Some uncertain generalized Shapley aggregation operators for multi-attribute group decision making

Abstract

This paper studies some uncertain generalized aggregation operators based on the Shapley function, two of which are named the uncertain induced generalized Shapley averaging (UIGSA) operator and the uncertain induced generalized hybrid Shapley averaging (UIGHSA) operator. These two operators consider the interactive characteristics between elements and that of their ordered positions. Meanwhile, some important cases and desirable properties are studied. To simplify the complexity of solving a fuzzy measure, the uncertain induced generalized λ-Shapley averaging (UIGλSA) operator and the uncertain induced generalized hybrid λ-Shapley averaging (UIGHλSA) operator based on λ-fuzzy measures are presented. Finally, two new approaches to uncertain multi-attribute group decision making are developed, and a numerical example is provided to illustrate the proposed procedures.

Keywords

Multi-attribute group decision making uncertain aggregation operator Shapley function

1 Introduction

Decision making is one of the most significant and omnipresent human activities in business, service, manufacturing and selection of products. The key of decision making is to find the proper way to aggregate decision makers’ preferences [10 , 53]. The ordered weighted averaging (OWA) operator proposed by Yager [44] is a useful tool to aggregate the real numbers for the arguments, whose fundamental aspect is a reordering step in which the input arguments are rearranged in descending order and the weight vector is merely associated with its ordered position. Since it was first proposed in 1988, the OWA operator has been widely used in decision making under uncertainty, including expert systems [3 , 37], neural networks [45], fuzzy systems [4 , 42] and controls [1], and many extending forms are developed, such as the induced generalized continuous OWA operator [5], the hybrid weighted averaging operators [16], the induced generalized intuitionistic fuzzy OWA operator [32], the induced continuous ordered weighted geometric operators [38], the uncertain OWA operator [39], the continuous interval OWA operator [46], the generalized OWA aggregation operators [47], the uncertain induced quasi-arithmetic OWA operator [25], the uncertain probabilistic ordered weighted averaging operator [26], the intuitionistic fuzzy ordered weighted geometric operator [41] and the continuous ordered weighted geometric operator [48]. Whilst some scholars applied the OWA operator to determine the weight vector [2 , 36].

However, in some situations, the input arguments take the form of uncertain variables rather thannumerical ones for time pressure, lack of knowledge, the decision-maker’s limited attention and information processing capabilities. Therefore, it is of necessity to pay attention to the aggregation operators under uncertain conditions. Xu and Da [39] proposed the uncertain OWA (UOWA) operator. Zhou et al. [52] introduced the uncertain generalized ordered weighted averaging (UGOWA) operator. To both consider the importance of the elements and that of their ordered positions, Xu and Da [40] defined the hybrid weighted averaging (HWA) operator on real numbers. Later, Lin and Jiang [16] noticed that the HWA operator satisfies neither boundary nor idempotency, and defined the hybrid weighted arithmetical averaging (HWAA) operator. Later, Zhou et al. [52] introduced the uncertain generalized hybrid averaging (UGHA) operator, which does not satisfy boundary or idempotency. To avoid the issue of the UGHA operator and deal with the situation where the elements in a set are correlative, this paper develops some uncertain aggregation operators by applying the well-known Shapley function [30] with respect to (w.r.t.) fuzzy measures [31], which can reflect the interactive phenomenon between elements and satisfy some reasonable properties.

The rest parts of this paper are organized as follows: In Section 2, some related concepts are reviewed. In Section 3, the UIGSA and UIGHSA operators are defined. Meantime, some important cases and desirable properties are researched. In Section 4, we first introduce the UIGλSA and UIGHλSA operators. Then, two approaches to uncertain multi-attribute group decision making are developed. Meantime, the corresponding example is provided to illustrate the developed procedure. The conclusion is made in the last section.

2 Some basic concepts

Since the interval numbers [18] are a very useful and simple technique for representing the uncertainty. It has been used in an astonishingly wide range of applications. By R₊, we denote the set of positive real numbers. Let $\bar{α} = [a, b]$ , if a ≤ b and a, b ∈ R₊, then $\bar{α}$ is said to be a positive interval number. Let ${\bar{R}}_{+}$ be the set of all positive interval numbers. In the following, we are going to review some basic interval number operations. Let $\bar{α}$ and $\bar{β}$ be two positive interval numbers, where $\bar{α} = [a_{1}, b_{1}]$ and $\bar{β} = [a_{2}, b_{2}]$ , then

$\bar{α} + \bar{β} = [a_{1} + b_{1}, a_{2} + b_{2}]$ ,

$\bar{α} - \bar{β} = [a_{1} - b_{2}, b_{1} - a_{2}]$ ,

$\bar{α} \times \bar{β} = [a_{1} a_{2}, b_{1} b_{2}]$ ,

$\bar{α} \div \bar{β} = [a_{1} / b_{2}, b_{1} / a_{2}]$ ,

$\bar{α} = [{sa}_{1}, {sb}_{1}] s \in R_{+},$

${\bar{α}}^{s} = [a_{1}^{s}, b_{1}^{s}] s \in R_{+} .$

Definition 1. [52] Let $\bar{α} = [a_{1}, b_{1}]$ , $\bar{β} = [a_{2}, b_{2}]$ $\in {\bar{R}}_{+}$ , then the degree of possibility of $\bar{α} \geq \bar{β}$ is defined as $P (\bar{α} \geq \bar{β}) = max {1 - \frac{1}{2} max (\frac{m_{b} - m_{a}}{w_{b} + w_{a}} + 1, 0), 0},$ where m_a = 0.5 (a₁ + b₁), m_b = 0.5 (a₂ + b₂), w_a = 0.5(b₁ – a₁) and w_b = 0.5(b₂ – a₂).

In a similar way to Xu and Da [40], Zhou et al. [52] defined the uncertain generalized hybrid averaging (UGHA) operator.

Definition 2. [52] An UGHA operator of dimension n is a mapping UGHA: ${\bar{R}}_{+}^{n} \to {\bar{R}}_{+}$ , which has an associated weight vector w = (w₁, w₂, …, w_n) ^T on N = {1, 2, …, n} such that w_j ∈ [0,1] and $\sum_{j = 1}^{n} w_{j} = 1$ and a parameter γ ∈ (- ∞ , + ∞) and γ ≠ 0, denoted by ${UGHA}_{w, ω} ({\bar{a}}_{1}, {\bar{a}}_{2}, \dots, {\bar{a}}_{n}) = {(\sum_{j = 1}^{n} w_{j} {\bar{b}}_{j}^{γ})}^{1 / γ},$ where ${\bar{b}}_{j}$ is the jth largest value of the weighted argument nω_ja_j (i = 1, 2, …, n), ω = (ω₁, ω₂, …, ω_n) ^T is the weight vector of ${\bar{a}}_{i} (i = 1, 2, \dots, n)$ with ω_i > 0 and $\sum_{i = 1}^{n} ω_{i} = 1$ , and n is the balancing coefficient.

