A group decision making approach in interval-valued intuitionistic hesitant fuzzy environment with confidence levels

Abstract

The main purpose of this paper is to research the multi-attribute group decision making (MAGDM) problems with interval-valued hesitant fuzzy numbers based on the confidence level. In this paper, we provide the confidence level of decision-making information which is applicable to put forward an optimized weight solution. Then we define the interval-valued intuitionistic hesitant fuzzy operator to aggregate decision information combing the φs transformation and Choquet integral. Taking the disadvantage of ranking by distance into account, we propose a ranking approach using the interval-valued intuitionistic hesitant fuzzy entropy. Finally, an illustrative example has been made to demonstrate the validity and feasibility of the proposed method.

Keywords

Interval-valued intuitionistic hesitant fuzzy numbers confidence levels Choquet integral

1 Introduction

As an important branch of decision making theory, multi-attribute group decision making has been applied to solve the numerous multi-attribute decision making (MADM) problems by means of experts’ selecting an optimal alternative through co-decision. Linguistic information and fuzzy sets are used to represent evaluation information [1, 2]. On account of the complex objective environment and the discrepant subjective human cognition, it is unavoidable that the experts cannot determine a common membership degree of an element. For instance, one expert assigns 0.3 to the membership degree of x to set A while the other assigns 0.5. They cannot convince each other and they do not want to compromise. In order to handle this situation, Torra [3] proposed the hesitant fuzzy sets which allowed the membership degree of an element to a set to be represented by different values such as {(0.3, 0.5)} [3]. Therefore, the hesitant fuzzy set (HFS) is generally introduced in MAGDM in view of the uncertainty in the process of decision making.

Wei [4] proposed several prioritized aggregation operators for hesitant fuzzy numbers and applied them to MADM problems with attributes in different priority levels [4]. Zhu et al. [5] used the geometric Bonferroni mean (GBM) to integrate hesitant fuzzy information, under hesitant fuzzy environment [5]. Zhang [6] discussed a number of aggregation operators dealing with hesitant fuzzy numbers and proved their properties and relationship [6]. Liu et al. [34] defined hesitant fuzzy relative entropy and symmetric cross entropy and proposed a novel similarity measure to construct a similarity coefficient matrix [7]. Hu and Zhou [8] proposed a generalized hesitant fuzzy ordered weighted averaging operator based on the Archimedean t-norm and s-norm [8]. With the increasing complexity in decision making, it is difficult to use a real number to represent the membership degree. So some extension sets based on hesitant fuzzy set are utilized in MADM, such as dual hesitant fuzzy sets [9], generalized hesitant fuzzy sets [10], higher order hesitant fuzzy sets [11] and hesitant fuzzy linguistic term sets [12]. Chen et al. [36] introduced the concept of the interval-valued hesitant fuzzy sets and considered preference relation to express the uncertainty in evaluation information [36]. Zhou et al. [14] presented hesitant-intuitionistic fuzzy numbers to indicate decision information, because compared with other preference relations, the proposed relations use hesitant fuzzy elements (HFEs) to express the priority intensities of decision makers [14]. With the popularity of intelligent algorithms, in recent study, many scholars studied on fuzzy decision making through granular computing [15]. Peters and Weber [16] put forward a framework for dynamic granular clustering [16]. Lingras et al. [17] proposed a concept of meta-clustering which aggregated clustering information from another or the same set of networked granule [17]. In order to increase the accuracies of rule-based classifiers, Antonelli et al. [18] developed a multi-objective evolutionary approach to the classifier generation which was borrowed from the multi-objective decision making [18]. Yao [19], Ciucci [20], Min [21] introduced the triarchic theory [19], Orthopairs [20], Semi-greedy heuristics [21] into granular computing respectively. Furthermore, some common problems in decision making were also discussed in the granular computing, such as the interaction among attributes [22, 23], the uncertainty in initial information [24, 25], the data granulation aggregation [26 –29].

Joshi and Kumar [30] proposed an interval-valued intuitionistic hesitant fuzzy operator which denotes the degrees of membership and non-membership by closed subintervals of the interval [0,1] [30]. This approach not only expands the ability of the HFS to handle uncertain information but also enhances its ability to solve practical decision-making problems [31]. But there are two main drawbacks in paper [31]. 1) After aggregation, the numbers of elements in each alternative are different, but Joshi and Kumar only used one reference point. 2) It is extremely subjective of experts to set fuzzy measure, and the results will be quite different when different fuzzy measures are used.

The above study shows that aggregation operators are effective tools to aggregate hesitant fuzzy numbers. But they all neglect the experts’ confidence levels denoting the degree to which the decision makers are familiar with the professional field. When an expert is unfamiliar with the attributes in a decision-making problem, the evaluation information given by him should not be as important as other information given by experts who are qualified. Xia et al. [32] presented an induced aggregation operator in consideration of confidence levels [32]. Yu [33] developed some confidence intuitionistic fuzzy weighted aggregation operators, and then proposed confidence intuitionistic fuzzy Einstein weighted averaging (CIFEWA) operator and the confidence intuitionistic fuzzy Einstein weighted geometric (CIFEWG) operator [33]. Liu et al. [34] proposed some formulas of hesitant fuzzy correlation measures based on the confidence levels and constructed a mathematical model based on correlation measures for a hesitant fuzzy MADM problem [34]. Although the confidence level is suitable for practical MADM problems, there are two defects in these papers. 1) Traditional aggregation operators do not consider the interdependency or interactive characteristics of attributes; the assumption of independency of attributes is too strong to be satisfied in many MADM problems. 2) It is unreasonable of some papers [15 , 18] to use closeness coefficient to rank alternatives. Closeness coefficient is based on distance. Different alternatives correspond to different ideal alternatives, so the results will be different according to the selection of ideal alternatives.

