MAGDM method based on interval grey linguistic correlated ordered geometric operators

Abstract

The multiple attribute group decision making (MAGDM) problem, in which both the attribute weights and the expert weights are correlated and the attribute values take the form of interval grey linguistic (IGL) variables, is investigated in this paper. First, based on the relative concepts of IGL variables and the operation rules between IGL variables, two new aggregation operators are developed, which are the interval grey linguistic correlated ordered geometric (IGLCOG) operator and the induced interval grey linguistic correlated ordered geometric (I-IGLCOG) operator. Then, some desirable properties of the I-IGLCOG operator are studied, such as commutativity, idempotency, monotonicity and boundedness. Next, the IGLCOG and I-IGLCOG operator-based approaches are developed to solve the abovementioned MAGDM problem. Finally, an illustrative examples are given to verify the developed approach and demonstrate its practicality and effectiveness.

Keywords

Multiple attribute group decision making interval grey linguistic variables interval grey linguistic correlated ordered geometric operator induced interval grey linguistic correlated ordered geometric operator

1 Introduction

Multiple attribute group decision making (MAGDM) has been extensively applied to various areas, such as society, economics, management, and the military [1 –10]. It is well known that the objective things are complex and uncertain and human thinking is ambiguous. The majority of decision making is also uncertain, and fuzziness is the major factor in the process. However, in dealing with the problem of incomplete information caused by poor information, decision-making also demonstrates its greyness. “Fuzzy” means those uncertain factors in the evaluation information that are caused by the fuzziness of human thinking, while “grey” means the objective uncertainty caused by the insufficient and incomplete information. Therefore, “fuzzy” and “grey” are different concepts. Since Zadeh introduced the concept of the fuzzy set [11] and Deng presented the grey system theory [12], grey fuzzy decision making has been widely investigated and applied to a variety of fields. And many scholars have studied grey fuzzy multiple attribute decision making that demonstrates its fuzziness and its greyness.

However, in many real-life decision-making problems, the linguistic variable more easily expresses fuzzy information and is closer to the actual condition [13 –15]. Therefore, Liu and Jin defined the concept of interval grey linguistic (IGL) variables whose fuzzy part and grey part take the form of linguistic variables and interval numbers, respectively. They further studied the operation rules of IGL variables and developed the MAGDM method based on the IGL variables [16]. Liu and Zhang proposed the interval grey linguistic weighted geometric (IGLWG) operator, the interval grey linguistic ordered weighted geometric (IGLOWG) operator and the interval grey linguistic hybrid weighted geometric (IGLHWG) operator, and then suggested a method for solving MAGDM problems based on these operators [17].

The existing grey fuzzy MAGDM method only considers the situation where all the elements in the grey fuzzy sets are independent. However, in many practical situations, the elements in the grey fuzzy sets are usually correlated. Therefore, it is necessary to find some new ways to address the situations in which the decision data in question are correlated and the weights are correlated. The Choquet integral is a very useful way of measuring the expected utility of an uncertain event, and it can be utilized to depict the correlations among the decision data under consideration [18 –21]. The aim of this paper is to investigate the MAGDM problem in which both the attribute weights and the expert weights are correlated and the attribute values take the form of IGL variables. Motivated by the correlation properties of the Choquet integral and the induced operators with uncertain linguistic variables, the interval grey linguistic correlated ordered geometric (IGLCOG) operator and the induced interval grey linguistic correlated ordered geometric (I-IGLCOG) operator with IGL variables are developed. The prominent characteristic of the operators is that they consider the importance of the elements or their ordered positions, and they also reflect the correlations among the elements or their ordered positions. Then, the proposed operators are developed to address the MAGDM problems with IGL information.

The remainder of this paper is as follows. In the next section, some basic concepts related to IGL variables and the operational laws of IGL variables are introduced. In Section 3, the IGLCOG operator and the I-IGLCOG operator are developed. In Section 4, an approach to MAGDM in which both the attribute weights and the expert weights are correlated and the attribute values take the form of IGL variables based on the IGLCOG operator and the I-IGLCOG operator is developed. In Section 5, an illustrative example for selecting a management information system (MIS) with IGL information is addressed. Section 6 concludes the paper and gives some remarks.

2 Preliminaries

In this section, some basic concepts to be used throughout the paper are reviewed briefly.

Definition 1. Let $\underset{\otimes}{\tilde{A}} (x) = (\tilde{A} (x), \underset{\otimes}{A} (x))$ be the grey fuzzy number. If its fuzzy part is a linguistic variable $\tilde{s} = [s_{α}, s_{β}]$ where $s_{α}, s_{β} \in \bar{S}$ , $\bar{S}$ is a finite and totally ordered discrete term set, and its grey part $\underset{\otimes}{A} (x)$ is a closed interval $[g_{A}^{L}, g_{A}^{U}]$ , then $\underset{\otimes}{\tilde{A}} (x)$ is called the IGL variable.

