A new method of obtaining the priority weights from an interval fuzzy preference relation
Introduction
The analytic hierarchy process (AHP) has been proposed by Saaty [21], [22] to reduce a complex system of multiple criteria decision-making or planning process to a hierarchy of criteria, sub-criteria or alternatives. The decision maker (DM) may give a multiplicative preference relation through a sequence of pairwise comparisons, then obtain the priority weights of the alternatives by using some techniques. In the classical Saaty’s priority theory, in order to obtain a reasonable solution, one of the key issues is the consistency or acceptable consistency of a multiplicative preference relation, which has been studied in detail [22]. In addition, Orlovski [18] gave the definition of fuzzy preference relations to reflect the fuzzy nature of a preference relation in the set of alternatives. The consistency of fuzzy preference relations was further studied by Tanino [24]. Hereafter many methods have been proposed for obtaining the priority weights from multiplicative and fuzzy preference relations respectively (see e.g., [4], [5], [6], [8], [9], [10], [12], [14], [25], [31], [32], [35], [38], [39]).
However, due to the complexity and uncertainty of real-world decision-making problems, it is very difficult to provide a precise numerical value for the DM’s level of preference. That is to say, it is more natural to adopt the fuzzy or interval ratios to estimate the DM’s judgements than to use the real number. Along this line, Van Laarhoven and Pedrycz [26] extended the Saaty’s priority theory [22] to consider the problem of choosing the best among a number of alternatives, where the opinions of the decision-makers were expressed as fuzzy numbers with triangular membership functions and the logarithmic least-squares method was utilized to obtain the fuzzy weights. Saaty and Vargas [23] proposed interval numbers as the judgements of the decision-makers and obtained the priority weights from the given interval multiplicative preference relations using a Monte Carlo simulation method. Now many methods of generating the priority weights from interval multiplicative preference relations have been developed such as various mathematical programing models (see, e.g., [2], [3], [11], [20], [27], [28]), the continuous ordered weighted geometric operator [37] and the convex combination method [16]. Among the above-stated literatures, it is noted that two different definitions of consistent interval multiplicative preference relations have been given by Liu [16] and Wang et al. [27] respectively. Moreover, it is also reasonable and natural to express the DM’s judgements as an interval fuzzy preference relation. Then a continuous ordered weighted averaging operator (COWA) was proposed by Yager [36] and utilized by Xu [33] to derive the priority weights. Xu and Chen [34] further defined additive consistent and multiplicative consistent interval fuzzy preference relations and gave some linear programing models to derive the priority weights. Based on the multiplicative or additive transitivity, many researchers have further focused their attention on the methods of generating the priority weights from interval fuzzy preference relations such as Genc et al. [7], Jiang et al. [13], Xia et al. [29], Yue et al. [40] and so on.
From the above-mentioned works, one can see that it may be effective and feasible to transform or adjust an initial preference relation given by a decision maker. For example, the transformation formulae between fuzzy and multiplicative preference relations have been proposed and applied to assess the priority vectors from fuzzy preference relations ([9], [31], [32]). Moreover the concept of information granularity was put forward by Pedrycz and Song [19]. Under the consideration of the admitted level of granularity, a multiplicative preference relation can be adjusted to own a level of consistency, then the level of consensus in group decision making can be increased. Similarly, since a decision maker may easily express her/his opinions as an interval fuzzy or multiplicative preference relation, it will be interesting and important to establish their relations, especially in group decision making with different kinds of preference relations [1]. To the best knowledge of the authors, the important issue is little to be addressed. This paper is focused on the transformation formulae between interval fuzzy and multiplicative preference relations, then the method of how to derive the priority weights from interval fuzzy preference relations is given. It is structured as follows. Section 2 introduces the preliminaries. In Section 3, a consistent interval fuzzy preference relation is defined and its properties are studied. The transformation formulae between interval fuzzy and multiplicative preference relations are further proposed. And a new algorithm of obtaining the priority weights from consistent or inconsistent interval fuzzy preference relations is presented. In Section 4, three examples are given to illustrate the effectiveness and feasibility of the proposed method. The main conclusions are shown in Section 5.
Section snippets
Preliminaries
In the classical AHP [22], a multiple criteria decision-making problem is structured hierarchically at different levels, which contain a set of alternatives such as X = {x1, x2, … , xn}. Making use of a pairwise comparison technique, we can determine a multiplicative preference relation A = (aij)n×n, where the element aij represents a multiplicative preference degree of alternative xi over xj and meets aijaji = 1 with 1/9 ⩽ aij ⩽ 9 for all i, j = 1, 2, … , n. The consistency and acceptable consistency of A have
Consistent interval fuzzy preference relations
In what follows, let us focus on a new definition of consistent interval fuzzy preference relation V. We assume that P = (pij)n×n and Q = (qij)n×n with:andwhere α ∈ [0, 1]. From (16), it is easy to see that R(0) = Q and R(1) = P. is a convex combination of the corresponding elements in P and Q. And it is a monotone continuous function with respect to αfor all i, j = 1, 2, … , n.
According to (15), (16), we
Numerical examples
In this section, we offer three numerical examples to illustrate the proposed approaches of generating the interval weight vector from interval fuzzy preference relations. In Example 1, the derived interval multiplicative preference relation is acceptably consistent, and those in Example 2, Example 3 are unacceptably consistent. Example 1 Consider the following interval fuzzy preference relation, which has been examined by Genc et al. [7]:
Conclusions
In the process of multiple criteria decision making, due to the complexity and uncertainty of real-life decision making problems, the DM’s preference information is expressed naturally as an interval fuzzy or multiplicative preference relation. Then of much interest are the consistency of the preference relations and the priority weights obtained by using some techniques to rank the given alternatives. In this paper, the definitions of consistent and acceptably consistent interval fuzzy
Acknowledgements
The work was supported by National Science Fund for Distinguished Young Scholars of China (No. 70825005), Guangxi Natural Science Fund (No. 0991029) and Guangxi University Science Foundation (XGL090002). The authors would like to thank the Editor-in-Chief, Professor Witold Pedrycz and the anonymous referees for their valuable comments and suggestions for improving the paper.
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