{"id":7237,"date":"2019-10-25T12:28:41","date_gmt":"2019-10-25T11:28:41","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=7237"},"modified":"2022-12-03T23:03:32","modified_gmt":"2022-12-03T22:03:32","slug":"lp-solutions-et-domaine-realisables","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-solutions-and-realizable-domain\/","title":{"rendered":"LP: Solutions and feasible area"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"7237\" class=\"elementor elementor-7237\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-767d6b3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"767d6b3\" 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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rel=\"noopener\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-321d7a80 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"321d7a80\" 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-119c2c77\" data-id=\"119c2c77\" 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-5577bc84 elementor-widget elementor-widget-text-editor\" data-id=\"5577bc84\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewbox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewbox=\"0 0 24 24\" version=\"1.2\" baseprofile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-solutions-and-realizable-domain\/#Domaine-realisable\" >Achievable domain<\/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-solutions-and-realizable-domain\/#Domaine-realisable-ou-domaine-de-definition\" >Achievable domain (or definition domain)<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Domaine-realisable\"><\/span>Achievable domain<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A solution of one <a href=\"https:\/\/complex-systems-ai.com\/en\/help-with-the-decision\/linear-modeling\/\">linear problem<\/a> is said to be feasible if all the constraints are satisfied. The feasible domain contains all the feasible solutions of the problem. The optimal solution is the \u201cbest\u201d feasible solution(s).<\/p>\n\n<p>To know if a solution is feasible, it suffices to test if all the constraints are satisfied, that can be done by hand or in matrix form.<\/p>\n\n<p><strong><em>By hand :<\/em><\/strong><\/p>\n\n<figure class=\"wp-block-image size-medium\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-7242 size-medium\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/optimisationlineaire-300x213.png\" alt=\"linear programming domain of definition realizable domain\" width=\"300\" height=\"213\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/optimisationlineaire-300x213.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/optimisationlineaire.png 534w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/figure>\n\n<p>Let&#039;s check if the solution (3, 1) is feasible.<\/p>\n\n<p>The first equation gives 3 * 1\/3 + 1 = 2, the constraint is satisfied.<br \/>The second inequality gives -2 * 3 + 5 * 1 = -1 \u2264 7, the constraint is satisfied.<br \/>The third inequality gives 3 + 1 = 4 \u2264 4, the constraint is satisfied, we say that it is saturated.<br \/>Both type constraints are satisfied.<\/p>\n\n<p>The solution is achievable. The value of the objective function is z = 3 - 1 = 2.<\/p>\n\n<p><em><strong>From a matrix point of view: <\/strong><\/em>we must multiply the matrix of the <a href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/\">linear program<\/a> by the solution vector and compare the result to the right members of the linear program<\/p>\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" class=\"alignnone wp-image-7244 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire5.png\" alt=\"linear programming domain of definition realizable domain\" width=\"238\" height=\"90\" title=\"\"><\/figure>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Domaine-realisable-ou-domaine-de-definition\"><\/span>Achievable domain (or definition domain)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>Each constraint can be likened to an equation dividing the plane in two. For example the equation a<sub>i<\/sub>* x<sub>1<\/sub> + b<sub>i<\/sub>* x<sub>2<\/sub> = c<sub>i<\/sub> divides the plane into two half-planes P<sub>1<\/sub> and P<sub>2<\/sub> equation:<\/p>\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" class=\"aligncenter wp-image-7246 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire6.png\" alt=\"linear programming domain of definition realizable domain\" width=\"402\" height=\"201\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire6.png 402w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire6-300x150.png 300w\" sizes=\"(max-width: 402px) 100vw, 402px\" \/><\/figure>\n\n<p>The less than or equal constraint will determine one half-plane, the greater or equal constraint will determine the other half-plane. To know in which half-plane are the feasible solutions for the constraints, it suffices to test a simple example and to determine if it is feasible or not.<\/p>\n\n<p>For example for the constraint: x<sub>1<\/sub> + x<sub>2<\/sub> \u2264 4, the solution (0,0) is feasible, so the origin is in the feasible half-plane.<\/p>\n\n<p>The intersection of all feasible half-planes constitutes the feasible domain. The latter can be bounded or unbounded.<\/p>\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-7248 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire7.png\" alt=\"linear programming domain of definition realizable domain\" width=\"557\" height=\"205\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire7.png 557w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/10\/lineaire7-300x110.png 300w\" sizes=\"(max-width: 557px) 100vw, 557px\" \/><\/figure>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Linear Programming Homepage Wiki Feasible domain A solution of a linear problem is said to be feasible if all the constraints are satisfied. The feasible domain contains... <\/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-7237","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7237","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=7237"}],"version-history":[{"count":4,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7237\/revisions"}],"predecessor-version":[{"id":17904,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/7237\/revisions\/17904"}],"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=7237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}