# Interval-Valued Pythagorean Hesitant Fuzzy Set and Its Application to Multiattribute Group Decision-Making

Authors: Maoyin Zhang; Tingting Zheng; Wanrong Zheng; Ligang Zhou
Journal: Complexity
 (2020)

---

## Abstract

Pythagorean hesitant fuzzy sets are widely watched because of their excellent ability to deal with uncertainty, imprecise and vague information. This paper extends Pythagorean hesitant fuzzy environments to interval-valued Pythagorean hesitant fuzzy environments and proposes the concept of interval-valued Pythagorean hesitant fuzzy set (IVPHFS), which allows the membership of each object to be a set of several pairs of possible interval-valued Pythagorean fuzzy elements. Furthermore, we develop a series of aggregation operators for interval-valued Pythagorean hesitant fuzzy information and apply them to multiattribute group decision-making (MAGDM) problems. Then, some desired operational laws and properties of IVPHFSs are studied. Especially, considering an interval-valued Pythagorean fuzzy element (IVPHFE) is formed by several pairs of interval values, this paper proposes the concepts of score function and accuracy function in the form of two interval numbers which can retain interval-valued Pythagorean fuzzy information as much as possible. Then, the relationship among these operators is discussed by comparing the interval numbers. Eventually, an illustrative example fully shows the feasibility, practicality, and effectiveness of the proposed approach.


---

## Body

## 1. Introduction

There are numerous uncertain, imprecise, and incomplete problems in the real world. Zadeh’s fuzzy set theory [1] is a successful and effective tool to solve many similar problems. The theory and its primary extensions, such as intuitionistic fuzzy sets [2], interval-valued intuitionistic fuzzy sets [3], type-2 fuzzy sets [4], interval type-2 fuzzy sets [5], and q-rung orthopair fuzzy sets [6], have been successfully applied to pattern recognition [7], medical diagnosis [8], fuzzy logic [9], decision-making [10–18], and so on.In many practical cases, it is difficult to define a membership function of a universe because people would face several possible membership degrees of one object to be chosen and hesitate about which one would be the most right one. Hesitant fuzzy set theory, introduced by Torra and Narukawa [19, 20], has provided successful results dealing with hesitant situations, which are not well managed by the previous tools [21]. It constructs possible membership degrees of an object as a set and keeps more information in real environment. Many scholars have focused on hesitant fuzzy sets and proposed diverse corresponding extensions, such as dual hesitant fuzzy sets [22], interval-valued dual fuzzy sets [23, 24], intuitionistic hesitant fuzzy sets [25], interval-valued hesitant fuzzy sets [26–28], and interval-valued intuitionistic fuzzy sets [29]. These theories have been applied to decision-making [30, 31], clustering analysis [32], and so on.As another extension of fuzzy set theory, Yager [33, 34] proposed another class of nonstandard fuzzy sets, called Pythagorean fuzzy sets. The sets are represented by pairs of two values μPx,νPx, which satisfies μP2x+νP2x≤1. Obviously, their application range is broader than that of intuitionistic fuzzy sets. These situations are more common in different real-world problems. So, Pythagorean fuzzy sets have been paid attention in a short period of time. Yager [35, 36], Zhang and Xu [37], Ren et al. [38], Liu et al. [39], and Teng et al. [39] have studied several kinds of Pythagorean fuzzy aggregation operators and applied them to decision-making problems. Furthermore, Peng and Yang [40] proposed the definition of interval-valued Pythagorean fuzzy set and Rahamn et al. [41, 42] developed group decision-making with interval-valued Pythagorean fuzzy environments. Yi et al. [43] applied it to multicriteria decision-making problems. Also, Liang et al. [44] introduced the interval-valued Pythagorean fuzzy extended Bonferroni mean operators.Recently, some scholars have tried to combine Pythagorean fuzzy sets and hesitant fuzzy sets, and all called them Pythagorean hesitant fuzzy sets (PHFSs), but their construction methods are controversial. LetA=x,ΓAx,ΨAxx∈U be a Pythagorean fuzzy set in U, where ΓAx=μAxμAx∈0,1 and ΨAx=νAxνAx∈0,1. Liu and He [45] defined that, for any x, any μAx∈ΓAx, and any νAx∈ΨAx, μAx2+νAx2≤1 holds. Khan et al. [46] defined that, for any x and for any μAx∈ΓAx, there is νAx∈ΨAx, such that μAx2+νAx2≤1. Also, for any νAx∈ΨAx, there is μAx∈ΓAx, such that μAx2+νAx2≤1. Wei et al.[47] also defined another Pythagorean fuzzy set P=x,hPxx∈U, where hPx is a set of some Pythagorean fuzzy elements in U. The three definitions have their own merits. Considering the pairing of possible membership degrees and possible nonmembership degrees, it is more conducive to the aggregation of hesitant fuzzy numbers. In this paper, we choose Wei’s definition as the definition of PHFS.From the above analysis, we can see that PHFSs are more convenient to deal with fuzzy information than hesitant fuzzy sets or Pythagorean fuzzy sets. However, for real multiattribute group decision-making (MAGDM) problems, it is difficult for decision makers to provide some exact and crisp fuzzy values to depict uncertain or insufficient alternatives because of the increasing complexity of social and economic life. The aim of this paper is to extend PHFSs to interval-valued Pythagorean hesitant fuzzy sets (IVPHFSs) and develop MAGDM approaches to interval-valued Pythagorean hesitant fuzzy environments based on newly constructed aggregation operators. In particular, since there is no one-to-one correspondence between interval numbers and real numbers, this paper directly uses interval numbers to define score functions and accuracy functions, which can preserve interval-valued Pythagorean fuzzy information as much as possible.The rest of this paper is organized as follows. In Section2, we review some basic concepts and results of hesitant fuzzy sets, interval-valued hesitant fuzzy sets, and interval-valued Pythagorean fuzzy sets. Section 3 proposes the definition of IVPHFS and discusses some basic operational laws. In particular, we propose the concepts of the score function and accuracy function which are both appeared as interval values. Some operators for aggregating interval-valued Pythagorean hesitant fuzzy information are studied and developed in section 4. Section 5 shows the application to MAGDMs in interval-valued Pythagorean hesitant fuzzy environments and illustrates the feasibility and applicability of the proposed method. Concluding remarks are made in Section 6.


## 2. Preliminaries

This section will briefly review the basic notations and results of hesitant fuzzy sets, interval-valued Pythagorean fuzzy sets.Hereinafter, without explanation, letU be a nonempty finite set, called the universe of discourse, and Dx,y be the set of all closed subintervals of the interval x,y.


### 2.1. Hesitant Fuzzy Set (HFS)

HFS was introduced by Torra and Narukawa [19] and Torra [20], which permits the membership being a set of possible values. It is very suitable for describing problems that are difficult to determine with only one membership.Definition 1 (see [20]).
A hesitant fuzzy set (HFS)E on U is described as E=x,hExx∈U, where hEx=μExμEx∈0,1 represents the set of possible membership degrees of E at x.
For convenience, Xia and Xu [30] proposed the concept of a hesitant fuzzy element (HFE), denoted by h=hEx, where h=γγ∈0,1. They gave some operations on HFEs.Definition 2 (see [19, 20, 30]).
Leth, h1, and h2 be three HFEs and λ>0, then(1)
hC=1−γγ∈h(2)
hλ=γλγ∈h(3)
λh=1−1−γλγ∈h(4)
h1⊕h2=γ1+γ2−γ1γ2γ1∈h1,γ2∈h2(5)
h1⊗h2=γ1γ2γ1∈h1,γ2∈h2(6)
h1∪h2=maxγ1,γ2γ1∈h1,γ2∈h2(7)
h1∩h2=minγ1,γ2γ1∈h1,γ2∈h2
Furthermore, to compare the HFEs, the following comparison laws are given.Definition 3 (see [30]).
For an HFEh, Sh=1/h∑γ∈hγ is called the score function of h with ⋅ denoting the cardinality here and below. For two HFEs, h1 and h2, if Sh1>Sh2, then h1 is superior to h2, denoted by h1≻h2; if Sh1=Sh2, then h1 is equivalent to h2, denoted by h1=h2.


### 2.2. Interval-Valued Hesitant Fuzzy Set (IVHFS)

Chen et al. [26, 27] generalized HFSs to interval-valued hesitant fuzzy sets (IVPHFSs), in which the membership degree of each object of the universe is denoted by several possible interval values.Definition 4 (see [26]).
An interval-valued hesitant fuzzy set (IVHFS)E˜ on U is described as E˜=x,h˜E˜xx∈U, where h˜E˜x=μ˜E˜xμ˜E˜x≜μE˜−x,μE˜+x∈D0,1 represents the set of possible membership intervals of E˜ at x.
For convenience,h˜E˜x is called an interval-valued hesitant fuzzy element (IVHFE), denoted by h˜=μ˜μ˜≜μ−,μ+. Especially, if μ−=μ+, then h˜ degenerates to an HFS.Definition 5 (see [26, 27]).
Leth˜, h˜1, and h˜2 be three IVHFEs and λ>0, then(1)
h˜C=1−μ+,1−μ−μ˜∈h˜(2)
h˜λ=μ−λ,μ+λμ˜∈h˜(3)
λh˜=1−1−μ−λ,1−1−μ+λμ˜∈h˜(4)
h˜1⊕h˜2=μ1−+μ2−−μ1−⋅μ2−,μ1++μ2+−μ1+⋅μ2+μ˜1∈h˜1,μ˜2∈h˜2(5)
h˜1⊗h˜2=μ1−⋅μ2−,μ1+⋅μ2+μ˜1∈h˜1,μ˜2∈h˜2(6)
h˜1∪h˜2=maxμ1−,μ2−,maxμ1+,μ2+μ˜1∈h˜1,μ˜2∈h˜2(7)
h˜1∩h˜2=minμ1−,μ2−,minμ1+,μ2+μ˜1∈h˜1,μ˜2∈h˜2Definition 6 (see [48]).
Leta=a−,a+ and b=b−,b+ be two interval numbers, λ>0. The interval arithmetic is defined as(1)
a+b=a−+b−,a++b+(2)
a−b=a−−b+,a+−b−(3)
λa=λa−,λa+(4)
ab=mina−b−,a−b+,a+b−,a+b+,maxa−b−,a−b+,a+b−,a+b+Definition 7 (see [49]).
Leta=a−,a+ and b=b−,b+ be two interval numbers and la=a+−a−andlb=b+−b−, then the possibility degree of a>b is defined as follows:(1)Pa≥b=max1−maxb+−a−la+lb,0,0.
The equation of the possibility degree is used to compare two interval numbers. IfPa≥b>0.5, then a is superior to b, denoted by a>b; if Pa≥b=0.5, then a is equivalent to b, denoted by a=b.
Since an IVHFE is formed by several interval numbers, the comparative analysis of IVHFEs is different from that of HFEs. Based on the possibility degree of interval numbers in [49], Chen et al. [26] gave the following comparison laws.Definition 8 (see [26]).
For an IVHFEh˜, Sh˜=1/h˜∑μ˜∈h˜μ˜ is called the score function of h˜ with ⋅ denoting the cardinality, and Sh˜ is an interval value belonging to 0,1. For two IVHFEs, h˜1 and h˜2, if PSh˜1≥Sh˜2>0.5, then h˜1 is superior to h˜2, denoted by h˜1≻h˜2.


### 2.3. Interval-Valued Pythagorean Fuzzy Set (IVPFS)

A Pythagorean fuzzy set (PFS) is introduced by Yager [33], which is characterized by a membership function and a nonmembership function, where the sum of the square of the membership degree and the nonmembership degree of x is less than or equal to 1, while an intuitionistic fuzzy set is also characterized by them, where the sum is less than or equal to 1. Obviously, PFSs are more general than intuitionistic fuzzy sets. A PFS has emerged as an effective tool to solve multiattribute decision-making problems [37].Definition 9 (see [33, 35]).
A Pythagorean fuzzy set (PFS)P on U is described as(2)P=x,μPx,νPxμP2x+νP2x≤1,μPx,νPx∈0,1,x∈U,where μPx and νPx represent the Pythagorean membership degree and the Pythagorean nonmembership degree of P at x, respectively.
Since people often find it difficult to exactly quantify their opinions facing with incomplete fuzzy decision-making problems, interval-valued fuzzy elements can provide a better solving way. Peng et al. [40] focused on interval-valued Pythagorean fuzzy sets (IVPFSs), whose ideas are similar to interval-valued intuitionistic fuzzy sets.Definition 10 (see [40]).
An interval-valued Pythagorean fuzzy set (IVPFS)P˜ on U is described as(3)P˜=x,μP˜−x,μP˜+x,νP˜−x,νP˜+xμP˜−x,μP˜+x,νP˜−x,νP˜+x∈D0,1,μP˜+x2+νP˜+x2≤1,x∈U,where μP˜−x,μP˜+x and νP˜−x,νP˜+x are the Pythagorean membership intervals and the Pythagorean nonmembership intervals of P˜ at x, respectively.
Each pairμP˜−x,μP˜+x,νP˜−x,νP˜+x is called an interval-valued Pythagorean fuzzy element (IVPFE), denoted by P˜=μ−,μ+,ν−,ν+. Obviously, for any IVPFE P˜ on U, μ−,μ+ and ν−,ν+ are both singletons, the IVPFS degenerates into a PFS.Definition 11 (see [40]).
LetP˜=μ−,μ+,ν−,ν+, P˜1=μ1−,μ1+,ν1−,ν1+, and P˜2=μ2−,μ2+,ν2−,ν2+ be three IVPFEs and λ>0. The operational laws of IVPFEs are defined as follows:(1)
P˜C=ν−,ν+,μ−,μ+(2)
P˜λ=μ−λ,μ+λ,1−1−ν−2λ,1−1−ν+2λ(3)
λP˜=1−1−μ−2λ,1−1−μ+2λ,ν−λ,ν+λ(4)
P˜1⊕P˜2=μ1−2+μ2−2−μ1−2μ2−2,μ1+2+μ2+2−μ1+2μ2+2,ν1−ν2−,ν1+ν2+(5)
P˜1⊗P˜2=μ1−μ2−,μ1+μ2+,ν1−2+ν2−2−ν1−2ν2−2,ν1+2+ν2+2−ν1+2ν2+2(6)
P˜1∪P˜2=maxμ1−,μ2−,maxμ1+,μ2+,minν1−,ν2−,minν1+,ν2+(7)
P˜1∩P˜2=minμ1−,μ2−,minμ1+,μ2+,maxν1−,ν2−,maxν1+,ν2+Definition 12 (see [40]).
For an IVPFEP˜=μ−,μ+,ν−,ν+, the score function of P˜ is defined as follows:(4)SP˜=12μ−2+μ+2−ν−2−ν+2.
The accuracy function ofP˜ is defined as follows:(5)HP˜=12μ−2+μ+2+ν−2+ν+2.
For two IVPFEsP˜1 and P˜2,(1)
IfSP˜1>SP˜2, then P˜1≻P˜2(2)
IfSP˜1=SP˜2, then(2.1)
IfHP˜1>HP˜2, then P˜1≻P˜2;(2.2)
IfHP˜1=HP˜2, then P˜1=P˜2.


## 2.1. Hesitant Fuzzy Set (HFS)

HFS was introduced by Torra and Narukawa [19] and Torra [20], which permits the membership being a set of possible values. It is very suitable for describing problems that are difficult to determine with only one membership.Definition 1 (see [20]).
A hesitant fuzzy set (HFS)E on U is described as E=x,hExx∈U, where hEx=μExμEx∈0,1 represents the set of possible membership degrees of E at x.
For convenience, Xia and Xu [30] proposed the concept of a hesitant fuzzy element (HFE), denoted by h=hEx, where h=γγ∈0,1. They gave some operations on HFEs.Definition 2 (see [19, 20, 30]).
Leth, h1, and h2 be three HFEs and λ>0, then(1)
hC=1−γγ∈h(2)
hλ=γλγ∈h(3)
λh=1−1−γλγ∈h(4)
h1⊕h2=γ1+γ2−γ1γ2γ1∈h1,γ2∈h2(5)
h1⊗h2=γ1γ2γ1∈h1,γ2∈h2(6)
h1∪h2=maxγ1,γ2γ1∈h1,γ2∈h2(7)
h1∩h2=minγ1,γ2γ1∈h1,γ2∈h2
Furthermore, to compare the HFEs, the following comparison laws are given.Definition 3 (see [30]).
For an HFEh, Sh=1/h∑γ∈hγ is called the score function of h with ⋅ denoting the cardinality here and below. For two HFEs, h1 and h2, if Sh1>Sh2, then h1 is superior to h2, denoted by h1≻h2; if Sh1=Sh2, then h1 is equivalent to h2, denoted by h1=h2.


