The cyclic compact open-shop scheduling problem

We consider a variation of the classical nonpreemptive open-shop scheduling problem. For the normal open shop, finding whether a schedule exists in T time units is NP-complete [2]. The main difference with the classical model is that we will consider so-called periodical or cyclic schedules; this means that the schedule obtained is repeated periodically. If the length of a period is k time units, then a job which is on a processor during the first p time units of a period and the last 4 time units of a period will be considered as being processed without preemptions on that processor during q +p consecutive time units. Such scheduling problems are sometimes called cylindrical (instead of cyclic); this name stems from the fact that we take the processor schedules (and the job schedules) for one period and make a cylinder by gluing together the t = 0 and t = k axes. Cyclic problems occur more and more frequently in automated production systems; some types of cyclic problems are discussed in [3]. Another difference from the classical models is that we will consider compact schedules only. This additional requirement is sometimes encountered in production systems. A job when started must never wait during its processing (some operations need, for instance, to be performed at a fixed temperature). In a similar way, when a processor starts working, there should be no interruption (because starting the processor may be a long or difficult task or it may consume much energy).