Weak transitivity of interval-valued fuzzy relations

In this paper, we define and study the weak transitivity of interval-valued fuzzy relations (IVFRs). We propose the weak transitivity index (WTI) to measure the transitivity consistency degree of an IVFR, which is to count the cycles of length 3 in the digraph. Afterwards, an algorithm is proposed to compute the WTI and to locate each cycle, as well as to find the inconsistent judgments in an IVFR. In order to resolve the intransitivities of an IVFR, another algorithm is developed to find and remove all the 3-cycles in the digraph. Our method can not only repair the weak intransitivity for an IVFR, but also preserve the initial preference information as much as possible. Finally, two examples are shown to illustrate the proposed method.