Sains Malaysiana 53(6)(2024): 1441-1461

http://doi.org/10.17576/jsm-2024-5306-17

A Robust Design for the Omnibus SPRT Control Chart Under Skewed Data Distributions

(Reka Bentuk Teguh untuk Carta Kawalan Omnibus SPRT di Bawah Taburan Data Pencong)

JING WEI TEOH¹, WEI LIN TEOH^1,2,*, ZHI LIN CHONG³, MING HA LEE⁴ & KHAI WAH KHAW⁵

¹School of Mathematical and Computer Sciences, Heriot-Watt University Malaysia, 62200 Putrajaya, Malaysia

²International Chair in DS & XAI, International Research Institute for Artificial Intelligence and Data Science, Dong A University, Danang, Vietnam

³Department of Electronic Engineering, Faculty of Engineering and Green Technology, Universiti Tunku Abdul Rahman, 31900 Kampar, Perak, Malaysia

⁴Faculty of Engineering, Computing and Science, Swinburne University of Technology Sarawak Campus, 93350 Kuching, Sarawak, Malaysia

⁵School of Management, Universiti Sains Malaysia, 11800 Gelugor, Pulau Pinang, Malaysia

Received: 18 December 2023/Accepted: 23 May 2024

Abstract

Control charts are widely used in manufacturing industries to ensure that production levels are stable and satisfactory. Recently, the omnibus sequential probability ratio test (OSPRT) control chart was developed for the purpose of monitoring the mean and variability of a process simultaneously. As the OSPRT chart was proposed for the first time in literature, its development relied entirely on the assumption that data follow the Normal distribution. Nonetheless, researchers are frequently reminded that the quality characteristics of manufacturing processes do not necessarily follow the Normal distribution, e.g., strengths of glass fibres, and lifetimes of products. In this paper, we investigate the extent to which the performances of the OSPRT chart designed for the Normal model deteriorate, in situations where the data distributions are Gamma and Lognormal. Results show that the in-control average run length (ARL) and standard deviation of the run length of the OSPRT chart designed for the Normal distribution deteriorate rapidly as skewness increases. To address this issue, we propose a robust design for the OSPRT chart by adjusting its control limits, known as the skewness correction method. It is shown that the skewness-corrected OSPRT chart enjoys a guaranteed in-control ARL, with a justifiable degradation in its out-of-control performances. Besides, we also show some insights into selecting the charting parameters for the skewness-corrected OSPRT chart in order to achieve an optimum out-of-control ARL performance over various shift sizes. The paper wraps up with an illustrative example of the skewness-corrected OSPRT chart for monitoring the weights of radial tyres.

Keywords: Average run length; joint monitoring control chart; sequential probability ratio test; skewed distributions; statistical process control

Abstrak

Carta kawalan telah digunakan secara meluas dalam sektor pembuatan untuk memastikan bahawa tahap pengeluaran adalah stabil dan memuaskan. Baru-baru ini, carta kawalan berdasarkan ujian nisbah kebarangkalian berjujukan (OSPRT) telah direka untuk tujuan memantau min dan variabiliti sesuatu proses industri secara serentak. Oleh sebab carta OSPRT baharu diusulkan, rekaannya bergantung sepenuhnya pada andaian bahawa data mengikuti taburan Normal. Walau bagaimanapun, para penyelidik sering diingatkan bahawa ciri mutu dalam proses pembuatan tidak semestinya mengikuti taburan Normal, seperti kekuatan serat kaca dan hayat produk. Dalam makalah ini, kami mengkaji sejauh mana prestasi carta OSPRT yang direka untuk taburan Normal merosot, dalam situasi yang mana taburan data adalah Gamma dan Lognormal. Hasil kajian menunjukkan bahawa purata panjang larian (ARL) dan sisihan piawai panjang larian carta OSPRT bagi kes terkawal merosot dengan laju apabila darjah pencongan meningkat. Untuk menyelesaikan masalah ini, kami membina reka bentuk yang teguh untuk carta OSPRT dengan mengubahsuaikan had kawalan, dikenali sebagai kaedah pembetulan pencongan. Carta OSPRT berdasarkan pembetulan pencongan didapati menghasilkan ARL terkawal yang terjamin, dan prestasinya dalam kes tidak terkawal juga kurang dijejaskan. Selain itu, kami memberikan beberapa garis panduan untuk memilih parameter carta OSPRT yang sesuai bagi mencapai prestasi ARL tidak terkawal yang optimum untuk pelbagai magnitud anjakan. Makalah ini diakhiri dengan contoh aplikasi carta OSPRT berdasarkan pembetulan pencongan untuk memantau berat tayar radial.

Kata kunci: Carta kawalan pemantauan serentak; kawalan proses statistik; purata panjang larian; taburan pencong; ujian nisbah kebarangkalian berjujukan

REFERENCES

Abd Razak, N., Zubairi, Y.Z. & Yunus, R.M. 2014. Imputing missing values in modelling the PM10 concentrations. Sains Malaysiana 43(10): 1599-1607.

Abu-Shawiesh, M.O. & Abdullah, M.B. 2001. A modified nonparametric univariate control chart for location based on the trimmed mean. Sains Malaysiana 30(1): 87-106.

