GRP method for multiple attribute group decision making under trapezoidal interval type-2 fuzzy environment

Abstract

Multiple attribute group decision making (MAGDM) is a very active research field in management sciences. Many practical MAGDM problems are often characterized by ambiguity and uncertainty. The aim of this paper is to develop an integrated MAGDM method with unknown weight information under trapezoidal interval type-2 fuzzy environment based on the grey relational projection (GRP) method. Firstly, to determine the comprehensive weights of attributes, a novel method is proposed by combining the analytic hierarchy process (AHP) technique under trapezoidal interval type-2 fuzzy environment and inter-attribute coefficient method. Secondly, the traditional GRP method is extended to solve MAGDM problems under trapezoidal interval type-2 fuzzy environment, i.e., the optimial alternative should have the largest grey relational projection on the trapezoidal interval type-2 fuzzy positive ideal solution (TIT2-FPIS) and smallest grey relational projection on the trapezoidal interval type-2 fuzzy negative ideal solution (TIT2-FNIS) simultaneously. Finally, an emergency medical department selection problem is taken as an illustrative example to demonstrate the calculation process of the proposed method.

Keywords

Trapezoidal interval type-2 fuzzy sets (TIT2FSs)analytic hierarchy process (AHP)grey relational projection (GRP)multiple attribute group decision making (MAGDM)

1 Introduction

Multiple attribute group decision making (MAGDM) is one of the most important topics in operations research, which has been widely studied by researchers. The purpose of MAGDM is to find the most desirable alternative(s) from a set of feasible alternatives based on the attribute values given by many decision makers (DMs) [1 –3]. Many practical decision making problems often confront such situations: lack of information, uncertainty of the decision making environment. Recently, many desirable methods have been proposed to solve MAGDM problems bases on the type-1 fuzzy sets (T1FSs) initially proposed by Zadeh [4]. For these desirable methods, interested readers can refer to literature [5]. To overcome the limitations of T1FSs, Zadeh [6] further proposed type-2 fuzzy sets (T2FSs). The main difference between T1FSs and T2FSs is that the latter involves more uncertainties than the former, and the memberships of T1FSs are exact numbers whereas the memberships of T2FSs are T1FSs [7]. To reduce the computational complexity of general T2FSs, Mendel et al. [8] proposed the concept of interval type-2 fuzzy sets (IT2FSs). We can classify these achievements about IT2FSs into the following categories:

Theoretical research. Wu and Mendel [9] studied uncertainty measures for IT2FSs and made a comparative study of ranking methods, similarity measures and uncertainty measures for IT2FSs. Greenfield et al. [10] presented a method to deal with the discretised interval type-2 fuzzy sets. Das et al. [11] proposed a defuzzification method to deal with trapezoidal type-2 fuzzy variable. Qin and Liu [12] developed a new method to handle MAGDM problems based on ranking value under interval type-2 fuzzy environment. Chen [13] proposed the concept of likelihoods for interval type-2 trapezoidal fuzzy preference relations. Chen and Lee [14] presented the arithmetic operations between interval type-2 fuzzy sets. To capture both intra-personal uncertainty and inter-personal uncertainty, Bozdag et al. [15] presented a new fuzzy failure mode and effects analysis (FMEA) approach based on IT2FSs. Mendel and Liu [16] formulated continuous versions of the Karnik–Mendel (KM) algorithm to compute the centroid of an IT2FS. Ulu et al. [17] proposed the centroid type reduction method for piecewise linear interval type-2 fuzzy sets based on geometrical approach. Zhang and Zhang [18] proposed the notion of trapezoidal interval type-2 fuzzy soft sets by combining trapezoidal interval type-2 fuzzy sets with soft sets. Livi et al. [19] discussed the distinguish ability of interval type-2 fuzzy sets by analyzing upper and lower membership functions.

Applied research. A number of studies have been conducted on the extended methods within interval type-2 fuzzy environment. For instances, Chen and Lee [20] presented an interval type-2 fuzzy technique for order of preference by similarity to ideal solution (TOPSIS) method to handle MAGDM problems. Otheman and Abdullah [21] extended TOPSIS method based on the cosine similarity measure of the IT2FSs. Dymova et al. [22] proposed an extension of the TOPSIS method based on the alpha cut representation of the interval type-2 fuzzy values. Chen et al. [23] developed an extended QUALIFLEX method based on IT2FSs to deal with medical decision problems. Chen [24] developed an ELECTRE-based outranking method for multiple criteria group decision-making within the environment of interval type-2 fuzzy sets. Hosseini and Tarokh [25] extended the DEMATEL method based on the interval type-2 fuzzy sets to obtain the weights of attributes for the MAGDM problems. Baležentis and Zeng [26] extended the MULTIMOORA method with IT2FSs for the group decision making of human resource management. Chen [27] developed a new multiple criteria decision-making method within the environment of interval type-2 fuzzy sets. Chen [28] extended the LINMAP method to solve multiple criteria decision making problems within the interval type-2 fuzzy environment. Qin et al. [29] extended the traditional VIKOR method based on the prospect theory to accommodate interval type-2 fuzzy circumstances. Celik et al. [30] developed an interval type-2 fuzzy multiple attribute decision making (MADM) method by combining TOPSIS technique and grey relational analysis (GRA) method. Heidarzade et al. [31] introduced a new formula to compute the distance between two interval type-2 fuzzy sets and proposed a hierarchical clustering-based method to solve a supplier selection problem. In addition to the extension methods mentioned above, classical AHP method has been extended to the interval type-2 fuzzy environment [32 –34]. Abdullah and Zulkifli [35] proposed an integration model based on the fuzzy AHP and fuzzy DEMATEL to obtain weights of criteria. Celik et al. [36] surveyed the multiple criteria decision making approaches based on interval type-2 fuzzy sets and pointed some potential research areas in the future. Nehi and Keikha [37] proposed a method for MAGDM problems with interval type-2 fuzzy numbers based on TOPSIS and Choquet integral. Celik et al. [38] gave a trapezoidal type-2 fuzzy MCDM method to evaluate critical success factors for humanitarian relief logistics management.

Aggregation operator research. Gong et al. [7] developed some trapezoidal interval type-2 fuzzy aggregation operators based on geometric Bonferroni mean. Hu et al. [39] defined some new operations of trapezoidal interval type-2 fuzzy numbers. Hu et al. [40] proposed the concept of interval type-2 hesitant fuzzy set (IT2-HFS), and further developed some operators.

Grey relational projection (GRP) method is initially proposed by Lv and Cui [41], which combines GRA method and projection method. It simultaneously considers the projections on both the positive ideal solution (PIS) and negative ideal solution (NIS), and a preference order is given according to their relative projection, i.e., the best alternative has simultaneously the largest projection on the PIS and the smallest projection on the NIS [42]. Recently, some researchers have extended the GRP method to various uncertain situations [43, 44].

The purpose of this paper is to develop a new methodology for solving MAGDM problems with unknown weight information under trapezoidal interval type-2 fuzzy environment based on the GRP method. In this study, we develop a novel method to deal with MAGDM problems with unknown weight information under trapezoidal interval type-2 fuzzy environment based on the traditional idea of GRP method. The proposed method consists of two parts: (1) An aggregation method is presented to determine the comprehensive weights of attributes. (2) An extended GRP method is developed to rank alternatives under trapezoidal interval type-2 fuzzy environment.

The remainder of this paper is organized as follows. Section 2 introduces briefly the basic concept of trapezoidal interval type-2 fuzzy set. In Section 3, we develop a method to determine the comprehensive weights of attributes under interval type-2 fuzzy environment by using fuzzy AHP technique and inter-attribute coefficient method. In Section 4, GRP method is extended to deal with MAGDM problems with unknown weight information under trapezoidal interval type-2 fuzzy environment. In Section 5, an emergency medical department selection problem is provided to illustrate the application of the proposed method. Conclusions are presented in Section 6.

2 Preliminaries

Definition 1. [45] Let X be the universe of discourse, a type-2 fuzzy set $\tilde{A}$ can be represented by a type-2 membership function $u_{\tilde{A}} (x, u)$ as follows: $\tilde{A} = {((x, u), u_{\tilde{A}} (x, u)) | \forall x \in X, \forall u \in J_{x}},$ (1) where J_x denotes an interval in [0,1]. Moreover, the type-2 fuzzy set can also be expressed in the following form: $\begin{matrix} \tilde{A} & = & \int_{x \in X} \int_{u \in J_{x}} u_{\tilde{A}} (x, u) / (x, u) \\ = & \int_{x \in X} (\int_{u \in J_{x}} u_{\tilde{A}} (x, u) / u) / x, \end{matrix}$ where J_x ⊆ [0, 1] is the primary membership at x, and $\int_{u \in J_{x}} u_{\tilde{A}} (x, u) / u$ indicates the second membership at x.

