2 ; A011;3 ¼ 35 x011;1 þ a11;1 þ a11;2 þ   a11;3 þ a11;4 þ a11;5 ; A011;4 ¼ 45 x011;1 þ a11;1 þ a11;2 þ a11;3 þ a11;4 þ a11;5 þ a11;6 þ  a11;7 ;      A12;1 ¼ 50 x012;1 þ a12;1 ; A12;2 ¼ 60 x012;1 þ a12;1 þ a12;2 þ a12;3 ; A12;3 ¼ 70 x012;1 þ   a12;1 þ a12;2 þ a12;3 þ a12;4 ; A12;4 ¼ 90 x012;1 þ a12;1 þ a12;2 þ a12;3 þ a12;4 þ a12;5 þ     a12;6 ; A012;1 ¼ 45x012;1 ; A012;2 ¼ 55 x012;1 þ a12;1 þ a12;2 ; A012;3 ¼ 75 x012;1 þ a12;1 þ   a12;2 þ a12;3 þ a12;4 þ a12;5 ; A012;4 ¼ 95 x012;1 þ a12;1 þ a12;2 þ a12;3 þ a12;4 þ a12;5 þ  a12;6 þ a12;7 ;      A21;1 ¼ 60 x021;1 þ a21;1 ; A21;2 ¼ 70 x021;1 þ a21;1 þ a21;2 þ a21;3 ; A21;3 ¼ 80 x021;1 þ   a21;1 þ a21;2 þ a21;3 þ a21;4 ; A21;4 ¼ 90 x021;1 þ a21;1 þ a21;2 þ a21;3 þ a21;4 þ a21;5 þ     a21;6 ; A021;1 ¼ 55x021;1 ; A021;2 ¼ 65 x021;1 þ a21;1 þ a21;2 ; A021;3 ¼ 85 x021;1 þ a21;1 þ   a21;2 þ a21;3 þ a21;4 þ a21;5 ; A021;4 ¼ 95 x021;1 þ a21;1 þ a21;2 þ a21;3 þ a21;4 þ a21;5 þ  a21;6 þ a21;7 ;      A22;1 ¼ 70 x022;1 þ a22;1 ; A22;2 ¼ 80 x022;1 þ a22;1 þ a22;2 þ a22;3 ; A22;3 ¼ 100 x022;1 þ   a22;1 þ a22;2 þ a22;3 þ a22;4 ; A22;4 ¼ 120 x022;1 þ a22;1 þ a22;2 þ a22;3 þ a22;4 þ a22;5 þ     a22;6 ; A022;1 ¼ 65x022;1 ; A022;2 ¼ 75 x022;1 þ a22;1 þ a22;2 ; A022;3 ¼ 115 x022;1 þ a22;1 þ   a22;2 þ a22;3 þ a22;4 þ a22;5 ; A022;4 ¼ 125 x022;1 þ a22;1 þ a22;2 þ a22;3 þ a22;4 þ a22;5 þ  a22;6 þ a22;7 :

4 JMD Approach for Solving Unbalanced Fully …

196

 P Pn  n 0 0 0 0 Step 6: Using relation A ; A ; A ; A ; A ; A ; A ; A i;1 i;2 i;3 i;4 i;1 i;2 i;3 i;4 ¼ i¼1 i¼1 Ai;1 ; Pn Pn Pn Pn P P P n n n 0 0 0 0 the i¼1 Ai;2 ; i¼1 Ai;3 ; i¼1 Ai;4 ; i¼1 Ai;1 ; i¼1 Ai;2 ; i¼1 Ai;3 ; i¼1 Ai;4 Þ, intuitionistic fully fuzzy linear programming problem (4.14.2.2) can be transformed into its equivalent intuitionistic fully fuzzy linear programming problem (4.14.2.3). Intuitionistic fully fuzzy linear programming problem (4.14.2.3)

 Minimize A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 i þ A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4 Subject to  x011;1 þ x012;1 þ x013;1 ; a11;1 þ a12;1 þ a13;1 ; a11;2 þ a12;2 þ a13;2 ; a11;3 þ a12;3 þ a13;3 ; a11;4 þ a12;4 þ a13;4 ; a11;5 þ a12;5 þ a13;5 ; a11;6 þ a12;6 þ a13;6 ; a11;7 þ a12;7 þ  a13;7 M ¼ ð50; 10; 10; 10; 20; 10; 15; 5ÞM ;  x021;1 þ x022;1 þ x023;1 ; a21;1 þ a22;1 þ a23;1 ; a21;2 þ a22;2 þ a23;2 ; a21;3 þ a22;3 þ a23;3 ; a21;4 þ a22;4 þ a23;4 ; a21;5 þ a22;5 þ a23;5 ; a21;6 þ a22;6 þ a23;6 ; a21;7 þ  a22;7 þ a23;7 M ¼ ð30; 10; 10; 10; 20; 10; 20; 30ÞM ;  x031;1 þ x032;1 þ x033;1 ; a31;1 þ a32;1 þ a33;1 ; a31;2 þ a32;2 þ a33;2 ; a31;3 þ a32;3 þ a33;3 ; a31;4 þ a32;4 þ a33;4 ; a31;5 þ a32;5 þ a33;5 ; a31;6 þ a32;6 þ a33;6 ; a31;7 þ a32;7 þ  a33;7 M ¼ ð0; 5; 35; 0; 0; 0; 15; 0ÞM ;  x011;1 þ x021;1 þ x031;1 ; a11;1 þ a21;1 þ a31;1 ; a11;2 þ a21;2 þ a31;2 ; a11;3 þ a21;3 þ a31;3 ; a11;4 þ a21;4 þ a31;4 ; a11;5 þ a21;5 þ a31;5 ; a11;6 þ a21;6 þ a31;6 ; a11;7 þ a21;7 þ  a31;7 M ¼ ð45; 10; 20; 10; 10; 10; 25; 5ÞM ;  x012;1 þ x022;1 þ x032;1 ; a12;1 þ a22;1 þ a32;1 ; a12;2 þ a22;2 þ a32;2 ; a12;3 þ a22;3 þ a32;3 ; a12;4 þ a22;4 þ a32;4 ; a12;5 þ a22;5 þ a32;5 ; a12;6 þ a22;6 þ a32;6 ; a12;7 þ  a22;7 þ a32;7 M ¼ ð15; 15; 35; 5; 20; 5; 25; 15ÞM ;

4.14

Illustrative Examples

197

 x013;1 þ x023;1 þ x033;1 ; a13;1 þ a23;1 þ a33;1 ; a13;2 þ a23;2 þ a33;2 ; a13;3 þ a23;3 þ a33;3 ; a13;4 þ a23;4 þ a33;4 ; a13;5 þ a23;5 þ a33;5 ; a13;6 þ a23;6 þ a33;6 ; a13;7 þ  ¼ ð20; 0; 0; 5; 10; 5; 0; 15ÞM ; M

a23;7 þ a33;7

  x0ij;1 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;5 ; aij;6 ; aij;7

M

is a non-negative trapezoidal intu-

itionistic fuzzy number. Step 7: Using the relation ða0 ; a1 ; a2 ; a3 ; a4 ; a5 ; a6 ; a7 ÞM ¼ ðb0 ; b1 ; b2 ; b3 ; b4 ; b5 ; b6 ; b7 ÞM ) a0 ¼ b0 , a1 ¼ b1 , a2 ¼ b2 , a3 ¼ b3 , a4 ¼ b4 , a5 ¼ b5 , a6 ¼ b6 , a7 ¼ b7 and the relation ða0 ; a1 ; a2 ; a3 ; a4 ; a5 ; a6 ; a7 ÞM is a non-negative trapezoidal intuitionistic fuzzy number ) a0  0, a1  0, a2  0; a3  0; a4  0; a5  0; a6  0; a7  0, the intuitionistic fully fuzzy linear programming problem (4.14.2.3) can be transformed into in its equivalent intuitionistic fuzzy linear programming problem (4.14.2.4). Intuitionistic fuzzy linear programming problem (4.14.2.4)

 Minimize A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ i A013;2 þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

Subject to x011;1 þ x012;1 þ x013;1 ¼ 50; a11;1 þ a12;1 þ a13;1 ¼ 10; a11;2 þ a12;2 þ a13;2 ¼ 10; a11;3 þ a12;3 þ a13;3 ¼ 10; a11;4 þ a12;4 þ a13;4 ¼ 20; a11;5 þ a12;5 þ a13;5 ¼ 10; a11;6 þ a12;6 þ a13;6 ¼ 15; a11;7 þ a12;7 þ a13;7 ¼ 15; x021;1 þ x022;1 þ x023;1 ¼ 30;

198

4 JMD Approach for Solving Unbalanced Fully …

a21;1 þ a22;1 þ a23;1 ¼ 10; a21;2 þ a22;2 þ a23;3 ¼ 10; a21;3 þ a22;3 þ a23;3 ¼ 10; a21;4 þ a22;4 þ a23;4 ¼ 20; a21;5 þ a22;5 þ a23;5 ¼ 10; a21;6 þ a22;6 þ a23;6 ¼ 20; a21;7 þ a22;7 þ a23;7 ¼ 30; x031;1 þ x032;1 þ x033;1 ¼ 0; a31;1 þ a32;1 þ a33;1 ¼ 5; a31;2 þ a32;2 þ a33;3 ¼ 35; a31;3 þ a32;3 þ a33;3 ¼ 0; a31;4 þ a32;4 þ a33;4 ¼ 0; a31;5 þ a32;5 þ a33;5 ¼ 0; a31;6 þ a32;6 þ a33;6 ¼ 15; a31;7 þ a32;7 þ a33;7 ¼ 0; x011;1 þ x021;1 þ x031;1 ¼ 45; a11;1 þ a12;1 þ a13;1 ¼ 10; a11;2 þ a12;2 þ a13;2 ¼ 20; a11;3 þ a12;3 þ a13;3 ¼ 10; a11;4 þ a12;4 þ a13;4 ¼ 10; a11;5 þ a12;5 þ a13;5 ¼ 10; a11;6 þ a12;6 þ a13;6 ¼ 25;

4.14

Illustrative Examples

199

a11;7 þ a12;7 þ a13;7 ¼ 5; x012;1 þ x022;1 þ x032;1 ¼ 15; a12;1 þ a22;1 þ a32;1 ¼ 15; a12;2 þ a22;2 þ a32;2 ¼ 35; a12;3 þ a22;3 þ a32;3 ¼ 5; a12;4 þ a22;4 þ a32;4 ¼ 20; a12;5 þ a32;5 þ a32;5 ¼ 5; a12;6 þ a22;6 þ a32;6 ¼ 25; a12;7 þ a22;7 þ a32;7 ¼ 15; x013;1 þ x023;1 þ x033;1 ¼ 20; a13;1 þ a23;1 þ a33;1 ¼ 0; a13;2 þ a23;2 þ a33;2 ¼ 0; a13;3 þ a23;3 þ a33;3 ¼ 5; a13;4 þ a23;4 þ a33;4 ¼ 10; a13;5 þ a33;5 þ a33;5 ¼ 5; a13;6 þ a23;6 þ a33;6 ¼ 0; a13;7 þ a23;7 þ a33;7 ¼ 15; x0ij;k ; aij;k  0:

Step 8: Using the proposed DAUGHTER approach for comparing trapezoidal intuitionistic fuzzy numbers, the intuitionistic fuzzy linear programming problem (4.14.2.4) can be transformed into its equivalent crisp multi-objective linear programming problem (4.14.2.5).

200

4 JMD Approach for Solving Unbalanced Fully …

Crisp multi-objective linear programming problem (4.14.2.5)

 Minimize D A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2 i þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize A A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2 i þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize U A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2 i þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize G A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2 i þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize H A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2 i þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize T A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2 i þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize E A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2 i þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

 Minimize R A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2 i þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

Subject to Constraints of the intuitionistic fuzzy linear programming problem (4.14.2.4).

4.14

Illustrative Examples

201

Step 9: According to DAUGHTER approach, there is a need to check that a unique crisp optimal solution exist for the crisp linear programming problem (4.14.2.6) or not. Crisp linear programming problem (4.14.2.6)

 Minimize D A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2 i þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4

Subject to Constraints of the intuitionistic fuzzy linear programming problem (4.14.2.4). where  D A11;1 þ A12;1 þ A13;1 þ A14;1 ; A11;2 þ A12;2 þ A13;2 þ A14;2 ; A11;3 þ A12;3 þ A13;3 þ A14;3 ; A11;4 þ A12;4 þ A13;4 þ A14;4 ; A011;1 þ A012;1 þ A013;1 þ A014;1 ; A011;2 þ A012;2 þ A013;2   þ A014;2 ; A011;3 þ A012;3 þ A013;3 þ A014;3 ; A011;4 þ A012;4 þ A013;4 þ A014;4 ¼ 200x011;1 þ   540x012;1 þ 0x013;1 þ 600x021;1 þ 750x022;1 þ 0x023;1 þ 195a11;1 þ 495a12;1 þ 0a13;1   þ 545a21;1 þ 685a22;1 þ 0a23;1 þ 185a11;2 þ 445a12;2 þ 0a13;2 þ 485a21;2   þ 615a22;2 þ 0a23;2 þ 170a11;3 þ 390a12;3 þ 0a13;3 þ 420a21;3 þ 540a22;3   þ 0a23;3 þ 150a11;4 þ 330a12;4 þ 0a13;4 þ 350a21;4 þ 460a22;4 þ 0a23;4    þ 120a11;5 þ 260a12;5 þ 0a13;5 þ 270a21;5 þ 360a22;5 þ 0a23;5 þ 85a11;6   þ 185a12;6 þ 0a13;6 þ 185a21;6 þ 245a22;6 þ 0a23;6 þ 45a11;7 þ 95a12;7 þ 0a13;7  þ 95a21;7 þ 125a22;7 þ 0a23;7 :

Step 10: It can be easily verified that on solving the crisp linear programming problem (4.14.2.6) the following unique crisp optimal solution is obtained. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

x011;1 ¼ 20, x012;1 ¼ 15, x013;1 ¼ 15, x021;1 ¼ 0, x022;1 ¼ 0, x023;1 ¼ 30. a11;1 ¼ 10, a12;1 ¼ 0, a13;1 ¼ 0, a21;1 ¼ 0, a22;1 ¼ 5, a23;1 ¼ 5. a11;2 ¼ 10, a12;2 ¼ 0, a13;2 ¼ 0, a21;2 ¼ 0, a22;2 ¼ 5, a23;2 ¼ 5. a11;3 ¼ 10, a12;3 ¼ 0, a13;3 ¼ 0, a21;3 ¼ 0, a22;3 ¼ 5, a23;3 ¼ 5. a11;4 ¼ 20, a12;4 ¼ 0, a13;4 ¼ 0, a21;4 ¼ 0, a22;4 ¼ 10, a23;4 ¼ 10. a11;5 ¼ 10, a12;5 ¼ 0, a13;5 ¼ 0, a21;5 ¼ 0, a22;5 ¼ 5, a23;5 ¼ 5. a11;6 ¼ 10, a12;6 ¼ 0, a13;6 ¼ 0, a21;6 ¼ 0, a22;6 ¼ 5, a23;6 ¼ 5. a11;7 ¼ 10, a12;7 ¼ 0, a13;7 ¼ 0, a21;7 ¼ 0, a22;7 ¼ 5, a23;7 ¼ 5.

4 JMD Approach for Solving Unbalanced Fully …

202

Step 11: Using Step 19 of the proposed JMD approach, (i) x11;1 ¼ 30, x11;2 ¼ 50, x11;3 ¼ 70; x11;4 ¼ 90; x011;1 ¼ 20; x011;2 ¼ 40; x011;3 ¼ 80; x011;4 ¼ 100: (ii) x12;2 ¼ 15, x12;2 ¼ 15, x12;3 ¼ 15; x12;4 ¼ 15; x012;1 ¼ 15; x012;2 ¼ 15; x012;3 ¼ 15; x012;4 ¼ 15: (iii) x13;1 ¼ 15, x13;2 ¼ 15, x13;3 ¼ 15; x13;4 ¼ 15; x013;1 ¼ 15; x013;2 ¼ 15; x013;3 ¼ 15; x013;4 ¼ 15: (iv) x21;1 ¼ 0, x21;2 ¼ 0, x21;3 ¼ 0; x21;4 ¼ 0; x021;1 ¼ 0; x021;2 ¼ 0; x021;3 ¼ 0; x021;4 ¼ 0: (v) x22;1 ¼ 5, x22;2 ¼ 15, x22;3 ¼ 25; x22;4 ¼ 35; x022;1 ¼ 0; x022;2 ¼ 10; x022;3 ¼ 30; x022;4 ¼ 0: (vi) x23;1 ¼ 0, x23;2 ¼ 45, x23;3 ¼ 55; x23;4 ¼ 65; x023;1 ¼ 30; x023;2 ¼ 40; x023;3 ¼ 60; x023;4 ¼ 70. Step 12: Using Step 20 of the proposed JMD approach, the obtained intuitionistic fuzzy optimal solution of the unbalanced fully trapezoidal intuitionistic fuzzy transportation problem, represented by Table 4.6, is ~xI11 ¼ ð30; 50; 70; 90; 20; 40; 80; 100Þ, ~xI12 ¼ ð15; 15; 15; 15; 15; 15; 15; 15Þ, ~xI21 ¼ ð0; 0; 0; 0; 0; 0; 0; 0Þ; ~xI22 ¼ ~xI13 ¼ ð15; 15; 15; 15; 15; 15; 15; 15Þ, I ð5; 15; 25; 35; 0; 10; 30; 40Þ, ~x23 ¼ ð35; 45; 55; 65; 30; 40; 60; 70Þ and the obtained intuitionistic fuzzy optimal value is ð3300; 5800; 9100; 13; 200; 2350; 4450; 11; 050; 15; 550Þ.

