MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2021/57352725735272Research ArticleTOPSIS Hybrid Multiattribute Group Decision-Making Based on Interval Pythagorean Fuzzy Numbershttps://orcid.org/0000-0002-8585-8916JunH. U.https://orcid.org/0000-0002-9103-5940JunminW. U.https://orcid.org/0000-0002-4070-7903JieW. U.FerraraMassimilianoSchool of Economic and ManagementJiangsu University of Science and TechnologyZhenjiangJiangsu 212003Chinajust.edu.cn2021312202120211810202141120211811202131220212021Copyright © 2021 H. U. Jun et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

National Natural Science Foundation of China71874073National Social Science Foundation of China19fglb029NSFC71771161
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  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  and interval fuzzy sets . In 2013, Yager  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  defined the similarity and distance measures of Pythagorean fuzzy numbers. Chen  extended the Pythagorean fuzzy environment to the VIKOR decision model. Ren et al.  combined a Pythagorean fuzzy set with the TODIM method. Garg , Liu Weifeng , 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  and Muhammad  defined interval Pythagorean fuzzy integration operators. Zhang , Li Na [14, 15], and Lin Wenhao  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.  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.  used the TOPSIS and analytic hierarchy process (AHP) methods to determine, prioritize, and assess the most effective desertification indices. Kacprzak et al.  presented a new approach for ranking the alternatives for group decision-making using the TOPSIS method based on ordered fuzzy numbers. Rezaei et al.  developed a method to solve the sustainable circular partner selection problem with completely unknown decision-making experts. Du Yingxue et al.  proposed a triangular Pythagorean fuzzy TOPSIS decision-making method. Qu Guohua  proposed an intuitionistic fuzziness λ-Shapley Choquet integral operator TOPSIS multiattribute group decision-making method. Zhao and others  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 Concepts2.1. Pythagorean Fuzzy SetsDefinition 1 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Suppose X is a given universe and P=x,PμPx,νPx|xX 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 0μPx,νPx1, and 0μP2x+νP2x1. The hesitation degree of x to P is defined as πPx=1μP2x+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 [<xref ref-type="bibr" rid="B6">6</xref>]).

For the Pythagorean fuzzy number α=μPx,νPx, the scoring function of α is defined as Sα=μP2xνP2x and the precision function α is defined as Hα=μP2x+νP2x, where Sα1,1 and Hα0,1.

Definition 3 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

For two Pythagorean fuzzy numbers αi=PμPix,νPixi=1,2, the comparison is as follows:

Suppose Sα1=Sα2. If Hα1=Hα2, then α1=α2, and if Hα1<Hα2, then α1<α2.

Suppose Sα1>Sα2. Then, α1>α2.

Definition 4 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

For two Pythagorean fuzzy numbers α1=PμP1x,νP1x and α2=PμP2x,νP2x, the following operation rules are satisfied:

α1α2=μP12x+μP22xμP12xμP22x,νP1xνP2x

α1α2=μP1xμP2x,νP12x+νP22xνP12xνP22x

λα=11μP2xλ,νPλx

αλ=μPλx,11νP2xλ

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 [<xref ref-type="bibr" rid="B13">13</xref>]).

Suppose X is a given universe and P=x,PμPx,μP+x,PνPx,νP+x|xX is an interval Pythagorean fuzzy set on X. μPx and μP+x are the lower and upper bounds of the membership degree, respectively, and νPx and νP+x are the lower and upper bounds of the nonmembership degree, respectively. The interval Pythagorean fuzzy set P satisfies 0μP+x2+νPx21. The hesitation degree of x to the interval Pythagorean fuzzy set P is defined as πPx=πPx,πP+x=1μP+x2νP+x2,1μPx2νPx2.

Definition 6 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

For the interval Pythagorean fuzzy number α=PμPx,μP+x,PνPx,νP+x, the scoring function α is defined as Sα=1/2μPx2νPx2+μP+x2νP+x2 and the precision function α is defined as Hα=1/2μPx2+νPx2+μP+x2+νP+x2, where Sα1,1 and Hα0,1.