Definition 2. [46] A trapezoidal interval type-2 fuzzy set (TIT2FS) is denoted as follows:

$\begin{matrix} {\tilde{A}}_{i} & = & ({\tilde{A}}_{i}^{U}, {\tilde{A}}_{i}^{L}) \\ = & < (a_{i 1}^{U}, a_{i 2}^{U}, a_{i 3}^{U}, a_{i 4}^{U}; H_{1} ({\tilde{A}}_{i}^{U}), H_{2} ({\tilde{A}}_{i}^{U})), \\ (a_{i 1}^{L}, a_{i 2}^{L}, a_{i 3}^{L}, a_{i 4}^{L}; H_{1} ({\tilde{A}}_{i}^{L}), H_{2} ({\tilde{A}}_{i}^{L})) >, \end{matrix}$ (2) where ${\tilde{A}}_{i}^{U}$ and ${\tilde{A}}_{i}^{L}$ are type-1 fuzzy set, the four parameters $a_{i 1}^{U}$ , $a_{i 2}^{U}$ , $a_{i 3}^{U}$ , $a_{i 4}^{U}$ associated with the upper membership function ${\tilde{A}}_{i}^{U}$ , taking membership values of 0, $H_{1} ({\tilde{A}}_{i}^{U})$ , $H_{2} ({\tilde{A}}_{i}^{U})$ and 0, respectively; whereas the four parameters $a_{i 1}^{L}$ , $a_{i 2}^{L}$ , $a_{i 3}^{L}$ , $a_{i 4}^{L}$ associated with the lower membership function ${\tilde{A}}_{i}^{L}$ , taking

membership values of 0, $H_{1} ({\tilde{A}}_{i}^{L})$ , $H_{2} ({\tilde{A}}_{i}^{L})$ and 0,respectively.

Definition 3. [14] For three trapezoidal interval type-2 fuzzy numbers $\begin{matrix} {\tilde{A}}_{1} & = & < (a_{11}^{U}, a_{12}^{U}, a_{13}^{U}, a_{14}^{U}; H_{1} ({\tilde{A}}_{1}^{U}), H_{2} ({\tilde{A}}_{1}^{U})), \\ (a_{11}^{L}, a_{12}^{L}, a_{13}^{L}, a_{14}^{L}; H_{1} ({\tilde{A}}_{1}^{L}), H_{2} ({\tilde{A}}_{1}^{L})) >, \\ {\tilde{A}}_{2} & = & < (a_{21}^{U}, a_{22}^{U}, a_{23}^{U}, a_{24}^{U}; H_{1} ({\tilde{A}}_{2}^{U}), H_{2} ({\tilde{A}}_{2}^{U})), \\ (a_{21}^{L}, a_{22}^{L}, a_{23}^{L}, a_{24}^{L}; H_{1} ({\tilde{A}}_{2}^{L}), H_{2} ({\tilde{A}}_{2}^{L})) >, \\ \tilde{A} & = & < (a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; H_{1} ({\tilde{A}}^{U}), H_{2} ({\tilde{A}}^{U})), \\ (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; H_{1} ({\tilde{A}}^{L}), H_{2} ({\tilde{A}}^{L})) >, \end{matrix}$ some arithmetic operations are defined asfollows: $\begin{matrix} (1) & {\tilde{A}}_{1} \oplus {\tilde{A}}_{2} = < (a_{11}^{U} + a_{21}^{U}, a_{12}^{U} + a_{22}^{U}, a_{13}^{U} \\ + a_{23}^{U}, a_{14}^{U} + a_{24}^{U}; min (H_{1} ({\tilde{A}}_{1}^{U}), H_{1} ({\tilde{A}}_{2}^{U})), \\ min (H_{2} ({\tilde{A}}_{1}^{U}), H_{2} ({\tilde{A}}_{2}^{U}))), (a_{11}^{L} + a_{21}^{L}, a_{12}^{L} \\ + a_{22}^{L}, a_{13}^{L} + a_{23}^{L}, a_{14}^{L} + a_{24}^{L}; min (H_{1} ({\tilde{A}}_{1}^{L}), \\ H_{1} ({\tilde{A}}_{2}^{L})), min (H_{2} ({\tilde{A}}_{1}^{L}), H_{2} ({\tilde{A}}_{2}^{L}))) > . \\ (2) & {\tilde{A}}_{1} \otimes {\tilde{A}}_{2} = < (a_{11}^{U} \times a_{21}^{U}, a_{12}^{U} \times a_{22}^{U}, a_{13}^{U} \\ \times a_{23}^{U}, a_{14}^{U} \times a_{24}^{U}; min (H_{1} ({\tilde{A}}_{1}^{U}), H_{1} ({\tilde{A}}_{2}^{U})), \\ min (H_{2} ({\tilde{A}}_{1}^{U}), H_{2} ({\tilde{A}}_{2}^{U}))), (a_{11}^{L} \times a_{21}^{L}, a_{12}^{L} \\ \times a_{22}^{L}, a_{13}^{L} \times a_{23}^{L}, a_{14}^{L} \times a_{24}^{L}; min (H_{1} ({\tilde{A}}_{1}^{L}), \\ H_{1} ({\tilde{A}}_{2}^{L})), min (H_{2} ({\tilde{A}}_{1}^{L}), H_{2} ({\tilde{A}}_{2}^{L}))) > . \end{matrix}$

$\begin{matrix} (3) & λ \tilde{A} = < (λ a_{1}^{U}, λ a_{2}^{U}, λ a_{3}^{U}, λ a_{4}^{U}; H_{1} ({\tilde{A}}^{U}), \\ H_{2} ({\tilde{A}}^{U})), (λ a_{1}^{L}, λ a_{2}^{L}, λ a_{3}^{L}, λ a_{4}^{L}; \\ H_{1} ({\tilde{A}}^{L}), H_{2} ({\tilde{A}}^{L})) >, λ > 0 . \end{matrix}$

Furthermore, Qin and Liu [29] and Kahraman et al. [33] proposed the following arithmetic operations, respectively. $\begin{matrix} (4) & (\tilde{A})^{λ} = < ((a_{1}^{U})^{λ}, (a_{2}^{U})^{λ}, (a_{3}^{U})^{λ}, (a_{4}^{U})^{λ}; \\ H_{1} ({\tilde{A}}^{U}), H_{2} ({\tilde{A}}^{U})), ((a_{1}^{L})^{λ}, (a_{2}^{L})^{λ}, (a_{3}^{L})^{λ}, \\ (a_{4}^{L})^{λ}; H_{1} ({\tilde{A}}^{L}), H_{2} ({\tilde{A}}^{L})) >, λ > 0 . \\ (5) & {\tilde{A}}_{1} / {\tilde{A}}_{2} = < (a_{11}^{U} / a_{24}^{U}, a_{12}^{U} / a_{23}^{U}, a_{13}^{U} / a_{22}^{U}, \\ a_{14}^{U} / a_{21}^{U}; min (H_{1} ({\tilde{A}}_{1}^{U}), H_{1} ({\tilde{A}}_{2}^{U})), \\ min (H_{2} ({\tilde{A}}_{1}^{U}), H_{2} ({\tilde{A}}_{2}^{U}))), (a_{11}^{L} / a_{24}^{L}, a_{12}^{L} / a_{23}^{L}, \\ a_{13}^{L} / a_{22}^{L}, a_{14}^{L} / a_{21}^{L}; min (H_{1} ({\tilde{A}}_{1}^{L}), H_{1} ({\tilde{A}}_{2}^{L})), \\ min (H_{2} ({\tilde{A}}_{1}^{L}), H_{2} ({\tilde{A}}_{2}^{L}))) > . \end{matrix}$

Definition 4. [39] Let ${\tilde{A}}_{i} = < (a_{i 1}^{U}, a_{i 2}^{U}, a_{i 3}^{U}, a_{i 4}^{U}; H_{1} ({\tilde{A}}_{i}^{U}), H_{2} ({\tilde{A}}_{i}^{U})), (a_{i 1}^{L}, a_{i 2}^{L}, a_{i 3}^{L}, a_{i 4}^{L}; H_{1} ({\tilde{A}}_{i}^{L}), H_{2} ({\tilde{A}}_{i}^{L})) > (i = 1, 2, \dots, n)$ be a collection of non-empty trapezoidal interval type-2 numbers and assume that $a_{i 1}^{U}, a_{i 2}^{U}, a_{i 3}^{U}, a_{i 4}^{U}, a_{i 1}^{L}, a_{i 2}^{L}, a_{i 3}^{L}, a_{i 4}^{L} > 0$ , then the trapezoidal interval type-2 arithmetic averaging (TIT2-AA) operator is defined as: $\begin{matrix} TIT 2 - AA ({\tilde{A}}_{1}, {\tilde{A}}_{2}, \dots, {\tilde{A}}_{n}) \\ = < (\frac{1}{n} \sum_{i = 1}^{n} a_{i 1}^{U}, \frac{1}{n} \sum_{i = 1}^{n} a_{i 2}^{U}, \frac{1}{n} \sum_{i = 1}^{n} a_{i 3}^{U}, \frac{1}{n} \sum_{i = 1}^{n} a_{i 4}^{U}; \\ 1 - \prod_{i = 1}^{n} (1 - H_{1} (A_{i}^{U}))^{1 / n}, 1 - \prod_{i = 1}^{n} (1 - H_{2} (A_{i}^{U}))^{1 / n}), \\ (\frac{1}{n} \sum_{i = 1}^{n} a_{i 1}^{L}, \frac{1}{n} \sum_{i = 1}^{n} a_{i 2}^{L}, \frac{1}{n} \sum_{i = 1}^{n} a_{i 3}^{L}, \frac{1}{n} \sum_{i = 1}^{n} a_{i 4}^{L}; \\ 1 - \prod_{i = 1}^{n} (1 - H_{1} (A_{i}^{L}))^{1 / n}, 1 - \prod_{i = 1}^{n} (1 - H_{2} (A_{i}^{L}))^{1 / n}) > . \end{matrix}$ (3)

