An Adaptive Exponentially Weighted Moving Average Control Chart for Monitoring Process Variances

Transcription

1 An Adaptive Exponentially Weighted Moving Average Control Chart for Monitoring Process Variances Lianjie Shu Faculty of Business Administration University of Macau Taipa, Macau Abstract The exponentially weighted moving average (EWMA) control chart is efficient in detecting small changes in process parameters but less efficient when the changes are relatively large, due to what is known as the inertia problem. To diminish the inertia, an adaptive EWMA (AEWMA) chart has been proposed for monitoring process locations to improve over the traditional EWMA charts. The basic idea of the AEWMA scheme is to dynamically weight the past observations according to a suitable function of the current prediction error. This paper extends the idea of the AEWMA chart for monitoring process locations to the case of monitoring process dispersion. A Markov chain model is established to analyze and design the suggested chart. It is shown that the AEWMA dispersion chart performs better than the EWMA and other dispersion charts in terms of its ability to perform relatively well at both small and large changes in process dispersion. Keywords: Average Run Length; Statistical Process Control; Markov Chain; Normalizing transformation; Dispersion Charts 1

2 1 Introduction The exponentially weighted moving average (EWMA) control chart has received a great deal of attention in the quality control literature since it was first introduced by Roberts (1959). Many researchers have contributed to the theory and practical use of the EWMA scheme as a monitoring tool, primarily to detect shifts in the mean level of a normal process, see, for example, Robinson and Ho (1978), Crowder (1987, 1989), Lucas and Saccucci (1990), and Apley and Lee (2003). In addition to detecting changes in the process mean, it is also important to keep the process variance in control since an increase of process variance results in an increased number of defective units while a decrease of process variance implies an improvement of process capability (Acosta-Mejia et al. 1999). Recently, there is gradually increasing attention given to the use of EWMA charts as a tool for monitoring process variability. A sample of this research include Wortham and Ringer (1971), Sweet (1986), Ng and Case (1989), Crowder and Hamilton (1992), MacGregor and Harris (1993), Lu and Reynolds (1999), Acosta-Mejia et al. (1999), and Chen et al. (2001). To detect changes in process dispersion, rational subgroups of observations are often formed, and the subgroup variance or range is then computed for estimating the underlying process variation. Basically, the dispersion charts can be grouped into two categories based on if a transformation is performed to normalize the observations. The first category directly applies the traditional control charts to the subgroup variance or range for monitoring changes in process dispersion without normalizing transformations. For example, Ng and Case (1989) proposed an EWMA dispersion chart based on the subgroup range. Note that the distribution of subgroup variance is not normal but rightly skewed, which is inconsistent with the normality assumption often made in statistical process control (SPC). For this reason, some authors proposed using nonlinear functions to transform the sample variance into a normal or an approximately normal distribution so that traditional control charts can be easily applied to the normalized data. This results in the second class of dispersion charts. One of the widely discussed normalizing transformations is based on the logarithmic function of the subgroup variance, ln(s 2 ), due to its simplicity and efficiency. This transformation was first recommended by Box, Hunter, and Hunter (1978) for making inferences about variance of normally distributed data and was later developed by Crowder and Hamilton (1992) and Chang and Gan (1995) for the problem of 2

3 monitoring process dispersion. Chen et al. (2001) recently discussed using the EWMA chart of ln(s 2 ) for joint monitoring process location and dispersion. The essential motivation of using a normalizing transformation is that the transformed data are much more normally distributed than the subgroup variance itself, as pointed out in Crowder and Hamilton (1992). Furthermore, after the transformation, the variance of the normalized observations is often independent of the true process variance and remains constant over time. A change in the process variance can be translated into a shift in the mean level of the normalized variable. As a result, monitoring process variances is essentially equivalent to monitoring mean levels of the transformed data. Due to these properties, the normalizing transformation has been considered to be an efficient technique for monitoring process dispersion. The literature on the efficiency and robustness of the EWMA chart for monitoring process means indicates that it can provide greater sensitivity to small shifts than the Shewhart chart, but is not as effective as the Shewhart chart when the shifts in the process mean level are relatively large. Yashchin (1995) investigated the estimation efficiency of the EWMA scheme in terms of an inertia function. He showed that the inertia increases as the magnitude of the mean shift increases and thus the EWMA statistic with a small smoothing constant is not efficient in estimating abrupt mean changes of moderate and large magnitudes. To overcome the inertia problem, Yashchin (1995) proposed a modified EWMA statistic, EWMA-C, which can dynamically adjust the weight on past observations according to a function of the prediction error. Capizzi and Mastrotto (2003) developed a control chart based on the EWMA-C statistic for monitoring process means, which has been referred to as the adaptive EWMA (AEWMA) chart. Although the inertia problem of the EWMA chart can be counteracted in part by using the combined Shewhart-EWMA chart, the introduction of Shewhart limits only ameliorates this problem, as pointed out in Capizzi and Mastrotto (2003). In contrast, the AEWMA chart, which can be viewed as a smooth combination of the Shewhart and EWMA charts, can diminish the inertia problem. It has been shown that the AEWMA chart is able to offer an overall good detection performance against mean shifts of different sizes and performs better than the combined Shewhart-EWMA chart. Like the EWMA charts for monitoring process means, the EWMA-type dispersion charts such as the EWMA charts based on ln(s 2 ) also suffer from the inertia problem when detecting 3

