Efficient Multiple Attribute Group Decision Making Models with Correlation Coefficient of Vague Sets

Efficient Multiple Attribute Group Decision Making Models with Correlation Coefficient of Vague Sets

John Robinson (P, PG & Research Department of Mathematics, Bishop Heber College, Tiruchirappalli, India) and Henry Amirtharaj (E.C., PG & Research Department of Mathematics, Bishop Heber College, Tiruchirappalli, India)
DOI: 10.4018/ijoris.2014070102
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Abstract

A new approach for multiple attribute group decision making (MAGDM) problems where the attribute weights and the expert weights are real numbers and the attribute values take the form of vague values, is presented in this paper. Since families of ordered weighted averaging (OWA) operators are available in the literature, and only a few available for vague sets, the vague ordered weighted averaging (VOWA) operator and the induced vague ordered weighted averaging (IVOWA) operator are introduced in this paper and utilized for aggregating the vague information. The correlation coefficient for vague sets is used for ranking the alternatives and a new MAGDM model is developed based on the IVOWA operator and the vague weighted averaging (VWA) operator. In addition to the proposed model, two different models are proposed based on Linguistic Quantifiers for the situation when the expert weights are completely unknown. An illustrative example is given and a comparison is made between the models to demonstrate the applicability of the proposed approach of MAGDM.
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Introduction

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 uU 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 uU and the evidence against uU 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.

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