A study of variable ewma controller - IEEE Xplore Digital Library

IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 16, NO. 4, NOVEMBER 2003

633

A Study of Variable EWMA Controller Sheng-Tsaing Tseng, Member, IEEE, Arthur B. Yeh, Fugee Tsung, and Yun-Yu Chan

Abstract—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.

I. INTRODUCTION

S

TATISTICAL process control (SPC) and automatic/engineering process control (APC/EPC) are two widely used process control techniques. The goal of both is to control a process so that it meets the target with minimal variability. Box and Luceño [2] gave a general introduction to both SPC and APC techniques. SPC mainly applies statistical control charts to monitor the process outputs, detect out-of-control signals, and then take corrective actions to remove the root causes. APC, on the other hand, implements either feedback or feedforward control schemes to actively adjust the process inputs (see Tsung et al. [11], Tsung et al. [13]). In recent years, many researchers and practitioners have looked into how to integrate and combine the strengths of SPC and APC (Tsung and Shi [12], and references therein). Among them, the exponentially weighted Manuscript received October 22, 2002; revised June 24, 2003. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Contract NSC-90-2118-M-007-009. The work of S.-T. Tseng was supported by RGC Competitive Earmarked Research Grant HKUST6011/01E and Grant HKUST6183/03E. The work of A. B. Yeh was supported in part by a Summer Research Grant from the College of Business Administration, Bowling Green State University, Bowling Green, OH. S.-T. Tseng and Y.-Y. Chan are with the National Tsing-Hua University, Institute of Statistics, Hsinchu, Taiwan, R.O.C. (e-mail: [email protected]). A. B. Yeh is with the Department of Applied Statistics and Operations Research, Bowling Green State University, Bowling Green, OH 43403 USA (e-mail: [email protected]). F. Tsung is with the Department of Industrial Engineering and Engineering Management, Hong Kong University of Science and Technology, Kowloon, Hong Kong (e-mail: [email protected]). Digital Object Identifier 10.1109/TSM.2003.818960

moving average (EWMA) run by run (RbR) feedback control scheme has received great attention. The idea of RbR feedback control is to control the process output to meet its target by adjusting the process input variables for every production run. Each adjustment is calculated based on the output deviations from the target of the previous run. One typical example of an RbR control application is the photolithography process in semiconductor manufacturing. As its process parameter settings cannot be changed/manipulated during runs, RbR control is usually applied to control the feature size (critical dimension) by adjusting the parameter such as the exposure dose (energy) on the wafers between each run. The RbR control techniques have been applied to other semiconductor manufacturing processes such as reactive ion etching (RIE), the chemical mechanical polishing (CMP) processes, etc. Ingolfsson and Sachs [6] proposed a procedure for adjusting the process by using a single EWMA controller. They investigated the stability and sensitivity of the process output under different process models. Butler and Stefani [3] showed how a double EWMA controller can be used to regulate the input variables of a polysilicon gate etch process to compensate for the effect of process drift on the process outputs. Del Castillo [4] and Tseng et al. [9] investigated the stability conditions, the long-run behavior, the transient performance, and the determination of the optimal discount factor of the double EWMA controller. All the aforementioned models are restricted to the case when the discount factor is a fixed constant. Although the EWMA controller with a small discount factor can guarantee long-term stability (under fairly regular conditions), it usually requires a moderately large number of runs to bring the process output to its target. This will lead to severe consequences particularly in small-batch fabrication, commonly seen in semiconductor manufacturing processes. The reason is that the output deviations from the target are usually very large at the first few production runs and, as a result, the process may be out of specifications. In order to reduce a possibly high rework rate, a variable discount factor is proposed to overcome the difficulty. Instead of considering a fixed discount factor, this paper focuses on the application of a variable discount factor for the RbR feedback control. It is organized as follows: a motivating example is used to demonstrate the advantage of the variable discount factor. It is seen that the EWMA controller with a variable discount factor has better performance than the conventional EWMA controller with a fixed discount factor. We then state the main results in which the stability conditions and the optimal variable discount factor of this EWMA controller are derived. An example is given to further demonstrate its performance. We also propose a heuristic to simplify the computation of the variable discount factor. It is seen that

0894-6507/03$17.00 © 2003 IEEE

634

IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 16, NO. 4, NOVEMBER 2003

Fig. 1. Comparison of fixed and variable discount factors.