## 2.2. Interval-Valued Hesitant Fuzzy Set (IVHFS)

Chen et al. [26, 27] generalized HFSs to interval-valued hesitant fuzzy sets (IVPHFSs), in which the membership degree of each object of the universe is denoted by several possible interval values.Definition 4 (see [26]).
An interval-valued hesitant fuzzy set (IVHFS)E˜ on U is described as E˜=x,h˜E˜xx∈U, where h˜E˜x=μ˜E˜xμ˜E˜x≜μE˜−x,μE˜+x∈D0,1 represents the set of possible membership intervals of E˜ at x.
For convenience,h˜E˜x is called an interval-valued hesitant fuzzy element (IVHFE), denoted by h˜=μ˜μ˜≜μ−,μ+. Especially, if μ−=μ+, then h˜ degenerates to an HFS.Definition 5 (see [26, 27]).
Leth˜, h˜1, and h˜2 be three IVHFEs and λ>0, then(1)
h˜C=1−μ+,1−μ−μ˜∈h˜(2)
h˜λ=μ−λ,μ+λμ˜∈h˜(3)
λh˜=1−1−μ−λ,1−1−μ+λμ˜∈h˜(4)
h˜1⊕h˜2=μ1−+μ2−−μ1−⋅μ2−,μ1++μ2+−μ1+⋅μ2+μ˜1∈h˜1,μ˜2∈h˜2(5)
h˜1⊗h˜2=μ1−⋅μ2−,μ1+⋅μ2+μ˜1∈h˜1,μ˜2∈h˜2(6)
h˜1∪h˜2=maxμ1−,μ2−,maxμ1+,μ2+μ˜1∈h˜1,μ˜2∈h˜2(7)
h˜1∩h˜2=minμ1−,μ2−,minμ1+,μ2+μ˜1∈h˜1,μ˜2∈h˜2Definition 6 (see [48]).
Leta=a−,a+ and b=b−,b+ be two interval numbers, λ>0. The interval arithmetic is defined as(1)
a+b=a−+b−,a++b+(2)
a−b=a−−b+,a+−b−(3)
λa=λa−,λa+(4)
ab=mina−b−,a−b+,a+b−,a+b+,maxa−b−,a−b+,a+b−,a+b+Definition 7 (see [49]).
Leta=a−,a+ and b=b−,b+ be two interval numbers and la=a+−a−andlb=b+−b−, then the possibility degree of a>b is defined as follows:(1)Pa≥b=max1−maxb+−a−la+lb,0,0.
The equation of the possibility degree is used to compare two interval numbers. IfPa≥b>0.5, then a is superior to b, denoted by a>b; if Pa≥b=0.5, then a is equivalent to b, denoted by a=b.
Since an IVHFE is formed by several interval numbers, the comparative analysis of IVHFEs is different from that of HFEs. Based on the possibility degree of interval numbers in [49], Chen et al. [26] gave the following comparison laws.Definition 8 (see [26]).
For an IVHFEh˜, Sh˜=1/h˜∑μ˜∈h˜μ˜ is called the score function of h˜ with ⋅ denoting the cardinality, and Sh˜ is an interval value belonging to 0,1. For two IVHFEs, h˜1 and h˜2, if PSh˜1≥Sh˜2>0.5, then h˜1 is superior to h˜2, denoted by h˜1≻h˜2.


## 2.3. Interval-Valued Pythagorean Fuzzy Set (IVPFS)

A Pythagorean fuzzy set (PFS) is introduced by Yager [33], which is characterized by a membership function and a nonmembership function, where the sum of the square of the membership degree and the nonmembership degree of x is less than or equal to 1, while an intuitionistic fuzzy set is also characterized by them, where the sum is less than or equal to 1. Obviously, PFSs are more general than intuitionistic fuzzy sets. A PFS has emerged as an effective tool to solve multiattribute decision-making problems [37].Definition 9 (see [33, 35]).
A Pythagorean fuzzy set (PFS)P on U is described as(2)P=x,μPx,νPxμP2x+νP2x≤1,μPx,νPx∈0,1,x∈U,where μPx and νPx represent the Pythagorean membership degree and the Pythagorean nonmembership degree of P at x, respectively.
Since people often find it difficult to exactly quantify their opinions facing with incomplete fuzzy decision-making problems, interval-valued fuzzy elements can provide a better solving way. Peng et al. [40] focused on interval-valued Pythagorean fuzzy sets (IVPFSs), whose ideas are similar to interval-valued intuitionistic fuzzy sets.Definition 10 (see [40]).
An interval-valued Pythagorean fuzzy set (IVPFS)P˜ on U is described as(3)P˜=x,μP˜−x,μP˜+x,νP˜−x,νP˜+xμP˜−x,μP˜+x,νP˜−x,νP˜+x∈D0,1,μP˜+x2+νP˜+x2≤1,x∈U,where μP˜−x,μP˜+x and νP˜−x,νP˜+x are the Pythagorean membership intervals and the Pythagorean nonmembership intervals of P˜ at x, respectively.
Each pairμP˜−x,μP˜+x,νP˜−x,νP˜+x is called an interval-valued Pythagorean fuzzy element (IVPFE), denoted by P˜=μ−,μ+,ν−,ν+. Obviously, for any IVPFE P˜ on U, μ−,μ+ and ν−,ν+ are both singletons, the IVPFS degenerates into a PFS.Definition 11 (see [40]).
LetP˜=μ−,μ+,ν−,ν+, P˜1=μ1−,μ1+,ν1−,ν1+, and P˜2=μ2−,μ2+,ν2−,ν2+ be three IVPFEs and λ>0. The operational laws of IVPFEs are defined as follows:(1)
P˜C=ν−,ν+,μ−,μ+(2)
P˜λ=μ−λ,μ+λ,1−1−ν−2λ,1−1−ν+2λ(3)
λP˜=1−1−μ−2λ,1−1−μ+2λ,ν−λ,ν+λ(4)
P˜1⊕P˜2=μ1−2+μ2−2−μ1−2μ2−2,μ1+2+μ2+2−μ1+2μ2+2,ν1−ν2−,ν1+ν2+(5)
P˜1⊗P˜2=μ1−μ2−,μ1+μ2+,ν1−2+ν2−2−ν1−2ν2−2,ν1+2+ν2+2−ν1+2ν2+2(6)
P˜1∪P˜2=maxμ1−,μ2−,maxμ1+,μ2+,minν1−,ν2−,minν1+,ν2+(7)
P˜1∩P˜2=minμ1−,μ2−,minμ1+,μ2+,maxν1−,ν2−,maxν1+,ν2+Definition 12 (see [40]).
For an IVPFEP˜=μ−,μ+,ν−,ν+, the score function of P˜ is defined as follows:(4)SP˜=12μ−2+μ+2−ν−2−ν+2.
The accuracy function ofP˜ is defined as follows:(5)HP˜=12μ−2+μ+2+ν−2+ν+2.
For two IVPFEsP˜1 and P˜2,(1)
IfSP˜1>SP˜2, then P˜1≻P˜2(2)
IfSP˜1=SP˜2, then(2.1)
IfHP˜1>HP˜2, then P˜1≻P˜2;(2.2)
IfHP˜1=HP˜2, then P˜1=P˜2.


## 3. Interval-Valued Pythagorean Hesitant Fuzzy Sets (IVPHFSs)

### 3.1. Interval-Valued Pythagorean Hesitant Fuzzy Elements

As mentioned earlier, in many practical problems, it is difficult for decision makers to determine precise membership degrees or nonmembership degrees, and the evaluation with relatively reasonable interval values often exists in decision-making. In order to better avoid the information loss and enhance the flexibility and applicability of the decision-making models in dealing with qualitative information, we propose the concept of interval-valued Pythagorean hesitant fuzzy set (IVPHFS).Definition 13.
An interval-valued Pythagorean hesitant fuzzy set (IVPHFS)P on U is described as(6)P=x,hPxx∈U,where(7)hPx=μ˜Px,ν˜Pxμ˜Px=μP−x,μP+x∈D0,1,ν˜Px=νP−x,νP+x∈D0,1,μP+x2+νP+x2≤1,where μ˜Px and ν˜Px are the possible Pythagorean membership intervals and the possible Pythagorean nonmembership intervals of P at x, respectively. The set of all IVPHFEs on U is denoted by Ω.
Obviously, for eachx∈U, if hPx includes only one pair of intervals, the IVPHFS degenerates into an IVPFS; if both μ˜Px and ν˜Px degenerate one singleton, the IVPHFS can be seen as a PHFS; if ν˜Px=0,0, the IVPHFS can be seen as an IVHFS; if μP+x+νP+x≤1, the IVPHFS can be seen as an interval-valued intuitionistic hesitant fuzzy set [29].
For convenience, we call each pairP˜=hPx as an interval-valued Pythagorean hesitant fuzzy element (IVPHFE), where P˜=μ˜,ν˜μ˜=μ−,μ+,ν˜=ν−,ν+.
Based on the operators of IVHFEs [27] and IVPFEs [40], the operational laws of IVPHFEs are defined as follows.Definition 14.
LetP˜=μ˜,ν˜μ˜=μ−,μ+,ν˜=ν−,ν+,P˜1=μ˜1,ν˜1μ˜1=μ1−,μ1+,ν˜1=ν1−,ν1+,andP˜2=μ˜2,ν˜2μ˜2=μ2−,μ2+,ν˜2=ν2−,ν2+ be three IVPHFEs and λ>0. The operational laws of IVPHFEs are defined as follows:(1)
P˜C=ν−,ν+,μ−,μ+μ˜,ν˜∈P˜(2)
P˜λ=μ−λ,μ+λ,1−1−ν−2λ,1−1−ν+2λμ˜,ν˜∈P˜(3)
λP˜=1−1−μ−2λ,1−1−μ+2λ,ν−λ,ν+λμ˜,ν˜∈P˜(4)
P˜1⊕P˜2=μ1−2+μ2−2−μ1−2μ2−2,μ1+2+μ2+2−μ1+2μ2+2,ν1−ν2−,ν1+ν2+μ˜i,ν˜i∈P˜i,i=1,2(5)
P˜1⊗P˜2=μ1−μ2−,μ1+μ2+ν1−2+ν2−2−ν1−2ν2−2,ν1+2+ν2+2−ν1+2ν2+2,μ˜i,ν˜i∈P˜i,i=1,2(6)
P˜1∪P˜2=maxμ1−,μ2−,maxμ1+,μ2+,minν1−,ν2−,minν1+,ν2+μ˜i,ν˜i∈P˜i,i=1,2(7)
P˜1∩P˜2=minμ1−,μ2−,minμ1+,μ2+,maxν1−,ν2−,maxν1+,ν2+μ˜i,ν˜i∈P˜i,i=1,2Proposition 1.
LetP˜,P˜1,andP˜2 be three IVPHFEs and λ>0, then P˜C, P˜λ, λP˜, P˜1⊕P˜2, P˜1⊗P˜2, P˜1∪P˜2, and P˜1∩P˜2 are all IVPHFESs.Proof.
Obviously,P˜C is an IVPHFE.
For anyμ˜,ν˜∈P˜, since μ+2+ν+2≤1,(8)μ+λ2+1−1−ν+2λ2=μ+2λ+1−1−ν+2λ≤μ+2λ+1−μ+2λ=1.
So,P˜λ is an IVPHFE. Similarly, λP˜ is also an IVPHFE.
As forP˜1⊕P˜2,(9)μ1+2+μ2+2−μ1+2μ2+2+ν1+2ν2+2≤μ1+2+μ2+2−μ1+2μ2+2+1−μ1+21−ν2+2=μ1+2+μ2+2−μ1+2μ2+2+1−μ1+2−μ2+2+μ1+2μ2+2=1.
So,P˜1⊕P˜2 is an IVPHFE. Similarly, P˜1⊗P˜2 is also an IVPHFE.
At last, we prove the last two claims. Assumeμ1+<μ2+.
Then,maxμ1+,μ2+2=μ2+2≤1−ν2+2≤1−minν1+,ν2+2. So, P˜1∪P˜2 is an IVPHFE. Similarly, P˜1∩P˜2 is also an IVPHFE. All the claims are proved.Proposition 2.
LetP˜,P˜1,P˜2, and P˜3 be four IVPHFEs and λ,λ1,λ2>0, then(1)
P˜1⊕P˜2=P˜2⊕P˜1 and P˜1⊗P˜2=P˜2⊗P˜1(2)
λP˜1⊕λP˜2=λP˜1⊕P˜2 and P˜1λ⊗P˜2λ=P˜1⊗P˜2λ(3)
λ1P˜⊕λ2P˜=λ1+λ2P˜ and P˜λ1⊗P˜λ2=P˜λ1+λ2(4)
P˜Cλ=λP˜C and λP˜C=P˜λC(5)
P˜1C⊕P˜2C=P˜1⊗P˜2C and P˜1C⊗P˜2C=P˜1⊕P˜2C(6)
P˜1⊕P˜2⊕P˜3=P˜1⊕P˜2⊕P˜3 and P˜1⊗P˜2⊗P˜3=P˜1⊗P˜2⊗P˜3Proof.
(1) Based on Definition14, claim (1) is obvious, so here the proof process is overleaped.(10)λP˜1⊕λP˜2=1−1−μ1−2λ,1−1−μ1+2λ,ν1−λ,ν1+λμ˜1,ν˜1∈P˜1⊕1−1−μ2−2λ,1−1−μ2+2λ,ν2−λ,ν2+λμ˜2,ν˜2∈P˜2=1−1−μ1−2λ1−μ2−2λ,1−1−μ1+2λ1−μ2+2λ,ν1−ν2−λ,ν1+ν2+λμ˜i,ν˜i∈P˜i,i=1,2,λP˜1⊕P˜2=λμ1−2+μ2−2−μ1−2μ2−2,μ1+2+μ2+2−μ1+2μ2+2,ν1−ν2−,ν1+ν2+μ˜i,ν˜i∈P˜i,i=1,2=λ1−1−μ1−21−μ2−2,1−1−μ1+21−μ2+2,ν1−ν2−,ν1+ν2+μ˜i,ν˜i∈P˜i,i=1,2=1−1−μ1−2λ1−μ2−2λ,1−1−μ1+2λ1−μ2+2λ,ν1−ν2−λ,ν1+ν2+λμ˜i,ν˜i∈P˜i,i=1,2.
So,λP˜1⊕λP˜2=λP˜1⊕P˜2 holds. Similarly, we have P˜1λ⊗P˜2λ=P˜1⊗P˜2λ.(11)λ1P˜⊕λ2P˜=1−1−μ−2λ1,1−1−μ+2λ1,ν−λ1,ν+λ1μ˜,ν˜∈P˜⊕1−1−μ−2λ2,1−1−μ+2λ2,ν−λ2,ν+λ2μ˜,ν˜∈P˜=1−1−μ−2λ1+λ2,1−1−μ+2λ1+λ2,ν−λ1+λ2,ν+λ1+λ2μ˜,ν˜∈P˜=λ1+λ2P˜.
Similarly, we haveP˜λ1⊗P˜λ2=P˜λ1+λ2.(12)P˜Cλ=ν−λ,ν+λ,1−1−μ−2λ,1−1−μ+2λμ˜,ν˜∈P˜=1−1−μ−2λ,1−1−μ+2λ,ν−λ,ν+λμ˜,ν˜∈P˜=λP˜C.
Similarly, we haveλP˜C=P˜λC.(13)P˜1C⊕P˜2C=ν1−,ν1+,μ1−,μ1+μ˜1,ν˜1∈P˜1⊕ν2−,ν2+,μ2−,μ2+μ˜2,ν˜2∈P˜2=ν1−2+ν2−2−ν1−2ν2−2,ν1+2+ν2+2−ν1+2ν2+2,μ1−μ2−,μ1+μ2+μ˜i,ν˜i∈P˜i,i=1,2=μ1−μ2−,μ1+μ2+,ν1−2+ν2−2−ν1−2ν2−2,ν1+2+ν2+2−ν1+2ν2+2μ˜i,ν˜i∈P˜i,i=1,2=P˜1⊗P˜2C.
Similarly,P˜1C⊗P˜2C=P˜1⊕P˜2C holds.(14)P˜1⊕P˜2⊕P˜3=μ1−2+μ2−2−μ1−2μ2−2,μ1+2+μ2+2−μ1+2μ2+2,ν1−ν2−,ν1+ν2+μ˜1,ν˜1∈P˜1,μ˜2,ν˜2∈P˜2⊕μ3−,μ3+,ν3−,ν3+μ˜3,ν˜3∈P˜3=μ1−2+μ2−2+μ3−2−μ1−2μ2−2−μ1−2μ3−2−μ2−2μ3−2+μ1−2μ2−2μ3−2,μ1+2+μ2+2+μ3+2−μ1+2μ2+2−μ1+2μ3+2−μ2+2μ3+2+μ1+2μ2+2μ3+2,ν1−ν2−ν3−,ν1+ν2+ν3+μ˜i,ν˜i∈P˜i,i=1,2,3=μ1−,μ1+,ν1−,ν1+μ˜1,ν˜1∈P˜1⊕μ2−2+μ3−2−μ2−2μ3−2,μ2+2+μ3+2−μ2+2μ3+2,ν2−ν3−,ν2+ν3+μ˜i,ν˜i∈P˜i,i=2,3=P˜1⊕P˜2⊕P˜3.
Similarly, we haveP˜1⊗P˜2⊗P˜3=P˜1⊗P˜2⊗P˜3.
All the claims of the proposition are proved.Proposition 3.
LetP˜,P˜1,P˜2, and P˜3 be four IVPHFEs and λ>0, then(1)
P˜1∪P˜2=P˜2∪P˜1 and P˜1∩P˜2=P˜2∩P˜1(2)
λP˜1∪λP˜2=λP˜1∪P˜2 and P˜1λ∩P˜2λ=P˜1∩P˜2λ(3)
P˜1C∪P˜2C=P˜1∩P˜2C and P˜1C∩P˜2C=P˜1∪P˜2C(4)
P˜1∪P˜2∪P˜3=P˜1∪P˜2∩P˜3 and P˜1∩P˜2∩P˜3=P˜1∩P˜2∩P˜3(5)
P˜1∪P˜2∩P˜2=P˜2 and P˜1∩P˜2∪P˜2=P˜2Proof.
They are trial. We omit them.