Definition 5. The distance between two trapezoidal interval type-2 fuzzy numbers $\tilde{A}$ and $\tilde{B}$ , denoted by $d (\tilde{A}, \tilde{B})$ , is defined as: $d (\tilde{A}, \tilde{B}) = \sqrt{(Δ^{U} + Δ^{L}) / - 12},$ (4) where $\begin{matrix} Δ^{U} & = & \sum_{i = 1}^{4} (a_{i}^{U} - b_{i}^{U})^{2} + \sum_{i = 1}^{2} [H_{i} ({\tilde{A}}^{U}) - H_{i} ({\tilde{B}}^{U})]^{2}, \\ Δ^{L} & = & \sum_{i = 1}^{4} (a_{i}^{L} - b_{i}^{L})^{2} + \sum_{i = 1}^{2} [H_{i} ({\tilde{A}}^{L}) - H_{i} ({\tilde{B}}^{L})]^{2} . \end{matrix}$

Definition 6. [33] For a trapezoidal interval type-2 fuzzy number $\tilde{A} = < (a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; H_{1} ({\tilde{A}}^{U}), H_{2} ({\tilde{A}}^{U})), (a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; H_{1} ({\tilde{A}}^{L}), H_{2} ({\tilde{A}}^{L})) >,$ the defuzzified trapezoidal type-2 (DTraT) approach is given as follows: $\begin{matrix} DTraT (\tilde{A}) \\ = \frac{(a_{4}^{U} - a_{1}^{U}) + (H_{1} ({\tilde{A}}^{U}) * a_{2}^{U} - a_{1}^{U}) + (H_{2} ({\tilde{A}}^{U}) * a_{3}^{U} - a_{1}^{U})}{8} \\ + \frac{(a_{4}^{L} - a_{1}^{L}) + (H_{1} ({\tilde{A}}^{L}) * a_{2}^{L} - a_{1}^{L}) + (H_{2} ({\tilde{A}}^{L}) * a_{3}^{L} - a_{1}^{L})}{8} \\ + \frac{a_{1}^{U}}{2} + \frac{a_{1}^{L}}{2} . \end{matrix}$ (5)

Definition 7. [14] For two trapezoidal interval type-2 fuzzy numbers ${\tilde{A}}_{1} = ({\tilde{A}}_{1}^{U}, {\tilde{A}}_{1}^{L}) = < (a_{11}^{U}, a_{12}^{U}, a_{13}^{U}, a_{14}^{U}; H_{1} ({\tilde{A}}_{1}^{U}), H_{2} ({\tilde{A}}_{1}^{U})), (a_{11}^{L}, a_{12}^{L}, a_{13}^{L}, a_{14}^{L}; H_{1} ({\tilde{A}}_{1}^{L}), H_{2} ({\tilde{A}}_{1}^{L})) >$ and ${\tilde{A}}_{2} = ({\tilde{A}}_{2}^{U}, {\tilde{A}}_{2}^{L}) = < (a_{21}^{U}, a_{22}^{U}, a_{23}^{U}, a_{24}^{U}; H_{1} ({\tilde{A}}_{2}^{U}), H_{2} ({\tilde{A}}_{2}^{U})), (a_{21}^{L}, a_{22}^{L}, a_{23}^{L}, a_{24}^{L}; H_{1} ({\tilde{A}}_{2}^{L}), H_{2} ({\tilde{A}}_{2}^{L})) >$ the fuzzy preference relations of ${\tilde{A}}_{1}^{U} ≻ {\tilde{A}}_{2}^{U}$ and ${\tilde{A}}_{1}^{L} ≻ {\tilde{A}}_{2}^{L}$ are defined as follows:

P ({\tilde{A}}_{1}^{U} ≻ {\tilde{A}}_{2}^{U}) = \max (1 - \max (\frac{\sum_{k = 1}^{4} \max (a_{2 k}^{U} - a_{1 k}^{U}, 0) + (a_{24}^{U} - a_{11}^{U}) + \sum_{k = 1}^{2} \max (H_{k} ({\tilde{A}}_{2}^{U}) - H_{k} ({\tilde{A}}_{1}^{U}), 0)}{\sum_{k = 1}^{4} | a_{2 k}^{U} - a_{1 k}^{U} | + (a_{24}^{U} - a_{21}^{U}) + (a_{14}^{U} - a_{11}^{U}) + \sum_{k = 1}^{2} | H_{k} ({\tilde{A}}_{2}^{U}) - H_{k} ({\tilde{A}}_{1}^{U}) |}, 0), 0),

P ({\tilde{A}}_{1}^{L} ≻ {\tilde{A}}_{2}^{L}) = \max (1 - \max (\frac{\sum_{k = 1}^{4} \max (a_{2 k}^{L} - a_{1 k}^{L}, 0) + (a_{24}^{L} - a_{11}^{L}) + \sum_{k = 1}^{2} \max (H_{k} ({\tilde{A}}_{2}^{L}) - H_{k} ({\tilde{A}}_{1}^{L}), 0)}{\sum_{k = 1}^{4} | a_{2 k}^{L} - a_{1 k}^{L} | + (a_{24}^{L} - a_{21}^{L}) + (a_{14}^{L} - a_{11}^{L}) + \sum_{k = 1}^{2} | H_{k} ({\tilde{A}}_{2}^{L}) - H_{k} ({\tilde{A}}_{1}^{L}) |}, 0), 0) .

Based on the fuzzy preference relations of ${\tilde{A}}_{1}^{U} ≻ {\tilde{A}}_{2}^{U}$ and ${\tilde{A}}_{1}^{L} ≻ {\tilde{A}}_{2}^{L}$ , the fuzzy preference relations of ${\tilde{A}}_{1} ≻ {\tilde{A}}_{2}$ is defined as follows [14]:

$P ({\tilde{A}}_{1} ≻ {\tilde{A}}_{2}) = \frac{P ({\tilde{A}}_{1}^{U} ≻ {\tilde{A}}_{2}^{U}) + P ({\tilde{A}}_{1}^{L} ≻ {\tilde{A}}_{2}^{L})}{2} .$ (6)

3 Determine the comprehensive weights of attributes

For a multiple attribute group decision making problem with trapezoidal interval type-2 fuzzy information, let A = {A₁, A₂, …, A_m} (m ≥ 2) be a finite set of feasible alternatives among which DMs have to choose, C = {C₁, C₂, …, C_n} (n ≥ 2) be a finite set of attributes with which alternative performance is measured, and D = {D₁, D₂, …, D_t} be a set of DMs. Suppose that the decision makers expect to use the linguistic variables to express their preferences with trapezoidal interval type-2 fuzzy information. The linguistic variables and their corresponding trapezoidal interval type-2 fuzzy numbers used to evaluate the ratings of attributes are shown in Table 1 [12].

Table 1
Linguistic terms and corresponding TIT2Ns for ratings of attributes [12]

Linguistic terms TIT2Ns

Very Low (VL) < (0, 0, 0, 0.1; 1, 1), (0, 0, 0, 0.05; 0.95, 0.95)>

Low (L) < (0, 0.1, 0.15, 0.3; 1, 1), (0.05, 0.1, 0.1, 0.20; 0.95, 0.95)>

Medium Low (ML) < (0.15, 0.3, 0.35, 0.5; 1, 1), (0.20, 0.25, 0.3, 0.4; 0.95, 0.95)>

Medium (M) < (0.3, 0.5, 0.55, 0.7; 1, 1), (0.4, 0.45, 0.50, 0.6; 0.95, 0.95)>

Medium High (MH) < (0.5, 0.7, 0.75, 0.90; 1, 1), (0.60, 0.65, 0.70, 0.8; 0.95, 0.95)>

High (H) < (0.7, 0.9, 0.95, 1; 1, 1), (0.80, 0.85, 0.90, 0.95; 0.95, 0.95)>

Very High (VH) < (0.9, 1, 1, 1; 1, 1), (0.95, 1, 1, 1; 0.95, 0.95)>

Linguistic terms	TIT2Ns
Very Low (VL)	< (0, 0, 0, 0.1; 1, 1), (0, 0, 0, 0.05; 0.95, 0.95)>
Low (L)	< (0, 0.1, 0.15, 0.3; 1, 1), (0.05, 0.1, 0.1, 0.20; 0.95, 0.95)>
Medium Low (ML)	< (0.15, 0.3, 0.35, 0.5; 1, 1), (0.20, 0.25, 0.3, 0.4; 0.95, 0.95)>
Medium (M)	< (0.3, 0.5, 0.55, 0.7; 1, 1), (0.4, 0.45, 0.50, 0.6; 0.95, 0.95)>
Medium High (MH)	< (0.5, 0.7, 0.75, 0.90; 1, 1), (0.60, 0.65, 0.70, 0.8; 0.95, 0.95)>
High (H)	< (0.7, 0.9, 0.95, 1; 1, 1), (0.80, 0.85, 0.90, 0.95; 0.95, 0.95)>
Very High (VH)	< (0.9, 1, 1, 1; 1, 1), (0.95, 1, 1, 1; 0.95, 0.95)>

According to the linguistic terms shown in Table 1, each decision maker can construct their own linguistic decision matrix ${\hat{R}}^{(k)} = ({\hat{r}}_{ij}^{(k)})_{m \times n}$ (k = 1, 2, …, t). The corresponding trapezoidal interval type-2 fuzzy decision matrix ${\tilde{R}}^{(k)} = ({\tilde{r}}_{ij}^{(k)})_{m \times n}$ given by the kth DM can be described as follows: ${\tilde{R}}^{(k)} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & \dots & C_{n} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} & [\begin{matrix} {\tilde{r}}_{11}^{(k)} & {\tilde{r}}_{12}^{(k)} & \dots & {\tilde{r}}_{1 n}^{(k)} \\ {\tilde{r}}_{21}^{(k)} & {\tilde{r}}_{22}^{(k)} & \dots & {\tilde{r}}_{2 n}^{(k)} \\ ⋮ & ⋮ & \dots & ⋮ \\ {\tilde{r}}_{m 1}^{(k)} & {\tilde{r}}_{m 2}^{(k)} & \dots & {\tilde{r}}_{mn}^{(k)} \end{matrix}] \end{matrix},$ (7) where ${\tilde{r}}_{ij}^{(k)}$ is in the form of trapezoidal interval type-2 fuzzy number provided by decision maker D_k for the alternative A_i with respect to the attribute C_j. In MAGDM problems, weights of attributes play an important role in ranking alternatives, which consists of two kinds: subjective weights and objective weights. In this section, we will develop a method to determine the comprehensive weights of attributes under trapezoidal interval type-2 fuzzyenvironment.

