The linear ordering problem with clusters: a new partial ranking

Abstract

The linear ordering problem is among core problems in combinatorial optimization. There is a squared non-negative matrix and the goal is to find the permutation of rows and columns which maximizes the sum of superdiagonal values. In this paper, we consider that columns of the matrix belong to different clusters and that the goal is to order the clusters. We introduce a new approach for the case when exactly one representative is chosen from each cluster. The new problem is called the linear ordering problem with clusters and consists of both choosing a representative for each cluster and a permutation of these representatives, so that the sum of superdiagonal values of the sub-matrix induced by the representatives is maximized. A combinatorial linear model for the linear ordering problem with clusters is given, and eventually, a hybrid metaheuristic is carefully designed and developed. Computational results illustrate the performance of the model as well as the effectiveness of the metaheuristic.

The target visitation arc routing problem

“…The aim of the TVARP is to find a tour that traverses all required edges at least once, which reflects a compromise between the preferences obtained by the order in which targets are serviced and the routing cost of the tour. Similar to the LOP with cluster (Alcaraz et al, 2020) the preferences are associated to clusters of targets. However, in the TVARP the elements of the cluster do not necessarily have to be serviced consecutively.…”