# TOPSIS Hybrid Multiattribute Group Decision-Making Based on Interval Pythagorean Fuzzy Numbers

Authors: H. U. Jun; W. U. Junmin; W. U. Jie
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Aiming at the mixed multiattribute group decision-making problem of interval Pythagorean fuzzy numbers, a weighted average (WA) operator model based on interval Pythagorean fuzzy sets is constructed. Furthermore, a decision-making method based on the technique for order preference by similarity to ideal solution (TOPSIS) method with interval Pythagorean fuzzy numbers is proposed. First, based on the completely unknown weights of decision-makers and attributes, interval Pythagorean fuzzy numbers are applied to TOPSIS group decision-making. Second, the interval Pythagorean fuzzy number WA operator is used to synthesize the evaluation matrices of multiple decision-makers into a comprehensive evaluation matrix, and the relative closeness of each scheme is calculated based on the TOPSIS decision-making method. Finally, an example is given to illustrate the rationality and effectiveness of the proposed method.


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## Body

## 1. Introduction

In a real decision-making process, the information obtained by decision-makers is often fuzzy and uncertain. Therefore, decision-making research based on fuzzy information is essential. Zadeh [1] first proposed the concept of a fuzzy set in 1965. Experts and scholars began to study fuzzy decision-making, and based on this, they carried out a series of studies, examining topics such as intuitionistic fuzzy sets [2–10] and interval fuzzy sets [11–16]. In 2013, Yager [3] first proposed the concept of a Pythagorean fuzzy number based on intuitionistic fuzzy sets. Experts and scholars began to apply Pythagorean fuzzy numbers to various decisions. Peng [4] defined the similarity and distance measures of Pythagorean fuzzy numbers. Chen [5] extended the Pythagorean fuzzy environment to the VIKOR decision model. Ren et al. [6] combined a Pythagorean fuzzy set with the TODIM method. Garg [7], Liu Weifeng [8], and Li Peng [9, 10] applied Pythagorean fuzzy numbers to geometric clustering operators, ordered weighting operators, generalized WOWA operators, and other operators.In the existing decision-making processes, it is often difficult for decision-makers to use an accurate value to evaluate the advantages and disadvantages of the scheme. Methods using interval Pythagorean fuzzy values are more conducive to expressing fuzzy information. Peng [11] and Muhammad [12] defined interval Pythagorean fuzzy integration operators. Zhang [13], Li Na [14, 15], and Lin Wenhao [16] applied an interval Pythagorean fuzzy environment to decision-making processes, such as the hierarchical qualitative flexible multiple criteria method (QUALIFLEX), AQM method, and VIKOR operator.The technique for order preference by similarity to ideal solution (TOPSIS) decision-making method is a widely used multiattribute decision-making method, and it is one the most frequently used techniques to deal with multicriteria group decision-making (MCGDM) conflicts. Umer et al. [17] expanded the TOPSIS method using a distance method with interval type-2 trapezoidal Pythagorean fuzzy numbers (IT2TrPFNs) and applied it for MCGDM dilemmas by considering the attitudes and perspectives of the decision-makers. Pishyar et al. [18] used the TOPSIS and analytic hierarchy process (AHP) methods to determine, prioritize, and assess the most effective desertification indices. Kacprzak et al. [19] presented a new approach for ranking the alternatives for group decision-making using the TOPSIS method based on ordered fuzzy numbers. Rezaei et al. [20] developed a method to solve the sustainable circular partner selection problem with completely unknown decision-making experts. Du Yingxue et al. [21] proposed a triangular Pythagorean fuzzy TOPSIS decision-making method. Qu Guohua [22] proposed an intuitionistic fuzziness λ-Shapley Choquet integral operator TOPSIS multiattribute group decision-making method. Zhao and others [23] proposed a new TOPSIS decision-making method based on intuitionistic fuzzy sets, interval fuzzy sets, and other evaluation information.In recent years, experts and scholars have conducted multidomain, in-depth research in the field of Pythagorean fuzzy numbers, extended Pythagorean fuzzy numbers to interval Pythagorean fuzzy numbers, and studied various operators and decision models in the environment of interval Pythagorean fuzzy numbers. However, the application of the TOPSIS decision method in an interval Pythagorean fuzzy environment is rare. Therefore, the weighted average (WA) operator of interval Pythagorean fuzzy numbers is defined, and a TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers is presented. The advantages of this method are as follows: (1) it alleviates the loss of information in the decision-making process, (2) it effectively solves different types of attribute information problems and fully utilizes the attribute advantages of each decision-maker, and (3) it enlarges the gap between the closeness of each scheme and the advantages and disadvantages, making the decision-making results more convincing.


## 2. Basic Concepts

### 2.1. Pythagorean Fuzzy Sets

Definition 1 (see [1]).
SupposeX is a given universe and P=x,PμPx,νPx|x∈X is a Pythagorean fuzzy set on X. μPx represents the membership degree of x belonging to P, and νPx indicates that x belongs to the nonmembership degree of P. μPx and νPx satisfy 0≤μPx,νPx≤1, and 0≤μP2x+νP2x≤1. The hesitation degree of x to P is defined as πPx=1−μP2x+νP2x. For simplicity, μPx,νPx is called a Pythagorean fuzzy number, and it is abbreviated as α=μPx,νPx in this paper.Definition 2 (see [6]).
For the Pythagorean fuzzy numberα=μPx,νPx, the scoring function of α is defined as Sα=μP2x−νP2x and the precision function α is defined as Hα=μP2x+νP2x, where Sα∈−1,1 and Hα∈0,1.Definition 3 (see [6]).
For two Pythagorean fuzzy numbersαi=PμPix,νPixi=1,2, the comparison is as follows:(1)
SupposeSα1=Sα2. If Hα1=Hα2, then α1=α2, and if Hα1<Hα2, then α1<α2.(2)
SupposeSα1>Sα2. Then, α1>α2.Definition 4 (see [6]).
For two Pythagorean fuzzy numbersα1=PμP1x,νP1x and α2=PμP2x,νP2x, the following operation rules are satisfied:(1)
α1⊕α2=μP12x+μP22x−μP12xμP22x,νP1xνP2x(2)
α1⊗α2=μP1xμP2x,νP12x+νP22x−νP12xνP22x(3)
λα=1−1−μP2xλ,νPλx(4)
αλ=μPλx,1−1−νP2xλ


