A capacitated three-echelon supply chain location-allocation problem in a continuous space with lp distance metrics

This paper investigates the location problem in a three-echelon supply network, considering primary suppliers, mid-tier distributors, and end consumers simultaneously; each layer with finite capacity and multiple sources. We discuss an often-overlooked characteristic of such a network with amenities located in continuous space. Unlike traditional models, the implemented universal l_p distance formulation represents diverse realistic transportation spatial structures with greater flexibility. The principal challenge is to develop a practical framework for real-scale networks with numerous customers, addressed through a hybrid approach, combining the greedy randomised adaptive search procedure with efficient local search based on variable neighbourhood search. A customised method is also extended for identifying the restricted candidate lists to boost efficiency. Various benchmark datasets, from small to large-sized, demonstrate the statistical robustness of the proposed framework under different distance functions. Results revealed a substantial superiority of the proposed approach by over 77%, outperforming four other heuristic algorithms and exact time-limited upper-bounds. Finally, we validate the framework using a comprehensive real‑world case study from Oman, where the continuous multi‑source model is employed to optimise a drone‑enabled medical distribution network. The results confirm the method’s effectiveness in handling complex, large‑scale spatial logistics systems.