A dynamic group grey target decision approach with group negotiation and Orness measure constraint

Abstract

Some factors such as expert information, behavior and time influence the decision-making results in dynamic group decision-making. In view of this, from the angle of group negotiation and system coordination, we exploit the Orness measure constraints and Nash bargaining model to establish a novel dynamic group grey target decision method. In this paper, we firstly define concepts of multi-index group measure matrix, stage bull’s-eye matrix and whole bull’s-eye matrix, which consists of the schemes, the objective, the experts, and the time. Then, based on the maximization of decision group satisfaction and the minimization of system coordination deviation, the asymmetric Nash bargaining model is used to construct a two-step optimization model. Last, the suggested optimization model is used to solve stage bull’s eye matrix. What’s more, we utilize the Orness measure constraints to establish a phase weight nonlinear programming model, and then use it to solve the weight of the stage and determine the entire bull’s-eye matrix. And we utilize the norm to measure the norm distance between group measure matrix and the stage and whole bull’s-eye matrix of the schemes, so that we can give the stage and whole schemes ranking. Finally, the case of emergency group decision making tests the validity and rationality of the proposed method.

Keywords

Bull’s-eye group negotiation Orness measure constraint dynamic norm distance

1. Introduction

Being an effective method to deal with decision-making problems, grey target decision method plays an important role in the grey system theory, and its development and innovation has important theoretical and practical significance for promoting the development of grey system theory and decision theory. Its basic idea is that the distance between the sequences is used to determine the scheme ranking [1]. Since the grey target decision has become an essential research hotspot in grey theory and has been widely used in the economic management [2, 3] and military decision [4, 5].

With the gradual democratization of management and the deepening of scientific decision-making, more and more decision problems need a method based on group decision-making to solve. Being an effective method to cope with them, group decision-making method plays a very important role both in theory and practice. Group decision method is to assemble the preference information from the experts or decision makers (DMs) based on their own knowledge and experience, and then give the consensus decision-making results. In the group decision-making process, each DM has an acceptable domain (satisfactory domain or the expected value, which can be called as a grey target) for each index, and then whether each scheme is selected according to the fact whether its attribute value falls into the grey target in the process of the evaluation. In comparison with the related literatures on grey target decision we can find that they mainly focus on establishing some improved grey target decision models [6 –8] and hybrid models [4 , 9–14] by combining with grey target and different methods such as the principal component analysis [4], TOPSIS [9], maximum entropy [5, 14]. The authors made the multi-objective, multi-index and multi-attribute grey target decision through discrete attribute values, and gave scheme ranking through the information aggregation of determining the index, attribute weight, time weight, and the comprehensive evaluation value. However, information distortions in the process of gathering information were unavoidable. Moreover, these useful decisions from various experts or DMs were not taken into account. In order to deal with group decision-making problems well and develop the application field and scope of the grey target decision making, various grey group decision-making ways have been proposed. These proposed models mainly contain group decision methods with the risk attitude and psychological behavior of the experts [15 –17] and grey target decision approaches based on the other soft computing methods [18 –23]. Although these models take into account the uncertain information of group decision making, they don’t consider the group behaviors of DMs or experts and discuses the setting mechanism of the group’s bull’s-eye in the actual decision process, and then the decision results maybe produce much larger deviation with the actual. Considering that schemes sorting of group decision making is influenced by the experts (DMS) and time, and the processes of group decision making group are negotiation and coordination, we construct a improved dynamic grey target group decision method, which is based on the asymmetric Nash bargaining model and Orness weighted nonlinear programming model for measure stage. A case study is carried out to validate its validity and rationality.

The remainder of this paper is organized as follows: Some concepts are defined in Section 2. In the Section 3, by exploiting the Orness measure constraints and Nash bargaining model, we construct a novel dynamic group grey target decision method to determine the weight of the stage, the stage bull’s-eye matrix, and the whole bull’s-eye matrix. A real-world case study is furnished in Section 4 to illustrate how the proposed method can be applied. The paper concludes with some remarks in Section 5.

2. Preliminaries

Dynamic group decision mainly refers to the five basic elements of events, schemes, time, experts, objective and decision values (decision information), which influence on the decision result. For a dynamic group decision system, there only exists an event, which is the group decision problem. In this section, some basic definitions on dynamic group grey target decision are introduced.

Assume that there exists a dynamic group decision system G ={ X, U, T, E, V }, in which X ={ x₁, …, x_i, …, x_n }, U ={ u₁, …, u_j, …, u_m }, T ={ t₁, …, t_l, …, t_T }, E ={ e₁, …, e_c, …, e_p }, and $V = {v_{ij}^{1 c}, \dots, v_{ij}^{lc}, \dots, v_{ij}^{Tc}}$ stands for the event or group decision problem, the decision schemes, objectives or objective, time stages, experts and decision values, respectively, for i = 1, 2, …, n; j = 1, 2, …, m; l = 1, 2, …, T; c = 1, 2, …, p, and $v_{ij}^{lc}$ expresses the decision value of the expert e_c over the scheme x_i under the objective u_j in the stage t_l.

Due to the fact that the objective u_j and the expert e_c takes on different importance in the dynamic group decision process, let π_j, w_c be the weight of the index u_j and the expert e_c, respectively, which satisfies $\sum_{j = 1}^{m} π_{j} = 1$ , and $\sum_{c = 1}^{p} w_{c} = 1$ .

Definition 1. For ∀x_i ∈ X, ∀ u_j ∈ U, ∀ t_l ∈ T; ∀ e_c ∈E, if $a_{ij}^{lc} (i = 1, 2, \dots, n; j = 1, 2, \dots, m; l = 1, 2, \dots, T; c = 1, 2, \dots, p)$ expresses the dimensionless measure value of $v_{ij}^{lc}$ , then $A_{i}^{l} = [\begin{matrix} a_{i 1}^{l 1} & \dots & a_{ij}^{l 1} & \dots & a_{im}^{l 1} \\ \dots & \dots & \dots & \dots & \dots \\ a_{i 1}^{lc} & \dots & a_{ij}^{lc} & \dots & a_{im}^{lc} \\ \dots & \dots & \dots & \dots & \dots \\ a_{i 1}^{lp} & \dots & a_{ij}^{lp} & \dots & a_{im}^{lp} \end{matrix}]$ (1) is called as the multi-objective group measure values matrix of the scheme x_i in the stage t_l.

