{"id":7377,"date":"2019-11-27T13:17:52","date_gmt":"2019-11-27T12:17:52","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=7377"},"modified":"2024-02-11T19:00:58","modified_gmt":"2024-02-11T18:00:58","slug":"lp-cas-particuliers","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-specific-cases-exercises-solutions\/","title":{"rendered":"3 Corrected Exercises for Special Cases in LP"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"7377\" class=\"elementor elementor-7377\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-42294c0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"42294c0\" 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-e8ef4aa elementor-widget elementor-widget-heading\" data-id=\"e8ef4aa\" 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_85 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 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Linear Programming<\/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-specific-cases-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-specific-cases-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-specific-cases-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-specific-cases-exercises-solutions\/#Exercice-3\" >Exercise 3<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercices-Corriges-de-Cas-Particuliers-Grand-M-en-Programmation-Lineaire\"><\/span>Corrected Exercises for Special Cases (Big M) in Linear Programming<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-9405ae1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9405ae1\" 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-4078d74\" data-id=\"4078d74\" 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-b7b6059 elementor-widget elementor-widget-text-editor\" data-id=\"b7b6059\" 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 exercises corrected on particular cases (degeneration, large M, two-phase simplex).<\/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=\"special cases\" 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-f429ff2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f429ff2\" 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-ec6ca00\" data-id=\"ec6ca00\" 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-7f5b421 elementor-widget elementor-widget-heading\" data-id=\"7f5b421\" 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-35e8fb3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"35e8fb3\" 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-9adfecf\" data-id=\"9adfecf\" 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-c2dfee0 elementor-widget elementor-widget-text-editor\" data-id=\"c2dfee0\" 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 and by the simplex.<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:heading {\"level\":4} --><!-- \/wp:heading --><!-- wp:paragraph --><\/p>\n<p>The linear program is as follows: X1 for the first batch and X2 for the second batch<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7368,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" class=\"alignnone wp-image-7368 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire44.png\" alt=\"special cases linear programming primal form corrected exercises\" width=\"224\" height=\"109\" title=\"\"><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>Which gives for graphic resolution:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7369,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img fetchpriority=\"high\" 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 --><!-- 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>\n<p><!-- \/wp:paragraph --><!-- wp:paragraph --><\/p>\n<p>For the resolution of the simplex it is necessary above all to reduce to the standard form.<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7375,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7375 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire46.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"762\" height=\"146\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire46.png 762w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire46-300x57.png 300w\" sizes=\"(max-width: 762px) 100vw, 762px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>It is necessary to solve the simplex in two phases in order to eliminate the artificial variables:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7385,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7385 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire48.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"444\" height=\"169\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire48.png 444w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire48-300x114.png 300w\" sizes=\"(max-width: 444px) 100vw, 444px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>And the result of the first phase. We notice that Z = 0 so there is a solution to the problem:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7388,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7388 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire49.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"485\" height=\"169\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire49.png 485w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire49-300x105.png 300w\" sizes=\"(max-width: 485px) 100vw, 485px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>For the second phase, row Z is recalculated after removing the columns of artificial variables. To do this, we take the good coefficients of the objective function (-1, -1, 0, 0):<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:paragraph --><\/p>\n<p>- (0) + (-1 * 6) + (-1 * 4) = -10 for column b (P0)<br \/>- (- 1) + (-1 * 1) + (-1 * 0) = 0 for column P1<br \/>- (- 1) + (-1 * 0) + (-1 * 1) = 0 for column P2<br \/>- (0) + (-1 * -1 \/ 6) + (-1 * 1\/9) = 1\/18 for column P3 <br \/>- (0) + (-1 * 1\/8) + (-1 * -1 \/ 6) = 1\/24 for column P4<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:paragraph --><\/p>\n<p>Which gives the following simplex:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7390,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7390 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire50.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"396\" height=\"170\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire50.png 396w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire50-300x129.png 300w\" sizes=\"(max-width: 396px) 100vw, 396px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>All the coefficients except column b are positive therefore the simplex already has the optimal solution which is Z = 10 with the vector (6, 4, 0, 0).<\/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-c322702 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c322702\" 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-e22a8a6\" data-id=\"e22a8a6\" 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-8fbee2e elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"8fbee2e\" 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-70ada31 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"70ada31\" 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-8552450\" data-id=\"8552450\" 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-41bbb6b elementor-widget elementor-widget-heading\" data-id=\"41bbb6b\" 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-bf15981 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bf15981\" 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-61aa7e5\" data-id=\"61aa7e5\" 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-c115d07 elementor-widget elementor-widget-text-editor\" data-id=\"c115d07\" 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 reseller promised his customers that at least 25% of his electricity would be of renewable origin. He calculated that for the coming year he will have a market of at most 18 TWh. He has also pre-selected three suppliers from whom he will buy his electricity wholesale. Here are the quantities (in TWh), the renewable electricity rate and the margin generated (in thousands of euros \/ TWh) that these three producers can provide. From which producers and in what quantity should this reseller buy his electricity to obtain the best possible profit? Solve the linear problem with the large M method using Excel.