Generalising optimal mean setting for any number and combination of serial and parallel manufacturing operations

Consider a production system where products are continuously manufactured and their features inspected for conformance within specification limits. If features are produced above or below the specification limits, they are either subject to rework or the product scrapped. Optimal mean setting may be applied to adjust the manufacturing means to influence the amount of rework or scrap produced, maximising profit. Within the production system, manufacturing and then inspecting each feature in turn is termed serial production, whereas manufacturing multiple features before inspection is termed parallel production. This paper develops a generalised expression to optimise the mean values of each feature (optimal mean setting), where n number of features are produced in any combination of serial and parallel operations. Previous literature is restricted to considering two features in parallel. The production of multiple features in combinations of serial and parallel operations is not fully considered. The new generalised expression is validated by showing it is consistent with specific cases from past literature. The approach is then applied to a practical example of a gearbox shaft, considering the expected profit of eight possible manufacturing sequences, as well as the deviation of the manufactured means relative to the design intent. The generalised expression is widely applicable in component design and manufacturing planning where the process capability index (Cpk) of features is below one. The generalised expression also forms the basis for trade-offs between profitability and minimising deviations of manufactured means, which is the subject of further development.