Sains Malaysiana 54(4)(2025): 1187-1207

http://doi.org/10.17576/jsm-2025-5404-18

 

Optimising Variable Sample Size Chart through Median Run Length with Estimated Process Parameters

(Mengoptimumkan Saiz Sampel Pemboleh Ubah Carta  melalui Panjang Larian Median dengan Parameter Proses Anggaran)

 

WEI LIN TEOH^1,2,*, KAI LE GOH¹, ZHI LIN CHONG³, XINYING CHEW⁴, MING HA LEE⁵ &

KHAI WAH KHAW⁶

 

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

²International Chair in Data Science & Explainable Artificial Intelligence, 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

⁴School of Computer Science, Universiti Sains Malaysia, 11800 Gelugor, Pulau Pinang, Malaysia

⁵Faculty of Engineering, Computing and Science, Swinburne University of Technology Sarawak Campus, 93350 Kuching, Sarawak, Malaysia

⁶School of Management, Universiti Sains Malaysia, 11800 Gelugor, Pulau Pinang, Malaysia

 

Diserahkan: 30 Ogos 2024/Diterima: 18 Disember 2024

 

Abstract

The classic charting procedures for designing the estimated process parameters-based variable sample size (VSS) chart rely on the average run length (ARL) criterion. Nevertheless, variations in the number of Phase-I samples and sample size, as well as the magnitude of the process mean shift affect the skewness of the run-length distribution for a control chart. Hence, we claim that the ARL can be a misleading metric when adopted in the estimated process parameters-based control charts. Instead, examining percentiles of the run-length distribution, which focus on the run-length behaviour, are more realistic and intuitive. From this point of view, this paper aims to develop two new optimal VSS charts using estimated process parameters, by minimising the (i) median run length (MRL) and (ii) expected MRL criteria, for known and unknown shift-size cases, respectively. Besides, the 5^th and 95^th percentiles are computed to closely examine the variability of the run length. In this paper, two VSS schemes that involve estimated process parameters are investigated extensively, i.e., the first sample size can be either small or large. Various practically manageable Phase-I sample sizes and magnitudes of process mean shift are implemented in the optimal design of the proposed charts. The results ascertain that the proposed optimal VSS  charts based on estimated process parameters not only provide a comprehensible interpretation for quality practitioners, but also give a low false-alarm rate. The proposed optimal charts are illustrated using real data from a wafer substrate manufacturing company.

Keywords: median run length; optimal design; percentile of the run-length distribution; process parameter estimation; variable sample size chart

 

Abstrak

Prosedur carta klasik untuk mereka carta bersaiz sampel berubah-ubah (VSS) berdasarkan anggaran parameter proses adalah bergantung pada kriteria panjang larian purata (ARL). Namun begitu, variasi dalam bilangan sampel Fasa-I dan saiz sampel, serta magnitud anjakan purata proses mempengaruhi kepencongan taburan panjang larian untuk carta kawalan. Oleh itu, kami berpendapat bahawa ARL merupakan metrik yang mengelirukan apabila digunakan dalam carta kawalan berdasarkan parameter proses yang dianggarkan. Sebaliknya, pemeriksaan percentil taburan panjang larian yang memberi tumpuan kepada tingkah laku panjang larian adalah lebih realistik dan intuitif. Dari sudut pandangan ini, kajian ini bertujuan untuk mencipta dua reka bentuk optimum baharu bagi carta VSS berdasarkan parameter proses yang dianggarkan dengan meminimumkan kriteria masing-masing untuk (i) panjang larian median (MRL) dan (ii) jangkaan MRL untuk kes perubahan saiz anjakan yang diketahui dan tidak diketahui. Selain itu, percentil ke-5 dan ke-95 dikira dengan teliti untuk mengukur variabiliti dalam panjang larian. Dalam kajian ini, dua skim VSS yang melibatkan parameter proses yang dianggarkan dikaji dengan secara mendalam, iaitu, saiz sampel pertama adalah sama dengan kecil atau besar. Pelbagai saiz sampel Fasa-I yang mudah diurus dan magnitud anjakan purata proses digunakan dalam reka bentuk optimum carta yang dicadangkan. Hasil kajian kami menunjukkan bahawa carta VSS optimum yang dicadangkan berdasarkan parameter proses yang dianggarkan bukan sahaja memberikan tafsiran yang komprehensif kepada pengamal kualiti, tetapi juga memberikan kadar amaran palsu yang rendah. Carta optimum yang dicadangkan diilustrasikan dengan menggunakan data sebenar daripada sebuah syarikat pembuatan substrat wafer.

Kata kunci: Anggaran proses parameter; carta bersaiz sampel berubah-ubah; panjang larian median; reka bentuk optimum; taburan peratusan panjang larian

 

RUJUKAN

Castagliola, P., Maravelakis, P.E. & Figueiredo, F.O. 2016. The EWMA median chart with estimated parameters. IIE Transactions 48(1): 66-74.

Castagliola, P., Zhang, Y., Costa, A. & Maravelakis, P. 2012. The variable sample size chart with estimated parameters. Quality and Reliability Engineering International 28(7): 687-699.

Chakraborti, S. 2007. Run length distribution and percentiles: The Shewhart chart with unknown parameters. Quality Engineering 19(2): 119-127.

Chong, Z.L., Tan, K.L., Khoo, M.B.C., Teoh, W.L. & Castagliola, P. 2022. Optimal designs of the exponentially weighted moving average (EWMA) median chart for known and estimated parameters based on median run length. Communications in Statistics – Simulation and Computation 51(7): 3660-3684.

