47 Analytical Results on the Evaluation of the Lead Time and the Fill Rate of Unreliable Production Systems (PSAM-0419)

A fluid modeling approach is adopted in this paper to evaluate the performance indices such as the time to produce a given order (called the lead time) and the probability of meeting a customer order on time (called the fill rate) for a failure-prone production system. Lead time and fill rate estimations are very important in the context of Advanced Resources Planning approach, which is an ideal high-level tuning and planning tool to be used in various planning environments like MRP, ERP, JIT, Load-Oriented Planning, Theory of Constraints, POLCA systems, and Finite Scheduling. We derive closed forms expressions of the lead time and the fill rate for constant and variable demands. In the case of a constant demand, the derivation of the lead time average and variance uses a well known result on the first passage time of a Wiener process to a constant boundary, and the derivation of the fill rate is based upon Shuster's derivation. In the case of a variable demand, the problem is solved by considering the first-passage time of Markov processes to moving boundaries as a first-exit time for a vector whose components include the process and the demand (i.e. the boundary). Thus, the theory of first-exit times for multidimensional processes is used to obtain differential equations for the moments and density of the first-passage time of the process to the boundary. Finally, we investigate the use of the theory of multidimensional diffusion processes to derive closed-forms expressions of the fill rate. An example of a cigarettes production line is used to illustrate the newly obtained analytical results.