{"id":7347,"date":"2019-11-13T14:41:00","date_gmt":"2019-11-13T13:41:00","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=7347"},"modified":"2024-02-11T20:47:59","modified_gmt":"2024-02-11T19:47:59","slug":"modelisation-lineaire-ex","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/linear-modeling-exercises-solutions\/","title":{"rendered":"6 Corrected Exercises Linear Modeling"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"7347\" class=\"elementor elementor-7347\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5363c42 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5363c42\" 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 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elementor-element-334bae5 elementor-widget elementor-widget-heading\" data-id=\"334bae5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<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-exercises-solutions\/#Exercices-Corriges-sur-la-modelisation-lineaire-a-partir-denonce\" >Corrected exercises on linear modeling from statements<\/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-exercises-solutions\/#Exercice-1\" >Exercise 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-exercises-solutions\/#Exercice-2\" >Exercise 2<\/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\/help-with-the-decision\/linear-modeling-exercises-solutions\/#Exercice-3\" >Exercise 3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/linear-modeling-exercises-solutions\/#Exercice-4\" >Exercise 4<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/linear-modeling-exercises-solutions\/#Exercice-5\" >Exercise 5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/linear-modeling-exercises-solutions\/#Exercice-6\" >Exercise 6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/linear-modeling-exercises-solutions\/#Exercice-7\" >Exercise 7<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercices-Corriges-sur-la-modelisation-lineaire-a-partir-denonce\"><\/span>Corrected exercises on linear modeling from statements<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7f834b4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7f834b4\" 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-100 elementor-top-column elementor-element elementor-element-0cec216\" data-id=\"0cec216\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-371a7c1 elementor-widget elementor-widget-text-editor\" data-id=\"371a7c1\" 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<p>Above corrected exercises of <a href=\"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/linear-modeling\/\">linear modeling<\/a>.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter wp-image-11096 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/09\/cropped-Capture.png\" alt=\"linear modeling\" width=\"97\" height=\"97\" title=\"\"><\/p>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-039cc63 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"039cc63\" 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-100 elementor-top-column elementor-element elementor-element-34e95b3\" data-id=\"34e95b3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7d48274 elementor-widget elementor-widget-heading\" data-id=\"7d48274\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-1\"><\/span>Exercise 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e2e4676 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e2e4676\" 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-100 elementor-top-column elementor-element elementor-element-0606959\" data-id=\"0606959\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2816055 elementor-widget elementor-widget-text-editor\" data-id=\"2816055\" 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<p>A company manufactures two products A and B, using a machine m and two raw materials p and q. We have 8 hours of m, 10 kg of p and 36 kg of q every day. We suppose that :<\/p>\n<ul>\n<li>the production of a unit of A requires 2 kg of p and 9 kg of q, and uses the machine m for 1 hour;<\/li>\n<li>the production of a unit of B requires 2 kg of p and 4 kg of q, and uses the machine m for 2 hours;<\/li>\n<li>the profits made are \u20ac 50 per unit of A and \u20ac 60 per unit of B<\/li>\n<\/ul>\n<p>The company&#039;s objective is to maximize the profit it will be able to derive, per day, from these 2 products by making the best use of its resources.<\/p>\n<ol>\n<li>Write a table to summarize the information<\/li>\n<li>Define the variables<\/li>\n<li>Define the constraints<\/li>\n<li>Define the objective function<\/li>\n<\/ol>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6d049ff elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6d049ff\" 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-100 elementor-top-column elementor-element elementor-element-5a3d81f\" data-id=\"5a3d81f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e646e4e elementor-widget elementor-widget-toggle\" data-id=\"e646e4e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2411\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2411\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2411\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2411\"><ol>\n<li>The following table summarizes the data pertaining to this production problem:<\/li>\n<\/ol>\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"334\" height=\"137\" class=\"wp-image-7218\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire1.png\" alt=\"\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire1.png 334w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire1-300x123.png 300w\" sizes=\"(max-width: 334px) 100vw, 334px\" \/><\/figure>\n<p>2. Here two data are unknown: the quantity of product A to produce and the quantity of product B to produce. Let us name them respectively x<sub>1<\/sub> and x<sub>2<\/sub>. We have two variables in our decision problems.<\/p>\n<p>3. Understand how the decision variables are constrained. For this, it is necessary to use the data in the specifications.<\/p>\n<ul>\n<li>Constraint of the machine m (first line): The time of use of the machine m to manufacture products A and B cannot exceed the 8 hours available: m\u22648.\n<ul>\n<li>The machine produces a unit of A in 1 hour<\/li>\n<li>The machine produces a unit of B in 2 hours<\/li>\n<li>Let us reformulate the constraint by: the quantity of A and B produced does not exceed 8h<\/li>\n<li>x<sub>1<\/sub> + 2 * x<sub>2<\/sub> \u22648<\/li>\n<\/ul>\n<\/li>\n<li>Constraint of material p: The quantity of material p cannot exceed 10kg\n<ul>\n<li>A unit of A consumes 2kg of p<\/li>\n<li>A unit of B consumes 2kg of p<\/li>\n<li>Formulation of the constraint: the quantity of product used by A and B does not exceed 10kg<\/li>\n<li>2 * x<sub>1<\/sub> + 2 * x<sub>2<\/sub> \u2264 10<\/li>\n<\/ul>\n<\/li>\n<li>Constraint of the material q: similarly we conclude that the quantity of product used by A (9kg per unit) and by B (4kg per unit) does not exceed 36kg\n<ul>\n<li>9 * x<sub>1<\/sub> + 4 * x<sub>2<\/sub> \u2264 36<\/li>\n<\/ul>\n<\/li>\n<li>Constraint of positivity: it is noted that the text does not give direct information on the decision variables. However, it is logical that these are positive or zero, it seems unlikely to produce -1 unit of A.