# Interval Pythagorean Fuzzy Decision Based on GWOWA Operator∗

Authors: Hu Jun; Wu Junmin; Wang Mengzhe
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

For the multiattribute group decision-making problem in an interval Pythagorean fuzzy environment, the existing experts and scholars have extended the weighted average (WA), ordered weighted average (OWA), generalized ordered weighted average (GOWA), weighted ordered weighted average (WOWA), and other operators to interval fuzzy environment, while the research on the application and promotion of interval Pythagorean fuzzy with generalized weighted ordered weighted average (GWOWA) operator has not been carried out, GWOWA operator not only retains the advantages of WOWA operator but also introduces artificial variables, which increases the ability of decision-makers to control the aggregation of fuzzy information. Therefore, the GWOWA operator model based on interval Pythagorean fuzzy sets is constructed. First, it is proved that interval Pythagorean fuzzy generalized weighted average operator (IVPFGWA) and interval Pythagorean fuzzy generalized ordered weighted average operator (IVPFGOWA) are special cases of IVPFGWOWA operator, and their idempotence, monotonicity, and boundedness are proved; second, a group decision-making method based on interval Pythagorean fuzzy GWOWA operator is presented. Finally, an example is given to illustrate the effectiveness and scientificity of this method. It is found that the interval Pythagorean fuzzy decision-making method of the GWOWA operator alleviates the loss of information in the decision-making process to a great extent. At the same time, with the increase in the value of artificial variables, the gap between the best scheme and other schemes continues to increase, making the decision-making results more obvious, scientific, and accurate.


---

## Body

## 1. Introduction

In the actual decision-making process, the decision-making information obtained by decision-making is often inaccurate and fuzzy. In this fuzzy information environment, the results of direct decision-making will deviate from the correct trajectory. Second, how to more effectively, more accurately, and more scientifically assemble the decision-making problems with multiple decision-making subjects and attributes is also a problem that cannot be ignored. Therefore, based on this fuzzy information environment, it is particularly important to study the corresponding decision-making methods. The existing studies have proposed a variety of aggregation operators (such as WA, OWA, GOWA, and WOWA) and extended them to intuitive fuzzy, Pythagorean fuzzy, and interval Pythagorean fuzzy environments. However, the existing research is more inclined to study the aggregation ability of operators, and the decision-makers’ control ability of information aggregation is weakened. Based on this, this paper studies the GWOWA operator by introducing artificial variables into the WOWA operator, extends it to the interval Pythagorean fuzzy environment, and verifies its effectiveness and scientificity for multi-decision-makers and multiattribute decision-making.At present, the academic circles have conducted a lot of research on fuzzy information decision-making and gradually extended it to Pythagorean fuzzy and interval Pythagorean fuzzy environments. For example, Zadeh [1] first proposed the concept of a fuzzy set in 1965. Since then, experts in the academic circles began to conduct a lot of targeted research on fuzzy mathematics. In 2013, Yager [2] first proposed the concept of Pythagorean fuzzy numbers based on intuitionistic fuzzy sets. Peng [3], Ting-Yu [4], Peijia [5], Garg [6], and others began to apply Pythagorean fuzzy numbers to various decisions. However, with the deviation of the membership degree of Pythagorean fuzzy numbers, experts and scholars gradually extended Pythagorean fuzzy to the interval Pythagorean fuzzy information environment, such as Peng and Yang [7] and Muhammad et al. [8], Zhang [9], Yager [10], and Herrera and Verdegay [11], which defined the interval Pythagorean fuzzy integration algorithm.The aggregation operator plays an irreplaceable role in the evaluation and decision-making of multiple decision-makers and multiple attributes in the fuzzy information environment. In 1988, Ashraf et al. [12] proposed the OWA operator and extended the application of the OWA operator to an intuitionistic fuzzy environment. Riaz et al. [13], Li et al. [14], Marzieh et al. [15], and Fei et al. [16] extended the OWA operator to a fuzzy environment and carried out relevant application verification. Torra [17], Yager [18], and others have proposed WOWA operator and GOWA operator based on the advantages of the OWA operator. Zhao et al. [19], Cai [20], Bi et al. [21], and others extended WOWA operators and GOWA operators to a Pythagorean fuzzy environment. In recent years, some experts and scholars [21, 22] have studied the generalized WOWA operator and extended it to the Pythagorean fuzzy environment. The generalized weighted ordered weighted (GWOWA) operator not only retains the advantages of the WOWA operator but also introduces artificial variables, which increases the ability of decision-makers to control information aggregation. Table 1 compares the research on aggregation operators in different fuzzy environments and points out the limitations of the current research.Table 1
Summary and comparison of decision-making methods in different fuzzy environments.
Fuzzy environmentsAggregation operatorApplication scenariosResearch limitationsResearch innovationsIntuitionistic fuzzyWA, OWA, GOWA, WOWA, and other operatorsMedical treatment, earthquake, and other dangerous situations [23–25]; enterprise management decision; location decision; risk assessment.(1) The sum of membership degree and nonmembership degree in intuitionistic fuzzy numbers must be less than or equal to 1, resulting in the loss of original data information.(2) The membership degree of Pythagorean fuzzy numbers has deviation.(3) The research on aggregation operator in interval Pythagorean fuzzy environment is not comprehensive enough.(1) From the perspective of fuzzy environment, both intuitionistic fuzzy and Pythagorean fuzzy information environment have certain limitations. The operator is extended to the interval Pythagorean fuzzy information environment, which enriches the fuzzy and uncertain expression of data information.(2) Considering the generalization of generalized wowa operator to interval Pythagorean fuzzy information environment, it is helpful to enrich the research of interval Pythagorean fuzzy sets and aggregation operators.Pythagorean fuzzyWA, OWA, GOWA, WOWA, GWOWA, Bonferroni, Heronian, and other operatorsInterval Pythagorean fuzzyWA, Bonferroni, WOWA, OWA, Heronian, and other operatorsFrom Table1, the research on decision-making in intuitive fuzzy and Pythagorean fuzzy environments has been relatively mature, and many scholars have applied fuzzy decision-making to various scenarios. However, due to the limitation of membership degree of intuitionistic fuzzy and Pythagorean fuzzy environment, the fuzzy decision-making in this case is not particularly accurate. In order to solve the membership limitation of intuitionistic fuzzy and Pythagorean fuzzy environment, the research community has proposed the concept of the interval Pythagorean fuzzy number. Many scholars have fused WA, Bonferroni, WOWA, OWA, Heronian, and other operators into the interval Pythagorean fuzzy information environment but have not considered the introduction of artificial variable GWOWA operator and the fusion of the interval Pythagorean fuzzy information environment.Therefore, this paper extends the GWOWA operator to the interval Pythagorean fuzzy environment, defines the generalized weighted ordered weighted average (IVPFGWOWA) operator based on interval Pythagorean fuzzy numbers, and gives a new interval Pythagorean fuzzy decision method based on the GWOWA operator. The advantages of this method are as follows: first, it eases the loss of information in the decision-making process to a great extent. Second, considering the objective weight and position weight, the comprehensive weight of the IVPFGWOWA operator is constructed. Third, artificial parameters are introduced. The increase of parameters widens the gap between the optimal scheme and other schemes, making the decision-making results more obvious, scientific, and accurate.


