An alternating heuristic for medianoid and centroid problems in the plane

Abstract

This paper develops two heuristics for solving the centroid problem on a plane with discrete demand points. The methods are based on the alternating step well known in location methods. Extensive computational testing with the heuristics reveals that they converge rapidly, giving good solutions to problems that are up to twice as large as those reported in the literature. The testing also provides some managerial insight into the problem and its solution. Scope and purpose When dealing with competitive location models, one popular solution concept is the Stackelberg solution. It assumes that one (group of) firm(s) acts as leader, while the other(s) act(s) as follower. In the locational context, the follower takes the locations of the leader as given and optimizes on that basis, whereas the leader will exercise foresight and take into account that a follower will subsequently locate additional facilities. It is commonly assumed that the leader knows how many facilities the follower will locate. In this bilevel programming problem, the leader's problem is called a centroid problem, whereas the follower faces a so-called medianoid problem. In both cases, the objective of the facility planner is to capture as much of the market as possible. Since centroid problems are inherently difficult, it is necessary to devise heuristic methods for all but the smallest models. This paper presents two such heuristics for the planar version of the centroid problem that are based on the repeated application of a medianoid solution, a much simpler problem. Computational results attesting to their performance are also included.