Incomplete Pairwise Comparison Matrix and Its Application to Ranking of Alternatives

A fuzzy preference matrix is the result of pairwise comparison - a powerful method in multi-criteria optimization. When comparing two elements, the decision maker assigns a value between 0 and 1 to any pair of alternatives representing the element of the fuzzy preference matrix. Here, we investigate relations between transitivity and consistency of fuzzy preference matrices and multiplicative preference ones. The results obtained are applied to decision situations where some elements of the fuzzy preference matrix are missing. We propose a new method for completing the fuzzy preference matrix with missing elements called the extension of the fuzzy preference matrix and investigate an important particular case of the fuzzy preference matrix with missing elements. Next, using the eigenvector of the transformed matrix we obtain the corresponding priority vector. Illustrative numerical examples are supplied.