4.15

Conclusions

Some limitations and a drawback of the existing approach [1] have been pointed out. Also, to overcome the limitations and to resolve the drawback, a new approach (named as DAUGHTER approach) for comparing trapezoidal intuitionistic fuzzy numbers as well as a new approach (named as JMD approach) for solving unbalanced fully trapezoidal intuitionistic fuzzy transportation problems has been proposed. Furthermore, to illustrate the proposed JMD approach, two unbalanced fully trapezoidal intuitionistic fuzzy transportations problems have been solved.

References

203

References 1. A. Ebrahimnejad, J.L. Verdegay, A new approach for solving fully intuitionistic fuzzy transportation problems. Fuzzy Optim. Decis. Making 17, 447–474 (2018) 2. W. Jianqiang, Z. Zhong, Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems. J. Syst. Eng. Electron. 20, 321–326 (2009) 3. A. Mahmoodirad, T. Allahviranloo, S. Niroomand, A new effective solution method for fully intuitionistic fuzzy transportation problem. Soft. Comput. 23, 4521–4530 (2019) 4. A. Mishra, A. Kumar, JMD method for transforming an unbalanced fully intuitionistic fuzzy transportation problem into a balanced fully intuitionistic fuzzy transportation problem. Soft. Comput. (2020). https://doi.org/10.1007/s00500-020-04889-6

Chapter 5

JMD Approach for Solving Unbalanced Fully Generalized Trapezoidal Intuitionistic Fuzzy Transportation Problems

Chakraborty et al. [1] proposed an approach for solving fully generalized trapezoidal intuitionistic fuzzy transportation problems (transportation problems in which each parameter is represented by a generalized trapezoidal intuitionistic fuzzy number). One may claim that Chakraborty et al.’s approach [1] can be used only to solve such fully generalized trapezoidal intuitionistic fuzzy transportation problems for which the aggregated value of the generalized intuitionistic fuzzy transportation cost, the generalized intuitionistic fuzzy availability and the generalized intuitionistic fuzzy demand, provided by all the decision-makers, is available. While Chakraborty et al.’s approach [1] cannot be used to solve such fully generalized trapezoidal intuitionistic fuzzy transportation problems for which, instead of the aggregated data, the data of each decision-maker is provided separately. To overcome this limitation, one may modify Chakraborty et al.’s approach [1] with the help of existing generalized trapezoidal intuitionistic fuzzy aggregation operator [2]. Also, one may use Chakraborty et al.’s approach [1] to solve real-life fully generalized trapezoidal intuitionistic fuzzy transportation problems. However, after a deep study, some limitations and a drawback have been observed in Chakraborty et al.’s approach [1]. The aim of this chapter is (i) To make the researchers aware about the observed limitations and a drawback of Chakraborty et al.’s approach [1]. (ii) To make the researchers aware about a drawback of the approach, used in Chakraborty et al.’s approach [1], for comparing generalized trapezoidal intuitionistic fuzzy numbers. (iii) To propose a valid approach (named as PRABHUS approach) for comparing generalized trapezoidal intuitionistic fuzzy numbers. (iv) To propose a valid approach (named as JMD (JAI MATA DI)) for solving unbalanced fully generalized trapezoidal intuitionistic fuzzy transportation problems. (v) To illustrate the proposed JMD approach with the help of a numerical example. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Mishra and A. Kumar, Aggregation Operators for Various Extensions of Fuzzy Set and Its Applications in Transportation Problems, Studies in Fuzziness and Soft Computing 399, https://doi.org/10.1007/978-981-15-6998-2_5

205

206

5.1

5 JMD Approach for Solving Unbalanced Fully Generalized …

Organization of the Chapter

This chapter is organized as follows: (i) In Sect. 5.2, some basic definitions have been presented. (ii) In Sect. 5.3, the tabular form of a fully generalized trapezoidal intuitionistic fuzzy transportation problem has been discussed. (iii) In Sect. 5.4, an existing approach for comparing generalized trapezoidal intuitionistic fuzzy numbers, used in Chakraborty et al.’s approach [1], has been discussed. (iv) In Sect. 5.4, the generalized intuitionistic fully fuzzy linear programming problem of a balanced fully generalized trapezoidal intuitionistic fuzzy transportation problem has been discussed. (v) In Sect. 5.5, Chakraborty et al.’s approach [1] for solving balanced fully generalized trapezoidal intuitionistic fuzzy transportation problems has been discussed in a brief manner. (vi) In Sect. 5.6, the origin of the generalized intuitionistic fully fuzzy linear programming problem, used in Chakraborty et al.’s approach [1], has been pointed out. (vii) In Sect. 5.7, some limitations of Chakraborty et al.’s approach [1] have been discussed. (viii) In Sect. 5.8, it is showed that Chakraborty et al. [1] have used a mathematical incorrect assumption in their proposed approach. Hence, Chakraborty et al.’s approach [1] is not valid. (ix) In Sect. 5.9, it is showed that the approach, used by Chakraborty et al. [1] for comparing generalized trapezoidal intuitionistic fuzzy numbers, is inappropriate. (x) In Sect. 5.10, a new approach (named as PRABHUS approach) has been proposed for comparing generalized trapezoidal intuitionistic fuzzy numbers. (xi) In Sect. 5.11, a new approach (named as JMD approach) has been proposed to solve unbalanced fully generalized trapezoidal intuitionistic fuzzy transportation problems. (xii) In Sect. 5.12, the proposed JMD approach has been illustrated with the help of a numerical problem. (xiii) Sect. 5.13 concludes the chapter.

5.2 Preliminaries

5.2

207

Preliminaries

In this section, some basic definitions have been presented [1]. nD E eI ¼ x; leI ð xÞ; meI ð xÞ : x 2 Definition 5.2.1 An intuitionistic fuzzy number A A A Rg is called a generalized intuitionistic fuzzy number, if the following properties hold, (i) There exist m 2 R such that leI ðmÞ ¼ w; meI ðmÞ ¼ 0 where 0  w  1. A A (ii) leI is continuous mapping from R to the interval ð0; w. A (iii) leI ð xÞ þ meI ð xÞ  1 8 x 2 R. A A e I is said to be a Definition 5.2.2 A generalized intuitionistic fuzzy number A generalized trapezoidal intuitionistic fuzzy number if its membership function leI ð xÞ and non-membership function meI ð xÞ are given by A

 8  xa1 > w ; > a a 2 1 > < w;  leI ð xÞ ¼ x A > w aa44a ; > > 3 : 0;

A

a1  x\a2 a2  x  a3 a3 \x  a4 otherwise

and

8  0  a x > > w a0 2a0 ; > > 1 2 < meI ð xÞ ¼ 0;  A > x > w aa33a ; > 0 > 4 : w;

a01  x\a2 a2  x  a3 a3 \x  a04 otherwise

e I may be denoted as A generalized trapezoidal intuitionistic fuzzy number A   e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a2 ; a3 ; a0 ; w . A 1 4 eI ¼ Definition 5.2.3 A generalized trapezoidal intuitionistic fuzzy number A   a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w is said to be a non-negative generalized trapezoidal intuitionistic fuzzy number if and only if a01  0. eI ¼ Definition 5.2.4 Two generalized trapezoidal intuitionistic fuzzy numbers A     e I ¼ b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; w2 are said a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w1 and B to be equal if and only if a1 ¼ b1 ; a2 ¼ b2 ; a3 ¼ b3 ; a4 ¼ b4 ; a01 ¼ b01 ; a02 ¼ b02 and w1 ¼ w2 .   e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a2 ; a3 ; a0 ; w 1 eI ¼ and B Definition 5.2.5 Let A 1 4   b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; w2 be two generalized trapezoidal intuitionistic fuzzy eI  B e I ¼ ða1 þ b1 ; a2 þ b2 ; a3 þ b3 ; a4 ; a01 þ b01 ; a2 þ b2 ; a3 þ numbers. Then, A 0 0 b3 ; a4 þ b4 ; minimumfw1 ; w2 gÞ .   e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a2 ; a3 ; a0 ; w1 and B e I ¼ ð b1 ; b 2 ; b3 ; Definition 5.2.6 Let A 1 4 0 0 b4 ; b1 ; b2 ; b3 ; b4 ; w2 Þ be two generalized trapezoidal intuitionistic fuzzy numbers. e I B e I ¼ ða1  b4 ; a2  b3 ; a3  b2 ; a4  b1 ; a01  b04 ; a2  b3 ; a3  b2 ; a04  Then, A 0 b1 ; minimumfw1 ; w2 gÞ .

5 JMD Approach for Solving Unbalanced Fully Generalized …

208

  e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a2 ; a3 ; a0 ; w1 and B e I ¼ ð b1 ; b2 ; b3 ; Definition 5.2.7 Let A 1 4 0 0 b4 ; b1 ; b2 ; b3 ; b4 ; w2 Þ be two non-negative generalized trapezoidal intuitionistic   eI  B e I ¼ a1 b1 ; a2 b2 ; a3 b3 ; a4 ; a01 b01 ; a2 b2 ; a3 b3 ; a04 b04 ; w , fuzzy numbers. Then, A where w ¼ minimumfw1 ; w2 g.   e I ¼ a1 ; a2 ; a3 ; a4 ; a0 ; a2 ; a3 ; a0 ; w1 be a generalized Definition 5.2.8 Let A 1 4 trapezoidal intuitionistic fuzzy number and k be a real number. Then, eI ¼ kA

5.3

  0 0 ka1 ; ka2 ; ka3 ; ka4 ; ka10 ; ka2 ; ka3 ; ka40 ; w1 ; k  0 ka4 ; ka3 ; ka2 ; ka1 ; ka4 ; ka3 ; ka2 ; ka1 ; w1 ; k\0

Tabular Representation of a Fully Generalized Trapezoidal Intuitionistic Fuzzy Transportation Problem

A fully generalized trapezoidal intuitionistic fuzzy transportation problem can be represented by Table 5.1 [1]. where (i) The generalized trapezoidal intuitionistic fuzzy number ~cIij represents the generalized intuitionistic fuzzy transportation cost for supplying the one  unit  quantity of the product from the ith source ðSi Þ to the jth destination Dj , (ii) The generalized trapezoidal intuitionistic fuzzy number ~ aIi represents the generalized intuitionistic fuzzy availability of the product at the ith source ðSi Þ,

Table 5.1 Tabular representation of fully generalized trapezoidal intuitionistic fuzzy transportation problem Sources

Destinations D1 D2

Dj



Dn

Generalized intuitionistic fuzzy availability

S1

~cI11

~cI12



~cI1j



~cI1n

~ aI1

.. . Si

.. .

.. .

.. .

.. .

~cI12

.. .

.. .

~cIi1

.. .

~cIin

~ aIi

.. .

.. . ~cIm2

.. . ~cmn

.. .

~cIm1

.. .

~bI 1

~bI 2



~ bIm

.. . Sm Generalized intuitionistic fuzzy demand

~cIij

.. .

.. . ~cImj



~bI j

~ aIm

5.3 Tabular Representation of a Fully Generalized Trapezoidal …

209

(iii) The generalized trapezoidal intuitionistic fuzzy number ~ bIj represents the intuitionistic fuzzy demand of the product at the jth destination generalized  Dj .

5.4

Existing Approach for Comparing Generalized Trapezoidal Intuitionistic Fuzzy Numbers

It is well-known fact that an optimal solution of a transportation problem will be that feasible solution corresponding to which the total transportation cost will be minimum. On the same direction, the generalized intuitionistic fuzzy optimal solution of the fully generalized trapezoidal intuitionistic fuzzy transportation problem, represented by Table 5.1, will be that generalized intuitionistic fuzzy feasible solution corresponding to which the total transportation cost will be minimum. Since, in the case of a fully generalized trapezoidal intuitionistic fuzzy transportation problem, the total cost will be a generalized trapezoidal intuitionistic fuzzy number. Therefore, to find a generalized trapezoidal intuitionistic fuzzy optimal solution of the fully generalized trapezoidal intuitionistic fuzzy transportation problem, represented by Table 5.1, there is a need to find the minimum of generalized trapezoidal intuitionistic fuzzy numbers, i.e., there is a need to compare generalized trapezoidal intuitionistic fuzzy numbers. In this section, the approach for comparing the generalized trapezoidal intuitionistic fuzzy numbers, used in Chakraborty et al.’s approach [1], has beendiscussed.   Let a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w1 and b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; w2 be two generalized trapezoidal intuitionistic fuzzy numbers. Then,     (i) a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w1 b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; w2 if H ða1 ;  a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; minimumfw1 ; w2 gÞ [ H b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; minimumfw1 ; w2 gÞ:     0 (ii) a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w1  b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; w2 if H ða1 ; b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; minimumfw1 ; w2 gÞ\ minimumfw1 ; w2 g:     0 0 0 0 (iii) a1 ; a2 ; a3 ; a4 ; a1 ; a2 ; a3 ; a4 ; w1 ¼ b1 ; b2 ; b3 ; b4 ; b1 ; b2 ; b3 ; b4 ; w2 if H ða1 ;  a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; minimumfw1 ; w2 gÞ ¼ H b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; minimumfw1 ; w2 gÞ: where   H a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; minimumfw1 ; w2 g    minimumfw1 ; w2 g  ¼ a1 þ a2 þ a3 þ a4 þ a01 þ a2 þ a3 þ a04 8

5 JMD Approach for Solving Unbalanced Fully Generalized …

210

and   H b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; minimumfw1 ; w2 g    minimumfw1 ; w2 g  b1 þ b2 þ b3 þ b4 þ b01 þ b2 þ b3 þ b04 : ¼ 8

5.5

Chakraborty et al.’s Approach for Solving Balanced Fully Generalized Trapezoidal Intuitionistic Fuzzy Transportation Problems

The aim of this chapter is to point out some limitations and a drawback of Chakraborty et al.’s approach [1]. Since, to achieve this aim, there is a need to discuss Chakraborty et al.’s approach [1]. Therefore, a brief review of Chakraborty et al.’s approach [1] has been presented in this section. Chakraborty et al. [1] proposed the following approach to solve generalized balanced fully trapezoidal intuitionistic fuzzy transportation problem represented by Table 5.1. Step 1: Write the generalized intuitionistic fuzzy linear programming problem (5.5.1) corresponding to the balanced fully generalized trapezoidal intuitionistic fuzzy transportation problem represented by Table 5.1. Generalized intuitionistic fuzzy linear programming problem (5.5.1) 2

00

n

o

13

1

6 C7 n BB 0 0 1 2 3 C Minimize4m i¼1 j¼1 @@cij;1 ; cij;2 ; cij;3 ; cij;4 ; cij;1 ; cij;2 ; cij;3 ; cij;4 ; min 1  i  m wij ; wi ; wj A  xij A5 1jn

Subject to 0 n X j¼1

n

o

1

1jn

0

i¼1

o

B C xij @ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m w1ij ; w2i ; w3j A; i ¼ 1; 2; . . .; m;

m X

n

1

B C xij @bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min 1  i  m w1ij ; w2i ; w3j A; j 1jn ¼ 1; 2; . . .; n;

xij is a non-negative real number.

5.5 Chakraborty et al.’s Approach for Solving Balanced Fully …

211

  Step 2: Using the existing multiplication k  a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w ¼   ka1 ; ka2 ; ka3 ; ka4 ; ka01 ; ka2 ; ka3 ; ka04 ; w ; k  0 , transform the generalized intuitionistic fuzzy linear programming problem (5.5.1) into its equivalent generalized intuitionistic fuzzy linear programming problem (5.5.2). Generalized intuitionistic fuzzy linear programming problem (5.5.2)  h n 0 0 Minimize m  i¼1 j¼1 cij;1 xij ; cij;2 xij ; cij;3 xij ; cij;4 xij ; cij;1 xij ; cij;2 xij ; cij;3 xij ; cij;4 xij ; n oi min w1ij ; w2i ; w3j 1im 1jn

Subject to Constraints of the generalized intuitionistic fuzzy linear programming problem (5.5.1). Step 3: Using the comparing approach, discussed in Sect. 5.4, transform the generalized intuitionistic fuzzy linear programming problem (5.5.2) into its equivalent crisp linear programming problem (5.5.3). Crisp linear programming problem (5.5.3) h h  n 0 0 Minimize H m  i¼1 j¼1 cij;1 xij ; cij;2 xij ; cij;3 xij ; cij;4 xij ; cij;1 xij ; cij;2 xij ; cij;3 xij ; cij;4 xij ; n o min w1ij ; w2i ; w3j 1im 1jn

Subject to

H

n X

0

!

