A New Monitoring Design for Uni-Variate Statistical ... - Semantic Scholar

A New Monitoring Design for Uni-Variate Statistical Quality Control Charts Mohammad Saber Fallah Nezhad, Ph.D. Assistant Professor of Industrial Engineering, Yazd University, Yazd, Iran Email: [email protected] Seyed Taghi Akhavan Niaki, Ph.D. 1 Professor of Industrial Engineering, Sharif University of Technology P.O. Box 11155-9414, Azadi Ave., Tehran, Iran 1458889694 Phone: (+9821) 66165740, Fax: (+9821) 66022702, Email: [email protected]

Abstract In this research, an iterative approach is employed to analyze and classify the states of univariate quality control systems. To do this, a measure (called the belief that process is incontrol) is first defined and then an equation is developed to update the belief recursively by taking new observations on the quality characteristic under consideration. Finally, the upper and the lower control limits on the belief are derived such that when the updated belief falls outside the control limits an out-of-control alarm is received. In order to understand the proposed methodology and to evaluate its performance, some numerical examples are provided by means of simulation. In these examples, the in and out-of-control average run lengths (ARL) of the proposed method are compared to the corresponding ARL's of the optimal EWMA, Shewhart EWMA, GEWMA, GLR, and CUSUM [11] methods within different scenarios of the process mean shifts. The simulation results show that the proposed methodology performs better than other charts for all of the examined shift scenarios. In addition, for an autocorrelated AR(1) process, the performance of the proposed control chart compared to the other existing residual-based control charts turns out to be promising.

1

Corresponding Author

Keywords: Statistical Quality Control; Process Monitoring; CUSUM Chart; EWMA Chart; Average Run Length

1.

Introduction and literature review

Traditional statistical process control (SPC) methods provide a group of statistical tests of a general hypothesis which maintains that the mean value of the quality characteristic of a process, or process mean in short, is consistently on its target level. A variety of graphical tools, such as Shewhart, cumulative sum (CUSUM), and exponentially weighted moving average (EWMA) charts, has been developed to monitor a process mean. Shewhart charts, first introduced by W.A. Shewhart [24], plot either the individual process measure or the average value of a small sample group (usually not more than five samples) against the target level as well as the control limits. Under the assumption that the plotted data are normally distributed around the process target value when the process is within statistical control, the possibility of observing a point that is out of the three-sigma control limits is less than 2.7 in a thousand. Therefore, spotting a point out-of-controllimits leads to an out-of-control alarm which in turn calls for investigating the process. Although it has been the common belief for many decades that a Shewhart chart is not the most effective tool for some common process errors, such as small shifts in the process mean, recent research has shown that the difference between Shewhart and CUSUM charts is not that significant. For instance, Nenes and Tagaras [20] compared the economic performance of CUSUM and Shewhart schemes in monitoring the process mean. The results of their study showed that the economic advantage of using a CUSUM chart over the simpler Shewhart scheme is substantial only when the sample size is one or it is constrained to low values. It is essential that the process mean be consistently maintained at its target level; however, random process errors, or random “shocks”, could shift the process mean to an

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unknown level. Furthermore, while a control chart is required to detect such shifts as soon as possible, it is also desired that it does not signal too many false alarms when the process mean is on target. These criteria are usually defined in terms of the average run length (ARL) of the control chart for in-control and out-of-control operations, i.e., in-control ARL ( ARL0 ) and out-of-control ARL ( ARL1 ) , respectively. At any sampling instant t , the EWMA control chart for the process mean uses the control statistic Yt   X t  (1   )Yt 1 , where X t is the sample mean, Y0 is the in-control

process mean ( 0 ), and  is the smoothing constant or the chart parameter. The chart signals a change in the process level when Yt exceeds control limits that are expressed by means of multiples of the asymptotic standard deviation of Yt . In other words, the lower and the upper control limits (LCL and UCL) of the charts are given by LCL  0  L Y and UCL  0  L Y , where

Y   0

 n(2   )

is the asymptotic

standard deviation of Yt ,  0 is the in-control process standard deviation, and L is the control limit parameter. We note that since the asymptotic standard deviation is not used in the initial samples, one needs to provide the exact standard deviation of the EWMA chart at these sampling epochs as  Yt   0



1  (1   ) 2t  . n(2   )

The EWMA chart parameter is chosen to impose some desired properties on the chart. In industrial applications, the commonly used value of

 is within the range of 0.1

to 0.5. Besides, the control limit parameter is chosen to achieve a desired probability of type-I error (or desired ARL0 ). Moreover, for an EWMA chart, the literature suggests detecting shifts of one-half to one standard deviation of the process mean [27].

