Pairwise comparison matrix with fuzzy elements on alo-group

Abstract

This paper deals with pairwise comparison matrices with fuzzy elements. Fuzzy elements of the pairwise comparison matrix are applied whenever the decision maker is not sure about the value of his/her evaluation of the relative importance of elements in question. In comparison with pairwise comparison matrices with crisp elements investigated in the literature, here we investigate pairwise comparison matrices with elements from abelian linearly ordered group (alo-group) over a real interval. We generalize the concept of reciprocity and consistency of crisp pairwise comparison matrices to matrices with triangular fuzzy numbers (PCFN matrices). We also define the concept of priority vector which is a generalization of the crisp concept. Such an approach allows for a generalization dealing both with the PCFN matrices on the additive, multiplicative and also fuzzy alo-groups. It unifies several approaches known from the literature. Moreover, we also deal with the problem of measuring the inconsistency of PCFN matrices by defining corresponding indexes. The first index called the consistency grade G is the maximal alpha of alpha-cut, such that the corresponding PCFN matrix is still alpha-consistent. On the other hand, the consistency index I of the PCFN matrix measures the distance of the PCFN matrix to the closest ratio matrix. If the PCFN matrix is crisp and consistent, then G is equal to 1 and the consistency index I is equal to the identity element e of the alo-group, otherwise, G is less than 1, or I is greater than e. Four numerical examples are presented to illustrate the concepts and derived properties. Finally, we show that the properties of reciprocity and consistency of PCFN matrices are saved by special transformations based on fuzzy extensions of isomorphisms between two alo-groups.