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‎README.md‎

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> **Note**: Any version numbered 0.0.x is likely to be buggy and it just preliminary before the 0.1.0 release.
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> **Note**: Any version numbered 0.0.x is likely to be buggy and just preliminary before the 0.1.0 release.
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# SimplexTableaux
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# Quick Start Instructions
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This is an overview. More extensive documentation is [housed here](https://docs.juliahub.com/General/SimplexTableaux/).
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## Set up the problem
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This example comes from [this video](https://www.youtube.com/watch?v=rzRZLGD_aeE).

‎docs/build/.documenter-siteinfo.json‎

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{"documenter":{"julia_version":"1.11.4","generation_timestamp":"2025-04-04T10:36:35","documenter_version":"1.10.1"}}
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{"documenter":{"julia_version":"1.11.4","generation_timestamp":"2025-04-07T11:43:50","documenter_version":"1.10.1"}}

‎docs/build/index.html‎

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<!DOCTYPE html>
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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>SimplexTableaux · SimplexTableaux</title><meta name="title" content="SimplexTableaux · SimplexTableaux"/><meta property="og:title" content="SimplexTableaux · SimplexTableaux"/><meta property="twitter:title" content="SimplexTableaux · SimplexTableaux"/><meta name="description" content="Documentation for SimplexTableaux."/><meta property="og:description" content="Documentation for SimplexTableaux."/><meta property="twitter:description" content="Documentation for SimplexTableaux."/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="search_index.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>SimplexTableaux</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>SimplexTableaux</a><ul class="internal"><li><a class="tocitem" href="#Creating-a-Tableau"><span>Creating a <code>Tableau</code></span></a></li><li><a class="tocitem" href="#Pivoting"><span>Pivoting</span></a></li><li><a class="tocitem" href="#Solving-the-LP"><span>Solving the LP</span></a></li><li><a class="tocitem" href="#The-restore!-Function"><span>The <code>restore!</code> Function</span></a></li><li><a class="tocitem" href="#Visualization"><span>Visualization</span></a></li><li><a class="tocitem" href="#Changing-the-Display-Format"><span>Changing the Display Format</span></a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>SimplexTableaux</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>SimplexTableaux</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/scheinerman/SimplexTableaux.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/scheinerman/SimplexTableaux.jl/blob/main/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SimplexTableaux"><a class="docs-heading-anchor" href="#SimplexTableaux">SimplexTableaux</a><a id="SimplexTableaux-1"></a><a class="docs-heading-anchor-permalink" href="#SimplexTableaux" title="Permalink"></a></h1><p>This is an illustration project for solving feasible optimization problems (Linear Programs) of the form <span>$\max c^t x$</span> subject to <span>$Ax ≤ b$</span> and <span>$x \ge 0$</span>.</p><p>For a quick introduction (and caveats), see the <a href="https://github.com/scheinerman/SimpleTableaux.jl/blob/main/README.md">README</a>.</p><h2 id="Creating-a-Tableau"><a class="docs-heading-anchor" href="#Creating-a-Tableau">Creating a <code>Tableau</code></a><a id="Creating-a-Tableau-1"></a><a class="docs-heading-anchor-permalink" href="#Creating-a-Tableau" title="Permalink"></a></h2><p>To set up a <code>Tableau</code> for the LP <span>$\max c^t x$</span> subject to <span>$Ax ≤ b$</span> and <span>$x \ge 0$</span> use <code>Tableau(A,b,c)</code>. </p><pre><code class="nohighlight hljs">A = [8 3; 1 1; 1 4]
