Modelling Economies of Scale in Hub and Spoke Systems: A Comparative Study

Abstract

Hub and spoke networks are essential to the design of efficient transportation and logistics systems, and the Single Allocation p-Hub Location Problem, known as SApHLP, has emerged as a key modelling framework to support such decisions. Earlier research has introduced multiple cost structures, including fixed discounts, flow-dependent, and hybrid models that combine fixed and variable cost elements, to capture economies of scale, defined as cost savings achieved by increasing the volume of flow through shared infrastructure. However, a systematic comparative evaluation of these formulations remains limited. This study seeks to determine which of the three SApHLPs most effectively represent economies of scale. The study analyses how each model performs across three key dimensions, which are flow distribution, total network cost, and computational effort. Each model was implemented under a uniform experimental design, as Mixed Integer Programming with the CPLEX solver, by employing three datasets, CAB, TR, and a synthetically generated dataset based on a uniform distribution for flow and cost values. Performance was assessed based on flow allocation patterns, cost efficiency, and runtime behaviour under varying numbers of hubs and network sizes. Findings reveal that the Hybrid Model consistently delivers the lowest total costs and the most realistic flow consolidation by integrating fixed and variable arc costs. The flow-dependent model provides a nuanced cost representation but suffers from substantial computation times, especially in larger instances. The fixed discount Model performs fastest computationally but lacks cost sensitivity, often resulting in suboptimal flow structures. This study contributes to the hub location literature by offering a structured comparison of established SApHLP models. It offers actionable guidance for logistics planners on model selection, emphasising the balance between realism and solvability in network design.