Definition 7 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

For two interval Pythagorean fuzzy numbers αi=PμPix,μPi+x,PνPix,νPi+xi=1,2, the comparison is as follows:

Suppose Sα1=Sα2. If Hα1=Hα2, then α1=α2, and if Hα1<Hα2, then α1<α2

Suppose Sα1>Sα2. Then, α1>α2

Definition 8 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

For two interval Pythagorean fuzzy numbers αi=PμPix,μPi+x,PνPix,νPi+xi=1,2, the following operations are satisfied:

α1α2=μP1x2+μP2x2μP1x2μP2x2,μP1+x2+μP2+x2μP1+x2μP2+x2,νP1xνP2x,νP1+xνP2+x

α1α2=μP1xμP2x,μP1+xμP2+x,νP1x2+νP2x2νP1x2νP2x2,νP1+x2+νP2+x2νP1+x2νP2+x2

λα=11μPx2λ,11μP+x2λ,νPxλ,νP+xλ

αλ=μPxλ,μP+xλ,11νPx2λ,11νP+x2λ

Definition 9 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

For two interval Pythagorean fuzzy numbers αi=PμPix,μPi+x,PνPix,νPi+xi=1,2, the Hamming distance dα1,α2 is defined as dα1,α2=1/4μP1xμP2x+μP1+xμP2+x+νP1xνP2x+νP1+xνP2+x.

2.3. Weighted Average (WA) Operator Based on Interval Pythagorean Fuzzy SetsDefinition 10 (see [<xref ref-type="bibr" rid="B24">24</xref>]).

There is a mapping WA: RnR, WAx1,x2,x3,,xn=i=1nwixi, where w=w1,w2,w3,,wnT is the weighted vector of WA, satisfying wi0,1 and i=1nwi=1. At this time, the function WA is called the weighted average operator.

Definition 11.

For the interval Pythagorean fuzzy number αi=PμPx,μP+x,PνPx,νP+x, wii=1,2,3,,n is the weight of the interval Pythagorean fuzzy number αi, satisfying wi0,1 and i=1nwi=1. The weighted average operator (IPVFWA) of the interval Pythagorean fuzzy number αi can be defined as follows:(1)IVPFWAα1,α2,α3,,αn=w1α1w2α2w3α3w4α4=1i=1n1μPix2wi,1i=1n1μPi+x2wi,i=1nνPxwi,i=1nνP+xwi.

Theorem 1.

If αi=PμPix,μPi+x,PνPix,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.

Because αi=PμPix,μPi+x,PνPix,νPi+x, according to Definition 8,(2)wiαi=11μPix2wi,11μPi+x2wi,νPixwi,νPi+xwii=1nwiαi=i=1n11μPix2wi,11μPi+x2wi,νPixwi,νPi+xwi=1i=1n1μPix2wi,1i=1n1μPi+x2wi,i=1nνPxwi,i=1nνP+xwi=IVPFWAα1,α2,α3,,αn.

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 is A=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=α11α12α1nα21α22α2nαm1αm2αmn,αij=μPijx,μPij+x,νPijx,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 matrix L+ and bilateral negative ideal matrices Le and Lf are expressed as follows:(4)L+=α11+α12+α1n+α21+α22+α2n+αm1+αm2+αmn+,Le=α11eα12eα1neα21eα22eα2neαm1eαm2eαmne,Lf=α11fα12fα1nfα21fα22fα2nfαm1fαm2fαmnf,where αij+=k=1Cαijk/C is the mean of the evaluated values of all decision-makers, αije=minαijk is the minimum value below the average value, and αije=maxα˜ijk is the maximum value above the average.

Step 2.

According to the positive and negative ideal matrices, the Hamming distance dαijk,L between each valuated value and the positive and negative ideal evaluated values can be obtained:(5)dαijk,αij+=14μPijkvxμPij+x+μPij+kxμPij++x+νPijkxνPij+x+νPij+kxνPij++x,dαijk,αije=14μPijkxμPijex+μPij+kxμPij+ex+νPijkxνPijex+νPij+kxνPij+ex,dαijk,αijf=14μPijkxμPijfx+μPij+kxμPij+fx+νPijkxνPijfx+νPij+kxνPij+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=dαijk,αije+dαijk,αijfdαijk,αij++dαijk,αije+dαijk,αijf.

Step 4.

The weights are determined as follows: wijk=rijk/k=1Crijk.

Step 5.

The weighted average operator (IPVFWA) of the interval Pythagorean fuzzy number αi 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=αijm×n and αij=wij1αij1wij2αij2wij3αij3wijtαijt 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 matrix M=αijm×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+=α1+,α2+,α3+,,αn+, where αj+=maxαij

Negative ideal solution: L=α1,α2,α3,,αn, where αj=minαij

3.1. Hamming Distance

(7)dαijk,αij+=14μPijkxμPij+x+μPij+kxμPij++x+νPijkxνPij+x+νPij+kxνPij++x,dαijk,αije=14μPijkxμPijex+μPij+kxμPij+ex+νPijkxνPijex+νPij+kxνPij+ex.

