An alternative weight sensitivity analysis for PROMETHEE II rankings

Most strategic decision problems involve the simultaneous optimization of several conflicting criteria. Among multicriteria decision aid methods, PROMETHEE II has earned some attention during the past decades. This method outputs a complete ranking of a set of alternatives, given some parameters provided by a decision maker, such as the weights of the criteria. In order to assess the stability of the ranking, weight stability intervals (WSI) have been developed. However, this method only focuses on one criterion at a time (changes are assumed to be applied uniformly to the other criteria in order to remain normalized). In this work, we analyze how weight stability intervals can be redefined while allowing any modifications on any criterion weight, by applying inverse optimization based on mixed integer linear programming (MILP). The problem formulation can be stated as follows: what would be the minimum modification on the weights such that a given alternative ai of rank n > 1 becomes first? This methodology has been applied on two case studies, and compared to the WSI. The results show that it is always possible to find narrower intervals while it is never required to modify all the weights simultaneously. Additionally, when using the WSI, only few alternatives can be ranked first. By taking the inverse optimization point of view, all (non-dominated) alternatives can be placed at the first position (at the price of large deviations). It is therefore possible to apply this methodology for a consensus search when different decision makers are involved.