## 2. Basic Concepts

### 2.1. Interval Pythagorean Fuzzy Sets

In decision-making, scholars have proposed the concept and algorithms of the 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 1 (see [11]).
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 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 2 (see [11]).
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 3 (see [11]).
For two-interval Pythagorean fuzzy numbersα˜i=P˜μP˜i−x,μP˜i+x,P˜νP˜i−x,νP˜i+xi=1,2, the comparison is as follows:(1)
SupposeSα˜1=Sα˜2. If Hα˜1=Hα˜2, then α˜1=α˜2; If Hα˜1<Hα˜2, then α˜1<α˜2;(2)
SupposeSα˜1>Sα˜2. Then, α˜1>α˜2.Definition 4 (see [11]).
For two-interval Pythagorean fuzzy numbersα˜i=P˜μP˜i−x,μP˜i+x,P˜νP˜i−x,νP˜i+xi=1,2, the following operations are satisfied:(1)
α˜1⊕α˜2=μP˜1−x2+μP˜2−x2−μP˜1−x2μP˜2−x2μP˜1+x2+μP˜2+x2−μP˜1+x2μP˜2+x2,νP˜1−xνP˜2−xνP˜1+xνP˜2+x(2)
α˜1⊗α˜2=μP˜1−xμP∼2−xμP˜1+xμP˜2+x,νP˜1−x2+νP˜2−x2−νP˜1−x2νP˜2−x2νP˜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 5 (see [11]).
For two-interval Pythagorean fuzzy numbersα˜i=P˜μP˜i−x,μP˜i+x,P˜νP˜i−x,νP˜i+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.


### 2.2. GWOWA Operator

Definition 6 (see [21]).
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 7 (see [21]).
there is a mapping OWA:Rn⟶R, OWAx1,x2,x3,…,xn=∑i=1nωixσi. Where ω=ω1,ω2,ω3,…,ωnT is the position weighting vector of OWA, satisfying ωi∈0,1 and ∑i=1nωi=1. At the same time, xσ1,xσ2,xσ3,…,xσn is the position exchange of x1,x2,x3,…,xn, which satisfies the existence of xσi>xσj for any i<j. At this time, the function OWA is called the ordered weighted average operator.Definition 8 (see [21]).
there is a mapping GWA:Rn⟶R, GWAx1,x2,x3,…,xn=∑i=1nwnxnλ1/λ. Where λ>0 and w=w1,w2,w3,…,wnT are weighted vectors of GWA, satisfying wi∈0,1 and ∑i=1nwi=1. At this time, the function GWA is called the generalized weighted average operator.Definition 9 (see [21]).
there is a mapping GOWA:Rn⟶R, GOWAx1,x2,x3,…,xn=∑i=1nwnxnλ1/λ. Where λ>0 and ω=ω1,ω2,ω3,…,ωnT are the position weighting vectors of GOWA, satisfying ωi∈0,1 and ∑i=1nωi=1. At the same time, xσ1,xσ2,xσ3,…,xσn is the position exchange of x1,x2,x3,…,xn, which satisfies the existence of xσi>xσj for any i<j. At this time, the function GOWA is called the generalized ordered weighted average operator.Definition 10 (see [21]).
there is a mapping GWOWA:Rn⟶R, GWOWAx1,x2,x3,…,xn=∑i=1nhnxσnλ1/λ. Where λ∈−∞,+∞, w=w1,w2,w3,…,wnT and ω=ω1,ω2,ω3,…,ωnT are the objective weight and position weight of x1,x2,x3,…,xn, respectively, which meet the requirements of wi∈0,1 and ∑i=1nwi=1; ωi∈0,1, ∑i=1nωi=1; h=h1,h2,h3,…,hnT is the weight vector of GWOWA operator, satisfying that hi=ω∗∑j≤iwσj−ω∗∑j<iwσj, ω∗ are monotonically increasing functions passing through point i/n,∑j≤iωj and point 0,0. At the same time, xσ1,xσ2,xσ3,…,xσn is the position exchange of x1,x2,x3,…,xn, which satisfies the existence of xσi>xσj for any i<j. At this time, the function GWOWA is called the generalized weighted ordered weighted average operator.
According tohi=ω∗∑j≤iwσj−ω∗∑j<iwσj, the weight vector h=h1,h2,h3,…,hnT of GWOWA operator can be obtained, which satisfies hi∈0,1 and ∑i=1nhi=1.


## 2.1. Interval Pythagorean Fuzzy Sets

In decision-making, scholars have proposed the concept and algorithms of the 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 1 (see [11]).
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 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 2 (see [11]).
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 3 (see [11]).
For two-interval Pythagorean fuzzy numbersα˜i=P˜μP˜i−x,μP˜i+x,P˜νP˜i−x,νP˜i+xi=1,2, the comparison is as follows:(1)
SupposeSα˜1=Sα˜2. If Hα˜1=Hα˜2, then α˜1=α˜2; If Hα˜1<Hα˜2, then α˜1<α˜2;(2)
SupposeSα˜1>Sα˜2. Then, α˜1>α˜2.Definition 4 (see [11]).
For two-interval Pythagorean fuzzy numbersα˜i=P˜μP˜i−x,μP˜i+x,P˜νP˜i−x,νP˜i+xi=1,2, the following operations are satisfied:(1)
α˜1⊕α˜2=μP˜1−x2+μP˜2−x2−μP˜1−x2μP˜2−x2μP˜1+x2+μP˜2+x2−μP˜1+x2μP˜2+x2,νP˜1−xνP˜2−xνP˜1+xνP˜2+x(2)
α˜1⊗α˜2=μP˜1−xμP∼2−xμP˜1+xμP˜2+x,νP˜1−x2+νP˜2−x2−νP˜1−x2νP˜2−x2νP˜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 5 (see [11]).
For two-interval Pythagorean fuzzy numbersα˜i=P˜μP˜i−x,μP˜i+x,P˜νP˜i−x,νP˜i+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.