From Definition 2, it is easy to know that the UGHA operator does not satisfy boundary or idempotency. Further, the UGHA operator is based on the assumption that the elements in a set are independent. However, in many situations, the elements in a set are usually inter-dependent [13–15 , 43]. All of them are defined on real numbers or (interval-valued) intuitionistic fuzzy sets, and there is less research about interval numbers. At present, we only find one uncertain aggregation operator called the uncertain generalized Choquet integral aggregation (UGCIA) operator [52] considers the interactions between elements. However, the Choquet integral aggregation operator cannot globally consider the interaction between elements.

Fuzzy measures [31] as an effective tool to measure the interactive phenomenon between elements have been deeply studied by many researchers [7 , 31].

Definition 3. [31] A fuzzy measure on finite set N = {1, 2, …, n} is a set function μ : P (N) → [0, 1]satisfying

μ (∅) =0, μ (N) =1,

If A, B ∈ P (N) and A ⊆ B, then μ (A) ≤ μ (B),

where P(N) is the power set of N.

As a key solution concept in game theory, the Shapley function [30] has been deeply studied by many scholars, which is based on several reasonable axioms, expressed by

$\begin{matrix} φ_{i} (μ, N) & = & \sum_{S \subseteq N ∖ i} \frac{(n - s - 1)! s!}{n!} (μ (S \cup i) - μ (S)) \\ \forall i \in N, \end{matrix}$ (1) where μ is a fuzzy measure on finite set N, s and n, respectively, denote the cardinalities of S and N.

Remark 1. In game theory, the value μ (S) of the coalition S ⊆ N can be any real number. When we use the Shapley function to measure the importance of the studying elements, then it gives us an expectation value of the globally marginal contributions between the element i and any coalition S ⊆ N ∖ i.

When we apply the Shapley function in the setting of decision making, where μ (N) =1. From the expression of the Shapley function, we know that φ_i (μ, N) ≥0 for each i ∈ N, and $\sum_{i = 1}^{n} φ_{i} (μ, N) = 1$ . Thus, {φ_i (μ, N)} _i∈N is a weight vector. Further, if there is no interaction between elements, then the fuzzy measure μ is an additive measure, namely, μ (S) = ∑_i∈Sμ (i) for all S ⊆ N. From Equation (1), we know the Shapley value of the element i degenerates to be μ (i). Thus, the Shapley values can be seen as an extension of the elements’ weights based on additive measures.

3 The uncertain generalized Shapley aggregation operators

In this section, we introduce the UIGSA and UIGHSA operators, which consider the interactive phenomenon between elements. Meantime, some desirable properties are studied.

3.1 The UIGSA operator

Definition 4. An UIGSA operator of dimension n is a mapping $UIGSA : {\bar{R}}_{+}^{n} \to {\bar{R}}_{+}$ defined on the set of second arguments of two tuples $< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >$ with a set of order- inducing variables u_i (i = 1, 2, …, n) and a parameter γ such that γ ∈ (- ∞ , + ∞) and γ ≠ 0, denoted by

$\begin{matrix} {UIGSA}_{μ} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {(\sum_{j = 1}^{n} φ_{j} (μ, N) {\bar{β}}_{j}^{γ})}^{1 / γ}, \end{matrix}$ (2) where φ_j (μ, N) is the Shapley value w.r.t. the associated fuzzy measure μ on N = {1, 2, …, n} for the jth index, u_j is the jth largest value of u_i (i = 1, 2, …, n), and ${{\bar{β}}_{1}, \dots, . .}$ is a permutation of ${{\bar{α}}_{1}, \dots, {\bar{α}}_{n}}$ .

Remark 2. If μ is additive, then the UIGSA operator reduces to be the uncertain induced generalized weighted averaging (UIGWA) operator $\begin{matrix} {UIGWA}_{μ} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {(\sum_{j = 1}^{n} w_{j} {\bar{β}}_{j}^{γ})}^{1 / γ}, \end{matrix}$ where w_j = μ(j) for all j = 1, 2, …, n.

Remark 3. If u_i = u_j for all i, j = 1, 2, …, n with i ≠ j, then the UIGSA operator reduces to be the uncertain generalized Shapley averaging (UGSA) operator $\begin{matrix} {UGSA}_{μ} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {(\sum_{j = 1}^{n} φ_{j} (μ, N) {\bar{α}}_{j}^{γ})}^{1 / γ} . \end{matrix}$ Remark 4. If μ is additive and u_i = u_j for all i, j = 1, 2, …, n with i ≠ j, then the UIGSA operator reduces to be the uncertain generalized weighted averaging (UGWA) operator [52].

Remark 5. If γ → 0, then the UIGSA operator degenerates to be the uncertain induced Shapley geometric mean (UISGM) operator $\begin{matrix} {UISGM}_{μ} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = Π_{j = 1}^{n} {\bar{β}}_{j}^{φ_{j} (μ, N)} . \end{matrix}$ Next, we investigate some desirable propositions of the UIGSA operator.

Proposition 1. (Commutativity ) Let ${\bar{α}}_{i} = [a_{i}, b_{i}] (i = 1, 2, \dots, n)$ be a collection of interval numbers in ${\bar{R}}_{+}$ , and ${\bar{α}}_{i}^{'} = [a_{i}^{'}, b_{i}^{'}] (i = 1, 2, \dots, n)$ be a permutation of ${\bar{α}}_{i}$ . Then, $\begin{matrix} {UIGSA}_{μ} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {UIGSA}_{μ} (< u_{1}, {\bar{α}}_{1}^{'} >, < u_{2}, {\bar{α}}_{2}^{'} >, \\ \dots, < u_{n}, {\bar{α}}_{n}^{'} >) . \end{matrix}$ Proposition 2. (Monotonicity) Let ${\bar{α}}_{i} = [a_{i}, b_{i}]$ and ${\bar{θ}}_{i} = [c_{i}, d_{i}] (i = 1, 2, \dots, n)$ be two collections of positive interval numbers in ${\bar{R}}_{+}$ , and μ be the associated fuzzy measure on {1, 2, …, n}. If c_i ≤ a_i and d_i ≤ b_i for all i = 1, 2, …, n, then $\begin{matrix} {UIGSA}_{μ} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ \geq {UIGSA}_{μ} (< u_{1}, {\bar{θ}}_{1} >, < u_{2}, {\bar{θ}}_{2} >, \dots, < u_{n}, {\bar{θ}}_{n} >) . \end{matrix}$ Proposition 3. (Idempotency) Let ${\bar{α}}_{i} = [a_{i}, b_{i}] (i = 1, 2, \dots, n)$ be a collection of positive interval numbers in ${\bar{R}}_{+}$ , and μ be the associated fuzzy measure on {1, 2, …, n}. If ${\bar{α}}_{i} = \bar{α} = [a, b]$ for all i = 1, 2, …, n, then $\begin{matrix} {UIGSA}_{μ} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = \bar{α} . \end{matrix}$ Proposition 4. (Boundary) Let ${\bar{α}}_{i} = [a_{i}, b_{i}] (i = 1, 2, \dots, n)$ be a collection of positive interval numbers in ${\bar{R}}_{+}$ , and μ be the associated fuzzy measure on N = {1, 2, …, n}, then $\begin{matrix} [min_{i} a_{i}, min_{i} b_{i}] \leq {UIGSA}_{μ} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \\ \dots, < u_{n}, {\bar{α}}_{n} >) \leq [max_{i} a_{i}, max_{i} b_{i}] . \end{matrix}$ Definition 5. Let μ be a fuzzy measure on N. An element i ∈ N is said to be inessential if μ (S ∪ i) = μ (S) for all S ⊆ N ∖ i.

