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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<html>
<HEAD>
<TITLE>Using lpsolve from Euler</TITLE>
<style TYPE="text/css"> BODY { font-family:verdana,arial,helvetica; margin:15; }
</style>
</HEAD>
<BODY>
<h1 align="left"><u>Using lpsolve from Euler</u></h1>
<a name="Euler"></a>
<h3>Euler?</h3>
<p>The Euler Mathematical Toolbox is a powerful, versatile, and open source
software for numerical and symbolic computations written and maintained by
<a href="http://www.rene-grothmann.de/">R. Grothmann</a> from the <a href="http://www.ku-eichstaett.de">University of Eichstätt</a>. Euler is similar to <a href="http://www.mathworks.com/access/helpdesk/help/techdoc/">MATLAB</a>, but uses an own style
and an own syntax. Euler supports symbolic mathematics using the open algebra system
<a href="http://maxima.sourceforge.net/">Maxima</a>. </p>
<p>The most recent version of Euler runs in Windows (98/XP/Vista), or under
Linux in Wine. The native Linux version is currently outdated. </p>
<p>We will not discuss the specifics of Euler here but instead refer the reader to the
<a href="http://mathsrv.ku-eichstaett.de/MGF/homes/grothmann/euler/">Euler</a> website.
</p>
<a name="Euler_and_lpsolve"></a>
<h3>Euler and lpsolve</h3>
<p>lpsolve is callable from Euler via a dynamic linked DLL function. As such, it looks like lpsolve is fully integrated
with Euler. Matrices can directly be transferred between Euler and lpsolve in both directions. The complete interface
is written in C so it has maximum performance. The whole lpsolve API is implemented with some extra's specific for
Euler (especially for matrix support). So you have full control to the complete lpsolve functionality via the eulpsolve
Euler driver. If you find that this involves too much work to solve an lp model then you can also work via higher-level
Euler files that can make things a lot easier. See further in this article.
</p>
<a name="Quickstart"></a>
<h3>Quickstart</h3>
<pre>
Compile and build eulpsolve:
----------------------------
1. Under Windows, the Microsoft Visual C/C++ compiler must be installed
and the environment variables must be active do that when a command prompt
is opened, the cl and nmake commands can be executed.
2. Go to directory lp_solve_5.5\extra\Euler
3. Edit cvc.bat and change the path of the Eulerpath environment variable to your path.
4. To compile and build eulpsolve, enter the following command:
cvc
Load the eulpsolve driver in the Euler memory space:
-------------------------------------------------------
1. Under Windows, make sure that the lpsolve55.dll file is somewhere in the path
(archive lp_solve_5.5.2.14_dev.zip)
2. A precompiled library is provided for Windows (eulpsolve.dll).
3. Start Euler
4. Enter the following command in Euler:
>dll("<path>/eulpsolve.dll", "eulpsolve", -1)
Or if this fails (because you use version an Euler version 7.0 or older):
>dll("<path>/eulpsolve.dll", "eulpsolve", 0)
>dll("<path>/eulpsolve.dll", "eulpsolve", 1);
>dll("<path>/eulpsolve.dll", "eulpsolve", 2);
>dll("<path>/eulpsolve.dll", "eulpsolve", 3);
>dll("<path>/eulpsolve.dll", "eulpsolve", 4);
>dll("<path>/eulpsolve.dll", "eulpsolve", 5);
>dll("<path>/eulpsolve.dll", "eulpsolve", 6);
>dll("<path>/eulpsolve.dll", "eulpsolve", 7);
For <path> use the path where the eulpsolve.dll is located.
These commands can be put in a file, for example loadlpsolve.en, with a regular editor like notepad.
They can then be opened as a notebook in Euler and executed.
</pre>
<p>Euler is ideally suited to handle linear programming problems.
These are problems in which you have a quantity, depending linearly on several variables,
that you want to maximize or minimize subject to several constraints that are expressed
as linear inequalities in the same variables. If the number of variables and the number
of constraints are small, then there are numerous mathematical techniques for solving a
linear programming problem.
Indeed these techniques are often taught in high school or university level
courses in finite mathematics. But sometimes these numbers are high, or even if low,
the constants in the linear inequalities or the object expression for the quantity
to be optimized may be numerically complicated in which case a software package like
Euler is required to effect a solution.</p>
<a name="Installation"></a>
<h3>Installation</h3>
<p>To make this possible, a driver program is needed: eulpsolve (eulpsolve.dll under Windows).
This driver must be loaded in Euler and Euler can call the eulpsolve solver.</p>
<p>This driver calls lpsolve via the lpsolve shared library (lpsolve55.dll under Windows).
This has the advantage that the eulpsolve driver doesn't have to
be recompiled when an update of lpsolve is provided. The shared library must be somewhere in the Windows path.</p>
<p>So note the difference between the Euler lpsolve driver that is called eulpsolve and the lpsolve library that implements the
API that is called lpsolve55.</p>
<p>There are also some Euler notebooks (.en) as a quick start.</p>
<p>The first thing that must be done, each time Euler is restarted and you want to use lpsolve is load
the eulpsolve driver into the Euler workspace. This is done via the dll command. Suppose that eulpsolve.dll
is installed in c:\Program Files\Euler\dll, then the following command must be used to load the driver:</p>
<pre>>dll("c:/Program Files/Euler/dll/eulpsolve.dll", "eulpsolve", -1)</pre>
<p>Note that this command is only accepted from Euler version 7.1 or newer.
On older Euler versions, -1 is not accepted on the third argument.
There is however a work-around for this. You can then use the following commands (all together):</p>
<pre>>dll("c:/Program Files/Euler/dll/eulpsolve.dll", "eulpsolve", 0)
>dll("c:/Program Files/Euler/dll/eulpsolve.dll", "eulpsolve", 1);
>dll("c:/Program Files/Euler/dll/eulpsolve.dll", "eulpsolve", 2);
>dll("c:/Program Files/Euler/dll/eulpsolve.dll", "eulpsolve", 3);
>dll("c:/Program Files/Euler/dll/eulpsolve.dll", "eulpsolve", 4);
>dll("c:/Program Files/Euler/dll/eulpsolve.dll", "eulpsolve", 5);
>dll("c:/Program Files/Euler/dll/eulpsolve.dll", "eulpsolve", 6);
>dll("c:/Program Files/Euler/dll/eulpsolve.dll", "eulpsolve", 7);</pre>
<p>These commands can be put in a file, for example loadlpsolve.en or loadlpsolve.e, with a regular editor like notepad.
The .en file can then be opened as a notebook in Euler and executed. The .e file can be loaded and executed via the load statement.</p>
<p>That is basically all you need to do. From now on, you can use the library. This until Euler is restarted.
Then this command must be given again to reload the library.</p>
<p>To test if everything is installed correctly, enter eulpsolve(); in the Euler command prompt.
If it gives the following, then everything is ok:</p>
<pre>>eulpsolve();</pre>
<pre>eulpsolve Euler Interface version 5.5.0.7
using lpsolve version 5.5.2.14
Usage: [ret1, ret2, ...] = eulpsolve("functionname", arg1, arg2, ...)
</pre>
<p>However, if you get a message box with the following:</p>
<pre>eulpsolve no function or variable, or wrong argument number!</pre>
<p>Then either the dll command that was previous given was unsuccessful (or not given at all) or something was misspelled after the ,</p>
<p>Check the dll command again. If it gives a message box with the following message:</p>
<pre>---------------------------
euler.exe - Unable To Locate Component
---------------------------
This application has failed to start because lpsolve55.dll was not found. Re-installing the application may fix this problem.
---------------------------
OK
---------------------------</pre>
<p>And then in Euler:</p>
<pre> Could not open the DLL library!
Could not find the function!
error in :
dll("<path>/eulpsolve.dll", "eulpsolve", 0)</pre>
<p>Then Euler can find the eulpsolve driver program, but the driver program cannot find the lpsolve library
that contains the lpsolve implementation. This library is called lpsolve55.dll and should be on your system
in a directory that in the PATH environment variable. This path can be shown via the command line command PATH.</p>
<p>The lpsolve55.dll files must be in one of these specified directories. It is common to place this in the WINDOWS\system32 folder.</p>
<p>If the message in Euler is shown without a message box of a missing lpsolve55.dll file then eulpsolve.dll cannot be found.</p>
<p>All this is developed and tested with Euler versions 7.0 and 7.1 beta. This is the minimum supported release.
