Ordered weighted average combinatorial optimization: Formulations and their properties

Multiobjective combinatorial optimization deals with problems considering more than one viewpoint or scenario. The problem of aggregating multiple criteria to obtain a globalizing objective function is of special interest when the number of Pareto solutions becomes considerably large or when a single, meaningful solution is required. Ordered weighted average or ordered median operators are very useful when preferential information is available and objectives are comparable since they assign importance weights not to specific objectives but to their sorted values. In this paper, ordered weighted average optimization problems are studied from a modeling point of view. Alternative integer programming formulations for such problems are presented and their respective domains studied and compared. In addition, their associated polyhedra are studied and some families of facets and new families of valid inequalities presented. The proposed formulations are particularized for two well-known combinatorial optimization problems, namely, shortest path and minimum cost perfect matching, and the results of computational experiments presented and analyzed. These results indicate that the new formulations reinforced with appropriate constraints can be effective for efficiently solving medium to large size instances.