Fair profit contract with soft time-windows for a carrier collaboration framework in a hub network: Dual Lexicographic max–min

This paper presents a novel and practical bi-objective hub location problem under a centralized carrier collaboration framework between one holding company and multiple carriers. In this paper, the holding company first establishes a hub-and-spoke network in order to locate p hubs and to assign the center nodes to the located hubs. Then, the holding company chooses some carriers and allocates the transportation routes of the hub network to them, i.e., which carrier for which transportation route should be selected. In contrast, the selected carriers should serve the transportation requests of the routes in the proposed hub network by appropriate vehicles based on a centralized transportation planning. The carriers are also able to meet the requests within a certain time window based on a soft time-window mechanism. Both the holding company and the carriers desire to maximize their profits with respect to the soft time-window mechanism. The aim of this paper is to design a fair profit contract between the holding company and the carriers to identify and exploit win–win situations among them. This aim is achieved using dual Lexicographic max–min approach in order to optimize their profits in a fair way such that their benefits are obtained as close to one another as possible. Finally, numerical experiments are presented to demonstrate the applicability of the proposed model and solution methodology. Sensitivity analyses and managerial insights about the soft time-windows are conducted as well.