### 2.2. Interval Pythagorean Fuzzy Sets

In decision-making, scholars have proposed the concept and algorithms of interval Pythagorean fuzzy numbers. An interval Pythagorean fuzzy number adds the upper and lower limits of the membership and nonmembership based on the Pythagorean fuzzy number.Definition 5 (see [13]).
SupposeX is a given universe and P∼=x,P∼μP∼−x,μP∼+x,P∼νP∼−x,νP∼+x|x∈X is an interval Pythagorean fuzzy set on X. μP∼−x and μP∼+x are the lower and upper bounds of the membership degree, respectively, and νP∼−x and νP∼+x are the lower and upper bounds of the nonmembership degree, respectively. The interval Pythagorean fuzzy set P∼ satisfies 0≤μP∼+x2+νP∼−x2≤1. The hesitation degree of x to the interval Pythagorean fuzzy set P∼ is defined as πP∼x=πP∼−x,πP∼+x=1−μP∼+x2−νP∼+x2,1−μP∼−x2−νP∼−x2.Definition 6 (see [13]).
For the interval Pythagorean fuzzy numberα∼=P∼μP∼−x,μP∼+x,P∼νP∼−x,νP∼+x, the scoring function α is defined as Sα∼=1/2μP∼−x2−νP∼−x2+μP∼+x2−νP∼+x2 and the precision function α is defined as Hα∼=1/2μP∼−x2+νP∼−x2+μP∼+x2+νP∼+x2, where Sα∼∈−1,1 and Hα∼∈0,1.Definition 7 (see [13]).
For two interval Pythagorean fuzzy numbersα∼i=P∼μPi∼−x,μPi∼+x,P∼νPi∼−x,νPi∼+xi=1,2, the comparison is as follows:(1)
SupposeSα∼1=Sα∼2. If Hα∼1=Hα∼2, then α∼1=α∼2, and if Hα∼1<Hα∼2, then α∼1<α∼2(2)
SupposeSα∼1>Sα∼2. Then, α∼1>α∼2Definition 8 (see [13]).
For two interval Pythagorean fuzzy numbersα∼i=P∼μPi∼−x,μPi∼+x,P∼νPi∼−x,νPi∼+xi=1,2, the following operations are satisfied:(1)
α∼1⊕α∼2=μP∼1−x2+μP∼2−x2−μP∼1−x2μP∼2−x2,μP∼1+x2+μP∼2+x2−μP∼1+x2μP∼2+x2,νP∼1−xνP∼2−x,νP∼1+xνP∼2+x(2)
α∼1⊗α∼2=μP∼1−xμP∼2−x,μP∼1+xμP∼2+x,νP∼1−x2+νP∼2−x2−νP∼1−x2νP∼2−x2,νP∼1+x2+νP∼2+x2−νP∼1+x2νP∼2+x2(3)
λα∼=1−1−μP∼−x2λ,1−1−μP∼+x2λ,νP∼−xλ,νP∼+xλ(4)
α∼λ=μP∼−xλ,μP∼+xλ,1−1−νP∼−x2λ,1−1−νP∼+x2λDefinition 9 (see [13]).
For two interval Pythagorean fuzzy numbersα∼i=P∼μPi∼−x,μPi∼+x,P∼νPi∼−x,νPi∼+xi=1,2, the Hamming distance dα∼1,α∼2 is defined as dα∼1,α∼2=1/4μP∼1−x−μP∼2−x+μP∼1+x−μP∼2+x+νP∼1−x−νP∼2−x+νP∼1+x−νP∼2+x.


### 2.3. Weighted Average (WA) Operator Based on Interval Pythagorean Fuzzy Sets

Definition 10 (see [24]).
There is a mapping WA:Rn⟶R, WAx1,x2,x3,…,xn=∑i=1nwixi, where w=w1,w2,w3,…,wnT is the weighted vector of WA, satisfying wi∈0,1 and ∑i=1nwi=1. At this time, the function WA is called the weighted average operator.Definition 11.
For the interval Pythagorean fuzzy numberα∼i=P∼μP∼−x,μP∼+x,P∼νP∼−x,νP∼+x, wii=1,2,3,…,n is the weight of the interval Pythagorean fuzzy number α∼i, satisfying wi∈0,1 and ∑i=1nwi=1. The weighted average operator (IPVFWA) of the interval Pythagorean fuzzy number α∼i can be defined as follows:(1)IVPFWAα∼1,α∼2,α∼3,…,α∼n=w1α∼1⊕w2α∼2⊕w3α∼3⊕⋯⊕w4α∼4=1−∏i=1n1−μPi∼−x2wi,1−∏i=1n1−μPi∼+x2wi,∏i=1nνP∼−xwi,∏i=1nνP∼+xwi.Theorem 1.
Ifα∼i=P∼μPi∼−x,μPi∼+x,P∼νPi∼−x,νPi∼+xi=1,2,3,…,n is a group of interval Pythagorean fuzzy numbers, then the IPVFWA operator is still an interval Pythagorean fuzzy number after integration.Proof.
Becauseα∼i=P∼μPi∼−x,μPi∼+x,P∼νPi∼−x,νPi∼+x, according to Definition 8,(2)wiα∼i=1−1−μPi∼−x2wi,1−1−μPi∼+x2wi,νPi∼−xwi,νPi∼+xwi⊕i=1nwiα∼i=⊕i=1n1−1−μPi∼−x2wi,1−1−μPi∼+x2wi,νPi∼−xwi,νPi∼+xwi=1−∏i=1n1−μPi∼−x2wi,1−∏i=1n1−μPi∼+x2wi,∏i=1nνP∼−xwi,∏i=1nνP∼+xwi=IVPFWAα∼1,α∼2,α∼3,…,α∼n.


