Hesitant pythagorean fuzzy hamacher aggregation operators and their application to multiple attribute decision making

Abstract

This article has been retracted, and the online PDF replaced with this retraction notice.

Keywords

Multiple attribute decision making (MADM)hesitant Pythagorean fuzzy values Hamacher aggregation operators hesitant Pythagorean fuzzy Hamacher hybrid average (HPFHHA) operator hesitant Pythagorean fuzzy Hamacher hybrid geometric (HPFHHG) operator

1 Introduction

Atanassov [1 –3] introduced the concept of intuitionistic fuzzy set (IFS), which is a generalization of the concept of fuzzy set [4]. Each element in the IFS is expressed by an ordered pair, and each ordered pair is characterized by a membership degree and a non-membership degree. The sum of the membership degree and the non-membership degree of each ordered pair is less than or equal to 1. The intuitionistic fuzzy set has received more and more attention since its appearance [5 –25]. More recently, Pythagorean fuzzy set (PFS) [26, 27] has emerged as an effective tool for depicting uncertainty of the MADM problems. The PFS is also characterized by the membership degree and the non-membership degree, whose sum of squares is less than or equal to 1, the PFS is more general than the IFS. In some cases, the PFS can solve the problems that the IFS cannot, for example, if a DM gives the membership degree and the non-membership degree as 0.8 and 0.6, respectively, then it is only valid for the PFS. In other words, all the intuitionistic fuzzy degrees are a part of the Pythagorean fuzzy degrees, which indicates that the PFS is more powerful to handle the uncertain problems. Zhang and Xu [28] provided the detailed mathematical expression for PFS and introduced the concept of Pythagorean fuzzy number(PFN). Meanwhile, they also developed a Pythagorean fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) for handling the MCDM problem within PFNs. Peng and Yang [29] proposed the division and subtraction operations for PFNs, and also developed a Pythagorean fuzzy superiority and inferiority ranking method to solve multicriteria group decision making problem with PFNs. Afterwards, Beliakov and James [30] focused on how the notion of “averaging” should be treated in the case of PFNs and how to ensure that the averaging aggregation functions produce outputs consistent with the case of ordinary fuzzy numbers. Reformat and Yager [31] applied the PFNs in handling the collaborative-based recommender system. Gou et al. [32] investigate the Properties of Continuous Pythagorean Fuzzy Information. Ren et al. [33] proposed the Pythagorean fuzzy TODIM approach to multi-criteria decision making. Garg [34] proposed the new generalized Pythagorean fuzzy information aggregation by using Einstein Operations. Zeng et al. [35] developed a hybrid method for Pythagorean fuzzy multiple-criteria decision making. Garg [36] studied a novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem.

Hamacher operations [37] include Hamacher product and Hamacher sum, which are good alternatives to the algebraic product and algebraic sum, respectively. Hamacher t-conorm and t-norm, which are the generalization of algebraic and Einstein t-conorm and t-norm [38 –41], are more general and more flexible. There is important significance to research aggregation operators based on Hamacher operations and their application to multiple attribute group decision making problems [42 –44].

In this paper, we define the concept of the hesitant Pythagorean fuzzy sets. Therefore, how to extend the Hamacher operations to aggregate the hesitant Pythagorean fuzzy information is a meaningful work, which is also the focus of this paper. To do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to hesitant Pythagorean fuzzy sets and some operational laws of hesitant Pythagorean fuzzy numbers. In Section 3 we have developed some hesitant Pythagorean fuzzy Hamacher aggregation operators: hesitant Pythagorean fuzzy Hamacher weighted average (HPFHWA) operator, hesitant Pythagorean fuzzy Hamacher weighted geometric (HPFHWG) operator, hesitant Pythagorean fuzzy Hamacher ordered weighted average (HPFHOWA) operator, hesitant Pythagorean fuzzy Hamacher ordered weighted geometric (HPFHOWG) operator, hesitant Pythagorean fuzzy Hamacher hybrid average (HPFHHA) operator and hesitant Pythagorean fuzzy Hamacher hybrid geometric (HPFHHG) operator and studied some desirable properties of the proposed operators.In Section 4, we have applied these operators to develop some models for multiple attribute decision making (MADM) problems based on the Hamacher aggregation operators with hesitant Pythagorean fuzzy information. In Section 5, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness. In Section 6, we conclude the paper and give some remarks.

2 Preliminaries

2.1 Pythagorean fuzzy set

The basic concepts of PFSs [26, 27] are briefly reviewed in this section. Afterwards, novel score and accuracy functions for PFNs are proposed. Furthermore, a new comparison method for PFNs is developed.

Definition 1. [26, 27] Let X be a fix set. A PFS is an object having the form $P = {〈 x, (μ_{P} (x), ν_{P} (x)) 〉 | x \in X}$ (1) where the function μ_P : X → [0, 1] defines the degree of membership and the function ν_P : X → [0, 1] defines the degree of non-membership of the element x ∈ X to P, respectively, and, for every x ∈ X, it holds that ${(μ_{p} (x))}^{2} + {(ν_{p} (x))}^{2} \leq 1 .$ (2)

Definition 2. [31] Let ${\tilde{a}}_{1} = (μ_{1}, ν_{1})$ , ${\tilde{a}}_{2} = (μ_{2}, ν_{2})$ , and $\tilde{a} = (μ, ν)$ be three Pythagorean fuzzy numbers, and some basic operations on them are defined as follows: $\begin{matrix} (1) & {\tilde{a}}_{1} \oplus {\tilde{a}}_{2} \\ = (\sqrt{{(μ_{1})}^{2} + {(μ_{2})}^{2} - {(μ_{1})}^{2} {(μ_{2})}^{2}}, ν_{1} ν_{2}); \\ (2) & {\tilde{a}}_{1} \otimes {\tilde{a}}_{2} \\ = (μ_{1} μ_{2}, \sqrt{{(ν_{1})}^{2} + {(ν_{2})}^{2} - {(ν_{1})}^{2} {(ν_{2})}^{2}}); \\ (3) & λ \tilde{a} = (\sqrt{1 - {(1 - μ^{2})}^{λ}}, ν^{λ}), λ > 0; \\ (4) & {(\tilde{a})}^{λ} = (μ^{λ}, \sqrt{1 - {(1 - ν^{2})}^{λ}}), λ > 0; \\ (5) & {\tilde{a}}^{c} = (ν, μ) . \end{matrix}$

2.2 Hesitant Pythagorean fuzzy set

In the following, we shall propose the concept of the hesitant Pythagorean fuzzy set (HPFS) on the basis of the Pythagorean fuzzy set (PFS) [26, 27] and hesitant fuzzy set(HFS) [45 –49].

Definition 3. Given a fixed set X, then a hesitant Pythagorean fuzzy set (HPFS) on X is in terms of a function that when applied to X returns a sunset of [0, 1]. The hesitant Pythagorean fuzzy set can be expressed by mathematical symbol: $P = {〈 x, (h_{P} (x)) 〉 | x \in X}$ (3) where h_P (x) is a set of some values in [0, 1], denoting the possible membership degree of the element x ∈ X to the set P, respectively, and, for every x ∈ X, it holds that (h_p (x)) ² ≤ 1.

For convenience, we call p = h_p (x) a hesitant Pythagorean fuzzy element(HPFE) and P the set of all HPFEs.