Where, the row vector and column vector of $A_{i}^{l}$ stands for the measured value of the expert e_c over the scheme x_i under the objectives in the stage t_l and the measured value of all the experts over the scheme x_i under the objective u_j in the stage t_l, respectively.

Due to the complexity and uncertainty of the empirical world and the limitation of human cognition, there exists a variety of uncertain information in the group decision problem. In order to well deal with the problem, by exploiting the basic principle of grey target decision, we seek to establish a novel multi-objective group grey target decision model. In the dynamic group decision process, the experts’ group behavior and time have much more important influence and effect on the decision results, and then the bull’s-eyes of the grey target should be based on the expert group game, the time compromise and system coordination to produce instead of the subjective setting. Depending on the above analysis, the whole bull’s-eye of the multi-objective dynamic group grey target decision should be determined and produced by the expert group, time stage, and system coordination. For each stage and the whole system in the multi-objective dynamic group grey target decision problem, there all exists bull’s-eyes. For the sake of convenience, the bull’s eye of the each stage and the whole system is termed as the stage and whole bull’s-eye, respectively.

Definition 2. Assume that $a_{0 j}^{lc}$ and $a_{0 j}^{lc} = (a_{01}^{lc}, \dots, a_{0 j}^{lc}, \dots, a_{0 m}^{lc})$ stands for the negotiated bull’s-eye of the expert e_c under the objective u_j and objectives U in the stage t_l, respectively, and $a_{0 j}^{l *}$ and $a_{0 j}^{l *} = (a_{01}^{l *}, \dots, a_{0 j}^{l *}, \dots, a_{0 m}^{l *})$ expresses the stage bull’s-eye under the objective u_j and objectives U in the stage t_l, respectively, for ∀x_i ∈ X, ∀ u_j ∈ U, ∀ t_l ∈ T, ∀ e_c ∈ E, if $a_{0 j}^{l *} = a_{0 j}^{l 1} = \dots = a_{0 j}^{lc} = \dots = a_{0 j}^{lp}$ , then $A_{+}^{l} = [\begin{matrix} a_{01}^{l *} & \dots & a_{0 j}^{l *} & \dots & a_{0 m}^{l *} \\ \dots & \dots & \dots & \dots & \dots \\ a_{01}^{l *} & \dots & a_{0 j}^{l *} & \dots & a_{0 m}^{l *} \\ \dots & \dots & \dots & \dots & \dots \\ a_{01}^{l *} & \dots & a_{0 j}^{l *} & \dots & a_{0 m}^{l *} \end{matrix}]$ (2) is called as the stage bull’s-eye measure value matrix in the stage t_l.

Definition 3. Assume that τ_l stands for the weight of the stage t_l, if γ expresses the Orness measure of τ_l, then it can be written as follows: $Orness (τ) = \frac{1}{T - 1} \sum_{l = 1}^{T} (T - l) τ_{l} = γ (0 \leq γ \leq 1)$ (3)

According to the definition 3, γ can reflect the emphasis degree of the experts on different stages. When γ = 0.5, it shows that the emphasis degree of the experts on different stages is equal; when γ is closer to 1, it means that the experts much prefer the old stages; when γ is closer to 0, the more experts pay attention to the recent stages.

Definition 4. Where $τ_{l}^{*} = (τ_{1}^{*}, τ_{2}^{*}, \dots, τ_{T}^{*})$ refers to the stage weight vector through the Orness measured in the multi-stage dynamic grey target decision process. Then, $a_{j +}^{l *} = (a_{1 +}^{l *}, \dots, a_{j +}^{l *}, \dots, a_{m +}^{l *})$ represents the negotiated bull’s-eye of the stage t_l under the objective u_j and objectives U, respectively. In addition, $a_{j +}^{*} = (a_{1 +}^{*}, \dots, a_{j +}^{*}, \dots, a_{m +}^{*})$ is the whole bull’s-eye the whole stages under the objective u_j and objectives U, respectively, Finally, the whole bulls-eye matrix is defined as A₊, which is obtained from the equation: $A_{+} = \sum_{l = 1}^{T} A_{+}^{l} \times t_{l}^{*} = [\begin{matrix} a_{1 +}^{*} & \dots & a_{j +}^{*} & \dots & a_{m +}^{*} \\ \dots & \dots & \dots & \dots & \dots \\ a_{1 +}^{*} & \dots & a_{j +}^{*} & \dots & a_{m +}^{*} \\ \dots & \dots & \dots & \dots & \dots \\ a_{1 +}^{*} & \dots & a_{j +}^{*} & \dots & a_{m +}^{*} \end{matrix}]$ (4)

Definition 5. Assume that there exist m objectives in the dynamic group decision, if $a_{0 j}^{l •}$ stands for the aggregation information of the experts under the objective u_j in the stage t_l, then

$\begin{matrix} R_{m}^{l} & = & {(a_{01}^{l *}, \dots, a_{0 m}^{l *}) | π_{1} (a_{01}^{l •} - a_{01}^{l *})^{2} + \dots \\ + π_{m} (a_{0 m}^{l •} - a_{0 m}^{l *})^{2} \leq R_{l}^{2}} \end{matrix}$ (5) is called as the multi-objective group grey target of the stage t_l with the stage bull’s-eye $a_{0 j}^{l *}$ and the radius R_l.

Definition 6. Assume that there exist m objectives in the dynamic group decision, if $a_{0 j}^{•}$ stands for the aggregation information of the experts under the objectives U in the stages T, then

$\begin{matrix} R_{m} & = & {(a_{1 +}^{*}, \dots, a_{m +}^{*}) | π_{1} (a_{01}^{•} - a_{01}^{*})^{2} + \dots \\ + π_{m} (a_{0 m}^{•} - a_{0 m}^{*})^{2} \leq R^{2}} \end{matrix}$ (6) is called as the group grey target of the stages T with the stage bull’s-eye $a_{j +}^{*}$ and the radius R².

