Stochastic multicriteria decision‐making approach based on SMAA‐ELECTRE with extended gray numbers

• DOI: http://doi.org/10.1111/itor.12380
• Author: Huan Zhou,Hong‐yu Zhang,Jian‐qiang Wang
• Date: 01 September 2019
• Published date: 01 September 2019

Intl. Trans. in Op. Res. 26 (2019) 2032–2052

DOI: 10.1111/itor.12380

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RESEARCH

Stochastic multicriteria decision-making approach based on

SMAA-ELECTRE with extended gray numbers

Huan Zhou, Jian-qiang Wang and Hong-yu Zhang

School of Business, Central South University, Changsha, 410083, China

E-mail: 27887182@qq.com [Zhou]; jqwang@csu.edu.cn [Wang]; Hyzhang@csu.edu.cn [Zhang]

Received 4 February2016; received in revised form 1 October 2016; accepted 13 November 2016

Abstract

Actual stochastic multicriteria decision-making (MCDM) problems usuallyexhibit two forms of information

loss: criteria value uncertainty and criteria weight uncertainty. In this paper, extended gray numbers(EGNs),

integrated with discrete gray numbers and interval gray numbers are used to express the uncertainty of

stochastic MCDM problems. Stochastic multicriteria acceptability analysis (SMAA) and ELECTRE III are

combined to solve stochastic MCDM problems with uncertain weight information. First, the outranking

relations on interval graynumbers and EGNs are defined. Then, a SMAA-ELECTRE model for dealing with

graystochastic MCDM problems is constructed. Finally, an illustrativeexample and two comparativeanalyses

are provided to verifythe feasibility and usability of the proposed approach. The proposed approachprovides

recommendations for alternatives based on uncertain preference information. It therefore contributes a new

way to solve stochastic MCDM problemswith uncertain, imprecise, and/or missing preference information.

Keywords:stochastic multicriteria decision-making; extended graynumbers; stochastic multicriteria acceptability analysis;

ELECTRE III

1. Introduction

Since the 1970s, multicriteria decision-making (MCDM) has been an active area of research. Its

purpose is to support decision-makers in selecting the most ideal options considering multiple

criteria. In recent years, MCDM has increasingly attracted the attention of many researchers,

leading to a productive output in relevant research literature (Hu et al., 2015; Wang et al., 2016b;

Yu et al., 2016; Zhou et al., 2016a, 2016b). According to the criteria evaluation information of each

alternative, MCDM problems can be classified into three types: fuzzy, gray, and stochastic. Among

these, stochastic MCDM problemsare highly common in real life, and havebeen extensively applied

to various areas (Liu et al., 2011; Tan et al., 2014; Wang et al., 2014b, 2016a; Cao et al., 2016).

In stochastic MCDM problems, criteria values are random variables with known or unknown

probability density functions, and criteria weights are usually uncertain. The following section

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H. Zhou et al. / Intl. Trans. in Op. Res. 26 (2019) 2032–2052 2033

provides an in-depth explanation of criteria values and weights in real-world stochastic decision-

making problems.

In traditional MCDM problems, decision-makers usually provide deterministic measurements

to express their preference. Owing to time limitations, decision-makers’ capability, and the increas-

ing complexity and uncertainty of decision problems, decision-makers frequently find it difficult

to provide deterministic measurements. In difficult cases, the use of random variables with cor-

responding probabilities is a good choice. Generally, the decision-maker specifies each random

variable’s upper and lower bounds. However, there is likely a number of decision-makers often

failing to reach consensus on evaluation information. Thus, random variables in the form of ex-

tended gray numbers (EGNs) (Yang, 2007) can be employed. For illustrative purposes, consider a

case where two experts provide scores in the range of [0,100] to evaluate a company’s innovation.

Perhaps one of the scores is an interval number [70,75], while the other one is [78,80]. In this situ-

ation, the two interval gray numbers cannot precisely express the overall evaluation. On the other

hand, an EGN [70,75] ∪[78,80] may be a better option for representing this problem. This is be-

cause extended gray random variables combine intervals and discrete sets of numbers and can thus

express uncertainty in a more powerful way. Gray random variables have been widely studied (Wang

et al., 2013; Li and Zhao, 2015; Zhou et al., 2015), and many of them are in the form of EGNs.

Wang et al. (2013) defined an expected probability degree and proposed a gray stochastic MCDM

method, in which alternatives’ criteria values are interval gray random variables. To deal with in-

terval gray stochastic MCDM problems, Li and Zhao (2015) proposed a prospect theory-based

VIKOR method. Zhou et al. (2015) proposed a gray stochastic MCDM approach based on regret

theory and TOPSIS, where the criteria values are extended gray random variables. In brief, gray

random variables, including extended gray random variables, are suitable for expressing evaluation

information in stochastic MCDM problems.

In addition to criteria value uncertainty, in stochastic MCDM decision-makers may have to

face the difficulty of obtaining exact preference information about criteria weights. First, due to

limitations in time, knowledge, and/or resources, decision-makers are unable to accurately specify

their preference. Second, decision-makers may be unwilling to provide accurate preferences. Third,

preference information may be changeable throughout the decision process and even after the

decision has been made. Therefore, the available weights information about the criteria is often

inaccurate, imprecise, or uncertain. We can divide the restrictions on criteria weights into the

following five forms (Soung and Chang, 1999; Lahdelma and Salminen, 2001): (1) complete or

partial ranking of the weights; (2) interval weight values; (3) interval ratios of weights; (4) linear

inequality constraints for weights; and (5) nonlinear inequality constraints for weights.

For handling stochastic MCDM problems where both criteria values and weights are uncer-

tain, Lahdelma et al. (1998) introduced the original stochastic multicriteria acceptability analysis

(SMAA). This original SMAA method is based on inverse weight analysis performed through

Monte Carlo simulation, and it explores the weight space that would make each alternative the

most highly preferred. In recent years, the initial SMAA method has been extended to several

versions and has been applied in a variety of fields (Lahdelma et al., 1998, 2003; Lahdelma and

Salminen, 2001, 2009; Tervonenet al., 2008; Durbach, 2009). Based on the original SMAA method,

Lahdelma and Salminen (2001) proposed the SMAA-2 method, extending the scope of anal-

ysis to consider each alternative’s acceptability for all ranks. Later on, Lahdelma et al. (2003)

proposed the SMAA-O method, this time extending SMAA-2 to deal with the decision

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