Generalized Orthopair Fuzzy Sets
TL;DR: It is noted that as q increases the space of acceptable orthopairs increases and thus gives the user more freedom in expressing their belief about membership grade, and introduces a general class of sets called q-rung orthopair fuzzy sets in which the sum of the ${\rm{q}}$th power of the support against is bonded by one.
Abstract: We note that orthopair fuzzy subsets are such that that their membership grades are pairs of values, from the unit interval, one indicating the degree of support for membership in the fuzzy set and the other support against membership. We discuss two examples, Atanassov's classic intuitionistic sets and a second kind of intuitionistic set called Pythagorean. We note that for classic intuitionistic sets the sum of the support for and against is bounded by one, while for the second kind, Pythagorean, the sum of the squares of the support for and against is bounded by one. Here we introduce a general class of these sets called q-rung orthopair fuzzy sets in which the sum of the ${\rm{q}}$ th power of the support for and the ${\rm{q}}$ th power of the support against is bonded by one. We note that as q increases the space of acceptable orthopairs increases and thus gives the user more freedom in expressing their belief about membership grade. We investigate various set operations as well as aggregation operations involving these types of sets.
Citations
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TL;DR: This work presented two new methods to deal with the multi‐attribute decision making problems under the fuzzy environment and used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.
Abstract: The q-rung orthopair fuzzy sets (q-ROFs) are an important way to express uncertain information, and they are superior to the intuitionistic fuzzy sets and the Pythagorean fuzzy sets. Their eminent characteristic is that the sum of the qth power of the membership degree and the qth power of the degrees of non-membership is equal to or less than 1, so the space of uncertain information they can describe is broader. Under these environments, we propose the q-rung orthopair fuzzy weighted averaging operator and the q-rung orthopair fuzzy weighted geometric operator to deal with the decision information, and their some properties are well proved. Further, based on these operators, we presented two new methods to deal with the multi-attribute decision making problems under the fuzzy environment. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.
567 citations
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TL;DR: A Fermatean fuzzy TOPSIS method is established to fix multiple criteria decision-making problem and an interpretative example is stated in details to justify the elaborated method and to illustrate its viability and usefulness.
Abstract: In this paper, we propose Fermatean fuzzy sets. We compare Fermatean fuzzy sets with Pythagorean fuzzy sets and intuitionistic fuzzy sets. We focus on complement operator of Fermatean fuzzy sets. We find out the fundamental set of operations for the Fermatean fuzzy sets. We define score function and accuracy function for ranking of Fermatean fuzzy sets. In addition, we also study Euclidean distance between two Fermatean fuzzy sets. Later, we establish a Fermatean fuzzy TOPSIS method to fix multiple criteria decision-making problem. Ultimately, an interpretative example is stated in details to justify the elaborated method and to illustrate its viability and usefulness.
346 citations
TL;DR: An approach to multiple attribute decision making based on q‐ROFGWHM (q‐ROFWGHM) operator is proposed and a practical example for enterprise resource planning system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.
Abstract: The generalized Heronian mean and geometric Heronian mean operators provide two aggregation operators that consider the interdependent phenomena among the aggregated arguments. In this paper, the generalized Heronian mean operator and geometric Heronian mean operator under the q‐rung orthopair fuzzy sets is studied. First, the q‐rung orthopair fuzzy generalized Heronian mean (q‐ROFGHM) operator, q‐rung orthopair fuzzy geometric Heronian mean (q‐ROFGHM) operator, q‐rung orthopair fuzzy generalized weighted Heronian mean (q‐ROFGWHM) operator, and q‐rung orthopair fuzzy weighted geometric Heronian mean (q‐ROFWGHM) operator are proposed, and some of their desirable properties are investigated in detail. Furthermore, we extend these operators to q‐rung orthopair 2‐tuple linguistic sets (q‐RO2TLSs). Then, an approach to multiple attribute decision making based on q‐ROFGWHM (q‐ROFWGHM) operator is proposed. Finally, a practical example for enterprise resource planning system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.
