A multivariate double EWMA process adjustment scheme for drifting processes

The “predictor-corrector” feedback controller, a process adjustment scheme proposed for semiconductor manufacturing run-to-run processes that drift, is extended to the multiple-input-multiple-output case. The controller is based on two coupled multivariate Exponentially-Weighted-Moving-Average (EWMA) equations, thus its performance depends on the choices of EWMA weight matrices. Stability conditions are given for a pure gain process adjusted with a MIMO double EWMA (EWMA) controller. It is shown that the stability conditions are invariant with respect to various realistic drift disturbance models. Recommendations on how to choose the EWMA weight matrices are given. An analysis is conducted to assess the impact of errors in the estimates of the process gains. The proposed MIMO EWMA feedback controller is compared to the common practice of using multiple single-input-single-ouput dEWMA controllers running in parallel.     

Online parameter estimation and run-to-run process adjustment using categorical observations

Categorical observations are frequently observed in run-to-run processes where obtaining accurate measurements of quality characteristics is difficult. In such circumstances, the use of categorical observations to estimate a process model and generate an adjustment recipe becomes inevitable. However, most conventional run-to-run controllers cannot be applied if no continuous observations are available; some parameter estimation methods that can handle categorical data only use historical dataset in an offline manner. In practice, it is common to see observations collected following a time sequence in a run-to-run process. Taking the lapping process in semiconductor manufacturing as an example, this paper develops an online approach for parameters estimation and run-to-run process adjustment using categorical observations. The proposed method optimises a penalised Maximum Likelihood (ML) function and updates parameters step by step when new categorical observations become available. A control strategy is also derived to generate receipts for process update between runs. The computational results of performance evaluation show that the proposed method is capable of estimating unknown parameters and control output quality online when initial bias exists.     

Variable EWMA run-to-run controller for drifted processes

In semiconductor production, the double Exponentially Weighted Moving Average (dEWMA) feedback controller is a popular model-based run-to-run controller for drift processes. Whilst a dEWMA controller with suitable discount factors can guarantee long-term stability under fairly regular conditions, it usually requires a moderately large number of runs to bring the output of a process to its target value. This is impractical for a process with small batches. To reduce a possibly high rework rate, we propose a variable discount factor to tackle the problem. The stability conditions and the optimal variable discount factor of the proposed EWMA controller are derived. The main advantage of the proposed controller is that it is very easy for implementing a run-to-run control scheme. In addition, our proposed controller achieves a better performance than that of a dEWMA controller unless the drift rate is poorly estimated. Hence, it provides us with an efficient tool to adjust a drifted run-to-run process.     

General run-to-run (R2R) control framework using self-tuning control for multiple-input multiple-output (MIMO) processes

During recent years, run-to-run (R2R) control techniques have been developed and used to control various semiconductor manufacturing processes. The R2R control methodology combines response surface modelling, engineering process control, and statistical process control. The main objective of such control is to manipulate the recipe to maintain the process output of each run as close to the nominal target as possible. The primary focus of this research is on the multiple-input multiple-output self-tuning control of R2R processes. A general control scheme is presented that can compensate for a variety of noise disturbances frequently encountered in semiconductor manufacturing. The controller can also compensate for various system dynamics, including autocorrelated responses, deterministic drifts, and varying process gains and offsets. Self-tuning controllers are developed to provide on-line parameter estimation and control. A recursive least squares algorithm is normally used to provide on-line parameter estimation to the controller. This type of control strategy used in the proposed self-tuning controller applies the principle of minimizing total cost (in the form of an expected off-target and controllable factors adjustment) to obtain a recipe for the next run. It is shown through the simulation study that even if the control model is non-linear, the self-tuning controller offers satisfactory control performance for R2R applications as compared with those of the control actions provided by the optimizing adaptive quality controller module. At last, a relevant application to chemical mechanical planarization in semiconductor manufacturing, a critical fabrication step involving two quality characteristics (removal rate and within-wafer non-uniformity), is used to illustrate the proposed controller. In this case study, a multivariate statistical process control technique via the Hotelling T?² statistic is also used as a dead-band for further investigation.     

