Multiattribute group decision making based on interval-valued intuitionistic fuzzy sets and analytically evidential reasoning methodology

Abstract

The existing multiattribute group decision making methods with interval-valued intuitionistic fuzzy (IVIF) sets only consider weights of the experts who participated in decision making, however the reliabilities of that are not considered at all. In order to solve this problem and make a much scientific decision, a transformation method from IVIF value to belief degree is defined based on the frame of discernment consisting of only excellent and non-excellent two hypotheses. Then the basic probability assignment (BPA) function computing method is introduced by discounting belief degrees with both weights and reliabilities. The derived BPA functions are combined by the analytically evidential reasoning (ER) methodology for two times, the one is to combine BPA functions on all attributes with respect to a specified expert, and the other is to combine the combined BPA functions for all experts with respect to a specified alternative. Finally, the relative coefficient is employed to rank the alternatives with IVIF sets. An example is proposed to illustrate the decision making process by the proposed method.

Keywords

Multiple attribute decision making group decision making interval-valued intuitionistic fuzzy sets evidential reasoning expert reliability

1 Introduction

Interval-valued intuitionistic fuzzy (IVIF) sets were introduced by Atanassov and Gargov [1], in which the membership degree and non-membership degree of each element were expressed with interval values. In recent years, IVIF sets were employed to solve the problems of multiattribute group decision making (MAGDM) and lots of methods had been presented. Boran et al. presented a fuzzy multicriteria group decision making method for supplier selection using the intuitionistic fuzzy TOPSIS (Technique for Order Performance by Similarity to an Ideal Solu-tion) [2]. Li presented a linear programming method for MAGDM with both ratings of alternatives on attributes and weights being expressed with IVIF sets [3]. Wang et al. presented an IVIF-MAGDM framework with incomplete preference over alternatives where qualitative and quantitative attribute values are furnished as linguistic variables and crisp numbers [4]. Meng et al. presented an IVIF-MAGDM based on cross entropy measure and choquet integral, in which the arithmetic IVIF choquet aggregation operator was defined to integrate decision information [5]. Huang et al. used a rough set model to extract rules in dominance-based IVIF information systems [6]. Wu et al. presented a new similarity measure of IVIF sets by considering its hesitancy degree and used the proposed measure to solve the MAGDM problem [7]. Jiang et al. presented a MAGDM with unknown experts’ weights information in the framework of interval intuitionistic trapezoidal fuzzy numbers [8]. Yu used the MAGDM with IVIF method to solve the problem of hydrogen production technologies evaluation [9]. Wan et al. presented an Atanassov’s intuitionistic fuzzy programming method for heterogeneous MAGDM with Atanassov’s intuitionistic fuzzy truth degrees [10]. Xu et al. used MAGDM method with IVIF sets to solve a cloud service selection problem [11].

The decision information in all of the methods as mentioned above are given by experts, and each expert should have different reliabilities since their knowledge, experience, background are not the same. The weight of expert is not equal to his/her reliability. The weight reflects the importance degree of information preferred in a person’s mind, and the reliability is used to measure the quality of information generated by an expert. In other words, the weight is subjective and depends on who makes the judgement, but the latter is objective and is independent of who may use the generated information. The existing MAGDM methods with IVIF sets only consider weights of the experts who participated in decision making, however the reliabilities of that are not considered at all.

Fortunately, the weight and the reliability of evidence (e.g., sensor, expert, or decision maker) can be both integrated into the decision process by discounting method in the evidential reasoning (ER) methodology [12]. The ER methodology proposed by Yang is a general approach for analyzing multiple criteria decision making problems under uncertainties, and use the ER rule to combine basic probability assignment (BPA) functions recursively. Because the ER rule is a recursive algorithm, sometimes it is hard to be modeled and computed, the analytical ER methodology is developed [13]. In order to solve the problem existing MAGDM methods with IVIF sets and make a much scientific decision, we propose a MAGDM method based on IVIF sets and analytical ER methodology. Note that, Chen et al. used the intuitionistic fuzzy sets and ER methodology to solve a fuzzy MAGDM problem, but their method was unsuitable for the IVIF situation and also unconsidered the reliability of expert.

The rest of this paper is organized as follows. In Section 2, we briefly review basic concepts of intuitionistic fuzzy sets, IVIF sets, and analytical ER methodology. In Section 3, we propose a MAGDM method based on IVIF sets and analytical ER methodology, in which experts’ weights and reliabilities are both considered. In Section 4, we use an example to illustrate the decision process of the proposed method. The conclusion is discussed in Section 5.

2 Preliminaries

This section reviews some fundamental notions such as intuitionistic fuzzy sets and IVIF sets according to reference [5 , 13].

Definition 1. Let a set X be fixed, an intuitionistic fuzzy set A in X is defined as follows: $A = {〈 x, u_{A} (x), v_{A} (x) 〉 | x \in X}$ (1) where u_A (x) : X → [0, 1] and v_A (x) : X → [0, 1] respectively determine the membership degree and non-membership degree of x ∈ X, 0 ≤ u_A (x) + v_A (x) ≤1. π_A (x) denotes hesitancy degree of element x ∈ X to the set A and is determined by $π_{A} (x) = 1 - u_{A} (x) - v_{A} (x),$ (2)π_A (x) ∈ [0, 1],∀x ∈ X. If |X|=1 meaning X includes only one element x, then Equation (1) is denoted by A = 〈u_A (x) , v_A (x) 〉 for short, which is also called intuitionistic fuzzy value.

Definition 2. Let a set X be fixed, an interval-valued intuitionistic fuzzy set $\tilde{A}$ in X is defined as follows: $\tilde{A} = {〈 x, u_{\tilde{A}} (x), v_{\tilde{A}} (x) 〉 | x \in X}$ (3) where $u_{\tilde{A}} (x) = [{\overset{ˇ}{u}}_{\tilde{A}} (x), {\hat{u}}_{\tilde{A}} (x)]$ is an interval membership degree and $v_{\tilde{A}} (x) = [{\overset{ˇ}{v}}_{\tilde{A}} (x), {\hat{v}}_{\tilde{A}} (x)]$ is an interval non-membership degree, $0 \leq u_{\tilde{A}} (x), v_{\tilde{A}} (x) \leq 1$ , x ∈ X. If |X|=1, Equation (3) is simplified to the form of IVIF value, i.e., $\tilde{A} = 〈 u_{\tilde{A}} (x), v_{\tilde{A}} (x) 〉$ .