3.1 Determine the subjective weights of attributes based on trapezoidal interval type-2 fuzzy AHP

In MAGDM problems, responses from decision makers are usually focused on opinions of the decision makers regarding the importance of attributes. The linguistic variables and their corresponding trapezoidal interval type-2 fuzzy numbers used to evaluate the importance of attributes are shown in Table 2 [33]. To determine subjective weights of attributes, Kahraman et al. [33] proposed interval type-2 fuzzy AHP (IT2F-AHP) based on the traditional idea of AHP [47]. In this subsection, we use the IT2F-AHP method to determine the subjective weights of attributes. The procedure of the IT2F-AHP method is summarized as follows [33]:

Table 2
Interval type-2 fuzzy scales of the linguistic variables for the importance of attributes

Linguistic terms TIT2Ns

Absolutely Strong (AS) < (7, 8, 9, 9; 1, 1), (7.2, 8.2, 8.8, 9; 0.8, 0.8)>

Very Strong (VS) < (5, 6, 8, 9; 1, 1), (5.2, 6.2, 7.8, 8.8; 0.8, 0.8)>

Fairly Strong (FS) < (3, 4, 6, 7; 1, 1), (3.2, 4.2, 5.8, 6.8; 0.8, 0.8)>

Slightly Strong (SS) < (1, 2, 4, 5; 1, 1),(1.2, 2.2, 3.8, 4.8; 0.8, 0.8)>

Exactly Equal (EE) < (1, 1, 1, 1; 1, 1), (1, 1, 1, 1; 1, 1)>

If factor i has one of the above linguistic variables assigned to it when compared with factor j, then factor j has the reciprocal value when compared with factor i Reciprocals of above

Linguistic terms	TIT2Ns
Absolutely Strong (AS)	< (7, 8, 9, 9; 1, 1), (7.2, 8.2, 8.8, 9; 0.8, 0.8)>
Very Strong (VS)	< (5, 6, 8, 9; 1, 1), (5.2, 6.2, 7.8, 8.8; 0.8, 0.8)>
Fairly Strong (FS)	< (3, 4, 6, 7; 1, 1), (3.2, 4.2, 5.8, 6.8; 0.8, 0.8)>
Slightly Strong (SS)	< (1, 2, 4, 5; 1, 1),(1.2, 2.2, 3.8, 4.8; 0.8, 0.8)>
Exactly Equal (EE)	< (1, 1, 1, 1; 1, 1), (1, 1, 1, 1; 1, 1)>
If factor i has one of the above linguistic variables assigned to it when compared with factor j, then factor j has the reciprocal value when compared with factor i	Reciprocals of above

Step 1. Construct the following pairwise comparison matrix among all attributes. $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n} = [\begin{matrix} \tilde{1} & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{1 n} \\ {\tilde{a}}_{21} & \tilde{1} & \dots & {\tilde{a}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{a}}_{n 1} & {\tilde{a}}_{n 2} & \dots & \tilde{1} \end{matrix}],$ (8) where $\begin{matrix} \tilde{1} & = & < (1, 1,, 1, 1; 1, 1), (1, 1,, 1, 1; 1, 1) >, \\ {\tilde{a}}_{ij} & = & < (a_{ij 1}^{U}, a_{ij 2}^{U}, a_{ij 3}^{U}, a_{ij 4}^{U}; H_{1} (a_{ij}^{U}), H_{2} (a_{ij}^{U})), \\ (a_{ij 1}^{L}, a_{ij 2}^{L}, a_{ij 3}^{L}, a_{ij 4}^{L}; H_{1} (a_{ij}^{L}), H_{2} (a_{ij}^{L})) > \end{matrix}$ is the scale of C_i comparing with C_j, and

$\begin{matrix} {\tilde{a}}_{ji} & = & ({\tilde{a}}_{ij})^{- 1} \\ = & < (\frac{1}{a_{ij 4}^{U}}, \frac{1}{a_{ij 3}^{U}}, \frac{1}{a_{ij 2}^{U}}, \frac{1}{a_{ij 1}^{U}}; H_{1} (a_{ij}^{U}), H_{2} (a_{ij}^{U})), \\ (\frac{1}{a_{ij 4}^{L}}, \frac{1}{a_{ij 3}^{L}}, \frac{1}{a_{ij 2}^{L}}, \frac{1}{a_{ij 1}^{L}}; H_{1} (a_{ij}^{L}), H_{2} (a_{ij}^{L})) >, \end{matrix}$ (9) is the scale of C_j comparing with C_i.

Step 2. Check the consistency of the pairwise comparison matrix in Equation (8).

We can use the method proposed by Kahraman et al. [33] to check the consistency of the pairwise comparison matrix $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ : If the defuzzified pairwise comparison matrix A = (a_ij) _n×n of $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is consistent, then it implies that the pairwise comparison matrix $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is also consistent. The defuzzified pairwise comparison matrix A = (a_ij) _n×n can be obtained by Equation (5), i.e., $a_{ij} = DTraT ({\tilde{a}}_{ij})$ .

Step 3. Calculate the geometric mean of each row of the pairwise comparison matrix in Equation (8) according to the operational laws (2) and (4) in Definition 3. ${\tilde{a}}_{j} = {[{\tilde{a}}_{j 1} \otimes \dots \otimes {\tilde{a}}_{jn}]}^{1 / n}, j = 1, 2, \dots, n .$ (10)

Step 4. Calculate the fuzzy weights of attributes according to the operational laws (1), (2) and (5) in Definition 3.

$\begin{matrix} {\tilde{w}}_{j} & = & {\tilde{a}}_{j} \otimes ({\tilde{a}}_{1} \oplus {\tilde{a}}_{2} \oplus \dots \oplus {\tilde{a}}_{n})^{- 1}, \\ j = 1, 2, \dots, n . \end{matrix}$ (11)

Step 5. Determine the subjective weights of attributes by Equation (12). $w_{j}^{s} = \frac{DTraT ({\tilde{w}}_{j})}{\sum_{j = 1}^{n} DTraT ({\tilde{w}}_{j})}, j = 1, 2, \dots, n,$ (12) where $DTraT ({\tilde{w}}_{j})$ can be calculated by Equation (5).

3.2 Determine the objective weights of attributes

For the MAGDM problem mentioned above, suppose that all decision makers have equal weights, that is, the weight vector is λ = (1/t, 1/t, …, 1/t) and t is the number of decision makers, then we can aggregate all individual trapezoidal interval type-2 fuzzy decision matrices in Equation (7) to obtain the following collective one:

$\tilde{R} = ({\tilde{r}}_{ij})_{m \times n} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & \dots & C_{n} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} & [\begin{matrix} {\tilde{r}}_{11} & {\tilde{r}}_{12} & \dots & {\tilde{r}}_{1 n} \\ {\tilde{r}}_{21} & {\tilde{r}}_{22} & \dots & {\tilde{r}}_{2 n} \\ ⋮ & ⋮ & \dots & ⋮ \\ {\tilde{r}}_{m 1} & {\tilde{r}}_{m 2} & \dots & {\tilde{r}}_{mn} \end{matrix}] \end{matrix},$ (13) where ${\tilde{r}}_{ij}$ can be calculated by Equation (3), i.e., ${\tilde{r}}_{ij} = TIT 2 - AA ({\tilde{r}}_{ij}^{(1)}, {\tilde{r}}_{ij}^{(2)}, \dots, {\tilde{r}}_{ij}^{(t)})$ .

For the collective trapezoidal interval type-2 fuzzy decision matrix $\tilde{R} = ({\tilde{r}}_{ij})_{m \times n}$ in Equation (13), we can calculate the defuzzified values of the elements ${\tilde{r}}_{ij} (i = 1, 2, \dots, m, j = 1, 2, \dots, n)$ , and further construct the following defuzzified collective decision matrix:

$F = (f_{ij})_{m \times n} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & \dots & C_{n} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} & [\begin{matrix} f_{11} & f_{12} & \dots & f_{1 n} \\ f_{21} & f_{22} & \dots & f_{2 n} \\ ⋮ & ⋮ & \dots & ⋮ \\ f_{m 1} & f_{m 2} & \dots & f_{mn} \end{matrix}] \end{matrix},$ (14) where f_ij can be calculated by Equation (5), i.e., $f_{ij} = DTraT ({\tilde{r}}_{ij}) (i = 1, 2, \dots, m, j = 1, 2, \dots, n)$ .

