Choquet-based multi-criteria decision making with objective and subjective information

Abstract

In this paper, the multi-criteria decision making in which criteria are assumed to be dependent with each other is investigated. Choquet integral is employed to aggregate the criteria evaluations of alternatives to reflect the interactions of the criteria. Objective and subjective information are integrated to derive the interactions of the criteria. The objective information of alternatives is expressed by decision matrix, whereas the subjective information is expressed by utilities or interval preference relations. Models are given to minimize the objective and subjective information. The proposed method can be considered as an extension of the existing methods, and can obtain smaller deviations and better overall evaluations of alternatives.

Keywords

Multi-criteria decision making objective information subjective information Choquet integral

1 Introduction

In decision making problems with multiple criteria, the information about alternatives can be expressed by using the decision matrix containing the performance of alternatives under each criterion, which is called subjective information, or can be expressed by using utilities or preference relations about alternatives or criteria, which is called objective information. Once the weight vector of the criteria is known, it is easy to obtain the overall evaluations of alternatives by employing some aggregation techniques based on the objective information. In real-life multi-criteria decision making, it is always hard to obtain the precise information about the weight vector of criteria [1, 21].

Determining the values of parameters of the aggregation function is a key problem in decision making [13]. In multi-criteria decision making, many methods [4] have been developed to elicit the values of parameters integrating objective and subjective information. Ma et al. [14] combined the subjective information about the criteria (the multiplicative preference relations about the criteria) with the objective information and gave an integrated approach to determine the criteria weight vector. Wang and Parkan [20] combined the subjective information about alternatives (the fuzzy preference relations about the alternatives) with the objective information and gave an integrated approach to determine criteria weight vector. Ma et al. [15] proposed a method to derive the ranking of alternatives in which the preference information about alternatives provided by decision makers can be represented by different formats. Xu [25] extended it into the interval environments. Xu and Chen [26] developed some linear models to deal with the situation where the decision makers have their preferences on alternatives.

The above mentioned methods assume that the criteria are independent, but sometimes there exist interactions among criteria [3] (Angilella et al., 2012, 2013). For example, supplier selection is an important issue. Product quality, offering price, delivery time, and service quality are four key criteria for supplier evaluation. From one side, delivery time and service quality are redundant criteria, because, in general, a supplier that has good service will deliver on time. Therefore, even if these two criteria can be very important for a supplier selection, their comprehensive importance is smaller than the sum of the importance of the two criteria. From the other side, the two criteria, product quality and offering price lead to a positive interaction because a supplier that supplies high quality and offers a low price is very well appreciated. Therefore, the comprehensive importance of quality and price should be greater than the sum of their importance.

Choquet integral [3 , 23] is an useful way to deal with such situation, and the application of Choquet integral in multi-criteria decision aid has attracted many attentions in recent years [8 , 22] (Angilella et al., 2012, 2013). In this paper, Choquet integral is employed to aggregate the objective information of alternatives. By combing the objective and subjective information, several models are developed to derive the interactions of the criteria. The remainder of the paper is constructed as: Section 2 introduces some basic concepts; Section 3 presents some linear models combining objective information and each kind of subjective information. Section 4 extends the methods proposed in Section 3 to the group decision making methods considering different kinds of subjective information.

2 Basic concepts

Let X be a finite set, a fuzzy measure μ on X is a function μ: P (X) → [0, 1], satisfying the axioms [18]: (i) μ (φ) =0; (ii) A ⊂ B ⊂ X implies μ (A) ≤ μ (B). We will assume here μ (X) =1 as usual, although this is not necessary in general.

The Möbius transform of μ is function on X defined as w_A = ∑_B⊂A (-1) ^|A∖B|μ (B), ∀A ⊂ X. In terms of Möbius representation [7], (i) and (ii) can be represented by: (iii) w_φ = 0; (iv) ∀i ∈ X, ∀S ⊆ X ∖ {i}, ∑_T⊆Sw_T∪{i} ≥ 0. Then, μ (X) =1 can be expressed as ∑_T⊆Xw_T = 1.

In a multi-criteria decision making, let X = (x₁, x₂, ⋯ , x_m) represent the set of alternatives and G = (G₁, G₂, ⋯ , G_n) the set of criteria, a_ij the performance of the alternative x_i under criteria G_j, then a decision matrix A = (a_ij) _m×n can be constructed. The criteria may be benefit or cost, to unify them, some techniques [12] can be used to normalize A = (a_ij) _m×n into R = (r_ij) _m×n.

Based on Choquet integral [3], the overall evaluation of alternative x_i can be expressed by [7]: $z_{i} (w) = \sum_{T \subseteq N} w_{T} \underset{j \in T}{\land} r_{ij}, i = 1, 2, \dots, m$ (1)where w_φ = 0, ∑_T⊆Nw_T = 1, ∀i ∈ N, ∀S ⊆ N ∖ {i}, ∑_T⊆Sw_T∪{i} ≥ 0, N = {1, 2, ⋯ , n}.

In expression, w is a 2ⁿ - 1 dimensional vector. w_T measures the interactions of the set of criteria c_j, j ∈ T (Marichal, 2004). If w_T > 0, then the set of criteria c_j, j ∈ T has positive interactions; if w_T < 0, then the set of criteria c_j, j ∈ T has negative interactions; otherwise there ae no interactions.

If w_T = 0, t ≥ 2, where t is the cardinality of the coalition T and t = |T|, then Equation reduces to the weighted averaging (WA) operator [11]: $z_{i} (w) = \sum_{j = 1}^{n} w_{j} r_{ij}, i = 1, 2, \dots, m$

Usually, it is very hard to obtain the precise information about the interaction of criteria in practical problems. Let H be the set of the known information about the interactions of the criteria, which can be expressed by using the following forms [5, 16]:

Weak ranking: {w_R ≥ w_S} , R ≠ S.

Strict ranking: {w_R - w_S ≥ ɛ_R} , ɛ_T > 0, R ≠ S.

