{"id":1193,"date":"2016-02-01T17:28:41","date_gmt":"2016-02-01T16:28:41","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=1193"},"modified":"2022-12-03T22:58:51","modified_gmt":"2022-12-03T21:58:51","slug":"methode-pert","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/planning-problem\/pert-method\/","title":{"rendered":"PERT method"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"1193\" class=\"elementor elementor-1193\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-95cf4c9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"95cf4c9\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-d84f2d8\" data-id=\"d84f2d8\" 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data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewbox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 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href=\"https:\/\/complex-systems-ai.com\/en\/planning-problem\/pert-method\/#Etape-1-construction-du-graphe-a-partir-de-lecheancier\" >Step 1: construction of the graph from the timeline<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/complex-systems-ai.com\/en\/planning-problem\/pert-method\/#Etape-2-determiner-les-dates-et-marges\" >Step 2: determine the dates and margins<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/complex-systems-ai.com\/en\/planning-problem\/pert-method\/#Aparte\" >Aside<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Methode-PERT\"><\/span>PERT method<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">As the <a href=\"https:\/\/complex-systems-ai.com\/en\/planning-problem\/gantt-chart\/\">Gantt chart<\/a>, the PERT method makes it possible to evaluate the duration of completion of a complex project and to detect the parts of this project that do not support any delay.<\/div>\n\n<p>The task information is summarized in a schedule like the following example:<\/p>\n\n<div class=\"standard\" style=\"text-align: justify;\">\n<table>\n<tbody>\n<tr>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">task<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">precedence<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">duration<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">TO<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">\u2013<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">6<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">B<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">\u2013<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">5<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">VS<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">TO<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">4<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">D<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">B<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">6<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">E<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">VS<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">5<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">F<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">A, D<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">6<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">G<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">E, F<\/div>\n<\/td>\n<td align=\"center\" valign=\"top\">\n<div class=\"plain_layout\">4<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n<h2 class=\"standard wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape-1-construction-du-graphe-a-partir-de-lecheancier\"><\/span>Step 1: construction of the graph from the timeline<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<ul class=\"wp-block-list\">\n<li style=\"text-align: justify;\"><strong><em>PERT method - Determination of task levels:<\/em><\/strong><\/li>\n<\/ul>\n\n<p>We will assign the level <strong>0<\/strong> to tasks that have no previous task.<\/p>\n\n<p>We will assign the level <strong>1<\/strong> to tasks whose previous tasks are level <strong>0<\/strong>.<\/p>\n\n<div style=\"padding: 5px; background-color: #ffdcd3; border: 2px solid #ff7964; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">We will thus determine the level of each task: level tasks <strong>k + 1<\/strong> will be the tasks whose previous tasks are lower level with at least one level k task among them.<\/div>\n\n<p>We will build the <a href=\"https:\/\/complex-systems-ai.com\/en\/graph-theory-2\/\">graph<\/a> by plotting the tasks in ascending order of level.<\/p>\n\n<ul class=\"wp-block-list\">\n<li style=\"text-align: justify;\"><strong><em>PERT method - Beginning, ending, convergent tasks:<\/em><\/strong><\/li>\n<\/ul>\n\n<p>Before embarking on the construction of the graph, it will often be useful to detect the so-called starting, ending or converging tasks.<\/p>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">Beginning tasks are tasks without a previous task, they start from the top <strong>1<\/strong> of the graph.<\/div>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">Completing tasks are tasks which are not a previous task, they arrive at the<br \/>terminal vertex of the graph.<\/div>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">Convergent tasks are tasks that we always meet together (ie never one without the other) in the previous tasks column; in the graph, they will have the same terminal vertex.<\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-1207 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert1.png\" alt=\"PERT method schedule levels of tasks critical path\" width=\"846\" height=\"331\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert1.png 846w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert1-300x117.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert1-768x300.png 768w\" sizes=\"(max-width: 846px) 100vw, 846px\" \/><\/figure>\n<\/div>\n\n<div style=\"padding: 3px; border: 2px dotted #a5a5a5; background-color: #f6f9fa;\">The ridge between <strong>2<\/strong> and <strong>5<\/strong> is called <em>dummy task<\/em> and is always of zero degree. Its purpose is to model the fact that the task <strong>TO<\/strong> must be completed to start the task <strong>F<\/strong>. The ridge between <strong>1<\/strong> and <strong>2<\/strong> means that the task <strong>TO<\/strong> starts as is <strong>1<\/strong> and ends as is <strong>2<\/strong>.