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

Authors: Abhishek Guleria; Rakesh Kumar Bajaj
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

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.


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## Body

## 1. 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 decision-making 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 decision-making 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 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 decision-making 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 R-norm intuitionistic fuzzy entropy and a weighted R-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 α-cut and (α,β)-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(R,S)-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 (R,S)-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 R and S have been studied and validated empirically. In Section 5, a new multicriteria decision-making algorithm is provided on the basis of the proposed (R,S)-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.


## 2. Preliminaries

In this section, we recall and present some fundamental concepts in connection with Pythagorean fuzzy set, which are well known in literature.Definition 1 (see [1]).
An intuitionistic fuzzy set (IFS)I in X (universe of discourse) is given by(1)I=x,μIx,νIx∣x∈Xwhere μI:X→[0,1] and νI:X→[0,1] denote the degree of membership and degree of nonmembership, respectively, and for every x∈X satisfy the condition(2)0≤μIx+νIx≤1and the degree of indeterminacy for any IFS I and x∈X is given by πI(x)=1-μI(x)-νI(x).Definition 2 (see [2]).
A Pythagorean fuzzy set (PFS)M in X (universe of discourse) is given by(3)M=x,μMx,νMx∣x∈Xwhere μM:X→[0,1] and νM:X→[0,1] denote the degree of membership and degree of nonmembership, respectively, and for every x∈X satisfy the condition(4)0≤μM2x+νM2x≤1and the degree of indeterminacy for any PFS M and x∈X is given by(5)πMx=1-μM2x-νM2x.In case of PFS, the restriction corresponding to the degree of membershipμM(x) and the degree of nonmembership νM(x) is(6)0≤μM2x+νM2x≤1,whereas the condition in case of IFS is(7)0≤μIx+νIx≤1,for μM(x),νM(x)∈[0,1]. This difference in constraint conditions gives a wider coverage for information span which can be geometrically shown in Figure 1.Figure 1
IFS versus PFS.Some of the important binary operations on PFSs are being presented below which are available in literature.Definition 3 (see [7]).
IfM and N are two Pythagorean fuzzy sets in X, then the operations can be defined as follows: (a)
Complement:Mc={x,νM(x),μM(x)∣x∈X}.(b)
Containment:M⊂Niff∀x∈X,μM(x)≤μN(x)andνM(x)≥νN(x).(c)
Union:M∪N={x,μM(x)∨μN(x),νM(x)∧νN(x)∣x∈X}.(d)
Intersection:M∩N={x,μM(x)∧μN(x),νM(x)∨νN(x)∣x∈X}.Definition 4 (see [10]).
LetM and N be two PFSs, then the Euclidean distance between M and N is defined as follows:(8)dM,N=12μMx2-μNx22+νMx2-νNx22+πMx2-πNx22.Definition 5 (see [5]).
LetM and N be two PFS, then the Hamming distance between M and N is defined as follows:(9)lM,N=12μMx2-μNx2+νMx2-νNx2+πMx2-πNx2.