Ranking with multiples: {w_R ≥ a_RSw_S} , 0 ≤ a_R ≤ 1, R ≠ S.

Ranking with multiples: {b_R ≤ w_R ≤ b_R + ɛ_R} , 0 ≤ b_R ≤ b_R + ɛ_R ≤ 1.

Ranking of differences: {w_R - w_S ≥ w_U - w_J} , R ≠ S ≠ U ≠ J.

Except for the criteria evaluations of alternatives, the decision makers may have their own preference information about these alternatives, which can be expressed by the following forms:

Interval utility values [19]: $\tilde{U} = {{\tilde{u}}_{1}, {\tilde{u}}_{2}, \dots, {\tilde{u}}_{m}}$ , where ${\tilde{u}}_{i} = [u_{i}^{-}, u_{i}^{+}]$ , $0 \leq u_{i}^{-} \leq u_{i}^{+} \leq 1$ . The bigger the value of ${\tilde{u}}_{i}$ is, the better the alternative x_i is.

Interval fuzzy preference relation [24]: $\tilde{P} = ({\tilde{p}}_{ij})_{m \times m}$ , where ${\tilde{p}}_{ij}$ indicates the interval degree that alternative x_i is priority to x_j and satisfies the following conditions: $\begin{matrix} {\tilde{p}}_{ij} = [p_{ij}^{-}, p_{ij}^{+}], p_{ij}^{+} \geq p_{ij}^{-} \geq 0, \\ p_{ij}^{-} + p_{ji}^{+} = p_{ji}^{-} + p_{ij}^{+} = 1, \\ p_{ii}^{-} = p_{ii}^{+} = 0.5, i, j = 1, 2, \dots, m \end{matrix}$

Interval multiplicative preference relation [17]: $\tilde{B} = ({\tilde{b}}_{ij})_{m \times m}$ , where ${\tilde{b}}_{ij}$ indicates the interval degree that alternative x_i is priority to x_j and satisfies the following conditions: $\begin{matrix} {\tilde{b}}_{ij} = [b_{ij}^{-}, b_{ij}^{+}], b_{ij}^{+} \geq b_{ij}^{-} \geq 0, b_{ij}^{-} b_{ji}^{+} = b_{ji}^{-} b_{ij}^{+} = 1, \\ b_{ii}^{-} = b_{ii}^{+} = 1, i, j = 1, 2, \dots, m \end{matrix}$

3 Models based on decision matrix and each kind of subjective information

To derive the weights of the criteria and the interactions of them, the objective information and subjective information should be combined. In this section, some models are given to integrate the decision matrix with each kind of subjective information.

1) Models based on decision matrix and interval utility

In this case, if the decision matrix R = (r_ij) _m×n is consistent with the utility values $\tilde{u} = [u_{i}^{-}, u_{i}^{+}]$ , i = 1, 2, ⋯ , m, then the overall evaluation z_i (w) of alternative x_i should belong to the interval $[u_{i}^{-}, u_{i}^{+}]$ , that is $u_{i}^{-} \leq \sum_{T \subseteq N} w_{T} \underset{j \in T}{\land} r_{ij} \leq u_{i}^{+}, j = 1, 2, \dots, n$

In practical problems, the above condition hardly holds, then the deviation variables $e_{i}^{-}$ and $e_{i}^{+}$ , i = 1, 2, ⋯ , n can be introduced to make sure that: $u_{i}^{-} - e_{i}^{-} \leq \sum_{T \subseteq N} w_{T} \underset{j \in T}{\land} r_{ij} \leq u_{i}^{+} + e_{i}^{+}, i = 1, 2, \dots, m$

Then the smaller the values of $e_{i}^{-}$ and $e_{i}^{+}$ , i = 1, 2, ⋯ , m, the better. To minimize the deviation variables, the following model is given: $\begin{matrix} (M 1) Min J_{1}^{*} = min \sum_{i = 1}^{n} (e_{i}^{-} + e_{i}^{+}) \\ s . t . u_{i}^{-} - e_{i}^{-} \leq \sum_{T \subseteq N} w_{T} \underset{j \in T}{\land} r_{ij} \leq u_{i}^{+} + e_{i}^{+}, \\ i = 1, 2, \dots, m \\ w_{φ} = 0, \sum_{T \subseteq N} w_{T} = 1, \forall i \in N, \forall S \subseteq N ∖ {i}, \\ \sum_{T \subseteq S} w_{T \cup {i}} \geq 0, w \in H \end{matrix}$

Suppose ${\dot{e}}_{i}^{-}$ and ${\dot{e}}_{i}^{+}$ , i = 1, 2, ⋯ , m are the optimal values in (M1). If ${\dot{e}}_{i}^{-} = {\dot{e}}_{i}^{+} = 0$ , i = 1, 2, ⋯ , m, then the information in decision matrix is consistent with the interval utilities; otherwise they are inconsistent. Based on the deviation variables ${\dot{e}}_{i}^{-}$ and ${\dot{e}}_{i}^{+}$ , i = 1, 2, ⋯ , m, we can roughly estimate criteria weight vector {w_T}, which satisfies $u_{i}^{-} - {\dot{e}}_{i}^{-} \leq \sum_{T \subseteq N} w_{T} \underset{j \in T}{\land} r_{ij} \leq u_{i}^{+} + {\dot{e}}_{i}^{+}, i = 1, 2, \dots, m$

Actually, there may be more than one criteria weight vector {w_T} satisfying the above conditions, which can be derived by the following model: $\begin{matrix} (M 2) Min / Max w_{T} \\ s . t . u_{i}^{-} - {\dot{e}}_{i}^{-} \leq \sum_{T \subseteq N} w_{T} \underset{j \in T}{\land} r_{ij} \leq u_{i}^{+} + {\dot{e}}_{i}^{+}, \\ i = 1, 2, \dots, m \\ w_{φ} = 0, \sum_{T \subseteq N} w_{T} = 1, \forall i \in N, \forall S \subseteq N ∖ {i}, \\ \sum_{T \subseteq S} w_{T \cup {i}} \geq 0, w \in H \end{matrix}$

2) Models based on decision matrix and interval fuzzy preference relation

If there exists a vector ω = (ω₁, ω₂, ⋯ , ω_m) ^T,ω_i ≥ 0, i = 1, 2, ⋯ , m, $\sum_{i = 1}^{m} ω_{i} = 1$ , such that $p_{ij}^{-} \leq 0.5 (ω_{i} - ω_{j} + 1) \leq p_{ij}^{+}, i, j = 1, 2, \dots, m$ (2)then $\tilde{P}$ is consistent [27].

