Introduction

In real life people usually take into account a number of factors when making decisions. Different factors may conflict with each other and people have to balance conflicting factors in their choice of action, such as cost and quality of products. When one of the factors is understood as the term “criterion”, action as alternative, and one person as a decision maker, such a situation is regarded as the context of multiple criteria decision making (MCDM). In MCDM, all criteria must be measurable, be capable of measuring different aspects of the problem considered, and distinguish between alternatives. As evaluating one alternative on multiple criteria is easier than evaluating the alternative in a comprehensive way, MCDM is beneficial for a decision maker to make a choice of alternatives when multiple criteria decision methods are used. The use of multiple criteria decision methods can help a decision maker relieve the cognitive overload caused by the large volume of information needed to be combined to create solutions to complex issues (Brownlow & Watson, 1987). To conduct MCDM, many different types of multiple criteria decision methods have been proposed, including multiple criteria utility function (MCUF) methods (Butler, Jia, & Dyer, 1997; Butler, Morrice, & Mullarkey, 2001; Keeney & Raiffa, 1993; Wakker, Jansen, & Stiggelbout, 2004), multiple criteria value function methods (Belton and Stewart, 2002; Chin, Fu, & Wang, 2015; Fischer, 1995; Fu & Xu, 2016; Fu, Xu, & Yang, 2016; Keeney, 2002; Lan, Chen, Ning, & Wang, 2015; Sari, 2017; Yan, Zhang, & Li, 2017; Zhang, Wang, Li, & Chen, 2017), distance based methods such as the extensions of TOPSIS method (Baykasoğlu & Gölcük, 2015; Wang, Liu, Li, & Niu, 2016) and VIKOR method (Madjid, Reza, Francisco, & Elahe, 2016; Qin, Liu, & Pedrycz, 2015), and outranking methods such as PROMETHEE methods (Chen, 2014a; Miłosz & Krzysztof, 2016) and ELECTRE methods (Chen, 2014b; Corrente, Greco, & Słowiński, 2016). Although there are different principles in different types of multiple criteria decision methods for using individual performances of each alternative on each criterion to generate solutions, criterion weights are always taken into account in the process of generating solutions. The weight of a criterion generally reflects the impact of the performance of the criterion on the overall performance (Butler et al., 2001). The weights of all criteria can accomplish the weighted trade-off between the performances of alternative on each criterion, which means that different sets of criterion weights may usually result in different solutions to the same decision problem considered. As a result, determining criterion weights is a key step of multiple criteria decision methods. To address the determination of criterion weights in MCDM, a large amount of research has been conducted in literature. When a decision maker is capable of providing subjective preferences for criterion weight assignment, many methods have been proposed to use the subjective preferences to determine criterion weights. Representative methods include point allocation method (Doyle, Green, & Bottomley, 1997; Roberts & Goodwin, 2002), direct rating method (Bottomley & Doyle, 2001; Roberts & Goodwin, 2002), eigenvector method (Saaty, 1977; Takeda, Cogger, & Yu, 1987), Delphi method (Hwang & Yoon, 1981), linear programming model (Horowitz & Zappe, 1995), and goal programming model (Shirland, Jesse, Thompson, & Iacovou, 2003). If there is flexibility in criterion weights in nature and a decision maker cannot provide reliable or credible subjective preferences for criterion weights, the performances of alternatives assessed on each criterion are used to objectively determine criterion weights.