Definition 2. An interval grey linguistic weighted geometric (IGLWG) operator of dimension n is a function Ωⁿ → Ω, which is associated with a set of weights or a weighting vector w = (ω₁, ω₂, …, ω_n) to it with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ . It is defined to aggregate a list of values ${\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n})}$ according to the following expression: $\begin{matrix} IGLWG (\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n})) \\ = \otimes_{j = 1}^{n} {(\underset{\otimes}{\tilde{A}} (x_{j}))}^{ω_{j}} = {[s_{\prod_{j = 1}^{n} (α_{j})^{ω_{j}}}, s_{\prod_{j = 1}^{n} (β_{j})^{ω_{j}}}] \\ [(1 - \prod_{j = 1}^{n} (1 - g_{j}^{L})), (1 - \prod_{j = 1}^{n} (1 - g_{j}^{U}))]} \end{matrix},$ (1)

where $\underset{\otimes}{\tilde{A}} (x_{j}) = ([{\tilde{s}}_{α_{j}}, {\tilde{s}}_{β_{j}}], [g_{j}^{L}, g_{j}^{U}])$ , (τ (1), τ (2), ⋯, τ (n)) is a permutation of the set (1, 2, ⋯, n) such that $\underset{\otimes}{\tilde{A}} (x_{τ (1)}) \geq \underset{\otimes}{\tilde{A}} (x_{τ (2)}) \geq \dots \geq \underset{\otimes}{\tilde{A}} (x_{τ (n)})$ . That is, $\underset{\otimes}{\tilde{A}} (x_{τ (j)})$ is the jth largest value in the set ${\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n})}$ .

Definition 3. An interval grey linguistic ordered weighted geometric (IGLOWG) operator of dimension n is a function Ωⁿ → Ω, which is associated with a set of weights or a weighting vector w = (ω₁, ω₂, ⋯, ω_n) to it with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ . It is defined to aggregate a list of values ${\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n})}$ according to the following expression: $\begin{matrix} IGLOWG (\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n})) \\ = \otimes_{j = 1}^{n} {(\underset{\otimes}{\tilde{A}} (x_{τ (j))})}^{ω_{j}} = {[s_{\prod_{j = 1}^{n} (α_{τ (j)})^{ω_{j}}}, s_{\prod_{j = 1}^{n} (β_{τ (j)})^{ω_{j}}}] \\ [(1 - \prod_{j = 1}^{n} (1 - g_{τ (j)}^{L})), (1 - \prod_{j = 1}^{n} (1 - g_{τ (j)}^{U}))]} \end{matrix},$ (2)

3 Interval grey linguistic correlated ordered geometric operators

In MAGDM, the considered attributes usually have different levels of importance, and thus need to be assigned different weights. Some operators have been introduced to aggregate the IGL variables together with independent weighted elements, but they only consider the addition of the importance of individual elements. However, in some practical situations, the elements in the IGL variables have some correlations with each other, and thus it is necessary to consider this issue. For real decision-making problems, there is always some degree of interdependent characteristics among attributes. Usually, there is interaction among the attributes of decision makers. However, this assumption is too strong to match the decision behaviors in the real world. The independence axiom generally cannot be satisfied. Thus, it is necessary to consider this issue.

Let m (x_i) (i = 1, 2, …, n) be the weight of the element x_i ∈ X (i = 1, 2, …, n), where m is a fuzzy measure. It is defined as follows.

Definition 4. [22] A fuzzy measure m on the set X is a set function m : θ (x) → [0, 1] satisfying the following axioms:

m (φ) =0, m (X) =1,

A ⊆ B implies m (A) ≤ m (B) for all A, B ⊆ X, and

m (A ∪ B) = m (A) + m (B) + ρm (A) m (B) for all A, B ⊆ X and A ∩ B = φ, where ρ ∈ (-1, ∞).

Especially, if ρ = 0, then condition (3) reduces to the axiom of additive measure that follows:

$\begin{matrix} m (A \cup B) = m (A) + m (B) \\ for all A, B \subseteq X and A \cap B = φ . \end{matrix}$ (3)

If all the elements in X are independent, then $m (A) = \sum_{x_{i} \in A} m ({x_{i}}) for all A \subseteq X .$ (4)

Based on Definition 3, the well-known Choquet integral is then used to develop an operator for aggregating the IGL variables with the correlated weights.

Definition 5. Let m be a fuzzy measure on X, and $\underset{\otimes}{\tilde{A}} (x_{j}) = ([s_{α_{j}}, s_{β_{j}}], [g_{j}^{L}, g_{j}^{U}])$ (j = 1, 2, …, n) be n IGL variables. Then, the interval grey linguistic correlated ordered geometric (IGLCOG) operator is defined as follows:

$(C) \int \underset{\otimes}{\tilde{A}} (x) dm = IGLCOG (\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n})) = {[s_{\prod_{j = 1}^{n} (α_{σ (j)})^{m (B (x_{τ (j)})) - m (B (x_{τ (j - 1)}))}}, s_{\prod_{j = 1}^{n} (β_{σ (j)})^{m (B (x_{τ (j)})) - m (B (x_{τ (j - 1)}))}},], [(1 - \prod_{j = 1}^{n} (1 - g_{σ (j)}^{L})), (1 - \prod_{j = 1}^{n} (1 - g_{σ (j)}^{U}))]}$ (5)

where $(C) \int \underset{\otimes}{\tilde{A}} dm$ denotes the Choquet integral and (τ (1), τ (2), ⋯, τ (n)) is a permutation of the set (1, 2, ⋯, n) such that $\underset{\otimes}{\tilde{A}} (x_{τ (1)}) \geq \underset{\otimes}{\tilde{A}} (x_{τ (2)}) \geq \dots \geq \underset{\otimes}{\tilde{A}} (x_{τ (n)})$ . That is, $\underset{\otimes}{\tilde{A}} (x_{τ (j)})$ is the jth largest value in the set ${\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n})}$ . Moreover, B (x_τ(j)) ={ x_τ(k)|k < j, k ≥ 1 } and B (x_τ(0)) = φ, whose aggregated value is also an IGL variable.