### 3.2. Score Function of IVPHFEs

To determine the priorities of the alternatives of an interval-valued Pythagorean hesitant fuzzy group decision-making problem, we need the concept of score functions for IVPHFEs. Since an IVPHFE includes several pairs formed by possible Pythagorean membership intervals and possible Pythagorean nonmembership intervals, if we use a method similar to Definition12, the intervals are represented by the average values of the intervals, which must lose some information because there is no one-to-one correspondence between an interval number and a value.To facilitate comparison of IVPHFEs, we shall give the following comparison laws.Definition 15.
LetP˜=μ˜,ν˜μ˜=μ−,μ+,ν˜=ν−,ν+ be an IVPHFE. The score function SP˜ is described as(15)SP˜=12P˜∑μ˜,ν˜∈P˜μ˜2−ν˜2=12P˜∑μ˜,ν˜∈P˜μ−2−ν+2,12P˜∑μ˜,ν˜∈P˜μ+2−ν−2.
The accuracy functionHP˜ is described as(16)HP˜=12P˜∑μ˜,ν˜∈P˜μ˜2+ν˜2=12P˜∑μ˜,ν˜∈P˜μ−2+ν−2,12P˜∑μ˜,ν˜∈P˜μ+2+ν+2.
The idea of the above concepts originated from Definition8 and used the definition of interval arithmetic in Definition 6. The score function of an IVPHFE is the mean of the difference between possible Pythagorean membership intervals and possible Pythagorean nonmembership intervals; also, the accuracy function reflects the overall accuracy degree of an IVPHFE. For keeping fuzzy information as much as possible, both the two functions are represented by interval values. For an IVPHFE P˜, the score function SP˜∈−1,1 and the accuracy function HP˜∈0,1 hold obviously. Based on Definition 7, we have the following definition.Definition 16.
LetP˜1andP˜2 be two IVPHFEs.(1)
IfPSP˜1>SP˜2<0.5, then we say P˜1≺P˜2(2)
IfPSP˜1>SP˜2=0.5, then(2.1)
IfPHP˜1>HP˜2<0.5, we say P˜1≺P˜2(2.2)
IfPHP˜1>HP˜2=0.5, we say P˜1=P˜2Proposition 4.
LetP˜1=μ˜1j,ν˜1jμ˜1j=μ1j−,μ1j+,ν˜1j=ν1j−,ν1j+,j=1,2,⋯,m and P˜2=μ˜2j,ν˜2jμ˜2j=μ2j−,μ2j+,ν˜2j=ν2j−,ν2j+,j=1,2,⋯,m be two IVPHFEs. If for any j=1,⋯,m, μ1j−≤μ2j−, μ1j+≤μ2j+, ν1j−≥ν2j−, and ν1j+≥ν2j+, then P˜1≼P˜2.Proof.
Based on Definition15, we have(17)SP˜1=12m∑j=1mμ1j−2−ν1j+2,μ1j+2−ν1j−2,SP˜2=12m∑j=1mμ2j−2−ν2j+2,μ2j+2−ν2j−2.
Suppose(18)L≜∑j=1mμ2j+2−ν2j−2+ν1j+2−μ1j−2∑j=1mμ1j+2−ν1j−2+ν1j+2−μ1j−2+μ2j+2−ν2j−2+ν2j+2−μ2j−2.
Hence, we obtainPSP˜1>SP˜2=max1−maxL,0,0 by Definition 7.
For anyj=1,⋯,m, μ1j−≤μ2j−, μ1j+≤μ2j+, ν1j−≥ν2j−, and ν1j+≥ν2j+, μ2j+2−μ1j−2+ν1j+2−ν2j−2≥μ1j+2−μ2j−2+ν2j+2−ν1j−2≥0; then, 0.5≤L≤1. So, 0≤1−maxL,0=1−L≤0.5. 'Hence, PSP˜1>SP˜2=1−L≤0.5. That means P˜1≼P˜2 holds.


## 3.1. Interval-Valued Pythagorean Hesitant Fuzzy Elements

As mentioned earlier, in many practical problems, it is difficult for decision makers to determine precise membership degrees or nonmembership degrees, and the evaluation with relatively reasonable interval values often exists in decision-making. In order to better avoid the information loss and enhance the flexibility and applicability of the decision-making models in dealing with qualitative information, we propose the concept of interval-valued Pythagorean hesitant fuzzy set (IVPHFS).Definition 13.
An interval-valued Pythagorean hesitant fuzzy set (IVPHFS)P on U is described as(6)P=x,hPxx∈U,where(7)hPx=μ˜Px,ν˜Pxμ˜Px=μP−x,μP+x∈D0,1,ν˜Px=νP−x,νP+x∈D0,1,μP+x2+νP+x2≤1,where μ˜Px and ν˜Px are the possible Pythagorean membership intervals and the possible Pythagorean nonmembership intervals of P at x, respectively. The set of all IVPHFEs on U is denoted by Ω.
Obviously, for eachx∈U, if hPx includes only one pair of intervals, the IVPHFS degenerates into an IVPFS; if both μ˜Px and ν˜Px degenerate one singleton, the IVPHFS can be seen as a PHFS; if ν˜Px=0,0, the IVPHFS can be seen as an IVHFS; if μP+x+νP+x≤1, the IVPHFS can be seen as an interval-valued intuitionistic hesitant fuzzy set [29].
For convenience, we call each pairP˜=hPx as an interval-valued Pythagorean hesitant fuzzy element (IVPHFE), where P˜=μ˜,ν˜μ˜=μ−,μ+,ν˜=ν−,ν+.
Based on the operators of IVHFEs [27] and IVPFEs [40], the operational laws of IVPHFEs are defined as follows.Definition 14.
LetP˜=μ˜,ν˜μ˜=μ−,μ+,ν˜=ν−,ν+,P˜1=μ˜1,ν˜1μ˜1=μ1−,μ1+,ν˜1=ν1−,ν1+,andP˜2=μ˜2,ν˜2μ˜2=μ2−,μ2+,ν˜2=ν2−,ν2+ be three IVPHFEs and λ>0. The operational laws of IVPHFEs are defined as follows:(1)
P˜C=ν−,ν+,μ−,μ+μ˜,ν˜∈P˜(2)
P˜λ=μ−λ,μ+λ,1−1−ν−2λ,1−1−ν+2λμ˜,ν˜∈P˜(3)
λP˜=1−1−μ−2λ,1−1−μ+2λ,ν−λ,ν+λμ˜,ν˜∈P˜(4)
P˜1⊕P˜2=μ1−2+μ2−2−μ1−2μ2−2,μ1+2+μ2+2−μ1+2μ2+2,ν1−ν2−,ν1+ν2+μ˜i,ν˜i∈P˜i,i=1,2(5)
P˜1⊗P˜2=μ1−μ2−,μ1+μ2+ν1−2+ν2−2−ν1−2ν2−2,ν1+2+ν2+2−ν1+2ν2+2,μ˜i,ν˜i∈P˜i,i=1,2(6)
P˜1∪P˜2=maxμ1−,μ2−,maxμ1+,μ2+,minν1−,ν2−,minν1+,ν2+μ˜i,ν˜i∈P˜i,i=1,2(7)
P˜1∩P˜2=minμ1−,μ2−,minμ1+,μ2+,maxν1−,ν2−,maxν1+,ν2+μ˜i,ν˜i∈P˜i,i=1,2Proposition 1.
LetP˜,P˜1,andP˜2 be three IVPHFEs and λ>0, then P˜C, P˜λ, λP˜, P˜1⊕P˜2, P˜1⊗P˜2, P˜1∪P˜2, and P˜1∩P˜2 are all IVPHFESs.Proof.
Obviously,P˜C is an IVPHFE.
For anyμ˜,ν˜∈P˜, since μ+2+ν+2≤1,(8)μ+λ2+1−1−ν+2λ2=μ+2λ+1−1−ν+2λ≤μ+2λ+1−μ+2λ=1.
So,P˜λ is an IVPHFE. Similarly, λP˜ is also an IVPHFE.
As forP˜1⊕P˜2,(9)μ1+2+μ2+2−μ1+2μ2+2+ν1+2ν2+2≤μ1+2+μ2+2−μ1+2μ2+2+1−μ1+21−ν2+2=μ1+2+μ2+2−μ1+2μ2+2+1−μ1+2−μ2+2+μ1+2μ2+2=1.
So,P˜1⊕P˜2 is an IVPHFE. Similarly, P˜1⊗P˜2 is also an IVPHFE.
At last, we prove the last two claims. Assumeμ1+<μ2+.
Then,maxμ1+,μ2+2=μ2+2≤1−ν2+2≤1−minν1+,ν2+2. So, P˜1∪P˜2 is an IVPHFE. Similarly, P˜1∩P˜2 is also an IVPHFE. All the claims are proved.Proposition 2.
LetP˜,P˜1,P˜2, and P˜3 be four IVPHFEs and λ,λ1,λ2>0, then(1)
P˜1⊕P˜2=P˜2⊕P˜1 and P˜1⊗P˜2=P˜2⊗P˜1(2)
λP˜1⊕λP˜2=λP˜1⊕P˜2 and P˜1λ⊗P˜2λ=P˜1⊗P˜2λ(3)
λ1P˜⊕λ2P˜=λ1+λ2P˜ and P˜λ1⊗P˜λ2=P˜λ1+λ2(4)
P˜Cλ=λP˜C and λP˜C=P˜λC(5)
P˜1C⊕P˜2C=P˜1⊗P˜2C and P˜1C⊗P˜2C=P˜1⊕P˜2C(6)
P˜1⊕P˜2⊕P˜3=P˜1⊕P˜2⊕P˜3 and P˜1⊗P˜2⊗P˜3=P˜1⊗P˜2⊗P˜3Proof.
(1) Based on Definition14, claim (1) is obvious, so here the proof process is overleaped.(10)λP˜1⊕λP˜2=1−1−μ1−2λ,1−1−μ1+2λ,ν1−λ,ν1+λμ˜1,ν˜1∈P˜1⊕1−1−μ2−2λ,1−1−μ2+2λ,ν2−λ,ν2+λμ˜2,ν˜2∈P˜2=1−1−μ1−2λ1−μ2−2λ,1−1−μ1+2λ1−μ2+2λ,ν1−ν2−λ,ν1+ν2+λμ˜i,ν˜i∈P˜i,i=1,2,λP˜1⊕P˜2=λμ1−2+μ2−2−μ1−2μ2−2,μ1+2+μ2+2−μ1+2μ2+2,ν1−ν2−,ν1+ν2+μ˜i,ν˜i∈P˜i,i=1,2=λ1−1−μ1−21−μ2−2,1−1−μ1+21−μ2+2,ν1−ν2−,ν1+ν2+μ˜i,ν˜i∈P˜i,i=1,2=1−1−μ1−2λ1−μ2−2λ,1−1−μ1+2λ1−μ2+2λ,ν1−ν2−λ,ν1+ν2+λμ˜i,ν˜i∈P˜i,i=1,2.
So,λP˜1⊕λP˜2=λP˜1⊕P˜2 holds. Similarly, we have P˜1λ⊗P˜2λ=P˜1⊗P˜2λ.(11)λ1P˜⊕λ2P˜=1−1−μ−2λ1,1−1−μ+2λ1,ν−λ1,ν+λ1μ˜,ν˜∈P˜⊕1−1−μ−2λ2,1−1−μ+2λ2,ν−λ2,ν+λ2μ˜,ν˜∈P˜=1−1−μ−2λ1+λ2,1−1−μ+2λ1+λ2,ν−λ1+λ2,ν+λ1+λ2μ˜,ν˜∈P˜=λ1+λ2P˜.
Similarly, we haveP˜λ1⊗P˜λ2=P˜λ1+λ2.(12)P˜Cλ=ν−λ,ν+λ,1−1−μ−2λ,1−1−μ+2λμ˜,ν˜∈P˜=1−1−μ−2λ,1−1−μ+2λ,ν−λ,ν+λμ˜,ν˜∈P˜=λP˜C.
Similarly, we haveλP˜C=P˜λC.(13)P˜1C⊕P˜2C=ν1−,ν1+,μ1−,μ1+μ˜1,ν˜1∈P˜1⊕ν2−,ν2+,μ2−,μ2+μ˜2,ν˜2∈P˜2=ν1−2+ν2−2−ν1−2ν2−2,ν1+2+ν2+2−ν1+2ν2+2,μ1−μ2−,μ1+μ2+μ˜i,ν˜i∈P˜i,i=1,2=μ1−μ2−,μ1+μ2+,ν1−2+ν2−2−ν1−2ν2−2,ν1+2+ν2+2−ν1+2ν2+2μ˜i,ν˜i∈P˜i,i=1,2=P˜1⊗P˜2C.
Similarly,P˜1C⊗P˜2C=P˜1⊕P˜2C holds.(14)P˜1⊕P˜2⊕P˜3=μ1−2+μ2−2−μ1−2μ2−2,μ1+2+μ2+2−μ1+2μ2+2,ν1−ν2−,ν1+ν2+μ˜1,ν˜1∈P˜1,μ˜2,ν˜2∈P˜2⊕μ3−,μ3+,ν3−,ν3+μ˜3,ν˜3∈P˜3=μ1−2+μ2−2+μ3−2−μ1−2μ2−2−μ1−2μ3−2−μ2−2μ3−2+μ1−2μ2−2μ3−2,μ1+2+μ2+2+μ3+2−μ1+2μ2+2−μ1+2μ3+2−μ2+2μ3+2+μ1+2μ2+2μ3+2,ν1−ν2−ν3−,ν1+ν2+ν3+μ˜i,ν˜i∈P˜i,i=1,2,3=μ1−,μ1+,ν1−,ν1+μ˜1,ν˜1∈P˜1⊕μ2−2+μ3−2−μ2−2μ3−2,μ2+2+μ3+2−μ2+2μ3+2,ν2−ν3−,ν2+ν3+μ˜i,ν˜i∈P˜i,i=2,3=P˜1⊕P˜2⊕P˜3.
Similarly, we haveP˜1⊗P˜2⊗P˜3=P˜1⊗P˜2⊗P˜3.
All the claims of the proposition are proved.Proposition 3.
LetP˜,P˜1,P˜2, and P˜3 be four IVPHFEs and λ>0, then(1)
P˜1∪P˜2=P˜2∪P˜1 and P˜1∩P˜2=P˜2∩P˜1(2)
λP˜1∪λP˜2=λP˜1∪P˜2 and P˜1λ∩P˜2λ=P˜1∩P˜2λ(3)
P˜1C∪P˜2C=P˜1∩P˜2C and P˜1C∩P˜2C=P˜1∪P˜2C(4)
P˜1∪P˜2∪P˜3=P˜1∪P˜2∩P˜3 and P˜1∩P˜2∩P˜3=P˜1∩P˜2∩P˜3(5)
P˜1∪P˜2∩P˜2=P˜2 and P˜1∩P˜2∪P˜2=P˜2Proof.
They are trial. We omit them.