3. A multi-objective dynamic group grey target decision approach

A multi-objective dynamic group decision problem involves four dimensions of the schemes, time, experts, objectives and decision values (decision information), and the decision result is mainly based on the four dimensions through the aggregation information to make the schemes ranking. In general, the schemes and the objectives influence the decision result through the decision information (attribute value), while the times and the experts can adjust and optimize the decision result through group game negotiation of the experts, the time stage compromise and system coordination. In view of this, in order to deal with the multi-objective dynamic group decision problem and make the schemes ranking, from the perspective of group negotiation and system coordination, by exploiting the asymmetric Nash bargaining model and Orness, we construct a novel multi-objective dynamic group grey target decision approach in this paper. Our work mainly consists of three parts, which are (1) establishing a two steps optimization model to calculate the stage bulls-eye; (2) improving the Orness measure model to solve the stage weights and whole bulls-eye; and (3) emploiting the Frobenius norm to measure the distance between the multi-objective group measure value matrix of the scheme, the stage bull’s-eye measure value matrix in the stage and the whole bulls-eye measure value matrix, and give out the evaluated schemes ranking in the stage and the stages.

3.1. Two steps optimization model for the stage bulls-eye

In the process of group grey target decision, an ideal scheme or bulls-eye is usually the result produced by the group negotiation [24 –26] and decision-making system coordination. In general, experts or DMs determine their ideal schemes through game negotiation with different bargaining power. However, negotiation schemes are not a consensus scheme accepted by all experts or DMs. In this case, founded on the stability of the decision-making system and the effectiveness of decision, decision system (coordinator) takes some actions to induce experts or DMs to change their schemes, so that it will produce a consensus ideal scheme or bulls-eye. In view of this, considering the existing problem of the grey target decision, we utilize the asymmetric Nash bargaining model and the thought of the minimization of system coordination deviation to construct a two steps optimization model to determine the stage bulls-eye.

In the group decision process, experts’ weights could be exploited to character and measure their bargaining power, let w_c be the weight of the expert e_c, and it satisfies $\sum_{c = 1}^{p} w_{c} = 1$ . Assume that $a_{0 j}^{lc}$ and $a_{0 j}^{l *}$ stands for the negotiation bulls-eye and the stage bulls-eye under the objective u_j, respectively. For ∀u_j, each expert has a minimum and maximization measure values to accept the schemes, which are denoted as $a_{1 j}^{lc}, a_{2 j}^{lc}$ , respectively. In general, the greater the actual evaluation measure value, the higher the expert’s satisfaction over the schemes, and then the definite evaluation measure should not be less than the minimum measure value. Accordingly, the ratio of the actual measured value and the maximization measure value and $a_{1 j}^{lc} / a_{2 j}^{lc}$ could be taken as the satisfaction and minimum satisfaction of the expert e_c under the objective u_j in the stage t_l, which means that the minimum satisfaction $a_{1 j}^{lc} / a_{2 j}^{lc}$ is the starting point for the expert e_k to negotiate. The greater the satisfaction of all experts, the more stable group decision system. However, due to the resource capability constraints, the experts’ ideal values over the different objectives are limited by the total amount of decision system resources, which could be taken as $\sum_{c = 1}^{p} a_{2 j}^{lc}$ . Let λ be the system bear coefficient of the decision system under an objective u_j in the stages, and it satisfies $max_{j} {\sum_{c = 1}^{p} a_{1 j}^{lc} / \sum_{c = 1}^{p} a_{2 j}^{lc}} < λ < 1 .$ (7)

According to the above discussion, experts can have achieved a relatively satisfactory result by maximizing the satisfaction deviation between the satisfied and the minimum satisfaction, and then an optimization model could be established to solve the negotiation bulls-eye. Because the negotiation bulls-eyes are not accepted by all experts, it is necessary for decision system or (coordinator) to generally adopt some coordination measures (such as the slightest deviation) in order to generate a stage bulls-eye for DMs. By the above analysis, we can construct a two-step optimization model to determine the stage bulls-eye from the perspective of group negotiation and system coordination, which is as follows: $\begin{matrix} max S_{l} = \sum_{j = 1}^{m} π_{j} \prod_{c = 1}^{p} (\frac{a_{0 j}^{lc}}{a_{2 j}^{lc}} - \frac{a_{1 j}^{lc}}{a_{2 j}^{lc}})^{w_{c}} \\ min D_{l} = \sum_{j = 1}^{m} π_{j} \sum_{c = 1}^{p} w_{c} (a_{0 j}^{l *} - a_{0 j}^{lc})^{2} \\ s . t {\begin{matrix} a_{1 j}^{lc} \leq a_{0 j}^{lc} \leq a_{2 j}^{lc} \\ \sum_{c = 1}^{P} a_{0 j}^{lc} = λ \sum_{c = 1}^{P} a_{2 j}^{lc} \\ a_{1 j}^{lc} = min_{i} {a_{ij}^{lc}} \\ a_{2 j}^{lc} = max_{i} {a_{ij}^{lc}} \\ \sum_{c = 1}^{p} w_{c} = 1 \\ \sum_{j = 1}^{m} π_{j} = 1 \\ max_{j} {\frac{\sum_{c = 1}^{p} a_{1 j}^{lc}}{\sum_{c = 1}^{p} a_{2 j}^{lc}}} < λ < 1 \\ i = 1, 2, \dots, n; j = 1, 2, \dots, m; \\ l = 1, 2, \dots, T; c = 1, 2, \dots, p \end{matrix} \end{matrix}$ (8)

Based on the above optimization model, we can obtain the stage bulls-eye $a_{0 j}^{l *} = (a_{01}^{l *}, \dots,$ $a_{0 j}^{l *}, \dots, a_{0 m}^{l *})$ , and then determine the stage bull’s-eye measure value matrix $A_{+}^{l}$ in the stage t_l.