## 3. New(R,S)-Norm Information Measure of PFS

Let△n={P=(p1,p2,…,pn),pi≥0,i=1,2,3,…,n and ∑pi=1} be the set of all probability distribution association with random variable X taking finite values x1,x2,…,xn. Joshi and Kumar [17] defined and studied a real valued function from △n to R+ as (R,S)-norm information measure of the distribution P for R and S∈R+ and given by(10)HRSP=R×SR-S∑i=1npiS1/S-∑i=1npiR1/R;where 0<S<1 and 1<R<∞. Or 0<S<1 and 1<R<∞.The most important property of this measure is that whenS=1 or R=1, then (10) becomes the R or S-norm entropy studied by Boekee and Lubbe [18] and if R=1 and S→1 or S=1 and R→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 H:X→[0,1], called entropy of Pythagorean fuzzy set M if and only if the following four axioms are satisfied:(i)
(PFS1) Sharpness:H(M)=0 iff M is a crisp set, i.e., μM(xi)=0,  νM(xi)=1; or μM(xi)=1,  νM(xi)=0;  ∀xi∈X.(ii)
(PFS2) Maximality:H(M) is maximum iff(11)μMxi=νMxi=πMxi=13∀xi∈X.(iii)
(PFS3) Symmetry:HM=H(Mc).(iv)
(PFS4) Resolution:H(M)≤H(N) iff M⊆N, i.e., μM(xi)≤μN(xi) and νM(xi)≥νN(xi) for μN(xi)≤νN(xi) or if μM(xi)≥μN(xi) and νM(xi)≤νN(xi) for μN(xi)≥νN(xi)∀xi∈X.In context with Pythagorean fuzzy information, we propose the following Pythagorean fuzzy entropy analogous to measure (10):(12)HRSM=R×SR-S∑i=1n1nμMxi2S+νMxi2S+πMxi2S1/S-μMxi2R+νMxi2R+πMxi2R1/R,whereR,S>0;either0<S<1and1<R<∞or0<S<1and1<R<∞,RnR-1∑i=1n1-μMxi2R+νMxi2R+πMxi2R1/R,whereS=1,R>0,R≠1,-1n∑i=1nμMxi2log⁡μMxi2+νMxi2log⁡νMxi2+πMxi2log⁡πMxi2,whereR=1andS→1orS=1andR→1Theorem 6.
The proposed entropy measureHRS(M) is a valid Pythagorean fuzzy information measure.Proof.
To prove this, we shall show that it satisfies all the axioms PFS1 to PFS4.(i)
(PFS1) (Sharpness): If HRS(M)=0, then(13)μMxi2S+νMxi2S+πMxi2S1/S-μMxi2R+νMxi2R+πMxi2R1/R=0.SinceR,S>1  R≠1≠S, it is possible only in the following cases:
(a)
EitherμM(xi)=1, i.e., νM(xi)=πM(xi)=0,(b)
νM(xi)=1 i.e., μM(xi)=πM(xi)=0,(c)
πM(xi)=1 i.e., μM(xi)=νM(xi)=0.These three cases implies thatM is a crisp set. Conversely, if M is a crisp set then HRS(M)=0 which is obvious.(ii)