Next, two special cases of the IGLCOG operator are discussed.

(1) If ρ = 0, then m (B ∪ C) = m (B) + m (C) and m ({ x_τ(j) }) = m (B (x_τ(j))) - m (B (x_τ(j-1))), j = 1, 2, …, n. In this case, the IGLCOG operator reduces to the IGLWG operator: $\begin{matrix} IGLWG (\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n})) \\ = \otimes_{j = 1}^{n} \underset{\otimes}{\tilde{A}} {((x_{j}))}^{m ({x_{j}})} = {[s_{\prod_{j = 1}^{n} (α_{j})^{m ({x_{j}})}}, s_{\prod_{j = 1}^{n} (β_{j})^{m ({x_{j}})}}] \\ [(1 - \prod_{j = 1}^{n} (1 - g_{j}^{L})), (1 - \prod_{j = 1}^{n} (1 - g_{j}^{U}))]} \end{matrix},$ (6)

Particularly, if $m ({x_{j}}) = \frac{1}{n}$ for all j = 1, 2, …, n, then the IGLWG operator reduces to the interval grey linguistic geometric (IGLG) operator:

$\begin{matrix} IGLG (\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n})) \\ = {[s_{\prod_{j = 1}^{n} (α_{j})^{\frac{1}{n}}}, s_{\prod_{j = 1}^{n} (β_{j})^{\frac{1}{n}}}], \\ [1 - \prod_{j = 1}^{n} (1 - g_{j}^{L}), 1 - \prod_{j = 1}^{n} (1 - g_{j}^{U})]} \end{matrix}$ (7)

(2) If $m (B) = \sum_{j = 1}^{| B |} ω_{j}$ for all B ⊆ X, where |B| is the number of elements in the set B, then ω_j = m (B (x_τ(j))) - m (B (x_τ(j-1))), j = 1, 2, …, n, where ω = (ω₁, ω₂, …, ω_n) ^T, ω_j ≥ 0, j = 1, 2, …, n, and $\sum_{j = 1}^{n} ω_{j} = 1$ . In this case, the IGLCOG operator reduces to the IGLOWG operator: $\begin{matrix} IGLOWG (\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n})) \\ = \otimes_{j = 1}^{n} \underset{\otimes}{\tilde{A}} {(x_{τ (j)})}^{ω_{j}} = {[s_{\prod_{j = 1}^{n} (α_{τ (j)})^{ω_{j}}}, s_{\prod_{j = 1}^{n} (β_{τ (j)})^{ω_{j}}}] \\ [(1 - \prod_{j = 1}^{n} (1 - g_{τ (j)}^{L})), (1 - \prod_{j = 1}^{n} (1 - g_{τ (j)}^{U}))]} \end{matrix},$ (8)

In particular, if $m (B) = \frac{| B |}{n}$ for all B ⊆ X, then both the IGLCOG operator and the IGLOWG operator reduce to the IGLG operator.

Definition 6. Let $\underset{\otimes}{\tilde{A}} (x_{j}) = ([{\tilde{s}}_{α_{j}}, {\tilde{s}}_{β_{j}}],$ $[g_{j}^{L}, g_{j}^{U}]) (j = 1, 2, \dots), n)$ be a collection of IGL variables, and μ be a fuzzy measure on X. An induced interval grey linguistic correlated ordered geometric (I-IGLCOG) operator is defined as follows:

$\begin{matrix} I - IGLCOG (〈 μ_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 μ_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \dots, \\ 〈 μ_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) \\ = \otimes_{j = 1}^{n} \underset{\otimes}{\tilde{A}} {(x_{σ (j)})}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} \\ = {[s_{\prod_{j = 1}^{n} {(α_{σ (j)})}^{(m (B (x_{σ (j - 1)})))}}, s_{\prod_{j = 1}^{n} {(β_{σ (j)})}^{(m (B (x_{σ (j - 1)})))}}], \\ [1 - \prod_{j = 1}^{n} (1 - g_{σ (j)}^{L}), 1 - \prod_{j = 1}^{n} (1 - g_{σ (j)}^{U})]} \end{matrix}$ (9)

where (σ (1), σ (2), …, σ (n)) is a permutation of the set (1, 2, …, n) such that $\underset{\otimes}{\tilde{A}} (x_{σ (1)}) \geq \underset{\otimes}{\tilde{A}} (x_{σ (2)}) \geq \dots \geq \underset{\otimes}{\tilde{A}} (x_{σ (n)})$ . That is, $〈 u_{j}, \underset{\otimes}{\tilde{A}} (x_{j}) 〉$ is the 2-tuple with u_σ(j) the jth largest values in the set {u₁, u₂, …, u_n}. u_j in $〈 u_{j}, \underset{\otimes}{\tilde{A}} (x_{j}) 〉$ is referred to as the order inducing variable and $\underset{\otimes}{\tilde{A}} (x_{j}) = ([s_{α_{j}}, s_{β_{j}}], [g_{j}^{L}, g_{j}^{U}])$ as the IGL value. B (x_σ(j)) ={ x_σ(k)|k ≤ j, k ≥ 1 } for k ≥ 1 and B (x_σ(0)) = φ.