## 2.1. Pythagorean Fuzzy Sets

Definition 1 (see [1]).
SupposeX is a given universe and P=x,PμPx,νPx|x∈X is a Pythagorean fuzzy set on X. μPx represents the membership degree of x belonging to P, and νPx indicates that x belongs to the nonmembership degree of P. μPx and νPx satisfy 0≤μPx,νPx≤1, and 0≤μP2x+νP2x≤1. The hesitation degree of x to P is defined as πPx=1−μP2x+νP2x. For simplicity, μPx,νPx is called a Pythagorean fuzzy number, and it is abbreviated as α=μPx,νPx in this paper.Definition 2 (see [6]).
For the Pythagorean fuzzy numberα=μPx,νPx, the scoring function of α is defined as Sα=μP2x−νP2x and the precision function α is defined as Hα=μP2x+νP2x, where Sα∈−1,1 and Hα∈0,1.Definition 3 (see [6]).
For two Pythagorean fuzzy numbersαi=PμPix,νPixi=1,2, the comparison is as follows:(1)
SupposeSα1=Sα2. If Hα1=Hα2, then α1=α2, and if Hα1<Hα2, then α1<α2.(2)
SupposeSα1>Sα2. Then, α1>α2.Definition 4 (see [6]).
For two Pythagorean fuzzy numbersα1=PμP1x,νP1x and α2=PμP2x,νP2x, the following operation rules are satisfied:(1)
α1⊕α2=μP12x+μP22x−μP12xμP22x,νP1xνP2x(2)
α1⊗α2=μP1xμP2x,νP12x+νP22x−νP12xνP22x(3)
λα=1−1−μP2xλ,νPλx(4)
αλ=μPλx,1−1−νP2xλ


## 2.2. Interval Pythagorean Fuzzy Sets

In decision-making, scholars have proposed the concept and algorithms of interval Pythagorean fuzzy numbers. An interval Pythagorean fuzzy number adds the upper and lower limits of the membership and nonmembership based on the Pythagorean fuzzy number.Definition 5 (see [13]).
SupposeX is a given universe and P∼=x,P∼μP∼−x,μP∼+x,P∼νP∼−x,νP∼+x|x∈X is an interval Pythagorean fuzzy set on X. μP∼−x and μP∼+x are the lower and upper bounds of the membership degree, respectively, and νP∼−x and νP∼+x are the lower and upper bounds of the nonmembership degree, respectively. The interval Pythagorean fuzzy set P∼ satisfies 0≤μP∼+x2+νP∼−x2≤1. The hesitation degree of x to the interval Pythagorean fuzzy set P∼ is defined as πP∼x=πP∼−x,πP∼+x=1−μP∼+x2−νP∼+x2,1−μP∼−x2−νP∼−x2.Definition 6 (see [13]).
For the interval Pythagorean fuzzy numberα∼=P∼μP∼−x,μP∼+x,P∼νP∼−x,νP∼+x, the scoring function α is defined as Sα∼=1/2μP∼−x2−νP∼−x2+μP∼+x2−νP∼+x2 and the precision function α is defined as Hα∼=1/2μP∼−x2+νP∼−x2+μP∼+x2+νP∼+x2, where Sα∼∈−1,1 and Hα∼∈0,1.Definition 7 (see [13]).
For two interval Pythagorean fuzzy numbersα∼i=P∼μPi∼−x,μPi∼+x,P∼νPi∼−x,νPi∼+xi=1,2, the comparison is as follows:(1)
SupposeSα∼1=Sα∼2. If Hα∼1=Hα∼2, then α∼1=α∼2, and if Hα∼1<Hα∼2, then α∼1<α∼2(2)
SupposeSα∼1>Sα∼2. Then, α∼1>α∼2Definition 8 (see [13]).
For two interval Pythagorean fuzzy numbersα∼i=P∼μPi∼−x,μPi∼+x,P∼νPi∼−x,νPi∼+xi=1,2, the following operations are satisfied:(1)
α∼1⊕α∼2=μP∼1−x2+μP∼2−x2−μP∼1−x2μP∼2−x2,μP∼1+x2+μP∼2+x2−μP∼1+x2μP∼2+x2,νP∼1−xνP∼2−x,νP∼1+xνP∼2+x(2)
α∼1⊗α∼2=μP∼1−xμP∼2−x,μP∼1+xμP∼2+x,νP∼1−x2+νP∼2−x2−νP∼1−x2νP∼2−x2,νP∼1+x2+νP∼2+x2−νP∼1+x2νP∼2+x2(3)
λα∼=1−1−μP∼−x2λ,1−1−μP∼+x2λ,νP∼−xλ,νP∼+xλ(4)
α∼λ=μP∼−xλ,μP∼+xλ,1−1−νP∼−x2λ,1−1−νP∼+x2λDefinition 9 (see [13]).
For two interval Pythagorean fuzzy numbersα∼i=P∼μPi∼−x,μPi∼+x,P∼νPi∼−x,νPi∼+xi=1,2, the Hamming distance dα∼1,α∼2 is defined as dα∼1,α∼2=1/4μP∼1−x−μP∼2−x+μP∼1+x−μP∼2+x+νP∼1−x−νP∼2−x+νP∼1+x−νP∼2+x.


