Rank Preservation and Propagation of Fuzziness in Pairwise-Comparison Methods for Multi-Criteria Decision Analysis

Abstract

We consider pairwise-comparison methods to rank and rate a finite number of stimuli. The decision makers express their preference ratios on a category scale: their responses are restricted to a set of categories labelled with a narrative degree of preference. We put these qualifications on numerical scales with geometric progression, and show that the rank order of the stimuli is preserved when the base number varies. If the preference ratios are expressed in fuzzy numbers with triangular membership functions, we find that the triangles usually exhibit a particular form of symmetry: the parameters satisfy the golden-section rule so that the logarithms have membership functions with isosceles triangles. Under the additional condition that the preference ratios have a uniform degree of fuzziness, we establish the membership functions of the decision criteria, the alternatives and the final scores, and we show that their degree of fuzziness depends on the hierarchical decision level only.