## 3.2. Score Function of IVPHFEs

To determine the priorities of the alternatives of an interval-valued Pythagorean hesitant fuzzy group decision-making problem, we need the concept of score functions for IVPHFEs. Since an IVPHFE includes several pairs formed by possible Pythagorean membership intervals and possible Pythagorean nonmembership intervals, if we use a method similar to Definition12, the intervals are represented by the average values of the intervals, which must lose some information because there is no one-to-one correspondence between an interval number and a value.To facilitate comparison of IVPHFEs, we shall give the following comparison laws.Definition 15.
LetP˜=μ˜,ν˜μ˜=μ−,μ+,ν˜=ν−,ν+ be an IVPHFE. The score function SP˜ is described as(15)SP˜=12P˜∑μ˜,ν˜∈P˜μ˜2−ν˜2=12P˜∑μ˜,ν˜∈P˜μ−2−ν+2,12P˜∑μ˜,ν˜∈P˜μ+2−ν−2.
The accuracy functionHP˜ is described as(16)HP˜=12P˜∑μ˜,ν˜∈P˜μ˜2+ν˜2=12P˜∑μ˜,ν˜∈P˜μ−2+ν−2,12P˜∑μ˜,ν˜∈P˜μ+2+ν+2.
The idea of the above concepts originated from Definition8 and used the definition of interval arithmetic in Definition 6. The score function of an IVPHFE is the mean of the difference between possible Pythagorean membership intervals and possible Pythagorean nonmembership intervals; also, the accuracy function reflects the overall accuracy degree of an IVPHFE. For keeping fuzzy information as much as possible, both the two functions are represented by interval values. For an IVPHFE P˜, the score function SP˜∈−1,1 and the accuracy function HP˜∈0,1 hold obviously. Based on Definition 7, we have the following definition.Definition 16.
LetP˜1andP˜2 be two IVPHFEs.(1)
IfPSP˜1>SP˜2<0.5, then we say P˜1≺P˜2(2)
IfPSP˜1>SP˜2=0.5, then(2.1)
IfPHP˜1>HP˜2<0.5, we say P˜1≺P˜2(2.2)
IfPHP˜1>HP˜2=0.5, we say P˜1=P˜2Proposition 4.
LetP˜1=μ˜1j,ν˜1jμ˜1j=μ1j−,μ1j+,ν˜1j=ν1j−,ν1j+,j=1,2,⋯,m and P˜2=μ˜2j,ν˜2jμ˜2j=μ2j−,μ2j+,ν˜2j=ν2j−,ν2j+,j=1,2,⋯,m be two IVPHFEs. If for any j=1,⋯,m, μ1j−≤μ2j−, μ1j+≤μ2j+, ν1j−≥ν2j−, and ν1j+≥ν2j+, then P˜1≼P˜2.Proof.
Based on Definition15, we have(17)SP˜1=12m∑j=1mμ1j−2−ν1j+2,μ1j+2−ν1j−2,SP˜2=12m∑j=1mμ2j−2−ν2j+2,μ2j+2−ν2j−2.
Suppose(18)L≜∑j=1mμ2j+2−ν2j−2+ν1j+2−μ1j−2∑j=1mμ1j+2−ν1j−2+ν1j+2−μ1j−2+μ2j+2−ν2j−2+ν2j+2−μ2j−2.
Hence, we obtainPSP˜1>SP˜2=max1−maxL,0,0 by Definition 7.
For anyj=1,⋯,m, μ1j−≤μ2j−, μ1j+≤μ2j+, ν1j−≥ν2j−, and ν1j+≥ν2j+, μ2j+2−μ1j−2+ν1j+2−ν2j−2≥μ1j+2−μ2j−2+ν2j+2−ν1j−2≥0; then, 0.5≤L≤1. So, 0≤1−maxL,0=1−L≤0.5. 'Hence, PSP˜1>SP˜2=1−L≤0.5. That means P˜1≼P˜2 holds.


## 4. Aggregation Operators for Interval-Valued Pythagorean Hesitant Fuzzy Information

In multiattribute decision-making problems, the selection of aggregation operators is a basis problem, which is also important in the interval-valued Pythagorean hesitant fuzzy environment. Considering an IVPHFE is regarded as the extension of an IVHFE, IVPFE, or PHFE, we propose a series of aggregation operators for interval-valued Pythagorean hesitant fuzzy information in this section based on the discussion of aggregation operators in [27, 45, 50, 51] and deduce some desirable properties.


### 4.1. The IVPHFWA, IVPHFWG, GIVPHFWA, and GIVPHFWG Operators

Definition 17.
LetP˜ii=1,2,⋯,n be a collection of some IVPHFEs and ω=ω1,ω2,⋯,ωnT be the weight vector of P˜ii=1,2,⋯,n, with ωi∈0,1, ∑i=1nωi=1, and λ>0.(1)
An interval-valued Pythagorean hesitant fuzzy weighted averaging (IVPHFWA) operator can be seen a mapIVPHFWA:Ωn⟶Ω, such that(19)IVPHFWAP˜1,⋯,P˜n=ω1P˜1⊕⋯⊕ωnP˜n=1−∏i=1n1−μi−2ωi,1−∏i=1n1−μi+2ωi,∏i=1nνi−ωi,∏i=1nνi+ωiμ˜i,ν˜i∈P˜i,i=1,⋯,n.(2)
An interval-valued Pythagorean hesitant fuzzy weighted geometric (IVPHFWG) operator can be seen a mapIVPHFWG:Ωn⟶Ω, such that(20)IVPHFWGP˜1,⋯,P˜n=P˜1ω1⊗⋯⊗P˜nωn=∏i=1nμi−ωi,∏i=1nμi+ωi,1−∏i=1n1−νi−2ωi,1−∏i=1n1−νi+2ωiμ˜i,ν˜i∈P˜i,i=1,⋯,n.(3)
A generalized interval-valued Pythagorean hesitant fuzzy weighted averaging (GIVPHFWA) operator can be seen a mapGIVPHFWA:Ωn⟶Ω, which satisfies(21)GIVPHFWAλP˜1,⋯,P˜n=ω1P˜1λ⊕⋯⊕ωnP˜nλ1/λ=1−∏i=1n1−μi−2λωi1/λ,1−∏i=1n1−μi+2λωi1/λ,1−1−∏i=1n1−1−νi−2λωi1/λ,1−1−∏i=1n1−1−νi+2λωi1/λμ˜i,ν˜i∈P˜i,i=1,⋯,n,where λ>0.(4)
A generalized interval-valued Pythagorean hesitant fuzzy weighted geometric (GIVPHFWG) operator can be seen a mapGIVPHFWG:Ωn⟶Ω, which satisfies(22)GIVPHFWGλP˜1,⋯,P˜n=1λλP˜1ω1⊗⋯⊗λP˜nωn=1−1−∏i=1n1−1−μi−2λωi1/λ,1−1−∏i=1n1−1−μi+2λωi1/λ,1−∏i=1n1−νi−2λωi1/λ,1−∏i=1n1−νi+2λωi1/λ,μ˜i,ν˜i∈P˜i,i=1,⋯,n,where λ>0. If λ=1, then the GIVPHFWA and GIVPHFWG operators reduce to the IVPHFWA and IVPHFWG operators, respectively.Example 1.
SupposeP˜1=0.3,0.5,0.2,0.4,(23)P˜2=0.1,0.4,0.3,0.6,0.1,0.4,0.5,0.7,P˜3=0.3,0.4,0.2,0.6,0.1,0.3,0.2,0.6,0.4,0.7,0.2,0.6,are three IVPHFEs, and their weight vector is ω=0.15,0.30,0.55T.
Then, by Definition17, we have(24)IVPHFWAP˜1,P˜2,P˜3=ω1P˜1⊕ω1P˜2⊕ωnP˜3=1−∏i=131−μi−2ωi,1−∏i=131−μi+2ωi,∏i=13νi−ωi,∏i=13νi+ωiμ˜i,ν˜i∈P˜i,i=1,2,3.=0.2583,0.4173,0.2259,0.5646,0.1498,0.3701,0.2259,0.5646,0.3270,0.6102,0.2259,0.5646,0.2583,0.4173,0.2633,0.5913,0.1498,0.3701,0.2633,0.5913,0.3270,0.6102,0.2633,0.5913;IVPHFWGP˜1,P˜2,P˜3=P˜1ω1⊗P˜3ω2⊗P˜3ω3=∏i=13μi−ωi,∏i=13μi+ωi,1−∏i=131−νi−2ωi,1−∏i=131−νi+2ωiμ˜i,ν˜i∈P˜i,i=1,2,3.=0.2158,0.4136,0.2351,0.5774,0.1179,0.3531,0.2351,0.5774,0.2528,0.5627,0.2351,0.5774,0.2158,0.4137,0.3294,0.6142,0.1179,0.3531,0.3294,0.6142,0.2528,0.5627,0.3294,0.6142;GIVPHFWA3P˜1,P˜2,P˜3=ω1P˜13⊕ω2P˜23⊕ω3P˜331/3=1−∏i=131−μi−2×3ωi1/3,1−∏i=131−μi+2×3ωi1/3,1−1−∏i=131−1−νi−23ωi1/3,1−1−∏i=131−1−νi+23ωi1/3,μ˜i,ν˜i∈P˜i,i=1,2,3,=0.2827,0.4243,0.2254,0.5589,0.2190,0.3982,0.2254,0.5589,0.3650,0.6418,0.2254,0.5589,0.2827,0.4243,0.2582,0.5805,0.2190,0.3982,0.2582,0.5805,0.3650,0.6418,0.2582,0.5805;GIVPHFWG3P˜1,P˜2,P˜3=133P˜1ω1⊗3P˜2ω2⊗3P˜3ω3=1−1−∏i=131−1−μi−23ωi1/3,1−1−∏i=131−1−μi+23ωi1/3,1−∏i=131−νi−2×3ωi1/3,1−∏i=131−νi+2×3ωi1/3,μ˜i,ν˜i∈P˜i,i=1,2,3,=0.2135,0.4129,0.2532,0.5858,0.1174,0.3508,0.2532,0.5858,0.2471,0.5437,0.2532,0.5858,0.2135,0.4129,0.4101,0.6298,0.1174,0.3508,0.4101,0.6298,0.2471,0.5437,0.4101,0.6298.
Based on Proposition1, we know the above four aggregation operators are all IVPHFEs. Also, we can obtain that the score values are(25)SIVPHFWAP˜1,P˜2,P˜3=−0.1344,0.0838;SIVPHFWGP˜1,P˜2,P˜3=−0.1569,0.0611;SGIVPHFWA3P˜1,P˜2,P˜3=−0.1188,0.0957;SGIVPHFWG3P˜1,P˜2,P˜3=−0.1649,0.0401;So,PSIVPHFWGP˜1,P˜2,P˜3≤SIVPHFWAP˜1,P˜2,P˜3=0.5518;PSIVPHFWGP˜1,P˜2,P˜3≤SGIVPHFWA3P˜1,P˜2,P˜3=0.5840;PSGIVPHFWG3P˜1,P˜2,P˜3≤SIVPHFWAP˜1,P˜2,P˜3=0.5876;Hence, IVPHFWGP˜1,P˜2,P˜3≼IVPHFWAP˜1,P˜2,P˜3;IVPHFWGP˜1,P˜2,P˜3≼GIVPHFWA3P˜1,P˜2,P˜3;GIVPHFWG3P˜1,P˜2,P˜3≼IVPHFWAP˜1,P˜2,P˜3.
In fact, we have the following.Lemma 1 (see [52]).
Letxi>0, ωi>0i=1,2,⋯,n, and ∑i=1nωi=1, then(26)∏i=1nxiωi≤∑i=1nωixi,with equality if and only if x1=x2=⋯=xn.Proposition 5.
LetP˜ii=1,2,⋯,n be a collection of IVPHFEs, ωi>0, and ∑i=1nωi=1, then(27)IVPHFWGP˜1,P˜2,⋯,P˜n≼IVPHFWAP˜1,P˜2,⋯,P˜n,with equality if and only if P˜1=P˜2=⋯=P˜n.Proof.
Supposeμi−,μi+,νi−,νi+∈P˜i, for any i. Based on Lemma 1, we have(28)∏i=1nμi−2ωi≤∑i=1nωiμi−2=1−∑i=1nωi1−μi−2≤1−∏i=1n1−μi−2ωi.
So,∏i=1nμi−ωi=∏i=1nμi−2ωi≤1−∏i=1n1−μi−2ωi. Similarly, we have(29)∏i=1nμi+ωi≤1−∏i=1n1−μi+2ωi,∏i=1nνi−ωi≤1−∏i=1n1−νi−2ωi,∏i=1nνi+ωi≤1−∏i=1n1−νi+2ωi.
By Definition17 and Proposition 4, PSP˜1ω1⊗⋯⊗P˜nωn≥Sω1P˜1⊕⋯⊕ωnP˜n≤0.5 holds, which means(30)IVPHFWGP˜1,P˜2,⋯,P˜n≼IVPHFWAP˜1,P˜2,⋯,P˜n.Proposition 6.
LetP˜ii=1,2,⋯,n be a collection of IVPHFEs, λ>0, ωi>0, and ∑i=1nωi=1, then(31)IVPHFWGP˜1,P˜2,⋯,P˜n≼GIVPHFWAλP˜1,P˜2,⋯,P˜n,GIVPHFWGλP˜1,P˜2,⋯,P˜n≼IVPHFWAP˜1,P˜2,⋯,P˜n.Proof.
Supposeμi−,μi+,νi−,νi+∈P˜i, for any i. Since ∏i=1nμi−2ωi=∏i=1nμi−2λωi1/λ≤∑i=1nωiμi−2λ1/λ=1−∑i=1nωi1−μi−2λ1/λ≤1−∏i=1n1−μi−2λωi1/λ, we have ∏i=1nμi−ωi=∏i=1nμi−2ωi≤1−∏i=1n1−μi−2λωi1/λ.
Similarly,∏i=1nμi+ωi=∏i=1nμi+2ωi≤1−∏i=1n1−μi+2λωi1/λ holds.
Also,(32)1−1−∏i=1n1−1−νi−2λωi1/λ≤1−1−∑i=1nωi1−1−νi−2λ1/λ=1−∑i=1nωi1−νi−2λ1/λ≤1−∏i=1n1−νi−2λωi1/λ=1−∏i=1n1−νi−2ωi.
Similarly,1−1−∏i=1n1−1−νi+2λωi1/λ≤1−∏i=1n1−νi+2ωi holds.
All imply thatIVPHFWGP˜1,⋯,P˜n≼GIVPHFWAλP˜1,⋯,P˜n.
In the same way, we haveGIVPHFWGλP˜1,⋯,P˜n≼IVPHFWAP˜1,⋯,P˜n and complete the proof of Proposition 6.Proposition 4.2 shows no matter how the parameterλλ>0 changes, the values obtained by IVPHFWG operators are not bigger than the ones obtained by GIVPHFWA operators and the values obtained by GIVPHFWG operators are not bigger than the ones obtained by IVPHFWA operators.Proposition 7.
LetP˜ii=1,2,⋯,n be a collection of IVPHFEs, λ2≥λ1>0, ωi>0, and ∑i=1nωi=1, then(33)GIVPHFWAλ1P˜1,P˜2,…,P˜n≼GIVPHFWAλ2P˜1,P˜2,…,P˜n,GIVPHFWGλ1P˜1,P˜2,…,P˜n≽GIVPHFWGλ2P˜2,P˜2,…,P˜n.Proof.
According to Theorem 3.8 in [31], we have(34)1−∏i=1n1−μi−2λ1ωi1/λ1≤1−∏i=1n1−μi−2λ2ωi1/λ2,1−∏i=1n1−μi+2λ1ωi1/λ1≤1−∏i=1n1−μi+2λ2ωi1/λ2.
Also, we can conclude that(35)1−1−∏i=1n1−1−νi−2λ1ωi1/λ1≥1−1−∏i=1n1−1−νi−2λ2ωi1/λ2.
Similarly,(36)1−1−∏i=1n1−1−νi+2λ1ωi1/λ1≥1−1−∏i=1n1−1−νi+2λ2ωi1/λ2.
Therefore, by Proposition4, we have(37)GIVPHFWAλ1P˜1,P˜2,…,P˜n≼GIVPHFWAλ2P˜1,P˜2,…,P˜n,GIVPHFWGλ1P˜1,P˜2,…,P˜n≽GIVPHFWGλ2P˜2,P˜2,…,P˜n.


