Keywords: Multiple criteria decision analysis, Interval type-2 fuzzy set, Likelihood index, Approximate ideal,Likelihood-based compromise index.
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INTRODUCTIONMultiple criteria decision analysis (MCDA) often takes place within a complex and uncertain environment and involve conflicting systems of criteria (Chen, 2017;Qin et al., 2017). Considering the decision maker's point of view and circumstances in the decision-making process, subjective opinions and judgments are inherently imprecise and involve many uncertainties (Chen, 2014a;Singh and Garg, 2017). In fuzzy community, Zadeh (1975) introduced fuzzy sets of Type-2, later shortened by others to type-2 fuzzy sets (T2 FSs) and fuzzy sets with interval-valued membership functions, which was later shortened by others to interval-valued fuzzy sets (Mendel, 2007;2010). The concept of T2 FSs is an extension of type-1 fuzzy sets (T1 FSs) and is characterized by a fuzzy membership function, where the degree of membership for any element in this set is a fuzzy number in the interval [0, 1] (Chen, 2013;Zhou et al., 2017). T2 FSs are better than T1 FSs for handling imprecision and uncertainties by modeling vagueness and unreliability of information (Chen, 2017;Zhou et al., 2017). Real-world decisions always require the use of more precise and accurate data (Lai and Chen, 2015); thus, T2 FSs are appropriate for addressing real-life problems if there is insufficient knowledge or experience (Chen, 2014a;2014b;Zhang and Zhang, 2013).However, the membership function of T2 FSs is usually difficult and troublesome to determine, which hinder their practical applications (Siminski, 2017). Resolving the difficulties in establishing and handling the secondary membership functions, interval type-2 fuzzy sets (IT2 FSs) are the most widely used of the higher order fuzzy sets because of their relative simplicity (Mendel, 2007;Siminski, 2017;Wu and Mendel, 2007). IT2 FSs are a simplified version of T2 FSs, and the membership grades of IT2 FSs are crisp intervals (Siminski, 2017;. The theory of IT2 FSs has been well developed in the literature and has been applied productively in the field related to MCDA under uncertainty (Celik and Taskin Gumus, 2016;Chen, 2014a;2014b; Lai and Chen, 2015;Singh and Garg, 2017;Wang and Chen, 2014;Zhang and Zhang, 2013;Zhong and Yao, 2017;Zhou et al., 2017). In particular, interval type-2 trapezoidal fuzzy numbers (IT2 TrFNs), as a special case of IT2 FSs, can efficiently express linguistic ratings and evaluations by objectively transforming them into numerical variables (Chen, 2017;Zhang and Zhang, 2013).