A Note on Two Methods for Estimating Missing Pairwise Preference Values

This note analyzes two methods for calculating missing values of an incomplete reciprocal fuzzy preference relation. The first method by Herrera-Viedma et al. appeared in the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS - PART B: CYBERNETICS [vol. 37, no. 1 (2007) 176-189], while the second one by Fedrizzi and Giove appeared later in the European Journal of Operational Research [vol. 183 (2007) 303-313]. The underlying concept driving both methods is the additive consistency property. We show that both methods, although different, are very similar. Both methods derive the same estimated values for the independent-missing-comparison case, while they differ in the dependent-missing-comparison case. However, it is shown that a modification of the first method coincides with the second one. Regarding the total reconstruction of an incomplete preference relation, it is true that the second method performs worse than the first one. When Herrera-Viedma et al.'s method is unsuccessful, Fedrizzi-Giove's method is as well. However, in those cases when Fedrizzi-Giove's method cannot guarantee the successful reconstruction of an incomplete preference relation, we have that Herrera-Viedma et al.'s method can. These results lead us to claim that both methods should be seen as complementary rather than competitors in their application, and as such, we propose a reconstruction policy of incomplete fuzzy preference relations using both methods. By doing this, the only unsuccessful reconstruction case is when there is a chain of missing pairwise comparisons involving each one of the feasible alternatives at least once.