A MACRO FOR MULTIVARIATE STATISTICAL PROCESS CONTROL

Transcription

1 A MACRO FOR MULTIVARIATE STATISTICAL PROCESS CONTROL Meng-Koon Chua, Arizona State University Abstract The usefulness of multivariate statistical process control techniques has been received a tremendous attention. However, the lack of computer tools presents one of the maior obstacles to the acceptance of these techniques. A macro is thus developed to provide two control charts: the multivariate exponentially weighted moving average control chart and the log lsi control chart. These control charts constitute a multivariate analog of univariate control charts and are useful in detecting small process mean shifts and process dispersions respectively. Some case studies under linear mean trend shift and covariance expansion situations are given to demonstrate the performance of the control charts. where x, Introduction tt is well known that the Hotelling r control chart is effective in detecting large process shifts, and the Multivariate Cumulative Sum (MCUSUM) and the Multivariate Exponentially Weighted Moving Average (MEWMA) control charts are effective in detecting small process shifts. The MEWMA control chart is however selected in this paper for the following reasons: (1) The EWMA control chart can be used as a process forecasting tool (Hunter [1986]). (2) The design of the EWMA control chart is easier than that of the CUSUM control chart (Montgomery, Gardiner, and Pizzano [1987]). (3) The Average Run Length (ARL) performance of the MEWMA control chart is better than that of the MCUSUM control chart as proposed by Crosier (1988) or Pignatiello and Runger (1988) (Lowry et al. [1989]). Due to the fact that the MEWMA control chart will not protect against changes in process variability as measured by the covariance matrix. a control chart to monitor process dispersions is necessary. Therefore. the log lsi control chart is suggested to use in conjunction with the MEWMA control chart. These control charts then constitute a statistical process control system which accounts for both process mean and process dispersion. MEWMA Control Chan Assume that x,. X 2.. x" are independent observation vectors. the Xi is: j = P k= n and the MEWMA mean vectors Zi can be defined as: where Z; = MEWMA mean vector wtth Zo = o. j m R = Diag (rl' r2,..., r p ). 0 <lj < 1, j = 1, 2,,p = Identity matrix X; = Sample mean vector Wrth an additional assumption that r 1 = r2 =... = r.e = r, the ARL performance of the MEWMA control chart will depend only on the value of the noncentral~y parameter. Then, the MEWMA mean vectors can be redefined as (Lowry et al. [1989]): and the sample covariance matrix for MEWMA is: r(i (I r) ) S7 S,, 2-r where r = Smoothing constant Sz. = Sample covariance matrix for sample i I formewma and 666

2 LCL = Y -Zc./2 Sy where Sijk is the covariance between variables j and k for sample i. Hence, any sample mean vector X. will be deemed out of control H the test statistic, I "f=nz; SZ;' Z; is greater than the constantlimk H where H > 0, and can be obtained from simulation based on a spechied Average Run Length (ARL). On the other hand, the process is in control H the test statistic is less than or equal to H. Log lsi Control Chart The log lsi control chart is recommended by Montgome'): and Wadsworth (1972) in conjunction wkh the Hotelling T' control chart. To use this control chart, the ISil is computed as: IS, I = where Sijk is the covariance between variables j and k for sample I and ISil is the determinant of the sample covariance matrix for sample i. Assume that then, and Yi =loglsil m - I L Y-- m Y. I ;'1 wheoe Zc./2 is the percentage point of a normal distribution. The use of normal distribution in the equations has been justhied by Gnanadesikan and Gupta (1970). Therefore, the standard "3 sigma" limks can be used to replace the Zan term as the usual practice in univariate case. Analogous to the relationship between Holelling "f and MEWMA control charts, a control chart for monitoring the covariance matrix of the MEWMA control chart is studied. This control chart is denoted as the log IS,I control chart and has the following relationship: where S, = O. The comparison between the log IS,I and o log lsi control charts will be studied in the following section. The Macro Two datasets are required to run the macro. The first dataset is a training dataset for setting up the control limits of the log lsi control chart, and the second dataset is a testing dataset for checking any abnormalkies or out-ofcontrol situations. Similar to the univariate control chart procedure, the training dataset must be in statistical control" before the testing dataset can be applied. The term "statistical control" implies that all samples in the training dataset must be within the control limits. In addition to the datasets, the control limits for the Hotelling T2 and MEWMA control charts need to be provided. The control limit lor the Hotelling "f control chart can be found by: p(n-i) -p-fila"" where p = number of variables n = sample size F a,p,lii' = F distribution with p and n-p degl99 of freedom and the control limit for the MEWMA control chart is obtained by simulation based on a specified ARL. In this case, a signhicant level of 0.01 is used to set up the ARL (lowly et al. [1989]). The macro is written in SAS Basics and incorporated with SAS/IML statements. The SAS/IML is used mainly for matrix manipulations and computations. Furthermore, the SAS normal random number generator RANNOR is used to generate the datasets. by: The upper and lower control limks are constructed Case Studies 667