<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7306,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7306 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire27.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"562\" height=\"143\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire27.png 562w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire27-300x76.png 300w\" sizes=\"(max-width: 562px) 100vw, 562px\" \/><\/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-3a3dfbd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3a3dfbd\" 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-07333c5\" data-id=\"07333c5\" 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-186893c elementor-widget elementor-widget-toggle\" data-id=\"186893c\" 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-2551\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2551\" 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-2551\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2551\"><p>After standard formatting, the problem is as follows, there are more constraints than variables, so we can solve the primal:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7396,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7396 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire51.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"602\" height=\"509\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire51.png 602w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire51-300x254.png 300w\" sizes=\"(max-width: 602px) 100vw, 602px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>The problem is solved in two phases because there is an artificial variable. Here is the table of the first phase:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7397,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7397 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire52.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"548\" height=\"250\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire52.png 548w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire52-300x137.png 300w\" sizes=\"(max-width: 548px) 100vw, 548px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>And the solution of the first phase, we notice that Z = 0 so there is a solution to the problem:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7398,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7398 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lienaire53.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"605\" height=\"250\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lienaire53.png 605w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lienaire53-300x124.png 300w\" sizes=\"(max-width: 605px) 100vw, 605px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>After recalculating the coefficients of Z, the second phase begins with the following table:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7399,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7399 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire54.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"567\" height=\"252\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire54.png 567w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire54-300x133.png 300w\" sizes=\"(max-width: 567px) 100vw, 567px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>And ends with the following table:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7400,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7400 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire55.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"606\" height=\"250\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire55.png 606w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire55-300x124.png 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/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-09a3d82 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"09a3d82\" 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-8dda0c9\" data-id=\"8dda0c9\" 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-b649e80 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"b649e80\" 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-68e3858 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"68e3858\" 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-70ccd20\" data-id=\"70ccd20\" 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-90714ed elementor-widget elementor-widget-heading\" data-id=\"90714ed\" 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-0666fd3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0666fd3\" 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-273d7a5\" data-id=\"273d7a5\" 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-fab7f25 elementor-widget elementor-widget-text-editor\" data-id=\"fab7f25\" 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 the following linear program (by Excel):<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7482,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7482 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire87.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"321\" height=\"142\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire87.png 321w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire87-300x133.png 300w\" sizes=\"(max-width: 321px) 100vw, 321px\" \/><\/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-1fab040 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1fab040\" 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-82936cf\" data-id=\"82936cf\" 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-c000f54 elementor-widget elementor-widget-toggle\" data-id=\"c000f54\" 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-2011\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2011\" 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-2011\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2011\"><p>It is noticed above all that there are more variables than constraints, it is thus necessary to solve the linear program while passing by the simplex and the complementary variations. The dual in canonical then standard form is as follows:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7484,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7484 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire88-1.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"566\" height=\"517\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire88-1.png 566w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire88-1-300x274.png 300w\" sizes=\"(max-width: 566px) 100vw, 566px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>We must therefore solve the problem of the large M in order to eliminate the artificial variables. The simplex to be solved is as follows:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7485,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7485 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire89.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"522\" height=\"224\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire89.png 522w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire89-300x129.png 300w\" sizes=\"(max-width: 522px) 100vw, 522px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>Which results in:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7487,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7487 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire90.