Dynamic multi-attribute group decision making method based on 4-dimensional matrix grey target model

Abstract

A new dynamic multi-attribute group decision making method based on matrix grey target decision model is proposed. The attributes’ data of alternatives about decision makers in different stages are represented by matrices, and they are considered as performance values in 4-dimensional space. The best, worst attributes’ values in other 3-dimensions formed the new matrices, which are defined as expected bull’s-eye, unexpected bull’s-eye, and then the deviations of alternatives and expected, unexpected bull’s eye are presented using matrix norm. The alternatives are ranked by the deviations. Finally, the examples are provided to illustrate the proposed method.

Keywords

Dynamic multi-attribute group decision making Matrix grey target model 4-dimensional space matrix norm

1 Introduction

The dynamic multi-attribute group decision making problems involve opinions of many decision makers, multi-stage information, and the alternatives with different attribute values. In recent years, dynamic multi-attribute group decision making problems have attracted increasing attention. The methods have been widely used in many fields, such as emergency decision making [1, 2], supplier selection [3], water resource allocation and scheduling [4], manager selection [5], and so on.

These problems are usually used in more complex situation, where the information is difficult to collect, and many decision makers are needed to express their preferences regarding the alternatives of attributes. In many cases, decision makers’ preferences cannot be assessed precisely in a quantitative form, so many scholars research the problems about different value types. About linguistic assessments, the dynamic linguistic weighted geometric operator is presented to rank alternatives [6], and mobile internet technologies are adopted to improve the user-system interaction through decision process [7]. About fuzzy numbers, a bibliometric-based review is worked to review the main contributions in this field [8], and dynamic intuitionistic fuzzy weighted averaging operator is proposed by employing TOPSIS method [9], the fuzzy aggregation operator, IOWA is used to aggregate their views in the group [10], prospect choquet integral operator and grey projection are used to pursuit dynamic cluster [11]. About interval-valued trapezoidal fuzzy numbers, the dynamic generalized weighted geometric aggregation operator is presented [12]. About grey numbers, grey relational analysis method and maximum entropy are used [13], grey target and prospect theory are both adopted [14]. About heterogeneous information, TOPSIS method is presented [15].

There are different methods presented, and the thoughts are different too. Several time-weighted averaging operators are given to aggregate multi-stage group preferences [16, 17]. When the decision makers’ preferences are given as judgment matrices, the weights of decision makers, stages are presented to aggregate group preference [18]. Consensus reaching models are widely applied in group decision making problems, within the group analytic hierarchy process, dynamic adaptive consensus reaching model is presented [19]. A systematic literature review on the recent evolution of consensus reaching models under dynamic environments is presented [20]. An approach with variable weights in a dynamic environment is suggested by synthesizing both data envelopment analysis and analytic network process [21]. The relationships between the preferences provided by the DMs are neglected, the collective preference is aggregated by using the Power Average (PA) operator to reflect consensus-reaching process [22]. If the set of alternatives is not fixed, mobile technologies are applied in the decision process to reflect alternatives changing throughout the decision making process [23].

Grey target thoughts are widely used in decision making problems, it is the content of grey system, which is widely used in prediction and decision making field [24 –26]. At present, the grey target models are almost based on positive and negative bull’s-eye [27], the other ideas are almost based on grey relational degree [28]. For decision attribute with satisfactory domain, Professor Liu [29] presents a new model, which projects attributes’ values into [-1, 1], and the values can reflect the attributes hit or miss the target. While, the grey target models are extended to dynamic decision making field [30, 31], but their thoughts are all aggregating data by weighted sum. In this paper, we will present a new method from angle of 4-dimensional (4-D) space, adopt grey target model and TOPSIS thoughts to establish the new matrix grey target model in 4-D space, it is not just aggregating data by weighted sum, while, the weighted sum method deals with the high-dimensional data by reducing to low dimensional data, and makes space information less adopted. The matrix grey target model takes full advantage of space information, adopts the information from horizontal and vertical axes.

The paper is set out as follows. Preliminaries are presented in Section 2. Section 3 presents 4-dimentional matrix grey target decision model. Examples are shown in Section 4, and finally, Section 5 draws our conclusions.

2 Preliminaries

Dynamic multi-attribute group decision making problem involves four dimensions: time, decision makers, attributes and alternatives. The data in the 4-dimensional space are taken as points, and they are denoted by coordinates of corresponding axis. The data corresponding to each attribute of alternative are represented by matrix series.

Definition 1. Let value of alternative s_i in the attribute axis u_j, time axis τ_t, decision makers axis v_h be denoted as, $a_{ij}^{tk}$ , and let $\begin{matrix} A_{ij} = [\begin{matrix} a_{ij}^{11} & a_{ij}^{12} & \dots & a_{ij}^{1 P} \\ a_{ij}^{21} & a_{ij}^{22} & \dots & a_{ij}^{2 P} \\ \dots & \dots & \dots & \dots \\ a_{ij}^{T 1} & a_{ij}^{T 2} & \dots & a_{ij}^{TP} \end{matrix}] \\ (i = 1, 2, \dots, m; j = 1, 2, \dots, n) \end{matrix}$ be attribute value matrix sequence of alternative s_i. Here, the row vector is decision makers’ dimension, and the column vector is stages’ dimension.

And A = {A₁₁, A₂₁, … , A_m1 ; A₁₂, A₂₂, …, A_m2 ;…, A_1n, A_2n, …, A_mn} is attribute value matrix sequence set of alternatives.

The actual complex decision problems usually require multi-stage information and multiple decision-makers involving to evaluate, such as complex equipment supplier selection problem, emergency events disposing, and so on. For dynamic group decision making problem, the matrix grey target model is presented, and it is improved to4-dimensional space.

3 4-dimentional matrix grey target decision model

Step 1. In general, the attributes are classified as benefit and cost types in decision problems, in order to compare different type attributes directly and eliminate dimension units, we standardize the attributes’ values as follows.

For benefit attributes, there is $b_{ij}^{tk} = \frac{a_{ij}^{tk} - min_{i} min_{k} min_{t} a_{ij}^{tk}}{max_{i} max_{k} max_{t} a_{ij}^{tk} - min_{i} min_{k} min_{t} a_{ij}^{tk}}$ (1)

For cost attributes, there is $b_{ij}^{tk} = \frac{max_{i} max_{k} max_{t} a_{ij}^{tk} - a_{ij}^{tk}}{max_{i} max_{k} max_{t} a_{ij}^{tk} - min_{i} min_{k} min_{t} a_{ij}^{tk}}$ (2)

If the attributes are interval numbers, they are normalized as follows. For benefit attributes, there is $\begin{matrix} [b_{ij}^{lt}, b_{ij}^{ut}] = [\frac{a_{ij}^{lt} - min_{i} min_{k} min_{t} a_{ij}^{lt}}{max_{i} max_{k} max_{t} a_{ij}^{ut} - min_{i} min_{k} min_{t} a_{ij}^{lt}}, \\ \frac{a_{ij}^{ut} - min_{i} min_{k} min_{t} a_{ij}^{lt}}{max_{i} max_{k} max_{t} a_{ij}^{ut} - min_{i} min_{k} min_{t} a_{ij}^{lt}}] \end{matrix}$ (3)

For cost attributes, there is $\begin{matrix} [b_{ij}^{lt}, b_{ij}^{ut}] = [\frac{max_{i} max_{k} max_{t} a_{ij}^{ut} - a_{ij}^{ut}}{max_{i} max_{k} max_{t} a_{ij}^{ut} - min_{i} min_{k} min_{t} a_{ij}^{lt}}, \\ \frac{max_{i} max_{k} max_{t} a_{ij}^{ut} - a_{ij}^{lt}}{max_{i} max_{k} max_{t} a_{ij}^{ut} - min_{i} min_{k} min_{t} a_{ij}^{lt}}] \end{matrix}$ (4)

Step 2. Obtaining expected bull’s-eye matrix, unexpected bull’s-eye of attribute u_j.