3.2. An improved Orness measure model for the stage weight and whole bulls-eye

In the multi-objective dynamic group grey target decision, the influences of the time stages on the group decision results mainly appear after the aggregation information, which take on different results, while the stage weight is the feature element and key variable of the multi-stage decision problem. Because the weight of the time stage set by the existed models is much more subjective, which easily leads to big deviation over the scheme ranking. There also exists huge difference of the schemes under different time stage, which increases the complexity and instability of decision. Thus, the stage deviation of the schemes need to be minimized in the multi-objective dynamic group grey target decision. In view of this, based on the minimum principle of the information deviation among the adjacent stages and the Orness measure model [27], by maximizing the contribution of the stage evaluation information to the whole stages evaluation information, we construct a novel nonlinear model shown as follows: $\begin{matrix} min D = \sum_{i = 1}^{n} \sum_{l = 2}^{T} {(d_{i}^{l} τ_{l} - d_{i}^{l - 1} τ_{l - 1})}^{2} \\ s . t . {\begin{matrix} Orness (τ) = \frac{1}{T - 1} \sum_{l = 1}^{T} (T - l) τ_{l} = γ, 0 \leq γ \leq 1 \\ τ_{l} = (τ_{1}, τ_{2}, \dots, τ_{T}) \\ \sum_{l = 1}^{T} τ_{l} = 1 \\ 0 \leq τ_{l} \leq 1 \end{matrix} \end{matrix}$ (9)

Based on the above optimization model, we can acquire the stage weight vector $τ_{l}^{*}$ , and then we can determine the whole bulls-eye matrix A₊ by aggregating stage evaluation information.

3.3. An improved Orness measure model for the stage weight and whole bulls-eye

Because grey target decision is essentially based on the proximity of the sequences to make the scheme ranking, the multi-objective dynamic group grey target decision in this work can utilize the distances between the multi-objective group measure value matrix of the scheme x_i and the stage bull’s-eye measure value matrix in the stage t_l. Consequently, the whole bulls-eye measure value matrix provides the schemes ranking in the stage t_l and the stages. Unfortunately, the traditional grey target decision methods can only deal with the two-dimensional decision problem. For the three-dimensional or multi-dimensional decision problem, we employ Frobenius norm [28] to measure the norm distance between group measure value matrix of the scheme x_i and the stage bull’s-eye measure value matrix and the whole bulls-eye measure value matrix. As a result, the schemes ranking in the stage t_l and the stages is obtained.

Definition 7. Assume that $A = {[(a_{ij}^{lc})]}_{n \times m}$ stands for a n × m matrix, then ${∥ A ∥}_{F} = \sqrt{tr (A^{T} A)} = \sqrt{\sum_{i = 1}^{n} \sum_{j = 1}^{m} a_{ij}^{lc 2}}$ (10) is called as the Frobenius of this matrix.

Definition 8. The stage bull’s-eye measure value matrix $A_{+}^{l}$ of the scheme x_i in the t_l time phase is called the phase bull’s-eye matrix of the three-dimensional data space. $A_{+}^{l} = [\begin{matrix} a_{01}^{l *} & \dots & a_{0 j}^{l *} & a_{0 m}^{l *} \\ \dots & \dots & \dots & \dots & \dots \\ a_{01}^{l *} & \dots & a_{0 j}^{l *} & \dots & a_{0 m}^{l *} \\ \dots & \dots & \dots & \dots & \dots \\ a_{01}^{l *} & \dots & a_{0 j}^{l *} & \dots & a_{0 m}^{l *} \end{matrix}]$ (11)

Where w_c and π_j are the weights of the row vector and the column vector in the matrix $A_{+}^{l}$ , respectively, and satisfy $0 # w_{c} 1, a_{c = 1}^{\circ p} w_{c} = 1, 0 # π_{j} 1, a_{j = 1}^{\circ^{m}} π_{j} = 1 .$

The matrix norm is used to describe the difference information between the matrices.

Definition 9. According to the definition above 8, the norm $∥ \prod_{j = 1}^{m} \prod_{c = 1}^{p} w_{c} π_{j} A_{+}^{l} - \prod_{j = 1}^{m} \prod_{c = 1}^{p} w_{c} π_{j} A_{i}^{l} ∥$ (12) is the comprehensive degree of separation between the matrix $A_{i}^{l}$ and $A_{+}^{l}$ , that is, the norm distance between the group measure values matrix $A_{i}^{l}$ and the stage bull’s-eye matrix $A_{+}^{l}$ , recorded as

$d_{i}^{l} = ∥ \prod_{j = 1}^{m} \prod_{c = 1}^{p} π_{j} w_{c} A_{+}^{l} - \prod_{j = 1}^{m} \prod_{c = 1}^{p} π_{j} w_{c} A_{i}^{l} ∥$ (13)

Obviously, when m = 1, the matrix degenerates into a sequence, and the matrix gray target degenerates into a vector gray target.

Then under the Frobenius norm $d_{i}^{l}$ is

$\begin{matrix} d_{i}^{l} & = & ∥ \prod_{j = 1}^{m} \prod_{c = 1}^{p} π_{j} w_{c} A_{+}^{l} - \prod_{j = 1}^{m} \prod_{c = 1}^{p} π_{j} w_{c} A_{i}^{l} ∥ \\ = & \sqrt{\sum_{j = 1}^{m} \sum_{c = 1}^{p} π_{j} w_{c} {(a_{0 j}^{l *} - a_{ij}^{lc})}^{2}} \end{matrix}$ (14)

Definition 10. The whole bulls-eye matrix B of the scheme A under all time sets is called the four-dimensional data space (dynamic) system overall bulls-eye matrix.

