Dual Hesitant Pythagorean Fuzzy Hamy Mean Operators in Multiple Attribute Decision Making

To fuse the information in dual-hesitant Pythagorean fuzzy sets (DHPFSs) more effectively, in this paper, some dual hesitant Pythagorean fuzzy Hamy mean (DHPFHM) operators, which can consider the relationships between being fused arguments, are defined and studied. Afterward, the defined aggregation operators are used to multiple attribute decision-making (MADM) with dual-hesitant Pythagorean fuzzy elements (DHPFEs), and the MADM decision-making model is developed. In accordance with the defined operators and the built model, the dual-hesitant Pythagorean fuzzy weighted Hamy mean (DHPFWHM) operator and the dual-hesitant Pythagorean fuzzy weighted dual Hamy mean (DHPFWDHM) operator are applied to deal with green supplier selection in supply chain management, and the availability and superiority of the proposed operators are analyzed by comparing with some existing approaches. The method presented in this paper can effectually solve the MADM problems, which the decision-making information is expressed by DHPFEs and the attributes are interactive.


I. INTRODUCTION
In real-life decision making environment, it's difficult for decision makers (DMs) to give evaluate information by using exact real numbers. In order to overcome this limitation, Zadeh [1] developed the fuzzy set (FS) theory which used the membership degree to depict decision making information instead of crisp results. In accordance of the studied of FS, Atanassov [2] further proposed another function which named non-membership degree as a supplementary. Thus, the intuitionistic fuzzy set (IFS) was found, each intuitionistic fuzzy set is characterized by the functions of membership degree and non-membership degree between 0 and 1, and the sum of them are limited to 1. After that, more and more scholars have studied about the IFS and its extensions in many multiple attribute decision making (MADM) problems [3]- [12]. Xu [13] studied the intuitionistic fuzzy weighted average operator, intuitionistic fuzzy ordered weighted average operator and intuitionistic The associate editor coordinating the review of this manuscript and approving it for publication was Alba Amato. fuzzy hybrid average operator. Xu and Yager [14] proposed some intuitionistic fuzzy weighted geometric (IFWG) aggregation operators. Wei and Wang [15] extended these IFWG operators to interval valued intuitionistic fuzzy environment. Hung and Yang [16] studied the similarity measures of IFS. Narayanamoorthy et al. [17] firstly developed the definition of interval-valued intuitionistic hesitant fuzzy entropy and built an extended VIKOR model for industrial robots selection. Zhang [18] combined the interval-valued intuitionistic fuzzy set and geometric Bonferroni means to define some new aggregation operators and applied them into multiple attribute decision making problems. Based on Shapley index, Zhou et al. [19] studied the Choquet integral correlation coefficient under extended intuitionistic fuzzy environment and applied to MADM. Zhang et al. [20] proposed some normal intuitionistic fuzzy Heronian mean operators based on Hamacher operation laws. Zhai et al. [21] defined the probabilistic interval-valued intuitionistic hesitant fuzzy sets and studied the distance and similarity measures of them. Based on the grey relational methods, the bidirectional projection and hesitant intuitionistic fuzzy linguistic information, VOLUME 7, 2019 This work is licensed under a Creative Commons Attribution 3.0 License. For more information, see http://creativecommons.org/licenses/by/3.0/ Zang et al. [22] developed a new approach to solve MADM problems. Yu et al. [23] presented some aggregation operators to fuse intuitionistic uncertain 2-tuple linguistic variables and applied them to MADM with heterogeneous relationship among attributes. Yao