Pythagorean Membership Grades, Complex Numbers, and Decision Making
01 May 2013-International Journal of Intelligent Systems (John Wiley & Sons, Ltd)-Vol. 28, Iss: 5, pp 436-452
TL;DR: It is shown that Pythagorean membership grades are a subclass of complex numbers called Π‐i numbers, and the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction is looked at.
Abstract: We describe the idea of Pythagorean membership grades and the related idea of Pythagorean fuzzy subsets. We focus on the negation and its relationship to the Pythagorean theorem. We look at the basic set operations for the case of Pythagorean fuzzy subsets. A relationship is shown between Pythagorean membership grades and complex numbers. We specifically show that Pythagorean membership grades are a subclass of complex numbers called Π-i numbers. We investigate operations that are closed under Π-i numbers. We consider the problem of multicriteria decision making with satisfactions expressed as Pythagorean membership grades, Π-i numbers. We look at the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction.
Citations
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24 Jun 2013
TL;DR: A new class of non-standard fuzzy subset called Pythagorean fuzzy subsets is introduced and the related idea of Pythgorean membership grades is introduced, with a focus on the negation operation and its relationship to the Pythagorian theorem.
Abstract: We introduce a new class of non-standard fuzzy subsets called Pythagorean fuzzy subsets and the related idea of Pythagorean membership grades. We focus on the negation operation and its relationship to the Pythagorean theorem. We compare Pythagorean fuzzy subsets with intuitionistic fuzzy subsets. We look at the basic set operations for the Pythagorean fuzzy subsets.
1,369 citations
TL;DR: Some novel operational laws of PFSs are defined and an extended technique for order preference by similarity to ideal solution method is proposed to deal effectively with them for the multicriteria decision‐making problems with PFS.
Abstract: Recently, a new model based on Pythagorean fuzzy set PFS has been presented to manage the uncertainty in real-world decision-making problems. PFS has much stronger ability than intuitionistic fuzzy set to model such uncertainty. In this paper, we define some novel operational laws of PFSs and discuss their desirable properties. For the multicriteria decision-making problems with PFSs, we propose an extended technique for order preference by similarity to ideal solution method to deal effectively with them. In this approach, we first propose a score function based comparison method to identify the Pythagorean fuzzy positive ideal solution and the Pythagorean fuzzy negative ideal solution. Then, we define a distance measure to calculate the distances between each alternative and the Pythagorean fuzzy positive ideal solution as well as the Pythagorean fuzzy negative ideal solution, respectively. Afterward, a revised closeness is introduced to identify the optimal alternative. At length, a practical example is given to illustrate the developed method and to make a comparative analysis.
1,084 citations
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.
1,056 citations
TL;DR: A Pythagorean fuzzy superiority and inferiority ranking method to solve uncertainty multiple attribute group decision making problem and its properties such as boundedness, idempotency, and monotonicity are investigated.
Abstract: Pythagorean fuzzy sets PFSs, originally proposed by Yager Yager, Abbasov. Int J Intell Syst 2013;28:436-452, are a new tool to deal with vagueness considering the membership grades are pairs µ,i¾? satisfying the condition µ2+i¾?2i¾?1. As a generalized set, PFSs have close relationship with intuitionistic fuzzy sets IFSs. PFSs can be reduced to IFSs satisfying the condition µ+i¾?i¾?1. However, the related operations of PFSs do not take different conditions into consideration. To better understand PFSs, we propose two operations: division and subtraction, and discuss their properties in detail. Then, based on Pythagorean fuzzy aggregation operators, their properties such as boundedness, idempotency, and monotonicity are investigated. Later, we develop a Pythagorean fuzzy superiority and inferiority ranking method to solve uncertainty multiple attribute group decision making problem. Finally, an illustrative example for evaluating the Internet stocks performance is given to verify the developed approach and to demonstrate its practicality and effectiveness.
657 citations
Additional excerpts
...SOME RESULTS FOR PYTHAGOREAN FUZZY SETS 1141 (8) (p1 ∩ p2) p3 = (p1 p3) ∩ (p2 p3), if μ3 ≤ min{μ1, μ2}, max{ν1, ν2} ≤ {ν3, ν3 π1 π3 , ν3 π2 π3 }, min{μ(2)1,μ(2)2}−μ(2)3 1−μ(2)3 + max{ν2 1 ,ν2 2 } ν2 3 ≤ 1; (9) (p1 ∪ p2) p3 = (p1 p3) ∪ (p2 p3), if ν3 ≤ min{ν1, ν2}, max{μ1, μ2} ≤ {μ3, μ3 π1 π3 , μ3 π2 π3 }, min{ν2 1 ,ν2 2 }−ν2 3 1−ν2 3 + max{μ(2)1,μ(2)2} μ(2)3 ≤ 1; (10) (p1 ∩ p2) p3 = (p1 p3) ∩ (p2 p3), if ν3 ≤ min{ν1, ν2}, max{μ1, μ2} ≤ {μ3, μ3 π1 π3 , μ3 π2 π3 }, max{ν2 1 ,ν2 2 }−ν2 3 1−ν2 3 + min{μ(2)1,μ(2)2} μ(2)3 ≤ 1....
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...In the following, we shall prove the (1), (3), (5), (7), (9) and the (2), (4), (6), (8), (10) can be proved analogously....
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...In the following, we shall prove (1), (3), (5), (7), (9) and (2), (4), (6), (8), (10) can be proved analogously....
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...(3) p1 ∪ p2 = p2 ∪ p1; (4) p1 ∩ p2 = p2 ∩ p1; (5) λ(p1 ∪ p2) = λp1 ∪ λp2; (6) (p1 ∪ p2) = p 1 ∪ p 2 ; (7) λ(p1 p2) = λp1 λp2, if μ1 ≥ μ2, ν1 ≤ min{ν2, ν2π1 π2 }; (8) (p1 p2) = p 1 p 2 , if μ1 ≤ min{μ2, μ2π1 π2 }, ν1 ≥ ν2; (9) λ1p λ2p = (λ1 − λ2)p, if λ1 ≥ λ2; (10) p1 p2 = p(λ1−λ2), if λ1 ≥ λ2....
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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
Cites background from "Pythagorean Membership Grades, Comp..."
...Normalize the decision matrix Since there are two types of attribute, that is, benefit type and cost type, in order to relieve the effect of the different attribute types, we need convert cost type to benefit one by the following formula (Note: The converted value is still expressed by r̃ij ): r̃ij = ( ũij , ṽij ) ={(uij , vij ) for benefit attribute Aj ( vij , uij ) for cost attribute Aj (26)...
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References
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13,376 citations
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
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7,131 citations
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
01 Jan 1988
3,030 citations
TL;DR: Establishing a small set of terms that let us easily communicate about type-2 fuzzy sets and also let us define such sets very precisely, and presenting a new representation for type- 2 fuzzy sets, and using this new representation to derive formulas for union, intersection and complement of type-1 fuzzy sets without having to use the Extension Principle.
Abstract: Type-2 fuzzy sets let us model and minimize the effects of uncertainties in rule-base fuzzy logic systems. However, they are difficult to understand for a variety of reasons which we enunciate. In this paper, we strive to overcome the difficulties by: (1) establishing a small set of terms that let us easily communicate about type-2 fuzzy sets and also let us define such sets very precisely, (2) presenting a new representation for type-2 fuzzy sets, and (3) using this new representation to derive formulas for union, intersection and complement of type-2 fuzzy sets without having to use the Extension Principle.
2,382 citations