¼ H @ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min

xij

n

ij

1im 1jn

j¼1

1 o w1 ; w2 ; w3 A; i

j

i ¼ 1; 2; . . .; m;

H

m X

0

! xij

¼ H @bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min

i¼1

1im 1jn

n

1 o w1 ; w2 ; w3 A; ij

i

j

j ¼ 1; 2; . . .; n; xij is a non-negative real number.    0 0 Step 4: Using the relations, H m a ; a ; a ; a ; a ; a ; a ; a ; w ¼ i;1 i;2 i;3 i;4 i;2 i;3 i¼1 i;1 i;4   P  P   Pm n n 0 0 , H i¼1 H ai;1 ; ai;2 ; ai;3 ; ai;4 ; ai;1 ; ai;2 ; ai;3 ; ai;4 ; w j¼1 xij ¼ j¼1 H xij and

5 JMD Approach for Solving Unbalanced Fully Generalized …

212

 Pm  Pm   H i¼1 xij ¼ i¼1 H xij , transform the crisp linear programming problem (5.5.3) into its equivalent crisp linear programming problem (5.5.4). Crisp linear programming problem (5.5.4) " Minimize

m X n  X H cij;1 xij ; cij;2 xij ; cij;3 xij ; cij;4 xij ; c0ij;1 xij ; cij;2 xij ; cij;3 xij ; c0ij;4 xij ; i¼1 j¼1

n oi min 1  i  m w1ij ; w2i ; w3j 1jn

Subject to  n n o X   H xij ¼ H ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m w1ij ; w2i ; w3j ; 1jn

j¼1

i ¼ 1; 2; . . .; m;  m n o X   0 0 1 2 3 H xij ¼ H bj;1 ; bj;2 ; bj;3 ; bj;4 ; bj;1 ; bj;2 ; bj;3 ; bj;4 ; min 1  i  m wij ; wi ; wj ; 1jn

i¼1

j ¼ 1; 2; . . .; n;

xij is a non-negative real number.     Step 5: Using the expressions H a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w ¼ w8 ða1 þ a2 þ   a3 þ a4 þ a01 þ a2 þ a3 þ a04 Þ and H xij ¼ xij , transform crisp linear programming problem (5.5.4) into its equivalent crisp linear programming problem (5.5.5). Crisp linear programming problem (5.5.5) n o1 min 1  i  m w1ij ; w2i ; w3j m n 6B C XX 1jn cij;1 xij þ cij;2 xij þ cij;3 xij Minimize4@ A 8 i¼1 j¼1 20

þ cij;4 xij þ c0ij;1 xij þ cij;2 xij þ cij;3 xij þ c0ij;4 xij

i

Subject to n X j¼1

n o1 0 min 1  i  m w1ij ; w2i ; w3j   1jn B C xij ¼ @ A ai;1 þ ai;2 þ ai;3 þ ai;4 þ a0i;1 þ ai;2 þ ai;3 þ a0i;4 ; 8

5.5 Chakraborty et al.’s Approach for Solving Balanced Fully …

213

i ¼ 1; 2; . . .; m; n o1 min 20c1  i  m w1ij ; w2i ; w3j   C B 1jn xij ¼ @ A bj; 1 þ bj; 2 þ bj;3 þ bj;4 þ b0j;1 þ bj;2 þ bj;3 þ b0j;4 ; 8 0

m X i¼1

j ¼ 1; 2; . . .; n;

xij is a non-negative real number.

 Step 6: Find the crisp optimal solution xij ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n of the crisp linear programming problem (5.5.5).

 Step 7: Using the crisp optimal solution xij ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n , obtained in Step 6, find the generalized intuitionistic fuzzy optimal value P P Pn Pn Pn Pn Pn Pn 0 Pn Pn n n i¼1 j¼1 cij;4 ; i¼1 j¼1 cij;1 ; i¼1 j¼1 cij;1 ; i¼1 j¼1 cij;2 ; i¼1 j¼1 cij;3 ;;  n o Pn Pn Pn Pn 0 Pn Pn 1 2 3 of the i¼1 j¼1 cij;2 ; i¼1 j¼1 cij;3 ; i¼1 j¼1 cij;4 ; min 1  i  m wij ; wi ; wj 1jn

generalized intuitionistic fuzzy linear programming problem (5.5.1).

5.6

Origin of the Generalized Intuitionistic Fuzzy Linear Programming Problem

The generalized intuitionistic fuzzy linear programming problem (5.5.1), used in Chakraborty et al.’s approach [1], has been obtained as follows: Step 1: To find an optimal solution of the balanced fully generalized trapezoidal intuitionistic fuzzy transportation problem, represented by Table 5.1, is equivalent to find an optimal solution of the generalized intuitionistic fully fuzzy linear programming problem (5.6.1). Generalized intuitionistic fully fuzzy linear programming problem (5.6.1) h  i n I ~ ~ Minimize m   x c ij i¼1 j¼1 ij Subject to n X

~xij ~aIi ;

i ¼ 1; 2; . . .; m;

~xij ~bIj ;

j ¼ 1; 2; . . .; n;

j¼1 m X i¼1

~xij is a non-negative generalized trapezoidal intuitionistic fuzzy number.

5 JMD Approach for Solving Unbalanced Fully Generalized …

214

Step 2: Replacing the parameters ~cIij , ~aIi , ~bIj and ~xij with the generalized trapezoidal   intuitionistic fuzzy numbers cij;1 ; cij;2 ; cij;3 ; cij;4 ; c0ij;1 ; cij;2 ; cij;3 ; c0ij;4 ; w1ij ,     ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; w2i , bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; w3j  n o 0 0 1 2 3 and xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj , respec1jn

tively, the generalized intuitionistic fully fuzzy linear programming problem (5.6.1) can be transformed into its equivalent generalized intuitionistic fully fuzzy linear programming problem (5.6.2). Generalized intuitionistic fully fuzzy linear programming problem (5.6.2) h   n 0 0 1 Minimize m i¼1 j¼1 ð cij;1 ; cij;2 ; cij;3 ; cij;4 ; cij;1 ; cij;2 ; cij;3 ; cij;4 ; wij  n o  xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j 1jn

Subject to  n o nj¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j 1jn   ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; w2i ; i ¼ 1; 2; . . .; m;  n o 0 0 1 2 3 xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj 1jn   bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; w3j ; j ¼ 1; 2; . . .; n; m i¼1

 n o 0 0 1 2 3 xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj 1jn

is

a

non-negative generalized trapezoidal intuitionistic fuzzy number.   Step 3: Using the relation, a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w1  ðb1 ; b2 ; b3 ; b4 ;    b01 ; b2 ; b3 ; b04 ; w2 Þ ¼ a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; minfw1 ; w2 g  b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; minfw1 ; w2 gÞ, the generalized intuitionistic fully fuzzy linear programming problem (5.6.2) can be transformed into its equivalent generalized intuitionistic fully fuzzy linear programming problem (5.6.3).

5.6 Origin of the Generalized Intuitionistic Fuzzy Linear …

215

Generalized intuitionistic fully fuzzy linear programming problem (5.6.3)  n o n 0 0 1 2 3  Minimize m  c ; c ; c ; c ; c ; c ; c ; c ; min w ; w ; w 1  i  m ij;1 ij;2 ij;3 ij;4 ij;2 ij;3 i¼1 j¼1 ij;1 ij;4 ij i j 1jn   n o xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j 1jn

Subject to Constraints of the generalized intuitionistic fully fuzzy linear programming problem (5.6.2). Step 4: Using the existing approach for comparing generalized trapezoidal intuitionistic fuzzy numbers, discussed in Sect. 5.4, the generalized intuitionistic fully fuzzy linear programming problem (5.6.3) can be transformed into its equivalent crisp linear programming problem (5.6.4). Crisp linear programming problem (5.6.4)  n o n 0 0 1 2 3 c Minimize H m  ; c ; c ; c ; c ; c ; c ; c ; min w ; w ; w  1  i  m ij;1 ij;2 ij;3 ij;4 ij;2 ij;3 i¼1 j¼1 ij;1 ij;4 ij i j 1jn   n o xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j 1jn

Subject to   n o H nj¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j ¼ 1jn  n o ; i ¼ 1; 2; . . .; m; H ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m w1ij ; w2i ; w3j 1jn

  n o m 0 0 1 2 3 H i¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj ¼ 1jn   n o ; j ¼ 1; 2; . . .; n; H bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min 1  i  m w1ij ; w2i ; w3j 1jn

 n o xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j

is a non-

1jn

negative generalized trapezoidal intuitionistic fuzzy number. h  i 0 0 1 Step 5: Using the relation, H m ¼ i¼1 ci;1 ; ci;2 ; ci;3 ; ci;4 ; ci;1 ; ci;2 ; ci;3 ; ci;4 ; wij

 n o 0 0 H c ; c ; c ; c ; c ; c ; c ; c ; min w1ij m 1  i  m ij;1 ij;2 ij;3 ij;4 ij;2 ij;3 i¼1 ij;1 ij;4 1jn

,

the

crisp

5 JMD Approach for Solving Unbalanced Fully Generalized …

216

linear programming problem (5.6.4) can be transformed into its equivalent crisp linear programming problem (5.6.5). Crisp linear programming problem (5.6.5) Minimize 

X  m X n n o H cij;1 ; cij;2 ; cij;3 ; cij;4 ; c0ij;1 ; cij;2 ; cij;3 ; c0ij;4 ; min 1  i  m w1ij ; w2i ; w3j  1jn

i¼1 j¼1

n

xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j

o

1jn

Subject to !  n n o X 0 0 1 2 3 H xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj ¼ 1jn

j¼1



n

H ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m w1ij ; w2i ; w3j

o

1jn

;

i ¼ 1; 2; . . .; m;

!  m n o X 0 0 1 2 3 H xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj 1jn

i¼1

 H

bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min 1  i  m 1jn

n

w1ij ; w2i ; w3j

o

;

j ¼ 1; 2; . . .; n;

 n o H xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j is a non1jn

negative real number.

  Step 6: Using the relation H a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w  ðb1 ; b2 ; b3 ; b4 ;    b01 ; b2 ; b3 ; b04 ; wÞ ¼ H a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w  H b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; wÞ,the crisp linear programming problem (5.6.5) can be transformed into its equivalent crisp linear programming problem (5.6.6). Crisp linear programming problem (5.6.6) " Minimize 

 m X n n o X H cij;1 ; cij;2 ; cij;3 ; cij;4 ; c0ij;1 ; cij;2 ; cij;3 ; c0ij;4 ; min 1  i  m w1ij ; w2i ; w3j  1jn

i¼1 j¼1

n

xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j 1jn

o

5.6 Origin of the Generalized Intuitionistic Fuzzy Linear …

217

Subject to !  n n o X 0 0 1 2 3 H xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj ¼ 1jn

j¼1

 n o H ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m w1ij ; w2i ; w3j ; 1jn

i ¼ 1; 2; . . .; m;

!  m n o X 0 0 1 2 3 ¼ H xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj 1jn

i¼1

 n o 0 0 1 2 3 H bj;1 ; bj;2 ; bj;3 ; bj;4 ; bj;1 ; bj;2 ; bj;3 ; bj;4 ; min 1  i  m wij ; wi ; wj ; 1jn

j ¼ 1; 2; . . .; n;

 n o H xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j is a non1jn

negative real number.  n o 0 0 1 2 3 Step 7: Since, H xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj 1jn  is a non-negative real number. So, assuming H xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; n o x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j ¼ xij , the crisp linear programming problem (5.6.6) 1jn

can be transformed into its equivalent crisp linear programming problem (5.6.7). Crisp linear programming problem (5.6.7) "  # m X n n o X 0 0 1 2 3  xij Minimize H cij;1 ; cij;2 ; cij;3 ; cij;4 ; cij;1 ; cij;2 ; cij;3 ; cij;4 ; min 1  i  m wij ; wi ; wj i¼1 j¼1

1jn

Subject to n X

 n o 0 0 1 2 3 xij ¼ H ai;1 ; ai;2 ; ai;3 ; ai;4 ; ai;1 ; ai;2 ; ai;3 ; ai;4 ; min 1  i  m wij ; wi ; wj ; 1jn

j¼1

i ¼ 1; 2; . . .; m; n X

 n o xij ¼ H bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min 1  i  m w1ij ; w2i ; w3j ;

j¼1

j ¼ 1; 2; . . .; n; xij is a non-negative real number.

1jn

218

5 JMD Approach for Solving Unbalanced Fully Generalized …

Step 8: Using the existing approach for comparing generalized trapezoidal intuitionistic fuzzy numbers, discussed in Sect. 5.4, the crisp linear programming problem (5.6.7) can be transformed into its equivalent generalized intuitionistic fuzzy linear programming problem (5.5.1).

5.7

Limitations of Chakraborty et al.’s Approach

In this section, some limitations of Chakraborty et al.’s approach [1] have been pointed out. (i) Chakraborty et al.’s approach [1] cannot be used to solve unbalanced fully generalized trapezoidal intuitionistic fuzzy transportation problems. (ii) To solve a real-life transportation problem, the opinion of two or more experts about the parameters is collected. Then, all the collected information is aggregated to obtain a single value of each parameter. Since, Chakraborty et al.’s approach [1] has been proposed by considering the assumption that the aggregated value of each parameter is available. Therefore, Chakraborty et al.’s approach [1] cannot be used to solve several real-life fully generalized trapezoidal intuitionistic fuzzy transportation problems. For example, Chakraborty et al.’s approach [1] cannot be used to solve the fully generalized trapezoidal transportation problem considered in Example 5.7.1. Example 5.7.1 Let us consider a product needs to be supplied from two sources to two destinations. For the same purpose, the information about each parameter is collected from two experts. If (i) Table 5.2 represents the generalized trapezoidal intuitionistic fuzzy transportation cost, the generalized trapezoidal intuitionistic fuzzy availability and the generalized trapezoidal intuitionistic fuzzy demand according to the first decision-maker. (ii) Table 5.3 represents the generalized trapezoidal intuitionistic fuzzy transportation cost, the generalized trapezoidal intuitionistic fuzzy availability and the generalized trapezoidal intuitionistic fuzzy demand according to the second decision-maker. Then, this fully generalized trapezoidal intuitionistic fuzzy transportation problem cannot be solved by Chakraborty et al.’s approach [1].

5.8 Invalidity of Chakraborty et al.’s Approach

219

Table 5.2 Generalized intuitionistic fuzzy data provided by the first decision-maker Sources

Destinations D1 

S1



S2 Generalized intuitionistic fuzzy demand



10; 30; 40; 50; 5; 30; 40; 55; 0:5

D2 

15; 30; 50; 80; 10; 30; 50; 90; 0:8 40; 60; 70; 90; 30; 60; 70; 95; 0:5

 

 

25; 50; 60; 80; 10; 50; 60; 90; 0:6



20; 40; 60; 80; 15; 40; 60; ; 85; 0:7   10; 45; 55; 70; 5; 45; 55; 80; 0:5



Generalized intuitionistic fuzzy availability   20; 60; 70; 80; 15; 60; 70; 85; 0:5   25; 45; 60; 70; 20; 45; 60; ; 80; 0:5

Table 5.3 Generalized intuitionistic fuzzy data provided by the second decision-maker Sources

S1 S2 Generalized intuitionistic fuzzy demand

5.8

Destinations D1

D2





 15; 35; 45; 55; 10; 35; 45; ; 60; 0:5   20; 35; 55; 85; 15; 35; 55; 95; 0:4   45; 65; 75; 95; 35; 65; 75; 100; 0:2

 

30; 55; 65; 85; 15; 55; 65; 95; 0:8 25; 45; 65; 85; 20; 45; 65; 90; 0:3 15; 50; 60; 75; 10; 50; 60; 85; 0:6

  

Generalized intuitionistic fuzzy availability   25; 65; 75; 85; 20; 65; 75; 90; 0:5   30; 50; 65; 75; 25; 50; 65; 85; 0:5

Invalidity of Chakraborty et al.’s Approach

In this section, it is showed that Chakraborty et al. [1] have used a mathematical incorrect assumption in their proposed approach. Hence, Chakraborty et al.’s approach [1] is not valid. et al. [1] have used

It is obvious from Step 6 that  Chakraborty   the relation H a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w  b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; w ¼ H ða1 ; a2 ;   a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; wÞ H b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; w to transform the crisp linear programming problem (5.6.5) into its equivalent crisp linear programming problem (5.6.6). While, the following example clearly indicates that this relation is not valid, i.e., the generalized crisp linear programming problem (5.6.5) cannot be transformed into the crisp linear programming problem (5.6.6). Hence, the generalized intuitionistic fuzzy linear programming problem (5.5.1) is not valid.   Let a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w ¼ ð2; 3; 4; 5; 1; 2; 4; 6; 0:5Þ and ðb1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; wÞ ¼ ð3; 4; 5; 6; 2; 4; 5; 7; 0:5Þ.

220

5 JMD Approach for Solving Unbalanced Fully Generalized …

    Then, H a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w  b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; w ¼ H ½ð2; 3; 4; 5; 1; 3; 4; 6; 0:5Þ  ð3; 4; 5; 6; 2; 4; 5; 7; 0:5Þ ¼ H ð6; 12; 20; 30; 3; 12; 20; 42; 0:5Þ ¼ 17:75.     While, H a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w  H b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ; b04 ; w ¼ H ð2; 3; 4; 5; 1; 2; 4; 6; 0:5Þ  H ð3; 4; 5; 6; 2; 4; 5; 7; 0:5Þ ¼ 1:6875  2:25 ¼ 3:796875.

5.9

Inappropriateness of the Existing Approach for Comparing Generalized Trapezoidal Intuitionistic Fuzzy Numbers

Chakraborty et al. [1] have used an existing approach, discussed in Sect. 5.4, for comparing generalized trapezoidal intuitionistic fuzzy numbers. However, the following example clearly indicates that it is inappropriate to use it for comparing generalized trapezoidal intuitionistic fuzzy numbers. It is obvious that ð20; 60; 80; 100; 10; 60; 80; 110; 0:8Þ and ð25; 55; 85; 95; 5; 55; 85; 115; 0:8Þ are two different generalized trapezoidal intuitionistic fuzzy numbers. Therefore, minimum of these generalized trapezoidal intuitionistic fuzzy numbers should be either ð20; 60; 80; 100; 10; 60; 80; 110; 0:8Þ or ð25; 55; 85; 95; 5; 55; 85; 115; 0:8Þ. While, as H ð20; 60; 80; 100; 10; 60; 80; 110; 0:8Þ ¼ H ð25; 55; 85; 95; 5; 55; 85; 115; 0:8Þ ¼ 52. Therefore, according to the existing approach for comparing generalized trapezoidal intuitionistic fuzzy numbers, discussed in Sect. 5.4, both the generalized trapezoidal intuitionistic fuzzy numbers ð20; 60; 80; 100; 10; 60; 80; 110; 0:8Þ and ð25; 55; 85; 95; 5; 55; 85; 115; 0:8Þ will represent the minimum value.

