# Bayesian methods for robustness in process optimization

Abstract (Summary)

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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
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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
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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.
Bibliographical Information:

Advisor:

School:Pennsylvania State University

School Location:USA - Pennsylvania

Source Type:Master's Thesis

Keywords:

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