{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:10:40Z","timestamp":1760145040306,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,12]],"date-time":"2024-06-12T00:00:00Z","timestamp":1718150400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"],"award-info":[{"award-number":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"]}]},{"name":"esearch on Humanities and Social Sciences of the Ministry of Education","award":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"],"award-info":[{"award-number":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"]}]},{"name":"Doctoral Research Start-up Fund of Liaoning Province","award":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"],"award-info":[{"award-number":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"]}]},{"name":"Education Department of Liaoning Province","award":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"],"award-info":[{"award-number":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"]}]},{"name":"Liaoning Provincial Department of Education Scientific Research Project","award":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"],"award-info":[{"award-number":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"]}]},{"name":"Research Project of Anshan Normal University","award":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"],"award-info":[{"award-number":["12201429","22YJC910009","2021- BS-142","JYTMS20230768","LJKZ1161","23kyxm029"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Monitoring the ratio of two correlated normal random variables is often used in many industrial manufacturing processes. At the same time, measurement errors inevitably exist in most processes, which have different effects on the performance of various charting schemes. This paper comprehensively analyses the impacts of measurement errors on the detection ability of the cumulative sum (CUSUM) charting schemes for the ratio of two correlated normal variables. A thorough numerical assessment is performed using the Monte Carlo simulation, and the results indicate that the measurement errors negatively impact the performance of the CUSUM scheme for the ratio of two correlated normal variables. Increasing the number of measurements per set is not a lucrative approach for minimizing the negative impact of measurement errors on the performance of the CUSUM charting scheme when monitoring the ratio of two correlated normal variables. We consider a food formulation as an example that illustrates the quality control problems involving the ratio of two correlated normal variables in an industry with a measurement error. The results are presented, along with some suggestions for further study.<\/jats:p>","DOI":"10.3390\/axioms13060393","type":"journal-article","created":{"date-parts":[[2024,6,12]],"date-time":"2024-06-12T08:45:50Z","timestamp":1718181950000},"page":"393","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Measurement Errors and Their Effects on the Cumulative Sum Schemes for Monitoring the Ratio of Two Correlated Normal Variables"],"prefix":"10.3390","volume":"13","author":[{"given":"Wei","family":"Yang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Liaoning University, Shenyang 110036, China"},{"name":"School of Mathematics and Information Science, Anshan Normal University, Anshan 114007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xueting","family":"Ji","sequence":"additional","affiliation":[{"name":"School of Economics and Law, University of Science and Technology Liaoning, Anshan 114051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiujun","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Liaoning University, Shenyang 110036, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,12]]},"reference":[{"key":"ref_1","unstructured":"Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons. [6th ed.]."},{"key":"ref_2","unstructured":"Qiu, P. (2014). Introduction to Statistical Process Control, Chapman & Hall\/CRC."},{"key":"ref_3","unstructured":"Shewhart, W.A. (1931). Economic Control of Quality of Manufactured Product, D. Van Nostrand Company."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1080\/00401706.1959.10489860","article-title":"Control chart tests based on geometric moving averages","volume":"1","author":"Roberts","year":"1959","journal-title":"Technometrics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1002\/qre.1748","article-title":"Design of a phase II control chart for monitoring the ratio of two normal variables","volume":"32","author":"Celano","year":"2016","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1080\/00224065.1990.11979203","article-title":"A control chart for ratios","volume":"22","author":"Spisak","year":"1990","journal-title":"J. Qual. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1080\/08982119308918717","article-title":"Statistical process control by the quotient of two correlated normal variables","volume":"6","author":"Oksoy","year":"1993","journal-title":"Qual. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1361","DOI":"10.1002\/qre.1558","article-title":"Statistical performance of a control chart for individual observations monitoring the ratio of two normal variables","volume":"30","author":"Celano","year":"2014","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1002\/qre.1783","article-title":"A synthetic control chart for monitoring the ratio of two normal variables","volume":"32","author":"Celano","year":"2016","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1670","DOI":"10.1080\/00207543.2015.1047982","article-title":"Monitoring the ratio of two normal varirables using run rules type control charts","volume":"54","author":"Tran","year":"2016","journal-title":"Int. J. Prod. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s00362-016-0769-4","article-title":"Monitoring the ratio of population means of a bivariate normal distribution using CUSUM type control charts","volume":"59","author":"Tran","year":"2016","journal-title":"Stat. Pap."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1853","DOI":"10.1002\/qre.1918","article-title":"Monitoring the ratio of two normal varirables using EWMA type control charts","volume":"32","author":"Tran","year":"2016","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1002\/qre.2259","article-title":"Steady-state ARL analysis of ARL-unbiased EWMA-RZ control chart monitoring the ratio of two normal variables","volume":"34","author":"Tran","year":"2018","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1002\/qre.2412","article-title":"Monitoring the ratio of two normal variables using variable sampling interval exponentially weighted moving average control charts","volume":"35","author":"Nguyen","year":"2019","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1002\/qre.2595","article-title":"CUSUM control charts with variable sampling interval for monitoring the ratio of two normal variables","volume":"36","author":"Nguyen","year":"2020","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4559","DOI":"10.1080\/03610926.2020.1818099","article-title":"Run sum control chart for monitoring the ratio of population means of a bivariate normal distribution","volume":"51","author":"Abubakar","year":"2022","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1080\/00224065.2001.11980068","article-title":"Effect of measurement error on Shewart control chart","volume":"33","author":"Linna","year":"2001","journal-title":"J. Qual. Technol."