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| 2 | +<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Bases · SimplexTableaux</title><meta name="title" content="Bases · SimplexTableaux"/><meta property="og:title" content="Bases · SimplexTableaux"/><meta property="twitter:title" content="Bases · 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script data-outdated-warner src="../assets/warner.js"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" 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href="../">Overview</a></li><li><a class="tocitem" href="../create/">Creating Tableaux</a></li><li class="is-active"><a class="tocitem" href>Bases</a><ul class="internal"><li><a class="tocitem" href="#Setting-a-Basis"><span>Setting a Basis</span></a></li><li><a class="tocitem" href="#Finding-a-Basis"><span>Finding a Basis</span></a></li></ul></li><li><a class="tocitem" href="../pivot/">Pivoting Tableaux</a></li><li><a class="tocitem" href="../solve/">Solving LPs</a></li><li><a class="tocitem" href="../other/">Other Functions</a></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>Bases</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Bases</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/bases.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="Bases"><a class="docs-heading-anchor" href="#Bases">Bases</a><a id="Bases-1"></a><a class="docs-heading-anchor-permalink" href="#Bases" title="Permalink"></a></h1><h2 id="Setting-a-Basis"><a class="docs-heading-anchor" href="#Setting-a-Basis">Setting a Basis</a><a id="Setting-a-Basis-1"></a><a class="docs-heading-anchor-permalink" href="#Setting-a-Basis" title="Permalink"></a></h2><p>Use <code>set_basis!(T, B)</code> to specify a starting basis for the tableau. Here, <code>B</code> is a list (<code>Vector</code>) of integers specifying the columns that are in the basis. </p><pre><code class="nohighlight hljs">julia> T |
| 3 | +┌──────────┬───┬─────┬─────┬─────┬─────┬─────┬─────┐ |
| 4 | +│ │ z │ x_1 │ x_2 │ x_3 │ x_4 │ x_5 │ RHS │ |
| 5 | +│ Obj Func │ 1 │ -25 │ -10 │ 0 │ 0 │ 0 │ 0 │ |
| 6 | +├──────────┼───┼─────┼─────┼─────┼─────┼─────┼─────┤ |
| 7 | +│ Cons 1 │ 0 │ 3 │ 10 │ -1 │ 0 │ 0 │ 100 │ |
| 8 | +│ Cons 2 │ 0 │ 5 │ 6 │ 0 │ -1 │ 0 │ 100 │ |
| 9 | +│ Cons 3 │ 0 │ 10 │ 2 │ 0 │ 0 │ -1 │ 100 │ |
| 10 | +└──────────┴───┴─────┴─────┴─────┴─────┴─────┴─────┘ |
| 11 | + |
| 12 | +julia> set_basis!(T,[1,4,5]) |
| 13 | +┌──────────┬───┬─────┬───────┬───────┬─────┬─────┬────────┐ |
| 14 | +│ │ z │ x_1 │ x_2 │ x_3 │ x_4 │ x_5 │ RHS │ |
| 15 | +│ Obj Func │ 1 │ 0 │ 220/3 │ -25/3 │ 0 │ 0 │ 2500/3 │ |
| 16 | +├──────────┼───┼─────┼───────┼───────┼─────┼─────┼────────┤ |
| 17 | +│ Cons 1 │ 0 │ 1 │ 10/3 │ -1/3 │ 0 │ 0 │ 100/3 │ |
| 18 | +│ Cons 2 │ 0 │ 0 │ 32/3 │ -5/3 │ 1 │ 0 │ 200/3 │ |
| 19 | +│ Cons 3 │ 0 │ 0 │ 94/3 │ -10/3 │ 0 │ 1 │ 700/3 │ |
| 20 | +└──────────┴───┴─────┴───────┴───────┴─────┴─────┴────────┘</code></pre><blockquote><p>Note: On the screen, the headings for the basis (in this case, <code>x_1</code>, <code>x_4</code>, and <code>x_5</code>) appear in green. </p></blockquote><p><img src="../color-tab.png" alt/></p><h2 id="Finding-a-Basis"><a class="docs-heading-anchor" href="#Finding-a-Basis">Finding a Basis</a><a id="Finding-a-Basis-1"></a><a class="docs-heading-anchor-permalink" href="#Finding-a-Basis" title="Permalink"></a></h2><p>The function <code>find_a_basis(T)</code> returns a feasible basis for <code>T</code> using the phase-one method.</p><pre><code class="nohighlight hljs">julia> T |
| 21 | +┌──────────┬───┬────────┬─────────┬─────┬─────┬─────┬───────┐ |
| 22 | +│ │ z │ x_1 │ x_2 │ x_3 │ x_4 │ x_5 │ RHS │ |
| 23 | +│ Obj Func │ 1 │ 61/200 │ 109/200 │ 0 │ 0 │ 0 │ 0 │ |
| 24 | +├──────────┼───┼────────┼─────────┼─────┼─────┼─────┼───────┤ |
| 25 | +│ Cons 1 │ 0 │ -7/10 │ -9/10 │ -1 │ 0 │ 0 │ -5000 │ |
| 26 | +│ Cons 2 │ 0 │ -1/10 │ -1/20 │ 0 │ -1 │ 0 │ -500 │ |
| 27 | +│ Cons 3 │ 0 │ 1 │ -1/2 │ 0 │ 0 │ -1 │ 0 │ |
| 28 | +└──────────┴───┴────────┴─────────┴─────┴─────┴─────┴───────┘ |
| 29 | + |
| 30 | + |
| 31 | +julia> find_a_basis(T) |
| 32 | +3-element Vector{Int64}: |
| 33 | + 1 |
| 34 | + 2 |
| 35 | + 4</code></pre><p>The function <code>find_all_bases(T)</code> returns a list of all feasible bases for <code>T</code>:</p><pre><code class="nohighlight hljs">julia> find_all_bases(T) |
| 36 | +6-element Vector{Vector{Int64}}: |
| 37 | + [1, 2, 4] |
| 38 | + [1, 2, 5] |
| 39 | + [1, 3, 4] |
| 40 | + [1, 3, 5] |
| 41 | + [2, 3, 4] |
| 42 | + [3, 4, 5]</code></pre><p>Note that <code>find_all_bases</code> is rather inefficient as it considers all possible <code>m</code>-element subsets of the columns.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../create/">« Creating Tableaux</a><a class="docs-footer-nextpage" href="../pivot/">Pivoting Tableaux »</a><div class="flexbox-break"></div><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 11 August 2025 17:29">Monday 11 August 2025</span>. Using Julia version 1.11.6.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
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