Heuristic methods for solving two discrete optimization problems
Cabezas García, José Xavier
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In this thesis we study two discrete optimization problems: Traffic Light Synchronization and Location with Customers Orderings. A widely used approach to solve the synchronization of traffic lights on transport networks is the maximization of the time during which cars start at one end of a street and can go to the other without stopping for a red light (bandwidth maximization). The mixed integer linear model found in the literature, named MAXBAND, can be solved by optimization solvers only for small instances. In this manuscript we review in detail all the constraints of the original linear model, including those that describe all the cyclic routes in the graph, and we generalize some bounds for integer variables which so far had been presented only for problems that do not consider cycles. Furthermore, we summarized the first systematic algorithm to solve a simpler version of the problem on a single street. We also propose a solution algorithm that uses Tabu Search and Variable Neighbourhood Search and we carry out a computational study. In addition we propose a linear formulation for the shortest path problem with traffic lights constraints (SPTL). On the other hand, the simple plant location problem with order (SPLPO) is a variant of the simple plant location problem (SPLP) where the customers have preferences on the facilities which will serve them. In particular, customers define their preferences by ranking each of the potential facilities. Even though the SPLP has been widely studied in the literature, the SPLPO has been studied much less and the size of the instances that can be solved is very limited. In this manuscript, we propose a heuristic that uses a Lagrangean relaxation output as a starting point of a semi-Lagrangean relaxation algorithm to find good feasible solutions (often the optimal solution). We also carry out a computational study to illustrate the good performance of our method. Last, we introduce the partial and stochastic versions of SPLPO and apply the Lagrangean algorithm proposed for the deterministic case to then show examples and results.