Vehicle Sharing System Optimization: Scenario-based approach

One-way Vehicle Sharing Systems (VSS), such as Velib, have a poor performance without regulation. We want to improve the efficiency of VSS using design strategies or pricing as incentive. Stochastic models are intractable for the size of systems we want to study. We therefore discuss a scenario-based approach, off-line deterministic optimization problem on a given stochastic realization ({scenario). This deterministic model could be used to provide a heuristic and a bound for on-line stochastic optimization. This approach raises a new constraint the First Come First Served constrained flow (FCFS flow). We propose a design problem, \textsc{Max FCFS Flow Station Capacities}, and two pricing problems: Max FCFS Flow Trip Pricing and Max FCFS Flow Station Pricing based on a FCFS flow. We show that they are all APX-Hard. We study the upper bound given by the Max Flow problem and prove its poor worst case ratio to the Max FCFS Flow.