Stochastic Kriging-assisted Controlled Random Search for Simulation Optimization and Its Application to Critical Dimension Measurement

Abstract

We consider an optimization problem with continuous variables, where the objective function must be evaluated via simulation. Our interests specifically focus on the situation where limited simulation observations (e.g., one or two) at each solution vector can be obtained, even though each observation includes random noise. To solve the problem, one may employ existing surrogate-assisted algorithms that typically update their surrogate models whenever a new observation is obtained. However, this updating process can face challenges such as out-of-memory issues, high computational costs, or imbalanced data due to excessive exploitation. In this study, we introduce a new algorithm called S tochastic K riging-assisted C ontrolled R andom S earch ( SKCRS ). At each iteration, it employs a controlled random search to sample multiple vectors and uses a stochastic kriging model to select the most promising candidate solution vector from those sampled vectors. If the model fails to provide useful information to identify a suitable solution vector, the algorithm defaults to functioning as the original controlled random search without using the model. We tested our algorithm and compared it with existing algorithms using numerical functions and a case study in semiconductor manufacturing. The results demonstrate that our algorithm empirically outperforms existing algorithms regarding computational efficiency and estimation accuracy. Note to Practitioners-One practical application of the target problem is estimating parameters of a simulation model. In this context, we provide a case study in semiconductor manufacturing. Critical dimension (CD) refers to the geometric length of a grating structure on a wafer, which can include measurements such as width or height. Accurately and efficiently evaluating CD values across different wafers is crucial for rapidly detecting and treating defects. We applied our algorithm to measure critical dimensions using optical techniques known as the optical critical dimension (OCD) measurement. Our approach focuses on solving a nonlinear parameter estimation problem to match a measured spectrum with a spectrum generated by a simulation model. However, simulation models often come with high computational costs, and thus it is desirable to reduce the number of simulation runs while still achieving accurate estimates. In our case study, we estimated CD values of a 2D high-aspect-ratio pattern using the OCD technique. The results demonstrate that the SKCRS algorithm outperforms existing heuristic and simulation optimization algorithms. Practitioners can directly apply our algorithm to parameter estimation problems in various applications, including the OCD measurement.