Analysis of fixed production cycles

In the analysis of the fixed production cycles we will study the situation with one machine, on which M different types of products will be manufactured. The machine is assumed to be perfect, that is to have no breakdowns, and the set-up cost and the set-up time are independent of both the previous type which is produced and the next type that will be produced. The set-up time will be deterministic. Due the production rule it is known exactly when a particular type will be produced. This knowledge can be used for determining exact due dates. Therefore the arriving orders will be scheduled according to the first-come-first-served rule (FCFS) and the due date will be based on this schedule. Since this schedule will not be changed, there will be no due date deviation and therefore no holding costs or penalty costs. The analysis of the production rule will be quite difficult if the last order at the end of a production interval is always finished, the so-called non-pre-emptive service discipline. Because of these difficulties we will assume that the work on the last order continues in the next production interval of the same type, the so-called pre-emptive service discipline. 

A lot of research has been done on problems with cyclic production rules. The difficulty of the analysis depends on the service discipline. Problems where service continues until a queue is empty, the so-called exhaustive service, have been treated by Eisenberg (1970), Swartz (1980), Watson (1984) and others. These authors have found general results on average delivery times and on the average amount of woik in process. Problems with other service disciplines, for instance where only one customer of each type is served per cycle (ordinary service), seem to present 

These two sets S and C are not the only important characteristics of the fixed cycle. The service discipline is also very important. During the production interval of a type we can only woik on orders of that type. The orders of that type will be produced according to the FCFS-rule. Because of the exact due dates this rule is the same rule as the earliest-due-date rule (FDD). If there are no (more) orders of a type we have to wait for new orders of this type or for the end of the production interval. This way of service, in which we stop working on orders from a certain type at the end of a production interval, is usually called gating service. We will consider two different ways of gating service. 

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