\n<ul>\n<li>x<sub>1<\/sub> , x<sub>2<\/sub> \u2265 0<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>4. The objective function is, as its name suggests, the industrialist&#039;s objective: to maximize his profit. We know that selling a unit of A earns $ 50 and selling a unit of B earns $ 60. Let z be the total profit, then the objective function is max z = 50 * x<sub>1<\/sub> + 60 * x<sub>2<\/sub> .<\/p>\n<p>the <a href=\"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/mathematical-modeling\/\">mathematical model<\/a> linear, noted (P), can be summarized as follows:<\/p>\n<figure class=\"wp-block-image size-large\"><img fetchpriority=\"high\" decoding=\"async\" width=\"371\" height=\"141\" class=\"wp-image-7221\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire2.png\" alt=\"\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire2.png 371w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire2-300x114.png 300w\" sizes=\"(max-width: 371px) 100vw, 371px\" \/><\/figure><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0252c98 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0252c98\" 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-100 elementor-top-column elementor-element elementor-element-0f6ccfa\" data-id=\"0f6ccfa\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cdccb41 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"cdccb41\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f94419f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f94419f\" 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-100 elementor-top-column elementor-element elementor-element-3e3cb81\" data-id=\"3e3cb81\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-54d7c74 elementor-widget elementor-widget-heading\" data-id=\"54d7c74\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-2\"><\/span>Exercise 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8c08955 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8c08955\" 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-100 elementor-top-column elementor-element elementor-element-a9b744a\" data-id=\"a9b744a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e136912 elementor-widget elementor-widget-text-editor\" data-id=\"e136912\" 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<p>A chassis manufacturing company plans to produce two new models using the capabilities of its three workshops. These are respectively an aluminum frame and a wooden frame.<\/p>\n<p>The first product requires going to the first workshop to make the aluminum frame and to the third workshop where the glass is mounted on the frame. While the second product requires the passage in the second workshop to manufacture the wooden frame and in the third workshop where the glass is mounted on the frame. The unit profits, the manufacturing times of each of the products in each of the workshops as well as the weekly capacities of these workshops are given in the following table:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7224,\"sizeSlug\":\"large\"} --><\/p>\n<figure><img decoding=\"async\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire4.png\" alt=\"\" title=\"\"><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Formulate the linear problem.<\/p>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f0e8e1f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f0e8e1f\" 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-100 elementor-top-column elementor-element elementor-element-854fad3\" data-id=\"854fad3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d86d93b elementor-widget elementor-widget-toggle\" data-id=\"d86d93b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2261\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2261\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2261\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2261\"><p>The constraints are on the first three lines while the objective function is on the last line. Which gives: X1 for product 1 and X2 for product 2<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7350,\"sizeSlug\":\"large\"} --><\/p>\n<figure><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire38.png?w=264\" alt=\"linear modeling corrected exercises\" width=\"264\" height=\"150\" title=\"\"><\/figure><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2dc3194 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2dc3194\" 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-100 elementor-top-column elementor-element elementor-element-323a4fa\" data-id=\"323a4fa\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fee0f81 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"fee0f81\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-919e503 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"919e503\" 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-100 elementor-top-column elementor-element elementor-element-5abe999\" data-id=\"5abe999\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-72db959 elementor-widget elementor-widget-heading\" data-id=\"72db959\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-3\"><\/span>Exercise 3<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-cd888a7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cd888a7\" 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-100 elementor-top-column elementor-element elementor-element-4f0fa00\" data-id=\"4f0fa00\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-00c0306 elementor-widget elementor-widget-text-editor\" data-id=\"00c0306\" 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<p>An electricity producer wants these two nuclear power plants P and Q to supply a quantity of energy to cities A, B and C. The minimum quantities of energy to be satisfied are 16 for A, 12 for B and 18 for C.<\/p>\n<p>When a reactor of P produces, it sends 2 units to A, 1 unit to B and 1 unit to C; and costs 20 \u20ac per day. When a reactor of Q produces, it sends 1 unit to A, 1 unit to B and 3 units to C; and it costs 40 \u20ac. The producer is looking for the cheapest combination of P and Q reactors that will meet the minimum consumption requirement for cities A, B, C. The data is summarized in the following table:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7223,\"sizeSlug\":\"large\"} --><\/p>\n<figure><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire-3.png?w=383\" alt=\"linear modeling corrected exercises\" width=\"383\" height=\"138\" title=\"\"><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Formulate the linear problem.