## 2.3. Weighted Average (WA) Operator Based on Interval Pythagorean Fuzzy Sets

Definition 10 (see [24]).
There is a mapping WA:Rn⟶R, WAx1,x2,x3,…,xn=∑i=1nwixi, where w=w1,w2,w3,…,wnT is the weighted vector of WA, satisfying wi∈0,1 and ∑i=1nwi=1. At this time, the function WA is called the weighted average operator.Definition 11.
For the interval Pythagorean fuzzy numberα∼i=P∼μP∼−x,μP∼+x,P∼νP∼−x,νP∼+x, wii=1,2,3,…,n is the weight of the interval Pythagorean fuzzy number α∼i, satisfying wi∈0,1 and ∑i=1nwi=1. The weighted average operator (IPVFWA) of the interval Pythagorean fuzzy number α∼i can be defined as follows:(1)IVPFWAα∼1,α∼2,α∼3,…,α∼n=w1α∼1⊕w2α∼2⊕w3α∼3⊕⋯⊕w4α∼4=1−∏i=1n1−μPi∼−x2wi,1−∏i=1n1−μPi∼+x2wi,∏i=1nνP∼−xwi,∏i=1nνP∼+xwi.Theorem 1.
Ifα∼i=P∼μPi∼−x,μPi∼+x,P∼νPi∼−x,νPi∼+xi=1,2,3,…,n is a group of interval Pythagorean fuzzy numbers, then the IPVFWA operator is still an interval Pythagorean fuzzy number after integration.Proof.
Becauseα∼i=P∼μPi∼−x,μPi∼+x,P∼νPi∼−x,νPi∼+x, according to Definition 8,(2)wiα∼i=1−1−μPi∼−x2wi,1−1−μPi∼+x2wi,νPi∼−xwi,νPi∼+xwi⊕i=1nwiα∼i=⊕i=1n1−1−μPi∼−x2wi,1−1−μPi∼+x2wi,νPi∼−xwi,νPi∼+xwi=1−∏i=1n1−μPi∼−x2wi,1−∏i=1n1−μPi∼+x2wi,∏i=1nνP∼−xwi,∏i=1nνP∼+xwi=IVPFWAα∼1,α∼2,α∼3,…,α∼n.


## 3. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) Decision-Making Based on Interval Pythagorean Fuzzy Numbers

This paper presents a new TOPSIS decision-making method based on interval Pythagorean fuzzy numbers.Suppose there is a mixed multiattribute decision-making problem. The scheme set isA=A1,A2,A3,…,An, the attribute set is B=B1,B2,B3,…,Bn, and the decision-maker set is C=C1,C2,C3,…,Cn. The evaluated value matrix given by the decision-maker K is(3)Dk=α∼11α∼12⋯α∼1nα∼21α∼22⋯α∼2n⋯⋯⋮⋯α∼m1α∼m2⋯α∼mn,α∼ij=μPij∼−x,μPij∼+x,νPij∼−x,νPij∼+x.Step 1.
The TOPSIS decision-making method is used to analyze interval Pythagorean fuzzy problems. First, the positive ideal matrix and negative ideal matrix of the interval Pythagorean fuzzy sets are defined, and multiple decision matrices are integrated into a comprehensive decision matrix.
According to the evaluated values of the all decision-makers, the positive ideal matrixL+ and bilateral negative ideal matrices Le− and Lf− are expressed as follows:(4)L+=α∼11+α∼12+⋯α∼1n+α∼21+α∼22+⋯α∼2n+⋯⋯⋮⋯α∼m1+α∼m2+⋯α∼mn+,Le−=α∼11−eα∼12−e⋯α∼1n−eα∼21−eα∼22−e⋯α∼2n−e⋯⋯⋮⋯α∼m1−eα∼m2−e⋯α∼mn−e,Lf−=α∼11−fα∼12−f⋯α∼1n−fα∼21−fα∼22−f⋯α∼2n−f⋯⋯⋮⋯α∼m1−fα∼m2−f⋯α∼mn−f,where α∼ij+=∑k=1Cα∼ijk/C is the mean of the evaluated values of all decision-makers, α∼ij−e=minα∼ijk is the minimum value below the average value, and α∼ij−e=maxα˜ijk is the maximum value above the average.Step 2.
According to the positive and negative ideal matrices, the Hamming distancedα∼ijk,L between each valuated value and the positive and negative ideal evaluated values can be obtained:(5)dα∼ijk,α∼ij+=14μPij∼−kvx−μPij∼−+x+μPij∼+kx−μPij∼++x+νPij∼−kx−νPij∼−+x+νPij∼+kx−νPij∼++x,dα∼ijk,α∼ije=14μPij∼−kx−μPij∼−ex+μPij∼+kx−μPij∼+ex+νPij∼−kx−νPij∼−ex+νPij∼+kx−νPij∼+ex,dα∼ijk,α∼ijf=14μPij∼−kx−μPij∼−fx+μPij∼+kx−μPij∼+fx+νPij∼−kx−νPij∼−fx+νPij∼+kx−νPij∼+fx.Step 3.
The closeness of each evaluated value is calculated according to the valuated values of the positive and negative ideal matrices:(6)rijk=dα∼ijk,α∼ije+dα∼ijk,α∼ijfdα∼ijk,α∼ij++dα∼ijk,α∼ije+dα∼ijk,α∼ijf.Step 4.
The weights are determined as follows:wijk=rijk/∑k=1Crijk.Step 5.
The weighted average operator (IPVFWA) of the interval Pythagorean fuzzy numberα∼i is mainly used to synthesize the decision matrix set of multiple decision-makers into a single comprehensive decision evaluation matrix by obtaining the evaluated value weight of each decision-maker. Therefore, after determining the evaluated value weight of each decision-maker in the fourth step, the evaluation matrix set of each decision-maker can be synthesized into comprehensive decision matrices M=α∼ijm×n and α∼ij=wij1α∼ij1⊕wij2α∼ij2⊕wij3α∼ij3⊕⋯⊕wijtα∼ijt through the obtained weight and interval Pythagorean fuzzy number WA operator (IPVFWA).Step 6.
The positive and negative ideal solutions of the interval Pythagorean fuzzy comprehensive decision matrixM=α∼ijm×n and the Hamming distance between each evaluated value and the positive and negative ideal evaluated values are determined as follows:Positive ideal solution:L+=α∼1+,α∼2+,α∼3+,…,α∼n+, where α∼j+=maxα∼ijNegative ideal solution:L−=α∼1−,α∼2−,α∼3−,…,α∼n−, where α∼j−=minα∼ij


