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<h1 class="chapter"> 64. simplex </h1>
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<tr><td align="left" valign="top"><a href="#SEC231">64.2 Definitions for simplex</a></td><td> </td><td align="left" valign="top">
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<h2 class="section"> 64.1 Introduction to simplex </h2>
<p><code>simplex</code> is a package for linear optimization using the simplex algorithm.
</p>
<p>Example:
</p>
<table><tr><td> </td><td><pre class="example">(%i1) load("simplex")$
(%i2) minimize_sx(x+y, [3*x+2*y>2, x+4*y>3]);
9 7 1
(%o2) [--, [y = --, x = -]]
10 10 5
</pre></td></tr></table>
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<h2 class="section"> 64.2 Definitions for simplex </h2>
<dl>
<dt><u>Option variable:</u> <b>epsilon_sx</b>
<a name="IDX1832"></a>
</dt>
<dd><p>Default value: <code>10^-8</code>
</p>
<p>Epsilon used for numerical computations in <code>linear_program</code>.
</p>
<p>See also: <code>linear_program</code>.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>linear_program</b><i> (<var>A</var>, <var>b</var>, <var>c</var>)</i>
<a name="IDX1833"></a>
</dt>
<dd><p><code>linear_program</code> is an implementation of the simplex algorithm.
<code>linear_program(A, b, c)</code> computes a vector <var>x</var> for which <code>c.x</code> is minimum
possible among vectors for which <code>A.x = b</code> and <code>x >= 0</code>. Argument
<var>A</var> is a matrix and arguments <var>b</var> and <var>c</var> are lists.
</p>
<p><code>linear_program</code> returns a list which contains the minimizing vector <var>x</var> and the
minimum value <code>c.x</code>. If the problem is not bounded, it returns "Problem not bounded!" and
if the problem is not feasible, it returns "Problem not feasible!".
</p>
<p>To use this function first load the <code>simplex</code> package with <code>load(simplex);</code>.
</p>
<p>Example:
</p>
<table><tr><td> </td><td><pre class="example">(%i2) A: matrix([1,1,-1,0], [2,-3,0,-1], [4,-5,0,0])$
(%i3) b: [1,1,6]$
(%i4) c: [1,-2,0,0]$
(%i5) linear_program(A, b, c);
13 19 3
(%o5) [[--, 4, --, 0], - -]
2 2 2
</pre></td></tr></table>
<p>See also: <code>minimize_sx</code>, <code>scale_sx</code>, and <code>epsilon_sx</code>.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>maximize_sx</b><i> (<var>obj</var>, <var>cond</var>, [<var>pos</var>])</i>
<a name="IDX1834"></a>
</dt>
<dd><p>Maximizes linear objective function <var>obj</var> subject to some linear constraints
<var>cond</var>. See <code>minimize_sx</code> for detailed description of arguments and return
value.
</p>
<p>See also: <code>minimize_sx</code>.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>minimize_sx</b><i> (<var>obj</var>, <var>cond</var>, [<var>pos</var>])</i>
<a name="IDX1835"></a>
</dt>
<dd><p>Minimizes a linear objective function <var>obj</var> subject to some linear
constraints <var>cond</var>. <var>cond</var> a list of linear equations or
inequalities. In strict inequalities <code>></code> is replaced by <code>>=</code>
and <code><</code> by <code><=</code>. The optional argument <var>pos</var> is a list of
decision variables which are assumed to be positive.
</p>
<p>If the minimum exists, <code>minimize_sx</code> returns a list which contains
the minimum value of the objective function and a list of decision variable
values for which the minimum is attained. If the problem is not bounded,
<code>minimize_sx</code> returns "Problem not bounded!" and if the problem
is not feasible, it returns "Ploblem not feasible!".
</p>
<p>The decision variables are not assumed to be nonegative by default. If all
decision variables are nonegative, set <code>nonegative_sx</code> to <code>true</code>.
If only some of decision variables are positive, list them in the optional
argument <var>pos</var> (note that this is more efficient than adding
constraints).
</p>
<p><code>minimize_sx</code> uses the simplex algorithm which is implemented in maxima
<code>linear_program</code> function.
</p>
<p>To use this function first load the <code>simplex</code> package with <code>load(simplex);</code>.
</p>
<p>Examples:
</p>
<table><tr><td> </td><td><pre class="example">(%i1) minimize_sx(x+y, [3*x+y=0, x+2*y>2]);
4 6 2
(%o1) [-, [y = -, x = - -]]
5 5 5
(%i2) minimize_sx(x+y, [3*x+y>0, x+2*y>2]), nonegative_sx=true;
(%o2) [1, [y = 1, x = 0]]
(%i3) minimize_sx(x+y, [3*x+y=0, x+2*y>2]), nonegative_sx=true;
(%o3) Problem not feasible!
(%i4) minimize_sx(x+y, [3*x+y>0]);
(%o4) Problem not bounded!
</pre></td></tr></table>
<p>See also: <code>maximize_sx</code>, <code>nonegative_sx</code>, <code>epsilon_sx</code>.
</p>
</dd></dl>
<dl>
<dt><u>Option variable:</u> <b>nonegative_sx</b>
<a name="IDX1836"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>If <code>nonegative_sx</code> is true all decision variables to <code>minimize_sx</code>
and <code>maximize_sx</code> are assumed to be positive.
</p>
<p>See also: <code>minimize_sx</code>.
</p>
</dd></dl>
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