5.10

PRABHUS Approach for Comparing Generalized Trapezoidal Intuitionistic Fuzzy Numbers

It is obvious from Sect. 5.9 that it is not appropriate to use the existing approach, discussed in Sect. 5.4, for comparing generalized trapezoidal intuitionistic fuzzy numbers. In this section, a new approach (named as PRABHUS approach) has been proposed for comparing generalized trapezoidal intuitionistic fuzzy numbers Using the proposed PRABHUS  approach, the generalized trapezoidal intuI e itionistic fuzzy numbers A 1 ¼ a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; w1 and   e I ¼ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a0 ; a2;2 ; a2;3 ; a0 ; w2 A 2;1 2;4 2 can be compared as follows:

5.10

PRABHUS Approach for Comparing Generalizsed …

221

  Step 1: Check that P a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [    or P a1;1 ; a1;2 ; a1;3 ; a1;4 ; P a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 g  P a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 gÞ\    minfw1 ; w2 gÞ or P a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ P a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ , where,    P a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ minfw1 ; w2 g a1;1 þ  a1;2 þ a1;3 þ a1;4 þ a01;1 þ a1;2 þ a1;3 þ a01;4 Þ and P a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ;  a2;3 ; a02;4 ; minfw1 ; w2 gÞ ¼ minfw1 ; w2 g a2;1 þ a2;2 þ a2;3 þ a2;4 þ a02;1 þ a2;2 þ a2;3 þ a02;4 Þ.

   Case (i): If P a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [ P a2;1 ; a2;2 ;  a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ;   a1;3 ; a01;4 ; w1 Þ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .    Case (ii): If P a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g \ P a2;1 ;  a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ;  a1;2 ; a1;3 ; a01;4 ; w1 Þ  a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 Þ.    Case (iii): If P a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ P a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, go to Step 2.  Step 2: Check that R a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 gÞ [    R a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 g or R a1;1 ; a1;2 ; a1;3 ; a1;4 ;  a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 gÞ\ R a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ;    minfw1 ; w2 gÞ or R a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ R a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ, where    R a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ minfw1 ; w2 g a1;1 þ  a1;2 þ a1;3 þ a1;4 þ a1;2 þ a1;3 þ a01;4 Þ and R a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ;   minfw1 ; w2 gÞ ¼ minfw1 ; w2 g a2;1 þ a2;2 þ a2;3 þ a2;4 þ a22 þ a2;3 þ a02;4 .

222

5 JMD Approach for Solving Unbalanced Fully Generalized …

   Case (i): If R a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [ R a2;1 ; a2;2 ;  a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ;  a1;3 ; a01;4 ; w1 Þ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 Þ.    Case (ii): If R a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g \ R a2;1 ; a2;2 ;  a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ;  a1;3 ; a01;4 ; w1 Þ  a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 Þ.    Case (iii): If R a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ R a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, go to Step 2.   Step 3: Check that A a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [    A a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 g or A a1;1 ; a1;2 ; a1;3 ; a1;4 ;  a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 gÞ\A a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; min    fw1 ; w2 gÞ or A a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ A a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ, where    A a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ minfw1 ; w2 g a1;2 þ  a1;3 þ a1;4 þ a1;2 þ a1;3 þ a01;4 Þ and A a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ;   minfw1 ; w2 gÞ ¼ minfw1 ; w2 g a2;2 þ a2;3 þ a2;4 þ a22 þ a2;3 þ a02;4 .    Case (i): If A a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [ A a2;1 ;  a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ;   a1;2 ; a1;3 ; a01;4 ; w1 Þ a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ; w2 .   Case (ii): If A a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 gÞ\A a2;1 ; a2;2 ;  a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then,   a1;3 ; a01;4 ; w1 Þ  a2;1 ; a2;2 ; a2;3 ; a02;1 ; a2;2 ; a02;3 ; w2 .    Case (iii): If A a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ A a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, go to Step 2.   Step 4: Check that B a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [    B a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 g or B a1;1 ; a1;2 ; a1;3 ; a1;4 ;

5.10

PRABHUS Approach for Comparing Generalizsed …

223

 a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 gÞ\B a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; min    fw1 ; w2 gÞ or B a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ B a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ, where    B a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ minfw1 ; w2 g a1;3 þ   a1;4 þ a1;3 þ a01;4 Þ and B a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 g ¼   minfw1 ; w2 g a2;3 þ a2;4 þ a2;3 þ a02;4 .    Case (i): If B a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [ B a2;1 ; a2;2 ;  a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ;   a1;3 ; a01;4 ; w1 Þ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .    Case (ii): If B a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g \B a2;1 ;  a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ;   a1;2 ; a1;3 ; a01;4 ; w1 Þ  a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .    Case (iii): If B a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ B a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, go to Step 2.   Step 5: Check that H a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [    H a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 g or H a1;1 ; a1;2 ; a1;3 ; a1;4 ;  a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 gÞ\H a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; min    fw1 ; w2 gÞ or H a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ H a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ, where   H a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ minfw1 ; w2 g a1;4 þ   a01;4 Þ and H a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 g ¼ minfw1 ; w2 g   a2;4 þ a02;4 .    Case (i): If H a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [ H a2;1 ;  a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ;   a1;2 ; a1;3 ; a01;4 ; w1 Þ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .

224

5 JMD Approach for Solving Unbalanced Fully Generalized …

   Case (ii): If H a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g \ H a2;1 ; a2;2 ;  a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ;   a1;3 ; a01;4 ; w1 Þ  a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .    Case (iii): If H a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ H a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then, go to Step 2.   Step 6: Check that U a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [    U a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 g or U a1;1 ; a1;2 ; a1;3 ; a1;4 ;  a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 gÞ\U a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; min    fw1 ; w2 gÞ or U a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ U a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ, where    U a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ minfw1 ; w2 g a01;4     and U a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 g ¼ minfw1 ; w2 g a02;4 .    Case (i): If U a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [ U a2;1 ; a2;2 ;  a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then,   a1;3 ; a01;4 ; w1 Þ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .    Case (ii): If U a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g \ U a2;1 ; a2;2 ;  a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then,   a1;3 ; a01;4 ; w1 Þ  a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .    Case (iii): If U a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ U a2;1 ;  a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then,   a1;2 ; a1;3 ; a01;4 ; w1 Þ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .   Step 7: Check that S a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [    S a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 g or S a1;1 ; a1;2 ; a1;3 ; a1;4 ;  S a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; min a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 gÞ\    fw1 ; w2 gÞ or S a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ S a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ,

5.10

PRABHUS Approach for Comparing Generalizsed …

225

where    S a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ w1 and S a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ ¼ w2 .    Case (i): If S a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g [ S a2;1 ; a2;2 ;  a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then,   a1;3 ; a01;4 ; w1 Þ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .    Case (ii): If S a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g \S a2;1 ; a2;2 ;  a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then,   a1;3 ; a01;4 ; w1 Þ  a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .    Case (iii): If S a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; minfw1 ; w2 g ¼ S a2;1 ; a2;2 ;  a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; minfw1 ; w2 gÞ. Then,   a1;3 ; a01;4 ; w1 Þ ¼ a2;1 ; a2;2 ; a2;3 ; a2;4 ; a02;1 ; a2;2 ; a2;3 ; a02;4 ; w2 .

5.11

Proposed JMD Approach

In this section, to overcome the limitations and to resolve a drawback of Chakraborty et al.’s approach [1], a new approach (named as JMD approach) has been proposed for solving unbalanced fully generalized trapezoidal intuitionistic fuzzy transportation problems. Step 1: Check that the aggregated value of the generalized trapezoidal intuitionistic fuzzy transportation cost, the generalized trapezoidal intuitionistic fuzzy availability and the generalized trapezoidal intuitionistic fuzzy demand, provided by all the decision-makers, is available or not. Case (i): If it is available, then go to Step 2. Case (ii): If it is not available, then find generalized trapezoidal intuitionistic fuzzy number ~cij ¼ P   Pp Pp Pp p k k k k wk  ~ckij ¼ k¼1 wk cij;1 ; k¼1 wk cij;2 ; k¼1 wk cij;3 ; k¼1 wk cij;4 ;  n o Pp Pp Pp Pp k k k k k1 w c ; w c : ; w c ; w c ; min w 1im ij k¼1 k ij;5 k¼1 k ij;2 k¼1 k ij;3 k¼1 k ij;6

(i) The

pk¼1

1jn

representing the aggregated value of the generalized intuitionistic fuzzy cost

226

5 JMD Approach for Solving Unbalanced Fully Generalized …

for supplying the one unit quantity of the product from the ith source to the jth destination. (ii) The generalized trapezoidal intuitionistic fuzzy number ~ ai ¼ P  P P P p  p p p p k k k k k¼1 wk  ~ aki ¼ k¼1 wk ai;1 ; k¼1 wk ai;2 ; k¼1 wk ai;3 ; k¼1 wk ai;4 ; Pp P p k k k¼1 wk ai;5 ; k¼1 wk ai;2 ; :

k2  Pp Pp k k representing the aggregated k¼1 wk ai;3 ; k¼1 wk ai;6 ; min1  i  m wi value of the generalized intuitionistic fuzzy availability of the product at the ith source. (iii) The generalized trapezoidal intuitionistic fuzzy number ~ bj ¼ P   P P P p p p p p k k k k k¼1 wk  ~bkj ¼ k¼1 wk bj;1 ; k¼1 wk bj;2 ; k¼1 wk bj;3 ; k¼1 wk bj;4 ; n o Pp Pp Pp Pp k k k k k3 repk¼1 wk bj;5 ; k¼1 wk bj;2 ; : k¼1 wk bj;3 ; k¼1 wk bj;6 ; min1  j  n wj resenting the aggregated value of the generalized intuitionistic fuzzy demand of the product at the jth destination. where (i) wk 2 ½0; 1 represents the normalized weight of the kth decision-maker.  (ii) The generalized trapezoidal intuitionistic fuzzy number ~ckij ¼ ckij;1 ; ckij;2 ; ckij;3 ; ckij;4 ; ckij;5 ; ckij;2 ; ckij;3 ; ckij;6 ; wk1 ij Þ represents the generalized trapezoidal intuitionistic fuzzy transportation cost for supplying the one unit quantity of the product from the ith source to the jth destination according to the kth decision-maker. 

~ki ¼ aki;1 ; aki;2 ; aki;3 ; (iii) The generalized trapezoidal intuitionistic fuzzy number a aki;4 ; aki;5 ; aki;2 ; aki;3 ; aki;6 ; wk2 i Þ represents the generalized trapezoidal intuitionistic fuzzy availability of the product at the ith source according to the kth decision-maker.  (iv) The generalized trapezoidal intuitionistic fuzzy number ~ bkj ¼ bkj;1 ; bkj;2 ; bkj;3 ; bkj;4 ; bkj;5 ; bkj;2 ; bkj;3 ; bkj;6 ; wk3 j Þ represents the generalized trapezoidal intuitionistic fuzzy demand of the product at the jth destination according to the kth decision-maker. and go to Step 2. For example, if in Example 5.7.1, the normalized weights of first and second decision-makers are 0.4 and 0.6, repsectively. Then, Table 5.13.1 will represent the aggregated generalized trapezoidal intuitionistic fuzzy cost for supplying the one unit quantity of the product, the generalized trapezoidal intuitionistic fuzzy availability and the generalized trapezoidal intuitionistic fuzzy demand (Table 5.4).

5.11

Proposed JMD Approach

227

Table 5.4 Aggregated generalized trapezoidal intuitionistic fuzzy data of decision-makers Source

S1 S2 Generalized intuitionistic fuzzy demand

Destinations D1   

13; 33; 43; 53; 8; 33; 43; 58; 0:5

D2 

18; 33; 53; 83; 13; 33; 53; 93; 0:4 43; 63; 73; 93; 33; 63; 73; 98; 0:2

 

 

28; 53; 63; 83; 13; 53; 63; 93; 0:6



 23; 43; 63; 83; 18; 43; 63; 88; 0:3   13; 48; 58; 73; 8; 48; 58; 83; 0:5

Generalized intuitionistic fuzzy availability   23; 63; 73; 83; 18; 63; 73; 88; 0:5   28; 48; 63; 73; 23; 48; 63; 83; 0:5

Step 2: Find the total generalized intuitionistic of the product, i.e.,  fuzzyavailability Pm Pm Pm m 0 0 2 i¼1 ai;1 ; ai;2 ; ai;3 ; ai;4 ; ai;1 ; ai;2 ; ai;3 ; ai;4 ; wi ¼ i¼1 ai;1 ; i¼1 ai;2 ; i¼1 ai;3 ;

2 Pm 0 Pm Pm Pm 0 Pm i¼1 ai;4 ; i¼1 ai;1 ; i¼1 ai;2 ; i¼1 ai;3 ; i¼1 ai;4 ;min1  i  m wi Þ and the total generalized intuitionistic demand of  the product, i.e.,  Pn Pn Pn n 0 0 3 j¼1 bj;1 ; bj;2 ; bj;3 ; bj;4 ; bj;1 ; bj;2 ; bj;3 ; bj;4 ; wj ¼ j¼1 bj;1 ; j¼1 bj;2 ; j¼1 bj;3 ; n o Pn Pn 0 Pn Pn Pn 0 3 j¼1 bj;4 ; j¼1 bj;1 ; j¼1 bj;2 ; j¼1 bj;3 ; j¼1 bj;4 ; min1  j  n wj Þ P Pm Pm Pm Pm 0 m Step 3: Check that i¼1 ai;1 ; i¼1 ai;2 ; i¼1 ai;3 ; i¼1 ai;4 ; i¼1 ai;1 ; P

2 Pm Pm Pm 0 P P n n n i¼1 ai;2 ; i¼1 ai;3 ; i¼1 ai;4 ;min1  i  m wi Þ ¼ j¼1 bj;1 ; j¼1 bj;2 ; j¼1 bj;3 ; n o Pn Pn 0 Pn Pn Pn 0 3 i.e., j¼1 bj;4 ; j¼1 bj;1 ; j¼1 bj;2 ; j¼1 bj;3 ; j¼1 bj;4 ; min1  j  n wj Þ, Pn Pm Pn Pm Pn Pm ai;1 ¼ j¼1 bj;1 , ai;2 ¼ bj;2 , i¼1 ai;3 ¼ j¼1 bj;3 , Pn Pm 0 i¼1 Pn 0 j¼1 Pm 0 P Pi¼1 m n 0 a ¼ b , a ¼ b , a ¼ b and i¼1 i;4 j¼1 j;4 i¼1 i;1 i¼1 i;4 j¼1 j;4 n o j¼1 j;1

2 min1  i  m wi ¼ min1  j  n w3j or not. P Pm Pm Pm Pm 0 Pm Pm m Case 1: If i¼1 ai;1 ; i¼1 ai;2 ; i¼1 ai;3 ; i¼1 ai;4 ; i¼1 ai;1 ; i¼1 ai;2 i¼1 ai;3 ; P

2 Pn Pn Pn Pn 0 Pm 0 n i¼1 ai;4 ; min1  i  m wi Þ ¼ j¼1 bj;1 ; j¼1 bj;2 ; j¼1 bj;3 ; j¼1 bj;4 ; j¼1 bj;1 ; n o Pn Pn 0 Pn 3 j¼1 bj;2 ; j¼1 bj;3 ; j¼1 bj;4 min1  j  n wj Þ: Then, the considered fully generalized trapezoidal intuitionistic fuzzy transportation problem is a balanced fully generalizedtrapezoidal intuitionistic fuzzy transportation problem. Go to Step 5. Pm Pm Pm Pm 0 Pm Pm Pm Case 2: If i¼1 ai;1 ; i¼1 ai;2 ; i¼1 ai;3 ; i¼1 ai;4 ; i¼1 ai;1 ; i¼1 ai;2 ; i¼1 ai;3 ; 

 Pn Pn Pn Pn 0 Pn Pm 0 2 i¼1 ai;4 ; min1  i  m wi Þ 6¼ j¼1 bj;1 ; j¼1 bj;2 ; j¼1 bj;3 ; j¼1 bj;4 ; j¼1 bj;1 ; n o Pn Pn 0 Pn 3 Then, the considered fully j¼1 bj;2 ; j¼1 bj;3 ; j¼1 bj;4 ; min1  j  n wj Þ:

5 JMD Approach for Solving Unbalanced Fully Generalized …

228

generalized trapezoidal intuitionistic fuzzy transportation problem is an unbalanced fully generalized trapezoidal intuitionistic fuzzy transportation problem. Go to Step 4. Step 4: Add a dummy source Sm þ 1 having generalized intuitionistic fuzzy avail I ability ~aðm þ 1Þ ¼ aðm þ 1Þ;1 ; aðm þ 1Þ;2 ; aðm þ 1Þ3 ; aðm þ 1Þ4 ; a0ðm þ 1Þ;1 ; aðm þ 1Þ;2 ; aðm þ 1Þ3 ; n n oo

 a0ðm þ 1Þ;4 :; min min1  i  m w2i ; min1  j  n w3j and consider the cost for supplying the one unit quantity of the product from the dummy source Sm þ 1 to all the destinations as a generalized trapezoidal intuitionistic fuzzy number ~0 ¼ ð0; 0; 0; 0; 0; 0; 0; 0; 1Þ. where ( 0

aðm þ 1Þ;1 ¼ max 0;

n X j¼1

aðm þ 1Þ;1 ¼

a0ðm þ 1Þ;1

b0j;1



(

m X

;

i¼1

þ max 0; (

aðm þ 1Þ;2 ¼ aðm þ 1Þ;1 þ max 0; ( aðm þ 1Þ;3 ¼ aðm þ 1Þ;2 þ max 0; ( aðm þ 1Þ;4 ¼ aðm þ 1Þ;3 þ max 0; ( a0ðm þ 1Þ;4

) a0i;1

¼ aðm þ 1Þ;4 þ max 0;

n X

bj1 

n X

j¼1

j¼1

n X

n X

bj2 

j¼1

j¼1

n X

n X

bj3 

j¼1

j¼1

n X

n X

bj4 

j¼1 n X j¼1

! b0j;1 ! 

bj1 !



bj2 !



n X j¼1



bj3

j¼1

b0j;4



m X

ai1 

i¼1

i¼1

m X

m X

ai2 

i¼1

i¼1

m X

m X

ai3 

i¼1

i¼1

m X

m X

ai4 

i¼1

! bj4

m X



m X i¼1

!) a0i;1 !) ;

ai1 !)