## 2.2. GWOWA Operator

Definition 6 (see [21]).
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 7 (see [21]).
there is a mapping OWA:Rn⟶R, OWAx1,x2,x3,…,xn=∑i=1nωixσi. Where ω=ω1,ω2,ω3,…,ωnT is the position weighting vector of OWA, satisfying ωi∈0,1 and ∑i=1nωi=1. At the same time, xσ1,xσ2,xσ3,…,xσn is the position exchange of x1,x2,x3,…,xn, which satisfies the existence of xσi>xσj for any i<j. At this time, the function OWA is called the ordered weighted average operator.Definition 8 (see [21]).
there is a mapping GWA:Rn⟶R, GWAx1,x2,x3,…,xn=∑i=1nwnxnλ1/λ. Where λ>0 and w=w1,w2,w3,…,wnT are weighted vectors of GWA, satisfying wi∈0,1 and ∑i=1nwi=1. At this time, the function GWA is called the generalized weighted average operator.Definition 9 (see [21]).
there is a mapping GOWA:Rn⟶R, GOWAx1,x2,x3,…,xn=∑i=1nwnxnλ1/λ. Where λ>0 and ω=ω1,ω2,ω3,…,ωnT are the position weighting vectors of GOWA, satisfying ωi∈0,1 and ∑i=1nωi=1. At the same time, xσ1,xσ2,xσ3,…,xσn is the position exchange of x1,x2,x3,…,xn, which satisfies the existence of xσi>xσj for any i<j. At this time, the function GOWA is called the generalized ordered weighted average operator.Definition 10 (see [21]).
there is a mapping GWOWA:Rn⟶R, GWOWAx1,x2,x3,…,xn=∑i=1nhnxσnλ1/λ. Where λ∈−∞,+∞, w=w1,w2,w3,…,wnT and ω=ω1,ω2,ω3,…,ωnT are the objective weight and position weight of x1,x2,x3,…,xn, respectively, which meet the requirements of wi∈0,1 and ∑i=1nwi=1; ωi∈0,1, ∑i=1nωi=1; h=h1,h2,h3,…,hnT is the weight vector of GWOWA operator, satisfying that hi=ω∗∑j≤iwσj−ω∗∑j<iwσj, ω∗ are monotonically increasing functions passing through point i/n,∑j≤iωj and point 0,0. At the same time, xσ1,xσ2,xσ3,…,xσn is the position exchange of x1,x2,x3,…,xn, which satisfies the existence of xσi>xσj for any i<j. At this time, the function GWOWA is called the generalized weighted ordered weighted average operator.
According tohi=ω∗∑j≤iwσj−ω∗∑j<iwσj, the weight vector h=h1,h2,h3,…,hnT of GWOWA operator can be obtained, which satisfies hi∈0,1 and ∑i=1nhi=1.


