Robust parameter design (RPD) is a quality improvement method to mitigate the effect of input noise such as random field deviations (random function type deviations) from nominal product shape profiles on system output via adjustment of control factors. In this thesis, we develop methodologies for three RPD problems in deterministic computer experiments, i.e., multi-functional target RPD, RPD with nonstandard quality loss functions, and RPD with multi-fidelity simulations. A review on RPD with computer experiments, research motivations, objectives, and contributions of this thesis are given in Chapter 1.

In Chapter 2, we consider RPD with multiple functional outputs and multiple target functions based on a time-consuming nonlinear simulator, which is a challenging problem rarely studied in the literature. The Joseph-Wu formulation of multi-target RPD as an optimization problem is extended to accommodate multiple functional outputs, and a Gaussian process (GP) model is used to estimate the optimal control factor setting based on the extended formulation. Due to the large amount of data, a computationally efficient approach to GP model fitting and expected loss function estimation is needed. In this chapter, a multi-functional output GP model is employed. The separable prior mean and covariance functions of the model and the Cartesian product structure of the data are exploited to derive computationally efficient formulas for the posterior means of the expected loss criteria used as objective functions for optimizing the signal and control factors. The separability of the model and the Cartesian data structure are also exploited to develop a fast Monte Carlo procedure for building credible intervals for the expected loss criteria. An example on robust design of a coronary stent used in treatment of narrowed arteries is given to demonstrate the proposed approach. Our approach allows the optimal control factor setting and optimal signal factor versus target function parameter profile to be obtained with low computational cost.

In Chapter 3, we propose a shifted log loss GP model for expected quality loss (EQL) prediction that is applicable to RPD based on any continuous loss function. RPD aims at reducing the effect of noise variation, such as random field deviations (random function type deviations) from nominal product shape profiles, on quality through achieving a small EQL. In RPD with time consuming computer simulations, GP models are used to predict the EQL. Three straightforward models for predicting the EQL include a GP model for the simulator output, a GP model for the quality loss, and a lognormal process model for the quality loss (the log quality loss is modeled as a GP). Each of these models has some drawbacks, as discussed in this chapter. We propose the shifted log loss GP model, which includes the lognormal process model for the quality loss and the GP model for the quality loss as special cases when the shift parameter varies from zero to infinity. The proposed model overcomes some of the limitations of the three existing models. It has a simple and accurate approximation for the posterior EQL distribution, and it gives accurate and precise predictions of the EQL. We illustrate the superior performance of the proposed model over the three existing models with a toy example and an RPD problem involving a steel beam.

In Chapter 4, we extend the proposed shifted log loss GP model in Chapter 3 to RPD with multifidelity simulations. In problems with time consuming computer simulations at multiple levels of fidelity, autoregressive GP emulators are used to improve computational efficiency. However, approaches for predicting the expected high fidelity (HF) quality loss (also called HF EQL) that take into account interpolation uncertainty are lacking for RPD with multifidelity simulations. Modelling the low fidelity (LF) and HF quality losses, and their log transforms, with an autoregressive GP emulator can yield similar shortcomings as a single fidelity GP model for the quality loss and a single fidelity lognormal process model for the quality loss respectively. To overcome some of the shortcomings of these straightforward modeling approaches, we model shifted log transforms of the HF and LF quality losses with the autoregressive GP emulator. The model has a simple and accurate approximation to its posterior HF EQL distribution, which eases computations of its predictions of the HF EQL. We illustrate the excellent performance of the proposed model for predicting the HF EQL compared to the alternative straightforward models described above with an example on RPD of a steel stepped shaft based on bi-fidelity simulations.