Definition 2. Let $A_{j +} = [\begin{matrix} a_{j +}^{11} & a_{j +}^{12} & \dots & a_{j +}^{1 P} \\ a_{j +}^{21} & a_{j +}^{22} & \dots & a_{j +}^{2 P} \\ \dots & \dots & \dots & \dots \\ a_{j +}^{T 1} & a_{j +}^{T 2} & \dots & a_{j +}^{TP} \end{matrix}]$ (5) be the expected bull’s-eye of attribute u_j in 4-dimensiona space. where in, $a_{j +}^{tk} = max_{1 \leq i \leq m} a_{ij}^{tk}$ , j = 1, 2, …, n; t = 1, 2, …, T ; k = 1, 2, …, P.

Definition 3. Let $A_{j -} = [\begin{matrix} a_{j -}^{11} & a_{j -}^{12} & \dots & a_{j -}^{1 P} \\ a_{j -}^{21} & a_{j -}^{22} & \dots & a_{j -}^{2 P} \\ \dots & \dots & \dots & \dots \\ a_{j -}^{T 1} & a_{j -}^{T 2} & \dots & a_{j -}^{TP} \end{matrix}]$ (6)

be the unexpected bull’s-eye of attribute u_j in 4-dimensional space. Wherein, $a_{j -}^{tk} = min_{1 \leq i \leq m} a_{ij}^{tk}$ , j = 1, 2, …, n; t = 1, 2, …, T ; K = 1, 2, …, P.

Step 3. Determining distance of alternative s_i and expected, unexpected bull’s-eye matrix about attribute u_j.

In the linear space C^m×n consisting of all m × n matrices, matrix norm represents the magnitude of matrix – induced variation, and it is used to define the distance between two points in linear space C^m×n.

Definition 4. Let the matrix norm $d_{ij}^{+} = ∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j +}^{11}) & λ_{2} τ_{1} (a_{ij}^{12} - a_{j +}^{12}) & \dots & λ_{P} τ_{1} (a_{ij}^{1 P} - a_{j +}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j +}^{21}) & λ_{2} τ_{2} (a_{ij}^{22} - a_{j +}^{22}) & \dots & λ_{P} τ_{2} (a_{ij}^{2 P} - a_{j +}^{2 P}) \\ \dots \dots \dots & \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j +}^{T 1}) & λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j +}^{T 2}) & \dots & λ_{P} τ_{T} (a_{ij}^{TP} - a_{j +}^{TP}) \end{matrix} ∥$ (7) be distance of alternative s_i and expected bull’s-eye A_j+ about attribute u_j when the values are real numbers.

When the values are interval numbers, Let the matrix norm $d_{ij}^{+} = ∥ \begin{matrix} λ_{1} τ_{1} d (a_{ij}^{11}, a_{j +}^{11}) & λ_{2} τ_{1} d (a_{ij}^{12}, a_{j +}^{12}) & \dots & λ_{P} τ_{1} (a_{ij}^{1 P}, a_{j +}^{1 P}) \\ λ_{1} τ_{2} d (a_{ij}^{21}, a_{j +}^{21}) & λ_{2} τ_{2} d (a_{ij}^{22}, a_{j +}^{22}) & \dots & λ_{P} τ_{2} (a_{ij}^{2 P}, a_{j +}^{2 P}) \\ \dots \dots \dots & \dots \\ λ_{1} τ_{r} d (a_{ij}^{T 1}, a_{j +}^{T 1}) & λ_{2} τ_{r} d (a_{ij}^{T 2}, a_{j +}^{T 2}) & \dots & λ_{P} τ_{T} (a_{ij}^{TP}, a_{j +}^{TP}) \end{matrix} ∥$ be distance of alternative s_i and expected bull’s-eye A_j+ about attribute u_j.

λ_k, T_t are respectively weights of column vector and row vector in the matrix, they are satisfied.

$0 \leq λ_{k} \leq 1, \sum_{k = 1}^{P} λ_{k} = 1$ , $0 \leq τ_{t} \leq 1, \sum_{t = 1}^{T} τ_{t} = 1$ .

In 4-dimensional space, the distance of alternative s_i and expected bull’s eye is $\begin{matrix} d_{i}^{+} = \sum_{j = 1}^{n} ω_{j} \cdot \\ ∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j +}^{11}) λ_{2} τ_{1} (a_{ij}^{12} - a_{j +}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P} - a_{j +}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j +}^{21}) λ_{2} τ_{2} (a_{ij}^{22} - a_{j +}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P} - a_{j +}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j +}^{T 1}) λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j +}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP} - a_{j +}^{TP}) \end{matrix} ∥ \end{matrix}$ (8)

Wherein, Column addition norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j +}^{11}) λ_{2} τ_{1} (a_{ij}^{12} - a_{j +}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P} - a_{j +}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j +}^{21}) λ_{2} τ_{2} (a_{ij}^{22} - a_{j +}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P} - a_{j +}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j +}^{T 1}) λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j +}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP} - a_{j +}^{TP}) \end{matrix} ∥}_{1} \\ = max_{1 \leq k \leq P} λ_{k} \cdot \sum_{t = 1}^{T} τ_{t} \cdot | a_{ij}^{tk} - a_{j +}^{tk} | \end{matrix}$ (9)

Row addition norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j +}^{11}) λ_{2} τ_{1} (a_{ij}^{12} - a_{j +}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P} - a_{j +}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j +}^{21}) λ_{2} τ_{2} (a_{ij}^{22} - a_{j +}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P} - a_{j +}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j +}^{T 1}) λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j +}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP} - a_{j +}^{TP}) \end{matrix} ∥}_{\infty} \\ = max_{1 \leq t \leq T} τ_{t} \cdot \sum_{k = 1}^{P} λ_{k} \cdot | a_{ij}^{tk} - a_{j +}^{tk} | \end{matrix}$ (10)

Column addition norm and row addition norm represent respectively the maximal addition distances on column vector and row vector.

