Online scheduling of car-sharing requests between two locations with many cars and flexible advance bookings

We study an on-line scheduling problem that is motivated by applications such as car-sharing, in which users submit ride requests, and the scheduler aims to accept requests of maximum total profit using k servers (cars). Each ride request specifies the pick-up time and the pick-up location (among two locations, with the other location being the destination). The scheduler has to decide whether or not to accept a request immediately at the time when the request is submitted (booking time). We consider two variants of the problem with respect to constraints on the booking time: In the fixed booking time variant, a request must be submitted a fixed amount of time before the pick-up time. In the variable booking time variant, a request can be submitted at any time during a certain time interval (called the booking horizon) that precedes the pick-up time. We present lower bounds on the competitive ratio for both variants and propose a balanced greedy algorithm (BGA) that achieves the best possible competitive ratio. We prove that, for the fixed booking time variant, BGA is 1.5-competitive if k = 3i (i ∈ N) and the fixed booking length is not less than the travel time between the two locations; for the variable booking time variant, BGA is 1.5-competitive if k = 3i (i ∈ N) and the length of the booking horizon is less than the travel time between the two locations, and BGA is 5/3-competitive if k = 5i (i ∈ N) and the length of the booking horizon is not less than the travel time between the two locations