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Top1. Introduction
Multiple attribute decision making (MADM) refers to making choice of the best alternative from among a finite set of decision alternatives in terms of multiple usually conflicting attributes (or called criteria) (Barron and Barrett, 1996; Altuzarra et al., 2010; Zhang et al., 2004; Yu and Lai, 2011; Wang, 1997; Wei and Wang, 2017; Ahn and Park, 2008; Pekelman and Sen, 1974; Sugihara, 2001; Wang and Parkan, 2006). In the recent years, MADM has received a great deal of attention from researchers in many disciplines (Tang et al., 2017; Wei et al., 2012; Zhao and Wei, 2013; Merigo and Casanovas, 2009; Wei et al., 2017; Zhao et al., 2010; Wang, 2010; Wei, 2015; Chen et al., 2011; Wei et al., 2010; Wei, 2011; Merigo and Cazanovas, 2010; Li and Sun, 2007; Xu, 2009). However, under many conditions, the decision information about alternatives is usually uncertain or fuzzy due to the increasing complexity of the socio-economic environment and the vagueness of inherent subjective nature of human thinking. Thus, numerical values are inadequate or insufficient to model real-life decision problems. So many aggregation operators and approaches have been developed to solve the multiple attribute decision-making problems with interval numbers (Zhang et al., 2005; Yue, 2011; Sayadi et al., 2009; Jiang et al., 2008; Jiang et al., 2008; Papadakis and Kaburlasos, 2010; Kuo and Liang, 2012; Jahan and Edwards, 2013; Hu et al., 2013; Luo and Wang, 2012; Baležentis and Zheng, 2013). Furthermore, Xu (2010) developed some uncertain Bonferroni mean operators. Xu (2002) investigated the uncertain OWA operator in which the associated weighting parameters cannot be specified. Merigo and Casanovas (2011) presented the uncertain induced quasi-arithmetic OWA (Quasi-UIOWA) operator. Merigo (2011) presented the uncertain probabilistic weighted average (UPWA) whose main advantage is that it unifies the probability and the weighted average. Merigo and Casanovas (2011) introduced the uncertain generalized OWA (UGOWA) operator which is an extension of the OWA operator that uses generalized means and uncertain information represented as interval numbers. Merigo and Casanovas (2011) developed some new extensions about the OWA operator such as the induced heavy OWA (IHOWA) operator, the uncertain heavy OWA (UHOWA) operator and the uncertain induced heavy OWA (UIHOWA) operator. Cao and Wu (2011) developed two extended continuous ordered weighted geometric (COWG) operators, such as the weighted geometric averaging COWG (WG-COWG) and ordered weighted geometric averaging COWG (OWG-COWG) operators and presented their application to multiple attribute group decision making (MAGDM) problems with interval numbers. Merigo and Wei (2011) presented the uncertain probabilistic ordered weighted averaging (UPOWA) operator which is an aggregation operator that used probabilities and OWA operators in the same formulation considering the degree of importance of each concept in the analysis. Zhou et al. (2012) provided a new class of operators called the uncertain generalized ordered weighted averaging (UGOWA) operator which provides a very general formulation that includes as special cases a wide range of aggregation operators and aggregates the input arguments taking the form of intervals rather than exact numbers and further generalizes the UGOWA operator to obtain the uncertain generalized hybrid averaging operator, the quasi uncertain ordered weighted averaging operator and the uncertain generalized Choquet integral aggregation operator. Merigo et al. (2012) proposed the uncertain induced ordered weighted averaging-weighted averaging (UIOWAWA) operator and studied some of the main advantages and properties of the new aggregation such as the uncertain arithmetic UIOWA (UA-UIOWA) and the uncertain arithmetic UWA (UAUWA).