Extension of the TOPSIS for Multi-Attribute Group Decision Making under Atanassov IFS Environments

Introduction

Multi-attribute group decision making (MAGDM) problems are of importance in most kinds of fields such as engineering, economics and management. It is obvious that much knowledge in the real world is fuzzy rather than precise. Imprecision comes from a variety of sources such as unquantifiable information (Chen & Hwang, 1992; Triantaphyllou & Lin, 1996; Chen, 2000; Chu, 2002a, 2002b; Braglia et al., 2003; Chu & Lin, 2003; Li, 2003; Chen & Tzeng, 2004; Zhao et al., 2006). In many situations decision makers have imprecise/vague information about alternatives with respect to attributes. One of the methods which describe imprecise cases is the fuzzy set (FS) introduced by Zadeh (1965). The main characteristic of an FS is that a membership degree is assigned to each element u in a universe of discourse and a non-membership degree automatically is equal to , i.e., this membership degree combines the evidence for u and the evidence against u. However, a human being who expresses the degree of membership of a given element in an FS very often does not express the corresponding degree of non-membership as the complement to 1. This reflects a well-known psychological fact that the linguistic negation is not always identified with the logical negation. Thus in 1983 Atanassov (1986, 1999) introduced the concept of an intuitionistic fuzzy set (IFS), which is called the Atanassov IFS for short in this paper. The Atanassov IFS is characterized by two functions expressing the degree of belongingness and the degree of non-belongingness, respectively. This idea is useful in decision making processes. Especially, the Atanassov IFS may give us a very natural tool for modeling preferences. Sometimes it seems to be more natural to describe imprecise and uncertain opinions not only by membership functions due to the fact that in some situations it is easier to describe our negative feeling than positive attitude. Even more, quite often one can easily specify alternatives (or objects) one dislikes, but simultaneously cannot specify clearly what are really wanted. Let us consider a situation observed in a real estate agency. Very often an interested customer looking for an apartment is not completely convinced on the location and considers several variants. It is obvious that some districts are more preferable than others whereas there are some districts that the customer dislikes. It seems to be much suitable for modeling such situations using the Atanassov IFSs. For example, it may happen that the customer asked about his/her favorite district in Warsaw cannot definitely determine whether it is Ochota, Mokotów or Zoliborz, but he/she feels sure that he/she hates Wola. Thus the preferences of the customer may be modeled using the Atanassov IFS, where the membership function expresses the degree of a given district being preferred by the customer while the non-membership function indicates the degree of the given district which is not preferred (Grzegorzewski & Mrówka, 2005). People maybe meet a similar situation when comparing preferences expressed by means of orderings which admit uncertainty due to imprecision, vagueness and hesitance. In this case the Atanassov IFS may give us a natural tool for modeling such improper orderings (Grzegorzewski & Mrówka, 2005; Deschrijver & Kerre, 2007).