the proposed method is quite robust when compared with the optimal variable discount factor. II. MOTIVATION AND PROBLEM DESCRIPTION In an RbR feedback control, assume denotes the value of the process quality characteristic (output) for batch (or run) denotes the input variable chosen at the number and end of each production run. Then, the following linear model is usually used to describe the process single input-single output (SISO) relationship (1) is the process disturbance, and and are unknown where and denote the parameters to be estimated from data. Let and are typinitial estimate of and , respectively. Here ically chosen to be the least square estimates (LSE) of and based on data. At the beginning of an RbR control, we set , where is the desired target value of the process output. Ingolfsson and Sachs [6] proposed the following EWMA equations to recursively update the estimate of the unknown parameter and input variable , for

and (2) is called the fixed discount factor. Note where that the key idea of EWMA control scheme is that for a predetermined , we only need to recursively update the value of intercept and the input variable. The expected output will then asymptotically converge to the desired target. Therefore, the associated adjustment with respect to the slope is not considered in current study. follows a nonAssume that the process disturbance stationary IMA(1, 1) series, a commonly used disturbance in control models (Box et al., [1]), that is (3)

where

Under the condition that , it can be shown that is the optimal fixed discount factor for the EWMA controller and the output of this optimal controller will asymptotically converge to its target. However, inaccurate estimates of the unknown parameters ( and ) lead to a large value and it usually requires several of initial bias runs for the conventional EWMA controller to bring the process output back to the target. Let us consider the following example. Example 1: By slightly modifying the illustrative example of Del Castillo [4], a numerical illustration is used to describe the feedback control scheme. Set the true parameters in (1) and , and . Assume (3) to be is and the target value that the LSE of . The solid line in Fig. 1 presents simulated is set to be outputs of an EWMA controller with the optimal discount factor . From Fig. 1, it is seen that the process output will gradually converge to its target value even though the slope is not precisely estimated. However, it also indicates that the in the first few runs are very large, which deviations may cause the process output to be out of specifications. In order to accelerate rate of convergence, a larger discount factor is needed at the first few runs of an RbR control. Consider the following control scheme with a variable (nonfixed) : discount factor, for all (4) where (5) Shown in Fig. 1 are the comparisons of the simulated process outputs by fixed (with solid line) and variable (with dotted line) . discount factors with It is seen that the variable EWMA controller is better at reducing the process variation, particularly in the first few production runs. This simple example raises the following interesting

TSENG et al.: STUDY OF VARIABLE EWMA CONTROLLER

635

issues. By adopting a variable discount factor, it is possible to further reduce the bias of the process output, thus bringing the output closer to the target, at the beginning of the production run. More importantly, such a reduction can be done at no loss of long-term process stability since the two processes in Fig. 1 are virtually identical after approximately 15 production runs. In modern semiconductor manufacturing environments where multiple products are typically manufactured in relatively short production runs, a control scheme that greatly reduces the process bias within the first few production runs could potentially be very beneficial to manufacturers. In the remaining sections, we intend to fully investigate the EWMA controller based on the variable discount factor. We attempt to answer several fundamental questions with regard to the proposed EWMA control scheme. 1) Under what conditions will the controller achieve long-term stability? 2) How can the optimal variable discount factor be analytically derived under finite production runs? 3) Compared to the conventional EWMA (with a fixed discount factor) controller, how is the performance of the variable EWMA controller? 4) How can the optimal variable discount factor be obtained in practical applications? III. STABILITY CONDITIONS For any control scheme to be of potential value, a fundamental requirement is that the process, as regulated by a chosen control scheme, achieves long-term process stability. In other words, the control scheme should lead the process under control to a stable state. More specifically, a process is said to be asymptotically stable if and

(6)