The analytical ER methodology is a non-recursive form of the ER methodology and it is more convenient for computing and modeling [13]. The analytical ER methodology regards evaluation grades as the frame of discernment, and uses belief degree to extract assessments for alternatives from evidence.

Definition 3. Let the grades for assessing alternatives be H = {H_n|n = 1, ⋯ , N}, the belief degree that an alternative is assessed to H_n be β_n derived from a piece of evidence, then the assessment is profiled by $B = {(H_{n}, β_{n}), n = 1, \dots, N; (H, β_{H})},$ (4) where $\sum_{n = 1}^{N} β_{n} \leq 1$ , β_n ≥ 0 for ∀n, $β_{H} = 1 - \sum_{n = 1}^{N} β_{n}$ represents the degree of global ignorance. If β_H = 0, the assessment is complete; otherwise, it is incomplete. For convenience, let (H_N+1, β_N+1) = (H, β_H), and the assessment as Equation (4) can be denoted as B = {(H_n, β_n) , n = 1, ⋯ , N, N + 1}.

Recently, a new discounting method is defined to discount evidence with both weight and reliability [12]. The set of discounted belief degree is seen as a BPA function, and is used to make combination with analytical ER rule. The discounting method and the analytical ER rule are defined as follows [12, 13].

Definition 4. Let the belief degree assessed by a piece of evidence be B as Equation (4), the weight and reliability of evidence be w and r, then the discounting method with both weight and reliability is defined to generate a BPA function as follows: $m = m (H_{n}) = {\begin{matrix} λ β_{n}, n = 1, \dots, N \\ 1 - λ \sum_{n = 1}^{N} β_{n}, n = N + 1 \end{matrix}$ (5) where λ = w/(1 + w - r) , $m_{H} = {\bar{m}}_{H} + {\tilde{m}}_{H}$ , ${\bar{m}}_{H} = 1 - λ$ , ${\tilde{m}}_{H} = λ (1 - \sum_{n = 1}^{N} β_{n})$ . ${\bar{m}}_{H}$ depicts the role extents of the evidence to be combined, ${\tilde{m}}_{H}$ represents the incompleteness of the assessment.

Definition 5. There are I pieces of evidence be combined, let the BPA function of the i^th one be $m_{n}^{i}$ generated by Equation (5), i = 1, ⋯ , I, n = 1, ⋯ , N + 1, then the combined belief degree for I pieces of evidence is calculated by analytical ER rule as follows: $B = B (H_{n}) = {\begin{matrix} \frac{m_{n}}{1 - {\bar{m}}_{H}}, n = 1, \dots, N \\ \frac{{\tilde{m}}_{n}}{1 - {\bar{m}}_{H}}, n = N + 1 \end{matrix}$ (6) where $\begin{matrix} m_{n} & = & δ [\prod_{i = 1}^{I} (m_{n}^{i} + {\bar{m}}_{H}^{i} + {\tilde{m}}_{H}^{i}) \\ - \prod_{i = 1}^{I} ({\bar{m}}_{H}^{i} + {\tilde{m}}_{H}^{i})], n = 1, \dots, N \\ {\tilde{m}}_{H} & = & δ [\prod_{i = 1}^{I} ({\bar{m}}_{H}^{i} + {\tilde{m}}_{H}^{i}) - \prod_{i = 1}^{I} {\bar{m}}_{H}^{i}] \\ {\bar{m}}_{H} & = & δ [\prod_{i = 1}^{I} {\bar{m}}_{H}^{i}] \\ δ & = & [\sum_{n = 1}^{N} \prod_{i = 1}^{I} (m_{n}^{i} + {\bar{m}}_{H}^{i} + {\tilde{m}}_{H}^{i}) \\ - (N - 1) {\prod_{i = 1}^{I} ({\bar{m}}_{H}^{i} + {\tilde{m}}_{H}^{i})]}^{- 1} \end{matrix}$

3 The proposed method

3.1 Transformation from IVIF value to belief degree

Suppose a set of experts E = {e_i|i = 1, ⋯ , I} is invited to evaluate a set of alternatives Y = {y_k|k = 1, ⋯ , K} on attribute set C = {c_j|j = 1, ⋯ , J}. Experts are invited to evaluate alternatives on each attribute with IVIF sets. Without loss of generality, let the assessment value of alternative y_k evaluated by expert e_i with respect to attribute c_j be $D_{i, j}^{k} = {〈 x, u_{\tilde{D}}^{i, j, k} (x), v_{\tilde{D}}^{i, j, k} (x) 〉 | x \in X}, \forall i, j, k$ (7) where $u_{\tilde{D}}^{i, j, k} (x) = [{\overset{ˇ}{u}}_{\tilde{D}}^{i, j, k} (x), {\hat{u}}_{\tilde{D}}^{i, j, k} (x)]$ is interval valued membership degree, $v_{\tilde{D}} (x) = [{\overset{ˇ}{v}}_{\tilde{D}}^{i, j, k} (x), {\hat{v}}_{\tilde{D}}^{i, j, k} (x)]$ is interval valued non-membership degree.