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The CUSUM chart uses one statistic for detecting a positive shift and another to identify a negative shift in the process mean. The statistics for detecting positive and



negative shifts are Yt  Max 0, Yt 1  Z t  k 







and Yt  Max 0, Yt 1  Z t  k 





respectively, where the starting values are Y0  Y0  0 , Z t  X t is the sample mean and k is the reference value or the chart parameter. The value of k is often chosen about

halfway between the target 0 and the out-of-control value of the process mean 1 that has to be detected quickly [18]. If either Yt  or Yt  exceeds the decision interval H , the process is considered to be out-of-control. In addition to the reference value, the decision interval is chosen so that to give the CUSUM chart some good properties. The value of H is often chosen as a multiple (five times) of the process standard deviation [18]. CUSUM charts were first introduced by Page [21]. When a shift in the process mean is to be detected and the size of the shift is known, then a CUSUM control chart is the most efficient method according to its ARL properties [2]. The CUSUM procedure can be seen as equivalent to applying a sequential likelihood ratio test to a shift in the process mean [1 & 9]. A CUSUM chart monitors the accumulated process deviation after the process is determined to be in an out-of-control state. The parameters of a CUSUM chart can be assigned such that it turns to be the optimal likelihood ratio test on a particular shift size. The EWMA [7 & 13] chart can be seen as a variation of the CUSUM [28] control scheme in the sense that they both have been used to improve the detection of small process shifts. Based on the notion that the most recent observed process deviation can have more information on process errors than the previous ones, different weights may be assigned to data according to their recorded times. An EWMA scheme lets the weights decrease exponentially with the age of data, while a CUSUM scheme keeps the same value

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for the weights. A Shewhart scheme, in contrast, assigns the total weight to the most recent observation and zero to others. Usually, an EWMA chart can be designed to have similar ARL properties to a CUSUM chart's through simulation studies [6]. Vargas et al. [27] presented a comparative study of the performance of CUSUM and EWMA control charts by simulation. The objective of their research was to verify when CUSUM and EWMA control charts provide the best control region to detect small changes in the process mean. They observed that the CUSUM control chart practically did not signal out-of-control points for the levels of variation between  1 standard deviation, whereas the EWMA control chart was more efficient. Among the parameters of EWMA control chart, the ones with   0.1 and 0.05 detected a larger number of altered positions. While both the optimal EWMA and CUSUM control charts are based on a given reference value, say δ, Han and Tsung [11] proposed a generalized EWMA control chart (GEWMA) which does not depend on δ. They theoretically compared the performance of GEWMA control chart with the optimal EWMA, CUSUM, Shewhart-EWMA (a combination of Shewhart and Optimal EWMA), and the generalized likelihood ratio (GLR) control charts. The results of the comparison, which considered the in-control average run length approaching infinity, showed that the GEWMA control chart was better than the optimal EWMA control chart in detecting a mean shift of any size. For the mean shifts not in the interval (0.7842δ, 1.3798δ), the GEWMA chart performed better than the CUSUM control chart. Furthermore, the GLR control chart had the best performance in detecting mean shifts among the five control charts except for detecting a particular mean shift δ, when incontrol average run length approaches infinity. Serel and Moskowitz [23] proposed an EWMA control chart to jointly monitor the mean and variance of a process. In this research, an EWMA cost minimization model was presented to design the joint control scheme based on purely economic or economic-

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statistical criteria. Through a computational study, they showed that the optimal sample sizes decrease as the magnitude of shifts in mean and/or variance is increased. Recently developed quality control schemes are mostly based on sequential analysis, which requires to analyze the data at hand in order to determine the necessary number of additional observations at the next stage. For example, Wu and Shamsuzzaman [30] proposed a scenario for continuously improving the X & S control charts, where the information collected from out-of-control cases in a manufacturing process was used to update the chart parameters. Moreover, Zhang et al. [32] employed a sequential sampling scheme in phase one of an exponential control chart to monitor the time between events. Adaptive charts may either be based on the Bayes’ rule for continuously updating the knowledge about the state of the process, or not [26]. Over the years, to simultaneously monitor the process mean and variance, numerous researchers have proposed different statistically designed adaptive charts. Examples of these research efforts are the X control chart of Lin and Chou [14], the joint X and R control charts of Costa and Rahim [5], and the CUSUM chart of Wu et al. [31]. However, the performance criterion that is increasingly used to measure the effectiveness of these adaptive charts, especially the Bayesian ones, is the minimization of total expected quality-related costs. Nenes and Tagaras [19] studied a model for the economic design of an adaptive X chart for short production runs that were subject to the occurrence of assignable causes. These causes may lead to either an increase or a decrease in the mean of the quality characteristic. At each sampling instance, the probabilities that the process operates under the effect of an assignable cause were updated using the Bayes’ theorem. All three chartparameters i.e., the time until the next sampling instance, the sample size, and the control limits were adaptive and depended on these probabilities.