4 large changes in process dispersion. Motivated by this, this paper extends the idea of AEWMA charts for monitoring process means to the case of monitoring process variability. A Markov chain model is developed to analyze the performance of the suggested chart. This simplifies the performance analysis without running a large number of simulations. The rest of the paper is organized as follows. In the next section, the AEWMA chart for monitoring process dispersion is introduced. Then, a two-stage design procedure is recommended for AEWMA dispersion charts. Next, the average run length (ARL) performance is compared with other control charts. After that, the performance of the AEWMA charts based on different normalizing transformations is discussed. Finally, some concluding remarks are given. 2 The AEWMA Chart Before discussing the AEWMA dispersion chart, the AEWMA chart for monitoring process means is first reviewed. Let µ 0 and σ 0 be the established standard values for the normal process mean and standard deviation, respectively, and denote Z t = (X t µ 0 )/(σ 0 / n), where X t is the tth subgroup mean of size n, i.e., X t = n i=1 X ti/n. The EWMA statistic for monitoring process means is Q t = (1 λ)q t 1 + λz t = Q t 1 + λ(z t Q t 1 ) where λ is a smoothing constant with 0 < λ 1. Yashchin (1995) showed that the EWMA scheme has optimality property when estimating mean shifts of small magnitudes within the class of linear estimators. However, the best estimating procedures cannot be linear if they are to adapt to changes of relatively large magnitude because the inertia increases as the magnitude of the shift increases. form To overcome the inertia problem, Yashchin (1995) suggested an AEWMA statistic of the Q t = Q t 1 + φ(e t ), where e t = Z t Q t 1 is the prediction error, and φ( ) is a score function motivated by some robust procedures such as Huber s function (Huber 1981) and Welsch s function (Holland and Welsch 1977). Note that, when e t 0, the AEWMA statistic can be rewritten as Q t = (1 φ(e t) e t )Q t 1 + φ(e t) e t Z t. 4

5 Clearly, the AEWMA statistic can adapt the weight on the past estimate Q t 1 according to the current prediction error to detect shifts of different sizes in a balanced way. Capizzi and Masarotto (2003) suggested using it for monitoring process means. For the sake of simplicity, this paper will limit the discussions on the following Huber s score function, e + (1 λ)γ, e < γ φ γ (e) = λe, e γ e (1 λ)γ, e > γ, while other score functions can also be similarly discussed. Note that when γ, φ γ (e) = λe, and the AEWMA statistic reduces to the EWMA scheme. Moreover, when λ = 1 or γ = 0, the AEWMA statistic performs essentially like a Shewhart statistic. Therefore, the AEWMA statistic can be considered as a smooth combination of the Shewhart statistic and the EWMA statistic. The following considers extensions of the AEWMA scheme to the case of monitoring process variability. Throughout the remaining of this paper, the following assumptions are made: (i) process observations are collected from an independent, and identically distributed (iid) normal process N(µ t, σ 2 t ); (ii) the only concern is to detect increases in the process variance, and the process mean is considered to be in control, i.e., µ t µ 0 ; and (iii) before an unknown time τ, σt 2 = σ0 2, and after τ, σ2 t = σ 2 > σ0 2. The objective is to detect an upward change in the process variance as soon as possible. For monitoring increases in the process standard deviation, Crowder and Hamilton (1992) proposed an EWMA-type dispersion chart of the form D t = max[0, (1 λ)d t 1 + λy t ], (1) where Y t is an estimate of the process variance/standard deviation or a simple function of them. Note that the EWMA statistic is reset to zero whenever it falls below zero for the purpose of quickly catching up with an upward change in the process standard deviation. The initial value of D t is usually set to the target value of zero, i.e., D 0 = 0. The EWMA chart declares an alarm when D t exceeds the upper control limit h. In Crowder and Hamilton (1992), the logarithmic function of the subgroup variance is used, namely, Y t = ln(st 2 /σ0 2 ), where n St 2 i=1 = (X ti X t ) 2. n 1 5

