An analysis of the impact of demand aggregation on the solution quality for facility location problems

Abstract

Inspired by real-world applications, this paper studies the impact on the quality of solutions for facility location problems in which demand points are aggregated to reduce the size of the underlying mathematical formulation. Two aggregation methods are analyzed and compared: demand points aggregated based on municipal boundaries or other similar administrative boundaries as usually done in practice and using the K-means clustering algorithm. Regarding a business-to-business (B2B) distribution context, two datasets comprising the location of thousands of drugstores in Brazil were generated, and 18 different instances of the fixed cost facility location problem were derived. The results show that solutions with aggregated demand points by municipality yield a maximum 0.43% difference in the objective function value in comparison with the respective disaggregated mode, while the difference using K-means algorithm did not exceed 0.03%. We also performed an in-depth analysis of the regions where the demand points were allocated to distinct selected facilities in the aggregated and disaggregated models. It was possible to observe that in the model with aggregated demand points by municipality, differences in transportation costs are greater than using the K-means clustering algorithm as the aggregation procedure. This suggests that aggregating demand points with the K-means clustering algorithm yields both better objective function values, and selected facilities closer to demand points in the cases where the resulting assignment of demand points to the selected facilities is not the same as the results of the unaggregated model.