From above analysis, we can see that hesitant fuzzy numbers could outdo fuzzy numbers when it comes to expressing the uncertainty. As the extension of hesitant fuzzy numbers, interval-valued intuitionistic hesitant fuzzy numbers (IVIHFN) provide complete and factual information of decision-making [14]. This paper utilizes interval-valued intuitionistic hesitant fuzzy numbers with confidence levels to represent the decision making information. In consideration of all drawbacks in above papers, firstly, this paper defines an interval-valued intuitionistic hesitant fuzzy entropy instead of distance functions to express uncertainty in interval-valued intuitionistic hesitant fuzzy numbers. Secondly, in order to avoid interdependency of attributes and subjectivity of fuzzy measure given by experts, we define the interval-valued intuitionistic hesitant fuzzy operator based on Choquet integral and φ_s transformation. Finally, results are calculated by the aggregation operator and the entropy, and then ranking results are obtained by entropy values.

2 Preliminaries

2.1 Interval-valued intuitionistic hesitant fuzzy set

Definition 2.1. [35] Let X be the universe of discourse. A hesitant fuzzy set (HFS) A is symbolized by $A = {(x, h_{A} (x)) | x \in X}$ where h_A (x), called a hesitant fuzzy element (HFE), is a set of some values in [0,1] representing the possible membership degrees of elements x ∈ X to A. A is referred to as a fuzzy set (FS) on condition that there is only one element in X.

Example 2.2. If X = {x₁, x₂, x₃} is the reference set, h (x₁) = {0.3, 0.4, 0.7} , h (x₂) = {0.7, 0.8} , h (x₃) = {0.3, 0.4, 0.6, 0.5} are the HFEs of x₁, x₂, x₃ to set A. Then A can be considered as an HFS represented by $\begin{matrix} A & = & {(x_{1} {0.3, 0.4, 0.7}), (x_{2} {0.7, 0.8}), \\ (x_{3} {0.3, 0.4, 0.6, 0.5})} \end{matrix}$

Definition 2.3. If X is a non-empty reference set, an interval-valued intuitionistic hesitant fuzzy set (IVIHFS) is defined as: $A = {(x, μ_{A} (x), ν_{A} (x)) | x \in X}$ where μ_A : X → Int [0, 1] and ν_A : X → Int [0, 1] represent the possible membership and non-membership degrees of elements x ∈ X to A, with the condition ∀x ∈ X, 0 ≤ sup(μ_A (x)) + sup(ν_A (x)) ≤ 1.

If μ_AL (x) = inf(μ_A (x)), ν_AL (x) = inf(ν_A (x)), μ_AU (x) = sup(μ_A (x)) and ν_AU (x) = sup(ν_A (x)), A is represented as follows: $\begin{matrix} A & = & {(x, [μ_{AL} (x), μ_{AU} (x)], \\ [ν_{AL} (x), ν_{AU} (x)]) | x \in X} \end{matrix}$

Definition 2.4. Let A = {(x, [μ_AL (x) , μ_AU (x)] , [ν_AL (x) , ν_AU (x)]) |x ∈ X}, B = {(x, [μ_BL (x) , μ_BU (x)] , [ν_BL (x) , ν_BU (x)]) |x ∈ X} be two IVIHFNs on X [13, 21]. Then, $\begin{matrix} (1) A^{c} & = & {([ν_{AL} (x), ν_{AU} (x)], \\ [μ_{AL} (x), μ_{AU} (x)]) | x \in X} \end{matrix}$ $\begin{matrix} (2) A \cap B = {[μ_{AL} (x) \land μ_{BL} (x), μ_{AU} (x) \land μ_{BU} (x)], \\ [ν_{AL} (x) \lor ν_{BL} (x), ν_{AU} (x) \lor ν_{BU} (x)] | x \in X} \end{matrix}$ $\begin{matrix} (3) A \cup B = {[μ_{AL} (x) \lor μ_{BL} (x), μ_{AU} (x) \lor μ_{BU} (x)], \\ [ν_{AL} (x) \land ν_{BL} (x), ν_{AU} (x) \land ν_{BU} (x)] | x \in X} \end{matrix}$ $\begin{matrix} (4) A \otimes B = {([μ_{AL} (x) μ_{BL} (x), μ_{AU} (x) μ_{BU} (x)], 1 pt 1 pt \\ [ν_{AL} (x) + ν_{BL} (x) - ν_{AL} (x) ν_{BL} (x), ν_{AU} (x) + ν_{BU} (x) \\ - ν_{AU} (x) ν_{BU} (x)]) | x \in X} \end{matrix}$ $\begin{matrix} (5) A \oplus B = {([μ_{AL} (x) + μ_{BL} (x) - μ_{AL} (x) μ_{BL} (x), \\ μ_{AU} (x) μ_{BU} (x) - μ_{AU} (x) μ_{BU} (x)], [ν_{AL} (x) ν_{BL} (x), \\ ν_{AU} (x) ν_{BU} (x)]) | x \in X} \end{matrix}$ $\begin{matrix} (6) τ A = {([1 - (1 - μ_{AL} (x))^{τ}, 1 - (1 - μ_{AU} (x))^{τ}], \\ [ν_{AL} (x)^{τ}, ν_{AU} (x)]^{τ}) | x \in X} τ > 0 \end{matrix}$ $\begin{matrix} (7) A^{τ} = {(x, [μ_{AL} (x)^{τ}, μ_{AU} (x)^{τ}], [1 - (1 - ν_{AL} (x))^{τ}, \\ 1 - (1 - ν_{AU} (x))^{τ}]) | x \in X} τ > 0 \end{matrix}$

Definition 2.5. Let A = {(x, μ_A (x) , ν_A (x)) |x ∈ X} be an IVIHFN. The score function is defined as follows [36]. $S (A) = \sum S (x) / # A$ where # A means the number of elements in A.

