Clara M. Campos Rodríguez

Multiple voting location problems

The facility voting location problems arise from the application of criteria derived from the voting processes concerning the location of facilities. The multiple location problems are those location problems in which the alternative solutions are sets of points. This paper extends previous results and notions on single voting location problems to the location of a set of facility points. The application of linear programming techniques to solve multiple facility voting location problems is analyzed. We propose an algorithm to solve Simpson multiple location problems from which the solution procedures for other problems are derived.

Relaxation of the Condorcet and Simpson conditions in voting location

Abstract A Condorcet point, in voting location, is a location point such that there is no other closer to more than half of the users. However, such Condorcet solution does not necessarily exist. This concept is based on two assumptions. First, two locations are indifferent only if they are at the same distance of the voter. Second, the number of voters needed to reject a location is more than half of them. We relax the Condorcet condition in two ways. First, by considering that two locations are indifferent for every user if the difference of the distances to them is within a positive threshold. Secondly, by considering that the proportion of users needed to reject a location is not one half. We consider the resulting new solution concepts that arise by applying both relaxations at the same time and develop algorithms for obtaining them in the finite case.

An exact procedure and LP formulations for the leader—follower location problem

The leader—follower location problem consists of determining an optimal strategy for two competing firms which make decisions sequentially. The leader optimisation problem is to minimise the maximum market share of the follower. The objective of the follower problem is to maximise its market share. We describe linear programming formulations for both problems and analyse the use of these formulations to solve the problems. We also propose an exact procedure based on an elimination process in a candidate list.

Two-Swarm PSO for Competitive Location Problems

Competitive location problems consist of determining optimal strategies for competing firms which make location decisions. The standard problem is the leader-follower location problem which consists of determining optimal strategies for two competing firms, the leader and the follower, which make decisions sequentially. The follower has the objective of maximizing its market share, given the locations chosen by the leader. The leader optimization problem is to minimize the maximum market share that the follower can get. We propose a two-swarm particle swarm optimization procedure in which each swarm contains locations for one of the two firms. We analyze the application of this procedure to the (r|p)-centroid problem in the plane. It is the leader-follower problem where the leader chooses p points and then the follower chooses r points.

Comparison of α-Condorcet points with median and center locations

The usual concept of solution in single voting location is the Condorcet point. A Condorcet solution is the location such that no other location is preferred by a strict majority of voters; i.e. a half of them. It is assumed that each user always prefers closer locations. Because a Condorcet point does not necessarily exist, the α-Condorcet point is defined in the same way but assuming that two locations are indifferent for a user if the distances to both differ at most in α. We give bounds for the value of the objective function in an α-Condorcet point in the median and center problems. These results, for a general graph and for a tree, extend previous bounds for the objective function in a Condorcet point. We also provide a set of instances where these bounds are asymptotically reached.