Pythagorean Fuzzy (𝑅, 𝑆) -Norm Information Measure for Multicriteria Decision-Making Problem

In the present communication, a parametric ( 𝑅 , 𝑆 )-norm information measure for the Pythagorean fuzzy set has been proposed with the proof of its validity. The monotonic behavior and maximality feature of the proposed information measure have been studied and presented. Further, an algorithm for solving the multicriteria decision-making problem with the help of the proposed information measure has been provided keeping in view of the different cases for weight criteria, when weights are unknown and other when weights are partially known. Numerical examples for each of the case have been successfully illustrated. Finally, the work has been concluded by providing the scope for future work.


Introduction
The concept of intuitionistic fuzzy set (IFS) (Atanassov) [1] has been widely studied and applied to deal with uncertainties and hesitancy inherent in practical circumstances.The prominent characteristic of an IFS is that it assigns a number from the unit interval [0, 1] to every element in the domain of discourse, a degree of membership, and a degree of nonmembership along with the degree of indeterminacy whose total sum equals unity.In literature, intuitionistic fuzzy sets comprehensively span applications in the field of decisionmaking problems, pattern recognition, sales analysis financial services, medical diagnosis, etc.
Pythagorean fuzzy set (PFS), proposed by Yager [2], is an efficient generalization of intuitionistic fuzzy set, characterized by a membership value and a nonmembership value satisfying the inequality that the squared sum of these values is less than or equal to 1. Yager and Abbasov [3] well stated that, in some practical multiple-criteria decisionmaking problems, it is viable that sum of the degree of the membership and the degree of nonmembership value of a particular alternative provided by a decision-maker may be in such a way that their sum is bigger than 1, where it would not be feasible to use intuitionistic fuzzy set.Therefore, PFS proves to be proficiently more capable of representing and handling vagueness, impreciseness, and uncertainties than IFS in various decision-making processes.It may be noted that PFS is more generalized than IFS as the span of membership degree of PFS is more than span of membership degree of IFS which enables wider applicability.
Various researchers theoretically developed the concept of Yager's Pythagorean fuzzy sets [4] and applied it in the field of decision-making problems, medical diagnosis, and pattern recognition and in other real-world problem.In order to deal the decision-making problem with PFSs, Zhang and Xu [5] proposed a comparison method based on a score function to identify the Pythagorean fuzzy positive ideal solution (PIS) and the Pythagorean fuzzy negative ideal solution (NIS).Further they extended the TOPSIS method to compute the distances between each alternative with PIS and NIS, respectively.Peng and Yang [6] proposed some basic operations for PFSs and provided Pythagorean fuzzy aggregation operators along with their important properties.In continuation, they developed a Pythagorean superiority and inferiority ranking algorithm to solve group decision-making problems in view of uncertainty.Further, Peng et al. [7] established the relationship between the distance measure, similarity measure, entropy, and the inclusion measure and suggested the systematic transformation of information measures for PFSs.Yager [8] introduced some of the basic set operations for PFSs Advances in Fuzzy Systems and established the relationship between Pythagorean membership values and complex number.In addition to this, the solutions of multicriteria decision-making with satisfactions through Pythagorean membership values have been carried out.A new method for Pythagorean fuzzy MCDM problems with the help of aggregation operators and distance measures has been developed by Zeng et al. [9].Further, they proposed the Pythagorean fuzzy ordered weighted averaging weighted average distance (PFOWAWAD) operator and developed a hybrid TOPSIS method.
Using PFSs, Ren et al. [10] had a simulation test to study the effect of the risk attitudes of the decision makers over the solutions of decision-making problems.Zhang [11] introduced a novel closeness index-based ranking method for Pythagorean fuzzy numbers and proposed interval valued Pythagorean fuzzy set with basic operations and important properties.In addition to this, the hierarchical multicriteria decision-making problems in Pythagorean fuzzy environment have been solved by developing a closeness index-based Pythagorean fuzzy QUALIFLEX method.Liu et al. [12] developed various types of Pythagorean fuzzy aggregation operators and used them to solve decisionmaking problems.Zeng [13] developed a Pythagorean fuzzy multiattribute group decision-making method on the basis of a new Pythagorean fuzzy probabilistic ordered weighted averaging (OWA) operator.Though various researchers have significantly contributed in the development of the theory of PFSs as deliberated above, a seldom study on the entropy of PFSs and its applications has been found in literature.Xue et al. [14] studied the linear programming technique for multidimensional analysis of preference (LINMAP) method under Pythagorean fuzzy environment to solve multiple attribute group decision-making problem by incorporating Pythagorean fuzzy entropy along with various other applications.Vital applications of entropy and information measures based on the IFS theory have been well known in the literature.In order to deal with real-world problems more efficiently and to cater the need of the hour, generalizations of the existing approaches play an important role as they contribute more flexibility in applications; e.g., parameters may characterize various factors such as time constraint, lack of knowledge, and environmental conditions, etc. Bajaj et al. [15] proposed a new -norm intuitionistic fuzzy entropy and a weighted -norm Intuitionistic fuzzy divergence measure with their computational applications in pattern recognition and image thresholding.Gandotra et al. [16] studied multiple-criteria decision-making problem with the help of parametric entropy under -cut and (, )cut based distance measures for different possible values of parameters and provided the ranking method for the available alternatives.
In this communication, we have proposed a new (, )norm information measure of Pythagorean fuzzy set and applied the information measure in an algorithm to solve multicriteria decision-making problem.In continuation, the implementation of the proposed algorithm by taking suitable examples has also been illustrated.The rest of this paper is organized as follows: in Section 2, we present some basic notions and preliminaries related to the proposed information measure.A new (, )-norm information measure of Pythagorean fuzzy set has been well proposed with the proof of its validity in Section 3. Further, in Section 4, the maximality and the monotonic behavior of the proposed information measure with respect to parameters  and  have been studied and validated empirically.In Section 5, a new multicriteria decision-making algorithm is provided on the basis of the proposed (, )-norm information measure of PFS in view of two cases of weights of criteria: one when weights are unknown and other when weights are partially known.In order to support and implement the proposed algorithm, an example for each case has been explicitly dealt in Section 6.The paper is finally concluded in Section 7.

