{"id":203316,"date":"2025-03-19T05:20:28","date_gmt":"2025-03-19T05:20:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=203316"},"modified":"2025-03-19T05:20:30","modified_gmt":"2025-03-19T05:20:30","slug":"fundamental-theorem-of-linear-programming","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/19\/fundamental-theorem-of-linear-programming\/","title":{"rendered":"Fundamental Theorem of Linear Programming"},"content":{"rendered":"\n<p>Fundamental Theorem of Linear Programming: Version 3 If the optimal value of the objective function in a linear programming problem exists, then that value must occur at one or more of the basic feasible solutions of the initial system.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Correct Answer:<\/h3>\n\n\n\n<p>The <strong>Fundamental Theorem of Linear Programming (Version 3)<\/strong> states that:<\/p>\n\n\n\n<p><em>&#8220;If the optimal value of the objective function in a linear programming problem exists, then that value must occur at one or more of the basic feasible solutions of the initial system.&#8221;<\/em><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>Linear programming (LP) is a mathematical method used to find the best possible outcome in a given mathematical model, subject to constraints. The fundamental theorem of linear programming ensures that if there is an optimal solution to an LP problem, it must be found at a <strong>basic feasible solution (BFS)<\/strong>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Key Concepts:<\/strong><\/h4>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Feasible Region:<\/strong> This is the set of all points that satisfy the given constraints of an LP problem.<\/li>\n\n\n\n<li><strong>Extreme Points (Corner Points):<\/strong> These are the points where constraints intersect, and they are candidates for optimality.<\/li>\n\n\n\n<li><strong>Basic Feasible Solutions (BFS):<\/strong> These are solutions that correspond to extreme points of the feasible region. They are obtained by setting some variables to zero (non-basic variables) and solving for the remaining basic variables.<\/li>\n\n\n\n<li><strong>Optimal Solution:<\/strong> The best possible value of the objective function, which maximizes or minimizes the function within the feasible region.<\/li>\n<\/ol>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Why Does the Optimal Solution Occur at a BFS?<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The objective function in an LP problem is linear, meaning it increases or decreases along straight lines.<\/li>\n\n\n\n<li>The feasible region, formed by linear constraints, is a convex polyhedron.<\/li>\n\n\n\n<li>Since a linear function attains its maximum or minimum on a convex polyhedron at extreme points, the optimal solution (if it exists) must be at one of these points.<\/li>\n<\/ul>\n\n\n\n<p>Thus, instead of checking every possible point in the feasible region, we only need to evaluate BFS, significantly reducing computational effort. The <strong>Simplex Method<\/strong> exploits this property by moving from one BFS to another until an optimal solution is found.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Fundamental Theorem of Linear Programming: Version 3 If the optimal value of the objective function in a linear programming problem exists, then that value must occur at one or more of the basic feasible solutions of the initial system. The correct answer and explanation is : Correct Answer: The Fundamental Theorem of Linear Programming (Version [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-203316","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203316","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=203316"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203316\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=203316"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=203316"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=203316"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}