$A_{+} = [\begin{matrix} a_{1 +}^{*} & \dots & a_{j +}^{*} & \dots & a_{m +}^{*} \\ \dots & \dots & \dots & \dots & \dots \\ a_{1 +}^{*} & \dots & a_{j +}^{*} & \dots & a_{m +}^{*} \\ \dots & \dots & \dots & \dots & \dots \\ a_{1 +}^{*} & \dots & a_{j +}^{*} & \dots & a_{m +}^{*} \end{matrix}]$ (15)

Where w_c, π_j and τ_l are the weights of the lateral, vertical, and longitudinal quantities in the matrix $A_{+}^{l}$ , respectively, and satisfy $\begin{matrix} 0 # w_{c} 1, a_{c = 1}^{\circ p} w_{c} = 1, 0 # π_{j} 1, a_{j = 1}^{\circ^{m}} π_{j} = 1, \\ 0 # w_{c} 1, a_{l = 1}^{\circ T} τ_{l} = 1 . \end{matrix}$

The matrix norm is used to describe the difference information between the matrices.

Definition 11. According to the definition above 10, the norm $∥ \prod_{j = 1}^{m} \prod_{c = 1}^{p} \prod_{l = 1}^{T} π_{j} w_{c} τ_{l} A_{+} - \prod_{j = 1}^{m} \prod_{c = 1}^{p} \prod_{l = 1}^{T} π_{j} w_{c} τ_{l} A_{i} ∥$ (16) is the comprehensive degree of separation between the matrices A_i and A₊, which is called the norm distance of group measure values matrix A_i and the whole bulls-eye matrix A₊, recorded as $d_{i} = ∥ \prod_{j = 1}^{m} \prod_{c = 1}^{p} \prod_{l = 1}^{T} π_{j} w_{c} τ_{l} A_{+} - \prod_{j = 1}^{m} \prod_{c = 1}^{p} \prod_{l = 1}^{T} π_{j} w_{c} τ_{l} A_{i} ∥$ (17)

Same as above, then under the Frobenius norm d_i is

$\begin{matrix} d_{i} & = & ∥ \prod_{j = 1}^{m} \prod_{c = 1}^{p} \prod_{l = 1}^{T} π_{j} w_{c} τ_{l} A_{+} - \prod_{j = 1}^{m} \prod_{c = 1}^{p} \prod_{l = 1}^{T} π_{j} w_{c} τ_{l} A_{i} ∥ \\ = & \sqrt{\sum_{j = 1}^{m} \sum_{c = 1}^{p} \sum_{l = 1}^{T} π_{j} w_{c} τ_{l} {(a_{j +}^{*} - a_{ij}^{lc})}^{2}} \end{matrix}$ (18)

According to $d_{i}^{l}$ and d_i, we can determine give the schemes ranking in the stage t_l and the stages.

3.4. Sensitivity analysis of the Orness measure parameter

In the multi-objective dynamic group grey target decision model, the Orness measure parameter γ is usually determined by the subjective preferences of DMs, which increases the decision risk. In order to measure the effect of the parameter on the decision result, we analyze the sensitivity of the value γ, which can provide for DMs a scientific reference by adjusting the parameters γ in a certain range to guarantee the current optimal scheme unchanged.

Assume that r (x₁, x₂, …, x_n), R stands for the schemes ranking based on the proposed model and the schemes ranking set, respectively, and r (x₁, x₂, …, x_n) ∈ R. For the multi-objective dynamic group decision problem, there may exist n ! = n (n - 1) …2 × 1 ranking results. According to the section 3.2, it is well known that the ranking results r (x₁, x₂, …, x_n) are the function of τ_l (l = 1, 2, …, T), which can be denoted as f (τ_l). In order to analyze the sensitivity of the value γ on the decision results, we construct an analytical model, which shown as follows:

$\begin{matrix} max / min \bar{γ} = \frac{1}{T - 1} \sum_{l = 1}^{T} (T - l) τ_{l} \\ s . t . {\begin{matrix} r (x_{1}, x_{2}, \dots, x_{n}) = f (τ_{l}) \\ \begin{matrix} min D = \sum_{i = 1}^{n} \sum_{l = 2}^{T} {(d_{i}^{l} τ_{l} - d_{i}^{l - 1} τ_{l - 1})}^{2} \\ s . t . {\begin{matrix} Orness (τ) = \frac{1}{T - 1} \sum_{l = 1}^{T} (T - l) τ_{l} \\ = γ_{0}, 0 \leq γ_{0} \leq 1 \\ τ_{l} = (τ_{1}, τ_{2}, . . ., τ_{T}) \\ \sum_{l = 1}^{T} τ_{l} = 1 \\ 0 \leq τ_{l} \leq 1 \end{matrix} \end{matrix} \end{matrix} \end{matrix}$ (19)

Where, γ₀ stands for the initial value of the measure parameter, while $\bar{γ}$ expresses the variation ranges of the measure parameter corresponding to the scheme ranking result r (x₁, x₂, …, x_n).

The DMs can judge whether the previously selected Orness measure parameter y is appropriate according to the range calculated by the model (13), so as to achieve the purpose of controlling the evaluation result and reducing the decision risk.

4. Illustrative example

As social and economic development, more and more public events have been appearing, so that it is an urgent task for the managers to make the emergency response, which can reduce the economic losses and casualties. However, there exists a variety of uncertainties in the evolution of the sudden public events, so that how to dynamically generate and adjust contingency decision schemes based on the judgment and experience of the experts and the evolution of an emergency situation is a difficult problem. In addition, emergencies often go through some stages consisting of the occurrence, development, inhibition, and elimination, and the emergency management of each stage has its own characteristics. Therefore, how to make dynamic emergency decision and choose an appropriate emergency decision scheme according to the crisis situation and the development stage is a worthy and urgent dynamic group decision problem to deal with. In view of this, we exploit the proposed model to solve it.