Based on the defuzzified collective decision matrix F = (f_ij) _m×n, we can determine the objective weights of attributes by using the inter-attribute correlation coefficient method. In statistics, correlation coefficient is a concept used to analyze the correlation of a data set. Generally, a big correlation coefficient represents high dependency of a given data set. The basic principle of the inter-attribute correlation coefficient method is that when correlation of a specific attribute with another attribute is high, less importance should be assigned to the correlated attributes. For the given attributes C_j and C_l, the inter-attribute correlation coefficient can be defined asfollows [48]:

$\begin{matrix} R_{jl} & = & \frac{\sum_{i = 1}^{m} (f_{ij} - {\bar{f}}_{j}) (f_{il} - {\bar{f}}_{l})}{\sqrt{\sum_{i = 1}^{m} (f_{ij} - {\bar{f}}_{j})^{2} \sum_{i = 1}^{m} (f_{il} - {\bar{f}}_{l})^{2}}}, \\ j, l = 1, 2, \dots, n, \end{matrix}$ (15) where $\begin{matrix} {\bar{f}}_{j} & = & \frac{1}{m} \sum_{i = 1}^{m} f_{ij}, j = 1, 2, \dots, n, \\ {\bar{f}}_{l} & = & \frac{1}{m} \sum_{i = 1}^{m} f_{il}, l = 1, 2, \dots, n . \end{matrix}$

Then, we can construct the inter-attribute correlation coefficient matrix $\hat{R} = (R_{jl})_{n \times n}$ and furthermore define the objective weights of attributes: $w_{j}^{o} = \frac{\sum_{l = 1}^{n} (1 - R_{jl})}{\sum_{j = 1}^{n} \sum_{l = 1}^{n} (1 - R_{jl})}, j = 1, 2, \dots, n .$ (16)

Obviously, the larger the inter-attribute coefficient R_jl is, the smaller the objective weight $w_{j}^{o}$ is. This objective weighting mechanism can avoid the subjectivity from the DM’s personal bias. Based on the subjective and objective weights, we can obtain the comprehensive weights of attributes by Equation (17). $w_{j} = \frac{(w_{j}^{s} w_{j}^{o})^{1 / 2}}{\sum_{j = 1}^{n} (w_{j}^{s} w_{j}^{o})^{1 / 2}}, j = 1, 2, \dots, n .$ (17)

4 Extension of GRP method for MAGDM problems under trapezoidal interval type-2 fuzzy environment

For the collective trapezoidal interval type-2 fuzzy decision matrix shown in Equation (13), we will extend the traditional GRP method to solve MAGDM problems under trapezoidal interval type-2 fuzzy environment. The specific procedure is briefly described as follows:

Step 1. Determine the trapezoidal interval type-2 fuzzy positive ideal solution (TIT2-FPIS) and the trapezoidal interval type-2 fuzzy negative ideal solution (TIT2-FNIS) of the collective trapezoidal interval type-2 fuzzy decision matrix in Equation (13), respectively. $R^{+} = ({\tilde{r}}_{1}^{+}, {\tilde{r}}_{2}^{+}, \dots, {\tilde{r}}_{n}^{+}),$ (18) $R^{-} = ({\tilde{r}}_{1}^{-}, {\tilde{r}}_{2}^{-}, \dots, {\tilde{r}}_{n}^{-}),$ (19) where ${\tilde{r}}_{j}^{+} = max_{i} {{\tilde{r}}_{ij}}$ and ${\tilde{r}}_{j}^{-} = min_{i} {{\tilde{r}}_{ij}}$ .

For the attribute C_j, to find ${\tilde{r}}_{j}^{+}$ , we can define a novel comparison matrix from the trapezoidal interval type-2 fuzzy number set ${{\tilde{r}}_{1 j}, {\tilde{r}}_{2 j}, \dots, {\tilde{r}}_{mj}}$ , which is shown as follows:

$\begin{matrix} P^{(j)} & = & (p_{il}^{(j)})_{m \times m} \\ = & \begin{matrix} \begin{matrix} {\tilde{r}}_{1 j} & {\tilde{r}}_{2 j} & \dots & {\tilde{r}}_{mj} \end{matrix} \\ \begin{matrix} {\tilde{r}}_{1 j} \\ {\tilde{r}}_{2 j} \\ ⋮ \\ {\tilde{r}}_{mj} \end{matrix} & [\begin{matrix} p_{11}^{(j)} & p_{12}^{(j)} & \dots & p_{1 m}^{(j)} \\ p_{21}^{(j)} & p_{22}^{(j)} & \dots & p_{2 m}^{(j)} \\ ⋮ & ⋮ & \dots & ⋮ \\ p_{m 1}^{(j)} & p_{m 2}^{(j)} & \dots & p_{mm}^{(j)} \end{matrix}] \end{matrix}, \end{matrix}$ (20) where $p_{il}^{(j)} = P ({\tilde{r}}_{ij} ≻ {\tilde{r}}_{lj})$ can be calculated by Equation (6) and $p_{ii}^{(j)} = 0.5, i = 1, 2, \dots, m .$

For the attribute C_j, to rank the elements of the trapezoidal interval type-2 fuzzy number set ${{\tilde{r}}_{1 j}, {\tilde{r}}_{2 j}, \dots, {\tilde{r}}_{mj}}$ , we add each row of the comparison matrix $P^{(j)} = (p_{il}^{(j)})_{m \times m}$ and obtain the following aggregated preference: $p_{i}^{(j)} = \sum_{l = 1}^{m} p_{il}^{(j)} .$ (21)

Obviously, $0.5 \leq p_{i}^{(j)} \leq m - 0.5$ .

Thus, for the attribute C_j, the maximum trapezoidal interval type-2 fuzzy number ${\tilde{r}}_{j}^{+}$ of the set ${{\tilde{r}}_{1 j}, {\tilde{r}}_{2 j}, \dots, {\tilde{r}}_{mj}}$ is the one with the highest aggregated preference, i.e., ${\tilde{r}}_{j}^{+} = {{\tilde{r}}_{lj} | p_{l}^{(j)} = max_{i} p_{i}^{(j)}} .$ (22)

Similarly, we can determine the TIT2-FNIS $R^{-} = ({\tilde{r}}_{1}^{-}, {\tilde{r}}_{2}^{-}, \dots, {\tilde{r}}_{n}^{-})$ , where ${\tilde{r}}_{j}^{-} = {{\tilde{r}}_{lj} | p_{l}^{(j)} = min_{i} p_{i}^{(j)}},$ (23) is the minimum trapezoidal interval type-2 fuzzy number of the set ${{\tilde{r}}_{1 j}, {\tilde{r}}_{2 j}, \dots, {\tilde{r}}_{mj}}$ .

Step 2. Construct the grey relational coefficient matrices.

For the collective trapezoidal interval type-2 fuzzy decision matrix in Equation (13), the grey relational coefficient between ${\tilde{r}}_{ij}$ and ${\tilde{r}}_{j}^{+}$ can be calculated by Equation (24). $ξ_{ij}^{+} = \frac{min_{i} min_{j} d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{+}) + ρ max_{i} max_{j} d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{+})}{d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{+}) + ρ max_{i} max_{j} d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{+})},$ (24) where ρ ∈ [0, 1] is distinguishing coefficient and $d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{+})$ is the distance between ${\tilde{r}}_{ij}$ and ${\tilde{r}}_{j}^{+}$ , which can be calculated by Equation (4). In general,ρ = 0.5.

Similarly, the grey relational coefficient between ${\tilde{r}}_{ij}$ and ${\tilde{r}}_{j}^{-}$ can be calculated by Equation (25). $ξ_{ij}^{-} = \frac{min_{i} min_{j} d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{-}) + ρ max_{i} max_{j} d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{-})}{d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{-}) + ρ max_{i} max_{j} d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{-})},$ (25) where $d ({\tilde{r}}_{ij}, {\tilde{r}}_{j}^{-})$ is the distance between ${\tilde{r}}_{ij}$ and ${\tilde{r}}_{j}^{-}$ , and it can be calculated by Equation (4).

Based on $ξ_{ij}^{+}$ and $ξ_{ij}^{-} (i = 1, 2, \dots, m, j = 1, 2, \dots, n)$ , the following two grey relational coefficient matrices can be constructed: $ξ^{+} = (ξ_{ij}^{+})_{m \times n} = [\begin{matrix} ξ_{11}^{+} & ξ_{12}^{+} & \dots & ξ_{1 n}^{+} \\ ξ_{21}^{+} & ξ_{22}^{+} & \dots & ξ_{2 n}^{+} \\ ⋮ & ⋮ & \dots & ⋮ \\ ξ_{m 1}^{+} & ξ_{m 2}^{+} & \dots & ξ_{mn}^{+} \end{matrix}],$ (26) $ξ^{-} = (ξ_{ij}^{-})_{m \times n} = [\begin{matrix} ξ_{11}^{-} & ξ_{12}^{-} & \dots & ξ_{1 n}^{-} \\ ξ_{21}^{-} & ξ_{22}^{-} & \dots & ξ_{2 n}^{-} \\ ⋮ & ⋮ & \dots & ⋮ \\ ξ_{m 1}^{-} & ξ_{m 2}^{-} & \dots & ξ_{mn}^{-} \end{matrix}],$ (27)

where $ξ^{+} = (ξ_{ij}^{+})_{m \times n}$ is the grey relational coefficient matrix between all alternatives and the TIT2-FPIS, and $ξ^{-} = (ξ_{ij}^{-})_{m \times n}$ is the grey relational coefficient matrix between all alternatives and theTIT2-FNIS.

