Re-aggregation Heuristic for Large P-median Problems

Abstract

When a location problem is too large to be computed in a reasonable time or if it is impossible to store it in the computer memory, an aggregation is commonly used tool that allows for transforming it to smaller size. Typically, an aggregation method is used only once, in the initial phase, before the solving process. An unavoidable consequence of the aggregation is the loss of optimality due to aggregation errors. The effects of aggregation errors might be significant, especially, when solving spatially large problems with huge number of potential customers. Here, we propose new simple re-aggregation approach, which minimizes sources of the aggregation errors. Our method aggregates the original problem to the size that can be solved by the used optimization algorithm, and in each iteration, the aggregated problem is adapted to achieve more precise location of facilities. The proposed approach is versatile and thus it can be easily extended to other location problems. To investigate the optimality error, we use benchmarks that can be computed exactly and to test the applicability of our approach, we study large benchmarks reaching 670 000 customers. Numerical experiments reveal that the re-aggregation heuristic works well for commonly used, as well as for extremely large sizes of the p-median problem. The approach allows for finding solutions of higher quality than the exact method that is applied to the aggregated problem.