{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:46:22Z","timestamp":1747197982346,"version":"3.40.5"},"reference-count":17,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04621\/2020","UIDP\/04621\/2020"],"award-info":[{"award-number":["UIDB\/04621\/2020","UIDP\/04621\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,12,1]]},"abstract":"Abstract<\/jats:title>\n In the statistical process control literature, counts of nonconforming items are frequently assumed to be independent and have a binomial distribution with parameters \n \n \n \n (<\/m:mo>\n n<\/m:mi>\n ,<\/m:mo>\n p<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n (n,p)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where \ud835\udc5b and \ud835\udc5d represent the fixed sample size and the fraction nonconforming.\nIn this paper, the traditional \n \n \n \n n<\/m:mi>\n \u2062<\/m:mo>\n p<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n np<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-chart with 3-\ud835\udf0e control limits is reexamined.\nWe show that, even if its lower control limit is positive and we are dealing with a small target value \n \n \n \n p<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n p_{0}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of the fraction nonconforming \n \n \n \n (<\/m:mo>\n p<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n (p)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, this chart average run length (ARL) function achieves a maximum to the left of \n \n \n \n p<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n p_{0}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nMoreover, the in-control ARL of this popular chart is also shown to vary considerably with the fixed sample size \ud835\udc5b.\nWe also look closely at the ARL function of the ARL-unbiased \n \n \n \n n<\/m:mi>\n \u2062<\/m:mo>\n p<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n np<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-chart proposed by Morais [An ARL-unbiased \n \n \n \n n<\/m:mi>\n \u2062<\/m:mo>\n p<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n np<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-chart,\nEcon. Qual. Control<\/jats:italic>\n 31<\/jats:bold> (2016), 1, 11\u201321], which attains a pre-specified maximum value in the in-control situation.\nThis chart triggers a signal at sample \ud835\udc61 with probability one if the observed number of nonconforming items, \n \n \n \n x<\/m:mi>\n t<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n x_{t}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, is beyond the lower and upper control limits (\ud835\udc3f and \ud835\udc48), probability \n \n \n \n \u03b3<\/m:mi>\n L<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n \\gamma_{L}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> (resp. \n \n \n \n \u03b3<\/m:mi>\n U<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n \\gamma_{U}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>) if \n \n \n \n x<\/m:mi>\n t<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n x_{t}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> coincides with \ud835\udc3f (resp. \ud835\udc48).\nA graphical display for the ARL-unbiased \n \n \n \n n<\/m:mi>\n \u2062<\/m:mo>\n p<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n np<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-chart is proposed, taking advantage of the qcc<\/jats:italic> package for the statistical software R.\nFurthermore, as far as we have investigated, its control limits can be obtained using three different search algorithms; their computation times are thoroughly compared.<\/jats:p>","DOI":"10.1515\/eqc-2022-0032","type":"journal-article","created":{"date-parts":[[2022,10,26]],"date-time":"2022-10-26T15:50:42Z","timestamp":1666799442000},"page":"107-116","source":"Crossref","is-referenced-by-count":1,"title":["The \ud835\udc5b\ud835\udc5d-Chart with 3-\ud835\udf0e Limits and the ARL-Unbiased \ud835\udc5b\ud835\udc5d-Chart Revisited"],"prefix":"10.1515","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2214-2910","authenticated-orcid":false,"given":"Manuel Cabral","family":"Morais","sequence":"first","affiliation":[{"name":"Department of Mathematics & CEMAT (Center for Computational and Stochastic Mathematics) , Instituto Superior T\u00e9cnico , Universidade de Lisboa , Av. Rovisco Pais, 1049-001 Lisboa , Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7151-8243","authenticated-orcid":false,"given":"Philipp","family":"Wittenberg","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics , Helmut Schmidt University , Holstenhofweg 85, 22043 Hamburg , Germany"}]},{"given":"Camila Jeppesen","family":"Cruz","sequence":"additional","affiliation":[{"name":"Instituto Superior T\u00e9cnico , Universidade de Lisboa , Av. Rovisco Pais, 1049-001 Lisboa , Portugal"}]}],"member":"374","published-online":{"date-parts":[[2022,10,27]]},"reference":[{"key":"2023033113280474529_j_eqc-2022-0032_ref_001","doi-asserted-by":"crossref","unstructured":"C. A. Acosta-Mej\u00eda,\nImproved p-charts to monitor process quality,\nIIE Trans. 