6 and The mean and variance of Y t = ln(st 2 /σ0 2 ) are approximated by µ Y = ln( σ2 t σ0 2 ) 1 n 1 1 3(n 1) (n 1) 4 (2) σ 2 Y = 2 n (n 1) (n 1) (n 1) 5, (3) respectively. From Equations (2) and (3), it is clear that the variance of Y t is independent of the true process variance σ 2 t and remains unchanged over the time, and that an increase in σ 2 t will lead to an increase in the mean level of Y t. More importantly, the transformed observations are much more normally distributed than the subgroup variance. These properties motivate the use of logarithmic transformation rather than the subgroup variance for monitoring process dispersion. Like the EWMA location chart, the EWMA dispersion chart in Equation (1) also encounters the inertia problem when reacting to large changes in process variability. problem, an AEWMA scheme can be applied to Y t, given by To diminish this W t = max[0, W t 1 + φ(e t )], (4) where e t = Y t W t 1 and W 0 = 0. The AEWMA chart signals an out-of-control situation when W t exceeds the upper control limit h. Decreases in process variability are also important. A one-sided lower chart for detecting decreases in process variability can be similarly defined as follows: L t = min[0, L t 1 + φ(e t )], To detect both increases and decreases in process dispersion, one can make the combined use of the two one-sided AEWMA charts. The one-sided upper AEWMA chart is chosen for discussion since it responds faster to increases in process variability, which is the main concern of this paper. Due to its simplicity and efficiency of the logarithmic transformation, the remaining of this paper will mainly focus on AEWMA charts based on Y t = ln(st 2 ) while the performance of AEWMA charts using other normalizing transformations will be also briefly discussed. 6

7 γ (σ Y ) 2 20 γ (σ Y ) λ λ (a) σ/σ 0 = 1.2 (b) σ/σ 0 = 2.0 Figure 1: Contour Plot of the Out-of-Control ARL Values of the AEWMA Chart based on ln(s 2 ) for n = 5 (In-Control ARL=200). 3 Design of AEWMA Charts The design of an AEWMA chart is a three-degree-of-freedom problem, which involves the selection of parameters λ, γ, and h. To get some implications on the choice of chart parameters, the parameter effects of λ and γ are first investigated. Then, a two-stage procedure is proposed to design an AEWMA chart for the purpose of providing an overall good detection performance at both a small increase and a large increase in the process standard deviation. 3.1 Effects of Parameters The choice of λ and γ has strong impacts on the performance of AEWMA charts. It is relatively easy to investigate the effect of λ and γ alone on the chart performance, which has been discussed in Yashchin (1995) and Capizzi and Mastrotto (2003). It is known that small values of λ and/or large values γ improve the detection performance of AEWMA charts at small mean shifts while large values of λ and/or small values of γ values improve the detection performance at large mean shifts. However, the joint effect of λ and γ on the run length performance is less clear since it depends on both parameters. To better understand the overall effect, Figure 1 shows contour 7

8 plots of the out-of-control ARL values of AEWMA charts based on ln(s 2 ) as a function of λ and γ for n = 5 when the zero-state in-control ARL is 200. From Figure 1 (a), when detecting a small increase (20%) in the process standard deviation, the ARL decreases as λ decreases or γ increases. From Figure 1 (b), when detecting a large increase (100%) in the process standard deviation, the ARL decreases as λ increases or γ decreases. Furthermore, some implications on the joint choice of AEWMA charting parameters aimed at balancing the sensitivity to both small and large changes in process dispersion can be observed from Figure 1. First, too small values of γ and too large values of λ should be avoided for this purpose. From Figure 1 (a), when γ is too small, say γ 0.5σ Y, the AEWMA chart provides a very poor performance at small increases in the process standard deviation, regardless of the values of λ. From Figure 1 (b), when λ is larger than 0.5, the out-of-control ARL of the AEWMA chart is nearly independent of the choice of γ. This implies that the AEWMA chart would not benefit from the introduction of parameter γ but suffer from the increased design complexity, if too large values of λ were used. Therefore, the selection of too small values of γ or too large values of λ cannot balance the protection against changes of different sizes. Consider the special cases when λ is chosen as large as λ = 1 and γ is set as small as γ = 0. As mentioned earlier, the AEWMA chart reduces to a Shewhart chart in these cases. Clearly, the Shewhart chart cannot provide a balanced performance at both small and large changes in process dispersion. Second, using a small value of λ and an appropriate value of γ can provide the AEWMA chart an overall good detection performance for both small and large changes in process dispersion. As shown from Figures 1 (a) and (b), the parameters within the range 0.1 λ 0.4 and 1.5σ Y γ 3σ Y seem to provide an acceptable detection performance for the AEWMA chart against both small and large changes in the process standard deviation. 3.2 Design Procedure Capizzi and Masarotto (2003) proposed a two-stage procedure to jointly search the optimal parameters of an AEWMA chart for efficiently detecting a small shift and a large shift in the process mean. Similarly, a two-stage procedure can be followed to design an AEWMA chart for monitoring process dispersion. However, note that Capizzi and Masarotto s procedure is a joint searching procedure. It requires extensive computation work and is thus complicated. Rather, 8

