  In this paper 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 theoretical and practical 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. In this paper we extend that list by considering new special cases
and obtain new complexity results and approximation algorithms. We show that
even if there is only a single non-renewable resource, and each job has unit
weight and requires only one unit from the resource, the problem is still
NP-hard, however, in our construction we need a high-multiplicity encoding of
the jobs in the input. We also propose an FPTAS for a variant in which the jobs
have arbitrary weights, and the number of supply time points is bounded by a
constant. Finally, we prove some non-trivial approximation guarantees for
simple greedy algorithms for some further variants of the problem.
