Double EWMA Controller Research Articles

ARL Evaluation of a DEWMA Control Chart for Autocorrelated Data: A Case Study on Prices of Major Industrial Commodities

The double exponentially weighted moving average (DEWMA) control chart, an extension of the EWMA control chart, is a useful statistical process control tool for detecting small shift sizes in the mean of processes with either independent or autocorrelated observations. In this study, we derived explicit formulas to compute the average run length (ARL) for a moving average of order q (MA(q)) process with exponential white noise running on a DEWMA control chart and verified their accuracy by comparison with the numerical integral equation (NIE) method. The results for both were in good agreement with the actual ARL. To investigate the efficiency of the proposed procedure on the DEWMA control chart, a performance comparison between it and the standard and modified EWMA control charts was also conducted to determine which provided the smallest out-of-control ARL value for several scenarios involving MA(q) processes. It was found that the DEWMA control chart provided the lowest out-of-control ARL for all cases of varying the exponential smoothing parameter and shift size values. To illustrate the efficacy of the proposed methodology, the presented approach was applied to datasets of the prices of several major industrial commodities in Thailand. The findings show that the DEWMA procedure performed well in almost all of the scenarios tested. Doi: 10.28991/ESJ-2023-07-05-020 Full Text: PDF

A New Distribution-Free Control Chart for Monitoring Process Median Based on the Statistic of the Sign Test

Abstract Control charts are used to improve the quality of outputs in manufacturing, industrial, and service processes. The parametric control charts produce more false alarms and unacceptable out-of-control signals when the underlying distribution of the process is not normal. Nonparametric/distribution-free control charts are efficient alternatives to overcome the said situation. In this article, the performance of a distribution-free double exponentially weighted moving average (EWMA) chart has been investigated based on the sign test statistic under simple and ranked set sampling schemes. The run-length properties of the proposed charts have been evaluated and compared with nonparametric EWMA sign, parametric EWMA, and parametric double EWMA control charts, using different run-length measures. The comparison reveals the efficiency of the proposed chart over its alternatives in detecting small and medium shifts in the process location. A real-data application using the substrate manufacturing process has been provided to show the implementation of the proposed chart.

Nonparametric Double EWMA Control Charts based on Mood Statistic for Process Variability

This research aims to propose a nonparametric double exponentially weighted moving average (NDEWMA) control chart for detecting shifts in a process using the Mood statistic. A Monte Carlo simulation study was used. The data had a normal distribution and a non-normal distribution where the magnitude of shift size (δ) was δ = 1, 1.05, 1.10 - 2.00 and 3.00, respectively. A performance comparison of the control charts was evaluated by using the Average Run Length (ARL). In the case of an in-control process, a large ARL value is desired, while a small ARL value is desired when the process is out-of-control. From the results of the simulation study, it was observed that the proposed NDEWMA control chart is effective in detecting small shifts in the process and gives better performance compared to the nonparametric exponentially weighted moving average control (NEWMA) control chart based on Mood statistics.

Design of a run‐to‐run controller based on discrete sliding‐mode observer

AbstractIn this study, a run‐to‐run controller based on a discrete sliding‐mode observer (DSMO) is presented for photolithography manufacturing processes. A second‐order DSMO is introduced to estimate the process disturbances caused by tool aging and preventive maintenance. The proposed controller is constructed by applying DSMO estimation to the deadbeat control law and the stable region is discussed. Numerical simulations and an industrial case demonstrated the performance of the proposed algorithm. It is observed that the proposed controller has a superior performance over the exponentially weighted moving average (EWMA) and double EWMA (dEWMA) controller under the first‐order integrated moving average disturbance with shift or drift. This superiority is maintained when the system contains a certain plant‐model mismatch. Moreover, the tolerance of the proposed controller for metrology delay is better than that of the dEWMA controller.

Run-to-run control of batch production process in manufacturing systems based on online measurement

Run-to-run control is an important and effective method for quality control of batch production process in manufacturing systems. Most of current run-to-run control methods are investigated based on offline sampled measurement due to the limitation of measurement technology. The sampled measurements cannot represent each run, thus lead to performance deterioration of run-to-run control methods. The development of new measurement technologies provides great opportunities to improve product quality, since all data of each run is obtained by online measurement. This paper presents a systematic run-to-run control method for batch production process in manufacturing systems based on online measurement. Firstly, a single-input-single-output process model of multiple workpieces (MSISO) is built and a MSISO double exponentially-weighted-moving-average (EWMA) controller is proposed. The proposed MSISO double EWMA controller updates control recipe of current run based on all output data of its previous run. Then, the stability analysis of the proposed MSISO double EWMA controller is conducted to obtain the sufficient conditions for the asymptotic stability of the control process in manufacturing systems. Mean square error (MSE) is used to evaluate the quality control performance. The results of simulation experiments and a case study show that the proposed method can effectively improve the control performance for batch production process in manufacturing systems when compared to other run-to-run-based methods.