;

ai2 !) ai3

i¼1

a0i;4



;

m X

; !)

ai4

;:

i¼1

Also,  add a dummy destination Dn þ 1 having intuitionistic fuzzy demand 0 0 ~bI ðn þ 1Þ ¼ bðn þ 1Þ;1 ; bðn þ 1Þ;2 ; bðn þ 1Þ3 ; bðn þ 1Þ4 ; bðn þ 1Þ;1 ; bðn þ 1Þ;2 ; bðn þ 1Þ3 ; bðn þ 1Þ;4 ; n n o o

 min min1  i  m w2i ; min1  j  n w3j and consider the cost for supplying the one unit quantity of the product from all the sources to the dummy destination as a generalized trapezoidal intuitionistic fuzzy number Dn þ 1 ~0 ¼ ð0; 0; 0; 0; 0; 0; 0; 0; 1Þ. where

5.11

Proposed JMD Approach

( b0ðn þ 1Þ;1

¼ max 0;

m X i¼1

bðn þ 1Þ;1 ¼

b0ðn þ 1Þ;1

229

a0i;limits1

) b0j;limits1

;

j¼1

(

þ max 0; (

bðn þ 1Þ;2 ¼ bðn þ 1Þ;1 þ max 0; ( bðn þ 1Þ;3 ¼ bðn þ 1Þ;2 þ max 0; ( bðn þ 1Þ;4 ¼ bðn þ 1Þ;3 þ max 0;

m X

ai1 

¼ bðn þ 1Þ;4 þ max 0;

m X

i¼1

i¼1

m X

m X

ai2 

i¼1

i¼1

m X

m X

ai3 

i¼1

i¼1

m X

m X

ai4 

i¼1

( b0ðn þ 1Þ;4



n X

m X

! a0i;limits1 ! 

ai1 !



ai2 ! ai3

i¼1



m X i¼1

bj1 

j¼1



i¼1

a0i;limits4



n X

! ai4

n X

bj2 

n X j¼1

n X

n X

bj3 

j¼1

j¼1

n X

n X



!) b0j;limits1

bj4 

!) ;

bj1 !)

;

bj2 !) bj3

;

j¼1 n X j¼1

;

j¼1

j¼1

j¼1

n X

b0j;limits4



n X

!) bj4

:

j¼1

Step 5: Write the generalized intuitionistic fully fuzzy linear programming problem (5.11.1) corresponding to the considered/transformed balanced fully generalized trapezoidal intuitionistic fuzzy transportation problem. Generalized intuitionistic fully fuzzy linear programming problem (5.11.1) h  i þ1 nþ1 I ~ ~ Minimize m   x c ij i¼1 j¼1 ij Subject to nX þ1

~xij ~aIi ;

i ¼ 1; 2; . . .; m þ 1;

j¼1 m þ1 X

~xij ~bIj ;

j ¼ 1; 2; . . .; n þ 1;

i¼1

~xij is a non-negative generalized trapezoidal intuitionistic fuzzy number. Step 6: Transform the generalized intuitionistic fully fuzzy linear programming problem (5.11.1) into its equivalent generalized intuitionistic fully fuzzy linear programming problem (5.11.2) by replacing the parameters ~cIij , ~ aI , ~ bI and ~xij with i j the generalized trapezoidal intuitionistic fuzzy numbers cij;1 ; cij;2 ; cij;3 ; cij;4 ;    c0ij;1 ; cij;2 ; cij;3 ; c0ij;4 ; w1ij Þ, ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; w2i , bj;1 ; bj;2 ; bj;3 ; bj;4 ;

5 JMD Approach for Solving Unbalanced Fully Generalized …

230

b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; w3j Þ

and

ðxij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; 1jn

w2i ; w3j Þ, respectively. Generalized intuitionistic fully fuzzy linear programming problem (5.11.2) h   i þ1 nþ1 0 0 1 Minimize m  c ; c ; c ; c ; c ; c ; c ; c ; w ij;1 ij;2 ij;3 ij;4 ij;2 ij;3 i¼1 j¼1 ij;1 ij;4 ij   n o xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j 1jn

Subject to þ1 nj¼1





xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m 1jn

 ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; w2i ;

n

w1ij ; w2i ; w3j

o



i ¼ 1; 2; . . .; m þ 1;

 n o 0 0 1 2 3 xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj 1jn   bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; w3j ; j ¼ 1; 2; . . .; n þ 1; þ1 m i¼1

 n o xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j is a non1jnþ1

negative generalized trapezoidal intuitionistic fuzzy number.    Step 7: Using the relation, a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; w1  b1 ; b2 ; b3 ; b4 ; b01 ; b2 ;    b3 ; b04 ; w2 Þ ¼ a1 ; a2 ; a3 ; a4 ; a01 ; a2 ; a3 ; a04 ; minfw1 ; w2 g  b1 ; b2 ; b3 ; b4 ; b01 ; b2 ; b3 ;   b04 ; minfw1 ; w2 gÞ ¼ a1 b1 ; a2 b2 ; a3 b3 ; a4 b4 ; a01 b01 ; a2 b2 ; a3 b3 ; a04 b04 ; minfw1 ; w2 g ,transform the generalized intuitionistic fully fuzzy linear programming problem (5.11.2) into its equivalent generalized intuitionistic fully fuzzy linear programming problem (5.11.3). Generalized intuitionistic fuzzy linear programming problem (5.11.3) h  þ1 nþ1 Minimize m cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;c0ij;1 x0ij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; c0ij;4 x0ij;4 ; i¼1 j¼1 n o 1 2 3 min 1  i  m þ 1 wij ; wi ; wj 1jnþ1

Subject to

5.11

 

Proposed JMD Approach

þ1 nj¼1



231

xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m þ 1 1jnþ1

 ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; w2i ;

n

w1ij ; w2i ; w3j

o



i ¼ 1; 2; . . .; m þ 1;

  n o mþ1 0 0 1 2 3 i¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m þ 1 wij ; wi ; wj 1jnþ1   bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; w3j ; j ¼ 1; 2; . . .; n þ 1;  n o 0 0 1 2 3 xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m þ 1 wij ; wi ; wj 1jnþ1

is

a

non-negative generalized trapezoidal intuitionistic fuzzy number. Step 8: Using the proposed PRABHUS approach for comparing generalized trapezoidal intuitionistic fuzzy numbers and  the relation  n o xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j is a non-negative 1jn

generalized trapezoidal intuitionistic fuzzy number implies 0  x0ij;1  xij;1  xij;2  xij;3 ;  xij;4  x0ij;4 , transform the generalized intuitionistic fully fuzzy linear programming problem (5.11.3) into its equivalent generalized intuitionistic fuzzy linear programming problem (5.11.4). Generalized intuitionistic fuzzy linear programming problem (5.11.4) h  þ1 nþ1 Minimize m cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;c0ij;1 x0ij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; c0ij;4 x0ij;4 ; i¼1 j¼1  n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

Subject to   n o þ1 ¼ P nj¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j 1jnþ1  n o ; i ¼ 1; 2; . . .; m þ 1; P ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j 1jnþ1

5 JMD Approach for Solving Unbalanced Fully Generalized … n o þ1 0 0 1 2 3 ¼ P m x ; x ; x ; x ; x ; x ; x ; x ; min w ; w ; w 1imþ1 ij;1 ij;2 ij;3 ij;4 ij;1 ij;2 ij;3 ij;4 i¼1 ij i j 1jnþ1   n o ; j ¼ 1; 2; . . .; n þ 1; P bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j

232





1jnþ1

  n o þ1 ¼ R nj¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j 1jnþ1   n o ; i ¼ 1; 2; . . .; m þ 1; R ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m w1ij ; w2i ; w3j 1jn





n o þ1 0 0 1 2 3 ¼ R m x ; x ; x ; x ; x ; x ; x ; x ; min w ; w ; w 1im ij;1 ij;2 ij;3 ij;4 ij;1 ij;2 ij;3 ij;4 i¼1 ij i j 1jn   n o ; j ¼ 1; 2; . . .; n þ 1; R bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j 1jnþ1

  n o þ1 ¼ A nj¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j 1jnþ1   n o ; i ¼ 1; 2; . . .; m; A ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m w1ij ; w2i ; w3j 1jn

  n o  þ1 0 0 1 2 3 A m x ; x ; x ; x ; x ; x ; x ; x ; min w ; w ; w ¼ 1imþ1 ij;1 ij;2 ij;3 ij;4 ij;1 ij;2 ij;3 ij;4 i¼1 ij i j 1jnþ1  n o A bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j ; j ¼ 1; 2; . . .; n þ 1; 1jnþ1

  n o nþ1 0 0 1 2 3 ¼ B j¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; xij;1 ; xij;2 ; xij;3 ; xij;4 ; min 1  i  m wij ; wi ; wj 1jn   n o B ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m w1ij ; w2i ; w3j ; i ¼ 1; 2; . . .; m þ 1; 1jn

  n o  þ1 0 0 1 2 3 B m x ; x ; x ; x ; x ; x ; x ; x ; min w ; w ; w ¼ 1  i  m þ 1 ij;1 ij;2 ij;3 ij;4 ij;2 ij;3 i¼1 ij;1 ij;4 ij i j 1jnþ1   n o B bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j ; j ¼ 1; 2; . . .; n þ 1; 1jnþ1

  n o þ1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j ¼ B nj¼1 1jnþ1  n o B ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j ; i ¼ 1; 2; . . .; m þ 1; 1jnþ1

5.11

Proposed JMD Approach 233  n o  mþ1 ¼ H i¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j 1jnþ1   n o H bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j ; j ¼ 1; 2; . . .; n þ 1;



1jnþ1





n o  þ1 ¼ U nj¼1 xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j 1jnþ1   n o ; i ¼ 1; 2; . . .; m þ 1; U ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j 



1jnþ1

n o  þ1 0 0 1 2 3 ¼ x ; x ; x ; x ; x ; x ; x ; x ; min w ; w ; w U m 1  i  m þ 1 ij;1 ij;2 ij;3 ij;4 ij;2 ij;3 i¼1 ij;1 ij;4 ij i j 1jnþ1   n o ; j ¼ 1; 2; . . .; n þ 1; U bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j 1jnþ1

0  x0ij;1  xij;1  xij;2  xij;3 ;  xij;4  x0ij;4 : h  i 0 0 a ; a ; a ; a ; a ; a ; a ; a ; w Step 9: Using the relations, P m ¼ i;1 i;2 i;3 i;4 i;2 i;3 i i¼1 i;1 i;4 hP  i

 m 0 0 , R m i¼1 ai;1 ; i¼1 P aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; min1  i  m fwi g hP  m 0 ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; wi Þ ¼ i¼1 R aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;1 ; aij;2 ; aij;3 ; h  i P m 0 0 a0ij;4 ; min1  i  m fwi gÞ, A m ¼ i¼1 ai;1 ; ai;2 ; ai;3 ; ai;4 ; ai;1 ; ai;2 ; ai;3 ; ai;4 ; wi i¼1 A  

 aij;1 ; aij;2 ; aij;3 ; aij;4 ; a0ij;1 ; aij;2 ; aij;3 ; a0ij;4 ; min1  i  m fwi g , B m i¼1 ai;1 ; ai;2 ; ai;3 ; hP  m 0 0 ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; wi Þ ¼ i¼1 B aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; h  i

Pm 0 0 min1  i  m fwi gÞ, H m ¼ i¼1 ai;1 ; ai;2 ; ai;3 ; ai;4 ; ai;1 ; ai;2 ; ai;3 ; ai;4 ; wi i¼1 H  

 aij;1 ; aij;2 ; aij;3 ; aij;4 ; a0ij;1 ; aij;2 ; aij;3 ; a0ij;4 ; min1  i  m fwi g  and U m i¼1 ai;1 ; ai;2 ; hP  m 0 0 ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; wi Þ ¼ i¼1 U aij;1 ; aij;2 ; aij;3 ; aij;4 ; aij;1 ; aij;2 ; aij;3 ; aij;4 ; min1  i  m fwi gÞ , transform the generalized intuitionistic fuzzy linear programming problem (5.11.4) into its equivalent generalized intuitionistic fuzzy linear programming problem (5.11.5). Generalized intuitionistic fuzzy linear programming problem (5.11.5) h  þ1 nþ1 Minimize m cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;c0ij;1 x0ij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; c0ij;4 x0ij;4 ; i¼1 j¼1 13 n o min w1ij ; w2i ; w3j A5 1imþ1 1jnþ1

Subject to

5 JMD Approach for Solving Unbalanced Fully Generalized …

234

0 @

nX þ1

0 P@xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

j¼1

0

P@ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; 0

m þ1 X

@

n min

1imþ1 1jnþ1

n min

1imþ1 1jnþ1

n

P@xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

min

1imþ1 1jnþ1

i¼1

0

0 @

nX þ1

min

1imþ1 1jnþ1

R@xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

j¼1

0

R@ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; 0

m þ1 X

@

n min

1imþ1 1jnþ1

n min

1imþ1 1jnþ1

min

i¼1

0

R@bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; 0 @

nX þ1

min

1imþ1 1jnþ1

A@xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

j¼1

0

A@ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; 0 @

m þ1 X

n min

1imþ1 1jnþ1

n min

i

0 A@xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

i¼1

0

A@bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ;

min

1imþ1 1jnþ1

n min

AA ¼

j ¼ 1; 2; . . .; n þ 1; 11

i ¼ 1; 2; . . .; m þ 1;

o

11

w1ij ; w2i ; w3j AA ¼

j ¼ 1; 2; . . .; n þ 1;

o

11

w1ij ; w2i ; w3j AA ¼

j

n

1imþ1 1jnþ1

1 o A;

1 o w1 ; w2 ; w3 A; ij

1imþ1 1jnþ1

11

w1ij ; w2i ; w3j AA ¼

1 n o w1ij ; w2i ; w3j A;

0

o

o

n 1imþ1 1jnþ1

i ¼ 1; 2; . . .; m þ 1;

w1ij ; w2i ; w3j

w1ij ; w2i ; w3j

0 R@xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

1 o A;

1 n o w1ij ; w2i ; w3j A;

0

11

w1ij ; w2i ; w3j AA ¼

w1ij ; w2i ; w3j

0

P@bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ;

o

i ¼ 1; 2; . . .; m þ 1;

o

11

w1ij ; w2i ; w3j AA ¼ o

1

w1ij ; w2i ; w3j A;

j ¼ 1; 2; . . .; n þ 1;

5.11

Proposed JMD Approach

0

nX þ1

@

0

B@xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

j¼1

0

B@ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; 0 @

m þ1 X

n 1imþ1 1jnþ1

n min

1imþ1 1jnþ1

B@xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

0

B@bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ;

nX þ1

@

H @ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ;

m þ1 X

n min

1imþ1 1jnþ1

n min

1imþ1 1jnþ1

min

1imþ1 1jnþ1

i¼1

0

H @bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ; 0

nX þ1

@

n min

1imþ1 1jnþ1

U @xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

0

U @ai;1 ; ai;2 ; ai;3 ; ai;4 ; a0i;1 ; ai;2 ; ai;3 ; a0i;4 ; 0 @

m þ1 X

n min

1imþ1 1jnþ1

n min

1imþ1 1jnþ1

U @xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

0

U @bj;1 ; bj;2 ; bj;3 ; bj;4 ; b0j;1 ; bj;2 ; bj;3 ; b0j;4 ;

min

1imþ1 1jnþ1

j ¼ 1; 2; . . .; n þ 1;

o

11 AA ¼

1 A;

o

i ¼ 1; 2; . . .; m þ 1;

o

11 AA ¼

1 A;

j ¼ 1; 2; . . .; n þ 1;

o

11

1

w1ij ; w2i ; w3j A;

n min

A;

w1ij ; w2i ; w3j AA ¼

n

1imþ1 1jnþ1

1

w1ij ; w2i ; w3j

o

0

i¼1

o

w1ij ; w2i ; w3j

0

j¼1

o

w1ij ; w2i ; w3j

n

H @xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

11

w1ij ; w2i ; w3j AA ¼

w1ij ; w2i ; w3j

0

i ¼ 1; 2; . . .; m þ 1;

o

w1ij ; w2i ; w3j

0

0

@

min

1imþ1 1jnþ1

min

j¼1

0

n

1imþ1 1jnþ1

H @xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ;

