Optimal Placement of Programmable Tooling Machines Considering Hierarchical Structure via Sparse Learning for Multistage Assembly Processes

End-of-line product dimensional quality assurance is crucial in multistage assembly processes (MAPs). Active control strategies involve the deployment of controllable, programmable tooling machines (PTs) to adjust part positions for in-process dimensional error compensation. In MAPs, multiple PTs may be deployed within a single stage or across various stages. To enable the normal operation of deployed PTs within a specific stage, an associated supporting platform (SP) is necessary. Consequently, the lower PT-level and the upper platform-level define a two-level hierarchical structure in MAPs. This paper aims to propose an optimal placement strategy of PTs, focusing on the final dimensional quality of the product and the total cost related to the number of accommodated PTs and required SPs. Based on the stream-of-variation (SOV) model of MAPs, we develop a novel sparse learning framework along with the corresponding parameter estimation algorithm to achieve the optimal placement. The case study demonstrates the effectiveness of our proposed method for the optimal placement of PTs in reducing dimensional variation in MAPs.