Chowdhury, S., Mukherjee, A. & Chakraborti, S. 2015. Distribution-free phase II CUSUM control chart for joint monitoring of location and scale. Quality and Reliability Engineering International 31(1): 135-151.

Dakhn, L.N.B., Bakar, M.A.A. & Ibrahim, K. 2023. Bayesian estimation of time to failure distributions based on skew normal degradation model: An application to GaAs laser degradation data. Sains Malaysiana 52(2): 641-653.

Diaz Pulido, A.J., Cordero Franco, A.E. & Tercero Gómez, V.G. 2023. A distribution-free control chart for joint monitoring of location and scale in finite horizon productions. Computational Statistics https://doi.org/10.1007/s00180-023-01361-5

Farouk, M., Aziz, N. & Zain, Z. 2020. The application of lognormal distribution on the new two-sided group chain sampling plan. Sains Malaysiana 49(5): 1145-1152.

Godase, D.G., Mahadik, S.B. & Rakitzis, A.C. 2022. The SPRT control charts for the Maxwell distribution. Quality and Reliability Engineering International 38(4): 1713-1728.

Haq, A., Brown, J. & Moltchanova, E. 2015. An improved maximum exponentially weighted moving average control chart for monitoring process mean and variability. Quality and Reliability Engineering International 31(2): 265-290.

Ho, Y.C., Kao, S.C. & Chou, C.F. 2021. Robustness of dispersion control charts in skewed distributions. International Journal of Industrial Engineering 28(4): 372-389.

Hossain, M.P., Omar, M.H., Riaz, M. & Arafat, S.Y. 2022. On designing a new control chart for Rayleigh distributed processes with an application to monitor glass fiber strength. Communications in Statistics-Simulation and Computation 51(6): 3168-3184.

Hou, S. & Yu, K. 2021. A non-parametric CUSUM control chart for process distribution change detection and change type diagnosis. International Journal of Production Research 59(4): 1166-1186.

Huberts, L.C., Schoonhoven, M., Goedhart, R., Diko, M.D. & Does, R.J. 2018. The performance of control charts for large non-normally distributed datasets. Quality and Reliability Engineering International 34(6): 979-996.

Khoo, M.B.C. 2004. An extension for the univariate exponentially weighted moving average control chart. MATEMATIKA 20(1): 43-48.

Lee, P.H., Torng, C.C., Lin, C.H. & Chou, C.Y. 2022. Control chart pattern recognition using spectral clustering technique and support vector machine under gamma distribution. Computers & Industrial Engineering 171(9): 108437.

Li, S.Y., Tang, L.C. & Ng, S.H. 2010. Nonparametric CUSUM and EWMA control charts for detecting mean shifts. Journal of Quality Technology 42(2): 209-226.

Mahadik, S.B. & Godase, D.G. 2023. The SPRT sign chart for process location. Communications in Statistics-Theory and Methods 52(7): 2276-2290.

Mehmood, R., Lee, M.H., Afzel, A., Bashir, S. & Riaz, M. 2020. Generalized skewness correction structure of X̅ control chart for unknown process parameters and skewed probability distributions. Journal of Statistical Computation and Simulation 90(8): 1349-1372.

Montgomery, D.C. 2019. Introduction to Statistical Quality Control. 8th ed. New York: John Wiley & Sons.

Nawaz, M.S., Azam, M. & Aslam, M. 2021. EWMA and DEWMA repetitive control charts under non-normal processes. Journal of Applied Statistics 48(1): 4-40.

Noorossana, R., Fathizadan, S. & Nayebpour, M.R. 2016. EWMA control chart performance with estimated parameters under non-normality. Quality and Reliability Engineering International 32(5): 1637-1654.

Qiu, P. 2018. Some perspectives on nonparametric statistical process control. Journal of Quality Technology 50(1): 49-65.

Riaz, M., Mehmood, R., Iqbal, M.R. & Abbasi, S.A. 2016. On efficient skewness correction charts under contamination and non-normality. Quality and Reliability Engineering International 32(3): 837-854.

Sabahno, H., Amiri, A. & Castagliola, P. 2021. A new adaptive control chart for the simultaneous monitoring of the mean and variability of multivariate normal processes. Computers & Industrial Engineering 151(1): 106524.

Stoumbos, Z.G. & Reynolds Jr., M.R. 1997. Control charts applying a sequential test at fixed sampling intervals. Journal of Quality Technology 29(1): 21-40.

Teh, S.Y., Khoo, M.B.C., Ong, K.H., Soh, K.L. & Teoh, W.L. 2015. A study on the S²-EWMA chart for monitoring the process variance based on the MRL performance. Sains Malaysiana 44(7): 1067-1075.

Teoh, J.W., Teoh, W.L., Khoo, M.B.C., Celano, G. & Chong, Z.L. 2023. Optimal designs of the omnibus SPRT control chart for joint monitoring of process mean and dispersion. International Journal of Production Research 62(12): 4202-4225.

Zwillinger, D., & Kokoska, S. 1999. CRC Standard Probability and Statistics Tables and Formulae. Boca Raton: CRC Press.

*Corresponding author; email: wei_lin.teoh@hw.ac.uk