Frobenius norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j +}^{11}) λ_{2} τ_{1} (a_{ij}^{12} - a_{j +}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P} - a_{j +}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j +}^{21}) λ_{2} τ_{2} (a_{ij}^{22} - a_{j +}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P} - a_{j +}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j +}^{T 1}) λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j +}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP} - a_{j +}^{TP}) \end{matrix} ∥}_{F} \\ = \sqrt{\sum_{t = 1}^{T} \sum_{k = 1}^{P} τ_{t} λ_{k} {(a_{ij}^{tk} - a_{j +}^{tk})}^{2}} \end{matrix}$ (11)

Frobenius norm (F-norm) represents the straight line distance between two points in 4-dimensional space.

If the attributes are interval numbers, the norms are respectively as follows.

Column addition norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} d (a_{ij}^{11}, a_{j +}^{11}) λ_{2} τ_{1} d (a_{ij}^{12}, a_{j +}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P}, a_{j +}^{1 P}) \\ λ_{1} τ_{2} d (a_{ij}^{21}, a_{j +}^{21}) λ_{2} τ_{2} d (a_{ij}^{22}, a_{j +}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P}, a_{j +}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} d (a_{ij}^{T 1}, a_{j +}^{T 1}) λ_{2} τ_{T} d (a_{ij}^{T 2}, a_{j +}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP}, a_{j +}^{TP}) \end{matrix} ∥}_{1} \\ = max_{1 \leq k \leq P} λ_{k} \cdot \sum_{t = 1}^{T} τ_{t} \cdot d (a_{ij}^{tk}, a_{j +}^{tk}) \end{matrix}$ (12)

Row addition norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} d (a_{ij}^{11}, a_{j +}^{11}) λ_{2} τ_{1} d (a_{ij}^{12}, a_{j +}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P}, a_{j +}^{1 P}) \\ λ_{1} τ_{2} d (a_{ij}^{21}, a_{j +}^{21}) λ_{2} τ_{2} d (a_{ij}^{22}, a_{j +}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P}, a_{j +}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} d (a_{ij}^{T 1}, a_{j +}^{T 1}) λ_{2} τ_{T} d (a_{ij}^{T 2}, a_{j +}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP}, a_{j +}^{TP}) \end{matrix} ∥}_{\infty} \\ = max_{1 \leq t \leq T} τ_{t} \cdot \sum_{k = 1}^{P} λ_{k} \cdot d (a_{ij}^{tk}, a_{j +}^{tk}) \end{matrix}$ (13)

F-norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} d (a_{ij}^{11}, a_{j +}^{11}) λ_{2} τ_{1} d (a_{ij}^{12}, a_{j +}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P}, a_{j +}^{1 P}) \\ λ_{1} τ_{2} d (a_{ij}^{21}, a_{j +}^{21}) λ_{2} τ_{2} d (a_{ij}^{22}, a_{j +}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P}, a_{j +}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} d (a_{ij}^{T 1}, a_{j +}^{T 1}) λ_{2} τ_{T} d (a_{ij}^{T 2}, a_{j +}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP}, a_{j +}^{TP}) \end{matrix} ∥}_{F} \\ = \sqrt{\sum_{t = 1}^{T} \sum_{k = 1}^{P} τ_{t} λ_{k} (d (a_{ij}^{tk}, a_{j +}^{tk}))^{2}} \end{matrix}$ (14)

Definition 5. Let the matrix norm $\begin{matrix} d_{ij}^{-} = \\ ∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j -}^{11}) λ_{2} τ_{1} (a_{ij}^{12} - a_{j -}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P} - a_{j -}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j -}^{21}) λ_{2} τ_{2} (a_{ij}^{22} - a_{j -}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P} - a_{j -}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j -}^{T 1}) λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j -}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP} - a_{j -}^{TP}) \end{matrix} ∥ \end{matrix}$ (15) be distance of alternative s_i and unexpected bull’s-eye matrix A_j- about attribute u_j when the values are real numbers.

When the values are interval numbers, let the ma-trix norm $\begin{matrix} d_{ij}^{-} = \\ ∥ \begin{matrix} λ_{1} τ_{1} d (a_{ij}^{11}, a_{j -}^{11}) λ_{2} τ_{1} d (a_{ij}^{12}, a_{j -}^{12}) \dots λ_{p} τ_{1} d (a_{ij}^{1 P}, a_{j -}^{1 P}) \\ λ_{1} τ_{2} d (a_{ij}^{21}, a_{j -}^{21}) λ_{2} τ_{2} d (a_{ij}^{22}, a_{j -}^{22}) \dots λ_{p} τ_{2} d (a_{ij}^{2 P}, a_{j -}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} d (a_{ij}^{T 1}, a_{j -}^{T 1}) λ_{2} τ_{T} d (a_{ij}^{T 2}, a_{j -}^{T 2}) \dots λ_{p} τ_{T} d (a_{ij}^{TP}, a_{j -}^{TP}) \end{matrix} ∥ \end{matrix}$ be distance of alternative s_i and unexpected bull’s-eye matrix A_j- about attribute u_j.

In 4-dimensional space, the distance of alternative s_i and unexpected bull’s eye is $\begin{matrix} d_{i}^{-} = \sum_{j = 1}^{n} ω_{j} \cdot \\ ∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j -}^{11}) λ_{2} τ_{1} (a_{ij}^{12} - a_{j -}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P} - a_{j -}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j -}^{21}) λ_{2} τ_{2} (a_{ij}^{22} - a_{j -}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P} - a_{j -}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j -}^{T 1}) λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j -}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP} - a_{j -}^{TP}) \end{matrix} ∥ \end{matrix}$ (16)

Wherein,

Column addition norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j -}^{11}) λ_{2} τ_{1} (a_{ij}^{12} - a_{j -}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P} - a_{j -}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j -}^{21}) λ_{2} τ_{2} (a_{ij}^{22} - a_{j -}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P} - a_{j -}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j -}^{T 1}) λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j -}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP} - a_{j -}^{TP}) \end{matrix} ∥}_{1} \\ = max_{1 \leq k \leq P} λ_{k} \cdot \sum_{t = 1}^{T} τ_{t} \cdot | a_{ij}^{tk} - a_{j -}^{tk} | \end{matrix}$ (17)

Row addition norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j -}^{11}) λ_{2} τ_{1} (a_{ij}^{12} - a_{j -}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P} - a_{j -}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j -}^{21}) λ_{2} τ_{2} (a_{ij}^{22} - a_{j -}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P} - a_{j -}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j -}^{T 1}) λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j -}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP} - a_{j -}^{TP}) \end{matrix} ∥}_{\infty} \\ = max_{1 \leq t \leq T} τ_{t} \cdot \sum_{k = 1}^{P} λ_{k} \cdot | a_{ij}^{tk} - a_{j -}^{tk} | \end{matrix}$ (18)

Frobenius norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j -}^{11}) λ_{2} τ_{1} (a_{ij}^{12} - a_{j -}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P} - a_{j -}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j -}^{21}) λ_{2} τ_{2} (a_{ij}^{22} - a_{j -}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P} - a_{j -}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j -}^{T 1}) λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j -}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP} - a_{j -}^{TP}) \end{matrix} ∥}_{F} \\ = \sqrt{\sum_{t = 1}^{T} \sum_{k = 1}^{P} τ_{t} λ_{k} {(a_{ij}^{tk} - a_{j -}^{tk})}^{2}} \end{matrix}$ (19)

If the attributes are interval numbers, the norms are respectively as follows.