1

w1ij ; w2i ; w3j A;

n

235

o w1ij ; w2i ; w3j AA ¼ o

0

i¼1

0

min

11

i ¼ 1; 2; . . .; m þ 1;

o

11

w1ij ; w2i ; w3j AA ¼ o

1

w1ij ; w2i ; w3j A;

j ¼ 1; 2; . . .; n þ 1;

0  x0ij;1  xij;1  xij;2  xij;3 ;  xij;4  x0ij;4 :   Step 10: Using the relations, P aij;1 ; aij;2 ; aij;3 ; aij;4 ; a0ij;1 ; aij;2 ; aij;3 ; a0ij;4 ; w ¼    w aij;1 þ aij;2 þ aij;3 þ aij;4 þ a0ij;1 þ aij;2 þ aij;3 þ a0ij;4 , R aij;1 ; aij;2 ; aij;3 ; aij;4 ; a0ij;1 ;

5 JMD Approach for Solving Unbalanced Fully Generalized …

236

   aij;2 ; aij;3 ; a0ij;4 ; wÞ ¼ w aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;2 þ aij;3 þ a0ij;4 , A aij;1 ; aij;2 ;    aij;3 ; aij;4 ; a0ij;1 ; aij;2 ; aij;3 ; a0ij;4 ; wÞ ¼ w aij;2 þ aij;3 þ aij;4 þ aij;2 þ aij;3 þ a0ij;4 , B aij;1 ;    aij;2 ; aij;3 ; aij;4 ; a0ij;1 ; aij;2 ; aij;3 ; a0ij;4 ; wÞ ¼ w aij;3 þ aij;4 þ aij;3 þ a0ij;4 , H aij;1 ; aij;2 ;    aij;3 ; aij;4 ; a0ij;1 ; aij;2 ; aij;3 ; a0ij;4 ; wÞ ¼ w aij;4 þ a0ij;4 and U aij;1 ; aij;2 ; aij;3 ; aij;4 ; a0ij;1 ;   aij;2 ; aij;3 ; a0ij;4 ; wÞ ¼ w a0ij;4 , transform the generalized intuitionistic fuzzy linear programming problem (5.11.5) into its equivalent crisp linear programming problem (5.11.6). Crisp linear programming problem (5.11.6) h  n 0 0 0 0 Minimize m i¼1 j¼1 cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; 13 n o min w1ij ; w2i ; w3j A5 1imþ1 1jnþ1

Subject to 0 @ 0 @

n min

1imþ1 1jnþ1

min

1imþ1 1jnþ1

@

n min

1imþ1 1jnþ1

o

1 A

nX þ1 

xij;1 þ xij;2 þ xij;3 þ xij;4 þ x0ij;1

þ xij;2 þ xij;3 þ x0ij;4

1 o   w1ij ; w2i ; w3j A ai;1 þ ai;2 þ ai;3 þ ai;4 þ a0i;1 þ ai;2 þ ai;3 þ a0i;4 ;

0

@

n min

1imþ1 1jnþ1

i ¼ 1; 2; . . .; m þ 1;

1imþ1 1jnþ1

w1ij ; w2i ; w3j

o

1 A

nX þ1 

j ¼ 1; 2; . . .; n þ 1;

xij;1 þ xij;2 þ xij;3 þ xij;4 þ xij;2 þ xij;3 þ x0ij;4



! ¼

j¼1

n min

¼

i¼1

1jnþ1

0

!

1 ! m þ1  o  X w1ij ; w2i ; w3j A xij;1 þ xij;2 þ xij;3 þ xij;4 þ x0ij;1 þ xij;2 þ xij;3 þ x0ij;4 ¼

 n o   bj;1 þ bj;2 þ bj;3 þ bj;4 þ b0j;1 þ bj;2 þ bj;3 þ b0j;4 ; min 1  i  m þ 1 w1ij ; w2i ; w3j

@



j¼1

n

0

w1ij ; w2i ; w3j

w1ij ; w2i ; w3j

1 o   A ai;1 þ ai;2 þ ai;3 þ ai;4 þ ai;2 þ ai;3 þ a0 ; i;4

i ¼ 1; 2; . . .; m þ 1;

5.11 Proposed JMD Approach 0 1 ! m þ1  n o  X @ min ¼ w1ij ; w2i ; w3j A xij;1 þ xij;2 þ xij;3 þ xij;4 þ xij;2 þ xij;3 þ x0ij;4 1imþ1 1jnþ1

0

n

@

min

1imþ1 1jnþ1

0 @ 0 @

min

1imþ1 1jnþ1

0 @ 0 @

w1ij ; w2i ; w3j

w1ij ; w2i ; w3j

n min

1imþ1 1jnþ1

min

0 @

w1ij ; w2i ; w3j

n min

1imþ1 1jnþ1

0 @

nX þ1 

xij;2 þ xij;3 þ xij;4 þ xij;2 þ xij;3 þ x0ij;4



! ¼

o

1 A

m þ1  X

i ¼ 1; 2; . . .; m þ 1;

xij;2 þ xij;3 þ xij;4 þ xij;2 þ xij;3 þ x0ij;4



!

min

1imþ1 1jnþ1

1 o   A bj;2 þ bj;3 þ bj;4 þ bj;2 þ bj;3 þ b0 ; j;4

w1ij ; w2i ; w3j

o

1 A

nX þ1 

xij;3 þ xij;4 þ xij;3 þ x0ij;4

j ¼ 1; 2; . . .; n þ 1;

! 

¼

1 o   w1ij ; w2i ; w3j A ai;3 þ ai;4 þ ai;3 þ a0i;4 ;

i ¼ 1; 2; . . .; m þ 1;

min

1 ! m þ1  n o  X 1 2 3 A 0 wij ; wi ; wj xij;3 þ xij;4 þ xij;3 þ xij;4 ¼

min

1 n o   w1ij ; w2i ; w3j A bj;3 þ bj;4 þ bj;3 þ b0j;4 ;

1imþ1 1jnþ1

1imþ1 1jnþ1

0 @ 0 @

¼

j¼1

n

0 @

A

1 o   A ai;2 þ ai;3 þ ai;4 þ ai;2 þ ai;3 þ a0 ; i;4

w1ij ; w2i ; w3j

0 @

1

i¼1

n 1imþ1 1jnþ1

o

j ¼ 1; 2; . . .; n þ 1;

j¼1

n min

i¼1

o   w1ij ; w2i ; w3j A bj;1 þ bj;2 þ bj;3 þ bj;4 þ bj;2 þ bj;3 þ b0j;4 ;

n

1imþ1 1jnþ1

1

237

i¼1

j ¼ 1; 2; . . .; n þ 1;

min

1 ! nX þ1  n o  1 2 3 A 0 wij ; wi ; wj xij;4 þ xij;4 ¼

min

1 n o   w1ij ; w2i ; w3j A ai;4 þ þ a0i;4 ;

1imþ1 1jnþ1

1imþ1 1jnþ1

j¼1

i ¼ 1; 2; . . .; m þ 1;

238

5 JMD Approach for Solving Unbalanced Fully Generalized …

0 @

min

1imþ1 1jnþ1

1 ! m þ1  n o  X 1 2 3 A 0 wij ; wi ; wj xij;4 þ xij;4 i¼1

0

¼@

min

1imþ1 1jnþ1

1 n o   1 2 3 A wij ; wi ; wj bj;4 þ b0j;4 ;

j ¼ 1; 2; . . .; n þ 1; 0 @

min

1imþ1 1jnþ1

1 ! 0 nX þ1  n o  w1ij ; w2i ; w3j A x0ij;4 ¼@ j¼1

min

1imþ1 1jnþ1

1 n o   w1ij ; w2i ; w3j A a0i;4 ;

i ¼ 1; 2; . . .; m þ 1; 0 @

n min

1imþ1 1jnþ1

o

1

w1ij ; w2i ; w3j A

m þ1  X

x0ij;4

i¼1

! 

0 ¼@

n min

1imþ1 1jnþ1

1 o   w1ij ; w2i ; w3j A b0j;4 ;

j ¼ 1; 2; . . .; n þ 1; 0  x0ij;1  xij;1  xij;2  xij;3 ;  xij;4  x0ij;4 : Step 11: Transform the generalized intuitionistic fuzzy linear programming problem (5.11.6) into its equivalent generalized intuitionistic fuzzy linear programming problem (5.11.7) by solving the constraints of the generalized intuitionistic fuzzy linear programming problem (5.11.6). Generalized intuitionistic fuzzy linear programming problem (5.11.7) h  þ1 nþ1 Minimize m cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;c0ij;1 x0ij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; c0ij;4 x0ij;4 ; i¼1 j¼1 n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

Subject to nP þ1

xij;1 ¼ ai;1 ,

j¼1 nP þ1 x0ij;4 j¼1

nP þ1

xij;2 ¼ ai;2 ,

j¼1

¼ a0i;4 ; i ¼ 1; 2; . . .; m þ 1;

nP þ1 j¼1

xij;3 ¼ ai;3 ,

nP þ1 j¼1

xij;4 ¼ ai;4 ,

nP þ1 j¼1

x0ij;1 ¼ a0i;1 ,

5.11

Proposed JMD Approach

mP þ1

xij;1 ¼ bj;1 ,

i¼1 mP þ1 x0ij;4 i¼1

mP þ1

xij;2 ¼ bj;2 ,

i¼1

239 mP þ1

xij;3 ¼ bj;3 ,

i¼1

mP þ1

xij;4 ¼ bj;4 ,

i¼1

mP þ1 i¼1

x0ij;1 ¼ b0j;1 ,

¼ b0j;4 ; j ¼ 1; 2; . . .; n þ 1; 0  x0ij;1  xij;1  xij;2  xij;3 ;  xij;4  x0ij;4 :

Step 12: Using the proposed PRABHUS approach, transform the generalized intuitionistic fuzzy linear programming problem (5.11.7) into its equivalent crisp multi-objective linear programming problem (5.11.8). Crisp linear programming problem (5.11.8) h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 P cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;  n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 R cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;  n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 A cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;  n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 B cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;  n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 H cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;  n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 U cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;  n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

5 JMD Approach for Solving Unbalanced Fully Generalized …

240

Subject to Constraints of the crisp linear programming problem (5.11.7). Step 13: Solve the crisp linear programming problem (5.11.9) and check that a unique crisp optimal solution exists for the crisp linear programming problem (5.11.9) or not. Crisp linear programming problem (5.11.9) h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 P cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

Subject to Constraints of the crisp linear programming problem (5.11.7). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (5.11.9), then go to Step 18. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (5.11.9), then go to Step 14. Step 14: Solve the crisp linear programming problem (5.11.10) and check that a unique crisp optimal solution exists for the crisp linear programming problem (5.11.10) or not. Crisp linear programming problem (5.11.10) h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 R cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;  n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

Subject to  þ1 nþ1 0 0 0 m i¼1 j¼1 P cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 n o x0ij;4 ;min 1  i  m þ 1 w1ij ; w2i ; w3j ¼ Crisp optimal value of the crisp linear pro1jnþ1

gramming problem (5.11.9) and Constraints of the crisp linear programming problem (5.11.9).

5.11

Proposed JMD Approach

241

Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (5.11.10), then go to Step 18. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (5.11.10), then go to Step 15. Step 15: Solve the crisp linear programming problem (5.11.11) and check that a unique crisp optimal solution exists for the crisp linear programming problem (5.11.11) or not. Crisp linear programming problem (5.11.11) h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 A cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

Subject to  þ1 nþ1 0 0 0 0 m i¼1 j¼1 R cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; n o min 1  i  m þ 1 w1ij ; w2i ; w3j ¼ Crisp optimal value of the crisp linear programming 1jnþ1

problem (5.11.10) and Constraints of the crisp linear programming problem (5.11.10). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (5.11.11), then go to Step 18. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (5.11.11), then go to Step 16. Step 16: Solve the crisp linear programming problem (5.11.12) and check that a unique crisp optimal solution exists for the crisp linear programming problem (5.11.12) or not. Crisp linear programming problem (5.11.12) h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 B cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;  n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

Subject to

5 JMD Approach for Solving Unbalanced Fully Generalized …

242

 þ1 nþ1 m c0ij;1 x0ij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; c0ij;4 i¼1 j¼1 A cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; n o x0ij;4 ;min 1  i  m þ 1 w1ij ; w2i ; w3j ¼ Crisp optimal value of the crisp linear pro1jnþ1

gramming problem (5.11.11) and Constraints of the crisp linear programming problem (5.11.11). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (5.11.12), then go to Step 18. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (5.11.12), then go to Step 17. Step 17: Solve the crisp linear programming problem (5.11.13) and check that a unique crisp optimal solution exists for the crisp linear programming problem (5.11.13) or not. Crisp linear programming problem (5.11.13) h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 H cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

Subject to  þ1 nþ1 0 0 0 0 m i¼1 j¼1 B cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; n o min 1  i  m þ 1 w1ij ; w2i ; w3j ¼ Crisp optimal value of the crisp linear programming 1jnþ1

problem (5.11.12) and Constraints of the crisp linear programming problem (5.11.12). Case (i): If a unique crisp optimal solution exists for the crisp linear programming problem (5.11.13), then go to Step 19. Case (ii): If more than one crisp optimal solution exists for the crisp linear programming problem (5.11.13), then go to Step 18. Step 18: Solve the crisp linear programming problem (5.11.14) and go to Step 19. Crisp linear programming problem (5.11.14) h  þ1 nþ1 0 0 0 0 Minimize m i¼1 j¼1 U cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;  n o w1ij ; w2i ; w3j min 1imþ1 1jnþ1

5.11

Proposed JMD Approach

243

Subject to  þ1 nþ1 0 0 0 0 m i¼1 j¼1 H cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ;cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; n o min 1  i  m þ 1 w1ij ; w2i ; w3j ¼ Crisp optimal value of the crisp linear programming 1jnþ1

problem (5.11.13) and Constraints of the crisp linear programming problem (5.11.13). n Step 19: Using the obtained crisp optimal solution xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; x0ij;4 ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; ng, find (i) The generalized intuitionistic fuzzy optimal solution ðxij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; min 1  i  m w1ij ; w2i ; w3j Þ; i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n of the 1jn

generalized intuitionistic fuzzy linear programming problem (5.11.1). (ii) The generalized intuitionistic fuzzy optimal value ðcij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; n o 0 0 0 0 1 2 3 cij;4 xij;4 ; cij;1 xij;1 ; cij;2 xij;2 ; cij;3 xij;3 ; cij;4 xij;4 ; min 1  i  m wij ; wi ; wj of the 1jn

generalized intuitionistic fuzzy linear programming problem (5.11.1).

5.12

Illustrative Example

Chakraborty et al. [1] solved the balanced fully generalized trapezoidal intuitionistic fuzzy transportation problem, represented by Table 5.5, to illustrate their proposed approach. In this section, the generalized intuitionistic fuzzy optimal solution of the same problem is obtained by the proposed JMD approach. Using the proposed JMD approach, the generalized intuitionistic fuzzy optimal solution of generalized intuitionistic fully fuzzy transportation problem, represented by Table 5.5, can be obtained as follows: Step 1: To find a generalized intuitionistic fuzzy optimal solution of the balanced fully generalized trapezoidal intuitionistic fuzzy transportation problem, represented by Table 5.5, is equivalent to find a generalized intuitionistic fuzzy optimal solution of the generalized intuitionistic fully fuzzy linear programming problem (5.12.1).

Generalized intuitionistic fuzzy demand ð~ bIj Þ

S3

S2

S1

Sources

1; 3; 4; 5; 0:5; 3; 4; 5; 0:6   3; 4; 5; 8; 2; 4; 5; 9; 0:7   6; 7; 8; 9; 5; 7; 8; 11; 1



 3; 5; 6; 7; 2; 5; 6; 8; 0:6





1; 2; 3; 4; 0:5; 2; 3; 5; 0:8   4; 5; 6; 7; 3; 5; 6; 8; 0:8





4; 6; 7; 8; 3; 6; 7; 9; 0:2





 2; 4; 5; 6; 1; 4; 5; 6; 0:5

D2

Destinations D1

Table 5.5 Fully generalized trapezoidal intuitionistic fuzzy transportation problem

 















2; 4; 5; 6; 0:5; 4; 5; 7; 0:6

2; 4; 5; 10; 1; 4; 5; 11; 0:2

2; 6; 7; 11; 1; 6; 7; 12; 0:4

3; 7; 8; 12; 2; 7; 8; 13; 0:3

D3

Generalized intuitionistic fuzzy availability ð~aIi Þ   4; 6; 8; 9; 2; 6; 8; 10; 0:6   0; 0:5; 1; 2; 0; 0:5; 1; 5; 0:7   8; 9:5; 10; 11; 6:5; 9:5; 10; 11; 0:8

244 5 JMD Approach for Solving Unbalanced Fully Generalized …

5.12

Illustrative Example

245

Generalized intuitionistic fully fuzzy linear programming problem (5.12.1)    Minimize ð2; 4; 5; 6; 1; 4; 5; 6; 0:5Þ  x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2    ð4; 6; 7; 8; 3; 6; 7; 9; 0:2Þ  x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2    ð3; 7; 8; 12; 2; 7; 8; 13; 0:3Þ  x13;1 ; x13;2 ; x13;3 ; x13;4 ; x013;1 ; x13;2 ; x13;3 ; x013;4 ; 0:2    ð1; 3; 4; 5; 0:5; 3; 4; 5; 0:6Þ  x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2    ð3; 5; 6; 7; 2; 5; 6; 8; 0:6Þ  x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2    ð2; 6; 7; 11; 1; 6; 7; 12; 0:4Þ  x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2    ð3; 4; 5; 8; 2; 4; 5; 9; 0:7Þ  x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2    ð1; 2; 3; 4; 0:5; 2; 3; 5; 0:8Þ  x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2    ð2; 4; 5; 10; 1; 4; 5; 11; 0:2Þ  x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2

Subject to     x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2  x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2    x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2 ð4; 6; 8; 9; 2; 6; 8; 10; 0:6Þ;     x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2  x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2    x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2 ð0; 0:5; 1; 2; 0; 0:5; 1; 5; 0:7Þ;     x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2  x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2    x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ð8; 9:5; 10; 11; 6:5; 9:5; 10; 11; 0:8Þ;     x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2  x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2    x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2 ð6; 7; 8; 9; 5; 7; 8; 11; 1Þ;     x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2  x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2    x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2 ð4; 5; 6; 7; 3; 5; 6; 8; 0:8Þ;     x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2  x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2    x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ð2; 4; 5; 6; 0:5; 4; 5; 7; 0:6Þ;

246

5 JMD Approach for Solving Unbalanced Fully Generalized …

  xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; 0:2 is a non-negative generalized trapezoidal intuitionistic fuzzy number.