On-line tuning system of multivariate dEWMA control based on a neural network approach

The double exponentially weighted moving average (EWMA) controller is a popular algorithm for on-line quality control of semiconductor manufacturing processes. The performance of the closed-loop system hinges on the adequacy of the two weight parameters of the double EWMA equations. In 2004, Su and Hsu presented an approach based on the neural technique for ‘on-line’ tuning the weight of the single EWMA equation in the single-input single-output (SISO) system. The present paper extends the neural network on-line tuning scheme to the double EWMA controller for the non-squared multiple-input multiple-output (MIMO) system, and validates the control performance by means of a simulated chemical–mechanical planarization (CMP) process in semiconductor manufacturing. Both linear and non-linear equipment models are considered to evaluate the proposed controller, coupling with the deterministic drift, the Gaussian noise and the first-order integrated moving average (IMA) disturbance. It has been shown from a variety of simulation studies that the proposed method exhibits quite competitive control performance as compared with the previous control system. The other merit of the proposed approach is that the tuning system, if sufficient training in a neural network is available, can be practicably applied to complex semiconductor processes without undue difficulty.     

Long run and transient analysis of a double EWMA feedback controller

The “predictor-corrector” feedback controller is a popular adjustment scheme proposed for the quality control of certain semiconductor manufacturing process steps. This controller is based on a double Exponentially-Weighted Moving Average (EWMA) scheme; thus the performance of the closed-loop system depends on the two weight parameters of the EWM A equations. In this paper, the conditions the weights must satisfy to ensure closed-loop stability are discussed. The optimal determination of the controller weights depends on a trade-off between long-run process variance and transient bias performance. It is shown that small weights, although they can guarantee stability, may result in severe, long transients, an important concern if fabrication is in small batches. An optimization model for finding the controller weights is given and numerically solved. An extension of this type of controllers to the multiple controllable factor case is described. The performance of the controller is illustrated with an application to Chemical Mechanical Polishing, a critical semiconductor manufacturing step.     

Integration of run-to-run control schemes and on-line experiment to deal with the changes in semiconducting dynamic processes

Recently, detailed investigations of various ‘run-to-run’ (R2R) control schemes for semiconductor manufacturing have been conducted. However, a pure R2R control scheme cannot ensure sufficient control quality when the process suddenly undergoes a larger change (herein, a larger change is defined when the output exceeds 2σ_ε ). In many semiconductor processes, sudden changes in the situation are often generated through the operation of different devices during the same process or changes in the control environment (e.g. process ageing or the influence of chemical concentrations). We propose an integrated R2R control system (IRCS) to alleviate this problem; the system includes an on-line experiment, R2R double exponentially weighted moving average (dEWMA) control, R2R triple EWMA (triEWMA) control and R2R self-tuning control. The IRCS initially employs a warning threshold to evaluate the process changes. When the process is changed, the system then uses the 2^k centre points design to amend the process model and selects a suitable R2R control scheme to execute the process control. We used polysilicon gate etching and chemical mechanical planarisation processes as case studies for verifying the proposed method. Based on the analysis results, the IRCS can reduce problems due to process changes. The IRCS is also an improvement in that R2R EWMA-like control does not deal with the problem of process changes in more complicated models.     

A New Exponentially Weighted Moving Average Control Chart for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely recognized as an advanced statistical process monitoring tool due to their excellent performance in detecting small to moderate shifts in process parameters. In this paper, we propose a new EWMA control chart for monitoring the process dispersion based on the best linear unbiased absolute estimator (BLUAE) obtained under paired ranked set sampling (PRSS) scheme, which we name EWMA‐PRSS chart. The performance of the EWMA‐PRSS chart is evaluated in terms of the average run length and standard deviation of run length, estimated using Monte Carlo simulations. These control charts are compared with their existing counterparts for detecting both increases and decreases in the process dispersion. It is observed that the proposed EWMA‐PRSS chart performs uniformly better than the EWMA dispersion charts based on simple random sampling and ranked set sampling (RSS) schemes. We also construct an EWMA chart based on imperfect PRSS (IPRSS) scheme, named EWMA‐IPRSS chart, for detecting overall changes in the process variability. It turns out that, with reasonable assumptions, the EWMA‐IPRSS chart outperforms the existing EWMA dispersion charts. A real data set is used to explain the construction and operation of the proposed EWMA‐PRSS chart. Copyright © 2014 John Wiley & Sons, Ltd.     