and Wang [24] defined the hesitant intuitionistic fuzzy entropy and hesitant intuitionistic fuzzy cross-entropy and studied their applications. Yang et al. [25] proposed an extended VIKOR model under linguistic hesitant intuitionistic fuzzy environment. However, the scope of evaluation information is limited by using intuitionistic fuzzy set, that is, the sum of membership and non-membership must be less or equal to 1. Thus, Pythagorean fuzzy set (PFS) [26], [27] has emerged to overcome this limitation. Similar to IFS, The Pythagorean fuzzy set (PFS) is also consisted of the function of membership degree and non-membership degree, and the sum of squares of them is restricted to 1, thus, it's clear that the PFS is more widespread than the IFS and can express more decision-making information. For instance, the membership is given as 0.6 and the non-membership is given as 0.8, it's obvious that this problem is only valid for PFS. In other words, all the intuitionistic fuzzy decision-making problems are the special case of Pythagorean fuzzy decision-making problems, which means that the PFS is more efficient to deal with MADM problems. In previous literature, some researching works have studied by a large amount of investigators. Zhang and Xu [28] proposed the Pythagorean fuzzy TOPSIS model to handle the MADM problems. Consider incomplete weight information, Khan et al. [29] studied MADM problems under Pythagorean hesitant fuzzy environment. Peng and Yang [30] primarily developed two Pythagorean fuzzy operations including the division and subtraction operations to better understand PFS. Reformat and Yager [31] dealed with the collaborative-based recommender system with Pythagorean fuzzy information. Based on traditional TOPSIS method, Khan et al. [32] developed an interval-valued Pythagorean TOPSIS method based on Choquet integral and Khan et al. [33] also studied the TOPSIS model under Pythagorean hesitant fuzzy environment. Gou et al. [34] reesearched some precious properties of continuous Pythagorean fuzzy information. Garg [35] gave the definition of some new Pythagorean fuzzy aggregation operators. Khan et al. [36] proposed some Pythagorean hesitant fuzzy Choquet integral aggregation operators. Wu and Wei [37] presented some Pythagorean fuzzy Hamacher aggregation operators to fuse Pythagorean fuzzy information. Zeng et al. [38] used the Pythagorean fuzzy ordered weighted averaging weighted average distance (PFOWAWAD) operator to study Pythagorean fuzzy MADM issues. Ren et al. [39] built the Pythagorean fuzzy TODIM model. Wei and Lu [40] developed some new Maclaurin symmetric mean(MSM) [41] operator based on Pythagorean fuzzy environment. Wei [42] developed some Pythagorean fuzzy interaction aggregation operators based on arithmetic and geometric operations. Wei and Lu [43] proposed some Pythagorean fuzzy power aggregation operators. Zeb et al. [44] presented some approaches to handle MADM problems with risk preference under extended Pythagorean fuzzy environment. Wei and Wei [45] defined ten cosine similarity measures under Pythagorean fuzzy environment. Liang et al. [46] investigated some Bonferroni mean operator with Pythagorean fuzzy information. Liang et al. [47] presented Pythagorean fuzzy Bonferroni mean aggregation operators based on geometric averaging (GA) operations. Khan and Abdullah [48] proposed the interval-valued Pythagorean fuzzy GRA method to solve MADM problems with incomplete weight information. Combined the PFSs [26], [27] and dual hesitant fuzzy sets(DHFSs) [49], [50], Wei and Lu [51] introduced the definition of the dual hesitant Pythagorean fuzzy sets (DHPFSs) and proposed some dual hesitant Pythagorean fuzzy Hamacher aggregation operators. Obviously, the DHPFSs have the advantages of considering the hesitance of DMs and expressing fuzzy information more effectively and reasonably. Khan et al. [52] investigated the Pythagorean hesitant fuzzy sets for group decision making with incomplete weight information. Liang and Xu [53] extended the TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets.
