An Extended Outranking Approach to Rough Stochastic Multi-criteria Decision-Making Problems

Abstract

Background/Introduction

Randomness and roughness commonly exist simultaneously in one decision-making problem in the real world; however, fewer systematic studies have been carried out for such problems.

Methods

In this study, we employ interval-valued rough random variables (IRRVs) and interval-valued rough numbers (IRNs) to process decision information. Additionally, we propose a more reasonable comparison method of IRNs. We combine and extend the stochastic dominance of IRRVs and the classic ELECTRE III method (a useful outranking method). Finally, we develop an extended ELECTRE III approach for rough stochastic multi-criteria decision-making (MCDM) problems. We provide examples concerning site selection and investment appraisal in order to demonstrate the feasibility of our proposed approach. We verify the applicability and advantages of our approach through comparative analyses with other existing methods.

Results

Illustrative and comparative analyses indicate that our proposed approach is feasible for many practical MCDM problems, and the final ranking results of the proposed approach are more accurate and consistent with those obtained in actual decision-making processes.

Conclusion

The IRNs and IRRVs are useful for dealing with rough stochastic decision information. The proposed approach is feasible and effective for solving rough stochastic MCDM problems, and retains the merits of IRRVs, SD relations, and outranking methods. The final outcomes from our approach are convincing and consist with actual decision-making. Thus, our proposed approach is more widely applicable.

Appendices

Appendix 1: Table of Acronyms

Abbreviation	Description	Symbols	Mathematical meanings
MCDM	Multi-criteria decision-making	$U$	Universe
RRV	Rough random variable	$\Re$	The set of real numbers
IRN	Interval-valued rough number	$\zeta$	Rough variable
IRRV	Interval-valued rough random variable	$\xi_{i}$	IRRV
ELECTRE	Elimination and choice translating reality	$\varOmega$	Sample space
SD	Stochastic dominance	$u$	Utility function
SDD	Stochastic dominance degree	$a_{i} Oa_{j}$	$a_{i}$ outranks $a_{j}$
${\text{SD}}_{1} ,\;{\text{SD}}_{2} ,\;{\text{SD}}_{3}$	First-, second-, and third-degree stochastic dominance

Appendix 2: Proof of Property 2 in “Stochastic Dominance Rules” section

• (1) If $a_{i} {\text{SD}}_{1} a_{j}$, it means $\forall x \in I_{0} , \, H_{1} (x) = F_{j} (x) - F_{i} (x) \ge 0,$ and $\, \exists x_{0} \in I_{0}$, $H_{1} (x_{0} ) > 0$ hold. Then $\forall x \in I_{0} ,$ there is $H_{2} (x) = \int_{a}^{x} {H_{1} (t)} dt \ge 0$, and $H_{2} (a) = 0, \, H_{2} (b) = 0$. Therefore, $\forall x \in I, \, H_{2} (x) \ge 0,$ and $\, \exists x_{0} \in I$, $H_{2} (x_{0} ) > 0$ hold, i.e., $a_{i} {\text{SD}}_{1} a_{j} \Rightarrow a_{i} {\text{SD}}_{2} a_{j}$. The proof of $a_{i} {\text{SD}}_{2} a_{j} \Rightarrow a_{i} {\text{SD}}_{3} a_{j}$ can be obtained in a similar way. According to the SD rules in Definition 14 and 15, the property ${\text{SD}}_{1} \Rightarrow {\text{SD}}_{2} \Rightarrow {\text{SD}}_{3}$ can be easily obtained.

• (2) If $a_{i} {\text{SD}}_{1} a_{j}$, then $\forall x \in I_{0} , \, H_{1} (x) = F_{j} (x) - F_{i} (x) \ge 0,$ and $\, \exists x_{0} \in I_{0}$, $H_{1} (x_{0} ) > 0$ hold. If $a_{j} SD_{1} a_{i}$ exists, then $\forall x \in I_{0} , \, H_{1}^{\prime } (x) = F_{i} (x) - F_{j} (x) \ge 0,$ and $\exists x_{0} \in I_{0}$, $H_{1}^{\prime } (x_{0} ) > 0$ hold. Equivalently, $\forall x \in I_{0}$, $H_{1} (x) = F_{j} (x) - F_{i} (x) \le 0$, and $\exists x_{0} \in I_{0}$, $H_{1} (x_{0} ) < 0$ hold, which is the opposite of case $a_{i} {\text{SD}}_{1} a_{j}$.

  Therefore, if $a_{i} {\text{SD}}_{1} a_{j}$, then there is no $a_{j} {\text{SD}}_{1} a_{i}$, and, according to Property 2(1), there is no $a_{j} {\text{SD}}_{2,3} a_{i}$. The proofs of the other cases can be obtained in a similar way. Thus, the asymmetry of SD rules can be proved.

• (3) If $a_{i} {\text{SD}}_{1} a_{j}$ and $a_{j} {\text{SD}}_{1} a_{k}$, then $\forall x \in I_{0} , \, H_{1} (x) = F_{j} (x) - F_{i} (x) \ge 0,$ and $\exists x_{0} \in I_{0}$, $H_{1} (x_{0} ) > 0$ hold; $\forall x \in I_{0} , \, H_{1}^{\prime } (x) = F_{k} (x) - F_{j} (x) \ge 0,$ and $\exists x_{0}^{\prime } \in I_{0} ,$ $\, H_{1}^{\prime } (x_{0}^{\prime } ) > 0$ hold. Thus, it can be obtained that $\forall x \in I_{0} , \, H_{1}^{\prime \prime } (x) = F_{k} (x) - F_{i} (x) \ge 0,$ and $\exists x_{0}^{\prime \prime } \in I_{0} ,$ $H_{1}^{\prime \prime } (x_{0}^{\prime \prime } ) > 0$ hold, then $a_{i} {\text{SD}}_{1} a_{k}$. The proofs of other cases are similar. If $a_{i} {\text{SD}}_{2} a_{j}$ and $a_{j} {\text{SD}}_{1} a_{k}$, and, according to Property 2(1), $a_{j} {\text{SD}}_{1} a_{k} \Rightarrow a_{j} {\text{SD}}_{2} a_{k}$, then $a_{i} {\text{SD}}_{2} a_{k}$ can be obtained.

  Suppose that $a_{i} {\text{SD}}_{h} a_{j} \;{\text{and}}\;a_{j} {\text{SD}}_{g} a_{k} { (}h > g),$ and then, based on Property 2(1), $a_{j} {\text{SD}}_{g} a_{k} \Rightarrow a_{j} {\text{SD}}_{h} a_{k}$. Finally, the result of $a_{i} {\text{SD}}_{h} a_{k}$ can be obtained. This proves the transitivity of SD rules.