Single machine scheduling with fixed lot-sizes and variable processing times

TLDR: The optimisation problem with quadratic cost function and nonlinear constraints is stated and formalised as a multi-stage optimal control problem; the optimal closed-loop control laws at each stage are obtained.

Abstract: The problem of optimally defining and controlling the behaviour of a single machine processing a certain number of jobs and modelled as a discrete event dynamic system is addressed. The number of jobs, their sizes, and their service sequence are fixed, whereas their timing is the matter of the optimisation problem. The objective function to be optimised is a weighted sum of the inventory cost of the quadratic deviations from the due-dates of jobs and their completion times, and the quadratic deviations between the unitary processing times of jobs and those specified by the regular system functioning. An optimisation problem with quadratic cost function and nonlinear constraints is stated and formalised as a multi-stage optimal control problem. The control problem is solved by a procedure making use of dynamic programming techniques; the optimal closed-loop control laws at each stage are thus obtained.