In the above definition, the reordering of the set of values to be aggregated, $(\underset{\otimes}{\tilde{A}} (x_{1}), \underset{\otimes}{\tilde{A}} (x_{2}), \dots, \underset{\otimes}{\tilde{A}} (x_{n}))$ , is induced by the reordering of the set of values {u₁, u₂, …, u_n} associated with them, which is based on their magnitudes. The main difference between the IGLCOG operator and the I-IGLCOG operator resides in the reordering step of the argument variable. In the case of the IGLCOG operator, this reordering is based on the magnitude of the IGL values to be aggregated, whereas in the case of the I-IGLCOG operator an order-inducing variable is used as the criterion to induce that reordering. Obviously, in Definition 6, if $u_{j} = \underset{\otimes}{\tilde{A}} (x_{j})$ for all j, then the I-IGLCOG operator reduces to the IGLCOG operator.

Similar to other operators, the I-IGLCOG operator has the following properties.

Theorem 1. (Commutativity). I - IGLCOG $(〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \dots, 〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) =$ $I - IGLCOG (〈 u_{1}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{1}) 〉, 〈 u_{2}^{'}, {\tilde{A}}_{\otimes}^{'} (x_{2}) 〉, \dots,$ $〈 u_{n}^{'}, {\tilde{A}}_{\otimes}^{'} (x_{n}) 〉), where (〈 u_{1}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{1}) 〉, 〈 u_{2}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{2}) 〉,$ $\dots, 〈 u_{n}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{n}) 〉)$ is any permutation of $(〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \dots, 〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉)$ .

Proof. Let $I - IGLCOG (〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉,$ $\dots, 〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉) = \otimes_{j = 1}^{n} {(\underset{\otimes}{\tilde{A}} (x_{σ (j)}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))},$ $I - IGLCOG (〈 u_{1}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{1}) 〉, 〈 u_{2}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{2}) 〉, \dots,$ $〈 u_{n}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{n}) 〉) = \otimes_{j = 1}^{n} {({\tilde{A}}_{\otimes}^{'} (x_{σ (j)}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} .$

Since $(〈 u_{1}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{1}) 〉, 〈 u_{2}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{2}) 〉, \dots,$ $〈 u_{n}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{n}) 〉)$ is any permutation of $(〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉,$

$〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \dots, 〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉)$ , it follows that $\underset{\otimes}{\tilde{A}} (x_{σ (j)}) = \underset{\otimes}{{\tilde{A}}^{'}} (x_{σ (j)}) (j = 1, 2, \dots, n)$ , and then

$\begin{matrix} I - IGLCOG (〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉,) \\ \dots, 〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) \\ = I - IGLCOG (〈 u_{1}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{1}) 〉, 〈 u_{2}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{2}) 〉,) \\ \dots, 〈 u_{n}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{n}) 〉) . \end{matrix}$

Theorem 2. (Idempotency). If $\underset{\otimes}{\tilde{A}} (x_{j}) = ([s_{α_{j}}, s_{β_{j}}],$ $[g_{j}^{L}, g_{j}^{U}]) = \underset{\otimes}{\tilde{A}} (x) = ([s_{α}, s_{β}], [g^{L}, g^{U}])$ for all j, then I - IGLCOG $(〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉,$ $〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \dots,$ $〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) = \underset{\otimes}{\tilde{A}} (x) = ([s_{α}, s_{β}], [g^{L}, g^{U}]) .$

Proof. Since $\underset{\otimes}{\tilde{A}} (x_{j}) = \underset{\otimes}{\tilde{A}} (x)$ for all j, it yields that $\begin{matrix} I - IGLCOG (〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \dots, \\ 〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) \\ = \otimes_{j = 1}^{n} \underset{\otimes}{\tilde{A}} {(x_{σ (j)})}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} \\ = \otimes_{j = 1}^{n} \underset{\otimes}{\tilde{A}} {(x)}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} = \underset{\otimes}{\tilde{A}} (x) . \end{matrix}$

Theorem 3. (Monotonicity). If $\underset{\otimes}{\tilde{A}} (x_{j}) \leq \underset{\otimes}{{\tilde{A}}^{'}} (x_{j})$ (j = 1, 2, …, n), then $\begin{matrix} I - IGLCOG (〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉 \\ , \dots, 〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) \\ \leq I - IGLCOG (〈 u_{1}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{1}) 〉, 〈 u_{2}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{2}) 〉 \\ , \dots, 〈 u_{n}^{'}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{n}) 〉) . \end{matrix}$

Proof. Let $I - IGLCOG (〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉,$ $〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \dots,$ $〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉)$ $= \otimes_{j = 1}^{n} {(\underset{\otimes}{\tilde{A}} (x_{σ (j)}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))},$

$I - IGLCOG (〈 u_{1}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{2}) 〉,$ $\dots, 〈 u_{n}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{n}) 〉)$ $= \otimes_{j = 1}^{n} {(\underset{\otimes}{{\tilde{A}}^{'}} (x_{σ (j)}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} .$ .