Proposition 5. Let ${\bar{α}}_{i} = [a_{i}, b_{i}] (i = 1, 2, \dots, n)$ be a collection of positive interval numbers in ${\bar{R}}_{+}$ , and μ be the associated fuzzy measure on {1, 2, …, n}. If p ∈ {1, 2, …, n} is an inessential element, then $\begin{matrix} {UIGSA}_{μ} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {UIGSA}_{μ} (\dots, < u_{p - 1}, {\bar{α}}_{p - 1} >, \\ < u_{p + 1}, {\bar{α}}_{p + 1} >, \dots) . \end{matrix}$ Proof. Since p is an inessential element, by Equation (1) we get φ_p (μ, N) =0. From Equation (2), it is not difficult to get the conclusion. □

According to the work of Beliakov [2], we can get the induced quasi uncertain Shapley aggregation (IQUSA) operator, which replaces an exponential function with a general continuous strictly monotone function.

Definition 6. An IQUSA operator of dimension n is a mapping IQUSA: ${\bar{R}}_{+}^{n} \to {\bar{R}}_{+}$ defined on the set of second arguments of two tuples $< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >,$ $\dots, < u_{n}, {\bar{α}}_{n} >$ with a set of order-inducing variables u_i (i = 1, 2, …, n), denoted by $\begin{matrix} {IQUSA}_{μ} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = g^{- 1} (\sum_{j = 1}^{n} φ_{j} (μ, N) g ({\bar{β}}_{j})), \end{matrix}$ where φ_j (μ, N) is the Shapley value w.r.t. the associated fuzzy measure μ on N = {1, 2, …, n} for the jth index, u_j is the jth largest value of u_i (i = 1, 2, …, n), g is a strictly continuous monotonic function, and ( ${\bar{β}}_{1}, \dots, . {\bar{β}}_{n}$ ) is a permutation of ( ${\bar{α}}_{1}, \dots, {\bar{α}}_{n}$ ).

3.2 The UIGHSA operator

From Definition 4, we know the UIGSA operator only considers the ordered positions of the arguments, while the importance of the arguments and the interactions between them do not taken into account. To consider the importance of the arguments and their ordered positions as well as the interactions between them, we define another uncertain aggregation operator called the uncertain induced generalized hybrid Shapley averaging (UIGHSA) operator.

Definition 7. An UIGHSA operator of dimension n is a mapping UIGHSA: ${\bar{R}}_{+}^{n} \to {\bar{R}}_{+}$ defined on the set of second arguments of two tuples $< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >$ with a set of order- inducing variables u_I (i = 1, 2, …, n) and a parameter γ such that γ ∈ (- ∞ , + ∞) and γ ≠ 0, denoted by $\begin{matrix} {UIGHSA}_{μ, v} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {(\frac{\sum_{j = 1}^{n} φ_{j} (μ, N) {\bar{β}}_{j}^{γ}}{\sum_{j = 1}^{n} φ_{j} (μ, N) n^{γ} φ_{j}^{γ} (v, A)})}^{1 / γ}, \end{matrix}$ (3) where φ_j (μ, N) is the Shapley value w.r.t. the associated fuzzy measure μ on N = {1, 2, …, n} for the jth index, u_j is the jth largest value of $u_{i} (i = 1, 2, \dots, n), {\tilde{β}}_{j}$ is the Shapley weighted interval number $n φ_{j} (v, A) {\tilde{α}}_{j} (j = 1, 2, \dots, n), φ_{j} (v, A)$ is the Shapley value w.r.t. the fuzzy measure v on $A = {{\tilde{α}}_{j}}_{j = 1, \dots, n}$ for ${\tilde{α}}_{j}$ (j = 1, 2, …, n), and n is the balancing coefficient.

Remark 6. If μ and v are both additive, then the UIGHSA operator reduces to be the uncertain induced generalized hybrid weighted averaging (UIGHWA) operator $\begin{matrix} {UIGHWA}_{w, ω} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \\ \dots, < u_{n}, {\bar{α}}_{n} >) = {(\frac{\sum_{j = 1}^{n} w_{j} {\bar{β}}_{j}^{γ}}{\sum_{j = 1}^{n} w_{j} n^{γ} ω_{j}^{γ}})}^{1 / γ}, \end{matrix}$ where w_j = μ(j) and ω_j = v (j) for all j = 1, 2, …, n.

Remark 7. If u_i = u_j for all i, j = 1, 2, …, n with i ≠ j, then the UIGHSA operator reduces to be the uncertain generalized hybrid Shapley averaging (UGHSA) operator $\begin{matrix} {UGHSA}_{μ, v} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {(\frac{\sum_{j = 1}^{n} φ_{j} (μ, N) {\bar{β}}_{j}^{γ}}{\sum_{j = 1}^{n} φ_{j} (μ, N) n^{γ} φ_{j}^{γ} (v, A)})}^{1 / γ} . \end{matrix}$ Remark 8. If μ is additive and u_i = u_j for all i, j = 1, 2, …, n with i ≠ j, then the UIGHSA operator reduces to be the uncertain generalized hybrid weighted averaging (UGHWA) operator $\begin{matrix} {UGHWA}_{μ, v} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {(\frac{\sum_{j = 1}^{n} w_{j} {\bar{α}}_{j}^{γ}}{\sum_{j = 1}^{n} w_{j} n^{γ} ω_{j}^{γ}})}^{1 / γ}, \end{matrix}$ where w_j = μ(j) and ω_j = v (j) for all j = 1, 2, …, n.

Similar to the UIGSA operator, it is not difficult to obtain that the UIGHSA operator satisfies Commutativity, C omonotonicity, I dempotency, and B oundary.

