Interval-Valued Intuitionistic Fuzzy Sets [1st ed. 2020] 978-3-030-32089-8, 978-3-030-32090-4

Table of contents :
Front Matter ....Pages i-xi
On Brouwer’s Intuitionism and Intuitionistic Fuzziness (Krassimir T. Atanassov)....Pages 1-7
On Interval Valued Intuitionistic Fuzzy Sets (Krassimir T. Atanassov)....Pages 9-25
Relations and Operations over IVIFSs (Krassimir T. Atanassov)....Pages 27-51
Operators over IVIFSs (Krassimir T. Atanassov)....Pages 53-122
Interval Valued Intuitionistic Fuzzy Pairs (Krassimir T. Atanassov)....Pages 123-129
Applications of IVIFSs (Krassimir T. Atanassov)....Pages 131-194
Conclusion and Remarks on Future Research (Krassimir T. Atanassov)....Pages 195-197
Back Matter ....Pages 199-200

Citation preview

Studies in Fuzziness and Soft Computing

Krassimir T. Atanassov

Interval-Valued Intuitionistic Fuzzy Sets

Studies in Fuzziness and Soft Computing Volume 388

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

Krassimir T. Atanassov

Interval-Valued Intuitionistic Fuzzy Sets

123

Krassimir T. Atanassov Department of Bioinformatics and Mathematical Modelling Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences Sofia, Bulgaria

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-3-030-32089-8 ISBN 978-3-030-32090-4 (eBook) https://doi.org/10.1007/978-3-030-32090-4 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

I dedicate this book to the 150th anniversary of the establishment of the Bulgarian Academy of Sciences—the oldest institution in my home country.

Preface

Intuitionistic Fuzzy Sets (IFSs) were introduced in 1983 as an extension of fuzzy sets of Lotfi Zadeh (1921–2017), and as soon as the first paper in which they were proposed, the justification of this extension was provided. Six years later, in 1989, Georgi Gargov (1947–1996) extended the IFSs to Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs). In the next years, the theory of ICIFS was enriched with new operations and operators, and covered in my 1999 monograph ‘Intuitionistic Fuzzy Sets: Theory and Applications’, published by Springer-Verlag. I started working on the present book some 20 years ago, at the beginning of the century, but it was only the results obtained in the last 2 years that urged me to complete it. In this book, I present my main results and their applications which in my opinion are the ones most interesting. In the last decade, a number of predominantly applied studies on IVIFS have appeared. I would encourage their authors to follow my example and collect their results in books. The presented research has been partially supported by the National Science Fund of Bulgarian under Grant No. Grant Ref No DN-02-10 ‘New Instruments for Knowledge Discovery from Data, and their Modelling’. I would like to express my deepest gratitude to my former Ph.D. students Velin Andonov and Peter Vassilev, and to my daughter Vassia Atanassova for their careful reading and correction of the manuscript. Sofia, Bulgaria June 2019

Krassimir T. Atanasov

vii

Contents

1 On Brouwer’s Intuitionism and Intuitionistic Fuzziness . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... ..... .....

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2 On 2.1 2.2 2.3

Interval Valued Intuitionistic Fuzzy Sets . . . . . . . . . . . . . Intuitionistic Fuzzy Sets and Interval Valued Fuzzy Sets . . . Definition of Interval Valued Intuitionistic Fuzzy Sets . . . . On Representability of Interval Valued Intuitionistic Fuzzy Sets by Pairs of Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . 2.4 Some Ways for Altering Experts’ Estimations . . . . . . . . . . 2.5 Norms and Metrics on IVIFS . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Relations and Operations over IVIFSs . . . . . . . . . . . 3.1 Relations over IVIFSs . . . . . . . . . . . . . . . . . . . . . 3.2 Operations over IVIFSs . . . . . . . . . . . . . . . . . . . 3.3 An IVIFS Whose Universe is an IFS or an IVIFS with Respect to Another Universe . . . . . . . . . . . . 3.4 IVIFSs over Different Universes . . . . . . . . . . . . . 3.5 The Extension Principle for the IVIFS-Case . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Operators over IVIFSs . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Interval Valued Intuitionistic Fuzzy Modal Operators of First Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Interval Valued Intuitionistic Fuzzy Modal Operators of Second Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Theorem for Equivalence of the Two Most Extended Modal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Topological Operators over IVIFSs . . . . . . . . . . . . . . . 4.5 Level and Modal-Level Operators on IVIFSs . . . . . . . . 4.6 Level Operator That Decrease the Number of Elements of the IVIFSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.7 Two Other Types of Operators . . . . . . . . . . . . . . . . . 4.8 Simplest and ðfi; flÞ-Shrinking Operators over IVIFSs . 4.9 Weight-Center Operator . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Interval Valued Intuitionistic Fuzzy Pairs . 5.1 Definition of an IVIFP . . . . . . . . . . . . 5.2 Relations over IVIFPs . . . . . . . . . . . . . 5.3 Opertions over IVIFPs . . . . . . . . . . . . 5.4 Operators over IVIFPs . . . . . . . . . . . . 5.5 On IVIF-Interpretation of Interval Data References . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Applications of IVIFSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Extended Interval Valued Intuitionistic Fuzzy Index Matrices . . 6.2 Interval Valued Intuitionistic Fuzzy Graphs . . . . . . . . . . . . . . . 6.3 Intuitionistic Fuzzy Neural Networks with Interval Valued Intuitionistic Fuzzy Conditions . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Interval Valued Intuitionistic Fuzzy Generalized Nets . . . . . . . . 6.5 Intercriteria Analysis with Interval-Valued Intuitionistic Fuzzy Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Interval Valued Intuitionistic Fuzzy Sets as Tools for Evaluation of Data Mining Processes—Possibilities for the Future . . . . . . . 6.6.1 IVIF-Estimations in Expert Systems, Data Bases, Data Warehouses, Big Data, OLAP-Structures . . . . . . . . 6.6.2 IVIF-Estimations of a Procedure for Inductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 IVIF-Estimations in Decision Making Procedures . . . . . 6.6.4 IVIFS-Estimations in Pattern Recognition Procedures . . 6.6.5 IF-Estimations in Neural Networks and Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Conclusion and Remarks on Future Research . . . . . . . . . . . . . . . . . 195 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

About the Author

Krassimir T. Atanassov is a Professor and the Head of the Department of ‘Bioinformatics and Mathematical Modelling’ at the Bulgarian Academy of Sciences. He received the title of Doctor of Mathematical Sciences, Doctor of Technical (Computer) Sciences, and completed a Ph.D. in Mathematical Sciences. In 2013, he received the Award ‘Pythagoras’ from the Ministry of Education, Youth and Science of Bulgaria, as an established researcher in the area of Technical Sciences. Since the same year, he has also been Fellow of the International Fuzzy Systems Association. His research interests cover the areas of Petri Nets, Fuzzy Sets and Number Theory. In the field of fuzzy sets, he developed the theory of intuitionistic fuzzy sets and showed their applications to artificial intelligence, systems theory, medicine, etc. He has been serving as E-i-C of different journals. He is the author of numerous monographs, including the book Intuitionistic Fuzzy Logics (Springer, 2017). He is a corresponding member of the Bulgarian Academy of Sciences.

xi

Chapter 1

On Brouwer’s Intuitionism and Intuitionistic Fuzziness

Aristotle made the first steps to the establishment of the mathematical logic by giving ideas for the concepts “sentence” and “predicate”; the fundamental logical functions as conjunction, disjunction and negation, the logical quantifiers, modal operators, and many others. He was the first who justified the need for axioms and presented the first examples of such. For 23 centuries one of his proposed axioms—the Law of Excluded Middle (LEM)—has been among the main tools for proving mathematical assertions. The first paradoxes appeared in Ancient Greece - Paradox “Achilles and the Tortoise”, “Arrow” and Epimenides’s Paradox “All Cretans are Liars”, that in simpler form is “I lie”. In the last quarter of XIX century, Georg Kantor developed set theory, but in 1888 he generated the first important mathematical and logical paradox. He did not publish it, because he had fears for the fate of his theory. Even 40 years later, David Hilbert published it under the name of Cantor’s paradox. Meantime, in the beginning of 20-th century, a lot of logical paradoxes appeared, which put in doubt the very foundations of mathematics. The first of them is the well-known Bertrand Russel’s Paradox of the Barber. In the first decade of the century, David Hilbert and Bertrand Russel proposed ideas how the crisis in mathematics might be overcome, but with his theorem from the beginning of the 1930s Kurt Gödel showed the inapplicability of such ideas. In 1912, Luitzen Egbertus Jan Brouwer proposed a new idea, called by him “intuitionism”. Brouwer’s idea for intuitionism was published in [8] (its English translation is given in [9]). It is: mathematical constructions to use only finite objects and to not use the Aristoteles’ LEM (see also [6, 7, 10]). In 1926, Jan Łukasiewicz for the first time proposed the sentences and predicates to be evaluated not by the the two values 0 (“false”) and 1 (“true”), as has been done since the times of Aristotle, but to add an additional value 21 (“uncertainty”). This revolutionary step led to the emergence of a completely new type of logic. It is a serious argument supporting Brouwer’s intuitionism. In 1956, Łukasiewicz generalized the proposed by him three-valued logic to n-valued (many-valued logic). © Springer Nature Switzerland AG 2020 K. T. Atanassov, Interval-Valued Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing 388, https://doi.org/10.1007/978-3-030-32090-4_1

1

2

1 On Brouwer’s Intuitionism and Intuitionistic Fuzziness

The next step in the development of this idea was made by Lotfi Zadeh in 1965 with the introduction of the concept “fuzzy set” [17]. Just several years after the introduction of fuzzy sets, they became an object of further generalizations: L-fuzzy sets (see [13]), rough sets (see [15, 16]), etc. In 1983, the intuitionistic fuzzy sets were defined, in which for the first time two degrees were proposed of membership, or validity, or correctness (μ) and of nonmembership, or non-validity, or non-correctness, etc (ν) [1] (see also [2]). Behind this definition is clearly seen Brouwer’s idea for intuitionism because every sentence, predicate, object, etc is evaluated not only as true (μ) or false (ν), but also by the degree of indeterminacy π. This was the reason the new sets were called intuitionistic fuzzy sets. Their definition does not preclude the possibility to define over such sets operations of the classical logic - for instance, negation and implication, however, it also provides the opportunity to define a wide class of non-classical operations: negation, implication, conjunction, disjunction, etc. In recent years, the literature devoted to fuzzy sets there appeared definitions of objects named with the word “intuitionistic” but without the word “fuzzy”. This is incorrect because the reader is left with the impression that these objects are from the area of Brouwer’s intuitionism, but not from the area of Zadeh’s fuzzy sets. After their introduction, intuitionistic fuzzy sets became by the second half of the 1980s an object of generalizations—intuitionistic L-fuzzy, interval valued intuitionistic fuzzy sets, intuitionistic fuzzy sets of second (and more generally n-th)-type, temporal intuitionistic fuzzy sets (see [4]). Let us have a fixed universe E and its subset A. The set A∗ = {x, μ A (x), ν A (x) | x ∈ E}, where 0 ≤ μ A (x) + ν A (x) ≤ 1

(1.1)

is called IFS and functions μ A : E → [0, 1] and ν A : E → [0, 1] represent the degree of membership (validity, etc.) and non-membership (non-validity, etc.) of element x ∈ E to a fixed set A ⊆ E. Thus, we can also define function π A : E → [0, 1] by means of π(x) = 1 − μ(x) − ν(x) and it corresponds to the degree of indeterminacy (uncertainty, etc.). For brevity, we shall write below A instead of A∗ , whenever this is possible. We must mention that some authors assert that they use a modification of IFSs for which functions μ A and ν A obtain a finite number of (fixed) values. But this is not a modification of the IFSs. In practice, these authors use standard IFSs, because from the above definition we see that functions μ A and ν A obtain values from interval [0, 1], but it nowhere says that these functions must obtain all values from this interval. For example, if these functions represent the weather√forecast for tomorrow, then the share of sunny time will not have the form π1 or 5 − 2. They will evaluate both functions as percentages or other rational numbers. Therefore, functions μ A and ν A

1 On Brouwer’s Intuitionism and Intuitionistic Fuzziness

3

can be defined in such a way that they obtain, e.g. values 0 or 1; 0 or 0,5 or 1; or 0, 1 2 9 or 10 or 10 or 1, etc. 10 The IFS A is an extension of the fuzzy set B over the fixed universe E, because the fuzzy set B is defined by B = {x, μ B (x) | x ∈ E}, where 0 ≤ μ B (x) ≤ 1 and function μ B : E → [0, 1] represents the degree of membership (validity, etc.) of element x ∈ E to fixed set B ⊆ E. Obviously, each fuzzy set can be represented in the form of an IFS: B = {x, μ B (x), 1 − μ B (x) | x ∈ E}. In the last 35 years, a lot of opertions, relations and operators from different types were introduced over IFSs (see, e.g. [4, 5]). For example, A ⊂ B iff (∀x ∈ E)((μ A (x) ≤ μ B (x) & ν A (x) > ν B (x)) ∨(μ A (x) < μ B (x) & ν A (x) ≥ ν B (x)) ∨(μ A (x) < μ B (x) & ν A (x) > ν B (x))); A ⊆ B iff (∀x ∈ E)(μ A (x) ≤ μ B (x) & ν A (x) ≥ ν B (x)); A = B iff (∀x ∈ E)(μ A (x) = μ B (x) & ν A (x) = ν B (x)); ¬A = {x, ν A (x), μ A (x)|x ∈ E}; A ∩ B = {x, min(μ A (x), μ B (x)), max(ν A (x), ν B (x))|x ∈ E}; A ∪ B = {x, max(μ A (x), μ B (x)), min(ν A (x), ν B (x))|x ∈ E}; A + B = {x, μ A (x) + μ B (x) − μ A (x).μ B (x), ν A (x).ν B (x) | x ∈ E}; A.B = {x, μ A (x).μ B (x), ν A (x) + ν B (x) − ν A (x).ν B (x) | x ∈ E}; B (x)) (ν A (x)+ν B (x)) A@B = {x, ( μ A (x)+μ , |x ∈ E}. 2 2 About 20 years ago, some of my colleagues, who opposed to the concept of IFS asserted that the IFSs are a trivial modification, and not extension of the fuzzy sets, because each IFS A can be represented by a pair of fuzzy sets B and C. I answered them that their idea is not new one, because 10 years before them, in 1988, Toader Buhaescu introduced it in [11, 12], but without the claim that the IFSs are a modification of fuzzy sets. In addition, I mentioned that the complex numbers are represented as pairs of real numbers, but it would be nonsensical to claim that the complex numbers are a trivial modification of the real numbers. First, we mention that in [3] two geometrical interpretation of the IFSs are given. The first geometrical interpretation of the IFS is shown in Fig. 1.1. Its analogue is given in Fig. 1.2. The second geometrical interpretation of the IFSs is given on Fig. 1.3. The set of the points of the interpretation triangle from Fig. 1.1 can be written as

4 Fig. 1.1 First form of the first geometrical interpretation of an IFS

1 On Brouwer’s Intuitionism and Intuitionistic Fuzziness

1

μA

-

0

νA

E Fig. 1.2 Second form of the first geometrical interpretation of an IFS

1

1 − νA

μA

-

0

E

Fig. 1.3 Second geometrical interpretation of an IFS

0, 1

νA (x)

0, 0

@ @ @

E @

• x

@ @

@ @ @  • @

μA (x)

@ @

@ 1, 0

1 On Brouwer’s Intuitionism and Intuitionistic Fuzziness

5

L = { p, q | p, q, p + q ∈ [0, 1]}. In [10], Brouwer wrote: “An immediate consequence was that for a mathematical assertion the two cases of truth and falsehood, formerly exclusively admitted, were replaced by the following three: (1) has been proved to be true; (2) has been proved to be absurd; (3) has neither been proved to be true nor to be absurd, nor do we know a finite algorithm leading to the statement either that is true or that is absurd.” Therefore, if we have a proposition A, we can state that either A is true, or A is false, or that we do not know whether A is true or false. On the level of first order logic, the proposition A ∨ ¬A is always valid. In the framework of G. Boole’s algebra this expression has truth value “true” (or 1). In the ordinary fuzzy logic of L. Zadeh, as well as in many-valued logics (starting with that of J.Łukasiewicz) the above expression can possess a value smaller than 1. The same is true in the intuitionistic fuzzy case, but here the situation occurs on semantical as well as on estimations’ level. Practically, we fuzzify our estimation in Brouwer’s sense, accounting for the three possibilities. This was Georgi Gargov’s reason to offer me to use the name “intuitionistic fuzzy set”. In [5], I described my first acquaintance with Prof. Lotfi Zadeh. It was in 2001 in Villa Real, Portugal, where Prof. Pedro Melo-Pinto organized a school on fuzzy sets. Prof. Zadeh was invited for a 3-h lecture, which he concluded with presentation of slides with articles by Samuel Kleene, Kurt Gödel and other luminaries of mathematical logic, who have written against the fuzzy sets. In the next years, I have been long tormented by the question why these mathematicians had opposed the fuzzy sets while they did not have anything to say against the three- and multi-valued logics of Jan Lukasiewicz. Thus I reached the conclusion that the reason for the then negative attitude towards fuzzy sets was hidden in the presence of the [0, 1] interval as the set of the fuzzy sets’ membership function (see, e.g, [14, 17, 18]). Indeed, the values of the membership function do belong to the [0, 1] interval, yet it does not mean that this function obtains all possible values in this interval! If an expert or a group of experts evaluates, for instance, the chances of a political party to win the elections, it is slightly ever probable (if not absurd) for them to use estimations like 1e √ or 2 − 1. For any unbiased man it is clear that the experts would not use anything more complex than decimal fractions with one or two digits after the decimal point, i.e. rational numbers. Rational and even integer numbers are those which we use to measure the sizes of objects, the daily temperature or the speed of the vehicles, which are often described by fuzzy sets. Yes, the contemporary mathematics is the mathematics of multiple integrals, topological spaces, arithmetic functions, yet all these objects are abstractions of objects, existing in reality, which in the end of the day are measured, i.e. certain mathematical estimations are constructed for them, hence these estimations are constructive! In addition, the estimated objects are finite in number. Obviously, no one can estimate infinite number of objects for all his life. Therefore, Brouwer’s idea mathematics to use only finite objects, has been exactly

6

1 On Brouwer’s Intuitionism and Intuitionistic Fuzziness

realized. In [5], it is shown that each IFS is a constructive object and all operations and operators defined over it keep this property. Now, we discuss the relations between ordinary fuzzy sets and IFSs from point of view of the LEM, from two aspects: geometrical and probabilistical. Initially, I would like to note that some authors discuss the fact that in the case of ordinary fuzzy sets μ ∨ ¬μ ≤ 1 as a manifest of the idea of intuitionism. Indeed, this inequality, in its algebraic interpretation of “∨” by “max”, does not satisfy the LEM. But this is not the situation in a geometrical interpretation. Having in mind that in fuzzy set theory ¬μ = 1 − μ, we obtain that the geometrical interpretation is as follows: 0

1 

 μ



 ¬μ



The situation in the IFS case is different and is as follows: 

0 



πA (x) 





1 

μA (x)





νA (x) μA (x)

1 − νA (x)

Now, the geometrical sums of both degrees can really be smaller than 1, i.e., LEM is not valid here. From probabilistic point of view, for the case of ordinary fuzzy sets, if μ ∧ ¬μ = 0, then the probability p(μ ∨ ¬μ) = p(μ) + p(¬μ) = 1, like in the geometrical case, while in the IFS case we have the inequality p(μ ∨ ¬μ) ≤ 1, which for proper IFS-elements will be strong. Therefore, the intuitionistic fuzziness corresponds exactly to Brouwer’s idea that the LEM is not valid. There are publications, in which the authors discussing some intuitionistic fuzzy objects (sets, values, pairs, etc), write the expression “intuitionistic objects” omitting the word “fuzzy” between the two words. This is important mistake, because in this

1 On Brouwer’s Intuitionism and Intuitionistic Fuzziness

7

way they direct their research not to the area of the intuitionistic fuzziness, but to Brouwer’s intuitionism. On the other hand, the author would like to note that as early as the first paper in which the concept of IFS was introduced, it was shown that they are an extention of the fuzzy sets. Several years later, IFSs were extended to Intuitionistic L-Fuzzy Sets, Intuitionistic Fuzzy Sets of Second Type (which nowadays some authors incorrectly have rebranded as Pythagorean Fuzzy Sets), Temporal Intuitionistic Fuzzy Sets, IVIFS and others. For each of these extensions it has been demonstrated that they are either equivalent to IFSs, (e.g., Intuitionistic Fuzzy Sets of Second Type, and in general Intuitionistic Fuzzy Sets of n-th Type, when n is a fixed positive number), or they are extensions of the IFSs. Unfortunately, as of the mid 1990s, some concepts have started appearing, which their inventors claim to be extensions of IFS, without providing formal proofs. For most of them, it has already been shown that are conceptually equivalent to IFSs, or are particular cases of IVIFSs, or are plainly misdefined. These would be the topic of the next book of the author.

References 1. Atanassov, K. Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, June 1983 (Deposed in Central Sci. - Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian). Reprinted: Int. J. Bioautomation, 2016, 20(S1), S1-S6 2. Atanassov, K.: Intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol. 20, Issue 1, pp. 87–96 (1986) 3. Atanassov K.: Geometrical interpretation of the elements of the intuitionistic fuzzy objects, Preprint IM-MFAIS-1-89, Sofia (1989). Reprinted: Int. J. Bioautom. 20(S1), S27–S42 (2016) 4. Atanassov, K.: Intuitionistic Fuzzy Sets. Springer, Heidelberg (1999) 5. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012) 6. van Atten, M.: On Brouwer. Wadsworth, Behnout (2004) 7. Benacerraf, P., Putnam, H. (eds.): Philosophy of Mathematics: Selected Readings, 2nd edn. Cambridge University Press, Cambridge (1983) 8. Brouwer, L.E.J.: Intuïtionisme en Formalisme. Clausen, Amsterdam (1912) 9. Brouwer, L.E.J.: In: Heyting, A. (ed.) Collected Works Vol. 1: Philosophy and Foundations of Mathematics. North Holland, Amsterdam (1975) 10. Brouwer, L.E.J.: In: van Dalen, D. (ed.) Brouwer’s Cambridge Lectures on Intuitionism. Cambridge University Press, Cambridge (1981) 11. Buhaescu, T.: On the convexity of intuitionistic fuzzy sets, pp. 137–144. Approximation and Convexity, Cluj-Napoca, Itinerant Seminar on Functional Equations (1988) 12. Buhaescu, T.: Some observations on intuitionistic fuzzy rerelations, pp. 111–118. Approximation and Convexity, Cluj-Napoca, Itinerant Seminar on Functional Equations (1989) 13. Goguen, J.: L-fuzzy sets. J. Math. Anal. Appl. 18(1), 145–174 (1967) 14. Kaufmann, A.: Introduction a la Theorie des Sour-Ensembles Flous. Masson, Paris (1977) 15. Pawlak, Z.: Rough functions, ICS, PAS Report 467 (1981) 16. Pawlak, Z.: Rough sets, ICS, PAS Report 431 (1981) 17. Zadeh, L.: Fuzzy sets. Inf. Control 8, 338–353 (1965) 18. Zadeh, L.: The Concept of a Linguistic Variable and its Application to Approximate Reasoning. American Elsevier Publ. Co., New York (1973)

Chapter 2

On Interval Valued Intuitionistic Fuzzy Sets

In this chapter, the basic definitions of the concepts of Interval Valued Fuzzy Sets (IVFSs) and Interval Valued Intuitionistic Fuzzy Sets (IVIFSs) will be introduced. The relation between IFSs and IVFSs will be discussed.

2.1 Intuitionistic Fuzzy Sets and Interval Valued Fuzzy Sets One of the extensions of the notion of fuzzy set is called IVFS (cf. e.g. [12, 15]). An IVFS A (over a universe set E) is specified by a function M A : E → P([0, 1]), where P(X ) is the set of all subsets of the set X , i.e. for all x ∈ E, M A (x) is an interval within [0, 1]. It has the form A = {x, M A (x) | x ∈ E}. The justification of both generalizations follows from Theorem 2.1.1 IFSs and IVFSs are equipollent generalizations of the notion of fuzzy set. Proof Initially we will construct two maps: • the map f assigns to every IVFS A an IFS B = f (A) given by: μ B (x) = inf M A (x) ν B (x) = sup M A (x); © Springer Nature Switzerland AG 2020 K. T. Atanassov, Interval-Valued Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing 388, https://doi.org/10.1007/978-3-030-32090-4_2

9

10

2 On Interval Valued Intuitionistic Fuzzy Sets

• the map g assigns to every IFS B an IVFS A = g(B) given by: M A (x) = [μ B (x), 1 − ν B (x)]. We will prove below that for every IVFS A: g( f (A)) = A and for all IFS B: f (g(B)) = B. Let A be an IVFS. Then for all x ∈ E: Mg( f (A)) (x) = [μ f (A) (x), 1 − ν f (A) (x)] = [inf M A (x), 1 − 1 + sup M A (x)) = M A (x). Let B be an IFS. Then for all x ∈ E: μ f (g(B)) (x) = inf Mg(B) (x) = inf[μ B (x), 1 − ν B (x)] = μ B (x); ν f (g(B)) (x) = 1 − sup Mg(B) (x) = 1 − sup[μ B (x), 1 − ν B (x)] = ν B (x). This completes our proof.

♦

2.2 Definition of Interval Valued Intuitionistic Fuzzy Sets The concept of an IVIFS, defined below is an extension of both IFS and IVFS. It was announced in [1], introduced in [7] and described in details in [2, 3, 11]. An IVIFS A over E is an object of the form: A = {x, M A (x), N A (x) | x ∈ E},

2.2 Definition of Interval Valued Intuitionistic Fuzzy Sets

16 HH XXX (( PPHH XX((((( XX  PP XXX P X P XX  XX XX hhh XX (P ( ( ( P ( ( XX P ( X( PP P 0

11

-

1 − inf NA (x) 1 − sup NA (x) sup MA (x) inf MA (x)

Fig. 2.1 First geometrical interpretation of an IVIFS element

where M A (x) ⊂ [0, 1] and N A (x) ⊂ [0, 1] are intervals and for all x ∈ E: sup M A (x) + sup N A (x) ≤ 1. This definition is analogous to the definition of IFS. It can be however rewritten to become an analogue of the definition from Sect. 2.1—namely, if M A and N A are interpreted as functions. Then, an IVIFS A (over a universe set E) is given by functions M A : E → P([0, 1]) and N A : E → P([0, 1]) and the above inequality. We must note that there is no difference in principle between the two approaches. And what is more, the same is true also in the ordinary fuzzy sets theory. The author originally used the first one influenced by the Kaufmann’s book [13]. Perhaps it was this approach that helped him develop the theory of operators over IFS in its present form. The same notation was used in 1987–88 in the research on IVIFSs, too (see [1, 3, 7]). The elements of an IVIFS have geometrical interpretations similar to, but more complex than these of the IFSs (cf. [3, 5]). The first of these geometrical interpretations is shown on Fig. 2.1. As in the IFS-case, we well as in the IVIFS-case, the second geometrical interpretation is the most important and informative (see Fig. 2.2). The author is not aware of a geometrical interpretation of an IVFS in the sense of Fig. 2.2. Now, we will do this. Obviously, each IVFS A can be represented by an IVIFS as A = {x, M A (x), N A (x) | x ∈ E} = {x, M A (x), [1 − sup M A (x), 1 − inf N A (x) | x ∈ E}.

12 Fig. 2.2 Second geometrical interpretation of an IVIFS element

2 On Interval Valued Intuitionistic Fuzzy Sets

0, 1 @ @ @

1 − sup MA r

@ @ @ @

sup NA r

@r @

@r @

inf NA r 0, 0

Fig. 2.3 Second geometrical interpretation of an IVFS element

r

r

r

@

@

@

inf MA sup MA 1 − sup NA

1, 0

0, 1 @ @ @q q sup MA = 1 − inf MA @ @ @ @ @ @ @ @ inf NA = 1 − sup MA q @q @ q 0, 0 inf M A

Fig. 2.4 Third geometrical interpretation of an IVIFS element

@

@ @ q @ 1, 0 sup MA


@ @
@
@ H
HH HH@
H@
α C α H@ β1  βH H
CC 2 C 1  2 @ α1 = π. inf MA (x), α2 = π. sup MA (x) β1 = π. inf NA (x), β2 = π. sup NA (x),

where π = 3.1415....

The geometrical interpretation of the IVFS A is shown on Fig. 2.3. It has the form of a section lying on the triangles hypothenuse. Other geometrical interpretations of the IVIFS elements are given on Figs. 2.4, 2.5, 2.6, 2.7 and 2.8.

2.2 Definition of Interval Valued Intuitionistic Fuzzy Sets

13

Fig. 2.5 Fourth geometrical interpretation of an IVIFS element

@ @ 1 − M2

N2 N1

@ @ @ l l @ l @ l @ l Q @ l Q @ l Q @ l Q @ l @ M1

1 − N2

M2

M1 = inf MA (x), M2 = sup MA (x) N1 = inf NA (x), N2 = sup NA (x) Fig. 2.6 Fifth geometrical interpretation of an IVIFS element

 

TT

T T  T sup NA (x)  "" "  T " P TN " T inf NA (x)  TT TT"""  T  T M inf M (x) sup M (x) A

A

Fig. 2.7 Three dimensional (sixth) geometrical interpretation of an IVIFS element

ν

6

@

@ @ @  @  @@ @  @ @ @  @ -μ             

π



14

2 On Interval Valued Intuitionistic Fuzzy Sets

Fig. 2.8 Seventh geometrical interpretation of an IVIFS element

The seventh geometrical interpretation of the IVIFS elements was introduced in [8]. As originally proposed in [8], the Fig. 2.8 represents a geometrical interpretation of an IVIFS in the form of a radar chart. It particularly shows a data series of 12 periods (months) with the averaged data of the minimal and maximal monthly temperatures over a 10-year period of time. This figure, as well as 2.1, shows the IVIFS as a whole, while the rest of the figures represent single elements of the IVIFS. Let us call the IVIFS A: • tautological set, if for each x ∈ E : inf M A (x) = sup M A (x) = 1 and inf N A (x) = sup N A (x) = 0 (therefore, E ∗ is a tautological set; • strong tautological IVIFS, if for each x ∈ E : inf M A (x) > 0.5 > sup N A (x); • tautological IVIFS, if for each x ∈ E : inf M A (x) ≥ sup N A (x); • weak tautological IVIFS, if for each x ∈ E : inf M A (x) ≥ inf N A (x) and sup M A (x) > sup N A (x).

2.3 On Representability of Interval Valued Intuitionistic Fuzzy Sets by Pairs of Intuitionistic Fuzzy Sets Following [6], let x ∈ E be an arbitrary element with degrees a and b (i.e., a, b, a + b ∈ [0, 1]), for the triple x, a, b. We say that the triple x, a, b is a nesting in the IVIFS A and denote x, a, bεA if and only if (iff) a ∈ M A (x), b ∈ N A (x).

2.3 On Representability of Interval Valued Intuitionistic Fuzzy Sets by Pairs of Intuitionistic Fuzzy Sets 15

We will say that the IFS C = {x, μC (x), νC (x) | x ∈ E} is nesting in the IVIFS A iff for each x ∈ E, the triple x, μC (x), νC (x)εA. Now, let us define for the fixed above IVIFS A, A = {x, inf M A (x), sup N A (x) | x ∈ E}, A = {x, sup M A (x), inf N A (x) | x ∈ E}. Obviously, for each IVIFS A, A ⊆ A. Moreover, sets A and A are IFSs and they are determined bijectively from A. Theorem 2.3.1 Let A be an IVIFS and B be an IFS. A ⊆ B ⊆ A iff B is nesting in A. Proof Let for the IVIFS A and IFS B, the inclusion A ⊆ B ⊆ A be valid. Therefore, for each x ∈ E: inf M A (x) ≤ μ B (x) ≤ sup M A (x) and inf N A (x) ≤ ν B (x) ≤ sup N A (x). Hence, for each x ∈ E: μ B (x) ∈ M A (x) and ν B (x) ∈ N A (x), i.e. IFS B is nesting in A. The opposite direction of the theorem and the next theorem can be proved in a similar way. We call the two IFSs A and A lower and upper boundary IFSs of the IVIFS A, respectively. Theorem 2.3.2 For every two IFSs B and C over universe E, such that for each x∈E sup M B (x) + sup NC (x) ≤ 1 and C ⊆ B, there exists a unique IVIFS A with upper and lower boundary IFSs A = B and A = C. From both theorems it follows that each IVIFS can be represented by a pair of IFSs.

16

2 On Interval Valued Intuitionistic Fuzzy Sets

2.4 Some Ways for Altering Experts’ Estimations In [3] and later, in [5], different ways for altering experts’ estimations are discussed for the case, when the universe is finite and values of functions μ and ν are given by experts, who can be wrong. These ways that are described in [5] can be transformed directly from IFS- to IVIFS-case and by this reason we will not discuss them. In [16] Peter Vassilev and in [9] Piotr Dworniczak introduced other ways that also can be transformed for the IVIFS-case. Having in mind [4, 5], the following problem is discussed: To find a continuous bijective transformation that transforms the unit square ABC D to the IFS interpretation triangle AB D (Fig. 2.9). The solution to this question, as it is shown in [5] gives the possibility: • to prove that each bi-lattice (see e.g., [10]) can be interpreted by an IFS; • to prove that each intuitionistic L-fuzzy set with universe L of the same power as the continuum (see [3]) can be interpreted by an IFS; • to construct a new algorithm for modifying incorrect expert estimations (see [3]). Here, we cite two theorems from [5] without proofs and will use them to transform wrong expert IVIF-estimations. Theorem 2.4.1 The transformation ⎧ 0, 0 if x = y = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  2  ⎪ ⎪ x , xy ⎪ if x, y ∈ [0, 1] and x ≥ y ⎪ ⎪ x + y x + y ⎨ and x + y > 0 F(x, y) = ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ xy y2 ⎪ ⎪ x + y , x + y if x, y ∈ [0, 1] and x ≤ y ⎪ ⎪ ⎪ ⎩ and x + y > 0 presents a solution to the Problem (see Fig. 2.9). Corollary 2.4.1 For every x, y ∈ [0, 1], 

 1 y x 1 F(1, y) = , , F(x, 1) = , , 1+y 1+y x +1 x +1  F(1, 1) = F



 1 1 , , F(x, 1 − x) = x 2 , x − x 2 , 2 2

1 1 , 2 2

 =

1 1 , , F(0, 0) = 0, 0. 4 4

The following theorem can be proved in a similar manner,

(2.4.1)

2.4 Some Ways for Altering Experts’ Estimations

17

D (0,1)

Fig. 2.9 Transformation F

@ @

C (1,1) •

•



@
• ◦

@

@ ◦ ◦@ •   @ ◦ • @  @  ◦

@

A (0,0)

B (1,0)

D (0,1)

Fig. 2.10 Transformation G

C (1,1) •

•

@ @ ◦ • @ @ ◦ @◦ @• @ @ ◦ • @ ◦

@

A (0,0)

B (1,0)

Theorem 2.4.2 The transformation ⎧ y ⎪ ⎪ x − 2, ⎨ G(x, y) =  ⎪ ⎪ ⎩ x,y − 2

y 2 x 2

 

if x, y ∈ [0, 1] and x ≥ y (2.4.2) if x, y ∈ [0, 1] and x ≤ y

(see Fig. 2.10). For every u, v ∈ (0, 1] such that u + v ≤ 1, G −1 (u, v) =

⎧ ⎨ u + v, 2v if u ≥ v ⎩

2u, u + v if u ≤ v

.

Corollary 2.4.2 For every x, y ∈ [0, 1],  x y y x G(1, y) = 1 − , , G(x, 1) = , 1 − , 2 2 2 2   1 1 3x − 1 1 − x 1 G(1, 1) = , , G(x, 1 − x) = , , if x ≥ , 2 2 2 2 2

18

2 On Interval Valued Intuitionistic Fuzzy Sets

D (0,1)

Fig. 2.11 An example of an IVIFS element—a parallelogram P Q R S with vertexes

 coordinates

 P 21 , 41 , Q 43 , 41 ,

3 17  1 17  R 4 , 40 , S 2 , 40

C (1,1)

@ @ @ @

A (0,0)

@ @ S @ R @ P @Q @ @ a B (1,0) b

D (0,1)

C (1,1)

d c

Fig. 2.12 The parallelogram P Q R S after F-transformation is a tetragon P Q R S with vertexes

coordinates

9 3 P 13 , 16 , Q 16 , ,  

16  51 10 , 17 , S , R 45 94 188 37 74

@ @ @ @

@ @

@ @ R S H H  @ ( Q (( Q @ P 

@

A (0,0)

 G(x, 1 − x) =

B (1,0)



 3x 1 1 1 1 1 x ,1 − , if x ≤ , G , = , , 2 2 2 2 2 4 4 G(0, 0) = 0, 0.

Now, we can transform the above results from IFS- to the IVIFS-case. Therefore, we must apply functions F and G over the vertexes of the rectangle P Q RS from Fig. 2.11 and let for brevity, the coordinates of its vertexes be a, c, b, c, b, d, a, d, respectively. As a result, we will obtain the tetragon P Q R S that is not a rectangle. For the function G we obtain a parallelogram. On Figs. 2.12 and 2.13 the results of applying of functions F and G over the vertexes of the rectangle P Q RS is shown. Hence, the functions F and G must be applied only over two of the vertexes of P Q RS—P and R and the remaining two vertexes will be constructed, using the calculated coordinates of P and R, keeping the rectangle form of the new figure P Q R S . For function F we must consider the following four cases. Case 1: a ≥ c, b ≥ d. Then  F(a, c) =

a2 ac , , a+c a+c

2.4 Some Ways for Altering Experts’ Estimations

19

D (0,1)

Fig. 2.13 The parallelogram P Q R S after G-transformation is a parallelogram P Q R S with coordinates 

vertexes  P 38 , 18 , Q 58 , 18 ,

43 17  23 17  , 80 , S 80 , 80 R 80

C (1,1)

@ @ @ @

d c

@ @ @

S C P

A (0,0)

a

@ @ CR  @ Q @

b

bd b2 , . F(b, d) = b+d b+d 

Therefore, the parallelogram P Q R S will have coordinates  2 ac ac a2 b , , b , c  = , a , c  = a+c a+c b+d a+c





b , d  =



 2 bd bd b2 a , , a , d  = , . b+d b+d a+c b+d

respectively. Case 2: a ≥ c, b ≤ d. Then a2 ac , F(a, c) = , a+c a+c 

 F(b, d) =

bd d2 , . b+d b+d

Therefore, the parallelogram P Q R S will have coordinates a , c  =



 ac ac a2 bd , , b , c  = , a+c a+c b+d a+c

 2 d2 d2 bd a , , a , d  = , . b , d  = b+d b+d a+c b+d

respectively.





B (1,0)

20

2 On Interval Valued Intuitionistic Fuzzy Sets

Case 3: a ≤ c, b ≥ d. Then c2 ac , , F(a, c) = a+c a+c 



b2 bd F(b, d) = , . b+d b+d Therefore, the parallelogram P Q R S will have coordinates a , c  = b , d  =





 2 c2 c2 ac b , , b , c  = , a+c a+c b+d a+c

 bd bd b2 ac , , a , d  = , . b+d b+d a+c b+d

Case 4: a ≤ c, b ≤ d. Then  F(a, c) =

ac c2 , , a+c a+c

d2 bd , . F(b, d) = b+d b+d 

Therefore, the parallelogram P Q R S will have coordinates  c2 c2 ac bd , , b , c  = , a , c  = a+c a+c b+d a+c





 d2 d2 bd ac , , a , d  = , . b , d  = b+d b+d a+c b+d





For function G we must consider also the following four cases. Case 1: a ≥ c, b ≥ d. Then  c c G(a, c) = a − , , 2 2  d d G(b, d) = b − , . 2 2 Therefore, the parallelogram P Q R S will have coordinates   c c d c a , c  = a − , , b , c  = b − , 2 2 2 2

2.4 Some Ways for Altering Experts’ Estimations

21

  d d c d , a , d  = a − , . b , d  = b − , 2 2 2 2 Case 2: a ≥ c, b ≤ d. Then  c c G(a, c) = a − , , 2 2  b b G(b, d) = , d − . 2 2 Therefore, the parallelogram P Q R S will have coordinates   c c b c a , c  = a − , , b , c  = , 2 2 2 2



 b b b c , a , d  = a − , d − . b , d  = , d − 2 2 2 2





Case 3: a ≤ c, b ≥ d. Then a

a , 2 2  d d G(b, d) = b − , . 2 2 G(a, c) =

,c −

Therefore, the parallelogram P Q R S will have coordinates  a a d , b , c  = b − , c − a , c  = , c − 2 2 2 2



a

  d d a d , a , d  = , . b , d  = b − , 2 2 2 2 Case 4: a ≤ c, b ≤ d. Then a

a , 2 2  b b G(b, d) = , d − . 2 2 G(a, c) =

,c −

22

2 On Interval Valued Intuitionistic Fuzzy Sets

Therefore, the parallelogram P Q R S will have coordinates  a a b , b , c  = , c − a , c  = , c − 2 2 2 2



a

 b b b a , a , d  = , d − . b , d  = , d − 2 2 2 2





In all cases, we transform the parallelogram P Q RS to a new one lying the interpretation triangle, i.e., the estimation now is correct.

2.5 Norms and Metrics on IVIFS Already, there are a lot of norms and metrics defined on IVIFSs, but here we include only a part of them. For an IVIFS A and for an element x ∈ E, following [3] we define: σ A,inf (x) σ A,sup (x) σ A (x) δ A,inf (x) δ A,sup (x)

= inf M A (x) + inf N A (x), = sup M A (x) + sup N A (x), = sup M A (x) − inf M A (x) + sup N A (x) − inf N A (x), = inf M A (x)2 + inf N A (x)2 , = sup M A (x)2 + sup N A (x)2 .

The functions below are analogous to the IFS’ π-function: π A,inf (x) = 1 − sup M A (x) − sup N A (x), π A,sup (x) = 1 − inf M A (x) − inf N A (x). In the IVIFS-case, the Hamming metrics have the forms: h A,inf (x, y) = 21 (| inf M A (x) − inf M A (y) | + | inf N A (x) − inf N A (y) |) h A,sup (x, y) = 21 (| sup M A (x) − sup M A (y) | + | sup N A (x) − sup N A (y) |) h A (x, y)

= h A,inf (X, Y ) + h A,sup (X, Y )

and the Euclidean metrics are:  e A,inf (x, y) = 21 ((inf M A (x) − inf M A (y))2 + (inf N A (x) − inf N A (y))2 )  e A,sup (x, y) = 21 ((sup M A (x) − sup M A (y))2 + (sup N A (x) − sup N A (y))2 )  e A (x, y) = 21 (e A,inf (x, y)2 + e A,sup (x, y)2 )

2.5 Norms and Metrics on IVIFS

23

There exist different versions of the Hamming’s distances: 1 (| inf M A (x) − inf M B (x) | + | inf N A (x) − inf N B (x) |), 2 x∈E

Hinf (A, B) =

Hsup (A, B) =

1 (| sup M A (x) − sup M B (x) | + | sup N A (x) − sup N B (x) |), 2 x∈E H (A, B) = Hinf (A, B) + H sup(A, B),

H (A, B) =

1 (| (sup M A (x) − inf M A (x)) − (sup M B (x) − inf M B (x)) | 2 x∈E

+ | (sup N A (x) − inf N A (x)) − (sup N B (x) − inf N B (x)) |) and of the Euclidean distances:  1 (inf M A (x) − inf M A (y))2 + (inf N A (x) − inf N A (y))2 , E inf (A, B) = 2 x∈E  E sup (A, B) =

1 (supM A (x)−supM A (y))2 + (supN A (x)−supN A (y))2 , 2 x∈E E(A, B) =



E inf (A, B)2 + E sup (A, B)2 .

In [14], Eulalia Szmidt and Janusz Kacprzyk introduced extension of the norms and metrics, described in [3], that in [5] were named after them. Now, we introduce their analogues for the IVIFS-case under name Szmidt-Kacprzyk norms and metrics, as follows: 1 h ∗A,inf (x, y) = (| inf M A (x) − inf M A (y) | + | inf N A (x) − inf N A (y) |) 2 + | π A,inf (x) − π A,inf (y) |), 1 h ∗A,sup (x, y) = (| sup M A (x) − sup M A (y) | + | sup N A (x) − sup N A (y) |) 2 + | π A,sup (x) − π A,sup (y) |), h ∗A (x, y) = h ∗A,inf (X, Y ) + h ∗A,sup (X, Y ), 1 e∗A,inf (x, y) = ( ((inf M A (x) − inf M A (y))2 + (inf N A (x) − inf N A (y))2 ) 2 1 + (π A,inf (x) − π A,inf (y))2 ) 2 ,

24

2 On Interval Valued Intuitionistic Fuzzy Sets

1 e∗A,sup (x, y) = ( ((sup M A (x) − sup M A (y))2 + (sup N A (x) − sup N A (y))2 ) 2 1 + (π A,sup (x) − π A,sup (y))2 ) 2 ,  1 e∗A (x, y) = (e A,inf (x, y)2 + e A,sup (x, y)2 ), 2 1 ∗ (A, B) = (| inf M A (x) − inf M B (x) | + | inf N A (x) − inf N B (x) | Hinf 2 x∈E + |π A,inf (x) − π B,inf (x)|), 1 ∗ Hsup (A, B) = (| sup M A (x)−sup M B (x) | + | sup N A (x)−sup N B (x) | 2 x∈E + |π A,sup (x) − π B,sup (x)|), H (A, B) = Hinf (A, B) + Hsup (A, B),  2 1 E inf (A, B) = (inf M A (x) − inf M A (y) + (inf N A (x) − inf N A (y))2 2 x∈E 1

+ (π A,inf (x) − π B,inf (x))2 ) 2 , E sup (A, B) =

1 2



2 (sup M A (x) − sup M A (y)

x∈E

1

+(sup N A (x) − sup N A (y))2 + (π A,sup (x) − π B,sup (x))2 ) 2 , E(A, B) =



E inf (A, B)2 + E sup (A, B)2 .

References 1. Atanassov, K.: Review and new results on intuitionistic fuzzy sets. Preprint IM-MFAIS-1-88, Sofia (1988) 2. Atanassov, K.: Operators over interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 64(2), 159–174 (1994) 3. Atanassov, K.: Intuitionistic Fuzzy Sets. Springer, Heidelberg (1999) 4. Atanassov, K.: Remark on a property of the intuitionistic fuzzy interpretation triangle. Notes Intuitionistic Fuzzy Sets 8(1), 34–36. http://ifigenia.org/wiki/issue:nifs/8/1/34-36 (2002) 5. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012) 6. Atanassov, K.: Intuitionistic fuzzy sets and interval valued intuitionistic fuzzy sets. Adv. Stud. Contemp. Math. 28(2), 167–176 (2018) 7. Atanassov, K., Gargov, G.: Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 343–349 (1989) 8. Atanassova, V., Angelova, N.: Representation of Interval-Valued Intuitionistic Fuzzy Data by Radar Charts. In: Atanassov, K.T., Kacprzyk, J., at al. (eds.) Uncertainty and Imprecision in Decision Making and Decision Support: Cross fertilization, New Models and Applications, Springer, Cham (2019) (in press)

References

25

9. Dworniczak, P.: Further remarks about the unconscientious experts’ evaluations in the intuitionistic fuzzy environment. Notes Intuitionistic Fuzzy Sets 19(1), 27–31 (2013) 10. Gargov, G.: Knowledge, uncertainty and ignorance in logic: bilattices and beyond. J. Appl. Non-Classical Logics 9(2–3), 195–283 (1999) 11. Georgiev, P., Atanassov, K.: Geometrical interpretations of the interval valued intuitionistic fuzzy sets. Notes Intuitionistic Fuzzy Sets 2(2), 1–10 (1996) 12. Gorzalczany, M.: Interval-valued fuzzy fuzzy inference method - some basic properties. Fuzzy Sets Syst. 31(2), 243–251 (1989) 13. Kaufmann, A.: Introduction a la Theorie des Sour-Ensembles Flous. Masson, Paris (1977) 14. Szmidt, E., Kacprzyk, J.: Distances between intuitionistic fuzzy sets. Fuzzy Sets and Syst. 114(3), 505–518 (2000) 15. Turksen, I.: Interval valued fuzzy sets based normal forms. Fuzzy Sets and Syst. 20(2), 191–210 (1986) 16. Vassilev, P.: On reassessment of expert evaluations in the case of intuitionistic fuzzines. Adv. Stud. Contemp. Math. 20(4), 569–574 (2010)

Chapter 3

Relations and Operations over IVIFSs

In this chapter, the basic definitions and properties of the relations and operations over IVIFSs will be discussed. We omit the majority of the proofs below, which are, in general, analogous to the proofs from [4, 5] for IFSs.

3.1 Relations over IVIFSs For every two IVIFSs A and B the following relations hold: A ⊂ ,inf B A ⊂ ,sup B A ⊂♦,inf B A ⊂♦,sup B A⊂ B A ⊂♦ B A⊂B A=B

iff (∀x ∈ E)(inf M A (x) ≤ inf M B (x)), iff (∀x ∈ E)(sup M A (x) ≤ sup M B (x)), iff (∀x ∈ E)(inf N A (x) ≥ inf N B (x)), iff (∀x ∈ E)(sup N A (x) ≥ sup N B (x)), iff A ⊂ ,inf B & A ⊂ ,sup B, iff A ⊂♦,inf B & A ⊂♦,sup B, iff A ⊂ B& A ⊂♦ B, iff A ⊂ B&B ⊂ A.

We can see directly that A ⊂ B iff (∀x ∈ E)(inf M A (x) ≤ inf M B (x) & sup M A (x) ≤ sup M B (x)), A ⊂♦ B iff (∀x ∈ E)(inf N A (x) ≥ inf N B (x) & sup N A (x) ≥ sup N B (x)), A ⊂ B iff (∀x ∈ E)(inf M A (x) ≤ inf M B (x) & sup M A (x) ≤ sup M B (x)), & inf N A (x) ≥ inf N B (x) & sup N A (x) ≥ sup N B (x)), A = B iff (∀x ∈ E)(inf M A (x) = inf M B (x) & sup M A (x) = sup M B (x), & inf N A (x) = inf N B (x) & sup N A (x) = sup N B (x)).

© Springer Nature Switzerland AG 2020 K. T. Atanassov, Interval-Valued Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing 388, https://doi.org/10.1007/978-3-030-32090-4_3

27

28

3 Relations and Operations over IVIFSs

3.2 Operations over IVIFSs For any two IVIFSs A and B the following operations are defined: ¬A = {x, N A (x), M A (x) | x ∈ E}, A ∩ B = {x, [min(inf M A (x), inf M B (x)), min(sup M A (x), sup M B (x))], [max(inf N A (x), inf N B (x)), max(sup N A (x), sup N B (x))] | x ∈ E}, A ∪ B = {x, [max(inf M A (x), inf M B (x)), max(sup M A (x) sup M B (x))], [min(inf N A (x), inf N B (x)), min(sup N A (x), sup N B (x))] | x ∈ E}, A + B = {x, [inf M A (x) + inf M B (x) − inf M A (x). inf M B (x), sup M A (x) + sup M B (x) − sup M A (x). sup M B (x)], [inf N A (x). inf N B (x), sup N A (x). sup N B (x)] | x ∈ E}, A.B = {x, [inf M A (x). inf M B (x), sup M A (x). sup M B (x)], [inf N A (x) + inf N B (x) − inf N A (x). inf N B (x), sup N A (x) + sup N B (x) − sup N A (x). sup N B (x)] | x ∈ E}, A@B = {x, [(inf M A (x) + inf M B (x))/2, (sup M A (x) + sup M B (x))/2], [(inf N A (x) + inf N B (x))/2, | x ∈ E}, (sup N A (x) + sup N B (x))/2]  √ (x), sup M A (x). sup M B (x)], A$B = {x, [ inf M A (x). inf M B √ [ infN A (x). inf N B (x), sup N A (x). sup N B (x)] | x ∈ E},  2. inf M A (x). inf M B (x) 2. sup M A (x). sup M B (x) , A# B = x, (inf M A (x) + inf M B (x)) (sup M A (x) + sup M B (x))   2. inf N A (x). inf N B (x) 2. sup N A (x). sup N B (x) , |x∈E , (inf N A (x) + inf N B (x)) (sup N A (x) + sup N B (x)) inf M A (x) + inf M B (x) , A ∗ B = x, [ 2.(inf M A (x). inf M B (x) +  1) sup M A (x) + sup M B (x) , 2.(sup M A (x). sup M B (x) + 1)  inf N A (x) + inf N B (x) , 2.(inf N A (x). inf N B (x) + 1)   sup N A (x) + sup N B (x) ] |x∈E . 2.(sup N A (x). sup N B (x) + 1) The correctness of the above definitions is checked in the same way as in [4] for IFSs. All assertions from IFSs still hold; the only necessary change is in the kind of sets involved. Here we will prove only the most important assertions, as well as these whose proofs differ substantially from the other. For example, the following equalities and inequalities are valid for every three IVIFSs A, B and C:

3.2 Operations over IVIFSs

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

29

A ∩ B = B ∩ A; A ∪ B = B ∪ A; A + B = B + A; A.B = B.A; A@B = B@A; A$B = B$A; A  B = B  A; A ∗ B = B ∗ A; (A ∩ B) ∩ C = A ∩ (B ∩ C); (A ∪ B) ∪ C = A ∪ (B ∪ C); (A + B) + C = A + (B + C); (A.B).C = A.(B.C); (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C); (A ∩ B) + C = (A + C) ∩ (B + C); (A ∩ B).C = (A.C) ∩ (B.C); (A ∩ B)@C = (A@C) ∩ (B@C); (A ∩ B)  C = (A  C) ∩ (B  C); (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C); (A ∪ B) + C = (A + C) ∪ (B + C); (A ∪ B).C = (A.C) ∪ (B.C); (A ∪ B)@C = (A@C) ∪ (B@C); (A ∪ B)  C = (A  C) ∪ (B  C); (A + B).C ⊂ (A.C) + (B.C); (A + B)@C ⊂ (A@C) + (B@C); (A.B) + C ⊃ (A + C).(B + C); (A.B)@C ⊃ (A@C).(B@C); (A@B) + C = (A + C)@(B + C); (A@B).C = (A.C)@(B.C); A ∩ A = A; A ∪ A = A; A@A = A; A$A = A; A  A = A; ¬(¬A ∩ ¬B) = A ∪ B; ¬(¬A ∪ ¬B) = A ∩ B; ¬(¬A + ¬B) = A.B; ¬(¬A . ¬B) = A + B; ¬(¬A@¬B) = A@B; ¬(¬A  ¬B) = A  B; ¬(¬A ∗ ¬B) = A ∗ B.

By the moment, only the first six operations have been practically used in real applications.

30

3 Relations and Operations over IVIFSs

It is suitable to define the three special IVIFSs: O ∗ = {x, [0, 0], [1, 1] | x ∈ E}, U ∗ = {x, [0, 0], [0, 0] | x ∈ E}, E ∗ = {x, [1, 1], [0, 0] | x ∈ E}. Obviously,

O ∗ ⊂ U ∗ ⊂ E ∗,

and for each IVIFS A:

A ∩ O ∗ = A.O ∗ = O ∗ , A ∪ O ∗ = A + O ∗ = A.