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href="assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>SimplexTableaux</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>SimplexTableaux</a><ul class="internal"><li><a class="tocitem" href="#Creating-a-Tableau"><span>Creating a <code>Tableau</code></span></a></li><li><a class="tocitem" href="#Pivoting"><span>Pivoting</span></a></li><li><a class="tocitem" href="#Solving-the-LP"><span>Solving the LP</span></a></li><li><a class="tocitem" href="#The-restore!-Function"><span>The <code>restore!</code> Function</span></a></li><li><a class="tocitem" href="#Visualization"><span>Visualization</span></a></li><li><a class="tocitem" href="#Changing-the-Display-Format"><span>Changing the Display Format</span></a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>SimplexTableaux</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>SimplexTableaux</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/scheinerman/SimplexTableaux.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/scheinerman/SimplexTableaux.jl/blob/main/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SimplexTableaux"><a class="docs-heading-anchor" href="#SimplexTableaux">SimplexTableaux</a><a id="SimplexTableaux-1"></a><a class="docs-heading-anchor-permalink" href="#SimplexTableaux" title="Permalink"></a></h1><p>This is an illustration project for solving feasible optimization problems (Linear Programs) of the form <span>$\max c^t x$</span> subject to <span>$Ax ≤ b$</span> and <span>$x \ge 0$</span>.</p><blockquote><p><strong>Note</strong>: Any version numbered 0.0.x is likely to be buggy and just preliminary before the 0.1.0 release. </p></blockquote><p>For a quick introduction (and caveats), see the <a href="https://github.com/scheinerman/SimplexTableaux.jl/blob/main/README.md">README</a>.</p><h2 id="Creating-a-Tableau"><a class="docs-heading-anchor" href="#Creating-a-Tableau">Creating a <code>Tableau</code></a><a id="Creating-a-Tableau-1"></a><a class="docs-heading-anchor-permalink" href="#Creating-a-Tableau" title="Permalink"></a></h2><p>To set up a <code>Tableau</code> for the LP <span>$\max c^t x$</span> subject to <span>$Ax ≤ b$</span> and <span>$x \ge 0$</span> use <code>Tableau(A,b,c)</code>. </p><pre><code class="nohighlight hljs">A = [8 3; 1 1; 1 4]
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T = Tableau(A, b, c)</code></pre><p>This is the result:</p><pre><code class="nohighlight hljs">4×7 DataFrame
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4 │ 0 -1/4 1/4 0 0 1 6</code></pre><h2 id="Solving-the-LP"><a class="docs-heading-anchor" href="#Solving-the-LP">Solving the LP</a><a id="Solving-the-LP-1"></a><a class="docs-heading-anchor-permalink" href="#Solving-the-LP" title="Permalink"></a></h2><h3 id="Solution-by-pivoting"><a class="docs-heading-anchor" href="#Solution-by-pivoting">Solution by pivoting</a><a id="Solution-by-pivoting-1"></a><a class="docs-heading-anchor-permalink" href="#Solution-by-pivoting" title="Permalink"></a></h3><p>The function <code>pivot_solve</code> produces the optimal solution to the LP. </p><pre><code class="nohighlight hljs">julia&gt; pivot_solve(T)
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4 │ 0 -1/4 1/4 0 0 1 6</code></pre><p>To manually solve a linear program, use <code>pivot!</code> at each step so the <code>Tableau</code> is modified by the operation. </p><h2 id="Solving-the-LP"><a class="docs-heading-anchor" href="#Solving-the-LP">Solving the LP</a><a id="Solving-the-LP-1"></a><a class="docs-heading-anchor-permalink" href="#Solving-the-LP" title="Permalink"></a></h2><h3 id="Solution-by-pivoting"><a class="docs-heading-anchor" href="#Solution-by-pivoting">Solution by pivoting</a><a id="Solution-by-pivoting-1"></a><a class="docs-heading-anchor-permalink" href="#Solution-by-pivoting" title="Permalink"></a></h3><blockquote><p><strong>Note</strong>: As of this release, this mostly works; however, it may reach a maximum number of iterations without finding an optimal solution. Also, if the LP is unbounded or infeasible, it will likely fail. </p></blockquote><p>The function <code>pivot_solve</code> produces the optimal solution to the LP. </p><pre><code class="nohighlight hljs">julia&gt; pivot_solve(T)
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4 │ obj -2 -1 0 0 0 1 0</code></pre><p>We now see that rows 1, 2, and 3 correspond to constraints and row 4 to the objective function. </p></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.10.1 on <span class="colophon-date" title="Friday 4 April 2025 10:36">Friday 4 April 2025</span>. Using Julia version 1.11.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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4 │ obj -2 -1 0 0 0 1 0</code></pre><p>We now see that rows 1, 2, and 3 correspond to constraints and row 4 to the objective function. </p></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.10.1 on <span class="colophon-date" title="Monday 7 April 2025 11:43">Monday 7 April 2025</span>. Using Julia version 1.11.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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