Based on Equation(2), z_i (w) can be transformed into a consistent fuzzy preference relation, that is: $p_{ij} = 0.5 (z_{i} (w) - z_{j} (w) + 1)$

which can be written as: $p_{ij} = 0.5 (\sum_{T \subseteq N} w_{T} (\underset{k \in T}{\land} r_{ik} - \underset{k \in T}{\land} r_{jk}) + 1)$

If the decision matrix is consistent with the interval fuzzy preference relation, then p_ij should be in the interval $[p_{ij}^{-}, p_{ij}^{+}]$ , that is $\begin{matrix} p_{ij}^{-} & \leq 0.5 (\sum_{T \subseteq N} w_{T} (\underset{k \in T}{\land} r_{ik} - \underset{k \in T}{\land} r_{jk}) + 1) \\ \leq p_{ij}^{+}, i, j = 1, 2, \dots, m \end{matrix}$ Equivalently, $\begin{matrix} p_{ij}^{-} & \leq 0.5 (\sum_{T \subseteq N} w_{T} (\underset{k \in T}{\land} r_{ik} - \underset{k \in T}{\land} r_{jk}) + 1) \\ \leq p_{ij}^{+}, i = 1, 2, \dots, m - 1, j = i + 1, \dots, m \end{matrix}$

If they are not consistent, then the deviation variables $e_{ij}^{-}$ and $e_{ij}^{+}$ are introduced, which make that: $\begin{matrix} p_{ij}^{-} - e_{ij}^{-} & \leq 0.5 (\sum_{T \subseteq N} w_{T} (\underset{k \in T}{\land} r_{ik} - \underset{k \in T}{\land} r_{jk}) + 1) \\ \leq p_{ij}^{+} + e_{ij}^{+}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m \end{matrix}$

To derive the minimization of $e_{ij}^{-}$ and $e_{ij}^{+}$ , the following model is given: $\begin{matrix} (M 3) J_{2}^{*} = min \sum_{i = 1}^{n} (e_{i}^{-} + e_{i}^{+}) \\ s . t . p_{ij}^{-} - e_{ij}^{-} \leq 0.5 (\sum_{T \subseteq N} w_{T} (\underset{k \in T}{\land} r_{ik} - \underset{k \in T}{\land} r_{jk}) + 1) \\ [2 pt] & \leq p_{ij}^{+} + e_{ij}^{+}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m \\ w_{φ} = 0, \sum_{T \subseteq N} w_{T} = 1, \forall i \in N, \forall S \subseteq N ∖ {i}, \\ \sum_{T \subseteq S} w_{T \cup {i}} \geq 0, w \in H . \end{matrix}$

Suppose ${\dot{e}}_{ij}^{-}$ and ${\dot{e}}_{ij}^{+}$ , i = 1, 2, ⋯ , m - 1, j = i + 1, ⋯ , m are the optimal values derived from the above model. If ${\dot{e}}_{ij}^{-} = {\dot{e}}_{ij}^{+} = 0$ , i = 1, 2, ⋯ , m - 1, j = i + 1, ⋯ , m, then the decision matrix is consistent with the interval fuzzy preference relation; otherwise they are inconsistent. Based on the deviation variables, the following model is given to derive the variation of interactions of criteria: $\begin{matrix} (M 4) Min / Max w_{T} \\ s . t . p_{ij}^{-} - {\dot{e}}_{ij}^{-} \leq 0.5 (\sum_{T \subseteq N} w_{T} (\underset{k \in T}{\land} r_{ik} - \underset{k \in T}{\land} r_{jk}) + 1) \\ \leq p_{ij}^{+} + {\dot{e}}_{ij}^{+}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m \\ w_{φ} = 0, \sum_{T \subseteq N} w_{T} = 1, \forall i \in N, \forall S \subseteq N ∖ {i}, \\ \sum_{T \subseteq S} w_{T \cup {i}} \geq 0, w \in H \end{matrix}$

3) Models based on decision matrix and interval multiplicative preference relation

An interval multiplicative preference $\tilde{B}$ is consistent [17], if there exists a vector ω = (ω₁, ω₂, ⋯ , ω_m) ^T, ω_i ≥ 0, i = 1, 2, ⋯ , m, $\sum_{i = 1}^{m} ω_{i} = 1$ , such that $b_{ij}^{-} \leq \frac{ω_{i}}{ω_{j}} \leq b_{ij}^{+}, i, j = 1, 2, \dots, m$ (3)

Based on Equation(3), z_i (w) can be transformed into a consistent multiplicative preference relation, then $b_{ij} = \frac{z_{i} (w)}{z_{j} (w)}, i, j = 1, 2, \dots, m$ or $b_{ij} = \frac{\sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{ik}}{\sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk}}, i, j = 1, 2, \dots, m$

If R is consistent with the interval multiplicative preference relation $\tilde{B}$ , then $b_{ij}^{-} \leq \frac{\sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{ik}}{\sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk}} \leq b_{ij}^{+}, i, j = 1, 2, \dots, n$

Namely, $\begin{matrix} b_{ij}^{-} \leq \frac{\sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{ik}}{\sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk}} \leq b_{ij}^{+}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m \end{matrix}$ or can be written as: $\begin{matrix} b_{ij}^{-} \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} \leq \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{ik} \leq b_{ij}^{+} \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk}, \\ i = 1, 2, \dots, m - 1, j = i + 1, \dots, m \end{matrix}$

