Hesitant Pythagorean fuzzy interaction aggregation operators and their application in multiple attribute decision-making

The aim of this paper is to develop hesitant Pythagorean fuzzy interaction aggregation operators based on the hesitant fuzzy set, Pythagorean fuzzy set and interaction between membership and non-membership. The new operation laws can overcome shortcomings of existing operation laws of hesitant Pythagorean fuzzy values. Several new hesitant Pythagorean fuzzy interaction aggregation operators have been developed including the hesitant Pythagorean fuzzy interaction weighted averaging operator, the hesitant Pythagorean fuzzy interaction weighted geometric averaging operator and the generalized hesitant Pythagorean fuzzy interaction weighted averaging operator. Using the Bonferroni mean, some hesitant Pythagorean fuzzy interaction Bonferroni mean operators have been developed including the hesitant Pythagorean fuzzy interaction Bonferroni mean operator, the hesitant Pythagorean fuzzy interaction weighted Bonferroni mean (HPFIWBM) operator, the hesitant Pythagorean fuzzy interaction geometric Bonferroni mean operator and the hesitant Pythagorean fuzzy interaction geometric weight Bonferroni mean (HPFIGWBM) operator. Some properties have been studied. A new multiple attribute decision-making method based on the HPFIWBM operator and the HPFIGWBM operator has been presented. Numerical example is presented to illustrate the new method.


