{"id":1177,"date":"2024-09-05T22:36:30","date_gmt":"2024-09-05T20:36:30","guid":{"rendered":"https:\/\/site.ellistat.com\/?p=1177"},"modified":"2024-10-23T15:22:06","modified_gmt":"2024-10-23T13:22:06","slug":"como-calcular-cp-pp-ppk","status":"publish","type":"post","link":"https:\/\/ellistat.com\/es\/comment-calculer-cp-pp-ppk\/","title":{"rendered":"C\u00f3mo calcular los \u00edndices de capacidad Cp, Pp, Ppk"},"content":{"rendered":"\n<p>Dans cette article nous allons voir comment calculer les diff\u00e9rents indices de capabilit\u00e9 et notamment les indices Cp, Pp et Ppk.<\/p>\n\n\n\n<p>Pour bien appr\u00e9cier la capabilit\u00e9 d\u2019un processus la notion de temps est particuli\u00e8rement importante car on diff\u00e9rencie deux types de variabilit\u00e9 :<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>La variabilit\u00e9 court-terme : lorsque l\u2019on produit deux pi\u00e8ces cons\u00e9cutivement, ces deux pi\u00e8ces ne seront pas tout \u00e0 fait \u00e9quivalentes \u00e0 causes des variations intrins\u00e8ques de la machine. Cette variabilit\u00e9 court-terme d\u00e9pend principalement de la machine.<\/li>\n\n\n\n<li>La variabilit\u00e9 long-terme : lorsque l\u2019on produit sur un m\u00eame process pendant une plus longue p\u00e9riode, la machine elle-m\u00eame va se d\u00e9r\u00e9gler, les changements de s\u00e9rie, le changement de lot mati\u00e8re etc\u2026 vont apporter de nouvelles sources de variation. Cette variabilit\u00e9 long-terme d\u00e9pend de la machine mais \u00e9galement de nombreuses sources de variabilit\u00e9 ext\u00e9rieures et de la mani\u00e8re de piloter le process.<br><\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Exemple<\/h2>\n\n\n\n<p>Pour illustrer ce propos, observons le sch\u00e9ma de production suivant :<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"627\" height=\"441\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Court-et-long-terme.png\" alt=\"\" class=\"wp-image-1179\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Court-et-long-terme.png 627w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Court-et-long-terme-300x211.png 300w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Court-et-long-terme-18x12.png 18w\" sizes=\"auto, (max-width: 627px) 100vw, 627px\" \/><\/figure>\n<\/div>\n\n\n<p>La production de la semaine est le r\u00e9sultat de la production des diff\u00e9rents jours de la semaine. Or comme l\u2019illustre le sch\u00e9ma ci-dessus, les conditions de production entre les diff\u00e9rents jours de la semaine ne sont pas \u00e9quivalentes. Le centrage du proc\u00e9d\u00e9 n\u2019est pas le m\u00eame entre la production de lundi, mardi, etc\u2026<\/p>\n\n\n\n<p>Par suite, on observe que la variabilit\u00e9 sur l\u2019ensemble de la semaine (courbe rouge) est plus importante que la variabilit\u00e9 que l\u2019on observe sur une faible p\u00e9riode de temps car elle prend en compte moins de ph\u00e9nom\u00e8nes pouvant provoquer de la variabilit\u00e9 comme un r\u00e9glage ou un changement de mati\u00e8re.<\/p>\n\n\n\n<p>Il y a donc deux mani\u00e8res de caract\u00e9riser la capabilit\u00e9 d\u2019un process, selon le type de variabilit\u00e9 que l\u2019on observe : le court terme cp et le long terme pp.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Indice de Capabilit\u00e9 court terme Cp<\/h2>\n\n\n\n<p><strong>Court terme Cp<\/strong>: La capabilit\u00e9 court terme permet de caract\u00e9riser l\u2019aptitude du process \u00e0 produire des pi\u00e8ces bonnes en ne prenant en compte que la variabilit\u00e9 intrins\u00e8que du process (la variabilit\u00e9 entre deux pi\u00e8ces cons\u00e9cutives. La capabilit\u00e9 court terme est not\u00e9e Cp et se calcule par :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Cp=\\frac{\\text{tolerance interval}}{6*\\text{short term dispersion}}=\u00a0\\frac{\\text{tolerance interval}}{6*\\sigma_{\\text{short