<\/p>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-518f166 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"518f166\" 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-100 elementor-top-column elementor-element elementor-element-9405b34\" data-id=\"9405b34\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e57d2d3 elementor-widget elementor-widget-toggle\" data-id=\"e57d2d3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2401\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2401\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2401\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2401\"><p>The constraints are on the first three lines while the objective function is on the last line. Which gives: X1 for producer P, X2 for producer Q<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7352,\"sizeSlug\":\"large\"} --><\/p>\n<figure><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire39.png?w=238\" alt=\"linear modeling corrected exercises\" width=\"238\" height=\"159\" title=\"\"><\/figure><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b71ad88 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b71ad88\" 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-100 elementor-top-column elementor-element elementor-element-7818b13\" data-id=\"7818b13\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4385564 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"4385564\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5bf2bbe elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5bf2bbe\" 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-100 elementor-top-column elementor-element elementor-element-4459f1e\" data-id=\"4459f1e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c417f9b elementor-widget elementor-widget-heading\" data-id=\"c417f9b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-4\"><\/span>Exercise 4<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-841e92c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"841e92c\" 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-100 elementor-top-column elementor-element elementor-element-0dd2755\" data-id=\"0dd2755\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-303271d elementor-widget elementor-widget-text-editor\" data-id=\"303271d\" 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<p>The following exercises are not independent, the constraints accumulate or change from one part to another.<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:heading {\"level\":4} --><\/p>\n<p><span style=\"color: var( --e-global-color-text ); font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-weight: var( --e-global-typography-text-font-weight ); font-size: 1.125rem;\">A car factory receives 100 tonnes of aluminum and 80 tonnes of steel every week. The factory produces three types of construction vehicle: Transport, Structural work, Crane; sold respectively 250k \u20ac, 300k \u20ac and 400k \u20ac.<\/span><\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The production of Transport requires 5 tonnes of aluminum and 3 tonnes of steel; the Structural Work needs 3 tonnes of aluminum and 5 tonnes of steel; the Crane requires 5 tons of each. Formulate the linear problem in order to maximize the gains of the plant.<\/p>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-06e81d8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"06e81d8\" 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-100 elementor-top-column elementor-element elementor-element-8e4196f\" data-id=\"8e4196f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-88d0ee5 elementor-widget elementor-widget-toggle\" data-id=\"88d0ee5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1431\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1431\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1431\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1431\"><p>the <a href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/\">linear program<\/a> is as follows: X1 for the Transport, X2 for the Shell and X3 for the Crane<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7354,\"sizeSlug\":\"large\"} --><\/p>\n<figure><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire40.png?w=318\" alt=\"linear modeling corrected exercises\" width=\"318\" height=\"145\" title=\"\"><\/figure><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2ff7eb4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2ff7eb4\" 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-100 elementor-top-column elementor-element elementor-element-e2cabb2\" data-id=\"e2cabb2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fe01a4b elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"fe01a4b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-54d33fc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"54d33fc\" 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-100 elementor-top-column elementor-element elementor-element-00101c3\" data-id=\"00101c3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ee40f7e elementor-widget elementor-widget-heading\" data-id=\"ee40f7e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-5\"><\/span>Exercise 5<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-eb217d6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"eb217d6\" 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-100 elementor-top-column elementor-element elementor-element-9d33db8\" data-id=\"9d33db8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-27d1a9d elementor-widget elementor-widget-text-editor\" data-id=\"27d1a9d\" 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<p>The Marketing team warns the plant that vehicle production is not keeping up with customer demand. After a study of sales statistics for the last year, financial analysts deduced the following data: an average of 10 Transport per week is sold, with a standard deviation of 2 vehicles (confidence of 99%); an average of 8 Big Work per week is sold, with a standard deviation of 4 vehicles (confidence of 99%); an average of 4 Cranes is sold per week with a standard deviation of 1 Crane (confidence of 99%). Formulate the problem to maximize the gains of the plant.<\/p>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-96f4a56 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"96f4a56\" 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-100 elementor-top-column elementor-element elementor-element-0b9c213\" data-id=\"0b9c213\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ee6a15e elementor-widget elementor-widget-toggle\" data-id=\"ee6a15e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2491\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2491\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2491\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2491\"><p>Here is the linear program with the new constraints:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7357,\"sizeSlug\":\"large\"} --><\/p>\n<figure><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire41.png?w=323\" alt=\"linear modeling corrected exercises\" width=\"323\" height=\"249\" title=\"\"><\/figure><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6632e8a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6632e8a\" 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-100 elementor-top-column elementor-element elementor-element-16a2afd\" data-id=\"16a2afd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fe1a076 