Extension of the maximum flow-covering location and service start time problem to allow flexible consumption

This paper extends the maximum flow-covering location and service start time problem (MFCLSTP) by additionally imposing a minimum stay at a facility before the service can be used. The original formulation determines the locations of facilities and the start times of fixed-duration services so as to maximally cover flows of commuters who access services on the way home from work. In MFCLSTP, commuters must stay at a facility from the start of the service until the end of the service in order to consume it. Examples of such services include movies, lectures, and baseball games. There are, however, many on-demand services, which can be consumed by commuters who access the providing facility for a fixed continuous duration of a hours during any of c (≥ a) open hours of the facility. To deal with this situation, we extend the definition of coverage in the original model and provide an integer programming formulation of the proposed problem. The model is applied to the Tokyo metropolitan railway network, using census data of commuter traffic for railway users in this area. Exact optimal solutions are obtained by using a mathematical optimization solver with both the original and extended problems, and the characteristics of the solutions are compared. The results show that the optimal solutions for the extended problem allow a much larger flow than the optimal solutions for the original problem do, even when a is very close to c. The locations selected for the optimal solutions of the proposed problem are seen to be much more spatially dispersed than those chosen for the original problem.