Skip to main content


Log in

Economic Design of Cumulative Conformance Count Charts with Variable Sampling Intervals

  • Research Article - Systems Engineering
  • Published:
Arabian Journal for Science and Engineering Aims and scope Submit manuscript


The cumulative conformance count (CCC) charts have been shown to be useful in high-quality processes. The CCC-charts regarded every individual item inspected as a sample and called the time between two successive samples the sampling interval. Typically, the length of sampling interval they employed is constant and fixed (FSI). Recently, the use of CCC-charts with the variable sampling interval (VSI) scheme has been proposed to enhance the efficiency of CCC-charts without increasing the rate of inspected items and false alarm occurrences. This paper presents an economic model for designing the VSI CCC-chart. In the economic design, a cost function is constructed, involving the cost of sampling and testing, the cost of false alarms, the cost to detect and remove the assignable cause, and the cost when the process is operating out-of-control. A heuristic method of finding optimal values of adaptive design parameters for minimizing the cost function is also presented. The VSI and FSI CCC-charts are compared with respect to the expected loss per unit time. In addition, the effects of process and cost parameters upon the operating cost and the design parameters for charting are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Duncan A.J.: The economic design of \({\bar {X}}\) charts used to maintain current control of a process. J. Am. Stat. Assoc. 51, 228–242 (1956)

    MATH  Google Scholar 

  2. Montgomery D.C.: The economic design of control charts: a review and literature survey. J. Qual. Technol. 12, 75–87 (1980)

    Google Scholar 

  3. Ho C., Case K.E.: Economic design of control charts: a literature review for 1981–1991. J. Qual. Technol. 26, 39–53 (1994)

    Google Scholar 

  4. Albers W.: The optimal choice of negative binomial charts for monitoring high-quality process. J. Stat. Plan. Infer. 140, 214–225 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhang C.W., Xie M., Goh T. N.: Economic design of cumulative count of conforming charts under inspection by samples. Int. J. Prod. Econ. 111, 93–104 (2008)

    Article  Google Scholar 

  6. Calvin T.W.: Quality control techniques for zero-defects. IEEE Trans. Compon. Hybrid Manuf. Technol. 6, 323–328 (1983)

    Article  Google Scholar 

  7. Goh T.N.: A control chart for very high yield processes. Qual. Assur. 13, 18–22 (1987)

    Google Scholar 

  8. Bourke P.D.: Detecting a shift in fraction nonconforming using run length control charts with 100 % inspection. J. Qual. Technol. 23, 225–238 (1991)

    Google Scholar 

  9. Liu J.Y., Xie M., Goh T.N., Liu Q.H., Yang Z.H.: Cumulative count of conforming chart with variable sampling intervals. Int. J. Prod. Econ. 101, 286–297 (2006)

    Article  Google Scholar 

  10. Reynolds M.R. Jr., Amin R.W., Arnold J.C., Nachlas J.A.: \({\bar {X}}\)-charts with variable sampling interval. Technometrics 30, 181–192 (1988)

    MathSciNet  Google Scholar 

  11. Reynolds M.R. Jr., Amin R.W., Arnold J.C.: CUSUM charts with variable sampling intervals. Technometrics 32, 371–384 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Runger G.C., Pignatiello J.J. Jr.: Adaptive sampling for process control. J. Qual. Technol. 23, 135–155 (1991)

    Google Scholar 

  13. Saccucci M.S., Amin R.W., Lucas J.M.: Exponentially weighted moving average control schemes with variable sampling intervals. Commun. Stat. Simul. Comput. 21, 627–657 (1992)

    Article  MathSciNet  Google Scholar 

  14. Vaughan T.S.: Variable sampling interval np process control chart. Commun. Stat. Theory Methods. 22, 147–167 (1993)

    MathSciNet  MATH  Google Scholar 

  15. Aparisi F., Haro C.L.: Hotelling’s T 2 control chart with variable sampling intervals. Int. J. Prod. Res. 39, 3127–3140 (2001)

    Article  MATH  Google Scholar 

  16. Xie M., Tang X.Y., Goh T.N.: On economic design of cumulative count of conforming chart. Int. J. Prod. Econ. 72, 89–97 (2001)

    Article  Google Scholar 

  17. Xie M., Goh T.N.: The use of probability limits for process control based on geometric distribution. Int. J. Qual. Reliab. Manag. 14, 64–73 (1997)

    Article  Google Scholar 

  18. Çinlar E.: Introduction to Stochastic Processes. Prentice Hall, Englewood Cliffs (1975)

    MATH  Google Scholar 

  19. Goldberg D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning Reading. Addison-Wesley, MA (1989)

    Google Scholar 

  20. Aparisi F., García-díaz J.C.: Optimization of univariate and multivariate exponentially weighted moving-average control charts using genetic algorithms. Comput. Oper. Res. 31, 437–1454 (2004)

    Article  Google Scholar 

  21. He D., Grigoryan A., Sigh M.: Design of double and triple-sampling X control charts using genetic algorithms. Int. J. Prod. Res. 40, 1387–1404 (2002)

    Article  MATH  Google Scholar 

  22. Chen Y.K.: Economic design of \({\bar {X}}\) control charts for non-normal data using variable sampling policy. Int. J. Prod. Econ. 92, 61–74 (2004)

    Article  Google Scholar 

  23. Xie M., Goh T.N., Xie W.: A study of economic design of control charts for cumulative count of conforming items. Commun. Stat. Simul. Comput. 26, 1009–1027 (1997)

    Article  MATH  Google Scholar 

  24. Bai D.S., Lee K.T.: An economic design of variable sampling interval X-bar control charts. Int. J. Prod. Econ. 54, 57–64 (1998)

    Article  Google Scholar 

  25. Costa A.F.B.: Economic design of \({\bar {X}}\) charts with variable parameter: the Markov chain approach. J. Appl. Stat. 28, 875–885 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Yan-Kwang Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, YK., Chen, CY. Economic Design of Cumulative Conformance Count Charts with Variable Sampling Intervals. Arab J Sci Eng 37, 2079–2088 (2012).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: