path: root/help/en_US/cbcintlinprog.xml

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<refentry version="5.0-subset Scilab" xml:id="cbcintlinprog" xml:lang="en"
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  <refnamediv>
    <refname>cbcintlinprog</refname>
    <refpurpose>Solves a mixed integer linear programming constrained optimization problem in intlinprog format.</refpurpose>
  </refnamediv>


<refsynopsisdiv>
   <title>Calling Sequence</title>
   <synopsis>
   xopt = cbcintlinprog(c,intcon,A,b)
   xopt = cbcintlinprog(c,intcon,A,b,Aeq,beq)
   xopt = cbcintlinprog(c,intcon,A,b,Aeq,beq,lb,ub)
   xopt = cbcintlinprog(c,intcon,A,b,Aeq,beq,lb,ub,options)
   xopt = cbcintlinprog('path_to_mps_file')
   xopt = cbcintlinprog('path_to_mps_file',options)
   [xopt,fopt,status,output] = cbcintlinprog( ... )
   
   </synopsis>
</refsynopsisdiv>

<refsection>
   <title>Input Parameters</title>
   <variablelist>
   <varlistentry><term>c :</term>
      <listitem><para> a vector of double, contains coefficients of the variables in the objective</para></listitem></varlistentry>
   <varlistentry><term>intcon :</term>
      <listitem><para> Vector of integer constraints, specified as a vector of positive integers. The values in intcon indicate the components of the decision variable x that are integer-valued. intcon has values from 1 through number of variable.</para></listitem></varlistentry>
   <varlistentry><term>A :</term>
      <listitem><para> A matrix of doubles, containing the coefficients of linear inequality constraints of size (m X n) where 'm' is the number of linear inequality constraints.</para></listitem></varlistentry>
   <varlistentry><term>b :</term>
      <listitem><para> A vector of doubles, related to 'A' and containing the the Right hand side equation of the linear inequality constraints of size (m X 1).</para></listitem></varlistentry>
   <varlistentry><term>Aeq :</term>
      <listitem><para> A matrix of doubles, containing the coefficients of linear equality constraints of size (m1 X n) where 'm1' is the number of linear equality constraints.</para></listitem></varlistentry>
   <varlistentry><term>beq :</term>
      <listitem><para> A vector of doubles, related to 'Aeq' and containing the the Right hand side equation of the linear equality constraints of size (m1 X 1).</para></listitem></varlistentry>
   <varlistentry><term>lb :</term>
      <listitem><para> A vector of doubles, containing the lower bounds of the variables of size (1 X n) or (n X 1) where 'n' is the number of variables.</para></listitem></varlistentry>
   <varlistentry><term>ub :</term>
      <listitem><para> A vector of doubles, containing the upper bounds of the variables of size (1 X n) or (n X 1) where 'n' is the number of variables.</para></listitem></varlistentry>
   <varlistentry><term>options :</term>
      <listitem><para> A list, containing the option for user to specify. See below for details.</para></listitem></varlistentry>
      </variablelist>
</refsection>
<refsection>
<title> Outputs</title>
 <variablelist>
   <varlistentry><term>xopt :</term>
      <listitem><para> A vector of doubles, containing the computed solution of the optimization problem.</para></listitem></varlistentry>
   <varlistentry><term>fopt :</term>
      <listitem><para> A double, containing the the function value at x.</para></listitem></varlistentry>
   <varlistentry><term>status :</term>
      <listitem><para> An integer, containing the flag which denotes the reason for termination of algorithm. See below for details.</para></listitem></varlistentry>
   <varlistentry><term>output :</term>
      <listitem><para> A structure, containing the information about the optimization. See below for details.</para></listitem></varlistentry>
   </variablelist>
</refsection>

<refsection>
   <title>Description</title>
   <para>
Search the minimum or maximum of a constrained mixed integer linear programming optimization problem specified by :
   </para>
   <para>
<latex>
\begin{eqnarray}
&amp;\mbox{min}_{x}
&amp; C^T⋅x \\
&amp; \text{Subjected to:} &amp; A⋅x \leq b \\
&amp; &amp; Aeq⋅x = beq \\
&amp; &amp; lb \leq x \leq ub \\
&amp; &amp; x_i \in \!\, \mathbb{Z}, i \in \!\, intcon\\
\end{eqnarray}
</latex>
</para>
   <para>
CBC, an optimization library written in C++, is used for solving the linear programming problems.