Older releases are unsupported.
Only from version 7.1 on, it is possible to print something in the Euler window from the external library.
So on older versions, when something must be printed, it will be shown in a messagebox.
If you are working with a lower version than 7.1 it is best that the verbose level of the <a href="set_verbose.htm">set_verbose</a> API call is less
than or equal to 3. Otherwise too much messageboxes will be given and it is unpractical to work with this.
If you need this verbose level, then use at least version 7.1
</p>
<a name="Solve_an_lp_model_from_Euler_via_eulpsolve"></a>
<h3>Solve an lp model from Euler via eulpsolve</h3>
<p>In the following text, > before the Euler commands is the Euler command line.
Only the text after > must be entered.
</p>
<p>To call an lpsolve function, the following syntax must be used:</p>
<pre>>{ret1, ret2, ...} = eulpsolve("functionname", arg1, arg2, ...)</pre>
<p>The return values are optional and depend on the function called. functionname must always be enclosed between double
quotes to make it alphanumerical and it is case sensitive. The number and type of arguments depend on the function called.
Some functions even have a variable number of arguments and a different behaviour occurs depending on the type of the argument.
functionname can be (almost) any of the lpsolve API routines (see <a href="lp_solveAPIreference.htm">lp_solve API reference</a>)
plus some extra Euler specific functions.
Most of the lpsolve API routines use or return an lprec structure. To make things more robust in Euler, this structure
is replaced by a handle or the model name. The lprec structures are maintained internally by the lpsolve driver.
The handle is an incrementing number starting from 0.
Starting from driver version 5.5.0.2, it is also possible to use the model name instead of the handle.
This can of course only be done if a name is given to the model. This is done via lpsolve routine
<a href="#set_lp_name">set_lp_name</a> or by specifying the model name in routine <a href="#read_lp">read_lp</a>.
See <a href="#Using_model_name_instead_of_handle">Using model name instead of handle</a>.
</p>
<p>Almost all callable functions can be found in the <a href="lp_solveAPIreference.htm">lp_solve API reference</a>.
Some are exactly as described in the reference guide, others have a slightly different syntax to make maximum
use of the Euler functionality. For example make_lp is used identical as described. But get_variables is slightly
different. In the API reference, this function has two arguments. The first the lp handle and the second the
resulting variables and this array must already be dimensioned. When lpsolve is used from Euler, nothing must
be dimensioned in advance. The eulpsolve driver takes care of dimensioning all return variables and they are
always returned as return value of the call to eulpsolve. Never as argument to the routine. This can be a single
value as for get_objective (although Euler stores this in a 1x1 matrix) or a matrix or vector as in get_variables.
In this case, get_variables returns a 4x1 matrix (vector) with the result of the 4 variables of the lp model.
</p>
<a name="An_example"></a>
<h3>An example</h3>
<p>(Note that you can execute this example by entering command per command as shown below or by opening and executing notebook example1.en)</p>
<pre>>lp=eulpsolve("make_lp", 0, 4);
>eulpsolve("set_verbose", lp, 3);
>eulpsolve("set_obj_fn", lp, [1, 3, 6.24, 0.1]);
>eulpsolve("add_constraint", lp, [0, 78.26, 0, 2.9], 2, 92.3);
>eulpsolve("add_constraint", lp, [0.24, 0, 11.31, 0], 1, 14.8);
>eulpsolve("add_constraint", lp, [12.68, 0, 0.08, 0.9], 2, 4);
>eulpsolve("set_lowbo", lp, 1, 28.6);
>eulpsolve("set_lowbo", lp, 4, 18);
>eulpsolve("set_upbo", lp, 4, 48.98);
>eulpsolve("set_col_name", lp, 1, "COLONE");
>eulpsolve("set_col_name", lp, 2, "COLTWO");
>eulpsolve("set_col_name", lp, 3, "COLTHREE");
>eulpsolve("set_col_name", lp, 4, "COLFOUR");
>eulpsolve("set_row_name", lp, 1, "THISROW");
>eulpsolve("set_row_name", lp, 2, "THATROW");
>eulpsolve("set_row_name", lp, 3, "LASTROW");
>eulpsolve("write_lp", lp, "a.lp");
>eulpsolve("get_mat", lp, 1, 2)
78.26
>eulpsolve("solve", lp)
0
>eulpsolve("get_objective", lp)
31.78275862069
>eulpsolve("get_variables", lp)
28.6
0
0
31.8275862069
>eulpsolve("get_constraints", lp)
92.3
6.864
391.2928275862
</pre>
<p>Note that there are some commands that return an answer. To see the answer, the command was not terminated with
a semicolon (;). If the semicolon is put at the end of a command, the answer is not shown. However it is also possible
to write the answer in a variable.