term}}}<\/span><\/p>\n\n\n\n<p>En g\u00e9n\u00e9ral, on souhaite :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Cp &gt; 2<\/span><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Indice de capabilit\u00e9 long terme Pp<\/h2>\n\n\n\n<p><strong>Long terme Pp<\/strong>: La capabilit\u00e9 long terme permet de caract\u00e9riser l\u2019aptitude du process \u00e0 produire des pi\u00e8ces bonnes sur une longue p\u00e9riode de temps, c\u2019est-\u00e0-dire en prenant en compte les r\u00e9glages et les modification de process qui peuvent avoir lieu. La capabilit\u00e9 long terme est not\u00e9e Pp et se calcule par :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Pp=\\frac{tolerance\u00a0interval}{6\u2217long\u00a0term\u00a0dispersion}=\\frac{tolerance\u00a0interval}{6\u2217\u03c3_{long\u00a0term}}<\/span><\/p>\n\n\n\n<p>En g\u00e9n\u00e9ral, on souhaite :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Pp &gt; 1.33<\/span><\/p>\n\n\n\n<p>La capabilit\u00e9 long-terme, prenant en compte plus de sources de variabilit\u00e9 que pour le court-terme, on a forc\u00e9ment :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\text{long term dispersion} &lt; \\text{short term dispersion}<\/span><\/p>\n\n\n\n<p>Et donc<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Pp &lt; Cp<\/span><\/p>\n\n\n\n<p>Si, ce n\u2019est pas le cas, cela ne signifie pas que la dispersion court terme est plus importante que la dispersion long terme, mais que la dispersion n\u2019est pas stable au cours du temps.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Les m\u00e9thodes pour calculer les indices de capabilit\u00e9 Cp, Pp, Ppk<\/h2>\n\n\n\n<p>Comme nous venons de le voir, on calcule le Cp et Pp selon la m\u00eame formule. On diff\u00e9rencie le Cp et le Pp avec les p\u00e9riodes de temps sur laquelle la variabilit\u00e9 est calcul\u00e9e. Il y a plusieurs m\u00e9thodes pour calculer Cp et Pp.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">M\u00e9thode 1 : En r\u00e9alisant plusieurs pr\u00e9l\u00e8vements \u00e0 intervalle r\u00e9gulier<\/h3>\n\n\n\n<p>La premi\u00e8re m\u00e9thode pour calculer les variabilit\u00e9s court-terme et long-terme est de r\u00e9aliser plusieurs pr\u00e9l\u00e8vements \u00e0 intervalle r\u00e9gulier.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"627\" height=\"365\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Illustration.png\" alt=\"\" class=\"wp-image-1180\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Illustration.png 627w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Illustration-300x175.png 300w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Illustration-18x10.png 18w\" sizes=\"auto, (max-width: 627px) 100vw, 627px\" \/><\/figure>\n<\/div>\n\n\n<p><strong>Variabilit\u00e9 court-terme :<\/strong>&nbsp;on calcule la variabilit\u00e9 court-terme en utilisant le calcul de l\u2019\u00e9cart-type intra-s\u00e9rie de l\u2019ensemble des pr\u00e9l\u00e8vements :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{short term}}=\\sigma_{\\text{intra sample}}<\/span><\/p>\n\n\n\n<p><strong>Variabilit\u00e9 long-terme :<\/strong>&nbsp;on pr\u00e9l\u00e8ve 50 pi\u00e8ces r\u00e9parties sur une p\u00e9riode caract\u00e9ristique du process, pour prendre en compte de multiples sources de variation du process comme des r\u00e9glages, des changements d\u2019outils, de mati\u00e8re, etc\u2026 La variabilit\u00e9 long-terme est calcul\u00e9e par :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{long term}}=\\sigma_{\\text{all sample}}=\\sqrt{\\sum_{}^{}(\\frac{^{(x_{i}-\\mu)^{2}}}{n-1})}<\/span><\/p>\n\n\n\n<p>Exemple : On r\u00e9alise le pr\u00e9l\u00e8vement suivant<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"184\" height=\"208\" src=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Illustration-2.png\" alt=\"\" class=\"wp-image-1181\" srcset=\"https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Illustration-2.png 184w, https:\/\/ellistat.com\/wp-content\/uploads\/2024\/09\/Illustration-2-11x12.png 11w\" sizes=\"auto, (max-width: 184px) 100vw, 184px\" \/><\/figure>\n<\/div>\n\n\n<p>L\u2019intervalle de tol\u00e9rance est [1 ;10].