A control chart based on likelihood ratio test for detecting patterned mean and variance shifts

Control charts based on generalized likelihood ratio test (GLRT) are attractive from both theoretical and practical points of view. Most of the existing works in the literature focusing on the detection of the process mean and variance are almost based on the assumption that the shifts remain constant over time. The case of the patterned mean and variance changes may not be well discussed. In this research, we propose a new control chart which integrates the exponentially weighted moving average (EWMA) procedure with the GLRT statistics to monitor the process with patterned mean and variance shifts. The attractive advantage of our control chart is its reference-free property. Due to the good properties of GLRT and EWMA procedures, our simulation results show that the proposed chart provides quite effective and robust detecting ability for various types of shifts. The implementation of our proposed control chart is illustrated by a real data example from chemical process control.     

Using geometric Poisson exponentially weighted moving average control schemes in a compound Poisson production environment

In this study, the variable to be controlled over time is the number of defects. Meanwhile, the underlying distribution of defects is the geometric Poisson distribution, a Poisson distribution compounded by a geometric distribution. For production process control, the exponentially weighted moving average (EWMA) control scheme based on the geometric Poisson process is addressed. Performance of the EWMA control scheme is assessed not only by both in-control and out-of-control average run lengths (ARL’s), but also by higher moments of the run length (RL) distribution. The run length distribution properties can be obtained from the probability transition matrix and implemented using the computer programs developed in this study. With proper ARL and variance of RL selected, any small shift in mean can be detected via the geometric Poisson EWMA control scheme.     

Enhancing the performance of CUSUM scale chart

Control charts act as the most important statistical process monitoring tool, widely used for the purpose of identifying unusual variations in process parameters. Researchers have implemented different rules to increase the sensitivity of Shewhart, CUSUM and EWMA control charts for the detection of small shifts in process location. However, for the monitoring of process scale, the use of such rules has been limited to Shewhart charts. This study proposes the implementation of sensitizing rules in CUSUM scale charts to enhance their ability to detect smaller changes in process variability. The performance of the proposed schemes is evaluated and compared with the simple scale CUSUM scheme, the EWMS chart, the M-EWMS chart and the COMB chart, in terms of run length characteristics such as average run length (ARL) and standard deviation of the run length distribution (SDRL). Control chart coefficients to set the ARL at the desired level are also provided. Two numerical examples are given to illustrate the application of the proposed schemes on practical data sets.     

Comparative study of the performance of the CuSum and EWMA control charts

This work presents a comparative study of the performance of the cumulative sum (CuSum), as well as the exponentially weighted moving average (EWMA) control charts. The objective of this research is to verify when CuSum and EWMA control charts do the best control region, in order to detect small changes in the process average. Starting from the data of a productive process, several series were simulated. CuSum and EWMA control charts were used to determine the average run length (ARL) to detect a condition out of control. ARL found by each chart which was then, compared. It was observed that the CuSum control chart practically did not sign points out of control for the levels of variation between ±1.0 standard deviation. For these variation levels the EWMA control chart was more efficient than CuSum. Among the parameters EWMA control chart the ones with constant λ=0.10 and 0.05, with the respective control limits L=2.814 and 2.625, were the ones that detected larger number of altered positions.     

Evaluation of fuzzy neural network run-to-run controller using numerical simulation analysis for SISO process

During the past decade, a variety of run-to-run (R2R) control techniques have been proposed and extensively used to control various semiconductor manufacturing processes. The R2R control methodology combines response surface modeling, engineering process control, and statistical process control, with the main objective of fine-tuning the recipe so that the process output of each run can be maintained as close to the nominal target as possible. In this paper, the single-input single-output (SISO) model is addressed. To overcome the shortcomings in the traditional R2R EWMA controller, a fuzzy neural network (FNN) control strategy is proposed. When a process has large autoregressive parameters, traditional EWMA control methods cannot establish stable SISO process control. To solve this problem, an SISO process control model based on an FNN was used to build an SISO process control procedure. The analysis results from a numerical simulation indicated that when the coefficient of autocorrelation  > 0.6, the MSE ratio when using the FNN controller was 97.11% lower than when using the EWMA controller and 61.12% lower than when using an adaptive EWMA controller. This showed that the FNN control method established better SISO process control than the EWMA and adaptive EWMA control methods.     

A new approach to detecting the process changes for multistage systems

The study aims to develop a new control chart model suitable for monitoring the process quality of multistage manufacturing systems.Considering both the auto-correlated process outputs and the correlation occurring between neighboring stages in a multistage manufacturing system, we first propose a new multiple linear regression model to describe their relationship. Then, the multistage residual EWMA and CUSUM control charts are used to monitor the overall process quality of multistage systems. Moreover, an overall run length (ORL) concept is adopted to compare the detecting performance for various multistage residual control charts. Finally, a numerical example with oxide thickness measurements of a three-stage silicon wafer manufacturing process is given to demonstrate the usefulness of our proposed multistage residual control charts in the Phase II monitoring. A computerized algorithm can also be written based on our proposed scheme for the multistage residual EWMA/CUSUM control charts and it may be further converted to an expert and intelligent system. Hopefully, the results of this study can provide a better alternative for detecting process change and serve as a useful guideline for quality practitioners when monitoring and controlling the process quality of multistage systems with auto-correlated data.     