The stability of a process ensures that the mean of the process output converges to the desired target, while its asymptotic variance remains bounded. The question naturally arises: under what conditions will a process regulated under a variable discount factor EWMA controller achieve long-term stability? The following lemma gives a very useful representation of the process output in terms of the initial bias, the variable discount factor, and the process disturbance (the proof is given in Appendix I). and as follows: Define the initial bias (7) and

where

(10) Assume that the process disturbance model, namely general ARIMA

in (1) follows a (11)

where

and where white noise series with variance ; back shift operator; positive integers. Now, the stability conditions for an EWMA controller with a variable discount factor can be stated as follows. Theorem 1 (Stability Conditions for the Variable Discount Factor): An EWMA controller with a variable discount factor and , for all . [as in (4)] is asymptotically stable if Theorem 1 (the proof can be found in Appendix II) ensures the stability of the EWMA controller with a variable discount factor for most stationary and nonstationary processes, including autoregressive moving-average (ARMA) processes and the popular first-order integrated moving average . From Theorem 1, the feasible (IMA(1, 1)) processes or , and region of a variable discount factor is (0, 1) if . The inequality indicates that when the slope is known , there must be a larger discount factor corresponding to a smaller for a quick convergence rate. is oversized, there is a risk that , and However, if the variation of process outputs may be undesirably enlarged. IV. OPTIMAL VARIABLE DISCOUNT FACTOR From a practical point of view, knowing that a process is asymptotically stable does not necessarily help in implementing an appropriate controller. This is equivalent to choosing a variable discount factor such that some predetermined criterion is met. When small batches are produced, the production cycle is not long enough for the long-term stability to take place. Under this situation, however, it would be preferable to find the optimal variable discount factor by minimizing the total mean square error (MSE) within the first N runs. Note that MSE at time is a function of . Thus, our goal is to solve the following decision problem:

(8) (12) Then, we have the following result. Lemma 1: The adjusted process output can be expressed as (9)

For a general process disturbance, it is not easy to derive an explicit form for the optimal variable discount factor. In the following, we will discuss the derivation of the optimal variable discount factor under two well-known cases.

636

IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 16, NO. 4, NOVEMBER 2003

1) Case A: follows an ARMA(1, 1) series. From (10), it can be seen that

, which minimizes can be obtained by

for any given

,

(13a) and is defined in the equation found at the bottom of the page. The proof of (13a) is given in Appendix III. Note that the first term of the right-hand side (RHS) of (13a) is the variation due to the initial bias, while the second term denotes the variation due . From Appendix IV, it can be shown that can be to expressed by the iterative formula, shown in (14a), at the bottom of the page, where

where

(15a) where we also have (16a), also shown at the bottom of the page. . Note that The proof of Theorem 2A is given in Appendix V. follows an series. 2) Case B: Similarly, from (10), we have (13b), shown at the bottom of can be obtained by the following the next page. Again, : iterative formula, for all

(14b) From (14a), we have the following result. follows an ARMA(1, 1) seTheorem 2A: Assume that ries. Then the sequence of optimal discount factors

The detailed proofs of (13b) and (14b) are very similar to (13a) and (14a) and are thus omitted. From (14b), we have the following result.

(14a)

(16a)

TSENG et al.: STUDY OF VARIABLE EWMA CONTROLLER

637

TABLE I OPTIMAL DISCOUNT FACTOR AND THE CORRESPONDING MSE

Fig. 2. Process output by the optimal variable discount factor.

Theorem 2B: Assume that follows an series. Then, the sequence of optimal discount factors , which minimizes for any given , can be obtained by (15b) where

Fig. 2 presents a simulated output of an EWMA controller with this optimal discount factor. It is seen that the total sum of squares of errors (TSSE) of the optimal variable EWMA controller is 7.89. Compared with the case of the conventional fixed EWMA controller , the TSSE of the optimal fixed discount factor EWMA controller can be approximately reduced by 76%. V. PERFORMANCES OF VARIABLE DISCOUNT FACTOR

. (16b) The proof of Theorem 2B is very similar to that of Theorem 2A and is omitted. Now, we use an example to demonstrate the efficiency of this variable discount factor. Example 2 (Example 1 Continued): From Example 1, under , the conditions that , it is easy to obtain that . and and can be obtained Then, by using (15b) and (16b), and the results are summarized in Table I.