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Chun and Rinks [4] applied Bayesian analysis to a single sampling plan in which the defective-proportion was assumed a random variable that follows a Beta distribution. Furthermore, Wu [29] applied Bayesian analysis to assess the process capability index in a sequential manner based on subsamples. Sachs et al. [22] used a sequential Bayesian statistical approach of feedback control in which the process was first divided into incontrol and out-of-control states. The out-of-control and in-control probability functions were then defined and the certainty of the shift was updated with each addition of new data. This sequential approach reflects the latest evidence supporting or discounting the occurrence of a disturbance. They assumed a prior probability distribution for the shift and obtained an expression that highlights the sequential nature of updating the shift distribution. Marcellus [17] presented a Bayesian analogue of the Shewhart X chart and compared it with CUSUM charts. He showed the advantage of changing from Shewhart or

CUSUM chart to Bayesian monitoring in situations where the required information about the process structure is obtained. Although implementing the Bayesian chart requires more detailed knowledge of the process structure than the best-known types of charts, acquiring this information can yield tangible benefits. For this case, a Bayesian monitoring system was defined for a standard production process model introduced by Duncan [8] and it was shown that the monitoring system to be equivalent to an adaptive Shewhart monitoring scheme. Some efforts have been made to take a fuzzy approach to control charting. The major contribution of fuzzy set theory lies in its capability of representing vague data. Fuzzy logic offers a systematic base to deal with situations that are ambiguous or not well defined. A number of papers on fuzzy control charts use defuzziffication methods in the early steps of their algorithms that make their approach similar to the conventional ways of

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control charting. However, Gülbay and Kahramana [10] proposed a new alternative approach of “direct fuzzy approach (DFA)” to control charting method. In contrast to the existing fuzzy control charts, the proposed approach does not require taking the defuzziffication step. This prevents the loss of information included in the samples and directly compares the linguistic data in fuzzy space without making any transformation. At the end, they provide some numerical examples to illustrate the performance of their method and to interpret the results. In this research, a recursive equation is first defined to update a statistic (called belief) in each iteration of the data gathering process. Then, similar to the well-known

CUSUM and EWMA methods, thresholds are derived for the updated values of the beliefs. When the updated belief is out of the derived threshold range, an out-of-control signal is issued. The rest of the paper is organized as follows: in section 2, the belief and the recursive method of its improvement are first defined. Then, we design the new univariate control charting method. Section 3 contains the results of some simulation experiments in which the performance of the proposed methodology is compared to the performance of the optimal EWMA, Shewhart EWMA, GEWMA, GLR, and CUSUM methods. This section also contains the results of a comparative study on an autocorrelated AR(1) process. The performance of the proposed scheme is further evaluated in a case study in section 4. Finally, the conclusions and recommendation for future research appear in section 5.

2. Belief and the approach to its improvement

For the sake of simplicity, assume only one single observation ( n  1 ) is gathered on the quality characteristic of interest in each iteration of the data gathering process. At the kth

iteration, let Ok  ( x1 , x2 ,..., xk ) be a vector of observations on the quality characteristic of

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the current and the previous k  1 iterations. After taking a new observation, xk , let B ( xk , Ok 1 ) define the belief in the process to be in an in-control state. In this iteration, our aim is to improve this belief based on the observation vector Ok 1 and the new observation xk . Assuming that the quality characteristic of interest follows a normal distribution with mean 0 and variance  02 and letting B (Ok 1 )  B( xk 1 , Ok  2 ) be the prior belief in an in-control state, in order to update the posterior belief B ( xk , Ok 1 ) , we define

B  x k ,O k 1   B O k  

B O k 1  e B O k 1  e

x k  0

0

x k  0

0

(1)

 1  B O k 1  

Then, by defining the statistic Zk 

B  x k ,O k 1 

1  B  x k ,O k 1 



B O k





(2)

Z k 1

(3)

1  B O k

the recursive equation will be B O k 1 

x k  0

Zk e

0

x k  0

1  B O k 1 

e

0

Hence, k

x k  0

Zk e

0

Z k 1  e

x k  0 x k 1  0 

0

0

 x i  k 0 i 1

Z k  2  .......  e

0

(4)

