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Systems of rail-mounted vehicles play a key role in many logistics applications, and the efficiency of their operation frequently has a significant impact on the overall performance of the surrounding production environment. In theory, assigning transport requests to the vehicles of such systems and scheduling their execution amounts to finding k tours on a common line, where tours may never cross each other in time--dynamic collision constraints need to be respected. The goal is to minimize the makespan for a given set of transport requests.
We establish a model capturing the core challenges in transport planning problems of this type and relate it to other models in literature. After proving NP-hardness for a basic version of the problem, the large part of the paper is dedicated to devising various fast heuristic algorithms suitable for practice. We present computational results regarding the performance of the algorithms proposed for several classes of problem instances.
We study two related problems in non-preemptive scheduling and packing of malleable tasks with precedence constraints to minimize the makespan. We distinguish the scheduling variant, in which we allow the free choice of processors, and the packing variant, in which a task must be assigned to a contiguous subset of processors.
For precedence constraints of bounded width, we completely resolve the complexity status for any particular problem setting concerning width bound and number of processors, and give polynomial-time algorithms with best possible performance. For both, scheduling and packing malleable tasks, we present an FPTAS for the NP-hard problem variants and exact algorithms for all remaining special cases. To obtain the positive results, we do not require the common monotonous penalty assumption on processing times, whereas our hardness results hold even when assuming this restriction.
With the close relation between contiguous scheduling and strip packing, our FPTAS
is the first (and best possible) constant factor approximation for (malleable) strip packing under special precedence constraints.