Column addition norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} d (a_{ij}^{11}, a_{j -}^{11}) λ_{2} τ_{1} d (a_{ij}^{12}, a_{j -}^{12}) \dots λ_{p} τ_{1} d (a_{ij}^{1 P}, a_{j -}^{1 P}) \\ λ_{1} τ_{2} d (a_{ij}^{21}, a_{j -}^{21}) λ_{2} τ_{2} d (a_{ij}^{22}, a_{j -}^{22}) \dots λ_{p} τ_{2} d (a_{ij}^{2 P}, a_{j -}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} d (a_{ij}^{T 1}, a_{j -}^{T 1}) λ_{2} τ_{T} d (a_{ij}^{T 2}, a_{j -}^{T 2}) \dots λ_{p} τ_{T} d (a_{ij}^{TP}, a_{j -}^{TP}) \end{matrix} ∥}_{1} \\ = max_{1 \leq k \leq P} λ_{k} \cdot \sum_{t = 1}^{T} τ_{t} \cdot d (a_{ij}^{tk}, a_{j -}^{tk}) \end{matrix}$ (20)

F-norm is $\begin{matrix} {∥ \begin{matrix} λ_{1} τ_{1} (a_{ij}^{11} - a_{j -}^{11}) λ_{2} τ_{1} (a_{ij}^{12} - a_{j -}^{12}) \dots λ_{p} τ_{1} (a_{ij}^{1 P} - a_{j -}^{1 P}) \\ λ_{1} τ_{2} (a_{ij}^{21} - a_{j -}^{21}) λ_{2} τ_{2} (a_{ij}^{22} - a_{j -}^{22}) \dots λ_{p} τ_{2} (a_{ij}^{2 P} - a_{j -}^{2 P}) \\ \dots \dots \dots \dots \\ λ_{1} τ_{T} (a_{ij}^{T 1} - a_{j -}^{T 1}) λ_{2} τ_{T} (a_{ij}^{T 2} - a_{j -}^{T 2}) \dots λ_{p} τ_{T} (a_{ij}^{TP} - a_{j -}^{TP}) \end{matrix} ∥}_{F} \\ = \sqrt{\sum_{t = 1}^{T} \sum_{k = 1}^{P} τ_{t} λ_{k} (d (a_{ij}^{tk}, a_{j -}^{tk}))^{2}} \end{matrix}$ (22)

Step 4. Ranking alternatives according to relative distance of alternatives and expected, unexpected bull’s-eye.

In decision process, we hope the best alternative is farther from unexpected bull’s-eye A_j- about every attribute u_j, and nearer to expected bull’s-eye A_j+ about every attribute u_j. So, the TOPSIS thought is used. It not only reflects the ranking of alternatives, but also reflects satisfaction degree of alternatives. It is presented as follows.

Definition 6. Relative distance of alternative s_i and expected, unexpected matrix bull’s-eye is $c_{i} = \frac{d_{i}^{-}}{d_{i}^{-} + d_{i}^{+}}$ (23)

While, the traditional weighted sum method ranks by comprehensive values of alternatives, which is calculated by dimension reduction, the data in whole space is projected to axes. This method loses space location information, and it reflects solely the ranking of alternatives, not reflecting satisfaction degree.

4-dimensional matrix grey target model can be used in complex decision making problems, which need multi-attributes’ evaluation of many decision makers’ and multi-stage information, such as stubborn disease diagnosis, enterprise operation evaluation, complex equipment supplier selection, and so on. From the angle of mathematics, matrix norm is the most accurate distance measure between alternatives in 4-D space, and the model presents the expected and unexpected alternative, ranks the selected alternatives through the relative distance of them and expected, unexpected bull’s eye, if decision makers focus on overall information, pursuit the best alternative in the whole space, and want to know the relatively ideal alternative, then this model is recommended to adopt. But this model is only suitable for decision making in 4-D space, while, the traditional weighted sum method can be used in the wider scope, in different application case, it can be changed flexibly through promoting or reducing dimension, but it only ranks the selected alternatives, not considering the ideal one.

