Heuristic and exact methods for the discrete (r|p)-centroid problem

Authors:
Ekaterina Alekseeva;Nina Kochetova;Yury Kochetov;Alexandr Plyasunov
Affiliations:
Sobolev Institute of Mathematics, Novosibirsk State University, Russia;Sobolev Institute of Mathematics, Novosibirsk State University, Russia;Sobolev Institute of Mathematics, Novosibirsk State University, Russia;Sobolev Institute of Mathematics, Novosibirsk State University, Russia
Venue:
EvoCOP'10 Proceedings of the 10th European conference on Evolutionary Computation in Combinatorial Optimization
Year:
2010

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Abstract

In the discrete (r |p)-centroid problem two decision makers, a leader and a follower, compete to attract clients from a given market. The leader opens p facilities, anticipating that the follower will react to the decision by opening his own r facilities. The decision makers try to maximize their own profits. This Stackelberg game is $\Sigma_2^P$-hard. So, we develop a hybrid memetic algorithm for it. A probabilistic tabu search heuristic is applied for improving the offspring. To obtain an upper bound, we reformulate the problem as a mixed integer program with an exponential number of constraints and variables. Selecting some of them, we get the desired upper bound. To find optimal solutions, we iteratively modify the subset of the constraints and variables. This approach is tested on the benchmarks from the library Discrete Location Problems. The optimal solutions are found for r=p=5, 100 clients, and 100 facilities.