For any two IVIHFNs A and B, the comparative rules are given by Xu [37] as follows. If

S (A) > S (B), then A > B

S (A) < S (B), then A < B

S (A) = S (B), then A = B

Example 2.6. Suppose A = {([0.4, 0.6] , [0.3, 0.4]) , ([0.6, 0.9] , [0.1, 0.1]) , ([0.2, 0.3] , [0.4, 0.7])} and B = {([0.2, 0.4] , [0.2, 0.4]) , ([0.8, 0.9] , [0.1, 0.1])} are two IVIHFNs, then their score functions are given as follows: $\begin{matrix} S (A) & = & 0.1667 \\ S (B) & = & 0.375 \end{matrix}$

According to Definition 2.4, A < B.

2.2 Interval-valued intuitionistic hesitant fuzzy entropy

Entropy is an important tool in coping with uncertain information. In intuitionistic fuzzy sets, the larger the amount of information is, the smaller the uncertainty and the entropy will be. So entropy is a decreasing function of the difference between membership and non-membership. Burillo and Bustince [38] firstly proposed intuitionistic fuzzy entropy which was applied to interval-valued intuitionistic fuzzy sets (IVIFS) [38]. Since the existing definition of entropy for interval-valued hesitant fuzzy sets (IVHFS) only reflected one type of uncertainty [39, 40], Quirós et al. [41] built the entropy by three different functions and proposed an entropy measure for IVHFS [41]. In view of little research on entropy for IVIHFS, this paper defines the interval-valued intuitionistic hesitant fuzzy (IVIHF) entropy based on [42].

Definition 2.7. Let X be the universe of discourse. An IVIHFS on X is defined as A = {(x, μ_A (x) , ν_A (x)) |x ∈ X} where $μ_{A} (x), ν_{A} (x) = {([μ_{AL}^{1} (x), μ_{AU}^{1} (x)],$ $[ν_{AL}^{1} (x), ν_{AU}^{1} (x)]), \dots, ([μ_{AL}^{n} (x), μ_{AU}^{n} (x)], [ν_{AL}^{n} (x), ν_{AU}^{n} (x)])}$ . The IVIHF entropy I (A) of A is expressed as follows:

$\begin{matrix} I (A) & = & \frac{1}{# A} \sum {1 - ({\bar{μ}}_{A} (x) + {\bar{ν}}_{A} (x)) \\ \times sin (\frac{{\bar{μ}}_{A} (x) + {\bar{ν}}_{A} (x)}{2} π)} \end{matrix}$ (1) where ${\bar{μ}}_{A} (x) = μ_{AL} (x) + ϑ (μ_{AU} (x) - μ_{AL} (x))$ and ${\bar{ν}}_{A} (x) = ν_{AL} (x) + ϑ (ν_{AU} (x) - ν_{AL} (x))$ , ϑ ∈ [0, 1]. The new entropy considers not only the membership and non-membership degree, but also the hesitant degree in IVIHFS. So it will be more reasonable.

Property 2.8.WhenAturns out to be a fuzzy set, I (A) =0.

Proof. When A is a fuzzy set, μ_Au (x) + ν_Au (x) =1, ∀x ∈ X, then we have I (A) =0.

Property 2.9.If and only ifμ_A (x) , ν_A (x) = {([0, 0] , [0, 0]) , …, ([0, 0] , [0, 0])}, I (A) =1.

Proof. $I (A) = 1 \Leftrightarrow ({\bar{μ}}_{A} (x) + {\bar{ν}}_{A} (x)) \times sin (\frac{{\bar{μ}}_{A} (x) + {\bar{ν}}_{A} (x)}{2} π) = 0$ , ∀ϑ ∈ (0, 1).

Suppose μ_AU (x) + ν_AU (x) >0, then ${\bar{μ}}_{A} (x) + {\bar{ν}}_{A} (x) > 0$ .

Since $\forall ϑ \in (0, 1), ({\bar{μ}}_{A} (x) + {\bar{ν}}_{A} (x)) \times sin (\frac{{\bar{μ}}_{A} (x) + {\bar{ν}}_{A} (x)}{2} π) > 0$ , which contradicts the given condition, the assumption is proven to be wrong. Hence, μ_A (x) , ν_A (x) = {([0, 0] , [0, 0]) , …, ([0, 0] , [0, 0])}.

Property 2.10.I (A) = I (A^c).

Property 2.11.IfA ≤ B, thenI (A) ≥ I (B).

Proof. Assume that # A = # B. If A ≤ B, then (μ_A (x) , ν_A (x)) ≤ (μ_B (x) , ν_B (x)) and we can have (1 − ϑ) (μ_BL (x) − μ_AL (x)) + ϑ (μ_BU (x) − μ_AU (x)) + (1 − ϑ) (ν_BL (x) − ν_AL (x)) + ϑ (ν_BU (x) − ν_AU (x)) ≥0. Since ∀ϑ ∈ [0, 1], μ_AL (x) + ϑ (μ_AU (x) − μ_AL (x)) + ν_AL (x) + ϑ (ν_AU (x) − ν_AL (x)) ≤ μ_BL (x) + ϑ (μ_BU (x) − μ_BL (x)) + ν_BL (x) + ϑ (ν_BU (x) − ν_BL (x)).

