An exact approach for the multi-depot electric vehicle scheduling problem

TL;DR

This paper introduces the Multi-Depot Electric Vehicle Scheduling Problem (MDEVSP) with partial recharging and presents a graph-based formulation that enforces time-feasible schedules. It develops a compact 3-index MILP solvable directly and a 2-index MILP requiring cutting-plane Pareto constraints (infeasible-path or connectivity constraints) to handle depot pairing, accompanied by valid inequalities. A branch-and-cut algorithm solves the 2-index model efficiently, especially with multiple depots, and computational experiments compare propulsion technologies (Diesel, BEB, FCEB) under various conditions, including cold temperatures and battery-preservation policies. The findings show that integrating multiple lines substantially reduces fleet size, BEBs impose the largest fleet requirements and charging loads, FCEBs exhibit robustness to weather, and the 2-index approach with connectivity cuts offers superior scalability for large, multi-depot settings, providing actionable guidance for transit operators and policy makers.

Abstract

The "avoid - shift - improve" framework and the European Clean Vehicles Directive set the path for improving the efficiency and ultimately decarbonizing the transport sector. While electric buses have already been adopted in several cities, regional bus lines may pose additional challenges due to the potentially longer distances they have to travel. In this work, we model and solve the electric bus scheduling problem, lexicographically minimizing the size of the bus fleet, the number of charging stops, and the total energy consumed, to provide decision support for bus operators planning to replace their diesel-powered fleet with zero emission vehicles. We propose a graph representation which allows partial charging without explicitly relying on time variables and derive 3-index and 2-index mixed-integer linear programming formulations for the multi-depot electric vehicle scheduling problem. While the 3-index model can be solved by an off-the-shelf solver directly, the 2-index model relies on an exponential number of constraints to ensure the correct depot pairing. These are separated in a cutting plane fashion. We propose a set of instances with up to 80 service trips to compare the two approaches, showing that, with a small number of depots, the compact 3-index model performs very well. However, as the number of depots increases the developed branch-and-cut algorithm proves to be of value. These findings not only offer algorithmic insights but the developed approaches also provide actionable guidance for transit agencies and operators, allowing to quantify trade-offs between fleet size, energy efficiency, and infrastructure needs under realistic operational conditions.

Figures (2)

• Figure 1: Graph for the instance $I_1$ with a single origin ($o_1$) and destination ($d_1$) depot.
• Figure 2: Graph for the instance $I_2$ with two origin depots ($o_1, o_2$) and two destination depots ($d_1, d_2$).