How much more efficient is the variable discount factor EWMA controller than the fixed discount factor EWMA controller? It needs to be discussed under two cases: 1) is known and 2) when is unknown. Note that when all comparisons are addressed under the assumption that . A. When

Is Known

It is seen that the optimal variable discount factor starts with a larger discount factor at the beginning of first few runs, and then it gradually converges to a stable discount factor. Note that in the earlier example, two process outputs have virtually identical behavior after approximately 15 production runs. Thus, we

if if

(13b)

638

IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 16, NO. 4, NOVEMBER 2003

TABLE II PERFORMANCE COMPARISON OF THE OPTIMAL VARIABLE EWMA CONTROLLER WITH THE CONVENTIONAL EWMA CONTROLLER

restrict the comparisons up to batch . Now, we define an index to compare the relative performance of optimal variable discount factor with respect to that of the optimal fixed discount factor as follows:

. In the following case, we set and the modified relative performance is defined as:

(17) denote the process outputs controlled by variwhere and able and fixed discount factors, respectively. and Under various combinations of , Table II shows a detailed comparison of the relative efficiency of the EWMA controller with two discount factors. It is seen that the proposed controller is more efficient than the and conventional controller. For example, when , it is seen that %, which means that the total MSE of the optimal variable discount factor is approximately 25% of the total MSE of the fixed discount factor. That is, the total MSE has a significant improvement up to 75%. For a given , the variable discount factor shows a more efficient improvement when there is a larger initial bias. On the other hand, for a given initial bias, the variable discount factor is more efficient when is larger. Significant improvements can be seen especially when and , since in these cases the fixed discount factor is smaller. B. When

Is Unknown

is seldom In practice, the exact value of the parameter known in advance. It is, therefore, important to understand how is on the variable discount factor sensitive a misspecified thus obtained and what effect it has on the performance of the EWMA controller. We consider a simulation study to investigate the performance of the variable discount factor with respect to the fixed discount factor, taking into account that the is unknown. For all , let and denote the th simulated outputs at time by using the optimal variable and fixed discount factors, respectively. Under the case is unknown, a reasonable estimator for is that

Set . Under various conditions that and , Table III lists the results of for , and From Table III, it is seen that is very close to one if is small and/or is small, a result consistent with in Table II. Based on the above comparison, the proposed variable discount factor controller is preferred when the initial bias and the are moderately large. In these situations, the provalue of posed controller may yield a significant improvement in efficiency, and such an improvement is quite robust to the misspecification of the initial process variance. VI. HEURISTIC

FOR DETERMINING DISCOUNT FACTOR

THE

VARIABLE

In order to obtain the optimal variable discount factor, we , an effort need to use an iterative formula to compute that requires a tremendous amount of computations. In practical applications, (5) provides an alternative to deriving the variable discount factor. Under the class of (5) (for any choice of ), we are interested in finding the optimal combina. Assume that the process disturbance follows an tion of model and the run size is . By using the grid can be obtained search method, the optimal settings of under various combinations of initial biases and . The result , while can be selected from reveals that . As shown in Fig. 3, the optimal Fig. 3 according to should be larger when the bias ratio is higher or is larger. For an illustrative purpose, assume that and . From Fig. 3, we can obtain that and the optimal variable discount factor is

TSENG et al.: STUDY OF VARIABLE EWMA CONTROLLER

639

TABLE III RELATIVE EFFICIENCY COMPARISONS OF THE OPTIMAL VARIABLE EWMA CONTROLLER WITH THE CONVENTIONAL EWMA CONTROLLER (UNDER err = 0:05 AND 0:10)

Fig. 3.

Optimal d with an IMA (1, 1) disturbance (under =b = 1:5).

The sequences of and are listed in Table IV. Fig. 4 (with solid line) and (with dotted shows the plots of line). It is seen that these two sequences are almost identical

. Note that in order to make a clearer presentation of after and , the axis on Fig. 4 is the difference between truncated at 0.15. From (17), the relative performance of these

640

Fig. 4.

IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 16, NO. 4, NOVEMBER 2003

The plots of f! g and f! g. TABLE IV COMPARISON OF OPTIMAL VARIABLE DISCOUNT FACTOR WITH RESPECT TO THAT OF THE PROPOSED METHOD

