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The double exponentially weighted moving average (dEWMA) feedback controller is a popular run-to-run (RTR) controller that is used to adjust semiconductor manufacturing processes that have a linear drift. Although this controller, with suitable fixed discount factors, can guarantee long-term stability, it usually requires a moderately large number of runs to bring the process output to its target value. To overcome this difficulty, an enhanced controller called the 'adaptive variable EWMA controller,' is proposed. This controller brings the process output to a desired target much more quickly. An analytical expression of the process output of this controller and its long-term stability conditions are derived. Furthermore, examples are presented that compare the performance of the proposed controller with several competing controllers. It is demonstrated that the proposed controller has the best performance of the considered controllers in terms of the reduction of total mean square errors. [Supplementary materials are available for this article. Go to the publisher's online edition of HE Transactions for the following supplemental resource: Appendix] Keywords: Run-to-run controller, drifted processes, EWMA controller, adaptive, variable EWMA controller, 1. Introduction On-line process control has been extensively adopted by semiconductor manufacturers. Statistical Process Control (SPC) and Engineering Process Control (EPC) are two fundamental process control techniques that can be [...]

The Exponentially Weighted Moving Average (EWMA) feedback controller is a popular model-based run-to-run controller which primarily uses data from previous process runs to adjust settings for the next run. The long-term stability conditions of EWMA controllers for this closed-loop system have received considerable attention in the literature. Most of the reported results are obtained under the assumption that the process I-O (Input-Output) relationship follows a static model. Generally speaking, the effect of the input recipe on the output response can be carried over several periods. In this paper, focusing on a first-order dynamic I-O model and assuming that the process disturbance follows a general ARIMA series, a systematic approach to address this control problem is proposed. First, the long-term stability conditions of a single EWMA controller are investigated. Then, the determination of sample size to allow the design of a single EWMA controller for dynamic models is considered. Under the assumption that the process I-O variables follow a bivariale normal distribution, a formula to calculate sample sizes that allow the stability condition to be met with a minimum probability protection is derived. Finally, the effects of dynamic parameters on the determination of sample size are considered. [Supplementary materials are available for this article. Go to the publisher's online edition of IIE Transactions for the following free supplementary resource: Appendix] Keywords: Run-to-run control, dynamic models, single EWMA controller, stability conditions, 1. Introduction Run-to-run (R2R) process control techniques have been extensively studied and are widely utilized in semiconductor manufacturing operations (Sachs et al., 1995; Hamby et al., 1998; Chen and Guo, [...]

In existing pH acceleration models, which are used to assess the shelf-life of liquid-phase nano-sol products, a mixture of normal distributions is commonly employed to describe the sizes of particles mixing from two populations. The Gaussian mixture model approach falls short when used to characterize the asymmetric distributions of particle sizes in subgroups. This work considers instead a broader class of a mixture of log-F distributions, to be embedded in a pH acceleration model. This study aims at understanding the impact of the new modeling approach, in the presence of model misspecification of the particle size distribution, on the accuracy and precision for making the shelf-life predictions. This study found that model misspecification indeed significantly affects the shelf-life predictions. The proposed method shows favorable finite sample performance on a simulated data set. Both the quantitative analysis of the impact due to model misspecification and the solution proposed herein could benefit practitioners in the long run. [ABSTRACT FROM AUTHOR]