Definition 4. For a HPFE h, $s (h) = \frac{1}{# h} \sum_{γ \in h} γ$ is called the score function of h, where # h is the number of the elements in h. For two HFEs h₁ and h₂, if s (h₁) > s (h₂), then h₁ > h₂; if s (h₁) = s (h₂), then h₁ = h₂.

2.3 Hamacher operations

T-norm and t-conorm are an important notion in fuzzy set theory, which are used to define a generalized union and intersection of fuzzy sets [39]. Roychowdhury and Wang [40] gave the definition and conditions of t-norm and t-conorm. Based on a t-norm (T) and t-conorm (T^*), a generalized union and a generalized intersection of intuitionistic fuzzy sets were introduced by Deschrijver and Kerre [41]. Further, Hamacher [37] proposed a more generalized t-norm and t-conorm. Hamacher operation [37] include the Hamacher product and Hamacher sum, which are examples of t-norms and t-conorms, respectively. They are defined as follows:

Hamacher product ⊗ is a t-norm and Hamacher sum ⊕ is a t-conorm, where

$\begin{matrix} T (a, b) = a \otimes b \\ = \frac{ab}{γ + (1 - γ) (a + b - ab)}, γ > 0 . \end{matrix}$ (4) $\begin{matrix} T * (a, b) = a \oplus b \\ = \frac{a + b - ab - (1 - γ) ab}{1 - (1 - γ) ab}, γ > 0 . \end{matrix}$ (5)

Especially, when γ = 1, then Hamacher t-norm and t-conorm will reduce to $T (a, b) = a \otimes b = ab$ (6) $T * (a, b) = a \oplus b = a + b - ab$ (7) which are the algebraic t-norm and t-conorm respectively; when γ = 2, then Hamacher t-norm and t-conorm will reduce to $T (a, b) = a \otimes b = \frac{ab}{1 + (1 - a) (1 - b)}$ (8) $T * (a, b) = a \oplus b = \frac{a + b}{1 + ab}$ (9) which are called the Einstein t-norm and t-conorm respectively [50].

2.4 Hamacher operations of hesitant Pythagorean fuzzy set

Motivated by the arithmetic aggregation operators [48 –53], the Hamacher product ⊗ and the Hamacher sum ⊕, then the generalized intersection and union on two HPFEs h₁ and h₂ become the Hamacher product (denoted by h₁ ⊗ h₂) and Hamacher sum (denoted by h₁ ⊕ h₂) of two HPFEs h₁ and h₂, γ > 0, respectively, as follows: $\begin{matrix} (1) & h_{1} \oplus h_{2} = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}} \\ {\sqrt{\frac{{(γ_{1})}^{2} + {(γ_{2})}^{2} - {(γ_{1})}^{2} {(γ_{2})}^{2} - (1 - γ) {(γ_{1})}^{2} {(γ_{2})}^{2}}{1 - (1 - γ) {(γ_{1})}^{2} {(γ_{2})}^{2}}}}; \\ (2) & h_{1} \otimes h_{2} = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}} \\ {\frac{γ_{1} γ_{2}}{\sqrt{γ + (1 - γ) ({(γ_{1})}^{2} + {(γ_{2})}^{2} - {(γ_{1})}^{2} {(γ_{2})}^{2})}}}; \\ (3) & λ h_{1} = \cup_{γ_{1} \in h_{1}} \\ {\sqrt{\frac{{(1 + (γ - 1) {(γ_{1})}^{2})}^{λ} - {(1 - {(γ_{1})}^{2})}^{λ}}{{(1 + (γ - 1) {(γ_{1})}^{2})}^{λ} + (γ - 1) {(1 - {(γ_{1})}^{2})}^{λ}}}}, \\ λ > 0; \\ (4) & {(h_{1})}^{λ} = \cup_{γ_{1} \in h_{1}} \\ {\frac{\sqrt{γ} {(γ_{1})}^{λ}}{\sqrt{{(1 + (γ - 1) (1 - {(γ_{1})}^{2}))}^{λ} + (γ - 1) {(γ_{1})}^{2 λ}}}}, \\ λ > 0 . \end{matrix}$

3 Hesitant Pythagorean fuzzy Hamacher aggregation operators

3.1 Hesitant Pythagorean fuzzy Hamacher arithmetic aggregation operators

In the following, we shall develop some hesitant Pythagorean fuzzy Hamacher arithmetic aggregation operator based on the operations of HPFEs and Hamacher operations.

Definition 5. Let h_j (j = 1, 2, ⋯ , n) be a collection of HPFEs, then we define the hesitant Pythagorean fuzzy Hamacher weighted average (HPFHWA) operator as follows: ${HPFHWA}_{ω} (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (ω_{j} h_{j})$ (10) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of h_j (j = 1, 2, ⋯ , n), and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

Based on Hamacher sum operations of the hesitant Pythagorean fuzzy values described, we can drive the Theorem 1.

Theorem 1. Let h_j (j = 1, 2, ⋯ , n) be a collection of HPFEs, then their aggregated value by using the HPFHWA operator is also a HFE, and

$\begin{matrix} {HPFHWA}_{ω} (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (ω_{j} h_{j}) \\ = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots, γ_{n} \in h_{n}} \\ {\sqrt{\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) {(γ_{j})}^{2})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - {(γ_{j})}^{2})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) {(γ_{j})}^{2})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - {(γ_{j})}^{2})}^{ω_{j}}}}} \end{matrix}$ (11)

where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of h_j (j = 1, 2, ⋯ , n), and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ , γ > 0.

Now, we can discuss some special cases of the HPFHWA operator with respect to the parameter γ.

When γ = 1, HPFHWA operator reduces to the hesitant Pythagorean fuzzy weighted average (HPFWA) operator as follows:

$\begin{matrix} {HPFWA}_{ω} (h_{1}, h_{2}, \dots, h_{n}) \\ = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots, γ_{n} \in h_{n}} \\ {\sqrt{1 - \prod_{j = 1}^{n} {(1 - {(γ_{j})}^{2})}^{ω_{j}}}} \end{matrix}$ (12)

When γ = 2, HPFHWA operator reduces to the hesitant Pythagorean fuzzy Einstein weighted average (HPFEWA) operator as follows: $\begin{matrix} {HPFEWA}_{ω} (h_{1}, h_{2}, \dots, h_{n}) \\ = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots, γ_{n} \in h_{n}} \\ {\sqrt{\frac{\prod_{j = 1}^{n} {(1 + {(γ_{j})}^{2})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - {(γ_{j})}^{2})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + {(γ_{j})}^{2})}^{ω_{j}} + \prod_{j = 1}^{n} {(1 - {(γ_{j})}^{2})}^{ω_{j}}}}} \end{matrix}$ (13)

Definition 6. Let h_j (j = 1, 2, ⋯ , n) be a collection of HPFEs, then we define the hesitant Pythagorean fuzzy Hamacher ordered weighted average (HPFHOWA) operator as follows: ${HPFHOWA}_{w} (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (w_{j} h_{σ (j)})$ (14) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that h_σ(j-1) ≥ h_σ(j) for all j = 2, ⋯ , n, and w = (w₁, w₂, ⋯ , w_n) ^T is the aggregation-associated weight vector such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ .

Based on Hamacher sum operations of the hesitant Pythagorean fuzzy values described, we can drive the Theorem 2.

Theorem 2. Let h_j (j = 1, 2, ⋯ , n) be a collection of HPFEs, then their aggregated value by using the HPFHOWA operator is also a HPFE, and $\begin{matrix} {HPFHOWA}_{w} (h_{1}, h_{2}, \dots, h_{n}) \\ = \oplus_{j = 1}^{n} (w_{j} h_{σ (j)}) \\ = \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\sqrt{\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) {(γ_{σ (j)})}^{2})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - {(γ_{σ (j)})}^{2})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) {(γ_{σ (j)})}^{2})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - {(γ_{σ (j)})}^{2})}^{ω_{j}}}}} \end{matrix}$ (15)

where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that h_σ(j-1) ≥ h_σ(j) for all j = 2, ⋯ , n, and w = (w₁, w₂, ⋯ , w_n) ^T is the aggregation-associated weight vector such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ , γ > 0.

Now, we can discuss some special cases of the HPFHOWA operator with respect to the parameter γ.

When γ = 1, HPFHOWA operator reduces to the hesitant Pythagorean fuzzy ordered weighted average (HFOWA)operator as follows:

$\begin{matrix} {HPFOWA}_{w} (h_{1}, h_{2}, \dots, h_{n}) \\ = \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\sqrt{1 - \prod_{j = 1}^{n} {(1 - {(γ_{σ (j)})}^{2})}^{w_{j}}}} \end{matrix}$ (16)

When γ = 2, HPFHOWA operator reduces to the hesitant Pythagorean fuzzy Einstein ordered weighted average (HPFEOWA) operator as follows: $\begin{matrix} {HPFEOWA}_{w} (h_{1}, h_{2}, \dots, h_{n}) \\ = \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\sqrt{\frac{\prod_{j = 1}^{n} {(1 + {(γ_{σ (j)})}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {(γ_{σ (j)})}^{2})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + {(γ_{σ (j)})}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {(γ_{σ (j)})}^{2})}^{w_{j}}}}} \end{matrix}$ (17)

From Definitions 5 and 6, we know that the HPFHWA operator weights the hesitant Pythagorean fuzzy argument itself, while the HPFHOWA operator weights the ordered positions of the hesitant Pythagorean fuzzy arguments instead of weighting the arguments themselves. Therefore, weights represent different aspects in both the HPFHWA and HPFHOWA operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose a hesitant Pythagorean fuzzy Hamacher hybrid average (HPFHHA) operator.

Definition 7. A hesitant Pythagorean fuzzy Hamacher hybrid average (HPFHHA) operator is defined as follows: ${HPFHHA}_{w, ω} (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (w_{j} {\dot{h}}_{σ (j)})$ (18) where w = (w₁, w₂, ⋯ , w_n) is the associated weighting vector, with w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ , and ${\dot{h}}_{σ (j)}$ is the j-th largest element of the hesitant Pythagorean fuzzy arguments ${\dot{h}}_{j} ({\dot{h}}_{j} = (n ω_{j}) h_{j}, j = 1, 2, \dots, n)$ , ω = (ω₁, ω₂, ⋯ , ω_n) is the weighting vector of hesitant Pythagorean fuzzy arguments h_j (j = 1, 2, ⋯ , n), with ω_i ∈ [0, 1], $\sum_{i = 1}^{n} ω_{i} = 1$ , and n is the balancing coefficient. Especially, if w = (1/ - n, 1/ - n, ⋯ , 1/ - n) ^T, then HPFHHA is reduced to the hesitant Pythagorean fuzzy Hamacher weighted average (HPFHWA) operator; if ω = (1/ - n, 1/ - n, ⋯ , 1/ - n), then HPFHHA is reduced to the hesitant Pythagorean fuzzy Hamacher ordered weighted average (HPFHOWA) operator.

Based on Hamacher sum operations of the hesitant Pythagorean fuzzy values described, we can drive the Theorem 3.

Theorem 3. Let h_j (j = 1, 2, ⋯ , n) be a collection of HPFEs, then their aggregated value by using the HPFHHA operator is also a HPFE, and $\begin{matrix} {HPFHHA}_{w, ω} (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (w_{j} {\dot{h}}_{σ (j)}) \\ = \cup_{{\dot{γ}}_{σ (1)} \in {\dot{h}}_{σ (1)}, {\dot{γ}}_{σ (2)} \in {\dot{h}}_{σ (2)}, \dots, {\dot{γ}}_{σ (n)} \in {\dot{h}}_{σ (n)}} \\ \sqrt{\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) {({\dot{γ}}_{σ (j)})}^{2})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - {({\dot{γ}}_{σ (j)})}^{2})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) {({\dot{γ}}_{σ (j)})}^{2})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - {({\dot{γ}}_{σ (j)})}^{2})}^{ω_{j}}}} \end{matrix}$ (19)

where w = (w₁, w₂, ⋯ , w_n) is the associated weighting vector, with w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ , and ${\dot{h}}_{σ (j)}$ is the j-th largest element of the hesitant Pythagorean fuzzy arguments ${\dot{h}}_{σ (j)} ({\dot{h}}_{σ (j)} = (n ω_{j}) h_{j}, j = 1, 2, \dots, n)$ , ω = (ω₁, ω₂, ⋯ , ω_n) is the weighting vector of hesitant Pythagorean fuzzy arguments h_j (j = 1, 2, ⋯ , n), with ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ , and n is the balancing coefficient, γ > 0.

Now, we can discuss some special cases of the HPFHHA operator with respect to the parameter γ.

When γ = 1, HPFHHA operator reduces to the hesitant Pythagorean fuzzy hybrid average (HPFHA)operator as follows:

$\begin{matrix} {HPFHA}_{ω, w} (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (w_{j} {\dot{h}}_{σ (j)}) \\ = \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\sqrt{1 - \prod_{j = 1}^{n} {(1 - {({\dot{γ}}_{σ (j)})}^{2})}^{w_{j}}}} \end{matrix}$ (20)

When γ = 2, HPFHHA operator reduces to the hesitant Pythagorean fuzzy Einstein hybrid average (HPFEHA) operator as follows:

$\begin{matrix} {HPFEHA}_{w, ω} (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (w_{j} {\dot{h}}_{σ (j)}) \\ = \cup_{{\dot{γ}}_{σ (1)} \in {\dot{h}}_{σ (1)}, {\dot{γ}}_{σ (2)} \in {\dot{h}}_{σ (2)}, \dots, {\dot{γ}}_{σ (n)} \in {\dot{h}}_{σ (n)}} \\ {\sqrt{\frac{\prod_{j = 1}^{n} {(1 + {({\dot{γ}}_{σ (j)})}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {({\dot{γ}}_{σ (j)})}^{2})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + {({\dot{γ}}_{σ (j)})}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - {({\dot{γ}}_{σ (j)})}^{2})}^{w_{j}}}}} \end{matrix}$ (21)

3.2 Hesitant Pythagorean fuzzy Hamacher geometric aggregation operators

Based on the hesitant Pythagorean fuzzy Hamacher arithmetic aggregation operators and the geometric mean [54, 55], here we define some hesitant Pythagorean fuzzy Hamacher geometric aggregation operators:

Definition 8. Let h_j (j = 1, 2, ⋯ , n) be a collection of HPFEs, then we define the hesitant Pythagorean fuzzy Hamacher weighted geometric (HPFHWG) operator as follows: $HPFHWG (h_{1}, h_{2}, \dots, h_{n}) = \otimes_{j = 1}^{n} {(h_{j})}^{ω_{j}}$ (22) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of h_j (j = 1, 2, ⋯ , n), and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

Based on Hamacher product operations of the hesitant Pythagorean fuzzy values described, we can drive the Theorem 4.