  Step 2: Using the multiplication, a1;1 ; a1;2 ; a1;3 ; a1;4 ; a01;1 ; a1;2 ; a1;3 ; a01;4 ; w1     b1;1 ; b1;2 ; b1;3 ; b1;4 ; b01;1 ; b1;2 ; b1;3 ; b01;4 ; w2 ¼ a1;1 b1;1 ; a1;2 b1;2 ; a1;3 b1;3 ; a1;4 b1;4 ; a01;1 b01;1 ; a1;2 b1;2 ; a1;3 b1;3 ; a01;4 b01;4 ; minfw1 ; w2 gÞ, the generalized intuitionistic fully fuzzy linear programming problem (5.12.1) can be transformed into its equivalent generalized intuitionistic fully fuzzy linear programming problem (5.12.2). Generalized intuitionistic fully fuzzy linear programming problem (5.12.2)   Minimize 2x11;1 ; 4x11;2 ; 5x11;3 ; 6x11;4 ; 1x011;1 ; 4x11;2 ; 5x11;3 ; 6x011;4 ; 0:2    4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2    3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2    1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2    3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2    2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2    3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2    x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2    2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2 Subject to Constraints of the generalized intuitionistic fully fuzzy linear programming problem (5.12.1). Step 3: Using the proposed PRABHUS approach for comparing generalized trapezoidal intuitionistic fuzzy numbers, the generalized intuitionistic fully fuzzy linear programming problem (5.12.2) can be transformed into its equivalent generalized intuitionistic fuzzy linear programming problem (5.12.3).

5.12

Illustrative Example

247

Generalized intuitionistic fuzzy linear programming problem (5.12.3)   Minimize 2x11;1 ; 4x11;2 ; 5x11;3 ; 6x11;4 ; 1x011;1 ; 4x11;2 ; 5x11;3 ; 6x011;4 ; 0:2    4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2    3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2    1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2    3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2    2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2    3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2    x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2    2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2 Subject to   P x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ P x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ P x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2 ¼ Pð4; 6; 8; 9; 2; 6; 8; 10; 0:2Þ;   P x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ P x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ P x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2 ¼ Pð0; 0:5; 1; 2; 0; 0:5; 1; 5; 0:2Þ;   P x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2   þ P x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2   þ P x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ Pð8; 9:5; 10; 11; 6:5; 9:5; 10; 11; 0:2Þ;

  P x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ P x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ P x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2 ¼ Pð6; 7; 8; 9; 5; 7; 8; 11; 0:2Þ;

248

5 JMD Approach for Solving Unbalanced Fully Generalized …

  P x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ P x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ P x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2 ¼ Pð4; 5; 6; 7; 3; 5; 6; 8; 0:2Þ;   P x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2   þ P x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2   þ P x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ Pð2; 4; 5; 6; 0:5; 4; 5; 7; 0:2Þ;   R x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ R x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ R x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2 ¼ Rð4; 6; 8; 9; 2; 6; 8; 10; 0:2Þ;   R x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ R x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ R x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2 ¼ Rð0; 0:5; 1; 2; 0; 0:5; 1; 5; 0:2Þ;   R x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2   þ R x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2   þ P x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ Rð8; 9:5; 10; 11; 6:5; 9:5; 10; 11; 0:2Þ;

  A x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ A x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ A x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2 ¼ Rð6; 7; 8; 9; 5; 7; 8; 11; 0:2Þ;

  A x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ A x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ A x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2 ¼ Að4; 5; 6; 7; 3; 5; 6; 8; 0:2Þ;

5.12

Illustrative Example

249

  A x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2   þ A x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2   þ A x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ Að2; 4; 5; 6; 0:5; 4; 5; 7; 0:2Þ;   A x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ A x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ A x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2 ¼ Að4; 6; 8; 9; 2; 6; 8; 10; 0:2Þ;   A x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ A x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ A x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2 ¼ Að0; 0:5; 1; 2; 0; 0:5; 1; 5; 0:2Þ;   A x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2   þ A x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2   þ A x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ Að8; 9:5; 10; 11; 6:5; 9:5; 10; 11; 0:2Þ;

  A x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ A x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ A x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2 ¼ Að6; 7; 8; 9; 5; 7; 8; 11; 0:2Þ;   A x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ A x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ A x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2 ¼ Að4; 5; 6; 7; 3; 5; 6; 8; 0:2Þ;   A x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2   þ A x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2   þ A x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ Að2; 4; 5; 6; 0:5; 4; 5; 7; 0:2Þ;

250

5 JMD Approach for Solving Unbalanced Fully Generalized …

  B x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ B x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ B x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2 ¼ Bð6; 7; 8; 9; 5; 7; 8; 11; 0:2Þ;

  B x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ B x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ B x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2 ¼ Bð4; 5; 6; 7; 3; 5; 6; 8; 0:2Þ;   B x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2   þ B x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2   þ B x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ Bð2; 4; 5; 6; 0:5; 4; 5; 7; 0:2Þ;   B x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ B x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ B x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2 ¼ Bð4; 6; 8; 9; 2; 6; 8; 10; 0:2Þ;   B x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ B x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ B x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2 ¼ Bð0; 0:5; 1; 2; 0; 0:5; 1; 5; 0:2Þ;   B x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2   þ B x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2   þ B x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ Bð8; 9:5; 10; 11; 6:5; 9:5; 10; 11; 0:2Þ;

  B x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ B x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ B x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2 ¼ Bð6; 7; 8; 9; 5; 7; 8; 11; 0:2Þ;

5.12

Illustrative Example

251

  B x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ B x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ B x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2 ¼ Bð4; 5; 6; 7; 3; 5; 6; 8; 0:2Þ;   B x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2   þ B x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2   þ B x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ Bð2; 4; 5; 6; 0:5; 4; 5; 7; 0:2Þ;   H x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ H x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ H x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2 ¼ H ð4; 6; 8; 9; 2; 6; 8; 10; 0:2Þ;   H x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ H x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ H x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2 ¼ H ð0; 0:5; 1; 2; 0; 0:5; 1; 5; 0:2Þ;   H x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2   þ H x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2   þ H x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ H ð8; 9:5; 10; 11; 6:5; 9:5; 10; 11; 0:2Þ;

  H x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ H x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ H x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2 ¼ Bð6; 7; 8; 9; 5; 7; 8; 11; 0:2Þ;   H x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ H x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ H x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2 ¼ H ð4; 5; 6; 7; 3; 5; 6; 8; 0:2Þ;

252

5 JMD Approach for Solving Unbalanced Fully Generalized …

  U x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ U x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ U x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2 ¼ U ð4; 6; 8; 9; 2; 6; 8; 10; 0:2Þ;   U x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ U x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ U x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2 ¼ U ð0; 0:5; 1; 2; 0; 0:5; 1; 5; 0:2Þ;   U x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2   þ U x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2   þ U x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ U ð8; 9:5; 10; 11; 6:5; 9:5; 10; 11; 0:2Þ;

  U x11;1 ; x11;2 ; x11;3 ; x11;4 ; x011;1 ; x11;2 ; x11;3 ; x011;4 ; 0:2   þ U x21;1 ; x21;2 ; x21;3 ; x21;4 ; x021;1 ; x21;2 ; x21;3 ; x021;4 ; 0:2   þ U x31;1 ; x31;2 ; x31;3 ; x31;4 ; x031;1 ; x31;2 ; x31;3 ; x031;4 ; 0:2 ¼ U ð6; 7; 8; 9; 5; 7; 8; 11; 0:2Þ;   U x12;1 ; x12;2 ; x12;3 ; x12;4 ; x012;1 ; x12;2 ; x12;3 ; x012;4 ; 0:2   þ U x22;1 ; x22;2 ; x22;3 ; x22;4 ; x022;1 ; x22;2 ; x22;3 ; x022;4 ; 0:2   þ U x32;1 ; x32;2 ; x32;3 ; x32;4 ; x032;1 ; x32;2 ; x32;3 ; x032;4 ; 0:2 ¼ U ð4; 5; 6; 7; 3; 5; 6; 8; 0:2Þ;   U x13;1 ; x13;2 ; x13;3 ; x12;4 ; x013;1 ; x13;2 ; x32;3 ; x013;4 ; 0:2   þ U x23;1 ; x23;2 ; x23;3 ; x23;4 ; x023;1 ; x23;2 ; x23;3 ; x023;4 ; 0:2   þ U x33;1 ; x33;2 ; x33;3 ; x33;4 ; x033;1 ; x33;2 ; x33;3 ; x033;4 ; 0:2 ¼ U ð2; 4; 5; 6; 0:5; 4; 5; 7; 0:2Þ;

  xij;1 ; xij;2 ; xij;3 ; xij;4 ; x0ij;1 ; xij;2 ; xij;3 ; x0ij;4 ; 0:2 is a non-negative generalized trapezoidal intuitionistic fuzzy number.

  Step 4: Using the relations, P aij;1 ; aij;2 ; aij;3 ; aij;4 ; a0ij;1 ; aij;2 ; aij;3 ; a0ij;4 ; w ¼    w aij;1 þ aij;2 þ aij;3 þ aij;4 þ a0ij;1 þ aij;2 þ aij;3 þ a0ij;4 , R aij;1 ; aij;2 ; aij;3 ; aij;4 ; a0ij;1 ;    aij;2 ; aij;3 ; a0ij;4 ; wÞ ¼ w aij;1 þ aij;2 þ aij;3 þ aij;4 þ aij;2 þ aij;3 þ a0ij;4 , A aij;1 ; aij;2 ;    aij;3 ; aij;4 ; a0ij;1 ; aij;2 ; aij;3 ; a0ij;4 ; wÞ ¼ w aij;2 þ aij;3 þ aij;4 þ aij;2 þ aij;3 þ a0ij;4 , B aij;1 ;

5.12

Illustrative Example

253

   aij;2 ; aij;3 ; aij;4 ; a0ij;1 ; aij;2 ; aij;3 ; a0ij;4 ; wÞ ¼ w aij;3 þ aij;4 þ aij;3 þ a0ij;4 , H aij;1 ; aij;2 ;    aij;3 ; aij;4 ; a0ij;1 ; aij;2 ; aij;3 ; a0ij;4 ; wÞ ¼ w aij;4 þ a0ij;4 and U aij;1 ; aij;2 ; aij;3 ; aij;4 ; a0ij;1 ;   aij;2 ; aij;3 ; a0ij;4 ; wÞ ¼ w a0ij;4 , the generalized intuitionistic fuzzy linear programming problem (5.12.3) can be transformed into its equivalent generalized intuitionistic fuzzy linear programming problem (5.12.4). Generalized intuitionistic fuzzy linear programming problem (5.12.4)   Minimize 2x11;1 ; 4x11;2 ; 5x11;3 ; 6x11;4 ; 1x011;1 ; 4x11;2 ; 5x11;3 ; 6x011;4 ; 0:2    4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2    3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2    1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2    3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2    2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2    3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2    x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2    2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2 Subject to   0:2 x11;1 þ x11;2 þ x11;3 þ x11;4 þ x011;1 þ x11;2 þ x11;3 þ x011;4   þ 0:2 x12;1 þ x12;2 þ x12;3 þ x12;4 þ x012;1 þ x12;2 þ x12;3 þ x012;4   þ 0:2 x13;1 þ x13;2 þ x13;3 þ x13;4 þ x013;1 þ x13;2 þ x13;3 þ x013;4 ¼ 0:2ð53Þ;   0:2 x21;1 þ x21;2 þ x21;3 þ x21;4 þ x021;1 þ x21;2 þ x21;3 þ x021;4   þ 0:2 x22;1 þ x22;2 þ x22;3 þ x22;4 ; x022;1 þ x22;2 þ x22;3 þ x022;4   þ 0:2 x23;1 þ x23;2 þ x23;3 þ x23;4 þ x023;1 þ x23;2 þ x23;3 þ x023;4 ¼ 0:2ð10Þ;   0:2 x31;1 þ x31;2 þ x31;3 þ x31;4 þ x031;1 þ x31;2 þ x31;3 þ x031;4   þ 0:2 x32;1 þ x32;2 þ x32;3 þ x32;4 þ x032;1 þ x32;2 þ x32;3 þ x032;4   þ 0:2 x33;1 þ x33;2 þ x33;3 þ x33;4 þ x033;1 þ x33;2 þ x33;3 þ x033;4 ¼ 0:2ð75:5Þ;

254

5 JMD Approach for Solving Unbalanced Fully Generalized …

  0:2 x11;1 þ x11;2 þ x11;3 þ x11;4 þ x011;1 þ x11;2 ; þ x11;3 þ x011;4   þ 0:2 x21;1 þ x21;2 þ x21;3 þ x21;4 þ x021;1 þ x21;2 þ x21;3 þ x021;4   þ 0:2 x31;1 þ x31;2 þ x31;3 þ x31;4 þ x031;1 þ x31;2 þ x31;3 þ x031;4 ¼ 0:2ð61Þ;   0:2 x12;1 þ x12;2 þ x12;3 þ x12;4 þ x012;1 þ x12;2 þ x12;3 þ x012;4   þ 0:2 x22;1 þ x22;2 þ x22;3 þ x22;4 þ x022;1 þ x22;2 þ x22;3 þ x022;4   þ 0:2 x32;1 þ x32;2 þ x32;3 þ x32;4 þ x032;1 þ x32;2 þ x32;3 þ x032;4 ¼ 0:2ð44Þ;   0:2 x13;1 þ x13;2 þ x13;3 þ x12;4 þ x013;1 þ x13;2 þ x32;3 þ x013;4   þ 0:2 x23;1 þ x23;2 þ x23;3 þ x23;4 þ x023;1 þ x23;2 þ x23;3 þ x023;4   þ 0:2 x33;1 þ x33;2 þ x33;3 þ x33;4 þ x033;1 þ x33;2 þ x33;3 þ x033;4 ¼ 0:2ð33:5Þ;   0:2 x11;1 þ x11;2 þ x11;3 þ x11;4 þ x11;2 þ x11;3 þ x011;4   þ 0:2 x12;1 þ x12;2 þ x12;3 þ x12;4 þ x12;2 þ x12;3 þ x012;4   þ 0:2 x13;1 þ x13;2 þ x13;3 þ x13;4 þ x13;2 þ x13;3 þ x013;4 ¼ 0:2ð51Þ;   0:2 x21;1 þ x21;2 þ x21;3 þ x21;4 þ x21;2 þ x21;3 þ x021;4   þ 0:2 x22;1 þ x22;2 þ x22;3 þ x22;4 þ þ x22;2 þ x22;3 þ x022;4   þ 0:2 x23;1 þ x23;2 þ x23;3 þ x23;4 þ þ x23;2 þ x23;3 þ x023;4 ¼ 0:2ð10Þ;   0:2 x31;1 þ x31;2 þ x31;3 þ x31;4 þ x31;2 þ x31;3 þ x031;4   þ 0:2 x32;1 þ x32;2 þ x32;3 þ x32;4 þ x32;2 þ x32;3 þ x032;4   þ 0:2 x33;1 þ x33;2 þ x33;3 þ x33;4 þ x33;2 þ x33;3 þ x033;4 ¼ 0:2ð69Þ;   0:2 x11;1 þ x11;2 þ x11;3 þ x11;4 þ x11;2 ; þ x11;3 þ x011;4   þ 0:2 x21;1 þ x21;2 þ x21;3 þ x21;4 þ x21;2 þ x21;3 þ x021;4   þ 0:2 x31;1 þ x31;2 þ x31;3 þ x31;4 þ x31;2 þ x31;3 þ x031;4 ¼ 0:2ð56Þ;   0:2 x12;1 ; þ x12;2 þ x12;3 þ x12;4 þ x12;2 þ x12;3 þ x012;4   þ 0:2 þ 0:2 x32;1 þ x32;2 þ x32;3 þ x32;4 þ x32;2 þ x32;3 þ x032;4 ¼ 0:2ð41Þ;

