Goal programming approaches to obtain the priority vectors from the intuitionistic fuzzy preference relations

Abstract

The priority method on the intuitionistic fuzzy preference relation (IFPR) is proposed. In order to avoid the operational difficulty in dealing with the intuitionistic sets, the equivalent interval matrices of the IFPR are introduced. Based on the multiplicative consistent definition of the fuzzy interval preference relation (FIPR), the goal programming models for deriving the priority vector of the IFPR have been put forward by analyzing the relation between the IFNPR and the IFPR. This goal programming method is generalized to the case of group decision making with the weight information defined by each DM. Two numerical examples are provided to illustrate the application of the proposed models.

Introduction

The pairwise comparison method is often utilized to rank a finite number of the alternatives from the best to the worst in multiple attribute decision making. Usually, the decision makers (DMs) express their pairwise comparison information in the form of fuzzy preference relation (Ma et al., 2006, Tanino, 1984, Wang and Xu, 1990) with entries being crisp numbers. However, in many situations, due to either the uncertainty of objective things or the vague nature of human judgment, the DMs may prefer imprecise judgment information which takes the forms of uncertainty numbers such as the fuzzy interval, the triangular fuzzy number or the trapezoid fuzzy number to precise judgment information. The preference relation has been extended to many kinds of uncertain formats such as fuzzy interval preference relation (FIPR), triangular fuzzy number preference relation (TFNPR) and trapezoid fuzzy number preference relation (TDFNPR) (Gogus and Boucher, 1998, Kwiesielewicz, 1996, Kwiesielewicz, 1998, Van Laarhoven and Pedrycs, 1983, Wang and Elhag, 2007, Wang and Chin, 2008, Wang, 2006, Wang and Fan, 2007, Wang et al., 2008, Wang and Elhag, 2006, Wang et al., 2006, Wei et al., 1994), etc.

Much research has been given to obtain the priority vectors of these different kinds of preference relations. Chang (1996) proposes an extent analysis method to derive a crisp priority vector from the TFNPR. Kwiesielewicz, 1996, Kwiesielewicz, 1998 applies the fuzzy logarithmic least squares method (LLSM) which originates from van Laarhoven and Pedrycz’s (1983) work to find the general solution of the TFNPR. Lots of limitations of the extent analysis method and the LLSM have been pointed out (Sugihara et al., 2004, Wang et al., 2006, Wang et al., 2008, Wang and Elhag, 2006), and lots of efforts to eliminate these weaknesses have been spent. For example, in literature (Wang et al., 2008), it is shown by examples that the priority vectors determined by the extent analysis method do not represent the relative importance of decision alternatives and that the misapplication of the extent analysis method to fuzzy AHP problems may lead to a wrong decision to be made. The correct normalization methods for interval and fuzzy weights are presented and relevant theorems in support of them are offered.

Intuitionistic fuzzy sets (Atanassov, 1986, Atanassov, 1999) introduced by Atanassov using membership degree, non membership degree and hesitation index to express the DMs’ subjective preference, is the generation of the classic fuzzy sets. The advantages of the intuitionistic sets (Pankowska & Wygralak, 2006) in dealing with the inevitably imprecise or not totally reliable judgment have been attached great importance by many scholars in fuzzy multiple attribute decision making fields: Li (2005) utilizes linear programming models to generate optimal weights for criteria in the context of intuitionistic fuzzy sets. Lin, Yuan, and Xia (2007) develop an easier way to handle this kind of problem. Szmidt and Kacprzyk, 2003, Szmidt and Kacprzyk, 2005 introduce the definition of the intuitionistic fuzzy preference relation (IFPR) to study the consensus-reaching process, and to analyze the extent of agreement in a group of experts. Little research has been made on the priority method of the IFPR due to the fact that the operation of the intuitionistic fuzzy set is too complex and has not been fully solved. Bustince and Burillo (1996) point out that the notion of vague set is actually that of the intuitionistic set. Therefore, by discussing the relation between the IFPR and the FIPR, we will derive a goal programming model to get the priority of the IFPR.

The paper is organized as follows. In Section 2, the definition of the IFPR will be firstly introduced, and then, the relation between the FIPR and the IFPR will be developed. In Section 3, the multiplicative consistent concepts of the IFPR will be given. Based on the discussion of priority method of the multiplicative consistent FIPR, a goal programming method of the inconsistent IFPR will be proposed, meanwhile, a goal programming method of the collective IFPRs will also be proposed. In Section 4, two numerical examples will be given to illustrate the validity and practicality of the proposed methods. At last, a short conclusion is given in Section 5.