S. Guillén-Burguete1, H. Sánchez-Larios*2, J.G. Vázquez-Vázquez3
1 Instituto de Ingeniería, Universidad Nacional Autónoma de México, México, D. F., México.
2 Facultad de Ingeniería, Universidad Nacional Autónoma de México, México, D. F., México. *herica.sanchez@ciencias.unam.mx.
3 Facultad de Ciencias, Universidad Nacional Autónoma de México, México, D.F., México.
Abstract
]]> Motivated by a problem faced by road construction companies, we develop a new model to obtain an optimal transportation schedule of mobile machines which have to travel to execute tasks. In this problem, each task is characterized by the location where it is to be executed, a work-content in terms of machine-time units, and one or more time intervals within which it can be performed. The machines can be transported from one location to another at any time, thus the problem has an indefinite number of variables. However, this indefinite number of variables can be reduced to a definite one because, as we prove, the problem has an optimal solution in which the arrivals of machines occur only at certain time instants. The objective is to minimize the total transportation cost such that all the tasks are executed within their time intervals. The constraints ensuring that the tasks are processed within their prescribed time intervals are nonlinear; nevertheless, due to the sets of the possible arrival times of the machines forming bounded convex polyhedra, our problem can be transformed into a mixed integer linear program by the same device used in the decomposition principle of Dantzig-Wolfe.Keywords: transportation schedule, generalized linear programming, bounded convex polyhedron, work-content.
Resumen
Motivados por un problema que enfrentan las compañías de la construcción, desarrollamos un modelo nuevo para obtener un calendario óptimo del transporte de máquinas móviles que tienen que viajar para realizar tareas. En este problema, cada tarea está caracterizada por el lugar donde ésta tiene que ser realizada, una carga de trabajo en términos de tiempo-máquina, y uno o más intervalos de tiempo dentro de los cuales la tarea puede ser procesada. Las máquinas se pueden transportar desde un lugar hasta otro en cualquier momento, por lo tanto el problema tiene un número indefinido de variables. Sin embargo, este número indefinido de variables se puede reducir a uno definido porque, como se demuestra, el problema tiene una solución óptima en la que las llegadas de las máquinas ocurren solamente en ciertos momentos. El objetivo es minimizar el costo total de transporte tal que todas las tareas sean ejecutadas dentro de sus intervalos de tiempo. Las restricciones que aseguran que cada tarea sea procesada dentro de sus intervalos de tiempo prescritos son no lineales; sin embargo, debido a que los conjuntos de los posibles tiempos de llegada de las máquinas forman poliedros convexos acotados, nuestro problema puede transformarse en un programa lineal entero mixto por el mismo artificio usado en el principio de descomposición de Dantzig-Wolfe.
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