As we mentioned, the above operations have analogues in the IFS theory. Now we define one more operation, which is a combination of the ∪ and ∩ operations defined over IVIFS that does not have an IFS-analogue: A ◦ B = {x, [min(inf M A (x), inf M B (x)), min(max(sup M A (x), sup M B (x)), 1 − max(sup N A (x), sup N B (x)))], [min(inf N A (x), inf N B (x)), min(max(sup N A (x), sup N B (x)), 1 − max(sup M A (x), sup M B (x)))] | x ∈ E} Theorem 3.2.1 For evert two IVIFSs A and B, A ◦ B is an IVIFS. Proof From the inequalities max(sup M A (x), sup M B (x)) ≥ min(inf M A (x), inf M B (x)), 1 − max(sup M A (x), sup M B (x)) = min(1 − sup M A (x), 1 − sup M B (x)) ≥ min(sup N A (x), sup N B (x)) it follows that 0 ≤ min(inf M A (x), inf M B (x)) ≤ min(max(sup M A (x), sup M B (x)), 1 − max(sup N A (x), sup N B (x))) ≤ 1.

3.2 Operations over IVIFSs

31

By analogy, we check that 0 ≤ min(inf N A (x), inf N B (x)) ≤ min(max(sup N A (x), sup N B (x)), 1 − max(sup M A (x), sup M B (x))) ≤ 1. Finally, from the definition of operation ◦ we see that 0 ≤ min(max(sup M A (x), sup M B (x)), 1 − max(sup N A (x), sup N B (x))) + min(max(sup N A (x), sup N B (x)), 1 − max(sup M A (x), sup M B (x))) ≤ max(sup M A (x), sup M B (x)) + 1 − max(sup M A (x), sup M B (x)) = 1, i.e., the definition of operation ◦ is correct. By similar way, we can check the correctness of the definitions of all operations, given above and can prove the following theorem. Theorem 3.2.2 For every two IVIFSs A and B: (a) A ◦ B = B ◦ A, (b) ¬(¬A ◦ ¬B) = A ◦ B, (c) A ∩ B ⊂ A ◦ B ⊂ A ∪ B. Theorem 3.2.3 For every IVIFS A: (a) A ◦ O ∗ = {x, [0, 0], [inf N A (x), 1 − sup M A (x)] | x ∈ E}, (b) A ◦ U ∗ = {x, [inf M A (x), 1 − sup N A (x)], [0, 0] | x ∈ E}, (c) A ◦ E ∗ = {x, [0, 0], [inf N A (x), 1 − sup M A (x)] | x ∈ E}. Below, we will present the geometrical interpretations of the operations ∪, ∩, +, ., @ and ◦. They are shown in Figs. 3.1a–g, 3.2a–g, 3.3a–g, 3.4a–g and 3.5a–g. If we denote i = 1, 2, . . . , 5, then Figs. 3.i (a) shows the geometrical interpretation of the truth-values of the element x ∈ E with respect to the IVIFSs A and B; Figs. 3.i (b)–3.i (g) demonstrate the respective geometrical interpretations of this element for the IVIFSs A ∪ B, A ∩ B, A + B, A.B, @ and A ◦ B for the corresponding configuration. In [5], 138 different implications over IFSs are defined. In [6], 185 implications over IFL are defined, so that, the first 138 implications are analogous of these from [5]. New five implications are introduced in the last two years. All these operations can be transformed to operations over IVIFSs. For example, the first implication from [5], called first Zadeh’s intuitionistic fuzzy implication: A →1 B = {x, max(ν A (x), min(μ A (x), μ B (x))), min(μ A (x), ν B (x)) |x ∈ E}, for two IFSs A and B, have the following IVIFS-form for two IVIFSs A and B:

32

3 Relations and Operations over IVIFSs

(a) 0, 1

N2,A N2,B N1,A N1,B 0, 0

(c) 0, 1

M1,A M2,A M1,B M2,B

(b) 0, 1

1, 0

N1,A∩B

@ @

M1,A∩B M2,A∩B

0, 0

(e) 0, 1

N1,A.B

M1,A.B M2,A.B

(f)

0, 1

N2,A@B @ @ @

N1,A@B 1, 0

(g) 0, 1

N2,A◦B N1,A◦B

@ @ @

@ @

@ @

@ @

                           

0, 0

@ @

0, 0 M1,A◦B M2,A◦B

@ @

M2,A∪B

@ @ @



@ 1, 0

@ @ @

M1,A+B M2,A+B

0, 0

@ @ @

N2,A.B

@

N2,A+B N1,A+B 1, 0

@ @ @  @       @ @

M1,A∪B

0, 0

@

@

@ @ @ @ EE E E E E E E E @ EEEEEEEEE @

N1,A∪B

@ @

(d) 0, 1

@ @          @ 

@ @ @ @

N2,A∪B

@ @ @ @

N2,A∩B

0, 0

@ @ @ @ @ @ @ @ A @ B @

1, 0

@ @ @

@ @ @ @ @   @ @ @ M1,A@B M2,A@B 1, 0

@ 1, 0

Fig. 3.1 a The element x ∈ E with respect to the IVIFSs A and B. b The element x ∈ E to the IVIFS A ∪ B = B. c The element x ∈ E to the IVIFS A ∩ B = A. d The element x ∈ E to the IVIFS A + B. e The element x ∈ E to the IVIFS A.B. f The element x ∈ E to the IVIFS A@B. g The element x ∈ E to the IVIFS A ◦ B

3.2 Operations over IVIFSs

(a) 0, 1

@ @

N2,A N1,A N2,B

33

(b) 0, 1 @ @ @ A

N1,B

@ @

@ @ B @

M1,A M2,A M1,B M2,B

0, 0

(c) 0, 1

N1,A∩B

M 0, 0 1,A∩BM2,A∩B

(e) 0, 1 N2,A.B N1,A.B

0, 0

M1,A.B M2,A.B

N2,A@B N1,A@B

@ @ 1, 0

(g) 0, 1

@ @

0, 0

(f) 0, 1

@ @ @

@      @  A @ @ @ @ B

@ @

@

N2,A+B N1,A+B 1, 0

@ @

@

1, 0

M1,A∪B

A @ @

@

M2,A∪B

0, 0

(d) 0, 1

@

@ @ EE E E E E E E E @ EEEEEEEEE @

N1,A∪B

@

@ @ @

N2,A∩B

@ @

N2,A∪B

1, 0

@ @          @ @  @ @

@ @ @

@ @ @ @ B

@

@

M1,A+BM2,A+B @ @

@ A

@

@

1, 0

@

 

@ @ @ B @

0, 0 M1,A@B M2,A@B

@ 1, 0

@ @

@ A @  @                  B @ @ N1,A◦B                 @ M2,A◦B M 0, 0 1,A◦B 1, 0 N2,A◦B

Fig. 3.2 a The element x ∈ E with respect to the IVIFSs A and B. b The element x ∈ E to the IVIFS A ∪ B ≡ B. c The element x ∈ E to the IVIFS A ∩ B ≡ A. d The element x ∈ E to the IVIFS A + B. e The element x ∈ E to the IVIFS A.B. f The element x ∈ E to the IVIFS A@B. g The element x ∈ E to the IVIFS A ◦ B

34

3 Relations and Operations over IVIFSs

(a) 0, 1

N2,A N N1,A2,B N1,B 0, 0

(c) 0, 1

N2,A∩B N1,A∩B

(b) 0, 1

@ @ @ @ @ @ A

@ @ B @

M1,A M2,A M1,B M2,B

M 0, 0M1,A∩B 2,A∩B

@ @

M1,A.B M2,A.B

0, 0

(f) 0, 1

@ @ @

N2,A◦B N1,A◦B 0, 0

@ @ @

N1,A@B

@ @

@ @

@ @ @

@

B

@

@

@ @

@

1, 0

@ @

@ @

@ @ A  @  B @

0, 0 M 1,A@B M2,A@B

@

@ 1, 0

@

@  @  @                    

M1,A◦B

@

M1,A+B M2,A+B 1, 0

N2,A@B

1, 0

(g) 0, 1

@

N2,A+B N1,A+B

(e) 0, 1

0, 0

@ @

A @ @

@

M M1,A∪B 2,A∪B

0, 0

(d) 0, 1

@

EE E E E E E E E EEEEEEEEE

N1,A∪B

1, 0

@ @ @ @ N2,A.B @  @  @ N1,A.B           @ A B @

@ @ @

N2,A∪B

1, 0

@ @ @ @ @ @ @          @  @

@ @

M2,A◦B

@ @ 1, 0

Fig. 3.3 a The element x ∈ E with respect to the IVIFSs A and B. b The element x ∈ E to the IVIFS A ∪ B ≡ B. c The element x ∈ E to the IVIFS A ∩ B ≡ A. d The element x ∈ E to the IVIFS A + B. e The element x ∈ E to the IVIFS A.B. f The element x ∈ E to the IVIFS A@B. g The element x ∈ E to the IVIFS A ◦ B

3.2 Operations over IVIFSs

(a) 0, 1

35

(b) 0, 1

@ @ @

@ @ @ A @ @ @ B

N2,A N2,B N1,B N1,A

M M 0, 0 M 1,A 2,AM 1,B 2,B

(c) 0, 1

@ @ 1, 0

N2,A.B

N1,A.B

@ @                  @                 @  N1,A∪B @ M1,A∪B M2,A∪B 0, 0 1, 0

(d) 0, 1

@ @ @

@ @

N2,A+B N1,A+B 0, 0

(f) 0, 1 @

@  @     @      A @   @ B      @

M 0, 0 1,A.B M

@ @ 1, 0

2,A.B

(g) 0, 1

@ @ @

N2,A∪B

@ @ @ A @ N2,A∩B @          @ B         @ N1,A∩B          @ M 0, 0 1,A∩BM2,A∩B 1, 0

(e) 0, 1

@ @ @

@ @ @

@ @ @ A @ @ B @                     @ @ M1,A+B M2,A+B

1, 0

@ @ @

@ @ A @ N2,A@B @  @             B @   @ N1,A@B @ 0, 0 M1,A@B M2,A@B 1, 0

@ @ @

@ @ A @ N2,A◦B @  @              B  @   @ N1,A◦B               @ M M 0, 0 1,A◦B 2,A◦B 1, 0

Fig. 3.4 a The element x ∈ E with respect to the IVIFSs A and B. b The element x ∈ E to the IVIFS A ∪ B. c The element x ∈ E to the IVIFS A ∩ B. d The element x ∈ E to the IVIFS A + B. e The element x ∈ E to the IVIFS A.B. f The element x ∈ E to the IVIFS A@B. g The element x ∈ E to the IVIFS A ◦ B

36

3 Relations and Operations over IVIFSs

(a) 0, 1

N2,A N2,B

@ @ @

(b) 0, 1 @ @

A

@ @ @ @ @ @ M2,A

B N1,B N1,A M1,A 0, 0 M 1,B M2,B

(c) 0, 1

@ @ @

N2,A∩B

1, 0

(e) 0, 1 N2,A.B

N1,A.B

@ @ A N2,A∪B @                 @  @                  N1,A∪B @ M M2,A∪B 0, 0 1,A∪B 1, 0

(d) 0, 1 @ @

@            A @  @            @  @ N1,A∩B @ M1,A∩B M2,A∩B 0, 0 1, 0

@ @ @ @ @

N2,A+B N1,A+B 0, 0

@ @ @ @ @ @ @ A @ B @

@                      @ M1,A+B M2,A+B

1, 0

(f) 0, 1 @ @ @

@     @      @     A @      B @   @

M 0, 0 1,A.B M

@ 1, 0

2,A.B

(g) 0, 1

@ @ @ @ @ @ A @ N2,A@B @                           @  @ N1,A@B               @ 0, 0 M1,A@B M2,A@B 1, 0

@ @ @

@ @ N2,A◦B @  @ @                   @  N1,A◦B                    @ M M2,A◦B 0, 0 1,A◦B 1, 0 Fig. 3.5 a The element x ∈ E with respect to the IVIFSs A and B. b The element x ∈ E to the IVIFS A ∪ B. c The element x ∈ E to the IVIFS A ∩ B. d The element x ∈ E to the IVIFS A + B. e The element x ∈ E to the IVIFS A.B. f The element x ∈ E to the IVIFS A@B. g The element x ∈ E to the IVIFS A ◦ B ≡ A

3.2 Operations over IVIFSs

37

A →1 B = {x, [max(inf N A (x), min(inf M A (x), inf M B (x))), max(sup N A (x), min(sup M A (x), sup M B (x)))], [min(inf M A (x), inf N B (x)), min(sup M A (x), sup N B (x))] |x ∈ E} We can see directly, that 0 ≤ max(inf N A (x), min(inf M A (x), inf M B (x))) ≤ max(sup N A (x), min(sup M A (x), sup M B (x))) ≤ 1, 0 ≤ min(inf M A (x), inf N B (x)) ≤ min(sup M A (x), sup N B (x)) ≤ 1, and max(sup N A (x), min(sup M A (x), sup M B (x))) + min(sup M A (x), sup N B (x)) ≤ max(sup N A (x), sup M A (x)) + min(sup M A (x), sup N B (x)) = sup M A (x), sup N A (x) ≤ 1. Therefore, the definition of the IVIFS implication →1 is correct and we will keep its name, but for the IVIFS-case. At the moment, an Open Problem is to define the IVIFS implications that are analogues of the existing (by the moment) 188 other IFS-implications and to study their basic properties. In some definitions below, we use functions sg and sg defined by, sg(x) =

⎧ ⎨ 1 if x > 0 ⎩

sg(x) =

0 if x ≤ 0

,

⎧ ⎨ 0 if x > 0 ⎩

1 if x ≤ 0

In [5], 34 different negations over IFSs are defined. In [6], 53 implications over IFL are defined, so that, the first 34 negations are analogous of these from [5]. A new negation is introduced in [7]. An Open Problem is all these negations to be transformed to operations over IVIFSs. For example, the first IVIFS implication →1 generates the first IVIFS negation, using the standard logical formula ¬ϕ = ϕ → f alse.

(3.2.1)

38

3 Relations and Operations over IVIFSs

Its IVIFS-form for implication →1 is the following ¬1 A = A →1 O ∗ = {x, M A (x), N A (x) | x ∈ E} →1 {x, [0, 0], [1, 1] | x ∈ E} = {x, [max(inf N A (x), min(inf M A (x), 0)), max(sup N A (x), min(sup M A (x), 0))], [min(inf M A (x), 1), min(sup M A (x), 1)] |x ∈ E} = {x, [max(inf N A (x), 0), max(inf N A (x), 0)], [min(inf M A (x), 1), min(sup M A (x), 1)] |x ∈ E} = {x, [inf N A (x), sup N A (x)], [inf M A (x), sup M A (x)] |x ∈ E} = {x, N A (x), M A (x) | x ∈ E}. Therefore, ¬1 coincides with the first negation, introduced for IVIFSs. The mostly used IFS implication has the following IVIFS-analogue A →4 B = {x, [max(inf N A (x), inf M B (x)), max(sup N A (x), sup M B (x))], [min(inf M A (x), inf N B (x)), min(sup M A (x), sup N B (x))] |x ∈ E}. It will be called Kleene-Dienes IVIFS-implication. Its generates also the negation ¬1 . The IVIFS-analogue of the second IFS-implication, called Gaines-Rescher intuitionistic fuzzy implication is A →2 B = {x, [sg(sup M A (x) − sup M B (x)), sg(inf M A (x) − inf M B (x))], [inf N B (x).sg(inf M A (x) − inf M B (x)), sup N B (x).sg(sup M A (x) − sup M B (x))] |x ∈ E}. Using formula (3.2.1), we obtain ¬2 A = A →2 O ∗ = {x, [sg(sup M A (x)), sg(inf M A (x))], [sg(inf M A (x)), sg(sup M A (x))] |x ∈ E}.

3.2 Operations over IVIFSs

39

Theorem 3.2.4 For each IVIFS A (a) (b) (c) (d) (e) (f) (g) (h)

A →1 ¬1 ¬1 A is a weak tautological IVIFS, A →2 ¬2 ¬2 A is a tautological set, A →4 ¬1 ¬1 A is a weak tautological IVIFS, ¬1 ¬1 A →1 A is a weak tautological IVIFS, ¬2 ¬2 A →2 A is not any form of a tautological IVIFS, ¬1 ¬1 A →4 A is a weak tautological IVIFS, ¬1 ¬1 ¬1 A = ¬1 A, ¬2 ¬2 ¬2 A = ¬2 A.

Proof Let the IVIFS A be given. First, we see that ¬1 ¬1 A = ¬1 {x, N A (x), M A (x) | x ∈ E} = {x, M A (x), N A (x) | x ∈ E} = A. From here, the validity of (a), (c), (d), (f) and (g) follows directly. For (b) we obtain A →2 ¬2 ¬2 A = A →2 ¬2 {x, [sg(sup M A (x)), sg(inf M A (x))], [sg(inf M A (x)), sg(sup M A (x))] |x ∈ E} = A →2 {x, [sg(sg(inf M A (x))), sg(sg(sup M A (x)))], [sg(sg(sup M A (x))), sg(sg(inf M A (x)))] |x ∈ E} (from sg(sg(r )) = sg(r ) and sg(sg(r )) = sg(r ) for every real number r ) = A →2 {x, [sg(inf M A (x)), sg(sup M A (x))], [sg(sup M A (x)), sg(inf M A (x))] |x ∈ E} = {x, [sg(sup M A (x) − sg(sup M A (x))), sg(inf M A (x) − sg(inf M A (x)))], [sg(sup M A (x)).sg(inf M A (x) − sg(inf M A (x))), sg(inf M A (x)).sg(sup M A (x) − sg(sup M A (x)))] |x ∈ E}. (from sg(r − sg(r )) = 1 and sg(r − sg(r )) = 0 for every real number r ) = {x, [1, 1], [0, 0] |x ∈ E} = E ∗ , i.e., A →2 ¬2 ¬2 A is a tautological set. For (e) we obtain

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3 Relations and Operations over IVIFSs

¬2 ¬2 A →2 A = {x, [sg(inf M A (x)), sg(sup M A (x))], [sg(sup M A (x)), sg(inf M A (x))] |x ∈ E} →2 A = {x, [sg(sg(sup M A (x)) − sup M A (x)), sg(sg(inf M A (x)) − inf M A (x))], [inf N A (x).sg(sg(inf M A (x)) − inf M A (x)), sup N A (x).sg(sg(sup M A (x)) − sup M A (x))] |x ∈ E}. This set is not any form of a tautological IVIFS. Really, for example, if A = {x, [0.5, 0.5], [0.5, 0.5] | x ∈ E}, then ¬2 ¬2 A →2 A = {x, [sg(sg(0.5) − 0.5), sg(sg(0.5) − 0.5)], [0.5sg(sg(0.5) − 0.5), 0.5sg(sg(0.5) − 0.5)] |x ∈ E} = {x, [sg(0.5), sg(0.5)], [0.5sg(0.5), 0.5sg(0.5)] |x ∈ E} = {x, [0, 0], [0.5, 0.5)] |x ∈ E}. Finally, for (h) we obtain ¬2 ¬2 ¬2 A = ¬2 {x, [sg(sg(inf M A (x))), sg(sg(sup M A (x)))], [sg(sg(sup M A (x))), sg(sg(inf M A (x)))] |x ∈ E} = {x, [sg(sg(sg(sup M A (x)))), sg(sg(sg(inf M A (x))))], [sg(sg(sg(inf M A (x)))), sg(sg(sg(sup M A (x))))] |x ∈ E} (from sg(sg(r )) = sg(r ) and sg(sg(r )) = sg(r ) for every real number r ) = {x, [sg(sup M A (x)), sg(inf M A (x))))], [sg(inf M A (x)), sg(sup M A (x))] |x ∈ E} = ¬2 A.

3.2 Operations over IVIFSs

41

In [1–3], to each one of the 189 intuitionistic fuzzy implications introduced in [5], 3 different intuitionistic fuzzy conjunctions and disjunctions are juxtaposed. In the beginning of the section, the standard interval-valued intuitionistic fuzzy conjunction and disjunction are given, and the Open Problem is to construct all other conjunctions and disjunctions. Finally, we introduce five different Cartesian products over two IVIFSs. Let A be an IVIFS over E 1 and B – over E 2 . We define: A ×1 B = {x, y , [inf M A (x). inf M B (y), sup M A (x). sup M B (y)], [inf N A (x). inf N B (y), sup N A (x). sup N B (y)] | x ∈ E 1 , y ∈ E 2 } A ×2 B = {x, y , [inf M A (x) + inf M B (y) − inf M A (x). inf M B (y), sup M A (x) + sup M B (y) − sup M A (x). sup M B (y)], [inf N A (x). inf N B (y), sup N A (x). sup N B (y)] | x ∈ E 1 , y ∈ E 2 } A ×3 B = {x, y , [inf M A (x). inf M B (y), sup M A (x). sup M B (y)], [inf N A (x) + inf N B (y) − inf N A (x). inf N B (y), sup N A (x) + sup N B (y) − sup N A (x). sup N B (y)] | x ∈ E 1 , y ∈ E 2 } A ×4 B = {x, y , [min(inf M A (x), inf M B (y)), min(sup M A (x), sup M B (y))], [max(inf N A (x), inf N B (y)), max(sup N A (x), sup N B (y))] | x ∈ E 1 , y ∈ E 2 } A ×5 B = {x, y , [max(inf M A (x), inf M B (y)), max(sup M A (x), sup M B (y))], [min(inf N A (x), inf N B (y)), min(sup N A (x), sup N B (y))] | x ∈ E 1 , y ∈ E 2 } The following theorems can be proved by analogy with their analogues from [4] for the IFS-case. Theorem 3.2.5 A ×1 B, A ×2 B, A ×3 B, A ×4 B, A ×5 B are IVIFSs. Proof We will prove that for every four real numbers a, b, c, d ∈ [0, 1], such that a ≤ c, b ≤ d, c + d ≤ 1, the inequality c + d − cd − a − b + ab ≥ 0 holds. Let α = c − a, β = d − b and let X ≡ c + d − cd − a − b + ab.

(3.2.2)

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3 Relations and Operations over IVIFSs

Then X = α + β − (a + α)(b + β) − ab = α + β − aβ − bα − αβ = α(1 − b) + β(1 − a) − αβ. If α ≥ β, then

X ≥ aα + β − aβ − αβ ≥ β − αβ ≥ 0.

If α ≤ β, then

X ≥ α − bα + bβ − αβ ≥ α − αβ ≥ 0.

From the validity of (3.2.2) it follows that sup M A (x) + sup M B (y) − sup M A (x) sup M B (y) ≥ inf M A (x) + inf M B (y) − inf M A (x) inf M B (y). On the other hand, for every four real numbers a, b, c, d ∈ [0, 1], such that a + b ≤ 1, c + d ≤ 1, it is valid that a + c − ac + bd ≤ a + c − ac + (1 − a)(1 − c) = 1, i.e., sup M A (x) + sup M B (y) − sup M A (x) sup M B (y) + sup N A (x) sup N B (y) ≤ 1 and hence A ×2 B is an IVIFS. For the four other products the checks are analogous. In [4], this Theorem is not formulated and proved. Theorem 3.2.6 For every three universes E 1 , E 2 and E 3 and four IVIFSs A, B (over E 1 ), C (over E 2 ) and D (over E 3 ): (a) (b) (c) (d) (e)

(A × C) × D = A × (C × D), (A ∪ B) × C = (A × C) ∪ (B × C), (A ∩ B) × C = (A × C) ∩ (B × C), C × (A ∪ B) = (C × A) ∪ (C × B), C × (A ∩ B) = (C × A) ∩ (C × B),

where × ∈ {×1 , ×2 , ×3 , ×4 , ×5 }.

3.2 Operations over IVIFSs

43

Theorem 3.2.7 For every two universes E 1 and E 2 and three IVIFSs A, B (over E 1 ) and C (over E 2 ): (a) (b) (c) (d) (e) (f)

(A + B) × C ⊂ (A × C) + (B × C), (A.B) × C ⊃ (A × C).(B × C), (A@B) × C = (A × C)@(B × C), C × (A + B) ⊂ (C × A) + (C × B), C × (A.B) ⊃ (C × A).(C × B), C × (A@B) = (C × A)@(C × B),

where × ∈ {×1 , ×2 , ×3 }. Theorem 3.2.8 If A is an IVIFS over E 1 and B is an IVIFS over E 2 , then for the standard negation (¬1 or for brevity ¬): (a) (b) (c) (d) (e)

¬(¬A) ×1 ¬B) = ¬(¬A) ×2 ¬B) = ¬(¬A) ×3 ¬B) = ¬(¬A) ×4 ¬B) = ¬(¬A) ×5 ¬B) =

A ×1 A ×3 A ×2 A ×5 A ×4

B, B, B, B, B.

Therefore, operations ×2 and ×3 ; ×4 and ×5 are dual and the ×1 operation is an autodual one.

3.3 An IVIFS Whose Universe is an IFS or an IVIFS with Respect to Another Universe In [4], the idea for an IFS whose universe is an IFS with respect to another universe, is discussed. Here we will consider this idea for the IVIFS-case. Let E be a fixed universe and let A be an IVIFS over E. Let F be another universe and let the set E be an IFS over F having the form: E = {y, μ E (y), ν E (y) |y ∈ F}. Therefore the element x ∈ E has the form: x = y, μ E (y), ν E (y) , i.e., x ∈ F × [0, 1] × [0, 1], A = {y, μ E (y), ν E (y) , M A (y, μ E (y), ν E (y) ), N A (y, μ E (y), ν E (y) ) |y, μ E (y), ν E (y) ∈ E}

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3 Relations and Operations over IVIFSs

and there exists a bijection between the E- and F-elements of x- and y-types, respectively. Thus we can use the symbol “y” for both y- and x-elements. Let A/E stand for “A is an IVIFS over E” and A/(E/F) – “A is an IVIFS over E that is an IFS over F”. If the degrees of membership and non-membership of an element y to a set A in the frames of a universe E are M A (y) and N A (y) and the element y, M A (y), N A (y) has degrees of membership and non-membership to the set E within the universe F are μ E (y) and ν E (y), then we can define six forms of the transformation of the universe of the IVIFS A: A/1 (E/F) = {y, [μ E (y) + inf M A (y) − μ E (y). inf M A (y), μ E (y) + sup M A (y) − μ E (y). sup M A (y)], [ν E (y). inf N A (y), ν E (y). sup N A (y)] |y ∈ F}, A/2 (E/F) = {y, [max(μ E (y), inf M A (y)), max(μ E (y), sup M A (y))], [min(ν E (y), inf N A (y)), min(ν E (y), sup M A (y))] |y ∈ F},  A/3 (E/F) = 

 y,

 μ E (y) + inf M A (y) μ E (y) + sup M A (y) , , 2 2

ν E (y) + inf N A (y) ν E (y) + sup N A (y) , 2 2



 |y ∈ F ,

A/4 (E/F) = {y, [min(μ E (y), inf M A (y)), min(μ E (y), sup M A (y))], [max(ν E (y), inf N A (y)), max(ν E (y), sup N A (y))] |y ∈ F}, A/5 (E/F) = {y, [μ E (y). inf M A (y), μ E (y). sup M A (y)], [ν E (y) + inf N A (y) − ν E (y). inf N A (y), ν E (y) + sup N A (y) − ν E (y). sup N A (y)] |y ∈ F}, A/6 (E/F) = {y, [μ E (y). inf M A (y), μ E (y). sup M A (y)], [ν E (y). inf N A (y), ν E (y). sup N A (y)] |y ∈ F}. We can call all these transformations, respectively, very optimistic, optimistic, average, pessimistic, very pessimistic, standard. The last name is given for the older transformation described for the IFS case in [4].

3.3 An IVIFS Whose Universe is an IFS or an IVIFS with Respect to Another Universe

45

Let F be a universe and let the set E be an IVIFS over F having the form: E = {y, M E (y), N E (y) |y ∈ F}. Therefore the element x ∈ E has the form: x = y, M E (y), N E (y) . Now, we can define six new forms of the transformation of the universe of the IVIFS A: A/1 (E/F) = {y, [inf M E (y) + inf M A (y) − inf M E (y). inf M A (y), sup M E (y) + sup M A (y) − sup M E (y). sup M A (y)], [inf N E (y). inf N A (y), sup N E (y). sup N A (y)] |y ∈ F}, A/2 (E/F) = {y, [max(inf M E (y), inf M A (y)), max(sup M E (y), sup M A (y))], [min(inf N E (y), inf N A (y)), min(sup N E (y), sup N A (y))] |y ∈ F},  A/3 (E/F) = 

 y,

 inf M E (y) + inf M A (y) sup M E (y) + sup M A (y) , , 2 2

inf N E (y) + inf N A (y) sup N E (y) + sup N A (y) , 2 2



 |y ∈ F ,

A/4 (E/F) = {y, [min(inf M E (y), inf M A (y)), min(sup M E (y), sup M A (y))], [max(inf N E (y), inf N A (y)), max(sup N E (y), sup N A (y))] |y ∈ F}, A/5 (E/F) = {y, [inf M E (y). inf M A (y), sup M E (y). sup M A (y)], [inf N E (y) + inf N A (y) − inf N E (y). inf N A (y), sup N E (y) + sup N A (y) − sup N E (y). sup N A (y)] |y ∈ F}, A/6 (E/F) = {y, [inf M E (y). inf M A (y), sup M E (y). sup M A (y)], [inf N E (y). inf N A (y), sup N E (y). sup N A (y)] |y ∈ F}. As above, we can call these transformations, respectively, very optimistic, optimistic, average, pessimistic, very pessimistic, standard.

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3 Relations and Operations over IVIFSs

3.4 IVIFSs over Different Universes Let E and F be two different universes, and let A E and B F be IVIFSs over E and F, respectively, i.e., A E = {x, M A (x), N A (x) |x ∈ E}, B F = {x, M B (x), N B (x) |x ∈ F}. We will call an IVIFS A defined over the universe E “an E-IVIFS”. First, we define operations over the IVIFSs A and B in a standard form as follows, where ∪∗ is the standard set-theoreticsl operation “union”: A E ∩ B F = {x, [min(inf M A (x), inf N B (x)), min(sup M A (x), sup N B (x))], [max(inf N A (x), inf N B (x)), max(sup N A (x), sup N B (x))] |x ∈ E ∪∗ F}, A E ∪ B F = {x, [max(inf M A (x), inf N B (x)), max(sup M A (x), sup N B (x))], [min(inf N A (x), inf N B (x)), min(sup N A (x), sup N B (x))] |x ∈ E ∪∗ F}, A E + B F = {x, [inf M A (x) + inf M B (x) − inf M A (x) inf M B (x), sup M A (x) + sup M B (x) − sup M A (x) sup M B (x)] [inf N A (x) inf N B (x), sup N A (x) sup N B (x)] |x ∈ E ∪ F}, A E .B F = {x, [inf M A (x) inf M B (x), sup M A (x) sup M B (x)] [inf N A (x) + inf N B (x) − inf N A (x) inf N B (x), sup N A (x) + sup N B (x) − sup N A (x) supN B (x)] ||x ∈ E ∪ F}, A E @B F =

where:



inf M A (x)+inf M B (x) sup M A (x)+sup M B (x) , , 2 2  inf N A (x)+inf N B (x) sup N A (x)+sup N B (x) |x ∈ E , 2 2

x,



M A (x), [0,  0], N A (x), N A (x) = [1, 1],

M A (x) =

and



M B (x), [0,  0], N B (x), N B (x) = [1, 1],

M B (x) =

 ∪F ,

if x ∈ E if x ∈ F − E if x ∈ E if x ∈ F − E if x ∈ F if x ∈ E − F if x ∈ F if x ∈ E − F

It can be seen directly that all assertions concerning IVIFSs still hold. Obviously, for every universe E, every IVIFS defined over E is an E-IFS. On the other hand, every E-IVIFS can be interpreted as an ordinary IFS over the universe E.

3.4 IVIFSs over Different Universes

47

Let Ai be an E i -IVIFS (i ∈ I and I is an index set). Then we can construct the universe  Ei , E∗ = i∈I

and the functions of membership and non-membership:  M Ai (x) =  N Ai (x) =

M Ai (x), if x ∈ E i [0, 0], otherwise N Ai (x), if x ∈ E i [1, 1], otherwise

and hence every E i -IVIFS Ai (i ∈ I ) will be an IVIFS over E ∗ . All operations defined above can be transformed over E ∗ . Introducing E-IVIFSs allows us to work with IVIFSs for which at least one of the inequalities sup M A (x) > 0 and inf N A (x) < 1 holds everywhere for a certain A ⊂ E, for every element x of the universe and functions M A and N A . Second, having in mind than in future we can introduce a lot of new operations over IVIFS, we can give another, more general form of the membership and nonmembership functions. Let us, for the IVIFSs A E and B F over E and F, respectively, define operation ∗. Let ϕinf , ϕsup , ψ inf , and ψ sup be four operations related to operation ∗ as follows A E ∗ B F = {x, M A∗B (x), N A∗B (x) |x ∈ E ∪∗ F} = {x, [ϕinf (inf M A (x), sup M A (x), inf N A (x), sup N A (x)), ϕsup (inf M A (x), sup M A (x), inf N A (x), sup N A (x))], [ψ inf (inf M A (x), sup M A (x), inf N A (x), sup N A (x)), ψ sup (inf M A (x), sup M A (x), inf N A (x), sup N A (x))] |x ∈ E ∪∗ F}, and M A∗B , N A∗B are defined over the two universes as discussed above.

3.5 The Extension Principle for the IVIFS-Case A part of L. Atanassova’s results from [8] will be given, because they are not well known, but they are the most general ones compared to similar research in other papers.

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3 Relations and Operations over IVIFSs

Let X, Y and Z be fixed universes and let f : X × Y → Z . Let A and B be IVIFSs over X and Y , respectively. Then we can construct the sets Di = A ×i B, where i = 1, 2, . . . , 5 and can obtain the sets Fi = f (Di ). For the IVIFS-case, the extension principle has the following 15 forms. Optimistic forms of the extension principle are: opt

F1

= {z, [ sup (inf M A (x). inf M B (y)), sup (sup M A (x). sup M B (y))], z= f (x,y)

z= f (x,y)

[ inf (inf N A (x). inf N B (y)), z= f (x,y)

inf (sup N A (x). sup N B (y))]

z= f (x,y)

|x ∈ E 1 &y ∈ E 2 }, opt

F2

= {z, [ sup (inf M A (x) + inf M B (y) − inf M A (x). inf M B (y)), z= f (x,y)

sup (sup M A (x) + sup M B (y) − sup M A (x). sup M B (y))],

z= f (x,y)

[ inf (inf N A (x). inf N B (y)), z= f (x,y)

inf (sup N A (x). sup N B (y))]

z= f (x,y)

| x ∈ E 1 , y ∈ E 2 }, opt

F3

= {z, [ sup (inf M A (x). inf M B (y)), sup (sup M A (x). sup M B (y))], z= f (x,y)

z= f (x,y)

[ inf (inf N A (x) + inf N B (y) − inf N A (x). inf N B (y)), z= f (x,y)

inf (sup N A (x) + sup N B (y) − sup N A (x). sup N B (y))] |x ∈ E 1 , y ∈ E 2 },

z= f (x,y)

opt

F4

= {z, [ sup (min(inf M A (x), inf M B (y))), z= f (x,y)

sup (min(sup M A (x), sup M B (y)))],

z= f (x,y)

[ inf (max(inf N A (x), inf N B (y))), z= f (x,y)

inf (max(sup N A (x), sup N B (y)))] | x ∈ E 1 , y ∈ E 2 },

z= f (x,y)

opt

F5

= {z, [ sup (max(inf M A (x), inf M B (y))), z= f (x,y)

sup (max(sup M A (x), sup M B (y)))],

z= f (x,y)

3.5 The Extension Principle for the IVIFS-Case

49

[ inf (min(inf N A (x), inf N B (y))), z= f (x,y)

inf (min(sup N A (x), sup N B (y)))] |x ∈ E 1 , y ∈ E 2 }.

z= f (x,y)

Pessimistic forms of the extension principle are: pes

F1

= {z, [ inf (inf M A (x). inf M B (y)), z= f (x,y)

inf (sup M A (x). sup M B (y))],

z= f (x,y)

[ sup (inf N A (x). inf N B (y)), sup (sup N A (x). sup N B (y))] z= f (x,y)

z= f (x,y)

|x ∈ E 1 &y ∈ E 2 }, pes

F2

= {z, [ inf (inf M A (x) + inf M B (y) − inf M A (x). inf M B (y)), z= f (x,y)

inf (sup M A (x) + sup M B (y) − sup M A (x). sup M B (y))],

z= f (x,y)

[ sup (inf N A (x). inf N B (y)), sup (sup N A (x). sup N B (y))] z= f (x,y)

z= f (x,y)

| x ∈ E 1 , y ∈ E 2 }, pes

F3

= {z, [ inf (inf M A (x). inf M B (y)), z= f (x,y)

inf (sup M A (x). sup M B (y))],

z= f (x,y)

[ sup (inf N A (x) + inf N B (y) − inf N A (x). inf N B (y)), z= f (x,y)

sup (sup N A (x) + sup N B (y) − sup N A (x). sup N B (y))] |x ∈ E 1 , y ∈ E 2 },

z= f (x,y)

pes

F4

= {z, [ inf (min(inf M A (x), inf M B (y))), z= f (x,y)

inf (min(sup M A (x), sup M B (y)))],

z= f (x,y)

[ sup (max(inf N A (x), inf N B (y))), z= f (x,y)

sup (max(sup N A (x), sup N B (y)))] | x ∈ E 1 , y ∈ E 2 },

z= f (x,y)

pes

F5

= {z, [ inf (max(inf M A (x), inf M B (y))), z= f (x,y)

inf (max(sup M A (x), sup M B (y)))],

z= f (x,y)

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3 Relations and Operations over IVIFSs

[ sup (min(inf N A (x), inf N B (y))), z= f (x,y)

sup (min(sup N A (x), sup N B (y)))] | x ∈ E 1 , y ∈ E 2 }.

z= f (x,y)

Let α=

1 , car d(E 1 × E 2 )

where car d(Z ) is the cardinality of set Z . Average forms of the extension principle are: 

F1ave = {z, [α

z= f (x,y)

[α





inf M A (x). inf M B (y), α

sup M A (x). sup M B (y)],

z= f (x,y)

inf N A (x). inf N B (y), α

z= f (x,y)



sup N A (x). sup N B (y)]

z= f (x,y)

|x ∈ E 1 &y ∈ E 2 }, F2ave = {z, [α



(inf M A (x) + inf M B (y) − inf M A (x). inf M B (y)),

z= f (x,y)

α



(sup M A (x) + sup M B (y) − sup M A (x). sup M B (y))],

z= f (x,y)



[α

(inf N A (x). inf N B (y)), α

z= f (x,y)



(sup N A (x). sup N B (y))]

z= f (x,y)

|x ∈ E 1 y ∈ E 2 }, F3ave = {z, [α



(inf M A (x). inf M B (y)), α

z= f (x,y)

[α





(sup M A (x). sup M B (y))],

z= f (x,y)

(inf N A (x) + inf N B (y) − inf N A (x). inf N B (y)),

z= f (x,y)

α



(sup N A (x) + sup N B (y) − sup N A (x). sup N B (y))]|x ∈ E 1 y ∈ E 2 },

z= f (x,y)

F4ave = {z, [α



(min(inf M A (x), inf M B (y))),

z= f (x,y)

3.5 The Extension Principle for the IVIFS-Case



α

51

(min(sup M A (x), sup M B (y))]),

z= f (x,y)

[α



(max(inf N A (x), inf N B (y))),

z= f (x,y)

α



(max(sup N A (x), sup N B (y)))] |x ∈ E 1 y ∈ E 2 },

z= f (x,y)



F5ave = {z, [α

(max(inf M A (x), inf M B (y))),

z= f (x,y)



α

(max(sup M A (x), sup M B (y)))],

z= f (x,y)

[α



(min(inf N A (x), inf N B (y))),

z= f (x,y)

α



(min(sup N A (x), sup N B (y)))] |x ∈ E 1 y ∈ E 2 }.

z= f (x,y)

References 1. Angelova, N., M. Stoenchev. Intuitionistic fuzzy conjunctions and disjunctions from first type. In: Annual of “Informatics” Section, Union of Scientists in Bulgaria, vol. 8, pp. 1–17 (2015/2016) 2. Angelova, N., Stoenchev, M., Todorov, V.: Intuitionistic fuzzy conjunctions and disjunctions from second type. Issues Intuitionistic Fuzzy Sets Gen. Nets 13, 143–170 (2017) 3. Angelova, N., Stoenchev, M.: Intuitionistic fuzzy conjunctions and disjunctions from third type. Notes Intuitionistic Fuzzy Sets 23(5), 29–41 (2017) 4. Atanassov, K.: Intuitionistic Fuzzy Sets. Springer, Heidelberg (1999) 5. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012) 6. Atanassov, K.: Intuitionistic Fuzzy Logics. Springer, Cham (2017) 7. Atanassov, K., Szmidt, E., Kacprzyk, J.: Properties of the intuitionistic fuzzy implication →188 . Notes Intuitionistic Fuzzy Sets 23(5), 1–6 (2019) 8. Atanassova, L.: On interval-valued intuitionistic fuzzy versions of L. Zadeh’s extension principle. Issues Intuitionistic Fuzzy Sets Gen. Nets 7, 13–19 (2008)

Chapter 4

Operators over IVIFSs

In this chapter, we discuss modal (of two types), level (of two types) and topological operators, defined over IVIFSs and study some of their basic properties. We prove only these assertions that do not have analogues in the IFS-case. The proofs of the remaining assertions are similar to the proofs, given, e.g., in [4, 6].

4.1 Interval Valued Intuitionistic Fuzzy Modal Operators of First Type The Interval Valued Intuitionistic Fuzzy Modal Operators of first type (IVIFMO1) are defined similarly to these, defined over IFSs, but for difference with the later, they have two forms: shorter and extended. Their shorter form is the following for each IVIFS A (Figs. 4.1, 4.2, 4.3, 4.4 and 4.5): A = {x, M A (x), [inf N A (x), 1 − sup M A (x)] | x ∈ E}, ♦A = {x, [inf M A (x), 1 − sup N A (x)], N A (x) | x ∈ E}, Dα (A) = {x, [inf M A (x), sup M A (x) + α.(1 − sup M A (x) − sup N A (x))], [inf N A (x), sup N A (x) + (1 − α).(1 − sup M A (x) − sup N A (x))] | x ∈ E}, Fα,β (A) = {x, [inf M A (x), sup M A (x) + α.(1 − sup M A (x) − sup N A (x))], [inf N A (x), sup N A (x) + β.(1 − sup M A (x) − sup N A (x))] | x ∈ E}, for α + β ≤ 1, G α,β (A) = {x, [α. inf M A (x), α. sup M A (x)], [β. inf N A (x), β. sup N A (x)] | x ∈ E},

© Springer Nature Switzerland AG 2020 K. T. Atanassov, Interval-Valued Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing 388, https://doi.org/10.1007/978-3-030-32090-4_4

53

54

4 Operators over IVIFSs

Fig. 4.1 Geometrical interpretation of operator for element x ∈ E

Fig. 4.2 Geometrical interpretation of operator ♦ for element x ∈ E

Hα,β (A) = {x, [α. inf M(x), α. sup M A (x)], [inf N A (x), sup N A (x) +β.(1 − sup M A (x) − sup N A (x))] | x ∈ E}, ∗ Hα,β (A) = {x, [α. inf M A (x), α. sup M A (x)], [inf N A (x), sup N A (x)

+β.(1 − α. sup M A (x) − sup N A (x))] | x ∈ E}, Jα,β (A) = {x, [inf M A (x), sup M A (x) + α.(1 − sup M A (x) − sup N A (x))], [β. inf N A (x), β. sup N A (x)] | x ∈ E}, ∗ (A) = {x, [inf M A (x), sup M A (x) + α.(1 − sup M A (x) Jα,β

−β. sup N A (x))], [β. inf N A (x), β. sup N A (x)] | x ∈ E}, where α, β ∈ [0, 1].

4.1 Interval Valued Intuitionistic Fuzzy Modal Operators of First Type

55

Fig. 4.3 Geometrical interpretation of operator Dα for element x ∈ E

Fig. 4.4 Geometrical interpretation of operator Fα,β for element x ∈ E

Obviously, as in the case of IFSs, the operator Dα is a particular case of the operator Fα,β . Theorem 4.1.1 For each IVIFS A and for all α, β ∈ [0, 1] : (a) Hα,β (A) = F0,β (A) ∩ G α,1 (A), (b) Jα,β (A) = Fβ,0 (A) ∪ G 1,α (A), ∗ (A) = F0,β (G α,1 (A)), (c) Hα,β

56

4 Operators over IVIFSs

Fig. 4.5 Geometrical interpretation of operator G α,β for element x ∈ E

∗ (d) Jα,β (A) = Fβ,0 (G 1,α (A)).

Now, we can extend these operators to the following (everywhere α, β, γ, δ ∈ [0, 1] such that α ≤ β and γ ≤ δ): F α,β,γ,δ (A) = {x, [inf M A (x) + α.(1 − sup M A (x) − sup N A (x)), sup M A (x) + β.(1 − sup M A (x) − sup N A (x))], [inf N A (x) + γ.(1 − sup M A (x) − sup N A (x)), sup N A (x) + δ.(1 − sup M A (x) − sup N A (x))] | x ∈ E} where β + δ ≤ 1, G α,β,γ,δ (A) = {x, [α. inf M A (x), β. sup M A (x)], [γ. inf N A (x), δ. sup N A (x)] | x ∈ E}, H α,β,γ,δ (A) = {x, [α. inf M A (x), β. sup M A (x)], [inf N A (x) + γ.(1 − sup M A (x) − sup N A (x)), sup N A (x) + δ.(1 − sup M A (x) − sup N A (x))] | x ∈ E} ∗

H α,β,γ,δ (A) = {x, [α. inf M A (x), β. sup M A (x)], [inf N A (x) + γ.(1 − β. sup M A (x) − sup N A (x)), sup N A (x) + δ.(1 − β. sup M A (x) − sup N A (x))] | x ∈ E}

4.1 Interval Valued Intuitionistic Fuzzy Modal Operators of First Type

57

J α,β,γ,δ (A) = {x, [inf M A (x) + α.(1 − sup M A (x) − sup N A (x)), sup M A (x) + β.(1 − sup M A (x) − sup N A (x))], [γ. inf N A (x), δ. sup N A (x)] | x ∈ E} ∗

J α,β,γ,δ (A) = {x, [inf M A (x) + α.(1 − δ. sup M A (x) − sup N A (x)), sup M A (x) + β.(1 − sup M A (x) − δ. sup N A (x))], [γ. inf N A (x), δ. sup N A (x)] | x ∈ E} Theorem 4.1.2 For every two IVIFSs A and B, and for every α, β, γ, δ ∈ [0, 1], such that α ≤ β, γ ≤ δ and β + δ ≤ 1: (a) (b) (c) (d) (e)

F α,β,γ,δ (A ∩ B) ⊂ F α,β,γ,δ (A) ∩ F α,β,γ,δ (B), F α,β,γ,δ (A ∪ B) ⊂ F α,β,γ,δ (A) ∪ F α,β,γ,δ (B), F α,β,γ,δ (A + B) ⊂ F α,β,γ,δ (A) + F α,β,γ,δ (B), F α,β,γ,δ (A.B) ⊃ F α,β,γ,δ (A).F α,β,γ,δ (B), F α,β,γ,δ (A@B) = F α,β,γ,δ (A)@F α,β,γ,δ (B).

Theorem 4.1.3 For each IVIFS A and for all α, β, γ, δ, α , β , γ , δ ∈ [0, 1] such that α ≤ β, γ ≤ δ, α ≤ β , γ ≤ δ , β + δ ≤ 1 and β + δ ≤ 1: F α,β,γ,δ (F α ,β ,γ ,δ )(A) = F α+α −α.β −α.δ ,β+β −β.β −β.δ ,γ+γ −γ.β −γ.δ ,δ+δ −δ.β −δ.δ (A). The IVIFSMO1 over IVIFSs with shorter form have the following representation by the IVIFSMO1 over IVIFSs with extended form for each IVIFS A and for all α, β, γ, δ ∈ [0, 1] such that α ≤ β, γ ≤ δ: (a) Fα,β (A) = F 0,α,0,β (A), for α + β ≤ 1, (b) G α,β (A) = G α,α,β,β (A), (c) Hα,β (A) = H α,α,0,β (A), ∗

∗ (A) = H α,α,0,β (A), (d) Hα,β

(e) Jα,β (A) = J 0,α,β,β (A), ∗ (f) Jα,β (A) = J 0,α,β,β (A). In [10], the following notation for the IVIFSMO1 with extended form is given: ∗ ∗ F  α γ  (A), G  α γ  (A), H  α γ  (A), H  α γ  (A), J  α γ  (A), J  α γ  (A). β δ

β δ

β δ

β δ

There, for the first time, the operator X  a1

β δ

b1 c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2



β δ

is introduced. After pub-

lishing of [10], the author saw that there are a lot of other different forms of this operator and in the cited paper is mentioned only one of them. In the simplest case, there are 4 different forms of this operator, in the more complex form, their number is 16, while in the most complex form this number is 256. In [10], there are some misprints, that here they are corrected.

58

4 Operators over IVIFSs

First, all forms of the X -operator are given and the conditions for the validity of each one of them is discussed and after this, we will reduce the more detailed research to its simplest case Let ext1 , ext2 , ext3 , ext4 , ext5 , ext6 , ext7 , ext8 ∈ {inf, sup}. Let 

ext1 ext2 ext3 ext4 ext5 ext6 ext7 ext8



X  a1

b1 c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

 (A)

≡ {x, [inf M X (x), sup M X (x)], [inf N X (x), sup N X (x)]|x ∈ E}, = {x, [a1 inf M A (x) + b1 (1 − ext1 M A (x) − c1 ext2 N A (x)), a2 sup M A (x) + b2 (1 − ext3 M A (x) − c2 ext4 N A (x))], [d1 inf N A (x) + e1 (1 − f 1 ext5 M A (x) − ext6 N A (x)), d2 sup N A (x) + e2 (1 − f 2 ext7 M A (x) − ext8 N A (x))]|x ∈ E}, where a1 , b1 , c1 , d1 , e1 , f 1 , a2 , b2 , c2 , d2 , e2 , f 2 ∈ [0, 1]. The definition will be correct, if 0 ≤ inf M X (x) ≤ sup M X (x) ≤ 1,

(4.1.1)

0 ≤ inf N X (x) ≤ sup N X (x) ≤ 1,

(4.1.2)

sup M X (x) + sup N X (x) ≤ 1.

(4.1.3)

Not the most complex (while not the simplest either) form of the operator is: 

ext1 ext2 ext3 ext4 ext1 ext2 ext3 ext4

X  a1



b1 c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

 (A)

= {x, [a1 inf M A (x) + b1 (1 − ext1 M A (x) − c1 ext2 N A (x)), a2 sup M A (x) + b2 (1 − ext3 M A (x) − c2 ext4 N A (x))], [d1 inf N A (x) + e1 (1 − f 1 ext1 M A (x) − ext2 N A (x)), d2 sup N A (x) + e2 (1 − f 2 ext3 M A (x) − ext4 N A (x))]|x ∈ E} ≡ X ( a11 b1 2c1 ext

ext

ext3 ext4

)  (A).

d1 e1 f 1 a2 b2 c2 d2 e2 f 2

For brevity, in the upper index of X we will write i and s instead of “inf” and “sup”, respectively. From the above records of the X -operator it is clear that in the first case it will have 256 different forms and in the second case—16 different forms.

4.1 Interval Valued Intuitionistic Fuzzy Modal Operators of First Type

59

The simplest form of the X -operator is 

ext1 ext1 ext2 ext2 ext1 ext1 ext2 ext2



X  a1

b1 c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

 (A)

= {x, [a1 inf M A (x) + b1 (1 − ext1 M A (x) − c1 ext1 N A (x)), a2 sup M A (x) + b2 (1 − ext2 M A (x) − c2 ext2 N A (x))], [d1 inf N A (x) + e1 (1 − f 1 ext1 M A (x) − ext1 N A (x)), d2 sup N A (x) + e2 (1 − f 2 ext2 M A (x) − ext2 N A (x))]|x ∈ E} ≡ X ( a11 b1 2c1) d1 ext

ext

e1 f 1 a2 b2 c2 d2 e2 f 2

 (A).

So, here will take into consideration the simplest X -operator, that, obviously, has only 4 cases that we will study sequentially. Let everywhere for the 4 cases, a1 , b1 , c1 , d1 , e1 , f 1 , a2 , b2 , c2 , d2 , e2 , f 2 ∈ [0, 1]. Then i i X ( a1 )b1 c1 d1 e1 f1  (A) a2 b2 c2 d2 e2 f 2

= {x, [a1 inf M A (x) + b1 (1 − inf M A (x) − c1 inf N A (x)), a2 sup M A (x) + b2 (1 − inf M A (x) − c2 inf N A (x))], [d1 inf N A (x) + e1 (1 − f 1 inf M A (x) − inf N A (x)), d2 sup N A (x) + e2 (1 − f 2 inf M A (x) − inf N A (x))]|x ∈ E} and to see that the operator is correct, we must find the conditions under which the inequalities (4.1.1)–(4.1.3) are valid. Below, we shall study the mentioned above four cases, each of which with four sub-cases. 1.1. We see that a1 inf M A (x) + b1 (1 − inf M A (x) − c1 inf N A (x)) ≥ a1 inf M A (x) + b1 (1 − inf M A (x) − inf N A (x)) ≥ a1 inf M A (x) ≥ 0. 1.2. We check a2 sup M A (x) + b2 (1 − inf M A (x) − c2 inf N A (x)) = a2 sup M A (x) − b2 inf M A (x) − b2 c2 inf N A (x) + b2 ≤ a2 sup M A (x) + b2 ≤ a2 + b2 . Therefore, the condition is a2 + b2 ≤ 1. Analogously, if d2 + e2 ≤ 1, then d2 sup N A (x) + e2 (1 − f 2 inf M A (x) − inf N A (x)) ≤ 1.

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4 Operators over IVIFSs

1.3. We obtain a2 sup M A (x) + b2 (1 − inf M A (x) − c2 inf N A (x)) −a1 inf M A (x) − b1 (1 − inf M A (x) − c1 inf N A (x)) = a2 sup M A (x) − a1 inf M A (x) + (b2 − b1 )(1 − inf M A (x)) +(b1 c1 − b2 c2 ) inf N A (x)) ≥ 0 for a2 ≥ a1 , b2 ≥ b1 and b1 c1 ≥ b2 c2 ; and d2 sup N A (x) + e2 (1 − f 2 inf M A (x) − inf N A (x)) −d1 inf N A (x) − e1 (1 − f 1 inf M A (x) − inf N A (x)) ≥ 0, for d2 ≥ d1 , e2 ≥ e1 and e1 f 1 ≥ e2 f 2 . 1.4. We have a2 sup M A (x) + b2 (1 − inf M A (x) − c2 inf N A (x)) +d2 sup N A (x) + e2 (1 − f 2 inf M A (x) − inf N A (x)) ≤ a2 sup M A (x) + d2 sup N A (x) + b2 + e2 ≤ max(a2 , d2 ) + b2 + e2 . Hence, the condition is max(a2 , d2 ) + b2 + e2 ≤ 1. Therefore, for this first case, the inequalities (4.1.1)–(4.1.3) have the concrete forms (4.1.4) a2 ≥ a1 , b2 ≥ b1 , d2 ≥ d1 , e2 ≥ e1 , a2 + b2 ≤ 1,

(4.1.5)

d2 + e2 ≤ 1,

(4.1.6)

b1 c1 ≥ b2 c2 ,

(4.1.7)

e1 f 1 ≥ e2 f 2 ,

(4.1.8)

max(a2 , d2 ) + b2 + e2 ≤ 1.