4 Examples

4.1 Example 1

For comparison, we adopt an example from Xu [16]. There are five companies s_i (i = 1, 2, ⋯ , 5) to be investigated. The attributes are respectively economic benefit u₁, social benefit u₂, environmental pollution u₃. The decision making needs several decision makers, here, there are three ones d_k (k = 1, 2, 3), and the weight vector of decision makers is λ = (0.5, 0.3, 0.2), the attribute weight vector is taken as ω = (0.43, 0.32, 0.25), the weight vector of stages is τ = (0.1, 0.3, 0.6). The companies are evaluated by decision makers, the attributes’ values in three years are given. Here, for computing conveniently, we list the normalized decision matrices according to definition 1, A_ij (i = 1, ⋯ , 5 ; j = 1, ⋯ , 3) show values of attribute u_j of company s_i given by decision makers in three years, the row vector is about years, and the column vector is about decision makers. The evaluation value matrices are listed as follows. $A_{11} = [\begin{matrix} [0.1685, 0.2000] [0.1839, 0.2152] [0.2022, 0.2317] \\ [0.1798, 0.2278] [0.1647, 0.2237] [0.1758, 0.2195] \\ [0.1868, 0.2169] [0.2000, 0.2405] [0.1758, 0.2048] \end{matrix}]$ $A_{12} = [\begin{matrix} [0.2000, 0.2250] [0.1839, 0.2222] [0.1628, 0.1923] \\ [0.2022, 0.2317] [0.1915, 0.2184] [0.1648, 0.2346] \\ [0.1828, 0.2410] [0.1739, 0.2099] [0.2093, 0.2375] \end{matrix}]$ $A_{13} = [\begin{matrix} [0.1372, 0.1946] [0.1748, 0.2557] [0.1943, 0.2519] \\ [0.1176, 0.2668] [0.1619, 0.2124] [0.1375, 0.2087] \\ [0.1563, 0.3150] [0.1370, 0.1869] [0.1596, 0.2760] \end{matrix}]$ $A_{21} = [\begin{matrix} [0.2022, 0.2375] [0.1954, 0.2405] [0.1798, 0.2195] \\ [0.1798, 0.2405] [0.1765, 0.2105] [0.1868, 0.2317] \\ [0.1868, 0.2169] [0.1778, 0.2532] [0.1868, 0.2169] \end{matrix}]$ $A_{22} = [\begin{matrix} [0.1647, 01875] [0.1839, 0.2099] [0.2093, 0.2436] \\ [0.1910, 0.2195] [0.1809, 0.2069] [0.1538, 0.1975] \\ [0.1720, 0.2289] [0.1848, 0.2222] [0.1744, 0.2125] \end{matrix}]$ $A_{23} = [\begin{matrix} [0.1829, 0.2432] [0.1271, 0.1790] [0.1413, 0.1763] \\ [0.1085, 0.1940] [0.2082, 0.2832] [0.1767, 0.2783] \\ [0.1758, 0.3150] [0.2055, 0.2991] [0.1824, 0.3311] \end{matrix}]$ $A_{31} = [\begin{matrix} [0.1685, 0.2125] [0.1724, 0.2152] [0.1910, 0.2195] \\ [0.1910, 0.2278] [0.1765, 0.2237] [0.1868, 0.2195] \\ [0.1758, 0.2289] [0.1556, 0.1899] [0.1758, 0.2289] \end{matrix}]$ $A_{32} = [\begin{matrix} [0.2000, 0.2250] [0.1954, 0.2222] [0.1628, 0.1923] \\ [0.1573, 0.1951] [0.1915, 0.2184] [0.1758, 0.2099] \\ [0.1828, 0.2169] [0.1630, 0.2346] [0.1628, 0.1875] \end{matrix}]$ $A_{33} = [\begin{matrix} [0.2057, 0.3243] [0.1398, 0.1989] [0.2221, 0.2923] \\ [0.1411, 0.3049] [0.1325, 0.1888] [0.1375, 0.2783] \\ [0.1278, 0.2100] [0.1233, 0.1869] [0.1418, 0.2070] \end{matrix}]$ $A_{41} = [\begin{matrix} [0.2022, 0.2500] [0.2069, 0.2405] [0.2022, 0.2439] \\ [0.1910, 0.2405] [0.1882, 0.2237] [0.1978, 0.2439] \\ [0.1978, 0.2289] [0.1667, 0.2405] [0.1868, 0.2289] \end{matrix}]$ $A_{42} = [\begin{matrix} [0.1882, 0.2125] [0.1954, 0.2222] [0.2093, 0.2439] \\ [0.2022, 0.2317] [0.1809, 0.2184] [0.1978, 0.2346] \\ [0.1828, 0.2289] [0.1957, 0.2469] [0.2093, 0.2375] \end{matrix}]$ $A_{43} = [\begin{matrix} [0.1829, 0.2432] [0.1998, 0.4474] [0.1943, 0.2519] \\ [0.1764, 0.5335] [0.2429, 0.3399] [0.1767, 0.4175] \\ [0.1563, 0.3150] [0.2466, 0.3739] [0.1596, 0.2760] \end{matrix}]$ $A_{51} = [\begin{matrix} [0.1573, 0.2125] [0.1494, 0.1899] [0.1461, 0.1707] \\ [0.1461, 0.1899] [0.1882, 0.2368] [0.1538, 0.1951] \\ [0.1648, 0.2048] [0.1778, 0.2152] [0.1868, 0.2169] \end{matrix}]$ $A_{52} = [\begin{matrix} [0.1882, 0.2125] [0.1724, 0.1975] [0.1628, 0.2308] \\ [0.1685, 0.2073] [0.1809, 0.2184] [0.1978, 0.2469] \\ [0.1720, 0.2048] [0.1630, 0.2222] [0.1744, 0.2000] \end{matrix}]$ $A_{53} = [\begin{matrix} [0.1372, 0.1769] [0.1398, 0.1989] [0.1295, 0.1603] \\ [0.1176, 0.2134] [0.1121, 0.1416] [0.1125, 0.1670] \\ [0.1278, 0.1890] [0.1121, 0.1662] [0.1277, 0.2070] \end{matrix}]$

4.1.1 Solution

The matrix sequences are in accordance with the normalized value by the new proposed method, the attribute value can be regarded as benefit type.

The expected bull’s-eye of attribute u₁ in 4-dimensional space is $A_{1 +} = [\begin{matrix} [0.5238, & 0.9701] & [0.5677, & 0.8814] & [0.5238, & 0.9132] \\ [0.4192, & 0.8814] & [0.3931, & 0.8469] & [0.4827, & 0.9132] \\ [0.4827, & 0.7731] & [0.5033, & 1] & [0.3800, & 0.7731] \end{matrix}]$

The unexpected bull’s-eye of attribute u₁ in 4- dimensional space is $A_{1 -} = [\begin{matrix} [0.1046, & 0.5033] & [0.0308, & 0.4089] & [0, & 0.2297] \\ [0, & 0.4090] & [0.1737, & 0.6013] & [0.0719, & 0.4575] \\ [0.1746, & 0.5481] & [0.0887, & 0.4089] & [0.2773, & 0.5481] \end{matrix}]$

Here, we use the Frobenius norm to compute distance, the distances of alternatives and expected bull’s-eye, unexpected bull’s-eye about attribute u₁ are respectively as follows. $\begin{matrix} d_{11}^{+} = 0.1663, d_{21}^{+} = 0.1767, d_{31}^{+} = 0.2581, d_{41}^{+} = 0.1668, \\ d_{51}^{+} = 0.3105 . \\ d_{11}^{-} = 0.2928, d_{21}^{-} = 0.3668, d_{31}^{-} = 0.2474, d_{41}^{-} = 0.3885, \\ d_{51}^{-} = 0.1330 . \end{matrix}$

The expected bull’s-eye of attribute u₂ in 4-dimensional space is $A_{2 +} = [\begin{matrix} [0.4962, 0.7647] [0.4468, 0.7347] [0.5961, 0.9677] \\ \begin{matrix} [0.5198, & 0.8367] & [0.4049, & 0.6939] & [0.4726 & 1] \end{matrix} \\ [\begin{matrix} 0.3115, & 0.9366] & [0.4500, & 1] & [0.5961 & 0.8990] \end{matrix} \end{matrix}]$

The unexpected bull’s-eye of attribute u₂ in 4- dimensional space is $A_{2 -} = [\begin{matrix} [0.1170, & 0.3619] & [0.1997, & 0.4693] & [0.0966, & 0.4135] \\ [0.0375, & 0.4436] & [0.2911, & 0.5704] & [0, & 0.4694] \\ [0.1955, & 0.5478] & [0.0988, & 0.6025] & [0.0966, & 0.3619] \end{matrix}]$

The distances of alternatives and expected bull’s-eye, unexpected bull’s-eye about attribute u₂ are respectively as follows. $\begin{matrix} d_{12}^{+} = 0.1773, d_{22}^{+} = 0.1887, d_{32}^{+} = 0.2877, d_{42}^{+} = 0.1134, \\ d_{52}^{+} = 0.3109 . \\ d_{12}^{-} = 0.3688, d_{22}^{-} = 0.2412, d_{32}^{-} = 0.1808, d_{42}^{-} = 0.3804, \\ d_{52}^{-} = 0.1821 . \end{matrix}$