(PFS2)(Maximality):In Section4, we have empirically proved that HRS(M) is maximum iff (14)μMxi=νMxi=πMxi=13.Analytically, we prove the concavity of theHRS(M) by calculating its hessian at the critical point, i.e., 1/3 with particular values of R and S. The Hessian of HRS(M) is as [R>1(=3)andS<1(=0.3)]:(15)HRSM=2n-10.45892.2328162.2328162.232816-10.45892.2328162.2328162.232816-10.4589.It may be noted thatHRS(M) is a negative semidefinite matrix for different possible values of R and S 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(16)HRSM=HRSMc.(iv)
(PFS4)(Resolution): We have(17)μMxi-13+νMxi-13+πMxi-13≥μNxi-13+νNxi-13+πNxi-13;and(18)μMxi-132+νMxi-132+πMxi-132≥μNxi-132+νNxi-132+πNxi-132;because ifμM(xi)≤μN(xi) and νM(xi)≤νN(xi) with max⁡μN(xi),νN(xi)≤1/3, then μM(xi)≤μN(xi)≤1/3;  νM(xi)≤νN(xi)≤1/3 and πM(xi)≥πN(xi)≥1/3 which implies that the above result holds. Similarly, if μM(xi)≥μN(xi) and νM(xi)≥νN(xi) with max⁡μM(xi),νM(xi)≥1/3, then also the above result holds.
Now, sinceHRS(M) is a concave function on the Pythagorean fuzzy set M, therefore, if max⁡μM(xi),νM(xi)≤1/3 then μM(xi)≤μN(xi) and νM(xi)≤νN(xi) imply πM(xi)≥πN(xi)≥1/3.
Therefore, by the above explained result, we conclude thatHRS(M) satisfies condition of resolution PFS4.
Similarly, ifmin⁡μM(xi),νM(xi)≥1/3, then μM(xi)≤μN(xi) and νM(xi)≥νN(xi). By using the above proved result, we conclude that HRS(M) satisfies the condition PFS4.
Hence,HRS(M) satisfies all the axioms of Pythagorean fuzzy entropy and, therefore, HRS(M) is a valid measure of Pythagorean fuzzy information.Theorem 7.
LetM and N be two PFSs defined in X=x1,x2,…,xn where M=xi,μM(xi),νM(xi)∣xi∈X and N=xi,μN(xi),νN(xi)∣xi∈X such that ∀xi∈X either M⊆N or N⊆M. Then(19)HRSM∪N+HRSM∩N=HRSM+HRSN.Proof.
DivideX into two parts X1 and X2 such that X1=xi∈X∣M⊆N, i.e., μM(xi)≤μN(xi),  νM(xi)≥νN(xi)  ∀xi∈X1;  X2=xi∈X∣N⊆M, i.e., μM(xi)≥μN(xi),νM(xi)≤νN(xi)  ∀xi∈X1. Now(20)HRSM∪N=R×SR-S∑i=1n1nμM∪Nxi2S+νM∪Nxi2S+πM∪Nxi2S1/S-μM∪Nxi2R+νM∪Nxi2R+πM∪Nxi2R1/R;which implies(21)HRSM∪N=R×SR-S∑X11nμNxi2S+νNxi2S+πNxi2S1/S-μNxi2R+νNxi2R+πNxi2R1/R+R×SR-S∑X21nμMxi2S+νMxi2S+πMxi2S1/S-μMxi2R+νMxi2R+πMxi2R1/R.Similarly,(22)HRSM∩N=R×SR-S∑X11nμMxi2S+νMxi2S+πMxi2S1/S-μMxi2R+νMxi2R+πMxi2R1/R+R×SR-S∑X21nμNxi2S+νNxi2S+πNxi2S1/S-μNxi2R+νNxi2R+πNxi2R1/R.On adding the above two terms, we get(23)HRSM∪N+HRSM∩N=HRSM+HRSN.Theorem 8.
For any Pythagorean fuzzy setM, we have(24)HRSM=HRSMc=HRSM∪Mc=HRSM∩Mc.Proof.
By definition, the proof is obvious.