   </para>
<para>
<title>Options</title>
The options allow the user to set various parameters of the Optimization problem. The syntax for the options is given by:
   </para>
   <para>
options= list("IntegerTolerance", [---], "MaxNodes",[---], "MaxIter", [---], "AllowableGap",[---] "CpuTime", [---],"gradobj", "off", "hessian", "off" );
<itemizedlist>
<listitem>IntegerTolerance : A Scalar, a number with that value of an integer is considered integer.</listitem>
<listitem>MaxNodes : A Scalar, containing the maximum number of nodes that the solver should search.</listitem>
<listitem>MaxTime : A scalar, specifying the maximum amount of CPU Time in seconds that the solver should take.</listitem>
<listitem>AllowableGap : A scalar, that specifies the gap between the computed solution and the the objective value of the best known solution stop, at which the tree search can be stopped.</listitem>
</itemizedlist>
   The default values for the various items are given as:
   </para>
   <para>
  options = list('integertolerance',1d-06,'maxnodes',2147483647,'cputime',1d10,'allowablegap')
   </para>
<para>
The exitflag allows the user to know the status of the optimization which is returned by OSI-CBC. The values it can take and what they indicate is described below:
<itemizedlist>
<listitem> 0 : Optimal Solution Found </listitem>
<listitem> 1 : Converged to a point of primal infeasibility.</listitem>
<listitem> 2 : Solution Limit is reached</listitem>
<listitem> 3 : Node Limit is reached. Output may not be optimal.</listitem>
<listitem> 4 : Numerical Difficulties.</listitem>
<listitem> 5 : Maximum amount of CPU Time exceeded. </listitem>
<listitem> 6 : Continuous Solution Unbounded.</listitem>
<listitem> 7 : Converged to a point of dual infeasibility.</listitem>
</itemizedlist>
   </para>
   <para>
For more details on exitflag, see the Bonmin documentation which can be found on http://www.coin-or.org/Cbc
   </para>
   <para>
</para>

</refsection>

<para>
A few examples displaying the various functionalities of cbcintlinprog have been provided below. You will find a series of problems and the appropriate code snippets to solve them.
   </para>

<refsection>
   <title>Example</title>
   <para>
 Here we solve a simple objective function, subjected to three linear inequality constraints.
   </para>
   <para>
Find x in R^8 such that it minimizes:
   </para>
   <para>
<latex>
  \begin{eqnarray}
\mbox{min}_{x}\ f(x) = 1750x_{1} + 990x_{2} + 1240x_{3} + 1680x_{4} + 500x_{5} + 450x_{6} + 400x_{7} + 100x_{8} \\
\end{eqnarray}\\
\text{Subjected to:}\\
\begin{eqnarray}
\hspace{70pt} &amp;6x_{1} + 4.25x_{2} + 5.5x_{3} + 7.75x_{4} + 3x_{5} + 3.25x_{6} + 3.5x_{7} + 3.75x_{8}&amp;\leq 100\\ 
\hspace{70pt} &amp;1.25x_{1} + 1.37x_{2} + 1.7x_{3} + 1.93x_{4} + 2.08x_{5} + 2.32x_{6} + 2.56x_{7} + 2.78x_{8}&amp;\leq 205\\ 
\hspace{70pt} &amp;1.15x_{1} + 1.34x_{2} + 1.66x_{3} + 1.99x_{4} + 2.06x_{5} + 2.32x_{6} + 2.58x_{7} + 2.84x_{8}&amp;\leq 409\\ 
\end{eqnarray}\\
\text{With integer constraints as: } 
\begin{eqnarray}
\begin{array}{cccccc}
[1 &amp; 2 &amp; 3 &amp; 4] \\
\end{array}
\end{eqnarray}
  </latex>
</para>
   <programlisting role="example"><![CDATA[
// Example 1:
// Reference: Westerberg, Carl-Henrik, Bengt Bjorklund, and Eskil Hultman. "An application of mixed integer
// programming in a Swedish steel mill." Interfaces 7, no. 2 (1977): 39-43. Modified acc. to requirements.
c = [350*5,330*3,310*4,280*6,500,450,400,100]';
A = [6,4.25, 5.5, 7.75, 3, 3.25, 3.5,3.75; 
    1.25,1.37,1.7,1.93,2.08,2.32,2.56,2.78;  
    1.15,1.34,1.66,1.99,2.06,2.32,2.58,2.84 ];
b = [100 ,205, 249 ];
//Defining the integer constraints
intcon = [1 2 3 4];
// Calling Symphony
[x,f,status,output] = cbcintlinprog(c,intcon,A,b)
// Press ENTER to continue