(4.1.9)

We must mention immediately, that the validity of conditions (4.1.5) and (4.1.6) follows directly from (4.1.9), i.e., these two conditions can be omitted.

4.1 Interval Valued Intuitionistic Fuzzy Modal Operators of First Type

61

The second X -operator is X ( a1 )b1 i s

c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

 (A)

= {x, [a1 inf M A (x) + b1 (1 − inf M A (x) − c1 inf N A (x)), a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x))], d1 inf N A (x) + e1 (1 − f 1 inf M A (x) − inf N A (x)), d2 sup N A (x) + e2 (1 − f 2 sup M A (x) − sup N A (x))]|x ∈ E}. 2.1. We see that this case coincides with case 1.1. 2.2. This case follows directly from a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x)) ≤ a2 sup M A (x) + b2 ≤ 1. Therefore, the conditions are a2 + b2 ≤ 1 and, respectively, for the second inequality d2 + e2 ≤ 1. 2.3. We obtain Z ≡ a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x)) −a1 inf M A (x) − b1 (1 − inf M A (x) − c1 inf N A (x)) = (a2 − b2 ) sup M A (x) − (a1 − b1 ) inf M A (x) + b2 − b1 −b2 c2 sup N A (x) + b1 c1 inf N A (x) ≥ (a2 − b2 − a1 + b1 ) inf M A (x) + b2 − b1 − b2 c2 sup N A (x) (from sup N A (x) ≤ 1 − sup M A (x) ≤ 1 − inf M A (x)) ≥ (a2 − b2 − a1 + b1 ) inf M A (x) + b2 − b1 − b2 c2 + b2 c2 inf M A (x) = (a2 − b2 − a1 + b1 + b2 c2 ) inf M A (x) + b2 − b1 − b2 c2 . If a2 − a1 ≥ b2 − b1 − b2 c2 , then Z ≥ b2 − b1 − b2 c2 . Therefore, Z ≥ 0 if a2 − a1 ≥ b2 − b1 − b2 c2 ≥ 0. If a2 − a1 ≤ b2 − b1 − b2 c2 , then Z 1 ≥ a2 − a1 . Therefore, the condition is min(a2 − a1 , b2 − b1 − b2 c2 ) ≥ 0.

62

4 Operators over IVIFSs

2.4. We check a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x))] +d2 sup N A (x) + e2 (1 − f 2 sup M A (x) − sup N A (x)) = (a2 − b2 − e2 f 2 ) sup M A (x) + (d2 − e2 − b2 c2 ) sup N A (x) + b2 + e2 ≤ max(a2 − b2 − e2 f 2 , d2 − e2 − b2 c2 ) + b2 + e2 , i.e., now the condition is max(a2 − b2 − e2 f 2 , d2 − e2 − b2 c2 ) + b2 + e2 ≤ 1. Therefore, for the second case, the inequalities (4.1.1)–(4.1.3) have the concrete forms (4.1.10) min(a2 − a1 , b2 − b1 − b2 c2 ) ≥ 0, min(d2 − d1 , e2 − e1 − e2 f 2 ) ≥ 0, max(a2 − b2 − e2 f 2 , d2 − e2 − b2 c2 ) + b2 + e2 ≤ 1.

(4.1.11) (4.1.12)

The third X -operator is X ( a1 )b1 s i

c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

 (A)

= {x, [a1 inf M A (x) + b1 (1 − sup M A (x) − c1 sup N A (x)), a2 sup M A (x) + b2 (1 − inf M A (x) − c2 inf N A (x))], [d1 inf N A (x) + e1 (1 − f 1 sup M A (x) − sup N A (x)), d2 sup N A (x) + e2 (1 − f 2 inf M A (x) − inf N A (x))]|x ∈ E}. 3.1. We obtain directly that a1 inf M A (x) + b1 (1 − sup M A (x) − c1 sup N A (x)) ≥ a1 inf M A (x) ≥ 0 and d1 inf N A (x) + e1 (1 − f 1 sup M A (x) − sup N A (x)) ≥ d1 inf N A (x) ≥ 0. 3.2. This case coincides with 1.2., 3.3. We have: a2 sup M A (x) + b2 (1 − inf M A (x) − c2 inf N A (x)) −a1 inf M A (x) − b1 (1 − sup M A (x) − c1 sup N A (x)) = b2 − b1 + (a2 + b1 ) sup M A (x) − (a1 + b2 ) inf M A (x) +b1 c1 sup N A (x) − b2 c2 inf N A (x) ≥ 0 for a2 ≥ a1 , b2 ≥ b1 , c1 ≥ c2 .

4.1 Interval Valued Intuitionistic Fuzzy Modal Operators of First Type

63

3.4. We check a2 sup M A (x) + b2 (1 − inf M A (x) − c2 inf N A (x)) +d2 sup N A (x) + e2 (1 − f 2 inf M A (x) − inf N A (x)) = b2 + e2 + a2 sup M A (x) + d2 sup N A (x) −(b2 + e2 f 2 ) inf M A (x) − (b2 c2 + e2 ) inf N A (x)) ≤ b2 + e2 + a2 sup M A (x) + d2 sup N A (x) ≤ max(a2 , d2 ) + b2 + e2 . Therefore, for the third case, the inequalities (4.1.1)–(4.1.3) have the concrete forms (4.1.13) a2 ≥ a1 , b2 ≥ b1 , c1 ≥ c2 , a2 + b2 ≤ 1,

(4.1.14)

d2 + e2 ≤ 1,

(4.1.15)

max(a2 , d2 ) + b2 + e2 ≤ 1.

(4.1.16)

The fourth X -operator is X ( a1 )b1 s s

c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

 (A)

= {x, [a1 inf M A (x) + b1 (1 − sup M A (x) − c1 sup N A (x)), a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x))], [d1 inf N A (x) + e1 (1 − f 1 sup M A (x) − sup N A (x)), d2 sup N A (x) + e2 (1 − f 2 sup M A (x) − sup N A (x))]|x ∈ E}. 4.1. This case coincides with 3.1. 4.2. This case coincides with 2.2. 4.3. We obtain directly that a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x)) −a1 inf M A (x) + b1 (1 − sup M A (x) − c1 sup N A (x)) ≥ 0 for a2 ≥ a1 , b2 ≥ b1 , c1 ≥ c2 . 4.4. This case coincides with 2.4. Therefore, for the fourth case, the inequalities (4.1.1)–(4.1.3) have the forms of (4.1.10)–(4.1.12), as for the third case. For all cases, one basic condition must be valid b2 + e2 ≤ 1.

(4.1.17)

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4 Operators over IVIFSs

Fig. 4.6 Interrelations between the modal type of operators

From all these checks it follow the validity of the following theorem. Theorem 4.1.4 For each IVIFS A and for each one of the four X -operators, ext ext X ( a1 1b1 c21 )d1 e1 f1  (A) is an IVIFS. a2 b2 c2 d2 e2 f 2

4.1 Interval Valued Intuitionistic Fuzzy Modal Operators of First Type ∗

All the above described simpler operators (Dα , . . . , J  α by the operator X ( a1 )b1 s s

c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2



γ β δ

65 

) can be represented

at suitably chosen values of its parameters (see

Fig. 4.6). These representations are the following: A = X ( 1 ♦A = X Dα (A)

s s) 0 r1 1 0 r2 (s s)  1 0 r1 1 1 1

1 0 s1 1 1 1 1 0 s1 1 0 s2

 (A),  (A),

= X ( 1 0) r1 s s

Fα,β (A) = X

1 α 1 1 0 r2 1 1 − α 1 (s s)   1 0 r1 1 0 s1 1 α 1 1 β 1

(A),

G α,β (A) = X ( α Hα,β (A) = X ∗ Hα,β (A) = X

Jα,β (A) = X

s s) 0 r1 α 0 r2 (s s)  α 0 r1 α 0 r2 (s s)  α 0 r1 1 0 r2 (s s)  1 0 r1 1 α 1

 (A),

β 0 s1 β 0 s2

 (A),

1 0 s1 1 β 1 α 0 s1 1 β α β 0 s1 β 0 s2

 (A),  (A),

∗ Jα,β (A) = X ( 1

s s)  0 r1 β 0 s1 1 α β β 0 s2 (s s)   1 α 1 1 γ 1 1 β 1 1 δ 1

F α

 (A)

=X

G α

 (A)

= X ( α

H α

 (A)

=X

γ β δ γ β δ γ β δ

∗

H α



Jα

 (A)

γ β δ

γ β δ

∗

Jα

γ β δ



(A) = X =X

 (A),

(A),

(A),

s s)  0 r1 β 0 s1 γ 0 r2 δ 0 s2 (s s)   α 0 r1 1 γ 1 β 0 r2 1 δ 1 (s s)   α 0 r1 1 γ 1 β 0 r2 1 δ β (s s)   1 α 1 γ 0 s1 1 β 1 δ 0 s2

(A),

(A), (A),

(A),

(A) = X ( 1 α) δ s s

γ 0 s1 1 β δ δ 0 s2

 (A),

where r1 , r2 , s1 , s2 are arbitrary real numbers in the interval [0, 1]. First, we will discuss one of the basic properties of the X -operators, and after this, we will use as examples two assertions, that by the moment are studied for one of the X -operators and as on Open Problem will be formulated the problem for checking of similar properties for the rest X -operators. Theorem 4.1.5 For each IVIFS A, for every two ext1 , ext2 ∈ {inf, sup}, for every a1 , b1 , d1 , e1 , a2 , b2 , d2 , e2 ∈ [0, 1] that satisfy the respective conditions related to ext1 , ext2 , it is valid that

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4 Operators over IVIFSs

¬X ( a11 b1 21) d1 ext

ext

e1 1 a2 b2 1 d2 e2 1

 (¬A)

= X ( d11 e1 21 )a1 ext

ext

b1 1 d2 e2 1 a2 b2 1

 (A).

Proof Let the two IVIFSs A and B be given, the two ext1 , ext2 ∈ {inf, sup} be fixed, and a1 , b1 , d1 , e1 , a2 , b2 , d2 , e2 ∈ [0, 1] satisfy the respective conditions related to ext1 , ext2 , and c1 = c2 = f 1 = f 2 = 1, then ¬X ( a11 b1 21 )d1 ext

ext

e1 1 a2 b2 1 d2 e2 1

= ¬X ( a11 b1 2c1) d1 ext

ext

e1 1 a2 b2 c2 d2 e2 1

 ({x, [inf

 (¬A)

N A (x), sup N A (x)],

[ inf M A (x), sup M A (x))] | x ∈ E}) = ¬{x, [a1 inf N A (x) + b1 (1 − ext1 N A (x) − ext1 M A (x)), a2 sup N A (x) + b2 (1 − ext2 N A (x) − ext2 M A (x))], [d1 inf M A (x) + e1 (1 − ext1 N A (x) − ext1 M A (x)), d2 sup M A (x) + e2 (1 − ext2 N A (x) − ext2 M A (x))]|x ∈ E} = {x, [d1 inf M A (x) + e1 (1 − ext1 M A (x) − ext1 N A (x)), d2 sup M A (x) + e2 (1 − ext2 M A (x) − ext2 N A (x))], [a1 inf N A (x) + b1 (1 − ext1 M A (x) − ext1 N A (x)), a2 sup N A (x) + b2 (1 − ext2 M A (x) − c2 ext2 N A (x))], |x ∈ E} = X ( d11 e1 21 )a1 ext

ext

b1 1 d2 e2 1 a2 b2 1

 (A).

The interrelation between all the modal type of operators is depicted in Fig. 4.6, where X → Y denotes the fact that operator X includes as a particular case operator Y. Theorem 4.1.6 For every two IVIFSs A and B, for every two ext1 , ext2 ∈ {inf, sup}, for every a1 , b1 , c1 , d1 , e1 , f 1 a2 , b2 , c2 , d2 , e2 , f 2 ∈ [0, 1] that satisfy (4.1.10)– (4.1.12), it is valid that (a) X ( a11 b1 2c1) ext

ext

d1 e1 f 1 a2 b2 c2 d2 e2 f 2

 (A

∩ B)

⊂ X ( a11 b1 2c1) ext

ext

d1 e1 f 1 a2 b2 c2 d2 e2 f 2

(b) X ( a11 b1 2c1) ext

ext

d1 e1 f 1 a2 b2 c2 d2 e2 f 2

 (A

 (A)

∩ X ( a11 b1 2c1)

 (B),

 (A)

∪ X ( a11 b1 2c1)

 (B).

ext

ext

d1 e1 f 1 a2 b2 c2 d2 e2 f 2

∪ B)

⊃ X ( a11 b1 2c1) ext

ext

d1 e1 f 1 a2 b2 c2 d2 e2 f 2

ext

ext

d1 e1 f 1 a2 b2 c2 d2 e2 f 2

Proof (b) Let a1 , b1 , c1 , d1 , e1 , f 1 , a2 , b2 , c2 , d2 , e2 , f 2 ∈ [0, 1] satisfy (4.1.10)– (4.1.12) and let A and B be fixed IVIFSs.

4.1 Interval Valued Intuitionistic Fuzzy Modal Operators of First Type

67

First, we obtain: Y = X ( a11 b1 2c1) d1 ext

ext

e1 f 1 a2 b2 c2 d2 e2 f 2

= X ( a11 b1 2c1) d1 ext

ext

e1 f 1 a2 b2 c2 d2 e2 f 2

 (A

 ({x, [max(inf

∪ B)

M A (x), inf M B (x)),

max(sup M A (x), sup M B (x))], [ min(inf N A (x), inf N B (x)), min(sup N A (x), sup N B (x))] | x ∈ E}) = {x, [a1 max(inf M A (x), inf M B (x)) + b1 (1 − max(inf M A (x), inf M B (x)) −c1 min(inf N A (x), inf N B (x))), a2 max(sup M A (x) sup M B (x)) + b2 (1 − max(sup M A (x) sup M B (x)) − c2 min(sup N A (x), sup N B (x)))], [d1 min(inf N A (x), inf N B (x)) + e1 (1 − f 1 max(inf M A (x), inf M B (x)) − min(inf N A (x), inf N B (x))), d2 min(sup N A (x), sup N B (x)) + e2 (1− f 2 max(sup M A (x) sup M B (x)) − min(sup N A (x), sup N B (x)))]|x ∈ E}. Second, we calculate: Z = X ( a11 b1 2c1) d1 ext

ext

e1 f 1 a2 b2 c2 d2 e2 f 2

 (A)

∪ X ( a11 b1 2c1) d1 ext

ext

e1 f 1 a2 b2 c2 d2 e2 f 2

 (B)

= {x, [a1 inf M A (x) + b1 (1 − inf M A (x) − c1 inf N A (x)), a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x))], [d1 inf N A (x) + e1 (1 − f 1 inf M A (x) − inf N A (x)), d2 sup N A (x) + e2 (1 − f 2 sup M A (x) − sup N A (x))]|x ∈ E} ∪{x, [a1 inf M B (x) + b1 (1 − inf M B (x) − c1 inf N B (x)), a2 sup M B (x) + b2 (1 − sup M B (x) − c2 sup N B (x))], [d1 inf N B (x) + e1 (1 − f 1 inf M B (x) − inf N B (x)), d2 sup N B (x) + e2 (1 − f 2 sup M B (x) − sup N B (x))]|x ∈ E}, = {x, [max(a1 inf M A (x) + b1 (1 − inf M A (x) − c1 inf N A (x)), a1 inf M B (x) + b1 (1 − inf M B (x) − c1 inf N B (x))), max(a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x)), a2 sup M B (x) + b2 (1 − sup M B (x) − c2 sup N B (x)))], [ min(d1 inf N A (x) + e1 (1 − f 1 inf M A (x) − inf N A (x)), d1 inf N B (x) + e1 (1 − f 1 inf M B (x) − inf N B (x))), min(d2 sup N A (x) + e2 (1 − f 2 sup M A (x) − sup N A (x)), d2 sup N B (x) + e2 (1 − f 2 sup M B (x) − sup N B (x)))] | x ∈ E}.

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4 Operators over IVIFSs

Let P = a1 max(inf M A (x), inf M B (x)) + b1 (1 − max(inf M A (x), inf M B (x)) −c1 min(inf N A (x), inf N B (x))) − max(a1 inf M A (x) + b1 (1 − inf M A (x) −c1 inf N A (x)), a1 inf M B (x) + b1 (1 − inf M B (x) − c1 inf N B (x))) = a1 max(inf M A (x), inf M B (x)) + b1 − b1 max(inf M A (x), inf M B (x)) −b1 c1 min(inf N A (x), inf N B (x))) − max((a1 − b1 ) inf M A (x) + b1 −b1 c1 inf N A (x), (a1 − b1 ) inf M B (x) + b1 − b1 c1 inf N B (x)) = a1 max(inf M A (x), inf M B (x)) − b1 max(inf M A (x), inf M B (x)) −b1 c1 min(inf N A (x), inf N B (x)) − max((a1 − b1 ) inf M A (x) − b1 c1 inf N A (x), (a1 − b1 ) inf M B (x) − b1 c1 inf N B (x)). Let inf M A (x) ≥ inf M B (x). Then P = (a1 − b1 ) inf M A (x) − b1 c1 min(inf N A (x), inf N B (x)) − max((a1 −b1 ) inf M A (x) − b1 c1 inf N A (x), (a1 − b1 ) inf M B (x) − b1 c1 inf N B (x)).

Let (a1 − b1 ) inf M A (x) − b1 c1 inf N A (x) ≥ (a1 − b1 ) inf M B (x) − b1 c1 inf N B (x). Then P = (a1 − b1 ) inf M A (x) − b1 c1 min(inf N A (x), inf N B (x)) − (a1 − b1 ) inf M A (x) +b1 c1 inf N A (x) = b1 c1 inf N A (x) − b1 c1 min(inf N A (x), inf N B (x)) ≥ 0.

If (a1 − b1 ) inf M A (x) − b1 c1 inf N A (x) < (a1 − b1 ) inf M B (x) − b1 c1 inf N B (x). Then P = (a1 − b1 ) inf M A (x) − b1 c1 min(inf N A (x), inf N B (x)) − (a1 − b1 ) inf M B (x) +b1 c1 inf N B (x)). = b1 c1 inf N B (x) − b1 c1 min(inf N A (x), inf N B (x)) ≥ 0.

Therefore, the inf-M A -component of IVIFS Y is higher or equal to the inf-M A component of IVIFS Z . In the same manner, it can be checked that the same inequality is valid for the sup-M A -components of these IVIFSs. On the other hand, we can check that that the inf-N A - and sup–N A -components of IVIFS Y are, respectively, lower or

4.1 Interval Valued Intuitionistic Fuzzy Modal Operators of First Type

69

equal to the inf-N A and sup-N A -components of IVIFS Z . Therefore, the inequality (c) is valid.

4.2 Interval Valued Intuitionistic Fuzzy Modal Operators of Second Type The Interval Valued Intuitionistic Fuzzy Modal Operators of second type (4IVIFMO2) are introduced for the first time in [8, 9]. They have also two forms: shorter (introduced in [8]) and extended (introduced in [9]). These IVIFO2s and their properties will be discussed below sequentially. Let the IVIFS A = {x, [inf M A (x), sup M A (x)], [inf N A (x), sup N A (x)] | x ∈ E}, be given. First, we discuss the IVIFMO2s having a shorter form. In the beginning, we introduce the first two IVIFMO2s having shorter form + and × that are defined by +A=





inf M A (x) sup M A (x) , 2 2



,

 inf N A (x)+1 sup N A (x)+1 |x ∈ E , , 2 2   × A = x, inf M A (x)+1 , sup M A (x)+1 , 2 2 

 inf N A (x) sup N A (x) |x ∈ E . , 2 2 

x,

All of their properties are valid for their immediate extensions, that for a given real number α ∈ [0, 1] and IVIFS A have the forms:

+ α A = {x, α inf M A (x), α sup M A (x) ,

α inf N A (x) + 1 − α, α sup N A (x) + 1 − α |x ∈ E},

× α A = {x, α inf M A (x) + 1 − α, α sup M A (x) + 1 − α ,

α inf N A (x), α sup N A (x) |x ∈ E}. Obviously, for every IVIFS A: + 0.5 A = + A, × 0.5 A = × A.

70

4 Operators over IVIFSs

Therefore, the new operators + α and × α are generalizations of the first ones. The following assertions hold for the first two types of the next operators. We give the proof of only one of them (Theorem 4.2.1 (b)), while the remaining assertions are proved in the same manner. Theorem 4.2.1 For every IVIFS A and for every α ∈ [0, 1]: (a) + α A ⊆ A ⊆ × α A, (b) ¬ + α ¬A = × α A, (c) + α + α A ⊆ + α A, (d) × α × α A ⊇ × α A. Proof Let A be an IVIFS and α ∈ [0, 1]. Then ¬ + α ¬A = ¬ + α ¬{x, [inf M A (x), sup M A (x)], [inf N A (x), sup N A (x)] | x ∈ E} = ¬ + α {x, [inf N A (x), sup N A (x)], [inf M A (x), sup M A (x)] | x ∈ E} = ¬{x, α inf N A (x), α sup N A (x) ,

α inf M A (x) + 1 − α, α sup M A (x) + 1 − α |x ∈ E},

= {x, α inf M A (x) + 1 − α, α sup M A (x) + 1 − α ,

α inf N A (x), α sup N A (x) |x ∈ E} = × α A. Theorem 4.2.2 For every two IVIFSs A and B: (a) + α (A + B) ⊆ + α A + + α B, (b) × α (A + B) ⊇ × α A + × α B, (c) + α (A.B) ⊇ + α A. + α B, (d) × α (A.B) ⊆ × α A. × α B. Moreover, the following assertions are also true. Theorem 4.2.3 For every IVIFS A and for every two real numbers α, β ∈ [0, 1]: (a) + α + β A = + β + α A, (b) × α × β A = × β × α A, (c) × α + β A ⊇ + β × α A.

4.2 Interval Valued Intuitionistic Fuzzy Modal Operators of Second Type

71

Theorem 4.2.4 For every IVIFS A and for every three real numbers α, β, γ ∈ [0, 1]: (a) + α Dβ (A) = Dβ ( + α A), (b) + α Fβ,γ (A) = Fβ,γ ( + α A), where β + γ ≤ 1, (c) + α G β,γ (A) ⊆ G β,γ ( + α A), (d) + α Hβ,γ (A) = Hβ,γ ( + α A), ∗ ∗ (A) = Hβ,γ ( + α A), (e) + α Hβ,γ

(f) + α Jβ,γ (A) = Jβ,γ ( + α A), ∗ ∗ (A) = Jβ,γ ( + α A), (g) + α Jβ,γ

(h) × α Dβ (A) = Dβ ( × α A), (i) × α Fβ,γ (A) = Fβ,γ ( × α A), where β + γ ≤ 1, (j) × α G β,γ (A) ⊆ G β,γ ( × α A), (k) × α Hβ,γ (A) = Hβ,γ ( × α A), ∗ ∗ (A) = Hβ,γ ( × α A), (l) × α Hβ,γ

(m) × α Jβ,γ (A) = Jβ,γ ( × α A), ∗ ∗ (A) = Jβ,γ ( × α A). (n) × α Jβ,γ

The second extension has the forms:

+ α,β A = {x, α inf M A (x), α sup M A (x) ,

α inf N A (x) + β, α sup N A (x) + β |x ∈ E},

× α,β A = {x, α inf M A (x) + β, α sup M A (x) + β ,

α inf N A (x), α sup N A (x) |x ∈ E}, where α, β, α + β ∈ [0, 1]. Obviously, for every IVIFS A: + A = + A0.5,0.5 , × A = × A0.5,0.5 , + Aα = + Aα,1−α , × Aα = × Aα,1−α . The following assertions hold for the new operators.

72

4 Operators over IVIFSs

Theorem 4.2.5 For every IVIFS A and for every α, β, α + β ∈ [0, 1]: (a) ¬ + α,β ¬A = × α,β A, (b) ¬ × α,β ¬A = + α,β A. Theorem 4.2.6 For every IFS A and for every α, β, γ, δ ∈ [0, 1] such that α + β, γ + δ ∈ [0, 1]: + α,β × γ,δ A ⊆ × γ,δ + α,β A. Now, we introduce the third extension of the above operators. They have the forms:

+ α,β,γ A = {x, α inf M A (x), α sup M A (x) ,

β inf N A (x) + γ, β sup N A (x) + γ |x ∈ E},

× α,β,γ A = {x, α inf M A (x) + γ, α sup M A (x) + γ ,

β inf N A (x), β sup N A (x) |x ∈ E}, where α, β, γ ∈ [0, 1] and max(α, β) + γ ≤ 1. Obviously, for every IVIFS A: + A = + A0.5,0.5,0.5 , × A = × A0.5,0.5,0.5 , + Aα = + Aα,α,1−α , × Aα = × Aα,1−α , + Aα,β = + Aα,α,β , × Aα,β = × Aα,α,β . The following assertions hold for the new operators. Theorem 4.2.7 For every IVIFS A and for every α, β, γ ∈ [0, 1], for which max(α + β) + γ ≤ 1: (a) ¬ + α,β,γ ¬A = × β,α,γ A, (b) ¬ × α,β,γ ¬A = + β,α,γ A. Theorem 4.2.8 For every two IVIFSs A and B and for every α, β, γ ∈ [0, 1], for which max(α + β) + γ ≤ 1: (a) + α,β,γ (A ∩ B) = + α,β,γ A ∩ + α,β,γ B, (b) × α,β,γ (A ∩ B) = × α,β,γ A ∩ × α,β,γ B,

4.2 Interval Valued Intuitionistic Fuzzy Modal Operators of Second Type

73

(c) + α,β,γ (A ∪ B) = + α,β,γ A ∪ + α,β,γ B, (d) × α,β,γ (A ∪ B) = × α,β,γ A ∪ × α,β,γ B, (e) + α,β,γ (A@B) = + α,β,γ A@ + α,β,γ B, (f) × α,β,γ (A@B) = × α,β,γ A@ × α,β,γ B. A natural extension of the last two operators ( + α,β,γ and × α,β,γ ) is the operator •



= {x, α inf M A (x) + γ, α sup M A (x) + γ ,

β inf N A (x) + δ, β sup N A (x) + δ |x ∈ E},

α,β,γ,δ A

where α, β, γ, δ ∈ [0, 1] and max(α, β) + γ + δ ≤ 1. It is the fourth type of operators from the current type. Obviously, for every IVIFS A: + A = • A0.5,0.5,0,0.5 , × A = • A0.5,0.5,0.5,0 , + Aα = • Aα,α,0,1−α , × Aα = • Aα,α,1−α,0 , + Aα,β = • Aα,α,0,β , × Aα,β = • Aα,α,β,0 . + Aα,β,γ = • Aα,β,0,γ , × Aα,β,γ = • Aα,β,γ,0 . The following assertions hold for the new operator. Theorem 4.2.9 For every IFS A and for every α, β, γ, δ ∈ [0, 1], for which max(α + β) + γ + δ ≤ 1 : (a) ¬ •

α,β,γ,δ ¬A

(b) •

α,β,γ,δ (

•

(c) •

α,β,γ,δ

(d) •

α,β,γ,δ ♦A

= •

β,α,δ,γ A,

ε,ζ,η,θ A)

A⊇ ⊆♦•

= •

•

αε,βζ,αη+γ,βθ+δ A,

α,β,γ,δ A,

α,β,γ,δ A.

Theorem 4.2.10 For every two IFSs A and B and for every α, β, γ, δ ∈ [0, 1], for which max(α + β) + γ + δ ≤ 1 :

74

4 Operators over IVIFSs

(a) •

α,β,γ,δ (A

∩ B) = •

α,β,γ,δ A

∩ •

α,β,γ,δ B,

(b) •

α,β,γ,δ (A

∪ B) = •

α,β,γ,δ A

∪ •

α,β,γ,δ B,

(c) •

α,β,γ,δ (A@B)

= •

α,β,γ,δ A@

•

α,β,γ,δ B.

In a series of papers (see, e.g., [15, 16]), Gökhan Çuvalcioˇglu introduced other operators from the second type, but having different forms. The author supposes that he will give for them interval-valued interpretations and for this reason, the author will not discuss them here. In [5], the extended forms of all above operators are introduced in the forms ◦ α,β,γ,δ,ε,ζ A

= {x, α inf M A (x) − ε inf N A (x) + γ, α sup M A (x) − ε inf N A (x) + γ , β inf N A (x) − ζ inf M A (x) + δ, β sup N A (x) − ζ inf M A (x) + δ |x ∈ E},

where α, β, γ, δ, ε, ζ ∈ [0, 1], γ ≥ ε, δ ≥ ζ and max(α − ζ, β − ε) + γ + δ ≤ 1,

(4.2.1)

min(α − ζ, β − ε) + γ + δ ≥ 0.

(4.2.2)

Obviously, for every IVIFS A: + A = ◦ 0.5,0.5,0,0.5,0,0 A, × A = ◦ 0.5,0.5,0.5,0,0,0 A, + α A = ◦ α,α,0,1−α,0,0 A, × α A = ◦ α,α,1−α,0,0,0 A, + α,β A = ◦ α,α,0,β,0,0 A, × α,β A = ◦ α,α,β,0,0,0 A, + α,β,γ A = ◦ α,β,0,γ,0,0 A, × α,β,γ A = ◦ α,β,γ,0,0,0 A, • α,β,γ,δ A = ◦ α,β,γ,δ,0,0 A. The following assertions hold for the operator ◦

α,β,γ,δ,ε,ζ .

Theorem 4.2.11 For every IVIFS A and for every α, β, γ, δ, ε, ζ ∈ [0, 1], for which (4.2.1) and (4.2.2) are valid, the equality ¬◦ holds.

α,β,γ,δ,ε,ζ ¬A

= ◦

β,α,δ,γ,ζ,ε A

4.2 Interval Valued Intuitionistic Fuzzy Modal Operators of Second Type

75

Theorem 4.2.12 For every two IVIFSs A and B and for every α, β, γ, δ, ε, ζ ∈ [0, 1], for which (4.2.1) and (4.2.2) are valid, the equality ◦

α,β,γ,δ,ε,ζ (A@B)

= ◦

α,β,γ,δ,ε,ζ A@

◦

α,β,γ,δ,ε,ζ B

holds. We must note that equalities ◦ ◦

∩ B) = ◦ ◦ α,β,γ,δ,ε,ζ (A ∪ B) = α,β,γ,δ,ε,ζ (A

which are valid for operator •

α,β,γ,δ ,

∩ ◦ ◦ α,β,γ,δ,ε,ζ A ∪ α,β,γ,δ,ε,ζ A

α,β,γ,δ,ε,ζ B, α,β,γ,δ,ε,ζ B,

are not always valid now.

Theorem 4.2.13 For every IVIFS A and for every α1 , β1 , γ1 , δ1 , ε1 , ζ1 , α2 , β2 , γ2 , δ2 , ε2 , ζ2 ∈ [0, 1] for which conditions that are similar to (4.2.1) and (4.2.2) are valid, the equality ◦ = ◦

α1 ,β1 ,γ1 ,δ1 ,ε1 ,ζ1 (

◦

α2 ,β2 ,γ2 ,δ2 ,ε2 ,ζ2 A)

α1 .α2 +ε1 .ζ2 ,β1 .β2 +ζ1 .ε2 ,α1 .γ2 −ε1 .δ2 +γ1 ,β1 .δ2 −ζ1 .γ2 +δ1 ,α1 .ε2 +ε1 .β2 ,β1 .ζ2 +ζ1 .α2 A

holds. Second, we discuss the IVIFMO2s having an extended form. The first two operators having an extended form— + and × coincide with the first two with simpler form and by this reason we will not discuss them and will start with the next two operators. Let the real numbers α1 , α2 ∈ [0, 1] be given and let α1 ≤ α2 . Then, following [8], we define:

+  α1  A = {x, α1 inf M A (x), α2 sup M A (x) , α2

α1 inf N A (x) + 1 − α1 , α2 sup N A (x) + 1 − α2 |x ∈ E},

×  α1  A = {x, α1 inf M A (x) + 1 − α1 , α2 sup M A (x) + 1 − α2 , α2

α1 inf N A (x), α2 sup N A (x) |x ∈ E}. Obviously, for every IVIFS A: +  0.5  A = + A, 0.5

×

0.5 0.5

A

= × A.

76

4 Operators over IVIFSs

Therefore, the new operators +  α1  and ×  α1  are generalizations of the first α2

α2

ones from [8] and of operators + , × , + α , × α . Theorem 4.2.14 For every IVIFS A and for every α1 , α2 ∈ [0, 1] and α1 ≤ α2 : (a) +  α1  A ⊆ A ⊆ ×  α1  A, α2

α2

(b) ¬ +  α1  ¬A = ×  α1  A, α2

α2

(c) ¬ ×  α1  ¬A = +  α1  A, α2

(d)

+

α1 α2



α2

+

α1 α2

A

⊆ +  α1  A, α2

(e) ×  α1  ×  α1  A ⊇ ×  α1  A. α2

α2

α2

Proof We only give the proof of (b), since the rest assertions are proved in the same manner. Let A be an IVIFS and α1 , α2 ∈ [0, 1] and α1 ≤ α2 . Then ¬ +  α1  ¬A α2

=

¬+

α1 α2

 ¬{x, [inf

M A (x), sup M A (x)], [inf N A (x), sup N A (x)] | x ∈ E}

= ¬ +  α1  {x, [inf N A (x), sup N A (x)], [inf M A (x), sup M A (x)] | x ∈ E} α2

= ¬{x, α1 inf N A (x), α2 sup N A (x) ,

α1 inf M A (x) + 1 − α1 , α2 sup M A (x) + 1 − α2 |x ∈ E},

= {x, α1 inf M A (x) + 1 − α1 , α2 sup M A (x) + 1 − α2 ,

α1 inf N A (x), α2 sup N A (x) |x ∈ E} = ×  α1  A. α2

Theorem 4.2.15 For every two IVIFSs A and B: (a) +  α1  (A + B) ⊆ +  α1  A + +  α1  B, α2

(b)

×

α1 α2

α2

 (A

+ B) ⊇

×

α1 α2

α2

A

+

×

α1 α2

 B,

(c) +  α1  (A.B) ⊇ +  α1  A. +  α1  B, α2

α2

α2

(d) ×  α1  (A.B) ⊆ ×  α1  A. ×  α1  B. α2

α2

α2

Moreover, the following assertions are also true. Theorem 4.2.16 For every IVIFS A and for every two real numbers α1 , α2 , β1 , β2 ∈ [0, 1] and α1 ≤ α2 , β1 ≤ β2 :

4.2 Interval Valued Intuitionistic Fuzzy Modal Operators of Second Type

77

(a) +  α1  +  β1  A = +  β1  +  α1  A, α2

β2

β2

α2

(b) ×  α1  ×  β1  A = ×  β1  ×  α1  A, α2

β2

β2

α2

(c) ×  α1  +  β1  A ⊇ +  β1  ×  α1  A. α2

β2

β2

α2

The second extension has the forms:

×  α1 α2



+  α1 β1  A = {x, α1 inf M A (x), α2 sup M A (x) , α β 2 2

α1 inf N A (x) + β1 , α2 sup N A (x) + β2 |x ∈ E},   A = {x, α inf M (x) + β , α sup M (x) + +  β1 α1 1 A 1 2 A β2 α2

α1 inf N A (x), α2 sup N A (x) |x ∈ E},

  β1 β2

β2

,

where α1 , α2 , β1 , β2 , α2 + β2 ∈ [0, 1] and α1 ≤ α2 , β1 ≤ β2 . Obviously, for every IVIFS A: +  α1  A = +  α1

r α2 1 − α2

α2

×  α1  A = ×  α2 α2

α2

1 − α2 r

 A,

 A,

where r ∈ [0, 1] is an arbitrary number. The following assertions hold for the new operators. Theorem 4.2.17 For every IVIFS A and for every α1 , α2 , β1 , β2 , α1 + β1 , α2 + β2 ∈ [0, 1] and α1 ≤ α2 , β1 ≤ β2 : (a) ¬ +  α1

 ¬A

= ×  α1

 A,

(b) ¬ ×  α1

 ¬A

= +  α1

 A.

β1 α2 β2 β1 α2 β2

β1 α2 β2 β1 α2 β2

Theorem 4.2.18 For every IVIFS A and for every α1 , α2 , β1 , β2 , γ1 , γ2 , δ1 , δ2 , α1 +β1 , α2 + β2 , γ1 + δ1 , γ2 + δ2 ∈ [0, 1] and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 ,: +  α1

β1 α2 β2



×  γ1

δ1 α2 β2

A

⊆ ×  γ1

δ1 α2 β2



+  α1

β1 α2 β2

 A.

Now, we introduce the third extensions of the above operators. They have the forms:

78

4 Operators over IVIFSs



+  α1 β1 γ1  A = {x, α1 inf M A (x), α2 sup M A (x) , α2 β2 γ2

β1 inf N A (x) + γ1 , β2 sup N A (x) + γ2 |x ∈ E},

×  α1 β1 γ1  A = {x, α1 inf M A (x) + γ1 , α2 sup M A (x) + γ2 , α2 β2 γ2

β1 inf N A (x), β2 sup N A (x) |x ∈ E}, where α1 , α2 , β1 , β2 , γ1 , γ2 ∈ [0, 1], max(αi , βi ) + γi ≤ 1 for i = 1, 2 and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 . Obviously, for every IVIFS A: +  α1

A

= +  α1

 A,

×  α1

A

= ×  α1

 A.

β1 α2 β2 β1 α2 β2

α1 β1 α2 α1 β2 α1 β1 α2 α1 β2

The following assertions hold for the new operators. Theorem 4.2.19 For every IVIFS A and for every α1 , α2 , β1 , β2 , γ1 , γ2 ∈ [0, 1], max(αi , βi ) + γi ≤ 1 for i = 1, 2 and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 : (a) ¬ +  α1

 ¬A

= ×  β1

 A,

(b) ¬ ×  α1

 ¬A

= +  β1

 A.

β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2

α1 γ1 β2 α2 γ2 α1 γ1 β2 α2 γ2

Theorem 4.2.20 For every two IVIFSs A and B and for every α1 , α2 , β1 , β2 , γ1 , γ2 ∈ [0, 1], max(αi , βi ) + γi ≤ 1 for i = 1, 2 and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 : (a) +  α1

 (A

∩ B) = +  α1

A

∩ +  α1

 B,

(b) ×  α1

 (A

∩ B) = ×  α1

A

∩ ×  α1

 B,

(c) +  α1

 (A

∪ B) = +  α1

A

∪ +  α1

 B,

(d) ×  α1

 (A

∪ B) = ×  α1

A

∪ ×  α1

 B,

(e) +  α1

 (A@B)

= +  α1

 A@ + 

(f) ×  α1

 (A@B)

= ×  α1

 A@ × 

β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2

β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2

β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2

β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2

α1 β1 γ1 α2 β2 γ2 α1 β1 γ1 α2 β2 γ2

 B,  B.

A natural extension of the last two operators is the operator •



α1 β1 γ1 δ1 α2 β2 γ2 δ2

A



= {x, α1 inf M A (x) + γ1 , α2 sup M A (x) + γ2 , β1 inf N A (x) + δ1 , β2 sup N A (x) + δ2 |x ∈ E},

4.2 Interval Valued Intuitionistic Fuzzy Modal Operators of Second Type

79

where α1 , α2 , β1 , β2 , γ1 , γ2 , δ1 , δ2 ∈ [0, 1], max(αi , βi ) + γi + δi ≤ 1 for i = 1, 2 and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 . It is the fourth type of operators of the current type. Obviously, for every IVIFS A: +  α1

A

= •



×  α1

A

= •



β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2

α1 β1 0 γ1 α2 β2 0 γ2 α1 β1 γ1 0 α2 β2 γ2 0

 A,  A.

The following assertions hold for the new operator. Theorem 4.2.21 For every IFS A and for every α1 , α2 , β1 , β2 , γ1 , γ2 , δ1 , δ2 ∈ [0, 1], max(αi , βi ) + γi + δi ≤ 1 for i = 1, 2 and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 : (a) ¬ • (b) •

(c) •

(d) •







(e) •



(f) •





α1 β1 γ1 δ1 α2 β2 γ2 δ2

α1 β1 γ1 δ1 α2 β2 γ2 δ2

α1 β1 γ1 δ1 α2 β2 γ2 δ2

α1 β1 γ1 δ1 α2 β2 γ2 δ2

α1 β1 γ1 δ1 α2 β2 γ2 δ2 α1 β1 γ1 δ1 α2 β2 γ2 δ2

 ¬A

 (A

= •



β1 α1 δ1 γ1 β2 α2 δ2 γ2

∩ B) = •



α1 β1 γ1 δ1 α2 β2 γ2 δ2

∩ •

 (A

∪ B) = •



= •



•

A⊇

 ♦A

⊆♦•









 B,

A

α1 β1 γ1 δ1 α2 β2 γ2 δ2

α1 β1 γ1 δ1 α2 β2 γ2 δ2

α1 β1 γ1 δ1 α2 β2 γ2 δ2

A

α1 β1 γ1 δ1 α2 β2 γ2 δ2

α1 β1 γ1 δ1 α2 β2 γ2 δ2

@ •





α1 β1 γ1 δ1 α2 β2 γ2 δ2

∪ •

 (A@B)

 A,

 B,

A

α1 β1 γ1 δ1 α2 β2 γ2 δ2

 B,

 A,

 A.

Theorem 4.2.22 For every IVIFS A and for every α1 , α2 , β1 , β2 , γ1 , γ2 , δ1 , δ2 ∈ [0, 1], max(αi , βi ) +γi + δi ≤ 1 for i = 1, 2 and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 ; ε1 , ε2 , ζ1 , ζ2 , η1 , η2 , θ1 , θ2 ∈ [0, 1], max(εi , ζi ) + ηi + θi ≤ 1 for i = 1, 2 and ε1 ≤ ε2 , ζ1 ≤ ζ2 , η1 ≤ η2 , θ1 ≤ θ2 : •



α1 β1 γ1 δ1 α2 β2 γ2 δ2

= •



(

•



ε1 ζ1 η1 θ1 ε2 ζ2 η2 θ2

 A)

α1 ε1 β1 ζ1 α1 η1 + γ1 β1 θ1 + δ1 α2 ε2 β2 ζ2 α2 η2 + γ2 β2 θ2 + δ2

 A.

Theorem 4.2.23 For every IVIFS A and for every α1 , α2 , β1 , β2 , γ1 , γ2 , δ1 , δ2 ∈ [0, 1], max(αi , βi ) +γi + δi ≤ 1 for i = 1, 2 and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 :

80

4 Operators over IVIFSs

(a) •



(b) •



α1 β1 γ1 δ1 α2 β2 γ2 δ2 α1 β1 γ1 δ1 α2 β2 γ2 δ2

 C(A)

= C( •

 I (A)

= I( •





α1 β1 γ1 δ1 α2 β2 γ2 δ2

α1 β1 γ1 δ1 α2 β2 γ2 δ2

 A),

 A).

The extended form of all above operators is the operator, introduced in [7] with the form: ◦





α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

A

= {x, α1 inf M A (x) − ε1 inf N A (x) + γ1 , α2 sup M A (x) − ε2 inf N A (x)

+ γ2 , β1 inf N A (x) − ζ1 inf M A (x) + δ1 , β2 sup N A (x) − ζ2 inf M A (x) + δ2 |x ∈ E}, where α1 , β1 , γ1 , δ1 , ε1 , ζ1 , α2 , β2 , γ2 , δ2 , ε2 , ζ2 ∈ [0, 1], and for i = 1, 2: max(αi − ζi , βi − εi ) + γi + δi ≤ 1,

(4.2.3)

max(αi , βi ) + γi + δi ≥ 0,

(4.2.4)

γi + δi ≤ 1,

(4.2.5)

γi ≥ εi δi ≥ ζi ,

(4.2.6)

α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 , ε1 ≥ ε2 , ζ1 ≤ ζ2 .

(4.2.7)

Obviously, for every IVIFS A: •



α1 β1 γ1 δ1 α2 β2 γ2 δ2

A

= ◦



α1 β1 γ1 δ1 0 0 α2 β2 γ2 δ2 0 0

The following assertions hold for the operator ◦

 A.





α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

(Fig. 4.7).

Theorem 4.2.24 For every IVIFS A and for every α1 , β1 , γ1 , δ1 , ε1 , ζ1 , α2 , β2 , γ2 , δ2 , ε2 , ζ2 ∈ [0, 1], and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 , ε1 ≥ ε2 , ζ1 ≤ ζ2 , for which (4.2.3)–(4.2.7) are valid, the equality ¬◦



α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε1 ζ1

 ¬A

= ◦



β1 α1 δ1 γ1 ζ1 ε1 β2 α2 δ2 γ2 ζ2 ε2

A

holds. Theorem 4.2.25 For every two IVIFSs A and B and for every α1 , β1 , γ1 , δ1 , ε1 , ζ1 , α2 , β2 , γ2 , δ2 , ε2 , ζ2 ∈ [0, 1], and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 , ε1 ≥ ε2 , ζ1 ≤ ζ2 , for which (4.2.3)–(4.2.7) are valid, the equality

4.2 Interval Valued Intuitionistic Fuzzy Modal Operators of Second Type

81

Fig. 4.7 Interrelations between the modal type of operators

◦ = ◦





α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

 A@

 (A@B)

◦



α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

B

holds. We must note that the equalities ◦ = ◦





α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

◦

= ◦





∩ ◦

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

which are valid for operator •

A

A

α,β,γ,δ ,

∪ ◦

 (A 

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

 (A 

∩ B)  B,

∪ B)

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

 B,

are not always valid now.

Theorem 4.2.26 For every IVIFS A and for every α1 , β1 , γ1 , δ1 , ε1 , ζ1 , α2 , β2 , γ2 , δ2 , ε2 , ζ2 ∈ [0, 1], and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 , ε1 ≥ ε2 , ζ1 ≤ ζ2 , for which (4.2.3)–(4.2.7) are valid, for every η1 , θ1 , ι1 , κ1 , λ1 , μ1 , η2 , θ2 , ι2 , κ2 , λ2 ,μ2 ∈

82

4 Operators over IVIFSs

[0, 1], and η1 ≤ η2 , θ1 ≤ θ2 , ι1 ≤ ι2 , κ1 ≤ κ2 , λ1 ≥ λ2 , μ1 ≤ μ2 , for which max(ηi − μi , θi − λi ) + ιi + κi ≤ 1, min(ηi − μi , θi − λi ) + ιi + κi ≥ 0 are valid, the equality ◦ = ◦



 

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2



◦

 

η1 θ1 ι1 κ1 λ1 μ1 η2 θ2 ι2 κ2 λ2 μ2

A

α1 η1 + ε1 μ1 β1 θ1 + ζ1 λ1 α1 ι1 − ε1 κ1 + γ1 β1 κ1 − ζ1 ι1 + δ1 α1 λ1 + ε1 θ1 β1 μ1 + ζ1 η1 α2 η2 + ε2 μ2 β2 θ2 + ζ2 λ2 α2 ι2 − ε2 κ2 + γ2 β2 κ2 − ζ2 ι2 + δ2 α2 λ2 + ε2 θ2 β2 μ2 + ζ2 η2

A

holds. Now, having in mind the four forms of the X -operator, here, following [11], we will introduce an extension of the operator ◦  α1 β1 γ1 δ1 ε1 ζ1  (see Fig. 4.7). Let again ext1 , ext2 ∈ {inf, sup}. We define ◦

( ext  1

α2 β2 γ2 δ2 ε2 ζ2

ext2 )  α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2

A

= {x, [α1 inf M A (x) − ε1 ext1 N A (x) + γ1 , α2 sup M A (x) − ε2 ext2 N A (x) + γ2 ], [β1 inf N A (x) − ζ1 ext1 M A (x) + δ1 , β2 sup N A (x) − ζ2 ext2 M A (x) + δ2 ] |x ∈ E}.

(4.2.8)

The components of this operator must satisfy the following conditions in a general form: 0 ≤ α1 inf M A (x) − ε1 ext1 N A (x) + γ1 ≤ α2 sup M A (x) − ε2 ext2 N A (x) + γ2 ≤ 1,

(4.2.9)

0 ≤ β1 inf N A (x) − ζ1 ext1 M A (x) + δ1 ≤ β2 sup N A (x) − ζ2 ext2 M A (x) + δ2 ≤ 1,

(4.2.10)

α2 sup M A (x) − ε2 ext2 N A (x) + γ2 + β2 sup N A (x) − ζ2 ext2 M A (x) + δ2 ≤ 1.

(4.2.11)

For example, for the case with operator ◦

( inf sup ) 

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ2 δ2 ε2 ζ2



the conditions are

max(αi − ζi , βi − εi ) + γi + δi ≤ 1,

(4.2.12)

α2 + γ2 ≤ 1, β2 + δ2 ≤ 1,

(4.2.13)

γi ≥ εi , δi ≥ ζi ,

(4.2.14)

4.2 Interval Valued Intuitionistic Fuzzy Modal Operators of Second Type

83

γ2 − γ1 − ε2 ≥ 0,

(4.2.15)

γ2 + δ2 ≥ 1,

(4.2.16)

α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 , ε1 ≥ ε2 , ζ1 ≤ ζ2 .

(4.2.17)

4.3 Theorem for Equivalence of the Two Most Extended Modal Operators In this section, following [11], we introduce and prove the following Theorem 4.3.1 The two most extended modal operators X ( a1

i s)  b1 c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

and ◦

(i s) 

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ1 δ2 ε1 ζ1



defined over a given IVIFS A are equivalent. Proof Let a1 , b1 , c1 , d1 , e1 , f 1 , a2 , b2 , c2 , d2 , e2 , f 2 ∈ [0, 1] and satisfy (4.1.10)– (4.1.12). Let for i = 1, 2: αi = ai − bi , βi = di − ei , γi = bi , δi = ei , εi = bi ci , ζi = ei f i . Also, let

X 1 ≡ α1 inf M A (x) − ε1 inf N A (x) + γ1 , Y1 ≡ β1 inf N A (x) − ζ1 inf M A (x) + δ1 , X 2 ≡ α2 sup M A (x) − ε2 sup N A (x) + γ2 , Y2 ≡ β2 sup N A (x) − ζ2 sup M A (x) + δ2 .

Then

X 1 = (a1 − b1 ) inf M A (x) − b1 c1 inf N A (x) + b1 , Y1 = (d1 − e1 ) inf N A (x) − e1 f 1 inf M A (x) + e1 , X 2 = (a2 − b2 ) sup M A (x) − b2 c2 sup N A (x) + b2 , Y2 = (d2 − e2 ) sup N A (x) − e2 f 2 sup M A (x) + e2 .

We obtain sequentially the following inequalities. If a1 ≥ b1 , then X 1 ≥ −b1 c1 + b1 = b1 (1 − c1 ) ≥ 0, if a1 ≤ b1 , then

84

4 Operators over IVIFSs

X 1 ≥ (a1 − b1 ) inf M A (x) − b1 c1 + b1 c1 inf M A (x) + b1 = (a1 − b1 + b1 c1 ) inf M A (x) − b1 c1 + b1 . If a1 − b1 + b1 c1 ≥ 0, then X 1 ≥ −b1 c1 + b1 ≥ 0, If a1 − b1 + b1 c1 ≤ 0, then X 1 ≥ a1 − b1 + b1 c1 − b1 c1 + b1 = a1 ≥ 0, i.e., in all cases X 1 ≥ 0. X 2 ≤ (a2 − b2 ) sup M A (x) + b2 . If a2 − b2 ≥ 0, then If a2 − b2 ≤ 0, then

X 2 ≤ a2 − b2 + b2 = a2 ≤ 1, X 2 ≤ b2 ≤ 1,

i.e. X 2 ≤ 1 and analogously, Y2 ≤ 1. X 2 − X 1 = (a2 − b2 ) sup M A (x) − b2 c2 sup N A (x) + b2 −(a1 − b1 ) inf M A (x) + b1 c1 inf N A (x) − b1 ≥ (a2 − b2 − a1 + b1 ) sup M A (x) − b2 c2 sup N A (x) + b2 − b1 = (a2 − b2 − a1 + b1 + b2 c2 ) sup M A (x) − b2 c2 + b2 − b1 . If a2 − a1 ≥ b2 − b1 − b2 c2 , then from (4.1.10) it follow X 2 ≥ b2 − b1 − b2 c2 ≥ 0. If a2 − a1 ≤ b2 − b1 − b2 c2 , then again from (4.1.10) it follow X 2 ≥ a2 − b2 − a1 + b1 + b2 c2 − b2 c2 + b2 − b1 ≥ a2 − a1 ≥ 0, i.e., always X 2 ≥ X 1 . By the same manner we check that Y2 ≥ Y1 . X 2 + Y2 = (a2 − b2 ) sup M A (x) − b2 c2 sup N A (x) + b2 +(d2 − e2 ) sup N A (x) − e2 f 2 sup M A (x) + e2 = (a2 − b2 − e2 f 2 ) sup M A (x) + (d2 − e2 − b2 c2 ) sup N A (x) + b2 + e2 . Now, there are four cases that we must study sequentially.

4.3 Theorem for Equivalence of the Two Most Extended Modal Operators

If a2 − b2 − e2 f 2 ≥ 0 and d2 − e2 − b2 c2 ≥ 0, then X 2 + Y2 ≤ (a2 − b2 − e2 f 2 − d2 + e2 + b2 c2 ) sup M A (x) +d2 − e2 − b2 c2 + b2 + e2 ≤ a2 − b2 − e2 f 2 − d2 + e2 + b2 c2 + d2 − e2 − b2 c2 + b2 + e2 (from (4.1.12)) = a2 − e2 f 2 + e2 ≤ 1. If a2 − b2 − e2 f 2 ≥ 0 and d2 − e2 − b2 c2 ≤ 0, then as above X 2 + Y2 ≤ (a2 − b2 − e2 f 2 ) sup M A (x) + b2 + e2 ≤ a2 − b2 − e2 f 2 + b2 + e2 = a2 − e2 f 2 + e2 ≤ max(a2 , d2 ) + b2 + e2 ≤ 1. If a2 − b2 − e2 f 2 ≤ 0 and d2 − e2 − b2 c2 ≥ 0, then X 2 + Y2 ≤ (d2 − e2 − b2 c2 ) sup N A (x) + b2 + e2 ≤ d2 − e2 − b2 c2 + b2 + e2 ≤ d2 − b2 c2 + b2 ≤ max(a2 , d2 ) + b2 + e2 ≤ 1 ≤ 1. If a2 − b2 − e2 f 2 ≤ 0 and d2 − e2 − b2 c2 ≤ 0, then from (4.1.12) X 2 + Y2 ≤ b2 + e2 ≤ 1, i.e., always X 2 + Y2 ≤ 1. Also, from (4.1.12) we obtain that for i = 1, 2: αi + γi − ζi + δi = (ai − bi ) + bi − ei f i + ei = ai − ei f i + ei ≤ a2 − e2 f 2 + e2 ≤ max(a2 − b2 − e2 f 2 , d2 − e2 − b2 c2 ) + b2 + e2 ≤ 1 and analogously βi + γi − εi + δi = (di − ei ) + bi − bi ci + ei = bi + di − bi ci ≤ 1. Hence, max(α1 − ζ1 , β1 − ε1 ) + γ1 − ε1 + δ1 ≤ 1, i.e. (4.2.12) is valid.