As shown in Definition 2, if |X|=1 then the IVIF sets can be simplified to IVIF values. Here we let X = {Excellent}, $D_{i, j}^{k}$ is simplified to IVIF value as $D_{i, j}^{k} = 〈 u_{i, j}^{k}, v_{i, j}^{k} 〉, \forall i, j, k$ (8) where $u_{i, j}^{k} = [{\overset{ˇ}{u}}_{i, j}^{k}, {\hat{u}}_{i, j}^{k}]$ , $v_{i, j}^{k} = [{\overset{ˇ}{v}}_{i, j}^{k}, {\hat{v}}_{i, j}^{k}]$ . The decision matrix $D_{i} = {[D_{i, j}^{k}]}_{K \times J}$ of expert e_i is constructed as follows. $D_{i} = {[〈 [{\overset{ˇ}{u}}_{i, j}^{k}, {\hat{u}}_{i, j}^{k}], [{\overset{ˇ}{v}}_{i, j}^{k}, {\hat{v}}_{i, j}^{k}] 〉]}_{K \times J}, \forall i$ (9)

As illustrated in the theory of original interpretation of intuitionistic fuzzy sets, $u_{i, j}^{k}$ and $v_{i, j}^{k}$ in Equation (8) can be regarded as the answer of “Is the alternative y_k excellent or not on attribute c_j for expert e_i?”. If we let the frame of discernment for the answer is H = {H₁, H₂}, H₁ = Excellent, H₂ = Non - excellent, then the IVIF value $B_{i, j}^{k} = {(H_{1}, u_{i, j}^{k}), (H_{2}, v_{i, j}^{k}); (H, π_{i, j}^{k})}$ for ∀i, j, k can be transferred into the degree of belief as follows. $B_{i, j}^{k} = {\begin{matrix} (H_{1}, [{\overset{ˇ}{u}}_{i, j}^{k}, {\hat{u}}_{i, j}^{k}]); \\ (H_{2}, [{\overset{ˇ}{v}}_{i, j}^{k}, {\hat{v}}_{i, j}^{k}]); \\ (H, [{\overset{ˇ}{π}}_{i, j}^{k}, {\hat{π}}_{i, j}^{k}])} . \end{matrix}$ (10) where ${\overset{ˇ}{π}}_{i, j}^{k} = \max {0, 1 - {\hat{u}}_{i, j}^{k} - {\hat{v}}_{i, j}^{k}}$ , ${\hat{π}}_{i, j}^{k} = 1 - {\overset{ˇ}{u}}_{i, j}^{k} - {\overset{ˇ}{v}}_{i, j}^{k}$ . Note that, $π_{i, j}^{k}$ for global ignorance should be a non-negative number, but ${\hat{u}}_{i, j}^{k} + {\hat{v}}_{i, j}^{k}$ may be greater than 1 resulting in $1 - {\hat{u}}_{i, j}^{k} - {\hat{v}}_{i, j}^{k}$ may be less than 0, so here we let ${\overset{ˇ}{π}}_{i, j}^{k} = \max {0, 1 - {\hat{u}}_{i, j}^{k} - {\hat{v}}_{i, j}^{k}}$ . The belief degree that alternative y_k is assessed to H₁ with respect to attribute c_j by expert e_i lies in $[{\overset{ˇ}{u}}_{i, j}^{k}, {\hat{u}}_{i, j}^{k}]$ , the belief degree assessed to H₂ lies in $[{\overset{ˇ}{v}}_{i, j}^{k}, {\hat{v}}_{i, j}^{k}]$ , and the degree of global ignorance lies in $[{\overset{ˇ}{π}}_{i, j}^{k}, {\hat{π}}_{i, j}^{k}]$ .

3.2 BPA function computation

Weight and reliability are two kinds of important influence factors in the MAGDM problems. Weight is the subjectively relative importance degree of an attribute rather than another with respect to a given problem and usually can be determined by some computing methods such as Analytic Hierarchy Process (AHP) [14], Analytic Network Process (ANP) [15], or etc. Let the derived weight of attribute c_j be w_j, w_j ≥ 0, j = 1, ⋯ , J, and $\sum_{j = 1}^{J} w_{j} = 1$ is usually defined but not necessary. Compared with weight, reliability is defined as an ability of evidence source such as expert to provide correct assessment/solution for the given problem, and the reliability of an evidence source should be estimated from statistics when available, or by other techniques. It is believed that experts have different knowledges, backgrounds, or experiences in reality, so their reliabilities should be different in MAGDM problems. Let the reliability of expert e_i on attribute c_j be r_ij, 0 ≤ r_ij ≤ 1, i = 1, ⋯ , I, j = 1, ⋯ , J. On attribute c_j, r_ij = 1 means expert e_i is absolutely reliable and r_ij = 0 means expert e_i is absolutely unreliable. The assessment given by each expert can be regarded as a piece of evidence in MAGDM problems. The weight w_j is the contribution degree of the information on attribute c_j for solving the given decision problem, and it can be seen as the weight of each expert on attribute c_j. According to the discounting method with both weight and reliability as definition 4, the overall discounting factor for expert e_i with respect to attribute c_j can be computed by λ_i,j = w_i/(1 + w_i - r_ij) . Take the discounting factor λ_i,j and Equation (10) into Equation (5), the BPA function $m_{i, j, k} = {(H_{1}, m_{i, j, k}^{1}), (H_{2}, m_{i, j, k}^{2}); (H, m_{i, j, k}^{H})}$ for ∀i, j, k can be obtained as follows. $m_{i, j, k} = {\begin{matrix} (H_{1}, [λ_{i, j} {\overset{ˇ}{u}}_{i, j}^{k}, λ_{i, j} {\hat{u}}_{i, j}^{k}]); \\ (H_{2}, [λ_{i, j} {\overset{ˇ}{v}}_{i, j}^{k}, λ_{i, j} {\hat{v}}_{i, j}^{k}]); \\ (H, [\max {0, 1 - λ_{i, j} ({\hat{u}}_{i, j}^{k} + {\hat{v}}_{i, j}^{k})}, \\ 1 - λ_{i, j} ({\overset{ˇ}{u}}_{i, j}^{k} + {\overset{ˇ}{v}}_{i, j}^{k})]) . \end{matrix}$ (11)

3.3 Combination models

The analytical ER rule is used to combine the derived BPA functions for two times. The first time is to combine BPA functions on all attributes with respect to a specified expert, i.e., m_i,k = m_i,1,k ⊕ ⋯ ⊕ m_i,J,k, ∀i, k, the symbol ⊕ denotes making combination with analytical ER rule. Obviously, m_i,k is the overall evaluation value for alternative y_k on all attributes given by expert e_i. The second time is to combine the combined BPAs for all experts with respect to a specified alternative, i.e., m_k = m_1,k ⊕ ⋯ ⊕ m_I,k, ∀k. m_k is the overall evaluation value for alternative y_k for all experts. Above process is described in Fig. 1.

