We study the problem of repositioning autonomous vehicles in a shared mobility system in order to simultaneously minimize the unsatisfied demand and the total operating cost. We first present a mixed integer linear programming formulation for the deterministic version of the problem, and based on that we develop an extended formulation that is easier to work with in the non-deterministic setting that we aim to explore. We then show how the travel time uncertainty can be incorporated into the extended deterministic formulation using chance-constraint programming. Finally, two new reformulations for the proposed chance-constraint program are developed. We show a critical result that the size of one of the reformulations (in terms of the number of variables and constraints) does not depend on the number of scenarios, and so it outperforms the other reformulation. Both reformulations are bi-objective mixed integer linear programs with finite number of nondominated points and so they can be solved directly by algorithms such as the balanced box method (Boland et al., 2015). A computational study demonstrates the efficacy of the proposed reformulations.
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