{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T11:08:53Z","timestamp":1776510533854,"version":"3.51.2"},"reference-count":24,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,24]],"date-time":"2023-11-24T00:00:00Z","timestamp":1700784000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Thailand Science Research and Innovation Fund (NSRF)","award":["KMUTNB-FF-67-B-11"],"award-info":[{"award-number":["KMUTNB-FF-67-B-11"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Statistical process control (SPC) is used for monitoring and detecting anomalies in processes in the areas of manufacturing, environmental studies, economics, and healthcare, among others. Herein, we introduce an innovative SPC approach via mathematical modeling and report on its application via simulation studies to examine its suitability for monitoring processes involving correlated data running on advanced control charts. Specifically, an approach for detecting small to moderate shifts in the mean of a process running on a homogenously weighted moving average (HWMA) control chart, which is symmetric about the center line with upper and lower control limits, is of particular interest. A mathematical model for the average run length (ARL) of a moving average process with exogenous variables (MAX) focused only on the zero-state performance of the HWMA control chart is derived based on explicit formulas. The performance of our approach was investigated in terms of the ARL, the standard deviation of the run length (SDRL), and the median run length (MRL). Numerical examples are given to illustrate the efficacy of the proposed method. A detailed comparative analysis of our method for processes on HWMA and cumulative sum (CUSUM) control charts was conducted for process mean shifts in many situations. For several values of the design parameters, the performances of these two control charts are also compared in terms of the expected ARL (EARL), expected SDRL (ESDRL), and expected MRL (EMRL). It was found that the performance of the HWMA control chart was superior to that of the CUSUM control chart for several process mean shift sizes. Finally, the applicability of our method on a HWMA control chart is provided based on a real-world economic process.<\/jats:p>","DOI":"10.3390\/sym15122112","type":"journal-article","created":{"date-parts":[[2023,11,24]],"date-time":"2023-11-24T10:10:39Z","timestamp":1700820639000},"page":"2112","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Analytical Explicit Formulas of Average Run Length of Homogenously Weighted Moving Average Control Chart Based on a MAX Process"],"prefix":"10.3390","volume":"15","author":[{"given":"Rapin","family":"Sunthornwat","sequence":"first","affiliation":[{"name":"Industrial Technology and Innovation Management Program, Faculty of Science and Technology, Pathumwan Institute of Technology, Bangkok 10330, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5248-8173","authenticated-orcid":false,"given":"Saowanit","family":"Sukparungsee","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, 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 North Bangkok, Bangkok 10800, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,24]]},"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":"1954","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 averages","volume":"42","author":"Roberts","year":"1959","journal-title":"Technometrics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1016\/j.cie.2018.05.009","article-title":"Homogeneously weighted moving average control chart with an application in substrate manufacturing process","volume":"20","author":"Abbas","year":"2018","journal-title":"Comput. 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Axioms, 10.","DOI":"10.3390\/axioms10030154"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/12\/2112\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:29:41Z","timestamp":1760131781000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/12\/2112"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,24]]},"references-count":24,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["sym15122112"],"URL":"https:\/\/doi.org\/10.3390\/sym15122112","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,24]]}}}