## 3. GWOWA Operator Based on Interval Pythagorean Fuzzy Numbers

Combining the advantages of GWA operator and GOWA operator, experts and scholars define the GOWA operator. GOWA operator is a generalized weighted ordered weighted average operator. Some experts and scholars have extended the WOWA operator and GOWA operator to the Pythagorean fuzzy environment. Based on the previous research of experts and scholars, this paper further extends the GOWA operator to the interval Pythagorean fuzzy environment and proposes a generalized weighted ordered weighted average (IVPFGWOWA) operator based on interval Pythagorean fuzzy numbers.Definition 11.
[21]: it is known that α˜1,α˜2,α˜3,…,α˜n is a group of interval Pythagorean fuzzy numbers, and w=w1,w2,w3,…,wnT and ω=ω1,ω2,ω3,…,ωnT are the objective weight and position weight of this group of interval Pythagorean fuzzy numbers, respectively, meeting wi∈0,1 and ∑i=1nwi=1; ωi∈0,1, ∑i=1nωi=1. Set the mapping IVPFGWOWA: Rn⟶R, then define as(1)IVPFGWOWAα˜1,α˜2,α˜3,…,α˜n=∑i=1nhnα˜σnλ1/λ,where λ∈−∞,+∞; h=h1,h2,h3,…,hnT is the weight vector of IVPFGWOWA operator, satisfying that hi=ω∗∑j≤iwσj−ω∗∑j<iwσj, ω∗ are monotonically increasing functions passing through point i/n,∑j≤iωj and point 0,0. At the same time, α˜σ1,α˜σ2,α˜σ(3),…,α˜σn is the position exchange of α˜1,α˜2,α˜3,…,α˜n, which satisfies the existence of α˜σi>α˜σj for any i<j. According to hi=ω∗∑j≤iwσj−ω∗∑j<iwσj, the weight vector h=h1,h2,h3,…,hnT of GWOWA operator can be obtained, which satisfies hi∈0,1 and ∑i=1nhi=1.
For interval Pythagorean fuzzy numberα˜i=P˜μP˜−x,μP˜+x,P˜νP˜−x,νP˜+x, wii=1,2,3,…,n is the objective weight of interval Pythagorean fuzzy number α∼i, satisfying wi∈0,1 and ∑i=1nwi=1; ω=ω1,ω2,ω3,…,ωnT is the position weight of interval Pythagorean fuzzy number α∼i, satisfying ωi∈0,1 and ∑i=1nωi=1. Then, the generalized weighted ordered weighted average operator (IVPFGWOWA) of interval Pythagorean fuzzy number α∼i can be defined as(2)IVPFGWOWAα˜1,α˜2,α˜3,…,α˜n=h1α˜σ1λ⊕h2α˜σ2λ⊕h3α˜σ3λ⊕⋯⊕hnα˜σnλ1/λ=1−∏i=1n1−μPσi∼−x2λhi1/λ,1−∏i=1n1−μPσi∼+x2λhi1/λ1−1−∏i=1nνPσi∼−xλhi21/λ,1−1−∏i=1nνPσi∼+xλhi21/λ.
The following theorem can be proved by studying the IVPFGWOWA operator:Theorem 1.
It is known thatα˜1,α˜2,α˜3,…,α˜n is a set of interval Pythagorean fuzzy numbers, α˜i=P˜μP˜−x,μP˜+x,P˜νP˜−x,νP˜+x. When the position weight ωi=1/n, equation (1) will degenerate into IVPFGWA operator.Proof.
Becauseω∗ is a monotonically increasing function passing through point i/n,∑j≤iωj and point 0,0, that is, ω∗x=n∑j≤iωj/ix. When ωi=1/n, n∑j≤iωj/i=1.
Soω∗x=n∑j≤iωj/ix=x, hi=ω∗∑j≤iwσj−ω∗∑j<iwσj=∑j≤iwσj−∑j<iwσj=wσi.
So(3)IVPFGWOWAα˜1,α˜2,α˜3,…,α˜n=1−∏i=1n1−μPσi∼−x2λwi1/λ,1−∏i=1n1−μPσi∼+x2λwi1/λ1−1−∏i=1nνPσi∼−xλwi21/λ,1−1−∏i=1nνPσi∼+xλwi21/λ=IVPFGWAα˜1,α˜2,α˜3,…,α˜n.Theorem 2.
It is known thatα˜1,α˜2,α˜3,…,α˜n is a set of interval Pythagorean fuzzy numbers, α˜i=(P˜(μP˜−x,μP˜+x),P˜νP˜−x,νP˜+x). When the position weight wi=1/n, equation (1) will degenerate into the IVPFGOWA operator.Proof.
Whenwi=1/n, hi=ω∗i/n−ω∗i−1/n.
Becauseω∗ is a monotonically increasing function through point i/n,∑j≤iωj and point 0,0.
Sohi=ω∗i/n−ω∗i−1/n=ωj.
So(4)IVPFGWOWAα˜1,α˜2,α˜3,…,α˜n=1−∏i=1n1−μPσi∼−x2λωi1/λ,1−∏i=1n1−μPσi∼+x2λωi1/λ1−1−∏i=1nνPσi∼−xλωi21/λ,1−1−∏i=1nνPσi∼+xλωi21/λ=IVPFGOWAα˜1,α˜2,α˜3,…,α˜n.Theorem 3.
After verification, whenλ=1 and ωi=1/n, IVPFGWOWA operator will degenerate into IVPFWA operator; when λ=1 and wi=1/n, IVPFGWOWA operator will degenerate into IVPFOWA operator.Theorem 4 (Idempotency).
Letα˜i=P˜μP˜−x,μP˜+x,P˜νP˜−x,νP˜+x=α˜, then IVPFGWOWAα˜1,α˜2,α˜3,…,α˜n=α˜.
Proof: whenα˜i=P˜μP˜−x,μP˜+x,P˜νP˜−x,νP˜+x=α˜, IVPFGWOWAα˜1,α˜2,α˜3,…,α˜n=∑i=1nhnα˜σnλ1/λ=∑i=1nhnα˜σλ1/λ=α˜.Theorem 5 (Monotonicity).
Suppose there is∀i and α˜Ai<α˜Bi, then there is IVPFGWOWAα˜A1,α˜A2,α˜A3,…,α˜An<IVPFGWOWAα˜B1,α˜B2,α˜B3,…,α˜Bn.Proof.
IVPFGWOWAα˜A1,α˜A2,α˜A3,…,α˜An=∑i=1nhnα˜σAnλ1/λ, IVPFGWOWAα˜B1,α˜B2,α˜B3,…,α˜Bn=∑i=1nhnα˜σBnλ1/λ.
Because there are∀i, α˜Ai<α˜Bi, so α˜σAnλ>α˜σBnλ.
SoIVPFGWOWAα˜A1,α˜A2,α˜A3,…,α˜An<IVPFGWOWAα˜B1,α˜B2,α˜B3,…,α˜Bn.Theorem 6 (Boundedness).
If there areα˜−=minα˜i, α˜+=maxα˜i, then α˜−≤IVPFGWOWAα˜1,α˜2,α˜3,…,α˜n≤α˜+.Proof.
According to Definition11, α˜−=minα˜i=α˜σn, α˜+=maxα˜i=α˜σ1.
According to the idempotency property of Theorem4, α˜σn=IVPFGWOWAα˜σn,α˜σn,α˜σn,…,α˜σn, α˜σ1=IVPFGWOWAα˜σ1,α˜σ1,α˜σ1,…,α˜σ1.
Becauseα˜−=minα˜i, α˜+=maxα˜i, α˜−≤α˜i≤α˜+.
According to the monotonicity property of Theorem5, IVPFGWOWAα˜σn,α˜σn,α˜σn,…,α˜σn≤IVPFGWOWAα˜1,α˜2,α˜3,…,α˜n≤IVPFGWOWAα˜σ1,α˜σ1,α˜σ1,…,α˜σ1.
Soα˜−≤IVPFGWOWAα˜1,α˜2,α˜3,…,α˜n≤α˜+.


## 4. Interval Fuzzy Decision Based on IVPFGWOWA Operator

### 4.1. Decision-Making Issues

Suppose there is a 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 interval Pythagorean fuzzy evaluation value matrix given by decision maker k is(5)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.