If R is not consistent with $\tilde{B}$ , then the deviation variables $e_{ij}^{-}$ and $e_{ij}^{+}$ are introduced: $\begin{matrix} b_{ij}^{-} \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} - e_{ij}^{-} \leq \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{ik} \leq b_{ij}^{+} \\ \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} + e_{ij}^{+}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m \end{matrix}$

By solving the following model, the optimal deviation variables ${\dot{e}}_{ij}^{-}$ and ${\dot{e}}_{ij}^{+}$ can be derived: $\begin{matrix} (M 5) J_{3}^{*} = min \sum_{i = 1}^{n} (e_{i}^{-} + e_{i}^{+}) \\ s . t . b_{ij}^{-} \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} - e_{ij}^{-} \leq \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{ik} \leq b_{ij}^{+} \\ \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} + e_{ij}^{+}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m \\ w_{φ} = 0, \sum_{T \subseteq N} w_{T} = 1, \forall i \in N, \forall S \subseteq N ∖ {i}, \\ \sum_{T \subseteq S} w_{T \cup {i}} \geq 0, w \in H \end{matrix}$

Suppose ${\dot{e}}_{ij}^{-}$ and ${\dot{e}}_{ij}^{+}$ , i = 1, 2, ⋯ , m–1, j = i + 1, ⋯ , m are the optimal values derived from (M5). If ${\dot{e}}_{ij}^{-} = {\dot{e}}_{ij}^{+} = 0$ , i = 1, 2, ⋯ , m - 1, j = i + 1, ⋯ , m, then the decision matrix R is consistent with the interval multiplicative preference relation $\tilde{B}$ , otherwise it is not consistent. Then the interval variation of w_T can be calculated by the following model: $\begin{matrix} (M 6) Min / Max w_{T} \\ s . t . b_{ij}^{-} \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} - {\dot{e}}_{ij}^{-} \leq \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{ik} \leq b_{ij}^{+} \\ \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} + {\dot{e}}_{ij}^{+}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m \\ w_{φ} = 0, \sum_{T \subseteq N} w_{T} = 1, \forall i \in N, \forall S \subseteq N ∖ {i}, \\ \sum_{T \subseteq S} w_{T \cup {i}} \geq 0, w \in H \end{matrix}$

4 Models based on decision matrix and several kinds of subjective information

Let x = {x₁, x₂, ⋯ , x_m} be defined as before, D = {d₁, d₂, ⋯ , d_t} be a finite set of decision makers, and λ = {λ₁, λ₂, ⋯ , λ_t} be the weight vector of decision makers, where λ_k ≥ 0, k = 1, 2, ⋯ , t, $\sum_{k = 1}^{t} λ_{k} = 1$ .

In group decision making problems, several decision makers provide their preference information about alternatives, then

1) The decision makers d_k (k = 1, 2, ⋯ , t₁) provide the interval utility values ${\tilde{u}}_{i}^{(k)} = [u_{i}^{- (k)}, u_{i}^{+ (k)}]$ , i = 1, 2, ⋯ , m, k = 1, 2, ⋯ , t₁ to express their preference information about m alternatives x_i (i = 1, 2, ⋯ , m).

2) The decision makers d_k (k = t₁ + 1, t₁ + 2, ⋯ , t₂) provide their preference information on alternatives x_j (j = 1, 2, ⋯ , n) by means of interval fuzzy preference relations ${\tilde{P}}^{(k)} = ({\tilde{p}}_{ij}^{(k)})_{n \times n}$ , where ${\tilde{p}}_{ij}^{(k)} = [p_{ij}^{- (k)}, p_{ij}^{+ (k)}]$ , i, j = 1, 2, ⋯ , m, k =t₁ + 1, t₁ + 2, ⋯ , t₂.

3) The decision makers d_k (k = t₂ + 1, t₁ + 2, ⋯ , t) provide their preference information on alternatives x_j (i = 1, 2, ⋯ , n) by means of interval fuzzy preference relations ${\tilde{B}}^{(k)} = ({\tilde{b}}_{ij}^{(k)})_{n \times n}$ , where ${\tilde{b}}_{ij}^{(k)} = [b_{ij}^{- (k)}, b_{ij}^{+ (k)}]$ , i, j = 1, 2, ⋯ , m, k =t₂ + 1, t₂ + 2, ⋯ , t.

To derive the minimum deviations between the objective and subjective information, the following model is given: $\begin{matrix} (M 7) J_{4}^{*} = min (J_{4}^{(1)} + J_{4}^{(2)} + J_{4}^{(3)}) \\ s . t . u_{i}^{- (l)} - e_{i}^{- (l)} \leq \sum_{T \subseteq N} w_{T} \underset{j \in T}{\land} r_{ij} \leq u_{i}^{+ (l)} + e_{i}^{+ (l)}, \\ i = 1, 2, \dots, m, l = 1, 2, \dots, t_{1} \\ p_{ij}^{- (l)} - e_{ij}^{- (l)} \leq 0.5 (\sum_{T \subseteq N} w_{T} (\underset{k \in T}{\land} r_{ik} - \underset{k \in T}{\land} r_{jk}) + 1) \\ \leq p_{ij}^{+ (l)} + e_{ij}^{+ (l)}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m, l = t_{1} + 1, \dots t_{2}, \\ b_{ij}^{- (l)} \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} - e_{ij}^{- (l)} \leq \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{ik} \leq b_{ij}^{+ (l)} \\ \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} + e_{ij}^{+ (l)}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m, l = t_{2} + 1, \dots t \\ w_{φ} = 0, \sum_{T \subseteq N} w_{T} = 1, \forall i \in N, \forall S \subseteq N ∖ {i}, \\ \sum_{T \subseteq S} w_{T \cup {i}} \geq 0, w \in H \end{matrix}$ where $\begin{matrix} J_{4}^{(1)} & = \sum_{l = 1}^{t_{1}} \sum_{i = 1}^{n} λ_{l} (e_{i}^{- (l)} + e_{i}^{+ (l)}), J_{4}^{(2)} \\ = \sum_{l = t_{1} + 1}^{t_{2}} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} λ_{l} (e_{ij}^{- (l)} + e_{ij}^{+ (l)}), J_{4}^{(3)} \\ = \sum_{l = t_{2} + 1}^{t_{3}} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} λ_{l} (e_{ij}^{- (l)} + e_{ij}^{+ (l)}) \end{matrix}$