4.1. The multi-objective group measures value matrixes

Assume that there exist a sudden public event just happened, in order to deal with the event, the relevant departments quickly set up four schemes denoted as X ={ x₁, x₂, x₃, x₄ }. To obtain the option scheme, the relevant departments invite five experts to make an evaluation of the schemes, which are denoted as E ={ e₁, e₂, e₃, e₄, e₅ }. According to the importance and the authority of the experts, their weights are 0.05, 0.1, 0.3, 0.35, 0.2, respectively. The objectives of feasibility, timeliness, economy, risk, and aftermath denoted as U ={ u₁, u₂, u₃, u₄, u₅ } through the expert consultation and questionnaire survey are selected to make evaluation of the schemes according to the goal and characteristics of the emergency decision and the characteristics of emergency decision, and their weights are obtained as follows π = (0.3, 0.25, 0.15, 0.2, 0.1). After the discussion of the experts, the development stage of this emergency can be divided into three stages, which are denoted as T ={ t₁, t₂, t₃ }. By collecting evaluation information of the experts, the group measure value matrixes $A_{i}^{1}, A_{i}^{2}, A_{i}^{3} (i = 1, 2, 3, 4)$ in the stage t_l can be obtained as follows: $\begin{matrix} A_{1}^{1} = [\begin{matrix} 0.86 & 0.87 & 0.92 & 0.94 & 0.95 \\ 0.9 & 0.93 & 0.92 & 0.93 & 0.94 \\ 0.89 & 0.86 & 0.9 & 0.92 & 0.95 \\ 0.88 & 0.82 & 0.89 & 0.93 & 0.96 \\ 0.87 & 0.93 & 0.92 & 0.92 & 0.97 \end{matrix}] & A_{2}^{1} = [\begin{matrix} 0.89 & 0.94 & 0.9 & 0.92 & 0.94 \\ 0.91 & 0.93 & 0.92 & 0.95 & 0.96 \\ 0.9 & 0.88 & 0.91 & 0.93 & 0.97 \\ 0.92 & 0.92 & 0.91 & 0.95 & 0.98 \\ 0.91 & 0.89 & 0.94 & 0.96 & 0.92 \end{matrix}] \\ A_{3}^{1} = [\begin{matrix} 0.92 & 0.95 & 0.94 & 0.96 & 0.97 \\ 0.91 & 0.84 & 0.91 & 0.93 & 0.95 \\ 0.93 & 0.92 & 0.92 & 0.95 & 0.98 \\ 0.9 & 0.87 & 0.96 & 0.94 & 0.92 \\ 0.95 & 0.94 & 0.93 & 0.98 & 0.94 \end{matrix}] & A_{4}^{1} = [\begin{matrix} 0.84 & 0.83 & 0.87 & 0.85 & 0.90 \\ 0.85 & 0.94 & 0.89 & 0.82 & 0.93 \\ 0.93 & 0.87 & 0.93 & 0.88 & 0.84 \\ 0.92 & 0.92 & 0.82 & 0.96 & 0.93 \\ 0.90 & 0.96 & 0.87 & 0.94 & 0.96 \end{matrix}] \\ A_{1}^{2} = [\begin{matrix} 0.83 & 0.85 & 0.88 & 0.90 & 0.90 \\ 0.88 & 0.87 & 0.87 & 0.95 & 0.86 \\ 0.92 & 0.85 & 0.93 & 0.98 & 0.91 \\ 0.89 & 0.86 & 0.88 & 0.94 & 0.92 \\ 0.89 & 0.94 & 0.95 & 0.84 & 0.90 \end{matrix}] & A_{2}^{2} = [\begin{matrix} 0.95 & 0.93 & 0.85 & 0.98 & 0.92 \\ 0.92 & 0.94 & 0.96 & 0.89 & 0.89 \\ 0.83 & 0.91 & 0.87 & 0.95 & 0.96 \\ 0.96 & 0.85 & 0.89 & 0.89 & 0.94 \\ 0.91 & 0.83 & 0.85 & 0.93 & 0.98 \end{matrix}] \\ A_{3}^{2} = [\begin{matrix} 0.80 & 0.93 & 0.87 & 0.92 & 0.98 \\ 0.94 & 0.85 & 0.84 & 0.93 & 0.94 \\ 0.84 & 0.88 & 0.90 & 0.85 & 0.91 \\ 0.9 & 0.86 & 0.98 & 0.83 & 0.83 \\ 0.96 & 0.95 & 0.92 & 0.91 & 0.87 \end{matrix}] & A_{4}^{2} = [\begin{matrix} 0.88 & 0.89 & 0.93 & 0.93 & 0.87 \\ 0.94 & 0.85 & 0.92 & 0.91 & 0.94 \\ 0.87 & 0.94 & 0.95 & 0.85 & 0.92 \\ 0.86 & 0.86 & 0.86 & 0.84 & 0.89 \\ 0.92 & 0.93 & 0.83 & 0.88 & 0.92 \end{matrix}] \\ A_{1}^{3} = [\begin{matrix} 0.90 & 0.85 & 0.94 & 0.87 & 0.85 \\ 0.93 & 0.89 & 0.90 & 0.84 & 0.90 \\ 0.82 & 0.88 & 0.88 & 0.84 & 0.93 \\ 0.86 & 0.94 & 0.89 & 0.96 & 0.95 \\ 0.83 & 0.86 & 0.89 & 0.88 & 0.84 \end{matrix}] & A_{2}^{3} = [\begin{matrix} 0.87 & 0.92 & 0.83 & 0.83 & 0.92 \\ 0.96 & 0.94 & 0.85 & 0.96 & 0.97 \\ 0.87 & 0.85 & 0.93 & 0.90 & 0.94 \\ 0.97 & 0.96 & 0.88 & 0.91 & 0.84 \\ 0.94 & 0.97 & 0.90 & 0.94 & 0.90 \end{matrix}] \\ A_{3}^{3} = [\begin{matrix} 0.96 & 0.96 & 0.79 & 0.91 & 0.88 \\ 0.93 & 0.87 & 0.84 & 0.94 & 0.89 \\ 0.90 & 0.86 & 0.98 & 0.87 & 0.97 \\ 0.85 & 0.93 & 0.91 & 0.92 & 0.95 \\ 0.91 & 0.94 & 0.92 & 0.89 & 0.95 \end{matrix}] & A_{4}^{3} = [\begin{matrix} 0.87 & 0.89 & 0.92 & 0.91 & 0.89 \\ 0.93 & 0.82 & 0.87 & 0.91 & 0.96 \\ 0.87 & 0.85 & 0.95 & 0.92 & 0.92 \\ 0.94 & 0.88 & 0.92 & 0.84 & 0.85 \\ 0.85 & 0.84 & 0.89 & 0.92 & 0.94 \end{matrix}] \end{matrix}$

