Analytic hierarchy process based on interval analysis

Analytic hierarchical process (AHP) is proposed to give the priority weight with respect to many items. The priority weights are obtained with a comparison matrix whose elements are given by a human being as crisp pairwise comparisons. However, the comparison matrix is based on human intuition so that its elements must be inconsistent. Therefore, to deal with inconsistency in the given matrix, we propose a new AHP, where the weight of an item is given as a fuzzy value. We assume that inconsistency for human intuition is represented as fuzzy weights. The purpose is to obtain fuzzy weights so that the estimated pairwise comparisons that are formed from their h-level sets include all the given ones. We also extend crisp pairwise comparisons to fuzzy ones because of the uncertainty of human´s judgement. To give the uncertain information as a fuzzy value is more rational than as a crisp value. In the case of a fuzzy pairwise comparison matrix, two kinds of fuzzy weights with respect to the upper and lower approximation models are considered. The upper fuzzy weights are obtained in order to include the h-level sets of fuzzy pairwise comparisons and the lower ones are obtained so as to be included in them.