Scheduling controllable processing time jobs in a deteriorating environment

In many real-life applications, job processing times are a function of the waiting time prior to their execution. In the most general setting, each job comprises of a basic processing time, which is independent of its start time, and a start time-dependent deterioration function. Some common examples of deteriorating systems include fire fighting, pollution containment, and medical treatments. To date, research has focused on scheduling models where the basic processing time of jobs is constant. However, job processing times are often controllable through the allocation of a limited non-renewable resource. We study a single-machine setting that combines these two models under the assumptions of general linear deterioration and convex resource functions. We develop a polynomial time solution for minimizing the makespan. For the total flowtime criterion, we compute the optimal resource allocation policy for a given job instance and show that the sequencing problem is at least as hard as the case with non-controllable jobs. We follow by discussing the properties of several special cases.