Dynamic Sampling for Risk Minimization in Semiconductor Manufacturing

To control the quality of their processes, manufacturers perform measurement operations on their products. In semiconductor manufacturing, measurement capacity is limited because metrology tools are expensive, thus only a limited number of products can be measured. Selecting the set of lots to control to minimize risk is called sampling. In this paper, the objective is to minimize the number of wafers at risk, i.e. the number of wafers processed on a machine between two lots that are controlled. The problem can be modeled as the maximization of a submodular set function subject to various capacity constraints. The resulting problems, which are NP-hard, can be modeled as integer linear programs. Greedy heuristics and an exchange procedure are also presented. Computational experiments on industrial and randomly generated instances show that the integer linear programs solve the problems optimally, and that the heuristics have sufficiently good approximation ratios for industrial implementation.