We consider single machine scheduling problems with additional non-renewable resource constraints. Examples for non-renewable resources include raw materials, energy, or money. Usually they have an initial stock and replenishments arrive over time at a-priori known time points and quantities. The jobs have some requirements from the resources and a job can only be started if the available quantity from each of the required resources exceeds the requirements of the job. Upon starting a job, it consumes its requirements which decreases the available quantities of the respective non-renewable resources. There is a broad background for this class of problems. Most of the literature concentrate on the makespan, and the maximum lateness objectives. This paper focuses on the total weighted completion time objective for which the list of the approximation algorithms is very short. We extend that list by considering new special cases and obtain new complexity results and approximation algorithms.

The NP-hard Material Consumption Scheduling Problem and closely related problems have been thoroughly studied since the 1980's. Roughly speaking, the problem deals with minimizing the makespan when scheduling jobs that consume non-renewable resources. We focus on the single-machine case without preemption: from time to time, the resources of the machine are (partially) replenished, thus allowing for meeting a necessary pre-condition for processing further jobs, each of which having individual resource demands. We initiate a systematic exploration of the parameterized computational complexity landscape of the problem, providing parameterized tractability as well as intractability results. Doing so, we mainly investigate how parameters related to the resource supplies influence the computational complexity. Thereby, we get a deepened understanding of this fundamental scheduling problem.