Designing an Effective Combined Shewhart-CUSUM Control Scheme with Exponentially Distributed Data
Department of Mathematics and Statistics, Dr. Shakuntala Misra National Rehabilitation University, Lucknow-226017, India.
Received on May 13, 2019
Accepted on July 24, 2019
In this paper, the Combined Shewhart-CUSUM control scheme has been proposed to monitor the production process when the quality characteristic follows exponential distribution to quickly detect the shift in the process. The simulated values of ARL are determined after the transformation of the data into approximate normal distribution by Nelson transformation method and adding Shewhart control limits to existing CUSUM Control Chart. Scheme parameters (value of k and h) and out of control ARL are calculated at various shift and at various in-control ARL. Parameters are also calculated to detect δ standard deviation shifts, which may be helpful to the quality control practitioners in designing the Combined Shewhart-CUSUM scheme when data is highly skewed.
Keywords- Combined Shewhart-CUSUM scheme, Exponential distribution, Average runs length, Monte Carlo simulation.
Tyagi, D. (2019). Designing an Effective Combined Shewhart-CUSUM Control Scheme with Exponentially Distributed Data. International Journal of Mathematical, Engineering and Management Sciences, 4(5), 1277-1286. https://dx.doi.org/10.33889/IJMEMS.2019.4.5-101.
Conflict of Interest
The authors confirm that there is no conflict of interest to declare for this publication.
The author would like to express his sincere thanks to the referees for their valuable suggestions towards the improvement of the paper. The author is also grateful to Prof. Bhupendra Singh, Department of Statistics, C. C. S. University, Meerut for continuous guidance and University Grants Commission, Government of India for financial support for the work.
Alwan, L.C. (2000). Designing an effective exponential CUSUM chart without the use of nomographs. Communication in Statistics: Theory and Methods, 29(12), 2879–2894.
Chan, L.Y., Xie, M., & Goh, T.N. (2000). Cumulative quantity control charts for monitoring production processes. International Journal of Production Research, 38(2), 397–408.
Duncan, A.J. (1986). Quality control and industrial statistics. 5th. Ed. Irwin, Homewood, IL: Richard D Irvin.
Gan, F.F. (1994). Design of optimal exponential CUSUM control charts. Journal of Quality Technology, 26(2), 109- 124.
Hawkins, D.M. (1992). Evaluation of the average run lengths of cumulative sum charts for an arbitrary data distribution. Communications in Statistics-Simulation and Computation, 21(4), 1001–1020.
Kittlitz, R.G. (1999). Transforming the exponential for SPC applications. Journal of Quality Technology, 31(3), 301–308.
Liu, J.Y., Xie, M., & Goh, T.N. (2006). CUSUM chart with transformed exponential data. Communications in Statistics - Theory and Methods, 35(10), 1829-1843
Lucas, J.M. (1982). Combined Shewhart-CUSUM quality control schemes. Journal of Quality Technology 14(2), 51–59.
Lucas, J.M. (1985). Counted data CUSUMs. Technometrics, 27(2), 129-144.
Lucas, J.M., & Crosier, R.B (1982). Fast initial response for CUSUM quality control schemes: give your CUSUM a head start. Technometrics, 24(3), 199–205.
Montgomery, D.C. (2018). Introduction to statistical quality control. (6th ed.). John Wiley and Sons, Inc.
Nelson, L.S. (1994). A control chart for parts-per-million nonconforming items. Journal of Quality Technology, 26(3), 239-240.
Page, E.S. (1954). Continuous inspection schemes. Biometrika, 41(1/2), 100-115.
Siegmund, D. (1985). Sequential analysis: test and confidence intervals. Springer-Verlag, New York.
Tyagi, D. (2014). On some problems in engineering statistics. Ph.D. Thesis, C.C.S. University, Meerut, India.
Vardeman, S., & Ray, D. (1985). Average run lengths for CUSUM schemes when observations are exponentially distributed. Technometrics, 27(2), 145–150.
Woodall, W.H., & Adams, B.M. (1993). The statistical design of CUSUM charts. Quality Engineering, 5(4), 559-570.