Since $\underset{\otimes}{\tilde{A}} (x_{j}) \leq \underset{\otimes}{{\tilde{A}}^{'}} (x_{j}) (j = 1, 2, \dots, n)$ , it follows that $\underset{\otimes}{\tilde{A}} (x_{σ (j)}) \leq \underset{\otimes}{\tilde{A}} (x_{σ (j)})$ (j = 1, 2, …, n), and then $\begin{matrix} I - IGLCOG (〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉 \\ , \dots, 〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) \\ \leq I - GLCOG (〈 u_{1}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{2}) 〉 \\ , \dots, 〈 u_{n}, \underset{\otimes}{{\tilde{A}}^{'}} (x_{n}) 〉) . \end{matrix}$

Theorem 4. (Bounded). $\underset{j}{Min} \underset{\otimes}{\tilde{A}} (x_{j}) \leq I - IGLCOG (〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \dots, 〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) \leq \underset{j}{Max} \underset{\otimes}{\tilde{A}} (x_{j})$

Proof. Let $I - IGLCOG (〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉,$ $〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \dots, 〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) = \otimes_{j = 1}^{n} \underset{\otimes}{\tilde{A}} {(x_{σ (j)})}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} .$

Since $\underset{j}{Min} \underset{\otimes}{\tilde{A}} (x_{j}) \leq \underset{\otimes}{\tilde{A}} (x_{j}) \leq \underset{j}{Max} \underset{\otimes}{\tilde{A}} (x_{j})$ (j = 1, 2, …, n), then $\begin{matrix} \otimes_{j = 1}^{n} {(\underset{j}{Min} \underset{\otimes}{\tilde{A}} (x_{j}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} \\ \leq \otimes_{j = 1}^{n} {(\underset{\otimes}{\tilde{A}} (x_{σ (j)}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} \\ \leq \otimes_{j = 1}^{n} {(\underset{j}{Max} \underset{\otimes}{\tilde{A}} (x_{j}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} \end{matrix}$ .

Since $\otimes_{j = 1}^{n} {(\underset{j}{Min} \underset{\otimes}{\tilde{A}} (x_{j}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} = \underset{j}{Min} \underset{\otimes}{\tilde{A}} (x_{j}) and \otimes_{j = 1}^{n} {(\underset{j}{Max} \underset{\otimes}{\tilde{A}} (x_{j}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} = \underset{j}{Max} \underset{\otimes}{\tilde{A}} (x_{j})$ , it yields that $\underset{j}{Min} \underset{\otimes}{\tilde{A}} (x_{j}) \leq I - IGLCOG (〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \dots,$ $〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) \leq \underset{j}{Max} \underset{\otimes}{\tilde{A}} (x_{j})$ .

4 An approach to MAGDM based on the IGLCOG operators

In this section, we develop an approach to MAGDM in which both the attribute weights and the expert weights are correlated and the attribute values take the form of (IGL variables. We apply the IGLCOG and I-IGLCOG operators to MAGDA based on the interval grey uncertain linguistic information.

Let A = {a₁, a₂, …, a_m} be a discrete set of alternatives, X = {x₁, x₂, …, x_n} be the set of attributes, and m be a fuzzy measure on X, where 0 ≤ m ({x₁, …, x_j}) ≤ 1, m ({X}) =1 and m ({φ}) =0. Assume that D = {d₁, d₂, …, d_t} is the set of decision makers and μ is a fuzzy measure on D, where 0 ≤ μ ({d₁, …, d_k}) ≤1, μ ({D}) =1 and μ ({φ}) =0. Suppose that ${\tilde{R}}_{k} = {(\underset{\otimes}{{\tilde{A}}^{(k)}} (x_{j}))}_{m \times n}$ is the IGL decision matrix, where ${\tilde{A}}_{\otimes}^{(k)} (x_{j})$ indicates the attribute value of the alternative a_i that satisfies the attribute x_j given by the decision maker d_k, $u_{j}^{(k)}$ indicates the inducing value of the decision maker d_k that satisfies the attribute x_j, i = 1, 2, …, m, j = 1, 2, …, n, and k = 1, 2, …, t.

Then, the IGLCOG and I-IGLCOG operators are applied to MAGDM with IGL information. The method involves the following steps.

Step 1. Utilize the decision information given in matrix ${\tilde{R}}_{k}$ and the I-IGLCOG operator, which has the associated weighting vector ω = (ω₁, ω₂, …, ω_n) $\begin{matrix} r_{i}^{(k)} = I - IGLCOG (〈 u_{1}, \underset{\otimes}{\tilde{A}} (x_{1}) 〉, 〈 u_{2}, \underset{\otimes}{\tilde{A}} (x_{2}) 〉, \\ \dots, 〈 u_{n}, \underset{\otimes}{\tilde{A}} (x_{n}) 〉) \\ = \otimes_{j = 1}^{n} \underset{\otimes}{\tilde{A}} {(x_{σ (j)})}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} \\ = {[s_{\prod_{j = 1}^{n} {(α_{σ (j)})}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))}}], \\ s_{\prod_{j = 1}^{n} {(β_{σ (j)})}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))}}], \\ [1 - \prod_{j = 1}^{n} (1 - g_{σ (j)}^{L}), 1 - \prod_{j = 1}^{n} (1 - g_{σ (j)}^{U})]}, \end{matrix}$

to aggregate all the decision matrices ${\tilde{R}}_{k} (k = 1, 2, \dots, t)$ into a collective decision matrix $\tilde{R} = {(r_{i}^{(k)})}_{m \times t}$ , where m is a fuzzy measure on X, 0 ≤ m ({ x₁, …, x_j }) ≤ 1, m ({X}) =1 and m ({φ}) =0 (j = 1, 2, …, n).

