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

Pythagorean hesitant fuzzy sets are widely watched because of their excellent ability to deal with uncertainty, imprecise and vague information. ,is 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. ,en, 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 intervalvalued Pythagorean fuzzy information as much as possible. ,en, 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.

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], intervalvalued hesitant fuzzy sets [26][27][28], and interval-valued intuitionistic fuzzy sets [29]. ese 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. e sets are represented by pairs of two values 〈μ P (x), ] P (x)〉, which satisfies μ 2 P (x) + ] 2 P (x) ≤ 1. Obviously, their application range is broader than that of intuitionistic fuzzy sets. ese 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 decisionmaking 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.
and 1] . Liu and He [45] defined that, for any x, any μ A (x) ∈ Γ A (x), and any ] A (x) ∈ Ψ A (x), (μ A (x)) 2 + (] A (x)) 2 ≤ 1 holds. Khan et al. [46] defined that, for any x and for any μ A (x) ∈ Γ A (x), there is ] A (x) ∈ Ψ A (x), such that (μ A (x)) 2 + (] A (x)) 2 ≤ 1. Also, for any ] A (x) ∈ Ψ A (x), there is μ A (x) ∈ Γ A (x), such that (μ A (x)) 2 + (] A (x)) 2 ≤ 1. Wei et al. [47] also defined another Pythagorean fuzzy set P � 〈x, h P (x)〉 | x ∈ U , where h P (x) is a set of some Pythagorean fuzzy elements in U. e 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. e aim of this paper is to extend PHFSs to interval-valued Pythagorean hesitant fuzzy sets (IVPHFSs) and develop MAGDM approaches to intervalvalued 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. e rest of this paper is organized as follows. In Section 2, we review some basic concepts and results of hesitant fuzzy sets, interval-valued hesitant fuzzy sets, and intervalvalued 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 intervalvalued 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.

Preliminaries
is section will briefly review the basic notations and results of hesitant fuzzy sets, interval-valued Pythagorean fuzzy sets.
Hereinafter, without explanation, let U be a nonempty finite set, called the universe of discourse, and D [x, y] be the set of all closed subintervals of the interval [x, y]. (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.

Hesitant Fuzzy Set
Definition 1 (see [20]). A hesitant fuzzy set (HFS) E on U is described as 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 ey gave some operations on HFEs.
Definition 3 (see [30]). For an HFE h, S(h) � 1/|h| c ∈ h c is called the score function of h with |·| denoting the cardinality here and below. For two HFEs, h 1

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.
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.

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 where μ P (x) and ] P (x) 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 decisionmaking problems, interval-valued fuzzy elements can provide a better solving way. Peng et al. [40] focused on intervalvalued Pythagorean fuzzy sets (IVPFSs), whose ideas are similar to interval-valued intuitionistic fuzzy sets.

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 where where μ P (x) and ] P (x) are the possible Pythagorean membership intervals and the possible Pythagorean nonmembership intervals of P at x, respectively. e set of all IVPHFEs on U is denoted by Ω.
Obviously, for each x ∈ U, if h P (x) includes only one pair of intervals, the IVPHFS degenerates into an IVPFS; if both μ P (x) and ] P (x) degenerate one singleton, the IVPHFS can be seen as a PHFS; if ] P (x) � [0, 0], the IVPHFS can be seen as an IVHFS; if μ + P (x) + ] + P (x) ≤ 1, the IVPHFS can be seen as an interval-valued intuitionistic hesitant fuzzy set [29]. For convenience, we call each pair P � h P (x) as an interval-valued Pythagorean hesitant fuzzy element Based on the operators of IVHFEs [27] and IVPFEs [40], the operational laws of IVPHFEs are defined as follows.
Proof. Obviously, P C is an IVPHFE.
(1) Based on Definition 14, claim (1) is obvious, so here the proof process is overleaped.
Similarly, we have P Similarly, we have λ(P C ) � (P λ ) C .
Similarly, we have ( All the claims of the proposition are proved. □ Proposition 3. Let P, P 1 , P 2 , and P 3 be four IVPHFEs and λ > 0, then ey are trial. We omit them.

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 Definition 12, 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.
be an IVPHFE. e score function S(P) is described as e accuracy function H(P) is described as e idea of the above concepts originated from Definition 8 and used the definition of interval arithmetic in Definition 6. e 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 S(P) ∈ [− 1, 1] and the accuracy function H(P) ∈ [0, 1] hold obviously. Based on Definition 7, we have the following definition.
Definition 16. Let P 1 and P 2 be two IVPHFEs.

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 intervalvalued Pythagorean hesitant fuzzy information in this section based on the discussion of aggregation operators in [27,45,50,51] and deduce some desirable properties.

Complexity
Based on Proposition 1, we know the above four aggregation operators are all IVPHFEs. Also, we can obtain that the score values are So, P SIVPHFWG P 1 , P 2 , P 3 ≤ S IVPHFWA P 1 , P 2 , P 3 � 0.5518; Hence, IVPHFWG P 1 , P 2 , P 3 ≼ IVPHFWA P 1 , P 2 , P 3 ; In fact, we have the following.
□ Proposition 4.2 shows no matter how the parameter λ (λ > 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.

