Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/pkg/fining/examples/include/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 27.6.2023 mit Größe 766 B image not shown  

Quelle  chap3_mj.html   Sprache: unbekannt

 
<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<script type="text/javascript"
  src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (CddInterface) - Chapter 3: Linear Programs</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap3"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap2_mj.html">[Previous Chapter]</a>    <a href="chap4_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap3.html">[MathJax off]</a></p>
<p><a id="X825271797BE64406" name="X825271797BE64406"></a></p>
<div class="ChapSects"><a href="chap3_mj.html#X825271797BE64406">3 <span class="Heading">Linear Programs</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X78544DEC7F939A89">3.1 <span class="Heading">Creating and solving linear programs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78C17DC0813644BE">3.1-1 Cdd_LinearProgram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79EB266A8289CE29">3.1-2 Cdd_SolveLinearProgram</a></span>
</div></div>
</div>

<h3>3 <span class="Heading">Linear Programs</span></h3>

<p><a id="X78544DEC7F939A89" name="X78544DEC7F939A89"></a></p>

<h4>3.1 <span class="Heading">Creating and solving linear programs</span></h4>

<p><a id="X78C17DC0813644BE" name="X78C17DC0813644BE"></a></p>

<h5>3.1-1 Cdd_LinearProgram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_LinearProgram</code>( <var class="Arg">P</var>, <var class="Arg">str</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <em>CddLinearProgram</em> Object</p>

<p>The function takes three variables. The first is a polyhedron <em>poly</em>, the second <em>str</em> should be "max" or "min" and the third <em>obj</em> is the objective function.</p>

<p><a id="X79EB266A8289CE29" name="X79EB266A8289CE29"></a></p>

<h5>3.1-2 Cdd_SolveLinearProgram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_SolveLinearProgram</code>( <var class="Arg">lp</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a list if the program is optimal, otherwise returns the value 0</p>

<p>The function takes a linear program. If the program is optimal, the function returns a list of two entries, the solution vector and the optimal value of the objective, otherwise it returns <var class="Arg">fail</var>.</p>

<p>To illustrate the using of these functions, let us solve the linear program given by:</p>

<p class="center">\[\textbf{Maximize}\;\;P(x,y)= 1-2x+5y,\;\mathrm{with}\]</p>

<p class="center">\[100\leq x \leq 200,80\leq y\leq 170,y \geq -x+200.\]</p>

<p>We bring the inequalities to the form <span class="SimpleMath">\(b+AX\geq 0\)</span> and get:</p>

<p class="center">\[-100+x\geq 0, 200-x \geq 0, -80+y \geq 0, 170 -y \geq 0,-200 +x+y \geq 0.\]</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Cdd_PolyhedronByInequalities( [ [ -100, 1, 0 ], [ 200, -1, 0 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ -80, 0, 1 ], [ 170, 0, -1 ], [ -200, 1, 1 ] ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">lp1:= Cdd_LinearProgram( A, "max", [1, -2, 5 ] );</span>
<Linear program>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( lp1 );</span>
Linear program given by:
H-representation
begin
   5 X 3  rational

   -100     1     0
    200    -1     0
    -80     0     1
    170     0    -1
   -200     1     1
end
max  [ 1, -2, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_SolveLinearProgram( lp1 );</span>
[ [ 100, 170 ], 651 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">lp2:= Cdd_LinearProgram( A, "min", [ 1, -2, 5 ] );</span>
<Linear program>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( lp2 );</span>
Linear program given by:
H-representation
begin
   5 X 3  rational

   -100     1     0
    200    -1     0
    -80     0     1
    170     0    -1
   -200     1     1
end
min  [ 1, -2, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_SolveLinearProgram( lp2 );</span>
[ [ 200, 80 ], 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Cdd_V_Rep( A );</span>
<Polyhedron given by its V-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( B );</span>
V-representation
begin
   5 X 3  rational

   1  100  170
   1  100  100
   1  120   80
   1  200   80
   1  200  170
end
</pre></div>

<p>So the optimal solution for <span class="SimpleMath">\(\texttt{lp1}\)</span> is <span class="SimpleMath">\((x=100,y=170)\)</span> with optimal value <span class="SimpleMath">\(p=1-2(100)+5(170)=651\)</span> and for <span class="SimpleMath">\(\texttt{lp2}\)</span> is <span class="SimpleMath">\((x=200,y=80)\)</span> with optimal value <span class="SimpleMath">\(p=1-2(200)+5(80)=1\)</span>.</p>


<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap2_mj.html">[Previous Chapter]</a>    <a href="chap4_mj.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>

99%


[ zur Elbe Produktseite wechseln0.15Quellennavigators  Analyse erneut starten  ]