A forecast‐based monitoring methodology for process transitions

Nonconforming parts are often produced when a process moves from one level to another due to transition events. Control charting, when applied to a stable state process, is an effective monitoring tool to continuously check for process shifts or upsets. However, the presence of transition events can impede the normal performance of traditional control chart with increased false alarms. The presence of autocorrelation also requires modification to the control charting procedure. We present a methodology for characterizing the process transition which involves a tracking signal statistic, based on the forecast‐based exponentially weighted moving average (EWMA). This test will supplement the forecast‐based EWMA control charting as a means of detecting when the transition event is complete. Such a procedure facilitates smooth application of the appropriate control chart by knowing when the transition is over. The transition characterization methodology also carries benefits in cost and material savings. We use a color transition process in plastic extrusion to illustrate a transition event and demonstrate our proposed methodology. Simulation is employed to evaluate the performance of the methodology. Copyright © 2001 John Wiley & Sons, Ltd.     

New Exponentially Weighted Moving Average Control Charts for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have received considerable attention for detecting small changes in the process mean or the process variability. Several EWMA control charts are constructed using logarithmic and normalizing transformations on unbiased sample variance for monitoring changes in the process dispersion. In this paper, we propose new EWMA control charts for monitoring process dispersion based on the best linear unbiased absolute estimators obtained under simple random sampling (SRS) and ranked set sampling (RSS) schemes, named EWMA‐SRS and EWMA‐RSS control charts. The performance of the proposed EWMA control charts is evaluated in terms of the average run length and standard deviation of run length, estimated by using Monte Carlo simulations. The proposed EWMA control charts are then compared with their existing counterparts for detecting increases and decreases in the process dispersion. It turns out that the EWMA‐RSS control chart performs uniformly better than its analogues for detecting overall changes in process dispersion. Moreover, the EWMA‐SRS chart significantly outperforms the existing EWMA charts for detecting increases in process variability. A real data set is also used to explain the construction and operations of the proposed EWMA control charts. Copyright © 2013 John Wiley & Sons, Ltd.     

The Negative Binomial Exponentially Weighted Moving Average Chart with Estimated Control Limits

The control chart based on the compound Poisson distribution (the negative binomial exponentially weighted moving average (EWMA) chart) has been shown to be more effective than the c‐chart to monitor the wafer nonconformities in semiconductor production process. The performance of the negative binomial EWMA chart is generally evaluated with the assumption that the process parameters are known. However, in many control chart applications, the process parameters are usually unknown and are required to be estimated. For an accurate parameter estimate, a very large sample size may be required, which is seldom available in the applications. This article investigates the effect of parameter estimation on the run length properties of the negative binomial EWMA charts. Using a Markov chain approach, we show that the performance of the negative binomial EWMA chart is affected when parameters are estimated compared with the known‐parameter case. We also provide recommendations regarding phase I sample sizes, smoothing constant and clustering parameter. The sample size must be quite large for the in‐control chart performance to be close to that for the known‐parameter case. Finally, a wafer process example has been used to highlight the practical implications of estimation error and to offer advice to practitioners when constructing/analysing a phase I sample. Copyright © 2013 John Wiley & Sons, Ltd.     

Joint EWMA charts for multivariate process control: Markov chain and optimal design

This paper deals with the optimal design of a set of univariate exponentially weighted moving average (EWMA) quality control charts to monitor several correlated variables. The aim is to find the values for the parameters of the set of charts that minimise the average run length (ARL) for a given process shift, according to three different problem formulations, in terms of the symmetry of the directions of shift that may be expected in the particular process. The first step to achieve this objective has been the development of a multidimensional Markov chain model to compute the ARLs of the set of EWMA control charts. A genetic algorithm has been employed for the optimisation. Finally, a (non-exhaustive) performance comparison is presented between the joint EWMA charts and the equivalent multivariate EWMA (MEWMA) control chart. No scheme uniformly outperforms the other one. In some cases, the univariate charts largely outperform the MEWMA chart for the shift they are optimised but perform much worse for other shifts. Therefore, the tools described in this paper help the user to make an informed decision considering the shifts that may be expected in each particular case.     