Solving (M7), the optimal values of deviation variables can be obtained. If the deviations variables are all equivalent to zero, then the objective information in the decision matrix is consistent with the subjective information provided by the decision makers; otherwise they are not consistent. Based on the optimal deviation variables, the variation of the interaction of criteria can be derived. $\begin{matrix} (M 8) Min / Max w_{T} \\ s . t . u_{i}^{- (l)} - {\dot{e}}_{i}^{- (l)} \leq \sum_{T \subseteq N} w_{T} \underset{j \in T}{\land} r_{ij} \leq u_{i}^{+ (l)} + {\dot{e}}_{i}^{+ (l)}, \\ i = 1, 2, \dots, m, l = 1, 2, \dots, t_{1} \\ p_{ij}^{- (l)} - {\dot{e}}_{ij}^{- (l)} \leq 0.5 (\sum_{T \subseteq N} w_{T} (\underset{k \in T}{\land} r_{ik} - \underset{k \in T}{\land} r_{jk}) + 1) \\ \leq p_{ij}^{+ (l)} + {\dot{e}}_{ij}^{+ (l)}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m, l = t_{1} + 1, \dots t_{2} \\ b_{ij}^{- (l)} \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} - {\dot{e}}_{ij}^{- (l)} \leq \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{ik} \leq b_{ij}^{+ (l)} \\ \sum_{T \subseteq N} w_{T} \underset{k \in T}{\land} r_{jk} + {\dot{e}}_{ij}^{+ (l)}, i = 1, 2, \dots, m - 1, \\ j = i + 1, \dots, m, l = t_{2} + 1, \dots t \\ w_{φ} = 0, \sum_{T \subseteq N} w_{T} = 1, \forall i \in N, \forall S \subseteq N ∖ {i}, \\ \sum_{T \subseteq S} w_{T \cup {i}} \geq 0, w \in H \end{matrix}$

Based on the interaction weight intervals $[w_{T}^{-}, w_{T}^{+}]$ , T ⊆ N derived from (M8), the following model is established to derive the optimal interaction weight vector which maximizes the overall evaluations of alternatives: $\begin{matrix} (M 9) Max \sum_{i = 1}^{n} z_{i} (w) = \sum_{i = 1}^{n} \sum_{T \subseteq N} w_{T} \underset{j \in T}{\land} r_{ij} \\ s . t . w_{φ} = 0, \sum_{T \subseteq N} w_{T} = 1, \forall i \in N, \forall S \subseteq N ∖ {i}, \\ \sum_{T \subseteq S} w_{T \cup {i}} \geq 0 w_{T} \in [w_{T}^{-}, w_{T}^{+}], T \subseteq N \end{matrix}$

Based on the above analyses, an approach can be developed:

Step 1. Solving (M7) to calculate the deviation variables, based on which the interval interaction of criteria can be obtained by solving (M8).

Step 2. Determine the optimal interaction of criteria vector {w^*} by solving (M9), and get the overall values z_i (w^*) of the alternatives and the ranking of alternatives.

Example 1. [26]. Considering four alternatives x_j (j = 1, 2, 3, 4) with four criteria G₁, G₂, G₃ and G₄. The normalized decision matrix R = (r_ij) _4×4 is given in Table 1.

Three decision makers d_k (k = 1, 2, 3) with the weight vector λ = (1/ 3, 1/ 3, 1/ 3) ^T provide their information about criteria weights given as: $\begin{matrix} H = {0.15 \leq w_{1} \leq 0.3, 0.1 \leq w_{2} \leq 0.2, w_{3} \geq 0.2, \\ w_{4} \leq 0.4, w_{3} - w_{2} \geq 0.1} \end{matrix}$

And their preference information about alternatives: $\begin{matrix} \begin{matrix} d_{1} : \tilde{U} = {, [0.6, 0.7], [0.7, 0.9], [0.5, 0.7]} \\ d_{2} : \tilde{P} = (\begin{matrix} [0.5, 0.5] & [0.6, 0.7] & [0.4, 0.6] & [0.6, 0.9] \\ [0.3, 0.4] & [0.5, 0.5] & [0.3, 0.5] & [0.5, 0.6] \\ [0.4, 0.6] & [0.5, 0.7] & [0.5, 0.5] & [0.7, 0.9] \\ [0.1, 0.4] & [0.4, 0.5] & [0.1, 0.3] & [0.5, 0.5] \end{matrix}) \\ d_{3} : \tilde{B} = (\begin{matrix} [1, 1] & [3, 5] & [1 / 3, 3] & [7, 9] \\ [1 / 5, 1 / 3] & [1, 1] & [1 / 3, 1] & [1 / 3, 3] \\ [1 / 3, 3] & [1, 3] & [1, 1] & [7, 9] \\ [1 / 9, 1 / 7] & [1 / 3, 1] & [1 / 9, 1 / 7] & [1, 1] \end{matrix}) \end{matrix} \end{matrix}$

The following steps are given to derive the ranking of alternatives:

