multimodal transportation p-hub location routing problem with simultaneous pick-ups and deliveries

centralizing and using proper transportation facilities cut down costs and traffic. hub facilities concentrate on flows to cause economic advantage of scale and multimodal transportation helps use the advantage of another transporter. a distinctive feature of this paper is proposing a new mathematical formulation for a three-stage p-hub location routing problem with simultaneous pick-ups and deliveries on time. a few studies have been devoted to this problem; however, many people are still suffering from the problems of commuting in crowded cities. the proposed formulation controlled the tumult of each node by indirect fixed cost. node-to-node traveling cost was followed by a vehicle routing problem between nodes of each hub. a couple of datasets were solved for small and medium scales by gams software. but, for large-scale instances, a meta-heuristic algorithm was proposed. to validate the model, datasets were used and the results demonstrated the performance suitability of the proposed algorithm.