From the viewpoint that the vagueness of a decision maker’s evaluation causes inconsistencies in a pairwise comparison matrix, interval weights have been estimated using the interval AHP. However, the estimated interval weights are often insufficient to express the vagueness of the decision maker’s evaluation. We propose three modified estimation methods for interval weights. The first is based on a relaxation of the optimality of estimated interval weights in the conventional method. The second employs a modified objective function and the third is based on a relaxation of the optimality with respect to the modified objective function. Two of the proposed methods include parameters with degrees of relaxation. Through numerical experiments with 100,000 pairwise comparison matrices generated from 100 true interval weight vectors, we demonstrate the advantages of the proposed methods over the conventional method, and determine the best method and the suitable degree of relaxation.

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The analytic hierarchy process (AHP) provides a systematic approach to the evaluation of alternatives based on pairwise comparison matrices (PCMs) under multiple criteria. As human evaluation is not always accurate and precise, each component of a PCM showing relative importance has been expressed by an interval or a fuzzy number. In this paper, we treat a PCM whose components are represented by twofold intervals. The twofold intervals are composed of inner and outer intervals showing the range of surely acceptable values and the complement of the range of surely unacceptable values for relative importance, respectively. One may apply a fuzzy AHP approach by building a trapezoidal fuzzy number from the inner and outer intervals. However, this approach does not always fit the given information. Because the twofold interval information ambiguously specifies the range of acceptable values for the relative importance. Then we appropriately translate this information into a set of intervals including the inner interval and included in the outer interval, assuming that the decision-maker evaluates the priority weights implicitly as intervals. We investigate the decision analysis based on the twofold interval PCM. Three conflict resolution methods are proposed for treating the inconsistency in the twofold interval PCM. Parametric methods for visualizing possible alternative orderings are proposed. Numerical examples are given to demonstrate the differences between the proposed approach and the previous approaches.

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