Fig.1

Combination process for two times.

Note that, the BPA functions are interval values and can not be directly combined by analytical ER rule. In order to solve this problem, we establish nonlinear optimization models based on analytical ER rule. Let the combined BPA functions of alternative y_k for expert e_i on all attributes be $m_{i, k} = {m_{i, k} (H_{1}), m_{i, k} (H_{2}), {\tilde{m}}_{i, k} (H), {\bar{m}}_{i, k} (H)},$ where $m_{i, k} (H) = {\tilde{m}}_{i, k} (H) + {\bar{m}}_{i, k} (H)$ , ∀i, k. Because of m_i,j,k to be combined are interval values, the combined result m_i,k should also be interval values. For each expert, the interval probability masses on all attributes are combined and transformed into overall interval belief degrees by solving the pair of nonlinear optimization models for H₁ and H₂ as Equation (12). $max / min β_{i, k} (H_{n}) = \frac{m_{i, k} (H_{n})}{1 - {\bar{m}}_{i, k} (H)}$ (12)

$s . t . {\begin{matrix} m_{i, k} (H_{1}) = δ_{i, k} [\prod_{j = 1}^{J} (m_{i, j, k} (H_{1}) + {\bar{m}}_{i, j, k} (H) \\ + {\tilde{m}}_{i, j, k} (H)) - \prod_{j = 1}^{J} ({\bar{m}}_{i, j, k} (H) + {\tilde{m}}_{i, j, k} (H))], \\ m_{i, k} (H_{2}) = δ_{i, k} [\prod_{j = 1}^{J} (m_{i, j, k} (H_{2}) + {\bar{m}}_{i, j, k} (H) \\ + {\tilde{m}}_{i, j, k} (H)) - \prod_{j = 1}^{J} ({\bar{m}}_{i, j, k} (H) + {\tilde{m}}_{i, j, k} (H))], \\ {\tilde{m}}_{i, k} (H) = δ_{i, k} [\prod_{j = 1}^{J} ({\bar{m}}_{i, j, k} (H) + {\tilde{m}}_{i, j, k} (H)) \\ - \prod_{j = 1}^{J} {\bar{m}}_{i, j, k} (H)], \\ {\bar{m}}_{i, k} (H) = δ_{i, k} [\prod_{j = 1}^{J} {\bar{m}}_{i, j, k} (H)], \\ δ_{i, k} = [\sum_{n = 1}^{2} \prod_{j = 1}^{J} (m_{i, j, k} (H_{n}) + {\bar{m}}_{i, j, k} (H) \\ + {\tilde{m}}_{i, j, k} (H)) - {\prod_{j = 1}^{J} ({\bar{m}}_{i, j, k} (H) + {\tilde{m}}_{i, j, k} (H))]}^{- 1}, \\ λ_{i, j} {\overset{ˇ}{u}}_{i, j}^{k} \leq m_{i, j, k} (H_{1}) \leq λ_{i, j} {\hat{u}}_{i, j}^{k}, \\ λ_{i, j} {\overset{ˇ}{v}}_{i, j}^{k} \leq m_{i, j, k} (H_{2}) \leq λ_{i, j} {\hat{v}}_{i, j}^{k}, \\ \max {0, 1 - λ_{i, j} {\hat{u}}_{i, j}^{k} - λ_{i, j} {\hat{v}}_{i, j}^{k}} \leq {\tilde{m}}_{i, k} (H) \\ + {\bar{m}}_{i, k} (H) \leq 1 - λ_{i, j} {\overset{ˇ}{u}}_{i, j}^{k} - λ_{i, j} {\overset{ˇ}{v}}_{i, j}^{k}, \\ 1 - m_{i, j, k} (H_{1}) - m_{i, j, k} (H_{2}) \geq 0, \forall j, \\ m_{i, k} (H_{1}) + m_{i, k} (H_{2}) + {\tilde{m}}_{i, k} (H) + {\bar{m}}_{i, k} (H) = 1, \\ m_{i, k} (H_{1}), m_{i, k} (H_{2}), {\tilde{m}}_{i, k} (H), {\bar{m}}_{i, k} (H) \geq 0 . \end{matrix}$

Let ${\overset{ˇ}{β}}_{i, k} (H_{n})$ and ${\hat{β}}_{i, k} (H_{n})$ respectively be the minimum and maximum optimal objective function values of Equation (12), n = 1, 2. They form an overall interval belief degree $[{\overset{ˇ}{β}}_{i, k} (H_{n}), {\hat{β}}_{i, k} (H_{n})]$ for n = 1, 2, which represents the combined assessment of alternative y_k assessed to the grade H₁ and H₂ by expert e_i. Obviously, $m_{i, k} (H) = [{\overset{ˇ}{β}}_{i, k} (H), {\hat{β}}_{i, k} (H)]$ is easy to be computed by ${\hat{β}}_{i, k} (H) = 1 - \sum_{n = 1}^{2} {\overset{ˇ}{β}}_{i, k} (H_{n})$ , ${\overset{ˇ}{β}}_{i, k} (H) = \max {0, 1 - \sum_{n = 1}^{2} {\hat{β}}_{i, k} (H_{n})}$ . The combined assessments are also interval-valued distribution, i.e., $m_{i, k} = {(H_{1}, [{\overset{ˇ}{β}}_{i, k} (H_{1}), {\hat{β}}_{i, k} (H_{1})]), (H_{2}, [{\overset{ˇ}{β}}_{i, k} (H_{2}), {\hat{β}}_{i, k} (H_{2})]); (H, [{\overset{ˇ}{β}}_{i, k} (H), {\hat{β}}_{i, k} (H)])}$ .

