Scheduling controllable processing time jobs with position-dependent workloads

In various real life applications job processing times are controllable through the allocation of a limited resource. To date research has been conducted under the assumption that the relationship between the amount of resource allocated to a job and its processing time is independent of the number of tasks processed previously. However, there exist many manufacturing and service systems where workers and machines acquire, develop and refine skills through the repetition of identical or similar operations. In this paper we consider a scheduling model where job processing times are a convex function of the amount of resource they are allocated. In addition, we assume that the parameters of this function are position-dependent, i.e., vary with the job׳s position in the sequence. This assumption reflects general processes of learning or aging, or a combination of both. We first focus on a single machine setting and the makespan and total flowtime criteria. We show that the combined problem of finding an optimal job sequence and an optimal resource allocation can be solved in O(n3) time. We show that our algorithm can be used to address a bicriteria objective comprising of a linear combination of makespan and the total flowtime criteria on a single machine. We then extend the results to a parallel machine setting for the total flowtime criteria.