Step 7.

The closeness coefficient is determined as follows:(8)rij=dαij,αijdαij,αij++dαij,αij.

Step 8.

The greater the closeness coefficient of the scheme is, the closer the scheme is to being optimal.

4. Example Analysis4.1. 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.

Evaluation matrix of three decision-makers.

B1B2B3
C1A1([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 ProcessStep 9.

According to the evaluated values of all the decision-makers, the positive ideal matrix L+ and bilateral negative ideal matrices Le and Lf were found, as shown in Table 2.

Positive and negative ideal matrices.

B1B2B3
L+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])
LeA1([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])
LfA1([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 distance dαijk,L between each evaluated value and the positive and negative ideal evaluated values can be obtained, as shown in Table 3.

Hamming distance dαijk,L between each evaluated value and positive and negative ideal evaluated values.

B1B2B3
C1dαijk,αij+A10.10000.05830.1083
A20.01670.09170.0500
A30.09170.05830.0417
dαijk,αijeA10.20000.07500.0750
A20.10000.10000.0500
A30.10000.12500.1000
dαijk,αijfA10.10000.10000.1250
A20.05000.12500.1000
A30.15000.05000.0250

C2dαijk,αij+A10.08330.06670.0917
A20.06670.05830.0500
A30.13330.05830.0167
dαijk,αijeA10.05000.05000.1250
A20.05000.05000.0500
A30.12500.02500.0750
dαijk,αijfA10.05000.02500.0750
A20.10000.12500.0000
A30.12500.05000.0000

C3dαijk,αij+A10.05000.02500.0667
A20.08330.06670.0333
A30.05830.01670.0583
dαijk,αijeA10.05000.02500.2000
A20.10000.17500.0000
A30.20000.05000.0000
dαijk,αijfA10.05000.05000.0000
A20.05000.00000.0500
A30.05000.02500.0750
Step 11.

The closeness rijk 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.

Closeness of each evaluated value.

B1B2B3
rij1A10.75000.75000.6486
A20.90000.71050.7500
A30.73170.75000.7500

rij2A10.54550.52940.6857
A20.69230.75000.5000
A30.65220.56250.8182

rij3A10.66670.75000.7500
A20.64290.72410.6000
A30.81080.81820.5625
Step 12.

The weight wijk of each evaluated value was determined, and the results are shown in Table 5.

Weight of each evaluated value.

B1B2B3
wij1A10.38220.36960.3112
A20.40270.32520.4054
A30.33340.35200.3520

wij2A10.27800.26090.3290
A20.30970.34330.2703
A30.29720.26400.3840

wij3A10.33980.36960.3598
A20.28760.33150.3243
A30.36940.38400.2640
Step 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 Step 4, and the evaluation matrix sets of all of the decision-makers are assembled into a comprehensive decision matrix M=αijm×n, as shown in Table 6.

Comprehensive evaluation matrix.

B1B2B3
A1([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 Table 7.

Positive and negative ideal matrices of the comprehensive matrix.

B1B2B3
L+([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])

dαij,αij+A10.09760.02830.0000
A20.09620.08660.1764
A30.02170.00960.0371

dαij,αijA10.03620.05830.1764
A20.03760.00000.0000
A30.11220.07700.1392

rijA10.27070.67341.0000
A20.28120.00000.0000
A30.83790.88950.7894

wijA10.13920.34640.5144
A21.00000.00000.0000
A30.33290.35340.3136
Step 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 Table 8.

According to the results in Table 8, 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  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:

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.

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.

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.

Weighted distance and closeness of the comprehensive matrix.

A1A2A3
wijdαij,αij+0.02340.09620.0223
wijdαij,αij0.11600.03760.1082
rij0.16780.71880.1705
Sort312
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.

Data Availability

The data analyzed by the example in this paper are interval values randomly given in line with the assumptions in this paper. The data are real and can be used as a reference for readers.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the General Project of National Natural Science Foundation of China, “Price Distortion of Energy Factors: Causes, Measurement and Appropriate Correction Strategy” (71874073); the project supported by the National Social Science Foundation of China, “Research on Knowledge Transfer Game and Innovation Performance of Industry University Research Alliance” (19fglb029); and NSFC, “Research on Coupling Mechanism and Promotion Strategy of Knowledge Subject Collaborative Behavior and Value Creation in Innovation Ecosystem” (71771161).

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