Step 1. Solving (M7), the optimal objective function value J^* = 5.5309, and the optimal deviation values can be obtained as follows: $\begin{matrix} e_{1}^{- (1)} & = e_{1}^{+ (1)} = 0, e_{2}^{- (1)} = 0.136 e_{2}^{+ (1)} = 0, \\ e_{3}^{- (1)} & = e_{3}^{+ (1)} = 0, e_{4}^{- (1)} = 0.0306, e_{4}^{+ (1)} = 0 \\ e_{12}^{- (2)} & = e_{12}^{+ (2)} = 0, e_{13}^{- (2)} = e_{13}^{+ (2)} = 0, \\ e_{14}^{- (2)} & = e_{14}^{+ (2)} = 0, e_{23}^{- (2)} = 0.0180 \\ e_{23}^{+ (2)} & = 0, e_{24}^{- (2)} = 0.0027, e_{24}^{+ (2)} = 0, e_{34}^{- (2)} = e_{34}^{+ (2)} = 0 \\ e_{12}^{- (3)} & = 0.532 e_{12}^{+ (3)} = 0, e_{13}^{- (3)} = e_{13}^{+ (3)} = 0, \\ e_{14}^{- (3)} & = 2.4258, e_{14}^{+ (3)} = 0 \\ e_{23}^{- (3)} & = e_{23}^{+ (3)} = 0, e_{24}^{- (3)} = e_{24}^{+ (3)} = 0, e_{34}^{- (3)} = 2.3858, \\ e_{34}^{+ (3)} & = 0 \end{matrix}$

It is worth noting that the optimal objective function value obtained by using Xu and Chen’s method (2008a) is J^* = 6.969, which is bigger than the one obtained by (M7).

By solving (M8), the interval interaction of criteria can be derived: $\begin{matrix} {\tilde{w}}_{1} = [0.2, 0.2], {\tilde{w}}_{2} = [0.1, 0.1], {\tilde{w}}_{3} = [0.2, 0.2], \\ {\tilde{w}}_{4} = [0.0, 0.4], {\tilde{w}}_{12} = [- 0.1, - 0.1], {\tilde{w}}_{13} = [- 0.2, - 0.2] \\ {\tilde{w}}_{14} = [0.0474, 0.4593], {\tilde{w}}_{23} = [- 0.1, - 0.1], \\ {\tilde{w}}_{24} = [- 0.1, 0.9], {\tilde{w}}_{34} = [- 0.2, 0.8], {\tilde{w}}_{123} = [0.1, 0.1] \\ {\tilde{w}}_{124} = [- 0.5593, 0.4526], {\tilde{w}}_{134} = [- 0.4593, 0.5526], \\ {\tilde{w}}_{234} = [- 0.9, 0.9], {\tilde{w}}_{1234} = [- 1.2467, 0.5593] \end{matrix}$

Step 2. By (M9), the optimal interaction of criteria can be obtained: $\begin{matrix} w^{*} & = (0.2, 0.1, 0.2, 0.4, - 0.1, - 0.2, 0.4, \\ - 0.1, 0.5, 0.4, 0.1, - 0.5, - 0.4, - 0.5, 0.5)^{T} \end{matrix}$ and the overall evaluations of alternatives: $\begin{matrix} z_{1} (w^{*}) & = 0.86, z_{2} (w^{*}) = 0.464, z_{3} (w^{*}) = 0.9952, \\ z_{4} (w^{*}) & = 0.4694 \end{matrix}$ which derives the ranking of alternatives: $x_{3} ≻ x_{1} ≻ x_{4} ≻ x_{2}$

If the deviation values calculated by using Xu and Chen’s method (2008a) are used, then $\begin{matrix} e_{1}^{- (1)} = e_{1}^{+ (1)} = 0, e_{2}^{- (1)} = e_{2}^{+ (1)} = 0, \\ e_{3}^{- (1)} = e_{3}^{+ (1)} = 0, e_{3}^{+ (1)} = 0.011, e_{4}^{- (1)} = e_{4}^{+ (1)} = 0 \\ e_{12}^{- (2)} = e_{12}^{+ (2)} = 0, e_{13}^{- (2)} = e_{13}^{+ (2)} = 0, \\ e_{14}^{- (2)} = e_{14}^{+ (2)} = 0, e_{23}^{- (2)} = e_{23}^{+ (2)} = 0 \\ e_{24}^{- (2)} = e_{24}^{+ (2)} = 0, e_{34}^{- (2)} = 0.024, e_{34}^{+ (2)} = 0 \\ e_{12}^{- (3)} = 3.020, e_{12}^{- (3)} = e_{12}^{+ (3)} = 0, e_{13}^{- (3)} = e_{13}^{+ (3)} = 0, \\ e_{14}^{- (3)} = e_{14}^{+ (3)} = 0 \\ e_{23}^{- (3)} = e_{23}^{+ (3)} = 0, e_{24}^{- (3)} = e_{24}^{+ (3)} = 0, e_{34}^{- (3)} = 2.994, \\ e_{34}^{+ (3)} = 0 \end{matrix}$

By (M8), the intervals of interactive of criteria are obtained as follows: $\begin{matrix} {\tilde{w}}_{1} = [0.2355, 0.3], {\tilde{w}}_{2} = [0.1, 0.2], {\tilde{w}}_{3} = [0.2, 0.4037], \\ {\tilde{w}}_{4} = [0.0, 0.4], {\tilde{w}}_{12} = [- 0.2, 0.2628] \\ {\tilde{w}}_{13} = [- 0.3, 0.1628], {\tilde{w}}_{14} = [- 0.1039, 0.4842], \\ {\tilde{w}}_{23} = [- 0.2, 0.1037], {\tilde{w}}_{24} = [- 0.2, 0.9] \\ {\tilde{w}}_{34} = [- 0.4, 0.8], {\tilde{w}}_{123} = [- 0.3610, 0.3340], \\ {\tilde{w}}_{124} = [- 0.9826, 0.5609], {\tilde{w}}_{134} = [- 0.8826, 0.7039] \\ {\tilde{w}}_{234} = [- 1.1037, 0.8522], {\tilde{w}}_{1234} = [- 1.3501, 1.3437] \end{matrix}$

By (M9), the optimal interaction of criteria is derived: $\begin{matrix} w^{*} & = (0.3, 0.2, 0.4037, 0.4, 0.1908, 0.1628, 0.3, \\ 0.1037, - 0.2, - 0.4, - 0.3610, - 0.1908, \\ - 0.1665, 0.4926, - 0.2353)^{T} \end{matrix}$

The overall values of alternatives are $\begin{matrix} z_{1} (w^{*}) = 0.8990, z_{2} (w^{*}) = 0.8724, \\ z_{3} (w^{*}) = 0.9952, z_{4} (w^{*}) = 0.6979 \end{matrix}$

In Xu and Chen (2008a)’s method, the overall values of alternatives are: $\begin{matrix} z_{1} (w^{*}) = 0.8842, z_{2} (w^{*}) = 0.6015, \\ z_{3} (w^{*}) = 0.9103, z_{4} (w^{*}) = 0.5579 \end{matrix}$

It is noted that the overall evaluations of alternatives obtained by Xu and Chen (2008a)’s method are smaller than the ones calculated by the proposed method.

