Abstract
Competitive facility location models consider two main strategies for increasing the market share captured by a chain subject to a budget constraint. One strategy is the improvement of existing facilities. The second strategy is the construction of new facilities. In this paper we analyse these two strategies as well as the joint strategy which is a combination of the two. All three strategies are formulated as a unified model. The best solution to an individual strategy is a feasible solution to the joint one. Therefore, the joint strategy must yield solutions that are at least as good as the solutions to each of the individual strategies. Based on the results of extensive experiments, we conclude that the increase in market share captured by a chain when the joint strategy is employed can be significantly higher than increases obtained by individual strategies. A branch and bound procedure and a tabu search heuristic are constructed for the solution of the unified model. Both algorithms performed very well on a set of test problems with up to 900 demand points. A total of 62% of the test problems were optimally solved by the branch and bound procedure.
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Appendix
Appendix
A.1. Nonlinear Integer Programming Formulation
To formulate the problem as a nonlinear binary program define the following sets and binary variables:
Also, y
ij
=0 when i∉I
j
. By these definitions r
j
=
and q
i
=∑
j=1
p
y
ij
. The nonlinear binary program is:
subject to:
Constraints (A.2) guarantee that at most one distance is selected for each facility. If for some j the sum is equal to 0, facility j is not improved. The right-hand side of constraints (A.3) is r j . Constraints (A.3) guarantee that y ij =0 if d ij >r j . Since the objective function increases if y ij increases, then if constraints (A.3) allow for y ij =1, then y ij =1 in the optimal solution. Constraint (A.4) is the limited budget constraint. Constraints (A.5) require binary variables which reduce the number of y ij variables by requiring that those not in I j are equal to 0. In the objective function (A.1) a special treatment is required for demand points where F i +C i =0. In this case the expression in the objective function (A.1) is w i min {∑ j=1 p y ij , 1} rather than the ratio expression.
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Drezner, T., Drezner, Z. & Kalczynski, P. Strategic competitive location: improving existing and establishing new facilities. J Oper Res Soc 63, 1720–1730 (2012). https://doi.org/10.1057/jors.2012.16
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DOI: https://doi.org/10.1057/jors.2012.16