On-line tuning of a single EWMA controller based on the neural technique

The exponentially weighted moving average (EWMA) controller has been proven to be an effective algorithm in the control the modern manufacturing system. The performance of the EWMA controlled process is based on choosing the correct EWMA gain. Most related research has focused on analysing the optimal EWMA gain in the static condition. The objective was to propose an approach based on the neural technique for on-line tuning of the single EWMA gain. The underlying approach indicated that the network learns very quickly when taking autocorrelation function and sample partial autocorrelation function patterns as the input features. It is shown that the sequence of the EWMA gains, generated by the proposed adaptive approach, converges close to the optimal controller value under several disturbance models, including IMA(1,1), and step and small ramp disturbances. In addition, the approach possesses a superior controlled output performance compared with the previous adaptive system.     

EWMA Control Chart Performance with Estimated Parameters under Non‐normality

Exponentially weighted moving average (EWMA) control charts can be designed to detect shifts in the underlying process parameters quickly while enjoying robustness to non‐normality. Past studies have shown that performance of various EWMA control charts can be adversely affected when parameters are estimated or observations do not follow a normal distribution. To the best of our knowledge, simultaneous effect of parameter estimation and non‐normality has not been studied so far. In this paper, a Markov chain approach is used to model and evaluate performance of EWMA control charts when parameter estimation is subject to non‐normality using skewed and heavy‐tailed symmetric distributions. Using standard deviation of the run length (SDRL), average run length (ARL), and percentiles of run lengths for various phase I sample sizes, we show that larger phase I sample sizes do not necessarily lead to a better performance for non‐normal observations. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal EWMA of linear combination of Poisson variables for multivariate statistical process control

In this paper, we propose a new process control chart for monitoring correlated Poisson variables, the EWMA LCP chart. This chart is the exponentially weighted moving average (EWMA) version of the recently proposed LCP chart. The latter is a Shewhart-type control chart whose control statistic is a linear combination of the values of the different Poisson variables (elements of the Poisson vector) at each sampling time. As a Shewhart chart, it is effective at signalling large process shifts but is slow to signal smaller shifts. EWMA charts are known to be more sensitive to small and moderate shifts than their Shewhart-type counterparts, so the motivation of the present development is to enhance the performance of the LCP chart by the incorporation of the EWMA procedure to it. To ease the design of the EWMA LCP chart for the end user, we developed a user-friendly programme that runs on Windows© and finds the optimal design of the chart, that is, the coefficients of the linear combination as well as the EWMA smoothing constant and chart control limits that together minimise the out-of-control ARL under a constraint on the in-control ARL. The optimization is carried out by genetic algorithms where the ARLs are calculated through a Markov chain model. We used this programme to evaluate the performance of the new chart. As expected, the incorporation of the EWMA scheme greatly improves the performance of the LCP chart.     

On mixed memory control charts based on auxiliary information for efficient process monitoring

Control charts are popular monitoring tools in statistical process control toolkit. These are used to identify assignable causes in the process parameters (location and/or dispersion). These assignable causes result in a shift in the process parameter(s). The shift can be categorized into three sizes (small, moderate, and large). Memory control charts such as the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts are effective for identifying small-to-moderate shift(s) in the process. Likewise, mixed memory control charts are useful for efficient process monitoring. In this study, we have proposed two new mixed memory control charts based on auxiliary information named MxMEC and MxMCE control charts to improve the efficiency of these mixed charts. The MxMEC chart is a merger of the auxiliary information based MxEWMA chart and the classical CUSUM chart. Likewise, the MxMCE chart integrates the auxiliary information based MxCUSUM with the classical EWMA chart. The proposed MxMEC and MxMCE charts are evaluated through famous performance measures including average run length, extra quadratic loss, relative average run length, and performance comparison index. The performance of the study proposals is compared with the existing counterparts such as the classical CUSUM and EWMA, MxCUSUM, MxEWMA, MEC, MCE, and runs rules-based CUSUM charts. The comparisons revealed the superiority of the proposed charts against other competing charts particularly for small-to-moderate shifts in the process location. Finally, a real-life data is used to show the implementation procedure of the proposed charts in practical situations.     

