Bayesian methods for robustness in process optimization

by 1976- Rajagopal, Ramkumar

iii The core objective of the research presented in this dissertation is to develop new methodologies based on Bayesian inference procedures for some problems occurring in manufacturing processes. The use of Bayesian methods of inference provides a natural framework to obtain solutions that are robust to various uncertainties in such processes as well as to assumptions made during the analysis. Specifically, the methods presented here aim to provide robust solutions to problems in process optimization, tolerance control and multiple criteria decision making. Traditional approaches for process optimization start by fitting a model and then optimizing the model to obtain optimal operating settings. These methods do not account for any uncertainty in the parameters of the model or in the form of the model. Bayesian approaches have been proposed recently to account for the uncertainty on the parameters of the model, assuming the model form is known. This dissertation presents a Bayesian predictive approach to process optimization that accounts for the uncertainty in the model form, also accounting for the uncertainty of the parameters given each potential model. Both single response and multiple response systems are considered. The objective here is to optimize the model-averaged posterior predictive density (MAP) of the response where the weighted average is taken using the model posterior probabilities as weights. The MAP is thus used to maximize the posterior probability of obtaining the responses within given specification limits. The Bayesian approach to model-robust process optimization is then extended to the iv case where noise factors and non-normal error terms are present. Traditionally, in process optimization, methods such as the Dual Response Surface methodology are used in the presence of noise factors, and methods such as Robust Regression are used when the error terms are not normally distributed. In this dissertation, the idea of model-robustness using the Bayesian posterior predictive density is extended to cases where there is uncertainty due to noise factors and due to non-normal error terms. The tolerance control problem is the inverse of the process optimization problem. Here, the objective is to find the specification or tolerance limits on the responses. We propose a Bayesian method to set tolerance or specification limits on one or more responses and obtain optimal values for a set of controllable factors. The dependence between the controllable factors and the responses is assumed to be captured by a regression model fit from experimental data, where the data is assumed to be available. The proposed method finds the optimal setting of the control factors (parameter design) and the corresponding specification limits for the responses (tolerance control) in order to achieve a desired posterior probability of conformance of the responses to their specifications. In addition to process optimization and tolerance control, a new Bayesian method is presented for the multiple criteria decision making problem (MCDM). The usual approach to solving the MCDM problem is by either using a weighted objective function based on each individual objective or by optimizing one objective while setting constraints on the others. These approaches try to find a point on the efficient frontier or the Pareto optimal set based on the preferences of the decision maker. Here, a new algorithm is proposed to solve certain MCDM problems based on a Bayesian methodology. At a first stage, it is assumed that there are process responses that are functions of certain controllable factors or v regressors. At a second stage, the responses in turn influence the utility function of one or more decision makers. Both stages are modelled with Bayesian regression techniques. The methodology is applied to engineering design problems, providing a rigorous formulation to popular “Design for Six Sigma” approaches. Although the research focusses on applications in process optimization, tolerance control and MCDM, some of the results can be directly applied to other applications such as process control. These and other ideas for further research are described in the concluding chapter of this dissertation.