Introduction
Fuzzy decision-making has been studied and applied extensively [1][2][3]. Pythagorean fuzzy set [4,5] is the extension of intuitionistic fuzzy set [6]. In intuitionistic fuzzy set, the sum of membership and non-membership is no more than 1, while in Pythagorean fuzzy set, the square sum of membership and non-membership is no more than 1. Hence, Pythagorean fuzzy set has larger feasible region than that of intuitionistic fuzzy set. Thus, it is more powerful and flexible in modeling fuzzy and uncertain information. Pythagorean fuzzy set has been studied and applied extensively [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Some aggregation operators have been developed in Pythagorean fuzzy environment. Liang et al. [22] proposed the Pythagorean fuzzy Bonferroni mean operator and the weighted Pythagorean fuzzy Bonferroni mean operator. Zhang et al. [23] presented some general-B Wei Yang yangweipyf@163.com; yangwei_lxy@xauat.edu.cn 1 Xi'an University of Architecture and Technology, Xi'an 710055, Shaanxi, People's Republic of China ized Pythagorean fuzzy Bonferroni mean operator and the generalized Pythagorean fuzzy Bonferroni geometric mean operator. Yang and Pang [24] developed some Pythagorean fuzzy interaction Maclaurin symmetric mean operators. Rahman et al. [25] defined some interval-valued Pythagorean fuzzy aggregation operators including the interval-valued Pythagorean fuzzy weighted geometric operator, the intervalvalued Pythagorean fuzzy ordered weighted geometric operator, and the interval-valued Pythagorean fuzzy hybrid geometric operator. Wei and Lu defined some Pythagorean fuzzy power aggregation operators in [26] and presented some dual hesitant Pythagorean fuzzy aggregation operators in [27]. Garg presented the generalized Pythagorean fuzzy Einstein weighted average operator and the generalized Pythagorean fuzzy Einstein ordered weighted average operator in [28] and developed the Pythagorean fuzzy geometric interactive aggregation operators using Einstein operations in [29]. Du et al. [30] defined interval-valued Pythagorean fuzzy linguistic variable set and defined interval-valued Pythagorean fuzzy linguistic ordered weighted averaging operator and generalized interval-valued Pythagorean fuzzy linguistic ordered weighted average operator. Wei [31] defined some Pythagorean fuzzy interaction aggregation operators and some Pythagorean fuzzy interaction geometric aggregation operators. Some multiple attribute decision-making methods in Pythagorean fuzzy environment have been developed. Zhang and Xu [32] extended the TOPSIS method to accommodate Pythagorean fuzzy values. Ren et al. [33] proposed Pythagorean fuzzy TODIM approach. Chen [34] presented Pythagorean fuzzy VIKOR methods based on the generalized Pythagorean fuzzy distance measure. Pythagorean fuzzy set has been extended to accommodate interval values [35,36], linguistic arguments [37,38], probabilistic information [39], etc.
Hesitant fuzzy set [40] is the extension of fuzzy set and intuitionistic fuzzy set. In hesitant fuzzy set, each membership may include several possible values. Hesitant fuzzy set has been extended to accommodate intuitionistic fuzzy set [41], linguistic arguments [42], linguistic intuitionistic fuzzy values [43][44][45][46][47]. Hesitant Pythagorean fuzzy sets was defined [48] and some hesitant Pythagorean fuzzy Hamacher aggregation operators have been developed including the hesitant Pythagorean fuzzy Hamacher weighted average operator, hesitant Pythagorean fuzzy Hamacher weighted geometric operator. Khan et al. [49] proposed maximizing deviation method for Pythagorean hesitant fuzzy numbers in which information about attribute weights is incomplete. Garg [50] defined some hesitant Pythagorean fuzzy weighted aggregation operators and hesitant Pythagorean fuzzy geometric aggregation operators. But in real decision-making process, there are still cases that can not be dealt with using existing methods. There is only 0 non-membership and the other two non-memberships are not 0, but they have no effect on the final results. To overcome this shortcoming, we propose some interaction operation laws for hesitant Pythagorean fuzzy values by considering interaction between membership and non-membership. Then, we first develop some aggregation operators including the hesitant Pythagorean fuzzy interaction weighted averaging (HPFIWA) operator, hesitant Pythagorean fuzzy interaction weighted geometric averaging (HPFIWGA) operator and the generalized hesitant Pythagorean fuzzy interaction weighted geometric averaging (GHPFIWA) operator. The Bonferroni mean was first introduced by Bonferroni [51], which can capture inter-relationship among arguments to be aggregated. Yager [52] provided an interpretation of Bonferroni mean as involving a product of each argument with the average of the other arguments. Beliakov et al. [53] developed generalized Bonferroni mean. Beliakov and James [54] extended the generalized Bonferroni mean to intuitionistic fuzzy environment. Zhu and Xu [55] proposed the hesitant fuzzy Bonferroni mean operator. Yang et al. [56] developed the Pythagorean fuzzy interaction partitioned Bonferroni mean operator. But Bonferroni mean for hesitant Pythagorean values considering interaction between membership and non-membership has not been studied yet. Yang et al. [57] proposed q-rung orthopair fuzzy partitioned Bonferroni mean operators. To model interaction among hesitant Pythagorean fuzzy values and interaction between membership and non-membership at the same time, we develop some hesitant Pythagorean fuzzy interaction Bonferroni mean operator including the hesitant Pythagorean fuzzy interaction Bonferroni mean (HPFIBM) operator, the hesitant Pythagorean fuzzy interaction weighted Bonferroni mean (HPFIWBM) operator, the hesitant Pythagorean fuzzy interaction geometric Bonferroni mean (HPFIGBM) operator and the hesitant Pythagorean fuzzy interaction weighted geometric Bonferroni mean (HPFIWGBM) aggregation operator.
The objective of the paper is to develop some hesitant Pythagorean fuzzy interaction Bonferroni mean operators. To do so, the structure of the paper is as follows. In "Preliminaries", some basic concepts on Pythagorean fuzzy set, hesitant fuzzy set have been reviewed. Some interaction operational laws for hesitant Pythagorean fuzzy values have been defined and some properties have been studied. In "Hesitant Pythagorean fuzzy interaction aggregation operators", some hesitant Pythagorean fuzzy interaction aggregation operators have been defined. In "Hesitant Pythagorean fuzzy interaction Bonferroni mean operators", some hesitant Pythagorean fuzzy interaction Bonferroni mean operators have been proposed. In "An approach to Pythagorean fuzzy multiple attribute decision-making based on new interaction aggregation operators", a new multiple attribute decision-making method based on the HPFIWBM operator and the HPFI-WGBM operator has been presented. In "An illustrative example", numerical example is presented to illustrate the new method. Conclusions are presented in the final section.

Preliminaries
Definition 1 [40] Let X be a fixed set. A hesitant fuzzy set (HFS) H on X in terms of a function that when applied to X returns a subset of [0, 1], where μ P (x) : X → [0, 1] is the membership function and ν P (x) : X → [0, 1] is the non-membership function. For each x ∈ X , it satisfies the following Definition 3 [32] Let α = (μ α , ν α ), α 1 = (μ 1 , ν 1 ) and α 2 = (μ 2 , ν 2 ) be three PFNs, the operations are as follows Definition 4 [48,50] Let X be a fixed set. A hesitant Pythagorean fuzzy setP on X can be represented as follows whereh P (x) = {μ i } is the set of all the possible memberships of element x ∈ X andg P (x) = {ν i } is the set of all the possible non-memberships of element x ∈ X , is called a hesitant Pythagorean fuzzy element (HPFE).
be three HPFEs, λ > 0. The hesitant Pythagorean fuzzy interaction operations can be defined as Thef 1 ⊕f 2 andf 1 ⊗f 2 can be rewritten as follows The results of above operations are still HPFEs. The proofs of (1) and (3) are given as follows and others can be proved similarly.