31 (1999), 509\u2013516.","DOI":"10.1080\/07408179908969854"},{"key":"2023033113280474529_j_eqc-2022-0032_ref_002","unstructured":"M. A. Argoti and A. C. Garc\u00eda,\nA novel approach for estimating the ARL-bias severity of Shewhart p-charts,\nInt. J. Qual. Res. 12 (2018), 209\u2013226."},{"key":"2023033113280474529_j_eqc-2022-0032_ref_003","unstructured":"C. J. Cruz,\nCartas com ARL sem vi\u00e9s para processos i.i.d. e AR(1) com marginais binomiais (On ARL-unbiased charts for i.i.d. and AR(1) binomial counts),\nMaster\u2019s thesis, Department of Mathematics, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 2019."},{"key":"2023033113280474529_j_eqc-2022-0032_ref_004","unstructured":"C. J. Geyer and G. D. Meeden,\nump: An r package for ump and umpu tests, 2004, https:\/\/CRAN.R-project.org\/package=ump."},{"key":"2023033113280474529_j_eqc-2022-0032_ref_005","doi-asserted-by":"crossref","unstructured":"C. J. Geyer and G. D. Meeden,\nFuzzy and randomized confidence intervals and \ud835\udc43-values,\nStatist. Sci. 20 (2005), no. 4, 358\u2013387.","DOI":"10.1214\/088342305000000340"},{"key":"2023033113280474529_j_eqc-2022-0032_ref_006","unstructured":"C. J. Geyer and G. D. Meeden,\nDesign of the ump package, 2017, https:\/\/cran.r-project.org\/web\/packages\/ump\/vignettes\/design.pdf."},{"key":"2023033113280474529_j_eqc-2022-0032_ref_007","unstructured":"E. L. Lehmann,\nTesting Statistical Hypotheses,\nJohn Wiley & Sons, New York, 1959."},{"key":"2023033113280474529_j_eqc-2022-0032_ref_008","doi-asserted-by":"crossref","unstructured":"M. C. Morais,\nAn ARL-unbiased \n \n \n \n n\n \u2062\n p\n \n \n \n np\n \n -chart,\nEcon. Qual. Control 31 (2016), no. 1, 11\u201321.","DOI":"10.1515\/eqc-2015-0013"},{"key":"2023033113280474529_j_eqc-2022-0032_ref_009","doi-asserted-by":"crossref","unstructured":"F. Pascual,\nEWMA charts for the Weibull shape parameter,\nJ. Qual. Technol. 42 (2010), 400\u2013416.","DOI":"10.1080\/00224065.2010.11917836"},{"key":"2023033113280474529_j_eqc-2022-0032_ref_010","doi-asserted-by":"crossref","unstructured":"S. Paulino, M. C. Morais and S. Knoth,\nAn ARL-unbiased c-chart,\nQual. Reliab. Eng. Int. 32 (2016), 2847\u20132858.","DOI":"10.1002\/qre.1969"},{"key":"2023033113280474529_j_eqc-2022-0032_ref_011","doi-asserted-by":"crossref","unstructured":"S. Paulino, M. C. Morais and S. Knoth,\nOn ARL-unbiased c-charts for INAR(1) Poisson counts,\nStatist. Papers 60 (2019), no. 4, 1021\u20131038.","DOI":"10.1007\/s00362-016-0861-9"},{"key":"2023033113280474529_j_eqc-2022-0032_ref_012","unstructured":"J. J. Pignatiello, Jr., C. A. Acosta and B. V. Rao,\nThe performance of control charts for monitoring process dispersion,\n4th Industrial Engineering Research Conference Proceedings,\nInstitute of Industrial and Systems Engineers, Peachtree Corners (1995), 320\u2013328."},{"key":"2023033113280474529_j_eqc-2022-0032_ref_013","unstructured":"R Core Team,\nR: A language and environment for statistical computing, R foundation for statistical computing, 2022, Vienna, https:\/\/www.R-project.org\/."},{"key":"2023033113280474529_j_eqc-2022-0032_ref_014","unstructured":"T. P. Ryan,\nStatistical Methods for Quality Improvement,\nJohn Wiley & Sons, New York, 1989."},{"key":"2023033113280474529_j_eqc-2022-0032_ref_015","doi-asserted-by":"crossref","unstructured":"T. P. Ryan,\nStatistical Methods for Quality Improvement, 3rd ed.,\nJohn Wiley & Sons, New York, 2011.","DOI":"10.1002\/9781118058114"},{"key":"2023033113280474529_j_eqc-2022-0032_ref_016","doi-asserted-by":"crossref","unstructured":"T. P. Ryan and N. C. Schwertman,\nOptimal limits for attribute control charts,\nJ. Qual. Technol. 29 (1997), 86\u201398.","DOI":"10.1080\/00224065.1997.11979728"},{"key":"2023033113280474529_j_eqc-2022-0032_ref_017","unstructured":"L. Scrucca,\nqcc: an R package for quality control charting and statistical process control,\nR News 4 (2004), 11\u201317."}],"container-title":["Stochastics and Quality Control"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/eqc-2022-0032\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/eqc-2022-0032\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T18:22:54Z","timestamp":1680286974000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/eqc-2022-0032\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,27]]},"references-count":17,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,12,1]]},"published-print":{"date-parts":[[2022,12,1]]}},"alternative-id":["10.1515\/eqc-2022-0032"],"URL":"https:\/\/doi.org\/10.1515\/eqc-2022-0032","relation":{},"ISSN":["2367-2390","2367-2404"],"issn-type":[{"type":"print","value":"2367-2390"},{"type":"electronic","value":"2367-2404"}],"subject":[],"published":{"date-parts":[[2022,10,27]]}}}