A versatile adaptive aggregation framework for spatially large discrete location-allocation problems

Abstract We propose a versatile concept of the adaptive aggregation framework for the facility location problems that keeps the problem size in reasonable limits. Most location-allocation problems are known to be NP-hard. Thus, if a problem reaches the critical size, the computation exceeds reasonable time limits, or all computer memory is consumed. Aggregation is a tool that allows for transforming problems into smaller sizes. Usually, it is used only in the data preparation phase, and it leads to the loss of optimality due to aggregation errors. This is particularly remarkable when solving problems with a large number of demand points. The proposed framework embeds the aggregation into the solving process and it iteratively adjusts the aggregation level to the high quality solutions. To explore its versatility, we apply it to the p-median and to the lexicographic minimax problems that lead to structurally different patterns of located facilities. To evaluate the optimality errors, we use benchmarks which can be computed exactly, and to explore the limits of our approach, we study benchmarks reaching 670,000 demand points. Numerical experiments reveal that the adaptive aggregation framework performs well across a large range of problem sizes and is able to provide solutions of higher quality than the state-of-the-art exact methods when applied to the aggregated problem.