### 3.1. Hamming Distance

(7)dα∼ijk,α∼ij+=14μPij∼−kx−μPij∼−+x+μPij∼+kx−μPij∼++x+νPij∼−kx−νPij∼−+x+νPij∼+kx−νPij∼++x,dα∼ijk,α∼ije=14μPij∼−kx−μPij∼−ex+μPij∼+kx−μPij∼+ex+νPij∼−kx−νPij∼−ex+νPij∼+kx−νPij∼+ex.Step 7.
The closeness coefficient is determined as follows:(8)rij=dα∼ij,α∼ij−dα∼ij,α∼ij++dα∼ij,α∼ij−.Step 8.
The greater the closeness coefficient of the scheme is, the closer the scheme is to being optimal.


## 3.1. Hamming Distance

(7)dα∼ijk,α∼ij+=14μPij∼−kx−μPij∼−+x+μPij∼+kx−μPij∼++x+νPij∼−kx−νPij∼−+x+νPij∼+kx−νPij∼++x,dα∼ijk,α∼ije=14μPij∼−kx−μPij∼−ex+μPij∼+kx−μPij∼+ex+νPij∼−kx−νPij∼−ex+νPij∼+kx−νPij∼+ex.Step 7.
The closeness coefficient is determined as follows:(8)rij=dα∼ij,α∼ij−dα∼ij,α∼ij++dα∼ij,α∼ij−.Step 8.
The greater the closeness coefficient of the scheme is, the closer the scheme is to being optimal.


## 4. Example Analysis

### 4.1. Evaluation Matrix

In this example, it is assumed that when an epidemic occurs in an area, there are three affected areas (A=A1,A2,A3 that need support and three experts evaluate the epidemic situation in an area. The attribute set of the evaluation includes the health status, epidemic prevention, and medical environment of the infected personnel in the affected area. The evaluation matrix of three decision-makers is given in Table 1.Table 1
Evaluation matrix of three decision-makers.
B1B2B3C1A1([0.7,0.8], [0.2,0.3])([0.7,0.8], [0.4,0.5])([0.4,0.6], [0.2,0.4])A2([0.5,0.6], [0.3,0.4])([0.5,0.6], [0.2,0.3])([0.4,0.5], [0.1,0.2])A3([0.6,0.8], [0.4,0.5])([0.7,0.9], [0.3,0.5])([0.6,0.7], [0.2,0.3])C2A1([0.4,0.5], [0.2,0.3])([0.5,0.7], [0.3,0.4])([0.7,0.8], [0.4,0.5])A2([0.5,0.6], [0.4,0.5])([0.6,0.7], [0.4,0.5])([0.5,0.6], [0.2,0.3])A3([0.4,0.6], [0.1,0.3])([0.5,0.7], [0.3,0.5])([0.6,0.7], [0.2,0.4])C3A1([0.5,0.6], [0.3,0.4])([0.6,0.7], [0.4,0.5])([0.7,0.8], [0.2,0.4])A2([0.4,0.5], [0.2,0.3])([0.7,0.8], [0.2,0.4])([0.4,0.5], [0.2,0.3])A3([0.7,0.8], [0.2,0.4])([0.6,0.8], [0.4,0.5])([0.6,0.7], [0.4,0.5])


