Development of Tukey Control Charts for Monitoring Linear Profiles in Phase II
Keywords:
Tukey control chart, simple linear profile, average run length, Phase IIAbstract
In many statistical process control problems, process quality or product performance can be described by the distribution of a quality characteristic and monitored using univariate control charts, or more generally, by the joint distribution of several quality characteristics and monitored using multivariate control charts. In some cases, product quality or process performance is described through the relationship between a response variable and one or more independent variables, a relationship that researchers refer to as a profile. One of the most widely used profiles is the simple linear profile, which has extensive applications in calibration; therefore, monitoring this type of profile is highly important. Profile monitoring is conducted in two phases, Phase I and Phase II. In Phase I, process stability is assessed and the process parameters are estimated, whereas in Phase II, the parameters are known and the objective is the rapid detection of changes in the process parameters. In this study, two proposed control charts—EWMA/R(Tukey) control chart and T²(Tukey) control chart—are designed for monitoring simple linear profiles in Phase II, using three distributions (normal, t, and gamma) in each method. To simulate process changes, three types of shifts are applied, including shifts in the intercept, shifts in the slope, and shifts in the standard deviation. Simulation results indicate that the performance of the proposed control charts is superior to that of classical control charts.
References
Adibi, A., Montgomery, D. C., & Borror, C. M. (2014). Phase II monitoring of linear profiles using a P-value approach. International Journal of Quality Engineering and Technology, 4(2), 97-106. https://doi.org/10.1504/IJQET.2014.060429
Adsız, S., & Aytaçoğlu, B. (2024). Tukey-EWMA control chart with variable sampling intervals for process monitoring. Quality Engineering, 36(2), 249-272. https://doi.org/10.1080/08982112.2023.2199824
Alemi, F. (2004). Tukey's control chart. Quality Management in Healthcare, 13(4), 216-221. https://doi.org/10.1097/00019514-200410000-00004
Chiang, J. Y., Lio, Y. L., & Tsai, T. R. (2017). MEWMA control chart and process capability indices for simple linear profiles with within‐profile autocorrelation. Quality and Reliability Engineering International, 33(5), 1083-1094. https://doi.org/10.1002/qre.2101
Haq, A., Bibi, M., & Shah, B. A. (2022). A novel approach to monitor simple linear profiles using individual observations. Communications in Statistics-Simulation and Computation, 51(11), 6269-6282. https://doi.org/10.1080/03610918.2020.1799229
Kazemi, S., Noorossana, R., Rasouli, M., Nayebpour, M. R., & Heidari, K. (2021). Monitoring therapeutic processes using risk‐adjusted multivariate Tukey's CUSUM control chart. Quality and Reliability Engineering International, 37(6), 2818-2833. https://doi.org/10.1002/qre.2891
Lee, P. H., & Chou, C. Y. (2024). An economic-statistical design of synthetic Tukey's control chart with Taguchi's asymmetric loss functions under log-normal distribution. Communications in Statistics-Theory and Methods, 53(21), 7604-7623. https://doi.org/10.1080/03610926.2023.2269448
Mahmood, T., Riaz, M., Hafidz Omar, M., & Xie, M. (2018). Alternative methods for the simultaneous monitoring of simple linear profile parameters. Quality Engineering. https://doi.org/10.1007/s00170-018-2149-9
Noorossana, R., Eyvazian, M., & Vaghefi, A. (2010). Phase II monitoring of multivariate simple linear profiles. Computers & Industrial Engineering, 58(4), 563-570. https://doi.org/10.1016/j.cie.2009.12.003
Noorossana, R., Fatemi, S. A., & Zerehsaz, Y. (2015). Phase II monitoring of simple linear profiles with random explanatory variables. Quality Engineering. https://doi.org/10.1007/s00170-014-6287-4
Raza, M. A., Nawaz, T., & Han, D. (2019). On designing new optimal synthetic Tukey's control charts. Journal of Statistical Computation and Simulation, 89(12), 2218-2238. https://doi.org/10.1080/00949655.2019.1615062