Theorem 4. Let h_j (j = 1, 2, ⋯ , n) be a collection of HPFEs, then their aggregated value by using the HPFHWG operator is also a HPFE, and $\begin{matrix} {HPFHWG}_{ω} (h_{1}, h_{2}, \dots, h_{n}) = \otimes_{j = 1}^{n} {(h_{j})}^{ω_{j}} \\ = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots, γ_{n} \in h_{n}} \\ {\frac{\sqrt{γ} \prod_{j = 1}^{n} γ_{j}^{ω_{j}}}{\sqrt{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - {(γ_{j})}^{2}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(γ_{j})}^{2 ω_{j}}}}} \end{matrix}$ (23)

where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of h_j (j = 1, 2, ⋯ , n), and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ , γ > 0.

Now, we can discuss some special cases of the HPFHWG operator with respect to the parameter γ.

When γ = 1, HPFHWG operator reduces to the hesitant Pythagorean fuzzy weighted geometric (HPFWG) operator as follows:

$\begin{matrix} {HPFWG}_{ω} (h_{1}, h_{2}, \dots, h_{n}) \\ = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots, γ_{n} \in h_{n}} {\prod_{j = 1}^{n} {(γ_{j})}^{ω_{j}}} \end{matrix}$ (24)

When γ = 2, HPFHWG operator reduces to the hesitant Pythagorean fuzzy Einstein weighted geometric (HPFEWG) operator as follows:

$\begin{matrix} {HPFEWG}_{ω} (h_{1}, h_{2}, \dots, h_{n}) \\ = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots, γ_{n} \in h_{n}} \\ {\frac{\sqrt{2} \prod_{j = 1}^{n} γ_{j}^{ω_{j}}}{\sqrt{\prod_{j = 1}^{n} {(2 - {(γ_{j})}^{2})}^{ω_{j}} + \prod_{j = 1}^{n} {(γ_{j})}^{2 ω_{j}}}}} \end{matrix}$ (25)

Definition 9. Let h_j (j = 1, 2, ⋯ , n) be a collection of HPFEs, then we define the hesitant Pythagorean fuzzy Hamacher ordered weighted geometric (HPFHOWG)operator as follows: $HPFHOWG (h_{1}, h_{2}, \dots, h_{n}) = \otimes_{j = 1}^{n} {(h_{σ (j)})}^{w_{j}}$ (26) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that h_σ(j-1) ≥ h_σ(j) for all j = 2, ⋯ , n, and w = (w₁, w₂, ⋯ , w_n) ^T is the aggregation-associated weight vector such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ .

Based on Hamacher product operations of the hesitant Pythagorean fuzzy values described, we can drive the Theorem 5.

Theorem 5. Let h_j (j = 1, 2, ⋯ , n) be a collection of HPFEs, then their aggregated value by using the HPFHHG operator is also a HPFE, and $\begin{matrix} {HPFHOWG}_{w} (h_{1}, h_{2}, \dots, h_{n}) = \otimes_{j = 1}^{n} {(h_{σ (j)})}^{w_{j}} \\ \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\frac{\sqrt{γ} \prod_{j = 1}^{n} {(γ_{σ (j)})}^{ω_{j}}}{\sqrt{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - {(γ_{σ (j)})}^{2}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(γ_{σ (j)})}^{2 ω_{j}}}}} \end{matrix}$ (27)

Now, we can discuss some special cases of the HPFHOWG operator with respect to the parameter γ.

When γ = 1, HPFHOWG operator reduces to the hesitant Pythagorean fuzzy ordered weighted geometric (HPFOWG) operator as follows:

$\begin{matrix} {HPFOWG}_{w} (h_{1}, h_{2}, \dots, h_{n}) \\ = \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\prod_{j = 1}^{n} {(γ_{σ (j)})}^{w_{j}}} \end{matrix}$ (28)

When γ = 2, HPFHOWG operator reduces to the hesitant Pythagorean fuzzy Einstein ordered weighted geometric (HPFEOWG) operator as follows:

$\begin{matrix} {HPFEOWG}_{w} (h_{1}, h_{2}, \dots, h_{n}) \\ = \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\frac{\sqrt{2} \prod_{j = 1}^{n} {(γ_{σ (j)})}^{ω_{j}}}{\sqrt{\prod_{j = 1}^{n} {(2 - {(γ_{σ (j)})}^{2})}^{ω_{j}} + \prod_{j = 1}^{n} {(γ_{σ (j)})}^{2 ω_{j}}}}} \end{matrix}$ (29)

From Definitions 8 and 9, we know that the HPFHWG operator weights the hesitant Pythagorean fuzzy argument itself, while the HPFHOWG operator weights the ordered positions of the hesitant Pythagorean fuzzy arguments instead of weighting the arguments themselves. Therefore, weights represent different aspects in both the HPFHWG and HPFHOWG operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose a hesitant Pythagorean fuzzy Hamacher hybrid geometric (HPFHHG) operator.

Definition 10. A hesitant Pythagorean fuzzy Hamacher hybrid geometric (HPFHHG) operator is defined as follows: ${HPFHHG}_{w, ω} (h_{1}, h_{2}, \dots, h_{n}) = \otimes_{j = 1}^{n} {({\dot{h}}_{σ (j)})}^{w_{j}}$ (30) where w = (w₁, w₂, ⋯ , w_n) is the associated weighting vector, with w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ , and ${\dot{h}}_{σ (j)}$ is the j-th largest element of the hesitant Pythagorean fuzzy arguments ${\dot{h}}_{σ (j)} ({\dot{h}}_{σ (j)} = {(h_{j})}^{n ω_{j}}, j = 1, 2, \dots, n)$ , ω = (ω₁, ω₂, ⋯ , ω_n) is the weighting vector of hesitant Pythagorean fuzzy arguments h_j (j = 1, 2, ⋯ , n), with ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ , and n is the balancing coefficient. Especially, if w = (1/ - n, 1/ - n, ⋯ , 1/ - n) ^T, then HPFHHG is reduced to the hesitant Pythagorean fuzzy Hamacher weighted geometric (HPFHWG) operator; if ω = (1/ - n, 1/ - n, ⋯ , 1/ - n), then HPFHHA is reduced to the hesitant Pythagorean fuzzy Hamacher ordered weighted geometric (HPFHOWG) operator.

Based on Hamacher product operations of the hesitant Pythagorean fuzzy values described, we can drive the Theorem 6.