### 4.2. The IVPHFOWA, IVPHFOWG, GIVPHFOWA, and GIVPHFOWG Operators

Definition 18.
LetP˜ii=1,2,⋯,n be a collection of some IVPHFEs, P˜σi be the ith largest of them, and κ=κ1,κ2,⋯,κnT be the associated vector such that κi∈0,1, ∑i=1nκi=1, and λ>0. Then,(1)
An interval-valued Pythagorean hesitant fuzzy ordered weighted averaging (IVPHFOWA) operator can be seen a mapIVPHOFWA:Ωn⟶Ω, such that(38)IVPHFOWAP˜1,⋯,P˜n=κ1P˜σ1⊕⋯⊕κnP˜σn=1−∏i=1n1−μσi−2κi,1−∏i=1n1−μσi+2κi,∏i=1nνσi−κi,∏i=1nνσi+κiμ˜σi,ν˜σi∈P˜σi,i=1,⋯,n.(2)
An interval-valued Pythagorean hesitant fuzzy ordered weighted geometric (IVPHFOWG) operator can be seen a mapIVPHFOWG:Ωn⟶Ω, such that(39)IVPHFOWGP˜1,⋯,P˜n=P˜σ1κ1⊗⋯⊗P˜σnκn=∏i=1nμσi−κi,∏i=1nμσi+κi,1−∏i=1n1−νσi−2κi,1−∏i=1n1−νσi+2κiμ˜σi,ν˜σi∈P˜σi,i=1,⋯,n.(3)
A generalized interval-valued Pythagorean hesitant fuzzy ordered weighted averaging (GIVPHFOWA) operator can be seen a mapGIVPHFOWA:Ωn⟶Ω, which satisfies(40)GIVPHFOWAλP˜1,⋯,P˜n=κ1P˜˙σ1λ⊕⋯⊕κnP˜˙σn1/λ=1−∏i=1n1−μ˙σi−2λκi1/λ,1−∏i=1n1−μ˙σi+2λκi1/λ,1−1−∏i=1n1−1−ν˙σi−2λκi1/λ,1−1−∏i=1n1−1−ν˙σi+2λκi1/λ,μ˜˙σi,ν˜˙σi∈P˜˙σi,i=1,⋯,n,where λ>0.(4)
A generalized interval-valued Pythagorean hesitant fuzzy ordered weighted geometric (GIVPHFOWG) operator can be seen a mapGIVPHFOWG:Ωn⟶Ω, which satisfies(41)GIVPHFOWGλP˜1,⋯,P˜n=1λλP˜¨σ1κ1⊗⋯⊗λP˜¨σnκn=1−1−∏i=1n1−1−μ¨σi−2λκi1/λ,1−1−∏i=1n1−1−μ¨σi+2λκi1/λ,1−∏i=1n1−ν¨σi−2λκi1/λ,1−∏i=1n1−ν¨σi+2λκi1/λ,μ˜¨σi,ν˜¨σi∈P˜¨σi,i=1,⋯,n,where λ>0. If λ=1, then the GIVPHFOWA and GIVPHFOWG operators reduce to the IVPHFOWA and IVPHFOWG operators, respectively.
By comparing Definition17 with Definition 18, we can find that the IVPHFOWA, IVPHFOWG, GIVPHFOWA, and GIVPHFOWG operators weight the ordered positions of the IVPHFEs instead of weighting the IVPHFEs themselves. Also, their characteristics lie in reordering original IVPHFEs according to a decreasing order and in aggregating through the associated weights of positions where the weight κi is associated with the ith position in the collection of the IVPHFEs during aggregation processes.Example 2.
Suppose(42)P˜1=0.3,0.5,0.2,0.4,P˜2=0.1,0.4,0.3,0.6,0.1,0.4,0.5,0.7,P˜3=0.3,0.4,0.2,0.6,0.1,0.3,0.2,0.6,0.4,0.7,0.2,0.6,are three IVPHFEs, and the aggregation-associated vector is κ=0.2,0.45,0.35T.
Then, by Definition15, we have the score values of P˜1,P˜2,andP˜3 as follows:(43)SP˜1=120.32−0.42,120.52−0.22=−0.035,0.045,SP˜2=140.12−0.62+0.12−0.72,140.42−0.32+0.42−0.52=−0.2075,−0.005,SP˜3=160.32−0.62+0.12−0.62+0.42−0.62,160.42−0.22+0.32−0.22+0.72−0.22=−0.1367,0.1033.
Based on Definition16, we can obtain that PSP˜1>SP˜3=0.5678andPSP˜3>SP˜2=0.7024 and then P˜σ1=P˜1, P˜σ2=P˜3, and P˜σ3=P˜2.
According to Definition18, we have(44)IVPHFOWAP˜1,P˜2,P˜3=κ1P˜σ1⊕κ1P˜σ2⊕κnP˜σ3=1−∏i=131−μσi−2κi,1−∏i=131−μσi+2κi,∏i=13νσi−κi,∏i=13νσi+κiμ˜σi,ν˜σi∈P˜σi,i=1,2,3.=0.2505,0.4229,0.2305,0.5533,0.2505,0.4229,0.2756,0.5839,0.1629,0.3856,0.2305,0.5533,0.1629,0.3856,0.2756,0.5839,0.3097,0.5865,0.2305,0.5930,0.3097,0.5865,0.2756,0.6259;IVPHFOWGP˜1,P˜2,P˜3=P˜σ1κ1⊗P˜σ2κ2⊗P˜σ3κ3=∏i=13μσi−κi,∏i=13μσi+κi,1−∏i=131−νσi−2κi,1−∏i=131−νσi+2κiμ˜σi,ν˜σi∈P˜σi,i=1,2,3.=0.2042,0.4183,0.2404,0.5694,0.2042,0.4183,0.3456,0.6130,0.1246,0.3675,0.2404,0.5694,0.1246,0.3675,0.3456,0.6131,0.2325,0.5380,0.2404,0.6244,0.2325,0.5380,0.3456,0.6607;GIVPHFOWA3P˜1,P˜2,P˜3=κ1P˜σ13⊕κ2P˜σ23⊕κ3P˜σ331/3=1−∏i=131−μσi−2×3κi1/3,1−∏i=131−μσi+2×3κi1/3,1−1−∏i=131−1−νσi−23κi1/3,1−1−∏i=131−1−νσi+23κi1/3,μ˜σi,ν˜σi∈P˜σi,i=1,2,3,=0.2793,0.4310,0.2299,0.5466,0.2793,0.4310,0.2697,0.5706,0.2296,0.4122,0.2300,0.5466,0.2296,0.4122,0.2697,0.5706,0.3547,0.6241,0.2299,0.5780,0.3547,0.6241,0.2697,0.6053;GIVPHFOWG3P˜1,P˜2,P˜3=133P˜σ1κ1⊗3P˜σ2κ2⊗3P˜σ3κ3=1−1−∏i=131−1−μσi−23κi1/3,1−1−∏i=131−1−μσi+23κi1/3,1−∏i=131−νσi−2×3κi1/3,1−∏i=131−νσi+2×3κi1/3,μ˜σi,ν˜σi∈P˜σi,i=1,2,3,=0.2020,0.4174,0.2583,0.5806,0.2020,0.4174,0.4206,0.6324,0.1239,0.3648,0.2583,0.5806,0.1239,0.3648,0.4206,0.6324,0.2275,0.5213,0.2583,0.6438,0.2275,0.5213,0.4206,0.6767.
Based on Definition14, we can obtain that(45)SIVPHFOWAP˜1,P˜2,P˜3=−0.1389,0.0796,SIVPHFOWGP˜1,P˜2,P˜3=−0.1670,0.0556,SGIVPHFOWA3P˜1,P˜2,P˜3=−0.1197,0.1001,SGIVPHFOWG3P˜1,P˜2,P˜3=−0.1776,0.0356.
Then,(46)PSIVPHFOWGP˜1,P˜2,P˜3≤SIVPHFOWAP˜1,P˜2,P˜3=0.5591,PSIVPHFOWGP˜1,P˜2,P˜3≤SGIVPHFOWA3P˜1,P˜2,P˜3=0.6038,PSGIVPHFOWG3P˜1,P˜2,P˜3≤SIVPHFOWAP˜1,P˜2,P˜3=0.5958.
So,(47)IVPHFOWGP˜1,P˜2,P˜3≼IVPHFOWAP˜1,P˜2,P˜3,IVPHFOWGP˜1,P˜2,P˜3≼GIVPHFOWA3P˜1,P˜2,P˜3,GIVPHFOWG3P˜1,P˜2,P˜3≼IVPHFOWAP˜1,P˜2,P˜3.
Similar to Proposition5–7, we easily obtain the following properties of ordered aggregation operators of IVPHFEs.Proposition 8.
LetP˜ii=1,2,⋯,n be a collection of IVPHFEs, λ>0,andλ2>λ1>0, then(48)IVPHFOWGP˜1,P˜2,⋯,P˜n≼IVPHFOWAP˜1,P˜2,⋯,P˜n,IVPHFOWGP˜1,P˜2,⋯,P˜n≼GIVPHFOWAλP˜1,P˜2,⋯,P˜n,GIVPHFOWGλP˜1,P˜2,⋯,P˜n≼IVPHFOWAP˜1,P˜2,⋯,P˜n,GIVPHFOWAλ1P˜1,P˜2,⋯,P˜n≼GIVPHFOWAλ2P˜1,P˜2,⋯,P˜n,GIVPHFOWGλ1P˜1,P˜2,⋯,P˜n≽GIVPHFOWGλ2P˜1,P˜2,⋯,P˜n.
From Definition17 and Definition 18, we know that the IVPHFWA, IVPHFWG, GIVPHFWA, and GIVPHFWG operators weight the interval-valued hesitant fuzzy argument itself and there is no consideration of the impact of the order of parameters, while the IVPHFOWA, IVPHFOWG, GIVPHFOWA, and GIVPHFOWG operators only weight the ordered position of each given IVPHFE and there is no consideration of the impact of the argument itself. Hence, we propose a new hybrid aggregation operator which can weight both all the given arguments and their ordered positions.


### 4.3. The IVPHFHA, IVPHFHG, GIVPHFHA, and GIVPHFHG Operators

Definition 19.
For a collection of some IVPHFEsP˜ii=1,2,⋯,n, ω=ω1,ω2,⋯,ωnT is the weight vector of them with ωi∈0,1 and ∑i=1nωi=1. Suppose κ=κ1,κ2,⋯,κnT is the associated vector such that κi∈0,1, ∑i=1nκi=1, and λ>0. Then,(1)
An interval-valued Pythagorean hesitant fuzzy hybrid averaging (IVPHFHA) operator can be seen a mapIVPHFHA:Ωn⟶Ω, such that(49)IVPHFHAP˜1,⋯,P˜n=κ1P˜˙σ1⊕⋯⊕κnP˜˙σn=1−∏i=1n1−μ˙σi−2κi,1−∏i=1n1−μ˙σi+2κi,∏i=1nν˙σi−κi,∏i=1nν˙σi+κiμ˜˙σi,ν˜˙σi∈P˜˙σi,i=1,⋯,n,where P˜˙σi is the largest ith of P˜˙k=nωkP˜kk=1,2,⋯,n.(2)
An interval-valued Pythagorean hesitant fuzzy hybrid geometric (IVPHFHG) operator can be seen a mapIVPHFHG:Ωn⟶Ω, such that(50)IVPHFHGP˜1,⋯,P˜n=P˜¨σ1κ1⊗⋯⊗P˜¨σnκn=∏i=1nμ¨σi−κi,∏i=1nμ¨σi+κi,1−∏i=1n1−ν¨σi−2κi,1−∏i=1n1−ν¨σi+2κiμ˜¨σi,ν˜¨σi∈P˜¨σi,i=1,⋯,n.where P˜¨σi is the largest ith of P˜¨k=P˜knωkk=1,2,⋯,n.(3)
A generalized interval-valued Pythagorean hesitant fuzzy hybrid averaging (GIVPHFHA) operator can be seen a mapGIVPHFHA:Ωn⟶Ω, which satisfies(51)GIVPHFHAλP˜1,⋯,P˜n=κ1P˜˙σ1λ⊕⋯⊕κnP˜˙σnλ1/λ=1−∏i=1n1−μ˙σi−2λκi1/λ,1−∏i=1n1−μ˙σi+2λκi1/λ,1−1−∏i=1n1−1−ν˙σi−2λκi1/λ,1−1−∏i=1n1−1−ν˙σi+2λκi1/λ,μ˜˙σi,ν˜˙σi∈P˜˙σi,i=1,⋯,n,where P˜˙σi is the largest ith of P˜˙k=nωkP˜kk=1,2,⋯,n.(4)
A generalized interval-valued Pythagorean hesitant fuzzy hybrid geometric (GIVPHFHG) operator can be seen a mapGIVPHFHG:Ωn⟶Ω, which satisfies(52)GIVPHFHGλP˜1,⋯,P˜n=1λλP˜¨σ1κ1⊗⋯⊗λP˜¨σnκn=1−1−∏i=1n1−1−μ¨σi−2λκi1/λ,1−1−∏i=1n1−1−μ¨σi+2λκi1/λ,1−∏i=1n1−ν¨σi−2λκi1/λ,1−∏i=1n1−ν¨σi+2λκi1/λ,μ˜˙σi,ν˜˙σi∈P˜¨σi,i=1,⋯,n,where P˜¨σi is the largest ith of P˜˙k=P˜knωkk=1,2,⋯,n.
Ifλ=1, then the GIVPHFHA and GIVPHFHG operators reduce to the IVPHFHA and IVPHFHG operators, respectively.Example 3.
SupposeP˜1=0.3,0.5,0.2,0.4,(53)P˜2=0.1,0.4,0.3,0.6,0.1,0.4,0.5,0.7,P˜3=0.3,0.4,0.2,0.6,0.1,0.3,0.2,0.6,0.4,0.7,0.2,0.6,are three IVPHFEs, the weight vector of the attributes is ω=0.15,0.30,0.55T, and the aggregation-associated vector is κ=0.2,0.45,0.35T. Then, we can obtain(54)P˜˙1=3×0.15⊗P˜1=0.2038,0.3485,0.4847,0.6621,P˜˙2=3×0.3⊗P˜2=0.0949,0.3811,0.3384,0.6314,0.0949,0.3811,0.5359,0.7254,P˜˙3=3×0.3⊗P˜3=0.3796,0.5000,0.0703,0.4305,0.1282,0.3796,0.0703,0.4305,0.5000,0.8190,0.0703,0.4305.P˜¨1=P˜13×0.15=0.5817,0.7320,0.1349,0.2747,P˜¨2=P˜23×0.3=0.1259,0.4384,0.2853,0.5751,0.1259,0.4384,0.4776,0.6741,P˜¨3=P˜33×0.3=0.1372,0.2205,0.2552,0.7219,0.0224,0.1372,0.2552,0.7219,0.2205,0.5552,0.2552,0.7219.SP˜˙1=−0.1984,−0.0567,SP˜˙2=−0.2267,−0.0278,SP˜˙3=−0.0242,0.1750,SP˜¨1=0.1315,0.2588,SP˜¨2=−0.1884,0.0187,SP˜¨3=−0.2493,0.0300.
By Definition16, we have PSP˜˙1≤SP˜˙2=1, PSP˜˙2≤SP˜˙3=0.5009, and PSP˜¨3≤SP˜¨2=0.5509, and PSP˜¨2≤SP˜¨1=1. So, we can get(55)P˜˙σ1=P˜˙3,P˜˙σ2=P˜˙2,P˜˙σ3=P˜˙1,P˜¨σ1=P˜¨1,P˜¨σ2=P˜¨2,P˜¨σ3=P˜¨3.
According to Definition19, we can obtain(56)IVPHFHAP˜1,P˜2,P˜3=κ1P˜˙σ1⊕κ1P˜˙σ2⊕κnP˜˙σ3=1−∏i=131−μ˙σi−2κi,1−∏i=131−μ˙σi+2κi,∏i=13ν˙σi−κi,∏i=13ν˙σi+κiμ˜˙σi,ν˜˙σi∈P˜˙σi,i=1,2,3=0.2209,0.3991,0.2802,0.5947,0.2209,0.3991,0.3446,0.6330,0.1483,0.3698,0.2802,0.5947,0.1483,0.3698,0.3446,0.6330,0.2713,0.5356,0.2802,0.5947,0.2713,0.5356,0.3446,0.6330;IVPHFHGP˜1,P˜2,P˜3=P˜¨σ1κ1⊗P˜¨σ2κ2⊗P˜¨σ3κ3=∏i=13μ¨σi−κi,∏i=13μ¨σi+κi,1−∏i=131−ν¨σi−2κi,1−∏i=131−ν¨σi+2κiμ˜¨σi,ν˜¨σi∈P˜¨σi,i=1,2,3=0.1762,0.3819,0.2517,0.6042,0.0934,0.3235,0.2517,0.6042,0.2080,0.5276,0.2517,0.6042,0.1762,0.3819,0.3659,0.6487,0.0934,0.3234,0.3659,0.6487,0.2080,0.5276,0.3659,0.6487;GIVPHFHA3P˜1,P˜2,P˜3=κ1P˜˙σ13⊕κ2P˜˙σ23⊕κ3P˜˙σ331/3=1−∏i=131−μ˙σi−2×3κi1/3,1−∏i=131−μ˙σi+2×3κi1/3,1−1−∏i=131−1−ν˙σi−23κi1/3,1−1−∏i=131−1−ν˙σi+23κi1/3,μ˜˙σi,ν˜˙σi∈P˜˙σi,i=1,2,3=0.2924,0.4155,0.2713,0.5854,0.2924,0.4155,0.3255,0.6153,0.1725,0.3709,0.2713,0.5854,0.1725,0.3709,0.3256,0.6153,0.3833,0.6438,0.2713,0.5854,0.3833,0.6438,0.3256,0.6153;GIVPHFHG3P˜1,P˜2,P˜3=133P˜¨σ1κ1⊗3P˜¨σ2κ2⊗3P˜¨σ3κ3=1−1−∏i=131−1−μ¨σi−23κi1/3,1−1−∏i=131−1−μ¨σi+23κi1/3,1−∏i=131−ν¨σi−2×3κi1/3,1−∏i=131−ν¨σi+2×3κi1/3,μ˜¨σi,ν˜¨σi∈P˜¨σi,i=1,2,3=0.1714,0.3659,0.2642,0.6384,0.0902,0.3050,0.2642,0.6384,0.2024,0.5174,0.2642,0.6384,0.1714,0.3659,0.4196,0.6734,0.0902,0.3050,0.4196,0.6734,0.2024,0.5174,0.4196,0.6734.