9 a two-stage sequential searching procedure is employed here for finding the nearly optimal parameters. The sequential searching procedure requires less computation work while providing a relatively good performance at both small and large changes in the process standard deviation. In Phase I of the two-stage sequential searching procedure, the optimal λ value of an EWMA chart for detecting a specified small change in the process standard deviation is first determined. Due to the attractive property that the parameter γ can be managed to improve the detection performance at large changes by a large percentage while causing a slight loss in the detection performance at small changes (refer to Table 4 for details and Capizzi and Masarotto 2003), this optimal λ value for the EWMA chart can be also considered approximately optimal for the AEWMA chart. In Phase II, based on the optimal λ value obtained in Phase I, the γ value is obtained to optimize the detection performance of the AEWMA chart at a specified large change in the process standard deviation. Details of the two-stage design procedure for the AEWMA chart are summarized below: (i) Choose a desired in-control ARL, and a small change (σ 1 ) and a large change (σ 2 ) in the process standard deviation. (ii) For the specified in-control ARL, find the value of λ giving the minimum ARL of an EWMA chart at σ 1. Some of the optimal λ values can be found in Crowder and Hamilton (1992). (iii) Select a small positive constant α (e.g., α = 0.05) to control the efficiency loss at σ 1 due to the introduction of the parameter γ. (iv) Based on the in-control ARL and the optimal λ value obtained in Step (ii), search the optimal γ value having the minimum ARL at σ 2 subject to the condition that the percentage increase in the out-of-control ARL at σ 1 is at most α. Tables 1 to 3 report a list of parameters of AEWMA charts based on ln(s 2 ) for different in-control ARL values. These parameters are adequate for most practical purposes, and are obtained using the forgoing strategy through a Markov chain approximation shown in Appendix A. As an example, suppose that the desired in-control ARL is 200, and that the small and large changes in process dispersion to be detected are σ 1 = 1.2σ 0 and σ 2 = 1.7σ 0. When the sample 9

10 size is n = 5, from Table 2 in Crowder and Hamilton (1992), the optimal value of λ of an EWMA chart at σ 1 = 1.2σ 0 is λ = 0.05, giving the minimum ARL of If α = 0.05, the optimal γ value is searched on the range 1σ Y γ 3σ Y with a step size of 0.025σ Y for minimizing the ARL at σ 2 = 1.7σ 0 subject to the condition that the out-of-control ARL of the AEWMA chart at σ 1 = 1.2σ 0 is no more than (i.e., ( ) 18.10). The value of γ = 1.6σ Y gives the minimum ARL at σ 2 = 1.7σ 0 of 3.44 and an out-of-control ARL of at σ 1 = 1.2σ 0. Thus, the combination (λ = 0.05, γ = 1.6σ Y, h = ) is considered to be optimal for detecting dispersion changes on the range [1.2σ 0, 1.7σ 0 ]. The same approach can be used to determine the optimal parameters of AEWMA charts for other cases. It can be observed from Tables 1 to 3 that a large value of λ or a small value of γ is more sensitive to large changes while a small value of λ or a large value of γ is more sensitive to small changes. This is consistent with the results in Yashchin (1995) and Capizzi and Masarotto (2003). Furthermore, examination of Tables 1 to 3 indicates a tendency that for a fixed in-control ARL, to detect changes in process dispersion on the same interval [σ 1, σ 2 ], the optimal values of λ and γ tend to increase as the sample size n increases. For example, from Table 1, to detect the changes in process dispersion on the interval [1.4σ 0, 2σ 0 ], the optimal λ value increases from 0.29 to 0.44 while the optimal γ value increases from 1.125σ Y to 1.325σ Y as the sample size increases from n = 3 to 8. This can be explained by changes in the mean and variance of Y t. Based on Equations (2) and (3), note that the mean of Y t tends to increase while the standard deviation of Y t tends to decrease as n increases. This in turn results in an increased signal-to-noise ratio (i.e., µ Y /σ Y ). Thus, given the same detection interval [σ 1, σ 2 ], the λ value chosen for optimizing the detection performance at the same σ 1 value will be larger for larger n values as a large value of λ is more sensitive to large changes. On the other hand, the increased signal-to-noise ratio of Y t due to increase of n requires the selection of a larger γ value in order not to cause much loss in the detection efficiency at σ 1. 4 Performance Comparisons In this section, the run length distribution of the AEWMA chart based on ln(s 2 ) is first discussed. This discussion will provide clear insights into the ARL performance. Run length is the number of observations until the first out-of-control signal triggered by a control chart. As 10