Similarly, we also establish a nonlinear optimization model based on analytical ER rule to combine the BPA functions for all experts. The model is as follows. $max / min β_{k} (H_{n}) = \frac{m_{k} (H_{n})}{1 - {\bar{m}}_{k} (H)}$ (13)

$s . t . {\begin{matrix} m_{k} (H_{1}) = δ_{k} [\prod_{i = 1}^{I} (m_{i, k} (H_{1}) + {\bar{m}}_{i, k} (H) \\ + {\tilde{m}}_{i, k} (H)) - \prod_{i = 1}^{I} ({\bar{m}}_{i, k} (H) + {\tilde{m}}_{i, k} (H))], \\ m_{k} (H_{2}) = δ_{k} [\prod_{i = 1}^{I} (m_{i, k} (H_{2}) + {\bar{m}}_{i, k} (H) \\ + {\tilde{m}}_{i, k} (H)) - \prod_{i = 1}^{I} ({\bar{m}}_{i, k} (H) + {\tilde{m}}_{i, k} (H))], \\ {\tilde{m}}_{k} (H) = δ_{k} [\prod_{i = 1}^{I} ({\bar{m}}_{i, k} (H) + {\tilde{m}}_{i, k} (H)) \\ - \prod_{i = 1}^{I} {\bar{m}}_{i, k} (H)], \\ {\bar{m}}_{k} (H) = δ_{k} [\prod_{i = 1}^{I} {\bar{m}}_{i, k} (H)], \\ δ_{k} = [\sum_{n = 1}^{2} \prod_{i = 1}^{I} (m_{i, k} (H_{n}) + {\bar{m}}_{i, k} (H) \\ + {\tilde{m}}_{i, k} (H)) - {\prod_{i = 1}^{I} ({\bar{m}}_{i, k} (H) + {\tilde{m}}_{i, k} (H))]}^{- 1}, \\ {\overset{ˇ}{β}}_{i, k} (H_{1}) \leq m_{i, k} (H_{1}) \leq {\hat{β}}_{i, k} (H_{1}), \\ {\overset{ˇ}{β}}_{i, k} (H_{2}) \leq m_{i, k} (H_{2}) \leq {\hat{β}}_{i, k} (H_{2}), \\ {\overset{ˇ}{β}}_{i, k} (H) \leq {\tilde{m}}_{k} (H) + {\bar{m}}_{k} (H) \leq {\hat{β}}_{i, k} (H) \\ 1 - m_{i, k} (H_{1}) - m_{i, k} (H_{2}) \geq 0, \forall j, \\ m_{k} (H_{1}) + m_{k} (H_{2}) + {\tilde{m}}_{k} (H) + {\bar{m}}_{k} (H) = 1, \\ m_{k} (H_{1}), m_{k} (H_{2}), {\tilde{m}}_{k} (H), {\bar{m}}_{k} (H) \geq 0 . \end{matrix}$

The combined assessment $[{\overset{ˇ}{β}}_{k} (H_{n}), {\hat{β}}_{k} (H_{n})]$ of alternative y_k is also interval valued distribution assessment vectors, which is derived by solving Equation (11) for n = 1, 2. Similarly, $m_{k} (H) = [{\overset{ˇ}{β}}_{k} (H), {\hat{β}}_{k} (H)]$ is capable of being computed by ${\hat{β}}_{k} (H) = 1 - \sum_{n = 1}^{2} {\overset{ˇ}{β}}_{k} (H_{n})$ , ${\overset{ˇ}{β}}_{k} (H) = \max {0, 1 - \sum_{n = 1}^{2} {\hat{β}}_{k} (H_{n})}$ . The final combination assessment for alternative y_k can be expressed as $m_{k} = {(H_{1}, [{\overset{ˇ}{β}}_{k} (H_{1}), {\hat{β}}_{k} (H_{1})]), (H_{2}, [{\overset{ˇ}{β}}_{k} (H_{2}), {\hat{β}}_{k} (H_{2})]); (H, [{\overset{ˇ}{β}}_{k} (H), {\hat{β}}_{k} (H)])}, \forall k$ .

3.4 Ranking method

Global ignorance may be included in the final combination assessment m_k(m_k (H) >0), which probably resulting in that it is difficult to compare alternatives directly by the ER methodology. Fortunately, the final combination assessment m_k can be easily transferred into IVIF values. As illustrated in Section 2, IVIF values are equal to BPA functions when X = {Excellent}, so the final combination assessment m_k is capable of being transferred into $Z_{k} = 〈 [{\overset{ˇ}{β}}_{k} (H_{1}), {\hat{β}}_{k} (H_{1})], [{\overset{ˇ}{β}}_{k} (H_{2}), {\hat{β}}_{k} (H_{2})] 〉, \forall k .$

The relative coefficient is usually employed to calculate the relative closeness degree to the IVIF positive ideal solution and generate ranking order of alternatives [3 , 17]. This paper also uses it to make comparison to IVIF sets and it is calculated as $C_{k} = \frac{d (y_{k}, y^{-})}{d (y_{k}, y^{-}) + d (y_{k}, y^{+})} = [C_{k}^{-}, C_{k}^{+}]$ (14) where y⁺ and y^- are the ideal and the anti-ideal alternatives respectively, y⁺ = ([1, 1] , [0, 0], $y^{-} = ([0, 0], [1, 1]) . C_{k}^{-}$ and $C_{k}^{+}$ are the lower and the upper bounds of C_k, and can be calculated as: ${\begin{matrix} C_{k}^{-} = \frac{d_{k, 1}^{-}}{d_{k, 1}^{-} + d_{k, 2}^{-}} \\ C_{k}^{+} = \frac{d_{k, 2}^{+}}{d_{k, 1}^{+} + d_{k, 2}^{+}} \end{matrix}$ (15) where $\begin{matrix} d_{k, 1}^{-} & = & \sqrt{{({\overset{ˇ}{β}}_{k} (H_{1}))}^{2} + {(1 - {\hat{β}}_{k} (H_{2}))}^{2}} \\ d_{k, 2}^{-} & = & \sqrt{{(1 - {\overset{ˇ}{β}}_{k} (H_{1}))}^{2} + {({\hat{β}}_{k} (H_{2}))}^{2}} \\ d_{k, 1}^{+} & = & \sqrt{{({\hat{β}}_{k} (H_{1}))}^{2} + {(1 - {\overset{ˇ}{β}}_{k} (H_{2}))}^{2}} \\ d_{k, 2}^{+} & = & \sqrt{{(1 - {\hat{β}}_{k} (H_{1}))}^{2} + {({\overset{ˇ}{β}}_{k} (H_{2}))}^{2}} \end{matrix}$