Consequently, $\begin{matrix} {\bar{μ}}_{A} (x) + {\bar{ν}}_{A} (x) \leq {\bar{μ}}_{B} (x) + {\bar{ν}}_{B} (x), \\ ({\bar{μ}}_{A} (x) + {\bar{ν}}_{A} (x)) \times sin (\frac{{\bar{μ}}_{A} (x) + {\bar{ν}}_{A} (x)}{2} π) \\ \leq ({\bar{μ}}_{B} (x) + {\bar{ν}}_{B} (x)) \times sin (\frac{{\bar{μ}}_{B} (x) + {\bar{ν}}_{B} (x)}{2} π), \\ I (A) \geq I (B) . \end{matrix}$

3 Interval-valued intuitionistic hesitant fuzzy integral operator

3.1 Fuzzy measure of attributes calculated by φ_s transformation

Definition 3.1. Suppose X = {x_k|k = 1, 2, …, n} is a finite set, P (X) is the power set of X, (X, P (X)) is a space, and g : P (X) → [0, 1] is a group of set functions. If the following conditions are met:

g (Ø) =0, g (X) =1

∀A, B ∈ P (X), if A ⊆ B, then g (A) ≤ g (B)

then define g as a fuzzy measure function, and if the conditions are still satisfied when ∀A, B∈ P (X) , A ∩ B = Ø, and λ > −1 then:

g (A \cup B) = g (A) + g (B) + λ g (A) g (B)

(2) and define g as a fuzzy measure function of λ, where λ indicates the degree of interaction among attributes.

The parameter λ determines the degree of interaction among attributes. If

λ = 0, then g is additive and A, B is independent respectively.

λ ≠ 0, then g is non-additive and A, B interact with each other.

λ > 0, then g (A ∪ B) > g (A) + g (B) and there is the active cooperation relationship between A and B.

λ < 0, then g (A ∪ B) < g (A) + g (B) and there is the passive cooperation relationship between A and B.

Let X = {x₁, x₂, …, x_n} be a collection of n attributes. Then for i, j = 1, 2, …, n and i≠ j, i ∩ j = Ø, g (x_i) is defined as a fuzzy measure of x_i if satisfying the following condition. $g (X) = \frac{1}{λ} (\prod_{i = 1}^{n} [1 + λ g (x_{i})] - 1)$ (3)

If g (X) =1, λ can be uniquely determined by the following expression. $λ + 1 = \prod_{i = 1}^{n} (1 + λ g_{i})$ (4)

Definition 3.2. [43] Let X = {x₁, x₂, …, x_n} be a non-empty finite set and f be a positive discrete function on X. Suppose f (x_i) ≤ f (x_i+1) and g is a λ-fuzzy measure on X. Choquet integral of f with respect to g is defined as

$\begin{matrix} {Choquet}_{g} (f) = \sum_{i = 1}^{n} [f (x_{i}) - f (x_{i - 1})] g (A_{i}) \\ or {Choquet}_{g} (f) = \sum_{i = 1}^{n} f (x_{i}) [g (A_{i}) - g (A_{i + 1})] \end{matrix}$ (5) where A_i = {x_i, x_i+1, …, x_n}.

We can see that discrete fuzzy integral indicates a new linear expression with regard to f (x₁) , f (x₂) , …, f (x_n), considers the interaction among attributes and determines the importance level of different attributes.

Definition 3.3. When

$\begin{matrix} φ_{s} (ξ, ω) \\ = {\begin{matrix} 1, & ξ = 1, ω > 0 \\ 0, & ξ = 1, ω = 0 \\ 1, & ξ = 0, ω = 1 \\ 0, & ξ = 0, ω < 0 \\ [(\frac{(1 - ξ)^{2}}{ξ^{2}})^{ω} - 1] / [\frac{(1 - ξ)^{2}}{ξ^{2}} - 1], & otherwise \end{matrix} \end{matrix}$ (6) define φ_s : [0, 1] × [0, 1] → [0, 1] as the φ_s transformation. Then the λ fuzzy measure of attribute set X is represented by: $g_{ξ} (A) = φ_{s} (ξ, \sum_{x_{i} \in A} ω (x_{i})), \forall A \in P (X))$ (7) where $ξ = \frac{1}{\sqrt{1 + λ + 1}}$ is the attribute interaction degree of X, ω (x_i) is the weight of x_i, ξ ∈ [0, …, 0.5, …, 1], λ ∈ [+ ∞ , …, 0, …, −1]. Figures 1 and 2 show the function φ_s (ξ, ω) and inverse function $φ_{s}^{- 1} (ξ, ω)$ respectively.

In Fig. 1, curves respectively represent function images when ξ ranges from 0.1 to 0.9. On the contrary, the curves in Fig. 2 respectively represent images when φ_s ranges from 0.9 to 0.1. Figure 2 shows that $ω (x_{k}) = φ_{s}^{- 1} (ξ, ω)$ is the inverse function of interaction degree ξ. The fuzzy measure formula (7) can be rewritten as: $g (A) = g (x_{i} \cup X) = φ_{s} (ξ, \sum_{x_{i} \cup X} ω (x_{k}))$ (8)ξ is the interaction degree of attributes in the decision matrix, as well as the interaction degree between x_i and X.

3.2 Fuzzy integral aggregation operator

Drawing experience from [30], Choquet integral can eliminate the correlation among the attributes, so this paper proposes an interval-valued intuitionistic hesitant fuzzy Choquet integral operator.