Step 2. Utilize the decision information given in matrix $\tilde{R}$ and the IGLCOG operator $\begin{matrix} {\tilde{z}}_{i} = IGLCOG ({\tilde{r}}_{i}^{(1)}, {\tilde{r}}_{i}^{(2)}, \dots, {\tilde{r}}_{i}^{(t)}) \\ = \otimes_{k = 1}^{t} {({\tilde{r}}_{i}^{(k)})}^{(μ (B (d_{τ (k)})) - μ (B (d_{τ (k - 1)})))}, i = 1, 2, \dots, m \end{matrix}$ (10)

to derive the collective overall preference values ${\tilde{z}}_{i} (i = 1, 2, \dots, m)$ of the alternative A_i, where μ is a fuzzy measure on D, 0 ≤ μ ({d₁, …, d_k}) ≤ 1, μ ({D}) =1 and μ ({φ}) =0.

Step 3. Compute the size of the collective overall preference values $\tilde{Q} ({\tilde{z}}_{i}) (i = 1, 2, \dots, m)$ : $\tilde{Q} ({\tilde{z}}_{i}) = [s_{α_{1}}, s_{β_{1}}] \times f_{p} ([(1 - g_{A}^{L}), (1 - g_{A}^{U})])$ (11)

Step 4. To rank the size of these collective overall preference values $\tilde{Q} ({\tilde{z}}_{i})$ (i = 1, 2, …, m), we first compare each $\tilde{Q} ({\tilde{z}}_{i})$ with all $\tilde{Q} ({\tilde{z}}_{i})$ (i = 1, 2, …, m) [23]. For simplicity, we let $p_{ij} = p (\tilde{Q} ({\tilde{z}}_{i}) \geq \tilde{Q} ({\tilde{z}}_{j}))$ , and then we develop a complementary matrix as P = (p_ij) _m×m, where p_ij ≥ 0, p_ij + p_ji = 1, $p_{ii} = \frac{1}{2}$ , and i, j = 1, 2 …, m. By summing all elements in each line of matrix P, we have $v_{i} = \frac{\sum_{j = 1}^{m} p_{ij} + \frac{m}{2} - 1}{m (m - 1)}$ , i = 1, 2, …, m. Then, we rank $\tilde{Q} ({\tilde{z}}_{i}) (i = 1, 2, \dots, m)$ in descending order in accordance with the values of v_i (i = 1, 2, …, m).

Step 5. The ranking of the alternatives can be gained and the best one can be discovered.

5 Illustrative example

To demonstrate the application of the developed method, we consider an example for MIS selection with IGL information. Suppose there are four possible MISs {A₁, A₂, A₃, A₄} to be evaluated by three decision makers {D₁, D₂, D₃} according to four attributes: system performance (x₁), internal processes (x₂), learning and growth (x₃), and social objectives (x₄). The inducing variables U are as shown in Table 1, and the decision matrices ${\tilde{R}}_{k} = {({\underset{\otimes}{\tilde{A}}}_{i}^{(k)} (x_{j}))}_{4 \times 4} (k = 1, 2, 3)$ are listed as follows.

$\begin{matrix} R_{1} = [\begin{matrix} ([s_{3}, s_{4}], [0.2, 0.5]) & ([s_{2}, s_{4}], [0.4, 0.5]) & ([s_{3}, s_{5}], [0.3, 0.4]) & ([s_{3}, s_{5}], [0.2, 0.4]) \\ ([s_{3}, s_{5}], [0.3, 0.4]) & ([s_{3}, s_{4}], [0.4, 0.5]) & ([s_{2}, s_{4}], [0.2, 0.5]) & ([s_{3}, s_{4}], [0.2, 0.5]) \\ ([s_{2}, s_{4}], [0.3, 0.5]) & ([s_{3}, s_{5}], [0.3, 0.4]) & ([s_{3}, s_{4}], [0.3, 0.5]) & ([s_{2}, s_{5}], [0.3, 0.5]) \\ ([s_{2}, s_{5}], [0.2, 0.4]) & ([s_{2}, s_{5}], [0.2, 0.4]) & ([s_{3}, s_{5}], [0.2, 0.4]) & ([s_{2}, s_{4}], [0.3, 0.4]) \end{matrix}] \end{matrix}$ $\begin{matrix} {\tilde{R}}_{2} = [\begin{matrix} ([s_{3}, s_{4}], [0.2, 0.4]) & ([s_{2}, s_{4}], [0.2, 0.5]) & ([s_{3}, s_{5}], [0.2, 0.4]) & ([s_{3}, s_{5}], [0.3, 0.5]) \\ ([s_{3}, s_{5}], [0.4, 0.5]) & ([s_{2}, s_{5}], [0.3, 0.4]) & ([s_{3}, s_{4}], [0.2, 0.5]) & ([s_{2}, s_{5}], [0.2, 0.4]) \\ ([s_{2}, s_{5}], [0.2, 0.5]) & ([s_{3}, s_{4}], [0.2, 0.4]) & ([s_{2}, s_{5}], [0.3, 0.4]) & ([s_{2}, s_{4}], [0.2, 0.5]) \\ ([s_{2}, s_{4}], [0.3, 0.4]) & ([s_{3}, s_{5}], [0.4, 0.5]) & ([s_{2}, s_{4}], [0.3, 0.5]) & ([s_{3}, s_{4}], [0.3, 0.4]) \end{matrix}] \end{matrix}$

