<!doctype html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="X-UA-Compatible" content="IE=Edge" />
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
<title>Creating a mathematical programming model — DOcplex.MP: Mathematical Programming Modeling for Python V2.16 documentation</title>
<link rel="stylesheet" href="_static/bizstyle.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
<script type="text/javascript" id="documentation_options" data-url_root="./" src="_static/documentation_options.js"></script>
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
<script type="text/javascript" src="_static/language_data.js"></script>
<script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script type="text/javascript" src="_static/bizstyle.js"></script>
<link rel="index" title="Index" href="genindex.html" />
<link rel="search" title="Search" href="search.html" />
<link rel="next" title="Examples of mathematical programming" href="samples.html" />
<link rel="prev" title="Setting up an optimization engine" href="getting_started.html" />
<meta name="viewport" content="width=device-width,initial-scale=1.0">
<!--[if lt IE 9]>
<script type="text/javascript" src="_static/css3-mediaqueries.js"></script>
<![endif]-->
</head><body>
<div class="related" role="navigation" aria-label="related navigation">
<h3>Navigation</h3>
<ul>
<li class="right" style="margin-right: 10px">
<a href="genindex.html" title="General Index"
accesskey="I">index</a></li>
<li class="right" >
<a href="py-modindex.html" title="Python Module Index"
>modules</a> |</li>
<li class="right" >
<a href="samples.html" title="Examples of mathematical programming"
accesskey="N">next</a> |</li>
<li class="right" >
<a href="getting_started.html" title="Setting up an optimization engine"
accesskey="P">previous</a> |</li>
<li class="nav-item nav-item-0"><a href="index.html">DOcplex.MP: Mathematical Programming Modeling for Python V2.16 documentation</a> »</li>
</ul>
</div>
<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
<div class="sphinxsidebarwrapper">
<h3><a href="index.html">Table of Contents</a></h3>
<ul>
<li><a class="reference internal" href="#">Creating a mathematical programming model</a><ul>
<li><a class="reference internal" href="#define-model-decision-variables">Define model decision variables</a></li>
<li><a class="reference internal" href="#build-model-expressions">Build model expressions</a></li>
<li><a class="reference internal" href="#aggregated-expressions">Aggregated expressions</a></li>
<li><a class="reference internal" href="#building-constraints">Building constraints</a></li>
<li><a class="reference internal" href="#build-a-model">Build a model</a><ul>
<li><a class="reference internal" href="#import-necessary-modules">Import necessary modules</a></li>
<li><a class="reference internal" href="#solving-parameters">Solving parameters</a></li>
</ul>
</li>
<li><a class="reference internal" href="#retrieve-results">Retrieve results</a></li>
<li><a class="reference internal" href="#generate-lp-file">Generate LP file</a></li>
</ul>
</li>
</ul>
<h4>Previous topic</h4>
<p class="topless"><a href="getting_started.html"
title="previous chapter">Setting up an optimization engine</a></p>
<h4>Next topic</h4>
<p class="topless"><a href="samples.html"
title="next chapter">Examples of mathematical programming</a></p>
<div id="searchbox" style="display: none" role="search">
<h3>Quick search</h3>
<div class="searchformwrapper">
<form class="search" action="search.html" method="get">
<input type="text" name="q" />
<input type="submit" value="Go" />
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
</div>
</div>
<script type="text/javascript">$('#searchbox').show(0);</script>
</div>
</div>
<div class="document">
<div class="documentwrapper">
<div class="bodywrapper">
<div class="body" role="main">
<div class="section" id="creating-a-mathematical-programming-model">
<h1>Creating a mathematical programming model<a class="headerlink" href="#creating-a-mathematical-programming-model" title="Permalink to this headline">¶</a></h1>
<p>Building a model requires:</p>
<blockquote>
<div><ul class="simple">
<li>defining decision variables and their scopes (what are the possible values for these variables),</li>
<li>creating constraints from variables to express interactions between variables and business limitations; only variable values which satisfy the constraints are possible,</li>
<li>adding constraints in a model, and</li>
<li>defining what is the objective to optimize. The objective is a numerical criterion which is used to rank possible solutions. Mathematical programming algorithms aim to return the best possible solution. This step is optional: if no objective is defined, the algorithm returns one feasible solution.</li>
</ul>
</div></blockquote>
<p>The folder <code class="docutils literal notranslate"><span class="pre">Examples</span></code> contains a set of <a class="reference internal" href="samples.html"><span class="doc">examples</span></a> that can be used as a starting point to create a new model.</p>
<p>The mathematical programming elements are implemented in the Python modules located in <code class="docutils literal notranslate"><span class="pre">docplex/mp</span></code>.
The factory used to create constraints, manipulate the expressions, and so on is described <a class="reference external" href="docplex.mp.model.html">in the DOcplex.MP reference manual</a>.</p>
<div class="section" id="define-model-decision-variables">
<h2>Define model decision variables<a class="headerlink" href="#define-model-decision-variables" title="Permalink to this headline">¶</a></h2>
<p>Decision variables are created using factory methods on the <cite>Model</cite> class. The <cite>Model</cite> can create single variables, lists of variables, and dictionaries of variables indexed by business objects.