Definition 3.4. Let g be a fuzzy measure on X and ${\tilde{h}}_{i} (i = 1, \dots, n)$ be a set of IVIHFNs. The interval-valued intuitionistic hesitant fuzzy Choquet integral (IVIHFCI) operator is calculated by $\begin{matrix} IVIHFCI ({\tilde{h}}_{i}) = \otimes_{i = 1}^{n} ({\tilde{h}}_{σ (i)})^{g (A_{σ (i)}) - g (A_{σ (i - 1)})} \\ = {\begin{matrix} ([\prod_{i = 1}^{n} (μ_{σ (i) L})^{g (A_{σ (i)}) - g (A_{σ (i - 1)})}, \prod_{i = 1}^{n} (μ_{σ (i) U})^{g (A_{σ (i)}) - g (A_{σ (i - 1)})}]), \\ [1 - \prod_{i = 1}^{n} (1 - ν_{σ (i) L})^{g (A_{σ (i)}) - g (A_{σ (i - 1)})}, 1 - \prod_{i = 1}^{n} (1 - ν_{σ (i) U})^{g (A_{σ (i)}) - g (A_{σ (i - 1)})}]) \end{matrix}} \end{matrix}$ where σ (1) , …, σ (n) is a permutation of 1, 2, …, n, ${\tilde{h}}_{σ (1)} > {\tilde{h}}_{σ (2)} > \dots > {\tilde{h}}_{σ (n)}$ and ${\tilde{h}}_{σ (0)} = Ø$ .

4 Proposed method

4.1 Confidence level based weight solution

Definition 4.1. [44] If $\tilde{h_{1}}$ , $\tilde{h_{2}}$ are two IVIHFNs, then let k₁, k₂ be the number of $\tilde{h_{1}}$ and $\tilde{h_{2}}$ respectively and l₁, l₂ be the confidence level of $\tilde{h_{1}}$ and $\tilde{h_{2}}$ . k = max {k₁, k₂} and l₁, l₂ ∈ [0, 1]. Then the confident hesitant fuzzy correlation degree of two IVIHFNs is defined by the following expression.

$\begin{matrix} R_{l}^{1} (\tilde{h_{1}}, \tilde{h_{2}}) \\ = \frac{\sum_{j = 1}^{k} (l_{1}^{j}, S (\tilde{h_{1}^{j}})) (l_{2}^{j}, S (\tilde{h_{2}^{j}}))}{max {\sum_{j = 1}^{k} (l_{1}^{j}, S (\tilde{h_{1}^{j}}))^{2}, \sum_{j = 1}^{k} (l_{2}^{j}, S (\tilde{h_{2}^{j}}))^{2}}} \end{matrix}$ (9) where $S (\tilde{h_{1}^{j}})$ and $l_{1}^{j}$ are the score function and the confidence degree of the j-th element in $\tilde{h_{1}}$ respectively.

Let Y = {Y₁, Y₂, …, Y_m} and C = {C₁, C₂, …, C_n} be sets of m alternatives and n attributes. Then the decision makers evaluate the performance of alternatives and generate an initial m × n decision matrix composed by IVIHFNs ${\tilde{h}}_{ij} = ([μ_{Lij}, μ_{Uij}], [ν_{Lij}, ν_{Uij}])$ . If two or more decision makers provide the same value of ${\tilde{h}}_{ij}$ , we only keep one. Thus, the interval-valued intuitionistic hesitant fuzzy decision matrix $\tilde{D} = ({\tilde{h}}_{ij})_{m \times n}$ is constructed.

The decision matrix reflects the different opinions of decision makers to some degree and ${\tilde{h}}_{ij}$ is considered as the objective preference of alternatives with respect to attributes. Then we normalize the decision matrix where the cost attribute is normalized by ${\tilde{h}}^{c}$ according to Definition 2.4. In this method, the subjective preference of alternatives is ${\hat{h}}_{i} = {(l_{i 1}, S_{i 1}), \dots, (l_{ix}, S_{ix})}$ for i = 1, 2, …, m where l_i1, S_i1 are the confidence level and the score function respectively. In order to maximize the correlation degree of objective and subjective preference, the weight solution is proposed as follows.

$\begin{matrix} max f_{i} (ω) \\ = \sum_{t = 1}^{n} R_{l}^{1} ({\tilde{h}}_{it}, {\hat{h}}_{i}) ω_{t} \\ = \sum_{t = 1}^{n} \frac{\sum_{j = 1}^{k} (l_{it}^{j} S ({\tilde{h}}_{it}^{j})) (l_{i}^{j} S ({\hat{h}}_{i}^{j}))}{max {\sum_{j = 1}^{k} (l_{it}^{j}, S (\tilde{h_{it}^{j}}))^{2}, \sum_{j = 1}^{k} (l_{i}^{j}, S ({\hat{h}}_{i}^{j}))^{2}}} ω_{t} \\ s . t \sum_{t = 1}^{n} ω_{t}^{2} = 1, ω_{t} > 0, t = 1, \dots, n \end{matrix}$ (10)

Then we construct Lagrangian function: $L (ω, λ) = \sum_{i = 1}^{m} \sum_{t = 1}^{n} R_{l}^{1} ({\tilde{h}}_{it}, {\hat{h}}_{i}) ω_{t} + \frac{λ}{2} (\sum_{t = 1}^{n} ω_{t}^{2} - 1)$

Next, we compute the partial derivatives with respect to ω_t and λ. Let ${\begin{matrix} \frac{\partial L (ω, λ)}{\partial ω_{t}} = \sum_{i = 1}^{m} R_{l}^{1} ({\tilde{h}}_{it}, {\hat{h}}_{i}) + λ ω_{t} = 0 \\ \frac{\partial L (ω, λ)}{\partial λ} = \sum_{t = 1}^{n} ω_{t}^{2} - 1 = 0 \end{matrix}$

The optimized weight is obtained by the following expression. $ω_{t}^{*} = \frac{\sum_{i = 1}^{m} R_{l}^{1} ({\tilde{h}}_{it}, {\hat{h}}_{i})}{\sqrt{\sum_{t = 1}^{n} (\sum_{i = 1}^{m} R_{l}^{1} ({\tilde{h}}_{it}, {\hat{h}}_{i}))^{2}}}$ (11)

Normalize the weight by using the following expression. $ω_{t} = \frac{\sum_{i = 1}^{m} R_{l}^{1} ({\tilde{h}}_{it}, {\hat{h}}_{i})}{\sum_{t = 1}^{n} (\sum_{i = 1}^{m} R_{l}^{1} ({\tilde{h}}_{it}, {\hat{h}}_{i}))}$ (12)

Then we have $ω_{t} = \frac{\sum_{t = 1}^{n} \frac{\sum_{j = 1}^{k} (l_{it}^{j} S ({\tilde{h}}_{it}^{j})) (l_{i}^{j} S ({\hat{h}}_{i}^{j}))}{max {\sum_{j = 1}^{k} (l_{it}^{j}, S (\tilde{h_{it}^{j}}))^{2}, \sum_{j = 1}^{k} (l_{i}^{j}, S ({\hat{h}}_{i}^{j}))^{2}}}}{\sum_{t = 1}^{n} \sum_{i = 1}^{m} \frac{\sum_{j = 1}^{k} (l_{it}^{j} S ({\tilde{h}}_{it}^{j})) (l_{i}^{j} S ({\hat{h}}_{i}^{j}))}{max {\sum_{j = 1}^{k} (l_{it}^{j}, S (\tilde{h_{it}^{j}}))^{2}, \sum_{j = 1}^{k} (l_{i}^{j}, S ({\hat{h}}_{i}^{j}))^{2}}}}$ (13)

In this method, we compute the weights of attributes based on the confidence level. Firstly, determine λ and calculate the fuzzy measure using φ_s. Then aggregate decision information using the fuzzy integral operator. Finally, determine the order of alternatives according to IVIHF entropy.

4.2 Decision making steps

Step 1. Invite experts to give the evaluation information and the confidence levels according to the decision-making problem. The interval-valued intuitionistic hesitant fuzzy decision matrix $\tilde{D} = ({\tilde{h}}_{ij})_{m \times n}$ is obtained, where ${\tilde{h}}_{ij} = ([μ_{Lij}, μ_{Uij}], [ν_{Lij}, ν_{Uij}])$ denotes an IVIHFE. Experts should also give preference information ${\hat{h}}_{i} (i = 1, 2 \dots, m)$ on alternatives and confidence levels l_i of ${\hat{h}}_{i}$ .

Step 2. Normalize the decision matrix where the information of cost attributes is transformed according to Definition 2.4.

Step 3. By Equation (13), the optimized weights of attributes ω (C_j) based on confidence level are calculated, where j = 1, …, n.

Step 4. In view of history references, set λ and calculate the fuzzy measures g (C_j) using Equations (6, 7), where j = 1, …, n

Step 5. The scores of alternatives are calculated by using the score function.

Step 6. Determine the preference of one permutation over other by the score function. So ${\tilde{h}}_{ij}$ is aggregated into ${\tilde{h}}_{i}$ of the alternative Y_i.

Step 7. Aggregate the information of each alternative using Definition 3.4, and then we get $\tilde{h} (Y_{i})$ , i = 1, 2 … , m.

Step 8. Calculate the IVIHF entropy of alternatives I (Y_i), i = 1, 2 … , m, and rank all alternatives according to the descending order of the entropy values. The best alternative is the one with the lowest value of the entropy.

5 A numerical example

An electric power corporation intends to evaluate the financing risk of four electric power engineering construction projects Y₁, Y₂, Y₃ and Y₄ on the basis of four benefit attributes: C₁ debt coverage ratio, C₂ debt bearing ratio, C₃ asset coverage ratio and C₄ risk aversion. The evaluation is based on experts’ knowledge and experience, they may not possess an accurate level of knowledge for the problem, and may not be confident enough in their evaluation. Therefore, to solve the fuzziness and hesitation, according to the above analysis, we use the IVIHFN and the confidence level to indicate projects’ information.

A committee of three experts provides decision-making information of alternatives with respect to attributes using IVIHFNs. If two or more experts give the same value of one alternative with respect to one attribute, then only one remains. Table 1 shows the decision-making information with confidence levels of alternatives with respect to attributes.

In Table 1, CL is short for confidence level. Three experts also give preference information to four alternatives: $\begin{matrix} {\hat{h}}_{1} & = & {([0.3, 0.4], [0.5, 0.5]), ([0.4, 0.6], \\ [0.2, 0.3]), ([0.6, 0.6], [0.3, 0.4])}, \\ {\hat{h}}_{2} & = & {([0.5, 0.7], [0.2, 0.2]), \\ ([0.5, 0.6], [0.1, 0.2])}, \\ {\hat{h}}_{3} & = & {([0.2, 0.6], [0.3, 0.4]), \\ ([0.4, 0.6], [0.1, 0.2])}, \\ {\hat{h}}_{4} & = & {([0.4, 0.4], [0.5, 0.6]), ([0.1, 0.3], \\ [0.5, 0.6]), ([0.6, 0.9], [0.1, 0.1])} . \end{matrix}$

l₁ = 0.2, 0.6, 0.7, l₂ = 0.5, 0.3, l₃ = 0.4, 0.1, l₄ = 0.3, 0.6, 0.8 are confidence levels of ${\hat{h}}_{1}$ , ${\hat{h}}_{2}$ , ${\hat{h}}_{3}$ , ${\hat{h}}_{4}$ .

According to paper [45], we define λ = 0.5. The weights of attributes on the basis of confidence level are calculated by Equation (13). ω (C₁) =0.1376, ω (C₂) =0.2428, ω (C₃) =0.2945, ω (C₄) =0.3251.

The fuzzy measures of four attributes are computed by Equations (6, 7). g (C₁) =0.1147, g (C₂) =0.2069, g (C₃) =0.2536, g (C₄) =0.2818.

Furthermore, $\begin{matrix} g (C_{1}, C_{2}) = 0.3334, g (C_{1}, C_{3}) = 0.3828, \\ g (C_{2}, C_{3}) = 0.4867, g (C_{1}, C_{4}) = 0.4126, \\ g (C_{2}, C_{4}) = 0.5178, \\ g (C_{3}, C_{4}) = 0.5711, g (C_{1}, C_{2}, C_{3}) = 0.6293, \\ g (C_{1}, C_{2}, C_{4}) = 0.6621, g (C_{2}, C_{3}, C_{4}) = 0.8371, \\ g (C_{1}, C_{2}, C_{3}, C_{4}) = 1 . \end{matrix}$

Then the scores of alternatives are calculated by using the score function. The results are asfollows. $\begin{matrix} S ({\tilde{h}}_{11}) & = & 0.175, S ({\tilde{h}}_{12}) = 0.35, \\ S ({\tilde{h}}_{13}) & = & 0.4, S ({\tilde{h}}_{14}) = - 0.225 . \\ S ({\tilde{h}}_{13}) & > & S ({\tilde{h}}_{12}) > S ({\tilde{h}}_{11}) > S ({\tilde{h}}_{14}) \\ h_{1 σ (1)} & = & h_{13}, h_{1 σ (2)} = h_{12}, \\ h_{1 σ (3)} & = & h_{11}, h_{1 σ (4)} = h_{14} \end{matrix}$

$\begin{matrix} g (A_{σ (1)}) - g (A_{σ (0)}) \\ = 0.2536 g (A_{σ (2)}) - g (A_{σ (1)}) \\ = 0.2331 \\ g (A_{σ (3)}) - g (A_{σ (2)}) \\ = 0.1426 g (A_{σ (4)}) - g (A_{σ (3)}) \\ = 0.3707 \end{matrix}$

Then we use the IVIHFCI operator to aggregate the interval-valued intuitionistic hesitant fuzzy information. $\begin{matrix} \tilde{h} (Y_{1}) & = & {([0.3885, 0.5450], [0.2591, 0.3937]), \\ ([0.3004, 0.544 9], [0.2591, 0.2954]), \\ ([0.3728, 0.5194], [0.2731, 0.3107]), \\ ([0.2883, 0.5194], [0.2731, 0.2954]), \\ ([0.3612, 0.5113], [0.2809, 0.4311]), \\ ([0.3467, 0.4874], [0.2945, 0.4435]), \\ ([0.2193, 0.5113], [0.2809, 0.3389]), \\ ([0.3258, 0.4695], [0.3315, 0.453]), \\ ([0.3128, 0.4475], [0.3441, 0.4648]), \\ ([0.2419, 0.4475], [0.3441, 0.3780]), \\ ([0.2520, 0.4694], [0.3315, 0.3642])} \end{matrix}$

Similarly, $\begin{matrix} \tilde{h} (Y_{2}) & = & {([0.2888, 0.5423], [0.3208, 0.3738]), \\ ([0.3812, 0.600 6], [0.2403, 0.3453]), \\ ([0.4183, 0.5662], [0.2687, 0.3163]), \\ ([0.317, 0.5113], [0.3462, 0.3462]), \\ ([0.3004, 0.5587], [0.3002, 0.3548]), \\ ([0.3963, 0.6188], [0.2173, 0.3548]), \\ ([0.350, 0.5834], [0.2465, 0.2956]), \\ ([0.3297, 0.5267], [0.3263, 0.3263])} \end{matrix}$ $\begin{matrix} \tilde{h} (Y_{3}) & = & {([0.3557, 0.4578], [0.1791, 0.4292]), \\ ([0.3632, 0.4661], [0.1679, 0.4204]), \\ ([0.3125, 0.4189], [0.2521, 0.4664]), \\ ([0.4696, 0.5735], [0.1308, 0.3434]), \\ ([0.4599, 0.5634], [0.1424, 0.3533]), \\ ([0.3061, 0.4116], [0.2621, 0.4745])} \end{matrix}$ $\begin{matrix} \tilde{h} (Y_{4}) & = & {([0.3338, 0.3729], [0.4131, 0.5181]), \\ ([0.3229, 0.3807], [0.4306, 0.5107]), \\ ([0.3083, 0.3634], [0.4564, 0.5364]), \\ ([0.2734, 0.3338], [0.4912, 0.5661]), \\ ([0.2873, 0.3664], [0.4685, 0.5236]), \\ ([0.3239, 0.3989], [0.4322, 0.4911]), \\ ([0.2961, 0.3424], [0.4741, 0.5490]), \\ ([0.3111, 0.3759], [0.4506, 0.5049]), \\ ([0.3508, 0.4093], [0.4131, 0.4711]), \\ ([0.3393, 0.4179], [0.4052, 0.4629]), \\ ([0.3010, 0.3838], [0.4432, 0.4973]), \\ ([0.2864, 0.3497], [0.4670, 0.5420])} \end{matrix}$

The values of IVIHF entropy are computed as follows. $\begin{matrix} I (Y_{1}) & = & 0.9396, I (Y_{2}) = 0.8649, \\ I (Y_{3}) & = & 0.8821, I (Y_{4}) = 0.9778 . \end{matrix}$

Finally, we rank the alternatives in descending order as Y₂ ≻ Y₃ ≻ Y₁ ≻ Y₄. Thus the best alternative is Y₂.

6 Comparative analysis

6.1 The comparison of distance measure

Some decision-making methods based on distance are not suitable for hesitant fuzzy numbers because the measured objects need to have equal length. To solve this problem, Liu et al. [34] complete the hesitant fuzzy number using the largest data [34], which is not applicable to this paper because it ignores the hesitant degree and the confidence level.

Joshi and Kumar [30] uses Hamming distance and TOPSIS method to evaluate alternatives [30]. According to their study, the positive-ideal solution and negative-ideal solution are h⁺ = {([1, 1] , [0, 0]) , ([1, 1] , [0, 0]) , ([0, 0] , [1, 1])} and h^- = {([0, 0] , [1, 1]) , ([0, 0] , [1, 1]) , ([1, 1] , [0, 0])} respectively. We take the calculation of the second alternative as an example. ${\tilde{h}}_{2} = {([0.16, 0.29], [0.53, 0.64]), ([0.2, 0.34], [0.53, 0.61])}$ . There are only two interval-valued intuitionistic numbers which make it infeasible to calculate the distance from ideal solutions. Suppose two interval-valued intuitionistic numbers are from the first and second attributes which are benefit attributes. h⁺ = {([1, 1] , [0, 0]) , ([1, 1] , [0, 0])} , h^- = {([0, 0] , [1, 1]) , ([0, 0] , [1, 1])} , $d_{1} ({\tilde{h}}_{2}, h^{+}) = 0.665,$ $d_{1} ({\tilde{h}}_{2}, h^{-}) = 0.335 .$ The closeness coefficient is Cc₂ = 0.335which is different from Cc₂ = 0.3039 in paper[30]. If we use h⁺ = {([1, 1] , [0, 0]) , ([1, 1] , [0, 0]) , ([0, 0] , [1, 1])} , we can’t obtain results by Hamming distance in paper [30]. So it is concluded that TOPSIS is not suitable for IVIHFN. The reason is that the interaction relationship among attributes and the otherness of decision-making information with respect to attributes is eliminated after aggregation, so the decision-making information after aggregation is no longer IVIHFN for attribute.

6.2 The comparison of aggregation operators

In order to compare with different aggregation operators, this paper utilizes different operators to aggregate the data in Table 1 and ranks the alternatives shown in Table 2. Note that some operators only apply to hesitant fuzzy numbers or interval-valued hesitant fuzzy numbers. Therefore, the data have to be processed according to the operators. For example, {([0.4, 0.7] , [0.2, 0.3]) , ([0.3, 0.5] , [0.3, 0.4])} is transformed into an interval-valued hesitant fuzzy number {, [0.3, 0.5]} or an intuitionistic hesitant fuzzy number {(0.4, 0.2) , (0.3, 0.3)}.

Wei [4] used the prioritized average operator [4] and obtained the same positive-ideal and negative-ideal alternatives as this paper’s results. However, the results obtained by using other operators are quite different because different operators address different relationships of the aggregated arguments. The weighted geometric operator proposed by Yu [33] is infeasible dealing with extreme values [33]. For example, when {(0.036, 0) , (0.112, 0.789)} and {(0.119, 0.554)} are aggregated, the non-membership degree function is always 0, which does not satisfy the requirements of group decision making and ignores the confidence level. The geometric Bonferroni mean put forward by Zhu et al. [5] takes into account the relationship among attributes and the importance of the data [5], but it can’t deal with big data, hence too much hesitant fuzzy data during the hesitant group decision process can distort the aggregated information.

7 Conclusion

The proposed method is more flexible for evaluating, reserving and aggregating decision information with confidence level of decision makers taken into account. In this paper, the main conclusions are as follows.

The IVIHFS with confidence levels is proposed to express the evaluation information and also expand the hesitant fuzzy sets in order that the group decision-making information can conform to the real needs.

We have applied the confidence level based on IVIHF operator to aggregate the IVIHFNs without changing any initial information and eliminate interaction phenomena among attributes.

The advantage of the proposed confidence level based on weight solution is that it satisfies the maximization principle of the correlation degree of subjective and objective preference and combines the fuzzy Choquet integral with the transformation function.

We define the IVIHF entropy which is proved to overcome the shortcoming of the distance measure.

Footnotes

Acknowledgments

The authors would like to acknowledge the supports by National Natural Science Foundation of China (No. 71271084).

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A group decision making approach in interval-valued intuitionistic hesitant fuzzy environment with confidence levels

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Interval-valued intuitionistic hesitant fuzzy set

2.2 Interval-valued intuitionistic hesitant fuzzy entropy

3.1 Fuzzy measure of attributes calculated by φ s transformation

4 Proposed method

4.1 Confidence level based weight solution

5 A numerical example

6 Comparative analysis

6.1 The comparison of distance measure

6.2 The comparison of aggregation operators

7 Conclusion

Footnotes

Acknowledgments

References

3.1 Fuzzy measure of attributes calculated by φ_s transformation