   ]]></programlisting>
</refsection>

<refsection>
   <title>Example</title>
   <para>
Here we build up on the previous example by adding upper and lower bounds to the variables.
We add the following bounds to the problem specified above:   </para>
   <para>
<latex>
\begin{eqnarray}
0 &amp;\leq x_{1} &amp;\leq 1\\
0 &amp;\leq x_{2} &amp;\leq 1\\
0 &amp;\leq x_{3} &amp;\leq 1\\
0 &amp;\leq x_{4} &amp;\leq 1\\
0 &amp;\leq x_{5} &amp;\leq \infty\\
0 &amp;\leq x_{6} &amp;\leq \infty\\
0 &amp;\leq x_{7} &amp;\leq \infty\\
0 &amp;\leq x_{8} &amp;\leq \infty 
\end{eqnarray}
  </latex>
</para>
   <programlisting role="example"><![CDATA[
// Example 2:
// Reference: Westerberg, Carl-Henrik, Bengt Bjorklund, and Eskil Hultman. "An application of mixed integer
// programming in a Swedish steel mill." Interfaces 7, no. 2 (1977): 39-43. Modified acc. to requirements.
  c = [350*5,330*3,310*4,280*6,500,450,400,100]';
  //Inequality constraints
  A = [6,4.25, 5.5, 7.75, 3, 3.25, 3.5,3.75; 
      1.25,1.37,1.7,1.93,2.08,2.32,2.56,2.78;  
      1.15,1.34,1.66,1.99,2.06,2.32,2.58,2.84 ];
  b = [100 ,205, 249 ];

  // Lower Bound of variable
lb = repmat(0,1,8);
// Upper Bound of variables
ub = [repmat(1,1,4) repmat(%inf,1,4)];
  //Integer Constraints
  intcon = [1 2 3 4];
  // Calling Symphony
  [x,f,status,output] = cbcintlinprog(c,intcon,A,b,[],[],lb,ub)
// Press ENTER to continue

   ]]></programlisting>
</refsection>

<refsection>
   <title>Example</title>
   <para>
In this example, we proceed to add the linear equality constraints to the objective function.
   </para>
   
   <para>
<latex>
\begin{eqnarray}
&amp;5x_{1} + 3x_{2} + 4x_{3} + 6x_{4} + x_{5} + x_{6} + x_{7} + x_{8}&amp;= 25\\ 
&amp;0.25x_{1} + 0.12x_{2} + 0.2x_{3} + 0.18x_{4} + 0.08x_{5} + 0.07x_{6} + 0.06x_{7} + 0.03x_{8}&amp;= 1.25\\ 
&amp;0.15x_{1} + 0.09x_{2} + 0.16x_{3} + 0.24x_{4} + 0.06x_{5} + 0.07x_{6} + 0.08x_{7} + 0.09x_{8}&amp;= 1.25\\ 
\end{eqnarray}
  </latex>
</para>
   <programlisting role="example"><![CDATA[
// Example 3:
// Reference: Westerberg, Carl-Henrik, Bengt Bjorklund, and Eskil Hultman. "An application of mixed integer
// programming in a Swedish steel mill." Interfaces 7, no. 2 (1977): 39-43. Modified acc. to requirements.
c = [350*5,330*3,310*4,280*6,500,450,400,100]';
//Inequality constraints
A = [6,4.25, 5.5, 7.75, 3, 3.25, 3.5,3.75; 
    1.25,1.37,1.7,1.93,2.08,2.32,2.56,2.78;  
    1.15,1.34,1.66,1.99,2.06,2.32,2.58,2.84 ];
b = [100 ,205, 249 ];
// Lower Bound of variable
lb = repmat(0,1,8);
// Upper Bound of variables
ub = [repmat(1,1,4) repmat(%inf,1,4)];
// Equality Constraints
Aeq = [5,3,4,6,1,1,1,1;
5*0.05,3*0.04,4*0.05,6*0.03,0.08,0.07,0.06,0.03;
5*0.03,3*0.03,4*0.04,6*0.04,0.06,0.07,0.08,0.09;]
beq = [ 25, 1.25, 1.25];
  //Integer Constraints
intcon = [1 2 3 4];
// Calling CBC
[x,f,status,output] = cbcintlinprog(c,intcon,A,b,Aeq,beq,lb,ub)
// Press ENTER to continue

