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Weekly Updates1. Introduction
Robust Parameter Design (RPD), a methodology introduced by Taguchi (1986), has been widely employed for designing products and processes that are robust against environmental effects. The method consists in determining the levels of a set of controllable factors that make the process least sensitive (most robust) to the variability transmitted by another set of uncontrollable factors, the so-called noise variables. The noise factors cannot be controlled during production or use of a product, but can be controlled in a laboratory environment according to a designed experiment. Environmental temperature and raw material variation are two typical examples of noise factors encountered in manufacturing.
In traditional RPD methods, the settings of the controllable factors are computed off-line (i.e., before production starts) and once fixed the control factor settings are not altered during production (Myers et al., 1992). It is commonly assumed that noise factors will vary during production following a known non-correlated (white noise) series. In many manufacturing systems, however, the noise factors can be measured during production. This implies that it is possible to use the additional noise factor information to continuously update the choice of the controllable factors as production takes place. Furthermore, if the noise factors have strong autocorrelation (environmental temperature and humidity are examples of this case), then the off-line RPD settings may be ineffective in achieving the desired levels of process variability and it may be necessary to recompute them on-line.
The purpose of this paper is to develop a Bayesian RPD methodology that utilizes on-line measurements of the noise factors. A Bayesian modeling approach to RPD naturally accounts for parameter uncertainties, therefore it provides a solution that is robust not only against the online variability of the noise factors but also against the uncertainty in the model parameters. The methods presented account for the uncertainty in the model parameters, so they are equivalent to what is referred to as Cautious Control in the adaptive control literature (Astrom and Wittenmark, 1989), a terminology used recently for quality control by Apley and Kim (2004).
The remainder of the paper is organized as follows. Section 2 contrasts the proposed methods with past efforts for RPD. In Sections 3, 4 and 5 the proposed robust control schemes are derived for single-response processes. Section 6 briefly reviews two existing approaches, Certainty Equivalence (CE) and the Dual Response (DR) approach, which are then compared numerically to the proposed Bayesian RPD control rules in Section 7 using two simulation examples. In Section 8, the Bayesian RPD scheme is extended to multiple response processes and this is illustrated with a plasma etch process example taken from the semiconductor manufacturing literature. Conclusions and directions for further research are given in Section 9.
2. Relation of proposed methods with previous approaches
A Bayesian approach to (off-line) RPD has been proposed recently by Miro-Quesada et al. (2004). Their method differs from all approaches to RPD discussed in the present paper in that it is neither based on the DR model (see Equation (1) below) nor it is based on a quadratic cost objective (Equation (2) below). The proposed control rules in the present paper represent an effort to extend the Bayesian predictive approach in Miro-Quesada et al. (2004) to derive closed-form control rules to be used in on-line RPD. This required a different model and cost function formulation, in addition to on-line estimation methods for the predictive density of the noise factors and the (closed-form) solution to the corresponding optimization problems.
Recently, there has been considerable interest in the statistical process control literature for methods that combine RPD with feedback or feedforward control. Dasgupta and Wu (2006) use response models to find off-line robust settings of control factors while simultaneously tuning a proportional-integral feedback controller, based on a single experiment. Joseph (2003) provides methodology to find optimal off-line control factor settings while simultaneously finding a feedforward law for a single additional control factor (the "controlling variable" or, for our purposes, the on-line controllable factor).
Rather than choosing arbitrarily a single factor as the on-line controllable factor, the proposed approaches presented in later sections treat all controllable factors in an unified way. Factors that are not on-line controllable can be assigned a large control cost (as is customary in control theory) and the optimal robust solution will call for not varying these factors with respect to their initial setting (see example 3 below for an illustration of this feature of the proposed methods). Similarly to Joseph (2003), the optimal feedforward control law is found with respect to the distribution (in our case, the posterior predictive distribution) of the noise factors, but in addition and unlike in Joseph (2003), this is found also with respect to the posterior distribution of the model parameters. All control factors are optimized together in a single step, and closed-form expressions for the optimal (feedforward) control settings are presented.