Proposition 6. Let ${\bar{α}}_{i} = [a_{i}, b_{i}] (i = 1, 2, \dots, n)$ be a collection of positive interval numbers . If ${\bar{α}}_{p}$ is an inessential element w.r.t. the fuzz measure v defined on $A = {{\bar{α}}_{j}}_{j = 1, \dots, n}$ , then $\begin{matrix} {UIGHSA}_{μ, v} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {UIGHSA}_{μ, v} (\dots, < u_{p - 1}, {\bar{α}}_{p - 1} >, \\ < u_{p + 1}, {\bar{α}}_{p + 1} >, \dots) . \end{matrix}$ Proof. Since ${\bar{α}}_{p}$ is an inessential element, by Equation (1) we get φ_p (v, A) =0. From Equation (3), it has $\begin{matrix} {UIGHSA}_{μ, v} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {(\frac{\sum_{j = 1}^{n} φ_{j} (μ, N) {\bar{β}}_{j}^{γ}}{\sum_{j = 1}^{n} φ_{j} (μ, N) n^{γ} φ_{j}^{γ} (v, A)})}^{1 / γ} \\ = {(\frac{\sum_{j = 1, j \neq p}^{n} φ_{j} (μ, N) {\bar{β}}_{j}^{γ}}{\sum_{j = 1, j \neq p}^{n} φ_{j} (μ, N) n^{γ} φ_{j}^{γ} (v, A)})}^{1 / γ} \\ = {UIGHSA}_{μ, v} (\dots, < u_{p - 1}, {\bar{α}}_{p - 1} >, < u_{p + 1}, \\ {\bar{α}}_{p + 1} >, \dots) . □ \end{matrix}$

Similar to Definition 6, we introduce the following definition for the induced quasi uncertain hybrid Shapley aggregation (IQUHSA) operator

Definition 8. An IQUHSA operator of dimension n is a mapping IQUHSA: ${\bar{R}}_{+}^{n} \to {\bar{R}}_{+}$ defined on the set of second arguments of two tuples $< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >,$ $\dots, < u_{n}, {\bar{α}}_{n} >$ with a set of order- inducing variables u_i (i = 1, 2, …, n), denoted by $\begin{matrix} {IQUHSA}_{μ, v} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = g^{- 1} (\frac{\sum_{j = 1}^{n} φ_{j} (μ, N) g ({\bar{β}}_{j})}{\sum_{j = 1}^{n} φ_{j} (μ, N) n φ_{j} (v, A)}), \end{matrix}$ where φ_j (μ, N) is the Shapley value w.r.t. the associated fuzzy measure μ on N = {1, 2, …, n} for the jth index, u_j is the jth largest value of $u_{i} (i = 1, 2, \dots, n), {\tilde{β}}_{j}$ is the Shapley weighted interval number $n φ_{j} (v, A) {\tilde{α}}_{j}$ (j = 1, 2, …, n), φ_j (v, A) is the Shapley value w.r.t. the fuzzy measure v on $A = {{\tilde{α}}_{j}}_{j \in N}$ for ${\tilde{α}}_{j} (j \in N)$ , g is a strictly continuous monotonic function, and n is the balancing coefficient.

4 New approaches to uncertain multi-attribute group decision-making

This section presents two new approaches to uncertain multi-attribute group decision making. Since the fuzzy measures are defined on the power set, it makes the problem exponentially complex. Thus, it is not easy to get the fuzzy measure of each combination in a set when it is large. To increase the practicability of the introduced operators, we first introduce a special kind of fuzzy measures named as λ-fuzzy measures [31], which much simplifies the complexity of solving a fuzzy measure, denoted by $g_{λ} (A \cup B) = g_{λ} (A) + g_{λ} (B) + λ g_{λ} (A) g_{λ} (B),$ (4) where λ > -1, and A, B ⊆ N with A∩ B = ∅. For finite set N, the λ-fuzzy measure g_λ, for any A ⊆ N, can be equivalently expressed by $g_{λ} (A) = {\begin{matrix} \frac{1}{λ} (\underset{i \in A}{Π} [1 + λ g_{λ} (i)] - 1) & if λ \neq 0 \\ \sum_{i \in A} g_{λ} (i) & if λ = 0 \end{matrix} .$ (5) From μ (N) =1, we know λ is determined by $\underset{i \in N}{Π} [1 + λ g_{λ} (i)] = 1 + λ .$ (6)

If each g_λ (i) is given, then we can get the value λ. From Equation (5), for a set N with n elements we only need n values to get a λ-fuzzy measure, which much simplifies the difficulty of solving a fuzzy measure.

From Definitions 5 and 8, we further define the UIGSA and UIGHSA operators w.r.t. λ-fuzzy measures called the UIGλSA and UIGHλSA operators.

Definition 9. An UIGλSA operator of dimension n is a mapping $UIG λ SA : {\bar{R}}_{+}^{n} \to {\bar{R}}_{+}$ defined on the set of second arguments of two tuples $< u_{1}, {\bar{α}}_{1} >,$ $< u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >$ with a set of order-inducing variables u_I (i = 1, 2, …, n) and a parameter γ such that γ ∈ (- ∞ , + ∞) and γ ≠ 0, denoted by

$\begin{matrix} {UIG λ SA}_{g_{λ}} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >) \\ = {(\sum_{j = 1}^{n} φ_{j} (g_{λ}, N) {\bar{β}}_{j}^{γ})}^{1 / γ}, \end{matrix}$ (7) where φ_j (g_λ, N) is the Shapley value w.r.t. the associated λ-fuzzy measure g_λ on N = {1, 2, …, n} for the jth index, u_j is the jth largest value of u_i (i = 1, 2, …, n), and ${{\bar{β}}_{1}, \dots, {\bar{β}}_{n}}$ is a permutation of ${{\bar{α}}_{1}, \dots, {\bar{α}}_{n}}$ .

Definition 10. An UIGHλSA operator of dimension n is a mapping UIGHλSA: ${\bar{R}}_{+}^{n} \to {\bar{R}}_{+}$ defined on the set of second arguments of two tuples $< u_{1}, {\bar{α}}_{1} >,$ $< u_{2}, {\bar{α}}_{2} >, \dots, < u_{n}, {\bar{α}}_{n} >$ with a set of order-inducing variables u_i (i = 1, 2, …, n) and a parameter γ such that γ ∈ (- ∞ , + ∞) and γ ≠ 0, denoted by

$\begin{matrix} {UIGH λ SA}_{g_{λ}^{'}, g_{λ}} (< u_{1}, {\bar{α}}_{1} >, < u_{2}, {\bar{α}}_{2} >, \dots, \\ < u_{n}, {\bar{α}}_{n} >) = {(\frac{\sum_{j = 1}^{n} φ_{j} (g_{λ^{'}}, N) {\bar{β}}_{j}^{γ}}{\sum_{j = 1}^{n} φ_{j} (g_{λ^{'}}, N) n^{γ} φ_{j}^{γ} (g_{λ}, A)})}^{1 / γ}, \end{matrix}$ (8) where φ_j (g_λ′, N) is the Shapley value w.r.t. the associated λ-fuzzy measure g_λ′ on N = {1, 2, …, n} for the jth index, u_j is the jth largest value of u_i (i = 1, 2, …, n), ${\tilde{β}}_{j}$ is the Shapley weighted interval number $n φ_{j} (g_{λ}, A) {\tilde{α}}_{j}$ (j = 1, 2, …, n), φ_j (g_λ, A) is the Shapley value w.r.t. the λ-fuzzy measure g_λ on $A = {{\tilde{α}}_{j}}_{j = 1, \dots, n}$ for ${\tilde{α}}_{j}$ (j = 1, 2, …, n), and n is the balancing coefficient.

