{"id":7367,"date":"2019-11-14T16:57:42","date_gmt":"2019-11-14T15:57:42","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=7367"},"modified":"2024-02-11T16:52:04","modified_gmt":"2024-02-11T15:52:04","slug":"forme-primale-exo","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-primal-form-exercises-solutions\/","title":{"rendered":"6 corrected exercises on Primal Form of linear programming"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"7367\" class=\"elementor elementor-7367\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-174ce8e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"174ce8e\" 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-2667ee5 elementor-widget elementor-widget-heading\" data-id=\"2667ee5\" 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\/linear-programming-2\/lp-primal-form-exercises-solutions\/#Exercices-corriges-de-programmation-lineaire-forme-primale\" >Corrected linear programming exercises (primal form)<\/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\/linear-programming-2\/lp-primal-form-exercises-solutions\/#Tutoriel\" >Tutorial<\/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\/linear-programming-2\/lp-primal-form-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-4\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-primal-form-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-5\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-primal-form-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-6\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-primal-form-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-7\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-primal-form-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-8\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-primal-form-exercises-solutions\/#Exercice-6\" >Exercise 6<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercices-corriges-de-programmation-lineaire-forme-primale\"><\/span>Corrected linear programming exercises (primal form)<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-8b10dce elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8b10dce\" 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-1fda209\" data-id=\"1fda209\" 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-b136a86 elementor-widget elementor-widget-text-editor\" data-id=\"b136a86\" 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>This tutorial offers various corrected exercises on modeling in primal form of linear programming.<\/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=\"primal form of linear programming\" 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-ec4d68b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ec4d68b\" 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-aeddf7d\" data-id=\"aeddf7d\" 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-18dbb12 elementor-widget elementor-widget-heading\" data-id=\"18dbb12\" 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=\"Tutoriel\"><\/span>Tutorial<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-2037f27 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2037f27\" 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-a79cb70\" data-id=\"a79cb70\" 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-a4326b8 elementor-widget elementor-widget-text-editor\" data-id=\"a4326b8\" 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 milling tool manufacturer makes two types of cutters: A and B. Type A cutters sell for 300 each, and are manufactured with 1 unit of steel, 2 units of removable carbide and 1 unit of diamond synthetic. Type B cutters sell for 200 each, and are manufactured with 2 units of steel, 1 units of removable carbide and 1 unit of synthetic diamond.\u00a0<\/p>\n<p>The stocks are 5 units of steel, 5 units of carbide and 2 units of diamond. The manufacturer wants to build at least 5 type A cutters and 5 type B cutters. The maintenance cost of the factory is 2500. How should the manufacturer allocate his production to maximize his profit, is it profitable?<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:heading {\"level\":4} --><\/p>\n<p><!-- \/wp:heading --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>First, the linear program must be carried out:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:list --><\/p>\n<ul>\n<li>Objective function: z = 300A + 200B-2500<\/li>\n<li>Constraints:\n<ul>\n<li>A + 2B \u2264 5<\/li>\n<li>2A + B \u2264 5<\/li>\n<li>A + B \u2264 2<\/li>\n<li>A \u2265 5<\/li>\n<li>B \u2265 5<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><!-- \/wp:list --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The program is not in canonical form, it must be rendered in this form before solving it.<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Let x<sub>1<\/sub> = A-5 and x<sub>2<\/sub> = B-5. The linear program becomes the following:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7295,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" class=\"alignnone wp-image-7295 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire23.png\" alt=\"linear programming primal form corrected exercises\" width=\"241\" height=\"139\" title=\"\"><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The linear program has 2 variables, so it is possible to solve it graphically. The constraints give the domain of the following solutions:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7296,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-7296 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire24.png\" alt=\"linear programming primal form corrected exercises\" width=\"313\" height=\"214\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire24.png 313w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire24-300x205.png 300w\" sizes=\"(max-width: 313px) 100vw, 313px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The gradient of the objective function is (3,2), which gives as the end point the intersection of the second and third constraint. We must therefore solve the system:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7300,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7300 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire25-1.png\" alt=\"linear programming primal form corrected exercises\" width=\"216\" height=\"97\" title=\"\"><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Which has for solution the vector (10, 2). This gives in the initial problem the vector (15, 7) and for objective function equal to 900. The operation is positive and therefore profitable for the factory.