The expected bull’s-eye of attribute u₃ in 4-dimensional space is $A_{3 +} = [\begin{matrix} [0.2287, & 0.5077] & [0.2148, & 0.7974] & [0.2673, & 0.4325 \\ [0.1598, & 1] & [0.3162, & 0.5444] & [0.1604, & 0.7271] \\ [0.1584, & 0.4859] & [0.3249, & 0.6244] & [0.1738, & 0.5238] \end{matrix}]$ The unexpected bull’s-eye of attribute u₃ in 4- dimensional space is $A_{3 -} = [\begin{matrix} [0.0675, & 0.1609] & [0.0438, & 0.1659] & [0.0494, & 0.1218] \\ [ & 0 & 0.2012] & [0.0084, & 0.0779] & [0.0094, & 0.1376] \\ [0.0454, & 0.1894] & [0.0085, & 0.1358] & [0.0452, & 0.2318] \end{matrix}]$

The distances of alternatives and expected bull’s-eye, unexpected bull’s-eye about attribute u₃ are respectively as follows. $\begin{matrix} d_{13}^{+} = 0.3526, d_{23}^{+} = 0.3499, d_{33}^{+} = 0.3596, d_{43}^{+} = 0.1023, \\ d_{53}^{+} = 0.4660 . \\ d_{13}^{-} = 0.1990, d_{23}^{-} = 0.2637, d_{33}^{-} = 0.1573, d_{43}^{-} = 0.4720, \\ d_{53}^{-} = 0.0195 . \end{matrix}$ The distances of alternatives and expected bull’s-eye are respectively as follows. $\begin{matrix} d_{1}^{+} = 0.1580, d_{2}^{+} = 0.1623, d_{3}^{+} = 0.2053, d_{4}^{+} = 0.0855, \\ d_{5}^{+} = 0.2462 . \end{matrix}$

The distances of alternatives and unexpected bull’s-eye are respectively as follows. $\begin{matrix} d_{1}^{-} = 0.1953, d_{2}^{-} = 0.1949, d_{3}^{-} = 0.2474, d_{4}^{-} = 0.3885, \\ d_{5}^{-} = 0.1330 . \end{matrix}$

The relative distances of alternatives computing by Frobenius norm are as follows. $\begin{matrix} c_{1} = 0.5757, c_{2} = 0.5727, c_{3} = 0.4100, c_{4} = 0.7528, \\ c_{5} = 0.2561 . \end{matrix}$

While, the relative distances of alternatives computing by column norm are: $\begin{matrix} c_{1} = 0.6257, c_{2} = 0.6078, c_{3} = 0.4253, c_{4} = 0.7775, \\ c_{5} = 0.1992 . \end{matrix}$

The relative distances of alternatives computing by row norm are: $\begin{matrix} c_{1} = 0.642, c_{2} = 0.5911, c_{3} = 0.3419, c_{4} = 0.7552, \\ c_{5} = 0.2033 . \end{matrix}$

4.1.2 Comparison

The comparison of results is given in Table 1.

Table 1
comparison of results

Methods Ranking

Xu [16] s₄ ≻ s₂ ≻ s₁ ≻ s₃ ≻ s₅

Column norm s₄ ≻ s₁ ≻ s₂ ≻ s₃ ≻ s₅

Row norm s₄ ≻ s₁ ≻ s₂ ≻ s₃ ≻ s₅

F-norm s₄ ≻ s₁ ≻ s₂ ≻ s₃ ≻ s₅

Methods	Ranking
Xu [16]	s₄ ≻ s₂ ≻ s₁ ≻ s₃ ≻ s₅
Column norm	s₄ ≻ s₁ ≻ s₂ ≻ s₃ ≻ s₅
Row norm	s₄ ≻ s₁ ≻ s₂ ≻ s₃ ≻ s₅
F-norm	s₄ ≻ s₁ ≻ s₂ ≻ s₃ ≻ s₅

There is not obvious change in the ranking, only the alternatives s₁ and s₂ are different. From the original data, we know, under attribute u₁ and u₂, the values of the alternatives given by all decision makers in the second stage reflect that s₁ ≻ s₂. And under u₂, in the first stage, two decision makers consider s₁ ≻ s₂. Under u₃, in the second stage, two decision makers consider s₁ ≻ s₂. In other situation, the values are very close. So, from the data, s₁ ≻ s₂ is more reasonable.

The ranking of other alternatives is the same, which is mainly based on the data of a certain alternative is obviously bigger or smaller than another, in this case, the matrix grey target model is unable to reflect superiority.

While, in reference [16], the comprehensive values are: r₁ = 0.1934, r₂ = 0.1958, r₃ = 0.1890, r₄ = 0.2118, r₅ = 0.1828. The values are too close, but the relative distances computing by matrix norm are relatively different, although the ranking is not obviously changing, the distinction degree about matrix norm is better than weighted sum method.

For revealing the advantages of this method over the traditional weighted sum method, another application example is given.

4.2 Example 2

Commercial aircraft industry in China is very important, C919 made its first successful flight on May 5, 2017. The vendor selection of components of commercial aircraft is quite significant. Vendor management of commercial aircraft is a complex and huge process, it needs multiple decision makers taken part in, and long-term process performance is considered too, the performance of vendor is evaluated by different attribute, so the vendor selection of commercial aircraft is a dynamic group decision making problem. Here, four vendors are to be evaluated, which are denoted as s₁ to s₄. Based on survey, four attributes are adopted, which includes quality u₁, price u₂, technology u₃, competitive power u₄, and three experts v₁, v₂, v₃ are invited to evaluate, the information in three years is collected. For stating conveniently, we give the standardized data as follows, the row vector is decision makers’ dimension, and the column vector is stages’ dimension. Here, the data is real number, the weights of decision makers are: λ₁ = 0.2843, λ₂ = 0.2189, λ₃ = 0.4968, the weights of stages are: τ₁ = 0.2859, τ₂ = 0.287, τ₃ = 0.4271, and the weights of ttributes are ω₁ = 0.3, ω₂ = 0.15, ω₃ = 0.25, ω₄ = 0.3. $A_{11} = (\begin{matrix} 0.8124 0.7633 0.9210 \\ 0.5225 0.6633 0.6667 \\ 0.6225 0.7186 0.7818 \end{matrix})$ $A_{12} = (\begin{matrix} 1 0.9196 0.8115 \\ 0.6382 0.7174 0.8136 \\ 0.5555 0.6168 0.9026 \end{matrix})$ $A_{13} = (\begin{matrix} 0.7134 0.6768 0.8009 \\ 0.3344 0.5555 0.4566 \\ 0.6518 0.9018 0.6018 \end{matrix})$ $A_{14} = (\begin{matrix} 0.6516 0.5555 0.2348 \\ 0.8018 0.9008 0.8 \\ 0.86 0.78 0.91 \end{matrix})$ $A_{21} = (\begin{matrix} 0.6618 0.7188 0.8234 \\ 0.5008 0 0.4465 \\ 0.9218 0.8366 0.8556 \end{matrix})$ $A_{22} = (\begin{matrix} 0.4335 0.7856 0.6115 \\ 0.8024 0.9 0.6 \\ 0.5656 0.6125 0.78 \end{matrix})$ $A_{23} = (\begin{matrix} 1 0.9 0.78 \\ 0.35 0.48 0.57 \\ 0.44 0.78 0.6 \end{matrix})$ $A_{24} = (\begin{matrix} 0.78 0.82 0.924 \\ 0.85 0.92 1 \\ 0.8 0.82 0.96 \end{matrix})$ $A_{31} = (\begin{matrix} 0.635 0.712 0.834 \\ 0.925 0.88 1 \\ 0.9125 0.95 0.9 \end{matrix})$ $A_{32} = (\begin{matrix} 0.5552 0.65 0.75 \\ 0.68 0.55 0.8 \\ 0.65 0.75 0.8 \end{matrix})$ $A_{33} = (\begin{matrix} 0.70 0.8 0.6 \\ 0.78 0.65 0.72 \\ 0.83 0.75 0.94 \end{matrix})$ $A_{34} = (\begin{matrix} 0.5 0.7 0.6 \\ 0.52 0.62 0.45 \\ 0.468 0.534 0.634 \end{matrix})$ $A_{41} = (\begin{matrix} 0.55 0.65 0.76 \\ 0.65 0.86 1 \\ 0.78 0.65 0.85 \end{matrix})$ $A_{42} = (\begin{matrix} 0.26 0 0.34 \\ 0.34 0.42 0.35 \\ 0.45 0.5 0.36 \end{matrix})$ $A_{43} = (\begin{matrix} 0.35 0 0.24 \\ 0.45 0.52 0.38 \\ 0.56 0.62 0.45 \end{matrix})$ $A_{44} = (\begin{matrix} 0 0.36 0.28 \\ 0.45 0.55 0.62 \\ 0.42 0.28 0.65 \end{matrix})$