Metrology delay is a natural problem in the implementation of advanced process control scheme in semiconductor manufacturing systems. It is very important to understand the effect of metrology delay on performance of advanced process control systems. In this paper, the influences of metrology delay on both the transient and asymptotic properties of the product quality are analyzed for the case when a linear system with an initial bias and a stochastic autoregressive moving average (ARMA) disturbance is under an exponentially weighted moving average (EWMA) run-to-run control. Tuning guidelines are developed based on the study of numerical optimization results of the analytical closed-loop output response. In addition, effective metrology delay of a variable time delay system is analyzed based on the resampling technique implemented to a randomized time delay system. A virtual metrology technique is a possible solution to tackle the problem of metrology delay. The tradeoff between additional error of virtual metrology and reduction in time delay is studied. The results are illustrated using an example of control of the tungsten deposition rate in a tungsten chemical-vapor deposition reactor. The basic conclusion is that metrology delay is only important for processes that experience nonstationary stochastic disturbance. In such a case, use of virtual metrology is justified if the error of the virtual metrology method is less than the error caused by stochastic process noise. The accuracy of the virtual metrology noise with respect to the traditional metrology is not critical, provided that the error due to metrology is much less than that due to process disturbances. Index Terms--Asymptotic mean square error (AMSE), exponentially weighted moving average (EWMA) controller, resampling, run-to-run (RtR) control, transient.

Many semiconductor manufacturing processes have, by nature, Multiple-Input and Multiple-Output (MIMO) variables. For a drifted MIMO process, the double Multivariate Exponentially Weighted Moving Average (dMEWMA) controller is a popular run-to-run controller for adjusting the process mean to a desired target. The stability conditions of a dMEWMA controller have received considerable attention in the literature. There are two limitations in the existed literature. First, these approaches only demonstrate that the asymptotic mean will meet a desired target; however, they do not show that mean square error matrix for a process is bounded. Second, they do not have an explicit expression of the process output of the dMEWMA controller for any production run t. Hence,by using a conventional state space formulation, it is impossible to address the process performance (such as process fallout rate of the dMEWMA controller) analytically. Using a different approach, this paper derives an analytical expression for the process output of a dMEWMA controller. The problems of the stability conditions and the performance of the dMEWMA controller can then be addressed successfully. Keywords: Run-to-run process control, multiple-input and multiple-output (MIMO), process disturbance, multivariate EWMA (MEWMA) controller, double MEWMA (dMEWMA) controller, stability conditions, optimal discount matrix, 1. Introduction The goal of process (or quality) control is to keep the process output as close to a desired target as possible. In the area of statistical process control, [...]

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. Keywords: Run-to-run controller, drifted processes, variable EWMA controller, 1. Introduction Statistical Process Control (SPC) and Engineering Process Control (EPC) are two widely used process control techniques. The goal of both of these techniques is to control a process [...]

The double exponentially weighted moving average (dEWMA) controller is a popular run-to-run controller for a drifted process. It uses the information collected from past process data to adjust the process parameters for the later runs. The dEWMA controller can guarantee long-term stability under suitably fixed discount factors and fairly regular conditions. However, it usually requires a large number of runs to bring the process output to meet its target and, thus, may leave the output out of its specification at the beginning of the first few runs. Hence, dEWMA controller is not suitable for the processes with short production runs. To reduce the possibility of high rework rate, we propose an enhanced dEWMA controller, which we refer to as the initial intercept iteratively adjusted (IIIA) controller, to further eliminate the off-target (nonrandom biases) of the process output. We derive an analytic expression of the process output of the IIIA controller, and present several examples to show that the IIIA reduces the process mean square error significantly, as well as the rework rate. Index Terms--Double exponentially weighted moving average (dEWMA) controller, initial intercept iteratively adjusted (IIIA) controller, recursive least squares (RLS), run to run (R2R) process control.

The double exponentially weighted moving average (dEWMA) feedback control scheme, a conventional run-to-run control scheme, can adjust certain semiconductor manufacturing processes with a linear drift. The long-term stability conditions for this closed-loop system have received considerable attention in literature. These stability conditions can be expressed in terms of the predicted model assuming that an initial process input-output (I-O) predicted model can be obtained successfully in advance. However, the predicted model is constructed by a random sample of I-O variables, and therefore the strength of the linear relationship between I-O variables plays a major role in determining the validation of these stability conditions. In order to design a stable dEWMA control scheme, the covariance (or correlation) structure of I-O variables and the number of experiments should be simultaneously considered. By controlling a guaranteed probability of stability, this study first derives the formula for an adequate sample size required to construct the predicted model in the case of single-input single-output and multiple-input single-output systems. Illustrative examples demonstrate the effectiveness of the covariance structure of I-O variables in determining the sample size. Implications for research on multiple-input multiple-output systems are also addressed. Index Terms--Double exponentially weighted moving average (dEWMA) control scheme, multiple correlation coefficient, run-to-run process control, sample size, stability conditions.

