In this paper we propose the problem of finding the cyclic sequence which best represents a set of cyclic sequences. Given a set of elements and a precedence cost matrix we look for the cyclic sequence of the elements which is at minimum distance from all the ranks when the permutation metric distance is the Kendall Tau distance. In other words, the problem consists of finding a robust cyclic rank with respect to a set of elements. This problem originates from the Rank Aggregation Problem for combining different linear ranks of elements. Later we define a probability measure based on dissimilarity between cyclic sequences based on the Kendall Tau distance. Next, we also introduce the problem of finding the cyclic sequence with minimum expected cost with respect to that probability measure. Finally, we establish certain relationships among some classical problems and the new problems that we have proposed.
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.
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