From Equation (24), we know that the grey relational coefficient vector between the TIT2-FPIS and itself is $ξ_{0}^{+} = (1, 1, \dots, 1)$ . Similarly, the grey relational coefficient vector between the TIT2-FNIS and itself is $ξ_{0}^{-} = (1, 1, \dots, 1)$ .

Step 3. Determine the weighted grey relational coefficient matrices.

Two weighted grey relational coefficient matrices $η^{+} = (η_{ij}^{+})_{m \times n}$ and $η^{-} = (η_{ij}^{-})_{m \times n}$ can be calculated by Equations (28) and (29), respectively. $η^{+} = (η_{ij}^{+})_{m \times n} = [\begin{matrix} η_{11}^{+} & η_{12}^{+} & \dots & η_{1 n}^{+} \\ η_{21}^{+} & η_{22}^{+} & \dots & η_{2 n}^{+} \\ ⋮ & ⋮ & \dots & ⋮ \\ η_{m 1}^{+} & η_{m 2}^{+} & \dots & η_{mn}^{+} \end{matrix}],$ (28) $η^{-} = (η_{ij}^{-})_{m \times n} = [\begin{matrix} η_{11}^{-} & η_{12}^{-} & \dots & η_{1 n}^{-} \\ η_{21}^{-} & η_{22}^{-} & \dots & η_{2 n}^{-} \\ ⋮ & ⋮ & \dots & ⋮ \\ η_{m 1}^{-} & η_{m 2}^{-} & \dots & η_{mn}^{-} \end{matrix}],$ (29)

where $η_{ij}^{+} = w_{j} ξ_{ij}^{+}$ , $η_{ij}^{-} = w_{j} ξ_{ij}^{-}$ , w_j is the comprehensive weight of the attribute C_j, and it can be obtained by Equation (17).

From Equation (24), we know that the weighted grey relational coefficient vector between the TIT2-FPIS and itself is $η_{0}^{+} = (w_{1}, w_{2}, \dots, w_{n})$ . Similarly, the weighted grey relational coefficient vector between the TIT2-FNIS and itself is $η_{0}^{-} = (w_{1}, w_{2}, \dots, w_{n})$ .

Step 4. Calculate the grey relational projections of each alternative A_i (i = 1, 2, …, m) on the TIT2-FPIS and TIT2-FNIS, respectively.

Each line in the weighted grey relation matrix $η^{+} = (η_{ij}^{+})_{m \times n}$ is considered as a row vector $η_{i}^{+} = (η_{i 1}^{+}, η_{i 2}^{+}, \dots, η_{in}^{+})$ , which corresponds to the alternative A_i. Therefore, for the alternative A_i, the grey relational projection of $η_{i}^{+} = (η_{i 1}^{+}, η_{i 2}^{+}, \dots, η_{in}^{+})$ on the weighted grey relational coefficient vector $η_{0}^{+} = (w_{1}, w_{2}, \dots, w_{n})$ can be calculated by Equation (30).

$\begin{matrix} ρ_{i}^{+} & = & ∥ η_{i}^{+} ∥ \times cos (η_{i}^{+}, η_{0}^{+}) \\ = & \sqrt{\sum_{j = 1}^{n} (w_{j} ξ_{ij}^{+})^{2}} \times \frac{\sum_{j = 1}^{n} (w_{j} ξ_{ij}^{+} \times w_{j})}{\sqrt{\sum_{j = 1}^{n} (w_{j} ξ_{ij}^{+})^{2}} \times \sqrt{\sum_{j = 1}^{n} w_{j}^{2}}} \\ = & \frac{\sum_{j = 1}^{n} (w_{j}^{2} ξ_{ij}^{+})}{\sqrt{\sum_{j = 1}^{n} w_{j}^{2}}}, i = 1, 2, \dots, m . \end{matrix}$ (30)

Similarly, the grey relational projection of the alternative A_i (i = 1, 2, …, m) on the weighted grey relational coefficient vector $η_{0}^{-} = (w_{1}, w_{2}, \dots, w_{n})$ can be calculated by Equation (31).

$\begin{matrix} ρ_{i}^{-} & = & ∥ η_{i}^{-} ∥ \times cos (η_{i}^{-}, η_{0}^{-}) \\ = & \frac{\sum_{j = 1}^{n} (w_{j}^{2} ξ_{ij}^{-})}{\sqrt{\sum_{j = 1}^{n} w_{j}^{2}}}, i = 1, 2, \dots, m . \end{matrix}$ (31)

Step 5. Calculate the relative grey relational projection of each alternative.

The relative grey relational projection of each alternative to the TIT2-FPIS $η_{0}^{+} = (w_{1}, w_{2}, \dots, w_{n})$ is defined as follows: $ρ_{i} = \frac{ρ_{i}^{+}}{ρ_{i}^{+} + ρ_{i}^{-}}, i = 1, 2, \dots, m .$ (32)

Step 6. Sort the values of ρ_i (i = 1, 2, …, m) in a descending sequence and select the most desirable alternative(s). If one alternative has the highest ρ_i, then, it is the most desirable alternative(s).

5 Illustrative example

In this section, to illustrate the feasibility of the proposed method, we investigate a case on emergency medical department selection (EMDS) in one airport, one of the most famous airports in China. In this numerical example, after preliminary screening, four emergency medical departments (EMDs) A₁, A₂, A₃ and A₄ remain for further evaluation. Three DMs, D₁, D₂ and D₃, were invited to evaluate the attributes of the EMDS problem: Two experienced senior managers from emergency center and one veteran staff frequently involving in rescue activities. Five attributes in accordance with characteristics of emergency medical department evaluation are considered: preparing capacity (C₁), rescuing capacity (C₂), recovering capacity (C₃), technical equipment (C₄) as well as responding time (C₅).

In phase 1 of this example, DMs were asked to provide ratings of the attributes using linguistic scale in Table 1. The individual linguistic decision matrices ${\hat{R}}^{(k)} = ({\hat{r}}_{ij}^{(k)})_{4 \times 5} (k = 1, 2, 3)$ are constructed according to the opinions of DMs, which are shown in Table 3. In phase 2, three DMs discussed about the importance of the attribute and then provided their collective linguistic pairwise comparison matrix using the linguistic scale in Table 2. The collective linguistic pairwise comparison matrix is shown in Table 4, where ‘FS’, ‘EE’ and ‘SS’ are linguistic variables shown in Table 2, while ‘1/SS’ and ‘1/FS’ can calculated by Equation (9). The detailed computations of this example are given in the following three subsections.

Table 3
Ratings of EMDs provided by three DMs

DMs EMDs Attributes

C ₁ C ₂ C ₃ C ₄ C ₅

D ₁ A ₁ VH H H MH H

A ₂ MH H VH VH MH

A ₃ H MH VH H MH

A ₄ VH H H MH VH

D ₂ A ₁ H VH MH H MH

A ₂ MH H H H VH

A ₃ MH VH VH H H

A ₄ H H VH VH MH

D ₃ A ₁ H VH MH H VH

A ₂ VH H H MH MH

A ₃ VH H VH H H

A ₄ MH H VH VH MH

DMs	EMDs	Attributes
D ₁	A ₁	VH	H	H	MH	H
	A ₂	MH	H	VH	VH	MH
	A ₃	H	MH	VH	H	MH
	A ₄	VH	H	H	MH	VH
D ₂	A ₁	H	VH	MH	H	MH
	A ₂	MH	H	H	H	VH
	A ₃	MH	VH	VH	H	H
	A ₄	H	H	VH	VH	MH
D ₃	A ₁	H	VH	MH	H	VH
	A ₂	VH	H	H	MH	MH
	A ₃	VH	H	VH	H	H
	A ₄	MH	H	VH	VH	MH

Table 4

Collective linguistic pairwise comparison matrix

	C ₁	C ₂	C ₃	C ₄	C ₅
C ₁	EE	SS	EE	SS	FS
C ₂	1/SS	EE	1/SS	EE	SS
C ₃	EE	SS	EE	SS	FS
C ₄	1/SS	EE	1/SS	EE	SS
C ₅	1/FS	1/SS	1/FS	1/SS	EE

5.1 Determine the subjective weights of attributes

Step 1. According to the pairwise comparison matrix among all attributes in Table 4, we can get its corresponding trapezoidal interval type-2 fuzzy matrix $\tilde{A} = ({\tilde{a}}_{ij})_{5 \times 5}$ by Equation (8). To save space, here we omit its corresponding trapezoidal interval type-2 fuzzy matrix.

Step 2. By Equation (5), we can get the defuzzified pairwise comparison matrix of $\tilde{A} = ({\tilde{a}}_{ij})_{5 \times 5}$ , which is shown in Table 5. From Table 5, we know that the defuzzified pairwise comparison matrix is consistent for its consistency ratio is under 0.1, which implies that the pairwise comparison matrix $\tilde{A} = ({\tilde{a}}_{ij})_{5 \times 5}$ is also consistent.