3 To demonstrate the performance 01 the control charts, some case studies under p=2 are provided as below: (1) One-variable linear trend mean vector shift case (2) Two-variable linear trend mean vector shift case (3) One-variable inflated variance case (4) Two-variable inllated covariance case Cases 1 and 2 are studied under the linear trend increment unit lor each sample.. 5pec~ically, only the first variable is incremented in case 1 and both the variables are incremented in case 2. Case 3 is conducted with an inllated factor of 10 for the first variable variance and case 4 is obtained by inflating the covariance matrix with a factor of 5 for each sample. These cases are shown in Figures 1 to 4 respectively. Figure 1 shows the rand MEWMA control charts under one-variable linear mean trend. It is lound that the r control chart detects the shift at sample 17 and the MEWMA control chart detects the shift at sample 9. The out-of-control sample 9 In the MEWMA control chart may be misleading as the sample is missing in the chart. Figure 2 displays the rand MEWMA control charts under iwo-variable linear mean trend. In this ligure, the T2 control chart detects the out-of-control sample at pos~ion 17 and the MEWMA control chart detects the sample at pos~ion 13. Obviously, the MEWMA control chart always detect the shifts laster than the r control chart as expected. Figure 3 illustrates the log 151 and log ISzl control charts for one-variable variance inllation case. On the contrary to the expectation, the log lsi control chart outperforms the log ISzl control chart significantly. From Figure 3, the log lsi control chart detects the changes at sample t 4 and the log IS I control chart does not detect any changes for 20 samples. Figure 4 shows the log lsi and log ISzl control charts under two-variable inflated covariance casso In this case, the log 151 control chart detects the process changes as early as at sample 3 but the log 15z1 control chart fails to detect any changes in the entire 20 samples. Although cases 3 and 4 illustrate only two of the possibilities that may occur in the process dispersion situations, it appears that the log lsi control chart is preferred to the log ISzl control chart since the emphasis here is to detect the out-of-control situations as soon as possible. Crosier, R. B. (1988). "Multivariate Generalizations of Cumulative Sum Quality Control Schemes." Techoometrics 30, pp Gnanadesikan, M. and Gupta,S. S. (1970). "A Selection Procedure for Multivariate Normal Distribution in Terms of the Generalized Variance.- Technometrics 12, pp. ti3-1t9. Hunter, J. S. (1986). "The Exponentially Weighted Moving Average." Journal of Quality T echoo/ogy 18, pp Lowry, C. A. et al. (t989). "A Muftivariate Exponentially Weighted Moving Average Control Chart." Presented at the 149th Annual Meeting 01 ASA, August 6-10, Washington, D.C. Montgomery, D. C. and Wadsworth, H. M. (1972). "Some Techniques lor Multivariate Qual~y Control Applications." ASQC Technical Conference Transactions 26, pp Montgomery, D. C., Gardiner, J. S. and Pizzano, B. A. (1987). "Statistical Process Control Methods for Detecting Small Process Shifts." Frontiets in Statistical Quality Control 3, Physica-Verlag, Heidelberg, FGR. Pignatiello, J. J., Jr. and Runger, G. C. (1988). "Comparisons 01 Muftivariate CUSUM Charts." Working paper , Systems and Industrial Engineering Department, University of Arizona. Conclusion A macro is developed for the multivariate statistical process control. This macro provides two control charts which are useful for small process shifts and process dispersions respectively. The use of these two control charts then constitutes a multivariate version of the statistical process control system. Some case studies under different mean shifts and inflated covariance situations are examined. The case studies show that the MEWMA and log lsi control charts should be used for monitoring process means and process dispersions. References 668

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