Pairwise Comparisons Matrices with Fuzzy and Intuitionistic Fuzzy Elements in Decision-Making

This chapter focuses on pairwise comparison matrices with fuzzy elements and intuitionistic fuzzy elements. “Fuzzy” and/or “Intuitionistic fuzzy” elements are appropriate whenever the decision-maker (DM) is uncertain about the value of his/her evaluation of the relative importance of elements in question, or, when aggregating crisp pairwise comparisons of a group of decision-makers in the group DM problem. We formulate the problem in a general setting investigating pairwise comparisons matrices (PCM) with elements from an abelian linearly ordered group (alo-group) over a real interval. Such an approach enables extensions of traditional multiplicative, additive or fuzzy approaches. We review the approaches known from the literature, then we propose our new order preservation concept based on alpha-cuts. We define the concept of weak $$\alpha $$-$$\odot $$-consistency of the PC matrix with fuzzy elements and a stronger version: $$\alpha $$-$$\odot $$-consistency of PCM. Then we derive necessary and sufficient conditions for weak consistency as well as consistency and extend the POP conditions for PCMs, together with some desirable properties and relationships. Finally, we deal with some of the consequences of the problem of ranking the alternatives. In some sense, this chapter is a continuation of the previous chapter. Moreover, we deal with the problem of measuring the inconsistency of fuzzy/intuitionistic fuzzy pairwise comparison matrices by defining corresponding inconsistency indexes. Numerical examples are presented to illustrate the concepts and derived properties.