The inclusion comparison probability p (a_k ⪰ a_k′) = p (C_k ⊇ C_k′) = p_kk′ that alternative y_k is not worse than y_k′ is defined as follows: $p_{k k^{'}} = \max (1 - \max (\frac{C_{k^{'}}^{+} - C_{k}^{-}}{C_{k}^{+} - C_{k}^{-} + C_{k^{'}}^{+} - C_{k^{'}}^{-}}, 0), 0)$ (16)

The judgement matrix P = (p_kk′) _K×K is constructed based on Equation (16). The ranking value s_k of alternative y_k is calculated as follows: $s_{k} = \frac{1}{K (K - 1)} (\sum_{k^{'} = 1}^{K} p_{k k^{'}} + \frac{K}{2} - 1)$ (17)

The larger the value of s_k, the better the preference order of alternative y_k.

4 Illustration example

In this section, an example is proposed to illustrate the decision making process by the proposed method. Assume that a MAGDM problem consists of three alternatives y₁, y₂, y₃, three attributes c₁, c₂, c₃, and three experts e₁, e₂, e₃. The weights for three attributes are 0.5, 0.3, 0.2. On attribute c₁, the reliabilities of three experts are 1.0, 0.8, 0.6; On attribute c₂, the reliabilities of three experts are 0.9, 0.8, 1.0; On attribute c₃, the reliabilities of three experts are 0.5, 1.0, 0.8. Each expert is asked by the question “Is the alternative y_k excellent or not on attribute c_j?” and his/her answer is given by IVIF values. Assume decision matrix D₁, D₂, D₃ are given by three experts as follows. $\begin{matrix} D_{1} = [\begin{matrix} 〈 [0.4, 0.5], [0.3, 0.4] 〉, 〈 [0.4, 0.6], [0.2, 0.4] 〉, 〈 [0.1, 0.3], [0.5, 0.6] 〉 \\ 〈 [0.4, 0.6], [0.2, 0.3] 〉, 〈 [0.6, 0.7], [0.1, 0.2] 〉, 〈 [0.4, 0.7], [0.1, 0.2] 〉 \\ 〈 [0.3, 0.5], [0.3, 0.4] 〉, 〈 [0.5, 0.6], [0.2, 0.3] 〉, 〈 [0.5, 0.6], [0.1, 0.3] 〉 \end{matrix}] \\ D_{2} = [\begin{matrix} 〈 [0.3, 0.5], [0.2, 0.4] 〉, 〈 [0.2, 0.6], [0.3, 0.4] 〉, 〈 [0.2, 0.3], [0.3, 0.5] 〉 \\ 〈 [0.5, 0.6], [0.1, 0.3] 〉, 〈 [0.4, 0.6], [0.2, 0.3] 〉, 〈 [0.6, 0.7], [0.1, 0.2] 〉 \\ 〈 [0.4, 0.5], [0.2, 0.4] 〉, 〈 [0.1, 0.4], [0.4, 0.5] 〉, 〈 [0.4, 0.6], [0.2, 0.3] 〉 \end{matrix}] \\ D_{3} = [\begin{matrix} 〈 [0.5, 0.7], [0.1, 0.2] 〉, 〈 [0.5, 0.6], [0.1, 0.2] 〉, 〈 [0.1, 0.6], [0.2, 0.4] 〉 \\ 〈 [0.3, 0.6], [0.2, 0.3] 〉, 〈 [0.4, 0.5], [0.2, 0.3] 〉, 〈 [0.4, 0.6], [0.1, 0.3] 〉 \\ 〈 [0.3, 0.5], [0.2, 0.4] 〉, 〈 [0.2, 0.3], [0.5, 0.6] 〉, 〈 [0.4, 0.5], [0.1, 0.3] 〉 \end{matrix}] \end{matrix}$

With respect to $D_{1, 1}^{1}$ in decision matrix D₁, the interval degree of global ignorance $[{\overset{ˇ}{π}}_{i, j}^{k}, {\hat{π}}_{i, j}^{k}]$ is able to be calculated by ${\hat{π}}_{1, 1}^{1} = 1 - {\overset{ˇ}{u}}_{1, 1}^{1} - {\overset{ˇ}{v}}_{1, 1}^{1} = 1 - 0.4 - 0.3 = 0.3$ , ${\overset{ˇ}{π}}_{1, 1}^{1} = \max {0, 1 - {\hat{u}}_{1, 1}^{1} - {\hat{v}}_{1, 1}^{1}} = 1 - 0.5 - 0.4 = 0.1$ , so its belief degreeis $B_{1, 1}^{1} = {(H_{1}, [0.4, 0.5]), (H_{2}, [0.3, 0.4]); (H, [0.1, 0.3])}$ . Similarly, the IVIF values in three decision matrices can be transferred into the belief degrees by Equation (10).

According to the discounting method with both weight and reliability as Definition 4, take the weight and the reliability into λ_i,j = w_i/(1 + w_i - r_i,j) and we obtain the discounting factors as follows. $\begin{matrix} λ_{1, 1} = 1.00, λ_{1, 2} = 0.75, λ_{1, 3} = 0.29, \\ λ_{2, 1} = 0.71, λ_{2, 2} = 0.60, λ_{2, 3} = 1.00, \\ λ_{3, 1} = 0.56, λ_{3, 2} = 1.00, λ_{3, 3} = 0.50 . \end{matrix}$

Take λ_i,j and $B_{i, j}^{k}$ into Equation (5), the lower and upper bounds of BPA functions m_i,j,k (∀i, j, k) are obtained (see Table 1). For example, from the 4^th row and the 3-6 columns we can derive the BPA function m_1,1,1 = {(H₁, [0.40, 0.50]) , (H₂, [0.30, 0.40]) ; (H, [0.10, 0.30])}.

