Multi-criteria decision making based on relative measures

Abstract

According to regret theory, decision makers may care more about the relative values than the absolute values of alternatives. Generally, the relative values of two alternatives can be reflected from different views. Based on Abelian linearly ordered group, the relative measure is firstly proposed to calculate the relative values from different views. Then two methods are developed to calculate the relative values of alternatives, and are proved to obtain the same result under some conditions on the aggregation operator and the relative measure. Furthermore, based on the relative values of each pair of alternatives, several models are established to identify four types of optimal alternatives. The relationship of these four types of optimal alternatives is also discussed. The proposed methods can be considered as a general framework of the existing ones and can be further extended to identify the most inferior alternatives. Several examples are given to illustrate the proposed methods.

Appendix

Proof of Theorem 1

Without loss of generality, we assume that there exist two scenarios $(w^{*},v^{*})$ and $(w^{{\prime }},v^{{\prime }})$ and two alternatives $x_{i}$ and $x_d$ such that $\gamma _{k}=\{c_{ki} (w^{*},v^{*})\}$ and $\mathop {\max }\limits _{h\in M} \gamma _{kh} =\{c_{kd} (w^{{\prime }},v^{{\prime }})\}$. Note that $\gamma _{k}\le \mathop {\max }\limits _{w\in W,v\in V} \{c_{ki} (w,v)\}=\gamma _{ki} \le \mathop {\max }\limits _{h\in M} \gamma _{kh}$, and

$$\begin{aligned} \mathop {\max }\limits _{h\in M} \gamma _{kh} \le \mathop {\max }\limits _{w\in W,v\in V} \{c_{kd} (w,v)\}\le \mathop {\max }\limits _{w\in W,v\in V} \mathop {\max }\limits _{h\in M} \{c_{kh} (w,v)\}=\gamma _{k} \end{aligned}$$

which follows that $\gamma _{k}=\mathop {\max }\limits _{i\in M} \gamma _{ki}$.

Since $\delta _{k}=\mathop {\min }\limits _{v\in V} \mathop {\max }\limits _{w\in W} \mathop {\max }\limits _{h\in M\backslash k} c_{kh} (w,v)$, and $\delta _{k}\ge \mathop {\min }\limits _{v\in V} \mathop {\max }\limits _{w\in W} c_{kh} (w,v)=\delta _{kh}$ for all $h\in M$, we have $\delta _{k}\ge \mathop {\max }\limits _{h\in M} \delta _{kh}$. Suppose $\delta _{k}=\mathop {\min }\limits _{v\in V} \mathop {\max }\limits _{w\in W} c_{ki} (w,v)$, then we have $\delta _{k}\le \mathop {\max }\limits _{h\in M\backslash k} \mathop {\min }\limits _{v\in V} \mathop {\max }\limits _{w\in W} c_{kh} (w,v)=\mathop {\max }\limits _{h\in M\backslash k} \delta _{kh}$, which follows that $\delta _{k}=\mathop {\max }\limits _{h\in M\backslash k} \delta _{kh}$.

Since $\beta _{k}=\mathop {\max }\limits _{v\in V} \mathop {\min }\limits _{w\in W} \mathop {\max }\limits _{h\in M} c_{kh} (w,v)$, we have $\beta _{k}\ge \mathop {\max }\limits _{v\in V} \mathop {\min }\limits _{w\in W} c_{kh} (w,v)=\beta _{kh}$ for all $h\in M$, then $\beta _{k}\ge \mathop {\max }\limits _{h\in M\backslash k} \beta _{kh}$. Suppose $\beta _{k}=\mathop {\max }\limits _{v\in V} \mathop {\min }\limits _{w\in W} c_{ki} (w,v)$, then we have

$$\begin{aligned} \beta _{k}\le \mathop {\max }\limits _{h\in M\backslash k} \mathop {\max }\limits _{v\in V} \mathop {\min }\limits _{w\in W} c_{kh} (w,v)=\mathop {\max }\limits _{h\in M\backslash k} \beta _{kh} \end{aligned}$$

which follows that $\beta _{k}=\mathop {\max }\limits _{h\in M\backslash k} \beta _{kh}$.

Let $\alpha _{k}=\mathop {\min }\limits _{v\in V} \mathop {\min }\limits _{w\in W} \mathop {\max }\limits _{h\in M\backslash k} \{c_{kh} (w,v)\}=\mathop {\min }\limits _{v\in V} \mathop {\min }\limits _{w\in W} \{c_{kh^{{\prime }}} (w,v)\}$, then $\alpha _{k}\ge \mathop {\max }\limits _{h\in M\backslash k} \alpha _{kh}$, since the following inequality holds for all $h$:

$$\begin{aligned} \alpha _{k}=\mathop {\min }\limits _{v\in V} \mathop {\min }\limits _{w\in W} \mathop {\max }\limits _{h\in M\backslash k} \{c_{kh} (w,v)\}\ge \mathop {\min }\limits _{v\in V} \mathop {\min }\limits _{w\in W} \{c_{kh} (w,v)\}=\alpha _{kh} \end{aligned}$$

In addition,

$$\begin{aligned} \mathop {\max }\limits _{h\in M} \alpha _{kh} =\mathop {\max }\limits _{h\in M\backslash k} \mathop {\min }\limits _{v\in V} \mathop {\min }\limits _{w\in W} \{c_{kh} (w,v)\}\ge \mathop {\min }\limits _{v\in V} \mathop {\min }\limits _{w\in W} \{c_{kh^{{\prime }}} (w,v)\}=\alpha _{k} \end{aligned}$$

which follows that $\alpha _{k}=\mathop {\min }\limits _{h\in M\backslash k} \alpha _{kh}$.

Proof of Theorem 2

Assume that alternative $x_{k}$ is $o$-ASRO and dominated by alternative $x_{h}$. Then we have$\gamma _{k}=\mathop {\min }\limits _{g\in M} \gamma _g$ and $c_{kh} (w,v)>o$ for any $w\in W$, $v\in V$, which derives that $c_{km} (w,v)>c_{hm} (w,v)$, $\forall m\in M$. Let $\gamma _{k}=c_{ki} (w^{*},v^{*})$ and $\gamma _{h}=c_{hq} (w^{{\prime }},v^{{\prime }})$ where alternatives $x_{i}$ and $x_q$ are not necessarily identical. Note that $\gamma _{k}\ge c_{kq} (w^{*},v^{*})>c_{kq} (w^{{\prime }},v^{{\prime }})>c_{kq} (w^{{\prime }},v^{{\prime }})=\gamma _{h}$, which contradicts the assumption that alternative $x_{k}$ is $o$-ASRO. Therefore, the $o$-ASRO alternative is $o$-non-dominated. Similarly, we can prove $o$-AWRO, $o$-ASPO and $o$-AWPO alternatives are all $o$-non-dominated.