## 4.1. The IVPHFWA, IVPHFWG, GIVPHFWA, and GIVPHFWG Operators

Definition 17.
LetP˜ii=1,2,⋯,n be a collection of some IVPHFEs and ω=ω1,ω2,⋯,ωnT be the weight vector of P˜ii=1,2,⋯,n, with ωi∈0,1, ∑i=1nωi=1, and λ>0.(1)
An interval-valued Pythagorean hesitant fuzzy weighted averaging (IVPHFWA) operator can be seen a mapIVPHFWA:Ωn⟶Ω, such that(19)IVPHFWAP˜1,⋯,P˜n=ω1P˜1⊕⋯⊕ωnP˜n=1−∏i=1n1−μi−2ωi,1−∏i=1n1−μi+2ωi,∏i=1nνi−ωi,∏i=1nνi+ωiμ˜i,ν˜i∈P˜i,i=1,⋯,n.(2)
An interval-valued Pythagorean hesitant fuzzy weighted geometric (IVPHFWG) operator can be seen a mapIVPHFWG:Ωn⟶Ω, such that(20)IVPHFWGP˜1,⋯,P˜n=P˜1ω1⊗⋯⊗P˜nωn=∏i=1nμi−ωi,∏i=1nμi+ωi,1−∏i=1n1−νi−2ωi,1−∏i=1n1−νi+2ωiμ˜i,ν˜i∈P˜i,i=1,⋯,n.(3)
A generalized interval-valued Pythagorean hesitant fuzzy weighted averaging (GIVPHFWA) operator can be seen a mapGIVPHFWA:Ωn⟶Ω, which satisfies(21)GIVPHFWAλP˜1,⋯,P˜n=ω1P˜1λ⊕⋯⊕ωnP˜nλ1/λ=1−∏i=1n1−μi−2λωi1/λ,1−∏i=1n1−μi+2λωi1/λ,1−1−∏i=1n1−1−νi−2λωi1/λ,1−1−∏i=1n1−1−νi+2λωi1/λμ˜i,ν˜i∈P˜i,i=1,⋯,n,where λ>0.(4)
A generalized interval-valued Pythagorean hesitant fuzzy weighted geometric (GIVPHFWG) operator can be seen a mapGIVPHFWG:Ωn⟶Ω, which satisfies(22)GIVPHFWGλP˜1,⋯,P˜n=1λλP˜1ω1⊗⋯⊗λP˜nωn=1−1−∏i=1n1−1−μi−2λωi1/λ,1−1−∏i=1n1−1−μi+2λωi1/λ,1−∏i=1n1−νi−2λωi1/λ,1−∏i=1n1−νi+2λωi1/λ,μ˜i,ν˜i∈P˜i,i=1,⋯,n,where λ>0. If λ=1, then the GIVPHFWA and GIVPHFWG operators reduce to the IVPHFWA and IVPHFWG operators, respectively.Example 1.
SupposeP˜1=0.3,0.5,0.2,0.4,(23)P˜2=0.1,0.4,0.3,0.6,0.1,0.4,0.5,0.7,P˜3=0.3,0.4,0.2,0.6,0.1,0.3,0.2,0.6,0.4,0.7,0.2,0.6,are three IVPHFEs, and their weight vector is ω=0.15,0.30,0.55T.
Then, by Definition17, we have(24)IVPHFWAP˜1,P˜2,P˜3=ω1P˜1⊕ω1P˜2⊕ωnP˜3=1−∏i=131−μi−2ωi,1−∏i=131−μi+2ωi,∏i=13νi−ωi,∏i=13νi+ωiμ˜i,ν˜i∈P˜i,i=1,2,3.=0.2583,0.4173,0.2259,0.5646,0.1498,0.3701,0.2259,0.5646,0.3270,0.6102,0.2259,0.5646,0.2583,0.4173,0.2633,0.5913,0.1498,0.3701,0.2633,0.5913,0.3270,0.6102,0.2633,0.5913;IVPHFWGP˜1,P˜2,P˜3=P˜1ω1⊗P˜3ω2⊗P˜3ω3=∏i=13μi−ωi,∏i=13μi+ωi,1−∏i=131−νi−2ωi,1−∏i=131−νi+2ωiμ˜i,ν˜i∈P˜i,i=1,2,3.=0.2158,0.4136,0.2351,0.5774,0.1179,0.3531,0.2351,0.5774,0.2528,0.5627,0.2351,0.5774,0.2158,0.4137,0.3294,0.6142,0.1179,0.3531,0.3294,0.6142,0.2528,0.5627,0.3294,0.6142;GIVPHFWA3P˜1,P˜2,P˜3=ω1P˜13⊕ω2P˜23⊕ω3P˜331/3=1−∏i=131−μi−2×3ωi1/3,1−∏i=131−μi+2×3ωi1/3,1−1−∏i=131−1−νi−23ωi1/3,1−1−∏i=131−1−νi+23ωi1/3,μ˜i,ν˜i∈P˜i,i=1,2,3,=0.2827,0.4243,0.2254,0.5589,0.2190,0.3982,0.2254,0.5589,0.3650,0.6418,0.2254,0.5589,0.2827,0.4243,0.2582,0.5805,0.2190,0.3982,0.2582,0.5805,0.3650,0.6418,0.2582,0.5805;GIVPHFWG3P˜1,P˜2,P˜3=133P˜1ω1⊗3P˜2ω2⊗3P˜3ω3=1−1−∏i=131−1−μi−23ωi1/3,1−1−∏i=131−1−μi+23ωi1/3,1−∏i=131−νi−2×3ωi1/3,1−∏i=131−νi+2×3ωi1/3,μ˜i,ν˜i∈P˜i,i=1,2,3,=0.2135,0.4129,0.2532,0.5858,0.1174,0.3508,0.2532,0.5858,0.2471,0.5437,0.2532,0.5858,0.2135,0.4129,0.4101,0.6298,0.1174,0.3508,0.4101,0.6298,0.2471,0.5437,0.4101,0.6298.
Based on Proposition1, we know the above four aggregation operators are all IVPHFEs. Also, we can obtain that the score values are(25)SIVPHFWAP˜1,P˜2,P˜3=−0.1344,0.0838;SIVPHFWGP˜1,P˜2,P˜3=−0.1569,0.0611;SGIVPHFWA3P˜1,P˜2,P˜3=−0.1188,0.0957;SGIVPHFWG3P˜1,P˜2,P˜3=−0.1649,0.0401;So,PSIVPHFWGP˜1,P˜2,P˜3≤SIVPHFWAP˜1,P˜2,P˜3=0.5518;PSIVPHFWGP˜1,P˜2,P˜3≤SGIVPHFWA3P˜1,P˜2,P˜3=0.5840;PSGIVPHFWG3P˜1,P˜2,P˜3≤SIVPHFWAP˜1,P˜2,P˜3=0.5876;Hence, IVPHFWGP˜1,P˜2,P˜3≼IVPHFWAP˜1,P˜2,P˜3;IVPHFWGP˜1,P˜2,P˜3≼GIVPHFWA3P˜1,P˜2,P˜3;GIVPHFWG3P˜1,P˜2,P˜3≼IVPHFWAP˜1,P˜2,P˜3.
In fact, we have the following.Lemma 1 (see [52]).
Letxi>0, ωi>0i=1,2,⋯,n, and ∑i=1nωi=1, then(26)∏i=1nxiωi≤∑i=1nωixi,with equality if and only if x1=x2=⋯=xn.Proposition 5.
LetP˜ii=1,2,⋯,n be a collection of IVPHFEs, ωi>0, and ∑i=1nωi=1, then(27)IVPHFWGP˜1,P˜2,⋯,P˜n≼IVPHFWAP˜1,P˜2,⋯,P˜n,with equality if and only if P˜1=P˜2=⋯=P˜n.Proof.
Supposeμi−,μi+,νi−,νi+∈P˜i, for any i. Based on Lemma 1, we have(28)∏i=1nμi−2ωi≤∑i=1nωiμi−2=1−∑i=1nωi1−μi−2≤1−∏i=1n1−μi−2ωi.
So,∏i=1nμi−ωi=∏i=1nμi−2ωi≤1−∏i=1n1−μi−2ωi. Similarly, we have(29)∏i=1nμi+ωi≤1−∏i=1n1−μi+2ωi,∏i=1nνi−ωi≤1−∏i=1n1−νi−2ωi,∏i=1nνi+ωi≤1−∏i=1n1−νi+2ωi.
By Definition17 and Proposition 4, PSP˜1ω1⊗⋯⊗P˜nωn≥Sω1P˜1⊕⋯⊕ωnP˜n≤0.5 holds, which means(30)IVPHFWGP˜1,P˜2,⋯,P˜n≼IVPHFWAP˜1,P˜2,⋯,P˜n.Proposition 6.
LetP˜ii=1,2,⋯,n be a collection of IVPHFEs, λ>0, ωi>0, and ∑i=1nωi=1, then(31)IVPHFWGP˜1,P˜2,⋯,P˜n≼GIVPHFWAλP˜1,P˜2,⋯,P˜n,GIVPHFWGλP˜1,P˜2,⋯,P˜n≼IVPHFWAP˜1,P˜2,⋯,P˜n.Proof.
Supposeμi−,μi+,νi−,νi+∈P˜i, for any i. Since ∏i=1nμi−2ωi=∏i=1nμi−2λωi1/λ≤∑i=1nωiμi−2λ1/λ=1−∑i=1nωi1−μi−2λ1/λ≤1−∏i=1n1−μi−2λωi1/λ, we have ∏i=1nμi−ωi=∏i=1nμi−2ωi≤1−∏i=1n1−μi−2λωi1/λ.
Similarly,∏i=1nμi+ωi=∏i=1nμi+2ωi≤1−∏i=1n1−μi+2λωi1/λ holds.
Also,(32)1−1−∏i=1n1−1−νi−2λωi1/λ≤1−1−∑i=1nωi1−1−νi−2λ1/λ=1−∑i=1nωi1−νi−2λ1/λ≤1−∏i=1n1−νi−2λωi1/λ=1−∏i=1n1−νi−2ωi.
Similarly,1−1−∏i=1n1−1−νi+2λωi1/λ≤1−∏i=1n1−νi+2ωi holds.
All imply thatIVPHFWGP˜1,⋯,P˜n≼GIVPHFWAλP˜1,⋯,P˜n.
In the same way, we haveGIVPHFWGλP˜1,⋯,P˜n≼IVPHFWAP˜1,⋯,P˜n and complete the proof of Proposition 6.Proposition 4.2 shows no matter how the parameterλλ>0 changes, the values obtained by IVPHFWG operators are not bigger than the ones obtained by GIVPHFWA operators and the values obtained by GIVPHFWG operators are not bigger than the ones obtained by IVPHFWA operators.Proposition 7.
LetP˜ii=1,2,⋯,n be a collection of IVPHFEs, λ2≥λ1>0, ωi>0, and ∑i=1nωi=1, then(33)GIVPHFWAλ1P˜1,P˜2,…,P˜n≼GIVPHFWAλ2P˜1,P˜2,…,P˜n,GIVPHFWGλ1P˜1,P˜2,…,P˜n≽GIVPHFWGλ2P˜2,P˜2,…,P˜n.Proof.
According to Theorem 3.8 in [31], we have(34)1−∏i=1n1−μi−2λ1ωi1/λ1≤1−∏i=1n1−μi−2λ2ωi1/λ2,1−∏i=1n1−μi+2λ1ωi1/λ1≤1−∏i=1n1−μi+2λ2ωi1/λ2.
Also, we can conclude that(35)1−1−∏i=1n1−1−νi−2λ1ωi1/λ1≥1−1−∏i=1n1−1−νi−2λ2ωi1/λ2.
Similarly,(36)1−1−∏i=1n1−1−νi+2λ1ωi1/λ1≥1−1−∏i=1n1−1−νi+2λ2ωi1/λ2.
Therefore, by Proposition4, we have(37)GIVPHFWAλ1P˜1,P˜2,…,P˜n≼GIVPHFWAλ2P˜1,P˜2,…,P˜n,GIVPHFWGλ1P˜1,P˜2,…,P˜n≽GIVPHFWGλ2P˜2,P˜2,…,P˜n.


## 4.2. The IVPHFOWA, IVPHFOWG, GIVPHFOWA, and GIVPHFOWG Operators

Definition 18.
LetP˜ii=1,2,⋯,n be a collection of some IVPHFEs, P˜σi be the ith largest of them, and κ=κ1,κ2,⋯,κnT be the associated vector such that κi∈0,1, ∑i=1nκi=1, and λ>0. Then,(1)
An interval-valued Pythagorean hesitant fuzzy ordered weighted averaging (IVPHFOWA) operator can be seen a mapIVPHOFWA:Ωn⟶Ω, such that(38)IVPHFOWAP˜1,⋯,P˜n=κ1P˜σ1⊕⋯⊕κnP˜σn=1−∏i=1n1−μσi−2κi,1−∏i=1n1−μσi+2κi,∏i=1nνσi−κi,∏i=1nνσi+κiμ˜σi,ν˜σi∈P˜σi,i=1,⋯,n.(2)
An interval-valued Pythagorean hesitant fuzzy ordered weighted geometric (IVPHFOWG) operator can be seen a mapIVPHFOWG:Ωn⟶Ω, such that(39)IVPHFOWGP˜1,⋯,P˜n=P˜σ1κ1⊗⋯⊗P˜σnκn=∏i=1nμσi−κi,∏i=1nμσi+κi,1−∏i=1n1−νσi−2κi,1−∏i=1n1−νσi+2κiμ˜σi,ν˜σi∈P˜σi,i=1,⋯,n.(3)
A generalized interval-valued Pythagorean hesitant fuzzy ordered weighted averaging (GIVPHFOWA) operator can be seen a mapGIVPHFOWA:Ωn⟶Ω, which satisfies(40)GIVPHFOWAλP˜1,⋯,P˜n=κ1P˜˙σ1λ⊕⋯⊕κnP˜˙σn1/λ=1−∏i=1n1−μ˙σi−2λκi1/λ,1−∏i=1n1−μ˙σi+2λκi1/λ,1−1−∏i=1n1−1−ν˙σi−2λκi1/λ,1−1−∏i=1n1−1−ν˙σi+2λκi1/λ,μ˜˙σi,ν˜˙σi∈P˜˙σi,i=1,⋯,n,where λ>0.(4)
A generalized interval-valued Pythagorean hesitant fuzzy ordered weighted geometric (GIVPHFOWG) operator can be seen a mapGIVPHFOWG:Ωn⟶Ω, which satisfies(41)GIVPHFOWGλP˜1,⋯,P˜n=1λλP˜¨σ1κ1⊗⋯⊗λP˜¨σnκn=1−1−∏i=1n1−1−μ¨σi−2λκi1/λ,1−1−∏i=1n1−1−μ¨σi+2λκi1/λ,1−∏i=1n1−ν¨σi−2λκi1/λ,1−∏i=1n1−ν¨σi+2λκi1/λ,μ˜¨σi,ν˜¨σi∈P˜¨σi,i=1,⋯,n,where λ>0. If λ=1, then the GIVPHFOWA and GIVPHFOWG operators reduce to the IVPHFOWA and IVPHFOWG operators, respectively.
By comparing Definition17 with Definition 18, we can find that the IVPHFOWA, IVPHFOWG, GIVPHFOWA, and GIVPHFOWG operators weight the ordered positions of the IVPHFEs instead of weighting the IVPHFEs themselves. Also, their characteristics lie in reordering original IVPHFEs according to a decreasing order and in aggregating through the associated weights of positions where the weight κi is associated with the ith position in the collection of the IVPHFEs during aggregation processes.Example 2.
Suppose(42)P˜1=0.3,0.5,0.2,0.4,P˜2=0.1,0.4,0.3,0.6,0.1,0.4,0.5,0.7,P˜3=0.3,0.4,0.2,0.6,0.1,0.3,0.2,0.6,0.4,0.7,0.2,0.6,are three IVPHFEs, and the aggregation-associated vector is κ=0.2,0.45,0.35T.
Then, by Definition15, we have the score values of P˜1,P˜2,andP˜3 as follows:(43)SP˜1=120.32−0.42,120.52−0.22=−0.035,0.045,SP˜2=140.12−0.62+0.12−0.72,140.42−0.32+0.42−0.52=−0.2075,−0.005,SP˜3=160.32−0.62+0.12−0.62+0.42−0.62,160.42−0.22+0.32−0.22+0.72−0.22=−0.1367,0.1033.
Based on Definition16, we can obtain that PSP˜1>SP˜3=0.5678andPSP˜3>SP˜2=0.7024 and then P˜σ1=P˜1, P˜σ2=P˜3, and P˜σ3=P˜2.
According to Definition18, we have(44)IVPHFOWAP˜1,P˜2,P˜3=κ1P˜σ1⊕κ1P˜σ2⊕κnP˜σ3=1−∏i=131−μσi−2κi,1−∏i=131−μσi+2κi,∏i=13νσi−κi,∏i=13νσi+κiμ˜σi,ν˜σi∈P˜σi,i=1,2,3.=0.2505,0.4229,0.2305,0.5533,0.2505,0.4229,0.2756,0.5839,0.1629,0.3856,0.2305,0.5533,0.1629,0.3856,0.2756,0.5839,0.3097,0.5865,0.2305,0.5930,0.3097,0.5865,0.2756,0.6259;IVPHFOWGP˜1,P˜2,P˜3=P˜σ1κ1⊗P˜σ2κ2⊗P˜σ3κ3=∏i=13μσi−κi,∏i=13μσi+κi,1−∏i=131−νσi−2κi,1−∏i=131−νσi+2κiμ˜σi,ν˜σi∈P˜σi,i=1,2,3.=0.2042,0.4183,0.2404,0.5694,0.2042,0.4183,0.3456,0.6130,0.1246,0.3675,0.2404,0.5694,0.1246,0.3675,0.3456,0.6131,0.2325,0.5380,0.2404,0.6244,0.2325,0.5380,0.3456,0.6607;GIVPHFOWA3P˜1,P˜2,P˜3=κ1P˜σ13⊕κ2P˜σ23⊕κ3P˜σ331/3=1−∏i=131−μσi−2×3κi1/3,1−∏i=131−μσi+2×3κi1/3,1−1−∏i=131−1−νσi−23κi1/3,1−1−∏i=131−1−νσi+23κi1/3,μ˜σi,ν˜σi∈P˜σi,i=1,2,3,=0.2793,0.4310,0.2299,0.5466,0.2793,0.4310,0.2697,0.5706,0.2296,0.4122,0.2300,0.5466,0.2296,0.4122,0.2697,0.5706,0.3547,0.6241,0.2299,0.5780,0.3547,0.6241,0.2697,0.6053;GIVPHFOWG3P˜1,P˜2,P˜3=133P˜σ1κ1⊗3P˜σ2κ2⊗3P˜σ3κ3=1−1−∏i=131−1−μσi−23κi1/3,1−1−∏i=131−1−μσi+23κi1/3,1−∏i=131−νσi−2×3κi1/3,1−∏i=131−νσi+2×3κi1/3,μ˜σi,ν˜σi∈P˜σi,i=1,2,3,=0.2020,0.4174,0.2583,0.5806,0.2020,0.4174,0.4206,0.6324,0.1239,0.3648,0.2583,0.5806,0.1239,0.3648,0.4206,0.6324,0.2275,0.5213,0.2583,0.6438,0.2275,0.5213,0.4206,0.6767.
Based on Definition14, we can obtain that(45)SIVPHFOWAP˜1,P˜2,P˜3=−0.1389,0.0796,SIVPHFOWGP˜1,P˜2,P˜3=−0.1670,0.0556,SGIVPHFOWA3P˜1,P˜2,P˜3=−0.1197,0.1001,SGIVPHFOWG3P˜1,P˜2,P˜3=−0.1776,0.0356.
Then,(46)PSIVPHFOWGP˜1,P˜2,P˜3≤SIVPHFOWAP˜1,P˜2,P˜3=0.5591,PSIVPHFOWGP˜1,P˜2,P˜3≤SGIVPHFOWA3P˜1,P˜2,P˜3=0.6038,PSGIVPHFOWG3P˜1,P˜2,P˜3≤SIVPHFOWAP˜1,P˜2,P˜3=0.5958.
So,(47)IVPHFOWGP˜1,P˜2,P˜3≼IVPHFOWAP˜1,P˜2,P˜3,IVPHFOWGP˜1,P˜2,P˜3≼GIVPHFOWA3P˜1,P˜2,P˜3,GIVPHFOWG3P˜1,P˜2,P˜3≼IVPHFOWAP˜1,P˜2,P˜3.
Similar to Proposition5–7, we easily obtain the following properties of ordered aggregation operators of IVPHFEs.Proposition 8.
LetP˜ii=1,2,⋯,n be a collection of IVPHFEs, λ>0,andλ2>λ1>0, then(48)IVPHFOWGP˜1,P˜2,⋯,P˜n≼IVPHFOWAP˜1,P˜2,⋯,P˜n,IVPHFOWGP˜1,P˜2,⋯,P˜n≼GIVPHFOWAλP˜1,P˜2,⋯,P˜n,GIVPHFOWGλP˜1,P˜2,⋯,P˜n≼IVPHFOWAP˜1,P˜2,⋯,P˜n,GIVPHFOWAλ1P˜1,P˜2,⋯,P˜n≼GIVPHFOWAλ2P˜1,P˜2,⋯,P˜n,GIVPHFOWGλ1P˜1,P˜2,⋯,P˜n≽GIVPHFOWGλ2P˜1,P˜2,⋯,P˜n.
From Definition17 and Definition 18, we know that the IVPHFWA, IVPHFWG, GIVPHFWA, and GIVPHFWG operators weight the interval-valued hesitant fuzzy argument itself and there is no consideration of the impact of the order of parameters, while the IVPHFOWA, IVPHFOWG, GIVPHFOWA, and GIVPHFOWG operators only weight the ordered position of each given IVPHFE and there is no consideration of the impact of the argument itself. Hence, we propose a new hybrid aggregation operator which can weight both all the given arguments and their ordered positions.