A G&P EWMA algorithm for high-mix semiconductor manufacturing processes

In mixed run processes, typical in semiconductor manufacturing and other automated assembly-line type process, products with different recipes will be produced on the same tool. Product based run-to-run control can be applied to improve the process capability. The effect of product-based controller on low frequency products is, however, minimal, due to inability to track tool variations. In this work, we propose a group and product based EWMA control scheme which combines adaptive k-means cluster method and run-to-run EWMA control to improve the performance of low frequency products in the mixed run process. Similar products could be classified into the same group adaptively and controlled by a group EWMA controller. The group controller is updated by both low frequency products and similar high frequency products; so that low frequency products can be improved by shared information from similar large frequency products. However, the high frequency products are controlled by individual product-based EWMA to avoid interference of the low frequency products. The advantages of proposed control scheme are demonstrated by benchmark simulation and reversed engineered industrial applications.     

A new exponentially weighted moving average sign chart using repetitive sampling

In this paper, a new nonparametric control chart based on the exponentially weighted moving average (EWMA) sign statistic is proposed using repetitive sampling. The control chart is proposed to effectively detect the process mean shift away from the target value without the distributional assumption on the quality characteristic. The proposed control chart is based on two pairs of upper and lower control limits having different control coefficients. The in-control and the out-of-control average run lengths of the proposed control chart are evaluated through the Monte Carlo simulation. The proposed control chart is shown to be more efficient than the existing EWMA sign control chart in terms of the average run length.     

Online monitoring of nonlinear multivariate industrial processes using filtering KICA–PCA

In this paper, a novel approach for processes monitoring, termed as filtering kernel independent component analysis–principal component analysis (FKICA–PCA), is developed. In FKICA–PCA, first, a method to calculate the variance of independent component is proposed, which is significant to make Gaussian features and non-Gaussian features comparable and to select dominant components legitimately; second, Genetic Algorithm is used to determine the kernel parameter through minimizing false alarm rate and maximizing detection rate; furthermore, exponentially weighted moving average (EWMA) scheme is used to filter the monitoring indices of KICA–PCA to improve monitoring performance. In addition, a novel contribution analysis scheme is developed for FKICA–PCA to diagnosis faults. The feasibility and effectiveness of the proposed method are validated on the Tennessee Eastman (TE) process.     

A Kalman filter-based R2R control system with parallel stochastic disturbance models for semiconductor manufacturing processes

A Kalman filter-based run-to-run control system has been proposed for minimum variance control of semiconductor manufacturing process. In the proposed control system, both gain- and bias-varying process models combined with different stochastic disturbance models were considered and identified in parallel. The best-fit model is selected and used for the R2R controller design. Sub-models of the ARIMA(1,1,1) process were considered for stochastic modeling of the bias and gain variation, and the Kalman filters are used to find the optimum model parameter estimation. The control performance was analyzed for each case of the disturbance model to investigate the expected benefit from the control system in comparison with the EWMA filter-based controller.     

An adaptive Shiryaev-Roberts procedure for monitoring dispersion

In this paper, the Shiryaev-Roberts (SR) procedure is examined and compared with the change point CUSUM (CPC) procedure for monitoring the dispersion of a normal process. It will be shown that the SR chart performs better than the CPC chart for the pre-specified dispersion shift. In practice, when the magnitude of a future dispersion shift is unknown, it is always desired to design a control chart to perform reasonably well over a range of shifts rather than to optimize the performance at detecting a particular level of shifts. Compared with SR based on a pre-specified dispersion shift, an adaptive SR (ASR) chart that continually adjusts its form to be efficient for signaling a smoothing exponentially weighted moving average (EWMA) statistic in deviation from its target value is proposed in this paper. It can be easily implemented and numerical results show that it balances protection against a broad range of shift sizes.     

Short run spc using dynamic control chart

A generalization of the EWMA control chart, referred to within as the Dynamic EWMA control chart, is developed. The control chart is based upon a first-order, constant variance, dynamic linear model. Its applicability for short production run SPC applications described.     

