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Description
1. Introduction
In recent years, there has been substantial growth in the use of automated in-process sensing and data-capture technologies. This proliferation of technology has resulted in many manufacturers collecting large numbers of measurements on the quality of each manufactured unit. For example, Apley and Shi (1998, 2001) note that optical coordinate measurement machines installed in autobody assembly operations may generate observations on as many as 150 product quality variables from each subassembly. Although the availability of such data has created opportunities for manufacturers to monitor the quality of their products and processes, the large number of product quality variables observed may make it difficult to detect quality changes. Methods of detection (i.e., statistical process monitoring schemes) generally become less effective as the number of variables grows, requiring more time on average to detect quality changes (see, e.g., Crosier (1988), Pignatiello and Runger (1990) and Lowry et al. (1992)). Rather thorough reviews of the methods of multivariate statistical process monitoring are provided by Alt and Smith (1988), Lowry and Montgomery (1995) and Alt (2001).
Another development of note is the recent trend towards analytic models of product quality. Such models have been proposed, for example, by Apley and Shi (1998), Jin and Shi (1999), Kang and Albin (2000), Ding, Ceglarek and Shi (2002) and Zhou, Huang and Shi (2003). In combination with data available from in-process sensing, these models create an opportunity for quality improvement by providing a foundation for methods that identify sources of product-to-product variation. In this paper, we show how such models may be used to develop a monitoring procedure that detects process quality changes more quickly than the standard methods of multivariate process monitoring. Compared to the standard methods, the procedure has the advantage of monitoring the process in a potentially reduced dimensional subspace of the product quality variables. This helps to alleviate the diminished sensitivity that results from an increase in the number of product quality variables.
In Section 2, we give a general model formulation and show that it encompasses several of the recent analytic models of product quality. Based on the generalized least squares estimator of this model's parameters, Section 3 proposes a monitoring procedure for detecting a shift in the mean vector of the product quality variables. We show how to design the proposed procedure to have the desired average run length (and false alarm rate) and quantify the procedure's performance in shift detection. The procedure is illustrated using an example from the recent literature. In Section 4, we conceptually compare the procedure with six other multivariate monitoring methods and reveal the interrelationships among the methods. This comparison enables a unifying view of methods for monitoring multivariate processes. Section 5 discusses some diagnostics for identifying the sources of out-of-control signals from the procedure. Section 6 extends the procedure to models with singularities and illustrates this extension for a multistation autobody assembly process. We conclude in Section 7 with a brief summary.
2. Model and related literature
Let [y.sub.j] be the n x 1 vector of product quality variables observed from the jth manufactured unit (e.g., [y.sub.2] is the vector from the second unit produced). [y.sub.j] has mean vector [[mu].sub.0] and covariance matrix [SIGMA] when the process is in control. The objective is to detect a departure from the in-control mean vector [[mu].sub.0] as quickly as possible. A standard approach is to monitor the observations on [y.sub.j] with a method such as the multivariate cumulative sums of Crosier (1988) or Pignatiello and Runger (1990) or the Multivariate Exponentially Weighted Moving Average (MEWMA) of Lowry et al. (1992). We discuss the MEWMA because it has been shown to perform well and appears to have been studied more thoroughly than most multivariate monitoring methods. To define the MEWMA, form the vector of EWMAs:
[z.sub.j] [equivalent to] [theta]([y.sub.j] - [[mu].sub.0]) + (1 - [theta])[z.sub.j-1], j = 1, 2,..., (1)
where [z.sub.0] is a vector of all zeros and the smoothing constant [theta] is a scalar such that < [theta] [less than or equal to] 1. The MEWMA signals that the mean vector has shifted when (and only when) the statistic:
[[chi square].sub.j] [equivalent to] [z'.sub.j][[SIGMA].sub.[z.sub.j].sup.-1][z.sub.j], (2)
exceeds a control limit h, where:
[[SIGMA].sub.z.sub.j] = {[theta][1 - (1 - [theta])[.sup.2j]]/(2 - [theta])}[SIGMA], (3)
is the covariance matrix of [z.sub.j]. Instead of [[SIGMA].sub.z.sub.j] in Equation (3), it is typically assumed in the literature that the asymptotic (as j [right arrow] [infinity]) covariance matrix:
{[theta]/(2 - [theta])}[SIGMA], (4)
is used to calculate the MEWMA statistic [[chi square].sub.j] in Equation (2). We refer to the MEWMA based on the exact covariance matrix in Equation (3) as well as that based on Equation (4) as direct MEWMA because both of these methods apply MEWMA directly to [y.sub.j].
Several recent articles (e.g., Apley and Shi (1998), Jin and Shi (1999), Kang and Albin (2000) and Ding, Ceglarek and Shi (2002)) propose linear models for [y.sub.j]. For the purposes of the current article, each of these models may be written in the general form:
[y.sub.j] = X[[beta].sub.j] + [[epsilon].sub.j], j = 1, 2,..., (5)
where X is an n x p matrix of known constants, [[beta].sub.j] is a p x 1 vector of unknown parameters, and [[epsilon].sub.j] ~ N(0, [SIGMA]) is an n x 1 vector of random variables, where [SIGMA] is a positive definite matrix. It is assumed that [[epsilon].sub.1][[epsilon].sub.2],..., [[epsilon].sub.j],... are mutually independent.
Under the model (5), the product quality vector [y.sub.j] is multivariate normal with mean vector [mu] = X[[beta].sub.j] and covariance matrix [SIGMA]. The process is considered to be in control if the parameter vector [[beta].sub.j] = [[beta].sub.0], in which case [y.sub.j] has mean vector [[mu].sub.0] = X[[beta].sub.0]. The process is out of control if [[beta].sub.j] [not equal to] [[beta].sub.0]. The objective is to detect an out-of-control process as quickly as possible. Since [[beta].sub.j] has p elements, each of which may shift from its in-control value, there are p possible faults. Any combination of these faults may be present simultaneously (i.e., zero, one or multiple elements of [[beta].sub.j] may be shifted from their in-control values); thus, under the model (5), the mean vector of [y.sub.j] may shift along any vector in the subspace spanned by columns of X. This allows for a multitude of potential shift directions or special causes. For the time being, assume that n [greater than or equal to] p and X has full column rank. In Section 6, we consider the case where the columns of X are linearly dependent.
Several articles on monitoring linear profiles (e.g., Kang and Albin (2000), Kim et al. (2003) and Mahmoud and Woodall (2004)) consider the special case of a simple linear regression model in which X in model (5) has p = 2 columns, one of which is all ones, and the error covariance matrix [SIGMA] equals a constant times an identity matrix. The premise of linear profile monitoring is that the state of some processes is characterized by a linear function. As an example, Kang and Albin (2000) discuss a Mass Flow Controller (MFC) used in the etch step of semiconductor production. If the MFC is in control, the pressure in the chamber is a linear function of the set point for the flow. A fault in the MFC may result in a departure from this linear function. The aforementioned articles on linear profile monitoring, similar to the current study, focus on detecting when the parameter vector [[beta].sub.j] shifts from its in-control value [[beta].sub.0].
Another model of note is the fixture-fault model developed by Apley and Shi (1998), which may be formulated in the form of the general linear model (5). The Apley--Shi model is derived analytically from the geometry of the panel and fixture layout in assembly systems. More recently, several articles have proposed state space models to describe the propagation of dimensional variation in multistation manufacturing processes. Such models have been developed for assembly and... |
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