Hesitant bipolar fuzzy aggregation operators in 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This article has been retracted, and the online PDF replaced with this retraction notice.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)bipolar fuzzy set hesitant bipolar fuzzy set hesitant bipolar fuzzy hybrid average (HBFHA) operator hesitant bipolar fuzzy hybrid geometric (HBFHG) operator software quality constructional engineering

1 Introduction

Atanassov [1, 2] introduced the concept of intuitionistic fuzzy set(IFS) characterized by a membership function and a non-membership function, which is a generalization of the concept of fuzzy set [3] whose basic component is only a membership function. Xu [4] developed the intuitionistic fuzzy arithmetic aggregating operators. Xu [5] developed some geometric aggregation operators with intuitionistic fuzzy information. Furthermore, Torra [6] proposed the hesitant fuzzy set which permits the membership having a set of possible values and discussed the relationship between hesitant fuzzy set and intuitionistic fuzzy set, and showed that the envelope of hesitant fuzzy set is an intuitionistic fuzzy set. Xia and Xu [7] gave an intensive study on hesitant fuzzy information aggregation techniques and their application in decision making. Xu et al. [8] developed several series of aggregation operators for hesitant fuzzy information with the aid of quasi-arithmetic means. Wei [9] developed some prioritized aggregation operators for aggregating hesitant fuzzy information. Wei et al. [10] proposed two hesitant fuzzy Choquet integral aggregation operators: hesitant fuzzy choquet ordered averaging (HFCOA) operator and hesitant fuzzy choquet ordered geometric (HFCOG) operator. Zhu et al. [11] explored the geometric Bonferroni mean (GBM) considering both the BM and the geometric mean (GM) under hesitant fuzzy environment. The hesitant fuzzy set has received more and more attention since its appearance [12 –30]. More recently, the bipolar fuzzy set (BFS) [31, 32] has emerged lately as an alternative tool to depict uncertainty in MADM problems. A pair of numbers, namely, the positive membership degree and the negative membership degree, is employed to define an object in a BFS. But different from the IFS, the range of membership degree of the bipolar fuzzy set is [–1,1]. BFSs have been applied in many research areas including but not limited to bipolar logical reasoning and set theory [33, 34], traditional Chinese medicine theory [35, 36], bipolar cognitive mapping [37, 38], computational psychiatry [39, 40], decision analysis and organizational modeling [41, 42], photonics [43], quantum computing [44, 45], biosystem regulation [46 –48], quantum cellular combinatorics [49], physics and philosophy [50] and graph theory [51 –55]. Recently, Gul [56] defined some bipolar fuzzy aggregations operators, such as, bipolar fuzzy averaging weighted aggregation operators and bipolar fuzzy geometric aggregations operators.

In this paper, we investigate the multiple attribute decision making (MADM) problem based on the aggregation operators with hesitant bipolar fuzzy information. Then, motivated by the ideal of arithmetic and geometric operation [57 –60], we have developed some aggregation operators for aggregating hesitant bipolar fuzzy information: hesitant bipolar fuzzy weighted average (HBFWA) operator, hesitant bipolar fuzzy weighted geometric (HBFWG) operator, hesitant bipolar fuzzy ordered weighted average (HBFOWA) operator, hesitant bipolar fuzzy ordered weighted geometric (HBFOWG) operator, hesitant bipolar fuzzy hybrid average (HBFHA) operator and hesitant bipolar fuzzy hybrid geometric (HBFHG) operator. Then, we have utilized these operators to develop some approaches to solve the hesitant bipolar fuzzy multiple attribute decision making problems. The remainder of this paper is organized as follows. In the next section, we briefly review the basic concepts of the HBFSs and the fundamental operational laws of HBFNs. In Section 3, we develop hesitant bipolar fuzzy arithmetic aggregation operators. In Section 4, we develop hesitant bipolar fuzzy geometric aggregation operators. In Section 5, models are developed that apply the proposed aggregation operators to solve MADM problems. Finally, an illustrative example for evaluating the constructional engineering software quality is then analyzed to illustrate the relevance and effectiveness of the proposed methodology in Section 6. Some remarks are given in Section 7 to conclude the paper.

2 Preliminaries

2.1 The bipolar fuzzy set

In this section, we present a short overview of BFSs [31, 32].

Definition 1. [31, 32]. Let X be a fix set. A BFS is an object having the form $B = {〈 x, (μ_{B}^{+} (x), ν_{B}^{-} (x)) 〉 | x \in X}$ (1) where the positive membership degree function $μ_{B}^{+} (x) : X \to [0, 1]$ denotes the satisfaction degree of an element x to the property corresponding to a BFS B and the negative membership degree function $ν_{B}^{-} (x) : X \to [- 1, 0]$ denotes satisfaction degree of an element x to some implicit counter property corresponding to a BFS B, respectively, and, for every x ∈ X. For convenience, the pair b (x) = (μ⁺ (x) , ν^- (x)) is called a bipolar fuzzy number (BFN) denoted by b = (μ⁺, ν^-) with the conditions: 0 ≤ μ⁺ ≤ 1, -1 ≤ ν^- ≤ 0.

Definition 2. [56] Some basic operations on BFNs are expressed as follows:

${\tilde{b}}_{1} \oplus {\tilde{b}}_{2} = (μ_{1}^{+} + μ_{2}^{+} - μ_{1}^{+} μ_{2}^{+}, - | ν_{1}^{-} | | ν_{2}^{-} |);$

${\tilde{b}}_{1} \otimes {\tilde{b}}_{2} = (μ_{1}^{+} μ_{2}^{+}, ν_{1}^{-} + ν_{2}^{-} - ν_{1}^{-} ν_{2}^{-});$

$λ \tilde{b} = (1 - {(1 - μ^{+})}^{λ}, - {| ν^{-} |}^{λ}), λ > 0;$

${(\tilde{b})}^{λ} = ({(μ^{+})}^{λ}, - 1 + {| 1 + ν^{-} |}^{λ}), λ > 0;$

${\tilde{b}}^{c} = (1 - μ^{+}, | ν^{-} | - 1);$

${\tilde{b}}_{1} \subseteq {\tilde{b}}_{2}, if and only if μ_{1}^{+} \leq μ_{2}^{+} and ν_{1}^{-} \geq ν_{2}^{-};$

${\tilde{b}}_{1} \cup {\tilde{b}}_{2} = (max {μ_{1}^{+}, μ_{2}^{+}}, min {ν_{1}^{-}, ν_{2}^{-}});$

${\tilde{b}}_{1} \cap {\tilde{b}}_{2} = (min {μ_{1}^{+}, μ_{2}^{+}}, max {ν_{1}^{-}, ν_{2}^{-}}) .$

Based on the Definition 2, we can introduce the Theorem 1 easily.

Theorem 1. [56] Let ${\tilde{b}}_{1} = (μ_{1}^{+}, ν_{1}^{-})$ and ${\tilde{b}}_{2} = (μ_{2}^{+}, ν_{2}^{-})$ be two BFNs, λ, λ₁, λ₂ > 0, then

${\tilde{b}}_{1} \oplus {\tilde{b}}_{2} = {\tilde{b}}_{2} \oplus {\tilde{b}}_{1};$

${\tilde{b}}_{1} \otimes {\tilde{b}}_{2} = {\tilde{b}}_{2} \otimes {\tilde{b}}_{1};$

$λ ({\tilde{b}}_{1} \oplus {\tilde{b}}_{2}) = λ {\tilde{b}}_{1} \oplus λ {\tilde{b}}_{2};$

${({\tilde{b}}_{1} \otimes {\tilde{b}}_{2})}^{λ} = {({\tilde{b}}_{1})}^{λ} \otimes {({\tilde{b}}_{2})}^{λ};$

$λ_{1} {\tilde{b}}_{1} \oplus λ_{2} {\tilde{b}}_{1} = (λ_{1} + λ_{2}) {\tilde{b}}_{1};$

${({\tilde{b}}_{1})}^{λ_{1}} \otimes {({\tilde{b}}_{1})}^{λ_{2}} = {({\tilde{b}}_{1})}^{(λ_{1} + λ_{2})};$

${({({\tilde{b}}_{1})}^{λ_{1}})}^{λ_{2}} = {({\tilde{b}}_{1})}^{λ_{1} λ_{2}} .$

2.2 Hesitant bipolar fuzzy set (HBFS)

In the following, motivated by the bipolar fuzzy set (BFS) [31, 32] and hesitant fuzzy set (HFS) [6], we shall propose the hesitant bipolar fuzzy set (HBFS).

Definition 3. Let X be a fixed set, then a hesitant bipolar fuzzy set (HBFS) on X is described as: $B^{*} = (〈 x, h_{B^{*} (x)} 〉 | x \in X)$ (2) where h_B^*(x) is a set of some bipolar fuzzy number (BFN) in B, $h_{B^{*} (x)} = \cup_{(μ_{B^{*} (x)}^{+}, ν_{B^{*} (x)}^{-}) \in h_{B^{*} (x)}} (μ_{B^{*} (x)}^{+}, ν_{B^{*} (x)}^{-})$ , and the posi-tive membership degree function $μ_{B^{*} (x)}^{+} : X \to [0, 1]$ denotes the possible satisfaction degree of an element x to the property corresponding to a HBFS B^* and the negative membership degree function $ν_{B^{*} (x)}^{-} : X \to [- 1, 0]$ denotes the possible satisfaction degree of an element x to some implicit counter property corresponding to a HBFS B^*, respectively, and, for every x ∈ X, with the conditions: $0 \leq μ_{B^{*} (x)}^{+} \leq 1, - 1 \leq ν_{B^{*} (x)}^{-} \leq 0$

For convenience, the pair $\tilde{h} (x) = {(μ^{+} (x), ν^{-} (x))}$ is called a hesitant bipolar fuzzy number (HBFN) denoted by $\tilde{h} = (μ^{+}, ν^{-})$ , with the conditions: 0 ≤ γ⁺ ≤ 1, - 1 ≤ η^- ≤ 0, (γ⁺, η^-) ∈ (μ⁺, ν^-) .