<\/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-240cea0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"240cea0\" 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-9acfca7\" data-id=\"9acfca7\" 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-a6cfb80 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"a6cfb80\" 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-45deb9e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"45deb9e\" 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-22cae59\" data-id=\"22cae59\" 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-eb86472 elementor-widget elementor-widget-heading\" data-id=\"eb86472\" 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-db83b55 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"db83b55\" 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-6089a84\" data-id=\"6089a84\" 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-6f46155 elementor-widget elementor-widget-text-editor\" data-id=\"6f46155\" 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 brooker needs, for its customers, 108 MWh of electricity for city 1 and 96 MWh of electricity for city 2. However, Kirchhoff&#039;s laws do not allow a distributor to target a single city for the energy transit. Two distributors serve these cities: Distributor A can send 12 MWh to city 1 and 8 MWh to city 2 per lot purchased; Distributor B can send 9 MWh to city 1 and 12 MWh to city 2 per lot purchased. The lots all have the same price. How many lots must the brooker buy to meet the energy demand of the two cities? Solve by graphical method.<\/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-9a8d15c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9a8d15c\" 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-fed16ee\" data-id=\"fed16ee\" 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-cbe2e59 elementor-widget elementor-widget-toggle\" data-id=\"cbe2e59\" 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-2131\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2131\" 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-2131\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2131\"><p>The linear program is as follows: X1 for the first batch and X2 for the second batch<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7368,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7368 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire44.png\" alt=\"linear programming primal form corrected exercises\" width=\"224\" height=\"109\" title=\"\"><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Which gives for graphic resolution:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7369,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7369 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire45.png\" alt=\"linear programming primal form corrected exercises\" width=\"865\" height=\"684\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire45.png 865w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire45-300x237.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire45-768x607.png 768w\" sizes=\"(max-width: 865px) 100vw, 865px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The gradient is (-1, -1) because it is a minimum (min f (x) = - max -f (x)). In the field of definition in green, point C is the only extremum being a global optimum. Solving the previous system in the form of equality gives the coordinates (6, 4) with Z = 10.<\/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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6230c97 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6230c97\" 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-9071ae8\" data-id=\"9071ae8\" 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-33e6b40 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"33e6b40\" 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-ff01de7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ff01de7\" 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-68fdbec\" data-id=\"68fdbec\" 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-7998bdb elementor-widget elementor-widget-heading\" data-id=\"7998bdb\" 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-228ae00 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"228ae00\" 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-33c9989\" data-id=\"33c9989\" 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-7d9a915 elementor-widget elementor-widget-text-editor\" data-id=\"7d9a915\" 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>Solve by the graphical method then by the simplex the following problems:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:list {\"ordered\":true} --><\/p>\n<ol>\n<li>A company that manufactures two types of products seeks to maximize its profit. Here is his roadmap.<\/li>\n<\/ol>\n<p><!-- \/wp:list --><\/p>\n<p><!-- wp:image {\"id\":7418,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7418 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire66.png\" alt=\"linear programming primal form corrected exercises\" width=\"375\" height=\"233\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire66.png 375w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire66-300x186.png 300w\" sizes=\"(max-width: 375px) 100vw, 375px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:list {\"ordered\":true,\"start\":2} --><\/p>\n<ol start=\"2\">\n<li>Likewise with the following linear program:<\/li>\n<\/ol>\n<p><!-- \/wp:list --><\/p>\n<p><!-- wp:image {\"id\":7424,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7424 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire70.png\" alt=\"linear programming primal form corrected exercises\" width=\"286\" height=\"96\" title=\"\"><\/figure>\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-7118ddf elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7118ddf\" 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-ae8f505\" data-id=\"ae8f505\" 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-f82145a elementor-widget elementor-widget-toggle\" data-id=\"f82145a\" 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-2601\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2601\" 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-2601\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2601\"><p>The methodology is always the same for the graphic resolution part. Here are the corrections of the simplexes with the initial state and the final state of the table for the first example.<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Problem 0:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7422,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7422 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire68.png\" alt=\"linear programming primal form corrected exercises\" width=\"362\" height=\"166\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire68.png 362w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire68-300x138.png 300w\" sizes=\"(max-width: 362px) 100vw, 362px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>And for solution:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7423,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7423 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire69.png\" alt=\"linear programming primal form corrected exercises\" width=\"391\" height=\"166\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire69.png 391w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire69-300x127.png 300w\" sizes=\"(max-width: 391px) 100vw, 391px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The objective function has for value Z = 1875 with P1 and P2 in base (therefore not zero) having respectively for value (25,15) as stipulated in column b represented by P0.