Table 1
Interval BPA functions for all experts

Attribute c₁ Attribute c₂ Attribute c₃

H ₁ H ₂ H ₁ H ₂ H ₁ H ₂

y ₁ [0.4, 0.5] [0.3, 0.4] [0.4, 0.6] [0.2, 0.4] [0.1, 0.3] [0.5, 0.6]

e ₁ y ₂ [0.4, 0.6] [0.2, 0.3] [0.6, 0.7] [0.1, 0.2] [0.4, 0.7] [0.1, 0.2]

y ₃ [0.3, 0.5] [0.3, 0.4] [0.5, 0.6] [0.2, 0.3] [0.5, 0.6] [0.1, 0.3]

y ₁ [0.3, 0.5] [0.2, 0.4] [0.2, 0.6] [0.3, 0.4] [0.2, 0.3] [0.3, 0.5]

e ₂ y ₂ [0.5, 0.6] [0.1, 0.3] [0.4, 0.6] [0.2, 0.3] [0.6, 0.7] [0.1, 0.2]

y ₃ [0.4, 0.5] [0.2, 0.4] [0.1, 0.4] [0.4, 0.5] [0.4, 0.6] [0.2, 0.3]

y ₁ [0.5, 0.7] [0.1, 0.2] [0.5, 0.6] [0.1, 0.2] [0.1, 0.6] [0.2, 0.4]

e ₃ y ₂ [0.3, 0.6] [0.2, 0.3] [0.4, 0.5] [0.2, 0.3] [0.4, 0.6] [0.1, 0.3]

y ₃ [0.3, 0.5] [0.2, 0.4] [0.2, 0.3] [0.5, 0.6] [0.4, 0.5] [0.1, 0.3]

		Attribute c₁	Attribute c₂	Attribute c₃
	y ₁	[0.4, 0.5]	[0.3, 0.4]	[0.4, 0.6]	[0.2, 0.4]	[0.1, 0.3]	[0.5, 0.6]
e ₁	y ₂	[0.4, 0.6]	[0.2, 0.3]	[0.6, 0.7]	[0.1, 0.2]	[0.4, 0.7]	[0.1, 0.2]
	y ₃	[0.3, 0.5]	[0.3, 0.4]	[0.5, 0.6]	[0.2, 0.3]	[0.5, 0.6]	[0.1, 0.3]
	y ₁	[0.3, 0.5]	[0.2, 0.4]	[0.2, 0.6]	[0.3, 0.4]	[0.2, 0.3]	[0.3, 0.5]
e ₂	y ₂	[0.5, 0.6]	[0.1, 0.3]	[0.4, 0.6]	[0.2, 0.3]	[0.6, 0.7]	[0.1, 0.2]
	y ₃	[0.4, 0.5]	[0.2, 0.4]	[0.1, 0.4]	[0.4, 0.5]	[0.4, 0.6]	[0.2, 0.3]
	y ₁	[0.5, 0.7]	[0.1, 0.2]	[0.5, 0.6]	[0.1, 0.2]	[0.1, 0.6]	[0.2, 0.4]
e ₃	y ₂	[0.3, 0.6]	[0.2, 0.3]	[0.4, 0.5]	[0.2, 0.3]	[0.4, 0.6]	[0.1, 0.3]
	y ₃	[0.3, 0.5]	[0.2, 0.4]	[0.2, 0.3]	[0.5, 0.6]	[0.4, 0.5]	[0.1, 0.3]

For expert e₁, the m_1,1,1, m_1,2,1, and m_1,3,1 can be transferred into below restrictions. $\begin{matrix} 0.40 \leq m_{1, 1, 1} (H_{1}) \leq 0.50, 0.30 \leq m_{1, 1, 1} (H_{2}) \leq 0.40, \\ 0.10 \leq m_{1, 1, 1} (H) \leq 0.30; 0.30 \leq m_{1, 2, 1} (H_{1}) \leq 0.45; \\ 0.15 \leq m_{1, 2, 1} (H_{2}) \leq 0.30, 0.25 \leq m_{1, 2, 1} (H) \leq 0.55, \\ 0.03 \leq m_{1, 3, 1} (H_{1}) \leq 0.09, 0.14 \leq m_{1, 3, 1} (H_{2}) \leq 0.17, \\ 0.74 \leq m_{1, 3, 1} (H) \leq 0.83 . \end{matrix}$

Take these restrictions into the nonlinear optimization model as Equation (12), the interval probability masses on all attributes can be combined and transformed into the overall interval belief degrees. When the optimal objective function is min β_1,1 (H₁), we derive ${\overset{ˇ}{β}}_{1, 1} (H_{1}) = 0.38$ ; when the optimal objective function is max β_1,1 (H₁), we derive ${\hat{β}}_{1, 1} (H_{1}) = 0.73$ ; when the optimal objective function is min β_1,1 (H₂), we derive ${\overset{ˇ}{β}}_{1, 1} (H_{2}) = 0.25$ ; when the optimal objective function is max β_1,1 (H₂), we derive ${\hat{β}}_{1, 1} (H_{2}) = 0.56$ . Similarly, the overall interval belief degrees for other experts can be derived (see Table 2).

Table 2

The optimal objective function values

	Expert e₁		Expert e₁		Expert e_e
	H ₁	H ₂	H ₁	H ₂	H ₁	H ₂
y ₁	[0.38, 0.73]	[0.25, 0.56]	[0.23, 0.69]	[0.27, 0.74]	[0.51, 0.91]	[0.08, 0.37]
y ₂	[0.58, 0.89]	[0.10, 0.31]	[0.66, 0.94]	[0.06, 0.25]	[0.45, 0.83]	[0.15, 0.43]
y ₃	[0.45, 0.77]	[0.20, 0.49]	[0.36, 0.79]	[0.20, 0.58]	[0.25, 0.57]	[0.36, 0.73]