$\begin{matrix} {\tilde{R}}_{3} = [\begin{matrix} ([s_{2}, s_{4}], [0.2, 0.4]) & ([s_{2}, s_{5}], [0.3, 0.4]) & ([s_{3}, s_{5}], [0.2, 0.5]) & ([s_{2}, s_{5}], [0.2, 0.5]) \\ ([s_{3}, s_{5}], [0.3, 0.5]) & ([s_{3}, s_{4}], [0.3, 0.5]) & ([s_{2}, s_{4}], [0.3, 0.5]) & ([s_{2}, s_{4}], [0.3, 0.4]) \\ ([s_{3}, s_{4}], [0.3, 0.4]) & ([s_{4}, s_{5}], [0.4, 0.6]) & ([s_{2}, s_{5}], [0.2, 0.4]) & ([s_{3}, s_{4}], [0.2, 0.4]) \\ ([s_{2}, s_{5}], [0.2, 0.5]) & ([s_{2}, s_{4}], [0.1, 0.5]) & ([s_{3}, s_{4}], [0.3, 0.4]) & ([s_{3}, s_{5}], [0.3, 0.5]) \end{matrix}] \end{matrix}$

The experts evaluate the MISs A_i (i = 1, 2, 3, 4) in relation to the attributes x_j (j = 1, 2, 3, 4). Suppose m (x₁) =0.3, m (x₂) =0.35, m (x₃) =0.30, m (x₄) =0.22, m (x₁, x₂) =0.7, m (x₁, x₃) =0.6, m (x₁, x₄) =0.55, m (x₂, x₃) =0.5, m (x₂, x₄) =0.45, m (x₃, x₄) =0.4, m (x₁, x₂, x₃) =0.82, m (x₁, x₂, x₄) =0.87, m (x₁, x₃, x₄) =0.75, m (x₂, x₃, x₄) =0.6, and m (x₁, x₂, x₃, x₄) =1.

Suppose the fuzzy measure of the weight vector of the decision makers D_k (k = 1, 2, 3) and the subsets of D are as follows: $\begin{matrix} μ (φ) = 0, μ (D_{1}) = 0.25, μ (D_{2}) = 0.35, \\ μ (D_{3}) = 0.3, μ (D_{1}, D_{2}) = 0.7, μ (D_{1}, D_{3}) = 0.65, \\ μ (D_{2}, D_{3}) = 0.5, μ (D_{1}, D_{2}, D_{3}) = 1 \end{matrix}$ (12)

Step 1. Utilize the decision information given in matrix ${\tilde{R}}_{k}$ (k = 1, 2, 3), the inducing variables U, and the I-IGLCOG operator that has the correlative weight vector as follows. $\begin{matrix} r_{i}^{(k)} = I - IGLCOG (〈 {u_{1}, \underset{\otimes}{\tilde{A}}}_{i}^{(k)} (x_{1}) 〉, 〈 {u_{2}, \underset{\otimes}{\tilde{A}}}_{i}^{(k)} (x_{2}) 〉, \\ 〈 {u_{3}, \underset{\otimes}{\tilde{A}}}_{i}^{(k)} (x_{3}) 〉, 〈 {u_{4}, \underset{\otimes}{\tilde{A}}}_{i}^{(k)} (x_{4}) 〉) \\ = \overset{4}{\otimes} {({\underset{\otimes}{\tilde{A}}}_{i}^{(k)} (x_{σ (j)}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} \end{matrix}$

where $u_{3}^{(1)} \geq u_{1}^{(1)} \geq u_{2}^{(1)} \geq u_{4}^{(1)}$ . Then, ${\underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{τ (1)}) = {\underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{3})$ , ${\underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{τ (2)}) = {\underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{1})$ , ${\underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{τ (3)}) =$ ${\underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{2})$ , and ${\underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{τ (4)}) = {\underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{4})$ . We obtain m (B (x_τ(1))) = m ({ x₃ }) = 0.3, m ( B (x_τ(2) )) = m ({x₁, x₃}) = 0.6, m (B (x_τ(3))) = m ( {x₁, x₂, x₃} ) = 0.82, m ( B (x_τ(4))) = m ({ x₁, x₂, x₃, x₄ }) = 1, and

Table 1

Inducing variables U

Experts	Attribute (x₁)	Attribute (x₂)	Attribute (x₃)	Attribute (x₄)
D ₁	17	15	22	12
D ₂	15	22	25	13
D ₃	16	21	25	28

$\begin{matrix} r_{1}^{(1)} = I - IGLCOG (〈 {u_{1}, \underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{1}) 〉, 〈 {u_{2}, \underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{2}) 〉, \\ 〈 {u_{3}, \underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{3}) 〉, 〈 u_{4}, {\underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{4}) 〉) \\ = \otimes_{j = 1}^{4} {({\underset{\otimes}{\tilde{A}}}_{1}^{(1)} (x_{σ (j)}))}^{(m (B (x_{σ (j)})) - m (B (x_{σ (j - 1)})))} \\ = ([s_{2.744}, s_{4.452}], [0.731, 0.910]) \end{matrix}$

