Optimal Control of Production Processes with Variable Execution Times

Abstract

The paper deals with the optimal control of production processes characterized by the possibility of performing operations (relevant to the processing of a set of jobs) with variable execution times. A production process relevant to a single machine is addressed first. An optimization problem with a quite general cost function is considered, and some properties of optimal solutions are derived. Then, a particular version of the problem is analyzed, in which the cost function is the weighted sum of the quadratic earliness and tardiness of jobs and of quadratic deviations between pre-defined nominal unitary processing times and the actual ones. The decision variables of the problem are the possible idle times inserted before job executions and the processing times of jobs. This single machine problem is stated as an optimal control problem and a closed-loop solution is derived. Then, a second production process is considered, in which multiple machines serve jobs in parallel, again with variable processing times and with different processing costs. With reference to this second production scheme, a significant decision problem refers to the splitting of jobs over the different machines. Then, on the basis of a sensitivity analysis of the single machine problem solution, some conditions to verify the optimality of a pre-defined splitting are derived. An on-line splitting scheme using such conditions is finally presented.