### 4.2. Decision-Making Process

Step 9.
According to the evaluated values of all the decision-makers, the positive ideal matrixL+ and bilateral negative ideal matrices Le− and Lf− were found, as shown in Table 2.Table 2
Positive and negative ideal matrices.
B1B2B3L+A1([0.5333, 0.6333], [0.2333, 0.3333])([0.6000, 0.7333], [0.3667, 0.4667])([0.6000, 0.7333], [0.2667, 0.4333])A2([0.4667, 0.5667], [0.3000, 0.4000])([0.6000, 0.7000], [0.2667, 0.4000])([0.4333, 0.5333], [0.1667, 0.2667])A3([0.5667, 0.7333], [0.2333, 0.4000])([0.6000, 0.8000], [0.3333, 0.5000])([0.6000, 0.7000], [0.2667, 0.4000])Le−A1([0.4,0.5], [0.3,0.4])([0.5,0.7], [0.4,0.5])([0.4,0.6], [0.4,0.5])A2([0.4,0.5], [0.4,0.5])([0.5,0.6], [0.4,0.5])([0.4,0.5], [0.2,0.3])A3([0.4,0.6], [0.4,0.5])([0.5,0.7], [0.4,0.5])([0.6,0.7], [0.4,0.5])Lf−A1([0.5,0.6], [0.2,0.3])([0.6,0.7], [0.3,0.4])([0.7,0.8], [0.2,0.4])A2([0.5,0.6], [0.2,0.3])([0.7,0.8], [0.2,0.4])([0.5,0.6], [0.2,0.3])A3([0.7,0.8], [0.1,0.3])([0.6,0.8], [0.3,0.5])([0.6,0.7], [0.2,0.4])Step 10.
According to the positive and negative ideal matrices, the Hamming distancedα∼ijk,L between each evaluated value and the positive and negative ideal evaluated values can be obtained, as shown in Table 3.Table 3
Hamming distancedα∼ijk,L between each evaluated value and positive and negative ideal evaluated values.
B1B2B3C1dα∼ijk,α∼ij+A10.10000.05830.1083A20.01670.09170.0500A30.09170.05830.0417dα∼ijk,α∼ijeA10.20000.07500.0750A20.10000.10000.0500A30.10000.12500.1000dα∼ijk,α∼ijfA10.10000.10000.1250A20.05000.12500.1000A30.15000.05000.0250C2dα∼ijk,α∼ij+A10.08330.06670.0917A20.06670.05830.0500A30.13330.05830.0167dα∼ijk,α∼ijeA10.05000.05000.1250A20.05000.05000.0500A30.12500.02500.0750dα∼ijk,α∼ijfA10.05000.02500.0750A20.10000.12500.0000A30.12500.05000.0000C3dα∼ijk,α∼ij+A10.05000.02500.0667A20.08330.06670.0333A30.05830.01670.0583dα∼ijk,α∼ijeA10.05000.02500.2000A20.10000.17500.0000A30.20000.05000.0000dα∼ijk,α∼ijfA10.05000.05000.0000A20.05000.00000.0500A30.05000.02500.0750Step 11.
The closenessrijk of each evaluated value according to the evaluated values of the positive and negative ideal matrices were calculated, and the results are shown in Table 4.Table 4
Closeness of each evaluated value.
B1B2B3rij1A10.75000.75000.6486A20.90000.71050.7500A30.73170.75000.7500rij2A10.54550.52940.6857A20.69230.75000.5000A30.65220.56250.8182rij3A10.66670.75000.7500A20.64290.72410.6000A30.81080.81820.5625Step 12.
The weightwijk of each evaluated value was determined, and the results are shown in Table 5.Table 5
Weight of each evaluated value.
B1B2B3wij1A10.38220.36960.3112A20.40270.32520.4054A30.33340.35200.3520wij2A10.27800.26090.3290A20.30970.34330.2703A30.29720.26400.3840wij3A10.33980.36960.3598A20.28760.33150.3243A30.36940.38400.2640Step 13.
At this time, the interval Pythagorean fuzzy number WA operator (IPVFWA) can be used to synthesize the evaluated value weight of each decision-maker obtained in Step4, and the evaluation matrix sets of all of the decision-makers are assembled into a comprehensive decision matrix M=α∼ijm×n, as shown in Table 6.Table 6
Comprehensive evaluation matrix.
B1B2B3A1([0.5763 ,0.6806], [0.2295 ,0.3308])([0.6218, 0.7427], [0.3711, 0.4717])([0.6359,0.7546], [0.2512,0.4305])A2([0.5618,0.5746], [0.2919,0.3946])([0.6124, 0.7147], [0.2537,0.3933])([0.4305,0.5305], [0.1510,0.2545])A3([0.7633, 0.7569], [0.2051, 0.3956])([0.6196, 0.8275], [0.3350,0.5000])( 0.6000,0.7000], [0.2402,0.3834])Step 14.
The positive and negative ideal solutions of the interval Pythagorean fuzzy comprehensive decision matrix, the Hamming distance between each evaluated value and the positive and negative ideal evaluated values, and the closeness degree and weight of each evaluated value were determined. The results are shown in Table7.Table 7
Positive and negative ideal matrices of the comprehensive matrix.
B1B2B3L+([0.7633,0.7569], [0.2919,0.3956])([0.6218, 0.8275], [0.3711, 0.5000])([0.6359, 0.7546], [0.2512, 0.4305])L−([0.5618,0.5746], [0.2051,0.3308])([0.6124, 0.7147], [0.2537, 0.3933])([0.4305,0.5305], [0.1510,0.2545])dα∼ij,α∼ij+A10.09760.02830.0000A20.09620.08660.1764A30.02170.00960.0371dα∼ij,α∼ij−A10.03620.05830.1764A20.03760.00000.0000A30.11220.07700.1392rijA10.27070.67341.0000A20.28120.00000.0000A30.83790.88950.7894wijA10.13920.34640.5144A21.00000.00000.0000A30.33290.35340.3136Step 15.
The weighted distance and closeness between the evaluated value of each scheme and the positive and negative ideal solutions were determined, and the results are shown in Table8.
According to the results in Table8, scheme A2 had the highest closeness and its scheme was the best. At the beginning of the decision-making process, each expert can be assigned a weight. However, weights were not given directly to the decision-makers in this example, but the Hamming distance between the evaluated value and the positive and negative ideal evaluated values was used to calculate the closeness of each evaluated value. The evaluated value weight of each decision-maker was determined through the closeness of the evaluated value, which ensured the objectivity of the whole decision-making process. At the same time, when the scheme score (0.5135, 0.6389, 0.5301) was calculated using the interval Pythagorean fuzzy geometric weighted Bonferroni average operator of Jiang Yingying [25] in Step 6, the finite ranking of each scheme was still (3, 1, 2) and the score results of each scheme were not much different. However, the TOPSIS decision-making method of the interval Pythagorean fuzzy numbers used in this paper show that the gap between the advantages and disadvantages of each scheme is very obvious, and as a result, it would be easier for decision-makers to make decisions.
Therefore, the TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers has three advantages:(1)
In the TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers, the decision-making method was used to provide the upper and lower limits of the membership and nonmembership and then decisions were made through the TOPSIS decision-making method. This method alleviates the loss of information in the decision-making process to a great extent, making the decision-making process more accurate and scientific.(2)
For mixed decision-making involving multiple decision-makers, when the decision-maker weight and attribute weight are unknown, the corresponding weight can be obtained from the decision matrix by using a TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers. Compared with the case where the weights of the decision-makers are known, this method can effectively solve the problem of different types of attribute information and utilize the attribute advantages of each decision-maker to the greatest extent.(3)
Through the interval Pythagorean fuzzy TOPSIS hybrid multiattribute decision-making method, the decision matrices of multiple decision-makers are collected into a comprehensive evaluation matrix, and then, the TOPSIS decision-making method is used to solve the optimal scheme. In the whole decision-making process, the closeness values of each scheme calculated by the TOPSIS decision-making method twice would be significantly different. This would allow decision-makers to more easily select the optimal scheme, making the decision result more convincing.Table 8
Weighted distance and closeness of the comprehensive matrix.
A1A2A3wijdα∼ij,α∼ij+0.02340.09620.0223wijdα∼ij,α∼ij−0.11600.03760.1082rij0.16780.71880.1705Sort312