### 4.2. Multiattribute Group Decision-Making Based on IVPFGWOWA Operator

For a multiattribute group decision-making problem, this paper presents a new decision-making method and TOPSIS decision-making method based on the IVPFGWOWA operator, and the flow chart is shown in Figure1.Step 2: The objective weight is determined. TOPSIS decision-making method is used to analyze interval Pythagorean fuzzy problems. First, the positive ideal matrix(L+) and bilateral negative ideal matrix(Le−, Lf−) of interval Pythagorean fuzzy sets are defined, obtain Hamming distance dα˜ijk,L between each evaluation value and positive and negative ideal evaluation value, so as to determine the weight of each evaluation value. Then, calculate the closeness rijk of each evaluation value according to the obtained positive and negative ideal matrix evaluation value, so as to determine the weight wijk of each evaluation value.(7)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,dα˜ijk,α˜ij+=14μP˜ij−kx−μP˜ij−+x+μP˜ij+kx−μP˜ij++x+νP˜ij−kx−νP˜ij−+x+νP˜ij+kx−νP˜ij++x,dα˜ijk,α˜ije=14μP˜ij−kx−μP˜ij−ex+μP˜ij+kx−μP˜ij+ex+νP˜ij−kx−νP˜ij−ex+νP˜ij+kx−νP˜ij+ex,dα˜ijk,α˜ijf=14μP˜ij−kx−μP˜ij−fx+μP˜ij+kx−μP˜ij+fx+νP˜ij−kx−νP˜ij−fx+νP˜ij+kx−νP˜ij+fx,rijk=dα˜ijk,α˜ije+dα˜ijk,α˜ijfdα˜ijk,α˜ij++dα˜ijk,α˜ije+dα˜ijk,α˜ijf,wijk=rijk∑k=1crijk.Step 3: The comprehensive weight of the IVPFGWOWA operator is determined. According to the position weight and objective weight obtained in the first and second steps, the comprehensive weighthijk of IVPFGWOWA operator is solved.(8)hi=ω∗∑j≤iwσj−ω∗∑j<iwσj.Step 4: The synthesis matrix is assembled. According to different values, the interval Pythagorean fuzzy evaluation matrix of multiple decision-makers is aggregated into a single comprehensive evaluation matrix by using the IVPFGWOWA operator. After determining the evaluation value weighthijk of each decision-maker in the third step, through the obtained comprehensive weight and the GWOWA operator of interval Pythagorean fuzzy number (IVPFGWOWA), the evaluation matrix set of each decision-maker can be synthesized into a comprehensive decision matrix M=α˜ijm×n according to equation (2).Step 5: According to the comprehensive decision matrixM=α˜ijm×n, the TOPSIS decision method is used to determine the evaluation value weight of the interval Pythagorean fuzzy comprehensive decision matrix.Step 6: The weighted distance and closeness between the evaluation value of each scheme and the positive and negative ideal solutions are determined. Regarding the closeness coefficient of the scheme, the larger the closeness coefficient, the better the scheme.Figure 1
Flow chart of TOPSIS decision method based on IVPFGWOWA operator.


## 4.1. Decision-Making Issues

Suppose there is a 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 interval Pythagorean fuzzy evaluation value matrix given by decision maker k is(5)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.


## 4.2. Multiattribute Group Decision-Making Based on IVPFGWOWA Operator

For a multiattribute group decision-making problem, this paper presents a new decision-making method and TOPSIS decision-making method based on the IVPFGWOWA operator, and the flow chart is shown in Figure1.Step 2: The objective weight is determined. TOPSIS decision-making method is used to analyze interval Pythagorean fuzzy problems. First, the positive ideal matrix(L+) and bilateral negative ideal matrix(Le−, Lf−) of interval Pythagorean fuzzy sets are defined, obtain Hamming distance dα˜ijk,L between each evaluation value and positive and negative ideal evaluation value, so as to determine the weight of each evaluation value. Then, calculate the closeness rijk of each evaluation value according to the obtained positive and negative ideal matrix evaluation value, so as to determine the weight wijk of each evaluation value.(7)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,dα˜ijk,α˜ij+=14μP˜ij−kx−μP˜ij−+x+μP˜ij+kx−μP˜ij++x+νP˜ij−kx−νP˜ij−+x+νP˜ij+kx−νP˜ij++x,dα˜ijk,α˜ije=14μP˜ij−kx−μP˜ij−ex+μP˜ij+kx−μP˜ij+ex+νP˜ij−kx−νP˜ij−ex+νP˜ij+kx−νP˜ij+ex,dα˜ijk,α˜ijf=14μP˜ij−kx−μP˜ij−fx+μP˜ij+kx−μP˜ij+fx+νP˜ij−kx−νP˜ij−fx+νP˜ij+kx−νP˜ij+fx,rijk=dα˜ijk,α˜ije+dα˜ijk,α˜ijfdα˜ijk,α˜ij++dα˜ijk,α˜ije+dα˜ijk,α˜ijf,wijk=rijk∑k=1crijk.Step 3: The comprehensive weight of the IVPFGWOWA operator is determined. According to the position weight and objective weight obtained in the first and second steps, the comprehensive weighthijk of IVPFGWOWA operator is solved.(8)hi=ω∗∑j≤iwσj−ω∗∑j<iwσj.Step 4: The synthesis matrix is assembled. According to different values, the interval Pythagorean fuzzy evaluation matrix of multiple decision-makers is aggregated into a single comprehensive evaluation matrix by using the IVPFGWOWA operator. After determining the evaluation value weighthijk of each decision-maker in the third step, through the obtained comprehensive weight and the GWOWA operator of interval Pythagorean fuzzy number (IVPFGWOWA), the evaluation matrix set of each decision-maker can be synthesized into a comprehensive decision matrix M=α˜ijm×n according to equation (2).Step 5: According to the comprehensive decision matrixM=α˜ijm×n, the TOPSIS decision method is used to determine the evaluation value weight of the interval Pythagorean fuzzy comprehensive decision matrix.Step 6: The weighted distance and closeness between the evaluation value of each scheme and the positive and negative ideal solutions are determined. Regarding the closeness coefficient of the scheme, the larger the closeness coefficient, the better the scheme.Figure 1
Flow chart of TOPSIS decision method based on IVPFGWOWA operator.