However, in practical MADM problems, there do exist some relationships between being fused arguments, it's obvious that the dual hesitant Pythagorean fuzzy Hamacher aggregation operators defined by Wei and Lu [51] don't take the relationships between being fused arguments into consideration. Thus, we need to find another more effective method to fuse dual hesitant Pythagorean fuzzy information. To date, the Hamy mean(HM) [54] operator, which can effectively take the interrelationship between arguments into account, has drawn large quantity of scholars' attention. Based on the intuitionistic fuzzy information, Hamy operator and Dombi operation laws, Li et al. [3] developed some intuitionistic fuzzy Dombi Hamy operators and applied these aggregation operators to the selection of a car supplier. Consider interval-valued intuitionistic fuzzy information, Wu et al. [4] further proposed some interval-valued intuitionistic fuzzy Dombi Hamy operators and discussed the application for evaluating the elderly tourism service quality in tourism destination by using these operators. To overcome the limitation of intuitionistic fuzzy set, Li et al. [55] defined some Pythagorean fuzzy Hamy operators to select most desirable green supplier. Deng et al. [56] combined the 2-tuple linguistic set (2TLS) and Pythagorean fuzzy set (PFS) to give the definition of 2-tuple linguistic Pythagorean fuzzy set (2TLPFS), and then some Hamy operators are presented under 2-tuple linguistic Pythagorean fuzzy environment. Based on linguistic neutrosophic set (LNS), Liu and You [57] proposed some linguistic neutrosophic Hamy operators for MADM problems. Liu et al. [58] defined some intuitionistic uncertain linguistic Hamy mean operators to select an appropriate and effective health-care waste treatment technology.
From above literature review, we can obtain that the dual hesitant Pythagorean fuzzy set is a meaningful tool to 86698 VOLUME 7, 2019 depict fuzzy and ambiguous information, the Hamy operator can consider the interrelationship between any number of being fused arguments. In this paper, motivated by traditional Hamy mean (HM) operator and the dual hesitant Pythagorean fuzzy set, we shall develop some dual hesitant Pythagorean fuzzy Hamy mean aggregation operators. The mainly contribution of this manuscript is to introduce some more reasonable aggregation operator for multiple attribute decision making (MADM) problems, and the way to express evaluation information we proposed is more scientific and effective. In addition, consider the decision maker's risk attitude, we can dynamic adjust to the parameter to derive different decision making results under different decision making environments.
In order to do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to Pythagorean fuzzy set (PFS), dual hesitant Pythagorean fuzzy set (DHPFS) and their operational laws. In Section 3, we shall propose some dual hesitant Pythagorean fuzzy Hamy mean aggregation operators such as: the dual hesitant Pythagorean fuzzy Hamy mean (DHPFHM) operator, the dual hesitant Pythagorean fuzzy weighted Hamy mean (DHPFWHM) operator, the dual hesitant Pythagorean fuzzy dual Hamy mean (DHPFDHM) operator and the dual hesitant Pythagorean fuzzy weighted dual Hamy mean (DHPFWDHM) operator. In Section 4, based on DHPFWHM and DHPFWDHM operators, we shall propose some models for multiple attribute decision making (MADM) problems with dual hesitant Pythagorean fuzzy information. In Section 5, we present a numerical example for supplier selection in supply chain management with dual hesitant Pythagorean fuzzy information in order to illustrate the method proposed in this paper. Section 6 concludes the paper with some remarks.