Chong, Z.L., Yeong, W.C., Khaw, K.W., Chew, X.Y. & Teoh, W.L. 2024. Optimal design of the side sensitive group runs double sampling (SSGRDS) scheme with estimated process parameters. Transactions of the Institute of Measurement and Control 47(3): 487-505.

Costa, A.F.B. 1994. charts with variable sample size. Journal of Quality Technology 26(3): 155-163.

Epprecht, E.K., Loureiro, L.D. & Chakraborti, S. 2015. Effect of the amount of phase I data on the phase II performance of S2 and S control charts. Journal of Quality Technology 47(2): 139-155.

Gan, F.F. 1993. An optimal design of EWMA control charts based on median run length. Journal of Statistical Computation and Simulation 45(3-4): 169-184.

Gao, H., Khoo, M.B.C., Teh, S.Y. & Teoh, W.L. 2019. A study on the median run length performance of the run sum S control chart. International Journal of Mechanical Engineering and Robotics Research 8(6): 885-890.

Hu, X.L., Castagliola, P., Sun, J.S. & Khoo, M.B.C. 2016. The performance of variable sample size chart with measurement errors. Quality and Reliability Engineering International 32(3): 969-983.

Jensen, W.A., Jones-Farmer, L.A., Champ, C.W. & Woodall, W.H. 2006. Effects of parameter estimation on control chart properties: A literature review. Journal of Quality Technology 38(4): 349-364.

Karimi, H., Khaw, K.W., Chew, X.Y., Chong, Z.L. & Lim, S.T. 2023. Variable sample size control charts for monitoring multivariate coefficient of variation based on median run length and expected median run length. Journal of Science and Technology 15(2): 54-80.

Lee, M.H., Khoo, M.B.C., Haq, A., Wong, D.M.L. & Chew, X.Y. 2023. Synthetic  charts with known and estimated process parameters based on median run length and expected median run length. Quality Technology & Quantitative Management 20(2): 168-183.

Lim, S.L., Yeong, W.C., Chong, Z.L., Gan, C.P. & Khoo, M.B.C. 2024. A variable sample size side-sensitive synthetic coefficient of variation chart. Transactions of the Institute of Measurement and Control 46(9): 1694-1712.

Montgomery, D.C. 2013. Statistical Quality Control: A Modern Introduction. 7th ed. New York: John Wiley & Sons.

Mim, F.N., Khoo, M.B.C., Saha, S. & Castagliola, P. 2022. Revised triple sampling control charts for the mean with known and estimated process parameters. International Journal of Production Research 60(16): 4911-4935.

Qiao, Y.L., Sun, J.S., Castagliola, P. & Hu, X.L. 2022. Optimal design of one-sided exponential EWMA charts based on median run length and expected median run length. Communications in Statistics - Theory and Method 51(9): 2887-2907.

Reynolds Jr., M.R. 1996. Variable-sampling-interval control charts with sampling at fixed times. IIE Transactions 28(6): 497-510.

Ryan, T.P. 2000. Statistical Methods for Quality Improvement. 2nd ed. New York: John Wiley & Sons.

Shepherd, D.K., Champ, C.W. & Rigdon, S.E. 2016. Properties of the Markov-dependent attribute control chart with estimated parameters. Quality and Reliability Engineering International 32(2): 485-498.

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 Malaysiana44(7): 1067-1075.

Teoh, J.W., Teoh, W.L., Khoo, M.B.C., Tran, K.P. & Lee, M.H. 2024. An integrated optimal-GICP design for the SPRT control chart with estimated process parameters based on the average number of observations to signal. Quality Technology & Quantitative Management 22(1): 84-104.

Teoh, W.L., Chong, J.K., Khoo, M.B.C., Castagliola, P. & Yeong, W.C. 2017. Optimal designs of the variable sample size chart based on median run length and expected median run length. Quality and Reliability Engineering International 33(1): 121-134.

Teoh, W.L., Khoo, M.B.C., Castagliola, P. & Lee, M.H. 2016. The exact run length distribution and design of the Shewhart chart with estimated parameters based on median run length. Communications in Statistics – Simulation and Computation 45(6): 2081-2103.

Teoh, W.L., Khoo, M.B.C., Castagliola, P. & Chakraborti, S. 2015. A median run length-based double sampling chart with estimated parameters for minimizing the average sample size. The International Journal of Advanced Manufacturing Technology 80(1-4): 411-426.

Teoh, W.L., Khoo, M.B.C., Castagliola, P. & Chakraborti, S. 2014. Optimal designs of the double sampling  chart with estimated parameters based on median run length. Computers & Industrial Engineering 67(1): 104-115.

Yeong, W.C., Tan, Y.Y., Lim, S.L., Khaw, K.W. & Khoo, M.B.C. 2022. A variable sample size run sum coefficient of variation chart. Quality and Reliability Engineering International 38(4): 1869-1885.

You, H.W., Khoo, M.B.C., Castagliola, P. & Qu, L. 2016. Optimal exponentially weighted moving average charts with estimated parameters based on median run length and expected median run length. International Journal of Production Research 54(17): 5073-5094.

Zhou, Q., Zou, C., Wang, Z. & Jiang, W. 2012. Likelihood-based EWMA charts for monitoring Poisson count data with time-varying sample sizes. Journal of the American Statistical Association 107(499): 1049-1062.

 

*Pengarang untuk surat-menyurat; email: wei_lin.teoh@hw.ac.uk