Theorem 6. Let h_j (j = 1, 2, ⋯ , n) be a collection of HPFEs, then their aggregated value by using the HPFHHG operator is also a HPFE, and $\begin{matrix} {HFHHG}_{w, ω} (h_{1}, h_{2}, \dots, h_{n}) = \otimes_{j = 1}^{n} {({\dot{h}}_{σ (j)})}^{w_{j}} \\ = \cup_{{\dot{γ}}_{σ (1)} \in {\dot{h}}_{σ (1)}, {\dot{γ}}_{σ (2)} \in {\dot{h}}_{σ (2)}, \dots, {\dot{γ}}_{σ (n)} \in {\dot{h}}_{σ (n)}} \\ {\frac{\sqrt{γ} \prod_{j = 1}^{n} {({\dot{γ}}_{σ (j)})}^{ω_{j}}}{\sqrt{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - {({\dot{γ}}_{σ (j)})}^{2}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {({\dot{γ}}_{σ (j)})}^{2 ω_{j}}}}} \end{matrix}$ (31)

where w = (w₁, w₂, ⋯ , w_n) is the associated weighting vector, with w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ , and ${\dot{h}}_{σ (j)}$ is the j-th largest element of the hesitant Pythagorean fuzzy arguments ${\dot{h}}_{j} ({\dot{h}}_{j} = {(h_{j})}^{n ω_{j}}, j = 1, 2, \dots, n)$ , ω = (ω₁, ω₂, ⋯ , ω_n) is the weighting vector of hesitant Pythagorean fuzzy arguments h_j (j = 1, 2, ⋯ , n), with with ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ , and n is the balancing coefficient, γ > 0.

Now, we can discuss some special cases of the HPFHHG operator with respect to theparameter γ.

When γ = 1, HPFHHG operator reduces to the hesitant Pythagorean fuzzy hybrid geometric (HPFHG) operator as follows:

$\begin{matrix} {HPFHG}_{ω, w} (h_{1}, h_{2}, \dots, h_{n}) \\ = \oplus_{j = 1}^{n} (w_{j} {\dot{h}}_{σ (j)}) \\ = \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\prod_{j = 1}^{n} {({\dot{γ}}_{σ (j)})}^{w_{j}}} \end{matrix}$ (32)

When γ = 2, HPFHHG operator reduces to the hesitant Pythagorean fuzzy Einstein hybrid geometric (HPFEHG) operator as follows:

$\begin{matrix} {HPFEHG}_{w, ω} (h_{1}, h_{2}, \dots, h_{n}) = \otimes_{j = 1}^{n} {({\dot{h}}_{σ (j)})}^{w_{j}} \\ = \cup_{{\dot{γ}}_{σ (1)} \in {\dot{h}}_{σ (1)}, {\dot{γ}}_{σ (2)} \in {\dot{h}}_{σ (2)}, \dots, {\dot{γ}}_{σ (n)} \in {\dot{h}}_{σ (n)}} \\ {\frac{\sqrt{2} \prod_{j = 1}^{n} {({\dot{γ}}_{σ (j)})}^{ω_{j}}}{\sqrt{\prod_{j = 1}^{n} {(2 - {({\dot{γ}}_{σ (j)})}^{2})}^{ω_{j}} + \prod_{j = 1}^{n} {({\dot{γ}}_{σ (j)})}^{2 ω_{j}}}}} \end{matrix}$ (33)

4 Models for multiple attribute decision making with hesitant Pythagorean fuzzy information

In this section, we shall utilize the hesitant Pythagorean Hamacher aggregation operators to multiple attribute decision making with hesitant Pythagorean fuzzy information.

The following assumptions or notations are used to represent the MADM problems for potential evaluation of emerging technology commercialization with hesitant Pythagorean fuzzy information. Let A ={ A₁, A₂, ⋯ , A_m } be a discrete set of alternatives, and G ={ G₁, G₂, ⋯ , G_n } be the state of nature. If the decision makers provide several values for the alternative A_i under the state of nature G_j with anonymity, these values can be considered as a hesitant Pythagorean fuzzy element h_ij. In the case where two decision makers provide the same value, then the value emerges only once in h_ij. Suppose that the decision matrix H = (h_ij) _m×n is the hesitant Pythagorean fuzzy decision matrix, where h_ij (i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n) are in the form of HPFEs.

In the following, we apply the HPFHWA (or HPFHWG) operator to the MADM problems for potential evaluation of emerging technology commercialization with hesitant Pythagorean fuzzy information.

Step 1. We utilize the decision information given in matrix H, and the HPFHWA operator $\begin{matrix} h_{i} & = & HPFHWA (h_{i 1}, h_{i 2}, \dots, h_{in}) = \oplus_{j = 1}^{n} (ω_{j} h_{ij}) \\ = & \cup_{γ_{i 1} \in h_{i 1}, γ_{i 2} \in h_{i 2}, \dots, γ_{in} \in h_{in}} \\ {\sqrt{\frac{\prod_{j = 1}^{n} {(1 + {(γ_{ij})}^{2})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - {(γ_{ij})}^{2})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + {(γ_{ij})}^{2})}^{ω_{j}} + \prod_{j = 1}^{n} {(1 - {(γ_{ij})}^{2})}^{ω_{j}}}}}, \\ i = 1, 2, \dots, m . \end{matrix}$ (34)

Or the hesitant Pythagorean fuzzy Hamacher weighted geometric (HPFHWG) operator: $\begin{matrix} h_{i} = HPFHWG (h_{i 1}, h_{i 2}, \dots, h_{in}) = \otimes_{j = 1}^{n} {(h_{ij})}^{ω_{j}} \\ = \cup_{γ_{i 1} \in h_{i 1}, γ_{i 2} \in h_{i 2}, \dots, γ_{in} \in h_{in}} \\ {\frac{\sqrt{γ} \prod_{j = 1}^{n} γ_{ij}^{ω_{j}}}{\sqrt{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - {(γ_{ij})}^{2}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(γ_{ij})}^{2 ω_{j}}}}} \end{matrix}$ (35)

to derive the overall preference values h_i (i = 1, 2, ⋯ , m) of the alternative A_i.

Step 2. Calculate the scores S (h_i) (i = 1, 2, ⋯ , m) of the overall hesitant Pythagorean fuzzy preference values h_i (i = 1, 2, ⋯ , m) to rank all the alternatives A_i (i = 1, 2, ⋯ , m) and then to select the best one(s).

Step 3. Rank all the alternatives A_i (i = 1, 2, ⋯ , m) and select the best one(s) in accordance with S (h_i) (i = 1, 2, ⋯ , m).

Step 4. End.

5 Numerical example

Thus, in this section we shall present a numerical example to show potential evaluation of emerging technology commercialization with hesitant Pythagorean fuzzy information in order to illustrate the method proposed in this paper. There is a panel with five possible emerging technology enterprises A_i (i = 1, 2, 3, 4, 5) to select. The experts selects four attribute to evaluate the five possible emerging technology enterprises: ➀₁ is the technical advancement; ➁₂ is the potential market and market risk; ➂₃ is the industrialization infrastructure, human resources and financial conditions; ➃₄ is the employment creation and the development of science and technology. In order to avoid influence each other, the decision makers are required to evaluate the four possible emerging technology enterprises A_i (i = 1, 2, 3, 4, 5) under the above four attributes in anonymity and the decision matrix H = (h_ij) _5×4 is presented in Table 1, where h_ij (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4) are in the form of HFEs.

