NSC 2010 Midterm report
Su-Fen YangDepartment of Statistics National Chengchi University
Taiwan
Control charts are important tools in statistical quality control (SPC) and are used to effectively monitor a process. In SPC, when dealing with a variables quality characteristic,
X -S or X-R control charts are used to monitor the process mean and the spread. While the X chart is used to monitor the process mean shift, the S (or R) chart is used to detect the variation change. However it is more advantageous to use a single chart to monitor both of the mean and the variation. The obvious advantages are (i) simple to use and interpret, (ii) cut down the time, resource, manpower, money and effort, (iii) easily detect the shift of the process mean, and/or the change of the process variability, and (iv) identify the cause of the changes – due to mean or/and variation (see Cheng and Thaga [1]). In recent years, some literatures were contributed to the design of various single control charts, like the T chart (Chen, Cheng and Li [2]), the single B chart (Grabov and Ingman [3]), the Likelihood Ratio chart (Sullivan and Woodall [4]), the Max chart (Chen and Cheng [5]), the MSE chart (Cheng and Spiring [6]), and the weighted-loss chart (Wu and Tian [7]). Cheng and Thaga [1] gave an overview of the single variables charts.
In industry, the quality and the loss of products are important factors among competing companies. Loss function is used to measure the loss due to poor quality (Spiring and Yeung [8]). Taguchi emphasized the importance of the target value in a process (Gopalakrishnan, Jaraiedi, Iskander and Ahmad [9]). Sullivan [10] gave examples that stress the importance of monitoring a target value and variability. In practice, the in-control process mean may not be the process target. Hence, a single control chart would
be effective if it could detect the deviation from process target and the shift in process variability with small loss. Moreover, many papers on the performance of adaptive control charts had suggested using adaptive control scheme to the fixed control scheme. However, some researches had been proposed to use the double adaptive charts or a single adaptive chart to simultaneously monitor the process mean and variance, like Chengular, Arnold and Reynolds [11], Costa [12], Reynolds and Stoumbos [13], Zhang and Wu [14], Costa and Rahim [15], Costa and Magalh [16], Yang and Chen [17], Yang and Yang [18], Yang and Yu [19], Wu, Wang and Wang [20] and Prajapati and Mahapatra [21]. So far, in a single process, no one had proposed a single variable sampling interval (VSI) loss function chart to detect both the deviation from process target and the shift in process variability with small loss. We propose to construct a single VSI loss control chart to monitor both the process target and the variability with smaller loss. The performance of the VSI average loss control chart is described. Finally, an example of the metallic film thickness of the computer connectors system is used to illustrate the application of the optimal VSI average loss chart. The performance comparison among the VSI, FSI average loss charts and
S
X − charts shows that VSI chart is much better than the latter.
The above approach is extended to VP loss control chart. We found that using VP loss chart may save much detection time and more sensitive in the shifts of process target and variance than VSI and FP loss charts. The VP loss chart is thus recommended.
To detect small and median shifts in target and variance and with smaller loss, EWMA loss control chart is constructed. The performance is measured by Average Adjustment Time to Signal (AATS). The AATS is derived by Markov chain approach. The numerical analyses illustrate the EWMA loss chart performs Shewhart-type loss chart. The approach of deriving EWMA loss chart is extended to VSI, VSSI and VP EWMA loss control charts. For VSSI and VP EWMA loss charts, the distributions of EWMA loss statistics cannot be
chart may save much detection time and more sensitive in the shifts of process target and variance than adaptive and FP loss charts. The adaptive EWMA loss charts are thus recommended.
All above study is for variables data. So far, for a single process the study of loss chart for attribute data with binomial distribution has not been studied. We derive the FP loss chart for binomial data, then extend the approach to VSI, VSSI and VP loss charts. Since the loss statistics are discrete not variable, their average run lengths (ARL) or average time to signal (ATS) are always not the required 370 when the process is in control and the ARL’s and ATS’s are always not decrease as sample size increases. Hence, transformation for the discrete loss statistics seems necessary.
Two of the reports had been submitted to SCI journals. Some other studies will be written and submitted to SCI journals in the coming summer.
