Application of Resampling Methods in Multivariate Cumulative Sum Control Charts

Subject Areas : Industrial Management

Abdol-Rasoul Mostajeran ¹ , Amirhoussin Aghajani ²

1 - Islamic Azad University, Branch of Shahinshahr, Department of Mathematics
2 - Management and Planning Organization

Received: 2015-12-10 Accepted : 2016-07-18 Published : 2016-08-25

Keywords:

Abstract :

One of the important tools for process control is control charts. Shewhart control charts only use the information about the process given by the last observed value of the control statistic and completely ignore any past information, thus they are memoryless and they cannot detect small and moderate shifts. Most multivariate Shewhart control charts are dependent to normality assumption. However, in many situations, this condition does not hold. Multivariate cumulative sum control chart (mcusum) one of the most widely used tools in multivariate statistical process control for quality control. The mcusum chart does not have the disadvantages of Shewhart control chart. Mcusum control charts have memory and they are more sensitive to small and moderate changes in the process. Determining of the exact and limiting distribution of mcusum control chart statistic is difficult even under the normality assumption. Therefore the distribution of mcusum control chart statistic can be obtained through simulation. A bootstrap control chart is based on resampling of the original observation and it does not require any knowledge about the underlying distribution of the observations. In this paper for the first time, we propose the application of resampling methods in mcusum. In the proposed study four different resampling algorithms are introduced. These algorithms were compared based on ARL0 criteria in simulation studies. R program was used to write and run the codes of the simulation study. Finally, the proposed control charts applied on a real dataset obtained from Isfahan Sugar Factory.  

References:

1. Page, E. S. (1954). Continuous Inspection Schemes, Biometrika. 41, 100-114.


2. Montgomey, C. D. (2006). Introduction to Statistical Quality Control, 6th Edition. John Wiley & Sons.


3. Chatterjee, S., and Qiu, P. (2009). Distribution-free cumulative sum control charts using bootstrap-based control limits. the Annals of Applied Statistics, 3(1), 349-369.


4. Mood, A. M., Graybill, F. A., and Boes, D. C. (1974). Introduction to the Theory of Statistics (3rd ed.), New York: McGraw-Hill.


5. Woodall, W. H., and Ncube, M. M. (1985). Multivariate CUSUM Quality-Control (3rd ed.), New York: McGraw-Hill.


6. Healy, j. D. (1987). A Note on Multivariate CUSUM Procedures, Technimetrics, 29 (4).


7. Crosier, Ronald. (1988). Multivariate Generalizations of Cumulative Sum Quality-Control Schemes, Technometrics, 30:3, 291-303.


8. Pignatiello, J. J., and Runger, G. C. (1990). Comparisons of Multivariate CUSUM charts, J. Qual. Technol. 22, pp. 173-186.


9. Nagi, H., and Zhang, J. (2001). Multivariate Cumulative Sum Control charts based on Projection pursuit, Statist. Sinica 11, pp. 747-766.


10. Duncan, A. J. (1974) Quality Control and Industrial Statistics (4^th ed.), Homewood, IL: Richard D. Irwin.


11. Hotelling, H. (1947). Multivariate Quality Control. Illustrated bythe Air Testing of Sample Bombsights, “in Techniques of Statistical Analysis, eds. C. Eisenhart, M. W. Hastay, and W. A. Wallis, New York: McGraw-Hill, 111-184.  


12. Mahmoud, M. A., and Maravelakis, P. E. (2013). The Performance of the Multivariate CUSUM Control charts with Estimated Parameters. Comm. Statist. Simulation Comput, 83. 72.


13. Qiu, P., and Hawkins, D. (2001). A Rank-Based Multivariate Cusum Procedure, Technometrics, 43:2, 120-132.