In other words, k

Ln  Z k  

x i 1

i

 k 0

0

k  x  0    i   N  0, k  0  i 1 

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(5)

For initial values of Z 0 and B (O 0 ) , note that for k  1 equation (4) shows that 1

 x i  0 i 1

Z1  e

0

x 1  0

e

0

x 1  0

. In this case, by equation (3) we have Z 1  e

0

Z 0 which means

that Z 0  1 . Hence B (O 0 )  0.5 . Now we define the upper and the lower control limits (UCL and LCL) for Ln  Z k  as

UCL Ln  Z k   c k

and

LCL Ln  Z k   c k

(6)

Where c is a multiple of the standard deviation of Ln  Z k  and is determined such that for a given probability of type-I error,  , we have





P c k  Ln  z k   c k  1  

(7)

Since we need to determine threshold values for recursive statistic B  xk , Ok 1  and since computing Ln  Z k  for small values of Z k is difficult in terms of computer limitations, we may instead determine the values of B  xk , Ok 1  and compare them to their lower and upper control limits as derived bellow. Substituting equation (2) for Z k in equation (7) results in    B  x k ,O k 1   P  c k  Ln   c k   1   1  B  x ,O     k k 1  

(8)

Or  P e c 

k



B  x k ,O k 1 

1  B  x k ,O k 1 

 ec

k

   1 

(9)

This means that  P e c 

k

1 

1  ec 1  B  x k ,O k 1 

k

  1  1   

10

(10)

which leads to a 100 (1   ) % confidence interval for B  x k ,O k 1  as given in equation (11).

 e c k ec k  B  x k ,O k 1   P  e c k  1 1 ec 







k



   1  

(11)

Furthermore, since in the initial stages of the data gathering process the false alarm rate may be high, the parameter l is also introduced in equation (11) for the upper and the lower control limits (UCL and LCL) of B  x k ,O k 1  to become UCL B  x k ,O k 1  



e c

k l

1  e c

k l



and

LCL B  x k ,O k 1  



ec

k l

1 ec

k l



(12)

in which the exponential terms are computed easier than the logarithmic terms needed in equation (6). It is worth noting that the values of c and l should be determined such that to ascertain reasonable properties for the proposed control charting method. In other words, for a desired in-control average run length, small values of out-of-control average run length in different scenarios of the process mean shifts are in order (the probability of both type-I and type-II error must be small). Following Han and Tsung [11] who compared the abilities of their proposed GEWMA control charts to the performance of the optimal EWMA, Shewhart EWMA, GLR, and CUSUM, in the next section we perform some simulation experiments in which the performance of the proposed methodology in terms of both in-control and out-of-control average run lengths criterion is compared with other control charts. Furthermore, in situations in which the collected data on the quality characteristic are auto-correlated, the performance of the proposed procedure is compared with the residual-based EWMA chart (Lu and Reynolds [15]), residual-based CUSUM chart (Lu and Reunolds [16]) and

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triggered CUSCORE chart (Shu et al. [25]) for different values of the autocorrelation coefficient in an AR(1) process.

3. Simulation experiments

Simulation experiments are performed for two classes of independent standard normal and autocorrelated AR(1) observations.

3.1 Independent standard normal process

Suppose that the quality characteristic of interest in different stages of the data gathering process are identically and independently distributed (IID) standard normal random variables. In order to simulate this process, pairs of independent uniform random variates ( (R i , R i 1 ) ; i  2k  1 ; k  1, 2,3,... )

are

first

generated

and

then

x k  2Ln (R i ) cos(2 R i 1 ) is employed to generate a standard normal observation in the kth iteration of the data gathering process [3]. In the next step, using equation (1) the belief ( B (O k ) ) is updated in that iteration. When B (O k ) is out of the interval  c k  l e c k l  e ,  e c k  l  1 1  e c k  l 







  , then an out-of-control signal is observed.  

Because the two-sided CUSUM [12 & 20], GEWMA, EWMA, and Shewhart EWMA charts [12] also incorporate past information, it is natural to compare their performance with the proposed control chart to see whether the new chart performs better in terms of out-of-control average run lengths (ARL1). Furthermore, the optimal EWMA, Shewhart EWMA, GEWMA, GLR, and CUSUM control charts are based on a given reference value, which for the CUSUM chart is the magnitude of a shift in the process mean that should be

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quickly detected. Similar to Han and Tsung [11], in this research a reference value of 1 for the optimal EWMA and CUSUM chart will be used in the simulation experiments. Results based on 10000 independent replications, each representing a series of observations that ends with a signal (i.e., in each replication we generate standard normal deviates until the defined statistic goes out of the derived control interval) are summarized in Table (1). The simulation results are also given for various values of the process mean shift defined as multiples of the process standard deviation. The values in the “SD” columns of Table (1) are the standard deviations of the run lengths. The parameters of different control charts are given in the last row of Table (1). The reference value for the optimal EWMA and CUSUM is taken to be 1. For the proposed method we pick c=1.5 and l  0 to ascertain an accessible value of ARL 0 .