## 4.3. The IVPHFHA, IVPHFHG, GIVPHFHA, and GIVPHFHG Operators

Definition 19.
For a collection of some IVPHFEsP˜ii=1,2,⋯,n, ω=ω1,ω2,⋯,ωnT is the weight vector of them with ωi∈0,1 and ∑i=1nωi=1. Suppose κ=κ1,κ2,⋯,κnT is the associated vector such that κi∈0,1, ∑i=1nκi=1, and λ>0. Then,(1)
An interval-valued Pythagorean hesitant fuzzy hybrid averaging (IVPHFHA) operator can be seen a mapIVPHFHA:Ωn⟶Ω, such that(49)IVPHFHAP˜1,⋯,P˜n=κ1P˜˙σ1⊕⋯⊕κnP˜˙σn=1−∏i=1n1−μ˙σi−2κi,1−∏i=1n1−μ˙σi+2κi,∏i=1nν˙σi−κi,∏i=1nν˙σi+κiμ˜˙σi,ν˜˙σi∈P˜˙σi,i=1,⋯,n,where P˜˙σi is the largest ith of P˜˙k=nωkP˜kk=1,2,⋯,n.(2)
An interval-valued Pythagorean hesitant fuzzy hybrid geometric (IVPHFHG) operator can be seen a mapIVPHFHG:Ωn⟶Ω, such that(50)IVPHFHGP˜1,⋯,P˜n=P˜¨σ1κ1⊗⋯⊗P˜¨σnκn=∏i=1nμ¨σi−κi,∏i=1nμ¨σi+κi,1−∏i=1n1−ν¨σi−2κi,1−∏i=1n1−ν¨σi+2κiμ˜¨σi,ν˜¨σi∈P˜¨σi,i=1,⋯,n.where P˜¨σi is the largest ith of P˜¨k=P˜knωkk=1,2,⋯,n.(3)
A generalized interval-valued Pythagorean hesitant fuzzy hybrid averaging (GIVPHFHA) operator can be seen a mapGIVPHFHA:Ωn⟶Ω, which satisfies(51)GIVPHFHAλP˜1,⋯,P˜n=κ1P˜˙σ1λ⊕⋯⊕κnP˜˙σnλ1/λ=1−∏i=1n1−μ˙σi−2λκi1/λ,1−∏i=1n1−μ˙σi+2λκi1/λ,1−1−∏i=1n1−1−ν˙σi−2λκi1/λ,1−1−∏i=1n1−1−ν˙σi+2λκi1/λ,μ˜˙σi,ν˜˙σi∈P˜˙σi,i=1,⋯,n,where P˜˙σi is the largest ith of P˜˙k=nωkP˜kk=1,2,⋯,n.(4)
A generalized interval-valued Pythagorean hesitant fuzzy hybrid geometric (GIVPHFHG) operator can be seen a mapGIVPHFHG:Ωn⟶Ω, which satisfies(52)GIVPHFHGλP˜1,⋯,P˜n=1λλP˜¨σ1κ1⊗⋯⊗λP˜¨σnκn=1−1−∏i=1n1−1−μ¨σi−2λκi1/λ,1−1−∏i=1n1−1−μ¨σi+2λκi1/λ,1−∏i=1n1−ν¨σi−2λκi1/λ,1−∏i=1n1−ν¨σi+2λκi1/λ,μ˜˙σi,ν˜˙σi∈P˜¨σi,i=1,⋯,n,where P˜¨σi is the largest ith of P˜˙k=P˜knωkk=1,2,⋯,n.
Ifλ=1, then the GIVPHFHA and GIVPHFHG operators reduce to the IVPHFHA and IVPHFHG operators, respectively.Example 3.
SupposeP˜1=0.3,0.5,0.2,0.4,(53)P˜2=0.1,0.4,0.3,0.6,0.1,0.4,0.5,0.7,P˜3=0.3,0.4,0.2,0.6,0.1,0.3,0.2,0.6,0.4,0.7,0.2,0.6,are three IVPHFEs, the weight vector of the attributes is ω=0.15,0.30,0.55T, and the aggregation-associated vector is κ=0.2,0.45,0.35T. Then, we can obtain(54)P˜˙1=3×0.15⊗P˜1=0.2038,0.3485,0.4847,0.6621,P˜˙2=3×0.3⊗P˜2=0.0949,0.3811,0.3384,0.6314,0.0949,0.3811,0.5359,0.7254,P˜˙3=3×0.3⊗P˜3=0.3796,0.5000,0.0703,0.4305,0.1282,0.3796,0.0703,0.4305,0.5000,0.8190,0.0703,0.4305.P˜¨1=P˜13×0.15=0.5817,0.7320,0.1349,0.2747,P˜¨2=P˜23×0.3=0.1259,0.4384,0.2853,0.5751,0.1259,0.4384,0.4776,0.6741,P˜¨3=P˜33×0.3=0.1372,0.2205,0.2552,0.7219,0.0224,0.1372,0.2552,0.7219,0.2205,0.5552,0.2552,0.7219.SP˜˙1=−0.1984,−0.0567,SP˜˙2=−0.2267,−0.0278,SP˜˙3=−0.0242,0.1750,SP˜¨1=0.1315,0.2588,SP˜¨2=−0.1884,0.0187,SP˜¨3=−0.2493,0.0300.
By Definition16, we have PSP˜˙1≤SP˜˙2=1, PSP˜˙2≤SP˜˙3=0.5009, and PSP˜¨3≤SP˜¨2=0.5509, and PSP˜¨2≤SP˜¨1=1. So, we can get(55)P˜˙σ1=P˜˙3,P˜˙σ2=P˜˙2,P˜˙σ3=P˜˙1,P˜¨σ1=P˜¨1,P˜¨σ2=P˜¨2,P˜¨σ3=P˜¨3.
According to Definition19, we can obtain(56)IVPHFHAP˜1,P˜2,P˜3=κ1P˜˙σ1⊕κ1P˜˙σ2⊕κnP˜˙σ3=1−∏i=131−μ˙σi−2κi,1−∏i=131−μ˙σi+2κi,∏i=13ν˙σi−κi,∏i=13ν˙σi+κiμ˜˙σi,ν˜˙σi∈P˜˙σi,i=1,2,3=0.2209,0.3991,0.2802,0.5947,0.2209,0.3991,0.3446,0.6330,0.1483,0.3698,0.2802,0.5947,0.1483,0.3698,0.3446,0.6330,0.2713,0.5356,0.2802,0.5947,0.2713,0.5356,0.3446,0.6330;IVPHFHGP˜1,P˜2,P˜3=P˜¨σ1κ1⊗P˜¨σ2κ2⊗P˜¨σ3κ3=∏i=13μ¨σi−κi,∏i=13μ¨σi+κi,1−∏i=131−ν¨σi−2κi,1−∏i=131−ν¨σi+2κiμ˜¨σi,ν˜¨σi∈P˜¨σi,i=1,2,3=0.1762,0.3819,0.2517,0.6042,0.0934,0.3235,0.2517,0.6042,0.2080,0.5276,0.2517,0.6042,0.1762,0.3819,0.3659,0.6487,0.0934,0.3234,0.3659,0.6487,0.2080,0.5276,0.3659,0.6487;GIVPHFHA3P˜1,P˜2,P˜3=κ1P˜˙σ13⊕κ2P˜˙σ23⊕κ3P˜˙σ331/3=1−∏i=131−μ˙σi−2×3κi1/3,1−∏i=131−μ˙σi+2×3κi1/3,1−1−∏i=131−1−ν˙σi−23κi1/3,1−1−∏i=131−1−ν˙σi+23κi1/3,μ˜˙σi,ν˜˙σi∈P˜˙σi,i=1,2,3=0.2924,0.4155,0.2713,0.5854,0.2924,0.4155,0.3255,0.6153,0.1725,0.3709,0.2713,0.5854,0.1725,0.3709,0.3256,0.6153,0.3833,0.6438,0.2713,0.5854,0.3833,0.6438,0.3256,0.6153;GIVPHFHG3P˜1,P˜2,P˜3=133P˜¨σ1κ1⊗3P˜¨σ2κ2⊗3P˜¨σ3κ3=1−1−∏i=131−1−μ¨σi−23κi1/3,1−1−∏i=131−1−μ¨σi+23κi1/3,1−∏i=131−ν¨σi−2×3κi1/3,1−∏i=131−ν¨σi+2×3κi1/3,μ˜¨σi,ν˜¨σi∈P˜¨σi,i=1,2,3=0.1714,0.3659,0.2642,0.6384,0.0902,0.3050,0.2642,0.6384,0.2024,0.5174,0.2642,0.6384,0.1714,0.3659,0.4196,0.6734,0.0902,0.3050,0.4196,0.6734,0.2024,0.5174,0.4196,0.6734.