## 4.1. Evaluation Matrix

In this example, it is assumed that when an epidemic occurs in an area, there are three affected areas (A=A1,A2,A3 that need support and three experts evaluate the epidemic situation in an area. The attribute set of the evaluation includes the health status, epidemic prevention, and medical environment of the infected personnel in the affected area. The evaluation matrix of three decision-makers is given in Table 1.Table 1
Evaluation matrix of three decision-makers.
B1B2B3C1A1([0.7,0.8], [0.2,0.3])([0.7,0.8], [0.4,0.5])([0.4,0.6], [0.2,0.4])A2([0.5,0.6], [0.3,0.4])([0.5,0.6], [0.2,0.3])([0.4,0.5], [0.1,0.2])A3([0.6,0.8], [0.4,0.5])([0.7,0.9], [0.3,0.5])([0.6,0.7], [0.2,0.3])C2A1([0.4,0.5], [0.2,0.3])([0.5,0.7], [0.3,0.4])([0.7,0.8], [0.4,0.5])A2([0.5,0.6], [0.4,0.5])([0.6,0.7], [0.4,0.5])([0.5,0.6], [0.2,0.3])A3([0.4,0.6], [0.1,0.3])([0.5,0.7], [0.3,0.5])([0.6,0.7], [0.2,0.4])C3A1([0.5,0.6], [0.3,0.4])([0.6,0.7], [0.4,0.5])([0.7,0.8], [0.2,0.4])A2([0.4,0.5], [0.2,0.3])([0.7,0.8], [0.2,0.4])([0.4,0.5], [0.2,0.3])A3([0.7,0.8], [0.2,0.4])([0.6,0.8], [0.4,0.5])([0.6,0.7], [0.4,0.5])