The results of Table (1) show that while the GEWMA control chart maintains a relative advantage over the optimal EWMA control chart (except in detecting the shifts of around 0.5 to 1.25 ), Shewhart EWMA control charts (except in detecting the shifts of around 0.5 to 1.25 ), the CUSUM chart (except in detecting the shifts of 0.75 to 1.25 ), and the GLR control chart (except in detecting the shifts of less than 0.25 ), the

performance of the proposed methodology is the best for shifts of different magnitudes in the process mean. Moreover, the standard deviations of the run lengths obtained by the proposed method are generally less than corresponding values of the other methods. We also note that the in-control average run length of the proposed control chart (548.00) is larger than the corresponding values in the other control charts. In other words, not only is the probability of type-I but also the probability of type-II error associated with the proposed method is less than their corresponding values in the other two methods (according to ARL1 

1 1 and ARL0  ). 1  

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Insert Table (1) about here

3.2 Auto-correlated AR(1) process The usual assumption of using a control chart to monitor a process is that the observations from the process output are independent. However, for many processes this assumption does not hold and the observations are autocorrelated. This autocorrelation can have a significant effect on the performance of the control chart. In this section, we assume that the observations on the quality characteristic in different stages of the data gathering process can be modeled as an AR(1) process plus a normally distributed random error term. A process{ y k } is said to be AR(1) if it is generated by y k  0    y k 1  0    k

(13)

where  is the autocorrelation coefficient satisfying    1,1 and  k is a sequence of IID normal error term, i.e.,  k  N  0,  2  ; k  Z . In this process, the variance of the observations is V ar  y k  

 2 and the residuals are defined as 1 2

 k  y k  0    y k 1  0  ; k  1, 2,...

(14)

Since the residuals are IID random variables, in order to update the posterior belief ( B ( y k ,O k 1 ) ) let define

B  y k ,O k 1  

B O k 1  e B O k 1  e

k 

k 

 1  B O k 1  

B O k 1  e



B O k 1  e

y k  0   y k 1  0 

y k  0   y k 1  0 





(15)

 1  B O k 1  

Then, a 100 (1   )% confidence interval for B  y k ,O k 1  is easily determined using

Equation (16).

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 e c k  l e c k l P  B  y k ,O k 1    e c k  l  1 1  e c k l 









   1  

(16)

 e c k  l e c k l , Hence, when B (O k ) is out of the interval   e c k  l  1 1  e c k  l 







  , an out-of-control  

signal is observed. In this section the performance of the proposed method in terms of both ARL 0 and ARL1 is compared to the performance of the residual-based EWMA chart [15], residual-

based CUSUM chart [16] and one-sided CUSCORE chart [25] for selected auto-correlation coefficients of 0.1, 0.5 and 0.90. For   0.5 , based on 10000 independent replications, the simulation results are summarized in Table (2). The results in Tables (3) and (4) correspond to   0.9 and

  0.1 , respectively.

Insert Table (2) about here Insert Table (3) about here Insert Table (4) about here

The results of Table (2) show that for a moderate-level of autocorrelation, the proposed method performs better than the other charts for all shifts of less than 2 . For larger shifts than that the CUSCORE chart is the best. Furthermore, the standard deviation of the out-of-control run length of the proposed method is the least among all the competing methods. For a highly correlated process, the results of Table (3) show that the residual-based EWMA chart is the best for shifts of less than 0.1 . However, while for the shifts between

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0.1 and 0.25 the CUSCORE chart enjoys the best performance, for shifts larger than

0.25 the proposed chart is the best. Moreover, standard deviation of the out-of-control

run length of the CUSCORE chart is the least for shifts of less than 0.75 . Nonetheless, for shifts larger than that the standard deviation of the out-of-control run length is the least for the proposed procedure. For a low-level correlated process, the results in Table (4) show that the proposed chart performs as well as the other methods with the least standard deviation for the out-ofcontrol run length.