85

86

4 Operators over IVIFSs

On the other hand, α2 + γ2 = a2 − b2 + b2 = a2 ≤ 1 and β2 + δ2 = d2 − e2 + e2 = d2 ≤ 1, i.e. (4.2.13) is valid. γi − εi = bi − bi ci ≥ 0 and δi − ζi = ei − ei f i ≥ 0, i.e. (4.2.14) is valid. From (4.1.10) γ2 − γ1 − ε2 = b2 − b1 − b1 c1 ≥ 0, i.e. (4.2.15) is valid. Inequality (4.1.17) follows directly from (4.2.17). (i s)  The rest conditions for the operator ◦ α1 β1 γ1 δ1 ε1

ζ1 α2 β2 γ1 δ2 ε1 ζ1

Thus, we obtain ◦

(i s) 

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ1 δ2 ε1 ζ1





are checked directly.

(A)

= {x, α1 inf M A (x)−ε1 inf N A (x) + γ1 , α2 sup M A (x) − ε2 sup N A (x) + γ2 ,

β1 inf N A (x)−ζ1 inf M A (x) + δ1 , β2 sup N A (x)−ζ2 sup M A (x) + δ2 |x ∈ E}, = {x, [(a1 − b1 ) inf M A (x) − b1 c1 inf N A (x) + b1 , (a2 − b2 ) sup M A (x) − b2 c2 sup N A (x) + b2 ], [(d1 − e1 ) inf N A (x) − e1 f 1 inf M A (x) + e2 , b

(d2 − e2 ) sup N A (x) − e2 f 2 sup M A (x) + e2 |x ∈ E} = {x, [a1 inf M A (x) + b1 (1 − inf M A (x) − c1 inf N A (x)), a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x))], [d1 inf N A (x) + e1 (1 − f 1 inf M A (x) − inf N A (x)), d2 sup N A (x) + e2 (1 − f 2 sup M A (x) − sup N A (x))]|x ∈ E}, X ( a1

i s)  b1 c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

(A).

Conversely, let α1 , β1 , γ1 , δ1 , ε1 , ζ1 , α2 , β2 , γ2 , δ2 , ε2 , ζ2 ∈ [0, 1], and let them satisfy (4.2.12)–(4.2.17). Then, let γi , δi > 0 and ai = αi + γi (≤ 1), bi = γi ,

4.3 Theorem for Equivalence of the Two Most Extended Modal Operators

87

εi (≤ 1), γi di = βi + δi (≤ 1), ei = δi , ζi (≤ 1). fi = δi ci =

Let

X 1 ≡ a1 inf M A (x) + b1 (1 − inf M A (x) − c1 inf N A (x)), Y1 ≡ d1 inf N A (x) + e1 (1 − f 1 inf M A (x) − inf N A (x)), X 2 ≡ a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x)), Y2 ≡ d2 sup N A (x) + e2 (1 − f 2 sup M A (x) − sup N A (x)).

Then X 1 = (α1 + γ1 ) inf M A (x) + γ1 (1 − inf M A (x) − Y1 = (β1 + δ1 ) inf N A (x) + δ1 (1 −

ζ1 δ1

Y2 = (β2 + δ2 ) sup N A (x) + δ2 (1 −

inf N A (x)),

inf M A (x) − inf N A (x))

X 2 = (α2 + γ2 ) sup M A (x) + γ2 (1 − sup M A (x) − ζ2 δ2

ε1 γ1 ε2 γ2

sup N A (x)),

sup M A (x) − sup N A (x)).

Using (4.2.13) and (4.2.14), we obtain sequentially: 0 ≤ γ1 − ε1 ≤ X 1 = α1 inf M A (x) + γ1 − ε1 inf N A (x) ≤ α1 + γ1 ≤ 1, 0 ≤ δ1 − ζ1 ≤ Y1 = β1 inf N A (x) + δ1 − ζ1 inf M A (x) ≤ β1 + δ1 ≤ 1, 0 ≤ γ2 − ε2 ≤ X 2 ≤ α2 sup M A (x) + γ2 ≤ α2 + γ2 ≤ 1, 0 ≤ δ2 − ζ2 ≤ Y2 ≤ β2 sup N A (x) + δ2 ≤ β2 + δ2 ≤ 1. X 2 − X 1 = (α2 + γ2 ) sup M A (x) + γ2 (1 − sup M A (x) − γε22 sup N A (x)) −(α1 + γ1 ) inf M A (x) − γ1 (1 − inf M A (x) −

ε1 γ1

inf N A (x))

= α2 sup M A (x) − α1 inf M A (x) + γ2 − γ1 − ε2 sup N A (x) + ε1 inf N A (x) ≥ α2 sup M A (x) − α1 sup M A (x) + γ2 − γ1 − ε2 + ε2 sup M A (x) = (α2 − α1 + ε2 ) sup M A (x) + γ2 − γ1 − ε2 (from (4.2.15)) ≥ γ2 − γ1 − ε2 ≥ 0. X 2 + Y2 = (α2 + γ2 ) sup M A (x) + γ2 (1 − sup M A (x) − +(β2 + δ2 ) sup N A (x) + δ2 (1 −

ζ2 δ2

ε2 γ2

sup N A (x))

sup M A (x) − sup N A (x))

= (α2 − ζ2 ) sup M A (x) + (β2 − ε2 ) sup N A (x) + γ2 + δ2

88

4 Operators over IVIFSs

(from (4.2.12)) ≤ max(α2 − ζ2 , β2 − ε2 ) + γ2 + δ2 ≤ 1. Also, (4.1.10) that is valid because from (4.2.17) it follows that a2 − a1 = α2 + γ2 − α1 − γ1 ≥ 0 and from (4.2.15) b2 − b1 − b2 c2 = γ2 − γ1 − γ2 .

ε2 = γ2 − γ1 − ε2 ≥ 0, γ2

i.e. min(a2 − a1 , b2 − b1 − b2 c2 ) ≥ 0. By analogy we check the validity of (4.1.11). For (4.1.12) we obtain max(a2 − b2 − e2 f 2 , d2 − e2 − b2 c2 ) + b2 + e2 = max(α2 + γ2 − γ2 − ζ2 , β2 + δ2 − δ2 − ε2 ) + γ2 + δ2 = max(α2 − ζ2 , β2 + ε2 ) + γ2 + δ2 ≤ 1 from (4.2.12). Inequality (4.2.17) follows directly from (4.1.17). Then, we obtain X ( a1

i s)  b1 c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

(A)

= {x, [a1 inf M A (x) + b1 (1 − inf M A (x) − c1 inf N A (x)), a2 sup M A (x) + b2 (1 − sup M A (x) − c2 sup N A (x))], [d1 inf N A (x) + e1 (1 − f 1 inf M A (x) − inf N A (x)), d2 sup N A (x) + e2 (1 − f 2 sup M A (x) − sup N A (x))]|x ∈ E}, = {x, [(a1 − b1 ) inf M A (x) − b1 c1 inf N A (x) + b1 , (a2 − b2 ) sup M A (x) − b2 c2 sup N A (x) + b2 ], [(d1 − e1 ) inf N A (x) − e1 f 1 inf M A (x) + e2 , b

(d2 − e2 ) sup N A (x) − e2 f 2 sup M A (x) + e2 |x ∈ E}



= {x, α1 inf M A (x) − ε1 inf N A (x) + γ1 , α2 sup M A (x) − ε2 sup N A (x) + γ2 ,

β1 inf N A (x)−ζ1 inf M A (x) + δ1 , β2 sup N A (x)−ζ2 sup M A (x) + δ2 |x ∈ E}, ◦

(i s) 

α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ1 δ2 ε1 ζ1

Therefore, the two operators are equivalent.



(A).

4.4 Topological Operators over IVIFSs

89

4.4 Topological Operators over IVIFSs In [4], for each IVIFS A, the following two topological operators are defined C(A) = {x, [K inf , K sup ], [L inf , L sup ] | x ∈ E}, I (A) = {x, [K inf , K sup ], [L inf , L sup ] | x ∈ E},

where:

K inf = sup inf M A (x), K sup = sup sup M A (x), x∈E

L inf

= inf inf N A (x), x∈E

x∈E

L sup

= inf sup N A (x), x∈E

= inf inf M A (x), K sup = inf sup M A (x), K inf x∈E

L inf

= sup inf N A (x), x∈E

x∈E

L sup

= sup sup N A (x). x∈E

Let us have an universe E = {x, y, z} with elements that have the geometrical interpretations from Fig. 4.8. The interpretations of operators I and C are shown on Figs. 4.9 and 4.10, respectively. Theorem 4.4.1 For every two IVIFSs A and B: (a) (b) (c) (d)

C(A) and I (A) are IVIFSs; I (A) ⊂ A ⊂ C(A); C(C(A)) = C(A); C(I (A)) = I (A);

Fig. 4.8 IVIF interpretation of elements x, y, z ∈ E

90

4 Operators over IVIFSs

Fig. 4.9 IVIF interpretation of operator I

Fig. 4.10 IVIF interpretation of operator C

(e) (f) (g) (h) (i) (j) (k) (l) (m)

I (C(A)) = C(A); I (I (A)) = I (A); C(A ∪ B) = C(A) ∪ C(B); C(A ∩ B) ⊂ C(A) ∪ C(B); C(A@B) ⊂ C(A)@C(B); I (A ∪ B) ⊃ I (A) ∪ I (B); I ( A ∩ B) = I (A) ∩ I (B); I (A@B) ⊃ I (A)@I (B); ¬I (¬A) = C(A);

4.4 Topological Operators over IVIFSs

(n) (o) (p) (q) (r) (w) (t)

¬C(¬A) = I (A); C(O ∗ ) = O ∗ ; C(E ∗ ) = E ∗ ; C(U ∗ ) = U ∗ ; I (O ∗ ) = O ∗ ; I (E ∗ ) = E ∗ , I (U ∗ ) = U ∗ ,

(u)

(C(A)) = C(

91

(A)),

(v) (I (A)) = I ( (A)), (w) ♦(C(A)) = C(♦(A)), (x) ♦(I (A)) = I (♦(A)). Proof We will prove one of these assertions, e.g., (g), while the rest assertions are proved in the same manner. C(A ∪ B) = C({x, [max(inf M A (x), inf M B (x)), max(sup M A (x) sup M B (x))], [min(inf N A (x), inf N B (x)), min(sup N A (x), sup N B (x))] | x ∈ E}) = {x, [supx∈E (max(inf M A (x), inf M B (x))), supx∈E (max(sup M A (x), sup M B (x)))], [inf x∈E (min(inf N A (x), inf N B (x))), inf x∈E (min(sup N A (x), sup N B (x)))]| x ∈ E}) = {x, [max(supx∈E inf M A (x), supx∈E inf M B (x))), max(supx∈E sup M A (x), supx∈E sup M B (x)))], [min(inf x∈E inf N A (x), inf x∈E inf N B (x))), min(inf x∈E sup N A (x), inf x∈E sup N B (x)))] | x ∈ E}) = {x, [supx∈E inf M A (x), supx∈E sup M A (x)], [inf x∈E inf N A (x), inf x∈E sup N A (x)] | x ∈ E}) ∪{x, [supx∈E inf M B (x), supx∈E sup M B (x)], [inf x∈E inf N B (x), inf x∈E sup N B (x)] | x ∈ E}) = C(A) ∪ C(B).

We must note that I (A) ⊆ C(B) follows from the inclusion A ⊆ B, but in the general case, C(A) ⊂ I (B) does not hold. Further, in Sect. 4.6 we will show a condition for the validity of the last relation. For brevity we will use C A and I A instead of C(A) and I (A), respectively, in the next theorem.

92

4 Operators over IVIFSs

Theorem 4.4.2 For every IFS A: (a) C A = ♦C A = ¬ I ♦¬A = ¬♦I ♦¬A = {x, [K inf , K sup ], [L inf , 1 − K sup ]|x ∈ E}, (b) C♦A = ♦C♦A = ¬ I ¬A = ¬♦I ¬A = {x, [K inf , 1 − L sup ], [L inf , L sup ]|x ∈ E}, (c) I

A = ♦I A = ¬ C♦¬A = ¬♦C♦¬A = {x, [K inf , K sup ], [L inf , 1 − K sup ]|x ∈ E},

(d) I ♦A = ♦I ♦A = ¬ C ¬A = ¬♦C ¬A = {x, [K inf , 1 − L sup ], [L inf , L sup ]|x ∈ E}, (e) C

¬A = ♦C ¬A = ¬ I ♦A = ¬♦I ♦A = {x, [L inf , L sup ], [K inf , 1 − L sup ]|x ∈ E},

(f) C♦¬A = ♦C♦¬A = ¬ I A = ¬♦I A = {x, [L inf , 1 − K sup ], [K inf , K sup ]|x ∈ E}, (g) I

¬A = ♦I =

{x, [L inf ,

¬A = ¬

C♦A = ¬♦C♦A

L sup ], [K inf ,1

− L sup ]|x ∈ E},

(h) I ♦¬A = ♦I ♦¬A = ¬

C

A = ¬♦C

A

= {x, [L inf , 1 − K sup ], [K inf , K sup ]|x ∈ E}.

4.4 Topological Operators over IVIFSs

93

Theorem 4.4.3 For every IVIFS A and for every α, β, γ ∈ [0, 1], for which max(α + β) + γ ≤ 1: (a) + α,β,γ C(A) = C( + α,β,γ A), (b) × α,β,γ C(A) = C( × α,β,γ A), (c) + α,β,γ I (A) = I ( + α,β,γ A), (d) × α,β,γ I (A) = I ( × α,β,γ A). Theorem 4.4.4 For every IVIFS A and for every α, β, γ, δ ∈ [0, 1], for which max(α + β) + γ + δ ≤ 1 : (a) •

α,β,γ,δ C(A)

= C( •

(b) •

α,β,γ,δ I (A)

= I( •

α,β,γ,δ A),

α,β,γ,δ A),

Theorem 4.4.5 For every IVIFS A and for every α1 , α2 , β1 , β2 , γ1 , γ2 ∈ [0, 1], max(αi , βi ) + γi ≤ 1 for i = 1, 2 and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 : (a) +  α1

 C(A)

= C( +  α1

 A),

(b) ×  α1

 C(A)

= C( ×  α1

 A),

(c) +  α1

 I (A)

= I ( +  α1

 A),

(d) ×  α1

 I (A)

= I ( ×  α1

 A).

β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2

β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2

β1 γ1 α2 β2 γ2 β1 γ1 α2 β2 γ2

An IVIFS A over the universe E is called proper if there exists at least one x ∈ E for which 1 − sup M A (x) − sup N A (x) > 0. Theorem 4.4.6 Let A and B be two proper IVIFSs for which there exist y, z ∈ E such that inf M A (y) > 0 and inf N B (z) > 0. If C(A) ⊆ I (B), then there exist real numbers α1 , β1 , γ1 , δ1 , α2 , β2 , γ2 , δ2 ∈ [0, 1] such that α1 ≤ β1 , γ1 ≤ δ1 , α2 ≤ β2 , γ2 ≤ δ2 and J α1 ,β1 ,γ1 ,δ1 (A) ⊆ H α1 ,β1 ,γ1 ,δ1 (B). Proof Let C(A) ⊆ I (B) and let there exists at least one u ∈ E for which 1 − sup M A (u) − sup N A (u) > 0 and at least one v ∈ E for which 1 − sup M B (v) − sup N B (v) > 0. Therefore, for i = 1, 2, 0 < inf M A (y) ≤ K i ≤ ki and L i ≥ li ≥ inf N B (z) > 0,

94

4 Operators over IVIFSs

where

K 1 = sup inf M A (y), L 1 = inf inf N A (y), y∈E

y∈E

K 2 = sup sup M A (y), L 2 = inf sup N A (y), y∈E

y∈E

k1 = inf inf M B (y), l1 = sup inf N B (y), y∈E

y∈E

k2 = inf sup M B (y), l2 = sup sup N A (y), y∈E

y∈E

Let a = supx∈E (1 − sup M A (x) − sup N A (x)) ≥ 1 − sup M A (u) − sup N A (u) > 0, b = supx∈E (1 − sup M B (x) − sup N B (x)) ≥ 1 − sup M B (v) − sup N B (v) > 0, because A and B are proper IVIFSs. Let     1 2 α1 = min 1, k1 −K , β1 = min 1, k2 −K , 2a 2a L 1 +l1 , 2L 1 K 1 +k1 α2 = 2k1 ,   min 1, L 12b−l1 ,

γ1 =

γ2 =

δ2 = β2 =

L 2 +l2 , 2L 2 K 2 +k2 , 2k2

 δ2 = min 1,

L 2 −l2 2b



.

Then J α1 ,β1 ,γ1 ,δ1 (A) = {x, [inf M A (x) + α1 (1 − sup M A (x) − sup N A (x)), sup M A (x) + β1 (1 − sup M A (x) − sup N A (x))], [γ1 . inf N A (x), δ1 sup N A (x)] | x ∈ E}    1 = x, inf M A (x) + min 1, k1 −K (1 − sup M A (x) − sup N A (x)), 2a  k2 −K 2 

sup M A (x) + min 1, 2a (1 − sup M A (x) − sup N A (x)) ,

  L 1 +l1 L 2 +l2 inf N (x), sup N (x) | x ∈ E A A 2L 1 2L 2 and H α2 ,β2 ,γ2 ,δ2 (B) = {x, [α2 inf M B (x), β2 sup M B (x)], [inf N B (x) + γ2 (1 − sup M B (x) − sup N B (x)), sup N B (x) + δ2 (1 − sup M B (x) − sup N B (x))] | x ∈ E} 1 +k1 2 +k2 = {x, [ K2k inf M B (x), K2k sup M B (x)], 2 1 L 1 −l1  [inf N B (x) + min 1, 2b (1 − sup M B (x) − sup N B (x)),   sup N B (x) + min 1, L 22b−l2 (1 − sup M B (x) − sup N B (x))] | x ∈ E}.

4.4 Topological Operators over IVIFSs

95

From

K 1 +k1 2k1

≥

K 1 +k1 inf M B (x) − inf M A (x) 1   k2k 1 −K 1 − min 1, 2a .(1 − sup M A (x) − sup N A (x))

inf M B (x) − inf M A (x) −

k1 −K 1 (1 2a

− sup M A (x) − sup N A (x))

(from k1 ≥ inf M B (x) and a ≥ 1 − sup M A (x) − sup N A (x)) K 1 +k1 1 − K 1 − k1 −K = 0, 2 2 K 2 +k2 sup M B (x) − sup M A (x) 2   2k k2 −K 2 − min 1, 2a .(1 − sup M A (x) − sup N A (x)) K 2 +k2 2 sup M B (x) − sup M A (x) − k2 −K (1 − sup M A (x) − 2k2 2a

≥

≥

sup N A (x))

(from k2 ≥ sup M B (x) and a ≥ 1 − sup M A (x) − sup N A (x)) K 2 + k2 k2 − K 2 − k2 − = 0, 2 2   inf N A (x) − inf N B (x) − min 1, L 12b−l1 .(1 − sup M B (x) − sup N B (x)) ≥

L 1 +l1 2L 1

≥

L 1 +l1 2L 1

inf N A (x) − inf N B (x) −

L 1 −l1 (1 2b

− sup M B (x) − sup N B (x))

(from l1 ≥ inf N B (x) and b ≥ 1 − sup M B (x) − sup N B (x)) L 1 − l1 L 1 + l1 − l1 − = 0, 2 2   sup N A (x) − sup N B (x) − min 1, L 22b−l2 (1 − sup M B (x) − sup N B (x)) ≥

L 2 +l2 2L 2

≥

L 2 +l2 2L 2

sup N A (x) − sup N B (x) −

L 2 −l2 (1 2b

− sup M B (x) − sup N B (x))

(from L 2 ≥ sup N B (x) and b ≥ 1 − sup M B (x) − sup N B (x)) ≥

L 2 + l2 L 2 − l2 − l2 − =0 2 2

it follows that J α1 ,β1 ,γ1 ,δ1 (A) ⊆ H α2 ,β2 ,γ2 ,δ2 (B). In [6], for the IFS-case, a lot of extensions of the two quantifiers are described. An Open Problem is to construct their IVIFS-analogues.

96

4 Operators over IVIFSs

4.5 Level and Modal-Level Operators on IVIFSs There are two types of level operators—standard and special, that decreases the number of elements of the IVIFSs. In the present section, we discuss the operators of the first type and in the next section—of the second type. In [4], the level operators from the first type are introduced, but here, some of their properties will be given for a first time. The level operators, defined over IVIFS A are defined for α, β ∈ [0, 1] and α + β ≤ 1 by: Pα,β (A) = {x, [max(α, inf M A (x)), max(α, sup M A (x))], [min(β, inf N A (x)), min(β, sup N A (x))] | x ∈ E}, Q α,β (A) = {x, [min(α, inf M A (x)), min(α, sup M A (x))], [max(β, inf N A (x)), max(β, sup N A (x))] | x ∈ E}. Their extended forms are introduced in [4] for four numbers α, β, γ, δ ∈ [0, 1] so that α ≤ β, γ ≤ δ and β + δ ≤ 1: Pα,β,γ,δ (A) = {x, [max(α, inf M A (x)), max(β, sup M A (x))], [min(γ, inf N A (x)), min(δ, sup N A (x))] | x ∈ E}, Q α,β,γ,δ (A) = {x, [min(α, inf M A (x)), min(β, sup M A (x))], [max(γ, inf N A (x)), max(δ, sup N A (x))] | x ∈ E}. Obviously, for each IVIFS A and for every two real numbers α, β ∈ [0, 1] such that α + β ≤ 1: Pα,β (A) = Pα,α,β,β (A), Q α,β (A) = Q α,α,β,β (A). Theorem 4.5.1 For each IVIFS A and for all α, β, γ, δ, α , β , γ , δ ∈ [0, 1], such that α ≤ β, γ ≤ δ, α ≤ β , γ ≤ δ , β + δ ≤ 1 and β + δ ≤ 1: (a) ¬Pα,β,γ,δ (¬A) = Q γ,δ,α,β (A), (b) ¬Q α,β,γ,δ (¬A) = Pγ,δ,α,β (A), (c) Pα,β,γ,δ (Q α ,β ,γ ,δ (A)) = Q max(α,α ),max(β,β ),min(γ,γ ),min(δ,δ ) (Pα,β,γ,δ (A)), (d) Q α,β,γ,δ (Pα,β ,γ ,δ (A)) = Pmin(α,α ),min(β,β ),max(γ,γ ),max(δ,δ ) (Q α,β,γ,δ (A)).

4.5 Level and Modal-Level Operators on IVIFSs

(e) (f) (g) (h)

97

Pα,β,γ,δ (A ∩ B) = Pα,β,γ,δ (A) ∩ Pα,β,γ,δ (B), Pα,β,γ,δ (A ∪ B) = Pα,β,γ,δ (A) ∪ Pα,β,γ,δ (B), Q α,β,γ,δ (A ∩ B) = Q α,β,γ,δ (A) ∩ Q α,β,γ,δ (B), Q α,β,γ,δ (A ∪ B) = Q α,β,γ,δ (A) ∪ Q α,β,γ,δ (B).

Theorem 4.5.2 For each IVIFS A and for α, β, γ, δ ∈ [0, 1], such that α ≤ β, γ ≤ δ, β + δ ≤ 1: (a) (b) (c) (d)

C(Pα,β (A)) = Pα,β (C(A)), I (Pα,β (A)) = Pα,β (I (A)), C(Q α,β (A)) = Q α,β (C(A)), I (Q α,β (A)) = Q α,β (I (A)).

Theorem 4.5.3 For every two IFSs A and B, C(A) ⊂ I (B), iff there exist two real numbers α1 , α2 , β1 , β2 ∈ [0, 1] such that for i = 1, 2: αi + βi ≤ 1 and Pα1 ,α2 ,β1 ,β2 (A) ⊂ Q α1 ,α2 ,β1 ,β2 (B). Proof Let C(A) ⊂ I (B). Therefore for i = 1, 2, K i ≤ ki and L i ≥ li , where

K 1 = sup inf M A (y), L 1 = inf inf N A (y), y∈E

y∈E

K 2 = sup sup M A (y), L 2 = inf sup N A (y), y∈E

y∈E

k1 = inf inf M B (y), l1 = sup inf N B (y), y∈E

y∈E

k2 = inf sup M B (y), l2 = sup sup N B (y), y∈E

Let

α1 = α2 =

y∈E

K 1 +k1 , 2 K 2 +k2 , 2

β1 = β2 =

L 1 +l1 , 2 L 2 +l2 . 2

Then α1 + β1 = From

K 1 + k1 + L 1 + l2 K 2 + k2 + L 2 + l2 ≤ α2 + β2 = ≤ 1. 2 2 sup inf M A (y) ≤ y∈E

K 1 +k1 2

≤ inf inf M B (y), y∈E

98

4 Operators over IVIFSs K 2 +k2 2

≤ sup inf M B (y),

inf inf N A (y) ≥

y∈E

L 1 +l1 2

≥ inf inf N B (y),

inf sup N A (y) ≥

L 2 +l2 2

≥ sup inf N B (y),

sup sup M A (y) ≤ y∈E

y∈E

y∈E y∈E

y∈E

and for  max(sup inf M A (y), α1 ) = max y∈E

sup inf M A (y), y∈E



K 1 +k1 2

K 1 +k2 2

 =

K 1 +k1 2



= min(inf inf M B (y), α1 ), y∈E   K 1 +k2 = K 1 2+k1 max(sup sup M A (y), α2 ) = max sup sup M A (y), 2 = min

y∈E

= min

inf inf M B (y),

y∈E

y∈E

 sup sup M B (y), y∈E

K 1 +k1 2



= min(sup sup M B (y), α1 ). y∈E



 min(inf inf N A (y), β1 ) = min inf inf N A (y), L 12+l1 = L 12+l1 y∈E y∈E   = max sup inf N A (y), L 12+l1 = max(inf inf N A (y), β), y∈E

y∈E



L 2 +l2 2



= L 22+l2 min(inf sup N A (y), β2 ) = min inf sup N A (y), y∈E y∈E   L 2 +l2 = max(inf sup N A (y), β). = max sup sup N A (y), 2 y∈E

y∈E

Therefore Pα1 ,α2 ,β1 ,β2 (A) = Q α1 ,α2 ,β1 ,β2 (B), i.e., Pα1 ,α2 ,β1 ,β2 (A) ⊂ Q α1 ,α2 ,β1 ,β2 (B). For the opposite case, let there be α1 , α2 , β1 , β2 ∈ [0, 1] such that for i = 1, 2: αi + βi ≤ 1 and let Pα1 ,α2 ,β1 ,β2 (A) ⊂ Q α1 ,α2 ,β1 ,β2 (B). Then, for each x ∈ E max(sup inf M A (y), α1 ) ≤ min(inf inf M B (y), α1 ), y∈E

y∈E

max(sup sup M A (y), α2 ) ≤ min(inf sup M B (y), α2 ), y∈E

y∈E

min(inf inf N A (y), β1 ) ≥ max(inf inf N B (y), β1 ), y∈E

y∈E

min(inf sup N A (y), β2 ) ≥ max(inf sup N B (y), β2 ). y∈E

y∈E

4.5 Level and Modal-Level Operators on IVIFSs

Therefore:

99

sup inf M A (y) ≤ max(sup inf M A (y), α1 ) y∈E

y∈E

≤ min(inf inf M B (y), α1 ) ≤ inf inf M B (y), y∈E

y∈E

sup sup M A (y) ≤ max(sup sup M A (y), α2 ) y∈E

y∈E

≤ min(inf sup M B (y), α2 ) ≤ inf sup M B (y), y∈E

y∈E

inf inf N A (y) ≥ min(inf inf N A (y), β1 )

y∈E

y∈E

≥ max(inf inf N B (y), β1 ) ≥ inf inf N B (y), y∈E

y∈E

inf sup N A (y) ≥ min(inf sup N A (y), β2 )

y∈E

y∈E

≥ max(inf sup N B (y), β2 ) ≥ inf sup N B (y). y∈E

y∈E

i.e., C(A) ⊂ I (B). Theorem 4.5.4 For every IVIFS A and for every α, β, γ, δ ∈ [0, 1], for which max(α + β) + γ + δ ≤ 1 : (a) •

α,β,γ,δ Pε,ζ ((A))

= Pαε+γ,βζ+δ ( •

α,β,γ,δ A),

(b) •

α,β,γ,δ Pε,ζ ((A))

= Q αε+γ,βζ+δ ( •

α,β,γ,δ A).

Following [12], we introduce the first two new operators, defined over a given IVIFS A. They have the forms: α,β (A) = {x, [α inf M A (x), α sup M A (x)], H [ inf N A (x) + β − β inf N A (x), sup N A (x) + β − β sup N A (x)|x ∈ E},  Jα,β (A) = {x, [inf M A (x) + α − α inf M A (x), sup M A (x) + α − α sup M A (x)], [β inf N A (x), β sup N A (x)]|x ∈ E}, where α, β ∈ [0, 1] and α + β ≤ 1. We check that 0 ≤ α inf M A (x) ≤ α sup M A (x) ≤ 1, 0 ≤ inf N A (x) + β − β inf N A (x) ≤ sup N A (x) + β − β sup N A (x) ≤ 1, Z ≡ α sup M A (x) + sup N A (x) + β − β sup N A (x) ≤ α(1 − sup N A (x)) + sup N A (x) + β(1 − sup N A (x)) ≤ α + β + (1 − α − β) sup N A (x) ≤ α + β + 1 − α − β = 1

100

4 Operators over IVIFSs

i.e., the first definition is correct. Analogously, we check that the second definition is also correct. After this, we see that for each IVIFS A and for every α, β, γ, δ, ∈ [0, 1], so that α + β ≤ 1 and γ + δ ≤ 1: α,β (A) ⊆ A ⊆ Jγ,δ (A) H and the inequalities are transformed to equalities if and only if α = δ = 1 and hence β = γ = 0. Similarly to the rest modal operators, defined over IVIFSs, the two new operators can be also extended to the forms  α γ  (A) = {x, [α inf M A (x), β sup M A (x)], H β δ

[ inf N A (x) + γ − γ inf N A (x), sup N A (x) + δ − δ sup N A (x)|x ∈ E}, J α γ  (A) = {x, [inf M A (x) + α − α inf M A (x), sup M A (x) + β − β sup M A (x)], β δ

[γ inf N A (x), δ sup N A (x)]|x ∈ E},

where α, β, γ, δ ∈ [0, 1], α ≤ β, γ ≤ δ, β + δ ≤ 1. We can see again that both operators are defined correctly and for them inequalities  α γ  (A) ⊆ A ⊆ J  (A) H α γ β δ

β δ

hold for every α, β, γ, δ, α , β , γ , δ ∈ [0, 1] so that α ≤ β, γ ≤ δ, β + δ ≤ 1, α ≤ β , γ ≤ δ , β + δ ≤ 1. Obviously, for each IVIFS A:  α α,β (A) = H H

β α β

Jα,β (A) = J α

β α β

 (A),

 (A).

By this reason, below, we will work only with the two extended operators. Theorem 4.5.5 For each IVIFS A and for α, β, γ, δ ∈ [0, 1] so that α ≤ β, γ ≤ δ, β + δ ≤ 1:  α γ  (¬A) = J γ δ  (A), ¬H β δ

¬ J α

γ β δ

 (¬A)

α β

 γ =H

δ α β

 (A).

Theorem 4.5.6 For each IVIFS A and for α, β, γ, δ, α , β , γ , δ ∈ [0, 1] so that α ≤ β, γ ≤ δ, β + δ ≤ 1 and α ≤ β , γ ≤ δ , β + δ ≤ 1:

4.5 Level and Modal-Level Operators on IVIFSs

 α H

γ β δ

( J 

α γ β δ

 (A))

101

⊆ J α

γ β δ

( H 

α γ β δ

 (A)).

Theorem 4.5.7 For every two IFSs A and B, and for every α, β, γ, δ ∈ [0, 1] so that α ≤ β, γ ≤ δ, β + δ ≤ 1:  α C( H

 (A))

 α =H

 (C(A)),

 α I (H

 (A))

 α =H

 (I (A)),

C( J α

 (A))

= J α

 (C(A)),

I ( J α

 (A))

= J α

 (I (A)).

γ β δ γ β δ γ β δ γ β δ

γ β δ γ β δ

γ β δ γ β δ

Proof Let A be a given IVIFS and let α, β, γ, δ ∈ [0, 1]. Then  α C( H

γ β δ

 (A))

= C({x, [α inf M A (x), γ sup M A (x)],

[ inf N A (x) + β − β inf N A (x), sup N A (x) + δ − δ sup N A (x)|x ∈ E}) = {x, [sup α inf M A (y), sup γ sup M A (y)], y∈E

y∈E

[ inf (inf N A (y) + β − β inf N A (y)), inf (sup N A (y) + δ − δ sup N A (y))|x ∈ E} y∈E

y∈E

= {x, [α sup inf M A (y), γ sup sup M A (y)], y∈E

y∈E

[β + inf ((1 − β) inf N A (y)), δ + inf ((1 − δ) sup N A (y))|x ∈ E} y∈E

y∈E

= {x, [α sup inf M A (y), γ sup sup M A (y)], y∈E

y∈E

[β + (1 − β) inf inf N A (y), δ + (1 − δ) inf sup N A (y)|x ∈ E} y∈E

y∈E

= {x, [α sup inf M A (y), γ sup sup M A (y)], y∈E

y∈E

[ inf inf N A (y) + β − β inf inf N A (y), y∈E

y∈E

inf sup N A (y) + δ − δ inf sup N A (y)|x ∈ E}

y∈E

y∈E

 α =H

γ β δ

 ({x, [sup inf y∈E

M A (y), sup sup M A (y)], y∈E

[ inf inf N A (y), inf sup N A (y)|x ∈ E}) y∈E

y∈E

 α =H

γ β δ

 (C(A)).

The other assertions are proved in the same way. Below, we will discuss the four interval valued intuitionistic fuzzy operator from three different points of view.

102

4 Operators over IVIFSs

First, we interpret the four new operators from the point of view of the first type of interval valued intuitionistic fuzzy modal operators. As we discussed in Sect. 4.1, the composition of two -, ♦-, D-, F- and Goperators can be represented by only one of them, while this is impossible for the remaining operators, but now, we see that the following assertion is valid. Theorem 4.5.8 For each IVIFS A and for α, β, γ, δ, α , β , γ , δ ∈ [0, 1] so that α ≤ β, γ ≤ δ, β + δ ≤ 1 and α ≤ β , γ ≤ δ , β + δ ≤ 1:  α H

( H 

J α

( J 

γ β δ γ β δ

α γ β δ α γ β δ

 (A))

 =H αα

 (A))

= J α + α − αα

γ + γ − γγ ββ δ + δ − δδ

γγ β + β − ββ δδ

 (A),

 (A).

Proof We check sequentially:  α H

γ β δ

 α =H

γ β δ

( H 

 ({x, [α

α γ β δ

 (A))

inf M A (x), β sup M A (x)],

[ inf N A (x) + γ − γ inf N A (x), sup N A (x) + δ − δ sup N A (x)|x ∈ E}) = {x, [αα inf M A (x), ββ sup M A (x)], [ inf N A (x) + γ − γ inf N A (x) + γ − γ(inf N A (x) + γ − γ inf N A (x)), sup N A (x) + δ − δ sup N A (x) + δ − δ(sup N A (x) + δ − δ sup N A (x))|x ∈ E}) = {x, [αα inf M A (x), ββ sup M A (x)], [ inf N A (x) + γ + γ − γγ − (γ + γ − γγ ) inf N A (x), sup N A (x) + δ + δ − δδ − (δ + δ − δδ ) sup N A (x)|x ∈ E})   =H (A). αα γ + γ − γγ ββ δ + δ − δδ

The remaining assertion as well as the below ones can be proved similarly. Therefore, we see that these two operators satisfy the property of F- and Goperators. Moreover, for each IVIFS A:  0 0,1 (A) = H H

1 0 1

 (A)

= {x, [0, 0], [1, 1]|x ∈ E} = O ∗ ,

J1,0 (A) = J 1 0  (A) = {x, [1, 1], [0, 0]|x ∈ E} = E ∗ , 1 0

∗ ∗ similarly to operators H0,1 and J1,0 , respectively.

Theorem 4.5.9 For every two IFSs A and B, and for every α, β, γ, δ ∈ [0, 1] so that α ≤ β, γ ≤ δ, β + δ ≤ 1:

4.5 Level and Modal-Level Operators on IVIFSs

α,β (A ∩ B) = H α,β (A ∪ B) = H

103

α,β (A) ∩ H α,β (B), H α,β (A) ∪ H α,β (B), H

Jα,β (A ∩ B) = Jα,β (A) ∩ Jα,β (B), Jα,β (A ∪ B) = Jα,β (A) ∪ Jα,β (B),   α γ  (A ∩ B) = H  α γ  (A) ∩ H  α γ  (B), H β δ

β δ

β δ

 (A

 α ∪ B) = H

J 1

 (A

∩ B) = J 1 0  (A) ∩ J 1 0  (B),

J 1

 (A

∪ B) = J 1 0  (A) ∪ J 1 0  (B).

 α H

γ β δ

0 1 0 0 1 0

γ β δ

 (A)

 α ∪H

γ β δ

1 0

 (B),

1 0

1 0

1 0

Theorem 4.5.10 For every two IFSs A and B, and for every α, β, γ, δ, α , β , γ , δ ∈ [0, 1] so that α ≤ β, γ ≤ δ, β + δ ≤ 1 and α ≤ β , γ ≤ δ , β + δ ≤ 1:  α H

 (G 

J α

 (G 

γ β δ γ β δ

α γ β δ

 (A))

⊆ G  α

( H 

 (A))

⊇ G  α

( J 

α γ β δ

γ β δ γ β δ

α γ β δ

α γ β δ

 (A)),

 (A)).

Let ext1 , ext2 , ext3 , ext4 , ext5 , ext6 , ext7 , ext8 ∈ {inf, sup}. The most extended intuitionistic fuzzy modal operator from a first type has the form: 

ext1 ext2 ext3 ext4 ext5 ext6 ext7 ext8

X  a1



b1 c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

 (A)

≡ {x, [inf M X (x), sup M X (x)], [inf N X (x), sup N X (x)]|x ∈ E}, = {x, [a1 inf M A (x) + b1 (1 − ext1 M A (x) − c1 ext2 N A (x)), a2 sup M A (x) + b2 (1 − ext3 M A (x) − c2 ext4 N A (x))], [d1 inf N A (x) + e1 (1 − f 1 ext5 M A (x) − ext6 N A (x)), d2 sup N A (x) + e2 (1 − f 2 ext7 M A (x) − ext8 N A (x))]|x ∈ E}, where a1 , b1 , c1 , d1 , e1 , f 1 , a2 , b2 , c2 , d2 , e2 , f 2 ∈ [0, 1] and 0 ≤ inf M X (x) ≤ sup M X (x) ≤ 1, 0 ≤ inf N X (x) ≤ sup N X (x) ≤ 1, sup M X (x) + sup N X (x) ≤ 1.

104

4 Operators over IVIFSs

Now, we see directly that 

ext1 ext2 ext3 ext4 ext5 inf ext6 sup

α,β (A) = X  α H Jα,β (A) = X

γ β δ

ext1 ext2 ext3 ext4 ext5 inf ext6 sup

 (A)

= X α

 (A)

=X

α γ β δ

J α



0 r1 1 β 0 α 0 r2 1 β 0   inf ext1 sup ext2 ext ext ext ext  3 4 5 6 1 α 0 β 0 r3 1 α 0 β 0 r4



 H





(A), (A), 

0 r1 1 γ 0 β 0 r2 1 δ 0   inf ext1 sup ext2 ext ext ext ext  3 4 5 6 1 α 0 γ 0 r3 1 β 0 δ 0 r4

(A), (A),

where r1 , r2 , r3 , r4 ∈ [0, 1] are arbitrary numbers and ext1 , ext2 , ext3 , ext4 , ext5 , ext6 ∈ {inf, sup} are one of the two symbols without sense which. Therefore, the new operators have similar X -representation as the rest of the extended modal-type operators from first type. Second, we interpret the four new operators from point of view of the second type of interval valued intuitionistic fuzzy modal operators. The two new (more extended) modal operators have the following representations:  α H

 (A)

= ◦

J α

 (A)

= ◦

γ β δ

γ β δ

( ext  1

ext2 )  α 1−γ 0 γ 0 0 β 1−δ 0 δ 0 0

( ext  1

ext2 )  1−α γ α 0 0 0 1−β δ β 0 0 0

A, A,

where ext1 , ext2 ∈ {inf, sup} are one of the two symbols without sense which. Third, we interprete the four new operators from point of view of the level operators. Obviously, for every IVIFS A and for α, β, γ, δ ∈ [0, 1], α ≤ β, γ ≤ δ and β + δ ≤ 1: P α,β,γ,δ (A) = A ∪ {x, [α, β], [γ, δ]|x ∈ E}, Q α,β,γ,δ (A) = A ∩ {x, [α, β], [γ, δ]|x ∈ E}, Q α,β,γ,δ (A) ⊂ A ⊂ P α,β,γ,δ (A). Therefore, it will be suitable to denote both operators as follows: ∪ Oα,β,γ,δ (A) = P α,β,γ,δ (A), ∩ Oα,β,γ,δ (A) = Q α,β,γ,δ (A).

Now, we can define two new intuitionistic fuzzy level operators on the basis of operations “+” and “×”, defined in Sect. 4.2: + (A) = A + {x, [α, β], [γ, δ]|x ∈ E}, Oα,β,γ,δ × (A) = A.{x, [α, β], [γ, δ]|x ∈ E}. Oα,β,γ,δ

4.5 Level and Modal-Level Operators on IVIFSs

105

We see immediately that: + (A) = J α Oα,β,γ,δ

 (A),

×  α Oα,β,γ,δ (A) = H

 (A),

γ β δ γ β δ

while

α,β (A) = O × H α,α,β,β (A), +  Jγ,δ (A) = O (A). α,α,β,β

Having in mind [1–3], for every intuitionistic fuzzy conjunction and disjunction we can define new interval valued intuitionistic fuzzy operation and at the moment, this is an Open Problem. Therefore, the four new operators have level operator’s behaviour. Theorem 4.5.11 For each IVIFS A, for α, β, γ, δ, ε, ζ, η, θ ∈ [0, 1], so that α ≤ β, γ ≤ δ, β + δ ≤ 1, ε ≤ ζ, η ≤ θ, ζ + θ ≤ 1:  α H

 (P

ε,ζ,η,θ (A))

 α = P αε,βζ,(1−γ)η+γ,(1−δ)θ+δ ( H

 (A))

 α H

 (Q

ε,ζ,η,θ (A))

 α = Q αε,βζ,(1−γ)η+γ,(1−δ)θ+δ ( H

 (A))

J α

 (P

ε,ζ,η,θ (A))

= P αε,βζ,(1−γ)η+γ,(1−δ)θ+δ ( J α

 (A))

J α

 (Q

ε,ζ,η,θ (A))

= Q αε,βζ,(1−γ)η+γ,(1−δ)θ+δ ( J α

 (A))

γ β δ

γ β δ γ β δ

γ β δ

γ β δ γ β δ

γ β δ γ β δ

Proof Let the IVIFS A and α, β, γ, δ, ε, ζ, η, θ be given, so that they satisfy the above conditions. Then for (a) we obtain:  α H

γ β δ

 α =H

γ β δ

 (P

 ({x, [max(ε, inf

ε,ζ,η,θ (A))

M A (x)), max(ζ, sup M A (x))],

[ min(η, inf N A (x)), min(θ, sup N A (x))] | x ∈ E}) = {x, [α max(ε, inf M A (x)), β max(ζ, sup M A (x))], [ min(η, inf N A (x)) + γ − γ min(η, inf N A (x)), min(θ, sup N A (x)) + δ − δ min(θ, sup N A (x))] | x ∈ E} = {x, [max(αε, α inf M A (x)), max(βζ, β sup M A (x))], [min((1 − γ)η + γ, (1 − γ) inf N A (x) + γ), min((1 − δ)θ + δ, (1 − δ) sup N A (x) + δ)] | x ∈ E} = P αε,βζ,(1−γ)η+γ,(1−δ)θ+δ ({x, [α inf M A (x), β sup M A (x)], [inf N A (x) + γ − γ inf N A (x), sup N A (x) + δ − δ sup N A (x)|x ∈ E})  α γ  (A)). = P αε,βζ,(1−γ)η+γ,(1−δ)θ+δ ( H β δ

106

4 Operators over IVIFSs

It can be directly seen that for each IVIFS A and for α, β, γ, δ ∈ [0, 1], α ≤ β, γ ≤ δ and β + δ ≤ 1:  α H

γ β δ

 (A)

⊆ Q α,β,γ,δ (A) ⊆ A ⊆ P α,β,γ,δ (A) ⊆ J α

γ β δ

 (A).

Now, we can calculate for every IVIFS A and for α, β, γ, δ ∈ [0, 1], α ≤ β, γ ≤ δ and β + δ ≤ 1:  α H

γ β δ

 (A)@ J 

α γ β δ

 (A)

= {x, [α inf M A (x), β sup M A (x)], [inf N A (x) + γ − γ inf N A (x), sup N A (x) + δ − δ sup N A (x)|x ∈ E}, @{x, [inf M A (x) + α − α inf M A (x), sup M A (x) + β − β sup M A (x)], [γ inf N A (x), δ sup N A (x)]|x ∈ E},   = x, α inf M A (x)+inf M A2(x)+α−α inf M A (x) , β sup M A (x)+sup M A (x)+β−β sup M A (x) , 2  inf N A (x)+γ−γ inf N A (x)+γ inf N A (x) , 2    supN A (x)+δ−δ sup N A (x)+δ sup N A (x) x ∈ E    2 = x, inf M A2(x)+α , sup M A2(x)+β ,    γ+γ inf N A (x) δ+δ sup N A (x)  x ∈ E ,  2 2 A@{x, [α, β], [γ, δ]|x ∈ E}. From the above discussion we see that the four new operators are simultaneously modal as well as level operators. By this reason, we really can call them modal-level operators. Finally, following the above notation, we can denote: A@{x, [α, β], [γ, δ]|x ∈ E} = O@α

γ β δ

Fig. 4.11 Interrelation between the N -operators

 (A).

4.6 Level Operator That Decrease the Number of Elements of the IVIFSs

107

4.6 Level Operator That Decrease the Number of Elements of the IVIFSs Let α, β ∈ [0, 1] be fixed numbers for which α + β ≤ 1 and let Nα1 (A) = {x, M A (A), N A (x) | x ∈ E & inf M A (x) ≥ α}, β

N1 (A) = {x, M A (A), N A (x) | x ∈ E & sup N A (x) ≤ β}, 1 Nα,β (A) = {x, M A (A), N A (x) | x ∈ E & inf M A (x) ≥ α& sup N A (x) ≤ β},

Nα2 (A) = {x, M A (A), N A (x) | x ∈ E & sup M A (x) ≥ α}, β

N2 (A) = {x, M A (A), N A (x) | x ∈ E & inf N A (x) ≤ β}, 2 Nα,β (A) = {x, M A (A), N A (x) | x ∈ E & sup M A (x) ≥ α& inf N A (x) ≤ β},

Nα3 (A) = {x, M A (A), N A (x) | x ∈ E & inf M A (x) ≤ α}, β

N3 (A) = {x, M A (A), N A (x) | x ∈ E & sup N A (x) ≥ β}, 3 Nα,β (A) = {x, M A (A), N A (x) | x ∈ E & inf M A (x) ≤ α& sup N A (x) ≥ β},

Nα4 (A) = {x, M A (A), N A (x) | x ∈ E & sup M A (x) ≤ α}, β

N4 (A) = {x, M A (A), N A (x) | x ∈ E & inf N A (x) ≥ β}, 4 Nα,β (A) = {x, M A (A), N A (x) | x ∈ E & sup M A (x) ≤ α& inf N A (x) ≥ β}.

We will call the above sets sets of α-, β- and (α, β )-level generated by A, respectively. From the above definitions it directly follows that for all IVIFS A and for all α, β ∈ [0, 1], such that α + β ≤ 1: i (A) ⊂ Nαi (A) ⊂ A Nα,β β

i (A) ⊂ Ni (A) ⊂ A Nα,β

for i = 1, 2, 3 and 4, where the relation “⊂” is a relation in the set-theoretical sense. Moreover, for all IVIFS A and for all α, β ∈ [0, 1]: β

i (A) = Nαi (A) ∩ Ni (A) Nα,β

for i = 1, 2, 3 and 4, and A = Nαi (A) ∪ Nαi+2 (A) β

β

= Ni (A) ∪ Ni+2 (A) i+2 i (A) ∪ Nα,β (A) = Nα,β

108

4 Operators over IVIFSs

for i = 1 and 2. Now, following [6], we introduce four different extensions of the above operator. It is seen directly that N -operator decreases the number of elements of the given IVIFS A, preserving only these elements of universe E that have degrees, satisfying the respective condition. By this reason, we call the new operator Level operator that decreases the number of elements of the IVIFSs. First, let A be an IVIFS and B be IFS defined over the same universe. Then, N B1 (A) = {x, M A (x), N A (x)|(x ∈ E)&(inf M A (x) ≥ μ B (x)) &(sup N A (x) ≤ ν B (x))}, N B2 (A) = {x, M A (x), N A (x)|(x ∈ E)&(sup M A (x) ≥ μ B (x)) N B3 (A)

&(inf N A (x) ≤ ν B (x))}, = {x, M A (x), N A (x)|(x ∈ E)&(inf M A (x) ≤ μ B (x)) &(sup N A (x) ≥ ν B (x))},

N B4 (A) = {x, M A (x), N A (x)|(x ∈ E)&(sup M A (x) ≤ μ B (x)) &(inf N A (x) ≥ ν B (x))}. Obviously, for each IVIFS A, A = N O1 ∗ (A) = N O2 ∗ (A) = N E3 ∗ (A) = N E4 ∗ (A), where sets O ∗ and E ∗ are defined in Sect. 3.2. Theorem 4.6.1 For every two IVIFSs A and B (a) A = N B1 (A) and A = N B2 (A) iff B ⊂ A, (b) A = N B3 (A) and A = N B4 (A) iff A ⊂ B. Theorem 4.6.2 For every three IVIFSs A, B, C, 1 (a) NC1 (N B1 (A)) = N B∪C (A), 2 (A), (b) NC2 (N B2 (A)) = N B∪C 3 (A), (c) NC3 (N B3 (A)) = N B∩C 4 (A). (d) NC4 (N B4 (A)) = N B∩C

Theorem 4.6.3 For every three IVIFSs A, B, C, (a) NC1 (A ∩ B) = NC1 (A) ∩∗ NC1 (B), (b) NC2 (A ∩ B) = NC2 (A) ∩∗ NC2 (B), (c) NC3 (A ∪ B) = NC3 (A) ∪∗ NC3 (B), (d) NC3 (A ∪ B) = NC3 (A) ∪∗ NC3 (B),

4.6 Level Operator That Decrease the Number of Elements of the IVIFSs

109

where here and below ∪∗ and ∩∗ are set theoretic operations “union” and “intersection”. Theorem 4.6.4 For every two IVIFSs A and B, N B∗ (A) = ¬N¬B (¬A). Theorem 4.6.5 For every two IVIFSs A and B, (a) NC(B) (A) = (b) N I∗(B) (A) =

∩∗

N P C(A),

P⊂C(B)

∪∗

I (B)⊂P

(c) N B (I (A)) = (d) N B∗ (C(A)) =

∪∗

N P∗ (A),

P⊂I (A)

N B (P),

∩∗ C(A)⊂P

N B∗ (P).

Second, let f 1 , f 2 , f 3 , f 4 : [0, 1] → [0, 1] be functions. Define f , f , f , f4

1 2 3 Nα,β,γ,δ

(A) = {x, M A (x), N A (x)|x ∈ E & f 1 (inf M A (x)) ≥ α

& f 2 (sup M A (x)) ≥ β & f 3 (inf N A (x)) ≤ γ & f 4 (sup N A (x)) ≤ β}. f ,f ,f ,f

1 2 3 4 Obviously, when f 1 , f 2 , f 3 , f 4 are the identity, Nα,β,γ,δ (A) coincides with 1 Nα,β,γ,δ (A). Third, we extend the second modification, replacing the f -functions with a predicate p, i.e.,

p

Nα,β,γ,δ (A) = {x, M A (x), N A (x)|x ∈ E & p((M A (x), N A (x)), (α, β, γ, δ))}. When predicate p has the form p(([a, b], [c, d]), (α, β, γ, δ)) = “ f 1 (a) ≥ α" & “ f 2 (b) ≥ β" & “ f 3 (c) ≥ γ" & “ f 4 (d) ≥ δ", p

f ,f ,f ,f

1 2 3 4 (A). then Nα,β,γ,δ (A) coincides with Nα,β,γ,δ In the second and third cases, numbers α, β, γ, δ are given in advance and fixed. Now, we introduce extensions for each of the second and third cases. As in the first case, substitute the four given constants α, β, γ and δ with elements of a given IVIFS B. The two new N -operators have the respective forms:

110

4 Operators over IVIFSs f , f2 , f3 , f4

N B1

(A) = {x, M A (x), N A (x)|x ∈ E

& f 1 (inf M A (x)) ≥ inf M B (x) & f 2 (sup M A (x)) ≥ sup M B (x) & f 3 (inf N A (x)) ≤ inf N B (x) & f 4 (sup N A (x)) ≤ sup N B (x)} and

p

N B (A) = {x, M A (x), N A (x)|x ∈ E & p((M A (x), N A (x)), (M B (x), N B (x)))}.

Theorem 4.6.6 For every three IVIFSs A, B and C, (a) if B ⊆ C, then N B (A) ⊇ NC (A), (b) if A ⊆ B, then

NC (A) ⊆ NC (B).

Theorem 4.6.7 For every two IVIFSs A and B, ¬p

p

A = N B (A) ∪ N B (A), where ¬ p is the negation of predicate p. For every two IVIFSs A and B, and for every n predicates p1 , p2 , . . . , pn , p ∨ p2 ∨...∨ pn

N B1

p & p2 &...& pn

N B1

(A) = (A) =

∪ N Bpi (A).

1≤i≤n

∩ N Bpi (A).

1≤i≤n

The same inclusions are valid for the particular cases, discussed above. Figure 4.11 describes the interrelation between the operator Nα,β,γ,δ and its generalizations. Here, the symbol X → Y means that the operator X generalizes operator Y .

4.7 Two Other Types of Operators In this section, we discuss four other types of operators. The first of them (four different operators) and the second of them (two operators) were described in [4], the third type—one operator, called “Shrinking operator”, is introduced by E. Szmidt, J. Kacprzyk and the author in [14] and the fourth type—one operator, called “Weight operator” is an analogue of an operator with the same name, defined over IFSs. The last operator is introduced here for the first time. First, we introduce the following four operators, each one of which maps an IFS to every IVIFS.