On control chart for monitoring exponentially distributed quality characteristic

The double exponentially weighted moving average (DEWMA) control chart has been observed to be more sensitive than the exponentially weighted moving average (EWMA) control chart for process monitoring assuming that the quality characteristic follows the normal distribution. In this paper, the DEWMA control chart is proposed for monitoring quality characteristics that follow the exponential distribution using variable transformation technique. The in-control and out-of-control average run lengths (ARLs) of the proposed control chart is obtained for equal and unequal smoothing constants. The performance of the proposed control chart with equal and unequal smoothing constants was investigated and compared with recent existing control charts in terms of the out-of-control average run lengths. Real life example is given to demonstrate the application of the proposed chart. The findings show that the performance of the proposed control chart outweighs existing control charts in the monitoring of process parameter when the quality variable follows exponential distribution for all shift sizes.

An Extended State Observer-Based Run to Run Control for Semiconductor Manufacturing Processes

In this paper, an extended state observer (ESO)-based run to run (RtR) controller is presented to suppress various stochastic disturbances emerging in the semiconductor manufacturing process. Using the internal model control theory, the relationship between the ESO-RtR controller and the exponentially weighted moving average (EWMA) controller, and the relationship between the ESO-RtR controller and double EWMA controller are discussed. The stable region of the proposed ESO-RtR controller is calculated and discussed by using the Jury’s test. Furthermore, the ESO-RtR controller is extended to a nonlinear form (i.e., nonlinear ESO-RtR controller) to improve its disturbance compensation capability. Finally, numerical simulations and an industrial example are provided to demonstrate the effectiveness of the proposed algorithm.

An enhanced approach for the progressive mean control charts

AbstractTo maintain and improve the quality of the processes, control charts play an important role for reduction of variation. To detect large shifts in the process parameters, Shewhart control charts are commonly applied but for small shifts, exponentially weighted moving averages (EWMA), cumulative sum (CUSUM), double exponentially weighted moving average (DEWMA), double CUSUM, moving average (MA), double moving average (DMA), and progressive mean (PM) control charts, are used. This study proposes double progressive mean (DPM) and optimal DPM control charts to enhance the performance of the PM chart. As the proposed DPM control charts use information sequentially, hence their performance is compared with natural competitors EWMA, CUSUM, DEWMA, double CUSUM, MA, DMA, and PM control charts. Run length and its different properties are evaluated to compare the performance of the proposed charts and counterparts. Results reveal that proposed optimal DPM outperforms the other charts. An example related to voltage on fixed capacitance level is also provided to illustrate the proposed charts.

Model‐based and data‐driven with multiscale sum of squares double EWMA control chart for fault detection in biological systems

AbstractThe objectives of this paper will be sought. First, an enhanced technique that can accurately model biological processes will be developed. To deal with scenarios where a process model is available, the particle filter method will be developed to better handle the nonlinear and high‐dimensional state estimation problem. Second, a multiscale sum of squares double exponentially weighted moving average (MS‐SS‐DEWMA) chart will be applied to the monitored residuals in order to enhance the fault detection abilities. The advantage of MS‐SS‐DEWMA chart is twofold: (1) The SS‐DEWMA chart uses the sum of squares statistics; it simultaneously monitors the process mean and variance in a single chart. It has presented better performance than the classical EWMA‐based charts. (2) The multiscale data representation can be used as an effective tool for reducing noise from a signal's time series. The effectiveness of the proposed strategy is validated using a synthetic and simulated Cad system in Escherichia coli (CSEC) data. When the simulated CSEC model is used, the developed approach is applied for monitoring some of the key variables involved in the CSEC model. The proposed strategy is also applied to detect diseases using genomic copy number data through better detection of aberrations in the genetic information of patients, which can help medical doctors make more accurate diagnosis of diseases.

Bayesian Monitoring of Linear Profiles Using DEWMA Control Structures With Random $X$

The process structures of manufacturing industry are efficiently modeled using linear profiles. Classical and Bayesian set-ups are two well-appreciated schemes for designing control charts for the monitoring of process structures. Mostly in profiles monitoring the independent variables along with the process parameters are assumed fixed. There are manufacturing processes where these conditions may not hold. The advancement in technology and day-to-day changes in process structures caused the parametric uncertainty along with variability in explanatory variables. This paper considered the case of random $X$ and assumes different conjugate and non-conjugate priors to handle parametric uncertainty using double exponentially weighted moving average (DEWMA) control charts. Three univariate DEWMA charts are designed for the monitoring of $Y$ -intercepts, slopes, and error variances. The average run length criterion has been used to evaluate the proposed and competing charts. The wide spread relative study identifies that the proposed Bayesian DEWMA control charts are better than the competing charts based on early detection of out-of-control profiles, particularly for smaller value shifts. The Bayesian DEWMA charts using conjugate priors are the quickest in all as they take less sample points to show out-of-control profile. A case study has been considered to further justify the superiority of Bayesian DEWMA charts over competing charts.