11 Table 1: Optimal Parameters for AEWMA Charts based on ln(s 2 ) When the In-Control ARL is Equal to 200 n = 3 n = 5 n = 8 σ 1 /σ 0 σ 2 /σ 0 λ γ (σ Y ) h λ γ (σ Y ) h λ γ (σ Y ) h Note: γ is given in the unit of standard deviation of Y t, σ Y. Table 2: Optimal Parameters for AEWMA Charts based on ln(s 2 ) When the In-Control ARL is Equal to 370 n = 3 n = 5 n = 8 σ 1 /σ 0 σ 2 /σ 0 λ γ (σ Y ) h λ γ (σ Y ) h λ γ (σ Y ) h Note: γ is given in the unit of standard deviation of Y t, σ Y. 11

12 Table 3: Optimal Parameters for AEWMA Charts based on ln(s 2 ) When the In-Control ARL is Equal to 500 n = 3 n = 5 n = 8 σ 1 /σ 0 σ 2 /σ 0 λ γ (σ Y ) h λ γ (σ Y ) h λ γ (σ Y ) h Note: γ is given in the unit of standard deviation of Y t, σ Y. shown in Appendix A, the run length distribution of the AEWMA chart based on ln(s 2 ) can be approximated by a Markov chain. Next, the ARL performance of the AEWMA chart based on ln(s 2 ) is compared with the traditional EWMA and combined Shewhart-EWMA charts. Moreover, the AEWMA chart is compared with the change-point CUSUM (CP-CUSUM) chart when the process mean µ 0 is known. 4.1 Run Length Distribution of AEWMA Charts As an illustrative example, Figure 2 plots the zero-state in-control probability mass function (PMF) of run length from T = 1 to 50 for the AEWMA chart of ln(s 2 ) with λ = 0.1, γ = 2σ Y, and n = 5. The control limit is set at h = to provide a zero-state in-control ARL of 200. The run length distribution of the traditional EWMA chart of ln(s 2 ) with the same values of λ = 0.1 and n = 5 is also plotted for comparison. From Figure 2, in the in-control situation, the false alarm rate curve of the AEWMA chart is close to that of the EWMA chart except a little bit large false alarm rate at short run lengths T < 3. For run lengths T > 3, the run length distribution curves of both charts are almost the same. 12

13 6 x 10 3 EWMA AEWMA 5 4 Probability Run Length Figure 2: In-Control Run Length Distributions of the AEWMA and EWMA Charts EWMA AEWMA EWMA AEWMA Probability 0.03 Probability Run Length Run Length (a) σ/σ 0 = 1.2 (b) σ/σ 0 = 2.0 Figure 3: Out-of-Control Run Length Distributions of the AEWMA and EWMA Charts. 13

14 Figures 3 (a) and (b) plot the zero-state out-of-control run length PMF of both charts under increases in the process standard deviation of sizes of 20% and 100%, respectively. As can be seen from Figure 3 (a), when detecting a small increase in the process standard deviation, the signal probability of the AEWMA chart is larger than the EWMA chart at short run lengths T 3 and almost the same at T > 3. Therefore, there is negligible difference in the detection power between the AEWMA and EWMA charts for signalling small changes in process variability. On the other hand, when detecting a large increase in the process standard deviation, the AEWMA chart has a much higher signal probability than the EWMA chart at the very short run length T = 1, as can be observed from Figure 3 (b). This implies a better sensitivity of the AEWMA chart to large increases in process variability than the EWMA chart. These observations indicate a good property of the AEWMA chart. That is, the AEWMA chart can be designed to have performance close to the EWMA chart at a small increase in process variability but better performance than the EWMA chart at a large increase in process variability. 4.2 Comparison with EWMA Charts Table 4 compares the ARL of AEWMA charts with the traditional EWMA chart for monitoring increases in the process standard deviation. Both zero-state and steady-state ARL results are provided. The AEWMA and EWMA charts are designed based on λ = 0.1 and n = 5. All charts are designed to have a zero-state in-control ARL of 200. The same values of control limits were used for computing both zero-state and steady-state out-of-control ARL values. Note that when γ 3σ Y, the zero-state/steady-state ARL performance of the AEWMA chart is the same as that of the EWMA chart. This is because when γ takes a large value, the estimation errors are very unlikely to exceed γ, and the AEWMA scheme performs essentially the same as the regular EWMA scheme. However, it can be observed from Table 4 that when γ decreases, the AEWMA chart can perform substantially better than the EWMA chart for detecting large increases in process dispersion and perform closely to the EWMA charts at small increases in process dispersion. For example, when γ decreases from 3σ Y to 1.5σ Y, the percentage decrease of the zero-state out-of-control ARL at σ = 3σ 0 is around 30.6% (( )/1.83) while the percentage increase in the ARL at σ = 1.2σ 0 is 6.55% (( )/18.25) only. To get some visual insights on this, Figure 4 further plots the ARL curves of the AEWMA chart 14

