Optimal control strategies for single-machine family scheduling with sequence-dependent batch setup and controllable processing times

The problem of scheduling jobs on an unreliable single machine is considered in this paper. The scheduling problem is characterized by the following features: jobs are grouped into classes of equivalent jobs; the generalized due date model is adopted for each class of jobs; it is possible to reduce the processing time of a job, at the price of the payment of an extra cost; a costly setup is required when switching between jobs of different classes. The scheduling problem is solved from a perspective which is different from the traditional determination of an optimal sequence of jobs; in fact, the objective of the paper is to determine optimal control strategies (functions of the system state) which allow generating the optimal decisions during the evolution of the system, taking into account the actual system state. In this way, optimal decisions can be promptly taken also in the presence of perturbations which affect the single machine (such as breakdowns and slowdowns). To this aim, a specific optimal control problem is stated and solved in the paper.