{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T19:05:05Z","timestamp":1771268705290,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T00:00:00Z","timestamp":1579132800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007345","name":"King Mongkut's University of Technology North Bangkok","doi-asserted-by":"publisher","award":["KMUTNB-61-GOV-03-23"],"award-info":[{"award-number":["KMUTNB-61-GOV-03-23"]}],"id":[{"id":"10.13039\/501100007345","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The aim of this study was to derive explicit formulas of the average run length (ARL) of a cumulative sum (CUSUM) control chart for seasonal and non-seasonal moving average processes with exogenous variables, and then evaluate it against the numerical integral equation (NIE) method. Both methods had similarly excellent agreement, with an absolute percentage error of less than 0.50%. When compared to other methods, the explicit formula method is extremely useful for finding optimal parameters when other methods cannot. In this work, the procedure for obtaining optimal parameters\u2014which are the reference value (   a   ) and control limit (   h   )\u2014for designing a CUSUM chart with a minimum out-of-control ARL is presented. In addition, the explicit formulas for the CUSUM control chart were applied with the practical data of a stock price from the stock exchange of Thailand, and the resulting performance efficiency is compared with an exponentially weighted moving average (EWMA) control chart. This comparison showed that the CUSUM control chart efficiently detected a small shift size in the process, whereas the EWMA control chart was more efficient for moderate to large shift sizes.<\/jats:p>","DOI":"10.3390\/sym12010173","type":"journal-article","created":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T07:39:02Z","timestamp":1579246742000},"page":"173","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Average Run Length on CUSUM Control Chart for Seasonal and Non-Seasonal Moving Average Processes with Exogenous Variables"],"prefix":"10.3390","volume":"12","author":[{"given":"Rapin","family":"Sunthornwat","sequence":"first","affiliation":[{"name":"Industrial Technology Program, Faculty of Science and Technology, Pathumwan Institute of Technology, Bangkok 10330, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5103-9867","authenticated-orcid":false,"given":"Yupaporn","family":"Areepong","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics, Faculty of Applied Science King Mongkut\u2019s University of Technology, Bangkok 10800, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1093\/biomet\/41.1-2.100","article-title":"Continuous inspection schemes","volume":"41","author":"Page","year":"2004","journal-title":"Biometrika"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1080\/00401706.1959.10489860","article-title":"Control chart tests based on geometric moving average","volume":"1","author":"Roberts","year":"1959","journal-title":"Technometrics"},{"key":"ref_3","unstructured":"Shewhart, W.A. (1931). Economic Control of Quality of Manufactured Product, D. Van Nostrand Company."},{"key":"ref_4","first-page":"190","article-title":"Average run length with a practical investigation of estimating parameters of the EWMA control chart on the long memory AFRIMA process","volume":"16","author":"Sunthornwat","year":"2018","journal-title":"Thail. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1080\/00401706.1994.10485393","article-title":"Run-length distributions of special cause control charts for correlated processes","volume":"36","author":"Wardell","year":"1994","journal-title":"Technometrics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1080\/00401706.1993.10484992","article-title":"Performance of CUSUM control schemes for serially correlated observations","volume":"35","author":"Yashchin","year":"1993","journal-title":"Technometrics"},{"key":"ref_7","unstructured":"Carey, R.G. (2003). Improving Healthcare with Control Charts: Basic and Advanced SPC Methods and Case Studies, ASQ Quality Press."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1136\/qshc.2006.022194","article-title":"Application of statistical process control in healthcare improvement: Systematic review","volume":"16","author":"Johan","year":"2007","journal-title":"Qual. Saf. Health Care"},{"key":"ref_9","first-page":"335","article-title":"Implementation of quality control charts in bottles manufacturing industry","volume":"5","author":"Saravanan","year":"2013","journal-title":"Int. J. Eng. Sci. Technol."},{"key":"ref_10","first-page":"77","article-title":"An Application of Control Charts in Manufacturing Industry","volume":"1","author":"Muhammad","year":"2012","journal-title":"J. Stat. Econom. Methods"},{"key":"ref_11","unstructured":"Keats, J.B., and Hubele, N.F. (1989). Statistical Process Control in Computer Integrated Manufacturing Environment, Statistical Process Control in Automated Manufacturing, Marcel Dekker."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Matias, R., Ara\u00fajo, L.B., and Ara\u00fajo, M.F.C. (2010, January 22\u201325). Quality monitoring of network traffic forecasts using statistical process control. Proceedings of the IEEE Symposium on Computers and Communications, Riccione, Italy.","DOI":"10.1109\/ISCC.2010.5546771"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1093\/biomet\/39.1-2.168","article-title":"The time intervals between industrial accidents","volume":"39","author":"Maguire","year":"1952","journal-title":"Biometrika"},{"key":"ref_14","first-page":"1183","article-title":"A CUSUM control chart to monitor wafer quality world academy of science","volume":"7","year":"2013","journal-title":"Eng. Technol. Int. J. Ind. Manuf. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1051\/alr\/2008058","article-title":"Detection of changes in time-series of indicators using CUSUM control charts","volume":"22","author":"Benoit","year":"2009","journal-title":"Aquat. Liv. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1080\/03610928808829693","article-title":"On AR (1) processes with exponential white noise","volume":"17","author":"Andel","year":"1988","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_17","first-page":"601","article-title":"Bayesian analysis of an autoregressive process with exponential white noise","volume":"4","year":"1990","journal-title":"Statistics"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1771","DOI":"10.1080\/03610929408831353","article-title":"A conjugate family for AR (1) processes with exponential errors","volume":"23","author":"Caro","year":"1994","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1080\/0233188031000078042","article-title":"Bayesian estimation of an AR (1) process with exponential white noise","volume":"37","author":"Ibazizen","year":"2003","journal-title":"Statistics"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1080\/02331889008802270","article-title":"Bayesian analysis of an autoregressive process with exponential white noise","volume":"21","author":"Turkman","year":"1990","journal-title":"Statistics"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1080\/00401706.1975.10489273","article-title":"Approximations to the average run length in cumulative sum control charts","volume":"17","author":"Reynolds","year":"1975","journal-title":"Technometrics"},{"key":"ref_22","unstructured":"Mititelu, G., Areepong, Y., Sukparungsee, S., and Novikov, A. (2010). Explicit analytical solutions for the average run length of CUSUM and EWMA charts. East-West J. Math., 253\u2013265. Available online: https:\/\/opus.lib.uts.edu.au\/bitstream\/10453\/14528\/1\/2010001745.pdf."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"141","DOI":"10.2306\/scienceasia1513-1874.2015.41.141","article-title":"Exact solution of the average run length for the cumulative sum chart for a moving average process of order q","volume":"41","author":"Petcharat","year":"2015","journal-title":"ScienceAsia"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s11784-007-0041-6","article-title":"A simple proof of the Banach contraction principle","volume":"2","author":"Richard","year":"2007","journal-title":"J. Fixed Point Theory Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/173\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:43:13Z","timestamp":1760362993000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/173"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,16]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["sym12010173"],"URL":"https:\/\/doi.org\/10.3390\/sym12010173","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,16]]}}}