4. A case study

Consider an in-control standard normal process ( 0  0 and  02  1 ), and suppose that at a certain time, the process shifts to the mean 1  0.1 . We collect 20 observations as given in Table (5), and compare the performance of the proposed method against the standard Shewhart, CUSUM, and EWMA charts. Figure (1) shows the charts of all competing methods, which are obtained using the Minitab computer software. We note while none of the conventional charts is able to detect a small shift of 0.1 in the process mean, the proposed method shows out-of-control signals at observations 15, 16, 17 and 18. This means that the proposed chart is more capable in detecting the mean shift than the conventional charts and is likely to be more practical. The most important finding of this case study is the high sensitivity of the proposed method to the shifts in the process mean for large and small number of observations.

Insert Figure (1) about here

16

5- Conclusions and recommendation for future research

In this paper, an iterative approach was employed to analyze and classify the states of univariate quality control systems. This approach starts out with defining a measure called belief, and subsequently the beliefs in the system to be in-control are updated by taking new observations on the quality characteristic under study. Then by means of a control charting method when the updated beliefs are out of the control limits, the process is determined to be in an out-of-control state. In the simulation experiments of an independent standard normal process, we concluded that applying the proposed method to the process mean monitoring would improve the performance of the charts and would result in reduced probability of both type-I and type-II errors. Moreover, the results of another simulation study of an AR(1) process revealed that the proposed control chart performs at least as well as the other residual-based control charts. In this research, the performance of the proposed method was compared against the other existing methods based on simulated observations of a standard normal process. In case where the observations are taken from a general normal or even a non-normal process, the results of the comparison study remain to be shown in future research. Furthermore, determining the control threshold of the beliefs for different values of autocorrelation coefficient in different autocorrelated processes is an interesting subject for future research.

6. Acknowledgment

The authors would like to thank the referees for their valuable comments and suggestions that improved the presentation of this paper.

17

7. References

[1]. Barnard, G.A., (1959), “Control charts and stochastic processes.” Journal of Royal Statistical Society, Series B, 21, 239–271.

[2]. Basseville, M., and Nikiforov, I.V., (1993), “Detection of abrupt changes: Theory and application.” Prentice Hall Inc., Upper Saddle River, NJ, USA.

[3]. Box, G.E.P., and Muler, M.E., (1958), “A note on the generation of random normal deviates.” Annals of Mathematical Statistics, 29, 610-611. [4]. Chun, Y.H. and Rinks, D.B., (1998), “Three types of producer's and consumer's risks in the single sampling plan.” Journal of Quality Technology, 30, 254-268. [5]. Costa, A.F.B., and Rahim, M.A., (2004), “Joint X and R charts with two-stage samplings.” Quality and Reliability Engineering International, 20, 699-708. [6]. Crowder, S.V., (1986), “Kalman filtering and statistical process control.” Ph.D. thesis, Statistics, Iowa State University. [7]. Crowder, S.V., (1987), “A simple method for studying run-length distributions of exponentially weighted moving average charts.” Technometrics, 29, 401-407. [8]. Duncan A.J., (1956), “The economic design of X charts used to maintain current control of a process.” Journal of the American Statistical Association, 51, 228-242. [9]. Ewan, W.D. and Kemp, K.W., (1960), “Sampling inspection of continuous processes with no autocorrelation between successive results.” Biometrika, 47, 363-380. [10].

Gülbay, M., and Kahramana, C., (2007), “An alternative approach to fuzzy control

charts: Direct fuzzy approach.” Information Sciences, 177, 1463-1480. [11].

Han, D. and Tsung, F., (2004), “A generalized EWMA control chart and its

comparison with the optimal EWMA, CUSUM and GLR schemes.” Annals of Statistics, 32, 316-339.

18

[12].

Hawkins D.M. and Olwell, D.H., (1998), “Cumulative sum charts and charting for

quality improvement.” Springer, New York.

[13].

Hunter, S.J., (1986), “The exponentially weighted moving average.” Journal of

Quality Technology, 18, 203-210.

[14].

Lin, Y.-C., and Chou, C.-Y., (2005), “Adaptive X control charts with sampling at

fixed times.” Quality and Reliability Engineering International, 21, 163-175. [15].

Lu, C.W. and Reynolds Jr., M.R., (1999), “EWMA control charts for monitoring

the mean of autocorrelated processes.” Journal of Quality Technology, 31, 166-188. [16].

Lu, C.W. and Reynolds Jr., M.R., (2001), “CUSUM charts for monitoring an

autocorrelated process.” Journal of Quality Technology, 33, 316-334. [17].

Marcellus,

R.L.,

(2008),

“Bayesian

statistical

process

control.” Quality

Engineering, 20, 113-127.

[18].