<\/div>\n\n<p>It is important to place the tasks in order of execution. Task <strong>F<\/strong> can only be placed after the task <strong>TO<\/strong> and <strong>D<\/strong> placed, and the task <strong>D<\/strong> can only be placed after the task <strong>B<\/strong>. This explains the fictitious edge between <strong>2<\/strong> and <strong>5<\/strong> (<strong>TO<\/strong> and at a distance <strong>1<\/strong> while <strong>F<\/strong> is in the distance <strong>3<\/strong> from the start).<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape-2-determiner-les-dates-et-marges\"><\/span>Step 2: determine the dates and margins<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>Once the graph has been constructed, we will determine the dates at the earliest and at the latest for the<br \/>different vertices and free and total margins for tasks.<\/p>\n\n<ul class=\"wp-block-list\">\n<li style=\"text-align: justify;\"><em><strong>PERT method - Earliest dates:<\/strong><\/em>\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">For a summit, the earliest date (noted: <strong>t<\/strong>) represents the minimum time required to reach this peak. It will be determined step by step, in ascending order of vertex, from the entry of the graph, thanks to Ford&#039;s algorithm for finding the longest path.<\/div>\n<\/li>\n<\/ul>\n\n<p>Thereby :<br \/>t<sub>1<\/sub> = 0 and t<sub>j<\/sub> = Max (t<sub>i<\/sub> + d<sub>ij<\/sub> ) on all <strong>i<\/strong> preceding <strong>j<\/strong> with<sub>ij<\/sub> = time between peak <strong>i<\/strong> and <strong>j<\/strong>.<br \/>In the example, t<sub>1<\/sub> = 0, t<sub>2<\/sub> = 0 + 6 = 6, t<sub>3<\/sub> = 0 + 5 = 5, t<sub>4<\/sub> = 6 + 4 = 10, t<sub>5<\/sub> = max (6 + 0, 5 + 6) = 11, t<sub>6<\/sub> = max (11 + 6, 10 + 5) = 17, t<sub>7<\/sub> = 17+4 = 21.<\/p>\n\n<p>The earliest date of the graph output represents the minimum duration achievable for<br \/>the whole project (in the example, t<sub>7<\/sub>= 21, so the project will last 21 days at best).<\/p>\n\n<ul class=\"wp-block-list\">\n<li style=\"text-align: justify;\"><strong><em>PERT method - Dates at the latest:<\/em><\/strong>\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">For a summit, the latest date (noted: <strong>T<\/strong>) concretely represents the date on which this state must be reached if we do not want to increase the total duration of the project. It will be determined in a manner analogous to <strong>t<\/strong>, but in descending order of vertex, from the output of the graph to the input.<\/div>\n<\/li>\n<\/ul>\n\n<p>Thereby :<br \/>T<sub>not<\/sub> = t<sub>not<\/sub> = Duration of the project and T<sub>i<\/sub> = Min (T<sub>j<\/sub> - d<sub>ij<\/sub> ) on all j preceding i.<br \/>In the example, T<sub>7<\/sub> = 21, T<sub>6<\/sub> = 21 - 4 = 17, T<sub>5<\/sub> = 17 - 6 = 11, T<sub>4<\/sub> = 17 - 5 = 12, T<sub>3<\/sub> = 11 - 6 = 5, T<sub>2<\/sub> = min (11-0, 12-4) = 8, T<sub>1<\/sub> = min (8-6, 5-5) = 0.<br \/>We will always have t<sub>1<\/sub> = T<sub>1<\/sub> = 0 and <strong>t<\/strong> less than or equal to<strong> T<\/strong> for any summit. We call Tt la<br \/>top float margin.<\/p>\n\n<ul class=\"wp-block-list\">\n<li style=\"text-align: justify;\"><em><strong>PERT method - Task margins:<\/strong><\/em>\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">The free margin of a task represents the maximum possible delay of a task without delaying the start of subsequent tasks, note <strong>ML<\/strong>. The total margin of a task represents the maximum possible delay for the completion of a task without delaying the entire project, it will be noted <strong>MT<\/strong> : ML<sub>ij<\/sub> = t<sub>j<\/sub> - t<sub>i<\/sub> - d<sub>ij<\/sub> and MT<sub>ij<\/sub> = T<sub>j<\/sub> - t<sub>i<\/sub> - d<sub>ij<\/sub>.<\/div>\n<\/li>\n<\/ul>\n\n<p>Taking into account the calculation mode, the margins will always be positive or zero and the free margin of a task will always be less than or equal to its total margin.<\/p>\n\n<p>We will qualify as critical, a task whose total margin is zero. A critical task should not be delayed if you do not want to increase the total duration of the project.<\/p>\n\n<p>If the duration of a non-critical task increases, part of this increase will be absorbed by the task margin, only the surplus will affect the duration of the project.<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-1220 size-medium\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert-300x37.png\" alt=\"PERT method schedule levels of tasks critical path\" width=\"300\" height=\"37\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert-300x37.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert-1024x125.png 1024w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert-768x94.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert.png 1047w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Aparte\"><\/span>Aside<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>Vertices can contain several pieces of information at the same time:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-1233 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert3.png\" alt=\"PERT method schedule levels of tasks critical path\" width=\"787\" height=\"238\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert3.png 787w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert3-300x91.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/pert3-768x232.png 768w\" sizes=\"(max-width: 787px) 100vw, 787px\" \/><\/figure>\n<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Planning problem Home page Wiki PERT method Like the Gantt chart, the PERT method allows to evaluate the duration of a complex project \u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":868,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1193","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/1193","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=1193"}],"version-history":[{"count":4,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/1193\/revisions"}],"predecessor-version":[{"id":17915,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/1193\/revisions\/17915"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/868"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=1193"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}