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"646\" height=\"224\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire90.png 646w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire90-300x104.png 300w\" sizes=\"(max-width: 646px) 100vw, 646px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>By eliminating the artificial variables and returning to the dual program we have the following table:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7489,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7489 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire91.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"506\" height=\"223\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire91.png 506w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire91-300x132.png 300w\" sizes=\"(max-width: 506px) 100vw, 506px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>The optimal solution has for vector (3\/7, 2\/7) with Z = 65\/7. Let&#039;s go through the complementary differences to find a solution to the primal.<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:paragraph --><\/p>\n<p>The two variables are not zero, therefore the constraints of the primal are saturated. The constraint 2 and 3 of the dual are unsaturated, so the second and third variable of the primal are zero. The following problem must therefore be solved:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:list --><\/p>\n<ul>\n<li>X1 + 5 X4 = 15<\/li>\n<li>2 X1 - 4 X4 = 10<\/li>\n<\/ul>\n<p><!-- \/wp:list --><!-- wp:paragraph --><\/p>\n<p>The solution vector of the primal is (55\/7, 0, 0, 10\/7) and Z = 65\/7. There is strong duality so it is an optimal solution of the primal.<\/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-92be164 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"92be164\" 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-4cb5ff2\" data-id=\"4cb5ff2\" 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-7708842 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"7708842\" 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-19a06a9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"19a06a9\" 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-6f3a20a\" data-id=\"6f3a20a\" 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-5817d18 elementor-widget elementor-widget-heading\" data-id=\"5817d18\" 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-e2ad90d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e2ad90d\" 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-8a50d77\" data-id=\"8a50d77\" 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-e3cf8c9 elementor-widget elementor-widget-text-editor\" data-id=\"e3cf8c9\" 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>When an eco-district produces more energy than it consumes, then it sells the surplus energy to the neighborhood and to the network. The eco-district therefore seeks to maximize its earnings based on offers from the local market. The market offers are as follows:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:table {\"hasFixedLayout\":true} --><\/p>\n<figure class=\"wp-block-table\">\n<table class=\"has-fixed-layout\">\n<tbody>\n<tr>\n<td>Offer n \u00b0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>Profit<\/td>\n<td>10<\/td>\n<td>8<\/td>\n<td>15<\/td>\n<td>4<\/td>\n<td>1<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>Energy (in kWh)<\/td>\n<td>5<\/td>\n<td>10<\/td>\n<td>10<\/td>\n<td>5<\/td>\n<td>1<\/td>\n<td>7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n<p><!-- \/wp:table --><!-- wp:paragraph --><\/p>\n<p>The problem is formulated as follows: the eco-district seeks to gain the maximum profit for a sale of energy of 25 kWh. The eco-district must sell all of the energy surplus.<\/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-fe78576 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fe78576\" 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-3fe9ecd\" data-id=\"3fe9ecd\" 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-8694f14 elementor-widget elementor-widget-toggle\" data-id=\"8694f14\" 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-1411\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1411\" 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-1411\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1411\"><p>To solve this problem, consider that each offer is represented by a variable (in percentage of acceptance of the offer) which will be between 0 and 1.<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:paragraph --><\/p>\n<p>The problem is formulated as follows:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7495,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7495 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire92.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"432\" height=\"236\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire92.png 432w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire92-300x164.png 300w\" sizes=\"(max-width: 432px) 100vw, 432px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>Which gives as standard form:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7496,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7496 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire93.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"765\" height=\"236\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire93.png 765w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire93-300x93.png 300w\" sizes=\"(max-width: 765px) 100vw, 765px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>Let&#039;s start by solving the big M:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7497,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7497 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire94.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"708\" height=\"314\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire94.png 708w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire94-300x133.png 300w\" sizes=\"(max-width: 708px) 100vw, 708px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>After solving the large M and returning to the standard form:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"id\":7499,\"sizeSlug\":\"large\"} --><\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7499 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire95.png\" alt=\"two-phase simplex linear programming degenerate big M method\" width=\"753\" height=\"306\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire95.png 753w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/11\/lineaire95-300x122.png 300w\" sizes=\"(max-width: 753px) 100vw, 753px\" \/><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>We notice that the value Z for the variable 1, 2, 3, 4, 5 are at zero so there is an infinite number of solutions. The solution obtained here is Z = 166\/5 with the vector (1, 9\/10, 1, 0, 1, 0).<\/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>Linear Programming Home Page Wiki Corrected Exercises for Special Cases (Big M) in Linear Programming This tutorial offers corrected exercises for special cases... <\/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-7377","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7377","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=7377"}],"version-history":[{"count":9,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7377\/revisions"}],"predecessor-version":[{"id":20337,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7377\/revisions\/20337"}],"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=7377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}