Riaz, M., Khaliq, Q. U. A., & Gul, S. (2017). Mixed Tukey EWMA-CUSUM control chart and its applications. Quality Technology & Quantitative Management, 14(4), 378-411. https://doi.org/10.1080/16843703.2017.1304034
Riaz, M., Mahmood, T., Abbas, N., & Abbasi, S. A. (2019). On improved monitoring of linear profiles under modified successive sampling. Quality and Reliability Engineering International, 35(7), 2202-2227. https://doi.org/10.1002/qre.2498
Saghaei, A., & Mehrjoo, M. (2010). Monitoring simple linear profiles using cumulative sum control charts. Amirkabir Journal of Mechanical Engineering, 41(2), 73-82. https://mej.aut.ac.ir/article_270_en.html?lang=en
Saghaei, A., Mehrjoo, M., & Amiri, A. (2009). A CUSUM-based method for monitoring simple linear profiles. Quality Engineering. https://doi.org/10.1007/s00170-009-2063-2
Sherwani, R. A. K., Qasim, H., Abbas, S., Abbas, T., & Aslam, M. (2023). Performance evaluation of DEWMA3 in phase-II for capturing changes in simple linear profiles based on run rule mechanism. Scientific reports, 13(1), 8609. https://doi.org/10.1038/s41598-023-35779-0
Shojaee, M., Noori, S., Jafarian-Namin, S., & Johannssen, A. (2024). Integration of production-maintenance planning and monitoring simple linear profiles via Hotelling's T2 control chart and particle swarm optimization. Quality Engineering. https://doi.org/10.1016/j.cie.2023.109864
Spirić, J. V., Stanković, S. S., & Dočić, M. B. (2016). Determining a set of suspicious electricity customers using statistical ACL Tukey's control charts method. Quality Engineering. https://doi.org/10.1016/j.ijepes.2016.04.035
Taboran, R., Sukparungsee, S., & Areepong, Y. (2020). A new nonparametric Tukey MA-EWMA control charts for detecting mean shifts. Quality Engineering. https://doi.org/10.1109/ACCESS.2020.3037293
Taboran, R., Sukparungsee, S., & Areepong, Y. (2021). Design of a new Tukey MA-DEWMA control chart to monitor process and its applications. Quality Engineering. https://doi.org/10.1109/ACCESS.2021.3098172
Talordphop, K., Sukparungsee, S., & Areepong, Y. (2022). Performance of new nonparametric Tukey modified exponentially weighted moving average-Moving average control chart. PLoS One, 17(9), e0275260. https://doi.org/10.1371/journal.pone.0275260
Talordphop, K., Sukparungsee, S., & Areepong, Y. (2023). Mixed Tukey Exponentially Weighted Moving Average-Modified Exponentially Weighted Moving Average Control Chart for Process Monitoring. Emerging Science Journal, 7(3), 854-866. https://doi.org/10.28991/ESJ-2023-07-03-014
Tercero-Gomez, V., Ramirez-Galindo, J., Cordero-Franco, A., Smith, M., & Beruvides, M. (2012). Modified Tukey's control chart. Communications in Statistics-Simulation and Computation, 41(9), 1566-1579. https://doi.org/10.1080/03610918.2011.606951
Torng, C. C., & Lee, P. H. (2008). ARL performance of the Tukey's control chart. Communications in Statistics-Simulation and Computation, 37(9), 1904-1913. https://doi.org/10.1080/03610910802263141
Wang, Y. H. T., & Huang, W. H. (2017). Phase II monitoring and diagnosis of autocorrelated simple linear profiles. Quality Engineering. https://doi.org/10.1016/j.cie.2017.08.006
Wong, W. K., Lv, Z., Espinosa, C., & Vieito, J. P. (2025). The crude oil spot and futures prices dynamics: cointegration, linear and nonlinear causality. Studies in Economics and Finance, 42(3), 532-552. https://doi.org/10.1108/SEF-12-2023-0738
Xu, B., & Lin, B. (2025). Green credit and green technology innovation: Impact mechanism and nonlinear effect test. Environmental Impact Assessment Review, 110, 107652. https://doi.org/10.1016/j.eiar.2024.107652
Zhang, J., Li, Z., & Wang, Z. (2009). Control chart based on likelihood ratio for monitoring linear profiles. Computational Statistics & Data Analysis, 53(4), 1440-1448. https://doi.org/10.1016/j.csda.2008.12.002
Zou, C., Tsung, F., & Wang, Z. (2007). Monitoring general linear profiles using multivariate exponentially weighted moving average schemes. Technometrics, 49(4), 395-408. https://doi.org/10.1198/004017007000000164
Downloads
Published
Submitted
Revised
Accepted
Issue
Section
License
Copyright (c) 2026 Reza Mazloumi Takmehdash, Mohammad Amin Bogheyri (Author)
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