Here is a table of the standard factory methods to create variables:</p>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="42%" />
<col width="58%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">Function</th>
<th class="head">Creates</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><em>binary_var()</em></td>
<td>Single binary variable</td>
</tr>
<tr class="row-odd"><td><em>binary_var_list()</em></td>
<td>List of binary variables</td>
</tr>
<tr class="row-even"><td><em>binary_var_dict()</em></td>
<td>Dictionary of binary variables</td>
</tr>
<tr class="row-odd"><td><em>binary_var_matrix()</em></td>
<td>Matrix of binary variables</td>
</tr>
<tr class="row-even"><td><em>integer_var()</em></td>
<td>Single integer variable</td>
</tr>
<tr class="row-odd"><td><em>integer_var_list()</em></td>
<td>List of integer variables</td>
</tr>
<tr class="row-even"><td><em>integer_var_dict()</em></td>
<td>Dictionary of integer variables</td>
</tr>
<tr class="row-odd"><td><em>integer_var_matrix()</em></td>
<td>Matrix of integer variables</td>
</tr>
<tr class="row-even"><td><em>continuous_var()</em></td>
<td>Single continuous variable</td>
</tr>
<tr class="row-odd"><td><em>continuous_var_list()</em></td>
<td>List of continuous variables</td>
</tr>
<tr class="row-even"><td><em>continuous_var_dict()</em></td>
<td>Dictionary of continuous variables</td>
</tr>
<tr class="row-odd"><td><em>continuous_var_matrix()</em></td>
<td>Matrix of continuous variables</td>
</tr>
</tbody>
</table>
</div></blockquote>
<p>There are three types of decision variables according to their scope of possible values: binary variables (0 or 1),
integer variables, or continuous variables. The detailed attributes for variables can be found in the class <cite>Var</cite> in
the module <a class="reference external" href="docplex.mp.linear.html">linear.py</a>.</p>
</div>
<div class="section" id="build-model-expressions">
<h2>Build model expressions<a class="headerlink" href="#build-model-expressions" title="Permalink to this headline">¶</a></h2>
<p>Constraints in mathematical programming are built with linear combinations of decision variables, sums
of elementary expressions of the form <cite>k *x</cite> where <cite>k</cite> is a number and <cite>x</cite> is a variable.</p>
<p>Python arithmetic operators (+,-,*,/) are overloaded to create expressions in a simple manner;
for example, if <cite>x</cite>, <cite>y</cite>, <cite>z</cite> are decision variables, <cite>3*x+5*y+7*z</cite> is an expression.</p>
</div>
<div class="section" id="aggregated-expressions">
<h2>Aggregated expressions<a class="headerlink" href="#aggregated-expressions" title="Permalink to this headline">¶</a></h2>
<p>DOcplex.MP allows the creation of large expressions over collections of variables by using the <cite>Model.sum</cite> method. Though Python’s built-in <cite>sum()</cite> function can also be used, <cite>Model.sum()</cite> is much faster for building larger expressions.
Aggregated expressions can also be used to build constraints.</p>
</div>
<div class="section" id="building-constraints">
<h2>Building constraints<a class="headerlink" href="#building-constraints" title="Permalink to this headline">¶</a></h2>
<p>To simplify the writing of a model, Python comparison operators (==,<=,>=) are also overloaded to
compare expressions and build constraints that must be satisfied by the decision variables.
For example, <cite>x+y+z == 1</cite> is a constraint that forces the sum of all three variables to be equal to 1.</p>
<p>Explicit methods are also available on the model object to ease their creation,
such as <em>eq_constraint</em>, <em>le_constraint</em>…</p>
</div>
<div class="section" id="build-a-model">
<h2>Build a model<a class="headerlink" href="#build-a-model" title="Permalink to this headline">¶</a></h2>
<p>The mathematical programming model itself is represented by the class <em>Model</em> implemented in the module <a class="reference external" href="docplex.mp.model.html">model.py</a>.</p>
<p>A constraint is added to the model by calling the method <em>add_constraint()</em> with the constraint as the parameter,
and, possibly, an optional string argument to name the constraint.
A constraint is active only if it has been added to the model.</p>
<div class="section" id="import-necessary-modules">
<h3>Import necessary modules<a class="headerlink" href="#import-necessary-modules" title="Permalink to this headline">¶</a></h3>
<p>The following is a condensed example of a sudoku problem that uses the default import policy.
