基于变折扣因子EWMA的混合产品批间控制方法

批间(run-to-run,简称R2R)控制现今已被广泛用于半导体生产行业.指数加权移动平均(exponet weighted moving average.EWMA)是R2R控制的一种重要算法.折扣因子是EWMA控制期的主要参数.本文在模型中考虑了实际生产过程中混合产品少量多样的特点,引入了基于产品的变折扣因子EWMA控制算法,解决了产品切换时制程输出收敛速度过慢的问题.变折扣因子的引入提高了制程输出的响应速率而并不影响制程输出的稳定性.对实际过程的模拟仿真检验了该控制算法的可行性和优越性.     

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.     

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.     

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.     

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.     

含随机测量时延的EWMA批间控制器设计

批间控制(RtR)是半导体晶圆生产过程控制的有效算法. 然而, 受测量手段与测量成本的限制, 难以实时检 测晶圆的品质数据, 即: 存在一定的测量时延, 通常该测量时延是随机, 时变的, 且直接影响批间控制器的性能. 为 此, 本文基于指数加权移动平均(EWMA)算法, 提出一种含随机测量时延的扰动估计方法. 在分析测量概率的基础 上, 建立包含测量时延概率的扰动估计表达式; 并采用期望最大化(EM)算法估计该测量时延的概率; 然后分析系统 可能存在的静差项, 给出相应的补偿算法; 最后讨论系统的稳定性. 仿真实例验证所提算法的有效性.     

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.     

Multiple-input multiple-output double exponentially weighted moving average controller using partial least squares

This paper applies the partial least squares (PLS) method to the multiple-input multiple-output (MIMO) semiconductor processes in the run-to-run (R2R) control practice. Due to the property of batch processing, the semiconductor manufacturing processes frequently exhibit high multicollinearity among input variables and dependency among output variables. These two effects will typically cause variance inflation of the regression coefficient estimates which are utilized in triggering or updating the R2R controller. Furthermore, the process nonlinearity is also likely to occur in some semiconductor processes. As the nonlinearity exists, the performance of the exponentially weighted moving average (EWMA) controller is not adequate and becomes aggravated after a few transient runs. The PLS method is, essentially, well suited for situations where multicollinearity is present among input variables. To rectify the aforementioned difficulties that might realistically take place in practice, the PLS method is considered in this paper a potential estimation alternative to the standard regression method. Three types of R2R simulation studies are conducted to verify the advantages of the PLS method. The simulation results show that using the PLS method as the model-building technique helps the EWMA controller to yield more consistent and robust control outputs than purely using the conventional EWMA controller.     

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.

     

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 multivariate control chart for simultaneously monitoring process mean and variability

Recently, monitoring the process mean and variability simultaneously for multivariate processes by using a single control chart has drawn some attention. However, due to the complexity of multivariate distributions, existing methods in univariate processes cannot be readily extended to multivariate processes. In this paper, we propose a new single control chart which integrates the exponentially weighted moving average (EWMA) procedure with the generalized likelihood ratio (GLR) test for jointly monitoring both the multivariate process mean and variability. Due to the powerful properties of the GLR test and the EWMA procedure, the new chart provides quite robust and satisfactory performance in various cases, including detection of the decrease in variability and individual observation at the sampling point, which are very important cases in many practical applications but may not be well handled by existing approaches in the literature. The application of our proposed method is illustrated by a real data example in ambulatory monitoring.     

Trend Autoregressive Model Exact Run Length Evaluation on a Two-Sided Extended EWMA Chart

The Extended Exponentially Weighted Moving Average (extended EWMA) control chart is one of the control charts and can be used to quickly detect a small shift. The performance of control charts can be evaluated with the average run length (ARL). Due to the deriving explicit formulas for the ARL on a two-sided extended EWMA control chart for trend autoregressive or trend AR(p) model has not been reported previously. The aim of this study is to derive the explicit formulas for the ARL on a two-sided extended EWMA control chart for the trend AR(p) model as well as the trend AR(1) and trend AR(2) models with exponential white noise. The analytical solution accuracy was obtained with the extended EWMA control chart and was compared to the numerical integral equation (NIE) method. The results show that the ARL obtained by the explicit formula and the NIE method is hardly different, but the explicit formula can help decrease the computational (CPU) time. Furthermore, this is also expanded to comparative performance with the Exponentially Weighted Moving Average (EWMA) control chart. The performance of the extended EWMA control chart is better than the EWMA control chart for all situations, both the trend AR(1) and trend AR(2) models. Finally, the analytical solution of ARL is applied to real-world data in the health field, such as COVID-19 data in the United Kingdom and Sweden, to demonstrate the efficacy of the proposed method.     

