Abstract
Varying three parameters of the Hotelling's T 2 control chart, namely, the sample size, the sampling interval length, and the action limit, results in a great improvement over the traditional T 2 control chart particularly for the small process shift. This paper presents the economic model for designing the variable parameters T 2 chart. In the economic design, a cost function is constructed, involving the cost of sampling and testing, the cost of false alarm, the cost to detect and remove the assignable cause, and the cost when the process is operating out-of-control. Also, a heuristic method of finding optimal values of adaptive design parameters for minimizing the cost function is presented. Variable and fixed parameters T 2 control charts are compared with respect to the expected loss per unit time. Furthermore, the effects of the initial sample number (used for estimating the parameters of F-distribution) upon the operating cost and adaptive design parameters are examined.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alt FB (1984). Multivariate quality control. In: Kotz S, Johnson NL, and Read CR (eds). The Encyclopedia of Statistical Sciences. John Wiley, New York, pp 110–122.
Anderson TW (1958). An Introduction to Multivariate Statistical Analysis. John Wiley & Sons Inc.: New York.
Aparisi F (1996). Hotelling's T 2 control chart with adaptive sample sizes. Int J Product Res 34 (10): 2853–2862.
Aparisi F and García-Díaz JC (2003). Optimization of univariate and multivariate exponentially weighted moving-average control charts using genetic algorithms. Comput Opns Res 31: 1437–1454.
Aparisi F and Haro CL (2001). Hotelling's T 2 control chart with sampling intervals. Int J Product Res 39 (14): 3127–3140.
Aparisi F and Haro CL (2003). A comparison of T 2 control charts with variable sampling schemes as opposed to MEWMA chart. Int J Product Res 41 (10): 2169–2182.
Chen YK (2004). Economic design of X̄ control charts for non-normal data using variable sampling policy. Int J Product Econom 92: 61–74.
Chen YK (2005). Adaptive sampling enhancement for Hotelling's T 2 charts. Proceedings of the 2005 International conference in Management Science and Decision Making, Tamkang University, Taiwan, pp 281–296.
Cinlar E (1975). Introduction to Stochastic Processes. Prentice Hall: Englewood Cliffs, NJ.
Costa AFB (1994). X̄ control chart with variable sample size. J Quality Technol 26: 155–163.
Costa AFB (1997). X̄ charts with variable sample sizes and sampling intervals. J Quality Technol 29: 197–204.
Costa AFB (1999). X̄ Charts with variable parameters. J Quality Technol 31: 408–416.
Costa AFB (2001). Economic design of X̄ charts with variable parameter: the Markov Chain approach. J Appl Stat 2: 875–885.
Davis L (ed) (1991). Handbook of Genetic Algorithms. Van Nostrand Reinhold: New York.
Duncan AJ (1956). The economic design of X̄ charts used to maintain current control of a process. J Am Stat Assoc 51: 228–242.
Goldberg DE (1989). Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley: Reading, MA.
He D and Grigoryan A (2002). Construction of double sampling S-control charts for agile manufacturing. Quality Reliab Eng Int 18: 343–355.
He D, Grigoryan A and Sigh M (2002). Design of double and triple-sampling X control charts using genetic algorithms. Int J Product Res 40: 1387–1404.
Ho C and Case KE (1994). Economic design of control charts: a literature review for 1981–1991. J Quality Technol 26: 39–53.
Hotelling H (1947). Multivariate quality control-illustrated by air testing of sample bombsights. In: Eisenhart C, Hastay MW and Wallis WA (eds). Techniques of Statistical Analysis. McGraw-Hill: New York, pp 111–184.
Jackson JE (1985). Multivariate quality control. Commun Statistics-Theory Methods 14: 2657–2688.
Karr CL and Gentry EJ (1993). Fuzzy control of pH using genetic algorithms. IEEE Transa Fuzzy Systems 1 (1): 46–53.
Lowry CA and Montgomery DC (1995). A review of multivariate control charts. IIE Transa 27: 880–810.
Montgomery DC (1980). The economic design of control charts: a review and literature survey. J Quality Technol 12: 75–87.
Montgomery DC and Klatt PJ (1972). Economic design of T 2 control charts to maintain current control of a process. Mngt Sci 19: 76–89.
Nedumaran G and Pignatiello Jr JJ (1999). On constructing T 2 control charts for on-line process monitoring. IIE Transac 31: 529–536.
Prabhu SS, Runger GC and Keats JB (1993). An adaptive sample size X̄ chart. Int J Product Res 31: 2895–2909.
Prabhu SS, Montgomery DC and Runger GC (1994). A combined adaptive sample size and sampling interval X̄ control chart. J Quality Technol 26: 164–176.
Quesenberry CP (1993). The effect of sample size on estimated limits for X̄ and X control charts. J Quality Technol 25: 237–247.
Reynolds Jr MR, Amin RW, Arnold JC and Nachlas JA (1988). X̄-charts with variable sampling interval. Technometrics 30: 181–192.
Reynolds Jr MR and Arnold JC (1989). Optimal one-side Shewhart control chart with variable sampling intervals. Sequential Analysis 8 (1): 51–77.
Ross SM (1970). Applied Probability Models with Optimization Applications. Holden-Day: San Francisco, CA.
Runger GC and Pignatiello Jr JJ (1991). Adaptive sampling for process control. J Quality Technol 23 (2): 133–155.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, YK. Economic design of an adaptive T 2 control chart. J Oper Res Soc 58, 337–345 (2007). https://doi.org/10.1057/palgrave.jors.2602138
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1057/palgrave.jors.2602138