Journal of Korean Society for Quality Management (품질경영학회지)

Korean Society for Quality Management (한국품질경영학회)
권호 표지
DOI QR코드

DOI QR Code

Economic Design of $\bar{X}$-Control Charts with Warning Limits under Weibull Failure Model

와이블 고장모형 하에서 경고한계를 고려한 $\bar{X}$ 관리도의 경제적 설계

  • Jeong, Dong-Wook (Economic Statistics Bureau, Statistics Korea) ;
  • Lee, Joo-Ho (Department of Information & Statistics, Chungnam National University)
  • Received : 2012.02.22
  • Accepted : 2012.05.13
  • Published : 2012.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

Since Duncan(1956) first proposed an economic design of $\bar{X}$-control charts, most of the succeeding works on economic design of control charts assumed the exponential failure model like Duncan. Hu(1984), however, assumed a more versatile Weibull failure model to develop an economic design of $\bar{X}$-control charts and Banerjee and Rahim(1988) further improved Hu's design by changing the assumption of fixed-length sampling intervals to variable-length ones. In this article we follow the approach of Banerjee and Rahim(1988) but include a pair of warning limits inside the control limits in order to search for a failure without stopping the process when the sample mean falls between warning and control limits. The computational results indicate that the proposed model gives a lower cost than Banerjee and Rahim's model unless the early failure probability of a Weibull distribution is relatively large. The reduction in cost is shown to become larger as the cost of production loss outweighs the cost of searches for a failure.

Keywords

References

  1. 이태훈, 이재훈, 이민구, 이주호(2009), "대용 변수를 이용한 $\overline{X}$ 관리도의 경제적 설계", 품질경영학회지, 37권 2호, pp. 46-57.
  2. Banerjee, P. K. and Rahim, M. A.(1988), "Economic Design of $\overline{X}$- Control Charts Under Weibull Shock Models", Technometrics, Vol. 30, No. 4, pp. 407-414.
  3. Chen, Y. S. and Yang, Y. M.(2002), "Economic Design of $\overline{X}$-Control Charts with Weibull in-control times When There are Multiple Assignable Cause", International Journal of Production Economics, Vol. 77, pp. 17-23. https://doi.org/10.1016/S0925-5273(01)00196-7
  4. Duncan, A. J.(1956), "The Economic Design of $\overline{X}$ Charts used to Maintain Current Control of a Process", Journal of the American Statistical Association, Vol. 51, No. 274, pp. 228-242.
  5. Gordon, G. R. and Weindling, J. I.(1975), "A Cost model for Economic Design of Warning Limit Control Chart Schemes", AIIE Transactions, Vol. 7, No. 3, pp. 319-329. https://doi.org/10.1080/05695557508975013
  6. Ho, C. and Case, K. E.(1994), "Economic Design of Control Charts: A Literature Review for 1981 -1991", Journal of Quality Technology, Vol. 26, No. 1, pp. 39-53. https://doi.org/10.1080/00224065.1994.11979497
  7. Hooke, R. and Jeeves, T. A.(1961), "Direct Search Solution of Numerical and Statistical Problems", Journal of the Association for Computing Machinery, Vol. 8, pp. 212-229. https://doi.org/10.1145/321062.321069
  8. Hu, P. W.(1984), "Economic Design of $\overline{X}$-Control Chart Under Non-Poisson Process Shift", Abstract, TIMS/ORSA Joint National Meeting, San Francisco, pp. 87.
  9. Montgomery, D. C.(1980), "The Economic Design of Control Charts: A Review and Literature Survey", Journal of Quality Technology, Vol. 12, No. 27, pp. 75-87. https://doi.org/10.1080/00224065.1980.11980940
  10. Montgomery, D. C.(2004), Introduction to Statistical Quality Control, 5th ed., John Wiley & Sons, New York.
  11. Page, E. S.(1962), "A Modified Control Chart with Warning Limits", Biometrika, Vol. 49. No. 2, pp. 171-176. https://doi.org/10.1093/biomet/49.1-2.171
  12. Rahim, M. A. and Banerjee, P. K.(1993), "A Generalized Model for the Economic Design of $\overline{X}$- Control Charts for Production Systems with Increasing Failure Rate and Early Replacement", Naval Research Logistics, Vol. 40, No. 6, pp. 787-809. https://doi.org/10.1002/1520-6750(199310)40:6<787::AID-NAV3220400605>3.0.CO;2-4
  13. Rahim, M. A. and Costa, A. F. B.(2000), "Joint Economic Design of $\overline{X}$ and $R$ Control Charts under Weibull Shock Models", International Journal of Production Research, Vol. 38, No. 13, pp. 2871 -2889. https://doi.org/10.1080/00207540050117341
  14. Ross, S. M.(1983), Stochastic Processes, John Wiley & Sons, New York.
  15. Yang, S. F. and Rahim, M. A.(2006), "Multivariate extension to the economical design of $\overline{X}$ control chart under Weibull shock model", Journal of Statistical Computation and Simulation, Vol. 76, No. 12, pp. 1035-1047. https://doi.org/10.1080/10629360600569188