# Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

# Statistical Process Control in Manufacturing

• O. Arda Vanli
• Enrique Del Castillo
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_258-1

## Abstract

Statistical process control has been successfully utilized for process monitoring and variation reduction in manufacturing applications. This entry aims to review some of the important monitoring methods. Topics discussed include: Shewhart’s model, $$\bar{X}$$ and R control charts, EWMA and CUSUM charts for monitoring small process shifts, process monitoring for autocorrelated data, and integration of statistical and engineering (or automatic) control techniques. The goal is to provide readers from control theory, mechanical engineering, and electrical engineering an expository overview of the key topics in statistical process control.

## Keywords

Shewhart control chart EWMA CUSUM Feedback control Time-series analysis

## Introduction

Variation control is an important goal in manufacturing. The main set of tools for variation control used in discrete-part manufacturing industries up to the 1960s was developed by W. Shewhart in the 1920s and is known today as statistical process control, or SPC (Shewhart 1939). Shewhart’s SPC model assumes that the process varies about a fixed mean and that consecutive observations from a process are independent, as follows:
$$Y _{t} =\mu _{0} +\epsilon _{t}$$
(1)
in which μ 0 is the in-control process mean and ε t is iid (independent identically distributed) white noise $$\epsilon \stackrel{\mathrm{iid}}{\sim }N(0,\sigma ^{2})$$. The Shewhart model can be used in distinguishing assignable cause variation from common cause variation. For example, a mean change from μ 0 to $$\mu _{1} =\mu _{0}+\delta$$ (where δ is the unknown magnitude of change) or a variance increase from σ 0 2 to σ 1 2 at an unknown point in time can be detected as assignable causes.

The objective of this entry is to highlight some of the important references in the SPC literature and to discuss similarities and joint applications SPC has with automatic process control. The literature on statistical process control and applications to engineering problems is vast; therefore, no effort is made for an exhaustive review. More complete reviews of the literature on statistical process control and adjustment methods can be found in texts including Montgomery (2013), Ryan (2011), and Del Castillo (2002).

## Shewhart Control Charts

Shewhart’s $$\bar{X}$$ and R control charts are used to distinguish between common cause and assignable causes of variation (Shewhart 1939) by monitoring, respectively, the process mean and process variance. “Common cause” variation is the natural variability of the process due to uncontrollable factors in the environment that is not avoidable without substantial changes to the process. “Assignable cause” variation is due to unwanted disturbances or upsets to the process that can be detected and removed to produce acceptable quality products. When only common cause variation exists, the process is said to be operating “in statistical control.” Assignable causes of variation include operator changes, machine calibration errors or raw material variation between suppliers.

Another concept that is closely related to the Shewhart’s model is process capability. Process capability indices are used to assess whether the process is operating in a satisfactory manner with respect to the engineering specifications. It is crucial to attain a stable process (eliminating all problematic causes) before undertaking such a capability analysis because only when the samples come from a stable probability distribution can the future behavior of the process be predicted “within probability limits determined by the common cause system” (Box and Kramer 1992).

Figure 1 illustrates the two main phases, referred to as Phase I and Phase II, in constructing Shewhart charts (Sullivan 2002), using semiconductor lithography process data given in Montgomery (2013). It is desired to establish a statistical control of the width of the resist using $$\bar{X}$$ and R charts. Twenty-five preliminary subgroups, each of size five wafers, were taken at one-hour intervals and the resist width is measured. In Phase I, “retrospective analysis,” the historical data from the process is analyzed to bring an initially out-of-control process into statistical control. Subgroups y 1, , y n of size n are taken, and subgroup average $$\bar{y}$$ is used to monitor process mean μ 0, and the subgroup range is used to monitor standard deviation of the process mean $$\sigma _{\bar{Y }} =\sigma /\sqrt{n}$$. The upper and lower control limits are found for the $$\bar{X}$$ chart as $$\{UCL,LCL\} =\mu _{0} \pm L\sigma _{\bar{Y }}$$ where L is a constant representing the width of the control limits. Commonly chosen three-sigma limits (i.e., L = 3) provide a probability p = 0. 0027 that a single point falls outside the limits when process is in control (“false alarm probability”). Points that fall outside the control limits are investigated, and if an assignable cause was identified, then this point is omitted and control limits are recalculated. This is repeated until no further points plot outside the limits. In Phase II these charts are used to detect shifts in the process mean and variability.