基于多重相关状态采样的多元EWMA控制图

为提高控制图的监测效率,提出了一种基于多重相关状态采样的多元EWMA控制图,并利用改进后的马尔可夫链方法计算控制图的平均运行长度。根据不同参数下控制图的平均运行长度,分析了控制图在失控和受控状态下的性能表现,并与其它多元EWMA控制图进行比较。模拟结果表明,该控制图具有良好的监测能力。最后用一组模拟数据来说明该方法的使用。     

A hybrid fuzzy adaptive sampling - Run rules for Shewhart control charts

In crisp run control rules, usually it is stated that a process moves very sharply from in-control condition to out-of-control act. This causes an increase in both false-alarm rate and control chart sensitivity. Moreover, the classical run control rules are not implemented on an intelligent sampling strategy that changes control charts’ parameters to reduce error probability when the process appears to have a shift in parameter values. This paper presents a new hybrid method based on a combination of fuzzified sensitivity criteria and fuzzy adaptive sampling rules, which make the control charts more sensitive and proactive while keeping false alarms rate acceptably low. The procedure is based on a simple strategy that includes varying control chart parameters (sample size and sample interval) based on current fuzzified state of the process and makes inference about the state of process based on fuzzified run rules. Furthermore, in this paper, the performance of the proposed method is examined and compared with both conventional run rules and adaptive sampling schemes.     

Design and optimization of EWMA control charts for in-control,indifference, and out-of-control regions

The design of quality control charts is normally carried out considering a process shift size that is considered important to be detected. The EWMA control chart is one of the best available options to use when good performance is needed to detect small process shifts. This paper presents a method for design of EWMA charts for control processes, in which the detection of small shifts is not necessary, and at the same time is effective in detecting important shifts. In such cases the EWMA control chart can also be designed successfully to deal with these requirements. A Markov chain approach is also applied to determine the ARL of the modified EWMA control chart. The implementation and interpretations are provided and numerical examples are used to illustrate the application procedure. We also investigate some basic properties of the proposed scheme. Genetic algorithms have been used to carry out this design.     

自适应EWMA控制图统计经济设计

针对自适应指数加权移动平均(AEWMA) 控制图统计经济设计问题, 给出AEWMA控制图统计经济设计模型, 提出一种在偏移区间上对AEWMA控制图进行设计的多目标优化方法. 针对不同的偏移区间优化了AEWMA控制图, 并将AEWMA控制图统计经济性能与指数加权移动平均(EWMA) 控制图相比较. 结果表明, 所提出方法优化设计的AEWMA控制图仍具有克服EWMA控制图的惯性问题的统计特性, AEWMA控制图的经济性能也优于EWMA控制图.

     

An EWMA chart for monitoring the process standard deviation when parameters are estimated

The EWMA chart for the standard deviation is a useful tool for monitoring the variability of a process quality characteristic. The performance of this chart is usually evaluated under the assumption of known parameters. However, in practice, process parameters are estimated from an in-control Phase I data set. A modified EWMA control chart is proposed for monitoring the standard deviation when the parameters are estimated. The Run Length properties of this chart are studied and its performance is evaluated by comparing it with the same chart but with process parameters assumed known.     

Stability analysis and optimal tuning of EWMA controllers – Gain adaptation vs. intercept adaptation

Exponentially weighted moving average (EWMA) controllers are the most commonly used run-to-run controllers in semiconductor manufacturing industry. An EWMA controller can be implemented in two different ways. One way is to keep the process gain as its off-line estimate and update the intercept term at each run, which is termed EWMA with intercept adaptation; the other is to keep the intercept term as its off-line estimate and update the process gain at each run, which is termed EWMA with gain adaptation. Despite the fact that gain variation and adaptation is typical in semiconductor industry, most EWMA formulations are for intercept adaptation and few results exist on the stability and sensitivity of EWMA with gain adaptation. In this paper, we propose a general formulation to analyze the stability of both EWMA controllers. The proposed state-space representation not only reveals the similarities and differences between two types of EWMA controllers, but also explains why the stability conditions for both types of EWMA controllers are independent of process disturbances. In addition, we propose a general framework that unifies the analysis of the optimal control performance for both types of EWMA controllers. The proposed framework is different from existing approaches in that it decouples the state estimation from the control law, and derives the optimal weighting based on the state estimation performance. The proposed framework significantly simplifies the analysis procedure, especially for EWMA with gain adaptation. Using this framework, we derive the optimal EWMA weighting through solving the discrete-time algebraic Riccati equation (DARE) for various process disturbances that are encountered in semiconductor manufacturing industry. Simulation examples are given to illustrate the optimality of the EWMA weighting derived using the framework. Some practical aspects of controller tuning are also discussed based on the simulation results.     

Smoothed additive estimators for non-error rates in multiple discriminant analysis

A generalization of smoothed additive estimators for non-error rates to the case of more than two groups is discussed. Several properties the smoothing should have are shown to be satisfied. The problem of choosing a smoothing parameter is considered and a parameter choice depending on the sample is proposed. In simulation experiments with normal, uniform and discrete distributions the smoothed additive estimators with fixed and variable smoothing parameter are compared to the leaving-one out method and the resubstitution method with respect to bias and variance.