## 5. Example Analysis

### 5.1. Evaluation Matrix

There are three serious epidemic areas (A=A1,A2,A3) needed to support COVID-19 in a certain area. Now, three experts are evaluating the epidemic situation in the heavily epidemic areas, and the expert weight vector is ω=0.2,0.5,0.3T. The evaluation attribute set includes the health status, epidemic prevention status, and medical environment of infected persons in the affected areas. Table 2 shows the evaluation matrix of the three experts.Table 2
Evaluation matrix of three experts.
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(a[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])


### 5.2. Decision-Making Process

Step 1: Assuming that the position weight vector of the attributeBi is ω=0.4,0.3,0.3T, it can be obtained according to equation (3):(9)ω∗=1.2x,0≤x≤1/3,0.9x+0.1,1/3≤x≤1.Step 2: According to the evaluation values given by all experts, the objective weightwijk of each evaluation value is obtained by the TOPSIS method, and the results are shown in Table 3.Step 3: According to the objective weight and position weight, the comprehensive weight of the IVPFGWOWA operator is obtained through equation (4), and the results are shown in Table 4.Step 4: Takeλ=1,2,3, respectively. According to the interval Pythagorean fuzzy number generalized weighted ordered weighted average operator (IVPFGWOWA), use the third step to obtain the comprehensive weight of the evaluation value of each decision-maker, and synthesize the evaluation matrix set of each decision-maker into the comprehensive decision matrix M=α˜ijm×n, and the results are shown in Table 5.Step 5: Determine the evaluation value weight of interval Pythagorean fuzzy comprehensive decision matrix by TOPSIS decision method, and the results are shown in Table6.Step 6: Determine the weighted distance and closeness between the evaluation value of each scheme and the positive and negative ideal solution, and the results are shown in Table7.It can be seen from the results in Table7 that the scheme A2 has the highest closeness, and its scheme is the best. Horizontally, the TOPSIS decision-making method based on the IVPFGWOWA operator is used in this paper, and the closeness between schemes is obviously large, so it is easier for decision-makers to choose the optimal decision-making scheme. Vertically, with the change of the value of the manual control variable λ, the advantages and disadvantages of each scheme also change slightly. However, with the increase of the value, the closeness of the optimal scheme increases significantly, and it is always the optimal scheme.Table 3
Objective weightwijk of each evaluation value.
B1B2B3wij1A10.38220.36960.3112A20.40270.32520.4054A30.33340.35200.3520wij2A10.27800.26090.3290A20.30970.34330.2703A30.29720.26400.3840wij3A10.33980.36960.3598A20.28760.33150.3243A30.36940.38400.2640Table 4
Comprehensive weighthijk of each evaluation value.
B1B2B3wij1A10.44400.33260.2234A20.46240.29270.2449A30.40010.31680.2831wij2A10.33360.25140.4150A20.37160.31610.3123A30.35660.24840.3949wij3A10.40580.33260.2615A20.34510.31210.3428A30.43250.34560.2219Table 5
Comprehensive evaluation matrix.
λB1B2B31A1([0.6143, 0.7200], [0.0308, 0.0731])([0.5992, 0.7208], [0.1613, 0.2508])([0.6246, 0.7393], [0.0981, 0.2313])A2([0.5092, 0.6131], [0.0547, 0.1115])([0.5941, 0.6959], [0.0800, 0.1800])([0.4183, 0.5152], [0.0393, 0.0939])A3([0.6423, 0.7973], [0.0231, 0.1102])([0.5961, 0.8055], [0.1361, 0.2829])([0.5751, 0.6741], [0.0751, 0.1803])2A1([0.6172, 0.7194], [0.0005, 0.0027])([0.6144, 0.7307], [0.0131, 0.0320])([0.6462, 0.7529], [0.0048, 0.0271])A2([0.4967, 0.5992], [0.0015, 0.0062])([0.6094, 0.7093], [0.0032, 0.0163])([0.4318, 0.5286], [0.0008, 0.0044])A3([0.6399, 0.7884], [0.0003, 0.0061])([0.6118, 0.8167], [0.0093, 0.0408])([0.5854, 0.6840], [0.0028, 0.0164])3A1([0.6280, 0.7269], [0.0000, 0.0001])([0.6251, 0.7367], [0.0014, 0.0053])([0.6592, 0.7612], [0.0003, 0.0041])A2([0.4935, 0.5948], [0.0001, 0.0005])([0.6205, 0.7188], [0.0002, 0.0019])([0.4405, 0.5367], [0.0000, 0.0003])A3([0.6450, 0.7865], [0.0000, 0.0004])([0.6227, 0.8243], [0.0008, 0.0076])([0.5898, 0.6885], [0.0001, 0.0020])Table 6
Evaluation value weight of comprehensive matrix.
λB1B2B31wijA10.26060.27990.4594A21.00000.00000.0000A30.36320.35940.27742wijA10.34550.17100.4835A21.00000.00000.0000A30.37630.36150.26233wijA10.38660.11520.4981A21.00000.00000.0000A30.37700.36770.2553Table 7
Weighted distance and closeness of comprehensive matrix.
λA1A2A31wijdα˜ij,α˜ij+0.01910.07930.0186wijdα˜ij,α˜ij−0.09900.01750.0869rij0.16160.81920.1765Sort3122wijdα˜ij,α˜ij+0.01240.08310.0100wijdα˜ij,α˜ij−0.07930.00120.0655rij0.13500.98580.1329Sort2133wijdα˜ij,α˜ij+0.01000.08580.0095wijdα˜ij,α˜ij−0.08230.00010.0622rij0.10860.99890.1324Sort312