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"fe1a076\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-687b835 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"687b835\" 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-100 elementor-top-column elementor-element elementor-element-48622e3\" data-id=\"48622e3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0843f68 elementor-widget elementor-widget-heading\" data-id=\"0843f68\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-6\"><\/span>Exercise 6<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-cb10e9a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cb10e9a\" 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-100 elementor-top-column elementor-element elementor-element-3baeddc\" data-id=\"3baeddc\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a6715c0 elementor-widget elementor-widget-text-editor\" data-id=\"a6715c0\" 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<p>Big mistake on your part, by always wishing to maximize profits, you did not take into account the working time and the comfort of the workers! The latter rightly claim that production meets their working conditions. These are divided into three workshops.<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The foundry has two tanks in operation 24\/7, 1 tonne of aluminum takes 30 minutes to melt and 1 tonne of steel takes 1 hour to melt (each tank is configured for one of the two metals). The assembly line is open 15 \/ 24h and 6 \/ 7d, a Transport is assembled in 5h, the Big Work in 10h and a crane in 10h. The finishing workshop is open 12 hours a day, 6 days a week, finishing takes 3 hours whatever the vehicle. Formulate the problem to maximize the gains of the plant.<\/p>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c28547e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c28547e\" 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-100 elementor-top-column elementor-element elementor-element-3294acc\" data-id=\"3294acc\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-17ca2ad elementor-widget elementor-widget-toggle\" data-id=\"17ca2ad\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2491\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2491\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2491\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2491\"><p>The linear program with the new constraints is:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7360,\"sizeSlug\":\"large\"} --><\/p>\n<figure><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire42.png?w=335\" alt=\"linear modeling corrected exercises\" width=\"335\" height=\"319\" title=\"\"><\/figure><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b859ce6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b859ce6\" 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-100 elementor-top-column elementor-element elementor-element-971bb57\" data-id=\"971bb57\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5194de5 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"5194de5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a430d1b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a430d1b\" 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-100 elementor-top-column elementor-element elementor-element-9138883\" data-id=\"9138883\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e2a2687 elementor-widget elementor-widget-heading\" data-id=\"e2a2687\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-7\"><\/span>Exercise 7<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-02fc328 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"02fc328\" 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-100 elementor-top-column elementor-element elementor-element-e4dc7ab\" data-id=\"e4dc7ab\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5b4e92d elementor-widget elementor-widget-text-editor\" data-id=\"5b4e92d\" 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<p>The factory development team has created a revolutionary new vehicle! This Test vehicle only needs 2 tons of aluminum and 1 ton of steel for an estimated selling price of 150k \u20ac. Market research shows that it can sell at least 10 units per week. This vehicle is assembled in 4 hours and the finish only lasts 2 hours. Formulate the problem to maximize the gains of the plant.<\/p>\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a91b062 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a91b062\" 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-100 elementor-top-column elementor-element elementor-element-00c0ac9\" data-id=\"00c0ac9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b65a463 elementor-widget elementor-widget-toggle\" data-id=\"b65a463\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1911\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1911\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1911\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1911\"><p><!-- wp:heading --><\/p>\n<p><!-- \/wp:list --><\/p>\n<p><!-- wp:heading {\"level\":4} --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:list {\"ordered\":true} --><\/p>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:heading --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:heading {\"level\":4} --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<figure><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:heading {\"level\":4} --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<figure><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:heading --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<figure><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:heading {\"level\":4} --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<figure><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:heading {\"level\":4} --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<figure><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:heading {\"level\":4} --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The linear program after adding a new variable X4<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7362,\"sizeSlug\":\"large\"} --><\/p>\n<figure><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire43.png?w=329\" alt=\"linear modeling corrected exercises\" width=\"329\" height=\"357\" title=\"\"><\/figure>\n<p><!-- \/wp:image --><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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 Home page Wiki Corrected exercises on linear modeling based on statements Above corrected linear modeling exercises. Exercise 1 \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-7347","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7347","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=7347"}],"version-history":[{"count":11,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7347\/revisions"}],"predecessor-version":[{"id":20436,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7347\/revisions\/20436"}],"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=7347"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}