<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The second linear program has for solution Z = 8.2 with the vector of basic variables (3.2, 1.8).<\/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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-277b5a0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"277b5a0\" 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-fc9aba6\" data-id=\"fc9aba6\" 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-927d564 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"927d564\" 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-15234aa elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"15234aa\" 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-d19da62\" data-id=\"d19da62\" 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-d7bddf6 elementor-widget elementor-widget-heading\" data-id=\"d7bddf6\" 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-558ea71 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"558ea71\" 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-b602d5d\" data-id=\"b602d5d\" 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-7d290c4 elementor-widget elementor-widget-text-editor\" data-id=\"7d290c4\" 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>BIM optimizes the construction, renovation and maintenance of buildings. Among the elements to be optimized there are the workers. In general, there are 4 types of workers depending on their weekend. The workers&#039; wages depend on these days of leave:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7307,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7307 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire28.png\" alt=\"linear programming primal form corrected exercises\" width=\"562\" height=\"115\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire28.png 562w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire28-300x61.png 300w\" sizes=\"(max-width: 562px) 100vw, 562px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The daily demands for workers are as follows:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7311,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7311 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire29.png\" alt=\"linear programming primal form corrected exercises\" width=\"577\" height=\"63\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire29.png 577w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire29-300x33.png 300w\" sizes=\"(max-width: 577px) 100vw, 577px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>How many people in each category must be employed in order to meet the demand and minimize the cost of personnel? Give the canonical form of the problem and solve it using Solver Excel.<\/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-f95ddfc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f95ddfc\" 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-6714618\" data-id=\"6714618\" 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-d3500f2 elementor-widget elementor-widget-toggle\" data-id=\"d3500f2\" 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-2211\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2211\" 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-2211\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2211\"><p>The linear program is as follows:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7402,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7402 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire56.png\" alt=\"linear programming primal form corrected exercises\" width=\"387\" height=\"254\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire56.png 387w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire56-300x197.png 300w\" sizes=\"(max-width: 387px) 100vw, 387px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The resolution by Excel is as follows:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7404,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7404 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire57.png\" alt=\"linear programming primal form corrected exercises\" width=\"596\" height=\"355\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire57.png 596w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire57-300x179.png 300w\" sizes=\"(max-width: 596px) 100vw, 596px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The result is the following:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7405,\"width\":863,\"height\":414,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7405 size-large\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire58-1024x491.png\" alt=\"linear programming primal form corrected exercises\" width=\"1024\" height=\"491\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire58-1024x491.png 1024w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire58-300x144.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire58-768x368.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire58.png 1163w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Note that the optimal solution is not unique. The solution gives integers, which is practical given the problem (it seems obvious that \u201cinteger\u201d individuals are needed).<\/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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-981a0df elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"981a0df\" 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-c05d706\" data-id=\"c05d706\" 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-64a1ced elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"64a1ced\" 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-721ee4b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"721ee4b\" 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-a104b33\" data-id=\"a104b33\" 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-3233dbf elementor-widget elementor-widget-heading\" data-id=\"3233dbf\" 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-df248f2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"df248f2\" 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-6ffa939\" data-id=\"6ffa939\" 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-261a3c5 elementor-widget elementor-widget-text-editor\" data-id=\"261a3c5\" 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 makes three types of products, A, B, and C. Each product requires raw materials and labor, as well as storage space. Here are the raw material, labor and storage needs for a unit of each of the 3 types of products:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:list --><\/p>\n<ul>\n<li>Product A: 4 kg of raw materials, 2 hours of labor, 1 unit of volume.<\/li>\n<li>Product B: 2 kg of raw materials, 1\/2 hour of labor 1 unit of volume.<\/li>\n<li>Product C: 1 kg of raw materials, 3 hours of labor, 1 unit of volume.<\/li>\n<\/ul>\n<p><!-- \/wp:list --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Each product brings in 6 euros if it is of type A, 2 euros if it is of type B, and 4 euros if it is of type C. The available resources are 6000 kg of raw materials, 4000 hours of labor. work, and 2,500 storage volume units.