4.7 Two Other Types of Operators

111

Fig. 4.12 Second geometrical interpretation of operators ∗1 , ∗2 , ∗3 , ∗4

Fig. 4.13 Third geometrical interpretation of operators ∗1 , ∗2 , ∗3 , ∗4

Let A be an IVIFS. Then we will define: ∗1 A = {x, inf M A (x), inf N A (x) | x ∈ E}, ∗2 A = {x, inf M A (x), sup N A (x) | x ∈ E}, ∗3 A = {x, sup M A (x), inf N A (x) | x ∈ E}, ∗4 A = {x, sup M A (x), sup N A (x) | x ∈ E}. The geometrical interpretation of these operators is shown on Figs. 4.12, 4.13 and 4.14.

112

4 Operators over IVIFSs

Fig. 4.14 Fourth geometrical interpretation of operators ∗1 , ∗2 , ∗3 , ∗4

Fig. 4.15 Second geometrical interpretation of operator S

Therefore, for all IVIFS A:

∗2 A ⊂ ∗1 A, ∗4 A ⊂ ∗3 A, ∗1 A ⊂ ∗4 A, ∗4 A ⊂♦ ∗1 A.

Theorem 4.7.1 For every two IVIFSs A and B and for 1 ≤ i ≤ 4: (a) (b) (c) (d) (e) (f)

∗i (A ∩ B) = ∗i A ∩ ∗i B, ∗i (A ∪ B) = ∗i A ∪ ∗i B, ∗i (A + B) = ∗i A + ∗i B, ∗i (A.B) = ∗i A. ∗i B, ∗i (A@B) = ∗i A@ ∗i B, ∗i (A ×1 B) = ∗i A ×1 ∗i B,

4.7 Two Other Types of Operators

(g) (h) (i) (j)

∗i (A ×2 ∗i (A ×3 ∗i (A ×4 ∗i (A ×5

B) = ∗i A ×2 ∗i B, B) = ∗i A ×3 ∗i B, B) = ∗i A ×4 ∗i B, B) = ∗i A ×5 ∗i B.

Proof (b) ∗1 (A ∪ B) = ∗1 {x, [max(inf M A (x), inf M B (x)), max(sup M A (x), sup M B (x))], [min(inf N A (x), inf N B (x)), min(sup N A (x), sup N B (x))] | x ∈ E} = {x, max(inf M A (x), inf M B (x)), min(inf N A (x), inf N B (x)) | x ∈ E} = {x, inf M A (x), inf N A (x) | x ∈ E} ∪ {x, inf M B (x)), inf N B (x) | x ∈ E} = ∗1 A ∪ ∗1 B. (a), (c)–(j) are proved analogously. Theorem 4.7.2 For each IVIFS A: (a) ∗1

A = ∗1 A,

(b) ∗2

A ⊂ ∗2 A,

(c) ∗3

A = ∗3 A,

(d) (e) (f) (g) (h) (i) (j) (k) (l)

∗4 A ⊂ ∗4 A, ∗1 ♦A = ∗1 A, ∗2 ♦A = ∗2 A, ∗3 ♦A ⊃ ∗3 A, ∗4 ♦A ⊃ ∗4 A, ¬ ∗1 ¬A = ∗1 A, ¬ ∗2 ¬A = ∗3 A, ¬ ∗3 ¬A = ∗2 A, ¬ ∗4 ¬A = ∗4 A.

Theorem 4.7.3 For each IVIFS A and for α, β, γ, δ ∈ [0, 1], α ≤ β, γ ≤ δ: (a) (b) (c) (d) (e)

∗1 Fα,β (A) = ∗1 A, for α + β ≤ 1, ∗2 Fα,β (A) ⊂ ∗2 A, for α + β ≤ 1, ∗3 Fα,β (A) ⊃ ∗3 A, for α + β ≤ 1, ∗4 Fα,β (A) = ∗4 A, for α + β ≤ 1, ∗1 F α,β,γ,δ (A) = ∗1 A, for β + δ ≤ 1,

113

114

4 Operators over IVIFSs

(f) ∗2 F α,β,γ,δ (A) ⊃

∗2 A, for β + δ ≤ 1,

(g) ∗2 F α,β,γ,δ (A) ⊂♦ ∗2 A, for β + δ ≤ 1, (h) ∗3 F α,β,γ,δ (A) ⊃ ∗3 A, for β + δ ≤ 1, (i) ∗3 F α,β,γ,δ (A) ⊂♦ ∗3 A, for β + δ ≤ 1, (j) ∗4 F α,β,γ,δ (A) = ∗4 A, for β + δ ≤ 1. Theorem 4.7.4 For each IVIFS A, for α, β, γ, δ ∈ [0, 1] and for 1 ≤ i ≤ 4: (a) (b) (c) (d) (e)

∗i G α,β (A) = G α,β (∗i A), ∗1 G α,β,γ,δ (A) = G α,γ (∗1 A), ∗2 G α,β,γ,δ (A) = G α,δ (∗2 A), ∗3 G α,β,γ,δ (A) = G β,γ (∗3 A), ∗4 G α,β,γ,δ (A) = G β,δ (∗4 A).

We must note that the first G-operator in the above relations is an operator over IVIFSs and the second one is an operator over IFSs. Theorem 4.7.5 For each IVIFS A, for α, β, γ, δ ∈ [0, 1] and for 1 ≤ i ≤ 4: (a) ∗i Hα,β (A) ⊂ ∗i A, ∗ (b) ∗i Hα,β (A) ⊂ ∗i A, (c) ∗i H α,β,γ,δ (A) ⊂ ∗i A, ∗

(d) ∗i H α,β,γ,δ (A) ⊂ ∗i A, (e) ∗i Jα,β (A) ⊃ ∗i A, ∗ (f) ∗i Jα,β (A) ⊃ ∗i A, (g) ∗i J α,β,γ,δ (A) ⊃ ∗i A, ∗

(h) ∗i J α,β,γ,δ (A) ⊃ ∗i A. Theorem 4.7.6 For each IVIFS A and for α, β, γ, δ ∈ [0, 1] so that α ≤ β, γ ≤ δ, β + δ ≤ 1.:  α (a) ∗1 H

 (A)

= Hα,γ (∗1 A),

 α (b) ∗2 H

 (A)

= Hα,δ (∗2 A),

 α (c) ∗3 H

 (A)

= Hβ,γ (∗3 A),

 α (d) ∗4 H

 (A)

= Hβ,δ (∗4 A),

γ β δ γ β δ

γ β δ γ β δ

(e) ∗1 J α

 (A)

= Jα,γ (∗1 A),

(f) ∗2 J α

 (A)

= Jα,δ (∗2 A),

γ β δ γ β δ

4.7 Two Other Types of Operators

(g) ∗3 J α

 (A)

= Jβ,γ (∗3 A),

(h) ∗4 J α

 (A)

= Jβ,δ (∗4 A).

γ β δ γ β δ

115

Theorem 4.7.7 For each IVIFS A, for all two α, β ∈ [0, 1] and for 1 ≤ i ≤ 4: (a) ∗i Pα,β (A) = Pα,β (∗i A), (b) ∗i Q α,β (A) = Q α,β (∗i A). Thus, operators which map an IFS to every IVIFS were constructed. On the other hand, let A be an IFS. Then the following two operators (of second type) can be defined: 1 (A) = {B | B = {x, M B (x), N B (x) | x ∈ E}& (∀x ∈ E)(sup M B (x) + sup N B (x) ≤ 1)& (∀x ∈ E)(inf M B (x) ≥ μ A (x) & sup N B (x) ≤ ν A (x))}, 2 (A) = {B | B = {x, M B (x), N B (x) | x ∈ E}& (∀x ∈ E)(sup M B (x) + sup N B (x) ≤ 1)& (∀x ∈ E)(sup M B (x) ≤ μ A (x) & inf N B (x) ≥ ν A (x))}. Theorem 4.7.8 For each IFS A: (a) 1 (A) = {B | A ⊂ ∗2 B} ⊂ {B | A ⊂ ∗1 B}, (b) 2 (A) = {B | ∗4 B ⊂ A} ⊂ {B | ∗3 B ⊂ A}. Proof (a) Let C ∈ 1 (A). Then: C = {x, MC (x), NC (x) | x ∈ E} and

(∀x ∈ E)(sup MC (x) + sup NC (x) ≤ 1)& (∀x ∈ E)(inf MC (x) ≥ μ A (x)& sup NC (x) ≤ ν A (x)).

Then ∗2 C is an IFS and A ⊂ ∗2 C, i.e. C ∈ {B | A ⊂ ∗2 B}. The opposite direction is proved analogously. The validity of the second relation ( ⊂ ) was shown above. (b) is proved analogously. Theorem 4.7.9 For each IFS A: (a) 1 (A) is a filter, (b) 2 (A) is an ideal.

116

4 Operators over IVIFSs

Proof (a) Following [17] we will prove that: • if B ∈ 1 (A) and B ⊂ C, then C ∈ 1 (A); • if B, C ∈ 1 (A) then B ∩ C ∈ 1 (A). Let B ∈ 1 (A) and B ⊂ C. Then A ⊂ ∗2 B ⊂ ∗2 C. Hence C ∈ 1 (A). Let B, C ∈ 1 (A). Then A ⊂ ∗2 B and A ⊂ ∗2 C. Hence A ⊂ ∗2 B ∩ ∗2 C. From Theorem 4.7.1 (a) there follows that A ⊂ ∗2 (B ∩ C), i.e. B ∩ C ∈ 1 (A). (b) Now we will prove that: • if B ∈ 2 (A) and C ⊂ B, then C ∈ 2 (A) • if B ∪ C ∈ 2 (A) then B, C ∈2 (A). Let B ∈ 2 (A) and C ⊂ B. Then ∗4 C ⊂ ∗4 B ⊂ A. Hence C ∈ 2 (A). Let B ∪ C ∈ 2 (A). Then ∗4 (B ∪ C) ⊂ A. From Theorem 4.7.1 (b) it follows that ∗4 B ∪ ∗4 C ⊂ A. Hence ∗4 B, ∗4 C ⊂ A, i.e. ∗4 B, ∗4 C ∈ 2 (A).

4.8 Simplest and (α, β)-Shrinking Operators over IVIFSs Let A be an IVIFS. Then we define

  M A (x) inf N A (x)+sup N A (x) |x ∈ E , S(A) = x, inf M A (x)+sup 2 2   inf M A (x)+sup M A (x) inf M A (x)+sup M A (x) , , = x, 2 2 

 inf N A (x)+sup N A (x) inf N A (x)+sup N A (x) |x ∈ E . , 2 2 Obviously, for each IFS A: S(A) = A. The three geometrical interpretations of the new operator are given on Figs. 4.15, 4.16 and 4.17. Fig. 4.16 Third geometrical interpretation of operator S

4.8 Simplest and (α, β)-Shrinking Operators over IVIFSs

117

Fig. 4.17 Fourth geometrical interpretation of operator S

Fig. 4.18 Geometrical interpretation of operators W ∗

Theorem 4.8.1 For each IVIFS A: (a) S(S(A)) = S(A), (b) ¬S(¬A) = S(A). Proof Let A be an IVIFS. Then for (a) we obtain S(S(A)) = S 



 x,

inf M A (x)+sup M A (x) inf M A (x)+sup M A (x) , 2 2

inf N A (x)+sup N A (x) inf N A (x)+sup N A (x) , 2 2



=





x,



|x ∈ E

inf M A (x)+sup M A (x) inf M A (x)+sup M A (x) , 2 2

inf N A (x)+sup N A (x) inf N A (x)+sup N A (x) , 2 2









,

,

|x ∈ E = S(A).

118

4 Operators over IVIFSs

For (b) we obtain ¬S(¬A) = ¬S (¬{x, N A (x), M A (x) | x ∈ E}) 

 N A (x) inf M A (x)+sup M A (x) = ¬ x, inf N A (x)+sup |x ∈ E , 2 2 

 M A (x) inf N A (x)+sup N A (x) = x, inf M A (x)+sup |x ∈ E = S(A). , 2 2 The proof of the next assertions is made in a similar way. Theorem 4.8.2 For every two IVIFSs A and B: (a) S(A ∪ B) ⊇ S(A) ∪ S(B), (b) S(A ∩ B) ⊆ S(A) ∩ S(B), (c) S(A@B) = S(A)@S(B). Theorem 4.8.3 For each IVIFS A: (a) S( A) = S(A), (b) S(♦A) = ♦S(A). We can see that S(Dα (A)) = S({x, [inf M A (x), sup M A (x) +α.(1 − sup M A (x) − sup N A (x))], [inf N A (x), sup N A (x) + (1 − α) .(1 − sup M A (x) − sup N A (x))] | x ∈ E})   M A (x)−sup N A (x)) , = x, inf M A (x)+sup M A (x)+α.(1−sup 2 inf M A (x)+sup M A (x)+α.(1−sup M A (x)−sup N A (x)) , 2  inf N A (x)+sup N A (x)+(1−α).(1−sup M A (x)−sup N A (x)) , 2

 inf N A (x)+sup N A (x)+(1−α).(1−sup M A (x)−sup N A (x)) |x∈E 2 =



 x,

inf M A (x)+sup M A (x) inf M A (x)+sup M A (x) , 2 2

 +α. 1 − 

inf M A (x)+sup M A (x) 2

−

inf N A (x)+sup N A (x) 2

inf N A (x)+sup N A (x) inf N A (x)+sup N A (x) , 2 2



  M A (x) inf M A (x)+sup M A (x) , = Dα ( x, inf M A (x)+sup , 2 2 

 inf N A (x)+sup N A (x) inf N A (x)+sup N A (x) |x ∈ E , 2 2 = Dα (S(A)).

,

4.8 Simplest and (α, β)-Shrinking Operators over IVIFSs

119

On the other hand, for the operator G α,β , defined (see, [4]) for every α, β ∈ [0, 1] by: G α,β (A) = {x, [α. inf M A (x), α. sup M A (x)], [β. inf N A (x), β. sup N A (x)] | x ∈ E}, the equality S(G α.β (A)) = G α.β (S(A)) is valid. Theorem 4.8.4 For each IVIFS A: (a) S(C(A)) ⊆ C(S(A)), (b) S(I (A)) ⊇ I (S(A)). Theorem 4.8.5 For each IVIFS A: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r)

∗1 A ⊂ ,inf S(A) ⊂♦,inf ∗1 A, ∗1 A ⊂ ,sup S(A) ⊂♦,sup ∗1 A, ∗1 A ⊂ S(A) ⊂♦ ∗1 A, ∗2 A ⊂ ,inf S(A), ∗2 A ⊂ ,sup S(A), ∗2 A ⊂ S(A), ∗2 A ⊂♦,inf S(A), ∗2 A ⊂♦,sup S(A), ∗2 A ⊂♦ S(A), S(A) ⊂ ,inf ∗4 A ⊂♦,inf S(A), S(A) ⊂ ,sup ∗4 A ⊂♦,sup S(A), S(A) ⊂ ∗4 A ⊂♦ S(A), S(A) ⊂ ,inf ∗3 A, S(A) ⊂ ,sup ∗3 A, S(A) ⊂ ∗3 A, S(A) ⊂♦,inf ∗3 A, S(A) ⊂♦,sup ∗3 A, S(A) ⊂♦ ∗3 A.

Let A be an IVIFS and α, β ∈ [0, 0.5]. Then we define the (α, β)-shrinking operator by Sα,β (A) = {x, α(inf M A (x) + sup M A (x)), β(inf N A (x) + sup N A (x)) |x ∈ E} = {x, [α(inf M A (x) + sup M A (x)), α(inf M A (x) + sup M A (x))], [β(inf N A (x) + sup N A (x)), β(inf N A (x) + sup N A (x))]|x ∈ E}. Obviously,

120

4 Operators over IVIFSs

S(A) = S0.5,0.5 (A). First, we must check that the definition is correct. Indeed, α(inf M A (x) + sup M A (x)) + β(inf N A (x) + sup N A (x)) ≤ 2α sup M A (x) + 2β sup N A (x) ≤ sup M A (x) + sup N A (x) ≤ 1. Theorem 4.8.6 For each IVIFS A and for every α, β ∈ [0, 0.5]: (a) Sα,β (Sα,β (A)) = S2α2 ,2β 2 (A), (b) ¬Sα,β (¬A) = Sβ,α (A). Theorem 4.8.7 For every two IVIFSs A and B, and for every α, β ∈ [0, 0.5]: (a) Sα,β (A ∪ B) ⊇ Sα,β (A) ∪ Sα,β (B), (b) Sα,β (A ∩ B) ⊆ Sα,β (A) ∩ Sα,β (B), (c) Sα,β (A@B) = Sα,β (A)@Sα,β (B). Theorem 4.8.8 For each IVIFS A and for every two real numbers α, β ∈ [0, 1]: (a) if α ≤ β, then Sα,β ( A) = Sα,β (A), (b) if α ≥ β, then Sα,β (♦A) = ♦Sα,β (A). Theorem 4.8.9 For each IVIFS A and for every α, β ∈ [0, 0.5]: (a) Sα,β (C(A)) ⊆ C(Sα,β (A)), (b) Sα,β (I (A)) ⊇ I (Sα,β (A)). Theorem 4.8.10 For each IVIFS A, for every α, β ∈ [0, 0.5], and for every γ, δ ∈ [0, 1] : Sα,β (G γ,δ (A)) = Sαγ,βδ (A) = G γ,δ (Sα,β (A)). Similar equality is not valid for the other modal type of operators.

4.9 Weight-Center Operator In [13], Adrian Ban and the author introduced “weight-center operator” over a given IFS A (see, also [6] by:

W (A) =

⎧  ⎪ ⎨ ⎪ ⎩

 x,



μ A (y)

y∈E

car d(E)

,

ν A (y) 

y∈E

car d(E)

|x ∈ E

⎫ ⎪ ⎬ ⎪ ⎭

,

where car d(E) is the number of the elements of a finite set E. For the continuous case, the “summation” may be replaced by integrals over E.

4.9 Weight-Center Operator

121

Now, we introduce this operator for IVIFS-case, as follows ⎧ ⎨

⎡ x, ⎣



inf M A (y)

sup M A (y)

⎤

⎦ , , car d(E) car d(E)  ⎫ ⎡ ⎤ inf N A (y) sup N A (y)  ⎬ y∈E y∈E ⎣ ⎦ |x ∈ E . , car d(E) car d(E) ⎭

W ∗ (A) =

⎩

y∈E

y∈E

Let us have an universe E = {x, y, z} with elements that have the geometrical interpretations from Fig. 4.8. The interpretations of the operator W ∗ is shown on Fig. 4.18. Obviously, for every IVIFS A, I (A) ⊆ W ∗ (A) ⊆ C(A). Theorem 4.9.1 For every IVIFS A: (a) (b) (c) (d) (e)

W ∗ (W ∗ (A)) = W ∗ (A), ¬W ∗ (¬A) = W ∗ (A), I (W ∗ (A)) = C(W ∗ (A)) = W ∗ (A), W ∗ (C(A)) = C(A), W ∗ (I (A)) = I (A).

Theorem 4.9.2 For every two IVIFSs A and B: (a) W ∗ (A ∪ B) ⊇ W ∗ (A) ∪ W ∗ (B), (b) W ∗ (A ∩ B) ⊆ W ∗ (A) ∩ W ∗ (B), (c) W ∗ (A@B) = W ∗ (A)@W ∗ (B). Theorem 4.9.3 For every IVIFS A: (a) W ∗ (A) = W ∗ ( A), (b) ♦W ∗ (A) = W ∗ (♦A). Theorem 4.9.4 For every two IFSs A and B, and for every α, β, γ, δ ∈ [0, 1] so that α ≤ β, γ ≤ δ, β + δ ≤ 1:  α W ∗( H

 (A))

 α =H

W ∗ ( J α

 (A))

= J α

γ β δ γ β δ

γ β δ

γ β δ

 (W ∗ (A)),

 (W ∗ (A)).

Theorem 4.9.5 For every IVIFS A: W (S(A)) = S(W ∗ (A)).

122

4 Operators over IVIFSs

References 1. Angelova, N., Stoenchev, M.: Intuitionistic fuzzy conjunctions and disjunctions from first type. Ann. “Inf.” Sect. Union Sci. Bulg. 8, 1–17 (2015/2016) 2. Angelova, N., Stoenchev, M., Todorov, V.: Intuitionistic fuzzy conjunctions and disjunctions from second type. Issues Intuit. Fuzzy Sets Gen. Nets 13, 143–170 (2017) 3. Angelova, N., Stoenchev, M.: Intuitionistic fuzzy conjunctions and disjunctions from third type. Notes Intuit. Fuzzy Sets 23(5), 29–41 (2017) 4. Atanassov, K.: Intuitionistic Fuzzy Sets. Springer, Heidelberg (1999) 5. Atanassov, K.: The most general form of one type of intuitionistic fuzzy modal operators. Part 2. In: Kacprzyk, J., Atanassov, K. (eds.) Proceedings of the Twelfth International Conference on Intuitionistic Fuzzy Sets, Sofia, vol. 1. Notes on Intuitionistic Fuzzy Sets, vol. 14, Issue 1, pp. 27–32. Accessed 17–18 May 2008 6. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012) 7. Atanassov, K.: On the two most extended modal types of operators defined over interval-valued intuitionistic fuzzy sets. Ann. Fuzzy Math. Inf. 16(1), 1–12 (2018) 8. Atanassov, K.: Intuitionistic fuzzy modal operators of second type over interval-valued intuitionistic fuzzy sets. Part 1. Notes Intuit. Fuzzy Sets 24(2), 8–17. https://doi.org/10.7546/nifs. 2018.24.2.8-17 (2018) 9. Atanassov, K.: Intuitionistic fuzzy modal operators of second type over interval-valued intuitionistic fuzzy sets. Part 2. Notes Intuit. Fuzzy Sets 24(3), 1–10. https://doi.org/10.7546/nifs. 2018.24.3.1-10 (2018) 10. Atanassov, K.: On the most extended modal operator of first type over interval-valued intuitionistic fuzzy sets. Mathematics 6, 123. https://doi.org/10.3390/math6070123 (2018) 11. Atanassov, K.: On the most extended interval valued intuitionistic fuzzy modal operators from both types. Notes Intuit. Fuzzy Sets 25(2), 1–14 (2019). https://doi.org/10.7546/nifs.2019.25. 2.1-14 12. Atanassov, K.: Four interval valued intuitionistic fuzzy modal-level operators. Notes Intuit. Fuzzy Sets 25, 3 2019 (In Press) 13. Atanassov K., Ban, A. : On an operator over intuitionistic fuzzy sets. Comptes Rendus de l’Academie bulgare des Sciences, Tome 53(5), 39–42 (2000) 14. Atanassov, K., Szmidt, E., Kacprzyk, J.: Shrinking operators over interval-valued intuitionistic fuzzy sets. Notes Intuit. Fuzzy Sets 24, 4 (2018). https://doi.org/10.7546/nifs.2018.24.4.2028,20-28 15. Çuvalcioˇglu, G.: Some properties of E α,β operator. Adv. Stud. Contem. Math. 14(2), 305–310 (2007) ω . Proc. Jangjeon Math. Soci. 16. Çuvalcioˇglu, G. Expand the model operator diagram with Z α,β 13(3), 403–412 (2010) 17. Kaufmann, A.: Introduction a la Theorie des Sour-Ensembles Flous. Masson, Paris (1977)

Chapter 5

Interval Valued Intuitionistic Fuzzy Pairs

The concept of an Interval Valued Intuitionistic Fuzzy Pair (IVIFP) was introduced in [3] by Peter Vassilev, Janusz Kacprzyk, Eulalia Szmidt and the author.

5.1 Definition of an IVIFP The IVIFP is an object with the form M, N , where M, N ⊆ [0, 1] are closed sets, M = [inf M, sup M], N = [inf N , sup N ] and sup M + sup N ≤ 1, that is used as an evaluation of some object or process and which components (M and N ) are interpreted as intervals of degrees of membership and non-membership, or intervals of degrees of validity and non-validity, or intervals of degree of correctness and non-correctness, etc. For the needs of the discussion below, we define the notion of Interval Valued Intuitionistic Fuzzy Tautological Pair (IVIFTP) by: x is an IFTP if and only if inf M ≥ sup N , while x is a Tautological Pair (TP) iff M = [1, 1] and N = [0, 0].

5.2 Relations over IVIFPs Let us have two IVIFPs x = M, N  and y = P, Q. We define the relations x
sup Q x < y iff inf M < inf P and sup M < sup P and inf N > inf Q and sup N > sup Q © Springer Nature Switzerland AG 2020 K. T. Atanassov, Interval-Valued Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing 388, https://doi.org/10.1007/978-3-030-32090-4_5

123

124

5 Interval Valued Intuitionistic Fuzzy Pairs

x≤

y iff inf M ≤ inf P and sup M ≤ sup P

x ≤♦ y iff inf N ≥ inf Q and sup N ≥ sup Q x ≤ y iff inf M ≤ inf P and sup M ≤ sup P and inf N ≥ inf Q and sup N ≥ sup Q y iff inf M > inf P and sup M > sup P x> x >♦ y iff inf N < inf Q and sup N < sup Q x > y iff inf M > inf P and sup M > sup P and inf N < inf Q and sup N < sup Q y iff inf M ≥ inf P and sup M ≥ sup P x≥ x ≥♦ y iff inf N ≤ inf Q and sup N ≤ sup Q x ≥ y iff inf M ≥ inf P and sup M ≥ sup P and inf N ≤ inf Q and sup N ≤ sup Q y iff inf M = inf P and sup M = sup P x= x =♦ y iff inf N = inf Q and sup N = sup Q x = y iff inf M = inf P and sup M = sup P and inf N = inf Q and sup N = sup Q.

5.3 Opertions over IVIFPs Now, there are 190 intuitionistic fuzzy implications that generate 54 intuitionistic fuzzy negations and about 500 intuitionistic fuzzy conjunctions and the same number intuitionistic fuzzy disjunctions. They are all introduced only for intuitionistic fuzzy case, but in a near future, they will be modified for interval valued intuitionistic fuzzy case, too. Here, we illustrate with two examples the process of this modification. The first intuitionistic fuzzy implication (called first Zadeh’s intuitionistic fuzzy implication) for IFP-case has the form x →1 y = max(b, min(a, c)), min(a, d), where (only here and in the next example) x = a, b, y = c, d, a, b, c, d, a + b, c + d ∈ [0, 1], which in IVIFP-form is x →1 y = [max(inf N , min(inf M, inf P)), max(sup N , min(sup M, sup P))], [min(inf M, inf Q), min(sup M, sup Q)]. The second intuitionistic fuzzy implication (called Gödel’s intuitionistic fuzzy implication) for IFP-case has the form x →2 y = 1 − sg(a − c), dsg(a − c).

5.3 Opertions over IVIFPs

125

In IVIFP-form, the second Gödel’s intuitionistic fuzzy implication is x →2 y = [1 − sg(inf M − inf P), 1 − sg(sup M − sup P)], [inf Q.sg(inf M − inf P), sup Q.sg(sup M − sup P)]. In the same manner we can construct the interval valued intuitionistic fuzzy conjunctions and disjunctions. For example, following [1], where the intuitionistic fuzzy conjunctions and disjunctions from the first type are described, we can construct the following operations: x&4 y = x&y = [min(inf M, inf P), min(sup M, sup N )], [max(inf N , inf Q), max(sup N , sup Q)] x ∨4 y = x ∨ y = [max(inf M, inf P), max(sup M sup N )], [min(inf N , inf Q), min(sup N , sup Q)] x&2 y = x + y = [inf M + inf P − inf M inf P, sup M + sup P − sup M sup P], [inf N inf Q, sup N sup Q] x ∨2 y = x.y = [inf M inf P, sup M sup P], [inf N + inf Q − inf N inf Q, sup N + sup Q − sup N sup Q]   P sup M+sup N x&3 y = x ∨3 y = x@y =  inf M+inf , , 2 2   inf N +inf Q supN +sup Q . , 2 2 Analogously, we construct the interval valued intuitionistic fuzzy negations. For example, the first two implications generate the following two interval valued intuitionistic fuzzy negations: ¬1 x = N , M, ¬2 x = [1 − sg(sup M), 1 − sg(inf M)], [sg(inf M), sg(sup M)].

5.4 Operators over IVIFPs There are three types of modal operators over IFPs. The first of them is an intuitionistic fuzzy form of the standard modal operators (see, e.g., [4]). Let as above, x = a, b be an IFP and let α, β ∈ [0, 1]. Then the modal type of operators defined over x have the forms:

126

5 Interval Valued Intuitionistic Fuzzy Pairs

x = M A (x), [inf N , 1 − sup M], ♦x = [inf M, 1 − sup N ], N , Dα (x) = [inf M, sup M + α(1 − sup M − sup N )], [inf N , sup N + (1 − α)(1 − sup M − sup N ] Fα,β (x) = [inf M, sup M + α(1 − sup M − sup N )], [inf N , sup N + β(1 − sup M − sup N )], for α + β ≤ 1, G α,β (x) = [α inf M, α sup M], [β inf N , β sup N ] Hα,β (x) = [α inf M, α sup M], [inf N , sup N + β(1 − sup M − sup N )], ∗ Hα,β (x) = [α inf M, α sup M], [inf N , sup N + β(1 − α sup M − sup N )], Jα,β (x) = [inf M, sup M + α(1 − sup M − sup N )], [β inf N , β sup N ], ∗ Jα,β (x) = [inf M, sup M + α(1 − sup M − β sup N )], [β inf N , β. sup N ], where α, β ∈ [0, 1]. F α

 (x)

γ β δ

= [inf M + α(1 − sup M − sup N ), sup M + β(1 − sup M − sup N )], [inf N + γ(1 − sup M − sup N ), sup N + δ(1 − sup M − sup N )], where β + δ ≤ 1;

G α

 (x)

= [α inf M, β sup M], [γ inf N , δ sup N ],

H α

 (x)

= [α inf M, β sup M],

γ β δ γ β δ

[inf N + γ(1 − sup M − sup N ), sup N + δ(1 − sup M − sup N )], ∗

H α

γ β δ



(x) = [α inf M, β sup M], [inf N + γ(1 − β sup M − sup N ), sup N + δ(1 − β sup M − sup N )],

Jα

γ β δ

 (x)

= [inf M + α(1 − sup M − sup N ), sup M + β(1 − sup M − sup N )], [γ inf N , δ sup N ],

∗

Jα

γ β δ



(x) = [inf M + α(1 − δ sup M − sup N ), sup M + β(1 − sup M − δ sup N )], [γ inf N , δ sup N ],

where α, β, γ, δ ∈ [0, 1] such that α ≤ β and γ ≤ δ.

5.4 Operators over IVIFPs

127

All these operators are partial cases of the following operator X  a1

b1 c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

 (x)

= x, [a1 inf M + b1 (1 − inf M − c1 inf N ), a2 sup M + b2 (1 − sup M − c2 sup N )],

[d1 inf N + e1 (1 − f 1 inf M − inf N ), d2 sup N + e2 (1 − f 2 sup M − sup N )], where a1 , b1 , c1 , d1 , e1 , f 1 , a2 , b2 , c2 , d2 , e2 , f 2 ∈ [0, 1]. The following four conditions are valid for i = 1, 2: ai + ei − ei f i ≤ 1, bi + di − bi .ci ≤ 1, bi + ei ≤ 1, a1 ≤ a2 , b1 ≤ b2 , c1 ≤ c2 , d1 ≤ d2 , e1 ≤ e2 , f 1 ≤ f 2 . The second type of operators is of another (similar to modal) type. Let αi , βi , γi , δi , εi , ζi ∈ [0, 1] for i = 1, 2. Then, we define     sup M sup N + 1 +x , inf N2 + 1 , , = inf2M , 2 2     sup M + 1 inf N , sup N , ×x , = inf M2 + 1 , 2 2 2  =  α1 inf M, α2 sup M , α2  α1 inf N + 1 − α1 , α2 sup N + 1 − α2 ,  ×  α1  x =  α1 inf M + 1 − α1 , α2 sup M + 1 − α2 , α2  α1 inf N , α2 sup N ,  +  α1 β1  x =  α1 inf M, α2 sup M , α2 β2  α1 inf N + β1 , α2 sup N + β2 ,  ×  α1 β1  x =  α1 inf M + β1 , α2 sup M + β2 , α2 β2  α1 inf N , α2 sup N  +  α1  x

where α2 + β2 ≤ 1 and α1 ≤ α2 , β1 ≤ β2 ,

128

5 Interval Valued Intuitionistic Fuzzy Pairs

+  α1

 =  α1 inf M, α2 sup M ,  β1 inf N + γ1 , β2 sup N + γ2 ,   γ1 x =  α1 inf M + γ1 , α2 sup M + γ2 , γ2  β1 inf N , β2 sup N ,

β1 γ1 α2 β2 γ2

×  α1

β1 α2 β2

x

where max(αi , βi ) + γi ≤ 1 for i = 1, 2 and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , •



α1 β1 γ1 δ1 α2 β2 γ1 δ2

x

 =  α1 inf M + γ1 , α2 sup M + γ2 ,  β1 inf N + δ1 , β2 sup N + δ2 ,

where max(αi , βi ) + γi + δi ≤ 1 for i = 1, 2 and α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ 2 , ◦



α1 β1 γ1 δ1 ε1 ζ1 α2 β2 γ1 δ2 ε1 ζ1

x

 =  α1 inf M − ε1 inf N + γ1 , α2 sup M − ε2 inf N + γ2 ,  β1 inf N − ζ1 inf M + δ1 , β2 sup N − ζ2 inf M + δ2 , where α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 , δ1 ≤ δ2 , ε1 ≥ ε2 , ζ1 ≤ ζ2 , and for i = 1, 2: max(αi − ζi , βi − εi ) + γi + δi ≤ 1, min(αi − ζi , βi − εi ) + γi + δi ≥ 0. All assertions, proved for the IFS- and for the IFP-cases are valid in IVIFP-case, too. For example, we can prove (cf. Sect. 4.3) Theorem 5.4.1 The interval valued intuitionistic fuzzy modal operators X  a1 b1 c1 d1 e1 f1  and ◦  α1 β1 γ1 δ1 ε1 ζ1  defined over IVIFPs are equivalent. a2 b2 c2 d2 e2 f 2

α2 β2 γ1 δ2 ε1 ζ1

5.5 On IVIF-Interpretation of Interval Data In [2], we show that when working with interval data, we can transform them to intuitionistic fuzzy form, interpreting them as points of the IFS-interpretation triangle. Now, we modify this transformation to the IVIFS-case. Let us have the set of intervals [a1 , b1 ],

5.5 On IVIF-Interpretation of Interval Data

129

[a2 , b2 ], ... [an , bn ]. Let the following condition: A = min ai < max bi = B 1≤i≤n

1≤i≤n

holds. Of course, it must be valid inequality A < B, because otherwise for all i: ai = bi . Now, for interval [a j , b j ] (1 ≤ j ≤ n) we can construct the two intervals M j and N j so that min ai − A i , inf M j = B−A a j − min ai i

sup M j = inf M j +

B−A

inf N j =

sup N j = inf N j +

=

aj − A , B−A

=

B − bj . B−A

B − max bi i

B−A

max bi − b j i

B−A

Obviously sup M j + sup N j =

B − bj B − A + aj − bj aj − A + = ≤ 1. B−A B−A B−A

Therefore, M j , N j  is an IVIFP that bijectively corresponds to the interval [a j , b j ].

References 1. Angelova, N., Stoenchev, M.: Intuitionistic fuzzy conjunctions and disjunctions from first type. Ann. “Inf.” Sect. Union Sci. Bulg. 8, 1–17 (2015/2016) 2. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012) 3. Atanassov, K., Vassilev, P., Kacprzyk, J., Szmidt, E.: On interval valued intuitionistic fuzzy pairs. J. Univ. Math. 1, 3, 261–268 (2018). ISSN:2618-5660, 4. Feys, R.: Modal Logics. Gauthier, Paris (1965)

Chapter 6

Applications of IVIFSs

Some applications of the IVIFSs will be discussed. In the period 1989–2000, as fas as the author knows, there were only 7 publications related to IVIFSs of other authors—Bustince and Burillo [61–66] and Hong [139]. In the new centure, more than 200 papers over IVIFSs were published. The biggest part of them are related to some IVIFS-applications. The theoretical research has been focused in the following areas: • definitions of new operations, relations and operators over IVIFSs—[3, 52, 61, 62, 83, 119, 120, 124, 170, 173, 175, 180, 181, 193, 205, 220, 282, 285, 287, 291, 294, 300, 307, 313, 314, 327]; • distances and measures over IVIFSs—[1, 65, 66, 139, 161, 176, 269, 289, 299, 310, 318, 322, 329, 330]; • extension principle of L. Zadeh for IVIFSs—[47, 150, 170] (see Sect. 3.5). The IVIFSs are used in some areas of mathematics: • logic: [73, 213]; • algebra: [7, 50, 128]; • information and entropy: [63, 64, 75, 80, 81, 88, 123, 139, 140, 171, 204, 211, 268, 283, 290, 295, 301, 308, 317, 323, 324, 326]; • topology: [179]; • comparing methods, correlation analysis and discriminant analysis, probability theory: [75, 99, 104, 110, 135, 194, 283, 284, 308, 330]; • linear programming: [166–168]. The largest areas of applications of the IVIFSs are related to the Artificial Intelligence. They are used in: • approximate reasoning: [318, 329]; • learning processes: [78, 87, 293, 308, 310, 313, 314, 317, 324, 325]; • decision making: [9, 70, 74–77, 79, 80, 82, 84, 86, 88–91, 94, 105, 109, 117– 119, 122, 129, 141, 142, 144–146, 162, 163, 165, 169, 171–174, 177, 178, 182, 187, 191, 194–197, 201, 202, 211, 212, 214, 216, 217, 240, 243, 265–267, 274, © Springer Nature Switzerland AG 2020 K. T. Atanassov, Interval-Valued Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing 388, https://doi.org/10.1007/978-3-030-32090-4_6

131

132

6 Applications of IVIFSs

279–281, 283, 284, 286, 288–290, 295, 296, 299, 303, 305, 306, 308–315, 317, 324, 325, 328]; • other sciences: [2, 4–6, 53, 54, 68, 69, 71, 72, 74, 85, 106–108, 111, 156, 158, 159, 164, 183, 189, 190, 199, 200, 215, 218, 261, 271, 297, 298, 302, 316, 331, 332, 336]. I will not discuss in more details the publications of the colleagues, because probably they will do this, soon.

6.1 Extended Interval Valued Intuitionistic Fuzzy Index Matrices The concept of an Index Matrix (IM) was introduced in 1987 in [33], but the first detailed description of the research over them was published exactly 30 years later in [39]. There, different extensions and modifications of the concept of an IM are described. Two of these are Intuitionistic Fuzzy IM (IFIM) and Extended IFIM (EIFIM). Here, following [28], we introduce the concepts of an Interval Valued IFIM (IVIFIM) and Extended IVIFIM (EIVIFIM). Let I be a fixed set. By IVIFIM with index sets K and L (K , L ⊂ I ), we denote the object: [K , L , {Mki ,l j , Nki ,l j }] l1 ... lj k1 Mk1 ,l1 , Nk1 ,l1  . . . Mk1 ,l j , Nk1 ,l j  .. .. .. . ... . . ≡ ki Mki ,l1 , Nki ,l1  . . . Mki ,l j , Nki ,l j  .. .. .. . ... . .

... ln . . . Mk1 ,ln , Nk1 ,ln  .. ... . . . . Mki ,ln , Nki ,ln  .. ... .

,

km Mkm ,l1 , Nkm ,l1  . . . Mkm ,l j , Nkm ,l j  . . . Mkm ,ln , Nkm ,ln 

where for every 1 ≤ i ≤ m, 1 ≤ j ≤ n: Mki ,l j , Nki ,l j ⊆ [0, 1], sup Mki ,l j + sup Nki ,l j ≤ 1 and K = {k1 , k2 , ..., km }, L = {l1 , l2 , ..., ln }. Now, for the above sets K and L, the EIVIFIM is defined by: [K ∗ , L ∗ , {Mki ,l j , Nki ,l j }]

6.1 Extended Interval Valued Intuitionistic Fuzzy Index Matrices

133

l1 , Al1 , Bl1  . . . ln , Aln , Bln  k1 , Ak1 , Bk1  Mk1 ,l1 , Nk1 ,l1  . . . Mk1 ,ln , Nk1 ,ln  .. .. .. . ... . . ≡ , ki , Aki , Bki  Mki ,l1 , Nki ,l1  . . . Mki ,ln , Nki ,ln  .. .. .. . ... . . km , Akm , Bkm  Mkm ,l1 , Nkm ,l1  . . . Mkm ,ln , Nkm ,ln 

where for every 1 ≤ i ≤ m, 1 ≤ j ≤ n: Mki ,l j , Nki ,l j ⊆ [0, 1], sup Mki ,l j + sup Nki ,l j ≤ 1, Aki , Bki ⊆ [0, 1], sup Aki + sup Bki ≤ 1, Al j , Bl j ⊆ [0, 1], sup Al j + sup Bl j ≤ 1 and here and below, K ∗ = {ki , Aki , Bki |ki ∈ K } = {ki , Aki , Bki |1 ≤ i ≤ m}, L ∗ = {l j , Al j , Bl j |l j ∈ L} = {l j , Al j , Bl j |1 ≤ j ≤ n}. Let K ∗ ⊂ P ∗ iff (K ⊂ P) & (∀ki = pi ∈ K )((Aki ⊂∗ A pi ) & (Bki ⊃∗ B pi )). K ∗ ⊆ P ∗ iff (K ⊆ P) & (∀ki = pi ∈ K )((Aki ⊆∗ A pi ) & (Bki ⊇∗ B pi )), where for two sets X and Y X ⊂∗ Y iff inf X < inf Y and sup X < sup Y, X ⊃∗ Y iff inf X > inf Y and sup X > sup Y, X ⊆∗ Y iff inf X ≤ inf Y and sup X ≤ sup Y, X ⊇∗ Y iff inf X ≥ inf Y and sup X ≥ sup Y. All operations and relations over EIVIFIM must be re-defined, because they have different forms from these in [39]. Obviously, the hierarchical operators are not applicable now.

134

6 Applications of IVIFSs

Below, we will work only with EIVIFIMs, because the IVIFIMs are partial cases. Let “∗” and “◦” be two fixed operations over IVIFPs and let M, N  ∗ P, Q = M ∗l P, N ∗r Q, where the forms of operations “∗l ” and “∗r ” are determined by the form of the operation “∗”, as it is shown in the examples. For the EIVIFIMs A = [K ∗ , L ∗ , {Mki ,l j , Nki ,l j }], B = [P ∗ , Q ∗ , {R pr ,qs , S pr ,qs }], with

K ∗ = {ki , Aki , Bki |ki ∈ K }, L ∗ = {l j , Al j , Bl j |l j ∈ L}, P ∗ = { pr , C pr , Dqs | pr ∈ P}, Q ∗ = {qs , C pr , Dqs |qs ∈ Q}

operations that are analogous to the usual matrix operations of addition and multiplication are defined, as well as other specific ones. Addition-(∗) A ⊕(∗) B = [T ∗ , V ∗ , {Φtu ,vw , Ψtu ,vw }], where

T ∗ = K ∗ ∪ P ∗ = {tu , Atu , Btu |tu ∈ K ∪ P}, V ∗ = L ∗ ∪ Q ∗ = {vw , Avw , Bvw |vw ∈ L ∪ Q}, ⎧ A ki , if tu = ki ∈ K − P ⎪ ⎪ ⎪ ⎪ ⎨ if tu = pr ∈ P − K Atu = A pr , , ⎪ ⎪ ⎪ ⎪ ⎩ Aki ∪ A pr , if tu = ki = pr ∈ K ∩ P

Bvw

and

⎧ Bl , if vw = l j ∈ L − Q ⎪ ⎪ ⎪ j ⎪ ⎨ if tw = qs ∈ Q − L = Bqs , , ⎪ ⎪ ⎪ ⎪ ⎩ Bvw ∩ Bqs , if tw = vw = qs ∈ L ∩ Q

6.1 Extended Interval Valued Intuitionistic Fuzzy Index Matrices

Φtu ,vw , Ψtu ,vw  =

⎧ ⎪ ⎪ Mki ,l j , Nki ,l j , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R pr ,qs , S pr ,qs , ⎪ ⎪ ⎨

135

if tu = ki ∈ K and vw = l j ∈ L − Q or tu = ki ∈ K − P and vw = l j ∈ L; if tu = pr ∈ P and vw = qs ∈ Q − L or tu = pr ∈ P − K and vw = qs ∈ Q;

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Mki ,l j ∗l R pr ,qs , if tu = ki = pr ∈ K ∩ P ⎪ ⎪ ⎪ ⎪ Nki ,l j ∗r S pr ,qs ), and vw = l j = qs ∈ L ∩ Q ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [0, 0], [1, 1], otherwise

Termwise multiplication-(*) A ⊗(∗) B = [T ∗ , V ∗ , {Φtu ,vw , Ψtu ,vw }], where

T ∗ = K ∗ ∩ P ∗ = {tu , Atu , Btu |tu ∈ K ∪ P}, V ∗ = L ∗ ∩ Q ∗ = {vw , Avw , Bvw |vw ∈ L ∪ Q}, Atu = Aki ∩ A pr , for tu = ki = pr ∈ K ∩ P, Bvw = Bvw ∪ Bqs , for vw = l j = qs ∈ L ∩ Q

and Φtu ,vw , Ψtu ,vw  = Mki ,l j ∗l R pr ,qs , Nki ,l j ∗r S pr ,qs . Multiplication-(◦, ∗) A (◦,∗) B = [T ∗ , V ∗ , {Φtu ,vw , Ψtu ,vw }], where

T ∗ = (K ∪ (P − L))∗ = {tu , Atu , Btu |tu ∈ K ∪ (P − L)}, V ∗ = (Q ∪ (L − P))∗ = {vw , Avw , Bvw |vw ∈ Q ∪ (L − P)}, Atu =

⎧ ⎨ Aki , if tu = ki ∈ K ⎩

A pr , if tu = pr ∈ P − L

,

136

6 Applications of IVIFSs

Bvw =

⎧ ⎨ Bl j , if vw = l j ∈ L − P ⎩

Bqs , if tw = qs ∈ Q

,

and Φtu ,vw , Ψtu ,vw  =

=

⎧ Mki ,l j , Nki ,l j , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R pr ,qs , S pr ,qs , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

if tu = ki ∈ K and vw = l j ∈ L − P − Q if tu = pr ∈ P − L − K and vw = qs ∈ Q

 ◦l (Mki ,l j ∗l R pr ,qs )), if tu = ki ∈ K ⎪ ⎪ ⎪ l j = pr ∈L∩P ⎪ ⎪ ⎪ and vw = qs ∈ Q ⎪ ⎪ ⎪ ◦r (Nki ,l j ∗r S pr ,qs ), ⎪ ⎪ l j = pr ∈L∩P ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [0, 0], [1, 1], otherwise

Structural subtraction A  B = [T ∗ , V ∗ , {Φtu ,vw , Ψtu ,vw }], where

T ∗ = (K − P)∗ = {tu , Atu , Btu |tu ∈ K − P}, V ∗ = (L − Q)∗ = {vw , Avw , Bvw |vw ∈ L − Q},

for the set–theoretic subtraction operation and Atu = Aki , for tu = ki ∈ K − P, βwv = Bl j , for vw = l j ∈ L − Q and Φtu ,vw , Ψtu ,vw  = Mki ,l j , Nki ,l j , for tu = ki ∈ K − P and vw = l j ∈ L − Q. Negation of an EIVIFIM ¬A = [T ∗ , V ∗ , {¬Mki ,l j , Nki ,l j }], where ¬ is one of the possible interval valued intuitionistic fuzzy negations that can be defined on the basis of the definitions of intuitionistic fuzzy negations from [38].

6.1 Extended Interval Valued Intuitionistic Fuzzy Index Matrices

137

Termwise subtraction A −◦,∗ B = A ⊕◦,∗ ¬B. Let two EIVIFIMs A and B be given. We shall introduce the following definitions where ⊂ and ⊆ denote the relations “strong inclusion” and “weak inclusion”, respectively. The strict relation “inclusion about dimension” is A ⊂d B iff (((K ∗ ⊂ P ∗ ) & (L ∗ ⊂ Q ∗ )) ∨ ((K ∗ ⊆ P ∗ ) & (L ∗ ⊂ Q ∗ )) ∨((K ∗ ⊂ P ∗ ) & (L ∗ ⊆ Q ∗ ))) & (∀k ∈ K )(∀l ∈ L)(Ak,l , Bk,l  = Ck,l , Dk,l ). The non-strict relation “inclusion about dimension” is A ⊆d B iff (K ∗ ⊆ P ∗ ) & (L ∗ ⊆ Q ∗ ) & (∀k ∈ K )(∀l ∈ L) (Ak,l , Bk,l  = Ck,l , Dk,l ). The strict relation “inclusion about value” is A ⊂v B iff (K ∗ = P ∗ ) & (L ∗ = Q ∗ ) & (∀k ∈ K )(∀l ∈ L) (Ak,l , Bk,l  ⊂∗ Ck,l , Dk,l ). The non-strict relation “inclusion about value” is A ⊆v B iff (K ∗ = P ∗ ) & (L ∗ = Q ∗ ) & (∀k ∈ K )(∀l ∈ L) (Ak,l , Bk,l  ⊆∗ Ck,l , Dk,l ). The strict relation “inclusion” is A ⊂∗ B iff (((K ∗ ⊂ P ∗ ) & (L ∗ ⊂ Q ∗ )) ∨ ((K ∗ ⊆ P ∗ ) & (L ∗ ⊂ Q ∗ )) ∨ ((K ∗ ⊂ P ∗ ) & (L ∗ ⊆ Q ∗ ))) & (∀k ∈ K )(∀l ∈ L)(Ak,l , Bk,l  ⊂∗ Ck,l , Dk,l ). The non-strict relation “inclusion” is A ⊆∗ B iff (K ∗ ⊆ P ∗ ) & (L ∗ ⊆ Q ∗ ) & (∀k ∈ K )(∀l ∈ L) (Ak,l , Bk,l  ⊆∗ Ck,l , Dk,l ). / K and l0 ∈ / L be two fixed indices. Let the EIVIFIM A be given and let k0 ∈ Now, we introduce the following two operations over it having a general form.

138

6 Applications of IVIFSs

Let “∗” be a fixed operation over IVIFPs. (∗)-row-aggregation ρ(∗) (A, k0 ) l1 , Al,1 , Bl1  =

k0 ,  ∗l

1≤i≤m

Aki , ∗r

Bki   ∗l

...

ln , Aln , Bln 

. . .  ∗l

Mki ,ln , ∗r

1≤i≤m

1≤i≤m

1≤i≤m

Mki ,l1 , ∗r

1≤i≤m

1≤i≤m

Nki ,ln 

... Nki ,l1  . . .

,

(∗)-column-aggregation l0 ,  ∗l

1≤i≤m

σ∗ (A, l0 ) =

k1 , Ak1 , Bk1   ∗l

1≤i≤m

.. .

Al j , ∗r

1≤i≤m

Mk1 ,l j , ∗r

1≤i≤m

Bl j 

Nk1 ,l j 

.

.. .

km , Akm , Bkm   ∗l

1≤i≤m

Mkm ,l j , ∗r

1≤i≤m

Nkm ,l j 

Here and below we use symbol “⊥” for lack of some component in the separate definitions. In some cases, it is suitable to change this symbol with “[0, 0], [1, 1]”. Now, we introduce operations (k, ⊥)- and (⊥, l)-reduction of a given IVIFIM A: A(k,⊥) = [(K − {k})∗ , L ∗ , {Ptu ,vw , Q tu ,vw }] where Ptu ,vw , Q tu ,vw  = Mki ,l j , Nki ,l j  for tu = ki ∈ K − {k} and vw = l j ∈ L and A(⊥,l) = [K ∗ , (L − {l})∗ , {Ptu ,vw , Q tu ,vw }], where Ptu ,vw , Q tu ,vw  = Mki ,l j , Nki ,l j  for tu = ki ∈ K and vw = l j ∈ L − {l}.

6.1 Extended Interval Valued Intuitionistic Fuzzy Index Matrices

139

Second, we define A(k,l) = (A(k,⊥) )(⊥,l) = (A(⊥,l) )(k,⊥) . Third, let R = {k1 , k2 , ..., ks } ⊆ K and S = {q1 , q2 , ..., qt } ⊆ L. Then, we define the following three operations: A(R,l) = (...((A(k1 ,l) )(k2 ,l) )...)(ks ,l) , A(k,S) = (...((A(k,l1 ) )(k,l2 ) )...)(k,lt ) , A(R,S) = (...((A( p1 ,S) )( p2 ,S) )...)( ps ,S) = (...((A(R,q1 ) )(R,q2 ) )...)(R,qt ) . Let M ⊆ K and N ⊆ L and A be an EIVIFIM. Then, pr M,N A = [M, N , {Ptu ,vw , Q tu ,vw }], where for each ki ∈ M and each l j ∈ N , Pki ,l j , Q ki ,l j  = Mki ,l j , Nki ,l j . Obviously, for every EIVIFIM A and sets M1 ⊆ M2 ⊆ K and N1 ⊆ N2 ⊆ L the equality pr M1 ,N1 pr M2 ,N2 A = pr M1 ,N1 A is valid and for M ⊆ K , N ⊆ L, the equalities pr M,N A = A(K −M,L−N ) , A M,N = pr K −M,L−N A hold. Let, as above, x = M, N  be an IVIFP and let O X1 , OY2 , O Z3 be three interval valued intuitionistic fuzzy operators with parameters X, Y, Z , respectively. In practice, the parameters X, Y, Z can be IFPs, IVIFPs and other objects. The three operators affect the K -, L-indices and Mki ,l j , Nki ,l j -elements, respectively. They can be applied over an EIVIFIM A sequentially, or simultaneously. In the first case, their forms are (O X1 , ⊥, ⊥)(A) l1 , Al1 , Bl1  k1 , O X1 (Ak1 , Bk1 ) Mk1 ,l1 , Nk1 ,l1  = .. .. . .

. . . ln , Aln , Bln  . . . Mk1 ,ln , Nk1 ,ln  , .. .. . . km , O X1 (Akm , Bkm ) Mkm ,l1 , Nkm ,l1  . . . Mkm ,ln , Nkm ,ln 

140

6 Applications of IVIFSs

(⊥, OY2 , ⊥)(A) l1 , OY2 (Al1 , Bl1 ) k1 , Ak1 , Bk1  Mk1 ,l1 , Nk1 ,l1  = .. .. . .

. . . ln , OY2 (Aln , Bln ) . . . Mk1 ,ln , Nk1 ,ln  , .. .. . .

km , Akm , Bkm  Mkm ,l1 , Nkm ,l1  . . . Mkm ,ln , Nkm ,ln  (⊥, ⊥, O Z3 )(A)

l1 , Al1 , Bl1  ... ln , Aln , Bln  k1 , Ak1 , Bk1  O Z3 (Mk1 ,l1 , Nk1 ,l1 ) . . . O Z3 (Mk1 ,ln , Nk1 ,ln ) = . .. .. .. .. . . . . km , Akm , Bkm  O Z3 (Mkm ,l1 , Nkm ,l1 ). . . O Z3 (Mkm ,ln , Nkm ,ln )

In the second case, the form of the triple of operators is (O X1 , OY2 , O Z3 )(A) l1 , OY2 (Al1 , Bl1 ) . . . ln , OY2 (Aln , Bln ) 3 Bk1 ) O Z (Mk1 ,l1 , Nk1 ,l1 ). . . O Z3 (Mk1 ,ln , Nk1 ,ln ) .. .. .. .. . . . . = . ki , O X1 (Aki , Bki ) O Z3 (Mki ,l1 , Nki ,l1 ). . . O Z3 (Mki ,ln , Nki ,ln ) .. .. .. .. . . . . k1 , O X1 (Ak1 ,

km , O X1 (Akm , Bkm ) O Z3 (Mkm ,l1 , Nkm ,l1 ). . .