two sequences equals to 0.9722. It indicates that (5) provides a very good approximation to the optimal variable discount factor. VII. CONCLUDING REMARKS We have proposed and studied the EWMA controller based on a variable discount factor. The stability conditions and the optimal discount factor are derived under various model disturbances. The results indicate that the variable EWMA controller performs better than the conventional EWMA controller (whose discount factor is fixed) at reducing the process variation particularly at the first few production runs, thereby making it possible to further reduce the bias and bring the process output closer to its target as soon as the process begins to operate. More importantly, such a reduction can be achieved at no loss of long-term stability. The advantage of the variable EWMA controller could be potentially useful to semiconductor manufactures that operate in an environment in which multiple products are produced in relatively small number of batches. Before closing the discussion, some concluding remarks are given. 1) To develop the proposed optimal variable EWMA con, and . troller, we need to know the values of Certainly, without knowing the information, the optimal variable discount factor cannot be precisely determined. Here, we briefly discuss some possible approach for handling this difficulty. At the beginning of production run , note that the system’s parameters , and can be estimated from a regression model or design of experiment (DOE). Thus, a suitable sample size (in a regression analysis) can be determined by the predetermined error bounds for the above values. 2) In order to implement the variable EWMA controller in practical applications, we also proposed a heuristic to simplify the computation of the variable discount factor. Our results indicate that the proposed method is easy to

implement and provides a good approximation to the optimal discount factor. 3) To the semiconductor manufacturers, the rework rate of a process is another very important quality index for evaluating any controller. The rework rate (RR) of a process can be defined as

where can be suitably defined as the asymptotic vari. ance of the process output, i.e., Again, using Example 1, it can easily be calculated that for the fixed discount factor, and for the variable discount factor. It is evident that the proposed variable EWMA controller can significantly reduce the rework rate, and consequently, the rework cost. 4) The proposed variable EWMA controller is studied under the assumption that the process follows a single inputsingle/single-output (SISO) model with no presence of process drift. If the process has a linear drift, how to design a double EWMA controller (refer to Del Castillo [4] and Tseng et al. [9]) with discount factors would be an interesting research topic. In addition, manufacturing processes have, by nature, multiple-input/multiple-output (MIMO) variables (refer to Tseng et al. [10]). Thus, a further investigation of the variable EWMA controller in the case of MIMO settings will also be valuable. 5) Recently, Patel and Jenkins [8] presented a method to continuously adjust the discount factor of an EWMA controller by incorporating the idea of signal-to-noise ratio. The discount factor is adjusted with the objective of minimizing the asymptotic variance of the process output. They showed that the sequence of discount factors derived from their approach converges to a value that can be made arbitrarily close to the optimal fixed discount factor. The EWMA controller using the sequence of discount factors derived from our proposed method should have similar asymptotic performance with that of Patel and Jenkins [8] since the sequence converges to the optimal fixed discount factor. On the other hand, it would be worthwhile to conduct a detailed study comparing the performance between the two methods under finite production runs.

TSENG et al.: STUDY OF VARIABLE EWMA CONTROLLER

APPENDIX I

641

Let

. Then

Define

Then by (1), (2), (4), and (8), we have

because and Hence,

is stationary and it implies that

Also from Lemma 1, we have APPENDIX III PROOF OF (13a) By substituting the above equation into (1), we obtain the adjusted process output

When

, we have

Thus

Define and Then, for

Thus, (10) can be reduced to

where APPENDIX II From (10), we have

and

Therefore, from (9), we have

.

642

IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 16, NO. 4, NOVEMBER 2003

APPENDIX IV PROOF OF (14a) When

, from (13a) we have

When , from (14a) when corresponding

is obtained and the is a decreasing function of

. Thus

For all

, it is easy to derive that

In general, by mathematical induction, it is easily seen that Thus

is attained when the optimal variable discount factor is

ACKNOWLEDGMENT The authors would like to thank the Associate Editor and two referees for their helpful suggestions that improve the presentation of the paper. REFERENCES

APPENDIX V PROOF OF THEOREM 2A is a fixed constant. Thus, we only need to . Let . Thus, consider the case that is also a function of . Now

When

, from (14a), it is seen that the minimization of can be obtained by setting i.e.,

