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In this paper we introduce the multi-period incremental service facility location problem where the goal is to set a number of new facilities over a finite time horizon so as to cover dynamically the demand of a given set of customers. We prove that the coefficient matrix of the allocation subproblem that results when fixing the set of facilities to open is totally unimodular. This allows to solve efficiently the Lagrangean problem that relaxes constraints requiring customers to be assigned to open facilities. We propose a solution approach that provides both lower and upper bounds by combining subgradient optimization to solve a Lagrangean dual with an ad hoc heuristic that uses information from the Lagrangean subproblem to generate feasible solutions. Numerical results obtained in the computational experiments show that the obtained solutions are very good. In general, we get very small percent gaps between upper and lower bounds with little computation effort. ᭧

a b s t r a c tThis paper presents the p-next center problem, which aims to locate p out of n centers so as to minimize the maximum cost of allocating customers to backup centers. In this problem it is assumed that centers can fail and customers only realize that their closest (reference) center has failed upon arrival. When this happens, they move to their backup center, i.e., to the center that is closest to the reference center. Hence, minimizing the maximum travel distance from a customer to its backup center can be seen as an alternative approach to handle humanitarian logistics, that hedges customers against severe scenario deteriorations when a center fails.For this extension of the p-center problem we have developed several different integer programming formulations with their corresponding strengthenings based on valid inequalities and variable fixing. The suitability of these formulations for solving the p-next center problem using standard software is analyzed in a series of computational experiments. These experiments were carried out using instances taken from the previous discrete location literature.

In this paper we present a Mixed Integer Nonlinear Programming model that we developed as part of a pilot study requested by the R&D company Metrolab R 1 in order to design tools for finding solutions for line planning and timetable situations in automated urban metro subway networks. Our model incorporates important factors in public transportation systems from both, a cost-oriented and a passenger-oriented perspective, as time-dependent demands, interchange stations, short-turns and technical features of the trains in use. The incoming flows of passengers are modeled by means of piecewise linear demand functions which are parameterized in terms of arrival rates and bulk arrivals. Decisions about frequencies, train capacities, short-turning and timetables for a given planning horizon are jointly integrated to be optimized in our model. Finally, a novel Math-Heuristic approach is proposed to solve the problem. The results of extensive computational experiments are reported to show its applicability and effectiveness to handle real-world subway networks.