For example:
</p>
<pre>>obj=eulpsolve("get_objective", lp)
31.78275862069
</pre>
<p>Or:</p>
<pre>>obj=eulpsolve("get_objective", lp);</pre>
<p>Both will write the result in variable obj. Without the semicolon, the result is also shown on screen.
get_variables and get_constraints return a vector with the result. This can also be put in a variable:</p>
<pre>>x=eulpsolve("get_variables", lp);
>b=eulpsolve("get_constraints", lp);
</pre>
<p>It is always possible to show the contents of a variable by just giving it as command:</p>
<pre>>x
28.6
0
0
31.8275862069
</pre>
<p>Don't forget to free the handle and its associated memory when you are done:</p>
<pre>>eulpsolve("delete_lp", lp);</pre>
<p>Note that for larger and/or complex models, solving time can be long.
If this must be interrupt then the ESC key can be pressed. Euler shows this also in the status bar.
lp_solve checks regularly if this key is pressed and if so, solve is interrupted.
If verbose level is set then this is also shown on screen.
Anyway, the return status will indicate that solve was aborted.</p>
<a name="Using_model_name_instead_of_handle"></a>
<h3>Using model name instead of handle</h3>
From driver version 5.5.0.2, it is possible to use the model name instead of the handle. From the moment the model
has a name, you can use this name instead of the handle. This is best shown by an example. Above example would look
like this:
<pre>>lp=eulpsolve("make_lp", 0, 4);
>eulpsolve("set_lp_name", lp, "mymodel");
>eulpsolve("set_verbose", "mymodel", 3);
>eulpsolve("set_obj_fn", "mymodel", [1, 3, 6.24, 0.1]);
>eulpsolve("add_constraint", "mymodel", [0, 78.26, 0, 2.9], 2, 92.3);
>eulpsolve("add_constraint", "mymodel", [0.24, 0, 11.31, 0], 1, 14.8);
>eulpsolve("add_constraint", "mymodel", [12.68, 0, 0.08, 0.9], 2, 4);
>eulpsolve("set_lowbo", "mymodel", 1, 28.6);
>eulpsolve("set_lowbo", "mymodel", 4, 18);
>eulpsolve("set_upbo", "mymodel", 4, 48.98);
>eulpsolve("set_col_name", "mymodel", 1, "COLONE");
>eulpsolve("set_col_name", "mymodel", 2, "COLTWO");
>eulpsolve("set_col_name", "mymodel", 3, "COLTHREE");
>eulpsolve("set_col_name", "mymodel", 4, "COLFOUR");
>eulpsolve("set_row_name", "mymodel", 1, "THISROW");
>eulpsolve("set_row_name", "mymodel", 2, "THATROW");
>eulpsolve("set_row_name", "mymodel", 3, "LASTROW");
>eulpsolve("write_lp", "mymodel", "a.lp");
>eulpsolve("get_mat", "mymodel", 1, 2)
78.26
>eulpsolve("solve", "mymodel")
0
>eulpsolve("get_objective", "mymodel")
31.78275862069
>eulpsolve("get_variables", "mymodel")
28.6
0
0
31.8275862069
>eulpsolve("get_constraints", "mymodel")
92.3
6.864
391.2928275862
</pre>
<p>So everywhere a handle is needed, you can also use the model name. You can even mix the two methods.
There is also a specific Euler routine to get the handle from the model name: <a href="#get_handle">get_handle</a>.<br>
For example:</p>
<pre>
>eulpsolve("get_handle", "mymodel")
0
</pre>
<p>Don't forget to free the handle and its associated memory when you are done:</p>
<pre>>eulpsolve("delete_lp", "mymodel");</pre>
<p>In the next part of this documentation, the handle is used. But if you name the model, the name could thus also be used.</p>
<a name="Matrices"></a>
<h3>Matrices</h3>
In Euler, all numerical data is stored in matrices; even a scalar variable. Euler also supports complex numbers.
eulpsolve can only work with real numbers.