Table 1
Hesitant Pythagorean fuzzy decision matrix

G₁ G₂ G₃ G₄

A₁ (0.3,0.4,0.6) (0.4,0.7) (0.5,0.6,0.8) (0.4,0.5)

A₂ (0.5,0.6,0.7) (0.4,0.5) (0.3,0.5,0.6) (0.3,0.4)

A₃ (0.5,0.7) (0.1,0.2,0.4) (0.3,0.5) (0.4,0.6)

A₄ (0.4,0.5,0.6) (0.3,0.4) (0.7,0.8) (0.2,0.3,0.4)

A₅ (0.5,0.6) (0.4,0.6) (0.6,0.7) (0.1,0.2,0.3)

G₁	G₂	G₃	G₄
A₁	(0.3,0.4,0.6)	(0.4,0.7)	(0.5,0.6,0.8)	(0.4,0.5)
A₂	(0.5,0.6,0.7)	(0.4,0.5)	(0.3,0.5,0.6)	(0.3,0.4)
A₃	(0.5,0.7)	(0.1,0.2,0.4)	(0.3,0.5)	(0.4,0.6)
A₄	(0.4,0.5,0.6)	(0.3,0.4)	(0.7,0.8)	(0.2,0.3,0.4)
A₅	(0.5,0.6)	(0.4,0.6)	(0.6,0.7)	(0.1,0.2,0.3)

The information about the attribute weights is known as follows: ω = (0.3, 0.2, 0.4, 0.1).

In the following, we utilize the approach developed to show potential evaluation of emerging technology commercialization of four possible emerging technology enterprises.

Step 1. We utilize the decision information given in matrix H, and the HPFHWA operator to obtain the overall preference values h_i of the emerging technology enterprises A_i (i = 1, 2, 3, 4, 5). Take alternative A₁ for an example, we have (suppose that γ = 3): $\begin{matrix} h_{1} & = & HPFHWA (h_{11}, h_{12}, h_{13}, h_{14}) \\ = & \oplus_{j = 1}^{4} (ω_{j} h_{ij}) \\ = & \cup_{γ_{i 1} \in h_{i 1}, γ_{i 2} \in h_{i 2}, γ_{i 3} \in h_{i 3}, γ_{i 4} \in h_{i 4}} \\ {\sqrt{\frac{\prod_{j = 1}^{4} {(1 + {(γ_{1 j})}^{2})}^{ω_{j}} - \prod_{j = 1}^{4} {(1 - {(γ_{1 j})}^{2})}^{ω_{j}}}{\prod_{j = 1}^{4} {(1 + {(γ_{1 j})}^{2})}^{ω_{j}} + \prod_{j = 1}^{4} {(1 - {(γ_{1 j})}^{2})}^{ω_{j}}}}} \\ = & HPFHWA {(0.3, 0.4, 0.6), (0.4, 0.7), \\ (0.5, 0.6, 0.8), (0.4, 0.5)} \\ = & {0.4176, 0.4282, 0.4425, 0.4526, 0.4674, \\ 0.4770, 0.4900, 0.4919, 0.4991, 0.5010, \\ 0.5061, 0.5134, 0.5149, 0.5221, 0.5350, \\ 0.5434, 0.5480, 0.5548, 0.5562, 0.5628, \\ 0.5690, 0.5769, 0.5886, 0.5961, 0.6060, \\ 0.6064, 0.6133, 0.6136, 0.6421, 0.6487, \\ 0.6524, 0.6578, 0.6589, 0.6642, \\ 0.6988, 0.7046} \end{matrix}$

Step 2. Calculate the scores s (h_i) (i = 1, 2, 3, 4, 5) of the overall hesitant Pythagorean fuzzy preference values h_i (i = 1, 2, 3, 4, 5): $\begin{matrix} s (h_{1}) & = & 0.5589, s (h_{2}) = 0.5043 \\ s (h_{3}) & = & 0.4645, s (h_{4}) = 0.5874 \\ s (h_{5}) & = & 0.5641 \end{matrix}$

Step 3. Rank all the emerging technology enterprises A_i (i = 1, 2, 3, 4, 5) in accordance with the scores s (h_i) (i = 1, 2, 3, 4, 5) of the overall hesitant Pythagorean fuzzy preference values: A₄ ≻ A₅ ≻ A₁ ≻ A₂ ≻ A₃, and thus the most desirable emerging technology enterprise is A₄.

Based on the HPFHWG operator, then, in order to select the most desirable alternative, we can develop an approach to multiple attribute decision making problems with hesitant Pythagorean fuzzy information, which can be described as following:

Step 1. Aggregate all hesitant Pythagorean fuzzy value h_ij (j = 1, 2, 3, 4) by using the hesitant Pythagorean fuzzy Hamacher weighted geometric (HPFHWG) operator to derive the overall hesitant Pythagorean fuzzy values h_i (i = 1, 2, ⋯ , 5) of the alternative A_i. Take emerging technology enterprise A₁ for an example, we have (suppose that γ = 3): $\begin{matrix} h_{1} & = & HPFHWG (h_{11}, h_{12}, h_{13}, h_{14}) \\ = & \otimes_{j = 1}^{4} {(h_{1 j})}^{ω_{j}} \\ = & \cup_{γ_{i 1} \in h_{i 1}, γ_{i 2} \in h_{i 2}, γ_{i 3} \in h_{i 3}, γ_{i 4} \in h_{i 4}} \\ {\frac{\sqrt{γ} \prod_{j = 1}^{4} γ_{1 j}^{ω_{j}}}{\sqrt{\prod_{j = 1}^{4} {(1 + (γ - 1) (1 - {(γ_{1 j})}^{2}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{4} {(γ_{1 j})}^{2 ω_{j}}}}} \\ = & HPFHWG {(0.3, 0.4, 0.6), (0.4, 0.7), \\ (05, 0.6, 0.8), (0.4, 0.5)} \\ = & {0.4033, 0.4127, 0.4372, 0.4381, 0.4471, \\ 0.4481, 0.4561, 0.4663, 0.4740, 0.4845, \\ 0.4929, 0.4940, 0.4963, 0.5036, 0.5047, \\ 0.5059, 0.5071, 0.5168, 0.5326, 0.5351, \\ 0.5438, 0.5462, 0.5463, 0.5564, \\ 0.5575, 0.5666, 0.5679, 0.5781, 0.5974, \\ 0.6091, 0.6092, 0.6117, 0.6209, 0.6236, \\ 0.6770, 0.6891 \end{matrix}$

Step 2. Calculate the scores s (h_i) (i = 1, 2, 3, 4, 5) of the overall hesitant Pythagorean fuzzy values h_i (i = 1, 2, 3, 4, 5) of the alternative A_i: $\begin{matrix} s (h_{1}) & = & 0.5294, s (h_{2}) = 0.4835 \\ s (h_{3}) & = & 0.4081, s (h_{4}) = 0.5318 \\ s (h_{5}) & = & 0.5247 . \end{matrix}$

Step 3. Rank all the emerging technology enterprise A_i (i = 1, 2, 3, 4, 5) in accordance with the scores s (h_i) (i = 1, 2, 3, 4, 5) of the overall hesitant Pythagorean fuzzy values h_i (i = 1, 2, ⋯ , 5): A₄ ≻ A₁ ≻ A₅ ≻ A₂ ≻ A₃ and thus the most desirable emerging technology enterprise is A₄.