REFERENCE
1. Cheng, S. W. and Thaga, K. (2006). On singlevariables control charts-An overview. Quality and Reliability Engineering International, 22(7), 811-820.
2. Chen, G., Cheng, S. W. and Li, G. (1996). A single variables control chart-T chart. Research Manuscript, University of Manitoba, Winnipeg, Canada.
3. Grabov P. and Ingman D (1996) Adaptive control limits for bivariate process monitoring. J Qual Technol 28:320–330.
4. Sullivan, J. H., and Woodall, W. H. (1996), “A Comparison of Multivariate Quality Control Charts for Individual Observations,” Journal of Quality Technology, 265 -278
5. Chen, G. and Cheng, S. W. (1998). Max chart: combining X-bar chart and S chart.
6. Cheng, S. W. and Spiring. F.A. (1998). An alternate variable control chart: the univariate and multivariate cases. Statistica Sinica, 8(1), 273-287.
7. Wu, Z. and Tian, Y. (2006). Weighted-loss-function control charts. International
Journal of Advanced Manufacturing Technology, 31, 107-115.
8. Spiring, FA. and Yeung, AS. (1998). A general class of loss functions with individual applications. J Qual Technol, 30, 152–162.
9. Gopalakrishnan, B., Jaraiedi, M. , Iskander, W.H., Ahmad, A. (2007). Tolerance synthesis based on Taguchi philosophy , International Journal of Industrial and Systems Engineering, 2( 3), 311 – 326.
10. Sullivan, L.P. (1984). Reducing variability: A new approach to quality. Quality Progress. July, 15-21.
11. Chengular, I., Arnold, J. and Reynolds, M.R. (1989)” Variable Sampling Intervals for Multiparameter Shewhart Charts,” Communications in Statistics—Theory and Methods 18, 1769-1792.
12. Costa, A. B. (1998). Joint X and R charts with variable parameters. IIE Trans 30(6):505–514.
13. Reynolds, M.R., JR. and Stoumbos, Z. G. (2001),”Monitoring the process mean and variance using individual observations and variable sampling intervals”, Journal of Quality Technology 33, 181-205.
14. Zhang, S. and Wu, Z. (2006). Monitoring the process mean and variance using a weighted loss function CUSUM scheme with variable sampling intervals. IIE Transactions, 38(4), 377 – 387.
15. Costa, A. F. B. and Rahim, M.A. (2006). EWMA Chart for Monitoring Process Mean and Process Variance. Quality Technology and Quantitative Management , 3(3), 295-305.
16. Costa, A. B. and Magalh, M.S. (2007). An adaptive chart for monitoring the process mean and variance. Qual. Reliab. Eng. Int,. 23(7), 821-831.
17. Yang, S. and Chen, Y. (2009). Controlling Over-adjusted Process Means and Variances Using VSI Cause Selecting Control Charts," Expert System with Application, 36, 7170-7182.
18. Yang, C and Yang S. (2009). Adaptive Control Scheme for a Process with Incorrect Adjustment. First Asian Conference on Intelligent Information and Database Systems,
Dong hoi, Quang binh, Vietnam , 404-409.
19. Yang, S and Yu, Y. (2009). Monitoring Cascade Processes, Journal of Chemometrics, 23,449-462.
20. Wu, Z., Wang, P. and Wang, Q. (2009). A loss function-based adaptive control chart for monitoring the process mean and variance. Int J Adv Manuf Technol , 40:948–959.
21. Prajapati, D.R. and Mahapatra, P.B. (2009). Control charts for variables to monitor the process mean and dispersion: a literature review. International Journal of Productivity and Quality Management. 4 (4), 476 – 520.
22. Duncan, A. J.（1956）. The Economic Design of -charts Used to Maintain Current Control of a Process. Journal of American Statistical Association, 51, 228-242. 23. Cinlar, E. (1975), Introduction to Stochastic Process. Prentice-Hall,
Englewood Liffs,NJ.
24. Braverman, J. D. (1981). Fundamentals of Statistical Quality Control. Reston Publishing Co., Inc. Reston, Virginia, U.S.A.