Performance Analysis of Double EWMA Controller Under Dynamic Models With Drift

This paper mainly focuses on the performance of the double exponentially weighted moving average (DEWMA) controller applied to a first-order dynamic input–output model with process drift. Under a general process disturbance, based on Jury test, the necessary and sufficient conditions involving model parameters, the estimated process gain, and weighting factors are established to ensure that the closed-loop system is asymptotically stable. In addition, we explore why we should use the DEWMA controller to adjust the process rather than the single exponentially weighted moving average controller when the process drifts considerably. Later, under the stability condition, the control performance analysis of the closed-loop system is performed in terms of the asymptotical variation and the transient deviation. Then, an optimization model is established to figure out the appropriate weighting factors for reducing the overall variation of the process output during the process of operation. Finally, extensive simulations are carried out to demonstrate the validity of the theoretical analysis in the context of chemical mechanical polishing process.

Bayesian monitoring of linear profile monitoring using DEWMA charts

Process monitoring is an essential element for an improved quality of final products. A variety of tools are used for it; control charts are one of these choices. Classical and Bayesian thoughts are 2 main aspects of statistics used in different areas of application. This study introduces an approach to existing theories in applied quality control: Bayesian double exponentially weighted moving average (DEWMA) control charts for monitoring the profiles of products and processes. Three novel univariate Bayesian DEWMA charting structures for the Y intercepts, slopes, and error variances are designed under phase 2 procedures. The performance of the designed structures of control charts is evaluated based on different run length measures. The comparative analysis revealed that Bayesian DEWMA control charts are efficient at identifying the sustainable shifts in the process parameters. Moreover, DEWMA control charts are more effective under classical and Bayesian methodologies for detecting smaller value shifts compared with exponentially weighted moving average charts. We have examined that acquiring extra information in the form of prior's about process parameters comes up with tangible benefits and enhances the detection potential of DEWMA charts for profiles monitoring. An example and case studies are provided to justify the above findings.

Statistically constrained economic design of maximum double EWMA control charts based on loss functions

A maximum double exponentially weighted moving average (EWMA) chart, called the MaxDEWMA chart, is used for fast detection of small and moderate shifts in the process mean and/or variance. A loss model that combines Lorenzen and Vance’s cost model with linear, quadratic and exponential loss functions is applied in this work to develop the statistical constraints of the MaxDEWMA chart. The optimal decision variables, namely sample size n, sampling interval time h, control limit width L and smoothing constant , are obtained by minimizing the expected cost function. Via simulations, the MaxDEWMA charts have smaller expected cost than the MaxEWMA charts based on different loss functions. In addition, adopting different loss functions impacts the optimal expected cost and decision variables. Sensitivity analyses reveal that a larger magnitude of the mean and/or variance shifts leads to a smaller sample size and a more frequent sampling interval.

Nonparametric Double EWMA Control Chart for Process Monitoring

<p>In monitoring process parameters, we assume normality of the quality characteristic of interest, which is an ideal assumption. In many practical sit- uations, we may not know the distributional behavior of the data, and hence, the need arises use nonparametric techniques. In this study, a nonparametric double EWMA control chart, namely the NPDEWMA chart, is proposed to ensure efficient monitoring of the location parameter. The performance of the proposed chart is evaluated in terms of different run length properties, such as average, standard deviation and percentiles. The proposed scheme is compared with its recent existing counterparts, namely the nonparametric EWMA and the nonparametric CUSUM schemes. The performance mea- sures used are the average run length (ARL), standard deviation of the run length (SDRL) and extra quadratic loss (EQL). We observed that the pro- posed chart outperforms the said existing schemes to detect shifts in the process mean level. We also provide an illustrative example for practical considerations.</p>

Stability and Control Performance Analysis of Double EWMA Controller With Metrology Delay

This paper mainly focuses on establishing the stability conditions of the double exponentially weighted moving average (d-EWMA) controller with metrology delay when the process experiences a general disturbance and analyzing its control performance under some typical types of process disturbance. The necessary and sufficient conditions for stability are established by Routh-Hurwitz criteria. It has been shown that if the process gain mismatch is greater than 4/3, metrology delay will make the feasible region of two weighting factors in the d-EWMA controller decrease, and the larger metrology delay is, the smaller the feasible region is. Moreover, given two weighting factors, it is found that metrology delay will weaken the robustness of the d-EWMA controller. The influence of metrology delay on the asymptotical control performance is evaluated based on asymptotical mean square error (AMSE). Optimal weighting factors and AMSE can be solved based on analytical expression of the AMSE, which is derived by using the Yule-Walker equations to avoid function summing limit operation. Through numerical simulations, it demonstrates that metrology delay may increase the variability of the process output under certain conditions. Finally, the chemical mechanical polishing process in semiconductor manufacturing is used to illustrate the validity of theoretical results.