The conventional burn-in test, which only collects go/no go (or time-to-failure) data over short testing periods becomes rather ineffective for highly reliable products. This decision problem can be solved if there exists a suitable quality characteristic whose degradation over time can be related to the product's lifetime. An innovative approach to determine the burn-in policy by using a suitable degradation model has been suggested in the literature. However, a major disadvantage of this procedure is that it only uses information contained in current degradation data, and it ignores any information given by the entire sequence of observations. Thus, it becomes relatively insensitive to detecting weak items if the degradation path of the product degrades very slowly. To remedy this weakness, similar to the CUSUM control charting scheme in statistical process control, we propose an integrated Wiener process to model the cumulative degradation path of the product's quality characteristic. Then the optimal burn-in policy can be easily obtained. In addition, we also use an analytic approach to compare the efficiency of our proposed procedure with the method that is previously reported in the literature., 1. Introduction In today's increasingly intense globally competitive marketplace, it is critical for manufacturers to design an optimal burn-in policy to eliminate latent failures or weak components before their products [...]

The exponentially weighted moving average (EWMA) feedback controller (with a fixed discount factor) is a popular run by run control scheme which primarily uses data from past process runs to adjust settings for the next run. Although the EWMA controller with a small discount factor can guarantee a 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. This is impractical for process with small batches. The reason is that the output deviations are usually very large at the beginning of the first few runs and, as a result, the output may be out of process specifications. In order to reduce a possibly high rework rate, the authors propose a variable discount factor to tackle the problem. They state the main results in which the stability conditions and the optimal variable discount factor of the proposed EWMA controller are derived. An example is given to demonstrate the performance. Moreover, a heuristic is proposed to simplify the computation of the variable discount factor. It is seen that the proposed method is easy to implement and provides a good approximation to the optimal variable discount factor. Index Terms--Run by run process control, single EWMA controller, variable discount factor.

A termination rule is proposed for identifying a degradation experiment's appropriate stopping time. Experiment duration is important for designing product mean time-to-failure tests. The rule, which was validated using an LED device, was found to be computationally simple. The method is ideal for reliability estimation since it provides product lifetime on-line real-time information after identification of termination time.

The traditional Economic Manufacturing Quantity (EMQ) model assumes that the production process (system) is perfect. However, owing to aging, the production process will shift from the 'in-control' state to the 'out-of-control' state and produce defective items. When the process shift distribution follows an Increasing Failure Rate (IFR) distribution, preventive (scheduled) maintenance policy is usually used to enhance its reliability. In this paper we incorporate a preventive maintenance policy into this deteriorating production system and derive an optimal preventive maintenance policy. Two well-known IFR distributions, Weibull and extreme-value distributions, are considered and some examples are used to illustrate the proposed model. Finally, the advantages of the proposed model are addressed in the conclusion., 1. Introduction The traditional Economic Manufacturing Quantity (EMQ) model assumes that the production process is perfect. However, owing to aging, the production process will deteriorate from the 'in-control' state to [...]

A comparison of type-I and type-II accelerated life tests for estimating the life of highly reliable products in a short time shows that the former is easier to use and requires fewer samples than the latter but takes a longer time. The most reliable product is selected using a simple rule both when the acceleration factor is known and when it is not known. The comparison factors are computational complexity, case of selection-rule application, product cost, overall and individual average test times.

A selection guide for choosing the best product or material from type-II censored accelerated life test (ALT) data using the Arrhenius model is presented. The rule uses the average ratio of time-saving during life testing, number of failures and sample size as gauges in selecting the design with optimal reliability. A numerical example is discussed to illustrate the rule's superiority over non-ALT selection guides.