<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Model the linear program and solve it by hand using the simplex.<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>A competitor, who lacks raw materials, offers to buy 300 kg from this company, at a price of 1.5 euros per kg. Do you think the company should accept this proposal? (Assume that she accepts it if it is profitable, solve the problem by hand with the simplex.)<\/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-75ee89f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"75ee89f\" 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-fdcb62e\" data-id=\"fdcb62e\" 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-12a3b27 elementor-widget elementor-widget-toggle\" data-id=\"12a3b27\" 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-1951\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1951\" 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-1951\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1951\"><p>The problem in standard form is as follows:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7411,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7411 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire61.png\" alt=\"linear programming primal form corrected exercises\" width=\"411\" height=\"155\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire61.png 411w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire61-300x113.png 300w\" sizes=\"(max-width: 411px) 100vw, 411px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>With the simplex solution:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7412,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7412 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire62.png\" alt=\"linear programming primal form corrected exercises\" width=\"466\" height=\"193\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire62.png 466w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire62-300x124.png 300w\" sizes=\"(max-width: 466px) 100vw, 466px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The redemption problem has additionally in the objective function 450. The first variable has for element b = 5700. In this case the optimal solution is (1310, 0, 460) with Z = 10150, so the proposition is profitable.<\/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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-00fe71d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"00fe71d\" 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-00549e4\" data-id=\"00549e4\" 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-cf75d46 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"cf75d46\" 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-d5bf124 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d5bf124\" 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-33292fb\" data-id=\"33292fb\" 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-b03d62d elementor-widget elementor-widget-heading\" data-id=\"b03d62d\" 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-a6be04a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a6be04a\" 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-3cea7db\" data-id=\"3cea7db\" 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-a87a16a elementor-widget elementor-widget-text-editor\" data-id=\"a87a16a\" 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 three types of batteries: dry type 1 (PS1), dry type 2 (PS2) and fuel (PC). The manufacturing process has three stages: assembly, quality test, insulation treatment. Only batteries which pass the quality test are subjected to the insulation treatment. Batteries that fail the quality test are discarded. Over the next month, the company will have 9,000 hours of machine time for assembly, 1,200 hours for quality testing and 8,500 hours for insulation processing. The following table summarizes the relevant information of the manufacturing process:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7312,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7312 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire30.png\" alt=\"linear programming primal form corrected exercises\" width=\"602\" height=\"141\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire30.png 602w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire30-300x70.png 300w\" sizes=\"(max-width: 602px) 100vw, 602px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>What is the optimal number of batteries of each type to manufacture next month if the company is sure to sell all of its production?<\/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-d58f0b0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d58f0b0\" 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-e687ed9\" data-id=\"e687ed9\" 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-c2bc6b7 elementor-widget elementor-widget-toggle\" data-id=\"c2bc6b7\" 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-2041\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2041\" 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-2041\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2041\"><p>Let X1, X2, X3 be the three variables representing the number of valid stacks of each type and X4, X5, X6 the three variables representing the number of invalid stacks of each type.<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The linear program in canonical form is as follows:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7408,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7408 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire59.png\" alt=\"linear programming primal form corrected exercises\" width=\"479\" height=\"215\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire59.png 479w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire59-300x135.png 300w\" sizes=\"(max-width: 479px) 100vw, 479px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:image {\"id\":7409,\"width\":952,\"height\":324,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7409 size-large\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire60-1024x349.png\" alt=\"linear programming primal form corrected exercises\" width=\"1024\" height=\"349\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire60-1024x349.png 1024w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire60-300x102.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire60-768x262.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire60-1200x409.png 1200w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire60.png 1301w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>This gives as a solution (419417.476, 0, 741176.471) for valid batteries and for objective function Z = 1278957.05. As the solution does not give an integer result, it is possible to truncate the floating part (be careful, this will no longer be an optimal solution!).