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Top1. Introduction
Due to the ever-increasing complexity of socioeconomic environment, it is very difficult for only one Decision Maker (DM) to consider all the important aspects of a problem. Thus, most decisions in an organization, whether public or private, are made by a group of people (Azadfallah, 2016). Azam et al. (2017) believe that, along with Operations Research (OR) techniques it may also deal with intricate problems to find appropriate solutions. Hence, in recent years, multi-criteria evaluation methods have been widely used in solving both theoretical and practical problems (Zavadskas et al., 2010). Multi-Attribute Decision Making (MADM) [also often called Multi-Criteria Decision Making or MCDM], is an important part of modern decision science, which contains multiple decision attributes and multiple decision alternatives. The aim is to help the decision maker take all important objective and subjective criteria/ attributes of the problem into consideration using a more explicit, rational and efficient decision process (Yue, 2012a). MCDM (or MADM) is related to the choice of the best solution from a set of alternatives in the presence of multiple conflicting criteria (Felfel et al., 2017). Essentially, MADM models are selector models that are used for evaluating, ranking, and selecting the most appropriate alternative from among several alternatives (Azadfallah, 2017). MADM mainly consists of the following two parts (Xu, 2012); 1. Collect decision information, and 2. Aggregate the decision information through some proper approaches and then rank and select the given alternatives. A MADM problem with m alternatives and n attributes can be expressed in matrix format as follows (Yue, 2013a):
(1)W= (w1, w2, …, wn)
where, A
1, A
2, …, A
m are feasible alternatives, U
1, U
2, …, U
n are evaluation attributes, X
ij is the performance rating of alternative A
i under attribute U
j, and W
j is the weight of attribute U
j.
Multi-Attribute Group Decision Making (MAGDM) is an important part of the MADM problems (Yue, 2013a). Simply speaking, MAGDM could be regarded as a combination of MADM and GDM [Group Decision Making] (He et al., 2016). In other words, when more than one person is interested in the same MADM problem, it then becomes a MAGDM problem (Yang et al., 2015). The definition of MAGDM is described specifically as follow: multi DMs make judgments or evaluations by virtue of respective knowledge, experience and preference for a decision space (i.e., a finite set of alternatives) under multi attributes to rank all the alternatives or give evaluation information of each alternative, and then decision results from each DM are aggregated to form an overall ranking result for all the alternatives (Pang & Liang, 2012). Yue (2012a) believe that, the MAGDM problems have three common characteristics: alternatives, multiple attributes with incommensurable units and multiple experts. More specifically, a MAGDM problem with t (t≥1) DMs, m alternatives and n attributes can be expressed in matrices format as follows:
(2)
where x
k and w
k (k=1, 2, …, t), respectively, are the decision matrix and weight vector of attributes, which are provided by k
th DM (Yue, 2013a). Moreover, in the process of MADM, the DMs are usually asked to provide their preference information on attributes, and the attribute values are not precisely known but value ranges can be obtained. Therefore, it is significative that we consider MAGDM problem with interval number.