},{"key":"ref_19","first-page":"307","article-title":"Effects of the two-component measurement error model on X\u00af control charts","volume":"71","author":"Cocchi","year":"2011","journal-title":"Statistica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1080\/02664763.2011.590188","article-title":"Measurement error effect on the CUSUM control chart","volume":"39","author":"Maravelakis","year":"2012","journal-title":"J. Appl. Stat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1134\/S1061934814110100","article-title":"Monitoring analytical measurements in presence of two component measurement error","volume":"69","author":"Abbasi","year":"2014","journal-title":"J. Anal. Chem."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.cie.2016.10.026","article-title":"Measurement errors in statistical process monitoring: A literature review","volume":"103","author":"Maleki","year":"2017","journal-title":"Comput. Ind. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1177\/0142331216656756","article-title":"On the effect of measurement errors in simultaneous monitoring of mean vector and covariance matrix of multivariate processes","volume":"40","author":"Amiri","year":"2018","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_24","first-page":"990","article-title":"Effects of measurement error on control chart for monitoring process variability","volume":"39","author":"Liu","year":"2020","journal-title":"J. Appl. Stat. Manag."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.1002\/qre.2621","article-title":"A variable parameters multivariate control chart for simultaneous monitoring of the process mean and variability with measurement errors","volume":"36","author":"Sabahno","year":"2020","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1177\/0142331220935293","article-title":"Combined effect of autocorrelation and measurement errors on the adaptive X\u00af monitoring schemes","volume":"43","author":"Shongwe","year":"2021","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1177\/0142331220973569","article-title":"The effect of measurement errors on the performance of the homogenously weighted moving average X\u00af monitoring scheme","volume":"43","author":"Thanwane","year":"2021","journal-title":"Trans. Inst. Meas. Control."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2702","DOI":"10.1080\/03610926.2021.1958848","article-title":"The effect of gauge measurement errors on double sampling X\u00af control chart","volume":"52","author":"Maleki","year":"2023","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"108900","DOI":"10.1016\/j.cie.2022.108900","article-title":"Effect of measurement uncertainty on combined quality control charts","volume":"175","author":"Munir","year":"2023","journal-title":"Comput. Ind. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"7504","DOI":"10.1080\/00207543.2016.1198507","article-title":"The performance of the Shewhart-RZ control chart in the presence of measurement error","volume":"54","author":"Tran","year":"2016","journal-title":"Int. J. Prod. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1731","DOI":"10.1002\/qre.2656","article-title":"Effect of the measurement errors on two one-sided Shewhart control charts for monitoring the ratio of two normal variables","volume":"36","author":"Nguyen","year":"2020","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/j.ejor.2020.11.042","article-title":"The effect of measurement errors on the performance of the exponentially weighted moving average control charts for the ratio of two normally distributed variables","volume":"293","author":"Nguyen","year":"2021","journal-title":"Eur. J. Oper. Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"442","DOI":"10.2307\/2342070","article-title":"The frequency distribution of the quotient of two normal variates","volume":"93","author":"Geary","year":"1930","journal-title":"J. R. Stat. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1338","DOI":"10.1287\/mnsc.21.11.1338","article-title":"A note on the ratio of two normally distributed variables","volume":"21","author":"Hayya","year":"1975","journal-title":"Manag. Sci."},{"key":"ref_35","first-page":"99","article-title":"The distribution of the ratio of jointly normal variables","volume":"1","author":"Cedilnik","year":"2004","journal-title":"Metod. Zv."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1569","DOI":"10.1080\/03610920600683689","article-title":"Density of the ratio of two normal random variables and applications","volume":"35","author":"Turkkan","year":"2006","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1080\/00224065.2001.11980084","article-title":"The performance of multivariate control charts in the presence of measurement error","volume":"33","author":"Linna","year":"2001","journal-title":"J. Qual. Technol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1093\/biomet\/59.3.539","article-title":"An approach to the probability distribution of cusum run length","volume":"59","author":"Brook","year":"1972","journal-title":"Biometrika"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1213","DOI":"10.1002\/qre.1828","article-title":"A control chart for the multivariate coefficient of variation","volume":"32","author":"Yeong","year":"2016","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_40","unstructured":"Neuts, M. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins University Press."},{"key":"ref_41","unstructured":"Latouche, G., and Ramaswami, V. (1987). Introduction to Matrix Analytic Methods in Stochastic Modelling, Society for Industrial Mathematics."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/393\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:57:23Z","timestamp":1760108243000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/393"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,12]]},"references-count":41,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["axioms13060393"],"URL":"https:\/\/doi.org\/10.3390\/axioms13060393","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,6,12]]}}}