Proof
Then, the λf is still an HPFE.
be three HPFEs, then we have Proof We only prove (1), (3) and (5), and others can be proved similarly. (1)f (3)f be an HPFE. The score off can be defined as The accuracy function can be defined as where lh is the number of memberships inh and lg is the number of non-memberships ing.
To define distance measure between HPFEs more accurately, the two HPFEs should have the same number of memberships and non-memberships. The HPFEs can be extended according to the risk attitude of decision-makers. If the decision-maker is risk seeking, the largest Pythagorean fuzzy value can be added; if decision-maker is risk averse, the smallest Pythagorean fuzzy value can be added; and if decision-maker is risk neutral, the average value of Pythagorean fuzzy values can be added.

Hesitant Pythagorean fuzzy interaction aggregation operators
Definition 10 Letf i (i = 1, 2, . . . , n) be a collection of HPFEs. The hesitant Pythagorean fuzzy interaction weighted averaging (HPFIWA) operator can be defined as

) be a collection of HPFEs. The aggregated value of the HPFIWA operator is still an HPFE, that is
Proof The theorem can be proved using mathematical induction.
Suppose Eq. (8) holds for n = l, that is If n = l+1, using the interaction operation laws of hesitant Pythagorean fuzzy value, we can get By mathematical induction, Eq. (8) holds for all n. Moreover, for each (μ, ν) in the HPFIWA operator, we have Then, the aggregated result of the HPFIWA operator is still an HPFE.
If the weight vector is taken as 1 n , 1 n , . . . , 1 n , the HPFIWA operator reduces to the hesitant Pythagorean fuzzy interaction averaging (HPFIA) operator as follows . . , n) be a collection of HPFEs. If all the HPFEs reduces tof = (h,g), the HPFIWA operator reduces to the following form . . , n) be a collection of HPFEs. Then, the aggregated result of the HPFIWGA operator is still an HPFE, which has the following form Proof If n = 2, HPFIWGA(f 1 ,f 2 ) =f w 1 1 ⊗f w 2 2 .

be a collection of HPFEs. Letf
If the weight vector is taken as ( 1 n , 1 n , . . . , 1 n ), the GHPFIWA λ operator reduces to the generalized Pythagorean fuzzy interaction averaging (GHPFIA) operator as follows

An approach to Pythagorean fuzzy multiple attribute decision-making based on new interaction aggregation operators
Suppose there is a multiple attribute decision-making problem. {A 1 , A 2 , . . . , A m } is the alternative set, {C 1 , C 2 , . . . , C n } is the attribute set. The experts evaluate alternatives with respect to attributes with Pythagorean fuzzy values. If they are familiar with the attributes, they can give evaluation values; if they are not familiar with attributes, they can refuse to give any evaluation values. Hence, the hesitant Pythagorean fuzzy decision matrix is formed. The proposed method based on the new hesitant Pythagorean fuzzy interaction aggregation operators is as follows.
Step 1. Decision-makers evaluate alternatives with respect to attributes with Pythagorean fuzzy values and hesitant Pythagorean fuzzy decision matrix is formed asD = (f i j ) m×n .
Step 2. Calculate alternatives' collective evaluation values using the HPFIWBM operator or the HPFIGWBM operator using the following equations. Table 1 Pythagorean fuzzy decision matrixD If interaction between membership and non-membership is not considered and we calculate the collective ones using the hesitant Pythagorean fuzzy weighted averaging (HPFWA) operator as follows We can calculate collective evaluation values using the HPFWA operator to getf 1   If interaction between memberships and non-memberships is not considered, the hesitant Pythagorean fuzzy weighted Bonferroni mean (HPFWBM) operator as follows is used in aggregating.
HPFWBM (f 1 ,f 2 , . . . ,f n ) The collective evaluation values can be calculated asf 1  The alternatives can be ranked as A 3 > A 1 > A 4 > A 2 and the optimal alternative is A 3 . Though ranking result is the same as that based on the HPFIWBM operator, but in aggregation process, the effect of non-memberships has been reduced since there is 0 of non-membership in the evaluation process. In other decisionmaking problems, we may get different ranking results.
The differences between the proposed method and the existing methods have been summarized in Table 2. In a word, our proposed method is based on the hesitant Pythagorean fuzzy values and the Bonferroni mean operator. Moreover, interaction between arguments to be aggregated is considered and interaction between membership and non-membership is also considered.

Conclusions
In this paper, we first define some hesitant Pythagorean fuzzy interaction aggregation laws for HPFEs, and then develop some hesitant Pythagorean interaction aggregation operators. Using the Bonferroni mean operator, we develop In the future, we will apply the new aggregation operators to other complicated decision problems and we will also develop new interaction aggregation operators for HPFEs.