A stochastic model for the optimal batch size in multi-step operations with process and product variability

Virtually all manufacturing processes are subject to variability, an inherent characteristic of most production processes. No two parts can ever be exactly the same in terms of their dimensions. For machining processes such as drilling, milling, and lathing, overall variability is caused in part by machine tools, tooling, fixtures and workpiece material. Since variability, which can be accumulated from tolerance stacking, can result in defective parts the number of parts produced in a batch is limited. When there are too many parts in a batch, the likelihood of producing all acceptable parts in a batch decreases due to the increased tolerances. On the other hand, too small a batch size incurs an increase of manufacturing costs due to frequent setups and tool replacements, whereas the likelihood of acceptable parts increases. To address this challenge, we present a stochastic model for determining the optimal batch size where we consider part-to-part variation in terms of tool wear, which tends to be proportional to batch size. In this paper, a mathematical model is constructed based on the assumption that the process used for producing preceding parts affects the state of subsequent parts in a probabilistic manner.