## 4. Monotonicity of(R,S)- Norm Information Measure of PFS

In this section, we carry an empirical study for investigating the maximality and monotonic nature of the proposed(R,S)-norm information measure of PFS. For this, we consider the following four Pythagorean fuzzy sets M1, M2, M3, and M4 over the universe of discourse X={x1,x2,x3}:(25)M1=x1,13,13,x2,13,13,x3,13,13;M2=x1,0.6,0.6,x2,0.7,0.7,x3,0.55,0.55;M3=x1,0.5,0.6,x2,0.2,0.9,x3,0.9,0.3;M4=x1,0.4,0.8,x2,0.9,0.4,x3,0.7,0.6.For various values of R and S and using equation (12), we compute and tabulate all the values of HRS(M). On the basis of the tabulated data and the plots given in Table 1 and Figure 2, it is quite clear that HRS(M) takes maximum value when μM(xi)=νM(xi)=πM(xi)=1/3;∀xi∈X and is a monotonically decreasing function of R and S.Table 1
Values of entropy for different values ofR and S.
SI. No. R S = 0 . 15 S = 0 . 25 S  = 1 S = 4 S = 15 S = 60 H R S ( M 1 ) H R S ( M 2 ) H R S ( M 3 ) H R S ( M 4 ) H R S ( M 1 ) H R S ( M 2 ) H R S ( M 3 ) H R S ( M 4 ) H R S ( M 1 ) H R S ( M 2 ) H R S ( M 3 ) H R S ( M 4 ) H R S ( M 1 ) H R S ( M 2 ) H R S ( M 3 ) H R S ( M 4 ) H R S ( M 1 ) H R S ( M 2 ) H R S ( M 3 ) H R S ( M 4 ) H R S ( M 1 ) H R S ( M 2 ) H R S ( M 3 ) H R S ( M 4 ) 1 0.2 254.676 218.665 184.776 194.201 54.000 46.744 39.669 41.643 20.000 17.381 14.767 15.498 16.960 14.743 12.500 13.123 16.346 14.208 12.040 12.637 16.186 14.068 11.919 12.511 2 0.3 147.744 127.021 107.410 112.867 21.030 18.328 15.608 16.369 5.134 4.526 3.855 4.044 4.068 3.590 3.017 3.171 3.864 3.408 2.854 2.996 3.811 3.360 2.811 2.951 3 0.5 107.670 92.610 78.329 82.303 12.000 10.497 8.952 9.385 2.000 1.785 1.495 1.577 1.464 1.309 1.030 1.096 1.366 1.218 0.943 0.998 1.341 1.193 0.920 0.973 4 0.8 93.073 80.064 67.717 71.154 9.340 8.180 6.972 7.311 1.264 1.134 0.906 0.970 0.877 0.790 0.534 0.584 0.809 0.722 0.468 0.503 0.792 0.701 0.451 0.483 5 1.2 86.508 74.420 62.939 66.134 8.263 7.241 6.162 6.464 1.004 0.902 0.673 0.736 0.675 0.611 0.335 0.375 0.618 0.549 0.281 0.301 0.604 0.528 0.267 0.283 6 1.7 83.052 71.448 60.420 63.489 7.727 6.773 5.753 6.038 0.884 0.796 0.552 0.614 0.584 0.529 0.236 0.261 0.532 0.467 0.192 0.192 0.519 0.444 0.179 0.176 7 2.5 80.576 69.319 58.613 61.592 7.356 6.449 5.467 5.740 0.805 0.726 0.468 0.525 0.524 0.474 0.177 0.178 0.476 0.406 0.141 0.115 0.464 0.381 0.128 0.101 8 5 78.100 67.190 56.806 59.692 6.996 6.133 5.188 5.446 0.731 0.659 0.393 0.433 0.469 0.413 0.135 0.103 0.424 0.327 0.102 0.050 0.413 0.296 0.087 0.037 9 7 77.420 66.604 56.309 59.170 6.899 6.048 5.113 5.366 0.712 0.640 0.374 0.409 0.455 0.391 0.125 0.087 0.411 0.294 0.091 0.033 0.400 0.261 0.073 0.021 10 10 76.917 66.172 55.942 58.784 6.828 5.985 5.058 5.308 0.698 0.625 0.361 0.392 0.444 0.370 0.117 0.074 0.401 0.263 0.081 0.019 0.391 0.231 0.060 0.009 11 20 76.339 65.674 55.520 58.340 6.746 5.912 4.995 5.241 0.682 0.605 0.346 0.372 0.433 0.340 0.107 0.059 0.390 0.224 0.067 0.005 0.380 0.203 0.035 0.001 12 40 76.053 65.427 55.311 58.120 6.706 5.877 4.963 5.208 0.674 0.595 0.338 0.362 0.427 0.325 0.100 0.053 0.385 0.213 0.052 0.003 0.375 0.199 0.013 0.000 13 50 75.996 65.378 55.270 58.076 6.698 5.869 4.957 5.201 0.673 0.593 0.337 0.361 0.426 0.323 0.098 0.052 0.384 0.211 0.048 0.002 0.374 0.199 0.009 0.000 14 70 75.931 65.322 55.222 58.026 6.689 5.861 4.950 5.194 0.671 0.591 0.335 0.358 0.425 0.320 0.096 0.050 0.383 0.210 0.044 0.002 0.372 0.199 0.005 0.000 15 100 75.882 65.280 55.187 57.989 6.682 5.855 4.945 5.188 0.670 0.589 0.333 0.357 0.424 0.317 0.094 0.050 0.382 0.209 0.041 0.002 0.372 0.198 0.003 0.000 16 200 75.826 65.231 55.145 57.945 6.675 5.848 4.939 5.182 0.668 0.587 0.332 0.355 0.423 0.315 0.092 0.049 0.381 0.208 0.037 0.002 0.371 0.198 0.002 0.000 17 500 75.792 65.202 55.120 57.919 6.670 5.844 4.935 5.178 0.667 0.586 0.331 0.354 0.422 0.314 0.091 0.048 0.380 0.208 0.036 0.002 0.370 0.198 0.001 0.000Figure 2
Monotonicity of the(R,S)-norm information measure.


