The Discrete Resource Allocation Problem in Flow Lines

In this paper we address the discrete resource allocation problem in a deterministic flow line. We assume that the processing times are convex and nonincreasing in the amount of resources allocated to the machines. We consider the resource allocation problem for a fixed sequence of jobs for various performance criteria (makespan, weighted sum of completion times, cycle time for cyclic schedules), and develop a formulation of the problem as a convex program, where the number of constraints grows exponentially with the number of jobs and machines. We also present a generalization of the formulation for resource allocation problems in acyclic directed graphs. We demonstrate that the problem is NP-complete in the strong sense and present an effective solution procedure. The solution procedure is an implicit enumeration scheme where a surrogate relaxation of the formulation is used to generate upper and lower bounds on the optimal objective function value. Finally, we address the simultaneous scheduling and resource allocation problem, and we present an approximate and iterative solution procedure for the problem.