Post-optimality analysis of priority vectors derived from interval comparison matrices by lexicographic goal programming

Abstract

In a recent paper by Wang [Y.M. Wang, On lexicographic goal programming method for generating weights from inconsistent interval pair-wise comparison matrices. Applied Mathematics and Computation 173 (2006) 985–991], it is shown that lexicographic goal programming method results in different rankings for upper- and lower-triangular interval judgments arranged in a positive reciprocal pair-wise comparison matrix. This was unexpected from a prioritization point of view, because upper- and lower-triangular entries in reciprocal interval pair-wise comparison matrices essentially provide the same judgment information. A research question was posed at the same article: “Which ranking is true or more reliable?”. Motivated by this question, in this paper, we provide a post-optimality analysis to detect the most reliable priority vector among m vectors derived by lexicographical goal programming for an inconsistent interval pair-wise comparison matrix of order n. The method is based on violations in the order of preferences among the resultant priorities and runs in O ( mn 2 ) time. When there exists a tie between some vectors, we also illustrate a tie-breaking algorithm running in O ( mn 4 ) time, based on violations in the order of the preference intensities among the resultant priorities. We re-visited the examples discussed in the afore-mentioned article and validate the practical usability of this method.