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<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>
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<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>
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<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>


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