In Xu and Chen’s paper, according to the deviation values calculated by their method, the decision makers provide the reevaluated preference information as follows: $\begin{matrix} d_{1} : \tilde{U} = {[0.8, 0.9], [0.6, 0.7], [0.7, 1.0], [0.5, 0.7]} \\ d_{2} : \tilde{P} = (\begin{matrix} [0.5, 0.5] & [0.6, 0.7] & [0.4, 0.6] & [0.6, 0.9] \\ [0.3, 0.4] & [0.5, 0.5] & [0.3, 0.5] & [0.5, 0.6] \\ [0.4, 0.6] & [0.5, 0.7] & [0.5, 0.5] & [0.6, 0.9] \\ [0.1, 0.4] & [0.4, 0.5] & [0.1, 0.4] & [0.5, 0.5] \end{matrix}) \\ d_{3} : \tilde{B} = (\begin{matrix} [1, 1] & [1, 3] & [1 / 3, 3] & [1, 5] \\ [1 / 3, 1] & [1, 1] & [1 / 3, 1] & [1 / 3, 3] \\ [1 / 3, 3] & [1, 3] & [1, 1] & [1, 5] \\ [1 / 5, 1] & [1 / 3, 1] & [1 / 5, 1] & [1, 1] \end{matrix}) \end{matrix}$

Base on the new provided preference information, the objective function value is zero calculated by (M7). Therefore, the subjective information is consistent with the objective information. Then by (M8), the interval interactions of criteria are derived as: $\begin{matrix} {\tilde{w}}_{1} = [0.15, 0.3], {\tilde{w}}_{2} = [0.1, 0.2], {\tilde{w}}_{3} = [0.2, 0.5318], \\ {\tilde{w}}_{4} = [0.0, 0.4], {\tilde{w}}_{12} = [- 0.2, 0.75] \\ {\tilde{w}}_{13} = [- 0.3, 0.65], {\tilde{w}}_{14} = [- 0.3, 0.85], \\ {\tilde{w}}_{23} = [- 0.2, 0.2318], {\tilde{w}}_{24} = [- 0.2, 0.9], {\tilde{w}}_{34} = [- 0.4, 0.8] \\ {\tilde{w}}_{123} = [- 0.8242, 0.9], {\tilde{w}}_{124} = [- 1.75, 1], \\ {\tilde{w}}_{134} = [- 1.65, 1], {\tilde{w}}_{234} = [- 1.2318, 0.9] \\ {\tilde{w}}_{1234} = [- 2.3437, 1.8242] \end{matrix}$

Solving (M9), the optimal interaction of criteria is obtained: $\begin{matrix} w^{*} & = (0.3, 0.2, 0.5318, 0.4, - 0.2, 0.1682, 0.3, \\ 0.2318, 0.4, 0.0682, - 0.2318, - 0.4, - 0.7682, \\ - 0.8318, 0.8318)^{T} \end{matrix}$ and the overall evaluations of alternatives are: $\begin{matrix} z_{1} (w^{*}) = 0.9041, z_{2} (w^{*}) = 0.9841, \\ z_{3} (w^{*}) = 0.9952, z_{4} (w^{*}) = 0.7314 \end{matrix}$

In Xu and Chen’s paper (2008a), the overall evaluations of alternatives are: $\begin{matrix} z_{1} (w^{*}) = 0.8689, z_{2} (w^{*}) = 0.7387, \\ z_{3} (w^{*}) = 0.8394, z_{4} (w^{*}) = 0.6261 \end{matrix}$ which are all smaller than the ones obtained by the proposed method.

If decision makers give their revised preference information according to the optimal deviation variables calculated by (M7), then $\begin{matrix} d_{1} : \tilde{U} = {[0.8, 0.9], [0.4, 0.5], [0.7, 0.9], [0.4, 0.5]} \\ d_{2} : \tilde{P} = (\begin{matrix} [0.5, 0.5] & [0.6, 0.7] & [0.4, 0.6] & [0.6, 0.9] \\ [0.3, 0.4] & [0.5, 0.5] & [0.2, 0.3] & [0.4, 0.5] \\ [0.4, 0.6] & [0.5, 0.7] & [0.5, 0.5] & [0.7, 0.9] \\ [0.1, 0.4] & [0.4, 0.5] & [0.1, 0.3] & [0.5, 0.5] \end{matrix}) \\ d_{3} : \tilde{B} = (\begin{matrix} [1, 1] & [1, 2] & [1 / 3, 3] & [1, 2] \\ [1 / 5, 1 / 3] & [1, 1] & [1 / 3, 1] & [1 / 3, 3] \\ [1 / 3, 3] & [1, 3] & [1, 1] & [1, 2] \\ [1 / 9, 1 / 7] & [1 / 3, 1] & [1 / 9, 1 / 7] & [1, 1] \end{matrix}) \end{matrix}$

If Xu and Chen’s method (2008a) is used, then the objective function value is J^* = 0.2601, which indicates that the objective and subjective information is not consistent. But if (M 7) is used, we have J^* = 0, which shows that these two kinds of the information are consistent. That is because (M 7) considers the interaction of criteria, while Xu and Chen’s method (2008a) doesn’t.