## 5. MCDM Algorithm with(R,S)-Norm Entropy

Suppose that there is a set ofm feasible alternatives, i.e., Z=z1,z2,…,zm and a set of n criteria O=o1,o2,…,on. The decision-making problem is to select the most suitable alternative out of these m alternatives. The appraisal values of an alternative zii=1,2,3,…,m with respect to the criteria ojj=1,2,3,…,n are given by zij=pij,qij, where pij is the degree to which the alternative zi satisfies criteria oj and qij is the degree to which the alternative zi does not satisfy attribute oj, satisfying 0≤pij≤1,  0≤qij≤1 and 0≤pij+qij≤1 with i=1,2,3,…,m and j=1,2,3,…,n. This problem can be modeled by representing it through the following Pythagorean fuzzy decision matrix:(26)R=pij,qijm×n=zij=o1o2⋯onz1p11,q11p12,q12⋯p1n,q1nz2p21,q21p22,q22⋯p2n,q2n⋮⋮⋮⋮⋮zmpm1,qm1pm2,qm2⋯pmn,qmnLet w=w1,w2,…,wnT be the weight vector of all the criteria where 0≤wj≤1 and ∑j=1nwj is the degree of importance of the jth 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(27)wj=1-ejn-∑j=1nej,j=1,2,…,n;where ej=1/m∑i=1mHRS(zij), and(28)HRSzij=R×SR-S∑i=1m1mμMxi2S+νMxi2S+πMxi2S1/S-μMxi2R+νMxi2R+πMxi2R1/Ris the proposed Pythagorean fuzzy entropy for zij=(pij,qij).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.
The overall entropy of the alternativezi is(29)Ezi=∑j=1nHRSzij=R×SR-S∑j=1n∑i=1m1mμMxi2S+νMxi2S+πMxi2S1/S-μMxi2R+νMxi2R+πMxi2R1/R;where R,S>0; R>1,S<1 or R<1,S>1.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:(30)min⁡E=∑i=1mwjEzi=∑i=1mwj∑j=1nHRSzij=R×SR-S∑j=1nwj∑i=1m1mμMxi2S+νMxi2S+πMxi2S1/S-μMxi2R+νMxi2R+πMxi2R1/RR , S > 0;  R>1,S<1 or R<1,S>1, subject to ∑j=1nwj=1.Finally, the procedure for implementing the proposed algorithm is being presented using Figure3.Figure 3
Flowchart of the proposed algorithm using PFS.The steps of the proposed methodology are enumerated and detailed as follows.Step 1.
We construct the decision matrixR=(pij,qij)m×n=oj(zi), where the elements oj(zi)  i=1,2,...,m;j=1,2,...,n are the appraisal of the alternative zi∈Z with respect to the criteria oj∈O.Step 2.
Determine the criteria weights by using (27) and (30).Step 3.
Define the most preferred solution(z+) and the least preferred solution (z-) as(31)z+=α1+,β1+,α2+,β2+,…,αn+,βn+;where (αj+,βj+)=(sup⁡μM(zi),inf⁡νM(zi)),  zi∈Z;  (j=1,2,…,n); and(32)z-=α1-,β1-,α2-,β2-,…,αn-,βn-;where (αj-,βj-)=(inf⁡μMzi,sup⁡νM(zi)),  zi∈Z;  (j=1,2,…,n), respectively.Step 4.
By using Definition5, the distance measures of zi′s from z+ and z- will be evaluated as follows:(33)lzi,z+=12∑j=1nwjαij2-αj+2+βij2-βj+2+πij2-πj+2;and(34)lzi,z-=12∑j=1nwjαij2-αj-2+βij2-βj-2+πij2-πj-2.Step 5.
Determine the relative degrees of closenessli′s as follows:(35)li=lzi,z-lzi,z-+lzi,z+.Step 6.
On the basis of the relative degree of closeness obtained in Step5, we determine the optimal ranking order of the alternatives. The alternative with the maximal degree of closeness l(zi) is supposed to be the best alternative.