To compare the HBFN, we shall give the following comparison laws:

Definition 4. Let ${\tilde{h}}_{i} = (μ_{i}^{+}, ν_{i}^{-}) (i = 1, 2)$ be any two HBFNs, $s ({\tilde{h}}_{i}) = \frac{1}{# {\tilde{h}}_{i}} \sum_{i = 1}^{# {\tilde{h}}_{i}} \frac{1 + γ_{i}^{+} + η_{i}^{-}}{2},$ is the score function of ${\tilde{h}}_{i} = (μ_{i}^{+}, ν_{i}^{-})$ , and $a (\tilde{h}) = \frac{1}{# h} \sum_{i = 1}^{# h} \frac{γ_{i}^{+} - η_{i}^{-}}{2}$ the accuracy function of ${\tilde{h}}_{i} = (μ_{i}^{+}, ν_{i}^{-})$ , where $# {\tilde{h}}_{i}$ is the numbers of the elements in ${\tilde{h}}_{i}$ , $(γ_{i}^{+}, η_{i}^{-}) \in {\tilde{h}}_{i}, i = 1, 2, \dots, # {\tilde{h}}_{i}$ , then

If $s ({\tilde{h}}_{1}) > s ({\tilde{h}}_{2})$ , then ${\tilde{h}}_{1}$ is superior to ${\tilde{h}}_{2}$ , denoted by ${\tilde{h}}_{1} ≻ {\tilde{h}}_{2}$ ;

If $s ({\tilde{h}}_{1}) = s ({\tilde{h}}_{2})$ , then

If $a ({\tilde{h}}_{1}) = a ({\tilde{h}}_{2})$ , then ${\tilde{h}}_{1}$ is equivalent to ${\tilde{h}}_{2}$ , denoted by ${\tilde{h}}_{1} \sim {\tilde{h}}_{2}$ ;

If $a ({\tilde{h}}_{1}) > a ({\tilde{h}}_{2})$ , then ${\tilde{h}}_{1}$ is superior to ${\tilde{h}}_{2}$ , denoted by ${\tilde{h}}_{1} ≻ {\tilde{h}}_{2}$ .

Then, we define some new operations on the HBFNs $\tilde{h}$ , ${\tilde{h}}_{1}$ and ${\tilde{h}}_{2}$ :

${\tilde{h}}^{λ} = \cup_{(γ^{+}, η^{-}) \in (μ^{+}, ν^{-})} {({(γ^{+})}^{λ}, - 1 + {| 1 + η^{-} |}^{λ})}, λ > 0;$

$λ \tilde{h} = \cup_{(γ^{+}, η^{-}) \in (μ^{+}, ν^{-})} {(1 - {(1 - γ^{+})}^{λ}, - {| η^{-} |}^{λ})}, λ > 0;$

${\tilde{h}}_{1} \oplus {\tilde{h}}_{2} = \cup_{(γ_{1}^{+}, η_{1}^{-}) \in (μ_{1}^{+}, ν_{1}^{-}), (γ_{2}^{+}, η_{2}^{-}) \in (μ_{2}^{+}, ν_{2}^{-})} {(γ_{1}^{+} + γ_{2}^{+} - γ_{1}^{+} γ_{2}^{+}, - | η_{1}^{-} | | η_{2}^{-} |)};$

${\tilde{h}}_{1} \otimes {\tilde{h}}_{2} = \cup_{(γ_{1}^{+}, η_{1}^{-}) \in (μ_{1}^{+}, ν_{1}^{-}), (γ_{2}^{+}, η_{2}^{-}) \in (μ_{2}^{+}, ν_{2}^{-})} {(γ_{1}^{+} γ_{2}^{+}, η_{1}^{-} + η_{2}^{-} - η_{1}^{-} η_{2}^{-})} .$

3 Hesitant bipolar fuzzy aggregation operators

3.1 Hesitant bipolar fuzzy arithmetic aggregation operators

Let ${\tilde{h}}_{j} = (μ_{j}^{+}, ν_{j}^{-}) (j = 1, 2, \dots, n)$ be a collection of HBFNs. We next establish hesitant bipolar fuzzy arithmetic aggregation operators.

Definition 5. The hesitant bipolar fuzzy weighted average (HBFWA) operator is ${DHBFWA}_{ω} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) = \oplus_{j = 1}^{n} (ω_{j} {\tilde{h}}_{j})$ (3) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T denotes the weight vector associated with ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

Theorem 2 can be shown by its definition and mathematical induction.

Theorem 2. The HBFWA operator returns a HBFN with

$\begin{matrix} {HBFWA}_{ω} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = \oplus_{j = 1}^{n} (ω_{j} {\tilde{h}}_{j}) \\ = \cup_{(γ_{j}^{+}, η_{j}^{-}) \in (μ_{j}^{+}, ν_{j}^{-})} {(\begin{matrix} 1 - \prod_{j = 1}^{n} {(1 - γ_{j}^{+})}^{ω_{j}}, \\ - \prod_{j = 1}^{n} {| η_{j}^{-} |}^{ω_{j}} \end{matrix})} \end{matrix}$ (4)

Definition 6. The hesitant bipolar fuzzy ordered weighted average (HBFOWA)operator is definedas

$\begin{matrix} {HBFOWA}_{w} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = \oplus_{j = 1}^{n} (w_{j} {\tilde{h}}_{σ (j)}) \\ = \cup_{(γ_{σ (j)}^{+}, η_{σ (j)}^{-}) \in (μ_{σ (j)}^{+} ν_{σ (j)}^{-})} \\ {(\begin{matrix} 1 - \prod_{j = 1}^{n} {(1 - γ_{σ (j)}^{+})}^{w_{j}}, \\ - \prod_{j = 1}^{n} {| η_{σ (j)}^{-} |}^{w_{j}} \end{matrix})} \end{matrix}$ (5) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{h}}_{σ (j - 1)} \geq {\tilde{h}}_{σ (j)}$ 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 $\sum_{j = 1}^{n} w_{j} = 1$ .

Definitions 5 and 6 suggest that the HBFWA operator and the HBFOWA operator weigh the bipolar fuzzy arguments and the ordered positions of the bipolar fuzzy arguments, respectively. A hesitant bipolar fuzzy hybrid average (HBFHA) operator is proposed below to combine the characteristics of the HBFWA operator and the HBFOWA operator together.

Definition 7. A hesitant bipolar fuzzy hybrid average (HBFHA) operator is defined as follows:

$\begin{matrix} {HBFHA}_{w, ω} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = \oplus_{j = 1}^{n} (w_{j} {\dot{\tilde{h}}}_{σ (j)}) \\ = \cup_{({\dot{γ}}_{σ (j)}^{+}, {\dot{η}}_{σ (j)}^{-}) \in ({\dot{μ}}_{σ (j)}^{+} {\dot{ν}}_{σ (j)}^{-})} \\ {(\begin{matrix} 1 - \prod_{j = 1}^{n} {(1 - {\dot{γ}}_{σ (j)}^{+})}^{w_{j}}, \\ - \prod_{j = 1}^{n} {| {\dot{η}}_{σ (j)}^{-} |}^{w_{j}} \end{matrix})} \end{matrix}$ (6) where w = (w₁, w₂, ⋯ , w_n) is the associated weighting vector, with w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ , ${\dot{\tilde{b}}}_{σ (j)}$ is the j-th largest element of the bipolar fuzzy arguments ${\dot{\tilde{h}}}_{j} ({\dot{\tilde{h}}}_{j} = (n ω_{j}) {\tilde{h}}_{j}, j = 1, 2, \dots, n)$ , ω = (ω₁, ω₂, ⋯ , ω_n) is the weighting vector of bipolar fuzzy arguments ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ , with ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ , and n is the balancing coefficient.

3.2 Hesitant bipolar fuzzy geometric aggregation operators

Applying the hesitant bipolar fuzzy arithmetic aggregation operators and the concept of geometric mean [59 , 61–70], we can define hesitant bipolar fuzzy geometric aggregation operators.

Definition 8. The hesitant bipolar fuzzy weighted geometric (HBFWG) operator is defined as ${HBFWG}_{ω} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) = \otimes_{j = 1}^{n} {({\tilde{h}}_{j})}^{ω_{j}}$ (7) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T is the weight vector of ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ with ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

By definition and mathematical induction, we can prove the following theorem.