Similarly, we can obtain $\begin{matrix} r_{2}^{(1)} = ([s_{2.656}, s_{4.277}], [0.731, 0.925]), \\ r_{3}^{(1)} = ([s_{2.469}, s_{4.373}], [0.760, 0.925]), \\ r_{4}^{(1)} = ([s_{2.259}, s_{4.803}], [0.642, 0.870]), \\ r_{1}^{(2)} = ([s_{2.766}, s_{4.452}], [0.642, 0.910]), \\ r_{2}^{(2)} = ([s_{2.572}, s_{4.676}], [0.731, 0.910]), \\ r_{3}^{(2)} = ([s_{2.169}, s_{4.594}], [0.642, 0.910]), \\ r_{4}^{(2)} = ([s_{2.333}, s_{4.183}], [0.794, 0.910]), \\ r_{1}^{(3)} = ([s_{2.151}, s_{4.573}], [0.642, 0.910]), \\ r_{2}^{(3)} = ([s_{2.551}, s_{4.373}], [0.760, 0.925]), \end{matrix}$

$\begin{matrix} r_{3}^{(3)} = ([s_{2.954}, s_{4.354}], [0.731, 0.914]), and \\ r_{4}^{(3)} = ([s_{2.352}, s_{4.594}], [0.698, 0.925]) . \end{matrix}$

Step 2. Utilize the decision information given in matrix $\tilde{R}$ and the IGLCOG operator ${\tilde{z}}_{i} = IGLCOG ({\tilde{r}}_{i}^{(1)}, {\tilde{r}}_{i}^{(2)}, {\tilde{r}}_{i}^{(3)}) = \otimes_{k = 1}^{3} {({\tilde{r}}_{τ (i)}^{(k)})}^{(μ (B (d_{τ (k)})) - μ (B (d_{τ (k - 1)})))}$ . The collective overall preference values ${\tilde{z}}_{i}$ of the decision makers D_k (k = 1, 2, 3) are obtained as follows: $\begin{matrix} {\tilde{z}}_{1} = ([s_{2.530}, s_{4.494}], [0.965, 0.999]), \\ {\tilde{z}}_{2} = ([s_{2.590}, s_{4.383}], [0.983, 0.999]), \\ {\tilde{z}}_{3} = ([s_{2.464}, s_{4.475}], [0.977, 0.999]), and \\ {\tilde{z}}_{1} = ([s_{2.321}, s_{4.601}], [0.978, 0.999]) . \end{matrix}$

Step 3. Suppose the basic unit-interval monotonic function is ρ (y) = y². By using Equation (11), the size of the collective overall preference values $Q ({\tilde{z}}_{i}) (i = 1, 2, 3, 4)$ are obtained as follows: $\begin{matrix} Q ({\tilde{z}}_{1}) = [s_{0.059}, s_{0.105}], Q ({\tilde{z}}_{2}) = [s_{0.030}, s_{0.051}], \\ Q ({\tilde{z}}_{3}) = [s_{0.038}, s_{0.070}], and Q ({\tilde{z}}_{4}) = [s_{0.035}, s_{0.070}] . \end{matrix}$

Step 4. To rank the size of these collective overall preference values $\tilde{Q} ({\tilde{z}}_{i})$ (i = 1, 2, 3, 4), we compare each $\tilde{Q} ({\tilde{z}}_{i})$ and construct a complementary matrix. $P = [\begin{matrix} 0.500 & 1.000 & 0.857 & 0.864 \\ 0.000 & 0.500 & 0.247 & 0.292 \\ 0.143 & 0.753 & 0.500 & 0.526 \\ 0.136 & 0.708 & 0.474 & 0.500 \end{matrix}]$

We have v₁ = 0.352, v₂ = 0.170, v₃ = 0.243, and v₄ = 0.235, so we obtain $\tilde{Q} ({\tilde{z}}_{1}) ≻ \tilde{Q} ({\tilde{z}}_{3}) ≻ \tilde{Q} ({\tilde{z}}_{4}) ≻ \tilde{Q} ({\tilde{z}}_{2})$ , and then ${\tilde{z}}_{1} ≻ {\tilde{z}}_{3} ≻ {\tilde{z}}_{4} ≻ {\tilde{z}}_{2} .$

Step 5. Rank all the alternatives x_i(i = 1, 2, 3, 4) in accordance with ${\tilde{z}}_{i} (i = 1, 2, 3, 4) : x_{1} ≻ x_{3} ≻ x_{4} ≻ x_{2}$ . Thus, the best alternative is x₁.

6 Conclusion

In this paper, the Choquet integral is used to develop the IGLCOG operator and the I-IGLCOG operator, which are used to address the correlative IGL information. The relations between these operators and other known operators are also analyzed. Then, an approach to MAGDM with correlative attribute weights and correlative expert weights is developed in which the attribute values are given in terms of the induced IGL variables, which are based on the IGLCOG operator and the I-IGLCOG operator. Finally, an illustrative example is given to illustrate the application of the proposed MAGDM method. The applications of these operators in other actual fields, such as decision making, pattern recognition and clustering analysis, are open questions for future research.

Footnotes

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China [No.71672154]; Humanities and Social Sciences Foundation of Ministry of Education of China [No.16YJA630038, No. 16YJA630005]; Major Project of Philosophy and Social Science Research of Sichuan Province [No. SC17A030]; Soft Science Research Project of Chengdu City [No. 2016-RK00-00266-ZF]; Philosophy and Social Science Research Project of Chengdu City [No. 2018L29].

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