### 5.3. Decision Comparison

In this paper, a generalized weighted ordered weighted average (IVPFGWOWA) operator based on interval Pythagorean fuzzy numbers is introduced, and a new TOPSIS decision-making method based on the IVPFGWOWA operator is proposed. The steps of this new decision-making method are given and verified by an example. Next, this paper uses the relevant decision-making methods studied by other scholars for example comparison, and the results are as follows:(1)
Whenλ=1 is taken, in the sixth step, the closeness of the scheme calculated by using the interval Pythagorean fuzzy geometric weighted Bonferroni average operator of Jiang and Duan [26] is (0.6588, 0.8335, 0.6731), and the ranking of the advantages and disadvantages of the scheme is (3, 1, 2). By comparison, it can be found that the results calculated by using Jiang Yingying’s interval Pythagorean fuzzy geometric weighted Bonferroni average operator are consistent with the IVPFGWOWA operator proposed in this paper. However, the difference is that there is little difference in the closeness of each scheme calculated by interval Pythagorean fuzzy geometric weighted Bonferroni average operator, while there is a great difference in the progress of each scheme calculated by IVPFGWOWA operator. It can be seen that the GWOWA operator of interval Pythagorean fuzzy numbers (IVPFGWOWA) proposed in this paper increases the gap between the advantages and disadvantages of each scheme, making it easier for decision-makers to make decisions.(2)
When using the interval Pythagorean fuzzy number WA operator (IVPFWA) of Jun et al. [27], the closeness of each scheme is calculated to be (0.1678, 0.7188, 0.1705), and the advantages and disadvantages of the calculated schemes are ranked as (3, 1, 2). It can be found that Jun uses the interval Pythagorean fuzzy number WA operator ((IVPFWA) to also increase the gap between the advantages and disadvantages of each scheme. However, compared with the GWOWA operator of interval Pythagorean fuzzy numbers (IVPFGWOWA) proposed in this paper, by changing the value, this paper can more obviously and dynamically find the changes in the advantages and disadvantages of the scheme.Through comparison, it can be found that TOPSIS multiattribute group decision-making method based on the IVPFGWOWA operator has the following two obvious advantages:(1)
Through the TOPSIS group decision-making method based on the IVPFGWOWA operator, the decision results are consistent with those of other scholars’ methods, which verifies the effectiveness of the method proposed in this paper. Secondly, the TOPSIS group decision-making method of the IVPFGWOWA operator proposed in this paper shows that there is a large gap between the advantages and disadvantages of the closeness between the various cases, which makes the decision-makers’ decision easier and verifies its scientificity.(2)
In this paper, the control ability of the decision-maker to the information aggregation is considered when studying the aggregation operator, and the artificial control variable is introduced. By changing the value of the IVPFGWOWA operator, we can more dynamically find the changes in the closeness of each scheme. With the increase of the value, the gap between the best scheme and other schemes continues to increase, and the decision-making results are obviously more accurate.


## 5.1. Evaluation Matrix

There are three serious epidemic areas (A=A1,A2,A3) needed to support COVID-19 in a certain area. Now, three experts are evaluating the epidemic situation in the heavily epidemic areas, and the expert weight vector is ω=0.2,0.5,0.3T. The evaluation attribute set includes the health status, epidemic prevention status, and medical environment of infected persons in the affected areas. Table 2 shows the evaluation matrix of the three experts.Table 2
Evaluation matrix of three experts.
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(a[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])


## 5.2. Decision-Making Process

Step 1: Assuming that the position weight vector of the attributeBi is ω=0.4,0.3,0.3T, it can be obtained according to equation (3):(9)ω∗=1.2x,0≤x≤1/3,0.9x+0.1,1/3≤x≤1.Step 2: According to the evaluation values given by all experts, the objective weightwijk of each evaluation value is obtained by the TOPSIS method, and the results are shown in Table 3.Step 3: According to the objective weight and position weight, the comprehensive weight of the IVPFGWOWA operator is obtained through equation (4), and the results are shown in Table 4.Step 4: Takeλ=1,2,3, respectively. According to the interval Pythagorean fuzzy number generalized weighted ordered weighted average operator (IVPFGWOWA), use the third step to obtain the comprehensive weight of the evaluation value of each decision-maker, and synthesize the evaluation matrix set of each decision-maker into the comprehensive decision matrix M=α˜ijm×n, and the results are shown in Table 5.Step 5: Determine the evaluation value weight of interval Pythagorean fuzzy comprehensive decision matrix by TOPSIS decision method, and the results are shown in Table6.Step 6: Determine the weighted distance and closeness between the evaluation value of each scheme and the positive and negative ideal solution, and the results are shown in Table7.It can be seen from the results in Table7 that the scheme A2 has the highest closeness, and its scheme is the best. Horizontally, the TOPSIS decision-making method based on the IVPFGWOWA operator is used in this paper, and the closeness between schemes is obviously large, so it is easier for decision-makers to choose the optimal decision-making scheme. Vertically, with the change of the value of the manual control variable λ, the advantages and disadvantages of each scheme also change slightly. However, with the increase of the value, the closeness of the optimal scheme increases significantly, and it is always the optimal scheme.Table 3
Objective weightwijk of each evaluation value.
B1B2B3wij1A10.38220.36960.3112A20.40270.32520.4054A30.33340.35200.3520wij2A10.27800.26090.3290A20.30970.34330.2703A30.29720.26400.3840wij3A10.33980.36960.3598A20.28760.33150.3243A30.36940.38400.2640Table 4
Comprehensive weighthijk of each evaluation value.
B1B2B3wij1A10.44400.33260.2234A20.46240.29270.2449A30.40010.31680.2831wij2A10.33360.25140.4150A20.37160.31610.3123A30.35660.24840.3949wij3A10.40580.33260.2615A20.34510.31210.3428A30.43250.34560.2219Table 5
Comprehensive evaluation matrix.
λB1B2B31A1([0.6143, 0.7200], [0.0308, 0.0731])([0.5992, 0.7208], [0.1613, 0.2508])([0.6246, 0.7393], [0.0981, 0.2313])A2([0.5092, 0.6131], [0.0547, 0.1115])([0.5941, 0.6959], [0.0800, 0.1800])([0.4183, 0.5152], [0.0393, 0.0939])A3([0.6423, 0.7973], [0.0231, 0.1102])([0.5961, 0.8055], [0.1361, 0.2829])([0.5751, 0.6741], [0.0751, 0.1803])2A1([0.6172, 0.7194], [0.0005, 0.0027])([0.6144, 0.7307], [0.0131, 0.0320])([0.6462, 0.7529], [0.0048, 0.0271])A2([0.4967, 0.5992], [0.0015, 0.0062])([0.6094, 0.7093], [0.0032, 0.0163])([0.4318, 0.5286], [0.0008, 0.0044])A3([0.6399, 0.7884], [0.0003, 0.0061])([0.6118, 0.8167], [0.0093, 0.0408])([0.5854, 0.6840], [0.0028, 0.0164])3A1([0.6280, 0.7269], [0.0000, 0.0001])([0.6251, 0.7367], [0.0014, 0.0053])([0.6592, 0.7612], [0.0003, 0.0041])A2([0.4935, 0.5948], [0.0001, 0.0005])([0.6205, 0.7188], [0.0002, 0.0019])([0.4405, 0.5367], [0.0000, 0.0003])A3([0.6450, 0.7865], [0.0000, 0.0004])([0.6227, 0.8243], [0.0008, 0.0076])([0.5898, 0.6885], [0.0001, 0.0020])Table 6
Evaluation value weight of comprehensive matrix.
λB1B2B31wijA10.26060.27990.4594A21.00000.00000.0000A30.36320.35940.27742wijA10.34550.17100.4835A21.00000.00000.0000A30.37630.36150.26233wijA10.38660.11520.4981A21.00000.00000.0000A30.37700.36770.2553Table 7
Weighted distance and closeness of comprehensive matrix.
λA1A2A31wijdα˜ij,α˜ij+0.01910.07930.0186wijdα˜ij,α˜ij−0.09900.01750.0869rij0.16160.81920.1765Sort3122wijdα˜ij,α˜ij+0.01240.08310.0100wijdα˜ij,α˜ij−0.07930.00120.0655rij0.13500.98580.1329Sort2133wijdα˜ij,α˜ij+0.01000.08580.0095wijdα˜ij,α˜ij−0.08230.00010.0622rij0.10860.99890.1324Sort312


