Multi-criteria Large-Scale Group Decision-Making in Linguistic Contexts: A Perspective of Conflict Analysis and Resolution

Abstract

As a universal phenomenon, conflicts exist widely in various fields such as politics, economic life, military, and culture. Group decision-making techniques that effectively identify and resolve conflict during the decision-making process will result in stronger group consensus, while existing studies rarely discuss the multi-criteria large-scale group decision-making (LSGDM) from the perspective of conflict analysis and resolution. This paper systematically studies conflict analysis and resolution approach to obtain consensus decision results. Conflicts among decision makers (DMs) in LSGDM are divided into two kinds: goal conflicts and cognitive conflicts. Based on Pawlak conflict analysis, we introduce three relations among DMs, i.e., conflict, neutrality, alliance into multi-criteria LSGDM in linguistic contexts. Based on linguistic assessment, an improved Pawlak conflict analysis is used to analyze goal conflicts, and the alliance of DMs and the weight of criteria are obtained. According to three cognitive conflict relations, a conflict coordination and feedback mechanism is designed to resolve cognitive conflicts between alliance pairs. Finally, an illustrative example is used to verify the effectiveness and applicability of the proposed model.

Appendices

Appendix A: Proof of Theorem 1

By the Table 2, the expected loss functions produced by three actions for DM y are shown below.

$$\begin{aligned} C^{x}(a_{C}\mid y)=&\rho ^{\diamond }(x,y)\cdot \lambda _{CC}+(1-\rho ^{\diamond }(x,y))\cdot \lambda _{CA};\\ N^{x}(a_{N}\mid y)=&\rho ^{\diamond }(x,y)\cdot \lambda _{NC}+(1-\rho ^{\diamond }(x,y))\cdot \lambda _{NA};\\ A^{x}(a_{A}\mid y)=&\rho ^{\diamond }(x,y)\cdot \lambda _{AC}+(1-\rho ^{\diamond }(x,y))\cdot \lambda _{AA}. \end{aligned}$$

According to minimum-risk principle of Bayesian decision, we have the following rules:

(C): If $S^{x}(a_{C}\mid y)\le W^{x}(a_{N}\mid y)$ and $S^{x}(a_{C}\mid y)\le N^{x}(a_{A}\mid y)$, then $y\in CO_{\diamond }^{(\nu , \mu )}(x)$;

(N): If $W^{x}(a_{N}\mid y)\le S^{x}(a_{C}\mid y)$ and $W^{x}(a_{N}\mid y)\le N^{x}(a_{A}\mid y)$, then $y\in NE_{\diamond }^{(\nu , \mu )}(x)$;

(A): If $N^{x}(a_{A}\mid y)\le S^{x}(a_{C}\mid y)$ and $N^{x}(a_{A}\mid y)\le W^{x}(a_{N}\mid y)$, then $y\in AL_{\diamond }^{(\nu , \mu )}(x)$.

We know $0\le \lambda _{CC}\le \lambda _{NC}\le \lambda _{AC}$ and $0\le \lambda _{AA}\le \lambda _{NA} \le \lambda _{CA}$, then rules (S), (W), and (N) can be simplified as:

(C): If $\rho ^{\diamond }(x,y)>\mu$ and $\rho ^{\diamond }(x,y)>\gamma$, then $y\in CO_{\diamond }^{(\nu , \mu )}(x)$;

(N): If $\rho ^{\diamond }(x,y)\le \mu$ and $\rho ^{\diamond }(x,y)\ge \nu$, then $y\in NE_{\diamond }^{(\nu , \mu )}(x)$;

(A): If $\rho ^{\diamond }(x,y)< \nu$ and $\rho ^{\diamond }(x,y)<\gamma$, then $y\in AL_{\diamond }^{(\nu , \mu )}(x)$, where

$$\begin{aligned} \mu =&\frac{\lambda _{CA}-\lambda _{NA}}{(\lambda _{CA}-\lambda _{NA})+(\lambda _{NC}-\lambda _{CC})},\\ \nu =&\frac{\lambda _{NA}-\lambda _{AA}}{(\lambda _{NA}-\lambda _{AA})+(\lambda _{AC}-\lambda _{NC})}, \\ \gamma =&\frac{\lambda _{CA}-\lambda _{AA}}{(\lambda _{CA}-\lambda _{AA})+(\lambda _{AC}-\lambda _{CC})}. \end{aligned}$$

Appendix B: Proof of Theorem 2

Without loss of generality, suppose the alliance pair $(X_{y},X_{z})$ has a maximal cognitive conflict satisfying $\rho _{yz}^{\circ t}> \psi$ in the t-th iteration. In the $(t+1)$-th iteration, we discuss in three situations.

