Comparative analysis between FAR and ARL based control charts with runs rules

Year 2021, Volume: 50 Issue: 1, 275 - 288, 04.02.2021

Rashid Mehmood Muhammad Hisyam Lee Iftikhar Ali Muhammad Rıaz

https://doi.org/10.15672/hujms.758925

Cited By: 1

Abstract

In this study, we have conducted comparative analysis between false alarm rate (FAR) and average run length (ARL) based control charts with runs rules. In this regard, we have considered various univariate and multivariate control charts which include mean, standard deviation, variance, Hotelling, and generalized variance. For evaluation purpose, we have used actual false alarm rate, power, in-control actual average run length, and out-of-control average run length as performance indicators. Furthermore, the performance indicators are calculated through Monte Carlo simulation procedures. Results revealed that performance order of runs rules with FAR based control charts are persistent whereas, performance order of runs rules with ARL based control charts are dependent on the circumstances, that is, sample size, size of shift, type of control chart, and side of control limit (upper-sided and lower-sided). Besides, we have provided a real life example using the data on electrical resistance of insulation. In this approach, we have determined that behavior of FAR and ARL based control charts using the real data is recorded similar to the behavior using the statistical performance indicators.

Keywords

Average run length, control chart, false alarm rate, performance indicators, power, probability of single point, runs rules

References

• [1] L.C. Alwan, Statistical Process Analysis, McGraw-Hill International Editions, Singapore, 2000.
• [2] S. Chakraborti and S. Eryilmaz, A nonparametric Shewhart-type signed-rank control chart based on runs, Comm. Statist. Simulation Comput. 36 (2), 335–356, 2007.
• [3] C.W. Champ and W.H. Woodall, Exact results for Shewhart control charts with supplementary runs rules, Technometrics 29 (4), 393–399, 1987.
• [4] M.B. Khoo, Design of runs rules schemes, Qual. Eng. 16 (1), 27–43, 2003.
• [5] M. Klein, Two alternatives to the shewhart x control chart, J. Qual. Technol. 32 (4), 427–431, 2000.
• [6] J.C. Malela-Majika, S.C. Shongwe and P. Castagliola, One-sided precedence monitoring schemes for unknown shift sizes using generalized 2-of-(h+1) and w-of-w improved runs-rules, Comm. Statist. Theory Methods, 1–35, 2020.
• [7] R. Mehmood, M.H. Lee, M. Riaz, B. Zaman and I. Ali, Hotelling T2 control chart based on bivariate ranked set schemes, Comm. Statist. Simulation Comput., 1–28, 2019.
• [8] R. Mehmood, M.S. Qazi and M. Riaz, On the performance of $\bar{X}$ control chart for known and unknown parameters supplemented with runs rules under different probability distributions, J. Stat. Comput. Simul. 88 (4), 675–711, 2018.
• [9] R. Mehmood, M. Riaz and R.J.M.M. Does, Control charts for location based on different sampling schemes, J. Appl. Stat. 40 (3), 483–494, 2013.
• [10] R. Mehmood, M. Riaz and R.J.M.M. Does, Efficient power computation for r out of m runs rules schemes, Comput. Statist. 28 (2), 667–681, 2013.
• [11] R. Mehmood, M. Riaz and R.J.M.M. Does, Quality quandaries: on the application of different ranked set sampling schemes, Qual. Eng. 26 (3), 370–378, 2014.
• [12] R. Mehmood, M. Riaz, T. Mahmood, S.A. Abbasi and N. Abbas, On the extended use of auxiliary information under skewness correction for process monitoring, Trans. Inst. Meas. Control. 39 (6), 883–897, 2017.
• [13] D.C. Montgomery, Introduction to Statistical Quality Control, John Wiley Sons, New York, 2009.
• [14] Jr, J.J. Pignatiello and G.C. Runger, Comparisons of multivariate cusum charts, J. Qual. Technol. 22 (3), 173–186, 1990.
• [15] M. Riaz, R. Mehmood and R.J.M.M. Does, On the performance of different control charting rules, Qual. Reliab. Eng. 27 (8), 1059–1067, 2011.
• [16] D.K. Shepherd, S.E. Rigdon and C.W. Champ, Using runs rules to monitor an attribute chart for a markov process, QTQM 9 (4), 383–406, 2012.
• [17] W.A. Shewhart, Economic Control of Quality of Manufactured Product, ASQ Quality Press, 1931.
• [18] S.C. Shongwe, On the design of nonparametric runs-rules schemes using the markov chain approach, Qual. Reliab. Eng. 36 (5), 1604–1621, 2020.
• [19] S.C. Shongwe, J.C. Malela-Majika and T. Molahloe, One-sided runs rules schemes to monitor autocorrelated time series data using a first-order autoregressive model with skip sampling strategies, Qual. Reliab. Eng. 35 (6), 1973–1997, 2019.
• [20] S. Shongwe, J.C. Malela-Majika and E. Rapoo, One-sided and two-sided w-of-w runsrules schemes: An overall performance perspective and the unified run-length derivations, J. Probab. Stat., 1–20, 2019.
• [21] E.C. Western, Statistical Quality Control Handbook, Western Electric Company, Indianapolis, 1956.
• [22] B. Zaman, M. Riaz and S.A. Abbasi, On the efficiency of runs rules schemes for process monitoring, Qual. Reliab. Eng. 32 (2), 663–671, 2016.