A nonlinear optimization model based on analytical ER rule to combine the BPA functions for all experts can also be similarly established as Equation (13). The final combination assessment for each alternative can be derived by solving the model, and the optimal objective function values are as follows. $\begin{matrix} {\overset{ˇ}{β}}_{1} (H_{1}) = 0.20, {\hat{β}}_{1} (H_{1}) = 0.97; {\overset{ˇ}{β}}_{1} (H_{2}) = 0.01, \\ {\hat{β}}_{1} (H_{2}) = 0.80; {\overset{ˇ}{β}}_{2} (H_{1}) = 0.71, {\hat{β}}_{2} (H_{1}) = 1.00; \\ {\overset{ˇ}{β}}_{2} (H_{2}) = 0.00, {\hat{β}}_{2} (H_{2}) = 0.28; {\overset{ˇ}{β}}_{3} (H_{1}) = 0.16, \\ {\hat{β}}_{3} (H_{1}) = 0.96; {\overset{ˇ}{β}}_{3} (H_{2}) = 0.04, {\hat{β}}_{3} (H_{2}) = 0.84 . \end{matrix}$

The final combination assessment m_k is capable of transferred into the IVIF forms. $\begin{matrix} Z_{1} = 〈 [0.20, 0.97], [0.01, 0.80] 〉, \\ Z_{2} = 〈 [0.71, 1.00], [0.00, 0.28] 〉, \\ Z_{3} = 〈 [0.16, 0.96], [0.04, 0.84] 〉 . \end{matrix}$

Take Z₁ - Z₃ into Equation (15) to compute the lower bound and upper bound of the relative closeness coefficient C_k and we have $C_{1}^{-} = 0.20$ , $C_{1}^{+} = 0.98$ , $C_{2}^{-} = 0.72$ , $C_{2}^{+} = 1.00$ , $C_{3}^{-} = 0.16$ , $C_{3}^{+} = 0.96$ . Take above parameters into Equation (16),P = (p_kk′) _3×3 the judgement matrix for inclusion comparison probability can be derived as follows. $P = [\begin{matrix} 0.50, 0.25, 0.52 \\ 0.75, 0.50, 0.77 \\ 0.48, 0.23, 0.50 \end{matrix}]$

Take the judgement matrix P into Equation (17), we derive ranking values of alternatives, i.e., s₁ = 0.63, s₂ = 1.00, s₃ = 0.60. As a result, the alternative ranking is y₂ ≻ y₁ ≻ y₃.

5 Conclusion

The existing MAGDM methods with IVIF sets only consider weights of the experts who participated in decision making, however the reliabilities of that are not considered at all. In this paper, we propose a multiattribute group decision making method based on interval-valued intuitionistic fuzzy sets and analytically evidential reasoning methodology. Firstly, a transformation method from IVIF value to belief degree is defined based on the frame of discernment only consisting of excellent and non-excellent two hypotheses. Secondly, the BPA function computing method is introduced by discounting belief degrees with both weights and reliabilities. Thirdly, the derived BPA functions are combined by the analytical ER rule for two times, the one is to combine BPA functions on all attributes with respect to a specified expert, and the other is to combine the combined BPA functions for all experts with respect to a specified alternative. Finally, the relative coefficient is employed to rank the alternatives with IVIF sets. An example is proposed to illustrate the decision making process by the proposed method. The innovation point of this paper mainly is the intuitionistic fuzzy sets and analytic ER methodology are intersected to solve the MAGDM problems, in which the weights and reliabilities of experts are both considered in the decision process.

Footnotes

Acknowledgments

This research was supported by National Natural Science Foundation of China (NSFC) under Grant No. 71462022 and the Fundamental Research Funds for the Central Universities under Grant No. 201762026.

References

Atanassov

and Gargov

, Interval valued intuitionistic fuzzy sets, Fuzzy Sets & Systems31(3) (1989), 343–349.

Boran

F.E.

, Gen

, Kurt

and Akay

, A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method, Expert Systems with Applications36(8) (2009), 11363–11368.

, Linear programming method for MADM with intervalvalued intuitionistic fuzzy sets, Expert Systems with Applications37(8) (2010), 5939–5945.

Wang

and Li

K.W.

, An interval-valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives, Expert Systems with Applications39(18) (2012), 13509–13516.

Meng

and Tang

, Interval-valued intuitionistic fuzzy multiattribute group decision making based on cross entropy measure and choquet integral, International Journal of Intelligent Systems28(12) (2013), 1172–1195.

Huang

, Wei

, Li

and Zhuang

, Using a rough set model to extract rules in dominance-based intervalvalued intuitionistic fuzzy information systems, Information Sciences221(2) (2013), 215–229.

, Luo

, Li

and Ren

, A new similarity measure of interval-valued intuitionistic fuzzy sets considering its hesitancy degree and applications in expert systems, Mathematical Problems in Engineering2 (2014), 1–16.

Jiang

and Wang

, Multiattribute group decision making with unknown decision expert weights information in the framework of interval intuitionistic trapezoidal fuzzy numbers, Mathematical Problems in Engineering3 (2014), 1–7.

, Hydrogen production technologies evaluation based on interval-valued intuitionistic fuzzy multiattribute decision making method, Journal of Applied Mathematics4 (2014), 1–10.

10.

Wan

and Li

, Atanassov’s intuitionistic fuzzy programming method for heterogeneous multiattribute group decision making with atanassov’s intuitionistic fuzzy truth degrees, IEEE Transactions on Fuzzy Systems22(2) (2014), 300–312.

11.

, Wan

and Xie

, A selection method based on MAGDM with interval-valued intuitionistic fuzzy sets, Mathematical Problems in Engineering2015 (2015), 1–13.

12.

Yang

and Xu

, Evidential reasoning rule for evidence combination, Artificial Intelligence205 (2013), 1–29.

13.

Wang

, Yang

, Xu

and Chin

, The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees, European Journal of Operational Research175(1) (2006), 35–66.

14.

Dong

and Cooper

, An orders-of-magnitude AHP supply chain risk assessment framework, International Journal of Production Economics182 (2016), 144–156.

15.

Wan

, Xu

and Dong

, Supplier selection using ANP and ELECTRE II in interval 2-tuple linguistic environment, Information Sciences386 (2017), 19–38.

16.

Amat

and Busquier

, After notes on Chebyshev’s iterative method, Applied Mathematics and Nonlinear Sciences2(1) (2017), 1–12.

17.

, You

and Du Busquier

, LAV path planning by enhanced fireworks algorithm on prior knowledge, Applied Mathematics and Nonlinear Sciences1(1) (2016), 65–78.