333 citations
TL;DR: The proposed methods based on q‐ ROFWBM and q‐ROFWGBM operators are very useful to deal with MAGDM problems and are developed based on these operators.
Abstract: In the real multi-attribute group decision making (MAGDM), there will be a mutual relationship between different attributes. As we all know, the Bonferroni mean (BM) operator has the advantage of considering interrelationships between parameters. In addition, in describing uncertain information, the eminent characteristic of q-rung orthopair fuzzy sets (q-ROFs) is that the sum of the qth power of the membership degree and the qth power of the degrees of non-membership is equal to or less than 1, so the space of uncertain information they can describe is broader. In this paper, we combine the BM operator with q-rung orthopair fuzzy numbers (q-ROFNs) to propose the q-rung orthopair fuzzy BM (q-ROFBM) operator, the q-rung orthopair fuzzy weighted BM (q-ROFWBM) operator, the q-rung orthopair fuzzy geometric BM (q-ROFGBM) operator, and the q-rung orthopair fuzzy weighted geometric BM (q-ROFWGBM) operator, then the MAGDM methods are developed based on these operators. Finally, we use an example to illustrate the MAGDM process of the proposed methods. The proposed methods based on q-ROFWBM and q-ROFWGBM operators are very useful to deal with MAGDM problems.
280 citations
TL;DR: A new multiple-attribute decision-making (MADM) method is developed based on $q\hbox{-}{ROFWABM}$ operator and the Bonferroni mean (BM) operator is extended.
Abstract: The theory of $q$ -rung orthopair fuzzy sets ( $q$ -ROFSs) proposed by Yager effectively describes fuzzy information in the real world. Because $q$ -ROFSs contain the parameter $q$ and can adjust the range of expressed fuzzy information, they are superior to both intuitionistic and Pythagorean fuzzy sets. Archimedean T-norm and T-conorm (ATT) is an important tool used to generate operational rules based on the q -rung orthopair fuzzy numbers ( $q$ -ROFNs). In comparison, the Bonferroni mean (BM) operator has an advantage because it considers the interrelationships between the different attributes. Therefore, it is an important and meaningful innovation to extend the BM operator to the $q$ -ROFNs based upon the ATT. In this paper, we first discuss $q$ -rung orthopair fuzzy operational rules by using ATT. Furthermore, we extend BM operator to the $q$ -ROFNs and propose the $q$ -rung orthopair fuzzy Archimedean BM $(q\hbox{-}{ROFABM})$ operator and the q -rung orthopair fuzzy weighted Archimedean BM $(q\hbox{-}{ROFWABM})$ operator and study their desirable properties. Then, a new multiple-attribute decision-making (MADM) method is developed based on $q\hbox{-}{ROFWABM}$ operator. Finally, we use a practical example to verify effectiveness and superiority by comparing to other existing methods.
274 citations
Cites background or methods from "Generalized Orthopair Fuzzy Sets"
...In order to better understand this paper, we will introduce some basic and useful concepts of q-ROFSs [43], ATT, and BM operator [4], [41] in this section....
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...1 [43]: A q-ROFS Q̃ in a finite universe of discourse Y is represented by...
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...In the following, in order to compare q-ROFNs, we give the following definitions inspired by the idea of comparing PFNs [43]....
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...With the development of theory, a new concept was presented again by Yager [43], the q-rung orthopair fuzzy set (q-ROFS), in which the sum of the qth power of MD and the qth power of NMD is restricted to one, i....
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References
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TL;DR: Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.
13,376 citations
"Generalized Orthopair Fuzzy Sets" refers background in this paper
...In [3] and [4], Atanassov noted a second type of IFSs, one in which the square of the support for against is bounded by one this idea has been followed up in [12]–[16]....
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Book•
01 Jan 1995TL;DR: Fuzzy Sets and Fuzzy Logic is a true magnum opus; it addresses practically every significant topic in the broad expanse of the union of fuzzy set theory and fuzzy logic.