Theorem 3. The HBFWG operator returns a HBFN, and

$\begin{matrix} {HBFWG}_{ω} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) = \otimes_{j = 1}^{n} {({\tilde{h}}_{j})}^{ω_{j}} \\ = \cup_{(γ_{j}^{+}, η_{j}^{-}) \in (μ_{j}^{+}, ν_{j}^{-})} \\ {(\begin{matrix} \prod_{j = 1}^{n} {(γ_{j}^{+})}^{ω_{j}}, \\ - 1 + \prod_{j = 1}^{n} {(1 + η_{j}^{-})}^{ω_{j}} \end{matrix})} \end{matrix}$ (8) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T is the weight vector of ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ with ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

Definition 9. The hesitant bipolar fuzzy ordered weighted geometric (HBFOWG) operator is defined as

$\begin{matrix} {HBFOWG}_{w} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = \otimes_{j = 1}^{n} {({\tilde{h}}_{σ (j)})}^{w_{j}} \\ = \cup_{(γ_{σ (j)}^{+}, η_{σ (j)}^{-}) \in (μ_{σ (j)}^{+} ν_{σ (j)}^{-})} \\ {(\begin{matrix} \prod_{j = 1}^{n} {(γ_{σ (j)}^{+})}^{w_{j}}, \\ - 1 + \prod_{j = 1}^{n} {(1 + η_{σ (j)}^{-})}^{w_{j}} \end{matrix})} \end{matrix}$ (9) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{h}}_{σ (j - 1)} \geq {\tilde{h}}_{σ (j)}$ 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 $\sum_{j = 1}^{n} w_{j} = 1$ .

Definitions 8 and 9 imply that the HBFWG operator and the HBFOWG operator target, respectively, the hesitant bipolar fuzzy argument itself and the ordered positions of the hesitant bipolar fuzzy arguments. To mix the features of these two operators together, we propose the hesitant bipolar fuzzy hybrid geometric (HBFHG) operator below.

Definition 10. The hesitant bipolar fuzzy hybrid geometric (HBFHG) operator is defined as

$\begin{matrix} {HBFHG}_{w, ω} ({\tilde{h}}_{1}, {\tilde{h}}_{2}, \dots, {\tilde{h}}_{n}) \\ = \otimes_{j = 1}^{n} {({\dot{\tilde{h}}}_{σ (j)})}^{w_{j}} \\ = \cup_{({\dot{γ}}_{σ (j)}^{+}, {\dot{η}}_{σ (j)}^{-}) \in ({\dot{μ}}_{σ (j)}^{+} {\dot{ν}}_{σ (j)}^{-})} \\ {(\begin{matrix} \prod_{j = 1}^{n} {({\dot{γ}}_{σ (j)}^{+})}^{w_{j}}, \\ - 1 + \prod_{j = 1}^{n} {(1 + {\dot{η}}_{σ (j)}^{-})}^{w_{j}} \end{matrix})} \end{matrix}$ (10) where w = (w₁, w₂, ⋯ , w_n) is the associated weighting vector, with w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ , ${\dot{\tilde{h}}}_{σ (j)}$ is the j-th largest element of the hesitant bipolar fuzzy arguments ${\dot{\tilde{h}}}_{j} ({\dot{\tilde{h}}}_{j} = {({\tilde{h}}_{j})}^{n ω_{j}}, j = 1, 2, \dots, n), ω = (ω_{1}, ω_{2}, \dots, ω_{n})$ is the weighting vector of bipolar fuzzy arguments ${\tilde{h}}_{j} (j = 1, 2, \dots, n)$ , with ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ , and n is the balancing coefficient.

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

We next apply the hesitant bipolar aggregation operators developed in the previous section to solve MADM problems with hesitant bipolar fuzzy information. Denote a discrete set of alternatives by A ={ A₁, A₂, ⋯ , A_m } and the set of attributes by G ={ G₁, G₂, ⋯ , G_n }. Let w = (w₁, w₂, ⋯ , w_n) be the weight vector of attributes, where w_j ≥ 0, j = 1, 2, ⋯ , n, $\sum_{j = 1}^{n} w_{j} = 1$ . Suppose that $\tilde{H} = {{\tilde{h}}_{ij}}_{m \times n} = {(μ_{ij}^{+}, ν_{ij}^{-})}_{m \times n}$ is the hesitant bipolar fuzzy decision matrix, where $μ_{ij}^{+}$ and $ν_{ij}^{-}$ indicate, respectively, the positive degree and negative degree assessed by the decision maker that the alternative A_i satisfies the attribute G_j, $μ_{ij}^{+} \in [0, 1]$ , $ν_{ij}^{-} \in [- 1, 0]$ , i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n.

The process of utilizing the HBFWA (or HBFWG) operator to solve a MADM problem is presented below.

Step 1. Applying the HBFWA operator to process the information in matrix $\tilde{H}$ , derive the overall values ${\tilde{h}}_{i} (i = 1, 2, \dots, m)$ of the alternative A_i.

