Mixed bus fleet location-routing-scheduling under range uncertainty

This paper develops a framework to address the multi-depot vehicle location-routing-scheduling problem with multi-vehicle types, including electric buses, under range uncertainty. Three major issues in bus routing and scheduling are addressed in this framework, i.e., uncertain driving range, refueling or charging need, and locating refueling or charging facilities. Mathematically, the problem is formulated as a two-stage stochastic program. Two types of services, regular services running on fixed schedules and ad hoc services to cover incomplete scheduled trips arising from energy shortage of certain buses, are considered in this paper. An adaptive time-space-energy network is developed to model the refueling issues and location problem. We introduce the notion of range reliability to decompose and solve the two-stage stochastic formulation under range uncertainty. The regular services schedule is determined in stage-one to cover the demand under a certain range reliability level. Upon realization of the random driving range, the deployment of ad hoc services is made in stage-two to address the occurrence of energy shortage of certain buses. A range reliability-based gradient algorithm is developed to minimize the expected total cost of the system. We then apply the proposed method to bus services in Hong Kong. The range reliability-based approach shows promising results, leading to substantial cost savings as compared with the traditional methods that ignore the effects of driving range uncertainty.