Montgomery, D. (2005), “Introduction to statistical quality control.” 5th Ed., John

Wiley and Sons Inc., New York. [19].

Nenes, G., and Tagaras, G., (2007), “The economically designed two-sided

Bayesian X control chart.” European Journal of Operational Research, 183, 263-277. [20].

Nenes, G., and Tagaras, G., (2008), “An Economic Comparison of CUSUM and

Shewhart Charts,” IIE Transactions, 40, 133-146. [21].

Page, E.S., (1954), “Continuous inspection schemes.” Biometrika, 14, 100-115.

[22].

Sachs, E.,

Hu, A., and Ingolfsson, A., (1995), “Run by run process control:

Combining SPC and feedback control.” IEEE Transactions on Semiconductor Manufacturing, 8, 26-43.

[23].

Serel, D.A., and Moskowitz, H., (2008), “Joint economic design of EWMA control

charts for mean and variance.” European Journal of Operational Research, 184, 157-168.

19

[24].

Shewhart, W.A.T., (1931), “Economic control of quality of manufactured product.”

Van Nostrand, New York, USA. [25].

Shu, L., Apley, D.W., and Tsung, F., (2002), “Autocorrelated process monitoring

using triggered CUSCORE charts.” Quality and Reliability Engineering International, 18, 411-421. [26].

Tagaras, G., (1998), “A survey of recent developments in the design of adaptive

control charts.” Journal of Quality Technology, 30, 212-231. [27].

Vargas, V.C.C., Lopes, L.F.D, and Souza, A.M, (2004), “Comparative study of the

performance of the CuSum and EWMA control charts.” Computers and Industrial Engineering, 46, 707-724.

[28].

Woodall, W.H., (1986), “The design of CUSUM quality control charts.” Journal of

Quality Technology, 18, 99-101.

[29].

Wu, C.W., (2006), “Assessing process capability based on Bayesian approach with

subsamples.” European Journal of Operational Research; 184, 207-228. [30].

Wu, Z. and Shamsuzzaman, M., (2006), “The updatable X & S control charts.”

Journal of Intelligent Manufacturing, 17, 243-250.

[31].

Wu, Z., Zhang, S., and Wang, P., (2006), “A CUSUM scheme with variable sample

sizes and sampling intervals for monitoring the process mean and variance.” Quality and Reliability Engineering International, 23, 157-170.

[32].

Zhang, C.W., Xie, M., and Goh, T.N., (2006), “Design of exponential control

charts using a sampling sequential scheme.” IIE Transactions, 38, 1105-1116.

20

Table (1): The results of ARL0 and ARL1 study for IID N(0,1) observations Optimal