15 with γ = 1.5σ Y and the EWMA chart. The ARL scale is chosen to be logarithmic. From Figure 4, it is clear that the AEWMA chart performs closely to the EWMA chart at small changes in the process standard deviation but better than the EWMA chart at relatively large changes. The above observations demonstrate the benefit of introducing the additional parameter γ for the AEWMA chart. In particular, the parameter γ can be managed to improve the performance of an AEWMA chart at larger changes in process dispersion with minor loss in the efficiency in detecting smaller changes in process dispersion. This agrees with the expectation of the superiority of AEWMA estimators over conventional EWMA statistics in catching up with large changes in the process mean level. From Table 4, it is observed that the zero-state ARL is larger than the steady-state ARL. The difference between the two ARL results depends on both values of γ and λ. From Table 4, it is clear that the difference between the zero and steady-state ARL values decreases as γ decreases. Moreover, it is also well known that this ARL difference decreases as λ increases (Lucas and Saccucci 1990). Note that when γ = 0 or λ = 1, the AEWMA chart reduces to a Shewhart chart. In both cases, the steady-state ARL would be the same as the zero-state ARL. As can be seen from Table 4, when γ = 0.5σ Y ( 0.4 for n = 5), the steady-state ARL is almost the same as the zero-state ARL. In practice, however, the difference is minor and not critical since both ARL results lead to qualitatively the same conclusions. Either one suffices for most practical purposes. 4.3 Comparison with Shewhart-EWMA Charts To improve the performance of the traditional EWMA charts for monitoring large process means, the combined Shewhart-EWMA chart has been suggested (Lucas and Saccucci 1990; Albin, Kang and Shea 1997). This scheme is achieved by adding Shewhart limits to an EWMA scheme. It is also straightforward to extend this idea to the dispersion monitoring problem. Figure 5 compares the ARL between the AEWMA and Shewhart-EWMA charts based on ln(s 2 ) for n = 5. The zero-state in-control ARL is maintained as 200. The considered AEWMA chart is designed for σ 1 = 1.2σ 0 and σ 2 = 2σ 0, and α = 0.05 (λ = 0.05, γ = 1.6σ Y, and h = as shown in Table 1). For the combined Shewhart-EWMA chart, the same value of λ = 0.05 is used, and two different Shewhart control limits of SCL = 2σ Y and SCL = 3σ Y are considered here. 15

16 Table 4: ARL Values of AEWMA Charts based on ln(s 2 ) When λ = 0.1 and n = 5 AEWMA γ (σ Y )= EWMA σ/σ 0 Type h= Zero-state Steady-state Zero-state Steady-state Zero-state Steady-state Zero-state Steady-state Zero-state Steady-state Zero-state Steady-state Zero-state Steady-state Zero-state Steady-state Zero-state Steady-state Zero-state Steady-state Zero-state Steady-state

17 200 EWMA AEWMA 200 EWMA AEWMA ARL ARL σ /σ σ /σ 0 (a) Zero-State (b) Steady-State Figure 4: ARL Comparisons between AEWMA and EWMA Charts base on ln(s 2 ) When λ = 0.1 and n = 5. Simulations were used to obtain the ARL of the combined Shewhart-EWMA chart of ln(s 2 ). From Figure 5, it is clear that the AEWMA chart often performs closely to or better than the Shewhart-EWMA charts for a wide range of changes in process dispersion. This result is consistent with that observed in the situation of monitoring process means in Capizzi and Mastrotto (2003). As they pointed out, the Shewhart-EWMA chart can only ameliorates the inertial problem of the EWMA chart in detecting large changes in process means while the AEWMA chart can diminish this inertia problem. Therefore, the AEWMA chart performs better than the Shewhart-EWMA chart for monitoring process means. 4.4 Comparison with CP-CUSUM Charts When µ 0 is Known The above comparisons assume that the target mean is unknown. When the target mean is known and is always in control, a more accurate estimator T 2 t = n i=1 (X ti µ 0 ) 2 n can be used to estimate the process variance. Although both St 2 and Tt 2 are unbiased estimators of σt 2, the former does not utilize the process mean information but rather estimate it. The mean estimation results in a loss of one degree of freedom in the distribution of St 2. Acosta-Mejia et 17

18 200 Shewhart EWMA (SCL=2σ Y ) Shewhart EWMA (SCL=3σ Y ) AEWMA 50 ARL σ /σ 0 Figure 5: Zero-State ARL Comparisons between AEWMA and Shewhart-EWMA Charts base on ln(s 2 ) When λ = 0.1 and n = CP CUSUM (K=1.193) CP CUSUM (K=1.848) AEWMA 50 ARL σ /σ 0 Figure 6: Zero-State ARL Comparisons between AEWMA and CP-CUSUM Charts When µ 0 is Known and n = 5. 18