4.2. The stage bulls-eye matrix

According to the analysis of Section 3.1 and the group measure value matrixes, we can determine the total amount of decision system resources under the objective u_j, which are shown as follows:

$\sum_{c = 1}^{5} a_{2 j}^{1 c} (j = 1, 2, \dots, 5) = (4.63, 4.69, 4.69, 4.8, 4.86)$ $\sum_{c = 1}^{5} a_{2 j}^{2 c} (j = 1, 2, \dots, 5) = (4.73, 4.62, 4.77, 4.78, 4.8)$ $\sum_{c = 1}^{5} a_{2 j}^{3 c} (j = 1, 2, \dots, 5) = (4.73, 4.71, 4.66, 4.69, 4.76)$

By calculating, we can obtain the range of the system bear coefficient of the decision system under an objective, that is, λ ∈ (0.9698, 1). Based on the proposed two steps optimization model, by taking λ as 0.97, we can obtain the stage bulls-eye matrixes $A_{+}^{1}, A_{+}^{2}, A_{+}^{3}$ in the stage t₁, t₂, t₃, which are shown respectively. $\begin{matrix} A_{+}^{1} = [\begin{matrix} 0.9165 & 0.9247 & 0.9220 & 0.9496 & 0.9565 \\ 0.9165 & 0.9247 & 0.9220 & 0.9496 & 0.9565 \\ 0.9165 & 0.9247 & 0.9220 & 0.9496 & 0.9565 \\ 0.9165 & 0.9247 & 0.9220 & 0.9496 & 0.9565 \\ 0.9165 & 0.9247 & 0.9220 & 0.9496 & 0.9565 \end{matrix}] \\ A_{+}^{2} = [\begin{matrix} 0.9450 & 0.9069 & 0.9515 & 0.9430 & 0.9415 \\ 0.9450 & 0.9069 & 0.9515 & 0.9430 & 0.9415 \\ 0.9450 & 0.9069 & 0.9515 & 0.9430 & 0.9415 \\ 0.9450 & 0.9069 & 0.9515 & 0.9430 & 0.9415 \\ 0.9450 & 0.9069 & 0.9515 & 0.9430 & 0.9415 \end{matrix}] \\ A_{+}^{3} = [\begin{matrix} 0.9213 & 0.9180 & 0.9238 & 0.9235 & 0.9322 \\ 0.9213 & 0.9180 & 0.9328 & 0.9235 & 0.9322 \\ 0.9213 & 0.9180 & 0.9328 & 0.9235 & 0.9322 \\ 0.9213 & 0.9180 & 0.9328 & 0.9235 & 0.9322 \\ 0.9213 & 0.9180 & 0.9328 & 0.9235 & 0.9322 \end{matrix}] \end{matrix}$

4.3. The Frobenius distance $d_{i}^{l}$ and scheme ranking in the stage t_l

According to the group measure value matrix of the scheme x_i and the stage bulls-eye matrixes $A_{+}^{1}, A_{+}^{2}, A_{+}^{3}$ in the stage t₁, t₂, t₃, by utilizing the expression (11), we can obtain the Frobenius distance between group measure value matrix of the scheme x_i and the stage bull’s-eye measure value matrix $d_{i}^{l}$ in the stage t_l, and the results are shown as follows:

$d_{1}^{1} = 0.0437$ , $d_{2}^{1} = 0.0193$ , $d_{3}^{1} = 0.0279$ , and $d_{4}^{1} = 0.0502$ ; $d_{1}^{2} = 0.0505$ , $d_{2}^{2} = 0.0561$ , $d_{3}^{2} =$ 0.0666 and $d_{4}^{2} = 0.0651$ ; $d_{1}^{3} = 0.0437$ , $d_{2}^{3} =$ 0.0492, $d_{3}^{3} = 0.0572$ , and $d_{4}^{3} = 0.0512$ .

Accoding to the caculation results, we can determine the scheme ranking in different stage. In the stage t₁, the ranking result is x₂ ≻ x₃ ≻ x₁ ≻ x₄; In the stage t₂, the ranking result is x₁ ≻ x₂ ≻ x₄ ≻ x₃; In the stage t₃, the ranking result is x₁ ≻ x₂ ≻ x₄ ≻ x₃.

After the occurrence of the emergencies, the scheme ranking of various stages of a decision scheme chain, as showed in Fig. 1. According to Fig. 1, we can see that the decision system of the emergency should choose the decision scheme chain t₁ : x₂ → t₂ : x₁ → t₃ : x₁.

Fig.1

The decision scheme chains.

4.4. The scheme ranking in the stages and the sensitivity analysis

Through the expert consultation, taking the Orness measure parameter γ as 0.3, based on the proposed model in Section 3.2, we construct a nonlinear model with the stage weight, which is as follows: $\begin{matrix} min D = \sum_{i = 1}^{4} \sum_{l = 2}^{3} {(d_{i}^{l} τ_{l} - d_{i}^{l - 1} τ_{l - 1})}^{2} \\ s . t . {\begin{matrix} Orness (τ) = \frac{1}{2} \sum_{l = 1}^{3} (3 - l) τ_{l} = γ \\ τ_{l} = (τ_{1}, τ_{2}, τ_{3}) \\ \sum_{l = 1}^{3} τ_{l} = 1 \\ 0 \leq τ_{l} \leq 1 \end{matrix} \end{matrix}$

By utilizing the MATLAB R2014a, when γ = 0.3, we can obtain $τ_{1}^{*} = 0.1552$ , $τ_{2}^{*} = 0.2897$ , $τ_{3}^{*} = 0.5551$ , and the minimum deviation is 0.001. According to the stage bull’s-eye measure value matrix, we can determine the whole bulls-eye matrix A₊ by aggregating bulls-eye information in different stages as follows: $A_{+} = [\begin{matrix} 0.9275 & 0.9159 & 0.9316 & 0.9333 & 0.9388 \\ 0.9275 & 0.9159 & 0.9316 & 0.9333 & 0.9388 \\ 0.9275 & 0.9159 & 0.9316 & 0.9333 & 0.9388 \\ 0.9275 & 0.9159 & 0.9316 & 0.9333 & 0.9388 \\ 0.9275 & 0.9159 & 0.9316 & 0.9333 & 0.9388 \end{matrix}]$

Based on the expression (12), we can obtain the Frobenius norm distance between the group measure value matrix of the scheme x_i and the whole bulls-eye measure value matrix, which are the following d₁ = 0.0464, d₂ = 0.0333, d₃ = 0.0368, d₄ = 0.0440. From the above calculation results with the Orness measure parameter γ = 0.3, we can determine the whole scheme ranking results shown as r (x₁, x₂, x₃, x₄) = x₂≻ x₃ ≻ x₄ ≻ x₁.

