فرآیند سلسله مراتب تحلیلی با عبارات اولویت فاصله ای  The analytic hierarchy process with interval preference statements

1. Introduction People frequently face the problem of choosing the best option from among several alternatives or of making a partial or full ranking of alternatives on the basis of multiple conflicting criteria. Such problems are termed multiple criteria decision-making (MCDM) problems and have received considerable attention in the decision science literature [15, 29, 30]. The analytic hierarchy process (AHP) provides a practical solution based on the divide and conquer principle, compared with other sophisticated MCDM methods. Since the introduction of AHP by Saaty [31], AHP has been successfully applied to a variety of realworld MCDM problems. See Vaidya and Kumar [36] for an extensive survey categorized by themes and areas of application. The AHP decision process consists of three main parts: decomposition, measurement of preferences, and synthesis. In this paper, we focus on the latter two parts of the process. In measuring preferences, pairwise comparison judgments are mostly elicited as point estimates on a ratio scale from 1/9 to 9. However, it is not uncommon for a decision-maker to be uncertain about his or her preferences, which can be attributed to two types of uncertainty: (a) uncertainty about the occurrence of events and (b) uncertainty about the range of judgments used to express preferences [32]. The first uncertainty is beyond the control of the decisionmaker, whereas the second uncertainty is a consequence of the amount of information available to the decision-maker and his or her understanding of the problem [32]. Moreover, situations such as time pressure, lack of domain knowledge, limited attention, and information processing capabilities can heighten the uncertainty of the problem at hand [38]. In these circumstances, many people, when asked for a subjective judgment about the parameters of a problem, are reluctant to specify a unique number and would prefer to specify an interval within which the true judgment lies. To capture a decision-maker’ uncertainty in making pairwise comparisons, many researchers have used interval ratio judgments to elicit the decision-maker’s preferences instead of adhering to precise ratio judgments. However, the specific analysis of interval ratios differs depending on the assumptions of the interval ratios themselves and the aggregation methods: a fuzzy set approach to the interval ratio [10, 11, 20, 25], a distribution function approach for the weights of feasible region constructed by interval judgments [15], a simulation-based approach [8, 13, 32], and a goal programming approach [17]. Our focus in this paper is deriving priority vectors for three types of interval ratios (the loose articulation, minimum number of interval ratios, and general interval ratios) via extreme points. In the loose articulation, the decision-maker specifies that one factor is at least n times as important as another and so on. A consistent interval pairwise comparison matrix (PCM) can always be constructed if at least (? − 1) ratio bounds, the minimum number of interval ratios, are given when considering ? factors. In the most general interval ratio case, a total of ?(? − 1)/2 ratio bounds are specified by the decision-maker. Judging from the range of preference formats, we deal with a variety of specifications compared with other established methods for using interval ratios. Furthermore, our proposed methods to derive the extreme points of interval ratios are distinct from previous ones. In the case of the loose articulation, cone theory and the inverse positivity property of a matrix provide the theoretical foundation for finding its extreme points. A dual programming technique provides another approach to obtain the same result. The analysis of the minimum number of interval ratios via the change of variables leads to their extreme points, which is easy to understand and apply. In the case of general interval ratios, we extract a minimum number of interval ratios, incorporate the remaining interval ratios into the extracted set one by one, and on each occasion modify the current extreme points until all remaining ones have been considered. This approach consistently results in the desired extreme points whereas Arbel’s method [7] successively finds priority vectors by applying a technique based on a pivoting operation in the linear programming, but it often fails to produce all vertices. It should be noted that all of these methods to derive extreme points are valid when the set of interval ratios is not empty; otherwise, a fuzzy preference programming, a simulation-based approach, or a goal programming approach is more suitable for obtaining the priority vector.