A. PYTHAGOREAN FUZZY SET
The fundamental definition of Pythagorean fuzzy sets (PFSs) [26], [27] are briefly introduced in this part. Afterwards, novel score and accuracy functions of Pythagorean fuzzy numbers (PFNs) are developed. Furthermore, the comparison laws of PFNs are proposed.

C. THE HAMY MEAN OPERATOR
Definition 6 [59]: The Hamy mean (HM) operator is defined as follows: where k is a parameter and k = 1, 2, . . . , n, i 1 , i 2 , . . . , i k are k integer values taken from the set {1, 2, . . . , n} of k integer values, C k n denotes the binomial coefficient and

III. DUAL HESITANT PYTHAGOREAN FUZZY HAMY MEAN OPERATORS
In the following, we shall propose some dual hesitant Pythagorean fuzzy Hamy mean (DHPFHM) operator based on the DHPFEs and Hamy mean (HM) operations.
In addition, some precious properties, such as idempotency, boundedness and monotonicity, are discussed.

A. THE DHPFHM AGGREGATION OPERATOR
Definition 7: Letp j = h j , g j (j = 1, 2, . . . , n) be a group of DHPFEs, and then we define the dual hesitant Pythagorean fuzzy HM (DHPFHM) operator as follows: According to the operation laws of the DHPFEs described in definition 5, we can obtain the Theorem 1.
Theorem 1: Letp j = h j , g j (j = 1, 2, . . . , n) be a group of DHPFEs, then their fused results by utilizing the DHPFHM operator is also a DHPFE, and (7), as shown at the top of the next page, holds.
Proof: Based on Definition 5, we can derive: Therefore, Furthermore, see (11) at the top of this page.
Thus, we have finished the proof. It can be easily proved that the DHPFHM operator satisfies the following properties.
Proof: Forp j =p for all j, that means α j = α and β j = β for all j, thus we can obtain (13), as shown at the bottom of the next page.
Thus, property 1 is maintained. Property 2 (Monotonicity): Letp j = h j , g j andp j = h j , g j , j = 1, 2, . . . , n, be two set of DHPFEs, if h j ≤ h j , g j ≥ g j , for all j, then VOLUME 7, 2019 Proof: For h j ≤ h j for all j, that means α j ≤ α j for all j, thus we can obtain . . . , n) be a collection of DHPFEs, and let Proof: From property 1, we can obtain

B. THE DHPFWHM AGGREGATION OPERATOR
According to Definition 7, we can obtain that the DHPFHM operator doesn't take the importance of being fused arguments into account. However, in many practical MADM problems, we should consider the weights of attribute.
To overcome the limitation of DHPFHM operator, we shall propose the dual hesitant Pythagorean fuzzy weighted Hamy mean (DHPFWHM) operator as follows. Definition 8: Assume thatp j = h j , g j (j = 1, 2, . . . , n) be a group of DHPFEs, and then we define the dual hesitant Pythagorean fuzzy weighted Hamy mean (DHPFWHM) operator as follows: According to the operation laws of the DHPFEs described in definition 5, we can obtain the Theorem 2.
Proof: Based on Definition 5, we can obtain: Thus, Therefore, (22), as shown at the top of this page, holds. Thereafter, (23), as shown at the top of the next page, holds. Therefore, (24), as shown at the top of the next page, holds. Thus, we have finished the proof. Similarly, we can obtain It can be easily proved that the DHPFWHM operator satisfies the following properties.

Property 4 (Monotonicity):
The proof is similar to DHPFHM operator, so it's omitted here. Property 5 (Boundedness): Letp j = h j , g j (j = 1, 2, . . . , n) be a collection of DHPFEs, and let Proof: Based on theorem 2, we can obtain (27) and (28), as shown at the top of this page.
From property 4, we can derivẽ