Now, consider a group decision making problem, by A = {a₁, a₂, …, a_m} and C = {c₁, c₂, …, c_n} we, respectively, denote the set of alternatives and the set of the experts. Assume that the interval number ${\bar{α}}_{ij}^{k} = [a_{ij}^{k}, b_{ij}^{k}]$ is the importance degree of the alternative a_i (i = 1, 2, …, m) w.r.t. the attribute c_j (j = 1, 2, …, n) given by the expert e_k (k = 1, 2, … , q). Then, we can obtain the interval decision matrix ${\bar{A}}^{k} = {({\bar{α}}_{ij}^{k})}_{m \times n}$ given by the expert e_k (k = 1, 2, …, q). In the following, we give two new approaches to uncertain fuzzy multi-attribute group decision making.

Based on the UIGλSA operator, the main decision procedure to get the most desirable alternative(s) is given as follows:

Normalize the interval decision matrix ${\bar{A}}^{k} = {({\bar{α}}_{ij}^{k})}_{m \times n}$ into ${\bar{R}}^{k} = {({\bar{r}}_{ij}^{k})}_{m \times n}$ (k = 1, 2, …, q), where

\begin{matrix} {\bar{r}}_{ij}^{k} = {\begin{matrix} [a_{ij}^{k} / \sum_{i = 1}^{m} b_{ij}^{k}, b_{ij}^{k} / \sum_{i = 1}^{m} a_{ij}^{k}] & for benefit attribute c_{j} \\ [1 / b_{ij}^{k} / \sum_{i = 1}^{m} 1 / a_{ij}^{k}, 1 / a_{ij}^{k} / \sum_{i = 1}^{m} 1 / b_{ij}^{k}] & for cost attribute c_{j} \end{matrix} \\ (i = 1, 2, \dots, m; j = 1, 2, \dots, n) . \end{matrix}

Confirm the fuzzy measure g_{λ
_j} of each element in Q = {1, 2, …, q} w.r.t. the attribute c_j (j = 1, 2, …, n), Use Equation (6) to calculate the value λ, and compute the λ-fuzzy measure of every coalition in Q using Equation (5).

Calculate the Shapley value φ (g_{λ
_j}, Q) of each element in Q w.r.t. the attribute c_j (j = 1, 2, …, n).

Use the UIGλSA operator to aggregate all interval decision matrices ${\bar{R}}^{k} = {({\bar{r}}_{ij}^{k})}_{m \times n}$ (k = 1, 2, …, q) into the comprehensive interval decision matrix $\bar{R} = {({\bar{r}}_{ij})}_{m \times n}$ , where ${\tilde{r}}_{ij} = [c_{ij,} d_{ij}]$ (i = 1, 2, …, m j = 1, 2, …, n).

Confirm the fuzzy measure g_λ of each element in N = {1, 2, …, n}, utilize Equation (6) to calculate the value λ, and compute the λ-fuzzy measure of every coalition in N using Equation (5).

Calculate the Shapley value φ (g_λ, N) of each element in N.

Use the UIGλSA operator to get the comprehensive interval value ${\bar{r}}_{i}$ of the alternative a_i (i = 1, 2, …, m).

Compare each interval argument ${\bar{r}}_{i}$ with all ${\bar{r}}_{j}$ by the formula [44]:

$P ({\bar{r}}_{i} \geq {\bar{r}}_{j}) = max {1 - \frac{1}{2} max (\frac{m_{j} - m_{i}}{w_{i} + w_{j}} + 1, 0), 0},$ and get the complementary matrix P = (p_ij) _m×m.

Synthesize all elements of matrix P by the formula [8]: $p_{i} = \frac{\sum_{j = 1}^{m} p_{ij} + \frac{n}{2} - 1}{n (n - 1)}$ (i = 1, 2, …, m), then to select the best one.

End.

Example 1. Let us suppose there is an investment company, which wants to invest a sum of money in the best option (adapted from Ref. [33]). There is a panel with four possible alternatives A = {a₁, a₂, a₃, a₄} to invest the money. The investment company must take a decision according to the following four attributes C = {c₁, c₂, c₃, c₄}: (1) the level of technology innovation c₁; (2) the control ability of flow c₂; (3) the ability of management c₃; (4) the level of service c₄.

The four possible alternatives a_i (i = 1, 2, 3, 4) are to be evaluated using the interval information by three decision makers e_k (k = 1, 2, 3) with scores ranging [0, 10], which are listed in the following matrices. ${\bar{A}}^{1} = (\begin{matrix} [4, 6] & [7, 9] & [7, 9] & [3, 4] \\ [7, 9] & [6, 7] & [3, 4] & [7, 8] \\ [6, 8] & [7, 8] & [7, 9] & [4, 6] \\ [6, 8] & [5, 7] & [3, 5] & [4, 6] \end{matrix}),$ ${\bar{A}}^{2} = (\begin{matrix} [4, 6] & [6, 8] & [7, 9] & [3, 5] \\ [3, 5] & [4, 6] & [3, 5] & [5, 6] \\ [7, 9] & [4, 6] & [6, 8] & [6, 8] \\ [3, 5] & [5, 7] & [3, 5] & [4, 6] \end{matrix}),$ ${\bar{A}}^{3} = (\begin{matrix} [3, 5] & [6, 8] & [6, 8] & [4, 6] \\ [6, 8] & [4, 6] & [5, 7] & [6, 8] \\ [6, 8] & [5, 7] & [2, 4] & [5, 6] \\ [3, 5] & [4, 6] & [4, 5] & [6, 7] \end{matrix}) .$

Assume that the fuzzy measure of every element in Q = {1, 2, 3} w.r.t. each attribute c_j (j = 1, 2, 3, 4) is given as Table 1, which is only determined by its ordered position.

Further, assume that the fuzzy measure of every element in N = {1, 2, 3, 4} is given as $g_{λ} (1) = 0.2, g_{λ} (2) = 0.1, g_{λ} (3) = 0.3, g_{λ} (4) = 0.25 .$

In the following, we can utilize the proposed procedure to get the most desirable investment company.

Since all attributes are benefit and the interval values are given in ranging [0, 10], there is no need to modify the decision matrices. Namely, ${\bar{A}}^{k} = {\bar{R}}^{k} (k = 1, 2, 3)$ .

From Table 1, we get the values of λ _j for Q w.r.t. each attribute c_j (j = 1, 2, 3, 4):

$λ_{1} = - 0.37, λ_{2} = - 0.71, λ_{3} = 0.77, λ_{4} = 0.45 .$ From the values λ _j (j = 1, 2, 3, 4), we get the λ-fuzzy measure of every combination in Q w.r.t. each attribute c_j (j = 1, 2, 3, 4) as listed in Table 2.