4.2.1 Solution

The computing steps are listed in example 1, so they are omitted here.

The relative distances of alternatives computing by column norm are: $c_{1} = 0.5709, c_{2} = 0.6764, c_{3} = 0.6344, c_{4} = 0.2 .$

The relative distances of alternatives computing by row norm are: $c_{1} = 0.594, c_{2} = 0.5907, c_{3} = 0.6586, c_{4} = 0.2706 .$

The relative distances of alternatives computing by F-norm are: $c_{1} = 0.589, c_{2} = 0.6277, c_{3} = 0.648, c_{4} = 0.2447 .$

And the comprehensive values computing by traditional weighted sum method are: $r_{1} = 0.713, r_{2} = 0.7366, r_{3} = 0.7302, r_{4} = 0.5193 .$

4.2.2 Comparison

The comparison of results is given in Table 2.

Table 2
comparison of results

Methods Ranking

Weighted sum method s₂ ≻ s₃ ≻ s₁ ≻ s₄

Colum norm s₂ ≻ s₃ ≻ s₁ ≻ s₄

Row norm s₃ ≻ s₁ ≻ s₂ ≻ s₄

F-norm s₃ ≻ s₂ ≻ s₁ ≻ s₄

Methods	Ranking
Weighted sum method	s₂ ≻ s₃ ≻ s₁ ≻ s₄
Colum norm	s₂ ≻ s₃ ≻ s₁ ≻ s₄
Row norm	s₃ ≻ s₁ ≻ s₂ ≻ s₄
F-norm	s₃ ≻ s₂ ≻ s₁ ≻ s₄

The data of s₄ is almost the smallest, so it is the last whatever methods use. The ranking of other alternatives are different, although the ranking using weighted sum method is the same as one using 4-D matrix grey target method by column norm, the distinction degrees are different, using weighted sum method, Δr = r₃ - r₂ = 0.0064, using 4-D grey target method by column norm, Δc = c₃ - c₂ = 0.042.

From the data of s₂ and s₃, under u₄, s₂ ≻ s₃, under u₁, in most cases, s₃ ≻ s₂, under u₂ and u₃, in some cases, s₂ ≻ s₃, in some cases, s₃ ≻ s₂, so, there are different rankings according to different methods. Column addition norm reflects the alternative’s maximal addition distance all the stages, it focuses on performance at all stages rather than expert opinion. Based on this norm and the weighted sum method, s₂ ≻ s₃, based on other norms, s₃ ≻ s₂.

There are not obvious superiorities about s₁ and s₂ from the data, row addition norm reflects the alternative’s maximal addition distance about all the experts’ evaluation, it focuses on whole expert opinion rather than the performance of different stages. Using this norm, s₃ ≻ s₁ ≻ s₂ ≻ s₄.

While, F-norm represents the straight-line distance between two points in 4-dimensional space, which both reflect expert opinion and performance in different stages. Using this norm, s₃ ≻ s₂ ≻ s₁ ≻ s₄.

5 Conclusions

This paper presents a 4-D matrix grey target model to solve dynamic multi-attribute group decision making problems. The attribute data are shown as matrices, the expected, unexpected bull’s eye are defined by matrices too. The distances of alternatives and expected, unexpected bull’s-eye are defined based on matrix norm, and the relative distance reflects the alternative near to ideal alternative and far from unexpected alternative. The main contributions are as follows.

This method is based on the whole space angle, expands to the depth of the vertical and horizontal, makes full use of all the information in every axis. It seeks the distance in the entire 4-dimensional space, not computing the distance by projecting the points to the axes, reflects the direct distance from a geometric angle.

This method extends grey target model to 4-dimension, and adopts the most accurate distance measure from the angle of mathematics, matrix norm, to compute the distance of alternatives and expected bull’s eye, unexpected bull’s eye.

This method reflects degree of alternatives near to expected bull’s eye and far away from expected bull’s eye, it is not only reflects the ranking of alternatives, but also reflects satisfaction degree of alternatives.

In the future, we will excavate the distinction and connection between 4-D grey target model and traditional weighted sum method through the formulae, compare the formulae and prove it using mathematical analysis and higher algebra.

Footnotes

Acknowledgments

This work is funded by Jiangsu postdoctoral research funding plan(1701100 C), the National Natural Science Foundation of China (71801085,71801060, 71871084); Industrial research project of Henan department of science and technology (172102210257), scientific research foundation for doctor of Henan university of science and technology; Guangxi District Natural Science Fund of China (2017GXNSFBA 198182).

References

Cai

C.G.

, Xu

X.H.

, Wang

, et al., A multi-stage conflict style large group emergency decision-making method, Soft Computing (2016), 1–14.

Wang

, Rodriguez

R.M.

, Wang

Y.-M.

, A dynamic multi-attribute group emergency decision making method considering experts’ hesitation, International Journal of Computational Intelligence Systems 11 (2018), 163–182.

Zhang

, Chen

, Hu

, et al., A dynamic fuzzy group decision making method for supplier selection, Journal of Applied Sciences 13(14) (2013), 2788–2794.

Huang

, Zhang

, Li

, et al., A Multi-layer Dynamic Model for Coordination Based Group Decision Making in Water Resource Allocation and Scheduling, Intelligent Computing and Information Science Springer Berlin Heidelberg, 2011, p. 148.

Ramezani

, Memariani

and Lu

, A dynamic fuzzy multi-criteria group decision support system for manager selection, 124 (2011), 265–274.

