{"id":7206,"date":"2019-10-24T09:26:04","date_gmt":"2019-10-24T08:26:04","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=7206"},"modified":"2022-12-03T23:03:30","modified_gmt":"2022-12-03T22:03:30","slug":"modelisation-lineaire","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/linear-modeling\/","title":{"rendered":"Linear modeling"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"7206\" class=\"elementor elementor-7206\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fc3462b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fc3462b\" 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 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elementor-widget-text-editor\" data-id=\"33b4b62c\" 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 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewbox=\"0 0 24 24\" version=\"1.2\" baseprofile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/linear-modeling\/#Modelisation-lineaire\" >Linear modeling<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/linear-modeling\/#Exemple-1\" >Example 1<\/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\/help-with-the-decision\/linear-modeling\/#Exemple-2\" >Example 2<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Modelisation-lineaire\"><\/span>Linear modeling<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let us take again the bases of the linear modelization within the framework of linear problem. The steps to follow are as follows:<\/p>\n\n<ol class=\"wp-block-list\">\n<li>What are the variables? their type can be integer, float or binary.<\/li>\n<li>What are the constraints? since we are in linear modeling, the variables are isolated (that is to say that only a coefficient can modify the variables, first order operations such as addition and subtraction relate the variables).<\/li>\n<li>What is the objective function? it can be a <a href=\"https:\/\/complex-systems-ai.com\/en\/language-theory\/minimization-dun-afd\/\">minimization<\/a> or a maximization; since we are in linear modeling, the variables are isolated.<\/li>\n<li>The problem of <a href=\"https:\/\/complex-systems-ai.com\/en\/algorithmic\/complexity-in-time\/\">complexity<\/a> and solution method will not be discussed in this chapter.<\/li>\n<\/ol>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exemple-1\"><\/span>Example 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>An industrialist has three plants suitable for the manufacture of two products. Each batch of product earns him a certain amount, and he knows the number of hours needed to manufacture each type of batch in his factories.<\/p>\n\n<figure class=\"wp-block-image size-large\"><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter wp-image-6773 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision1.png\" alt=\"linear modeling\" width=\"521\" height=\"190\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision1.png 521w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision1-300x109.png 300w\" sizes=\"(max-width: 521px) 100vw, 521px\" \/><\/figure>\n\n<p>The industrialist wanting to maximize his profit, it is therefore necessary to find the best possible production.<\/p>\n\n<p>Let us set decision variables:<\/p>\n\n<ul class=\"wp-block-list\">\n<li>x<sub>1<\/sub> = the number of batches of product 1<\/li>\n<li>x<sub>2<\/sub> = the number of batches of product 2<\/li>\n<\/ul>\n\n<p>Let us set the constraints:<\/p>\n\n<ul class=\"wp-block-list\">\n<li>x<sub>1<\/sub> \u2264 4 (second row of the table)<\/li>\n<li>2 x<sub>2<\/sub> \u2264 12 (third row of the table)<\/li>\n<li>3 x<sub>1<\/sub> + 2 x_2 \u2264 18 (fourth row of the table)<\/li>\n<li>x<sub>1<\/sub> \u2265 0 and x<sub>2<\/sub> \u2265 0 (the number of batches is always positive or zero)<\/li>\n<\/ul>\n\n<p>Let us set the objective function:<\/p>\n\n<ul class=\"wp-block-list\">\n<li>z = total profit (in thousands of euros)<\/li>\n<li>z = 3 x<sub>1<\/sub> + 5 x<sub>2<\/sub> (from the last row of the table)<\/li>\n<li>max z, i.e. we are looking for the maximum value that z can take<\/li>\n<\/ul>\n\n<p>Which gives the <a href=\"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/mathematical-modeling\/\">mathematical model<\/a> next :<\/p>\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" class=\"aligncenter wp-image-6774 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision2.png\" alt=\"linear modeling\" width=\"258\" height=\"181\" title=\"\"><\/figure>\n\n<p>This can be represented from a graphical point of view by (the choice space is in gray):<\/p>\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" class=\"aligncenter wp-image-6775 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision3.png\" alt=\"linear modeling\" width=\"492\" height=\"380\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision3.png 492w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision3-300x232.png 300w\" sizes=\"(max-width: 492px) 100vw, 492px\" \/><\/figure>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exemple-2\"><\/span>Example 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>Now that the industrialist knows how to optimize his profit, he is trying to minimize his expenses. The latter consist solely of employees&#039; wages and working hours. The manufacturer to estimate the minimum number of employees (MinEmp) to be assigned during each period of the day. Each employee must work shifts in order to maximize his presence time, a day has four shifts and these require special remuneration. All the data is described in the following table:<\/p>\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6776 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision4.png\" alt=\"linear modeling\" width=\"820\" height=\"469\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision4.png 820w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision4-300x172.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision4-768x439.png 768w\" sizes=\"(max-width: 820px) 100vw, 820px\" \/><\/figure>\n\n<p>Let us set decision variables:<\/p>\n\n<ul class=\"wp-block-list\">\n<li>x<sub>1<\/sub>= the number of employees on the first shift<\/li>\n<li>x<sub>2<\/sub>= the number of employees on the second shift<\/li>\n<li>x<sub>3<\/sub>= the number of employees on the third shift<\/li>\n<li>x<sub>4<\/sub>= the number of employees on the fourth shift<\/li>\n<li>x<sub>5<\/sub>= the number of employees on the fifth shift<\/li>\n<\/ul>\n\n<p>Let us set the constraints:<\/p>\n\n<ul class=\"wp-block-list\">\n<li>x<sub>1\u00a0<\/sub>\u2265 48 (second row of the table)<\/li>\n<li>x<sub>1\u00a0<\/sub>+ x<sub>2\u00a0<\/sub>\u2265 79 (third row of the table)<\/li>\n<li>etc.<\/li>\n<\/ul>\n\n<p>Let us set the objective function:<\/p>\n\n<ul class=\"wp-block-list\">\n<li>Z = the total cost<\/li>\n<li>Z = 170 x<sub>1\u00a0<\/sub>+ 160 x<sub>2\u00a0<\/sub>+ 175 x<sub>3<\/sub>+ 180 x<sub>4<\/sub>+ 195 x<sub>5\u00a0<\/sub>(from the last row of the table)<\/li>\n<li>min Z, i.e. we are looking for the minimum value that Z can take<\/li>\n<\/ul>\n\n<p>Which gives the following mathematical model:<\/p>\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6777 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision5.png\" alt=\"linear modeling\" width=\"745\" height=\"587\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision5.png 745w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/03\/decision5-300x236.png 300w\" sizes=\"(max-width: 745px) 100vw, 745px\" \/><\/figure>\n\n<p>\u00a0<\/p>\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>Decision support Wiki homepage Linear modeling Let&#039;s go back to the basics of linear modeling in the framework of a linear problem. The steps to\u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":1799,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-7206","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7206","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=7206"}],"version-history":[{"count":7,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7206\/revisions"}],"predecessor-version":[{"id":17846,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7206\/revisions\/17846"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/1799"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=7206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}