From Tables 1 and 2, we get the Shapley value of each element in Q w.r.t. the attribute c_j (j = 1, 2, 3, 4) as listed in Table 3.

Suppose the fuzzy measure of each expert w.r.t. each attribute given as listed in Table 4.

When γ = 2 and $u_{k} = w_{e_{k}}^{j}$ (k = 1, 2, 3), by the UIGλSA operator we get the comprehensive interval decision matrix $\bar{R} = {({\bar{r}}_{ij})}_{4 \times 4}$ as follows: $\bar{R} = (\begin{matrix} [3.5, 5.6] & [6.4, 8.4] & [6.8, 8.8] & [3.3, 5] \\ [5.5, 7.4] & [4.8, 6.4] & [3.6, 5.2] & [6.2, 7.5] \\ [6.3, 8.3] & [5.6, 7.2] & [5.8, 7.7] & [4.9, 6.5] \\ [3.9, 5.8] & [4.6, 6.6] & [3.3, 5] & [4.7, 6.3] \end{matrix}) .$

From the fuzzy measure of every element in N = {1, 2, 3, 4}, we get λ = 0.54. By Equation (5), we get the following λ-fuzzy measure of each combination in N.

\begin{matrix} g_{λ} (1, 2) = 0.31, g_{λ} (1, 3) = 0.53, g_{λ} (1, 4) = 0.48, \\ g_{λ} (2, 3) = 0.42, g_{λ} (2, 4) = 0.36, g_{λ} (3, 4) = 0.59, \\ g_{λ} (1, 2, 3) = 0.66, g_{λ} (1, 2, 4) = 0.6, \\ g_{λ} (1, 3, 4) = 0.85, g_{λ} (2, 3, 4) = 0.72, g_{λ} (N) = 1 . \end{matrix}

From Step 5, we obtain the following Shapley values φ₁ (g_λ, N) =0.23, φ₂ (g_λ, N) =0.15, φ₃ (g_λ, N) =0.34, φ₄ (g_λ, N) =0.29.

Suppose that the fuzzy measure of each attribute in C is given as (w _c
₁, w _c
₂, w _c₃,w _c
₄) = (0.2, 0.3, 0.1, 0.4).

When γ=2 and u_j = w _{c
_j} (j = 1, 2, 3, 4), by the UIGλSA operator we get

\begin{matrix} {\bar{r}}_{1} & = & [5.3, 7.2], {\bar{r}}_{2} = [5.1, 6.7], \\ {\bar{r}}_{3} & = & [5.7, 7.5], {\bar{r}}_{4} = [4.1, 5, 9] . \end{matrix}

Step I- 8: By $P ({\bar{r}}_{i} \geq {\bar{r}}_{j}) = max {1 - \frac{1}{2} max (\frac{m_{j} - m_{i}}{w i + w j} + 1, 0), 0}$ , we get the complementary matrix P = (p_ij) _4×4 as follows: $P = (\begin{matrix} 0.5 & 0.6 & 0.41 & 0.84 \\ 0.4 & 0.5 & 0.29 & 0.76 \\ 0.59 & 0.71 & 0.5 & 0.94 \\ 0.16 & 0.24 & 0.06 & 0.5 \end{matrix}) .$

Step I-9: From $p_{i} = \frac{\sum_{j = 1}^{m} p_{ij} + \frac{n}{2} - 1}{n (n - 1)}$ , we have $p_{1} = 0.28, p_{2} = 0.25, p_{3} = 0.31, p_{4} = 0.16 .$ Thus, the alternative a₃ is the best investment company.

Based on the UIGHλSA operator, the main decision procedure to get the most desirable investment company is given as follows:

Normalize the interval decision matrix ${\bar{A}}^{k} = {({\bar{α}}_{ij}^{k})}_{m \times n}$ into ${\bar{R}}^{k} = {({\bar{r}}_{ij}^{k})}_{m \times n} (k = 1, 2, \dots, q)$ , where

$\begin{matrix} {\bar{r}}_{ij}^{k} = {\begin{matrix} [a_{ij}^{k} / \sum_{i = 1}^{m} b_{ij}^{k}, b_{ij}^{k} / \sum_{i = 1}^{m} a_{ij}^{k}] & for benefit attribute c_{j} \\ [1 / b_{ij}^{k} / \sum_{i = 1}^{m} 1 / a_{ij}^{k}, 1 / a_{ij}^{k} / \sum_{i = 1}^{m} 1 / b_{ij}^{k}] & for cost attribute c_{j} \end{matrix} \\ (i = 1, 2, \dots, m; j = 1, 2, \dots, n) . \end{matrix}$

Confirm the fuzzy measure g_{λ
_j} (e_k) of each expert e_k (k = 1, 2, …, q) w.r.t. each attribute c_j (j = 1, 2, …, n), use Equation (6) to calculate the value λ w.r.t. the expert set E, and compute the λ-fuzzy measure of every combination in E using Equation (5).

Calculate the Shapley value φ_{e
_k} (g_{λ
_j}, E) of each expert e_k (k = 1, 2, …, q) w.r.t. each attribute c_j (j = 1, 2, …, n).

Calculate the Shapley weighted interval value ${\bar{χ}}_{ij}^{k} = q φ_{e_{k}} (g_{λ_{j}}, E) {\bar{γ}}_{ij}^{k}$ (i = 1, 2, …, m; j = 1, 2, …, n; k = 1, 2, …, q), we obtain the Shapley weighted interval fuzzy decision matrices ${\bar{B}}^{k} = ({\bar{χ}}_{ij}^{k})_{m \times n}$ (k = 1, 2, …, q).

Confirm the fuzzy measure $g_{λ_{j}^{'}}$ of every element in Q = {1, 2, …, q} w.r.t. each attribute c_j (j = 1, 2, …, n), utilize Equation (6) to calculate the value λ w.r.t. Q, and compute the λ-fuzzy measure of every combination in Q using Equation (5).

Calculate the Shapley value $φ_{k} (g_{λ_{j}^{'}}, Q)$ of each element in Q w.r.t. each attribute c_j (j = 1, 2, …, n).

Use the UIGHλSA operator to aggregate all Shapley weighted interval decision matrices ${\bar{B}}^{k} = {({\bar{χ}}_{ij}^{k})}_{m \times n} (k = 1, 2, \dots, q)$ into a comprehensive interval decision matrix $\bar{H} = {({\bar{χ}}_{ij})}_{m \times n}$ , where ${\bar{χ}}_{ij} = [c_{ij,} d_{ij}] (i = 1, 2, \dots, m; j = 1, 2, \dots, n$ ).

Confirm the fuzzy measure g_λ (c_j) of each attribute c_j (j = 1, 2, …, n), use Equation (6) to calculate the value λ w.r.t. the attribute set C, and compute the λ-fuzzy measure of every attribute combination in C using Equation (5).

