<![CDATA[The hub location problem (HLP) is widely used to design hub-and-spoke networks by consolidating flows through selected hub facilities. However, standard formulations may produce solutions in which some hubs are connected to very few spokes, while others serve a disproportionately large number of spokes, resulting in inefficient hub utilization. This study aims to investigate the impact of imposing explicit lower bound and combined lower and upper bounds on hub–spoke allocations to improve hub utilization. The contribution of this study is to introduce two bounded spoke allocation constraints and systematically evaluating their impact on computational performance, total cost, network structure, and hub location decisions. A flow-based single allocation hub location formulation is adopted, and computational experiments are conducted on the CAB and TN benchmark datasets under various parameter settings. The results show that the lower bound eliminates hubs with very few spokes, while the upper bound limits excessive concentration of spokes, leading to more controlled hub and spoke configurations. The proposed constraints may also alter hub location decisions. The impact on total cost is small, with average increases of about 1.23% and 0.99% for the lower bound model, and 1.38% and 1.52% for the combined lower and upper bounds model in the CAB and TN datasets, respectively. Although computational effort increases for larger instances, the proposed approach provides decision-makers with a practical mechanism for controlling hub utilization.]]>
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