Dionisio Pérez-Brito

Variable neighborhood search for the linear ordering problem

Given a matrix of weights, the linear ordering problem (LOP) consists of finding a permutation of the columns and rows in order to maximize the sum of the weights in the upper triangle. This NP-complete problem can also be formulated in terms of graphs, as finding an acyclic tournament with a maximal sum of arc weights in a complete weighted graph. In this paper, we first review the previous methods for the LOP and then propose a heuristic algorithm based on the variable neighborhood search (VNS) methodology. The method combines different neighborhoods for an efficient exploration of the search space. We explore different search strategies and propose a hybrid method in which the VNS is coupled with a short-term tabu search for improved outcomes. Our extensive experimentation with both real and random instances shows that the proposed procedure competes with the best-known algorithms in terms of solution quality, and has reasonable computing-time requirements.Variable neighborhood search (VNS) is a metaheuristic method that has recently been shown to yield promising outcomes for solving combinatorial optimization problems. Based on a systematic change of neighborhood in a local search procedure, VNS uses both deterministic and random strategies in search for the global optimum.In this paper, we present a VNS implementation designed to find high quality solutions for the NP-hard LOP, which has a significant number of applications in practice. The LOP, for example, is equivalent to the so-called triangulation problem for input-output tables in economics. Our implementation incorporates innovative mechanisms to include memory structures within the VNS methodology. Moreover we study the hybridization with other methodologies such as tabu search.

Variable neighbourhood search for bandwidth reduction

The problem of reducing the bandwidth of a matrix consists of finding a permutation of rows and columns of a given matrix which keeps the non-zero elements in a band as close as possible to the main diagonal. This NP-complete problem can also be formulated as a vertex labelling problem on a graph, where each edge represents a non-zero element of the matrix. We propose a variable neighbourhood search based heuristic for reducing the bandwidth of a matrix which successfully combines several recent ideas from the literature. Empirical results for an often used collection of 113 benchmark instances indicate that the proposed heuristic compares favourably to all previous methods. Moreover, with our approach, we improve best solutions in 50% of instances of large benchmark tests.

A polynomial algorithm for the p‐centdian problem on a tree

The most common problems studied in network location theory are the p-median and the p-center models. The p-median problem on a network is concerned with the location of p points (medians) on the network, such that the total (weighted) distance of all the nodes to their respective nearest points is minimized. The p-center problem is concerned with the location of p-points (centers) on the network, such that the maximum (weighted) distance of all the nodes to their respective nearest points is minimized. To capture more real-world problems and obtain a good way to trade-off minisum (efficiency) and minimax (equity) approaches, Halpern introduced the centdian model, where the objective is to minimize a convex combination of the objective functions of the center and the median problems. In this paper, we studied the p-centdian problem on tree networks and present the first polynomial time algorithm for this problem.

The Centdian subtree on tree networks

Abstract This paper describes an O (n log n) algorithm for finding the optimal location of a tree shaped facility of a specified size in a tree network with n nodes, using the centdian criterion: a convex combination of the weighted average distance and the maximum weighted distance from the facility to the demand points (nodes of the tree). These optimization criteria introduced by Halpern, combine the weighted median and weighted center objective functions. Therefore they capture more real-world problems and provide good ways to trade-off minisum (efficiency) and minimax (equity) approaches.

A modified variable neighborhood search for the discrete ordered median problem

This paper presents a modified Variable Neighborhood Search (VNS) heuristic algorithm for solving the Discrete Ordered Median Problem (DOMP). This heuristic is based on new neighborhoods’ structures that allow an efficient encoding of the solutions of the DOMP avoiding sorting in the evaluation of the objective function at each considered solution. The algorithm is based on a data structure, computed in preprocessing, that organizes the minimal necessary information to update and evaluate solutions in linear time without sorting. In order to investigate the performance, the new algorithm is compared with other heuristic algorithms previously available in the literature for solving DOMP. We report on some computational experiments based on the well-known N-median instances of the ORLIB with up to 900 nodes. The obtained results are comparable or superior to existing algorithms in the literature, both in running times and number of best solutions found.

THE 2-FACILITY CENTDIAN NETWORK PROBLEM

Abstract The p -facility centdian network problem consists of finding the p points that minimize a convex combination of the p -center and p -median objective functions. The vertices and local centers constitute a dominating set for the 1-facility centdian; i.e., it contains an optimal solution for all instances of the problem. Hooker et al. (1991) give a theoretical result to extend the dominating sets for the 1-facility problems to the corresponding p -facility problems. They claim that the set of vertices and local centers is also a dominating set for the p -facility centdian problem. We give a counterexample and an alternative finite dominating set for p =2. We propose a solution procedure for a network that improves the complexity of the exhaustive search in the dominating set. We also provide a very efficient algorithm that solves the 2-centdian on a tree network with complexity O( n 2 ).

Center location problems on tree graphs with subtree-shaped customers

We consider the p-center problem on tree graphs where the customers are modeled as continua subtrees. We address unweighted and weighted models as well as distances with and without addends. We prove that a relatively simple modification of Handlers classical linear time algorithms for unweighted 1- and 2-center problems with respect to point customers, linearly solves the unweighted 1- and 2-center problems with addends of the above subtree customer model. We also develop polynomial time algorithms for the p-center problems based on solving covering problems and searching over special domains.

Thep-facility ordered median problem on networks

In this paper we deal with the ordered median problem: a family of location problems that allows us to deal with a large number of real situations which does not fit into the standard models of location analysis. Moreover, this family includes as particular instances many of the classical location models. Here, we analyze thep-facility version of this problem on networks and our goal is to study the structure of the set of candidate points to be optimal solutions.