Pledger (1996) and Jin and Ding (2004) also incorporate feedforward control in RPD by using on-line noise factor observations. The method discussed by Jin and Ding (2004) allows also for uncertainty in the measurements of the noise factors. However, neither of these studies considers uncertainty in the model parameters. Apley and Kim (2004) present a cautious minimum variance (feedback) controller that takes into account parameter uncertainties, but does not consider noise factors and hence does not represent a solution to the RPD problem. In contrast with the methods proposed in the present paper, all of the aforementioned previous approaches discuss only the single-response case.
In this study, two new robust on-line control schemes will be presented. By "robust" we mean insensitive to two different kinds of uncertainty: (i) uncertainty in the model parameters; and (ii) uncertainty in the noise factors. Evidently, different model parameters and noise factor values will require different control factor settings. Hence, in robust design we seek control factor settings that change little, or are as robust as possible, with respect to variations in either model parameters or noise factors.
The first proposed on-line RPD scheme, referred to as the Bayesian Response model robust control law (hereafter, BR law), uses the predictive density of the response and hence considers the parameter uncertainty of the response model only. The second proposed scheme, referred to as the Bayesian Response and Noise model robust control law (BRN control law), uses the joint predictive density of the response and the noise factors and hence considers the parameter uncertainty of both the response and the noise factor models simultaneously.
Although we consider both BR and BRN laws, it will be shown that the BRN approach results in the best performance and hence it is the recommended approach. The performance of the proposed approaches will be compared to existing CE control (Astrom and Wittenmark, 1989) and DR surface methods (Myers et al. 1992; Miro-Quesada and Del Castillo, 2004), which are frequentist approaches. Table 1 lists the proposed and existing approaches that will be considered in this study. Only procedures that are based on the Box-Jones-Myers dual-response model approach to RPD (Equation (1) below) optimized with respect to a quadratic cost objective (Equation (2), below) are considered in this work. This excludes the Miro-Quesada et al. (2004) approach to off-line RPD.
The proposed on-line RPD control schemes are non-adaptive; that is, the posterior distribution of the model parameters (hence the parameter estimates) are obtained from the off-line data and they are not updated with the on-line observations. Instead, the posterior predictive densities of the response and noise factors are updated given the on-line measurements of the noise factors. This avoids the "lack of excitation" and "bursting" problems associated with adaptive controllers (Apley and Kim, 2004).
3. Process assumptions
We consider initially a single-response process model of the form:
[y.sub.t] = [[beta].sub.0] + [beta]'[u.sub.t] + [gamma]'[z.sub.t] + [z'.sub.t][DELTA][u.sub.t] + [[epsilon].sub.t], (1)
where t is the time index or run number, y is the response variable, u = ([u.sub.1], ..., [u.sub.k])' is a vector of k controllable factors, z = ([z.sub.1], ..., [z.sub.r])' is a vector of r noise factors, and [member of] is an error term assumed independently and identically distributed N(0, [[sigma].sup.2]). The model parameters are the scalar [[beta].sub.0], the [[beta].sub.0] vector [beta], the (r x l) vector 7, and the (r x k) matrix [DELTA] that contains the control x noise interaction parameters.
Equation (1) is a first-order response surface model with control x noise interactions. It has been commonly used in the DR surface methodology for the RPD problem (Box and Jones, 1990; Myers et al., 1992) when the quadratic effects of the control factors are not significant.
We assume noise factors can be measured (and predicted) on-line during production at discrete, equidistant points in time. It is assumed that during the production period, [z.sub.t] follows a vector Autoregressive-Moving Average (ARMA) time series model (Reinsel, 1997). Following the standard basic assumption used in RPD, the noise factors are assumed to be controllable during an off-line experiment when model (1) is assumed to hold. We further assume the control objective is to keep the process output around a target value T with as low variability as possible ("Target is best" case) during production. According to model (1) the control factor levels [u.sub.t] are set at the beginning of run t, however, the measurements [y.sub.t] and [z.sub.t], are taken at the end of run t. At the end of each run t, the robust control factor settings [~.u] = [u.sub.t+1] for the next run are computed by minimizing the expected value of the assumed quadratic loss function:
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