More comments are available in the files in the directory <code class="docutils literal notranslate"><span class="pre">docplex/mp/examples</span></code>.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">docplex.mp.model</span> <span class="k">import</span> <span class="n">Model</span>
<span class="n">myInput</span> <span class="o">=</span><span class="p">[[</span><span class="mi">8</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Model</span><span class="p">(</span><span class="s2">"sudoku"</span><span class="p">)</span>
<span class="n">R</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="n">idx</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">R</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">R</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">R</span><span class="p">]</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">binary_var_dict</span><span class="p">(</span><span class="n">idx</span><span class="p">,</span> <span class="s2">"X"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">R</span><span class="p">:</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">R</span><span class="p">:</span>
<span class="k">if</span> <span class="n">myInput</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">model</span><span class="o">.</span><span class="n">add_constraint</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">myInput</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">R</span><span class="p">:</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">R</span><span class="p">:</span>
<span class="n">model</span><span class="o">.</span><span class="n">add_constraint</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">R</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">R</span><span class="p">:</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">R</span><span class="p">:</span>
<span class="n">model</span><span class="o">.</span><span class="n">add_constraint</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">R</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">R</span><span class="p">:</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">R</span><span class="p">:</span>
<span class="n">model</span><span class="o">.</span><span class="n">add_constraint</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">R</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">solution</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
<span class="n">solution</span><span class="o">.</span><span class="n">print_information</span><span class="p">()</span>
</pre></div>
</div>
<p>The <em>solve()</em> method returns an object of class <a class="reference external" href="docplex.mp.solution.html#docplex.mp.solution.SolveSolution">SolveSolution</a> that contains the result of solving, or None if the model has no solution.
This object is described in the section “Retrieve results”.</p>
<p>The method <em>print_information()</em> prints a default view of the status of the solve and the values of all variables.
The object <em>SolveSolution</em> contains all the necessary accessors to create a customized solution output.</p>
</div>
<div class="section" id="solving-parameters">
<h3>Solving parameters<a class="headerlink" href="#solving-parameters" title="Permalink to this headline">¶</a></h3>
<p>Solving parameters can be adjusted using the “parameters” attribute of the model. Parameters implement a hierarchical tree of attributes reflecting the parameter hierarchy of CPLEX.
For example, use <em>model.parameters.mip.tolerances.mip_gap = 0.05</em> to set the MIP gap to 5% before solve.</p>
</div>
</div>
<div class="section" id="retrieve-results">
<h2>Retrieve results<a class="headerlink" href="#retrieve-results" title="Permalink to this headline">¶</a></h2>
<p>Results from the solve are returned in a data structure of the class <em>SolveSolution</em>, implemented in the module <cite>SolveSolution</cite>.
This object contains:</p>
<blockquote>
<div><ul class="simple">
<li>global model information, such as status of the search, value of the objective, and</li>
<li>the value of each variable</li>
</ul>
</div></blockquote>
<p>Many shortcuts are available to write simpler code.</p>
<blockquote>
<div><ul class="simple">
<li>As <cite>solve()</cite> returns None if the model has no solution, one can test directly if a solution is present.</li>
<li><dl class="first docutils">
<dt>A simplified Python value for each object is directly accessible by using square brackets (<em>msol[vname]</em>). The result is:</dt>
<dd><ul class="first last">
<li>an integer for integer variables and</li>
<li>a float for continuous variables.</li>
</ul>
</dd>
</dl>
</li>
</ul>
</div></blockquote>
<p>The following code is an example of solution printing for the NQueen example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sys</span> <span class="k">import</span> <span class="n">stdout</span>
<span class="k">if</span> <span class="n">msol</span><span class="p">:</span>
<span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s2">"Solution:"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
<span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s2">" "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">msol</span><span class="p">[</span><span class="n">v</span><span class="p">]))</span>
<span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s2">"Solve status: "</span> <span class="o">+</span> <span class="n">msol</span><span class="o">.</span><span class="n">get_solve_status</span><span class="p">()</span> <span class="o">+</span> <span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="generate-lp-file">
<h2>Generate LP file<a class="headerlink" href="#generate-lp-file" title="Permalink to this headline">¶</a></h2>
<p>The generation of the LP file corresponding to a model is made available by calling the method <em>export_as_lp()</em>,
as demonstrated in the following example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">mdl</span> <span class="o">=</span> <span class="n">Model</span><span class="p">()</span>
<span class="o">.</span> <span class="o">.</span> <span class="o">.</span> <span class="o">.</span> <span class="o">.</span>
<span class="o"><</span><span class="n">Construction</span> <span class="n">of</span> <span class="n">the</span> <span class="n">model</span><span class="o">></span>
<span class="o">.</span> <span class="o">.</span> <span class="o">.</span> <span class="o">.</span> <span class="o">.</span>
<span class="n">mdl</span><span class="o">.</span><span class="n">export_as_lp</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="clearer"></div>
</div>
<div class="related" role="navigation" aria-label="related navigation">
<h3>Navigation</h3>
<ul>
<li class="right" style="margin-right: 10px">
<a href="genindex.html" title="General Index"
>index</a></li>
<li class="right" >
<a href="py-modindex.html" title="Python Module Index"
>modules</a> |</li>
<li class="right" >
<a href="samples.html" title="Examples of mathematical programming"
>next</a> |</li>
<li class="right" >
<a href="getting_started.html" title="Setting up an optimization engine"
>previous</a> |</li>
<li class="nav-item nav-item-0"><a href="index.html">DOcplex.MP: Mathematical Programming Modeling for Python V2.16 documentation</a> »</li>
</ul>
</div>
<div class="footer" role="contentinfo">
© Copyright 2016-2019, IBM®.
</div>
</body>
</html>