O Z3 (Mkm ,ln , Nkm ,ln )

6.2 Interval Valued Intuitionistic Fuzzy Graphs The concept of the IFG was introduced by Anthony Shannon and the author in 1994 in [239]. During the last years, it was essentially extended and it had found different applications. Here, we introduce two groups each one of which contains four different types of Interval Valued Intuitionistic Fuzzy Graphs (IVIFGs). Let us have a (fixed) set of vertices V. An (◦)-IFG G (over V) will be the ordered pair G = (V ∗ , A∗ ), where V ⊂ V, V ∗ = {v, μV (v), νV (v)|v ∈ V }, A ⊂ V × V,

6.2 Interval Valued Intuitionistic Fuzzy Graphs

141

A∗ = {x, y, μ A (x, y), ν A (x, y)|x, y ∈ V × V } and functions μV : V → [0, 1] and νV : V → [0, 1] define the degree of membership and the degree of non-membership, respectively, of the element v ∈ V to the set V ; functions μ A : E 1 × E 2 → [0, 1] and ν A : E 1 × E 2 → [0, 1] define the degree of membership and the degree of non-membership, respectively, of the element x, y ∈ E 1 × E 2 to the set A ⊆ E 1 × E 2 ; these functions have the forms of the corresponding components of the ◦-Cartesian product over IFSs, where ◦ ∈ {×1 , ×2 , ..., ×5 } is an operation over IFSs, and for all x, y ∈ E 1 × E 2 , 0 ≤ μV (x) + νV (x) ≤ 1, 0 ≤ μ A (x, y) + ν A (x, y) ≤ 1. The above definition is old (see, e.g., [37]), while the following three types of IVIFGs are introduced for a first time in [41]. Let us call the first definition (◦)(IFS,IFS)-IFG. Now, we introduce the following three new definitions. Let us have a (fixed) set of vertices V. An (◦)-(IFS,IVIFS)-IFG G (over V) will be the ordered pair G = (V ∗ , A∗ ), where V ⊂ V, V ∗ = {v, μV (v), νV (v)|v ∈ V }, A ⊂ V × V, A∗ = {x, y, M A (x, y), N A (x, y)|x, y ∈ V × V } and functions μV : V → [0, 1] and νV : V → [0, 1] define the degree of membership and the degree of non-membership, respectively, of the element v ∈ V to the set V ; functions M A : E 1 × E 2 → P([0, 1]) and N A : E 1 × E 2 → P([0, 1]) define the degree of membership and the degree of non-membership, respectively, of the element x, y ∈ E 1 × E 2 to the set A ⊆ E 1 × E 2 , where for each set Z , P(Z ) is the set of the subsets of Z ; these functions have the forms of the corresponding components of the ◦-Cartesian product over IVIFSs, where ◦ ∈ {×1 , ×2 , ..., ×5 } is an operation over IVIFSs, and for all x, y ∈ E 1 × E 2 , 0 ≤ μV (x) + νV (x) ≤ 1, 0 ≤ sup M A (x) + sup N A (x) ≤ 1. Let us have a (fixed) set of vertices V. An (◦)-(IVIFS,IFS)-IFG G (over V) will be the ordered pair G = (V ∗ , A∗ ), where

142

6 Applications of IVIFSs

V ⊂ V, V ∗ = {v, MV (v), N V (v)|v ∈ V }, A ⊂ V × V, A∗ = {x, y, μ A (x, y), ν A (x, y)|x, yV × V } and functions MV : V → P([0, 1]) and N V : V → P([0, 1]) define the degree of membership and the degree of non-membership, respectively, of the element v ∈ V to the set V ; functions μ A : E 1 × E 2 → [0, 1] and ν A : E 1 × E 2 → [0, 1] define the degree of membership and the degree of non-membership, respectively, of the element x, y ∈ E 1 × E 2 to the set A ⊆ E 1 × E 2 ; these functions have the forms of the corresponding components of the o-Cartesian product over IFSs, where ◦ ∈ {×1 , ×2 , ..., ×5 } is an operation over IFSs, and for all x, y ∈ E 1 × E 2 , 0 ≤ sup MV (x) + sup N V (x) ≤ 1, 0 ≤ μ A (x, y) + ν A (x, y) ≤ 1. Let us have a (fixed) set of vertices V. An (◦)-(IVIFS,IVIFS)-IFG G (over V) will be the ordered pair G = (V ∗ , A∗ ), where V ⊂ V, V ∗ = {v, MV (v), N V (v)|v ∈ V }, A ⊂ V × V, A∗ = {x, y, M A (x, y), N A (x, y)|x, y ∈ V × V } and functions MV : V → P([0, 1]) and N V : V → P([0, 1]) define the degree of membership and the degree of non-membership, respectively, of the element v ∈ V to the set V ; functions M A : E 1 × E 2 → P([0, 1]) and ν A : E 1 × E 2 → P([0, 1]) define the degree of membership and the degree of non-membership, respectively, of the element x, y ∈ E 1 × E 2 to the set A ⊆ E 1 × E 2 ; these functions have the forms of the corresponding components of the ◦-Cartesian product over IVIFSs, where ◦ ∈ {×1 , ×2 , ..., ×5 } is an operation over IVIFSs, and for all x, y ∈ E 1 × E 2 , 0 ≤ sup MV (x) + sup N V (x) ≤ 1, 0 ≤ sup M A (x, y) + sup N A (x, y) ≤ 1. As in [149] and by analogy with [37], we illustrate the last of the above definitions by an example of a Berge’s graph (see Fig. 6.1; the labels of the vertices and arcs

6.2 Interval Valued Intuitionistic Fuzzy Graphs

143

[0.4, 0.5], [0.3, 0.4]  A  h

[0.0, 0.0], [1.0, 1.0] [0.4, 0.5], [0.1, 0.4]

-



U



  h C 6  [0.1, 0.1], [0.7, 0.9] [1.0, 1.0], [0.0, 0.0]

M

[0.0, 0.0], [1.0, 1.0] [1.0, 1.0], [0.0, 0.0]

 h B ? 6  [0.2, 0.5], [0.0, 0.4]

[0.4, 0.5], [0.2, 0.3] Fig. 6.1 Berge’s IVIFG

show the corresponding degrees). Let the following IM giving M- and N -values be defined for its A-values (for example, the data can be obtained as a result of some observations). A B C M A , N A  A [0.4, 0.5], [0.3, 0.4] [1.0, 1.0], [0.0, 0.0] [0.0, 0.0], [1.0, 1.0] [0.0, 0.0], [1.0, 1.0] [0.2, 0.5], [0.0, 0.4] [0.4, 0.5], [0.2, 0.3] B [0.4, 0.5], [0.1, 0.4] [1.0, 1.0], [0.0, 0.0] [0.1, 0.1], [0.7, 0.9] C Having in mind that each real number r can be represented as an interval [r, r ], we see that the first three types of graphs are partial cases of the fourth type. These four types of IVIFGs are analogues of the idea for IFGs, a partial case of which was discussed in [37]. The graphs from the second four types of IVIFGs have similar to the above form, but without the condition for the forms of their μG and νG , or MG and NG elements. So, their definitions are the following. Let us have a (fixed) set of vertices V. An (IFS,IFS)-IFG G (over V) will be the ordered pair G = (V ∗ , A∗ ), where V ⊂ V,

144

6 Applications of IVIFSs

V ∗ = {v, μV (v), νV (v)|v ∈ V }, A ⊂ V × V, A∗ = {x, y, μ A (x, y), ν A (x, y)|x, y ∈ V × V } and functions μV : V → [0, 1] and νV : V → [0, 1] define the degree of membership and the degree of non-membership, respectively, of the element v ∈ V to the set V ; functions μ A : E 1 × E 2 → [0, 1] and ν A : E 1 × E 2 → [0, 1] define the degree of membership and the degree of non-membership, respectively, of the element x, y ∈ E 1 × E 2 to the set A ⊆ E 1 × E 2 ; and for all x, y ∈ E 1 × E 2 , 0 ≤ μV (x) + νV (x) ≤ 1, 0 ≤ μ A (x, y) + ν A (x, y) ≤ 1. Let us have a (fixed) set of vertices V. An (IFS,IVIFS)-IFG G (over V) will be the ordered pair G = (V ∗ , A∗ ), where V ⊂ V, V ∗ = {v, μV (v), νV (v)|v ∈ V }, A ⊂ V × V, A∗ = {x, y, M A (x, y), N A (x, y)|x, y ∈ V × V } and functions μV : V → [0, 1] and νV : V → [0, 1] define the degree of membership and the degree of non-membership, respectively, of the element v ∈ V to the set V ; functions M A : E 1 × E 2 → P([0, 1]) and ν A : E 1 × E 2 → P([0, 1]) define the degree of membership and the degree of non-membership, respectively, of the element x, y ∈ E 1 × E 2 to the set A ⊆ E 1 × E 2 ; and for all x, y ∈ E 1 × E 2 , 0 ≤ μV (x) + νV (x) ≤ 1, 0 ≤ sup M A (x) + sup N A (x) ≤ 1. Let us have a (fixed) set of vertices V. An (IVIFS,IFS)-IFG G (over V) will be the ordered pair G = (V ∗ , A∗ ), where V ⊂ V, V ∗ = {v, MV (v), N V (v)|v ∈ V }, A ⊂ V × V,

6.2 Interval Valued Intuitionistic Fuzzy Graphs

145

A∗ = {x, y, μ A (x, y), ν A (x, y)|x, y ∈ V × V } and functions MV : V → P([0, 1]) and N V : V → P([0, 1]) define the degree of membership and the degree of non-membership, respectively, of the element v ∈ V to the set V ; functions μ A : E 1 × E 2 → [0, 1] and ν A : E 1 × E 2 → [0, 1] define the degree of membership and the degree of non-membership, respectively, of the element x, y ∈ E 1 × E 2 to the set A ⊆ E 1 × E 2 ; and for all x, y ∈ E 1 × E 2 , 0 ≤ sup MV (x) + sup N V (x) ≤ 1, 0 ≤ μ A (x, y) + ν A (x, y) ≤ 1. Let us have a (fixed) set of vertices V. An (IVIFS,IVIFS)-IFG G (over V) will be the ordered pair G = (V ∗ , A∗ ), where V ⊂ V, V ∗ = {v, MV (v), N V (v)|v ∈ V }, A ⊂ V × V, A∗ = {x, y, M A (x, y), N A (x, y)|x, y ∈ V × V } and functions MV : V → P([0, 1]) and N V : V → P([0, 1]) define the degree of membership and the degree of non-membership, respectively, of the element v ∈ V to the set V ; functions M A : E 1 × E 2 → P([0, 1]) and ν A : E 1 × E 2 → P([0, 1]) define the degree of membership and the degree of non-membership, respectively, of the element x, y ∈ E 1 × E 2 to the set A ⊆ E 1 × E 2 ; and for all x, y ∈ E 1 × E 2 , 0 ≤ sup MV (x) + sup N V (x) ≤ 1, 0 ≤ sup M A (x, y) + sup N A (x, y) ≤ 1. Obviously, the first four definitions are partial cases of the new four definitions, respectively. From Sect. 6.1, it is clear that in the general case, if V = {v1 , v2 , ..., vn }, then the IM of the first, second, fifth and sixth graphs can have the form v1 v1 a1,1 A = v2 a2,1 .. . ... vn an,1

v2 . . . vn a1,2 . . . a1,n a2,2 . . . a2,n ... ... ... an,2 . . . an,n

146

6 Applications of IVIFSs

where n is the cardinality of set V and ai, j = μi, j , νi, j  ∈ [0, 1] × [0, 1] (1 ≤ i, j ≤ n), 0 ≤ μi, j + νi, j ≤ 1, while the third, fourth, seventh and eighth graphs can have the form of the same IM, but now ai, j = Mi, j , Ni, j  ⊆ [0, 1] × [0, 1] (1 ≤ i, j ≤ n), 0 ≤ sup Mi, j + sup Ni, j ≤ 1. Now, we can represent each one of the four types of graphs in IM-form as G = [V ∗ , V ∗ , {ai, j }]. Following ideas from [39], it can be easily seen that the above IM can be modified to the following form: ∗

∗

G = [VI∗ ∪ V , V ∪ VO∗ , {ai, j }], ∗

where VI∗ , VO∗ and V are respectively the sets of the input, output and internal vertices of the graph. At least one arc leaves every vertex of the first type, but none enters; at least one arc enters each vertex of the second type but none leaves it; every vertex of the third type has at least one arc ending in it and at least one arc starting from it. Obviously, the IM of this graph will have a smaller dimension than the ordinary graph matrix. Moreover, it can be non-square, unlike the ordinary graph matrices. As in the ordinary case, the vertex v p ∈ V has a loop if and only if a p, p = μ p, p , ν p, p  for the vertex v p and μ p, p > 0 (hence ν p, p < 1). Let us write below for brevity G instead of G ∗ and V instead of V ∗ . Let the graphs G 1 and G 2 be given and let G s = [Vs , Vs , {ai,s j }], where s = 1, 2 and Vs and Vs are the sets of the graph vertices (input and internal, and output and internal, respectively). Then, using the apparatus of the IMs, we construct the graph which is a union of the graphs G 1 and G 2 . The new graph has the description G = G 1 ∪ G 2 = [V1 ∪ V2 , V1 ∪ V2 , {a i, j }], where a i, j is determined by the respective IM-formulas for operation ∪ from Sect. 3.2.

6.2 Interval Valued Intuitionistic Fuzzy Graphs

147

Analogously, we can construct a graph which is the intersection of the two given graphs G 1 and G 2 . It would have the form G = G 1 ∩ G 2 = [V1 ∩ V2 , V1 ∩ V2 , {a i, j }], where a i, j is determined by the respective IM-formulas for operation ∩ from Sect. 3.2.

6.3 Intuitionistic Fuzzy Neural Networks with Interval Valued Intuitionistic Fuzzy Conditions During the last 25 years some research over the concept of an intuitionistic fuzzy neural network were published—see [24, 130, 131, 155, 246] and some other tools, related to intuitionistic fuzziness, e.g., the intercriteria analysis were used for estimation of the results of neural network functioning (see [251, 252]). In [24, 246] intuitionistic fuzzy feedforward network was constructed by combining feed forward neural networks and intuitionistic fuzzy logic. Some operations and two types of transferring functions involved in the working process of these nets were introduced. There are a few papers [131, 160] that combine ideas from intuitionistic fuzziness and artificial neural networks. In [131] is introduced an Intuitionistic fuzzy RBF network. In [160] is produced Intuitionistic fuzzy model of a neural network. Here, an extension of the concept of a neural network will be introduced. It will be based of the apparatus of the IFSs, IVIFSs, IFPs and IVIFPs. Let is have a Neural Network with s layers. Let the neurons in i-th layer be denoted by Vi, j , where 0 ≤ i ≤ s, 1 ≤ j ≤ wi , where wi is the number of neurons in i-th layer. Let to each arc connecting neurons Vi, j and Vi+1,k , where 1 ≤ k ≤ wi+1 , the pair of pairs Mi, j,k , Ni, j,k , ϕi, j , ψi, j  be juxtaposed, where Mi, j,k , Ni, j,k ⊆ [0, 1] be intervals, so that Mi, j,k = [inf Mi, j,k , sup Mi, j,k ], Ni, j,k = [inf Ni, j,k , sup Ni, j,k ] and sup Mi, j,k + sup Ni, j,k ≤ 1, and ϕi, j,k , ψi, j,k , ϕi, j,k + ψi, j,k ∈ [0, 1]. Pair ϕi, j,k , ψi, j,k  are the intuitionistic fuzzy values for neuron Vi, j , that are calculated by the formula (6.3.1), given below. The initial values of the neurons V0, j for 1 ≤ j ≤ w0 are ϕ0, j , ψ0, j  and they are given in the beginning. Pair Mi, j,k , Ni, j,k  corresponds to the condition that determines whether the neuron Vi, j can send signal to neuron Vi+1,k or not. This condition is given by predicate  P(Vi, j , Vi+1,k ) =

1, if ϕi, j,k ∈ Mi, j,k and ψi, j,k ∈ Ni, j,k . 0, otherwise

(6.3.1)

The existence of the condition P is the reason that we called the described neuron network an Intuitionistic Fuzzy Neural Networks with Interval Valued Intuitionistic Fuzzy Conditions (IFNN-IVIFC). Formally, it is defined by

148

6 Applications of IVIFSs

w0 , w1 , ..., ws , ϕ0,1 , ψ0,1 , ..., ϕ0,w0 , ψ0,w0 , {Vi, j |0 ≤ i ≤ s; 1 ≤ j ≤ wi }, [{Vi,1 , ..., Vi,wi }, {Vi+1,1 , ..., Vi+1,wi+1 }, {Mi, j,k , Ni, j,k }], where w0 , w1 , ..., ws are the numbers of neurons in the different layers; ϕ0,1 , ψ0,1 , ..., ϕ0,w0 , ψ0,w0  are the intuitionistic fuzzy values that the input neurons obtain; {Vi, j |0 ≤ i ≤ s; 1 ≤ j ≤ wi } is the set of all neurons; [{Vi,1 , ..., Vi,wi }, {Vi+1,1 , ..., Vi+1,wi+1 }, {Mi, j,k , Ni, j,k }]

=

... Vi+1,k . . . Mi,1,k , Ni,1,k  .. .. . . Mi, j,1 , Ni, j,1  . . . Mi, j,k , Ni, j,k  .. .. .. . . .

Vi+1,1 Vi,1 Mi,1,1 , Ni,1,1  .. .. . . Vi, j .. .

... Vi+1,wi+1 . . . Mi,1,wi+1 , Ni,1,wi+1  .. .. . . , . . . Mi, j,wi+1 , Ni, j,wi+1  .. .. . .

Vi,wi Mi,wi ,1 , Ni,wi ,1 . . .Mi,wi ,k , Ni,wi ,k . . .Mi,wi ,wi+1 , Ni,wi ,wi+1 

is an IVIFIM which elements—the pairs Mi, j,k , Ni, j,k  that are defined above. Before the IFNN-IVIFC can start functioning, the user must determine the two most suitable intuitionistic fuzzy conjunction and disjunction among operations &1 , &2 , & + 3, ∨1 , ∨2 , ∨3 (given in Sect. 3.2) or others, that are IVIFSanalogues of these, introduced in [12, 13, 18] for the IFS-case. For example, ◦, ∗ ∈ {∨1 , &1 , ∨2 , &2 , ∨3 , &3 . The results of the five logical operations can be interpreted by: ∨2 —strong optimistic result, ∨1 —optimistic result, ∨3 (&3 )—average result, &1 —pessimistic result, &2 —strong pessimistic result, Let us denote the fixed operations by pair ◦, ∗ and let us assume that  P(Vi, j , Vi+1,k )ϕi, j , ψi, j  =

ϕi, j , ψi, j , if P(Vi, j , Vi+1,k ) = 1 ⊥,

otherwise

,

(6.3.2)

where symbol ⊥ denotes a lack of an expression. Let use define for operations ◦ and ∗ for each IFP a, b: a, b∗ ⊥= a, b =⊥ ∗a, b, a, b∗ ⊥=⊥=⊥ ∗a, b, and especially for operation @, the denominator must be decreased with the number of arguments that are exactly ⊥. For example,

6.3 Intuitionistic Fuzzy Neural Networks with Interval Valued …

@(1, ⊥, 2, 3, ⊥) =

149

1+2+3 = 2. 3

On the first time-moment of the IFNN-IVIFC functioning, the signals enter input neurons with the respective intuitionistic fuzzy values. After i time-steps (for i ≥ 1, these signals from i-th layer (as in the standard case of the neuron networks) go to the neurons from the (i + 1)-st layer, but now, having in mind formula (6.3.2), only these values that satisfy predicate (6.3.1) participate for the determination of the new values of the neurons from the (i + 1)-st layer. These values are calculated by formula wi

◦ ((P(Vi, j , Vi+1,k )ϕi, j , ψi, j ) ∗ Mi, j,k , Ni, j,k .

(6.3.3)

j=1

In practice, formula (6.3.3) is new one for neural networks theory. In the theory, a function f is defined that changes the values that are similar to (6.3.3). Here, function f is changed with one of the operators over IVIFPs. Let Oα,β be a fixed operator and α, β ∈ [0, 1] are its fixed arguments. Then wi

ϕi+1,k , ψi+1,k  = Oα,β ( ◦ (P(Vi, j , Vi+1,k )ϕi, j , ψi, j ) ∗ Mi, j,k , Ni, j,k ). j=1

6.4 Interval Valued Intuitionistic Fuzzy Generalized Nets The concept of a Generalized Net (GN, see [8, 31, 35]) was introduced in 1982 as an extension of the Petri nets and their extensions and modifications. In 1985, in [32], two of GN-extensions were introduced and in 2001 in [19], these extensions were extended again. In [32] were defined the Intuitionistic Fuzzy GNs of first and second types (IFGN1s and IFGN2s) and in [19]—the Intuitionistic Fuzzy GNs of third and fourth types (IFGN3s and IFGN4s). The IFGN1 and IFGN2 are the first types of fuzzy Petri nets, published some years before Looney’s paper [184]. Similarly to Petri nets, the GNs in all their modifications contain transitions, places and tokens. n The GN-places are depicted as the symbol and the GN-transition (more precisely, this is the graphical structure of a transition) are depicted as the symbol A
indicates the transition’s conditions. A GN, as well as other nets, contains tokens which transfer from place to place. But for difference with all other nets, every GNtoken enters the net with an initial characteristic and during each transfer, it receives new characteristics. Every place has at most one arc entering and one arc exiting.

150

6 Applications of IVIFSs

Fig. 6.2 An IVIFGN1-transition

l1

m

.. . li

m

.. .  lm

m

r A
.. . . .. .. . . .. -



- m l1 .. .



- m lj .. .

 - mln

The places with no entering arcs are called input places of the net and those without exiting arcs are output places of the net. Each transition has at least one input and one output place. When tokens enter the input places of a transition, it becomes potentially fired (activated) and at the moment of their transfer towards the transition’s output places, it is fired. The transition is activated at a given time-moment and then becomes active again at some subsequent time-moment which is pre-determined. The first basic difference between GNs and the ordinary Petri nets is the “place— transition” relation. Here, transitions are objects of a more complex nature. Every transition contains m input and n output places, where m, n ≥ 1. Every IVIFGN1-transition is given by a seven-tuple (see Fig. 6.2): Z = L  , L  t1 , t2 , r, M,

,

where (a) L  and L  are finite, non-empty sets of places (the transition’s input and output places, respectively); (b) t1 is the current time-moment of the transition’s firing; (c) t2 is the current value of the duration of its activity; (d) r is the transition’s condition determining the tokens which will transfer from the transition’s inputs to its outputs; it has the form of an Index matrix: l1

r=

l1 . . . l j . . . ln

.. ri, j . li (ri, j − predicates) .. . (1 ≤ i ≤ m, 1 ≤ j ≤ n) lm

6.4 Interval Valued Intuitionistic Fuzzy Generalized Nets

151

where ri, j denotes the element of the IM, which corresponds to the i-th input and j-th output places. These elements are predicates and when the truth value of the (i, j)-th element f (ri, j ) = Mi, j , Ni, j  is valid according one of the 15 condition given below, the token from i-th input place can be transferred to j-th output place; otherwise, this is not possible, where Mi, j , Ni, j ⊆ [0, 1] and sup Mi, j + sup Ni, j ≤ 1 and function f is defined in point (e) of the definition of the IVIFGN1 E (see below). The conditions for interval valued intuitionistic fuzzy validity of the predicates are the following • • • • • • • • • • • • • • •

(C1) inf M = 1 (therefore sup M = 1 and inf N = sup N = 0), (C2) sup M = 1 (therefore inf N = sup N = 0), (C3) 21 < inf M ≤ 1 and sup N = 0, (C4) inf M > 21 > sup N , (C5) inf M ≥ 21 ≥ sup N , (C6) sup M > 21 > sup N , (C7) sup M ≥ 21 ≥ sup N , (C8) inf M > sup N , (C9) inf M ≥ sup N , (C10) sup M > sup N , (C11) sup M ≥ sup N , (C12) sup M > 0, (C13) inf M > 0, (C14) sup N < 1, (C15) inf N < 1.

Therefore, the tokens’s transfer from i-th input to j-th output places is possible for a fixed criterion, when the components of Mi, j , Ni, j  satisfy the respective criterion; (e) M is an IM of the capacities of transition’s arcs: l1

M=

l1 . . . l j . . . ln

.. Mi, j . li (Mi, j ≥ 0 − natural numbers ) .. (1 ≤ i ≤ m, 1 ≤ j ≤ n) . lm

(f) is an object whose form is similar to a propositional formula. It may contain symbols among the transition’s input places labels as variables, and the Boolean operations ∧ and ∨. We assign the following semantics to this formula: ∧(li1 , li2 , . . . , liu )—every place li1 , li2 , . . . , liu must contain at least one token, ∨(li1 , li2 , . . . , liu )—there must be at least one token in all places li1 , li2 , . . . , liu where {li1 , li2 , . . . , liu } ⊂ L  . When the value of a type, evaluated as a Boolean expression, is tr ue, the transition can become active, otherwise it cannot.

152

6 Applications of IVIFSs

The ordered four-tuple E = A, π A , π L , c, f, θ1 , θ2 , K , π K , θ K , T, t o , t ∗ , X, Φ, b is called an Interval Valued Intuitionistic Fuzzy Generalized Net of first type (IVIFGN1) if: (a) A is a set of transitions; (b) π A is a function giving the priorities of the transitions, i. e., πA : A → N , where N = {0, 1, 2, . . . } ∪ {∞}; (c) π L is a function giving the priorities of the places, i.e., π L : L → N , where L = pr1 A ∪ pr2 A, and pri X is the i-th projection of the n-dimensional set, where n ∈ N , n ≥ 1 and 1 ≤ k ≤ n (where L is the set of all GN-places); (d) c is a function giving the capacities of the places, i.e., c : L → N ; (e) f is a function which evaluates the truth values of the transition’s conditions predicates ri, j in the form Mi, j , Ni, j , where Mi, j , Ni, j ⊆ [0, 1] are closed sets, and sup Mi, j + sup Ni, j ≤ 1. (f) θ1 is a function giving the next time-moment when a transition can become active, i.e., θ1 (t) = t  , where t, t  ∈ [T, T + t ∗ ] and t ≤ t  . The value of this function is calculated at the moment when a transition terminates its active state; (g) θ2 is a function giving the duration of the active state of a transition, i.e., θ2 (t) = t  , where t ∈ [T, T + t ∗ ] and t  ≥ 0. The value of this function is calculated at the moment of activation; (h) K is the set of the GN’s tokens. (i) π K is a function giving the priorities of the tokens, i.e., π K : K → N ; (j) θ K is a function giving the time-moment when a token can enter the net, i.e., θ K (α) = t, where α ∈ K and t ∈ [T, T + t ∗ ]; (k) T is the time-moment when the GN starts functioning. This moment is determined with respect to a fixed (global) time-scale; (l) t o is an elementary time-step related to the fixed (global) time-scale; (m) t ∗ is the duration of the functioning of the net; (n) X is the set of all initial characteristics the tokens can receive when they enter the net; (o) Φ is a characteristic function which gives a new characteristic to every token when it moves from an input to an output place of a given transition; (p) b is a function giving the maximum number of characteristics a given token can receive, i.e., b : K → N . If for a certain token α, b(α) = 1, the token will enter the net with an initial characteristic (as a zero-characteristic). After this, it will keep only its current characteristic. When b(α) = ∞, the token α will receive all possible characteristics.

6.4 Interval Valued Intuitionistic Fuzzy Generalized Nets

153

When b(α) = k < ∞, except its zero-characteristic, the token α will keep the last k as its characteristics (older characteristics will be “forgotten”). Hence, in general, every token α has b(α) + 1 characteristics. In [20, 31, 35], two algorithms for the tokens transfer are given. Here, we modify them. The algorithm for tokens transfer after the time moment t1 = T I M E (here and below, we denote by T I M E the current time moment of the GN), denoted by algorithm A, takes into consideration the possibility of merging and splitting tokens. A token can be transferred from an input place to a certain output place, even when the capacity of this output place is reached, if the token can be merged with one or more tokens in this output place. This action will not affect the capacity of the output place. The actual number of tokens in the output place will not exceed the maximum number of tokens allowed. In order for a token α to be merged with one or more tokens {β1 , ..., βk } in an output place, these tokens have to be specified in the initial characteristics of the α token. A symbol S ∈ {Y S , N S } can be added as a second component of the initial characteristics of the α token. Y S means that the token can split and N S —that it cannot split. For example, the expression x0α = “{β1 , ..., βk }, Y S , x0α,∗ ”, denotes that the α token can split and can be merged with the tokens in the set {β1 , ..., βk }. The rest of the information of the token’s initial characteristics is stored in x0α,∗ . When the criterion Cn for 1 ≤ n ≤ 15 is fixed, the algorithm A is described in 12 steps, as follows: • (A01) Sort the input and output places of the transitions by their priorities. • (A02) Form two lists of tokens in each input place. The first list contains those of the tokens that can be transferred to a certain output place during the current time moment. Sort these tokens by their priorities. The second list is an empty one at first. These two lists shall be denoted with P1 (l) and P2 (l), respectively. • (A03) Generate an empty IM R that corresponds to the IM of the predicates r . Assign the value [0, 0], [1, 1] to all elements Ri, j of R which: – (A03a) are in a row that corresponds to an empty input place, i.e. there are no tokens in the input place that can be transferred to an output place of the current transition; – (A03b) are in a column that corresponds to a full output place which no tokens can be transferred to; – (A03c) are placed in (i, j) place for which the predicate ri, j is set as false or m i, j = 0, i.e. the current capacity of the arc between the i-th input place and the j-th output place is 0.

154

6 Applications of IVIFSs

Assign the value [1, 1], [0, 0] to those elements Ri, j of R that are placed in a place (i, j) for which the predicate ri, j is set as true. • (A04) Iterate through the input places in the order set by their priorities, starting with the place with highest priority for which no token has been transferred during the current time step and which has at least one token in it. The token that will be transferred, if possible, is the one with the highest priority in the P1 list of the current input place. Perform consequently the following steps in order to determine if and where to transfer the current token. – (A04a) Check the next Ri, j value of R. If the value of Ri, j has not been set yet, go to step (A04b). If Ri, j = [1, 1], [0, 0], go to step (A04c). If Ri, j = [0, 0], [1, 1], check if the corresponding output place has a token that can be merged with the one being transferred. If so, go to step (A04c). Otherwise, go to step (A04d). – (A04b) Evaluate the truth value of the corresponding predicate ri, j of the IM r . If ri, j satisfy criterion Cn, set the Ri, j value of R to Mi, j , Ni, j  and go to step (A04c). Otherwise, set the Ri, j value to [0, 0], [1, 1] and go to step (A04d). – (A04c) The current token is transferred to the corresponding jth output place and moved to its P2 list. The token is merged with specified tokens in the output place, if there are such. If the transferred token is not merged with other tokens or there are no other tokens left in the new host that the transferred one can be possibly merged with, evaluate the characteristic function of this output place. Assign this value as a characteristic of the transferred token. If there are tokens in the input places of the current transition that can be merged with the transferred one, it will wait in the output place until these tokens are moved. The new characteristics of the transferred token will be assigned after the token is merged with the last of the suitable tokens in the input places or right at the end of the transition functioning. Tokens, that have entered an input place of the current transition after its activation, are moved to the P2 list of the input place. – (A04d) If the current token cannot be split or all the predicates on the corresponding row are checked, go to step (A05). Otherwise, go to step (A04a). If the splitting of the current token is not allowed, the evaluation of the predicates stops with the first one evaluated as tr ue. The token then will be moved to the highest priority output place amongst those the token can be transferred to. If the splitting of the current token is allowed, then the token is split into as many tokens as the number of the Ri, j elements with evaluations satisfying criterion Cn. These new tokens are transferred to the corresponding output places. The characteristic functions of the output places are evaluated. The new characteristics are then assigned to the corresponding tokens upon entering the output places. For difference with the ordinary GNs, the tokens characteristics of the IVIFGN1s contain not only the evaluated values by the respective characteristic functions, but also the IVIFPs that are evaluations of the respective predicates.

6.4 Interval Valued Intuitionistic Fuzzy Generalized Nets

155

• (A05) If the highest priority token cannot be transferred during the current time step, move the token to the P2 list of the input place. • (A06) Increase by 1 the current number of tokens in each output place to which a token has been transferred if the token has not been merged with any of the other tokens in the host. Do not change the current number of tokens in the output place otherwise. • (A07) Decrease by 1 the current number of tokens in each input place from which a token has been transferred. If the current number of tokens in such an input place becomes 0, set to [0, 0], [1, 1] all the elements in the corresponding row of the IM R. • (A08) Decrease by 1 the capacities of all the arc through which a token has been transferred. If the current capacity of an arc becomes 0, assign [0, 0], [1, 1] to this element of the IM R that corresponds to the arc. • (A09) If there are more input places with lower priority from which no token has been transferred to an output place, go to step (A04). Otherwise, go to step (A10). • (A10) If the value of the current time is less than or equals t1 + t2 (the time components of the considered transition), go to (A04). Otherwise, go to step (A11). • (A11) End of the transition’s functioning. The general algorithm for the GN’s functioning, denoted by algorithm B, is described next. The concept of an Abstract Transition (AT) is introduced for the purpose of this algorithm as the union of all GN-transitions that are active at a given time moment. The algorithm B can be described as follows: • (B01) Put all α tokens for which θ K (α) ≤ T into the corresponding input places of the net. • (B02) Construct the GN’s AT. Initially it is empty. • (B03) Check if the value of the current time is less or equal to T + t ∗ . • (B04) If the answer to the question in (B03) is “no”, go to step (B12). Otherwise, go to step (B05). • (B05) Find those transitions for which t1 is greater than or equal to the current time. • (B06) Check the transitions’ types of all transitions determined on step (B05). The method used for the evaluation of the transitions’ types is as follows: – (B06a) replace the names of all places used as variables in the Boolean expression of the transition type with the value 0, if the corresponding place has no tokens in it at the current moment, and with the value 1, otherwise; – (B06b) calculate the truth value of the Boolean expression, result of (B06a). • (B07) Add to the AT those transitions, the transition types of which are evaluated as tr ue on step (B06b). • (B08) Apply algorithm A over the AT. • (B09) Remove from the AT those transitions which are inactive at the current time moment.

156

6 Applications of IVIFSs

• (B10) Increase the current time with t 0 . • (B11) Go to step (B03). • (B12) End of the GN’s functioning. Obviously, every standard GN is an IVIFGN. By analogy with Theorem 4.1.1, [31], we have proved the following Theorem 6.4.1 The functioning and the results of the work of every IVIFGN can be described by some ordinary GN. Proof Let the IVIFGN1 E be given. We shall construct a new, already ordinary, GN and we shall prove that both nets function equally. In the proof, we use the validity of Theorem 5.3.1, [35] that gives us possibility to research on the functioning and results of the works of the corresponding transitions in both nets (the given IVIFGN1 E and the ordinary GN G corresponding to it, that we will construct below). Let the components of IVIFGN1 E be marked by index E and these of GN G—by index G. Let GN G have the same graphical structure as GN E and let all its other components, without the function f G and the characteristic function ΦG , be the same in both nets. For example, the sets X E and X G are equal, i.e., the (equal) tokens in both nets will have equal initial characteristics and the transition condition predicates are also equal in both nets. Below, we shall construct the new E-components. Let for the real number x: ⎧ ⎨ 1 if x > 0 sg(x) = ⎩ 0 if x ≤ 0 and sg(x) =

⎧ ⎨ 0 if x > 0 ⎩

1 if x ≤ 0

Function f G is defined in respect of the choice of criterion Cn, as follows: • • • • • • • • • •

in case (C1): f G (ri, j ) = sg(inf Mi, j ).sg(sup Ni, j ); in case (C2): f G (ri, j ) = sg(sup Mi, j ).sg(sup Ni, j ); in case (C3): f G (ri, j ) = sg(inf Mi, j − 21 ).sg(sup Ni, j ); in case (C4): f G (ri, j ) = sg(inf Mi, j − 21 ).sg( 21 − sup Ni, j ); in case (C5): f G (ri, j ) = sg( 21 − inf Mi, j ).sg(sup Ni, j − 21 ); in case (C6): f G (ri, j ) = sg(sup Mi, j − 21 ).sg( 21 − sup Ni, j ); in case (C7): f G (ri, j ) = sg( 21 − sup Mi, j ).sg(sup Ni, j − 21 ); in case (C8): f G (ri, j ) = sg(inf Mi, j − sup Ni, j ); in case (C9): f G (ri, j ) = sg(inf Mi, j − sup Ni, j ); in case (C10): f G (ri, j ) = sg(sup Mi, j − sup Ni, j );

6.4 Interval Valued Intuitionistic Fuzzy Generalized Nets

• • • • •

in case (C11): in case (C12): in case (C13): in case (C14): in case (C15):

157

f G (ri, j ) = sg(sup Mi, j − sup Ni, j ); f G (ri, j ) = sg(sup Mi, j ); f G (ri, j ) = sg(inf Mi, j ); f G (ri, j ) = sg(1 − sup Ni, j ); f G (ri, j ) = sg(1 − inf Ni, j ).

Therefore, function f G gives as a result value f G (ri, j ) =

⎧ ⎨ 1, if criterion Cn is satisfied ⎩

0, otherwise

Let the characteristic function ΦG be defined by ΦG (α) = “Φ E (α), f E (ri, j )”. Thus, we described the new components of the GN G. Now, we must show that both nets function equally. Let Z and Z ∗ be corresponding transitions in the two nets. We shall show that they function equally. The fact that they will have equal tokens in equal input places if they are transitions from first level in the sense of Theorem 5.3.1, [35], is obvious and by induction we will admit that they have equal tokens with equal current characteristics ∗ in equal time moments t1Z and t1Z . Then following the sequentially steps of the Algorithm A for token’s transfer and its IVIFGN1’s modification, we see that in the case of the IVIFGN E the IM is constructed l1

l1 . . . l j . . . ln

.. f E (ri, j ) . li (ri, j − predicates) .. . (1 ≤ i ≤ m, 1 ≤ j ≤ n) lm where f E (ri, j ) ∈ P([0, 1]) × P([0, 1]) and for the GN G—the IM is l1

l1 . . . l j . . . ln

.. f G (ri, j ) . , li (ri, j − predicates) .. . (1 ≤ i ≤ m, 1 ≤ j ≤ n) lm where for every set Y , P(Y ) is the set of the subsets of set Y .

158

6 Applications of IVIFSs

Therefore, from the above we see that the corresponding elements of both IMs are equal. Therefore, the tokens in both transitions will made equal transitions from input to output transition places. There, they will receive similar new characteristics. The tokens from the new net will receive more values, but for the proof it is important that they will receive all the values which the tokens will receive from the first net. Hence, both transitions really function equally. The ordered four-tuple E = A, π A , π L , c, f, θ1 , θ2 , K , π K , θ K , T, t o , t ∗ , X, Φ, b is called an Interval Valued Intuitionistic Fuzzy Generalized Net of third type (IVIFGN3) if: (a) A is a set of transitions defined in the definition of an IVIFGN1-transition; points (b)–(n) and (p) from the definition of an IVIFGN1, are valid again, while point (o) has the form (o’) Φ is a characteristic function which gives a new characteristic to every token when it moves from an input to an output place of a given transition and its interval valued intuitionistic fuzzy evaluation Mi,∗ j , N ∗i, j , where Mi,∗ j , N ∗i, j ⊆ [0, 1] and sup Mi,∗ j + sup N ∗i, j ≤ 1. There exist some changes in the first algorithm for the tokens transfer, described in previous Section. The following assertions hold of IVIFGN3 similarly to IVIFGN1. Theorem 6.4.2 For every IVIFGN3 there exists a standard GN which represents it. In [14, 19, 26, 31, 32, 35, 37], the tokens of the new type of GNs, that we will discuss below, are called “quantities”, but this word here has another sense. By this reason here and in future, the word “quantity” will be replaced with the word “fluid” (cf. point (h) of Definition 2). Of course, we will continue to use word “token” instead of “fluid”, whenever this is possible. Every IVIFGN2-transition is given by a seven-tuple (it has the same form as the IVIFGN1-transition from Fig. 6.2): Z = L  , L  t1 , t2 , r, M,

,

where (a) L  and L  are finite, non-empty sets of places (the transition’s input and output places, respectively); (b) t1 is the current time-moment of the transition’s firing; (c) t2 is the current value of the duration of its activity; (d) r is the transition’s condition determining the fluid which will transfer from the transition’s inputs to its outputs; it has the form of an Index matrix:

6.4 Interval Valued Intuitionistic Fuzzy Generalized Nets

l1 . . . l j . . . ln

l1

r=

159

.. ri, j . li (ri, j − predicates) .. . (1 ≤ i ≤ m, 1 ≤ j ≤ n) lm

where ri, j denotes the element of the IM, which corresponds to the i-th input and j-th output places. These elements are predicates and when the truth value of the (i, j)-th element f (ri, j ) = Mi, j , Ni, j , that here and below is an IVIFP, is valid according one of the 15 condition (C1)–(C15) given above, the fluid from i-th input place can be transferred to j-th output place; otherwise, this is not possible, where Mi, j , Ni, j ⊆ [0, 1] and sup Mi, j + sup Ni, j ≤ 1 and function f is defined in point e) of the definition of an IVIFGN1. The conditions for interval valued intuitionistic fuzzy validity of the predicates are the same, as in the cases of IVIFGN1s and IVIFGN3s. Therefore, the fluid’s transfer from i-th input to j-th output places is possible for a fixed criterion, when the components of Mi, j , Ni, j  satisfy the respective criterion. In the case of IVIFGN2s and IVIFGN4s, the following additional conditions must be valid: if the fluid in input place li splits to s parts that must enter output places l j1 , l j2 , ..., l js , then s  sup Mi, jk ≤ 1, k=1 s 

inf Ni, jk ≥ 0;

k=1

(e) M is an IM of the capacities of transition’s arcs: l1

M=

l1 . . . l j . . . ln

.. . m i, j , li (m i, j ∈ R+ , .. . 1 ≤ i ≤ m, 1 ≤ j ≤ n) lm

where R+ is the set of real non-negative numbers (f) is an object whose form is a Boolean expression that contains identifiers of the transition’s input places as variables, and the Boolean operations ∧ and ∨. We assign the following semantics to this formula:

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6 Applications of IVIFSs

∧(li1 , li2 , . . . , liu )—every place li1 , li2 , . . . , liu must contain fluid, ∨(li1 , li2 , . . . , liu )—there must be least one fluid in all places li1 , li2 , . . . , liu where {li1 , li2 , . . . , liu } ⊂ L  . When the value of a type, evaluated as a Boolean expression, is tr ue, the transition can become active, otherwise it cannot. The ordered four-tuple E = A, π A , π L , c, f, θ1 , θ2 , K , π K , θ K , T, t o , t ∗ , X, Φ is called an Interval Valued Intuitionistic Fuzzy Generalized Net of second type (IVIFGN2) if: (a) A is a set of the transitions; (b) π A is a function giving the priorities of the transitions, i. e., πA : A → N , where N = {0, 1, 2, . . . } ∪ {∞}; (c) π L is a function giving the priorities of the places, i.e., π L : L → N , where L = pr1 A ∪ pr2 A, and pri X is the i-th projection of the n-dimensional set, where n ∈ N , n ≥ 1 and 1 ≤ k ≤ n (where L is the set of all GN-places); (d) c is a function giving the capacities of the places, i.e., c : L → R+ ; (e) f is a function which evaluates the truth values of the transition’s conditions predicates ri, j in the form Mi, j , Ni, j , where Mi, j , Ni, j ⊆ [0, 1] are closed sets, and sup Mi, j + sup Ni, j ≤ 1. (f) θ1 is a function giving the next time-moment when a transition can become active, i.e., θ1 (t) = t  , where t, t  ∈ [T, T + t ∗ ] and t ≤ t  . The value of this function is calculated at the moment when a transition terminates its active state; (g) θ2 is a function giving the duration of the active state of a transition, i.e., θ2 (t) = t  , where t ∈ [T, T + t ∗ ] and t  ≥ 0. The value of this function is calculated at the moment of activation; (h) K is the set of the fluids that enter and flow through the GN. Similarly to IFGN2s and IFGN4s, the IVIFGN2-fluids have only initial characteristics with the form “type and quantity of the ‘fluid”’ (element of set X , defined below) and do not have current characteristics; (i) π K is a function giving the priorities of the fluids, i.e., π K : K → N ; (j) θ K is a function giving the time-moment when fluid can enter the net, i.e., θ K (α) = t, where α ∈ K and t ∈ [T, T + t ∗ ]; (k) T is the time-moment when the GN starts functioning. This moment is determined with respect to a fixed (global) time-scale; (l) t o is an elementary time-step related to the fixed (global) time-scale; (m) t ∗ is the duration of the functioning of the net; (n) X is the set of all initial characteristics that the fluids can receive when they enter the net;

6.4 Interval Valued Intuitionistic Fuzzy Generalized Nets

161

Fig. 6.3 An example

R1

S g1 @ @ @ @ @ @

S g2

R2

(o) Φ is a characteristic function which gives a characteristic to each place when the fluid enters it. Now, places collect all their characteristics and therefore, an analogous of function b from the GN-definition is superfluous. We shall illustrate the idea for the IVIFGN2 by the following example, showing on Fig. 6.3. Let us have two reservoirs R1 and R2 connected with a tube that has on its two boundaries two sensors S1 and S2 . Let reservoir R1 contains quantity Q of some fluid. When the predicate P = “the fluid from reservoir R1 flows to reservoir R2 ” has an evaluation f (P) = M1,2 , N1,2 , then this corresponds to the case in which in the reservoir R1 stay quantity Q inf N1,2 of the fluid, in the reservoir R2 enter quantity Q inf M1,2 of the fluid, in the tube stay quantity Q(1 − sup M1,2 − sup N1,2 ) of the fluid, the sensor S1 has a tolerance sup N1,2 − inf N1,2 and the sensor S2 has a tolerance sup M1,2 − inf M1,2 . In [20, 31, 35], two algorithms for the fluids transfer are given. Here, we modify them. The algorithm for fluid’s transfer after the time moment t1 = T I M E (here and below, we denote by T I M E the current time moment of the GN), denoted by algorithm A, takes into consideration the possibility of merging and splitting of the fluids. When the criterion Cn for 1 ≤ n ≤ 15 is fixed, the algorithm A is described in 12 steps, as follows: • (A01) Sort the input and output places of the transitions by their priorities. • (A02) If the transition contains input places that are inputs for the IVIFGN2, the fluids that can enter the respective places enter them and places obtain as characteristics the initial fluid characteristics. • (A03) Generate an empty IM R that corresponds to the IM of the predicates r . Assign the value [0, 0], [1, 1] to all elements Ri, j of R which:

162

6 Applications of IVIFSs

– (A03a) are in a row that corresponds to an empty input place, i.e. there are no fluids in the input place that can be transferred to an output place of the current transition; – (A03b) are in a column that corresponds to a full output place which no fluids can be transferred to; – (A03c) are placed in (i, j) place for which the predicate ri, j is set as false or m i, j = 0, i.e. the current capacity of the arc between the i-th input place and the j-th output place is 0. Assign the value [1, 1], [0, 0] to those elements Ri, j of R that are placed in a place (i, j) for which the predicate ri, j is set as true. • (A04) Iterate through the input places in the order set by their priorities, starting with the place with highest priority for which no fluid has been transferred during the current time step and which has fluid in it. Perform consequently the following steps in order to determine if and where to transfer the current fluid. – (A04a) Check the current Ri, j value of R. If the value of Ri, j has not been set yet, go to step (A04b). If Ri, j = [1, 1], [0, 0], go to step (A04c). Otherwise, go to step (A04d). – (A04b) Evaluate the truth value of the corresponding predicate ri, j of the IM r . If ri, j satisfy criterion Cn, set the Ri, j value of R to Mi, j , Ni, j  and go to step (A04c). Otherwise, set the Ri, j value to [0, 0], [1, 1] and go to step (A04d). – (A04c) The current fluid is transferred from input place li to the corresponding output (one or more) place l j . The fluid is merged with specified parts in the output places, if there are such. After this, the characteristic function of the output place is evaluated, adding to it the type of the fluid in the place and its quantity that is determined by formula qi, j = min(m i, j , c(li , T I M E) inf Mi, j , c(l j ) − c(l j , T I M E)). – (A04d) If the current fluid cannot be split or all the predicates on the corresponding row are checked, go to step (A05). Otherwise, go to step (A04a). If the splitting of the current fluid is not allowed, the evaluation of the predicates stops with the first one which evaluation satisfy condition Cn for the fixed number n. The fluid then will be moved to the highest priority output place amongst those, where the fluid can be transferred too. If the splitting of the current fluid is allowed, then the fluid is split into as many parts as the number of the Ri, j elements with evaluations satisfying criterion Cn. These new fluids are transferred to the corresponding output places. The characteristic functions of the output places are evaluated as above. The new characteristics are then assigned to the corresponding output places. For difference with the ordinary GNs, the fluids’ characteristics of the IVIFGN2s contain not only the evaluated values by the respective characteristic functions, but also the IVIFPs that are evaluations of the respective predicates.

6.4 Interval Valued Intuitionistic Fuzzy Generalized Nets

163

• (A05) If the highest priority fluid cannot be transferred during the current time step, it continues to stay in the input place. • (A06) Increase by qi, j the current quantity of the fluid in each output place to which fluid has been transferred. s inf Ni, j the quantity of the fluid in place li from which • (A07) Decrease by qi, j + k=1

fluid has been transferred to s output places . If the quantity of the fluid in an input place becomes 0, set to [0, 0], [1, 1] all the elements in the corresponding row of the IM R. s • (A08) Decrease by c(li , T I M E) (1 − sup Mi, j − sup Ni, j ) the capacities of all k=1

• • • •

the arc through which a fluid has been transferred. If the current capacity of an arc becomes 0, assign [0, 0], [1, 1] to this element of the IM R that corresponds to the arc. (A09) If there are more input places with lower priority from which no fluid has been transferred to an output place, go to step (A04). Otherwise, go to step (A10). (A10) Add t 0 to the current model time. (A11) If the value of the current time is less than or equals t1 + t2 (the time components of the considered transition), go to (A04). Otherwise, go to step (A12). (A12) End of the transition’s functioning.

As above, the general algorithm for the GN’s functioning, denoted by algorithm B, is described as follows: • (B01) Put all α fluids for which θ K (α) ≤ T into the corresponding input places of the net. • (B02) Construct the GN’s AT. Initially it is empty. • (B03) Check if the value of the current time is less or equal to T + t ∗ . • (B04) If the answer to the question in (B03) is “no”, go to step (B12). Otherwise, go to step (B05). • (B05) Find those transitions for which t1 is greater than or equal to the current time. • (B06) Check the transitions’ types of all transitions determined on step (B05). The method used for the evaluation of the transitions’ types is as follows: – (B06a) replace the names of all places used as variables in the Boolean expression of the transition type with the value 0, if the corresponding place has no fluid in it at the current moment, and with the value 1, otherwise; – (B06b) calculate the truth value of the Boolean expression, result of (B06a). • (B07) Add to the AT those transitions, the transition types of which are evaluated as tr ue on step (B06b). • (B08) Apply algorithm A over the AT. • (B09) Remove from the AT those transitions which are inactive at the current time moment. • (B10) Increase the current time with t 0 . • (B11) Go to step (B03). • (B12) End of the GN’s functioning.

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6 Applications of IVIFSs

By analogy with Theorem 6.4.1, we can proved the following Theorem 6.4.3 The functioning and the results of the work of every IVIFGN2 can be described by some ordinary GN. The ordered four-tuple E = A, π A , π L , c, f, θ1 , θ2 , K , π K , θ K , T, t o , t ∗ , X, Φ is called an Interval Valued Intuitionistic Fuzzy Generalized Net of fourth type (IVIFGN4) if: (a) A is a set of transitions defined in the definition of an IVIFGN1-transition; Points (b)–(n) and (p) from the definition of an IVIFGN1, are valid again, while point o) has the form o’) Φ is a characteristic function which gives a new characteristic to every place when in it enters fluid and its interval valued intuitionistic fuzzy evaluation Mi,∗ j , N ∗i, j , where Mi,∗ j , N ∗i, j ⊆ [0, 1] and sup Mi,∗ j + sup N ∗i, j ≤ 1. There exist small addition in the first algorithm for the fluid transfer, described in previous Section, related to the new form of place characteristics. The following assertion holds. Theorem 6.4.4 For every IVIFGN4 there exists a standard GN which represents it. Different operations and relations are defined in the algebraic aspect of GN theory [31, 35]. These operations have significant practical importance. For example, if two different processes which flow parallelly in time are modelled by two GNs, then by the union operation on these nets a new GN can be constructed to describe the whole process, and it is a union of the two nets. For other types of nets, this is not directly possible. The defined already operators also have an important place in GN theory. In the near future, similar operations, relations and operators will be defined for the IVIFGN1s—IVIFGN4s. Following the ideas from [29, 40, 334], in future, we will discuss the possible applications of the IVIFGN1s and IVIFGN3s as tools in the Data Mining instruments. Also, different processes, described by GNs will be re-modelled by IVIFGNs from one of the four types and especially in the cases, when the processes flow in uncertainty.