For example:
<pre>>eulpsolve("add_constraint", lp, [0.24, 0, 11.31, 0], 1, 14.8);</pre>
<p>Most of the time, variables are used to provide the data:</p>
<pre>>eulpsolve("add_constraint", lp, a1, 1, 14.8);</pre>
<p>Where a1 is a matrix variable.</p>
<p>Matrices with too few or too much elements gives an 'invalid vector.' error.</p>
<p>Most of the time, eulpsolve needs vectors (rows or columns).
In all situations, it doesn't matter if the vectors are row or column vectors. The driver accepts them both.
For example:</p>
<pre>>eulpsolve("add_constraint", lp, [0.24; 0; 11.31; 0], 1, 14.8);</pre>
<p>Which is a column vector, but it is also accepted.</p>
<p>An important final note. Several lp_solve API routines accept a vector where the first element (element 0) is not used.
Other lp_solve API calls do use the first element. In the Euler interface, there is never an unused element in the matrices.
So if the lp_solve API specifies that the first element is not used, then this element is not in the Euler matrix.</p>
<a name="Maximum_usage_of_matrices_with_eulpsolve"></a>
<h3>Maximum usage of matrices with eulpsolve</h3>
<p>Because Euler is all about matrices, all lpsolve API routines that need a column or row number to get/set information for that
column/row are extended in the eulpsolve Euler driver to also work with matrices. For example set_int in the API can
only set the integer status for one column. If the status for several integer variables must be set, then set_int
must be called multiple times. The eulpsolve Euler driver however also allows specifying a vector to set the integer
status of all variables at once. The API call is: return = eulpsolve("set_int", lp, column, must_be_int). The
matrix version of this call is: return = eulpsolve("set_int", lp, [must_be_int]).
The API call to return the integer status of a variable is: return = eulpsolve("is_int", lp, column). The
matrix version of this call is: [is_int] = eulpsolve("is_int", lp)<br>
Also note the get_mat and set_mat routines. In Euler these are extended to return/set the complete constraint matrix.
See following example.
</p>
<p>Above example can thus also be done as follows:<br>
(Note that you can execute this example by entering command per command as shown below or by opening and executing notebook example2.en)</p>
<pre>>lp=eulpsolve("make_lp", 0, 4);
>eulpsolve("set_verbose", lp, 3);
>eulpsolve("set_obj_fn", lp, [1, 3, 6.24, 0.1]);
>eulpsolve("add_constraint", lp, [0, 78.26, 0, 2.9], 2, 92.3);
>eulpsolve("add_constraint", lp, [0.24, 0, 11.31, 0], 1, 14.8);
>eulpsolve("add_constraint", lp, [12.68, 0, 0.08, 0.9], 2, 4);
>eulpsolve("set_lowbo", lp, [28.6, 0, 0, 18]);
>eulpsolve("set_upbo", lp, [1.0e30, 1.0e30, 1.0e30, 48.98]);
>eulpsolve("set_col_name", lp, 1, "COLONE");
>eulpsolve("set_col_name", lp, 2, "COLTWO");
>eulpsolve("set_col_name", lp, 3, "COLTHREE");
>eulpsolve("set_col_name", lp, 4, "COLFOUR");
>eulpsolve("set_row_name", lp, 1, "THISROW");
>eulpsolve("set_row_name", lp, 2, "THATROW");
>eulpsolve("set_row_name", lp, 3, "LASTROW");
>eulpsolve("write_lp", lp, "a.lp");
>eulpsolve("get_mat", lp)
0 78.26 0 2.9
0.24 0 11.31 0
12.68 0 0.08 0.9
>eulpsolve("solve", lp)
0
>eulpsolve("get_objective", lp)
31.78275862069
>eulpsolve("get_variables", lp)
28.6
0
0
31.8275862069
>eulpsolve("get_constraints", lp)
92.3
6.864
391.2928275862
</pre>
<p>Note the usage of 1.0e30 in set_upbo. This stands for "infinity". Meaning an infinite upper bound.
It is also possible to use -1.0e30 to express minus infinity. This can for example be used to create a free variable.</p>
<p>To show the full power of the matrices, let's now do some matrix calculations to check the solution.
