Bilevel facility location problems: theory and applications.

In this doctoral thesis we focus on studying facility location problems considering customer preferences. In these problems, there is a set of customers or users who demand a service or product that must be supplied by one or more facilities. By facilities it is understood some object or structure that offers some service to customers. One of the most important assumptions is that customers have established their own preferences over the facilities and should be taken into account in the customer-facility assignment. In real life, customers choose facilities based on costs, preferences, a predetermined contract, or a loyalty coefficient, among others. That is, they are free to choose the facilities that will serve them. The situation described above is commonly modeled by bilevel programming, where the upper level corresponds to location decisions to optimize a predefined criteria, such as, minimize location and distribution costs or maximize the demand covered by the facilities; and the lower level is associated to -customer allocation- to optimize customer preferences. The hierarchy among both levels is justified because the decision taken in the upper level directly affects the decision’s space in the lower level

Analyzing the Performance of a Hybrid Heuristic for Solving a Bilevel Location Problem under Different Approaches to Tackle the Lower Level

The problem addressed here is a combinatorial bilevel programming problem called the uncapacitated facility location problem with customer’s preferences. A hybrid algorithm is developed for solving a battery of benchmark instances. The algorithm hybridizes an evolutionary algorithm with path relinking; the latter procedure is added into the crossover phase for exploring the trajectory between both parents. The proposed algorithm outperforms the evolutionary algorithm already existing in the literature. Results show that including a more sophisticated procedure for improving the population through the generations accelerates the convergence of the algorithm. In order to support the latter statement, a reduction of around the half of the computational time is obtained by using the hybrid algorithm. Moreover, due to the nature of bilevel problems, if feasible solutions are desired, then the lower level must be solved for each change in the upper level’s current solution. A study for illustrating the impact in the algorithm’s performance when solving the lower level through three different exact or heuristic approaches is made

Analyzing the Performance of a Hybrid Heuristic for Solving a Bilevel Location Problem under Different Approaches to Tackle the Lower Level

The problem addressed here is a combinatorial bilevel programming problem called the uncapacitated facility location problem with customer’s preferences. A hybrid algorithm is developed for solving a battery of benchmark instances. The algorithm hybridizes an evolutionary algorithm with path relinking; the latter procedure is added into the crossover phase for exploring the trajectory between both parents. The proposed algorithm outperforms the evolutionary algorithm already existing in the literature. Results show that including a more sophisticated procedure for improving the population through the generations accelerates the convergence of the algorithm. In order to support the latter statement, a reduction of around the half of the computational time is obtained by using the hybrid algorithm. Moreover, due to the nature of bilevel problems, if feasible solutions are desired, then the lower level must be solved for each change in the upper level’s current solution. A study for illustrating the impact in the algorithm’s performance when solving the lower level through three different exact or heuristic approaches is made

El problema de la p-mediana con orden: reformulaciones y algoritmos.

En el presente trabajo de tesis se analiza un modelo de programación binivel para el problema de la -Mediana donde consideramos las preferencias de los clientes. Tomar en cuenta dichas preferencias es de suma importancia debido a la competencia actual del mercado. El problema que analizamos es una extensión del problema simple de localización de plantas con orden, dicha extensión se basa en que asumimos que un número predeterminado de plantas debe ser adecuadamente localizado. Suponemos además que una planta puede abastecer la demanda de varios clientes pero un cliente debe ser abastecido por una planta y que los clientes establecen una lista ordenada de preferencias indicando sus deseos de ser servidos por las plantas abiertas. El problema se formula como un modelo de programación binivel donde el objetivo del líder es minimizar los costos de instalación y de distribución; y el seguidor quiere optimizar las preferencias ordenadas de los clientes. El objetivo del problema es abrir un número conocido de plantas que minimicen el costo total (instalación y distribución) de localización y, a su vez, minimicen las preferencias ordenadas de los clientes. Nosotros proponemos dos reformulaciones del problema estudiado debido a la dificultad derivada de resolver los problemas de programación binivel. La primer reformulación se basa en las relaciones primal-dual del problema del nivel inferior y la segunda en la adición de un conjunto de restricciones que asegura que se sigue considerando las preferencias de los clientes. Se llevó a cabo experimentación numérica y se mostró que las reformulaciones no son capaces de resolver las instancias de tamaño mediano en un tiempo computacional razonable. Este hecho nos llevó a desarrollar un algoritmo basado en la metaheurística Scatter Search donde se considera el equilibrio de Stackelberg durante el proceso. Durante el procedimiento iterativo de construcción de soluciones del problema binivel, el hecho de considerar la solución óptima del seguidor para cada solución del líder refleja la intención de encontrar el equilibrio de Stackelberg. Dicho algoritmo heurístico obtiene soluciones de buena calidad para todas las instancias analizadas en tiempos menores que el requerido para la solución de las reformulaciones de un solo nivel

Optimizing a Biobjective Production-Distribution Planning Problem Using a GRASP

This paper addresses a biobjective production-distribution planning problem. The problem is formulated as a mixed integer programming problem with two objectives. The objectives are to minimize the total costs and to balance the total workload of the supply chain, which consist of plants and depots, considering that it represents a company vertically integrated. In order to solve the model, we propose an adapted biobjective GRASP to obtain an approximation of the Pareto front. To evaluate the performance of the proposed algorithm, numerical experimentations are conducted over a set of instances used for similar problems. Results indicate that the proposed GRASP obtains a relatively small number of nondominated solutions for each tested instance in very short computational time. The approximated Pareto fronts are discontinuous and nonconvex. Moreover, the solutions clearly show the compromise between both objective functions