The $$\bar{X}$$ and R charts from Phase I data in Fig. 1a indicate statistical control; hence the computed control limits can be used for Phase II monitoring. Twenty additional subgroups (also of size 5) are taken in Phase II while the control charts are in use. The Phase II charts shown in Fig. 1b indicate that process variability is stable but the process mean has shifted at subgroup 18. The general trend in the $$\bar{X}$$ chart indicates that process mean probably has shifted earlier around subgroup 13.

## EWMA, CUSUM, and Changepoint Estimation

Shewhart charts can detect large magnitude process upsets reasonably well; however, they are relatively slow to detect small shifts. In order to reduce the reaction time for smaller shifts, a set of “runs” rules (e.g., two out of three runs beyond 2σ limits or four out of five runs beyond 1σ limits) has been proposed Western Electric (1956). A more systematic method is to accumulate information over successive observations using CUSUM and EWMA statistics rather than basing the detection on a single sample. In the cumulative sum (CUSUM) chart, a running total $$\sum _{i=1}^{t}(\bar{Y }_{t} -\mu _{0})$$ is plotted against subgroup number t, and a shift from the in-control mean μ 0 is signaled by an upward or downward linear trend in the plot. A two-sided CUSUM is defined as Woodall and Adams (1993):
$$S_{t}^{\pm } =\max \{ \pm Z_{ t} - k + S_{t-1}^{\pm },0\}\text{ for }t = 1,2,\ldots$$
(2)
where S t + and S t are the one-sided upper and lower cusums, respectively, $$Z_{t} = (\bar{Y _{t}} -\mu _{0})/\sigma _{\bar{Y }}$$ is the standardized subgroup average, $$k = \vert \mu _{1} -\mu _{0}\vert /(2\sigma )$$ is the reference value, and μ 1 is the level of process mean to be detected. An out-of-control signal is given at the first t for which S t > h where h is a suitably chosen threshold, usually selected based on the desired average number of samples to signal an alarm, also called the average run length (ARL). The recommended value for the threshold h is 4 or 5 (corresponding to four or five times the process standard deviation σ), and the value for the reference k is almost always taken as 0.5 (corresponding to shift size $$\vert \mu _{1} -\mu _{0}\vert=\sigma$$) (Montgomery 2013).
Another chart that accumulates deviations over several samples is the exponentially weighted moving average (EWMA) which is based on the statistic (Lucas and Saccucci 1990)
$$Z_{t} =\lambda \bar{ Y }_{t} + (1-\lambda )Z_{t-1}$$
(3)
where 0 < λ < 1 is a smoothing constant. Smaller λ provides large smoothing (similar to a large subgroup size n in the Shewhart charts). The starting value is the in-control mean Z 0 = μ 0. It can be shown that Z t is a weighted average of all previous sample means, where the weights decrease geometrically with the age of the subgroup mean. The EWMA statistic is plotted against the control limits $$\mu _{0} \pm L\sigma _{\bar{Y }}\sqrt{(\lambda /(2-\lambda ))[1 - (1-\lambda )^{2t } ]}$$. Shewhart charts that are effective for large shifts are more useful for Phase I, and CUSUM or EWMA charts that are effective for small shifts are more appropriate for Phase II.
We illustrate in Fig. 2 how to monitor with CUSUM and EWMA charts with the lithography data. The in-control process mean and standard deviation μ 0 and σ are found from the Phase I data. CUSUM upper and lower statistics S t ± computed with Phase II data are plotted in Fig. 2a (reference value k = 0. 5 and threshold h = 4 are used.). The upper cusum statistic S t + crosses the upper control limit indicating an upward shift at subgroup 15. The EWMA statistic applied with λ = 0. 2 on Phase II data, shown Fig. 2b, crosses the upper control limit at subgroup 16. Both charts have improved the reaction times of the Shewhart chart.
When a control chart signals an assignable cause, it does not indicate when the process change actually occurred. Estimating the instant of the change, or changepoint estimation, is especially useful in Phase I analysis where little is known about the process, and it is important to identify and remove the out-of-control samples from consideration (Hawkins et al. 2003; Basseville and Nikiforov 1993; Pignatiello and Samuel 2001). The process is modeled as
$$\displaystyle\begin{array}{rcl} Y _{i}& \sim & N(\mu _{1},\sigma ^{2})\text{ for }i = 1,2,\ldots,\tau \\ Y _{i}& \sim & N(\mu _{2},\sigma ^{2})\text{ for }i =\tau +1,\ldots,n\end{array}$$
(4)
where τ is the unknown changepoint, at which the in-control mean μ 1 is assumed to shift to a new value μ 2 assuming μ 1, σ are known but μ 2 is unknown. A generalized likelihood ratio (GLR) test statistic $$\Lambda _{t} =\sum _{ i=1}^{t}\log f_{2}(y_{i})/f_{1}(y_{i})$$ is used to test the hypothesis of a changepoint against the null hypothesis that there is no change. Assuming normality $$f(y) = 1/\sqrt{2\pi \sigma }\exp [-(y-\mu )^{2}/(2\sigma ^{2})]$$ is the probability density function of the quality characteristic. The changepoint model is equivalent to the CUSUM chart when all parameters μ 1, μ 2 and σ are known a priori. For the lithography Phase II data in Fig. 1b, it can be shown that the changepoint can be estimated as subgroup 13.