Preliminaries
In this section, we recall and present some fundamental concepts in connection with Pythagorean fuzzy set, which are well known in literature.
Definition 2 (see [2]).A Pythagorean fuzzy set (PFS)  in  (universe of discourse) is given by where   :  → [0, 1] and ]  :  → [0, 1] denote the degree of membership and degree of nonmembership, respectively, and for every  ∈  satisfy the condition and the degree of indeterminacy for any PFS  and  ∈  is given by In case of PFS, the restriction corresponding to the degree of membership   () and the degree of nonmembership ]  () is whereas the condition in case of IFS is for   (), ]  () ∈ [0, 1].This difference in constraint conditions gives a wider coverage for information span which can be geometrically shown in Figure 1.Some of the important binary operations on PFSs are being presented below which are available in literature.Definition 3 (see [7]).If  and  are two Pythagorean fuzzy sets in , then the operations can be defined as follows: Definition 4 (see [10]).Let  and  be two PFSs, then the Euclidean distance between  and  is defined as follows: Definition 5 (see [5]).Let  and  be two PFS, then the Hamming distance between  and  is defined as follows:
The most important property of this measure is that when =1 or =1, then (10) becomes the  or -norm entropy studied by Boekee and Lubbe [18] and if  = 1 and  → 1 or  = 1 and  → 1, then it gives Shannon's [19] entropy.
Based on the axiomatic definition of entropy for intuitionistic fuzzy set, proposed by Hung and Yang (2006) [20], we analogously define a real valued function  :  → [0, 1], called entropy of Pythagorean fuzzy set  if and only if the following four axioms are satisfied: (i) (PFS1) Sharpness: (ii) (PFS2) Maximality: () is maximum iff In context with Pythagorean fuzzy information, we propose the following Pythagorean fuzzy entropy analogous to measure (10): where ,  > 0; either 0 <  < 1 and 1 <  < ∞ or 0 <  < 1 and 1 <  < ∞,   ( − 1) where  = 1 and  → 1 or  = 1 and  → 1 Advances in Fuzzy Systems Theorem 6.The proposed entropy measure    () is a valid Pythagorean fuzzy information measure.Proof.To prove this, we shall show that it satisfies all the axioms PFS1 to PFS4.
Since ,  > 1 ( ̸ = 1 ̸ = ), it is possible only in the following cases: These three cases implies that  is a crisp set.Conversely, if  is a crisp set then    () = 0 which is obvious.(ii) (PFS2)(Maximality): In Section 4, we have empirically proved that    () is maximum iff Analytically, we prove the concavity of the    () by calculating its hessian at the critical point, i.e., 1/ √ 3 with particular values of  and .The Hessian of    () is as [ > 1(= 3) and  < 1(= 0. It may be noted that    () is a negative semidefinite matrix for different possible values of  and  which shows that it is a concave function.Hence, the concavity of the function establishes the maximality property.(iii) (PFS3)(Symmetry): It is obvious from the definition that and Advances in Fuzzy Systems 5 which implies Similarly, On adding the above two terms, we get Theorem 8.For any Pythagorean fuzzy set , we have Proof.By definition, the proof is obvious.