<\/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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-13744c0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"13744c0\" 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-61aaa8e\" data-id=\"61aaa8e\" 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-c13d079 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"c13d079\" 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-e9ebb66 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e9ebb66\" 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-ca29287\" data-id=\"ca29287\" 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-d1e90c7 elementor-widget elementor-widget-heading\" data-id=\"d1e90c7\" 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-4e7f597 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4e7f597\" 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-0bc5ccc\" data-id=\"0bc5ccc\" 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-718e4f1 elementor-widget elementor-widget-text-editor\" data-id=\"718e4f1\" 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>In an electricity grid, the amount of energy that can pass from power plants to cities is limited by the capacities of very high voltage lines. The problem is described as follows:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:list --><\/p>\n<ul>\n<li>Sources, represented by vertices, produce a quantity of energy<\/li>\n<li>Wells, represented by vertices, receive a quantity of energy<\/li>\n<li>Lines, represented by edges between a pair of vertices, through which the energy passes.<\/li>\n<\/ul>\n<p><!-- \/wp:list --><\/p>\n<p><!-- wp:image {\"id\":7322,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7322 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/dinicalgorithm.png\" alt=\"linear programming primal form corrected exercises\" width=\"499\" height=\"240\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/dinicalgorithm.png 499w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/dinicalgorithm-300x144.png 300w\" sizes=\"(max-width: 499px) 100vw, 499px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>This graph reads as follows:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:list --><\/p>\n<ul>\n<li>S is a fictitious source, which describes the production of the energy sources represented by vertex 1 and vertex 2.<\/li>\n<li>T is a fictitious well, which describes the energy reception of the cities represented by vertex 3 and vertex 4.<\/li>\n<li>The lines are oriented, that is to say that the energy only passes in one direction. For example the link (1, 3) can circulate up to 4 units of energy between vertex 1 and vertex 3.<\/li>\n<li>The links between S and the sources describe the energy production of the source. For example the link (S, 1) says that source 1 can produce up to 10 units of energy.<\/li>\n<li>The links between the wells and T describe the energy demand of the well. For example the link (3, T) says that city 3 requires 10 units of energy.<\/li>\n<\/ul>\n<p><!-- \/wp:list --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The objective function of this type of problem is:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:list --><\/p>\n<ul>\n<li>maximize the amount of energy that can be made to transit from S to T. The solution will provide the amount of energy passing through each line. To calculate this flow, an arc is added between the fictitious well and the unconstrained fictitious source. The optimal solution is equal to the flow passing through this arc.<\/li>\n<\/ul>\n<p><!-- \/wp:list --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>This problem is subject to two types of constraints:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:list --><\/p>\n<ul>\n<li>the sum of the energies entering a vertex minus the sum of the energies exiting the same vertex is zero. That is to say that there are no energy losses in the network<\/li>\n<li>the energy passing through a line cannot be greater than the capacity of the line.<\/li>\n<li>the energy passing through a line is positive or zero.<\/li>\n<\/ul>\n<p><!-- \/wp:list --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>The variables are therefore:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:list --><\/p>\n<ul>\n<li>for each row, a variable describes the amount of energy passing through it.<\/li>\n<\/ul>\n<p><!-- \/wp:list --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Formulate the linear problem for the graph given above and solve it via Excel.<\/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-5acc14d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5acc14d\" 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-5314929\" data-id=\"5314929\" 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-38d5c36 elementor-widget elementor-widget-toggle\" data-id=\"38d5c36\" 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-5951\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-5951\" 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-5951\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-5951\"><p>The linear problem is written in Excel as follows:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7416,\"width\":823,\"height\":402,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7416 size-large\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire64-1024x501.png\" alt=\"linear programming primal form corrected exercises\" width=\"1024\" height=\"501\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire64-1024x501.png 1024w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire64-300x147.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire64-768x376.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire64.png 1092w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<p><!-- \/wp:image --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>Which gives as result:<\/p>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:image {\"id\":7417,\"width\":826,\"height\":412,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7417 size-large\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire65-1024x511.png\" alt=\"linear programming primal form corrected exercises\" width=\"1024\" height=\"511\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire65-1024x511.png 1024w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire65-300x150.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire65-768x383.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire65.png 1072w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/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<\/div>","protected":false},"excerpt":{"rendered":"<p>Linear programming Home page Wiki Corrected linear programming exercises (primal form) This tutorial offers various corrected exercises on modeling in primal form\u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":486,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-7367","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7367","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=7367"}],"version-history":[{"count":6,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7367\/revisions"}],"predecessor-version":[{"id":20317,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7367\/revisions\/20317"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/486"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=7367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}