## 4.2. Decision-Making Process

Step 9.
According to the evaluated values of all the decision-makers, the positive ideal matrixL+ and bilateral negative ideal matrices Le− and Lf− were found, as shown in Table 2.Table 2
Positive and negative ideal matrices.
B1B2B3L+A1([0.5333, 0.6333], [0.2333, 0.3333])([0.6000, 0.7333], [0.3667, 0.4667])([0.6000, 0.7333], [0.2667, 0.4333])A2([0.4667, 0.5667], [0.3000, 0.4000])([0.6000, 0.7000], [0.2667, 0.4000])([0.4333, 0.5333], [0.1667, 0.2667])A3([0.5667, 0.7333], [0.2333, 0.4000])([0.6000, 0.8000], [0.3333, 0.5000])([0.6000, 0.7000], [0.2667, 0.4000])Le−A1([0.4,0.5], [0.3,0.4])([0.5,0.7], [0.4,0.5])([0.4,0.6], [0.4,0.5])A2([0.4,0.5], [0.4,0.5])([0.5,0.6], [0.4,0.5])([0.4,0.5], [0.2,0.3])A3([0.4,0.6], [0.4,0.5])([0.5,0.7], [0.4,0.5])([0.6,0.7], [0.4,0.5])Lf−A1([0.5,0.6], [0.2,0.3])([0.6,0.7], [0.3,0.4])([0.7,0.8], [0.2,0.4])A2([0.5,0.6], [0.2,0.3])([0.7,0.8], [0.2,0.4])([0.5,0.6], [0.2,0.3])A3([0.7,0.8], [0.1,0.3])([0.6,0.8], [0.3,0.5])([0.6,0.7], [0.2,0.4])Step 10.
According to the positive and negative ideal matrices, the Hamming distancedα∼ijk,L between each evaluated value and the positive and negative ideal evaluated values can be obtained, as shown in Table 3.Table 3
Hamming distancedα∼ijk,L between each evaluated value and positive and negative ideal evaluated values.
B1B2B3C1dα∼ijk,α∼ij+A10.10000.05830.1083A20.01670.09170.0500A30.09170.05830.0417dα∼ijk,α∼ijeA10.20000.07500.0750A20.10000.10000.0500A30.10000.12500.1000dα∼ijk,α∼ijfA10.10000.10000.1250A20.05000.12500.1000A30.15000.05000.0250C2dα∼ijk,α∼ij+A10.08330.06670.0917A20.06670.05830.0500A30.13330.05830.0167dα∼ijk,α∼ijeA10.05000.05000.1250A20.05000.05000.0500A30.12500.02500.0750dα∼ijk,α∼ijfA10.05000.02500.0750A20.10000.12500.0000A30.12500.05000.0000C3dα∼ijk,α∼ij+A10.05000.02500.0667A20.08330.06670.0333A30.05830.01670.0583dα∼ijk,α∼ijeA10.05000.02500.2000A20.10000.17500.0000A30.20000.05000.0000dα∼ijk,α∼ijfA10.05000.05000.0000A20.05000.00000.0500A30.05000.02500.0750Step 11.
The closenessrijk of each evaluated value according to the evaluated values of the positive and negative ideal matrices were calculated, and the results are shown in Table 4.Table 4
Closeness of each evaluated value.
B1B2B3rij1A10.75000.75000.6486A20.90000.71050.7500A30.73170.75000.7500rij2A10.54550.52940.6857A20.69230.75000.5000A30.65220.56250.8182rij3A10.66670.75000.7500A20.64290.72410.6000A30.81080.81820.5625Step 12.
The weightwijk of each evaluated value was determined, and the results are shown in Table 5.Table 5
Weight of each evaluated value.
B1B2B3wij1A10.38220.36960.3112A20.40270.32520.4054A30.33340.35200.3520wij2A10.27800.26090.3290A20.30970.34330.2703A30.29720.26400.3840wij3A10.33980.36960.3598A20.28760.33150.3243A30.36940.38400.2640Step 13.
At this time, the interval Pythagorean fuzzy number WA operator (IPVFWA) can be used to synthesize the evaluated value weight of each decision-maker obtained in Step4, and the evaluation matrix sets of all of the decision-makers are assembled into a comprehensive decision matrix M=α∼ijm×n, as shown in Table 6.Table 6
Comprehensive evaluation matrix.
B1B2B3A1([0.5763 ,0.6806], [0.2295 ,0.3308])([0.6218, 0.7427], [0.3711, 0.4717])([0.6359,0.7546], [0.2512,0.4305])A2([0.5618,0.5746], [0.2919,0.3946])([0.6124, 0.7147], [0.2537,0.3933])([0.4305,0.5305], [0.1510,0.2545])A3([0.7633, 0.7569], [0.2051, 0.3956])([0.6196, 0.8275], [0.3350,0.5000])( 0.6000,0.7000], [0.2402,0.3834])Step 14.
The positive and negative ideal solutions of the interval Pythagorean fuzzy comprehensive decision matrix, the Hamming distance between each evaluated value and the positive and negative ideal evaluated values, and the closeness degree and weight of each evaluated value were determined. The results are shown in Table7.Table 7
Positive and negative ideal matrices of the comprehensive matrix.
B1B2B3L+([0.7633,0.7569], [0.2919,0.3956])([0.6218, 0.8275], [0.3711, 0.5000])([0.6359, 0.7546], [0.2512, 0.4305])L−([0.5618,0.5746], [0.2051,0.3308])([0.6124, 0.7147], [0.2537, 0.3933])([0.4305,0.5305], [0.1510,0.2545])dα∼ij,α∼ij+A10.09760.02830.0000A20.09620.08660.1764A30.02170.00960.0371dα∼ij,α∼ij−A10.03620.05830.1764A20.03760.00000.0000A30.11220.07700.1392rijA10.27070.67341.0000A20.28120.00000.0000A30.83790.88950.7894wijA10.13920.34640.5144A21.00000.00000.0000A30.33290.35340.3136Step 15.
The weighted distance and closeness between the evaluated value of each scheme and the positive and negative ideal solutions were determined, and the results are shown in Table8.
According to the results in Table8, scheme A2 had the highest closeness and its scheme was the best. At the beginning of the decision-making process, each expert can be assigned a weight. However, weights were not given directly to the decision-makers in this example, but the Hamming distance between the evaluated value and the positive and negative ideal evaluated values was used to calculate the closeness of each evaluated value. The evaluated value weight of each decision-maker was determined through the closeness of the evaluated value, which ensured the objectivity of the whole decision-making process. At the same time, when the scheme score (0.5135, 0.6389, 0.5301) was calculated using the interval Pythagorean fuzzy geometric weighted Bonferroni average operator of Jiang Yingying [25] in Step 6, the finite ranking of each scheme was still (3, 1, 2) and the score results of each scheme were not much different. However, the TOPSIS decision-making method of the interval Pythagorean fuzzy numbers used in this paper show that the gap between the advantages and disadvantages of each scheme is very obvious, and as a result, it would be easier for decision-makers to make decisions.
Therefore, the TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers has three advantages:(1)
In the TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers, the decision-making method was used to provide the upper and lower limits of the membership and nonmembership and then decisions were made through the TOPSIS decision-making method. This method alleviates the loss of information in the decision-making process to a great extent, making the decision-making process more accurate and scientific.(2)
For mixed decision-making involving multiple decision-makers, when the decision-maker weight and attribute weight are unknown, the corresponding weight can be obtained from the decision matrix by using a TOPSIS hybrid multiattribute decision-making method based on interval Pythagorean fuzzy numbers. Compared with the case where the weights of the decision-makers are known, this method can effectively solve the problem of different types of attribute information and utilize the attribute advantages of each decision-maker to the greatest extent.(3)
Through the interval Pythagorean fuzzy TOPSIS hybrid multiattribute decision-making method, the decision matrices of multiple decision-makers are collected into a comprehensive evaluation matrix, and then, the TOPSIS decision-making method is used to solve the optimal scheme. In the whole decision-making process, the closeness values of each scheme calculated by the TOPSIS decision-making method twice would be significantly different. This would allow decision-makers to more easily select the optimal scheme, making the decision result more convincing.Table 8
Weighted distance and closeness of the comprehensive matrix.
A1A2A3wijdα∼ij,α∼ij+0.02340.09620.0223wijdα∼ij,α∼ij−0.11600.03760.1082rij0.16780.71880.1705Sort312


## 5. Conclusion

In this paper, a mixed multiattribute decision-making problem was studied in which the decision-maker’s weight and attribute weight are completely unknown and the attribute value is an interval Pythagorean fuzzy number. A TOPSIS mixed multiattribute decision-making method based on interval Pythagorean fuzzy numbers is proposed. The advantages of this method are as follows: First, in the decision-making process, by giving the upper and lower limits of the membership and nonmembership of the evaluated value, the information loss of decision-makers in the decision-making process is alleviated to the greatest extent. Second, the decision-makers are no longer given a decision weight directly in decision-making, fully utilizing the attribute advantages of each decision-maker and effectively solving the problems of different types of attribute information. Third, compared with other decision-making methods, the TOPSIS hybrid multiattribute decision-making method using interval Pythagorean fuzzy numbers increases the gap between the advantages and disadvantages of each scheme. As a result, the decision-making results are more convincing and it is easier for decision-makers to select the optimal scheme. The TOPSIS hybrid multiattribute decision-making method using interval Pythagorean fuzzy numbers not only reduces the subjectivity of the weight determination to a great extent but also uses interval Pythagorean fuzzy numbers to improve the risk impact of the deviation between membership and nonmembership on the decision-making results.



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*Source: 5735272-2021-12-03.xml*