Calculate the Shapley value φ_{c
_j} (g_λ, C) of each attribute c_j (j = 1, 2, …, n).

Calculate the Shapley weighted interval value ${\bar{π}}_{ij} = n φ_{c_{j}} (g_{λ}, C) {\bar{χ}}_{ij} (i = 1, 2, \dots, m; j = 1, 2, \dots, n)$ , we get the comprehensive Shapley weighted interval decision matrix $\bar{G} = ({\bar{π}}_{ij})_{m \times n}$ .

Confirm the fuzzy measure of every element in N = {1, 2, …, n}, utilize Equation (6) to calculate the value λ w.r.t. N, and compute the λ-fuzzy measure of every combination in N using Equation(5).

Calculate the Shapley value φ_j (g_λ′, N) of each element in N.

Use the UIGHλSA operator to get the comprehensive interval value ${\bar{π}}_{i} (i = 1, 2, \dots, m)$ of the alternative a_i (i = 1, 2, …, m).

Compare each interval argument ${\bar{π}}_{i}$ with all ${\bar{π}}_{j}$ by the formula [44]:

$P ({\bar{π}}_{i} \geq {\bar{π}}_{j}) = max {1 - \frac{1}{2} max (\frac{m_{j} - m_{i}}{w i + w j} + 1, 0), 0}$ and get the complementary matrix P = (p_ij) _m×m.

Synthesize all elements of matrix P by the formula [8]: $p_{i} = \frac{\sum_{j = 1}^{m} p_{ij} + \frac{n}{2} - 1}{n (n - 1)}$ (i = 1, 2, …, m),

then to select the best one.

End.

Example 2 (cf. Example 1). There is a panel with four possible alternatives A = {a₁, a₂, a₃, a₄} to invest the money. The investment company must take a decision according to the following four attributes C = {c₁, c₂, c₃, c₄}: (1) the level of technology innovation c₁; (2) the control ability of flow c₂; (3) the ability of management c₃; (4) the level of service c₄.

The four possible alternatives a_i (i = 1, 2, 3, 4) are to be evaluated using the interval information by three decision makers e_k (k = 1, 2, 3) with scores ranging [0, 10], which are listed in the following matrices. $\begin{matrix} {\bar{A}}^{1} = (\begin{matrix} [4, 6] & [7, 9] & [7, 9] & [3, 4] \\ [7, 9] & [6, 7] & [3, 4] & [7, 8] \\ [6, 8] & [7, 8] & [7, 9] & [4, 6] \\ [6, 8] & [5, 7] & [3, 5] & [4, 6] \end{matrix}), \\ {\bar{A}}^{2} = (\begin{matrix} [4, 6] & [6, 8] & [7, 9] & [3, 5] \\ [3, 5] & [4, 6] & [3, 5] & [5, 6] \\ [7, 9] & [4, 6] & [6, 8] & [6, 8] \\ [3, 5] & [5, 7] & [3, 5] & [4, 6] \end{matrix}), \\ {\bar{A}}^{3} = (\begin{matrix} [3, 5] & [6, 8] & [6, 8] & [4, 6] \\ [6, 8] & [4, 6] & [5, 7] & [6, 8] \\ [6, 8] & [5, 7] & [2, 4] & [5, 6] \\ [3, 5] & [4, 6] & [4, 5] & [6, 7] \end{matrix}) . \end{matrix}$

Assume that the fuzzy measure g_{λ
_j} (e_k) of each expert in E = {e₁, e₂, e₃} w.r.t. each attribute c_j (j = 1, 2, 3, 4) is given as Table 5.

And the fuzzy measure $g_{λ_{j}^{'}}$ of every element in Q = {1, 2, 3} w.r.t. each attribute c_j (j = 1, 2, 3, 4) is given as Table 6, which is only determined by its ordered position.

Further, assume that the fuzzy measure g_λ (c_j) of each attribute c_j (j = 1, 2, 3, 4) is given by $\begin{matrix} g_{λ} (c_{1}) & = & 0.3, g_{λ} (c_{2}) = 0.2, g_{λ} (c_{3}) = 0.25, \\ g_{λ} (c_{4}) & = & 0.2, \end{matrix}$ and the fuzzy measure g_λ′ of each element in N = {1, 2, 3, 4} is defined by $\begin{matrix} g_{λ^{'}} (1) & = & 0.2, g_{λ^{'}} (2) = 0.1, g_{λ^{'}} (3) = 0.3, \\ g_{λ^{'}} (4) & = & 0.25, \end{matrix}$ which is determined by their ordered positions. To get the best alternative(s), the following steps are involved:

See Step I-1;

From Table 5, we get the following values w.r.t. the expert set E and each attribute c_j (j = 1, 2, 3, 4)

λ_{1} = - 0.53, λ_{2} = - 0.37, λ_{3} = 0.61, λ_{4} = 0.17 .

From the values of λ_j (j = 1, 2, 3, 4), we get the λ-fuzzy measure of every combination in E w.r.t. each attribute c_j (j = 1, 2, 3, 4) as shown in Table 7.

From Tables 5 and 7, we get the experts’ Shapley values w.r.t. each attribute c_j (j = 1, 2, 3, 4) as listed in Table 8.

Calculate the Shapley weighted interval value ${\bar{χ}}_{ij}^{k} = 3 φ_{e_{k}} (g_{λ_{j}}, E) {\bar{γ}}_{ij}^{k}$ (i = 1, 2, 3, 4 ;j = 1, 2, 3, 4 ; k = 1, 2, 3), we get the Shapley weighted interval fuzzy decision matrices ${\bar{B}}^{k} = ({\bar{χ}}_{ij}^{k})_{m \times n}$ (k = 1, 2, 3) as listed follows:

\begin{matrix} {\bar{B}}^{1} = (\begin{matrix} [4.9, 7.4] & [6.3, 8.1] & [7.4, 9.5] & [2.9, 3.8] \\ [8.6, 11.1] & [5.4, 6.3] & [3.2, 4.2] & [6.7, 7.7] \\ [7.4, 9.8] & [6.3, 7.2] & [7.4, 9.5] & [3.8, 5.8] \\ [7.4, 9.8] & [4.5, 6.3] & [3.2, 5.3] & [3.8, 5.8] \end{matrix}), \\ {\bar{B}}^{2} = (\begin{matrix} [2.8, 4.1] & [7.9, 10.6] & [8.4, 10.8] & [2.4, 4.1] \\ [2.1, 3.5] & [5.3, 7.9] & [3.6, 6] & [4.1, 4.9] \\ [4.8, 6.2] & [5.3, 7.9] & [7.2, 9.6] & [4.9, 6.5] \\ [2.1, 3.5] & [6.6, 9.2] & [3.6, 6] & [3.2, 4.9] \end{matrix}), \\ {\bar{B}}^{3} = (\begin{matrix} [3.2, 5.4] & [4.5, 6] & [4.3, 5.8] & [5, 7.6] \\ [6.5, 8.6] & [3, 4.5] & [3.6, 5] & [7.6, 10.1] \\ [6.5, 8.6] & [3.8, 5.3] & [1.4, 2.9] & [6.3, 7.6] \\ [3.2, 5.4] & [3, 4.5] & [2.9, 3.6] & [7.6, 8.8] \end{matrix}) . \end{matrix}

See Step I-2.