(4) A generalized interval-valued Pythagorean hesitant fuzzy ordered weighted geometric (GIVPHFOWG)
operator can be seen a map GIVPHFOWG: Ω n ⟶ Ω, which satisfies where λ > 0. If λ � 1, then the GIVPHFOWA and GIVPHFOWG operators reduce to the IVPHFOWA and IVPHFOWG operators, respectively. By comparing Definition 17 with Definition 18, we can find that the IVPHFOWA, IVPHFOWG, GIVPH-FOWA, 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 κ i is associated with the i th position in the collection of the IVPHFEs during aggregation processes.
According to Definition 18, we have Complexity 13 So, IVPHFOWG P 1 , P 2 , P 3 ≼ IVPHFOWA P 1 , P 2 , P 3 , Similar to Proposition 5-7, we easily obtain the following properties of ordered aggregation operators of IVPHFEs.

Complexity
If λ � 1, then the GIVPHFHA and GIVPHFHG operators reduce to the IVPHFHA and IVPHFHG operators, respectively.

Complexity
According to Definition 19, we can obtain    (56)

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. us, 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. e specific details of the MAGDM method is shown in following.
Let Y � Y i | i � 1, 2, · · · , m be a finite set of alternatives, C � C j | j � 1, 2, · · · , n be a set of attributes, and D � D k | k � 1, 2, · · · , l be a set of l decision makers. en, ij ) m×n is an interval-valued Pythagorean hesitant fuzzy decision matrix (IVPHFDM) of the k th decision maker, where P ij , ] (k) ij 〉 , (i � 1, 2, · · · , m, j � 1, 2, · · · , n) is an IVPHFE given by the decision maker D K , in which μ (k) ij indicates the possible membership intervals that the alternative Y i satisfies the attribute C j and ] (k) ij indicates the possible nonmembership intervals that the alternative Y i nonmembership intervals not satisfy the attribute C j .
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 IVPHFDM M (k) � (P (k) ij ) m×n into the corresponding IVPHFDM N (k) � (P (k) ij ) m×n by the method in [53] where for bene fit attribute C j , for cost attribute C j , ij . Based on the above discussion and analysis, we develop an approach for multiattribute decision-making in intervalvalued Pythagorean hesitant fuzzy environments. e algorithm involves the following steps: Step 1: construct the IVPHFDM M (k) � (P (k) ij ) m×n and transform M (k) into the corresponding normalized matrix N (k) � (P (k) ij ) m×n .
Step 5: get the priority of the alternatives Y i by ranking S(P i ) (i � 1, 2, · · · , m) based on Definition 16, End.  Tables 1-3, respectively, where P (k) ij is an IVPHFE given by the decision maker D k (Tables 4-7).
Step 3: aggregate all the preference values P ij (j � 1, 2, 3) in the i th line of N based on the GIVPHFHA operators, whose associated weighting vector is κ � (0.25, 0.65, 0.1) T .
Step 4: compute the three score values as follows: (63) Step 5: get the priority of the alternatives by ranking the score functions. We can get the ranking order of all alternatives: Y 1 ≻ Y 3 ≻ Y 2 . So, the optimal scheme is Y 1 .
Furthermore, we study the change of the GIVPHFHA and GIVPHFHG operators with the parameter λ.
In Table 8, we can find that the score values obtained by the GIVPHFHA operators become bigger as the parameter λ Table 1: IVPHFDM M (1) .   . 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 Figure 1, we can see that all of S(P i )(i � 1, 2, 3) increase with the increase of λ. When λ � 0.7, the ranking order of the three alternatives is Y 2 ≻ Y 3 ≻ Y 1 and the best choice is Y 2 ; when λ ∈ [0.8, 1.5], the ranking order of the three alternatives is Y 3 ≻ Y 2 ≻ Y 1 and the best choice is Y 3 ; when λ ∈ [1.6, 11.4], the ranking order of the three alternatives is Y 1 ≻ Y 3 ≻ Y 2 and the best choice is Y 1 ; when λ ∈ [11.5, 50], the ranking order of the three alternatives is Y 2 ≻ Y 1 ≻ Y 3 and the best choice is Y 2 .
In Table 9, we use the GIVPHFHG operators to aggregate the values of the alternatives. We find the score Table 4: IVPHFDM N (1) .  .         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 Figure 2, we can see that all of S(P i )(i � 1, 2, 3) decrease with the increase of λ. When λ ∈ [0.7, 1.2], the ranking order of the three alternatives is Y 1 ≻ Y 2 ≻ Y 3 and the best choice is Y 1 ; when λ ∈ [1. 3,50], the ranking order of the three alternatives is Y 2 ≻ Y 1 ≻ Y 3 and the best choice is Y 2 .

Conclusions
is 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. en, 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. e 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. e 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.