   ]]></programlisting>
</refsection>

<refsection>
   <title>Example</title>
   <para>
Primal Infeasible Problems: Find x in R^8 such that it minimizes:
  </para>
   <para>
Find x in R^8 such that it minimizes:
   </para>
   <para>
<latex>
  \begin{eqnarray}
\mbox{min}_{x}\ f(x) = 1750x_{1} + 990x_{2} + 1240x_{3} + 1680x_{4} + 500x_{5} + 450x_{6} + 400x_{7} + 100x_{8} \\
\end{eqnarray}\\
\text{Subjected to:}\\
\begin{eqnarray}
\hspace{70pt} &amp;6x_{1} + 4.25x_{2} + 5.5x_{3} + 7.75x_{4} + 3x_{5} + 3.25x_{6} + 3.5x_{7} + 3.75x_{8}&amp;\leq 26.333\\ 
\hspace{70pt} &amp;1.25x_{1} + 1.37x_{2} + 1.7x_{3} + 1.93x_{4} + 2.08x_{5} + 2.32x_{6} + 2.56x_{7} + 2.78x_{8}&amp;\leq 3.916\\ 
\hspace{70pt} &amp;1.15x_{1} + 1.34x_{2} + 1.66x_{3} + 1.99x_{4} + 2.06x_{5} + 2.32x_{6} + 2.58x_{7} + 2.84x_{8}&amp;\leq 5.249\\ 
\end{eqnarray}\\
\text{With integer constraints as: } 
\begin{eqnarray}
\begin{array}{cccc}
[1 &amp; 2 &amp; 3 &amp; 4] 
\end{array}
\end{eqnarray}
  </latex>
</para>
   <programlisting role="example"><![CDATA[
// Example 4:
// Reference: Westerberg, Carl-Henrik, Bengt Bjorklund, and Eskil Hultman. "An application of mixed integer 
// programming in a Swedish steel mill." Interfaces 7, no. 2 (1977): 39-43. Modified acc. to requirements.
c = [350*5,330*3,310*4,280*6,500,450,400,100]';
//Inequality constraints 
A = [6,4.25, 5.5, 7.75, 3, 3.25, 3.5,3.75; 
    1.25,1.37,1.7,1.93,2.08,2.32,2.56,2.78;  
    1.15,1.34,1.66,1.99,2.06,2.32,2.58,2.84 ]; 
b = [26.333 ,3.916 ,5.249 ];
  //Integer Constraints
intcon = [1 2 3 4];
 
// Calling CBC
 
[x,f,status,output] = cbcintlinprog(c,intcon,A,b)
// Press ENTER to continue

   ]]></programlisting>
</refsection>

<refsection>
   <title>Example</title>
   <para>
Unbounded Problems. Find x in R^8 such that it minimizes:
   </para>
   <para>
<latex>
 \begin{eqnarray}
\mbox{min}_{x}\ f(x) = 1750x_{1} + 990x_{2} + 1240x_{3} + 1680x_{4} + 500x_{5} + 450x_{6} + 400x_{7} + 100x_{8} \\
\end{eqnarray}\\
\text{Subjected to:}\\
\begin{eqnarray}
\hspace{70pt} &amp;5x_{1} + 3x_{2} + 4x_{3} + 6x_{4} + x_{5} + x_{6} + x_{7} + x_{8}&amp;= 25\\ 
\hspace{70pt} &amp;0.25x_{1} + 0.12x_{2} + 0.2x_{3} + 0.18x_{4} + 0.08x_{5} + 0.07x_{6} + 0.06x_{7} + 0.03x_{8}&amp;= 1.25\\ 
\hspace{70pt} &amp;0.15x_{1} + 0.09x_{2} + 0.16x_{3} + 0.24x_{4} + 0.06x_{5} + 0.07x_{6} + 0.08x_{7} + 0.09x_{8}&amp;= 1.25\\
\end{eqnarray}\\
\text{With integer constraints as: } 
\begin{eqnarray}
\begin{array}{cccccc}
[1 &amp; 2 &amp; 3 &amp; 4] \\
\end{array}
\end{eqnarray}
  </latex>
</para>
   <programlisting role="example"><![CDATA[
// Example 5:
// Reference: Westerberg, Carl-Henrik, Bengt Bjorklund, and Eskil Hultman. "An application of mixed integer
// programming in a Swedish steel mill." Interfaces 7, no. 2 (1977): 39-43. Modified acc. to requirements.
c = [350*5,330*3,310*4,280*6,500,450,400,100]';
//Inequality constraints
A = [];
b = [];
// Equality Constraints
Aeq = [5,3,4,6,1,1,1,1;
5*0.05,3*0.04,4*0.05,6*0.03,0.08,0.07,0.06,0.03;
5*0.03,3*0.03,4*0.04,6*0.04,0.06,0.07,0.08,0.09;]
beq = [ 25, 1.25, 1.25];
  //Integer Constraints
intcon = [1 2 3 4];
// Calling CBC
[x,f,status,output] = cbcintlinprog(c,intcon,A,b,Aeq,beq)
// Press ENTER to continue

   ]]></programlisting>
</refsection>

<refsection>
   <title>Authors</title>
   <simplelist type="vert">
   <member>Akshay Miterani and Pranav Deshpande</member>
   </simplelist>
</refsection>
</refentry>