For $\xi _{t}^{k},\xi _{t}^{l}\in (0,1)$, if both alliance $X_{k}$ and $X_{l}$ accept adjustment strategies, then

$$\begin{aligned} \rho _{yz}^{\circ t+1}&=\frac{1}{2gnm } \sum _{i=1}^{n}\sum _{j=1}^{m}\left|\Delta ^{-1} v_{ij(t+1)}^{X_{y}}-\Delta ^{-1} v_{ij(t+1)}^{X_{z}}\right|\\&=\frac{1}{2gnm } \sum _{i=1}^{n}\sum _{j=1}^{m} \Big \vert \left( \xi _{t}^{y} \Delta ^{-1} v_{ij(t)}^{X_{y}} + (1-\xi _{t}^{y})\Delta ^{-1} v_{ij(t)}^{X_{z}} \right) \\&\qquad \qquad \qquad -\left( \xi _{t}^{z} \Delta ^{-1} v_{ij(t)}^{X_{z}} + (1-\xi _{t}^{z})\Delta ^{-1} v_{ij(t)}^{X_{y}} \right) \Big \vert \\&=\frac{\left|\xi _{t}^{y} + \xi _{t}^{z}-1 \right|}{2gnm } \sum _{i=1}^{n}\sum _{j=1}^{m} \left|\Delta ^{-1} v_{ij(t)}^{X_{y}} -\Delta ^{-1} v_{ij(t)}^{X_{z}}\right|\\&\le \frac{1}{2gnm } \sum _{i=1}^{n}\sum _{j=1}^{m}\left|\Delta ^{-1} v_{ij(t)}^{X_{y}}-\Delta ^{-1} v_{ij(t)}^{X_{z}}\right|=\rho _{yz}^{\circ t}. \end{aligned}$$

For any $X_{k}\in AL, k\ne y,z$, we can obtain $\rho _{yk}^{\circ t+1},\rho _{zk}^{\circ t+1} < \rho _{yz}^{\circ t}$. Then,

$$\begin{aligned} \max _{X_{k},X_{l}\in AL}\left\{ \rho _{kl}^{\circ t+1} \right\}&=\max \Bigg \{\rho _{yz}^{\circ t+1}, \max _{\begin{array}{c} X_{k},X_{l}\in AL, \\ k,l\ne y,z \end{array}}\left\{ \rho _{kl}^{\circ t+1} \right\} ,\max _{\begin{array}{c} X_{k}\in AL, \\ k\ne y,z \end{array}} \left\{ \rho _{yk}^{\circ t+1} \right\} , \max _{\begin{array}{c} X_{k}\in AL, \\ k\ne y,z \end{array}} \left\{ \rho _{zk}^{\circ t+1} \right\} \Bigg \} \\&< \max \Biggl \{ \max _{\begin{array}{c} X_{k},X_{l}\in AL, \\ k,l\ne y,z \end{array}}\left\{ \rho _{kl}^{\circ t} \right\} , \rho _{yz}^{\circ t} \Biggr \} =\rho _{yz}^{\circ t}=\max _{X_{k},X_{l}\in AL}\left\{ \rho _{kl}^{\circ t} \right\} . \end{aligned}$$

If one alliance accepts while the other rejects. Suppose alliance $X_{k}$ accepts and $X_{l}$ rejects modification strategy, then we have

$$\begin{aligned} \max _{X_{k},X_{l}\in AL}\left\{ \rho _{kl}^{\circ t+1} \right\}&=\max \Bigg \{\max _{\begin{array}{c} X_{k},X_{l}\in AL, \\ k,l\ne y,z \end{array}}\left\{ \rho _{kl}^{\circ t+1} \right\} ,\max _{X_{k}\in AL} \left\{ \rho _{yk}^{\circ t+1} \right\} \Bigg \} \\&< \max \Biggl \{ \max _{\begin{array}{c} X_{k},X_{l}\in AL, \\ k,l\ne y,z \end{array}}\left\{ \rho _{kl}^{\circ t} \right\} , \rho _{yz}^{\circ t} \Biggr \} =\rho _{yz}^{\circ t}=\max _{X_{k},X_{l}\in AL}\left\{ \rho _{kl}^{\circ t} \right\} . \end{aligned}$$

If both alliance $X_{y}$ and $X_{z}$ reject adjustment strategies, then $v_{ij(t+1)}^{X_{k}}=v_{ij(t)}^{X_{k}}$. So, $\max _{X_{k},X_{l}\in AL}\left\{ \rho _{kl}^{\circ t+1} \right\} \le \max _{X_{k},X_{l}\in AL}\left\{ \rho _{kl}^{\circ t} \right\}$.

In summary, it can be proved that $\max _{X_{k},X_{l}\in AL}\left\{ \rho _{kl}^{\circ t+1} \right\} \le \max _{X_{k},X_{l}\in AL}\left\{ \rho _{kl}^{\circ t} \right\}$ is established.