A fresh view on the Discrete Ordered Median Problem based on partial monotonicity

Abstract

This paper presents new results for the Discrete Ordered Median Problem (DOMP). It exploits properties of k-sum optimization to derive specific formulations for the monotone DOMP (MDOMP), that arises when the λ weights are non-decreasing monotone, and new formulations for the general non-monotone DOMP. The main idea in our approach is to express ordered weighted averages as telescopic sums whose terms are k-sums, with positive and negative coefficients. Formulations of k-sums with positive coefficients derive from the linear programming representations obtained by Ogryczack and Tamir (2003) and Blanco, Ali, and Puerto (2014). Valid formulations for k-sums with negative coefficients are more elaborated and we present 4 different approaches, all of them based on mixed integer programming formulations. An extensive computational experience based on a collection of well-known instances shows the usefulness of the new formulations to solve difficult problems such as trimmed and anti-trimmed mean.