Z.S.

, Multi-period multi-attribute group decision-making under linguistic assessments, International Journal of General Systems 38(8) (2009), 823–850.

Pérez

I.J.

, Cabrerizo

F.J.

, Herrera-Viedma

, Group decision making problems in a linguistic and dynamic context, Expert Systems with Applications 38(3) (2011), 1675–1688.

Blanco-Mesa

, Merigó

J.M.

, Gil-Lafuente

A.M.

, Fuzzy decision making: A bibliometric-based review, Journal of Intelligent & Fuzzy Systems 32(3) (2017), 2033–2050.

Z.X.

, Chen

M.Y.

, Xia

G.P.

, et al., An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making, Expert Systems with Applications 38(12) (2011), 15286–15295.

10.

Gupta

, Mohanty

B.K.

, An algorithmic approach to group decision making problems under fuzzy and dynamic environment, Expert Systems with Applications 55(15) (2016), 118–132.

11.

Yuan

J.H.

, Li

C.B.

, Intuitionistic trapezoidal fuzzy group decision-making based on prospect choquet integral operator and grey projection pursuit dynamic cluster, Mathematical Problems in Engineering 12 (2017), 1–13.

12.

Lin

, Yan

, Shi

, Dynamic multi-attribute group decision making model based on generalized interval- valued trapezoidal fuzzy numbers, Cybernetics & Information Technologies 14(4) (2015), 11–28.

13.

Luo

, Li

, Multi-stage and multi-attribute risk group decision-making method based on grey information, Grey Systems 5(2) (2015), 222–233.

14.

Yan

S.L.

, Liu

S.F.

, Multi-stage group risk decision making with grey numbers based on grey target and prospect theory, Grey Systems: Theory and Application 6(1) (2016), 64–79.

15.

Lourenzutti

, Krohling

R.A.

, A generalized TOPSIS method for group decision making with heterogeneous information in a dynamic environment, Information Sciences 330 (2016), 1–18.

16.

Z.S.

, Approach to multi-stage multi-attribute group decision making, International Journal of Information, Technology & Decision Making 10(1) (2011), 121–146.

17.

Kacprzyk

, Yager

R.R.

, Merigó

J.M.

, Towards human centric aggregation via the ordered weighted aggregation operators and linguistic data summaries: A new perspective on Zadeh’s inspirations, IEEE Computational Intelligence Magazine 14(1) (2019), 16–30.

18.

Zhu

J.J.

, Liu

S.F.

, Li

H.W.

, et al., Aggregation approach to multiple-stage multiple-judgment preference styles in group decision making, Control & Decision 23(7) (2008), 730–734.

19.

Dong

, Cooper

, A peer-to-peer dynamic adaptive consensus reaching model for the group AHP decision making, European Journal of Operational Research 250(2) (2016), 521–530.

20.

Pérez

I.J.

, Cabrerizo

F.J.

, Alonso

, et al., On dynamic consensus processes in group decision making problems, Information Sciences 459 (2018), 20–35.

21.

Sun

, Duan

, Li

C.H.

, et al. Multi-criteria group decision making approach with variable weights in dynamic environment, Computer Engineering & Applications 51(2) (2015), 1–6.

22.

Zhang

, Ignatius

, Lim

C.P.

, et al., A two-stage dynamic group decision making method for processing ordinal information, Knowledge-Based Systems 70(C) (2014), 189–202.

23.

Cabrerizo

F.J.

, Herrera

V.E.

, A mobile decision support system for dynamic group decision-making problems, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans 40(6) (2010), 1244–1256.

24.

Zeng

, Liu

S.F.

, A self-adaptive intelligence gray prediction model with the optimal fractional order accumulating operator and its application, Mathematical Methods in the Applied Sciences 23(1) (2017), 1–15.

25.

Zeng

, Li

, Forecasting the natural gas demand in China using a self-adapting intelligent grey model, Energy (112)(2016), 810–825.

26.

Zeng

, Li

, Improved multi-varible grey forecasting mod- el with a dynamic background vaue coefficient and its applica-tion, Computers & Industrial Engineering (118)(2018), 278–290.

27.

Luo

, Wang

, The multi-attribute grey target decision method for attribute value within three-parameter interval grey number, Applied Mathematical Modeling 36 (2012), 1957–1963.

28.

Wang

, Dang

Y.G.

, Multi-index grey target decision method based on adjustment coefficient, Journal of Intelligent & Fuzzy Systems 29(2) (2015), 769–775.

29.

Liu

S.F.

, Yuan

W.F.

, Sheng.

K.Q.

, Multi-attribute intelligent grey target decision model, Control and Decision 25(8) (2010), 1159–1163.

30.

Yan

S.L.

, Liu

S.F.

, Liu

J.F.

, Wu

L.F.

, Dynamic grey target decision making method with grey numbers based on existing state and future development trend of alternatives, Journal of Intelligent & Fuzzy Systems 28 (2015), 2159–2168.

31.

Zhu

J.J.

, Hipel

K.W.

, Multiple stages grey target decision making method with incomplete weight based on multi-granularity linguistic label, Information Sciences 212(12) (2012), 15–32.

Dynamic multi-attribute group decision making method based on 4-dimensional matrix grey target model

Abstract

Keywords

1 Introduction

2 Preliminaries

3 4-dimentional matrix grey target decision model

4.1 Example 1

4.1.1 Solution

4.1.2 Comparison

Table 1 comparison of results Methods Ranking Xu [16] s4 ≻ s2 ≻ s1 ≻ s3 ≻ s5 Column norm s4 ≻ s1 ≻ s2 ≻ s3 ≻ s5 Row norm s4 ≻ s1 ≻ s2 ≻ s3 ≻ s5 F-norm s4 ≻ s1 ≻ s2 ≻ s3 ≻ s5

4.2.1 Solution

4.2.2 Comparison

Table 2 comparison of results Methods Ranking Weighted sum method s2 ≻ s3 ≻ s1 ≻ s4 Colum norm s2 ≻ s3 ≻ s1 ≻ s4 Row norm s3 ≻ s1 ≻ s2 ≻ s4 F-norm s3 ≻ s2 ≻ s1 ≻ s4

Footnotes

Acknowledgments

References

Table 1
comparison of results

Methods Ranking

Xu [16] s₄ ≻ s₂ ≻ s₁ ≻ s₃ ≻ s₅

Column norm s₄ ≻ s₁ ≻ s₂ ≻ s₃ ≻ s₅

Row norm s₄ ≻ s₁ ≻ s₂ ≻ s₃ ≻ s₅

F-norm s₄ ≻ s₁ ≻ s₂ ≻ s₃ ≻ s₅

Table 2
comparison of results

Methods Ranking

Weighted sum method s₂ ≻ s₃ ≻ s₁ ≻ s₄

Colum norm s₂ ≻ s₃ ≻ s₁ ≻ s₄

Row norm s₃ ≻ s₁ ≻ s₂ ≻ s₄

F-norm s₃ ≻ s₂ ≻ s₁ ≻ s₄