In order to fully exploit the sensitivity of the value γ on the decision results f (τ_l) and determine its range, based on the expression (13), we can establish the following analytical model: $\begin{matrix} max / min \bar{γ} = \frac{1}{2} \sum_{l = 1}^{3} (3 - l) τ_{l} \\ s . t . {\begin{matrix} f (τ_{l}) = r (x_{1}, x_{2}, x_{3}, x_{4}) = x_{2} ≻ x_{3} ≻ x_{4} ≻ x_{1} \\ \begin{matrix} min D = \sum_{i = 1}^{4} \sum_{l = 2}^{3} {(d_{i}^{l} τ_{l} - d_{i}^{l - 1} τ_{l - 1})}^{2} \\ s . t . {\begin{matrix} Orness (τ) = \frac{1}{2} \sum_{l = 1}^{3} (3 - l) τ_{l} = 0.3 \\ τ_{l} = (τ_{1}, τ_{2}, τ_{3}) \\ \sum_{l = 1}^{3} τ_{l} = 1 \\ 0 \leq τ_{l} \leq 1 \end{matrix} \end{matrix} \end{matrix} \end{matrix}$

By calculating the above optimization model, we can acquire the range of the Orness measure parameter to keep the same scheme ranking results, which is $\bar{γ} \in [0.2638, 0.6584]$ . For example, when γ = 0.3, 0.5, the scheme ranking results are all x₂ ≻ x₃ ≻ x₄ ≻ x₁; while when γ = 0.7, the scheme ranking result is x₂ ≻ x₃ ≻ x₁ ≻ x₄, and they are different. For the sake of the observation, we can plot the change curve of the normal distances of the schemes over the Orness measure parameter γ, as showed in Fig. 2. According to the Fig. 2, we can see that the result of the scheme ranking is not influenced by the changing of the Orness measure parameter in the range, once over exceeds the range, the scheme ranking results will change.

Fig.2

The change curve of the norm distances over the Orness measure parameter.

4.5. Comparison

In order to further validate the validity and rationality of the multi-objective grey target decision model, the results based on the models involved in the literature [10, 12] are made. Based on the method from the literature [10], x₃ is the option scheme, which is significantly different from the ranking based on the proposed model. The consequence is caused by the reason that the method from the literature [10] utilizes the positive and negative bulls-eye distance of the schemes to make the ranking by subjectively setting its bulls-eye, and ignores the relationships among the objectives. Based on the method involved in the literature [12], its result is same as this paper. However, compared with the method involved in the literature [12], the proposed model is relatively simple in the calculation process by exploiting the normal distance to solve the scheme ranking and fully considers the group behavior. According to the comparative analysis based on different methods, it is well known that the proposed model possesses the advantages of fully considering the group behavior and scientifically and reasonable solving the stage bull’s-eye by analyzing and discussing the setting mechanism of bull’s-eye.

According to the above discussion and analysis, the proposed model makes full use of the experts’ information and interaction time. It effectively extract the hidden related information and rule about group decision making. It can also select the schemes for various stages and accident plots scientifically and rationally, which makes up for the shortcomings of simply considering selecting or generating decision plans of previous researches. The method is more suitable in practical and more effective in the group decision making.

5. Conclusion

In this paper, we study the dynamic grey target decision making problem, and propose a dynamic group grey target decision approach based on Orness measure constraints and Nash bargaining model, which enriches and develops the application field and scope of the grey target decision. Through the model and case analysis, the model contains the following advantages.

The model takes full account of the group behavioral characteristics, gives a relatively scientific and reasonable method of solving the stage bull’s-eye by analyzing and discussing the setting mechanism of bull’s-eye from the perspective of the group negotiation and the system coordination, which could help improve the stability and adaptability of the decision-making.

The model takes full account of the group behavior information and the time information, properly deals with the group decision-making problem by effectively excavating the related information and rules hidden in the group decision making, which can make the evaluation result be more realistic with reality.

The model can make a scientific and rational choice of the schemes under various stages and scenarios as the time develops and new information changes, which can make up for the shortcomings of the previous research in a simple choice.

However, there are also shortcomings in the model. For example, only using the method of assembly to set the overall bull’s-eye needs further improvement, which is the direction of our later efforts.

Footnotes

Acknowledgments

This work is partially funded by the National Natural Science Foundation of China (71503103); Natural Science Foundation of Jiangsu Province (BK20150157);Soft Science Foundation of Jiangsu Province (BR2018005); Social Science Foundation of Jiangsu Province (14GLC008); Ministry of Education Humanities and Social Science Fund Project (17YJC640223); Key Projects of Philosophy and Social Science Research in Jiangsu Province (2017ZDIXM034); Henan Science and Technology Department of Science and Technology Program Industry Research Project (172102210257); Special Fund for Fundamental Scientific Research Business Fees in Central Universities (2017JDZD06).

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A dynamic group grey target decision approach with group negotiation and Orness measure constraint

Abstract

Keywords

1. Introduction

2. Preliminaries

3.1. Two steps optimization model for the stage bulls-eye

4.1. The multi-objective group measures value matrixes

4.2. The stage bulls-eye matrix

4.3. The Frobenius distance d i l and scheme ranking in the stage t l

5. Conclusion

Footnotes

Acknowledgments

References

4.3. The Frobenius distance $d_{i}^{l}$ and scheme ranking in the stage t_l