6.5 Intercriteria Analysis with Interval-Valued Intuitionistic Fuzzy Evaluations During the last five years, the idea of Intercriteria Analysis (ICA) has been developed (see, e.g., [25, 42–46, 58, 101, 102, 186, 233, 272, 278, 335]) and a lot of its applications were described in industry [21, 121, 262, 263, 333], economics [48, 49, 100, 103, 147, 273], education [57, 59, 60, 154, 257–259], medical and biotechnological

6.5 Intercriteria Analysis with Interval-Valued Intuitionistic …

165

processes [17, 157, 207, 221–223, 225, 227, 231, 234, 270, 276, 320, 321], genetic algorithms and metaheuristics [11, 15, 16, 112–115, 207–210, 219, 224, 226, 229, 230, 232, 235], neural networks [247, 248, 254]. The software implementation of the ICA-algorithms is described in [143, 228]. The ICA was based on the theories of IMs and IFSs. Here, following paper [34] written by Pencho Mariniv, Vassia Atanassova and the author, a variant of ICA will be proposed, based on one of the IVIFSs and IVIFPs . Let us have an IM

C1 .. . Ck A= . .. Cl .. . Cm

O1 ... Oi aC1 ,O1 ... aC1 ,Oi .. .. .. . . . aCk ,O1 ... aCk ,Oi .. .. .. . . . aCl ,O1 ... aCl ,Oi .. .. .. . . . aCm ,O1 ... aCm ,Oi

... O j ... aC1 ,O j .. .. . . ... aCk ,O j .. .. . . ... aCl ,O j .. .. . . ... aCm ,O j

... On ... aC1 ,On .. .. . . ... aCk ,On .. .. , . . ... aCl ,On .. .. . . ... aCm ,On

where for every p, q, (1 ≤ p ≤ m, 1 ≤ q ≤ n): • C p is a criterion, • Oq is an object, • aC p ,Oq is a real number or another object, including the empty place in the matrix, marked by ⊥, that is comparable about relation R with the other a-objects, so that for each i, j, k: R(aCk ,Oi , aCk ,O j ) is defined. Let R be the opposite relation of R in the sense that if R is satified, then R is not satisfied and vice versa. For example, if “R” is the relation “”, and vice versa. In the research until the moment we have discussed the case when the relation between two objects aCk ,Oi and aCk ,O j , evaluated by real numbers, was a relation of equality (=) with the degree of uncertainty. The reason was that if we work by the first digit after decimal point and if both objects are evaluated, e.g., as 2.1, then we can adopt them as equal, but if we work with two digits after decimal ponit and if the first object has an estimation 2.15, then the probability the second object to have the same estimation is only 10%. We must note that if we have real numbers in fixed-point format, i.e. precision of k digits after the decimal point, we can multiply all numbers by 10k and then all a-arguments will be integers. In the general case for the real numbers in floatingpoint format we may assume some precision ε and two numbers c1 and c2 coinside if |c1 − c2 | ≤ ε. When the evaluations of the ICA-procedure is based on the apparatus of the IFSs, we must have all evaluations of the objects. If there are omitted values (denoted by ⊥), then we must: (a) ignore the row with the omitted evaluation, or (b) ignore the

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6 Applications of IVIFSs

column with the omitted evaluation, or (c) work with these objects as in the case of equality. ⊥ ⊥ ⊥ Let the real numbers αoL , αoR , β Lo , β oR , α⊥ L , α R , β L , β R be fixed and let them satisfy the inequalities 0 ≤ αoL ≤ αoR ≤ 1,

0 ≤ β Lo ≤ β oR ≤ 1,

⊥ 0 ≤ α⊥ L ≤ α R ≤ 1,

0 ≤ β L⊥ ≤ β ⊥ R ≤ 1,

0 ≤ αoL + αoR + β Lo + β oR ≤ 1, ⊥ ⊥ ⊥ 0 ≤ α⊥ L + α R + β L + β R ≤ 1.

There are some ICA-algorithms with respect to the type of a-elements. We will discuss them sequentially. We start with the general case. Let μ

• SCk ,Cl be the number of cases in which R(aCk ,Oi , aCk ,O j ) and R(aCl ,Oi , aCl ,O j ) are simultaneously satisfied and the a-arguments are different from ⊥. • SCν k ,Cl be the number of cases in which R(aCk ,Oi , aCk ,O j ) and R(aCl ,Oi , aCl ,O j ) are simultaneously satisfied and the a-arguments are different from ⊥. • SC⊥k ,Cl be the number of cases in which none of the relations R(aCk ,Oi , aCk ,O j ) and R(aCk ,Oi , aCk ,O j ) or none of the relations R(aCl ,Oi , aCl ,O j ) and R(aCl ,Oi , aCl ,O j ) is satisfied, because at least one of the elements aCk ,Oi and aCk ,O j or aCl ,Oi and aCl ,O j is ⊥ (is empty). • SCo k ,Cl be the number of cases in which aCk ,Oi = aCk ,O j , or aCl ,Oi = aCl ,O j , and all a-arguments are different from ⊥. Let

SCπk ,Cl = SC⊥k ,Cl + SCo k ,Cl Np =

n(n − 1) . 2

Obviously, μ

ν π SCk ,Cl + Sk,l + Sk,l =

n(n − 1) = N p. 2

Let us assume everywhere below that each interval X has the form [inf X, sup X ].

6.5 Intercriteria Analysis with Interval-Valued Intuitionistic …

167

Now, for every k, l, such that 1 ≤ k < l ≤ m and for n ≥ 2, we define μ

inf MCk ,Cl =

⊥ SCk ,Cl + αoL SCo k ,Cl + α⊥ L SCk ,Cl

Np

,

(6.5.1)

,

(6.5.2)

,

(6.5.3)

.

(6.5.4)

μ

sup MCk ,Cl =

inf NCk ,Cl =

sup NCk ,Cl =

SCk ,Cl + αoR SCo k ,Cl + α⊥R SC⊥k ,Cl Np SCν k ,Cl + β Lo SCo k ,Cl + β L⊥ SC⊥k ,Cl Np ⊥ SCν k ,Cl + β oR SCo k ,Cl + β ⊥ R SCk ,Cl

Np

Hence, we can construct the intervals MCk ,Cl = [inf MCk ,Cl , sup MCk ,Cl ]

(6.5.5)

NCk ,Cl = [inf NCk ,Cl , sup NCk ,Cl ],

(6.5.6)

and

so that sup MCk ,Cl + sup NCk ,Cl μ

=

SCk ,Cl + αoR SCo k ,Cl + α⊥R SC⊥k ,Cl

=

Np

+

⊥ SCν k ,Cl + β oR SCo k ,Cl + β ⊥ R SCk ,Cl

Np

1 μ ⊥ (S + SCν k ,Cl + (αoR + β oR )SCo k ,Cl + (α⊥R + β ⊥ R )SCk ,Cl ) N p Ck ,Cl μ

≤

SCk ,Cl + SCν k ,Cl + SCo k ,Cl + SC⊥k ,Cl Np

≤ 1.

Now, we construct the interval PCk ,Cl = [inf PCk ,Cl , sup PCk ,Cl ] = [1 − sup MCk ,Cl − sup NCk ,Cl , 1 − inf MCk ,Cl − inf NCk ,Cl ] . The simplest case, in which there are no intervals, is obtained for

(6.5.7)

168

6 Applications of IVIFSs ⊥ ⊥ ⊥ αoL = αoR = β Lo = β oR = α⊥ L = α R = β L = β R = 0.

In the simple form of the optimistic approach the α- and β-constants are 1 , 2

α⊥ L = 0,

α⊥R =

1 , 2

β Lo = β oR = 0,

β L⊥ = 0,

β⊥ R =

1 . 2

αoL = αoR =

Then

μ

inf MCk ,Cl =

SCk ,Cl + 21 SCo k ,Cl Np

,

μ

sup MCk ,Cl =

SCk ,Cl + 21 SCo k ,Cl + 21 SC⊥k ,Cl Np

inf NCk ,Cl =

sup NCk ,Cl =

SCν k ,Cl Np

,

,

SCν k ,Cl + 21 SC⊥k ,Cl Np

.

In the simple form of the pessimistic approach the α- and β-constants are αoL = αoR = 0,

α⊥ L = 0,

α⊥R =

1 , 2

1 , 2

β L⊥ = 0,

β⊥ R =

1 . 2

β Lo = β oR = Then

μ

inf MCk ,Cl =

SCk ,Cl Np

,

μ

sup MCk ,Cl =

inf NCk ,Cl =

sup NCk ,Cl =

SCk ,Cl + 21 SC⊥k ,Cl Np SCν k ,Cl + 21 SCo k ,Cl Np

,

,

SCν k ,Cl + 21 SCo k ,Cl + 21 SC⊥k ,Cl Np

.

6.5 Intercriteria Analysis with Interval-Valued Intuitionistic …

169

The general case has two forms: standard and uniform. For the standard (α − β)-approach, let α, β ∈ [0, 1] be two fixed numbers, so that α + β ≤ 1 and let αoL = αoR = α

⊥ α⊥ L = 0 , αR =

1 2

β Lo = β oR = β ,

β L⊥ = 0 , β ⊥ R =

1 2

Then

μ

inf MCk ,Cl =

SCk ,Cl + αSCo k ,Cl Np

,

μ

sup MCk ,Cl =

SCk ,Cl + αSCo k ,Cl + 21 SC⊥k ,Cl Np

inf NCk ,Cl =

sup NCk ,Cl =

SCν k ,Cl + β SCo k ,Cl Np

,

,

SCν k ,Cl + β SCo k ,Cl + 21 SC⊥k ,Cl Np

.

 For the uniform (α − β)-approach, let α, β ∈ 0, 41 be two fixed numbers, so that α + β ≤ 21 and let ⊥ αoL = β Lo = α⊥ L = β L = α, αoR = β oR = α⊥R = β ⊥ R = β. Then

μ

inf MCk ,Cl =

SCk ,Cl + αSCo k ,Cl + αSC⊥k ,Cl Np

,

μ

sup MCk ,Cl =

inf NCk ,Cl =

sup NCk ,Cl =

SCk ,Cl + β SCo k ,Cl + β SC⊥k ,Cl Np SCν k ,Cl + αSCo k ,Cl + αSC⊥k ,Cl Np SCν k ,Cl + β SCo k ,Cl + β SC⊥k ,Cl Np

,

,

.

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6 Applications of IVIFSs

Using the above values for pairs MCk ,Cl , NCk ,Cl , we can construct the final form of the IM that determines the degrees of correspondence between criteria C1 , ..., Cm : ... Cm . . . MC1 ,Cm , NC1 ,Cm  . .. .. . . Cm MCm ,C1 , NCm ,C1  . . . MCm ,Cm , NCm ,Cm  C1 C1 MC1 ,C1 , NC1 ,C1  .. .. . .

If we know which criteria are more complex, or that their evaluation is more expensive, or it needs longer time, then we can omit these criteria keeping the simpler, cheaper or those requiring less time. Now, we discuss a procedure for simplifying the IM that determines the degrees of correspondence between criteria. Let γ, δ ∈ [0, 1] be given, so that γ + δ ≤ 1. We say that criteria Ck and Cl are in • strong (γ, δ)-positive consonance, if inf MCk ,Cl > γ and sup NCk ,Cl < δ; • weak (γ, δ)-positive consonance, if sup MCk ,Cl > γ and inf NCk ,Cl < δ; • strong (γ, δ)-negative consonance, if sup MCk ,Cl < γ and inf NCk ,Cl > δ; • weak (γ, δ)-negative consonance, if inf MCk ,Cl < γ and sup NCk ,Cl > δ; • (γ, δ)-dissonance, otherwise. Analogically, we can compare the objects, determining which of them are in strong (γ, δ)-positive, weak (γ, δ)-positive, strong (γ, δ)-negative, weak (γ, δ)-negative consonance, or in (γ, δ)-dissonance.

6.6 Interval Valued Intuitionistic Fuzzy Sets as Tools for Evaluation of Data Mining Processes—Possibilities for the Future Following [40], we ask: “What is Data Mining”? The answer of this question is so unclear, as well as the answer of the question for the areas of the AI. Again, there are

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171

different answers in respect of the opinions of the specialists, giving answers. For example: “The aim of DM is to make sense of large amounts of mostly unsupervised data, in some domain” [96]; “The aim of DM is to extract implicit, previously unknown and potentially useful (or actionable) patterns from data. DM consists of many up-to-date techniques such as classification (decision trees, naive Bayes classifier, k-nearest neighbor, NNs), clustering (k-means, hierarchical clustering, density-based clusteering), association (one-dimensional, multi-dimensional, multilevel association, constraint-based association)” [264]; “DM stands at the confluence of the fields of statistics and machine learning” [241]; “DM is a term that covers a broad range of techniques being used in a variety of industries” [237]; “DM is the core of the knowledge discovery in databases process, involving the inferring of algorithms that explore the data, develop the model and discover previously unknown patterns” [185]; The Data Mining is a process of finding reasonable correlations, repeating patterns and trends in large DBs and Big Data (BD). There are a lot of papers and books, devoted to DM. As a basis of our research, we use the publications [10, 51, 55, 56, 67, 95–98, 125–127, 132–134, 136–138, 148, 151, 152, 185, 188, 192, 198, 203, 206, 236–238, 241, 242, 260, 264, 292, 304]. In the literature, different areas of the AI are determined as components of the DM. For example, the algorithms of decision making, pattern recognition, neural networks, genetic algorithms, etc. Extending and modifying the text from [30, 40], here we make a review of some of the problems related to the above ones—those already existing, and those planned for future research. Everywhere we emphasize on: • the way of the IVIFS-estimation of the process (object) up to now (if such already exists); • other ways for IVIFS-realization of this; • possible extensions or generalizations of already existing IFL-estimations of the corresponding processes (objects) and ways for their modifications.

6.6.1 IVIF-Estimations in Expert Systems, Data Bases, Data Warehouses, Big Data, OLAP-Structures As the author mentioned in [40], “A lot of colleagues already assert that the Expert Systems (ESs)are dying. The author supports the idea that they will live their “Renaissance”, obtaining a special place in the instrumentation of DM. Preserving their basic purpose to generate a new knowledge by answering to hypotheses, we can essentially extend the area of their possibilities. When some unclear situation arises

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6 Applications of IVIFSs

in a process controlled by DM-tools, and when some hypotheses for its future development are generated, then the new type of ESs can help.” In [36], the concept of an Intuitionistic Fuzzy ES (IFES) was introduced. It was essentially extended in [30, 92, 93, 153]. In these ESs, each fact F has IFestimations μ(F), ν(F), determining its degrees of validity and non-validity. So, the answer whether a given hypothesis is valid or not, obtains essentially more exact evaluation. In near future, we will introduce an extension of the IFES whose facts will have the IVIF-estimations M(F), N (F), where M(F), N (F) ⊆ [0, 1]and sup M(x) + sup N (x) ≤ 1. So, we eilll define Interval Valued IFES (IVIFES). A next step of the extensions will be introducing of facts that contain moments of time, when they started to be valid, and moments in which they finished being valid (a sequence of time-moments t1 , t2 , ..., tn ). Then (cf. [30]), on the one hand we can answer to questions related to the time (“at the moment”, “once”’, “sometimes”, “for long/short time”, “often”, “rarely”, “for short period”, “for long period”, etc.). On the other hand, the IVIFES rules can have essentially complex forms, containing different logical operations (conjunction, disjunction, implication, negation, ...), quantifiers (“for existence” and “for all”) and modal operators in their antecedents. In addition, the facts and rules can have priorities that will determine whether a given fact or rule can stay in the DB or must be changed with another one. In future, the ES-answers can be extended additionally, so they can have optimistic, pessimistic, neutral or other form. Similar directions for extensions of the Data Bases (DBs), Data Warehouses (DWs), Big Data (BD), OLAP-structures, etc. can be realized. As we assumed in [40] writting for IFESs and now—for IVIFESs, the solving of each of the above problems or, of course, all of them, will promote not only the theory and application of IVIFSs, but also the research in the area of DM, too. In the next section, an example is given that can be used as an illustration for determining of M- and N -evaluations of the facts.

6.6.2 IVIF-Estimations of a Procedure for Inductive Reasoning As it is mentioned in [127], “the rule induction is one of the fundamental tools of DM. Usually rules are expressions of the form if (attribute1 , value1 )&(attribute2 , value2 )& ... &(attributen , valuen ) then (decision, value). If we use the IFLs tools, we obtain sequentially (attribute1 , value1 , μ1 , ν1 ),

6.6 Interval Valued Intuitionistic Fuzzy Sets as Tools for Evaluation …

173

(attribute2 , value2 , μ2 , ν2 ), ... (attributen , valuen , μn , νn ), where, in the simplest case ⎧ ⎨ 1, if valuei is anticipated (expected, correct, etc.) μi = 0, if valuei is not anticipated (not expected, incorrect, etc.) ⎩ ∗, if there is not an information for attributei and νi = 1 − μi , but in a more general case μi , νi ∈ [0, 1] and μi + νi ≤ 1. Now, the final estimation can have the IVIF-form (decision, value, M, N ), where M, N ⊆ [0, 1] and sup M + sup N ≤ 1, and Let p be the number of degrees μi that are equal to 1, q be the number of degrees νi that are equal to 1, r be the number of degrees μi that satisfy 1 > μi > 21 , s be the number of degrees νi that satisfy 1 > νi > 21 . Obviously, p + q + r + s ≤ n. Now, the final estimation can have the IVIF-form (decision, value, M, N ), where

p p+r , , M= n n

N=

q q +s , . n n

Therefore, M, N ⊆ [0, 1] and sup M + sup N ≤ 1. Hence, we obtain a more precise estimation for the validity of the procedure for inductive reasoning than the cases of standard, fuzzy and intuitionistic fuzzy inductive reasoning. If in the beginning we determine some treshold of validity tv , then we can assert that a decision is positive sufficiently valid, if sup M > tv and it is strongly positive sufficiently valid, if inf M > tv . On the other hand, if we determine some treshold of non-validity tn then we can assert that a decision is negative sufficiently valid, if inf N < tn and it is is strongly negative sufficiently valid, if sup N < tn .

174

6 Applications of IVIFSs

6.6.3 IVIF-Estimations in Decision Making Procedures The procedures for decision making include mulsti-criterial decision making procedures, that can be re-organized so that they use IVIF-estimations. For example, let us have s experts who must estimate some object or process. Let m of them estimate it as “perfect”, “the best” or “very good”; n of them—as “worst” or “very bad”; r —as “good”, “suitable” or “useful”; and s are “bad”, “non-suitable” or “non-useful”, then we can estimate the object or process by IVIF-estimations. In [40], it is discussed a new type of decision making procedure, based on the apparatus of the intercriteria analysis (see, e.g., [42, 43]). It is called intercriterial decision making. Its aim is to search dependences among the used criteria. For example, it is very suitable when separate experts offer for use in concrete procedure different criteria. Now, after finishing of the procedure, we can determine whether there are connections between some of these criteria. In the IFS-case, this procedure is discussed in [27], while for the IVIFS-case similar research has just appeared. The new method is based on the apparatus of the index matrices (see [33, 39]).

6.6.4 IVIFS-Estimations in Pattern Recognition Procedures The apparatus of the IVIFSs is suitable for estimation of different pattern recognition procedures. Here, we give the following two short examples, inspired by [22]. First, let us have the original pattern—in our example, triangle ABC that must be compare with other pattern—e.g., triangle AF G (see Fig. 6.4). Let the section BC be fuzzified, i.e., it be modified to the region BC E D. Let us denote by # X the surface of region X and let s = # AB H G, a = # AF I E, b = #F D I, c = # D B H I, d = #C H G, e = #C E I H. Obviously, a + b + c + d + e = s. Therefore, the IVIF-degree of coincidence of the second pattern with the original pattern will be

6.6 Interval Valued Intuitionistic Fuzzy Sets as Tools for Evaluation …

175

G A A A

Fig. 6.4 Example 1

A C A @ @ A E @ @@ A @@ A @@A H @@ A @ A@ I A@@ A @@ A @@ A @@ A @@ A F D B





 a a+e b b+c , , , . s s s s

This simple example shows the IVIF-possibility to estimate more detail than the fuzzy set or IFS-tools, because in the fuzzy set case, the estimation had to be only ab   . and in the IFS-case, it had to be as , b+c s

 The degree of uncertainty is determined as the interval 0, ds . More complex is the following example (see Fig. 6.5). Let us have the original pattern—in the new example, again triangle ABC that must be compare with other pattern—now triangle AG I . Let the sections BC and G I be fuzzified, i.e., they are modified to the regions BC E D and F G I H . Let s = # AB L I, a = # AF K E, b = #F G M K , c = #G D M, d = # D B L M, e = #E K J C, f = #C J H, g = #I H J L , h = #J K M L.

176 Fig. 6.5 Example 2

6 Applications of IVIFSs I A H A AA A AA C AA @ @A A E @ @@A A J AA @@ @A@A L K@ A@ A A@ A@ M A A@@ A A @@ A A @@ A A @@ AA @@ A F G D B

Obviously, a + b + c + d + e + f + g + h = s. Now, the IVIF-degree of coincidence of the second pattern with the original pattern will be 



 a a+b+e+h c c+d , , , . s s s s This simple example shows the IVIF-possibility to estimate more detail than the fuzzy set or IFS-tools, because in the fuzzy set case, the estimation had to be only ab   . and in the IFS-case, it had to be as , b+c s   The degree of uncertainty is determined as the interval 0, f +g . s

6.6.5 IF-Estimations in Neural Networks and Evolutionary Algorithms The first results related to the IF-estimations in neural networks date back to the year 1990 [131] and they are continued in [23, 24, 244–246, 249, 250, 253, 255, 256, 275, 277]. These estimations are related to the initial values in the input vectors that now, will have the form Mi , Ni  for the i-th input neuron, where Mi , Ni ⊆ [0, 1] and sup Mi + sup Ni ≤ 1, and the weight coefficients of the connections between the nodes with the form Vi, j , Wi, j  for the i-th and j-th neurons lying in sequential layers, where Vi, j , Wi, j ⊆ [0, 1] and sup Vi, j + sup Wi, j ≤ 1. When some of these coefficients have IF-truth-value [0, 0], [1, 1], then we can interpret that the respective object (nodes or arcs between nodes) does not exists. So, we can modify the neural network structure in time. Other IF-estimations can be calculated for a given neural network parameters, using the (standard or extended) modal operators. In [116], it is mentioned that “The paradigm of Evolutionary Algorithms (EAs) consists of stochastic search algorithms inspired by the process of neo-Darwinian evolution. ... There are several kinds of EAs, such as Genetic Algorithms, Genetic

6.6 Interval Valued Intuitionistic Fuzzy Sets as Tools for Evaluation … Fig. 6.6 Geometrical interpretation of an IVIFS-element z

0, 1 @ @ @ @ @ sup N (z) q

177

@ @ @

inf N (z) q 0, 0

q

z @ @

q

@ @

inf M (z)sup M (z)

1, 0

Programming, Classifier Systems, Evolution Strategies, Evolutionary Programming, Estimation of Distribution Algorithms, etc.” In this direction of research, in near future the focus will be orientated to the mentioned above EAs. In [40], some other areas of DM that can use IF-estimations, are described. All they can use IVIF-estimations, too. Some of these areas are the following. • • • •

machine and e-learning clusterisation and classification of data knowledge discovery processes processes for imputation (filling in) of missing data

and others. In the conclusion, we mention that in [319], an idea for a new direction in AI is formulated by L. Zadeh, based on the concept of a granule. But, by the moment there is not a good formal definition of this concept. Probably, the estimations of the IVIFS-elements can be used for a model. Really, the geometrical interpretation of an IVIFS-element z is given in Fig. 6.6. Let E be an universe and A = {x, M(x), N (x)|x ∈ E} be an IVIFS, and z ∈ E is a fixed element of the IVIFS. A granule can be defined as the set G crisp (z) = {y|y ∈ E&μ(y) ∈ M(z) & ν(y) ∈ N (z)} (crisp form) or G I F S (z) = {y, μ(y), ν(y)|y ∈ E & μ(y) ∈ M(z) & ν(y) ∈ N (z)} (intuitionistic fuzzy form). Of course, this is only a first step of the development of this idea that probably, will be developed in the future.

178

6 Applications of IVIFSs

References 1. Abdullah, L., Ismail, W.K.W.: Hamming distance in intuitionistic fuzzy sets and interval valued intuitionistic fuzzy sets: a comparative analysis. Adv. Comput. Math. Appl. 1(1), 7–11 (2012) 2. Abdullah, S., Ayub, S., Hussain, I., Bedregal, B., Khan, M.: Analyses of S-boxes based on interval valued intuitionistic fuzzy sets and image encryption. Int. J. Comput. Intell. Syst. 10, 851–865 (2017). https://doi.org/10.2991/ijcis.2017.10.1.57 3. Adak, A.K., Bhowmik, M.: Interval cut-set of interval-valued intuitionistic fuzzy sets. Afr. J. Math. Comput. Sci. Res. 4(4), 192–200 (2011) 4. Ahn, J.Y., Han, K.S., Oh, S.Y., Lee, C.D., An application of interval-valued intuitionistic fuzzy sets for medical diagnosis of headache. Int. J. Innov. Comput. Inf. Control 7(5 B), 2755–2762 (2011) 5. Aikhuele, D.O., Turan, F.B.M.: An integrated fuzzy dephi and interval-valued intuitionistic fuzzy M-Topsis model for design concept selection. Pak. J. Stat. Oper. Res. 13(2), 425–438 (2017). https://doi.org/10.18187/pjsor.v13i2.1413 6. Aikhuele, D.O., Turan, F.M., Odofin, S.M., Ansah, R.H.: Interval-valued Intuitionistic Fuzzy TOPSIS-based model for troubleshooting marine diesel engine auxiliary system. Trans. R. Instit. Nav. Arch. Part A: Int. J. Marit. Eng. 159, 107–114 (2017). https://doi.org/10.3940/ rina.ijme.2016.al.402 7. Akram, M., Dudek, W.: Interval-valued intuitionistic fuzzy Lie ideals of Lie algebras. World Appl. Sci. J. 7(7), 812–819 (2009) 8. Alexieva, J., Choy, E., Koycheva, E.: Review and bibloigraphy on generalized nets theory and applications. In: Choy, E., Krawczak, M., Shannon, A., Szmidt, E. (eds.) A Survey of Generalized Nets. Raffles KvB Monograph No. 10, pp. 207–301 (2007) 9. An, X., Wang, Z., Li, H., Ding, J.: Project delivery system selection with interval-valued intuitionistic fuzzy set group decision-making method. Group Decis. Negot. 27(4), 689–707 (2018). https://doi.org/10.1007/s10726-018-9581-y 10. Angelov, P., Filev, D., Kasabov, N.: Evolving Intelligent Systems. Wiley, Hoboken (2010) 11. Angelova, M., Pencheva, T.: Intercriteria analysis approach for comparison of simple and multi-population genetic algorithms performance. In: Fidanova, S. (ed.) Recent Advances in Computational Optimization. Studies in Computational Intelligence, vol. 795, pp. 117–130 (2019) 12. Angelova, N., Stoenchev, M.: Intuitionistic fuzzy conjunctions and disjunctions from first type. Ann. “Inf.” Sect. Union Sci. Bulg. 8, 1–17 (2015/2016) 13. Angelova, N., Stoenchev, M.: Intuitionistic fuzzy conjunctions and disjunctions from third type. Notes Intuit. Fuzzy Sets 23(5), 29–41 (2017) 14. Angelova, N., Zoteva, D., Atanassov, K.: Interval valued intuitionistic fuzzy generalized nets. Part 2. In: Proceedings of the 12th International Conference “Information Systems and Grid Technologies” ISGT’2018 (in press) 15. Angelova, M., Pencheva, T.: Intercriteria analysis of multi-population genetic algorithms performance. Ann. Comput. Sci. Inf. Syst. 13, 77–82 (2017) 16. Angelova, M., Roeva, O., Pencheva, T.: Intercriteria analysis of crossover and mutation rates relations in simple genetic algorithm. Ann. Comput. Sci. Inf. Syst. 5, 419–424 (2015) 17. Angelova, M., Roeva, O., Pencheva, T.: Intercriteria analysis of a cultivation process model based on the genetic algorithm population size influence. Notes Intuit. Fuzzy Sets 21(4), 90–103 (2015) 18. Angelova, N., Stoenchev, M., Todorov, V.: Intuitionistic fuzzy conjunctions and disjunctions from second type. Issues Intuit. Fuzzy Sets Gen. Nets 13, 143–170 (2017) 19. Atanassov K., Nikolov, N.: Intuitionistic fuzzy generalized nets: definitions, properties, applications. In: Melo-Pinto, P., Teodorescu, H.-N., Fukuda, T. (eds.) Systematic Organization of Information in Fuzzy Systems. IOS Press, Amsterdam, pp. 161–175 (2003) 20. Atanassov K., Sotirova, E.: Generalized Nets Theory, “Prof. M. Drinov”. Academic Publishing House, Sofia (2017) (in Bulgarian)

References

179

21. Atanassov, K., Atanassova, V., Chountas, P., Mitkova, M., Sotirova, E., Sotirov, S., Stratiev, D.: Intercriteria analysis over normalized data. In: Proceedings of the 8th IEEE Conference Intelligent Systems, Sofia, pp. 136–138 (2016). Accessed 4–6 Sept 2016 22. Atanassov, K., Intercriteria Analysis over Patterns. In: Sgurev, V., Piuri, V., Jotsov, V. (eds.) Learning Systems: From Theory to Practice, Studies in Computational Intelligence, Vol. 7 Mathematics 2018, 6, 123, 61–71. Springer, Cham (2018) https://doi.org/10.3390/ math607012356 23. Atanassov, K., Sotirov, S., Kodogiannis, V.: Intuitionistic fuzzy estimations of the Wi-Fi connections. In: First International Workshop on Intuitionistic Fuzzy Sets, Generalized Nets, and Knowledge Engineering, London, pp. 75–80 (2006). Accessed 6–7 Sept 2006 24. Atanassov, K., Sotirov, S., Krawczak, M.: Generalized net model of the intuitionistic fuzzy feed forward neural network. Notes Intuit. Fuzzy Sets 15(2), 18–23 (2009) 25. Atanassov, K., Sotirova, E., Andonov, V.: Generalized net model of multicriteria decision making procedure using intercriteria analysis. In:- Advances in Fuzzy Logic and Technology, vol. 1, pp. 99–111. Springer, Cham (2017) 26. Atanassov, K., Zoteva, D., Angelova, N.: Interval valued intuitionistic fuzzy generalized nets. Part 1. Noten Intuit. Fuzzy Sets 24(3), 111–123 (2018) 27. Atanassov, K., E. Szmidt, J. Kacprzyk, V., Atanassova. An approach to a constructive simplification of multiagent multicriteria decision making problems via intercriteria analysis. Comptes Rendus de l’Academie Bulgare des Sciences 70(8), 1147–1156 (2017) 28. Atanassov, K.: Extended interval valued intuitionistic fuzzy index matrices. In: Atanassov, K.T., Kacprzyk, J. et al. (eds.) Uncertainty and Imprecision in Decision Making and Decision Support: Cross fertilization, New Models and Applications. Springer, Cham (2019) 29. Atanassov, K.: Generalized nets as a tool for the modelling of data mining processes. In: Sgurev, V., Yager, R., Kacprzyk, J., Jotsov, V.: Innovative Issues in Intelligent Systems, pp. 161–215. Springer, Cham (2016) 30. Atanassov, K.: Generalized Nets in Artificial Intelligence. Vol. 1: Generalized Nets and Expert Systems, “Prof. M. Drinov”. Academic Publishing House, Sofia (1998) 31. Atanassov, K.: On Generalized Nets Theory, “Prof. M. Drinov” Academic Publishing House, Sofia (2007) 32. Atanassov, K.: The Generalized nets and the other graphical means for modelling. AMSE Rev. 2(1), 59–64 (1985) 33. Atanassov, K. Generalized index matrices. – Compt. Rend. de l’Academie Bulgare des Sciences 40(11), 15–18 (1987) 34. Atanassov, K, Marinov, P., Atanassova, V.: Intercriteria analysis with interval valued intuitionistic fuzzy evaluations: symmetric case. In: 13th International Conference on Flexible Query Answering Systems, Amantea, Italy (submitted) (2019). Accessed 2–5 July 2019 35. Atanassov, K.: Generalized Nets. World Scientific, Singapore (1991) 36. Atanassov, K.: Remark on intuitionistic fuzzy expert systems. BUSEFAL 59, 71–76 (1994) 37. Atanassov, K.: Intuitionistic Fuzzy Sets. Springer, Heidelberg (1999) 38. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012) 39. Atanassov, K.: Index Matrices: Towards an Augmented Matrix Calculus. Springer, Cham (2014) 40. Atanassov, K.: Intuitionistic fuzzy logics as tools for evaluation of data mining processes. Knowl.-Based Syst. 80, 122–130 (2015) 41. Atanassov, K.: Interval valued intuitionistic fuzzy graphs. Notes Intuit. Fuzzy Sets 25(1), 21–31 (2019) 42. Atanassov, K., Mavrov, D., Atanassova, V.: Intercriteria decision making: a new approach for multicriteria decision making, based on index matrices and intuitionistic fuzzy sets. Issues Intuit. Fuzzy Sets Gen. Nets 11, 1–8 (2014) 43. Atanassov, K., Atanassova, V., Gluhchev, G.: Intercriteria analysis: ideas and problems. Notes Intuit. Fuzzy Sets 21(1), 81–88 (2015) 44. Atanassova V., Roeva, O.: Computational complexity and influence of numerical precision on the results of intercriteria analysis in the decision making process. Notes Intuit. Fuzzy Sets 24(3), 53–63 (2018)

180

6 Applications of IVIFSs

45. Atanassova, V., Doukovska, L., Michalikova, A., Radeva, I.: Intercriteria analysis: from pairs to triples. Notes Intuit. Fuzzy Sets 22, 5 (2016). ISSN:Print ISSN 1310-4926; Online ISSN 2367-8283, 98-110 46. Atanassova, V.: Interpretation in the Intuitionistic Fuzzy Triangle of the Results, Obtained by the InterCriteria Analysis, 16th World Congress of the International Fuzzy Systems Association (IFSA), 9th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), 30. 06-03. 07. 2015, Gijon, Spain, pp. 1369–1374 (2015). https://doi.org/10. 2991/ifsa-eusflat-15.2015.193 47. Atanassova, L.: On interval-valued intuitionistic fuzzy versions of L. Zadeh’s extension principle. Issues Intuit. Fuzzy Sets Gen. Nets 7, 13–19 (2008) 48. Atanassova, V., Doukovska, L., de Tre, G., Radeva, I.: Intercriteria analysis and comparison of innovation-driven and efficiency-to-innovation driven economies in the European Union. Notes Intuit. Fuzzy Sets 23(3), 54–68 (2017) 49. Atanassova, V., Doukovska, L., Kacprzyk, A., Sotirova, E., Radeva, I., Vassilev, P.: Intercriteria analysis of the global competitiveness reports: from efficiency- to innovation-driven economies. J. Mult.-Valued Logic Soft Comput. 31(5–6), 469–494 (2018) 50. Aygüno´glu, A., Varol, B.P., Cetkin, V., Aygün, H.: Interval valued intuitionistic fuzzy subgroups based on interval valued double t norm. Neural Comput. Appl. 21(SUPPL. 1), 207–214 (2012) 51. Berti-Equille, L.: Measuring and modelling data quality for quality-awareness in data mining. In: Guillet, F., Hamilton, H. (eds.) Quality Measures in Data Mining, pp. 101–126. Springer, Berlin (2007) 52. Bhowmik, M., Pal, M.: Some results on generalized interval valued intuitionistic fuzzy sets. Int. J. Fuzzy Syst. 14 (2), 193–203 (2012) 53. Biswas, A., Kumar, S.: An integrated TOPSIS approach to MADM with interval-valued intuitionistic fuzzy settings. Adv. Intell. Syst. Comput. 706, 533–543 (2018). https://doi.org/ 10.1007/978-981-10-8237-5_52 54. Bolturk, E., Kahraman, C. Interval-valued intuitionistic fuzzy CODAS method and its application to wave energy facility location selection problem. J. Intell. Fuzzy Syst. 35(4), 4865–4877 (2018). https://doi.org/10.3233/JIFS-18979 55. Bramer, M.: Principles of Data Mining. Springer, London (2013) 56. Bull, L., Ester, B.-M., Holmes, J.: Learning classifier systems in data mining: an introduction, In: Bull, L., Ester, B.-M., Holmes, J. (eds.) Learning Classifier Systems in Data Mining, pp. 1–15. Springer, Berlin (2008) 57. Bureva, V., Michalìkovà, A., Sotirova, E., Popov, S., Rieˇcan, B., Roeva, O.: Application of the InterCriteria Analysis to the universities rankings system in the Slovak Republic. Notes Intuit. Fuzzy Sets 23(2), 128–140 (2017) 58. Bureva, V., Sotirova, E., Atanassova, V., Angelova, N., Atanassov, K.: Intercriteria Analysis over Intuitionistic Fuzzy Data. Lecture Notes in Computer Science, vol. 10665, pp. 333–340. Springer, Berlin (2018). https://doi.org/10.1007/978-3-319-73441-5_35 59. Bureva, V., Sotirova, E., Panayotov, H.: The intercriteria decision making method to Bulgarian university ranking system. Ann. Inf. Sect., Union Sci. Bulg. 8, 54–70 (2015–2016) 60. Bureva, V., Sotirova, E., Sotirov, S., Mavrov, D.: Application of the intercriteria decision making method to Bulgarian universities ranking. Notes Intuit. Fuzzy Sets 21(2), 111–117 (2015) 61. Burillo P., Bustince H., Two operators on interval-valued intuitionistic fuzzy sets: Part II, Comptes rendus de l’Academie bulgare des Sciences, Tome 48, 1995, No. 1, 17-20 62. Burillo P., Bustince H., Two operators on interval-valued intuitionistic fuzzy sets: Part I, Comptes rendus de l’Academie bulgare des Sciences, Tome 47, 1994, No. 12, 9-12 63. Burillo, P., Bustince, H.: Informational energy on intuitionistic fuzzy sets and on intervalvalues intuitionistic fuzzy sets (Φ-fuzzy). Relationship between the measures of information. In: Lakov, D. (ed.) Proceedings of the First Workshop on Fuzzy Based Expert Systems, Sofia, pp. 46–49 (1994). Accessed 28–30 Sept. 1994

References

181

64. Burillo, P., Bustince, H.: Entropy on intuitionistic fuzzy sets and on interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 78(3), 305–316 (1996) 65. Bustince, H.: Numerical information measurements in interval-valued intuitionistic fuzzy sets (IVIFS). In: Lakov, D. (ed.) Proceedings of the First Workshop on Fuzzy Based Expert Systems, Sofia, pp. 50–52. Accessed 28–30 Sept. 1994 66. Bustince, H., Burillo, P.: Correlation of interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 74(2), 237–244 (1995) 67. Büyüközkan, G., Feyaioˇglu, O.: Accelerating the new product introduction with intelligent data mining, In: Ruan, D., Chen, G., Kerre, E., Wets, G. (eds.) Intelligent Data Mining: Techniques and Applications, pp. 337–354. Springer, Berlin (2005) 68. Büyüközkan, G., Göcer, F., Feyzioˇglu, O.: Cloud computing technology selection based on interval-valued intuitionistic fuzzy MCDM methods. Soft Comput. 22(15), 5091–5114 (2018). https://doi.org/10.1007/s00500-018-3317-4 69. Büyüközkan, G., Göcer, F.: An extension of ARAS methodology under Interval Valued Intuitionistic Fuzzy environment for Digital Supply Chain. Appl. Soft Comput. J. 69, 634–654 (2018). doi: https://doi.org/10.1016/j.asoc.2018.04.040 70. Büyüközkan, G., Göcer, F.: An extension of MOORA approach for group decision making based on interval valued intuitionistic fuzzy numbers in digital supply chain. In: Fuzzy Systems Association and 9th International Conference on Soft Computing and Intelligent Systems (IFSA-SCIS), 2017 Joint 17-th World Congress of International, INSPEC Accession Number: 17151170, pp. 1–6 (2017). https://doi.org/10.1109/IFSA-SCIS.2017.8023358 71. Büyüközkan, G., Göcer, F.: Smart medical device selection based on interval valued intuitionistic fuzzy VIKOR. Adv. Intell. Syst. Comput. 641, 306–317 (2018). https://doi.org/10.1007/ 978-3-319-66830-7_28 72. Büyüközkan, G., Göcer, F., Feyzioˇglu, O.: Cloud computing technology selection based on interval valued intuitionistic fuzzy COPRAS. Adv. Intell. Syst. Comput. 641, 318–329 (2018). https://doi.org/10.1007/978-3-319-66830-7_29 73. Callejas Bedrega, B., Visintin, L., Reiser, R.H.S.: Index, expressions and properties of intervalvalued intuitionistic fuzzy implications. Trends Appl. Comput. Math. 14(2), 193–208 (2013) 74. Cao, Y.-X., Zhou, H., Wang, J.-Q.: An approach to interval-valued intuitionistic stochastic multi-criteria decision-making using set pair analysis. Int. J. Mach. Learn. Cybern. 9(4), 629–640 (2018). https://doi.org/10.1007/s13042-016-0589-9 75. Chen, X.-H., Dai, Z.-J., Liu, X.: Approach to interval-valued intuitionistic fuzzy decision making based on entropy and correlation coefficient. Xi Tong Gong Cheng Yu Dian Zi Ji Shu/Syst. Eng. Electron. 35(4), 791–795 (2013) 76. Chen, S.M., Lee, L.W., Liu, H.C., Yang, S.W.: Multiattribute decision making based on interval valued intuitionistic fuzzy values. Expert Syst. Appl. 39(12), 10343–10351 (2012) 77. Chen, S.-M., Lee, L.-W.: A new method for multiattribute decision making using intervalvalued intuitionistic fuzzy values In: Proceedings—International Conference on Machine Learning and Cybernetics, vol. 1, art. no. 6016695, pp. 148–153 (2011) 78. Chen, S. M., Li, T. S., Yang, S. W., Sheu, T. W. A new method for evaluating students’ answerscripts based on interval valued intuitionistic fuzzy sets (2012) Proceedings International Conference on Machine Learning and Cybernetics, 4, art. no. 6359580, 1461-1467 79. Chen, S.-M., Yang, M.-W., Liau, C.-J.: A new method for multicriteria fuzzy decision making based on ranking interval-valued intuitionistic fuzzy values. In: Proceedings—International Conference on Machine Learning and Cybernetics, vol. 1, art. no. 6016698, pp. 154–159 (2011) 80. Chen, X., Yang, L., Wang, P., Yue, W.: A fuzzy multicriteria group decision-making method with new entropy of interval-valued intuitionistic fuzzy sets. J. Appl. Math. art. no. 827268 (2013) 81. Chen, X., Yang, L., Wang, P., Yue, W.: An effective interval-valued intuitionistic fuzzy entropy to evaluate entrepreneurship orientation of online P2P lending platforms. Adv. Math. Phys. art. no. 467215 (2013)

182

6 Applications of IVIFSs

82. Chen, S.M., Yang, M.W., Yang, S.W., Sheu, T.W., Liau, C.J.: Multicriteria fuzzy decision making based on interval valued intuitionistic fuzzy sets. Expert Syst. Appl. 39(15), 12085– 12091 (2012) 83. Chen, T.-Y.: Data construction process and qualiflex-based method for multiple-criteria group decision making with interval-valued intuitionistic fuzzy sets. Int. J. Inf. Technol. Decis. Mak. 12(3), 425–467 (2013) 84. Chen, T.-Y.: An interval-valued intuitionistic fuzzy LINMAP method with inclusion comparison possibilities and hybrid averaging operations for multiple criteria group decision making. Knowl.-Based Syst. 45, 134–146 (2013) 85. Chen, S.-M., Han, W.-H.: An improved MADM method using interval-valued intuitionistic fuzzy values. Inf. Sci. 467, 489–505 (2018). https://doi.org/10.1016/j.ins.2018.07.062 86. Chen, S.-M., Han, W.-H.: A new multiattribute decision making method based on multiplication operations of interval-valued intuitionistic fuzzy values and linear programming methodology. Inf. Sci. 429, 421–432 (2018). https://doi.org/10.1016/j.ins.2017.11.018 87. Chen, S.-M., Li, T.-S.: Evaluating students’ answerscripts based on interval-valued intuitionistic fuzzy sets. Inf. Sci. 235, 308–322 (2013) 88. Chen, Q., Xu, Z., Liu, S., et al.: A method based on interval-valued intuitionistic fuzzy entropy for multiple attribute decision making. Inf. Int. Interdiscip. J. 13(1), 67–77 (2010) 89. Chen, T.-Y., Wang, H.-P., Lu, Y.-Y.: A multicriteria group decision-making approach based on interval-valued intuitionistic fuzzy sets: a comparative perspective. Expert Syst. Appl. 38(6), 7647–7658 (2011) 90. Chen, S.-M., Kuo, L.-W., Zou, X.-Y.: Multiattribute decision making based on Shannon’s information entropy, non-linear programming methodology, and interval-valued intuitionistic fuzzy values. Inf. Sci. 465, 404–424 (2018). https://doi.org/10.1016/j.ins.2018.06.047 91. Cheng, S.-H.: Autocratic multiattribute group decision making for hotel location selection based on interval-valued intuitionistic fuzzy sets. Inf. Sci. 427, 77–87 (2018). https://doi.org/ 10.1016/j.ins.2017.10.018 92. Chountas, P., Kolev, B., Rogova, E., Tasseva, V., Atanassov, K.: Generalized Nets in Artificial Intelligence. Vol. 4: Generalized nets, Uncertain Data and Knowledge Engineering. “Prof. M. Drinov”. Academic Publishing House, Sofia (2007) 93. Chountas, P., Sotirova, E., Kolev, B., Atanassov, K.: On intuitionistic fuzzy expert systems with temporal components. In: Computational Intelligence, Theory and Applications, pp. 241–249. Springer, Berlin (2006) 94. Chu, J., Liu, X., Wang, L., Wang, Y.: A group decision making approach based on newly defined additively consistent interval-valued intuitionistic preference relations. Int. J. Fuzzy Syst. 20(3), 1027–1046 (2018). https://doi.org/10.1007/s40815-017-0353-7 95. Cios, K., Pedrycz, W., Swiniarski, R.: Data Mining Methods for Knowledge Discovery. Kluwer (1998) 96. Cios, K., Pedrycz, W., Swiniarski, R., Kurgan, L.: Data Mining. A Knowledge Discovery Approach. Springer, New York (2007) 97. Cox, E.: Fuzzy Modeling and Genetic Algorithms for Data Mining and Exploration. Elsevier, Amsterdam (2005) 98. Dahan, H., Cohen, S., Rokach, L., Maimon, O.: Proactive Data Mining with Decision Trees. Springer, New York (2014) 99. Deschrijver, G., Kerre, E.: Aggregation operators in interval-valued fuzzy and Atanassov’s intuitionistic fuzzy set theory. In: Bustince, H., Herrera, F., Montero, J. (eds.) Fuzzy Sets and Their Extensions: Representation, Aggregation and Models, pp. 183–203. Springer, Berlin (2008) 100. Doukovska, L., Atanassova, V., Shahpazov, G., Capkovic, F.: InterCriteria analysis applied to variuos EU enterprises. In: Proceedings of the Fifth International Symposium on Business Modeling and Software Design, Milan, Italy, pp. 284–291 (2015) 101. Doukovska, L., Atanassova, V., Sotirova, E., Vardeva, I., Radeva, I.: Defining consonance thresholds in intercriteria analysis: an overview. In Intuitionistic Fuzziness and Other Intelligent Theories and Their Applications, pp. 161–179. Springer, Cham (2019)

References

183

102. Doukovska, L., Atanassova, V.: InterCriteria Analysis approach in radar detection threshold analysis. Notes Intuit. Fuzzy Sets 21(4), 129–135 (2015). ISSN: 1310-4926 103. Doukovska, L., Shahpazov, G., Atanassova, V.: Intercriteria analysis of the creditworthiness of SMEs. A case study. Notes Intuit. Fuzzy Sets 22(2), 108–118 (2016) 104. Dymova, L., Sevastjanov, P., Tikhonenko, A.: A new method for comparing interval valued intuitionistic fuzzy values. Lecture Notes in Computer Science, vol. 7267 LNAI (PART 1), pp. 221–228 (2012) 105. Dymova, L., Sevastjanov, P., Tikhonenko, A.: Two-criteria method for comparing real-valued and interval-valued intuitionistic fuzzy values. Knowl.-Based Syst. 45, 166–173 (2013) 106. El Alaoui, M., Ben-Azza, H., El Yassini, K.: Optimal weighting method for interval-valued intuitionistic fuzzy opinions. Notes Intuit. Fuzzy Sets 24(3), 106–110 (2018) 107. Eyoh, I., John, R., De Maere, G.: Interval type-2 intuitionistic fuzzy logic for regression problems. IEEE Trans. Fuzzy Syst. 26(4), Article number 8115302, 2396–2408 (2018) 108. Eyoh, I., John, R., De Maere, G.: Time series forecasting with interval type-2 intuitionistic fuzzy logic systems. In: 2017 IEEE International Conference on Fuzzy Systems (FUZZIEEE), INSPEC Accession Number: 17137939 (2017). https://doi.org/10.1109/FUZZ-IEEE. 2017.8015463 109. Fan, L., Lei, Y.-J.: Probability of interval-valued intuitionistic fuzzy events and its general forms. Xi Tong Gong Cheng Yu Dian Zi Ji Shu/Syst. Eng. Electron. 33(2), 350–355 (2011) 110. Fan, L., Lei, Y.-J., Duan, S.-L.: Interval-valued intuitionistic fuzzy statistic adjudging and decision-making. Xitong Gongcheng Lilun yu Shijian/Syst. Eng. Theory Pract. 31(9), 1790– 1797 (2011) 111. Feyzioˇglu, O., Fethullah, G., Gulcin, B.: “Interval-valued intuitionistic fuzzy MULTIMOORA approach for new product development.” In: World Scientific Proceedings Series on Computer Engineering and Information Science: Volume 11. Data Science and Knowledge Engineering for Sensing Decision Support, pp. 1066–1073 (2018). https://doi.org/10.1142/ 9789813273238_0135 112. Fidanova, S., Roeva, O., Paprzycki, M., Gepner, P.: InterCriteria analysis of ACO start startegies. In: Proceedings of the Federated Conference on Computer Science and Information Systems, pp. 547–550 (2016) 113. Fidanova, S., Roeva, O.: Comparison of different metaheuristic algorithms based on intercriteria analysis. J. Comput. Appl. Math. 340, 615–628 (2018) 114. Fidanova, S., Roeva, O.: Intercriteria analysis of different variants of ACO algorithm for wireless sensor network positioning. Lect. Notes Comput. Sci. 11189, 88–96 (2019) 115. Fidanova, S., Roeva, O., Mucherino, A., Kapanova, K.: Intercriteria analysis of ant algorithm with environment change for GPS surveying problem. Lect. Notes Comput. Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 9883, 271–278 (2016) 116. Freitas, A.A.: A review of evolutionary algorithms for data mining. In: Maimon, O., Rokach, L. (eds.) Data and Knowledge Discovery Handbook, 2nd edn. pp. 371–400. Springer, New York (2010) 117. Fu, S. et al.: Interval-valued intuitionistic fuzzy multi-attribute decision-making method based on prospect theory and grey correlation. Recent Pat. Comput. Sci. 11(3), 215–221 (2018) 118. Garg, H., Kumar, K.: Group decision making approach based on possibility degree measure under linguistic interval-valued intuitionistic fuzzy set environment. J. Ind. Manag. Optim. 466–482 (2018). https://doi.org/10.3934/jimo.2018162 119. Garg, H.: Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process. J. Ind. Manag. Optim. 14(1), 283–308 (2018). https://doi.org/10.3934/jimo.2017047 120. Garg, H., Agarwal, N., Tripathi, A.: Some improved interactive aggregation operators under interval-valued intuitionistic fuzzy environment and their application to decision making process. Sci. Iran. E 24(5), 2581–2604 (2017). https://doi.org/10.24200/sci.2017.4386 121. Georgieva, V., Angelova, N., Roeva, O., Pencheva, T.: Intercriteria analysis of wastewater treatment quality. J. Int. Sci. Publ.: Ecol. Saf. 10, 365–376 (2016)

184

6 Applications of IVIFSs

122. Gong, Z., Ma, Y.: Multicriteria fuzzy decision making method under interval-valued intuitionistic fuzzy environment. In: International Conference on Machine Learning and Cybernetics, Baoding, 12–15 July 2009, vols. 1–6, 728–731 (2009) 123. Gong, Z.-T., Xie, T., Shi, Z.-H., Pan, W.-Q.: A multiparameter group decision making method based on the interval-valued intuitionistic fuzzy soft set. In: Proceedings—International Conference on Machine Learning and Cybernetics, vol. 1, art. no. 6016727, pp. 125–130 (2011) 124. Gou, X., Xu, Z.: Exponential operations for intuitionistic fuzzy numbers and interval numbers in multi-attribute decision making. Fuzzy Optim. Decis. Mak. 16(2), 183–204 (2017). https:// doi.org/10.1007/s10700-016-9243-y 125. Granichin, O., Volkovich, Z., Toledano-Kitai, D.: Randomized Algorithms in Automatic Control and Data Mining. Springer, Berlin (2015) 126. Grosan, V., Abraham, A.: Intelligent Systems—A Modern Approach. Springer, Berlin (2011) 127. Grzymala-Busse, J.W.: Rule induction. In: Maimon, O., Rokach, L. (eds.) Data Mining and Knowledge Discovery Handbook, 2nd edn, pp. 249–265. Springer, New York (2010) 128. Gu, X., Wang, Y., Yang, B.: Method for selecting the suitable bridge construction projects with interval-valued intuitionistic fuzzy information. Int. J. Digit. Content Technol. Appl. 5(7), 201–206 (2011) 129. Gupta, P., Mehlawat, M.K., Grover, N., Pedrycz, W.: Multi-attribute group decision making based on extended TOPSIS method under interval-valued intuitionistic fuzzy environment. Appl. Soft Comput. J. 69, 554–567 (2018). https://doi.org/10.1016/j.asoc.2018.04.032 130. Hadjyisky L., Atanassov K., Generalized net model of the intuitionistic fuzzy neural networks. Advances in Modelling & Analysis, vol. 23, No. 2, pp. 59–64. AMSE Press (1995) 131. Hadjyisky, L., Atanassov, K.: Intuitionistic fuzzy model of a neural network. BUSEFAL 54, 36–39 (1993) 132. Han, J., Kamber, M.: Data Mining: Concepts and Techniques. Morgan Kaufmann (2006) 133. Hand, D., Mannila, H., Smyth, P.: Principles of Data Mining. MIT Press (2001) 134. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning—Data Mining, Inference and Prediction. Springer, New York (2001) 135. Hedayati, H.: Interval-valued intuitionistic fuzzy subsemimodules with (S; T)-norms. Ital. J. Pure Appl. Math. 27, 157–166 (2010) 136. Hilderman, R., Peckham, T.: Statistical methodologies from mining potentially interesting contrast sets. In: Guillet, F., Hamilton, H. (eds.) Quality Measures in Data Mining, pp. 153– 177. Springer, Berlin (2007) 137. Holmes, D., Tweedale, J., Jain, L.: Data mining techniques in clustering, association and classification. In: Holmes, D., Jain, L. (eds.) Data Mining: Foundations and Intelligent Paradigms, Vol. 1: Clustering, Association and Classification, pp. 1–6. Springer, Berlin (2012) 138. Hong, T.-P., Chen, C.-H., Wu, Y.-L., Tseng, V.S.: Fining active membership functions in fuzzy data mining. In: Lin, T.Y., Xie, Y., Wasilewska, A., Liau, C.-J. (eds.) Data Mining: Foundations and Practice, pp. 179–196. Springer, Berlin (2008) 139. Hong, D.H.: A note on correlation of interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 95(1), 113–117 (1998) 140. Hu, L.: An approach to construct entropy and similarity measure for interval valued Intuitionistic fuzzy sets. In: Proceedings of the 2012 24th Chinese Control and Decision Conference, CCDC 2012, art. no. 6244285, pp. 1777–1782 (2012) 141. Hui, F., Shi, X., Yang, L. A novel efficient approach to evaluating the security of wireless sensor network with interval valued intuitionistic fuzzy information. Adv. Inf. Sci. Serv. Sci. 4(7), 83–89 (2012) 142. Huo, L., Liu, B., Cheng, L.: Method for interval-valued intuitionistic fuzzy multicriteria decision making with gray relation analysis. In: International Conference on Engineering and Business Management, Chengdu, 25–27 Mar 2010, pp. 1342–1345 (2010) 143. Ikonomov, N., Vassilev, P., Roeva, O.: ICrAData—Software for InterCriteria analysis. Int. J. Bioautomation 22(1), 1–10 (2018) 144. Izadikhah, M., Group DecisionMaking process for supplier selection with TOPSISMethod under interval-valued intuitionistic fuzzy numbers. Adv. Fuzzy Syst. 2012, Article ID 407942, 14 pages. https://doi.org/10.1155/2012/407942