Table 5
Pairwise comparison matrix for attributes

C ₁ C ₂ C ₃ C ₄ C ₅

C ₁ 1.00 2.85 1.00 2.85 4.75

C ₂ 0.45 1.00 0.45 1.00 2.85

C ₃ 1.00 2.85 1.00 2.85 4.75

C ₄ 0.45 1.00 0.45 1.00 2.85

C ₅ 0.21 0.45 0.21 0.45 1.00

	C ₁	C ₂	C ₃	C ₄	C ₅
C ₁	1.00	2.85	1.00	2.85	4.75
C ₂	0.45	1.00	0.45	1.00	2.85
C ₃	1.00	2.85	1.00	2.85	4.75
C ₄	0.45	1.00	0.45	1.00	2.85
C ₅	0.21	0.45	0.21	0.45	1.00

Step 3. The geometric mean of each row of the pairwise comparison matrix can be calculated by Equation (10). $\begin{matrix} {\tilde{a}}_{1} & = & < (1 . 25, 1 . 74, 2 . 49, 2 . 81; 1 . 00, 1 . 00), \\ (1 . 36, 1 . 83, 2 . 42, 2 . 75; 0 . 8, 0 . 8) >, \\ {\tilde{a}}_{2} & = & < (0 . 53, 0 . 66, 1 . 00, 1 . 38; 1 . 00, 1 . 00), \\ (0 . 55, 0 . 69, 0 . 95, 1 . 27; 0 . 8, 0 . 8) >, \\ {\tilde{a}}_{3} & = & < (1 . 25, 1 . 74, 2 . 49, 2 . 81; 1 . 00, 1 . 00), \\ (1 . 36, 1 . 83, 2 . 42, 2 . 75; 0 . 8, 0 . 8) >, \\ {\tilde{a}}_{4} & = & < (0 . 53, 0 . 66, 1 . 00, 1 . 38; 1 . 00, 1 . 00), \\ (0 . 55, 0 . 69, 0 . 95, 1 . 27; 0 . 8, 0 . 8) >, \\ {\tilde{a}}_{5} & = & < (0 . 24, 0 . 28, 0 . 44, 0 . 64; 1 . 00, 1 . 00), \\ (0 . 25, 0 . 29, 0 . 41, 0 . 58; 0 . 8, 0 . 8) > . \end{matrix}$

Step 4. The fuzzy weights of attributes can be calculated by Equation (11). $\begin{matrix} {\tilde{w}}_{1} & = & < (0 . 14, 0 . 23, 0 . 49, 0 . 74; 1 . 00, 1 . 00), \\ (0 . 16, 0 . 25, 0 . 46, 0 . 68; 0 . 80, 0 . 80) >, \\ {\tilde{w}}_{2} & = & < (0 . 06, 0 . 09, 0 . 20, 0 . 36; 1 . 00, 1 . 00), \\ (0 . 06, 0 . 10, 0 . 18, 0 . 31; 0 . 80, 0 . 80) >, \\ {\tilde{w}}_{3} & = & < (0 . 14, 0 . 23, 0 . 49, 0 . 74; 1 . 00, 1 . 00), \\ (0 . 16, 0 . 25, 0 . 46, 0 . 68; 0 . 80, 0 . 80) >, \\ {\tilde{w}}_{4} & = & < (0 . 06, 0 . 09, 0 . 20, 0 . 36; 1 . 00, 1 . 00), \\ (0 . 06, 0 . 10, 0 . 18, 0 . 31; 0 . 80, 0 . 80) >, \\ {\tilde{w}}_{5} & = & < (0 . 03, 0 . 04, 0 . 09, 0 . 17; 1 . 00, 1 . 00), \\ (0 . 03, 0 . 04, 0 . 08, 0 . 14; 0 . 80, 0 . 80) > . \end{matrix}$

Step 5. The subjective weights of attributes can be calculated by Equation (12): $w_{1}^{s} = 0.3264, w_{2}^{s} = 0.1417, w_{3}^{s} = 0.3264, w_{4}^{s} = 0.1417, w_{5}^{s} = 0.0637 .$

5.2 Determine the objective weights of attributes

Step 1. Aggregate all individual trapezoidal interval type-2 fuzzy decision matrices ${\tilde{R}}^{(k)} = ({\tilde{r}}_{ij}^{(k)})_{4 \times 5}$ into a collective one $\tilde{R} = ({\tilde{r}}_{ij})_{4 \times 5}$ by Equation (13), which is shown in Table 6.

Table 6
Collective trapezoidal interval type-2 fuzzy decision matrix

EMDs C ₁ C ₂ C ₃ C ₄ C ₅

A ₁ < (0.77,0.93,0.97,1.00; 1.00,1.00), (0.85,0.90,0.93,0.97; 0.95,0.95)> < (0.83,0.97,0.98,1.00; 1.00,1.00), (0.90,0.95,0.97,0.98; 0.95,0.95)> < (0.57,0.77,0.82,0.93;1.00,1.00), (0.67,0.72,0.77,0.85; 0.95,0.95)> < (0.63,0.83,0.88,0.97;1.00,1.00), (0.73,0.78,0.83,0.90; 0.95,0.95)> < (0.70,0.87,0.90,0.97;1.00,1.00), (0.78,0.83,0.87,0.92; 0.95,0.95)>

A ₂ < (0.63,0.80,0.83,0.93; 1.00,1.00), (0.72,0.77,0.80,0.87; 0.95,0.95)> < (0.70,0.90,0.95,1.00; 1.00,1.00), (0.80,0.85,0.90,0.95; 0.95,0.95)> < (0.77,0.93,0.97,1.00;1.00,1.00), (0.85,0.90,0.93,0.97; 0.95,0.95)> < (0.70,0.87,0.90,0.97;1.00,1.00), (0.78,0.83,0.87,0.92; 0.95,0.95)> < (0.63,0.80,0.83,0.93; 1.00,1.00), (0.72,0.77,0.80,0.87; 0.95,0.95)>

A ₃ < (0.70,0.87,0.90,0.97; 1.00,1.00), (0.78,0.83,0.87,0.92; 0.95,0.95)> < (0.70,0.87,0.90,0.97; 1.00,1.00), (0.78,0.83,0.87,0.92; 0.95, 0.95)> < (0.90,1.00,1.00,1.00;1.00,1.00), (0.95,1.00,1.00,1.00; 0.95,0.95)> < (0.70,0.90,0.95,1.00;1.00,1.00), (0.80,0.85,0.90,0.95; 0.95,0.95)> < (0.63,0.83,0.88,0.97; 1.00,1.00), (0.73,0.78,0.83,0.90; 0.95,0.95)>

A ₄ < (0.70,0.87,0.90,0.97; 1.00,1.00), (0.78,0.83,0.87,0.92; 0.95,0.95)> < (0.70,0.90,0.95,1.00; 1.00,1.00), (0.80,0.85,0.90,0.95; 0.95,0.95)> < (0.83,0.97,0.98,1.00;1.00,1.00), (0.90,0.95,0.97,0.98; 0.95,0.95)> < (0.77,0.90,0.97,0.97;1.00,1.00), (0.83,0.88,0.90,0.93; 0.95,0.95)> < (0.63,0.80,0.83,0.93;1.00,1.00), (0.72,0.77,0.80,0.87; 0.95,0.95)>

EMDs	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	< (0.77,0.93,0.97,1.00; 1.00,1.00), (0.85,0.90,0.93,0.97; 0.95,0.95)>	< (0.83,0.97,0.98,1.00; 1.00,1.00), (0.90,0.95,0.97,0.98; 0.95,0.95)>	< (0.57,0.77,0.82,0.93;1.00,1.00), (0.67,0.72,0.77,0.85; 0.95,0.95)>	< (0.63,0.83,0.88,0.97;1.00,1.00), (0.73,0.78,0.83,0.90; 0.95,0.95)>	< (0.70,0.87,0.90,0.97;1.00,1.00), (0.78,0.83,0.87,0.92; 0.95,0.95)>
A ₂	< (0.63,0.80,0.83,0.93; 1.00,1.00), (0.72,0.77,0.80,0.87; 0.95,0.95)>	< (0.70,0.90,0.95,1.00; 1.00,1.00), (0.80,0.85,0.90,0.95; 0.95,0.95)>	< (0.77,0.93,0.97,1.00;1.00,1.00), (0.85,0.90,0.93,0.97; 0.95,0.95)>	< (0.70,0.87,0.90,0.97;1.00,1.00), (0.78,0.83,0.87,0.92; 0.95,0.95)>	< (0.63,0.80,0.83,0.93; 1.00,1.00), (0.72,0.77,0.80,0.87; 0.95,0.95)>
A ₃	< (0.70,0.87,0.90,0.97; 1.00,1.00), (0.78,0.83,0.87,0.92; 0.95,0.95)>	< (0.70,0.87,0.90,0.97; 1.00,1.00), (0.78,0.83,0.87,0.92; 0.95, 0.95)>	< (0.90,1.00,1.00,1.00;1.00,1.00), (0.95,1.00,1.00,1.00; 0.95,0.95)>	< (0.70,0.90,0.95,1.00;1.00,1.00), (0.80,0.85,0.90,0.95; 0.95,0.95)>	< (0.63,0.83,0.88,0.97; 1.00,1.00), (0.73,0.78,0.83,0.90; 0.95,0.95)>
A ₄	< (0.70,0.87,0.90,0.97; 1.00,1.00), (0.78,0.83,0.87,0.92; 0.95,0.95)>	< (0.70,0.90,0.95,1.00; 1.00,1.00), (0.80,0.85,0.90,0.95; 0.95,0.95)>	< (0.83,0.97,0.98,1.00;1.00,1.00), (0.90,0.95,0.97,0.98; 0.95,0.95)>	< (0.77,0.90,0.97,0.97;1.00,1.00), (0.83,0.88,0.90,0.93; 0.95,0.95)>	< (0.63,0.80,0.83,0.93;1.00,1.00), (0.72,0.77,0.80,0.87; 0.95,0.95)>

Step 2. According to the collective trapezoidal interval type-2 fuzzy decision matrix shown in Table 6, we can get the defuzzified collective decision matrix by Equation (14), which is shown in Table 7.