Based on the above analysis, we can find that considering the interactions of the criteria not only can reduce the deviation of the objective information and subjective information, but also can increase the overall evaluations of alternatives. Especially, if Choquet integral reduces to the WA operator, the proposed methods reduce to the ones given by Xu and Chen [26].

5 Conclusion remarks

In this paper, the multi-criteria decision making have been investigated combing objective information and subjective information, the former is expressed by using decision matrix, while the latter can be expressed by using interval utilities or interval preference relations. To reflect the interactions of the criteria, Choquet integral has been used to aggregate the objective information in the decision matrix. To integrate the subjective information and objective information, the aggregated information of alternatives have been transformed into different kinds of subjective information. Several linear models have been developed to minimize the deviations, based on which, the interval interactions of the criteria have been derived. The model has been given to find the optimal interactions of the criteria under the condition that the overall evaluation of alternatives can attain the maximum one. The comparison has been given, which shows that the proposed method can obtain smaller deviations and better overall evaluations of alternatives. The proposed method can be considered as a generalization of the existing ones.

Footnotes

Acknowledgments

The authors would like to express their sincere thanks to the editors, and the anonymous reviewers for their constructive comments and suggestions that have led to an improved version of this paper. This work was supported by the Ministry of Education Foundation of Humanities and Social Sciences (No. 13YJC630185), and the National Natural Science Foundation of China under Grant Nos. (71532002 and 71501010).

References

Anandalingam

and Olsson

C.E.

, A multi-stage multi-attribute decision model for project selection, European Journal of Operational Research 43 (1989), 271–283.

Angilella

, Corrente

, Greco

and Slowiński

, MUSA-INT: Multicriteria customer satisfaction analysis with interacting criteria, Omega 42 (2014), 189–200.

Choquet

, Theory of capacities, Annales de l’Institut Fourier 54 (1953), 131–295.

Corrente

, Greco

and Slowiński

, Multi-criteria hierarchy process with ELECTRE and PROMETHEE, Omega 41 (2013), 820–846.

Eum

Y.S.

, Park

K.S.

and Kim

S.H.

, Establishing dominance and potential optimality in multi-criteria analysis with imprecise weight and value, Computers & Operations Research 28 (2001), 397–409.

Fujimoto

, Kojadinovic

and Marichal

J.L.

, Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices, Games and Economic Behavior 55 (2006), 72–99.

Grabisch

, K-order additive discrete fuzzy measures and their representation, Fuzzy Sets and Systems 92 (1997), 167–189.

Grabisch

, Kojadinovic

and Meyer

, A review of methods for capacity identification in Choquet integral based multi-attribute utility theory, European Journal of Operational Research 186 (2008), 766–785.

Grabisch

and Labreuche

, A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid, 4or: A Quarterly Journal of Operations Research 6 (2008), 1–44.

10.

Grabisch

, Marichal

J.L.

, Mesiar

and Pap

, Aggregation functions (Vol. 127). Cambridge, U.K: Cambridge University Press, 2009.

11.

Harsanyi

J.C.

, Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility, Journal of Political Economy 63 (1955), 309–321.

12.

Hwang

C.L.

and Yoon

, Multiple attribute decision making: Methods and applications. Berlin, Germany: Springer-Verlag, 1981.

13.

Keeney

R.L.

and Kirkwood

C.W.

, Group decision making using cardinal social welfare functions, Management Science 22 (1975), 430–437.

14.

, Fan

Z.P.

and Huang

L.H.

, A subjective and objective integrated approach to determine attribute weights, European Journal of Operational Research 112 (1999), 397–404.

15.

, Fan

Z.P.

, Jiang

Y.P.

and Mao

J.Y.

, An optimization approach to multiperson decision making based on different formats of preference information, IEEE Transactions on Systems, Man, and Cybernetics 36 (2006), 876–889.

16.

Park

K.S.

, Mathematical programming models for characterizing dominance and potential optimality when multicriteria alternative values and weights are simultaneously incomplete, IEEE Transactions on Systems, Man, and Cybernetics 34 (2004), 601–614.

17.

Saaty

T.L.

and Vargas

L.G.

, Uncertainty and rank order in the analytic hierarchy process, European Journal of Operational Research 32 (1987), 107–117.

18.

Sugeno

, Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Institute of Technology, Tokyo, 1974.

19.

Schubert

, On ρ in a decision-theoretic apparatus of Dempster-Shafer theory, International Journal of Approximate Reasoning 13 (1995), 185–200.

20.

Wang

Y.M.

and Parkan

, Multiple attribute decision making based on fuzzy preference information on alternatives: Ranking and weighting, Fuzzy Sets and Systems 153 (2005), 331–346.

21.

Weber

, Decision making with incomplete information, European Journal of Operational Research 28 (1987), 44–57.

22.

J.Z.

, Zhang

, Du

and Dong

, Compromise principle based methods of identifying capacities in the framework of multicriteria decision analysis, Fuzzy Sets and Systems 246 (2014), 91–106.

23.

J.Z.

and Zhang

, 2-order additive fuzzy measure identification method based on diamond pairwise comparison and maximum entropy principle, Fuzzy Optimization and Decision Making 9 (2010), 435–453.

24.

Z.S.

, On compatibility of interval fuzzy preference matrices, Fuzzy Optimization and Decision Making 3 (2003), 217–225.

25.

Z.S.

, Multiple-attribute group decision making with different formats of preference information on attributes, IEEE Transactions on Systems, Man, and Cybernetics 37 (2007), 1500–1511.

26.

Z.S.

and Chen

, MAGDM linear-programming models with distinct uncertain preference Structures, IEEE Transactions on Systems, Man, an, Cybernetics 38 (2008a), 1356–1370.

27.

Z.S.

and Chen

, Some models for deriving the priority weights from interval fuzzy preference relations, European Journal of Operational Research 184 (2008b), 266–280.