## 5. The Application of the Aggregation Operators of IVPHFEs for MAGDM

In multiattribute group decision-making (MAGDM) problems, because the different decision makers usually come from different research directions and backgrounds, they usually have diverging opinions. Thus, how to construct an optimal model that can be obtained with the maximum degree of consensus or agreement of these experts for the given alternatives is an important and interesting research content in MAGDM. In this section, we propose a new MAGDM method based on the interval-valued Pythagoras hesitant fuzzy integration operators. Furthermore, the ranking of integrated interval-valued Pythagorean hesitant fuzzy numbers is accomplished by adopting the score function. The specific details of the MAGDM method is shown in following.LetY=Yii=1,2,⋯,m be a finite set of alternatives, C=Cjj=1,2,⋯,n be a set of attributes, and D=Dkk=1,2,⋯,l be a set of l decision makers. Then, let Mk=P˜ijkm×n is an interval-valued Pythagorean hesitant fuzzy decision matrix (IVPHFDM) of the kth decision maker, where P˜ijk=μ˜ijk,ν˜ijk,i=1,2,⋯,m,j=1,2,⋯,n is an IVPHFE given by the decision maker DK, in which μ˜ijk indicates the possible membership intervals that the alternative Yi satisfies the attribute Cj and ν˜ijk indicates the possible nonmembership intervals that the alternative Yi nonmembership intervals not satisfy the attribute Cj.In the actual MAGDM problem, we will encounter some attributes are benefit (i.e., the bigger the attribute values, the better) and the other attributes are cost (i.e., the smaller the attribute values, the better). In such cases, transform the cost attribute values into the benefit attribute values and normalize the IVPHFDMMk=P˜ijkm×n into the corresponding IVPHFDM Nk=P˜ijkm×n by the method in [53] where(57)P˜ijk=P˜ijk,for bene fit attributeCj,P˜ijkC,for cost attributeCj,where P˜ijkC is the complement of P˜ijk.Based on the above discussion and analysis, we develop an approach for multiattribute decision-making in interval-valued Pythagorean hesitant fuzzy environments. The algorithm involves the following steps:Step 1: construct the IVPHFDMMk=P˜ijkm×n and transform Mk into the corresponding normalized matrix Nk=P˜ijkm×n.Step 2: utilize the GIVPHFWA operator (or the GIVPHFWG operator) to aggregate all the IVPHFDMsNk=P˜ijkm×nk=1,2,⋯,l into the collective IVPHFDM N=P˜ijm×n. where ξ=ξ1,ξ2,⋯,ξlT is the weight vector of the decision makers Dkk=1,2,⋯,l, and the specific integration operation is as follows:(58)P˜ij=GIVPHFWAλP˜ij1,⋯,P˜ijl=1−∏k=1l1−μP˜ijk−2λξk1/λ,1−∏k=1l1−μP˜ijk+2λξk1/λ,1−1−∏k=1l1−1−νP˜ijk−2λξk1/λ,1−1−∏k=1l1−1−νP˜ijk+2λξk1/λ,μ˜P˜ijk,ν˜P˜ijk∈P˜ijk,i=1,⋯,m,j=1,⋯,n.or(59)P˜ij=GIVPHFWGλP˜ij1,⋯,P˜ijl=1−1−∏k=1l1−1−μP˜ijk−2λξk1/λ,1−1−∏k=1l1−1−μP˜ijk+2λξk1/λ,1−∏k=1l1−νP˜ijk−2λξk1/λ,1−∏k=1l1−νP˜ijk+2λξk1/λ,μ˜P˜ijk,ν˜P˜ijk∈P˜ijk,i=1,⋯,m,j=1,⋯,n.Step 3: utilize the GIVPHFHA operator (or the GIVPHFHG operator) to aggregate all the preference valuesP˜i again, where κ=κ1,κ2,⋯,κnT is the associated weight vector. The specific integration operation is as follows:(60)P˜i=GIVPHFHAλP˜i1,⋯,P˜in=1−∏j=1n1−μ˙P˜iσj−2λκj1/λ,1−∏j=1n1−μ˙P˜iσj+2λκj1/λ,1−1−∏j=1n1−1−ν˙P˜iσj−2λκj1/λ,1−1−∏j=1n1−1−ν˙P˜iσj+2λκj1/λ,μ˜˙P˜iσj,ν˜˙P˜iσj∈P˜˙iσj,i=1,⋯,m,j=1,⋯,n.or P˜i=GIVPHFHGλP˜i1,⋯,P˜in(61)=1−1−∏j=1n1−1−μ¨P˜iσj−2λκj1/λ,1−1−∏j=1n1−1−μ¨P˜iσj+2λκj1/λ,1−∏j=1n1−ν¨P˜iσj−2λκj1/λ,1−∏j=1n1−ν¨P˜iσj+2λκj1/λ,ν˜¨P˜iσj,ν˜¨P˜iσj∈P˜¨iσj,i=1,⋯,m,j=1,⋯,n.Here, letω=ω1,ω2,⋯,ωnT be the weight vector of the attributes Cjj=1,2,⋯,n, where ωj∈0,1j=1,2,⋯,n and ∑j=1nωj=1. So, we can get P˜˙iσj orP˜¨iσj defined by(62)P˜˙ij=n×ωj⊗P˜ij=1−1−μP˜ij−2n×ωj,1−1−μP˜ij+2n×ωj,νP˜ij−n×ωj,νP˜ij+n×ωjμ˜P˜ij,ν˜P˜ij∈P˜ij,i=1,⋯,m,j=1,⋯,n,P˜¨ij=P˜ijn×ωj=μP˜ij−n×ωj,μP˜ij+n×ωj,1−1−νP˜ij−2n×ωj,1−1−νP˜ij+2n×ωjμ˜P˜ij,ν˜P˜ij∈P˜ij,i=1,⋯,m,j=1,⋯,n.Step 4: compute the score valuesSP˜i and the accuracy values HP˜i of P˜ii=1,2,⋯,m by Definition 15.Step 5: get the priority of the alternativesYi by ranking SP˜ii=1,2,⋯,m based on Definition 16, End.Example 4.
A company needs to select a new site for buildings. Three alternativesYi i=1,2,3 are available, and the three decision makers Dk k=1,2,3 consider three attributes to decide which project to choose: (1) C1 (price), (2) C2 (environment), and (3) C3 (location). Among the considered attributes, C1 is of cost type and C2andC3 are of benefit type. The weight vector of the decision makers Dkk=1,2,3 is ξ=0.1,0.6,0.3T. The weight vector of the attributes Cjj=1,2,3 is ω=0.5,0.2,0.3T. Suppose that the decision makers provide their own IVPHFDM Mk=P˜ijk3×3 as listed in Tables 1–3, respectively, where P˜ijk is an IVPHFE given by the decision maker Dk (Tables 4–7).Step 1: transform the decision matrixMk into the corresponding normalized matrix Nk=P˜ijk3×3k=1,2,3 (in Tables 4–6).Step 2: supposeλ=5, and utilize the GIVPHFWA operator to aggregate the three IVPHFDMs Nk=P˜ijk3×3k=1,2,3 into the collective IVPHFDM N=P˜ij3×3 (in Table 7).Step 3: aggregate all the preference valuesP˜ijj=1,2,3 in the ith line of N based on the GIVPHFHA operators, whose associated weighting vector is κ=0.25,0.65,0.1T.Step 4: compute the three score values as follows:(63)SP˜1=0.2736,0.2833,SP˜2=0.2142,0.2988,SP˜3=0.2398,0.2741.Step 5: get the priority of the alternatives by ranking the score functions. We can get the ranking order of all alternatives:Y1≻Y3≻Y2. So, the optimal scheme is Y1.
Furthermore, we study the change of the GIVPHFHA and GIVPHFHG operators with the parameterλ.
In Table8, we can find that the score values obtained by the GIVPHFHA operators become bigger as the parameter λ increases for the same aggregation arguments, and the decision makers can choose the values of λ according to their preferences, see that, as the parameter λ changes, we have different results. Suppose λ=0.1,0.2,⋯,50, then we analyze the impact of the role of λ on the aggregation results, and see the trend in Figure 1. Because of space limitations, we gave the image within the range of λ∈0.7,11.9, the same hereinafter.
From Figure1, we can see that all of SP˜ii=1,2,3 increase with the increase of λ. When λ=0.7, the ranking order of the three alternatives is Y2≻Y3≻Y1 and the best choice is Y2; when λ∈0.8,1.5, the ranking order of the three alternatives is Y3≻Y2≻Y1 and the best choice is Y3; when λ∈1.6,11.4, the ranking order of the three alternatives is Y1≻Y3≻Y2 and the best choice is Y1; when λ∈11.5,50, the ranking order of the three alternatives is Y2≻Y1≻Y3 and the best choice is Y2.
In Table9, we use the GIVPHFHG operators to aggregate the values of the alternatives. We find the score values obtained by the GIVPHFHG operators become smaller as the parameter λ increases for the same aggregation arguments, and the decision makers can choose the values of λ according to their preferences. See the details in Figure 2.
From Figure2, we can see that all of SP˜ii=1,2,3 decrease with the increase of λ. When λ∈0.7,1.2, the ranking order of the three alternatives is Y1≻Y2≻Y3 and the best choice is Y1; when λ∈1.3,50, the ranking order of the three alternatives is Y2≻Y1≻Y3 and the best choice is Y2.Table 1
IVPHFDMM1.
C
1
C
2
C
3
Y
1
0.4
,
0.5
,
0.3
,
0.5
0.2
,
0.3
,
0.6
,
0.8
0.6
,
0.8
,
0.1
,
0.3
,
0.6
,
0.7
,
0.3
,
0.3
,
0.7
,
0.9
,
0.1
,
0.1
Y
2
0.1
,
0.1
,
0.8
,
0.9
,
0.2
,
0.4
,
0.5
,
0.6
0.4
,
0.5
,
0.5
,
0.5
,
0.6
,
0.7
,
0.1
,
0.2
,
0.6
,
0.7
,
0.1
,
0.2
0.8
,
0.9
,
0.1
,
0.1
Y
3
0.2
,
0.3
,
0.6
,
0.6
,
0.1
,
0.4
,
0.6
,
0.7
0.8
,
0.9
,
0.1
,
0.1
0.3
,
0.5
,
0.4
,
0.4Table 2
IVPHFDMM2.
C
1
C
2
C
3
Y
1
0.7
,
0.9
,
0.1
,
0.2
,
0.7
,
0.8
,
0.1
,
0.2
,
0.6
,
0.8
,
0.1
,
0.1
0.4
,
0.6
,
0.3
,
0.5
,
0.5
,
0.7
,
0.2
,
0.3
0.9
,
0.9
,
0.1
,
0.2
Y
2
0.7
,
0.8
,
0.1
,
0.2
0.8
,
0.9
,
0.1
,
0.3
0.5
,
0.8
,
0.2
,
0.3
,
0.6
,
0.8
,
0.1
,
0.2
Y
3
0.2
,
0.3
,
0.5
,
0.6
0.6
,
0.8
,
0.2
,
0.2
,
0.7
,
0.9
,
0.1
,
0.1
0.1
,
0.3
,
0.6
,
0.7
,
0.2
,
0.2
,
0.7
,
0.8
,
0.3
,
0.4
,
0.6
,
0.6,Table 3
IVPHFDMM3.
C
1
C
2
C
3
Y
1
0.3
,
0.5
,
0.4
,
0.5
0.2
,
0.3
,
0.5
,
0.6
,
0.1
,
0.2
,
0.8
,
0.9
0.7
,
0.8
,
0.2
,
0.2
Y
2
0.4
,
0.6
,
0.3
,
0.4
,
0.4
,
0.5
,
0.5
,
0.5
0.1
,
0.3
,
0.6
,
0.9
0.3
,
0.5
,
0.5
,
0.5
,
0.1
,
0.2
,
0.7
,
0.7
,
0.3
,
0.4
,
0.5
,
0.6
Y
3
0.8
,
0.9
,
0.1
,
0.1
,
0.5
,
0.7
,
0.1
,
0.2
0.3
,
0.6
,
0.4
,
0.4
,
0.4
,
0.5
,
0.4
,
0.5
,
0.3
,
0.5
,
0.4
,
0.5
0.9
,
0.9
,
0.1
,
0.1Table 4
IVPHFDMN1.
C
1
C
2
C
3
Y
1
0.3
,
0.5
,
0.4
,
0.5
0.2
,
0.3
,
0.6
,
0.8
0.6
,
0.8
,
0.1
,
0.3
,
0.6
,
0.7
,
0.3
,
0.3
,
0.7
,
0.9
,
0.1
,
0.1
Y
2
0.8
,
0.9
,
0.1
,
0.1
,
0.5
,
0.6
,
0.2
,
0.4
0.4
,
0.5
,
0.5
,
0.5
,
0.6
,
0.7
,
0.1
,
0.2
,
0.6
,
0.7
,
0.1
,
0.2
0.8
,
0.9
,
0.1
,
0.1
Y
3
0.6
,
0.6
,
0.2
,
0.3
,
0.6
,
0.7
,
0.1
,
0.4
0.8
,
0.9
,
0.1
,
0.1
0.3
,
0.5
,
0.4
,
0.4Table 5
IVPHFDMN2.
C
1
C
2
C
3
Y
1
0.1
,
0.2
,
0.7
,
0.9
,
0.1
,
0.2
,
0.7
,
0.8
,
0.1
,
0.1
,
0.6
,
0.8
0.4
,
0.6
,
0.3
,
0.5
,
0.5
,
0.7
,
0.2
,
0.3
0.9
,
0.9
,
0.1
,
0.2
Y
2
0.1
,
0.2
,
0.7
,
0.8
0.8
,
0.9
,
0.1
,
0.3
0.5
,
0.8
,
0.2
,
0.3
,
0.6
,
0.8
,
0.1
,
0.2
Y
3
0.5
,
0.6
,
0.2
,
0.3
0.6
,
0.8
,
0.2
,
0.2
,
0.7
,
0.9
,
0.1
,
0.1
0.1
,
0.3
,
0.6
,
0.7
,
0.2
,
0.2
,
0.7
,
0.8
,
0.3
,
0.4
,
0.6
,
0.6Table 6
IVPHFDMN3.
C
1
C
2
C
3
Y
1
0.4
,
0.5
,
0.3
,
0.5
0.2
,
0.3
,
0.5
,
0.6
,
0.1
,
0.2
,
0.8
,
0.9
0.7
,
0.8
,
0.2
,
0.2
Y
2
0.3
,
0.4
,
0.4
,
0.6
,
0.5
,
0.5
,
0.4
,
0.5
0.1
,
0.3
,
0.6
,
0.9
0.3
,
0.5
,
0.5
,
0.5
,
0.1
,
0.2
,
0.7
,
0.7
,
0.3
,
0.4
,
0.5
,
0.6
Y
3
0.1
,
0.1
,
0.8
,
0.9
,
0.1
,
0.2
,
0.5
,
0.7
0.3
,
0.6
,
0.4
,
0.4
,
0.4
,
0.5
,
0.4
,
0.5
,
0.3
,
0.5
,
0.4
,
0.5
0.9
,
0.9
,
0.1
,
0.1Table 7
IVPHFDMN.
C
1
C
2
C
3
Y
1
0.3553
,
0.4562
,
0.4579
,
0.6046
,
0.3559
,
0.4562
,
0.4579
,
0.6014
,
0.3553
,
0.4562
,
0.4404
,
0.6014
0.3801
,
0.5702
,
0.3631
,
0.5396
,
0.4751
,
0.6655
,
0.2778
,
0.3778
,
0.3801
,
0.5702
,
0.3836
,
0.5614
,
0.4751
,
0.6655
,
0.2913
,
0.3870
0.8648
,
0.8744
,
0.1226
,
0.2079
,
0.8648
,
0.8722
,
0.1363
,
0.2079
,
0.8654
,
0.8818
,
0.1226
,
0.1862,
Y
2
0.6387
,
0.7283
,
0.4172
,
0.4696
,
0.3979
,
0.4791
,
0.4643
,
0.6207
,
0.6403
,
0.7288
,
0.4172
,
0.4511
,
0.4562
,
0.4959
,
0.4643
,
0.5790
0.7619
,
0.8622
,
0.1814
,
0.3798
,
0.7625
,
0.8630
,
0.1568
,
0.3451
,
0.7620
,
0.8630
,
0.1568
,
0.3451
0.6419
,
0.7985
,
0.2351
,
0.3016
,
0.6566
,
0.7988
,
0.1530
,
0.2351
,
0.6419
,
0.7983
,
0.2447
,
0.3150
,
0.6566
,
0.7983
,
0.1588
,
0.2447
,
0.6419
,
0.7983
,
0.2351
,
0.3104
,
0.6566
,
0.7983
,
0.1530
,
0.2414
Y
3
0.4767
,
0.4767
,
0.5428
,
0.6111
,
0.4767
,
0.5567
,
0.4758
,
0.6616
,
0.4767
,
0.4767
,
0.4998
,
0.6014
,
0.4767
,
0.5567
,
0.4447
,
0.6475
0.6566
,
0.7995
,
0.2247
,
0.2247
,
0.6998
,
0.8742
,
0.1466
,
0.1466
,
0.6567
,
0.7985
,
0.2247
,
0.2351
,
0.6998
,
0.8738
,
0.1466
,
0.1530
,
0.6566
,
0.7985
,
0.2247
,
0.2351
,
0.6998
,
0.8738
,
0.1466
,
0.1530
0.8094
,
0.8095
,
0.2913
,
0.2997
,
0.8094
,
0.8095
,
0.2997
,
0.3029
,
0.8094
,
0.8095
,
0.2913
,
0.2913Table 8
Score values based on the GIVPHFHA operators.
λ
=
0.1
λ
=
10
λ
=
20
Y
1
−
0.4036
,
−
0.3452
0.3207
,
0.3240
0.3519
,
0.3530
Y
2
0.0264
,
0.0736
0.2865
,
0.3506
0.3352
,
0.3988
Y
3
−
0.0184
,
0.0027
0.2979
,
0.3073
0.3388
,
0.3396
Ranking
Y
2
≻
Y
3
≻
Y
1
Y
1
≻
Y
2
≻
Y
3
Y
2
≻
Y
1
≻
Y
3
λ
=
30
λ
=
40
λ
=
50
Y
1
0.3634
,
0.3641
0.3694
,
0.3699
0.3730
,
0.3734
Y
2
0.3533
,
0.4179
0.3627
,
0.4279
0.3684
,
0.4339
Y
3
0.3543
,
0.3544
0.3624
,
0.3624
0.3673
,
0.3673
Ranking
Y
2
≻
Y
1
≻
Y
3
Y
2
≻
Y
1
≻
Y
3
Y
2
≻
Y
1
≻
Y
3Figure 1
Variation of the score functions of the GIVPHFHA operators by the parameterλ.Table 9
Score values based on the GIVPHFHG operators.
λ
=
0.1
λ
=
10
λ
=
20
Y
1
0.392
,
0.4493
−
0.3225
,
−
0.2264
−
0.3633
,
−
0.2567
Y
2
−
0.0184
,
0.0047
−
0.2448
,
−
0.1454
−
0.2774
,
−
0.1621
Y
3
−
0.0148
,
0.0178
−
0.3424
,
−
0.2848
−
0.3950
,
−
0.3340
Ranking
Y
1
≻
Y
2
≻
Y
3
Y
2
≻
Y
1
≻
Y
3
Y
2
≻
Y
1
≻
Y
3
λ
=
30
λ
=
40
λ
=
50
Y
1
−
0.3793
,
−
0.2681
−
0.3877
,
−
0.2344
−
0.3928
,
−
0.2011
Y
2
−
0.2899
,
−
0.0968
−
0.2217
,
−
0.0990
−
0.2244
,
−
0.1006
Y
3
−
0.4149
,
−
0.3524
−
0.4254
,
−
0.3619
−
0.4318
,
−
0.3677
Ranking
Y
2
≻
Y
1
≻
Y
3
Y
2
≻
Y
1
≻
Y
3
Y
2
≻
Y
1
≻
Y
3Figure 2
Variation of the score functions of the GIVPHFHG operators by the parameterλ.


## 6. Conclusions

This paper extends IVHFs and IVPFs to IVPHFSs. Firstly, the new score functions and accuracy functions of IVPHFSs are introduced to compare the size of IVPHFEs based on the comparison of interval numbers. Then, we focus on a new series of interval-valued Pythagorean hesitant fuzzy aggregation operators, including the IVPHFWA, IVPHFWG, GIVPHFWA, GIVPHFWG, IVPHFOWA, IVPHFOWG, GIVPHFOWA, GIVPHFOWG, IVPHFHA, IVPHFHG, GIVPHFHA, and GIVPHFHG operators. The relations of these operators are developed. Furthermore, a new approach has been developed based on the proposed operators to solve the MAGDM problems under the IVPHF environment. Finally, a numerical example is used to illustrate the effectiveness and feasibility of our proposed method.The following work is to enhance the study of the aggregation operators of weighted IVPHFSs; we shall develop the new potential application of the proposed operators in another field such as clustering analysis, image processing, pattern recognition, and so on. We hope that these will enrich and provide more new ideas and new methods for these fields under the interval-valued Pythagorean hesitant fuzzy environment.



---

*Source: 1724943-2020-02-13.xml*