Abstract: Fuzzy Sets and Fuzzy Logic is a true magnum opus. An enlargement of Fuzzy Sets, Uncertainty,
and Information—an earlier work of Professor Klir and Tina Folger—Fuzzy Sets and Fuzzy Logic
addresses practically every significant topic in the broad expanse of the union of fuzzy set theory
and fuzzy logic. To me Fuzzy Sets and Fuzzy Logic is a remarkable achievement; it covers its vast
territory with impeccable authority, deep insight and a meticulous attention to detail.
To view Fuzzy Sets and Fuzzy Logic in a proper perspective, it is necessary to clarify a point
of semantics which relates to the meanings of fuzzy sets and fuzzy logic.
A frequent source of misunderstanding fias to do with the interpretation of fuzzy logic. The
problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow
sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization
of classical multivalued logics. But in a wider sense, fuzzy logic, FL^ is almost synonymous
with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLW is much
broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different
from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is
usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLW
In Fuzzy Sets and Fuzzy Logic, fuzzy logic is interpreted in a sense that is close to FLW. However,
to avoid misunderstanding, the title refers to both fuzzy sets and fuzzy logic.
Underlying the organization of Fuzzy Sets and Fuzzy Logic is a fundamental fact, namely,
that any field X and any theory Y can be fuzzified by replacing the concept of a crisp set in X and Y
by that of a fuzzy set. In application to basic fields such as arithmetic, topology, graph theory, probability
theory and logic, fuzzification leads to fuzzy arithmetic, fuzzy topology, fuzzy graph theory,
fuzzy probability theory and FLn. Similarly, hi application to applied fields such as neural network
theory, stability theory, pattern recognition and mathematical programming, fuzzification leads to
fuzzy neural network theory, fuzzy stability theory, fuzzy pattern recognition and fuzzy mathematical
programming. What is gained through fuzzification is greater generality, higher expressive
power, an enhanced ability to model real-world problems and, most importantly, a methodology for
exploiting the tolerance for imprecision—a methodology which serves to achieve tractability,
7,131 citations
"Generalized Orthopair Fuzzy Sets" refers background in this paper
...In anticipation of this generalization we shall say something about the logical negation or complement operator [21]....
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03 Jan 1988
TL;DR: A type of operator for aggregation called an ordered weighted aggregation (OWA) operator is introduced and its performance is found to be between those obtained using the AND operator and the OR operator.
Abstract: The author is primarily concerned with the problem of aggregating multicriteria to form an overall decision function. He introduces a type of operator for aggregation called an ordered weighted aggregation (OWA) operator and investigates the properties of this operator. The OWA's performance is found to be between those obtained using the AND operator, which requires all criteria to be satisfied, and the OR operator, which requires at least one criteria to be satisfied. >
6,534 citations
"Generalized Orthopair Fuzzy Sets" refers background in this paper
...Another important example of an aggregation operator is the OWA operator [29]–[31]....
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01 Jan 1988
3,030 citations
TL;DR: The issue of having to choose a best alternative in multicriteria decision making leads the problem of comparing Pythagorean membership grades to be considered, and a variety of aggregation operations are introduced for these Pythagorian fuzzy subsets.
Abstract: We first look at some nonstandard fuzzy sets, intuitionistic, and interval-valued fuzzy sets. We note both these allow a degree of commitment of less then one in assigning membership. We look at the formulation of the negation for these sets and show its expression in terms of the standard complement with respect to the degree of commitment. We then consider the complement operation. We describe its properties and look at alternative definitions of complement operations. We then focus on the Pythagorean complement. Using this complement, we introduce a class of nonstandard Pythagorean fuzzy subsets whose membership grades are pairs, (a, b) satisfying the requirement a
2
+ b
2
≤ 1. We introduce a variety of aggregation operations for these Pythagorean fuzzy subsets. We then look at multicriteria decision making in the case where the criteria satisfaction are expressed using Pythagorean membership grades. The issue of having to choose a best alternative in multicriteria decision making leads us to consider the problem of comparing Pythagorean membership grades.
1,706 citations