Problemas de localización de instalaciones no fiables

Abstract

The aim of facility location problems consists on deciding where to optimally locate several plants or facilities (like industries, warehouses, schools, hospitals, distribution centers, datawarehouse, etc.) as well as how to optimally assign the clients to these facilities satisfying their demand. These decisions are usually determined considering the opening costs, and the costs of serving the clients. In the literature, it is frequently assumed that the open facilities are always availaible. Nevertheles, in practice some facilities can interrupt their service and become unavailaible. A lot of location problems of the most recent literature include the failing distribution of the plants in their models, and this type of problems are known like reliability facility location problems. The three contributions collected in the present dissertation are referred to reliability facility location problems. The objective of this dissertation is to analyze the properties of the so called Reliability Fixed-Charge Location Problem. Paper 1.- In this first paper, we discuss the previously known formulation of the Reliability Fixed-Charge Location Problem, in which the number of open facilities has to be decided and depends on the opening costs and it is assumed that some of the facilities, called failable, may fail with a given probability, which is identical for all of them. We reformulate the original mathematical programming model as a set packing problem and we study some polyhedral properties of this type of problems. Then, conditions for optimal solutions are introduced. We propose an improved compact formulation for the problem and we check the performance through an extensive computational study. Paper 2.- In this paper, we analyze which allocation variables in the Reliability Fixed-Charge Location Problem formulation can be linearized so that the optimal value match the optimal value of the binary problem.We prove that we can relax the integrality of all the allocation variables associated to non-failable facilities or all the allocation variables associated to failable facilities, but not both simultaneously. We also demonstrate that we can relax the integrality of all the allocation variables whenever a family of valid inequalities is added to the set of constraints or whenever the parameters of the problem satisfy certain conditions. Finally, on solving the instances in a data set, we discuss which integrality relaxation or which modification of the problem performs better in terms of resolution time, and we illustrate that to inappropriately relax the integrality of the allocation variables can incur in a high difference at the objective value. Paper 3.- In the last paper we propose and discuss different models to include capacity constraints into the Reliability Fixed-Charge Location Problem. In all cases, the proposed models represent a trade off between the extreme models that can be found in the literature, where a priori assignments are either fixed, or can be fully modified at each scenario. By several computational experiments, we analyze the obtained solutions of introducing capacity constraints according to the different proposed models, in terms of computational burden and in terms of solution cost.

La finalidad de los problemas de localizaci´on de plantas consiste en decidir d´onde ubicar de forma ´optima plantas o instalaciones (industrias, almacenes, escuelas, hospitales, centros de distribuci´on, centros de datos, etc.) as´ı como la asignaci´on ´optima de los clientes a estas instalaciones de forma que se satisfaga su demanda. Estas decisiones se toman habitualmente considerando los costes relativos al establecimiento de las plantas o costes de apertura, as´ı como los costes relativos al servicio de la demanda o costes de transporte. En la literatura cl´asica frecuentemente se asume que las plantas abiertas siempre est´an disponibles, sin embargo, en la pr´actica, las plantas pueden fallar interrumpiendo su prestaci´on de servicio y no estar disponibles. Problemas de localizaci´on de la literatura m´as recientes consideran en sus modelos distribuciones de probabilidad para el fallo de las plantas, es decir, la fiabilidad de las plantas. Las tres contribuciones recogidas en la presente tesis se refieren a este tipo de modelos. El objetivo de esta tesis es analizar propiedades del problema de localizaci´on de instalaciones no fiables, en ingl´es Reliability Fixed-Charge Location Problem. Esta tesis se estructura en torno a tres art´ıculos: Art´ıculo 1.- En esta primera aportaci´on los autores reformulan el problema de localizaci´on de instalaciones no fiables desde el enfoque de un problema de empaquetamiento de conjuntos, estudiando determinadas propiedades poli´edricas referidas al pol´ıtopo que conforma la regi´on factible e identificando todas las facetas asociadas a cliques. Consecuentemente, se estudian ciertas condiciones para las soluciones ´optimas. Se propone adem´as una formulaci´on compacta, probando su eficacia mediante un amplio estudio computacional. Art´ıculo 2.- En este segundo trabajo se estudia bajo qu´e condiciones podr´ıan relajarse el car´acter entero de todas las variables de asignaci´on, probando que puede ser relajado el car´acter entero para todas las variables de asignaci´on relativas a las instalaciones completamente fiables; o para las variables de asignaci´on relativas a las plantas cuyo servicio puede presentar interrupciones; mas no para ambas simult´aneamente. Finalmente, se propone una familia de desigualdades v´alidas cuya introducci´on en el modelo permite la relajaci´on del car´acter entero de todas las variables de asignaci´on.Art´ıculo 3.- En el tercer trabajo los autores proponen y estudian diferentes modelos que incluyen restricciones de capacidad en el modelo de localizaci´on de instalaciones no fiables. Para ello a˜naden a la formulaci´on ya existente diversas restricciones de capacidad encaminadas a acotar la sobrecarga de demanda con la que se enfrentan algunas instalaciones en caso de producirse fallos en las plantas. Adem´as, mediante una serie de pruebas computacionales se analizan las soluciones obtenidas y la bondad de los diferentes m´etodos empleados.