Accelerated life tests (ALTs) are widely used in industry and many optimal ALT test plans are developed in literature. To achieve an efficient optimal test plan, one needs to make sure that the planning values used in the test planning are not far away from the true parameters, which is challenging because lifetime data are not collected yet in the planning stage to estimate the true parameters. To overcome the weakness, sequential designs are developed and the parameter estimates are updated during the sequential testing process. Due to the sequential nature, one drawback of the sequential strategy is that it usually requires a long total testing time (TTT). Motivated by industrial needs, we propose batch sequential design (BSD) in ALT setting, in which multiple units are tested simultaneously while sequential parameter updates are also made. The proposed BSD for ALT can not only reduce the TTT but also preserves the advantage of sequential designs. After proposing two types of BSD strategies, we use a comprehensive simulation study to demonstrate the advantages of the proposed BSD in ALT setting. We also develop a method to find the optimal batch size which is essential for the deployment of the BSD in practice. [ABSTRACT FROM AUTHOR]

The performance of reliability inference strongly depends on the modeling of the product's lifetime distribution. Many products have complex lifetime distributions whose optimal settings are not easily found. Practitioners prefer to use simpler lifetime distribution to facilitate the data modeling process while knowing the true distribution. Therefore, the effects of model mis-specification on the product's lifetime prediction is an interesting research area. This article presents some results on the behavior of the relative bias (RB) and relative variability (RV) of pth quantile of the accelerated lifetime (ALT) experiment when the generalized Gamma (GG3) distribution is incorrectly specified as Lognormal or Weibull distribution. Both complete and censored ALT models are analyzed. At first, the analytical expressions for the expected log-likelihood function of the misspecified model with respect to the true model is derived. Consequently, the best parameter for the incorrect model is obtained directly via a numerical optimization to achieve a higher accuracy model than the wrong one for the end-goal task. The results demonstrate that the tail quantiles are significantly overestimated (underestimated) when data are wrongly fitted by Lognormal (Weibull) distribution. Moreover, the variability of the tail quantiles is significantly enlarged when the model is incorrectly specified as Lognormal or Weibull distribution. Precisely, the effect on the tail quantiles is more significant when the sample size and censoring ratio are not large enough. for this article are available online. [ABSTRACT FROM AUTHOR]

Shelf-life prediction of liquid-type (such as nano-sol) products is an interesting research topic. Recently, a pH acceleration method has been proposed in the literature to address the shelf-life prediction of nano-sol. The time evolution of particle-size distribution was obtained and modelled. There are two approaches for modelling the particle-size distribution, either by using a parametric approach (mixture-normal distribution) or a distribution-free approach. The main goal of this study is to quantify the seriousness of model misspecification on the shelf-life prediction when the true particle-size distribution follows a mixture-normal distribution, but wrongly treated it by a distribution-free model. The results demonstrated that the relative bias of shelf-life prediction may be under-estimated up to 13.66%, while its relative variability may be seriously inflated up to 13.35 times by using a distribution-free approach. The study shows that these effects of model misspecification are moderately large and cannot be neglected even when the sample size and the measurement times are small. [ABSTRACT FROM AUTHOR]

Consider a product that has several quality attributes for inspection where the inspection of some attributes may have carryover effect(s) on subsequent inspections of other attributes. In this case, not all the attributes can be inspected on the same unit and therefore more than one unit may be required to complete one inspection cycle of all the attributes. Furthermore, if the inspection time of each attribute is considered, the number of attributes and their inspection sequence on each inspected unit may affect the time-span of an inspection cycle. In this paper, heuristic procedures are developed to find the inspection sequences of all the attributes that minimize the total number of inspected units and to balance the inspection time of each inspected unit such that the time-span of an inspection cycle is minimized. An integrated circuit card inspection example is given to illustrate the implementation of these procedures. [ABSTRACT FROM AUTHOR]