19 al. (1999) proposed a CP-CUSUM chart based on T 2 t statistic derived from the sequential probability ratio test (SPRT), given by C t = max[0, C t 1 + T 2 t σ 2 0 k]. An out-of-control signal is triggered when C t exceeds a threshold h C. The CP-CUSUM chart using a SPRT reference value k = ln(σ2 /σ 2 0 ) 1 1/(σ 2 /σ 2 0 ) is proved to be optimal at detecting a variance of size σ 2 in Moustakides (1986). Figure 6 compares the zero-state ARL between the AEWMA chart based on ln T 2 t and the CP-CUSUM chart when µ 0 is known and n = 5. Two reference values of k = and k = are considered for the CP-CUSUM chart for providing an optimal detection performance at σ = 1.2σ 0 and σ = 2σ 0, respectively. The corresponding control limits are set as h C = and h C = 8.16 for providing an in-control zero-state ARL of 200. The parameters for the AEWMA chart are chosen as λ = 0.05, γ = 1.7σ Y, and h = aimed at optimizing the detecting performance on the interval [1.2σ 0, 2σ 0 ], which are obtained based on the foregoing design strategy. From Figure 6, it can be observed that a single CUSUM chart cannot perform well at both a small change and a large change in process dispersion while the AEWMA chart can provide a relatively good detection performance at both small and large changes. The AEWMA chart always provides an ARL either the shortest or close to the shortest for all the changes considered here. 5 AEWMA Charts based on Other Transformations The logarithmic transformation is simple to implement but it only provides an approximate normal distribution. Other more complicated transformation could be employed to generate even more normally distributed observations. One such transformation is based on the inverse normal distribution. Quesenberry (1995) showed that statistic ( )) (n 1)S P S,t = Φ (G 1 2 t has a standard normal distribution, where G ( ) is the cumulative distribution function of a chi-square distribution with n 1 degrees of freedom, and Φ 1 is the inverse of a standard σ

20 normal distribution. Besides these transformations, Wilson and Hilferty (1931) proposed a transformation based on the cubic root of a chi-square distribution to normalize the sample variance. They showed that 3 χ 2 n is approximately normally distributed with mean 1 2/(9n) and variance 2/(9n). Thus, χ S,t = ( (S 2 t σ 2 0 ) 1/3 ( ) ) / (n 1) 9(n 1) will have an approximate standard normal distribution when σ = σ 0. Acosta-Mejia et al. (1999) showed that the CUSUM charts based on χ S and P S have almost the same performance. That is the evidence that the distribution of χ S,t is almost identical to a standard normal distribution. Acosta-Mejia et al. (1999) compared the performance of CUSUM charts based on the three different normalizing transformations: ln(s 2 ), χ S and P S. They showed that the CUSUM charts based on χ S and P S have similar performance and both perform better than the CUSUM chart based on ln(s 2 ). However, note that the statistic ln(s 2 ) is not standardized into zero mean and unit variance while P S,t and χ S,t are standardized in Acosta-Mejia et al. (1999). Therefore, their comparison result is a little bit unfair. To be fair, ln(st 2 /σ0 2 ) is also standardized by using its mean and variance given by Equations (2) and (3) respectively in the following performance comparison of AEWMA charts based on different transformations. Table 5 reports the ARL values of AEWMA charts based on the above three standardized transformations. The ARL values of the AEWMA charts based on χ S and P S are also approximated using a Markov chain approach, details of which are shown in Appendix B. From Table 5, for the same value of γ, the AEWMA charts based on χ S and P S have nearly the same performance. This is consistent with that of Acosta-Mejia et al. (1999). Moreover, it can be observed that the performance of the AEWMA chart based on the standardized ln(s 2 ) is competitive to the charts based on χ S and P S. Clearly, the AEWMA chart based on the standardized ln(s 2 ) performs better than the AEWMA charts based on χ S and P S when γ = 1.5 but worse when γ = 2.5. When γ = 2, the AEWMA chart based on the standardized ln(s 2 ) performs better than the AEWMA charts based on χ S and P S for small increases in the process standard deviation σ 1.3σ 0 but worse for σ > 1.3σ 0. 20

21 Table 5: Zero-State ARL Values of AEWMA Charts based on Other Normalizing Transformation Functions When λ = 0.1 and n = 5 Using ln(s 2 ) Using χ S Using P S γ = σ/σ 0 h = Concluding Remarks This paper extends the idea of AEWMA charts for monitoring process means to the case of monitoring process dispersion. A Markov chain model is established to evaluate the performance of AEWMA dispersion charts, which simplifies the analysis without a large number of simulation replicates. A two-stage design procedure is suggested for the AEWMA charts aimed at providing a balanced detection performance against changes in the process standard deviation of different sizes. The comparison results show that the suggested control chart offers a more balanced protection performance against changes of different sizes in process dispersion than the traditional EWMA chart, the combined Shewhart-EWMA chart, and the CP-CUSUM chart. This paper mainly focus on the AEWMA chart based on the logarithmic transformation of the subgroup variance due to its simplicity and efficiency while other different transformations can also be used in the AEWMA chart. Although detecting decreases in process dispersion is not discussed in detail in this paper, one can easily extend the AEWMA chart to this case. Furthermore, joint monitoring of process location and dispersion has attracted a lot of attentions 21