From the above analysis, it is easily seen that although the overall rating values of the alternatives are different by using two operators respectively, the ranking orders of the alternatives are slightly different. However, the most desirable emerging technology enterprise is A₄.

6 Conclusion

In this paper, we investigate the multiple attribute decision making (MADM) problem based on the Hamacher aggregation operators with hesitant Pythagorean fuzzy information. Then, motivated by the ideal of Hamacher operation [42], we have developed some Hamacher aggregation operators for aggregating hesitant Pythagorean fuzzy information: hesitant Pythagorean fuzzy Hamacher weighted average (HPFHWA) operator, hesitant Pythagorean fuzzy Hamacher weighted geometric (HPFHWG) operator, hesitant Pythagorean fuzzy Hamacher ordered weighted average (HPFHOWA) operator, hesitant Pythagorean fuzzy Hamacher ordered weighted geometric (HPFHOWG) operator, hesitant Pythagorean fuzzy Hamacher hybrid average (HPFHHA) operator and hesitant Pythagorean fuzzy Hamacher hybrid geometric (HPFHHG) operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the hesitant Pythagorean fuzzy multiple attribute decision making problems. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness. In our future study, we shall extend the proposed models to other domain, such as, pattern recognition, risk analysis, supplier selection, and so on [59 –76].

Footnotes

Acknowledgements

The work was supported by the National Natural Science Foundation of China under Grant No. 71571128, 61174149 and the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China under Grant No.16YJA630033 and the Sciences Foundation of Sichuan Normal University (14yb18).

References

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

Atanassov

, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989), 37–46.

Atanassov

, Two theorems for intuitionistic fuzzy sets, Fuzzy Sets and Systems 110 (2000), 267–269.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–356.

Z.S.

, Intuitionistic fuzzy aggregation operators, IEEE Transations on Fuzzy Systems 15(6) (2007), 1179–1187.

Z.S.

and Yager

R.R.

, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General System 35 (2006), 417–433.

Tang

, Wen

L.L.

and Wei

G.W.

, Approaches to multiple attribute group decision making based on the generalized Dice similarity measures with intuitionistic fuzzy information, International Journal of Knowledge-Based and Intelligent Engineering Systems 21(2) (2017), 85–95.

Wang

J.C.

and Chen

T.Y.

, Likelihood-based assignment methods for multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets, Fuzzy Optimization and Decision Making 14(4) (2015), 425–457.

Chen

T.Y.

, The inclusion-based TOPSIS method with interval-valued intuitionistic fuzzy sets for multiple criteria group decision making, Applied Soft Computing 26 (2015), 57–73.

10.

Zhao

X.F.

and Wei

G.W.

, Some intuitionistic fuzzy einstein hybrid aggregation operators and their application to multiple attribute decision making, Knowledge-Based Systems 37 (2013), 472–479.

11.

D.F.

, Decision and Game Theory in Management With Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing, 308, Springer, 2014, pp. 1–441. ISBN978-3-642-40711-6.

12.

Zhu

Y.J.

and Li

D.F.

, A new definition and formula of entropy for intuitionistic fuzzy sets, Journal of Intelligent and Fuzzy Systems 30(6) (2016), 3057–3066.

13.

Wei

G.W.

, Maximizing deviation method for multiple attribute decision making in intuitionistic fuzzy setting, Knowledge-Based Systems 21 (2008), 833–836.

14.

Wei

G.W.

, Some geometric aggregation functions and their application to dynamic multiple attribute decision making in intuitionistic fuzzy setting, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 17 (2009), 179–196.

15.

Wei

G.W.

, GRA method for multiple attribute decision making with incomplete weight information in intuitionistic fuzzy setting, Knowledge-Based Systems 23 (2010), 243–247.

16.

Wei

G.W.

, Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making, Applied Soft Computing 10 (2010), 423–431.

17.

Wei

G.W.

, Wang

H.J.

and Lin

, Application of correlation coefficient to interval-valued intuitionistic fuzzy multiple attribute decision-making with incomplete weight information, Knowledge and Information Systems 26 (2011), 337–349.

18.

, Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment, European Journal of Operational Research 205 (2010), 202–204.

19.

, Similarity measures of intuitionistic fuzzy sets based on cosine function for the decision making of mechanical design schemes, Journal of Intelligent and Fuzzy Systems 30(1) (2016), 151–158.

20.

Nayagam

V.L.G.

, Muralikrishnan

and Sivaraman

, Multi-criteria decision-making method based on interval-valued intuitionistic fuzzy sets, Expert Systems with Applications 38 (2011), 1464–1467.

21.

Wei

G.W.

and Zhao

X.F.

, Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making, Expert Systems with Applications 39(2) (2012), 2026–2034.

22.

Szmidt

, Kacprzyk

and Bujnowski

, How to measure the amount of knowledge conveyed by Atanassov’s intuitionistic fuzzy sets, Information Sciences 257 (2014), 276–285.

23.

Szmidt

and Kacprzyk

, Dealing with typical values via Atanassov’s intuitionistic fuzzy sets, International Journal of General Systems 39 (2010), 489–506.

24.

Wei

G.W.

, Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information, International Journal of Fuzzy Systems 17(3) (2015), 484–489.

25.

Wei

G.W.

, Alsaadi

F.E.

, Hayat

, Alsaedi

, A linear assignment method for multiple criteria decision analysis with hesitant fuzzy sets based on fuzzy measure, International Journal of Fuzzy Systems 19(3) (2017), 607–614.

26.

Yager

R.R.

, Pythagorean fuzzy subsets, in: Proceeding of The Joint IFSA Wprld Congress and NAFIPS Annual Meeting, Edmonton, Canada, 2013, pp. 57–61.

27.

Yager

R.R.

, Pythagorean membership grades in multicriteria decision making, IEEE Trans Fuzzy Syst 22 (2014), 958–965.

28.

Zhang

X.L.

and Xu

Z.S.

, Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets, Int J Intell Syst 29 (2014), 1061–1078.

29.

Peng

and Yang

, Some results for Pythagorean fuzzy sets, Int J Intell Syst 30 (2015), 1133–1160.

30.

Beliakov

and James

, Averaging aggregation functions for preferences expressed as Pythagorean membership grades and fuzzy orthopairs, Fuzz-IEEE (2014), 298–305.

31.

Reformat

and Yager

R.R.

, Suggesting recommendations using pythagorean fuzzy sets illustrated using netflix movie data, IPMU, (1) (2014), 546–556.

32.

Gou

, Xu

and Ren

, The properties of continuous pythagorean fuzzy information, Int J Intell Syst 31(5) (2016), 401–424.

33.

Ren

, Xu

and Gou

, Pythagorean fuzzy TODIM approach to multi-criteria decision making, Appl Soft Comput 42 (2016), 246–259.

34.

Garg

, A new generalized pythagorean fuzzy information aggregation using einstein operations and its application to decision making, Int J Intell Syst 31(9) (2016), 886–920.

35.

Zeng

, Chen

and Li

, A hybrid method for pythagorean fuzzy multiple-criteria decision making, International Journal of Information Technology and Decision Making 15(2) (2016), 403–422.

36.

Garg

, A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem, Journal of Intelligent and Fuzzy Systems 31(1) (2016), 529–540.

37.