5.12

Illustrative Example

  0:2 x13;1 þ x13;2 þ x13;3 þ x12;4 þ x13;2 þ x32;3 þ x013;4   þ 0:2 x23;1 þ x23;2 þ x23;3 þ x23;4 þ x23;2 þ x23;3 þ x023;4   þ 0:2 x33;1 þ x33;2 þ x33;3 þ x33;4 þ x33;2 þ x33;3 þ x033;4 ¼ 0:2ð33Þ;   0:2 x11;2 þ x11;3 þ x11;4 þ x11;2 þ x11;3 þ x011;4   þ 0:2 x12;2 þ x12;3 þ x12;4 þ x12;2 þ x12;3 þ x012;4   þ 0:2 x13;1 þ x13;2 þ x13;3 þ x13;4 þ x13;2 þ x13;3 þ x013;4 ¼ 0:2ð45Þ;   0:2 x21;2 þ x21;3 þ x21;4 þ x21;2 þ x21;3 þ x021;4   þ 0:2 x22;2 þ x22;3 þ x22;4 þ þ x22;2 þ x22;3 þ x022;4   þ 0:2 x23;2 þ x23;3 þ x23;4 þ x23;2 þ x23;3 þ x023;4 ¼ 0:2ð10Þ;   0:2 x31;2 þ x31;3 þ x31;4 þ x31;2 þ x31;3 þ x031;4   þ 0:2 x32;2 þ x32;3 þ x32;4 þ x32;2 þ x32;3 þ x032;4   þ 0:2 x33;2 þ x33;3 þ x33;4 þ x33;2 þ x33;3 þ x033;4 ¼ 0:2ð61Þ;   0:2 x11;2 þ x11;3 þ x11;4 þ x11;2 ; þ x11;3 þ x011;4   þ 0:2 x21;2 þ x21;3 þ x21;4 þ x21;2 þ x21;3 þ x021;4   þ 0:2 x31;2 þ x31;3 þ x31;4 þ x31;2 þ x31;3 þ x031;4 ¼ 0:2ð41Þ;   0:2 x12;2 þ x12;3 þ x12;4 þ x12;2 þ x12;3 þ x012;4   þ 0:2 x22;2 þ x22;3 þ x22;4 þ x22;2 þ x22;3 þ x022;4   þ 0:2 x32;2 þ x32;3 þ x32;4 þ x32;2 þ x32;3 þ x032;4 ¼ 0:2ð42Þ;   0:2 x13;2 þ x13;3 þ x12;4 þ x13;2 þ x32;3 þ x013;4   þ 0:2 x23;2 þ x23;3 þ x23;4 þ x23;2 þ x23;3 þ x023;4   þ 0:2 x33;2 þ x33;3 þ x33;4 þ x33;2 þ x33;3 þ x033;4 ¼ 0:2ð31Þ;     0:2 x11;3 þ x11;4 þ x11;3 þ x011;4 þ 0:2 x12;3 þ x12;4 þ x12;3 þ x012;4   þ 0:2 x13;1 þ x13;3 þ x13;4 þ x13;3 þ x013;4 ¼ 0:2ð47Þ;

255

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    0:2 x21;3 þ x21;4 þ x21;3 þ x021;4 þ 0:2 x22;3 þ x22;4 þ x22;3 þ x022;4   þ 0:2 x23;3 þ x23;4 þ x23;3 þ x023;4 ¼ 0:2ð10Þ;     0:2 x31;3 þ x31;4 þ x31;3 þ x031;4 þ 0:2 x32;3 þ x32;4 þ x32;3 þ x032;4   þ 0:2 x33;3 þ x33;4 þ x33;3 þ x033;4 ¼ 0:2ð42Þ;     0:2 x11;3 þ x11;4 þ x11;3 þ x011;4 þ 0:2 x21;3 þ x21;4 þ x21;3 þ x021;4   þ 0:2 x31;3 þ x31;4 þ x31;3 þ x031;4 ¼ 0:2ð36Þ;     0:2 x12;3 þ x12;4 þ x12;3 þ x012;4 þ 0:2 x22;3 þ x22;4 þ x22;3 þ x022;4   þ 0:2 x32;3 þ x32;4 þ x32;3 þ x032;4 ¼ 0:2ð27Þ;     0:2 x13;3 þ x12;4 þ x32;3 þ x013;4 þ 0:2 x23;3 þ x23;4 þ x23;3 þ x023;4   þ 0:2 x33;3 þ x33;4 þ x33;3 þ x033;4 ¼ 0:2ð23Þ;       0:2 x11;4 þ x011;4 þ 0:2 x12;4 þ x012;4 þ 0:2 x13;4 þ x013;4 ¼ 0:2ð19Þ;       0:2 x21;4 þ x021;4 þ 0:2 x22;4 þ x022;4 þ 0:2 x23;4 þ x023;4 ¼ 0:2ð7Þ;       0:2 x31;4 þ x031;4 þ 0:2 x32;4 þ x032;4 þ 0:2 x33;4 þ x033;4 ¼ 0:2ð21Þ;       0:2 x11;4 þ x011;4 þ 0:2 x21;4 þ x021;4 þ 0:2 x31;4 þ x031;4 ¼ 0:2ð20Þ;       0:2 x12;4 þ x012;4 þ 0:2 x22;4 þ x022;4 þ 0:2 x32;4 þ x032;4 ¼ 0:2ð15Þ;       0:2 x12;4 þ x013;4 þ 0:2 x23;4 þ x023;4 þ 0:2 x33;4 þ x033;4 ¼ 0:2ð13Þ       0:2 x011;4 þ 0:2 x012;4 þ 0:2 x013;4 ¼ 0:2ð51Þ;       0:2 x021;4 þ 0:2 x022;4 þ 0:2 x023;4 ¼ 0:2ð5Þ;       0:2 x031;4 þ 0:2 x032;4 þ 0:2 x033;4 ¼ 0:2ð11Þ;       0:2 x011;4 þ 0:2 x021;4 þ 0:2 x031;4 ¼ 0:2ð11Þ;

5.12

Illustrative Example

257

      0:2 x012;4 þ 0:2 x022;4 þ 0:2 x032;4 ¼ 0:2ð8Þ;       0:2 x013;4 þ 0:2 x023;4 þ 0:2 x033;4 ¼ 0:2ð7Þ Step 5: Solving the constraints of the generalized intuitionistic fuzzy linear programming problem (5.12.4), it can be transformed into its equivalent generalized intuitionistic fuzzy linear programming problem (5.12.5). Generalized intuitionistic fuzzy linear programming problem (5.12.5) h  Minimize 4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2    3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2    1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2    3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2    2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2    3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2    x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2   i 2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2 ; Subject to x11;1 þ x121 þ x13;1 ¼ 4; x11;2 þ x122 þ x13;2 ¼ 6; x11;3 þ x123 þ x13;3 ¼ 8; x11;4 þ x124 þ x13;4 ¼ 9; x011;1 þ x012;1 þ x013;1 ¼ 2; x011;4 þ x012;4 þ x013;4 ¼ 10; x21;1 þ x221 þ x23;1 ¼ 0; x21;2 þ x222 þ x23;2 ¼ 0:5; x21;3 þ x223 þ x23;3 ¼ 1; x21;4 þ x224 þ x23;4 ¼ 2; x021;1 þ x022;1 þ x023;1 ¼ 0; x021;4 þ x022;4 þ x023;4 ¼ 5; x31;1 þ x321 þ x33;1 ¼ 8; x31;2 þ x322 þ x33;2 ¼ 9:5; x31;3 þ x323 þ x33;3 ¼ 10; x31;4 þ x324 þ x33;4 ¼ 11; x031;1 þ x032;1 þ x033;1 ¼ 6:5; x031;4 þ x032;4 þ x033;4 ¼ 11; x11;1 þ x21;1 þ x31;1 ¼ 6; x11;2 þ x21;2 þ x31;2 ¼ 7; x11;3 þ x21;3 þ x31;3 ¼ 8; x11;4 þ x21;4 þ x31;4 ¼ 9; x011;1 þ x021;1 þ x031;1 ¼ 5; x011;4 þ x021;4 þ x031;4 ¼ 11;

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5 JMD Approach for Solving Unbalanced Fully Generalized …

x12;1 þ x22;1 þ x32;1 ¼ 4; x12;2 þ x22;2 þ x32;2 ¼ 5; x12;3 þ x22;3 þ x32;3 ¼ 6; x12;4 þ x22;4 þ x32;4 ¼ 7; x012;1 þ x022;1 þ x032;1 ¼ 3; x012;4 þ x022;4 þ x032;4 ¼ 8; x13;1 þ x23;1 þ x33;1 ¼ 2; x13;2 þ x23;2 þ x33;2 ¼ 4; x13;3 þ x23;3 þ x33;3 ¼ 5; x13;4 þ x23;4 þ x33;4 ¼ 6; x013;1 þ x023;1 þ x033;1 ¼ 0:5; x013;4 þ x023;4 þ x033;4 ¼ 7; Step 6: Using the proposed PRABHUS approach, the generalized intuitionistic fuzzy linear programming problem (5.12.5) can be transformed into its equivalent crisp multi-objective linear programming problem (5.12.6). Crisp multi-objective linear programming problem (5.12.6) h   Minimize P 4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2   þ P 3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2   þ P 1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2   þ P 3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2   þ P 2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2   þ P 3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2   þ P x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2  i þ P 2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2 h   Minimize R 4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2   þ R 3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2   þ R 1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2   þ R 3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2   þ R 2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2   þ R 3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2   þ R x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2  i þ R 2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2

5.12

Illustrative Example

h   Minimize A 4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2   þ A 3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2   þ A 1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2   þ A 3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2   þ A 2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2   þ A 3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2   þ A x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2  i þ A 2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2 h   Minimize B 4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2   þ B 3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2   þ B 1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2   þ B 3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2   þ B 2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2   þ B 3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2   þ B x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2  i þ B 2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2 h   Minimize H 4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2   þ H 3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2   þ H 1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2   þ H 3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2   þ H 2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2   þ H 3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2   þ H x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2  i þ H 2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2

259

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5 JMD Approach for Solving Unbalanced Fully Generalized …

h   Minimize U 4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2   þ U 3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2   þ U 1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2   þ U 3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2   þ U 2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2   þ U 3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2   þ U x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2  i þ U 2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2 Subject to Constraints of the generalized intuitionistic fuzzy linear programming problem (5.12.5). Step 7: According to the proposed JMD approach, there is a need to check that a unique crisp optimal solution exists for the crisp linear programming problem (5.12.7) or not, i.e., there is a need to check a unique crisp optimal solution exists for the crisp linear programming problem (5.12.8) or not. Crisp linear programming problem (5.12.7) h   Minimize P 4x12;1 ; 6x12;2 ; 7x12;3 ; 8x12;4 ; 3x012;1 ; 6x12;2 ; 7x12;3 ; 9x012;4 ; 0:2   þ P 3x13;1 ; 7x13;2 ; 8x13;3 ; 12x13;4 ; 2x013;1 ; 7x13;2 ; 8x13;3 ; 13x013;4 ; 0:2   þ P 1x21;1 ; 3x21;2 ; 4x21;3 ; 5x21;4 ; 0:5x021;1 ; 3x21;2 ; 4x21;3 ; 5x021;4 ; 0:2   þ P 3x22;1 ; 5x22;2 ; 6x22;3 ; 7x22;4 ; 2x022;1 ; 5x22;2 ; 6x22;3 ; 8x022;4 ; 0:2   þ P 2x23;1 ; 6x23;2 ; 7x23;3 ; 11x23;4 ; x023;1 ; 6x23;2 ; 7x23;3 ; 12x023;4 ; 0:2   þ P 3x31;1 ; 4x31;2 ; 5x31;3 ; 8x31;4 ; 2x031;1 ; 4x31;2 ; 5x31;3 ; 9x031;4 ; 0:2   þ P x32;1 ; 2x32;2 ; 3x32;3 ; 4x32;4 ; 0:5x032;1 ; 2x32;2 ; 3x32;3 ; 5x032;4 ; 0:2  i þ P 2x33;1 ; 4x33;2 ; 5x33;3 ; 10x33;4 ; 1x033;1 ; 4x33;2 ; 5x33;3 ; 11x033;4 ; 0:2 Subject to Constraints of the generalized intuitionistic fuzzy linear programming problem (5.12.5).

5.12

Illustrative Example

261

Crisp linear programming problem (5.12.8) h   Minimize 0:2 4x12;1 þ 6x12;2 þ 7x12;3 þ 8x12;4 þ 3x012;1 þ 6x12;2 þ 7x12;3 þ 9x012;4   þ 0:2 3x13;1 þ 7x13;2 þ 8x13;3 þ 12x13;4 þ 2x013;1 þ 7x13;2 þ 8x13;3 þ 13x013;4   þ 0:2 1x21;1 þ 3x21;2 þ 4x21;3 þ 5x21;4 þ 0:5x021;1 þ 3x21;2 þ 4x21;3 þ 5x021;4   þ 0:2 3x22;1 ; 5x22;2 þ 6x22;3 þ 7x22;4 þ 2x022;1 þ 5x22;2 þ 6x22;3 þ 8x022;4 ;   þ 0:2 2x23;1 þ 6x23;2 ; 7x23;3 þ 11x23;4 þ x023;1 þ 6x23;2 þ 7x23;3 þ 12x023;4   þ 0:2 3x31;1 þ 4x31;2 þ 5x31;3 þ 8x31;4 þ 2x031;1 þ 4x31;2 þ 5x31;3 þ 9x031;4   þ 0:2 x32;1 þ 2x32;2 þ 3x32;3 þ 4x32;4 þ 0:5x032;1 þ 2x32;2 þ 3x32;3 þ 5x032;4  i þ 0:2 2x33;1 þ 4x33;2 þ 5x33;3 þ 10x33;4 þ 1x033;1 þ 4x33;2 þ 5x33;3 þ 11x033;4 Subject to Constraints of the crisp linear programming problem (5.12.5). Step 6: It can be easily verified that on solving the crisp linear programming problem (5.12.8), the following unique crisp optimal solution is obtained. (i). (ii). (iii). (iv). (v). (vi). (vii). (viii). (ix).

x11;1 x12;1 x13;1 x21;1 x22;1 x23;1 x31;1 x32;1 x33;1

¼ 3; ¼ 0; ¼ 1; ¼ 0; ¼ 0; ¼ 0; ¼ 3; ¼ 4; ¼ 1;

x11;2 x12;2 x13;2 x21;2 x22;2 x23;2 x31;2 x32;2 x33;2

¼ 3:5; x11;3 ¼ 4; x11;4 ¼ 4; x011;1 ¼ 2; x011;4 ¼ 4 ¼ 0; x12;3 ¼ 0:5; x12;4 ¼ 0:5; x012;1 ¼ 0; x012;4 ¼ 0:5 ¼ 2:5; x13;3 ¼ 3:5; x13;4 ¼ 4:5; x013;1 ¼ 0; x013;4 ¼ 5:5 ¼ 0:5; x21;3 ¼ 1; x21;4 ¼ 2; x021;1 ¼ 0; x021;4 ¼ 4 ¼ 0; x22;3 ¼ 0; x22;4 ¼ 0; x022;1 ¼ 0; x022;4 ¼ 1 ¼ 0; x23;3 ¼ 0; x23;4 ¼ 0; x023;1 ¼ 0; x023;4 ¼ 0 ¼ 3; x31;3 ¼ 3; x31;4 ¼ 3; x031;1 ¼ 3; x031;4 ¼ 3 ¼ 5; x32;3 ¼ 5:5; x32;4 ¼ 6:5; x032;1 ¼ 3; x022;4 ¼ 6:5 ¼ 1:5; x33;3 ¼ 1:5; x33;4 ¼ 1:5; x033;1 ¼ 1:5; x033;4 ¼ 1:5

Step 7: Using the crisp optimal solution, obtained in Step 6, the generalized intuitionistic fuzzy optimal solution of the generalized intuitionistic fuzzy linear programming problem (5.12.1) is (i). (ii). (iii). (iv). (v). (vi).

~x11 ~x12 ~x13 ~x21 ~x22 ~x23

¼ ð3; 3:5; 4; 4; 2; 3:5; 4; 4; 0:2Þ ¼ ð0; 0; 0:5; 0:5; 0; 0; 0:5; 0:5; 0:2Þ ¼ ð1; 2:5; 3:5; 4:5; 0; 2:5; 3:5; 5:5; 0:2Þ ¼ ð0; 0:5; 1; 2; 0; 0:5; 1; 4; 0:2Þ ¼ ð0; 0; 0; 0; 0; 0; 0; 1; 0:2Þ ¼ ð0; 0; 0; 0; 0; 0; 0; 0; 0:2Þ

5 JMD Approach for Solving Unbalanced Fully Generalized …

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(vii). ~x31 ¼ ð3; 3; 3; 3; 3; 3; 3; 3; 0:2Þ (viii). ~x32 ¼ ð4; 5; 5:5; 6:5; 3; 5; 5:5; 6:5; 0:2Þ (ix). ~x33 ¼ ð1; 1:5; 1:5; 1:5; 1:5; 1:5; 1:5; 1:5; 0:2Þ and the generalized intuitionistic fuzzy optimal value of the generalized intuitionistic fuzzy linear programming problem (5.12.1) is ð24; 61; 94:5; 157; 11; 61; 94:5; 203:5; 0:2Þ.

5.13

Conclusions

Some limitations and a drawback of the existing approach [1] for solving balanced fully generalized trapezoidal intuitionistic fuzzy transportation problems are pointed out. Also, to overcome the limitations and to resolve the drawback, a new approach (named as JMD approach) has been proposed for solving unbalanced fully generalized trapezoidal intuitionistic fuzzy transportation problems. Furthermore, the exact generalized intuitionistic fuzzy optimal solution of the existing balanced fully generalized trapezoidal intuitionistic fuzzy transportation problem [1] has been obtained by the proposed JMD approach.

References 1. D. Chakraborty, D.K. Jana, T.K. Roy, Arithmetic operations on generalized intuitionistic fuzzy number and its applications to transportation problem. Opsearch 52, 431–471 (2015) 2. W. Jianqiang, Z. Zhong, Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems. J. Syst. Eng. Electron. 20, 321–326 (2009)