References

185

145. Joshi, D., Kumar, S.: Improved accuracy function for interval-valued intuitionistic fuzzy sets and its application to multi–attributes group decision making. Cybern. Syst. 49(1), 64–76 (2018). https://doi.org/10.1080/01969722.2017.1412890 146. Joshi, D.K., Bisht, K., Kumar, S.: Interval-valued intuitionistic uncertain linguistic information-based TOPSIS method for multi-criteria group decision-making problems. Adv. Intell. Syst. Comput. 696, 305–315 (2018). https://doi.org/10.1007/978-981-10-7386-1_27 147. Kacprzyk, A., Sotirov, S., Sotirova, E., Shopova, D., Georgiev, P.: Application of intercriteria analysis in the finance and accountancy positions. Notes Intuit. Fuzzy Sets 23(4), 84–90 (2017) 148. Kasabov, N.: Evolving Connectionist Systems. Springer, London (2007) 149. Kaufmann, A.: Introduction a la Theorie des Sour-Ensembles Flous. Masson, Paris (1977) 150. Kavita, Y., Kumar, S.: A multi-criteria interval-valued intuitionistic fuzzy group decision making for supplier selection with TOPSIS method. Lect. Notes Artif. Intell. 5908, 303–312 (2009) 151. Kecman, V.: Learning and Soft Computing. MIT Press (2001) 152. Klosgen, W., Zytkow, J. (eds.): Handbook of Data Mining and Knowledge Discovery. Oxford University Press, New York (2002) 153. Kolev, B., El-Darzi, E., Sotirova, E., Petronias, I., Atanassov, K., Chountas, P., Kodogianis, V.: Generalized Nets in Artificial Intelligence. Vol. 3: Generalized nets, Relational Data Bases and Expert Systems. “Prof. M. Drinov” Academic Publishing House, Sofia (2006) 154. Krawczak, M., Bureva, V., Sotirova, E., Szmidt, E.: Application of the InterCriteria decision making method to universities ranking. Novel Developments in Uncertainty Representation and Processing, Advances in Intelligent Systems and Computing, pp. 365–372. Springer (2016) 155. Krawczak, M., El-Darzi, E., Atanassov, K., Tasseva, V.: Generalized net for control and optimization of real processes through neural networks using intuitionistic fuzzy estimations. Notes Intuit. Fuzzy Sets 13(2), 54–60 (2007) 156. Krishankumar, R., Ravichandran, K., Ramprakash, R.: A scientific decision framework for supplier selection under interval valued intuitionistic fuzzy environment. Math. Probl. Eng. 2017, Article ID 1438425, 18 pages (2017) 157. Krumova, S., Todinova, S., Mavrov, D., Marinov, P., Atanassova, V., Atanassov, K., Taneva, S.: Intercriteria analysis of calorimetric data of blood serum proteome. Biochimica et Biophysica Acta-Gen. Subj. 2017, 409–417 (1861) 158. Kumar, S., Biswas, A.: TOPSIS based on linear programming for solving MADM problems in interval-valued intuitionistic fuzzy settings. In: Proceedings of the 4th IEEE International Conference on Recent Advances in Information Technology, RAIT 2018, pp. 1–6 (2018). https://doi.org/10.1109/RAIT.2018.8389071 159. Kumar, K., Garg, H.: TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment. Comput. Appl. Math. 37(2), 1319– 1329 (2018). https://doi.org/10.1007/s40314-016-0402-0 160. Kuncheva L., Atanassov, A.: An intuitionistic fuzzy RBF network. In: Proceedings of EUFIT’96, Aachen, 2–5 Sept 1996, pp. 777–781 (1996) 161. Li, J., Chen, W., Yang, Z., Li, C., Sellers, J.S.: Dynamic interval-valued intuitionistic normal fuzzy aggregation operators and their applications to multi-attribute decision-making. J. Intell. Fuzzy Syst. 35(4), 3937–3954 (2018). https://doi.org/10.3233/JIFS-169717 162. Li, J., Lin, M., Chen, J.: ELECTRE method based on interval valued intuitionistic fuzzy number. Appl. Mech. Mater. 220–223, 2308–2312 (2012) 163. Li, P., Liu, S.F., Fang, Z.G.: Interval valued intuitionistic fuzzy numbers decision making method based on grey incidence analysis and MYCIN certainty factor. Kongzhi yu Juece/Control Decis. 27(7), 1009–1014 (2012) 164. Li, Y., Yin, J., Wu, G.: Model for evaluating the computer network security with interval valued intuitionistic fuzzy information. Int. J. Digit. Content Technol. Appl. 6(6), 140–146 (2012)

186

6 Applications of IVIFSs

165. Li, J., Zhang, X.-L., Gong, Z.-T.: Aggregating of interval-valued intuitionistic uncertain linguistic variables based on archimedean t-norm and it applications in group decision makings. J. Comput. Anal. Appl. 24(5), 874–885 (2018) 166. Li, D.-F.: Linear programming method for MADM with interval-valued intuitionistic fuzzy sets. Expert Syst. Appl. 37(8), 5939–5945 (2010) 167. Li, D.-F.: TOPSIS-Based nonlinear-programming methodology for multiattribute decision making with interval-valued intuitionistic fuzzy sets. IEEE Trans. Fuzzy Syst. 18(2), 299– 311 (2010) 168. Li, D.-F.: Extension principles for interval-valued intuitionistic fuzzy sets and algebraic operations. Fuzzy Optim. Decis. Mak. 10(1), 45–58 (2011) 169. Li, D.-F.: Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information. Appl. Soft Comput. J. 11(4), 3402–3418 (2011) 170. Li, D.-F.: Extension principles for interval-valued intuitionistic fuzzy sets and algebraic operations. Fuzzy Optim. Decis. Mak. 10, 45–58 (2011) 171. Li, P., Liu, S.-F.: Interval-valued intuitionistic fuzzy numbers decision-making method based on grey incidence analysis and D-S theory of evidence. Acta Automatica Sinica 37(8), 993– 998 (2011) 172. Li, K.W., Wang, Z.: Notes on “multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment”. J. Syst. Sci. Syst. Eng. 19(4), 504–508 (2010) 173. Li, B., Yue, X., Zhou, G.: Model for supplier selection in interval-valued intuitionistic fuzzy setting. J. Converg. Inf. Technol. 6(11), 12–16 (2011) 174. Lin, J., Zhang, Q.: Some continuous aggregation operators with interval-valued intuitionistic fuzzy information and their application to decision making. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 20(2), 185–209 (2012) 175. Lin, J., Zhang, Q.: Note on continuous interval-valued intuitionistic fuzzy aggregation operator. Appl. Math. Model. 43, 670–677 (2017). https://doi.org/10.1016/j.apm.2016.09.003 176. Liu, D., Chen, X., Peng, D.: Interval-Valued intuitionistic fuzzy ordered weighted cosine similarity measure and its application in investment decision-making. Complexity 2017, Article ID 1891923, 11 pages (2017). https://doi.org/10.1155/2017/1891923 177. Liu, J., Deng, X., Wei, D., Li , Y., Deng, Y.: Multi attribute decision making method based on interval valued intuitionistic fuzzy sets and D S theory of evidence. In: 24th Chinese, Date of Conference Control and Decision Conference (CCDC), 23–25 May 2012, pp. 2651–2654 (2012) 178. Liu, J., Li, Y., Deng, X., Wei, D., Deng, Y.: Multi attribute Decision making based on interval valued intuitionistic fuzzy sets. J. Inf. Comput. Sci. 9(4), 1107–1114 (2012) 179. Liu, J., Li, Y., Deng, X., Wei, D., Deng, Y.: Multi-attribute decision-making based on intervalvalued intuitionistic fuzzy sets. J. Inf. Comput. Sci. 9(4), 1107–1114 (2012) 180. Liu, Z., Teng, F., Liu, P., Ge, Q.: Interval-valued intuitionistic fuzzy power Maclaurin symmetric mean aggregation operators and their application to multiple attribute group decision-making. Int. J. Uncertain. Quantif. 8(3), 211–232 (2018). https://doi.org/10.1615/ Int.J.UncertaintyQuantification.2018020702 181. Liu, P., Wang, P.: Some interval-valued intuitionistic fuzzy Schweizer–Sklar power aggregation operators and their application to supplier selection. Int. J. Syst. Sci. 49(6), 1188–1211. https://doi.org/10.1080/00207721.2018.1442510 182. Liu, P.: Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators. Comput. Ind. Eng. 108, 199–212 (2017). https://doi.org/10.1016/j.cie.2017.04.033 183. Liu, Y., Bi, J., Fan, Z.: A method for ranking products through online reviews based on sentiment classification and interval-valued intuitionistic fuzzy TOPSIS. J. Inf. Technol. Decis. Mak. 16(06), 1497–1522 (2017). https://doi.org/10.1142/S021962201750033X 184. Looney, C.: Fuzzy Petri Nets for rule-based decision making. IEEE Trans. Syst. Man Cybern. 22, 178–183 (1988)

References

187

185. Maimon, O., Rokach, L.: Introduction to knowledge discovery and data mining. In: Maimon, O., Rokach, L. (eds.) Data Mining and Knowledge Discovery Handbook, 2nd edn., pp. 1–15. Springer, New York (2010) 186. Marinov, P., Fidanova, S.: Intercriteria and correlation analyses: similarities, differences and simultaneous use. Ann. “Inf.” Sect. Union Sci. Bulg. 8, 45–53 (2016) 187. Meng, F., Tang, J., Wang, P., Chen, X.: A programming-based algorithm for interval-valued intuitionistic fuzzy group decision making. Knowl.-Based Syst. 144, 122–143 (2018). https:// doi.org/10.1016/j.knosys.2017.12.033 188. Meyer-Nieberg, S., Beyer, H.-G.: Self-adaptation in evolutionary algorithms. In: Lobo, F., Lima, C., Michalewicz, Z. (eds.) Parameter Setting in Evolutionary Algorithms, Studies in Computational Intelligence, vol. 02007, No. 54, pp. 47–75. Springer, Berlin 189. Mishra, A.R., Rani, P.: Biparametric information measures-based TODIM technique for interval-valued intuitionistic fuzzy environment. Arab. J. Sci. Eng. 43(6), 3291–3309 (2018). https://doi.org/10.1007/s13369-018-3069-6 190. Mishra, A.R., Rani, P.: Interval-Valued intuitionistic fuzzy WASPAS method: application in reservoir flood control management policy. Group Decis. Negot. 27(6), 1047–1078 (2018). https://doi.org/10.1007/s10726-018-9593-7 191. Mondal, T.K., Samanta, S.: Topology of interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 119(3), 483–494 (2001) 192. Moyle, S.: Collaborative data mining. In: Maimon, O., Rokach, L. (eds.) Data Mining and Knowledge Discovery Handbook, 2nd edn., pp. 1029–1039. Springer, New York (2010) 193. Mu, Z., Zeng, S., Liu, Some, Q.: interval-valued intuitionistic fuzzy zhenyuan aggregation operators and their application to multi-attribute decision making. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 26(4), 633–653 (2018). https://doi.org/10.1142/S0218488518500290 194. Nayagam, L.G., Sivaraman, G.: Ranking of interval-valued intuitionistic fuzzy sets. Appl. Soft Comput. J. 11(4), 3368–3372 (2011) 195. Nayagam, L.G., Muralikrishnan, S., Sivaraman, G.: Multi-criteria decision-making method based on interval-valued intuitionistic fuzzy sets. Expert Syst. Appl. 38(3), 1464–1467 (2011) 196. Nayagam, V., Jeevaraj, S., Dhanasekaran, P.: An intuitionistic fuzzy multi-criteria decisionmaking method based on non-hesitance score for interval-valued intuitionistic fuzzy sets. Soft Comput. 21(23), 7077–7082 (2017). https://doi.org/10.1007/s00500-016-2249-0 197. Nguyen, H.: A new generalized knowledge measure in multi-attribute group decision making under interval-valued intuitionistic fuzzy environment. In: Proceedings of the 2nd International Conference on Machine Learning and Soft Computing, pp. 156–163 (2018). https:// doi.org/10.1145/3184066.3184067. ISBN: 978-1-4503-6336-5 198. Orriols-Puig, A., Bernado-Mansilla, E.: Mining imbalanced data with learning classifier systems. In: Bull, L., Ester, B.-M., Holmes, J. (eds.) Learning Classifier Systems in Data Mining, pp. 123–145. Springer, Berlin (2008) 199. Oztaysi, B., Onar, S., Kahraman, C., Yavuz, M.: Multi-criteria alternative-fuel technology selection using interval-valued intuitionistic fuzzy sets. Transp. Res. Part D: Transp. Environ. 53, 128–148 (2017). https://doi.org/10.1016/j.trd.2017.04.003 200. Park, C.: ([r, s], [t, u])-Interval-valued intuitionistic fuzzy alpha generalized continuous mappings. Korean J. Math. 25(2), 261–278 (2017) 201. Park, D., Kwun, Y.C., Park, J.H., et al.: Correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multiple attribute group decision making problems. Math. Comput. Model. 50(9–10), 1279–1293 (2009) 202. Park, J.H., Park, I.Y., Kwun, Y.C., Tan, X.: Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment. Appl. Math. Model. 35(5), 2544–2556 (2011) 203. Pechenizkiy, M., Puuronen, S., Tsymbal, A.: Does relevance matter to data mining research? In: Lin, T.Y., Xie, Y., Wasilewska, A., Liau, C.-J. (eds.) Data Mining: Foundations and Practice, pp. 251–275. Springer, Berlin (2008) 204. Pei, Z., Lu, J. S., Zheng, L.: Generalized interval valued intuitionistic fuzzy numbers with applications in workstation assessment. Xitong Gongcheng Lilun yu Shijian/Syst. Eng. Theory Pract. 32(10), 2198–2206 (2012)

188

6 Applications of IVIFSs

205. Pekala, B., Balicki, K.: Interval-valued intuitionistic fuzzy sets and similarity measure. Iran. J. Fuzzy Syst. Article 6, 14(4), 87–98 (2017). https://doi.org/10.22111/IJFS.2017.3327 206. Pena-Ayala, A. (ed.): Educational Data Mining. Springer, Cham (2014) 207. Pencheva, T., Angelova, M., Vassilev, P., Roeva, O.: InterCriteria analysis approach to parameter identification of a fermentation process model, novel developments in uncertainty representation and processing, Part V. Advances in Intelligent Systems and Computing, pp. 385–397. Springer (2016) 208. Pencheva, T., Angelova, M.: InterCriteria analysis of simple genetic algorithms performance, chapter. In: Georgiev, K., Todorov, M., Georgiev, I. (eds.) Advanced Computing in Industrial Mathematics, Studies in Computational Intelligence, vol. 681, pp. 147–159. Springer (2017) 209. Pencheva, T., Roeva, O., Angelova, M.: Investigation of genetic algorithm performance based on different algorithms for intercriteria relations calculation, chapter. In: Lirkov, I., Margenov, S. (eds.) Large-Scale Scientific Computing, Vol. 10665 of Lecture Notes in Computer Science, pp. 390–398 (2018) 210. Pencheva, T., Angelova, M., Atanassova, V., Roeva, O.: Intercriteria analysis of genetic algorithm parameters in parameter identification. Notes Intuit. Fuzzy Sets 21(2), 99–110 (2015) 211. Qi, F.: Research on the comprehensive evaluation of sports management system with interval valued intuitionistic fuzzy information. Int. J. Adv. Comput. Technol. 4(6), 288–294 (2012) 212. Qi, X.-W., Liang, C.-Y., Zhang, E.-Q., Ding, Y.: Approach to interval-valued intuitionistic fuzzy multiple attributes group decision making based on maximum entropy. Xitong Gongcheng Lilun yu Shijian/Syst. Eng. Theory Pract. 31(10), 1940–1948 (2011) 213. Qi, X.-W., Liang, C.-Y., Cao, Q.-W., Ding, Y.: Automatic convergent approach in intervalvalued intuitionistic fuzzy multi-attribute group decision making. Xi Tong Gong Cheng Yu Dian Zi Ji Shu/Syst. Eng. Electron. 33(1), 110–115 (2011) 214. Qin, Y., Liu, Y., Liu, J.: A novel method for interval-value intuitionistic fuzzy multicriteria decision-making problems with immediate probabilities based on OWA distance operators. Math Probl. Eng. 2018, Art. no. 1359610 (2018). https://doi.org/10.1155/2018/1359610 215. Rajesh, K., Srinivasan, R.: Application of interval-valued intuitionistic fuzzy sets of second type in pattern recognition. Notes Intuit. Fuzzy Sets 24(1), 80–86 (2018) 216. Rani, P., Jain, D., Hooda, D.S.: Shapley function based interval-valued intuitionistic fuzzy VIKOR technique for correlative multi-criteria decision making problems. Iran. J. Fuzzy Syst. 15(1), 25–54 (2018). https://doi.org/10.22111/ijfs.2018.3577 217. Reiser, R., Bedregal, B., Bustince, H., Fernandez, J.: Generation of interval valued intuitionistic fuzzy implications from K operators, fuzzy implications and fuzzy coimplications. Commun. Comput. Inf. Sci. CCIS (PART 2) 298, 450–460 (2012) 218. Ren, H., Chen, H., Fei, W., Li, D.: A MAGDM method considering the amount and reliability information of interval-valued intuitionistic fuzzy sets. Int. J. Fuzzy Syst. 19(3), 715–725 (2017). https://doi.org/10.1007/s40815-016-0179-8 219. Ribagin, S., Zaharieva, B., Radeva, I., Pencheva, T.: Generalized net model of proximal humeral fractures diagnosing. Int. J. Bioautomation 22(1), 11–20 (2018). https://doi.org/10. 7546/ijba.2018.22.1.11-20 220. Robinson, J.: Contrasting correlation coefficient with distance measure in interval valued intuitionistic trapezoidal fuzzy MAGDM Problems (Chapter 60). Fuzzy Syst. Concepts Methodol. Tools Appl. 1448–1479 (2017). https://doi.org/10.4018/978-1-5225-1908-9.ch060 221. Roeva O., Pencheva, T., Angelova, M., Vassilev, P.: Intercriteria analysis by pairs and triples of genetic algorithms application for models identification, chapter. In: Fidanova, S. (ed.) Recent Advances in Computational Optimization, Vol. 655 of Studies in Computational Intelligence, pp. 193–218 (2016) 222. Roeva O., Vassilev, P., Angelova, M., Pencheva, T.: Intercriteria analysis of parameters relations in fermentation processes models, chapter. In: Nunes, M., Nguyen, N., Camacho, D., Trawinski, B. (eds.) Computational Collective Intelligence, vol. 9330 of Lecture Notes in Artificial Intelligence, pp. 171–181 (2015) 223. Roeva O., Zoteva, D.: Knowledge discovery from data: InterCriteria Analysis of mutation rate influence. Notes Intuit. Fuzzy Sets 24(1), 120–130 (2018)

References

189

224. Roeva, O., Fidanova, S., Paprzycki, M.: InterCriteria analysis of ACO and GA hybrid algorithms. Studies in Computational Intelligence, pp. 107–126. Springer (2016) 225. Roeva, O., Fidanova, S., Vassilev, P., Gepner, P.: InterCriteria analysis of a model parameters identification using genetic algorithm. In: Proceedings of the Federated Conference on Computer Science and Information Systems, Annals of Computer Science and Information Systems, vol. 5, pp. 501–506 (2015) 226. Roeva, O., Pencheva, T., Angelova, M., Vassilev, P.: InterCriteria analysis by pairs and triples of genetic algorithms application for models identification. Recent Advances in Computational Optimization, Studies in Computational Intelligence, pp. 193–218. Springer (2016) 227. Roeva, O., Vassilev, P., Angelova, M., Su, J., Pencheva, T.: Comparison of different algorithms for interCriteria relations calculation. In: IEEE 8th International Conference on Intelligent Systems, Sofia, Bulgaria, 4–6 Sept 2016, pp. 567–572 (2016) 228. Roeva, O., Vassilev, P., Ikonomov, N., Angelova, M., Su, J., Pencheva, T.: On different algorithms for interCriteria relations calculation, chapter. In: Hadjiski, M., Atanassov, K.T. (eds.) Intuitionistic Fuzziness and Other Intelligent Theories and Their Applications, Vol. 757 of Studies in Computational Intelligence, pp. 143–160 (2019) 229. Roeva, O., Vassilev, P.: InterCriteria analysis of generation gap influence on genetic algorithms performance. In: Atanassov, K.T., Castillo, O., Kacprzyk, J., Krawczak, M., Melin, P., Sotirov, S., Sotirova, E., Szmidt, E., De Tre, G., Zadro˙zny, S. (eds.) Novel Developments in Uncertainty Representation and Processing, Part V, Advances in Intelligent Systems and Computing, vol. 401, pp. 301–313 (2016) 230. Roeva, O., Fidanova, S.: InterCriteria analysis of relations between model parameters estimates and ACO performance. Adv. Comput. Ind. Math. Stud. Comput. Intell. 681, 175–186 (2017) 231. Roeva, O., Vassilev, P.: InterCriteria analysis of generation gap influence on genetic algorithms performance. Adv. Intell. Syst. Comput. 401, 301–313 (2016) 232. Roeva, O., Vassilev, P., Fidanova, S., Paprzycki, M.: InterCriteria analysis of genetic algorithms performance. Stud. Comput. Intell. 655, 235–260 (2016) 233. Roeva, O., Vassilev, P., Chountas, P.: Application of topological operators over data from InterCriteria analysis, FQAS 2017. LNAI 10333(215–225), 19 (2017). https://doi.org/10. 1007/978-3-319-59692-1 234. Roeva, O., Fidanova, S., Paprzycki, M.: Comparison of different ACO start strategies based on InterCriteria analysis. Stud. Comput. Intell. Book Ser. 717, 53–72 (2018) 235. Roeva, O., Fidanova, S., Luque, G., Paprzycki, M.: Intercriteria analysis of ACO performance for workforce planning problem. Recent Adv. Comput. Optim. 795, 47–67 (2019) 236. Rokach, L.: A survey of clustering algorithms, In: Maimon, O., Rokach, L. (eds.) Data Mining and Knowledge Discovery Handbook, 2nd edn., pp. 269–298. Springer, New York (2010) 237. Rud, O.P.: Data Mining Cookbook. Wiley, Danvers (2001) 238. Seifert, J.: Data Mining: An Overview, CRS Report for Congress, Order Code RL31798, Dec 2004 239. Shannon, A., Atanassov, K.: A first step to a theory of the intuitionistic fuzzy graphs. In: Lakov, D. (ed.) Proceedings of the First Workshop on Fuzzy Based Expert Systems, Sofia, 28–30 Sept 1994, pp. 59–61 (1994) 240. Shen, L., Li, G., Liu, W.: Method of multi-attribute decision making for interval-valued intuitionistic fuzzy sets based on mentality function. Appl. Mech. Mater. 44–47, 1075–1079 (2011) 241. Shmueli, G., Patel, N., Bruce, P.: Data Mining for Business Intelligence. Wiley, Hoboken (2007) 242. Simovici, D., Djeraba, C.: Mathematical Tools for Data Mining, 2nd edn. Springer, London (2014) 243. Singh, S., Garg, H.: Symmetric triangular interval type-2 intuitionistic fuzzy sets with their applications in multi criteria decision making. Symmetry 10(9), art. no. 401 (2018). https:// doi.org/10.3390/sym10090401

190

6 Applications of IVIFSs

244. Sotirov S., Vardeva, I., Krawczak, M.: Iintuitionistic fuzzy multilayer perceptron as a part of integrated systems for early forest-fire detection. Intuitionistic Fuzzy Sets, Sofia, vol. 19, N: 3, pp. 81–89 (2013) 245. Sotirov, S., Atanassov, K.: Generalized nets in artificial intelligence. vol. 6: Generalized nets and Supervised Neural Networks. “Prof. M. Drinov” Academic Publishing House, Sofia (2012) 246. Sotirov, S., Atanassov, K.: Intuitionistic fuzzy feed forward neural network. Cybern. Inf. Technol. 9(2), 62–68 (2009) 247. Sotirov, S., Atanassova, V., Sotirova, E., Bureva, V., Mavrov, D.: Application of the intuitionistic fuzzy InterCriteria analysis method to a neural network preprocessing procedure. In: 9th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), 30.06-03.07.2015, Gijon, Spain, pp. 1559–1564 (2015). https://doi.org/10.2991/ifsa-eusflat15.2015.222 248. Sotirov, S., Atanassova, V., Sotirova, E., Doukovska, L., Bureva, V., Mavrov, D., Tomov, J.: Application of the intuitionistic fuzzy InterCriteria analysis method with triples to a neural network preprocessing procedure. Computational Intelligence and Neuroscience, Hindawi, vol. 2017, Article ID 2157852 (2017) 249. Sotirov, S., Kodogiannis, V., Blessing, R.E.: Intuitionistic fuzzy estimations for connections with Low Rate Wireless personal area networks. In: First International Workshop on on Intuitionistic Fuzzy Sets, Generalized Nets, and Knowledge Engineering, London, 6-7 Sept 2006, pp. 81–87 (2006) 250. Sotirov, S., Method for determining of intuitionistic fuzzy sets in discovering water floods by neural networks. Issue on Intuitionistic Fuzzy Sets and Generalized Nets, Warszawa, vol. 4, pp. 9–14 (2007) 251. Sotirov, S., Sotirova, E., Atanassova, V., Atanassov, K., Castillo, O., Melin, P., Petkov, T., Surchev, S.: A hybrid approach for modular neural network design using intercriteria analysis and intuitionistic fuzzy logic. Complexity 2018, Article ID 3927951, 11 pages. https://doi. org/10.1155/2018/3927951 252. Sotirov, S., Sotirova, E., Melin, P., Castillo, O., Atanassov, K.: Modular neural network preprocessing procedure with intuitionistic fuzzy intercriteria analysis method. In: Andreasen, T. at al. (eds.) Flexible Query Answering Systems 2015, pp. 175–186. Springer, Cham (2016) 253. Sotirov, S.: Determining of intuitionistic fuzzy sets in estimating probability of spam in the e-mail by the help of the neural network. Issue on Intuitionistic Fuzzy Sets and Generalized Nets, Warszawa, vol. 4, pp. 43–47 (2007) 254. Sotirov, S.: Opportunities for application of the intercriteria analysis method to neural network preprocessing procedures. Notes Intuit. Fuzzy Sets 21(4), 143–152 (2015) 255. Sotirov, S.: Intuitionistic fuzzy estimations for connections of the transmit routines of the bluetooth interface. Adv. Stud. Contemp. Math. 15(1), 99–108 (2007) 256. Sotirov, S., Dimitrov, A.: Neural network for defining intuitionistic fuzzy estimation in petroleum recognition. Issue Intuit. Fuzzy Sets Gen. Nets 8, 74–78 (2010) 257. Sotirova, E., Bureva, V., Chountas, P., Krawczak, M.: An application of intercriteria decision making method to the rankings of universities in the United Kingdom. Notes Intuit. Fuzzy Sets 22(3), 112–119 (2016) 258. Sotirova, E., Bureva, V., Sotirov, S.: A generalized net model for evaluation process using intercriteria analysis method in the university. Imprecision and Uncertainty in Information Representation and Processing, Studies in Fuzziness and Soft Computing, pp. 389–399. Springer (2015) 259. Sotirova, E., Shannon, A.: Application of InterCriteria decision making to the rankings of Australian universities. Notes Intuit. Fuzzy Sets 21(4), 136–142 (2015). ISSN 1310-4926 260. Spurgin, A., Petkov, G.: Advances simulator data mining for operators’ performance assessment, In: Ruan, D., Chen, G., Kerre, E., Wets, G. (eds.) Intelligent Data Mining: Techniques and Applications, pp. 487–514. Springer, Berlin (2005) 261. Stanujki´c, D., Meidut˙e-Kavaliauskien˙e, I.: An approach to the production plant location selection based on the use of the Atanassov interval-valued intuitionistic fuzzy sets. Transport 33(3), 835–842 (2018)

References

191

262. Stratiev, D., Nedelchev, A., Shishkova, I., Ivanov, A., Sharafutdinov, I., Nikolova, R., Mitkova, M., Yordanov, D., Rudnev, N., Belchev, Z., Atanassova, V., Atanassov, K.: Dependence of visbroken residue viscosity and vacuum residue conversion in a commercial visbreaker unit on feedstock quality. Fuel Process. Technol. 138, 595–604 (2015) 263. Stratiev, D., Sotirov, S., Shishkova, I., Nedelchev, A., Sharafutdinov, I., Vely, A., Mitkova, M., Yordanov, D., Sotirova, E., Atanassova, V., Atanassov, K., Stratiev, D.D., Rudnev, N., Ribagin, S.: Investigation of relationships between bulk properties and fraction properties of crude oils by application of the intercriteria analysis. Pet. Sci. Technol. 34(13), 1113–1120 (2015) 264. Sumathi, S., Sivanandam, S.: Introduction to Data Mining and Applications, Berlin (2006) 265. Tan, C.: A multi-criteria interval-valued intuitionistic fuzzy group decision making with Choquet integral-based TOPSIS. Expert Syst. Appl. 38(4), 3023–3033 (2011) 266. Tang, J., Meng, F., Zhang, Y.: Decision making with interval-valued intuitionistic fuzzy preference relations based on additive consistency analysis. Inf. Sci. 467, 115–134 (2018). https:// doi.org/10.1016/j.ins.2018.07.036 267. Tao, Z., Liu, X., Chen, H., Zhou, L.: Ranking interval-valued fuzzy numbers with intuitionistic fuzzy possibility degree and its application to fuzzy multi-attribute decision making. Int. J. Fuzzy Syst. 19(3), 646–658 (2017) 268. Tian, H., Li, J., Zhang, F., Xu, Y., Cui, C., Deng, Y., Xiao, S.: Entropy analysis on intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets and its applications in mode assessment on open communities. J. Adv. Comput. Intell. Intell. Inform. 22(1), 147–155 (2018). https:// doi.org/10.20965/jaciii.2018.p0147 269. Tiwari, P., Gupta, P.: Entropy, distance and similarity measures under interval valued intuitionistic fuzzy environment. Informatica 42(4), 617–627 (2018). https://doi.org/10.31449/ inf.v42i4.1303 270. Todorova, L., Vassilev, P., Surchev, J.: Using Phi coefficient to interpret results obtained by InterCriteria analysis. In: Novel Developments in Uncertainty Representation and Processing, Vol. 401, Advances in Intelligent Systems and Computing, pp. 231–239. Springer (2106). https://doi.org/10.1007/978-3-319-26211-6_20 271. Tooranloo, H.S., Ayatollah, A.S., Karami, M.: Analysis of causal relationship between factors affecting the successful implementation of enterprise resource planning using intuitionistic fuzzy: DEMATEL. Int. J. Bus. Inf. Syst. 29(4), 436–458 (2018). https://doi.org/10.1504/ IJBIS.2018.096032 272. Traneva, V., Tranev, S., Szmidt, E., Atanassov, K.: Three dimensional interCriteria analysis over intuitionistic fuzzy data. In: Advances in Fuzzy Logic and Technology’ 2017, vol. 3, 1, pp. 442–449. Springer, Cham (2018) 273. Traneva, V., Tranev, S.: Intuitionistic fuzzy InterCriteria approach to the assessment in a fast food restaurant. In: Advances in Intelligent Systems and Computing, Intelligent and Fuzzy Techniques in Big Data Analytics and Decision Making—Proceedings of the INFUS Conference, 23–25 July 2019, Istanbul, Turkey, vol. 1003 (2019). ISBN 978-3-030-23755-4. https://doi.org/10.1007/978-3-030-23756-1 274. Tu, C.C., Chen, L.H.: Novel score functions for interval valued intuitionistic fuzzy values. In: Proceedings of the SICE Annual Conference, art. no. 6318743, pp. 1787–1790 (2012) 275. Valchev D., Sotirov, S.: Intuitionistic fuzzy detection Of signal availability In multipath wireles chanels, 14-th International Conference on Intuitionistic Fuzzy Sets, Sofia, pp. 24–29 (2009) 276. Vankova, D., Sotirova, E., Bureva, V.: An application of the InterCriteria method approach to health-related quality of life. Notes Intuit. Fuzzy Sets 21(5), 40–48 (2015) 277. Vardeva I., Sotirov, S.: Intuitionistic fuzzy estimation of damaged packets with multilayer perceptron. In: Proceedings of the Tenth International Workshop on Generalized Nets, IWGN’2009, Sofia, pp. 63–69 (2009) 278. Vassilev, P., Todorova, L., Andonov, V.: An auxiliary technique for InterCriteria analysis via a three dimensional index matrix. Notes Intuit. Fuzzy Sets 21(2), 71–76 (2015) 279. Wan, S., Wang, F., Dong, J.: A three-phase method for group decision making with intervalvalued intuitionistic fuzzy preference relations. IEEE Trans. Fuzzy Syst. 26(2), 998–1010 (2018). https://doi.org/10.1109/TFUZZ.2017.2701324

192

6 Applications of IVIFSs

280. Wan, S.-P.: Multi-attribute decision making method based on interval-valued intuitionistic trapezoidal fuzzy number. Kongzhi yu Juece/Control Decis. 26(6), 857-860+866 (2011) 281. Wang, Z., Li, K.W., Xu, J.: A mathematical programming approach to multi-attribute decision making with interval-valued intuitionistic fuzzy assessment information. Expert Syst. Appl. 38(10), 12462–12469 282. Wang, Z.J., Li, K.W.: An interval valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives. Expert Syst. Appl. 39(18), 13509–13516 (2012) 283. Wang, J.-Q., Li, K.-J.; Zhang, H.-Y.U.: Interval-valued intuitionistic fuzzy multi-criteria decision-making approach based on prospect score function. Knowl.-Based Syst. 27, 119–125 (2012) 284. Wang, W., Liu, X., Qin, Y.: Interval valued intuitionistic fuzzy aggregation operators. J. Syst. Eng. Electron. 23(4), 574–580 (2012) 285. Wang, Q., Sun, H. Interval-Valued intuitionistic fuzzy einstein geometric choquet integral operator and its application to multiattribute group decision-making. Math. Probl. Eng. 2018, Art. no. 9364987 (2018). https://doi.org/10.1155/2018/9364987 286. Wang, J.Q., Li, K.J., Zhang, H.Y.: Interval valued intuitionistic fuzzy multi criteria decision making approach based on prospect score function. Knowl. Based Syst. 27, 119–125 (2012) 287. Wang, P., Xu, X., Wang, J., Cai, C.: Some new operation rules and a new ranking method for interval-valued intuitionistic linguistic numbers. J. Intell. Fuzzy Syst. 32(1), 1069–1078 (2017). https://doi.org/10.3233/JIFS-16644 288. Wang, P., Xu, X., Wang, J., Cai, C.: Interval-valued intuitionistic linguistic multi-criteria group decision-making method based on the interval 2-tuple linguistic information. J. Intell. Fuzzy Syst. 33(2), 985–994 (2017). https://doi.org/10.3233/JIFS-162291 289. Wei, G.-W., Wang, H.-J., Lin, R.: Application of correlation coefficient to interval-valued intuitionistic fuzzy multiple attribute decision-making with incomplete weight information. Knowl. Inf. Syst. 26(2), 337–349 290. Wei, C.-P., Wang, P., Zhang, Y.-Z.: Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications. Inf. Sci. 181(19), 4273–4286 291. Wei, G., Zhao, X.: An approach to multiple attribute decision making with combined weight information in interval-valued intuitionistic fuzzy environment. Control Cybern. 41(1), 97– 112 (2012) 292. Witten, H., Frank, E.: Data Mining: Practical Machine Learning Tools and Techniques. Morgan Kaufmann (2005) 293. Wu, J., Chiclana, F.: Non dominance and attitudinal prioritisation methods for intuitionistic and interval valued intuitionistic fuzzy preference relations. Expert Syst. Appl. 39(18), 13409– 13416 (2012) 294. Wu, L., Wei, G., Gao, H., Wei, Y.: Some interval-valued intuitionistic fuzzy Dombi Hamy mean operators and their application for evaluating the elderly tourism service quality in tourism destination. Mathematics 6(12), Art. no. 294 (2018). https://doi.org/10.3390/math6120294 295. Wu, Y., Yu, D.: Evaluation of sustainable development for research oriented universities based on interval valued intuitionistic fuzzy set. Dongnan Daxue Xuebao (Ziran Kexue Ban)/J. Southeast Univ. (Natural Science Edition), 42(4), 790–796 (2012) 296. Xian, S., Dong, Y., Liu, Y., Jing, N.: A novel approach for linguistic group decision making based on generalized interval-valued intuitionistic fuzzy linguistic induced hybrid operator and TOPSIS. Int. J. Intell. Syst. 33(2), 288–314 (2018). https://doi.org/10.1002/int.21931 297. Xu, K., Zhou, J., Gu, R., Qin, H.: Approach for aggregating interval-valued intuitionistic fuzzy information and its application to reservoir operation. Expert Syst. Appl. 38(7), 9032–9035 298. Xu, G.-L.: A consensus reaching model with minimum adjustments in interval-valued intuitionistic MAGDM. Math. Probl. Eng. 2018, art. no. 9070813 (2018). https://doi.org/10.1155/ 2018/9070813 299. Xu, Z.: Methods for aggregation interval-valued intuitionistic fuzzy information and their Application to decision making. Control Decis. 22(2), 215–219 (2007)

References

193

300. Xu, Z.: A method based on distance measure for interval-valued intuitionistic fuzzy group decision making. Inf. Sci. 180(1), 181–190 (2010) 301. Xu, Z., Cai, X.: Incomplete interval-valued intuitionistic fuzzy preference relations. Int. J. Gen. Syst. 38(8), 871–886 (2009) 302. Xu, G., Duan, X., Lü, H.: Target priority determination methods by interval-valued intuitionistic fuzzy sets with unknown attribute weights. J. Shanghai Jiaotong Univ. (Science) 22(5), 624–632 (2017). https://doi.org/10.1007/s12204-017-1880-y 303. Yang, Y., Liang, C., Ji, S.: Comments on “Fuzzy multicriteria decision making method based on the improved accuracy function for interval-valued intuitionistic fuzzy sets” by Ridvan Sahin. Soft Comput. 21(11), 3033–3035 (2017). https://doi.org/10.1007/s00500-015-19887 304. Yao, Y., Zhong, N., Zhao, Y.: A conceptual framework of data mining, In: Lin, T.Y., Xie, Y., Wasilewska, A., Liau, C.-J. (eds.) Data Mining: Foundations and Practice, pp. 501–515. Springer, Berlin (2008) 305. Ye, D., Liang, D., Hu, P.: Three-way decisions with interval-valued intuitionistic fuzzy decision-theoretic rough sets in group decision-making. Symmetry 10(7), art. no. 281 (2018). https://doi.org/10.3390/sym10070281 306. Ye, J.: Generalized Dice measures for multiple attribute decision making under intuitionistic and interval-valued intuitionistic fuzzy environments. Neural Comput. Appl. 30(12), 3623– 3632 (2018). https://doi.org/10.1007/s00521-017-2947-2 307. Ye, J.: Multicriteria decision making method using the Dice similarity measure based on the reduct intuitionistic fuzzy sets of interval valued intuitionistic fuzzy sets. Appl. Math. Model. 36(9), 4466–4472 (2012) 308. Ye, J.: Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment. Expert Syst. Appl. 36(3), 6899–6902 (2009) 309. Ye, F.: An extended TOPSIS method with interval-valued intuitionistic fuzzy numbers for virtual enterprise partner selection. Expert Syst. Appl. 37(10), 7050–7055 (2010) 310. Ye, J.: Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets. Appl. Math. Modell. 34(12), 3864– 3870 (2010) 311. Yin, S., Yang, Z., Chen, S.: Interval-valued intuitionistic fuzzy multiple attribute decision making based on the improved fuzzy entropy. Xi Tong Gong Cheng Yu Dian Zi Ji Shu/Syst. Eng. Electron. 40(5), 1079–1084 (2018). https://doi.org/10.3969/j.issn.1001-506X.2018.05. 18 312. Yu, G., Li, D., Qiu, J.: Interval-Valued intuitionistic fuzzy multi-attribute decision making based on satisfactory degree. Theoretical and Practical Advancements for Fuzzy System Integration, vol. 49, p. 23 (2017). https://doi.org/10.4018/978-1-5225-1848-8.ch003 313. Yu, D., Wu, Y.: Interval-valued intuitionistic fuzzy Heronian mean operators and their application in multi-criteria decision making. Afr. J. Bus. Manag. 6(11), 4158–4168 (2012) 314. Yu, D., Wu, Y., Lu, T.: Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making. Knowl.-Based Syst. 30, 57–66 (2012) 315. Yu, G., Li, D., Qiu, J., Ye, Y.: Application of satisfactory degree to interval-valued intuitionistic fuzzy multi-attribute decision making. J. Intell. Fuzzy Syst. 32(1), 1019–1028 (2017). https:// doi.org/10.3233/JIFS-16557 316. Yu, L., Wang, L., Bao, Y.: Technical attributes ratings in fuzzy QFD by integrating intervalvalued intuitionistic fuzzy sets and Choquet integral. Soft Comput. 22(6), 2015–2024 (2018). https://doi.org/10.1007/s00500-016-2464-8 317. Yue, Z.: Deriving decision maker’s weights based on distance measure for interval-valued intuitionistic fuzzy group decision making. Expert Syst. Appl. 38(9), 11665–11670 318. Yue, Z. An approach to aggregating interval numbers into interval-valued intuitionistic fuzzy information for group decision making. Expert Syst. Appl. 38 (5), 6333–6338 319. Zadeh, L.: Fuzzy sets. Inf. Control 8, 338–353 (1965) 320. Zaharieva, B., Doukovska, L., Ribagin, S., Mihalicova, A., Radeva, I.: Intercriteria analysis of Behterev’s Kinesitherapy program. Notes on Intuit. Fuzzy Sets 23(3), 69–80 (2017)

194

6 Applications of IVIFSs

321. Zaharieva, B., Doukovska, L., Ribagin, S., Radeva, I.: InterCriteria approach to Behterev’s disease analysis. Notes Intuit. Fuzzy Sets 23(2), 119–127 (2017) 322. Zeng, W., Zhao, Y.: Approximate, reasoning of interval valued fuzzy sets based on interval valued similarity measure set: ICIC Express Letters. Part B: Appl. 3(4), 725–732 (2012) 323. Zhang, Y.J., Ma, P.J., Su, X.H., Zhang, C.P.: Entropy on interval-valued intuitionistic fuzzy sets and its application in multi-attribute decision making. In: Fusion 2011–14th International Conference on Information Fusion, art. no. 5977465 324. Zhang, Y.J., Ma, P.J., Su, X.H., Zhang, C.P.: Multi attribute decision making with uncertain attribute weight information in the framework of interval valued intuitionistic fuzzy set. Zidonghua Xuebao/Acta Automatica Sinica 38(2), 220–228 (2012) 325. Zhang, J.L., Qi, X.W.: Induced interval valued intuitionistic fuzzy hybrid aggregation operators with TOPSIS order inducing variables. J. Appl. Math. 2012, Art. no. 245732 (2012) 326. Zhang, Q., Yao, H., Zhang, Z.: Some similarity measures of interval-valued intuitionistic fuzzy sets and application to pattern recognition. Appl. Mech. Mater. 44–47, 3888–3892 327. Zhang, Q.-sh., Jiang, S., Jia, B. et al.: Some information measures for interval-valued intuitionistic fuzzy sets. Inf. Sci. 180(24), 5130–5145 (2011) 328. Zhang, Z.: Approaches to group decision making based on interval-valued intuitionistic multiplicative preference relations. Neural Comput. Appl. 28(8), 2105–2145 (2017). https://doi. org/10.1007/s00521-016-2183-1 329. Zhang, Q., Jiang, S.: Relationships between entropy and similarity measure of interval-valued intuitionistic fuzzy sets. Int. J. Intell. Syst. 25(11), 1121–1140 (2010) 330. Zhang, H., Yu, L.: MADM method based on cross-entropy and extended TOPSIS with intervalvalued intuitionistic fuzzy sets. Knowl.-Based Syst. 30, 115–120 (2012) 331. Zhang, Z., Hu, Y., Ma, C., Xu, J., Yuan, S., Chen, Z.: Incentive-punitive risk function with interval valued intuitionistic fuzzy information for outsourced software project risk assessment. J. Intell. Fuzzy Syst. 32(5), 3749–3760 (2017). https://doi.org/10.3233/JIFS-169307 332. Zhao, H., Ni, M., Liu, H. A class of new interval valued intuitionistic fuzzy distance measures and their applications in discriminant analysis. Appl. Mech. Mater. 182–183, 1743–1745 (2012) 333. Zoteva D., Roeva, O., Delkov, A., Tsakov, H.: Intercriteria analysis of forest fire risk. In: Proceedings of the 4th International Conference on Numerical and Symbolic Computation— Developments and Applications (SYMCOMP 2019), pp. 215–229 (2019) 334. Zoteva, D., Krawczak, M.: Generalized Nets as a Tool for the Modelling of Data Mining Processes. A Survey. Issues in Intuitionistic Fuzzy Sets and Generalized Nets, vol. 13, pp. 1–60 (2017) 335. Zoteva, D., Roeva, O.: InterCriteria analysis results based on different number of objects. Notes Intuit. Fuzzy Sets 24(1), 110–119 (2018) 336. Zou, P., Liu, Y.: Model for evaluating the security of wireless sensor network in interval valued intuitionistic fuzzy environment. Int. J. Adv. Comput. Technol. 4(4), 254–260 (2012)

Chapter 7

Conclusion and Remarks on Future Research

There are a lot of directions of studying and extending the IVIFSs. In Sect. 3.4, we discussed one of these directions, describing the IVIFS over different universes. Below, we shortly discuss another IVIFS-extension. Let E be a universe, and T be a non-empty set. We call the elements of T “timemoments”. Based on the definition of Temporal IFS (TIFS), we define another type of an IVIFS (see, e.g., [1, 2]). We define a Temporal IVIFS (TIVIFS) as the following: A(T ) = {x, M A (x, t), N A (x, t)|x, t ∈ E × T }, where (a) A ⊂ E is a fixed set, (b) M A (x, t), N A (x, t) ⊂ [0, 1] and sup M A (x, t) + sup N A (x, t) ≤ 1 for every x, t ∈ E × T , (c) M A (x, t) and N A (x, t) are intervals of the degrees of membership and non-membership, respectively, of the element x ∈ E at the time-moment t ∈ T . Obviously, every ordinary IVIFS can be regarded as a TIVIFS for which T is a singleton set, while every ordinary TIFS can be regarded as a TIVIFS for which the intervals of the two degrees are singleton sets. All operations and operators on the IVIFSs can be defined for the TIVIFSs. Suppose that we have two TIVIFSs: A(T  ) = {x, M A (x, t), N A (x, t)|x, t ∈ E × T  }, and

B(T  ) = {x, M B (x, t), N B (x, t)|x, t ∈ E × T  },

© Springer Nature Switzerland AG 2020 K. T. Atanassov, Interval-Valued Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing 388, https://doi.org/10.1007/978-3-030-32090-4_7

195

196

7 Conclusion and Remarks on Future Research

where T  and T  have finite number of distinct time-elements or they are timeintervals. Then we can define the above IVIFS-operations (∩, ∪, etc.), the topological (C and I ) and modal ( and ♦) operators. For example, A(T  ) ∪ B(T  ) = {x, M A(T  )∪B(T  ) (x, t), N A(T  )∪B(T  ) (x, t)|x, t ∈ E × T  }, where x, M A(T  )∪B(T  ) (x, t), N A(T  )∪B(T  ) (x, t) ⎧ x, M A (x, t  ), N A (x, t  ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x, M B (x, t  ), N A (x, t  ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x, [max(inf M A (x, t  ), inf M B (x, t  )), = max(sup M A (x, t  ), sup M B (x, t  ))], ⎪ ⎪ ⎪ ⎪ min(inf N A (x, t  ), inf N B (x, t  )), ⎪ ⎪ ⎪ ⎪ min(sup N A (x, t  ), sup N B (x, t  ))], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x, [0, 0], [1, 1],

if t = t  ∈ T  − T  if t = t  ∈ T  − T  if t = t  = t  ∈ T  ∩ T 

otherwise

Two of the specific operators over TIVIFSs are: C ∗ (A(T )) = {x, [sup inf M A(T ) (x, t), sup sup M A(T ) (x, t)], t∈T

t∈T

[inf inf N A(T ) (x, t), inf sup N A(T ) (x, t)]|x ∈ E}, t∈T

t∈T

I ∗ (A(T )) = {x, [inf inf M A(T ) (x, t), inf sup M A(T ) (x, t)], t∈T

t∈T

[sup inf N A(T ) (x, t), sup sup N A(T ) (x, t)]|x ∈ E}. t∈T

t∈T

We have the following important equalities for every TIVIFS A(T ): (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

C ∗ (C ∗ (A(T ))) = C ∗ (A(T )), C ∗ (I ∗ (A(T ))) = I ∗ (A(T )), I ∗ (C ∗ (A(T ))) = C ∗ (A(T )), I ∗ (I ∗ (A(T ))) = I ∗ (A(T )), C(C ∗ (A(T ))) = C ∗ (C(A(T ))), I(I ∗ (A(T ))) = I ∗ (I(A(T ))), ¬C ∗ (¬A(T )) = I ∗ (A(T )), ¬I ∗ (¬A(T )) = C ∗ (A(T )), C ∗ ( A(T )) = C ∗ (A(T )), C ∗ (♦A(T )) = ♦C ∗ = (A(T )), I ∗ ( A(T )) = I ∗ (A(T )),

7 Conclusion and Remarks on Future Research

197

(l) I ∗ (♦A(T )) = ♦I ∗ = (A(T )). For every two TIFSs A(T  ) and B(T  ): (a) (b) (c) (d)

C ∗ (A(T  ) ∩ B(T  )) ⊂ C ∗ (A(T  )) ∩ C ∗ (B(T  )), C ∗ (A(T  ) ∪ B(T  )) = C ∗ (A(T  )) ∪ C ∗ (B(T  )), I ∗ (A(T  ) ∩ B(T  )) = I ∗ (A(T  )) ∩ I ∗ (B(T  )), I ∗ (A(T  ) ∪ B(T  )) ⊃ I ∗ (A(T  )) ∪ I ∗ (B(T  )).

Large part of the research on IVIFSs and on TIFSs can be transformed to TIVIFSs. Some of them will be object of future author’s research. For example, in next research we will describe the multi-dimensional IVIFSs as extensions of the TIVIFSs by analogy with the multi-dimensional IFSs that are extensions of the TIFS (see [2]). Quite possibly, as in previous books of the author, despite the multiple proofreading, there again will be some misprints, but the author hopes that the benevolent reader will be looking for the new ideas and the unsolved problems, presented in this book, and will endeavor to further develop these ideas, promoting the theory of IVIFS to a complete form.

References 1. Atanassov, K.: Intuitionistic Fuzzy Sets. Springer, Heidelberg (1999) 2. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012)

Index

E Extended interval valued intuitionistic fuzzy index matrices, 133 Extension principle for IVIFS average form, 50 optimistic form, 48 pessimistic form, 49

I Intercriteria analysis with interval-valued intuitionistic fuzzy evaluations, 165 Interpretation triangle, 3 Interval valued fuzzy set (IVFS), 9 Interval valued intuitionistic fuzzy generalized net of first type, 152 of fourth type, 164 of second type, 160 of third type, 158 Interval valued intuitionistic fuzzy graph (◦)-(IFS,IFS)-IFG, 141 (◦)-(IFS,IVIFS)-IFG, 141 (◦)-(IVIFS,IFS)-IFG, 141 (◦)-(IVIFS,IVIFS)-IFG, 142 Interval valued intuitionistic fuzzy index matrices, 132 Interval valued intuitionistic fuzzy pair (IVIFP), 123 Interval valued intuitionistic fuzzy set (IVIFS), 10 Intuitionistic fuzzy neural network with interval valued intuitionistic fuzzy conditions, 147 Intuitionistic fuzzy set (IFS), 2

M Metrix, 22 Modal operator Dα , 53 Fα,β , 53 G α,β , 53 ∗ , 53 Hα,β Hα,β , 53 ∗ , 55 Jα,β , 53 Jα,β 

ext1 ext2 ext3 ext4 ext5 ext6 ext7 ext8

X a



1 b1 c1 d1 e1 f 1

,

58

,

59

,

58

a b c d e f  2 2 2 2 2 2 ext1 ext1 ext2 ext2 ext1 ext1 ext2 ext2

X a

1 b1 c1 d1 e1 f 1

a b c d e f  2 2 2 2 2 2 ext1 ext2 ext3 ext4 ext1 ext2 ext3 ext4

X a

1 b1 c1 d1 e1 f 1 a2 b2 c2 d2 e2 f 2

♦, 53

© Springer Nature Switzerland AG 2020 K. T. Atanassov, Interval-Valued Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing 388, https://doi.org/10.1007/978-3-030-32090-4

, 53 ◦

α,β,γ ,δ,ε,ζ ,

◦



◦

α1 β1 γ1 α2 β2 γ2 ( ext1 ext2 )  α1 β1 γ1 α2 β2 γ2

74

δ1 ε1 ζ1 δ2 ε2 ζ2 δ1 ε1 ζ1 δ2 ε2 ζ2

, 

80

, 82

+ , 69 + α,β,γ , 72 + α,β , 71 + α , 69 + α

1 β1 α2 β2

+ α

,

1 β1 γ1 α2 β2 γ2

77 ,

77 199

200

Index + α

1 α2

,

75

× , 69 × α,β,γ , 72 × α,β , 71 × α , 69 × α

1 β1 γ1 α2 β2 γ2

× α

1 β1 α2 β2

× α

1 α2

,

,

,

78

77

75

•

α,β,γ ,δ ,

•



73

α1 β1 γ1 δ1 α2 β2 γ2 δ2

,

79

F α,β,γ ,δ , 56 G α,β,γ ,δ , 56 ∗ H α,β,γ ,δ , 56 H α,β,γ ,δ , 57 ∗ J α,β,γ ,δ , 57 J α,β,γ ,δ , 57 N Non-strict relation over EIVIFIM inclusion, 137 inclusion about dimension, 137 inclusion about value, 137 Norm, 22 O Operation +, 28 @, 28 ., 28 #, 28 $, 28 ∗, 28 ∩, 28 ◦, 30 ∪, 28 ¬, 28 ¬1 , 38 ¬2 , 39 sg, 37 →1 , 31 →2 , 38 →4 , 38 sg, 37

×1 , 41 ×2 , 41 ×3 , 41 ×4 , 41 ×5 , 41 Operation over EIVIFIM −◦,∗ , 137 ¬, 136 (◦,∗) , 135 , 136 ⊕(∗) , 136 ⊗(∗) , 135 (∗)-column-aggregation, 138 (∗)-row-aggregation, 138 Operator weight-center operator (W ), 120 Operator over temporal IVIFS C ∗ , 196 I ∗ , 196

P Proper IVIFS, 93

R Relation =, 27 ⊂♦,inf , 27 ⊂♦,sup , 27 ⊂♦ , 27 ⊂ ,inf , 27 ⊂ ,sup , 27 ⊂ , 27 ⊂, 27 S Special IVIFS E ∗ , 30 Q ∗ , 30 U ∗ , 30 Strict relation over EIVIFIM inclusion, 137 inclusion about dimension, 137 inclusion about value, 137

T Temporal IVIFS (TIFS), 195 Topological operator C, 89 I , 89