$\begin{matrix} {\tilde{h}}_{i} & = & (μ_{i}^{+}, ν_{i}^{-}) \\ = & {HBFWA}_{ω} ({\tilde{h}}_{i 1}, {\tilde{h}}_{i 2}, \dots, {\tilde{h}}_{in}) \\ = & \oplus_{j = 1}^{n} (ω_{j} {\tilde{h}}_{ij}) \\ = & \cup_{(γ_{ij}^{+}, η_{ij}^{-}) \in (μ_{ij}^{+}, ν_{ij}^{-})} \\ {(\begin{matrix} 1 - \prod_{j = 1}^{n} {(1 - γ_{ij}^{+})}^{ω_{j}}, \\ - \prod_{j = 1}^{n} {| η_{ij}^{-} |}^{ω_{j}} \end{matrix})} \end{matrix}$ (11)

If the HBFWG operator is chosen instead, we have

$\begin{matrix} {\tilde{h}}_{i} & = & (μ_{i}^{+}, ν_{i}^{-}) \\ = & {HBFWG}_{ω} ({\tilde{h}}_{i 1}, {\tilde{h}}_{i 2}, \dots, {\tilde{h}}_{in}) \\ = & \otimes_{j = 1}^{n} {({\tilde{h}}_{ij})}^{ω_{j}} \\ = & \cup_{(γ_{ij}^{+}, η_{ij}^{-}) \in (μ_{ij}^{+}, ν_{ij}^{-})} \\ {(\begin{matrix} \prod_{j = 1}^{n} {(γ_{ij}^{+})}^{ω_{j}}, \\ - 1 + \prod_{j = 1}^{n} {(1 + η_{ij}^{-})}^{ω_{j}} \end{matrix})} \end{matrix}$ (12)

Step 2. Calculate the scores $S ({\tilde{h}}_{i}) (i = 1, 2, \dots, m)$ .

Step 3. Rank all the alternatives A_i (i = 1, 2, ⋯ , m) in terms of $s ({\tilde{h}}_{i}) (i = 1, 2, \dots, m)$ . If there is no difference between two scores $s ({\tilde{h}}_{i})$ and $s ({\tilde{h}}_{j})$ , then calculate the accuracy degrees $a ({\tilde{h}}_{i})$ and $a ({\tilde{h}}_{j})$ to rank the alternatives A_i and A_j.

Step 4. Select the best alternative(s).

5 Numerical example

With the rapid development and the increasingly widespread application of information technology, the software becomes more and more important. Also, because of the increasing size and the complexity of software, the constructional engineering software quality has become difficult to control and manage. Improving the quality of software has become the focus of software industry. Constructional engineering software quality assurance becomes an important approach for improving constructional engineering software quality, which provides developers and managers with the information reflecting the product quality through monitoring the execution of software producing task by independent review. In this section, we present an empirical case study of evaluating the constructional engineering software quality. The project’s aim is to evaluate the best constructional engineering software quality from the different software systems, which provide alternatives of software systems to university. The constructional engineering software quality of five possible software systems A_i (i = 1, 2, 3, 4, 5) is evaluated. A software selection problem can be calculated as a multiple attribute group decision making problem in which alternatives are the software packages to be selected and criteria are those attributes under consideration. A computer center in a university desires to select a new information system in order to improve work productivity. After preliminary screening, five constructional engineering software systems A_i (i = 1, 2, ⋯ , 5) have remained in the candidate list. Three decision makers (experts) form a committee to act as decision makers. The computer center in the university must take a decision according to the following four attributes: ➀₁ is the costs of hardware/software investment; ➁₂ is the contribution to organization performance; ➂₃ is the effort to transform from current system; ➃₄ is the outsourcing software developer reliability. The five possible constructional engineering software system A_i (i = 1, 2, ⋯ , 5) are to be evaluated by the decision maker using the HBFNs according to the four attributes (whose weighting vector ω = (0.2, 0.1, 0.3, 0.4) ^T). The ratings are presented in the Table 1.