## 6. Numerical Examples

Based on two different cases considered in the proposed algorithm, we present two different examples as follows.Example 1 (unknown weights).
Suppose an automobile company produces four different cars, say,z1, z2, z3, and z4. Suppose a customer wants to buy a car based on the four given criteria, say, comfort o1, good mileage o2, safety o3, and interiors o4. Assume the appraisal values of the alternatives with respect to each criterion provided by the expert are represented by PFS as(36)o1o2o3o4z10.9,0.30.7,0.60.5,0.80.6,0.3z20.4,0.70.9,0.20.8,0.10.5,0.3z30.8,0.40.7,0.50.6,0.20.7,0.4z40.7,0.20.8,0.20.8,0.40.6,0.6Then the calculations for the ranking procedure are as follows:(1)
Calculate the criteria weight vector using (27):(37)w=w1,w2,w3,w4T=0.272107,0.263037,0.34878,0.116077T.(2)
The most preferred solution(z+) and the least preferred solution (z-) are given by(38)z+=0.9,0.3,0.9,0.2,0.8,0.1,0.7,0.4and(39)z-=0.4,0.7,0.7,0.6,0.5,0.8,0.6,0.6respectively.(3)
The distances between each ofzi′s from z+ and z- are given by(40)lz1,z+=0.040622,lz2,z+=0.186515,lz3,z+=0.0.06623,lz4,z+=0.048795,lz1,z-=0.209804,lz2,z-=0.13179,lz3,z-=0.177491,lz4,z+=0.116968.(4)
The values of relative degree of closeness are as follows:(41)l1=0.837788,l2=0.414036,l3=0.728256,l4=0.705633.(5)
The ranking of the alternatives as per the relative degree of closeness isz1≻z3≻z4≻z2 and z1 is the best available alternative. It may be noted that the above ranking is with respect to the specific values of R = 3 and S = 0.3.The consistency of the ranking procedure for different values of parametersR and S may also be observed and studied by making a simulation study over the varying values of the parameters depending on the requirement.Example 2 (partially known weights).
Suppose there are 1000 students in a college. On the basis of three selected criteria, say,o1 (personality), o2 (intelligence), and o3 (communication skills), the administration wants to select a college representative. Let there be three candidates, say, z1, z2, and z3. The PFS decision matrix for the above problem is(42)o1o2o3z10.8,0.50.6,0.60.8,0.2z20.6,0.50.7,0.40.8,0.4z30.5,0.70.7,0.60.9,0.3Let the information about the criteria weight be partially given in the following form{0.10≤w1≤0.30,0.35≤w2≤0.60,0.25≤w3≤0.70}. The calculation for the ranking procedure for the above decision-making problem is presented as follows:(1)
Using (30), we determine the criteria weights by following linear programming model:(43)minE=0.609037w1+0.641365w2+0.590874w3subject  to0.10≤w1≤0.30,0.35≤w2≤0.60,0.25≤w3≤0.70,w1+w2+w3=1.By solving this linear programming problem using MATLAB software, we obtained the criteria weight vector as follows:(44)w=0.10,0.35,0.55T.(2)
The most preferred solution(z+) and the least preferred solution (z-) are given by z+={(0.8,0.5),(0.7,0.4),(0.9,0.2)} and z-={(0.5,0.7),(0.6,0.6),(0.8,0.4)}, respectively.(3)
The distances between each ofzi′s from z+ and z- are given by(45)lz1,z+=0.013843,lz2,z+=0.015888,lz3,z+=0.068163,lz1,z-=0.052213,lz2,z-=0.026855,lz3,z-=0.049273(4)
The values of relative degree of closeness are(46)l1=0.79044,l2=0.628297,l3=0.419573.(5)
The ranking of the alternatives as per the relative degree of closeness isz1≻z2≻z3 and z1 is the best available alternative. It may be noted that the above ranking is with respect to the specific values of R = 3 and S = 0.3.The consistency of the ranking procedure for different values of the parametersR and S may also be observed and studied by making a simulation study over the varying values of the parameters.


## 7. Conclusions and Future Work

We have successfully proposed a new parametric(R,S)-norm information measure for Pythagorean fuzzy set along with the proof of its validity and discussed its maximality and the monotonic behavior with respect to parameters under consideration. Further, an algorithm for multicriteria decision-making problem has been well proposed and successfully implemented with the help of two different kind of numerical examples when weights are unknown and other when weights are partially known.In the area of pattern recognition, the directed divergence measure/symmetric divergence measure explains dissimilarity between pairs of probability distribution which is generally utilized for the procedure of factual surmising. It might be noticed that these difference measures and similarity measures are dual ideas. The similarity measure might be characterized as a diminishing capacity of the difference measures, particularly when the scope of divergence measures is[0,1].The proposed parametric(R,S)-norm information measure can further be extended to the concept of the parametric directed divergence measure/symmetric divergence measure for Pythagorean fuzzy sets. Various applications including total ambiguity and information improvement measures based on the proposed divergence measures may also be discussed in detail, e.g., taking a monotonic decreasing function into account, the upper bound of the symmetric divergence measure can be calculated and the similarity measure can be subsequently defined between two PFSs. The concept of similarity based clustering method (SCM) can also be investigated and the structure of the considered data set might be examined with the assistance of the similarity measure.



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