{"id":205852,"date":"2025-03-25T18:46:21","date_gmt":"2025-03-25T18:46:21","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=205852"},"modified":"2025-03-25T18:46:23","modified_gmt":"2025-03-25T18:46:23","slug":"define-the-canonical-form-of-an-lpp","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/25\/define-the-canonical-form-of-an-lpp\/","title":{"rendered":"Define the Canonical form of an LPP"},"content":{"rendered":"\n

Define the Canonical form of an LPP. What is the difference between the slack variable and the surplus variable?<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

Canonical Form of a Linear Programming Problem (LPP)<\/strong><\/h3>\n\n\n\n

The canonical form<\/strong> of an LPP is a standard way of representing a linear programming problem where:<\/p>\n\n\n\n

    \n
  1. The objective function is in the maximization<\/strong> form.<\/li>\n\n\n\n
  2. All constraints are in the form of equalities<\/strong> by introducing slack or surplus variables.<\/li>\n\n\n\n
  3. All decision variables are non-negative<\/strong> (i.e., ( x_i \\geq 0 )).<\/li>\n<\/ol>\n\n\n\n

    A Canonical LPP<\/strong> is typically written as:
    [
    \\text{Maximize } Z = c_1x_1 + c_2x_2 + \\dots + c_nx_n
    ]
    Subject to:
    [
    a_{11}x_1 + a_{12}x_2 + \\dots + a_{1n}x_n = b_1
    ]
    [
    a_{21}x_1 + a_{22}x_2 + \\dots + a_{2n}x_n = b_2
    ]
    [
    \\vdots
    ]
    [
    a_{m1}x_1 + a_{m2}x_2 + \\dots + a_{mn}x_n = b_m
    ]
    [
    x_i \\geq 0, \\quad \\forall i = 1, 2, \u2026, n.
    ]<\/p>\n\n\n\n

    Slack Variable vs. Surplus Variable<\/strong><\/h3>\n\n\n\n

    In linear programming, slack<\/strong> and surplus<\/strong> variables are used to convert inequalities into equalities:<\/p>\n\n\n\n

      \n
    • Slack Variable<\/strong>: Introduced in \u2264 (less than or equal to) constraints<\/strong> to convert them into equalities. It represents unused resources.<\/li>\n\n\n\n
    • Example: ( 2x_1 + 3x_2 \\leq 8 ) \u2192 ( 2x_1 + 3x_2 + s = 8 ) (where ( s \\geq 0 )).<\/li>\n\n\n\n
    • Surplus Variable<\/strong>: Introduced in \u2265 (greater than or equal to) constraints<\/strong> to convert them into equalities. It represents excess over a requirement.<\/li>\n\n\n\n
    • Example: ( 4x_1 + 2x_2 \\geq 10 ) \u2192 ( 4x_1 + 2x_2 – s = 10 ) (where ( s \\geq 0 )).<\/li>\n<\/ul>\n\n\n\n

      Thus, a slack variable<\/strong> adds to the left-hand side of the inequality, while a surplus variable<\/strong> subtracts from it.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      Define the Canonical form of an LPP. What is the difference between the slack variable and the surplus variable? The correct answer and explanation is : Canonical Form of a Linear Programming Problem (LPP) The canonical form of an LPP is a standard way of representing a linear programming problem where: A Canonical LPP is […]<\/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-205852","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/205852","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=205852"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/205852\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=205852"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=205852"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=205852"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}