Fuzzy Pairwise Comparison Matrices
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This chapter is concerned with the fuzzy extension of the multi-criteria decision making methods based on pairwise comparison and answers the following research question: “Based on a fuzzy pairwise comparison matrix of objects, how should fuzzy priorities of these objects be determined so that they reflect properly all preference information available in the fuzzy pairwise comparison matrices?” Three types of fuzzy pairwise comparison matrices are studied—fuzzy multiplicative pairwise comparison matrices, fuzzy additive pairwise comparison matrices with additive representation, and fuzzy additive pairwise comparison matrices with multiplicative representation. The chapter provides a comprehensive literature review of the fuzzy pairwise comparison methods based on the fuzzy extension of well-known methods originally developed for crisp pairwise comparison matrices. The review includes a detailed study of differences and analogies between the methods and a detailed description of their drawbacks illustrated by numerous numerical examples. Two key properties of the fuzzy pairwise comparison matrices and the related methods—reciprocity of the related pairwise comparisons and the invariance of the methods under permutation of objects—are studied in detail, and it is shown that all reviewed methods violate at least one of these properties. A novel approach to fuzzy extension of the pairwise comparison methods based on constrained fuzzy arithmetic is introduced in order to prevent the drawbacks. Subsequently, suitable methods are introduced for all three types of fuzzy pairwise comparison matrices, showcased on numerous numerical examples, and critically compared with the original methods. Finally, transformations between the approaches for the three types of fuzzy pairwise comparison matrices are studied.