Hamachar

, Uber logische verknunpfungenn unssharfer “Aussagen undderen Zugenhorige Bewertungsfunktione”, in: Trappl, Klir, Riccardi (Eds.), Progress in Cyatics and systems research, vol. 3, Hemisphere, Washington DC, 1978, pp. 276–288.

38.

Beliakov

, Pradera

and Calvo

, Aggregation Functions: A Guide for Practitioners, Springer, Heidelberg, Berlin, New York, 2007.

39.

Deschrijver

, Cornelis

and Kerre

E.E.

, On the representation of intuitionistic fuzzy t-norms and t-conorms, IEEE Transactions on Fuzzy Systems 12 (2004), 45–61.

40.

Roychowdhury

and Wang

B.H.

, On generalized Hamacher families of triangular operators, International Journal of Approximate Reasoning 19 (1998), 419–439.

41.

Deschrijver

and Kerre

E.E

, A generalization of operators on intuitionistic fuzzy sets using triangular norms and conorms, Notes on Intuitionistic Fuzzy Sets 8 (2002), 19–27.

42.

Zhou

L.Y.

, Zhao

X.F.

and Wei

G.W.

, Hesitant fuzzy hamacher aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 26(6) (2014), 2689–2699.

43.

Liu

P.D.

, Some hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making, IEEE Trans Fuzzy Systems 22(1) (2014), 83–97.

44.

Tan

, Yi

and Chen

, Hesitant fuzzy Hamacher aggregation operators for multicriteria decision making, Appl Soft Comput 26 (2015), 325–349.

45.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010), 529–539.

46.

Zhang

and Wei

G.W.

, Extension of VIKOR method for decision making problem based on hesitant fuzzy set, Applied Mathematical Modelling 37 (2013), 4938–4947.

47.

Wei

G.W.

, Hesitant Fuzzy prioritized operators and their application to multiple attribute group decision making, Knowledge-Based Systems 31 (2012), 176–182.

48.

Xia

M.M.

, Xu

Z.S.

and Zhu

, Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm, Knowledge-Based Systems 31 (2012), 78–88.

49.

Xia

and Xu

Z.S.

, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52(3) (2011), 395–407.

50.

Wang

W.Z.

and Liu

X.W.

, Intuitionistic fuzzy geometric aggregation operators based on einstein operations, International Journal of Intelligent Systems 26 (2011), 1049–1075.

51.

Yager

R.R.

, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Transactions on Systems Man and Cybernetics 18 (1988), 183–190.

52.

Merigó

J.M.

and Gil-Lafuente

A.M.

, Induced 2-tuple linguistic generalized aggregation operators and their application in decision-making, Information Sciences 236(1) (2013), 1–16.

53.

Merigo

J.M.

and Casanovas

, Induced aggregation operators in decision making with the Dempster-Shafer belief structure, International Journal of Intelligent Systems 24 (2009), 934–954.

54.

Merigo

J.M.

, A unified model between the weighted average and the induced OWA operator, Expert Systems with Applications 38 (2011), 11560–11572.

55.

Merigo

J.M.

, Fuzzy multi-person decision making with fuzzy probabilistic aggregation operators, International Journal of Fuzzy Systems 13 (2011), 163–174.

56.

Merigo

J.M.

and Casanovas

, Fuzzy generalized hybrid aggregation operators and its application in fuzzy decision making, International Journal of Fuzzy Systems 12 (2010), 15–24.

57.

Z.S.

and Da

Q.L.

, An overview of operators for aggregating information, International Journal of Intelligent System 18 (2003), 953–969.

58.

Chiclana

, Herrera

, Herrera-Viedma

, The ordered weighted geometric operator: Properties and application, Proc of 8th Int Conf on Information Processing and Management of Uncertainty in Knowledge-based Systems, Madrid, 2000, pp. 985–991.

59.

Wei

G.W.

, Picture fuzzy cross-entropy for multiple attribute decision making problems, Journal of Business Economics and Management 17(4) (2016), 491–502.

60.

Wei

G.W.

, Alsaadi

F.E.

, Hayat

and Alsaedi

, Hesitant fuzzy linguistic arithmetic aggregation operators in multiple attribute decision making, Iranian Journal of Fuzzy Systems 13(4) (2016), 1–16.

61.

Wei

G.W.

, Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making, International Journal of Machine Learning and Cybernetics 7(6) (2016), 1093–1114.

62.

Chen

T.Y.

and Li

C.H.

, Determining objective weights with intuitionistic fuzzy entropy measures, A comparative analysis, Information Sciences 180 (2010), 4207–4222.

63.

and Wei

G.W.

, Models for multiple attribute decision making with dual hesitant fuzzy uncertain linguistic information, International Journal of Knowledge-based and Intelligent Engineering Systems 20(4) (2016), 217–227.

64.

Wei

G.W.

, Xu

X.R.

and Deng

D.X.

, Interval-valued dual hesitant fuzzy linguistic geometric aggregation operators in multiple attribute decision making, International Journal of Knowledge-based and Intelligent Engineering Systems 20(4) (2016), 189–196.

65.

Rodriguez

R.M.

, Martinez

and Herrera

, Hesitant fuzzy linguistic term sets for decision making, IEEE Transactions on Fuzzy Systems 20 (2012), 109–119.

66.

Wei

G.W.

, Wang

J.M.

and Chen

, Potential optimality and robust optimality in multiattribute decision analysis with incomplete information: A comparative study, Decision Support Systems 55(3) (2013), 679–684.

67.

Wei

G.W.

and Wang

J.M.

, A comparative study of robust efficiency analysis and data envelopment analysis with imprecise data, Expert Systems with Applications 81 (2017), 28–38.

68.

Wei

G.W.

, Chen

and Wang

, Stochastic efficiency analysis with a reliability consideration, Omega 48 (2014), 1–9.

69.

Lin

, Zhao

X.F.

and Wei

G.W.

, Models for selecting an ERP system with hesitant fuzzy linguistic information, Journal of Intelligent and Fuzzy Systems 26(5) (2014), 2155–2165.

70.

Zhao

X.F.

, Lin

and Wei

G.W.

, Hesitant triangular fuzzy information aggregation based on Einstein operations and their application to multiple attribute decision making, Expert Systems with Applications 41(4) (2014), 1086–1094.

71.

Wei

G.W.

, Zhao

X.F.

and Lin

, Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making, Knowledge-Based Systems 46 (2013), 43–53.

72.

Wei

G.W.

and Zhao

X.F.

, Induced hesitant interval-valued fuzzy einstein aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 24 (2013), 789–803.

73.

Wei

G.W.

, Zhao

X.F.

, Lin

and Wang

H.J.

, Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making, Applied Mathematical Modelling 37 (2013), 5277–5285.

74.

Wang

H.J.

, Zhao

X.F.

and Wei

G.W.

, Dual hesitant fuzzy aggregation operators in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 26(5) (2014), 2281–2290.

75.

Jiang

X.P.

and Wei

G.W.

, Some Bonferroni mean operators with 2-tuple linguistic information and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 27 (2014), 2153–2162.

76.

Wei

G.W.

and Zhao

X.F.

, Some dependent aggregation operators with 2-tuple linguistic information and their application to multiple attribute group decision making, Expert Systems with Applications 39 (2012), 5881–5886.