Shifts

Proposed Method

SD

0.00

548.00

21.370

437.00

4.34

0.10

63.50

0.880

297.00

0.25

23.01

0.230

0.50

8.55

0.75 1.00

Shewhart

SD

GEWMA

SD

GLR

SD

CUSUM

SD

430.00

4.28

438.00

4.24

439.00

4.35

434.00

4.36

4.30

294.00

2.85

304.00

2.75

295.00

2.67

326.00

3.23

110.00

1.02

109.00

1.02

105.00

0.79

108.00

0.80

132.00

1.23

0.070

32.40

0.25

32.40

0.25

34.90

0.23

36.20

0.23

37.20

0.30

4.66

0.030

15.70

0.10

15.70

0.10

17.40

0.10

18.10

0.11

16.70

0.11

3.15

0.020

9.95

0.05

9.92

0.05

10.70

0.06

11.10

0.06

10.30

0.05

1.25

2.51

0.010

7.24

0.03

7.19

0.03

7.36

0.04

7.58

0.04

7.34

0.03

1.50

1.93

0.015

5.37

0.02

5.67

0.02

5.41

0.03

5.59

0.03

5.70

0.02

2.00

1.40

0.009

4.03

0.01

3.91

0.01

3.41

0.02

3.54

0.02

3.98

0.01

3.00

1.04

0.007

2.63

0.01

2.29

0.01

1.85

0.01

1.91

0.01

2.55

0.01

Parameters

c  1.5

  0.128

l 0

L  2.82

EWMA

SD

EWMA

  0.128 C  2.82 L  3.9

21

L  3.29

L  3.45

H  4.94

Table (2): The results of ARL0 and ARL1 study for AR(1) with   0.5

SD

Residualbased EWMA

SD

CUSCORE

SD

Residualbased CUSUM

SD

451.00

17.37

421.00

4.17

420.00

4.35

430.00

4.26

0.10

93.50

1.24

257.00

2.59

295.00

2.67

301.00

2.83

0.25

32.01

0.28

134.47

1.22

144.532

1.30

185.201

1.73

0.50

15.55

0.1

67.328

0.55

67.282

0.5

89.979

0.80

0.75

10.66

0.06

36.8727

0.30

36.677

0.26

48.2835

0.41

1.00

7.27

0.03

23.682

0.16

26.22

0.15

29.3704

0.23

1.25

5.51

0.02

17.4242

0.11

17.944

0.1

19.8704

0.14

Shifts

Proposed Method

0.00

1.50

4.93

0.015

13.1241

0.07

14.493

0.07

14.2967

0.09

2.00

3.60

0.008

4.93

0.04

3.9

0.04

8.98

0.05

3.00

2.4

0.005

3.32

0.02

1.8

0.02

4.05

0.02

Parameters

c  1.2 l  20

  0.1

L  3.5

L  2.51

k  0.25 r  0.5

22

H  4.25 k  0.5

Table (3): The results of ARL0 and ARL1 study for AR(1) with   0.9

SD

Residualbased EWMA

SD

CUSCORE

SD

Residualbased CUSUM

SD

451.00

17.37

418.00

4.25

443.00

4.05

426.00

4.19

394.63

11.24

386.00

3.97

393.00

3.5

391.00

3.83

0.25

291.24

6.77

330.35

3.34

285.02

3.22

358.11

3.48

0.50

178.13

3.34

268.02

2.69

201.48

2.55

301.27

3.01

0.75

127.09

1.90

211.28

2.12

146.79

2.09

256.89

2.49

1.00

91.84

1.22

177.96

1.69

115.40

1.70

217.42

2.09

1.25

72.14

0.85

144.45

1.31

88.62

1.35

183.75

1.79

1.50

60.18

0.63

116.64

1.11

72.97

1.10

160.25

1.52

2.00

38.1

0.41

85.93

0.72

82.9

0.47

118.98

1.13

3.00

28.2

0.23

51.32

0.39

55.8

0.26

63.05

0.63

Shifts

Proposed Method

0.00 0.10

Parameters

c  1.2 l  20

L  1.45

  0.1

k  0.05 r  0.1

L  2.51

23

H  4.25 k  0.5

Table (4): The results of ARL0 and ARL1 study for AR(1) with   0.1

SD

Residualbased EWMA

SD

CUSCORE

SD

Residualbased CUSUM

SD

451.00

17.37

425.00

4.37

445.00

4.05

428.00

4.17

0.10

103.63

1.54

189.4.00

1.87

232.00

2.22

391.00

3.83

0.25

40.04

0.36

69.01

0.62

100.30

0.40

101.25

0.97

0.50

17.68

0.11

27.51

0.21

33.63

0.14

35.33

0.30

0.75

10.87

0.05

15.83

0.08

16.84

0.08

17.22

0.12

1.00

8.29

0.04

10.57

0.05

10.54

0.05

10.44

0.06

1.25

6.30

0.03

7.70

0.03

7.64

0.03

7.49

0.04

1.50

5.25

0.02

6.43

0.03

5.93

0.03

5.74

0.03

Shifts

Proposed Method

0.00

2.00

4

0.01

4.39

0.01

4.15

0.01

3.98

0.01

3.00

2.67

0.01

2.84

0.01

2.67

0.01

2.54

0.01

Parameters

c  1.2 l  20

L  4.2

  0.1

k  0.45 r  0.9

L  2.51

24

H  4.25 k  0.5

Table (5): The process data No.

Obs.

No.

Obs.

1

-0.4037

11

-0.9202

2

1.6336

12

0.4118

3

-0.2238

13

0.7242

4

0.7271

14

1.6376

5

-1.3756

15

1.8857

6

-0.7735

16

0.6630

7

1.8541

17

0.4673

8

0.5053

18

1.8906

9

-0.1224

19

-2.5656

10

0.5932

20

-0.5820

0.8 3

3.0SL=3.000

2

EWMA

1

3.0SL=0.6831

0.6

Individual Value

0.4 0.2

0

X=0.000

X=0.000

0.0 -0.2

-1

-0.4

-2 -3

-0.6

-3.0SL=-3.000

-3.0SL=-0.6831

-0.8 0

10

20

0

Observation Number

Sample Number

Chart1: Standard Shewhart

Cumulative Sum

Chart2: Standard EWMA

Upper CUSUM

4

10

4

3 2 1 0 -1 -2 -3 -4

-4

Lower CUSUM

0

10

20

Subgroup Number

Chart3: Standard CUSUM

Chart4: The Proposed Method

Figure (1): The Control Charts of the Case Study

25

20