C. THE DHPFDHM AGGREGATION OPERATOR
In the following, based on the dual operation laws, Wu et al. [54] extended the Hamy mean (HM) operator to the dual Hamy mean (DHM) operator which can be depicted as follows.
Definition 9 [54]: The DHM operator is defined as follows: where k is a parameter and k = 1, 2, . . . , n, i 1 , i 2 , . . . , i k are k integer values taken from the set {1, 2, . . . , n} of k integer values, C k n denotes the binomial coefficient and C k n = n! k!(n−k)!
In this section, we shall study the dual Hamy mean (DHM) operator with dual hesitant Pythagorean fuzzy information. VOLUME 7, 2019 DHPFDHM (k) (p 1 ,p 2 , . . . ,p n ) According to Definition 5, we give the definition of the dual hesitant Pythagorean fuzzy dual Hamy mean (DHPFDHM) operator as follows. Definition 10: Assume thatp j = h j , g j (j = 1, 2, . . . , n) be a collection of DHPFEs, and then the dual hesitant Pythagorean fuzzy dual Hamy mean (DHPFDHM) operator can be defined as: According to the operation laws of the DHPFEs described in definition 5, we can obtain the Theorem 3.
Theorem 3: Assume thatp j = h j , g j (j = 1, 2, . . . , n) be a group of DHPFEs, then their fused results by utilizing the DHPFDHM operator is also a DHPFE, and (32), as shown at the top of this page, holds.
Proof: Based on Definition 5, we can derive: Therefore, (35), as shown at the top of this page, holds. Furthermore, (36), as shown at the top of this page, holds. Thus, we have finished the proof.
For the non-membership degree function β, the fused results are shown in (VI) at the top of this page.
Similarly, we can obtain It can be easily proved that the DHPFDHM operator satisfies the following properties. The proof is similar to DHPFHM operator, so it's omitted here.
Property 6 (Idempotency): If allp j = h j , g j (j = 1, 2, . . . , n) are equal, i.e.p j =p for all j, then DHPFDHM (k) (p 1 ,p 2 , . . . ,p n ) =p (37) Property 7 (Boundedness): Letp j = h j , g j (j = 1, 2, . . . , n) be a collection of DHPFEs, and let According to Definition 10, we can obtain that the DHPFDHM operator doesn't take the importance of being fused arguments into account. However, in many practical MADM problems, we should consider the weights of attribute. To overcome the limitation of DHPFDHM operator, we shall propose the dual hesitant Pythagorean fuzzy weighted dual Hamy mean (DHPFWDHM) operator as follows. Definition 11: Assume thatp j = h j , g j (j = 1, 2, . . . , n) be a collection of DHPFEs, and then the dual hesitant Pythagorean fuzzy weighted dual Hamy mean (DHPFWDHM) operator can be defined as: According to the operation laws of the DHPFEs described in definition 5, we can obtain the Theorem 4.
Proof: Based on Definition 5, we can obtain: Thus, Therefore, (44), as shown at the top of this page, holds. Thereafter, (45), as shown at the top of this page, holds. Therefore, (46), as shown at the top of the next page, holds. Thus, we have finished the proof.
It can be easily proved that the DHPFWDHM operator satisfies the following properties. The proof is similar to DHPFWHM operator, so it's omitted here.
Property 9 (Boundedness): Letp j = h j , g j (j = 1, 2, . . . , n) be a collection of DHPFEs, and let  In what follows, we apply the DHPFWHM or DHPFWDHM operator to the MADM problems for supplier selection in supply chain management with dual hesitant Pythagorean fuzzy information.
Step 1: We aggregate the decision information given in matrixP = p ij m×n by the DHPFWHM operator (49), as shown at the top of this page. Or the DHPFWDHM operator (50), as shown at the top of this page, to obtain the overall fused results p i (i = 1, 2, . . . , m).
Step 2: Compute the scores values S (p i ) (i = 1, 2, . . . , m) 1, 2, . . . , m). if there is no difference between any two scores S (p i ) and S p j , then we need to compute the accuracy values H (p i ) and H p j ofp i andp j , respectively, and then determine the ordering of all the alternatives A i and A j based on the accuracy results H (p i ) and H p j .
Step 3: Determine the ordering of all the alternatives A i (i = 1, 2, . . . , m) and select the best one(s) according to the scores values S (p i ) (i = 1, 2, . . . , m).

V. NUMERICAL EXAMPLE AND COMPARATIVE ANALYSIS A. NUMERICAL EXAMPLE
Supplier selection in supply chain management is classical MADM [62][63][64][65][66][67]. Thus, in this section we shall present a numerical example for supplier selection in supply chain management with DHPFEs in order to demonstrate the method proposed in this paper. Suppose there is a problem to deal with the supplier selection in supply chain management which is classical MADM problems. There are five prospect suppliers η i (i = 1, 2, 3, 4, 5) for four attributes δ j (j = 1, 2, 3, 4). The four attributes include product quality (δ 1 ), service (δ 2 ), delivery (δ 3 ) and price (δ 4 ), respectively. In order to avoid influence each other, the decision makers are required to evaluate the five suppliers η i (i = 1, 2, 3, 4, 5) under the above four attributes in anonymity and the decision matrixP = p ij 5×4 is presented in Table 1 In what follows, we can utilize our developed methods to deal with the supplier selection in supply chain management with DHPFEs.
Step 1: We aggregate the DHPFEs given in matrix by utilizing the DHPFWHM operator to obtain the overall preference valuesp i of the supplier in supply chain management η i (i = 1, 2, 3, 4, 5). Take alternative η 1 for an example (here, we take k = 2), we have (IX), as shown at the bottom of the next page.
Step Similarly, if we utilize the DHPFWDHM operator to solve this MADM, the decision making steps can be described as follows.
Step and it's clear that the most desirable supplier in supply chain management is η 4 . According to the above analysis, we can easily find that although the overall rating values of the alternatives are slightly different by using two operators respectively. However, the most desirable supplier in supply chain management is η 4 .