## SPC on Controlled and Autocorrelated Processes

It is well known that automatic control performance relies heavily on the accuracy of the process models. An active field of research in recent years is the monitoring of controlled systems using SPC charts (Box and Kramer 1992) in order to reduce the effect of model accuracy. Shewhart charts can be used to monitor the output of a feedback-controlled process; however, as the controller effectively corrects the shift, only a short window of opportunity is provided to detect the shift (Vander Wiel et al. 1992). Tsung and Tsui (2008) showed that monitoring the control actions gives better run-length performance than monitoring the output for small- and medium-size shifts, and monitoring the output gives better performance for large shifts. In monitoring controlled processes, measurements taken at short intervals with positive autocorrelation usually inflate the rate of false alarms (Harris and Ross 1991). Widening the control limits and monitoring the residuals of a time-series model fitted to the observations are some of the strategies employed to reduce the number of false alarms (Alwan and Roberts 1988).

To illustrate the effects of autocorrelation, we consider simulated data from an autoregressive moving average ARMA(1,1) time-series disturbance process $$D_{t} = 0.8D_{t-1} +\epsilon _{t} - 0.3\epsilon _{t-1}$$ (Box et al. 1994) defined with the white noise process $$\epsilon _{t}\stackrel{\mathrm{iid}}{\sim }N(0,1^{2})$$ (with in-control mean μ 0 = 0 and variance σ D 2 = 1. 694). Figure 3a shows a realization of the process monitored with a Shewhart chart (control limits at μ 0 ± 3σ D ). Due to autocorrelation, false alarms are signaled at samples 81–83. Figure 3b shows the control chart monitoring of the residuals of an ARMA(1,1) model. Residuals (standard normal with mean 0 and variance 1) are not autocorrelated, so the Shewhart chart for residuals does not signal any false alarms.
We illustrate monitoring of controlled processes with simulated data from a transfer function model $$Y _{t} = 2X_{t-1} + D_{t}$$ where X t are the adjustments made on the process. A proportional integral control rule $$X_{t} = -0.1Y _{t} - 0.15\sum _{i=1}^{t}Y _{i}$$ is employed, and the disturbance D t is assumed to follow the ARMA model considered earlier. As an assignable cause, the disturbance mean has shifted at sample 100 by a magnitude of 3σ D . Figure 4 shows the Shewhart charts monitoring the output Y t and the input X t . The effect of assignable cause (at sample 100) on the output is quickly removed by the controller; however, a sustained shift remains in the control input. The control chart for the input Fig. 4b signals the first alarm at sample 101 (much quicker) than the control chart for the output Fig. 4a which signals at sample 110.

## Summary and Future Directions

In this entry we reviewed some of the commonly used statistical process monitoring methods for manufacturing systems. Due to space limitations, only several important topics including Phase I and Phase II monitoring with Shewhart, EWMA, and CUSUM charts were discussed, highlighting main applications with numerical examples. Other current research areas include multivariate methods for monitoring processes with multiple quality characteristics taking advantage of relationships among them (Lowry and Montgomery 1992), profile monitoring for processes that generate functional data (Woodall et al. 2004), multistage monitoring for processes with multiple processing steps and variation transmission (Tsung et al. 2008), and run-to-run EWMA control for semiconductor manufacturing processes that require handling of multiple types of products, operators, and machine tools (Butler and Stefani 1994).

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