考虑Taguchi损失函数时自适应指数加权平滑控制图的经济统计设计

为解决指数加权平滑(EWMA)控制图惯性问题而提出的自适应EWMA(adaptive EWMA,AEWMA)控制图的统计特性已经被广泛研究,但AEWMA控制图经济特性的研究却从未见有成果发表.针对该问题,在考虑Taguchi损失函数的基础上,给出了AEWMA控制图经济统计设计的模型.提出了一种在偏移区间上对AEWMA控制图进行优化设计的方法,用该方法优化设计的AEWMA控制图与针对固定偏移优化设计的EWMA控制图进行了比较.结果表明该方法设计的AEWMA控制图仍然保持其解决EWMA控制图惯性问题的特性,AEWMA控制图的经济特性同样优于EWMA控制图.分析了AEWMA控制图经济统计设计的参数灵敏度,总结了AEWMA控制图的参数变化与损失、平均链长以及最优参数组合之间的关系.     

New EWMA control charts for monitoring process dispersion

There exist two EWMA-type dispersion charts for monitoring dispersion increases in the literature. One resets the EWMA statistic to zero whenever it is below zero. The other one truncates negative normalized observations to zero in the EWMA statistic. This paper proposes two one-sided EWMA charts for detecting dispersion increases and decreases, respectively, and one two-sided EWMA chart for monitoring dispersion increases or decreases simultaneously. Simulation studies show that the proposed upper-sided EWMA chart performs better than the two existing counterparts for detecting increases in dispersion, and that the proposed lower-sided EWMA chart significantly outperforms the two lower-sided EWMA charts developed similar to their two existing upper-sided EWMA charts for detecting decreases in dispersion. Moreover, the proposed two-sided EWMA chart provides much better sensitivity than the two two-sided EWMA charts generalized from the two existing upper-sided EWMA charts for detecting overall changes in dispersion.     

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.     

Petroleum demand forecasting for Taiwan using modified fuzzy‐grey algorithms

In this paper, we adopt the exponentially weighted moving average (EWMA) method to develop the residual modification EWMA grey forecasting model REGM(1,1) and combines it with fuzzy theory to derive the fuzzy REGM or the FREGM(1,1) model. The proposed model is used to forecast annual petroleum demand in Taiwan. The experimental results show that the mean absolute percentage errors, median absolute percentage error, and symmetric mean absolute percentage error of FREGM(1,1) model are higher by 23.71, 12.26, and 23.06% respectively, compared with those obtained using the traditional GM(1,1) model.     

Cycle forecasting EWMA (CF-EWMA) approach for drift and fault in mixed-product run-to-run process

In semiconductor manufacturing processes, mixed-products are usually fabricated on the same set of process tool with different recipes. Run-to-run controllers which based on the exponential weighted moving average (EWMA) statistic are probably the most frequently used in industry for the quality control of certain semiconductor manufacturing process steps. However, for mixed-product drifted process, if the break length of a product is large, then the process output at the beginning runs of each cycle will far deviate from the target value which will lead to a possible high rework rate and lots of waste wafers. Therefore, this study aims to develop a new approach named cycle forecasting EWMA (CF-EWMA) approach to deal with the problem of large deviations in the first few runs of each cycle. Furthermore, a common fault, i.e., the step fault, is also considered in this paper, and fault tolerant cycle forecasting EWMA (FTCF-EWMA) approach is proposed. Simulation study shows that the proposed approaches are effective.     

Optimization of univariate and multivariate exponentially weighted moving-average control charts using genetic algorithms

Exponentially weighted moving-average (EWMA) and multivariate EWMA (MEWMA) process control charts can be applied to detect small changes in statistical process control efficiently. This paper presents a software program developed in Windows environment for the optimal design of the EWMA and MEWMA chart parameters, to protect the process in the case of shifts of given size. Optimization has been done using genetic algorithms.     

基于多元线性自回归模型的流量预测

为使流量预测模型具有自适应性和相关性,以时间点为基础进行建模,结合时间序列与流量序列,引入多元线性自回归(AR)思想进行参数估算,对多次估算所得参数值建立指数加权移动平均数模型进行二次估算,在此基础上,建立多元线性自回归模型。实验结果证明,与AR模型、ARMA模型相比,基于多元线性AR模型的预测结果更准确。