The corresponding optimal

is

[1] G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994. [2] G. Box and A. Luceño, Statistical Control by Monitoring and Feedback Adjustment. New York: Wiley, 1997. [3] S. W. Butler and J. A. Stefani, “Supervisory run-to-run control of a polysilicon gate etch using in situ ellipsometry,” IEEE Trans. Semiconduct. Manuf., vol. 7, pp. 193–201, 1994. [4] E. Del Castillo, “Long run and transient analysis of a double EWMA feedback controller,” IIE Trans., vol. 31, pp. 1157–11 169, 1999. [5] E. Del Castillo and A. Hurwitz, “Run to run process control: A literature review and some extensions,” J. Quality Technol., vol. 29, pp. 184–196, 1997. [6] A. Ingolfsson and E. Sachs, “Stability and sensitivity of an EWMA controller,” J. Quality Technol., vol. 25, pp. 271–287, 1993. [7] D. C. Montgomery, J. B. Keats, G. C. Runger, G. C. Runger, and W. S. Messina, “Integrating statistical process control and engineering process control,” J. Quality Technol., vol. 26, pp. 79–87, 1994. [8] N. S. Patel and S. T. Jenkins, “Adaptive optimization of run-to-run controllers: The EWMA example,” IEEE Trans. Semiconduct. Manuf., vol. 13, pp. 97–107, 2000. [9] S. T. Tseng, R. J. Chou, and S. P. Lee, “Statistical design of a double EWMA controller,” Appl. Stochastic Models Business Indust., vol. 18, no. 3, pp. 313–322, 2002a. [10] , “A study of multivariate EWMA controller,” IIE Trans., vol. 34, no. 6, pp. 541–549, 2002b. [11] F. Tsung, H. Wu, and V. N. Nair, “On the efficiency and robustness of discrete proportional-integral control schemes,” Technometrics, vol. 40, pp. 214–222, 1998. [12] F. Tsung and J. Shi, “Integrated design of run-to-run PID controller and SPC monitoring for process disturbance rejection,” IIE Trans., vol. 31, pp. 517–527, 1999. [13] F. Tsung, J. Shi, and C. F. J. Wu, “Joint monitoring of PID-controlled processes,” J. Quality Technol., vol. 31, pp. 275–285, 1999.

TSENG et al.: STUDY OF VARIABLE EWMA CONTROLLER

Sheng-Tsaing Tseng (M’92) received the B.S. degree in business mathematics from Soochow University, R.O.C., the M.S. degree in applied mathematics from Tsing-Hua University, Taiwan, R.O.C., and the Ph.D. degree in management science from Tamkang University, R.O.C. He is a Professor at the Institute of Statistics at Tsing-Hua University. His current research interests include quality and reliability improvement, and statistical decision methodology. His articles have appeared in numerous technical journals. Dr. Tseng is an elected member of ISI and a member of ASQ. Currently, he is an Associate Editor of IEEE TRANSACTIONS ON RELIABILITY.

Arthur B. Yeh is an Associate Professor of Statistics in the Department of Applied Statistics and Operations Research, Bowling Green State University. He has conducted and published research in several areas of industrial statistics, including optimal experimental designs, univariate and multivariate control charts, multivariate process capability indexes, and multivariate run-by-run process control. He has also worked as a consultant for various local and international companies in both traditional and modern high-tech manufacturing environments. Dr. Yeh currently serves as an Associate Editor for The American Statistician. He is also the President of the Northwest Ohio Chapter of the American Statistical Association and the Chair-Elect of the Toledo Section of the American Society for Quality.

643

Fugee Tsung received both the M.S. and Ph.D. degrees in industrial and operations engineering from the University of Michigan, Ann Arbor, and the B.Sc. in mechanical engineering from National Taiwan University, R.O.C. He is an Associate Professor of Industrial Engineering and Engineering Management at the Hong Kong University of Science and Technology, Hong Kong. He worked for Ford Motor Company and Rockwell International and did his post-doctoral research with Chrysler. His current research interests include quality engineering and management, process control and monitoring, and Six Sigma implementation. Dr. Tsung is the Chair-Elect of the Quality, Statistics, and Reliability (QSR) Section of the Institute for Operations Research and the Management Sciences (INFORMS). He is also a senior member of IIE and ASQ, and is an ASQ Certified Six Sigma Black Belt.

Yun-Yu Chan received the B.S. degree in applied mathematics from Fu-Jen University and the M.S. degree in statistics from Tsing-Hua University, Hsinchu, Taiwan. She is currently working toward the Ph.D. degree at the Institute of Statistics, Tsing-Hua University. She served as an Engineer of the Engineering Information Management (EIM) Department, ProMOS IC Company. Her research interests include process control and monitoring as well as industrial statistics.