B. INFLUENCE OF THE PARAMETER ON THE FINAL RESULT
The parameter k plays an important role for the final ranking of alternatives. We may obtain different ordering results by assigning different values to k. By altering the values of k, the different ranking results are shown in Table 2 and Table 3 concretely. Therefore, the DHPFWHM and DHPFWDHM operators are considerably flexible by using a parameter vector. Table 2 shows that the ranking results decrease and become steady with the increase of values in parameter vector, Table 2 shows that the ranking results increase and become steady with the increase of values in parameter vector, That is, the final results become increasingly objective by considering the interrelationship

C. COMPARATIVE ANALYSIS
The prominent characteristic of the DHPFWHM and DHPFWDHM operators is that they can consider the interrelationship among any number of DHFNs. Next, we shall compare our developed methods with dual hesitant Pythagorean fuzzy weighted average (DHPFWA) and dual hesitant Pythagorean fuzzy weighted geometric (DHPFWG) operators [51], the comparative analysis results are listed as follows.
According to Table 1 and attribute weights, the fused values by DHPFWA operator are shown in (XI) at the top of the this page.   Then based on the score function of dual hesitant Pythagorean fuzzy elements (DHPFEs), we can obtain the score results ofp i as: Rank all the suppliers in supply chain management η i (i = 1, 2, 3, 4, 5) in accordance with the scores s (p i ) (i = 1, 2, 3, 4, 5) of the overall DHPFEsp i (i = 1, 2, . . . , 5):η 4 η 5 η 3 η 2 η 1 and thus the most desirable supplier in supply chain management is η 4 .
According to Table 1 and attribute weights, the fused values by DHPFWG operator are shown in (XII) at the top of the previous page.
Then based on the score function of DHPFEs, we can obtain the score results ofp i as:  p i (i = 1, 2, . . . , 5):η 4 η 5 η 2 η 1 η 3 and thus the most desirable supplier in supply chain management is η 4 .
According to Table 4, we can easily obtain that the ordering are slightly different and the best alternative are some, however, our defined operators are mainly characteristic of the advantages that can take the interrelationship between any number of being fused arguments into consideration and consider the human's hesitance in practical MADM problems, obviously, the DHPFWA and DHPFWG operators defined by Wei and Lu [51] cannot consider the interrelationship between the being fused arguments. In addition, in complicated decision-making environment, the decision maker's risk attitude is an important factor to think about, our methods can make this come true by altering the parameters k whereas the DHPFWA and DHPFWG operators presented by Wei and Lu [51] don't have the ability that dynamic adjust to the parameter according to the decision maker's risk attitude, so it is difficult to solve the risk multiple attribute decision making in real practice.

VI. CONCLUSION
The DHPFEs have applied the advantages of DHFSs and PFSs. They can flexibly denote decision-making information as well as effectively characterize the reliability of information. Thus, it is meaningful to study MADM problems with DHPFEs. In this paper, based on the Hamy mean (GHM) operator and dual Hamy mean (DHM) operator, we develop some dual hesitant Pythagorean fuzzy Hamy mean aggregation operators: the dual hesitant Pythagorean fuzzy Hamy mean (DHPFGHM) operator, the dual hesitant Pythagorean fuzzy weighted Hamy mean (DHPFWHM) operator, the dual hesitant Pythagorean fuzzy dual Hamy mean (DHPFDHM) operator and the dual hesitant Pythagorean fuzzy weighted dual Hamy mean (DHPFWDHM) operator. Of course, the precious merits of these defined operators are investigated. Moreover, we have adopted DHPFWHM and DHPFWDHM operators to build the decision-making model for MADM problems. In the end, we take a concrete instance for appraising the suppliers selection in supply chain management to demonstrate our defined model and to testify its accuracy and scientific. However, the scope of the evaluation information which is expressed by DHPFEs is still limited, it must satisfies the sum square of maximum membership and maximum non-membership is less or equal to 1. Thus, how to overcome this limitation need to be further studied. In the future, we shall continue studying the MADM problems with the application and extension of the developed operators to other domains [66]- [73].