Robust Vehicle Repositioning with Entropic Risk Measure

We consider a general multi-period repositioning problem in vehicle-sharing networks such as bicycle, scooter, and car-sharing systems. This problem involves uncertainties along multiple dimensions, including demand, travel time, and repositioning duration. It is also subject to various operational constraints, such as service level targets and capacity limits, which can be affected by future uncertainties. To tractably incorporate the system dynamics under real-life uncertainties into the optimization model and account for the risk of violating future operational constraints, we propose an entropic robust optimization (ERO) model. In practice, one can implement it in a rolling horizon manner. The ERO model minimizes the entropic risk of the total repositioning and penalty cost. It also safeguards future operational constraints under distributional ambiguity via entropic robust constraints evaluated by the entropic risk measure. We propose an integral affine recourse adaptation to approximate future recourse decisions. With this recourse adaptation and a mild approximation, we show that our ERO model can be reformulated as a mixed-integer convex optimization problem. It is further reduced to a mixed-integer linear optimization problem when we adopt the static recourse adaptation. To our knowledge, this work is the first to incorporate the various time-dependent uncertainties in vehicle repositioning. Extensive simulation studies and a case study on a real-world dataset demonstrate that our model achieves satisfactory performance in various settings and is computationally efficient. Compared with existing benchmarks, such as the fluid-based optimization model, the proposed model can achieve a higher average service level for a similar repositioning cost and lead to a lower total cost