Table 1
Hesitant bipolar fuzzy decision matrix

G₁ G₂

A₁ {(0.3,–0.5), (0.2,–0.4)} {(0.4,–0.7),(0.6,–0.3),

(0.8,–0.2)}

A₂ {(0.5,–0.3), (0.6,–0.2)} {(0.6,–0.6), (0.7,–0.5)}

A₃ {(0.7,–0.2), (0.8,–0.3)} {(0.5,–0.4), (0.6,–0.3)}

A₄ {(0.6,–0.4), (0.7,–0.3)} {(0.7,–0.4), (0.8,–0.5)}

A₅ {(0.4,–0.2), (0.5,–0.4)} {(0.6,–0.3), (0.8,–0.5)}

G₃ G₄

A₁ {(0.5,–0.6), (0.7,–0.3)} {(0.6,–0.2),(0.8,–0.4)}

A₂ {(0.4,–0.3), (0.6,–0.7)} {(0.5,–0.3), (0.7,–0.6)}

A₃ {(0.6,–0.2), (0.8,–0.5), {(0.4,–0.5), (0.6,–0.4)}

(0.9,–0.2)}

A₄ {(0.4,–0.2), (0.6,–0.4)} {(0.6,–0.1), (0.7,–0.2),

(0.8,–0.4)}

A₅ {(0.7,–0.5), (0.8,–0.4)} {(0.4,–0.6), (0.5,–0.3)}

	G₁	G₂
A₁	{(0.3,–0.5), (0.2,–0.4)}	{(0.4,–0.7),(0.6,–0.3),
		(0.8,–0.2)}
A₂	{(0.5,–0.3), (0.6,–0.2)}	{(0.6,–0.6), (0.7,–0.5)}
A₃	{(0.7,–0.2), (0.8,–0.3)}	{(0.5,–0.4), (0.6,–0.3)}
A₄	{(0.6,–0.4), (0.7,–0.3)}	{(0.7,–0.4), (0.8,–0.5)}
A₅	{(0.4,–0.2), (0.5,–0.4)}	{(0.6,–0.3), (0.8,–0.5)}
	G₃	G₄
A₁	{(0.5,–0.6), (0.7,–0.3)}	{(0.6,–0.2),(0.8,–0.4)}
A₂	{(0.4,–0.3), (0.6,–0.7)}	{(0.5,–0.3), (0.7,–0.6)}
A₃	{(0.6,–0.2), (0.8,–0.5),	{(0.4,–0.5), (0.6,–0.4)}
	(0.9,–0.2)}
A₄	{(0.4,–0.2), (0.6,–0.4)}	{(0.6,–0.1), (0.7,–0.2),
		(0.8,–0.4)}
A₅	{(0.7,–0.5), (0.8,–0.4)}	{(0.4,–0.6), (0.5,–0.3)}

\begin{matrix} {\tilde{h}}_{1} & = & {HBFWA}_{ω} ({\tilde{h}}_{11}, {\tilde{h}}_{12}, {\tilde{h}}_{13}, {\tilde{h}}_{14}) \\ = & \oplus_{j = 1}^{4} (ω_{j} {\tilde{h}}_{1 j}) \\ = & \cup_{(γ_{1 j}^{+}, η_{1 j}^{-}) \in (μ_{1 j}^{+}, ν_{1 j}^{-})} {(1 - \prod_{j = 1}^{4} {(1 - γ_{1 j}^{+})}^{ω_{j}}, - \prod_{j = 1}^{4} {| η_{1 j}^{-} |}^{ω_{j}})} \\ = & {HBFWA}_{ω} {\begin{matrix} {(0.3, - 0.5), (0.2, - 0.4)}, {(0.4, - 0.7), (0.6, - 0.3), (0.8, - 0.2)} \\ {(0.5, - 0.6), (0.7, - 0.3)}, {(0.6, - 0.2), (0.8, - 0.4)} \end{matrix}} \\ = & {\begin{matrix} (0.5109, - 0.3786), (0.6225, - 0.4995), (0.5726, - 0.3075), (0.6761, - 0.4085) \\ (0.5217, - 0.3478), (0.6375, - 0.4590), (0.5896, - 0.2825), (0.6890, - 0.3728) \\ (0.5537, - 0.3340), (0.6618, - 0.4407), (0.6171, - 0.2713), (0.7098, - 0.3580) \\ (0.4884, - 0.3621), (0.6123, - 0.4777), (0.5611, - 0.2941), (0.6674, - 0.3880) \\ (0.5087, - 0.3326), (0.6277, - 0.4389), (0.5785, - 0.2702), (0.6806, - 0.3565) \\ (0.5416, - 0.3194), (0.6526, - 0.4215), (0.6067, - 0.2595), (0.7020, - 0.3424) \end{matrix}} \end{matrix}

The information about the attribute weights is known as follows: ω = (0.20, 0.10, 0.30, 0.40).

In the following, we utilize the approach developed for evaluating the constructional engineering software quality with hesitant bipolar fuzzy information.

Step 1. We utilize the decision information given in matrix $\tilde{D}$ , and the HBFWA operator to obtain the overall preference values ${\tilde{h}}_{i}$ of the engineering software systems A_i (i = 1, 2, 3, 4, 5). Take engineering software system A₁ for an example, we have

Step 2. Calculate the scores $s ({\tilde{h}}_{i}) (i = 1, 2, 3, 4, 5)$ of the overall hesitant bipolar fuzzy numbers ${\tilde{h}}_{i} (i = 1, 2, 3, 4, 5)$ : $\begin{matrix} s ({\tilde{h}}_{1}) & = & 0.6221, s ({\tilde{h}}_{2}) = 0.5837, s ({\tilde{h}}_{3}) = 0.6673 \\ s ({\tilde{h}}_{4}) & = & 0.6874, s ({\tilde{h}}_{5}) = 0.5985 \end{matrix}$

Step 3. Rank all the engineering software systems A_i (i = 1, 2, 3, 4, 5) in accordance with the scores $s ({\tilde{h}}_{i}) (i = 1, 2, 3, 4, 5)$ of the overall hesitant bipolar fuzzy numbers: A₄ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₂, and thus the most desirable engineering software system is A₄.