## 5.3. Decision Comparison

In this paper, a generalized weighted ordered weighted average (IVPFGWOWA) operator based on interval Pythagorean fuzzy numbers is introduced, and a new TOPSIS decision-making method based on the IVPFGWOWA operator is proposed. The steps of this new decision-making method are given and verified by an example. Next, this paper uses the relevant decision-making methods studied by other scholars for example comparison, and the results are as follows:(1)
Whenλ=1 is taken, in the sixth step, the closeness of the scheme calculated by using the interval Pythagorean fuzzy geometric weighted Bonferroni average operator of Jiang and Duan [26] is (0.6588, 0.8335, 0.6731), and the ranking of the advantages and disadvantages of the scheme is (3, 1, 2). By comparison, it can be found that the results calculated by using Jiang Yingying’s interval Pythagorean fuzzy geometric weighted Bonferroni average operator are consistent with the IVPFGWOWA operator proposed in this paper. However, the difference is that there is little difference in the closeness of each scheme calculated by interval Pythagorean fuzzy geometric weighted Bonferroni average operator, while there is a great difference in the progress of each scheme calculated by IVPFGWOWA operator. It can be seen that the GWOWA operator of interval Pythagorean fuzzy numbers (IVPFGWOWA) proposed in this paper increases the gap between the advantages and disadvantages of each scheme, making it easier for decision-makers to make decisions.(2)
When using the interval Pythagorean fuzzy number WA operator (IVPFWA) of Jun et al. [27], the closeness of each scheme is calculated to be (0.1678, 0.7188, 0.1705), and the advantages and disadvantages of the calculated schemes are ranked as (3, 1, 2). It can be found that Jun uses the interval Pythagorean fuzzy number WA operator ((IVPFWA) to also increase the gap between the advantages and disadvantages of each scheme. However, compared with the GWOWA operator of interval Pythagorean fuzzy numbers (IVPFGWOWA) proposed in this paper, by changing the value, this paper can more obviously and dynamically find the changes in the advantages and disadvantages of the scheme.Through comparison, it can be found that TOPSIS multiattribute group decision-making method based on the IVPFGWOWA operator has the following two obvious advantages:(1)
Through the TOPSIS group decision-making method based on the IVPFGWOWA operator, the decision results are consistent with those of other scholars’ methods, which verifies the effectiveness of the method proposed in this paper. Secondly, the TOPSIS group decision-making method of the IVPFGWOWA operator proposed in this paper shows that there is a large gap between the advantages and disadvantages of the closeness between the various cases, which makes the decision-makers’ decision easier and verifies its scientificity.(2)
In this paper, the control ability of the decision-maker to the information aggregation is considered when studying the aggregation operator, and the artificial control variable is introduced. By changing the value of the IVPFGWOWA operator, we can more dynamically find the changes in the closeness of each scheme. With the increase of the value, the gap between the best scheme and other schemes continues to increase, and the decision-making results are obviously more accurate.


## 6. Conclusion

In this paper, a generalized weighted ordered weighted averaging (IVPFGWOWA) operator based on interval Pythagorean fuzzy numbers is introduced, the definition of IVPFGWOWA operator is given, and it is verified that interval Pythagorean fuzzy generalized weighted averaging (IVPFGWA) and interval Pythagorean fuzzy generalized ordered weighted averaging (IVPFGOWA) operators are special cases of IVPFGWOWA operators, and their idempotency, monotonicity, and boundedness are proved. Secondly, a new TOPSIS decision-making method based on the IVPFGWOWA operator is proposed, and the steps of this new decision-making method are given. Through the example verification of TOPSIS group decision-making method based on IVPFGWOWA operator and the comparison with the decision-making methods proposed by other scholars, we can intuitively find the advantages of this method: first, this method is based on interval Pythagorean fuzzy, which avoids the membership deviation of Pythagorean fuzzy numbers and alleviates the loss of information in the decision-making process to a great extent; secondly, the calculation of the comprehensive weight of this scheme comprehensively considers the position weight and objective weight. At the same time, the decision-makers are not directly given weight in the decision-making process but give full play to the attribute advantages of each decision-maker to the greatest extent; thirdly, through examples and comparison, the TOPSIS group decision-making method based on IVPFGWOWA operator proposed in this paper is consistent with the decision-making results of other scholars, but this method increases the closeness of each scheme, making the decision-maker more effective in decision-making. Fourthly, the introduction and enlargement of artificial variables make the advantages and disadvantages of each scheme more obvious, making the decision-making results more scientific and accurate.The TOPSIS group decision-making method based on the IVPFGWOWA operator proposed in this paper can be used to solve the fuzzy decision-making problems with multiple decision-makers, multiple attributes, and unknown weights of decision-makers. It can be applied to risk assessment decision-making and dangerous situation (epidemic situation and earthquake) decision-making. But there are also some limitations: (1) the position weight of the decision-maker’s evaluation value should be given; (2) this method cannot be extended to other uncertain environments for scheme evaluation and selection, such as the evaluation of key engineering characteristics of complex products [28–31]. In future research, we will focus on whether we can use some method to position the weight according to the given evaluation value, so the decision-making should be more scientific.



---

*Source: 9308502-2022-08-26.xml*