Table 7

Defuzzified collective decision matrix

EMDs	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	0.9031	0.9359	0.7512	0.8107	0.8436
A ₂	0.7840	0.8703	0.9031	0.8436	0.7840
A ₃	0.8436	0.8436	0.9688	0.8703	0.8107
A ₄	0.8436	0.8703	0.9359	0.8764	0.7840

Step 3. We can obtain the inter-attribute correlation coefficient matrix $\hat{R} = (R_{jl})_{5 \times 5}$ by Equation (15), which is shown in Table 8.

Table 8

The inter-attribute correlation coefficient matrix

	C ₁	C ₂	C ₃	C ₄	C ₅
C ₁	1.0000	0.6808	–0.6449	–0.4475	0.8596
C ₂	0.6808	1.0000	–0.9871	–0.8854	0.7051
C ₃	–0.6449	–0.9871	1.0000	0.9463	–0.7524
C ₄	–0.4475	–0.8854	0.9463	1.0000	–0.7148
C ₅	0.8596	0.7051	–0.7524	–0.7148	1.0000

Step 4. By Equation (16), we can calculate the following objective weights of the attributes: $w_{1}^{o} = 0.1580$ , $w_{2}^{o} = 0 . 1996$ , $w_{3}^{o} = 0 . 2419$ , $w_{4}^{o} = 0 . 2269$ , $w_{5}^{o} = 0 . 1736$ .

Step 5. The comprehensive weights of attributes can be calculated by Equation (17): w₁ = 0 . 2364, w₂ = 0 . 1750, w₃ = 0 . 2925, w₄ = 0 . 1866, w₅ = 0 . 1095.

5.3 Utilize the extended GRP method to select the most desirable EMDs

Step 1. Determine the TIT2-FPIS and TIT2-FNIS by Equations (18) and (19), respectively. $\begin{matrix} R^{+} & = & {< (0 . 77, 0 . 93, 0 . 97, 1 . 00; 1 . 00, 1 . 00), \\ (0 . 85, 0 . 90, 0 . 93, 0 . 97; 0 . 95, 0 . 95) >, \\ < (0 . 83, 0 . 97, 0 . 98, 1 . 00; 1 . 00, 1 . 00), \\ (0 . 90, 0 . 95, 0 . 97, 0 . 98; 0 . 95, 0 . 95) >, \\ < (0 . 90, 1 . 00, 1 . 00, 1 . 00; 1 . 00, 1 . 00), \\ (0 . 95, 1 . 00, 1 . 00, 1 . 00; 0 . 95, 0 . 95) >, \\ < (0 . 77, 0 . 90, 0 . 92, 0 . 97; 1 . 00, 1 . 00), \\ (0 . 83, 0 . 88, 0 . 90, 0 . 93; 0 . 95, 0 . 95) >, \\ < (0 . 70, 0 . 87, 0 . 90, 0 . 97; 1 . 00, 1 . 00), \\ (0 . 78, 0 . 83, 0 . 87, 0 . 92; 0 . 95, 0 . 95) >}, \\ R^{-} & = & {< (0 . 63, 0 . 80, 0 . 83, 0 . 93; 1 . 00, 1 . 00), \\ (0 . 72, 0 . 77, 0 . 80, 0 . 87; 0 . 95, 0 . 95) >, \\ < (0 . 70, 0 . 87, 0 . 90, 0 . 97; 1 . 00, 1 . 00), \\ (0 . 78, 0 . 83, 0 . 87, 0 . 92; 0 . 95, 0 . 95) >, \\ < (0 . 57, 0 . 77, 0 . 82, 0 . 93; 1 . 00, 1 . 00), \\ (0 . 67, 0 . 72, 0 . 77, 0 . 85; 0 . 95, 0 . 95) >, \\ < (0 . 63, 0 . 83, 0 . 88, 0 . 97; 1 . 00, 1 . 00), \\ (0 . 73, 0 . 78, 0 . 83, 0 . 90; 0 . 95, 0 . 95) >, \\ < (0 . 63, 0 . 80, 0 . 83, 0 . 93; 1 . 00, 1 . 00), \\ (0 . 72, 0 . 77, 0 . 80, 0 . 87; 0 . 95, 0 . 95) >} . \end{matrix}$

Step 2. The grey relational coefficient matrices can be constructed by Equations (26) and (27),respectively. $\begin{matrix} ξ^{+} & = & [\begin{matrix} 1.0000 & 1.0000 & 0.3333 & 0.6002 & 1.0000 \\ 0.4883 & 0.6003 & 0.6003 & 0.7502 & 0.6561 \\ 0.6562 & 0.5440 & 1.0000 & 0.7762 & 0.7501 \\ 0.6562 & 0.6003 & 0.7502 & 1.0000 & 0.6561 \end{matrix}], \\ ξ^{-} & = & [\begin{matrix} 0.4883 & 0.5440 & 1.0000 & 1.0000 & 0.6561 \\ 1.0000 & 0.7932 & 0.4241 & 0.7501 & 1.0000 \\ 0.6561 & 1.0000 & 0.3333 & 0.6561 & 0.7932 \\ 0.6561 & 0.7932 & 0.3736 & 0.6002 & 1.0000 \end{matrix}] . \end{matrix}$

Step 3. The weighted grey relational coefficient matrices can be determined by Equations (28) and (29), respectively. $\begin{matrix} η^{+} & = & [\begin{matrix} 0.2364 & 0.1750 & 0.0975 & 0.1120 & 0.1095 \\ 0.1154 & 0.1051 & 0.1756 & 0.1400 & 0.0718 \\ 0.1551 & 0.0952 & 0.2925 & 0.1448 & 0.0821 \\ 0.1551 & 0.1051 & 0.2194 & 0.1866 & 0.0718 \end{matrix}], \\ η^{-} & = & [\begin{matrix} 0.1154 & 0.0952 & 0.2925 & 0.1866 & 0.0718 \\ 0.2364 & 0.1388 & 0.1241 & 0.1400 & 0.1095 \\ 0.1551 & 0.1750 & 0.0975 & 0.1224 & 0.0869 \\ 0.1551 & 0.1388 & 0.1093 & 0.1120 & 0.1095 \end{matrix}] . \end{matrix}$

Step 4. The grey relational projections of each emergency medical department A_i (i = 1, 2, 3, 4) on the TIT2-FPIS and TIT2-FNIS can be calculated by Equations (30) and (31), respectively. $\begin{matrix} ρ_{1}^{+} = 0.3162, ρ_{2}^{+} = 0.2800, \\ ρ_{3}^{+} = 0.3739, ρ_{4}^{+} = 0.3461, \\ ρ_{1}^{-} = 0.3680, ρ_{2}^{-} = 0.3304, \\ ρ_{3}^{-} = 0.2740, ρ_{4}^{-} = 0.2689 . \end{matrix}$

Step 5. The relative grey relational projection of each emergency medical department can be calculated by Equation (32): ρ₁ = 0.4621, ρ₂ = 0.4588, ρ₃ = 0.5771, ρ₄ = 0.5627, which means the ranking of emergency medical departments is A₃ ≻ A₄ ≻ A₁ ≻ A₂.

6 Conclusions

In this paper, we propose a new integration method for handling MAGDM problems under trapezoidal interval type-2 fuzzy environment. The key characteristics of this paper consist of the following aspects: (1) Propose an integration method to determine the comprehensive weights of attributes based on fuzzy AHP technique under trapezoidal interval type-2 fuzzy environment and inter-attribute correlation coefficient method. The merit of the proposed integration method is that it not only considers the subjective weights of attributes, but also considers the objective weights of attributes. (2) Extend the traditional GRP method to accommodate the trapezoidal interval type-2 fuzzy environment. The main advantage of the extension is that its ability to solve the MAGDM problems in the situations where the attribute values are represented by TIT2FSs and its ability to accommodate additional impreciseness and uncertainty in process of decision making analysis. In the future studies, we will apply the developed integration method to solve some other decision making problems where the information about DMs and attributes is complete unknown.

Footnotes

Acknowledgments

The authors are thankful to the associate editor Dr. Jose Merigo Lindahl and anonymous reviewers for their constructive comments and suggestions. This research is supported by Program for New Century Excellent Talents in University (NCET-13-0037), Humanities and Social Sciences Foundation of Ministry of Education of China (14YJA630019) and Natural Science Foundation of China (Nos. 91224007 and 71401012).

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GRP method for multiple attribute group decision making under trapezoidal interval type-2 fuzzy environment

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 5 Pairwise comparison matrix for attributes C 1 C 2 C 3 C 4 C 5 C 1 1.00 2.85 1.00 2.85 4.75 C 2 0.45 1.00 0.45 1.00 2.85 C 3 1.00 2.85 1.00 2.85 4.75 C 4 0.45 1.00 0.45 1.00 2.85 C 5 0.21 0.45 0.21 0.45 1.00

6 Conclusions

Footnotes

Acknowledgments

References

Table 5
Pairwise comparison matrix for attributes

C ₁ C ₂ C ₃ C ₄ C ₅

C ₁ 1.00 2.85 1.00 2.85 4.75

C ₂ 0.45 1.00 0.45 1.00 2.85

C ₃ 1.00 2.85 1.00 2.85 4.75

C ₄ 0.45 1.00 0.45 1.00 2.85

C ₅ 0.21 0.45 0.21 0.45 1.00