A Novel Method to Assign Weights to Decision Makers for each Criterion in Group Decision Making Under Multiple Criteria with Crisp and Interval Data

1. Introduction

Decision-making is concerned mainly with the question which alternative or course of action should be undertaken under a specific situation by considering many aspects, including the degree of importance of each criterion (Kalbande & Thampi, 2009). Therefore, many real world decision-making problems involve multiple criteria (Roszkowska, 2013). In recent years, multi-criteria evaluation methods have been widely used in solving both theoretical and practical problems. Actually, these methods are universal (Zavadskas, Turskis, Ustinovichius & Shevchenko, 2010). Multi Criteria Decision Making (MCDM) refers to screening, prioritizing, ranking or selecting the alternatives based on human judgment from among a finite set of decision alternatives in terms of multiple usually conflicting criteria (Roszkowska, 2013). They allow us to quantitatively evaluate any complicated object described by a set of criteria. Another advantage of these methods is their ability to combine both maximizing and minimizing attributes expressed in various dimensions into one integrated criterion (Zavadskas, Turskis, Ustinovichius & Shevchenko, 2010). If more than one person is involved in the decision, the decision problem becomes a group decision making (Yang, Kuo, Parker & Chen, 2015). Moving from single decision makers (DMs), setting to group members setting would lead to a great deal of complexity of the analysis (Yang & Du, 2015).

Simply speaking, Multiple Attribute Group Decision Making (MAGDM, often called Multiple Criteria Group Decision Making or MCGDM) could be regarded as a combination of MADM (or MCDM) and GDM (Group Decision Making) (He, Xu & Chen, 2016). MAGDM problems may be defined as decision situations where: (1) there are two or more experts (or DMs), who are characterized by their own ideas, attitudes, motivations and knowledge, (2) there is a problem to be solved, and (3) they try to achieve a common solution. More specifically, a MAGDM problem with t (t≥1) DMs, m alternatives and n attributes can be expressed in matrices format as follows (Yue, 2013_a):

(1)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. 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.

First, we let

(2)

For all kεT,

Be decision matrix of the k_th (kεT) DM, in which each of the elements is characterized by interval number (Yue, 2011_b).