A run-to-run control algorithm based on timely and delayed mixed-resolution information

In certain run-to-run (R2R) processes, timely accurate measurements are difficult to obtain due to slow laboratory measurement operations. Instead, only low-resolution categorical observations are observed online for important quality variables; continuous measurements for the same variables are provided after a specific amount of delay. Currently, most conventional R2R controllers cannot be applied if no continuous observations are available. It is therefore important to develop online algorithms for R2R process control based on mixed-resolution information that is partially timely and partially delayed. In this study, we take the lapping process in semiconductor manufacturing as an example and propose parameter estimation models with these mixed-resolution data for processes with the first-order autoregressive, AR(1), disturbance series. We also derive control strategies to generate recipes between production runs for better process control. The computational results of a performance evaluation show that the control performance of the proposed method is competitive compared to existing methods that are based on accurate measurements.     

New Exponentially Weighted Moving Average Control Charts for Monitoring Process Mean and Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely accepted because of their excellent performance in detecting small to moderate shifts in the process parameters. In this paper, we propose new EWMA control charts for monitoring the process mean and the process dispersion. These EWMA control charts are based on the best linear unbiased estimators obtained under ordered double ranked set sampling (ODRSS) and ordered imperfect double ranked set sampling (OIDRSS) schemes, named EWMA‐ODRSS and EWMA‐OIDRSS charts, respectively. We use Monte Carlo simulations to estimate the average run length, median run length, and standard deviation of run length of the proposed EWMA charts. We compare the performances of the proposed EWMA charts with the existing EWMA charts when detecting shifts in the process mean and in the process variability. It turns out that the EWMA‐ODRSS mean chart performs uniformly better than the classical EWMA, fast initial response‐based EWMA, Shewhart‐EWMA, and hybrid EWMA mean charts. The EWMA‐ODRSS mean chart also outperforms the Shewhart‐EWMA mean charts based on ranked set sampling (RSS) and median RSS schemes and the EWMA mean chart based on ordered RSS scheme. Moreover, the graphical comparisons of the EWMA dispersion charts reveal that the proposed EWMA‐ODRSS and EWMA‐OIDRSS charts are more sensitive than their counterparts. We also provide illuminating examples to illustrate the implementation of the proposed EWMA mean and dispersion charts. Copyright © 2014 John Wiley & Sons, Ltd.     

A New Nonparametric EWMA Control Chart for Monitoring Process Variability

The exponentially weighted moving average (EWMA) control chart is one of a potentially powerful process monitoring tool of the statistical process control. The EWMA chart has now been widely used because of its excellent ability to detect small to moderate shifts in the process parameter(s). In this study, we propose a new nonparametric/distribution‐free EWMA chart for efficiently monitoring the changes in the process variability. We use extensive Monte Carlo simulations to compute the run length profiles of the proposed EWMA chart. For a better performance comparison, the proposed EWMA chart is compared with a recent existing EWMA chart that has already shown to have better performance than the existing control charts. It turns out that the proposed EWMA chart performs substantially and uniformly better than the existing powerful EWMA chart. The working and implementation of the proposed and existing EWMA charts with the help of an illustrative example are also included in this study. Copyright © 2017 John Wiley & Sons, Ltd.     

Comparisons of Some Exponentially Weighted Moving Average Control Charts for Monitoring the Process Mean and Variance

An exponentially weighted moving average (EWMA) control chart for monitoring the process mean μ may be slow to detect large shifts in μ when the EWMA tuning parameter λ is small. An additional problem, sometimes called the inertia problem, is that the EWMA statistic may be in a disadvantageous position on the wrong side of the target when a shift in μ occurs, which may significantly delay detection of a shift in μ. Options for improving the performance of the EWMA chart include using the EWMA chart in combination with a Shewhart chart or in combination with an EWMA chart based on squared deviations from target. The EWMA chart based on squared deviations from target is designed to detect increases in the process standard deviation σ, but it is also very effective for detecting large shifts inμ. Capizzi and Masarotto recently proposed the option of an adaptive EWMA control chart in which λ is a function of the data. With the adaptive feature, the EWMA chart behaves like a standard EWMA chart when the current observation is close to the previous EWMA statistic, and like a Shewhart chart otherwise. Here we extend the use of the adaptive feature to EWMA charts based on squared deviations from target, and also consider an alternate way of defining the adaptive feature. We discuss performance measures that we believe are appropriate for assessing the effects of inertia, and compare the performance of various charts and combinations of charts. Standard practice is to simultaneously monitor both μ and σ, so we consider control chart performance when the objective is to detect small or large changes in μ or increases in σ. We find that combinations of EWMA control charts that include a chart based on squared deviations from target give good overall performance whether or not these charts have the adaptive feature.