On the performance of EWMA and DEWMA control charts for censored data

Exponentially weighted moving average (EWMA) control charts are very widely used for the detection of small shifts. Another similar charting structure is double EWMA (DEWMA) control chart for the improved detection of the shifts. Many interesting features of EWMA and DEWMA have been described in the literature. This study intends to investigate EWMA and DEWMA control charts under Type-I censoring for gamma-distributed lifetimes. The idea of conditional expected values is used to monitor the mean level. The performance evaluations are carried out using average run length as a measure in this study. The optimum sample size comparisons for the specified and unspecified parameter are also part of the study. To assess the overall performance of the control charts, we also used extra quadratic loss and it is found DEWMA is an efficient chart for the detection of shift in scale parameter. Moreover, an illustrative example for practical considerations is included in the study. It is observed that varying censoring rates affect the performance of the chart depending upon the type of chart, the method of estimation, and the amount of shift.

Design of a self-tuning double EWMA controller for MIMO processes

The design of time-varying multivariate discount factors for the double EWMA controller is essential for ensuring the consistency of output qualities. In this research, a self-tuning double EWMA controller for MIMO processes is proposed. First, the process estimator is used to obtain the online process parameters at each run based on the recursive least square method. Then, based on the estimated process model, the double EWMA controller is developed to generate the control inputs for compensating the variations of the process at each run. A genetic algorithm searches for the approximately optimal discount factors within the stable region at each run and the objective is to minimise the cost of the outputs deviating from targets and the adjustment amounts of the control inputs between each run. The obtained discount factors are automatically adjusted to adapt to the process variations and disturbances and to maintain the process running at controlled conditions.

Robustness of DEWMA versus EWMA Control Charts to Non-Normal Processes

Exponentially weighted moving average (EWMA) and double EWMA (DEWMA) control charts were designed under the normality assumption. This study considers various skewed (Gamma) and symmetric non-normal (t) distributions to examine the effect of non-normality on the average run length (ARL) performance of EWMA and DEWMA. ARL performances were investigated and compared using Monte Carlo simulations. Results show that DEWMA charts can be designed to be robust to non-normality, that the ARL performances of EWMA and DEWMA charts were more robust to t distributions and DEWMA was more robust to non-normality for larger values of the smoothing parameter.

Modified Double EWMA Approach for Mixed Product Run-to-Run CMP Process Control

Advanced process control (APC) has been recognized as a proper tool for maximizing profitability of semiconductor manufacturing facilities by improving efficiency and product quality. Run-to-run (RtR) process control with good quality and reliable performance for APC applications are most applicable. Chemical mechanical planarization (CMP) is part of critical processing module in semiconductor manufacturing. This paper proposes a new RtR control scheme, modified double EWMA controller (m-dEWMA), which is adaptive to the mixed product CMP processes. The m-dEWMA controller combined thread dEWMA with the drift compensation scheme to deal with the drift and shift disturbances caused by tool and products, respectively. Comparing recently mixed product control schemes: threaded EWMA controller, JADE control and threaded prediction correction controller (threaded PCC), the simulation results show that the m-dEWMA has better control performance than other controllers. The experiment results revealed that the m-dEWMA controller improves the wafer uniformity significantly in mixed product CMP process.

Double EWMA controller using neural network-based tuning algorithm for MIMO non-squared systems

The double exponentially weighted moving average (dEWMA) control method is a popular algorithm for adjusting a process from run to run in semiconductor manufacturing. For MIMO non-squared statistic systems, the singular value decomposition (SVD) method is used for decoupling and the SVD-based dEWMA control scheme is treated as a MIMO extension of dEWMA control design. To enhance the performance and robustness of the linear system in the presence of ramp disturbances and white noises, the neural network-based adaptive algorithm is used to automatically tune the dEWMA controller parameters. Under the specified input patterns, the early stop criterion for the training-validation neural networks, and the stability constraints added in the tuning mechanism, the simulations show that the proposed control technique can effectively improve the means and standard deviations of the process outputs.