The vehicle scheduling problem (VSP) is a traditional problem in public transport. One of the main assumptions is that buses can be operated the whole day without any interruption for refueling etc. Recently, new technological innovations have led to the introduction of electric vehicles (EVs). For these new vehicles, we cannot ignore the need of refueling during the day, as the range of an electric bus is severely limited, because of the capacity of the batteries. In this paper, we study the electric VSP (e-VSP), where we use EVs with a limited range. During the day the batteries can be charged; in this paper we assume that a battery cannot be replaced/substituted. We present two models that differ in the level of detail resembling the actual processes. In our first model, we assume a linear charging process, work with a constant price of electricity during the day, and do not take the effect of the depth-of-discharge on the lifetime of the battery into account. Our second model resembles practice much better: we allow any type of charging process, work with the actual electricity prices, and take the depreciation cost of the battery into account. To keep this model tractable, however, we approximate the exact value of the charge by discretizing it. The refined model can be solved to optimality using integer linear programming for instances of small/medium size, and therefore, we describe two other solution methods based on column generation that find good, but not necessarily optimal, solutions for large instances. We have tested our algorithms on real-world instances.

In power system studies the unit commitment problem (UC) is solved to support market decisions and assess system adequacy. Simplifications are made to solve the UC faster, but they are made without considering the consequences on solution quality. In this study we thoroughly investigated the impacts of simplifications on solution quality and computation time on a benchmark set consisting of almost all the available instances in the literature. We found that omitting the minimum up- and downtime and simplifying the startup cost resulted in a significant quality loss without reducing the computation time. Omitting reserve requirements, ramping limits and transmission limits reduced the computation time, but degraded the solution significantly. However, the linear relaxation resulted in less quality loss with a significant speed-up and resulted in no difference when unserved energy was minimized. Finally, we found that the average and maximum capacity factor difference is large for all model variants.

Minimizing the number of late jobs on a single machine is a classic scheduling problem, which can be used to model the situation that from a set of potential customers, we have to select as many as possible whom we want to serve, while selling no to the other ones. This problem can be solved by Moore–Hodgson’s algorithm, provided that all data are deterministic. We consider a stochastic variant of this problem, where we assume that there is a small probability that the processing times differ from their standard values as a result of some kind of disturbance. When such a disturbance occurs, then we must apply some recovery action to make the solution feasible again. This leads us to the area of recoverable robustness, which handles this uncertainty by modeling each possible disturbance as a scenario; in each scenario, the initial solution must then be made feasible by applying a given, simple recovery algorithm to it. Since we cannot accept previously rejected customers, our only option is to reject customers that would have been served in the undisturbed case. Our problem therefore becomes to find a solution for the undisturbed case together with a feasible recovery to every possible disturbance. Our goal hereby is to maximize the expected number of served customers; we assume here that we know the probability that a given scenario occurs. In this respect, our problem falls outside the area of the ‘standard’ recoverable robustness, which contains the worst-case recovery cost as a component of the objective. Therefore, we consider our approach as a combination of two-stage stochastic programming and recoverable robustness. We show that this problem is \(\mathcal{NP}\)-hard in the ordinary sense even if there is only one scenario, and we present some sufficient conditions that allow us to find a part of the optimal solution in polynomial time. We further evaluate several solution methods to find an optimal solution, among which are dynamic programming, branch-and-bound, and branch-and-price.