Based on the HBFWG operator, then, in order to select the most desirable engineering software system, we can develop another approach to multiple attribute decision making problems for evaluating the constructional engineering software quality with hesitant bipolar fuzzy information, which can be described as following:

Step 1^′. Aggregate all the hesitant bipolar fuzzy numbers in the Table 1 by using the hesitant bipolar fuzzy weighted geometric (HBFWG) operator to derive the overall hesitant bipolar fuzzy numbers ${\tilde{h}}_{i} (i = 1, 2, \dots, 5)$ of the engineering software system A_i. Take engineering software system A₁ for an example, we have $\begin{matrix} {\tilde{h}}_{1} & = & {HBFWG}_{ω} ({\tilde{h}}_{11}, {\tilde{h}}_{12}, {\tilde{h}}_{13}, {\tilde{h}}_{14}) \\ = & \otimes_{j = 1}^{4} {({\tilde{h}}_{1 j})}^{ω_{j}} \\ = & \cup_{(γ_{1 j}^{+}, η_{1 j}^{-}) \in (μ_{1 j}^{+}, ν_{1 j}^{-})} {(\prod_{j = 1}^{4} {(γ_{1 j}^{+})}^{ω_{j}}, - 1 + \prod_{j = 1}^{4} {(1 + η_{1 j})}^{ω_{j}})} \\ = & {HBFWG}_{ω} {\begin{matrix} {(0.3, - 0.5), (0.2, - 0.4)}, {(0.4, - 0.7), (0.6, - 0.3), (0.8, - 0.2)} \\ {(0.5, - 0.6), (0.7, - 0.3)}, {(0.6, - 0.2), (0.8, - 0.4)} \end{matrix}} \\ = & {\begin{matrix} (0.4749, - 0.4638), (0.5328, - 0.5220), (0.5253, - 0.3657), (0.5894, - 0.4347) \\ (0.4945, - 0.4163), (0.5548, - 0.4798), (0.5471, - 0.3097), (0.6138, - 0.3847) \\ (0.5090, - 0.4085), (0.5710, - 0.4728), (0.5630, - 0.3004), (0.6317, - 0.3764) \\ (0.4379, - 0.4438), (0.4913, - 0.5043), (0.4844, - 0.3422), (0.5435, - 0.4137) \\ (0.4560, - 0.3947), (0.5116, - 0.4605), (0.5044, - 0.2840), (0.5660, - 0.3618) \\ (0.4693, - 0.3865), (0.5266, - 0.4532), (0.5192, - 0.2744), (0.5825, - 0.3533) \end{matrix}} \end{matrix}$

Step 2^′. Calculate the scores $s ({\tilde{h}}_{i}) (i = 1, 2, 3, 4, 5)$ of the overall hesitant bipolar fuzzy numbers ${\tilde{h}}_{i} (i = 1, 2, 3, 4, 5)$ of the engineering software system A_i: $\begin{matrix} s ({\tilde{h}}_{1}) & = & 0.5644, s ({\tilde{h}}_{2}) = 0.5499, s ({\tilde{h}}_{3}) = 0.6263 \\ s ({\tilde{h}}_{4}) & = & 0.6587, s ({\tilde{h}}_{5}) = 0.5604 . \end{matrix}$

Step 3^′. Rank all the engineering software system A_i (i = 1, 2, 3, 4, 5) in accordance with the scores $s ({\tilde{h}}_{i}) (i = 1, 2, 3, 4, 5)$ of the overall hesitant bipolar fuzzy numbers ${\tilde{h}}_{i} (i = 1, 2, \dots, 5)$ : A₄ ≻ A₃ ≻ A₁ ≻ A₂ ≻ A₅ and thus the most desirable engineering software system is A₄.

From the above analysis, it is easily seen that although the overall rating values of the alternatives are slightly different by using two operators respectively. However, the most desirable engineering software system is A₄.

6 Conclusion

In this paper, we investigate the multiple attribute decision making (MADM) problem based on the aggregation operators with hesitant bipolar fuzzy information. Then, motivated by the ideal of arithmetic and geometric operation [52 –55], we have developed some aggregation operators for aggregating hesitant bipolar fuzzy information: hesitant bipolar fuzzy weighted average (HBFWA) operator, hesitant bipolar fuzzy weighted geometric (HBFWG) operator, hesitant bipolar fuzzy ordered weighted average (HBFOWA) operator, hesitant bipolar fuzzy ordered weighted geometric (HBFOWG) operator, hesitant bipolar fuzzy hybrid average (HBFHA) operator and hesitant bipolar fuzzy hybrid geometric (HBFHG) operator. Then, we have utilized these operators to develop some approaches to solve the hesitant bipolar fuzzy multiple attribute decision making problems. Finally, an illustrative example for evaluating the constructional engineering software quality 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 and other environments, such as, pattern recognition, risk analysis, supplier selection, and so on [71 –76].

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