Strong consistency in pairwise comparisons matrix with fuzzy elements on alo-group

The decision making problem considered in this paper is to rank n alternatives from the best to the worst, using the information given by the decision maker in the form of an n by n pairwise comparisons matrix. Here, we deal with pairwise comparison matrices with fuzzy elements. Fuzzy elements of the pairwise comparisons 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. We investigate pairwise comparisons matrices with elements from abelian linearly ordered group (alo-group) over a real interval. The concept of reciprocity and consistency of pairwise comparisons matrices with fuzzy elements have been already studied in the literature. Here, we define stronger concepts, namely the strong reciprocity and strong consistency of pairwise comparisons matrices with fuzzy intervals as the matrix elements (PCF matrices), derive the necessary and sufficient conditions for strong reciprocity and strong consistency and investigate their properties as well as some consequences to the problem of ranking the alternatives.