From Step I-3, we get the Shapley values of the elements in Q w.r.t. each attribute c_j (j = 1, 2, 3, 4) as listed in Table 9.

When γ=2 and u_k = φ_{e
_k} (g_{λ
_j}, E) (k = 1, 2, 3 ; j = 1, 2, 3, 4), by the UIGHλSA operator we get the comprehensive interval decision matrix $\bar{H} = {({\bar{χ}}_{ij})}_{4 \times 4}$ as follows:

\bar{H} = (\begin{matrix} [3.1, 4.6] & [5.5, 7.4] & [5.6, 7.2] & [3.4, 5.2] \\ [4.5, 6] & [3.7, 5.5] & [2.8, 4.4] & [5.2, 6.8] \\ [4.9, 6.4] & [4, 5.8] & [4.5, 6.1] & [4.7, 5.9] \\ [3.9, 5.4] & [4.3, 6.2] & [2.6, 4.1] & [5, 6.1] \end{matrix}) .

From the fuzzy measure g_λ (c_j) of each attribute c_j (j = 1, 2, 3, 4), we obtain λ = 0.15. By Equation (9), we get the following λ-fuzzy measure of each attribute combination in C:

\begin{matrix} g_{λ} (c_{1}, c_{2}) = g_{λ} (c_{1}, c_{4}) = 0.51, g_{λ} (c_{1}, c_{3}) = 0.56, \\ g_{λ} (c_{2}, c_{3}) = g_{λ} (c_{3}, c_{4}) = 0.46, g_{λ} (c_{2}, c_{4}) = 0.41, \\ g_{λ} (c_{1}, c_{2}, c_{3}) = g_{λ} (c_{1}, c_{3}, c_{4}) = 0.78, g_{λ} (c_{1}, c_{2}, c_{4}) \\ = 0.72, g_{λ} (c_{2}, c_{3}, c_{4}) = 0.67, g_{λ} (C) = 1 . \end{matrix}

From Step II-8, we get the following attributes’ Shapley values

\begin{matrix} φ_{c_{1}} (g_{λ}, C) & = & 0.314, φ_{c_{2}} (g_{λ}, C) = φ_{c_{4}} (g_{λ}, C) = 0.211, \\ φ_{c_{3}} (g_{λ}, C) & = & 0.264 . \end{matrix}

Calculate the Shapley weighted interval value n ${\bar{π}}_{ij} = 4 φ_{c_{j}} (g_{λ}, C) {\bar{χ}}_{ij} (i = 1, 2, 3, 4; j = 1, 2, 3, 4)$ . We get the comprehensive Shapley weighted interval decision matrix $\bar{G} = ({\bar{π}}_{ij})_{4 \times 4}$ as follows:

\bar{G} = (\begin{matrix} [3.9, 5.8] & [4.6, 6.2] & [5.9, 7.6] & [2.8, 4.4] \\ [5.6, 7.5] & [3.1, 4.6] & [3, 4.7] & [4.4, 5.7] \\ [6.2, 8.1] & [3.4, 4] & [4.8, 6.4] & [4, 5] \\ [4.9, 6.8] & [3.7, 5.2] & [2.8, 4.3] & [4.2, 5.1] \end{matrix}) .

See Step I-5.

From Step I-6, we get the Shapley value of each element in N as listed follows: φ₁ (g_λ′, N) =0.23, φ₂ (g_λ′, N) =0.15, φ₃ (g_λ′, N) =0.34, φ₄ (g_λ′, N) =0.29.

When γ= 2 and u_j = φ_{c
_j} (g_λ, C) (j = 1, 2, 3, 4), by the UIGHλSA operator we get

\begin{matrix} {\bar{π}}_{1} & = & [3.6, 5.2], {\bar{π}}_{2} = [4.1, 5.5], \\ {\bar{π}}_{3} & = & [4.2, 5.3], {\bar{π}}_{4} = [3.5, 5, 4] . \end{matrix}

By $P ({\bar{π}}_{i} \geq {\bar{π}}_{j}) = max {1 - \frac{1}{2} max (\frac{m_{j} - m_{i}}{w i + w j}$ + 1, 0) , 0 }, we get the complementary matrix P = (p_ij) _4×4 as follows:

P = (\begin{matrix} 0.5 & 0.36 & 0.36 & 0.48 \\ 0.64 & 0.5 & 0.52 & 0.61 \\ 0.64 & 0.48 & 0.5 & 0.6 \\ 0.52 & 0.39 & 0.4 & 0.5 \end{matrix}) .

From $p_{i} = \frac{\sum_{j = 1}^{m} p_{ij} + \frac{n}{2} - 1}{n (n - 1)}$ , we obtain

p_{1} = 0.225, p_{2} = 0.273, p_{3} = 0.268, p_{4} = 0.234 .

Thus, the alternative a₂ is the best investment company.

From the provided example, we know the different optimal choices are obtained using the UIGλSA and UIGHλSA operators. The main reason is that the UIGλSA operator does not consider the importance of the attributes or the interactions between them. To consider more information, we recommend the experts use the UIGHλSA operator to make decision in real problems.

5 Conclusion

Based on the Shapley function, we have defined several uncertain aggregation operators, where the interactive phenomenon between elements is considered. Some desirable properties are studied, which are very important to their application. To simplify the complexity of solving a fuzzy measure, the UIGλSA and UIGHλSA operators are defined using λ-fuzzy measures. As a series of development, two new methods to uncertain multi-attribute group decision making with interactive conditions are developed. Both of them consider the importance of the experts’ ordered positions w.r.t. each attribute and reflect the interactions between them usingλ-fuzzy measures. Furthermore, the second method gives the importance of experts and attributes that considers more information than the first one. There are two main differences between the new methods and the existing ones. One is: the former is based on fuzzy measures, while the latter is based on additive measures. The other is: the former gives the importance of experts and (or) their ordered positions w.r.t. each attribute, while the latter considers their importance being equal w.r.t. different attributes. However, both of them are based on the uncertain Shapley arithmetic weighted operator, and it will be interesting to develop new methods using other hybrid aggregation operators, such as the weighted OWA operator [35], the importance OWA [49], the induced OWA-weighted average operator [27], the immediate weights [29, 50] and the probability OWA operators [28].

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 71201089, 71201110, 27127117 and 71271029), the Funds for Creative Research Groups of China (No. 71221061), the Projects of Major International Cooperation NSFC (No. 71210003), the Natural Science Foundation Youth Project of Shandong Province, China (ZR2012GQ005), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20111101110036), and the Program for New Century Excellent Talents in University of China (No. NCET-12-0541).

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