Article Preview
TopIntroduction
Since in the real world there is vague information about different applications, fuzzy set theory has long been introduced to handle inexact and imprecise data (Zadeh, 1965). In fuzzy set theory, each object u ∈ U is assigned a single real value, called the grade of membership, between zero and one. (Here U is a classical set of objects, called the universe of discourse). Gau & Buehrer (1994) pointed out that the drawback of using the single membership value in fuzzy set theory is that the evidence for u ∈ U and the evidence against u ∈ U are in fact mixed together. In order to tackle this problem, Gau & Buehrer (1994) proposed the notion of allowing interval-based membership instead of using point-based membership as in fuzzy sets. The interval-based membership generalization in vague sets is more expressive in capturing vagueness of data. However, vague sets are shown to be equivalent to that of Intuitionistic Fuzzy Sets (IFSs) (Bustince & Burillo, 1996). For this reason, the interesting features for handling vague data that are unique to vague sets are largely ignored. We find that there are many interesting features of vague sets from a data modelling point of view. Atanassov (1986,1989) introduced the concept of intuitionistic fuzzy sets, which is also a generalization of the concept of fuzzy set. Gau & Buehrer (1994) introduced the concept of vague set. But Bustince & Burillo (1996) showed that vague sets are intuitionistic fuzzy sets. But later, in the literature Lu,A & Ng,W (2004, 2005) pointed out the differences between intuitionistic fuzzy sets and vague sets, and concluded in their work that, only vague sets can handle vague information better than intuitionistic fuzzy sets. Essentially, due to the fact that a vague set corresponds to a more intuitive graphical view of data sets, it is much easier to define and visualize the relationship of vague data objects.
Decision-making is the process of finding the best option from all of the feasible alternatives. Sometimes, decision-making problems considering several criteria are called multi-criteria decision-making (MCDM) problems. The MCDM problems may be divided into two kinds. One is the classical MCDM problems (Hwang & Yoon,1981), among which the ratings and the weights of criteria are measured in crisp numbers. Another is the fuzzy multiple criteria decision-making (FMCDM) problems, among which the ratings and the weights of criteria evaluated on imprecision and vagueness are usually expressed by linguistic terms, fuzzy numbers or intuition fuzzy numbers. As the vague set (Bustince & Burillo,1996) took the membership degree, non-membership degree and hesitancy degree into account, and has more ability to deal with uncertain information than traditional fuzzy set, lots of scholars pay attentions to the research of vague sets. Atanassov & Gargov (1989) extended the intuition vague set and proposed the concept of interval intuition vague set, also named interval vague set. Interval vague set has more ability to express vagueness and uncertainty.