Optimal strategies for real-time determination of the next job's class in a single machine with setup times and controllable processing times

In this paper, a single machine allowed to process jobs belonging to different classes is considered, and the problem of determining the class of the next job to be processed is dealt with. The single machine is characterized by the following features: (i) two jobs belonging to the same class are considered as completely equivalent; (ii) a sequence of due-dates is specified for each class of jobs, and the serviced jobs, for each class, are assigned to the due-dates according to the EDD rule (generalized due-dates framework); the service time of any job of a given class has to be selected within an interval of possible values (controllable processing times); (iv) a setup time is required by the machine when switching between different classes. A deviation of the actual service time from the nominal value determines a cost which has a linear dependence on such a deviation. The optimal (closed-loop) strategies are determined through a four-steps procedure. Such strategies provide the optimal values of the control variables (next job's class and processing time) as functions of the system state.