22 recently in SPC research (Chen et al. 2001; Reynolds and Stoumbos 2004). The proposed technique can be developed for this purpose. Some research work is doing in this way. Appendix A: ARL Computations for AEWMA Charts of ln(s 2 ) Similar to the method of Brook and Evans (1972) and Lucas and Saccucci (1990), the incontrol region [0, h] can be divided into sub-regions to obtain a discretized Markov chain for approximating the run length distribution of an AEWMA chart based on ln(s 2 ). Let m be the number of sub-intervals along the W -axis on the interval [0, h]. Thus, the width of each subinterval is ω = (2h)/(2m 1), except that the width of the first one is ω/2. These sub-intervals are labelled as i = 1, 2,, m, and are approximated by their mid values, (i 1)ω. Let P (i, j) be the transition probability of W t in Equation (4) from state i to state j. For i = 1, 2,..., m and j 1, P (i, j) = Pr{ W t in state j W t 1 in state i} = Pr{(j 1)ω 0.5ω < (i 1)ω + φ(e t ) (j 1)ω + 0.5ω} = Pr{(i 1)ω + φ 1 [(j i 0.5)ω] < Y t (i 1)ω + φ 1 [(j i + 0.5)ω]} = Pr{a 1 < Y t a 2 }, where a 1 = (i 1)ω + φ 1 [(j i 0.5)ω], and a 2 = (i 1)ω + φ 1 [(j i + 0.5)ω]. The function φ 1 ( ) is the inverse of the Huber score function given by (Capizzi and Masarotto 2003) φ 1 γ (v) = v (1 λ)γ, v/γ v + (1 λ)γ v < λγ λγ v λγ v > λγ. Note that S 2 t /σ 2 0 has a gamma distribution with shape parameter α = n

23 and scale parameter 2σ 2 β = (n 1)σ0 2. It follows that Pr{a 1 < Y t a 2 } = F (exp(a 2 ), α, β) F (exp(a 1 ), α, β), where F (, α, β) is the cumulative distribution function of a gamma distribution with shape parameter α and scale parameter β defined above. For i = 1, 2,..., m and j = 1, P (i, j) = F (exp(a 2 ), α, β). Let the matrix R m m be the transition probability matrix, which contains the probabilities of going from one transient state to other states. Then, the probability mass function of run length T and the ARL are given by Pr(T = t) = p ini (R t 1 R t ) 1, and ARL = p ini (I R) 1 1, respectively, where p ini is any initial probability vector of states, and 1 is a column vector of ones. In the zero-state analysis, the initial probability vector p ini corresponds to W 0 = 0. To obtain the steady-state ARL, Lucas and Saccucci (1990) suggested using the cyclical steady-state probability vector, p ss, which is solved from p ss = P 1 p ss subject to 1 p ss = 1, where R (I R)1 P 1 = The cyclical steady-state ARL of the AEWMA chart is obtained in a similar way, given by ARL = p ss (I R) 1 1. The ARL of the EWMA chart base on ln S 2 was obtained using integral equation approach in Crowder and Hamilton (1992). Note that the EWMA scheme is a special case of the AEWMA scheme with γ +. Thus, the above Markov chain for the AEWMA chart can be modified to compute the ARL of the EWMA chart by taking a sufficiently large value of γ. Within our investigations, both the Markov chain approach and the integral equation approach can produce nearly the same results. In this paper, the Markov chain approach was used to evaluate the ARL of an EWMA chart of ln S 2. 23

24 Appendix B: ARL Computations for AEWMA Charts based on χ S and P S The transition probability matrix of the AEWMA charts based on χ S and P S can be obtained in a way similar to that of the AEWMA chart based on ln(s 2 ), which is summarized below. For i = 1, 2,..., m and j 1, the transition probability of the AEWMA chart based on χ S statistic is given by P (i, j) = Pr{a 1 < χ S,t a 2 } = F (b 2, α, β) F (b 1, α, β) where and b 1 = b 2 = (a 1 (a 2 2 9(n 1) + 2 9(n 1) + ( 1 ( 1 ) ) 3 2 9(n 1) ) ) (n 1) The values of a 1, a 2, α, β and F (, α, β) are defined as above. When j = 1, P (i, j) = F (b 2, α, β). For i = 1, 2,..., m and j 1, the transition probability of the AEWMA chart based on P S statistic is given by P (i, j) = Pr{a 1 < P S,t a 2 } = F (b 4, α, β) F (b 3, α, β), where and ( ) b 3 = G 1 Φ(a1 ) n 1 ( ) b 4 = G 1 Φ(a2 ). n 1 G 1 ( ) represents the inverse of a chi-square distribution with n 1 degrees of freedom. When j = 1, P (i, j) = F (b 4, α, β). 24

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