MCDM Algorithm with (𝑅,𝑆)-Norm Entropy
Suppose that there is a set of  feasible alternatives, i.e.,  = { 1 ,  2 , . . .,   } and a set of  criteria  = { 1 ,  2 , . . .,   }.The decision-making problem is to select the most suitable alternative out of these  alternatives.The appraisal values of an alternative   ( = 1, 2, 3, . . ., ) with respect to the criteria   ( = 1, 2, 3, . . ., ) are given by   = (  ,   ), where   is the degree to which the alternative   satisfies criteria   and   is the degree to which the alternative   does not satisfy attribute   , satisfying 0 ≤   ≤ 1, 0 ≤   ≤ 1 and 0 ≤   +   ≤ 1 with  = 1, 2, 3, . . .,  and  = 1, 2, 3, . . ., .This problem can be modeled by representing it through the following Pythagorean fuzzy decision matrix: Let  = ( 1 ,  2 , . . .,   )  be the weight vector of all the criteria where 0 ≤   ≤ 1 and ∑  =1   is the degree of importance of the th criteria.Sometimes this criteria weight is completely unknown and sometimes it is partially known because of the lack of knowledge, time, data, and the limited expertise of the problem domain.In this section, we discuss and devise two methods to determine the weights of criteria by using the proposed entropy (12).Case 1 (unknown weights).When the criteria weights are completely unknown, then we calculate the weights by using the proposed PFS entropy as where   = (1/) ∑  =1    (  ), and is the proposed Pythagorean fuzzy entropy for   = (  ,   ).

Advances in Fuzzy Systems
Case 2 (partially known weights).In case the weights are partially known for a multiple-criteria decision-making problem, we use the minimum entropy principle (Wang and Wang [21]) to determine the weight vector of the criteria by constructing the programming model as follows.
Since there are fair competitive environments between each of the alternatives, the weight coefficient with respect to the same criteria should also be equal.Further, in order to get the ideal weight, we construct the following accompanying model: ,  > 0;  > 1,  < 1 or  < 1,  > 1, subject to ∑  =1   = 1.
Finally, the procedure for implementing the proposed algorithm is being presented using Figure 3.
The steps of the proposed methodology are enumerated and detailed as follows.
Step 3. Define the most preferred solution ( + ) and the least preferred solution ( − ) as and Step 5. Determine the relative degrees of closeness     as follows: Step 6.On the basis of the relative degree of closeness obtained in Step 5, we determine the optimal ranking order of the alternatives.The alternative with the maximal degree of closeness (  ) is supposed to be the best alternative.Determine the optimal ranking order of the alternatives Then the calculations for the ranking procedure are as follows:

Numerical Examples
(1) Calculate the criteria weight vector using ( 27 (5) The ranking of the alternatives as per the relative degree of closeness is  1 ≻  3 ≻  4 ≻  2 and  1 is the best available alternative.It may be noted that the above ranking is with respect to the specific values of  = 3 and  = 0.3.
The consistency of the ranking procedure for different values of parameters  and  may also be observed and studied by making a simulation study over the varying values of the parameters depending on the requirement.

Figure 3 :
Figure 3: Flowchart of the proposed algorithm using PFS.

Table 1 :
Values of entropy for different values of  and .