Computing the weights of criteria with interval-valued fuzzy sets for MCDM problems

Decision making is one of the most complex administrative processes in management area (Ashtiani et al., 2009). In some situations, because of time pressure, lack of knowledge, and the decision maker’s limited attention and information processing capabilities (Xu, 2006), the decision maker cannot make the optimal decision easily. In fact, both qualitative and quantitative criteria should be considered simultaneously (Erol and Ferrell Jr, 2003) in the decision making process. It calls multiple criteria decision making (MCDM) process. Saaty (1980) proposed analytic hierarchy process (AHP) method for determining the weights of criteria in a MCDM problem by a hierarchical structure and systematization. According to the pairwise comparison result, the weight of each criterion can be computed for a hierarchical problem structure.

Due to the subjective judgments of the decision makers are often uncertain in determining the pairwise comparisons, the fuzzy AHP are proposed to overcome the drawback (Buckley, 1985; Buckley et al., 2001). The fuzzy AHP allowed the experts to use fuzzy ratios in place of exact ratios to make the pairwise comparisons. Based on the concepts, there are various fuzzy AHP methods have been developed such as the geometric mean method (Buckley, 1985), the Lambda-Max method (Csutora and Buckley, 2001), fuzzy logarithmic least squares method (Wang et al., 2006), the linear goal programming method (Jung, 2011), and the logarithmic fuzzy preference programming method (Rezaei et al., 2013).

However, it is often difficult for experts to exactly quantify their opinions as a crisp number in decision making process. Therefore, the reasonable way is to represent the degree of certainty by an interval (Ashtiani et al., 2009; Vahdani et al., 2010). The membership value express as an interval value can more fit the real world situations. Interval-valued fuzzy numbers have been applied in many fields such as R&D leader selection (Ashtiani et al., 2009), risk analysis (Wei and Chen, 2009), enterprise partner selection (Ye, 2010), traditional supplier selection (Chen, 2011), bus enterprise performance evaluation (Kuo and Liang, 2012) etc. Therefore, interval-valued fuzzy sets are suitable used to represent the fuzziness of membership value of subjective opinions. In this paper, a novel method is proposed by combining interval-valued fuzzy set with fuzzy AHP to determine the fuzzy weights of each criterion.

The organization of this paper is described as follows. In section 2, we will present some notations of basic definitions and operators such as triangular fuzzy number, α-cut of fuzzy number, defuzzified method and interval valued fuzzy sets. The proposed method will be discussed at the section 3. For illustrating the proposed method clearly, an example will be implemented. Finally, the conclusion and future research are summarized at the end of this paper.