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The solution of the Multi-Depot Vehicle Scheduling Problem (MDVSP) can oftenbe improved substantially by incorporating Trip Shifting (TS) as a modelfeature. By allowing departure times to deviate a few minutes from the originaltimetable, new combinations of trips may be carried out by the same vehicle,thus leading to more efficient scheduling. However, explicit modeling of eachpotential trip shift quickly causes the problem to get prohibitively large forcurrent solvers, such that researchers and practitioners were obligated toresort to heuristic methods to solve large instances. In this paper, we developa Dynamic Discretization Discovery algorithm that guarantees an optimalcontinuous-time solution to the MDVSP-TS without explicit consideration of alltrip shifts. It does so by iteratively solving and refining the problem on apartially time-expanded network until the solution can be converted to afeasible vehicle schedule on the fully time-expanded network. Computationalresults demonstrate that this algorithm outperforms the explicit modelingapproach by a wide margin and is able to solve the MDVSP-TS even when manydeparture time deviations are considered.

Dynamic Discretization Discovery for the Multi-Depot Vehicle Scheduling Problem with Trip Shifting