Inmaculada Rodríguez Martín

The Ring Star Problem: Polyhedral Analysis and Exact Algorithm

In the Ring Star Problem, the aim is to locate a simple cycle through a subset of vertices of a graph with the objective of minimizing the sum of two costs: a ring cost proportional to the length of the cycle and an assignment cost from the vertices not in the cycle to their closest vertex on the cycle. The problem has several applications in telecommunications network design and in rapid transit systems planning. It is an extension of the classical location–allocation problem introduced in the early 1960s, and closely related versions have been recently studied by several authors. This article formulates the problem as a mixed-integer linear program and strengthens it with the introduction of several families of valid inequalities. These inequalities are shown to be facet-defining and are used to develop a branch-and-cut algorithm. Computational results show that instances involving up to 300 vertices can be solved optimally using the proposed methodology.

Variable neighborhood tabu search and its application to the median cycle problem

Abstract The variable neighborhood tabu search consists of a constructive phase and a series of local searches that use tabu tools. When the local search stops at a nontabu local minimum, a shake procedure starts a new local search. The local searches and the shake procedure are based on a series of standard moves. The local search consists of applying the best possible move until no such move exists. The shake procedure consists of applying a number of random moves. Since the shake could provide an infeasible solution, the local search considers as possible moves those providing a feasible solution or reducing the infeasibility. A location/allocation problem consists of selecting the location for some facilities and the allocation of the users to them. Two functions to be minimized are considered: the length of the solution, as a measure of the set of locations, and the total cost of the allocations. The standard moves for these problems are the add, drop and add/drop moves. The heuristic procedure is tested on the median cycle problem.

Locating median cycles in networks

Abstract In the median cycle problem the aim is to determine a simple cycle through a subset of vertices of a graph involving two types of costs: a routing cost associated with the cycle itself, and the cost of assigning vertices not on the cycle to visited vertices. The objective is to minimize the routing cost, subject to an upper bound on the total assignment cost. This problem arises in the location of a circular-shaped transportation and telecommunication infrastructure. We present a mixed integer linear model, and strengthen it with the introduction of additional classes of non-trivial valid inequalities. Separation procedures are developed and an exact branch-and-cut algorithm is described. Computational results on instances from the classical TSP library and randomly generated ones confirm the efficiency of the proposed algorithm. An application related to the city of Milan (Italy) is also solved within reasonable computation time.