The effect of workload constraints in mathematical programming models for production planning

Linear and mixed integer programming models for production planning incorporate a model of the manufacturing system that is necessarily deterministic. Although these eterministic models are the current-state-of-art, it should be recognized that they are used in an environment that is inherently stochastic. This fact should be kept in mind, both when making modeling choices and when setting the parameters of the model. In this paper we study the relation between workload constraints that reflect the finite capacity of the manufacturing system, and the use of planned lead times. It is a common practice in rolling schedule based production planning to limit the periodic output to the average production rate. If lead times are not modeled explicitly, this also implies a restricition on the periodic releases to the average production rate. We demonstrate that this common practice results in inefficient use of the production capacity and show that the use of planned lead times leads to a better trade-off between efficiency and reliability. We analyze a stylized model of a manufacturing system with a single exponential server and two queues in series: an admission queue and a work-in-progress (WIP) queue. The admission queue represents the pool of unreleased orders that is virtually present in the state variables of the planning model. Periodically, jobs from the admission queue are released to the WIP queue such that the number of jobs in WIP and in service does not exceed the workload constraint. We present a simple formula for the maximum utilization rate of such a system, characterize the stationary queue-length distribution by its generating function, and give the distribution of the sojourn time of a job. We use the results to compare various settings of the workload constraint and the planned lead time.