<\/p>\n\n\n\n<p>On calcule l\u2019\u00e9cart-type intra-s\u00e9rie :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{short term}}=\\sigma_{\\text{intra sample}}=1.7321<\/span><\/p>\n\n\n\n<p>On calcule l\u2019\u00e9cart-type de toutes les pi\u00e8ces :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{long term}}=\\sigma_{\\text{all sample}}=\\sqrt{\\sum_{}^{}(\\frac{^{(x_{i}-\\mu)^{2}}}{n-1})}=2.6904<\/span><\/p>\n\n\n\n<p>On en d\u00e9duit donc :<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Cp=\\frac{\\text{tolerance interval}}{6*\\text{short term dispersion}}= \\frac{\\text{tolerance interval}}{6*\\sigma_{\\text{short term}}}=\\frac{9}{6*1.7321}=0.87<\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Pp=\\frac{\\text{tolerance interval}}{6*\\text{long term dispersion}}= \\frac{\\text{tolerance interval}}{6*\\sigma_{\\text{long term}}}=\\frac{9}{6*2.6904}=0.56<\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">Ppk=\\frac{Min(\\text{tolerance max}-\\mu,mu-\\text{tolerance min})}{3*\\text{long term dispersion}}= \\frac{4.33}{3*2.6904}=0.54<\/span><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">M\u00e9thode 2 : En r\u00e9alisant deux pr\u00e9l\u00e8vements distincts<\/h3>\n\n\n\n<p>La seconde m\u00e9thode pour calculer les variabilit\u00e9s court-terme et long-terme est de r\u00e9aliser deux pr\u00e9l\u00e8vements distincts.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Variabilit\u00e9 court-terme : on pr\u00e9l\u00e8ve 50 pi\u00e8ces cons\u00e9cutives sans r\u00e9glage pour permettre de calculer la variabilit\u00e9 court-terme du process. La variabilit\u00e9 court-terme est calcul\u00e9e par :<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{short term}}=\\sigma_{\\text{all sample}}=\\sqrt{\\sum_{}^{}(\\frac{^{(x_{i}-\\mu)^{2}}}{n-1})}<\/span><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Variabilit\u00e9 long-terme : on pr\u00e9l\u00e8ve 50 pi\u00e8ces r\u00e9parties sur une p\u00e9riode caract\u00e9ristique du process, pour prendre en compte de multiples sources de variation du process comme des r\u00e9glages, des changements d\u2019outils, de mati\u00e8re, etc\u2026 La variabilit\u00e9 long-terme est calcul\u00e9e par :<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\sigma_{\\text{long term}}=\\sigma_{\\text{all sample}}=\\sqrt{\\sum_{}^{}(\\frac{^{(x_{i}-\\mu)^{2}}}{n-1})}<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Dans cette article nous allons voir comment calculer les diff\u00e9rents indices de capabilit\u00e9 et notamment les indices Cp, Pp et Ppk. Pour bien appr\u00e9cier la capabilit\u00e9 d\u2019un processus la notion de temps est particuli\u00e8rement importante car on diff\u00e9rencie deux types de variabilit\u00e9 : Exemple Pour illustrer ce propos, observons le sch\u00e9ma de production suivant : [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":1178,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7,4],"tags":[],"class_list":["post-1177","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-data-analysis","category-spc"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Comment calculer les indices de capabilit\u00e9 Cp, Pp, Ppk - Ellistat<\/title>\n<meta name=\"description\" content=\"Quelles sont les diff\u00e9rentes m\u00e9thodes pour calculer les indices de capabilit\u00e9s Cp, Pp et Ppk \u00e0 partir de diff\u00e9rents types de pr\u00e9l\u00e8vements ?\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/ellistat.com\/es\/como-calcular-cp-pp-ppk\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Comment calculer les indices de capabilit\u00e9 Cp, Pp, Ppk - 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