It works further on above example:</p>
<pre>>A=eulpsolve("get_mat", lp);
>X=eulpsolve("get_variables", lp);
>B = A . X
92.3
6.864
391.2928275862
</pre>
<p>So what we have done here is calculate the values of the constraints (RHS) by multiplying the constraint matrix
with the solution vector. Now take a look at the values of the constraints that lpsolve has found:</p>
<pre>>eulpsolve("get_constraints", lp)
92.3
6.864
391.2928275862
</pre>
<p>Exactly the same as the calculated B vector, as expected.</p>
<p>Also the value of the objective can be calculated in a same way:</p>
<pre>>C=eulpsolve("get_obj_fn", lp);
>X=eulpsolve("get_variables", lp);
>obj = C . X
31.78275862069
</pre>
<p>So what we have done here is calculate the value of the objective by multiplying the objective vector
with the solution vector. Now take a look at the value of the objective that lpsolve has found:</p>
<pre>>eulpsolve("get_objective", lp)
31.78275862069
</pre>
<p>Again exactly the same as the calculated obj value, as expected.</p>
<a name="Using_string_constants"></a>
<h3>Using string constants</h3>
From driver version 5.5.2.14 on, it is possible to use string constants
everywhere an lp_solve constant is needed or returned. This is best shown by an example.
In the above code we had:
<pre>>lp=eulpsolve("make_lp", 0, 4);
>eulpsolve("set_verbose", lp, 3);
>eulpsolve("add_constraint", lp, [0, 78.26, 0, 2.9], 2, 92.3);
>eulpsolve("add_constraint", lp, [0.24, 0, 11.31, 0], 1, 14.8);
>eulpsolve("add_constraint", lp, [12.68, 0, 0.08, 0.9], 2, 4);</pre>
<p>Note the 3rd parameter on set_verbose and the 4th on add_constraint. These are
lp_solve constants. One could define all the possible constants in Euler and
then use them in the calls, but that has several disadvantages. First there
stays the possibility to provide a constant that is not intended for that
particular call. Another issue is that calls that return a constant are still
returning it numerical.</p>
<p>Both issues can now be handled by string constants. The above code can be done as
following with string constants:</p>
<pre>>lp=eulpsolve("make_lp", 0, 4);
>eulpsolve("set_verbose", lp, "IMPORTANT");
>eulpsolve("add_constraint", lp, [0, 78.26, 0, 2.9], "GE", 92.3);
>eulpsolve("add_constraint", lp, [0.24, 0, 11.31, 0], "LE", 14.8);
>eulpsolve("add_constraint", lp, [12.68, 0, 0.08, 0.9], "GE", 4);</pre>
<p>This is not only more readable, there is much lesser chance that mistakes are
being made. The calling routine knows which constants are possible and only
allows these. So unknown constants or constants that are intended for other
calls are not accepted. For example:</p>
<pre>>eulpsolve("set_verbose", lp, "blabla");
eulpsolve returned an error:
BLABLA: Unknown.
>eulpsolve("set_verbose", lp, "GE");
eulpsolve returned an error:
GE: Not allowed here.</pre>
<p>Note the difference between the two error messages. The first says that the
constant is not known, the second that the constant cannot be used at that
place.</p>
<p>Constants are case insensitive. Internally they are always translated to upper
case. Also when returned they will always be in upper case.</p>
<p>The constant names are the ones as specified in the documentation of each API
routine. There are only 3 exceptions, extensions actually. "LE", "GE" and "EQ" in
<a href="add_constraint.htm">add_constraint</a> and <a href="is_constr_type.htm">is_constr_type</a>
can also be "<", "<=", ">", ">=", "=". When returned however, "GE", "LE", "EQ"
will be used.</p>
<p>Some constants can be a combination of multiple constants. For example
<a href="set_scaling.htm">set_scaling</a>:</p>
<pre>>eulpsolve("set_scaling", lp, 3+128);</pre>
<p>With the string version of constants this can be done as following:</p>
<pre>>eulpsolve("set_scaling", lp, "SCALE_MEAN|SCALE_INTEGERS");</pre>
<p>| is the OR operator used to combine multiple constants. There may optinally be
spaces before and after the |.</p>
<p>Not all OR combinations are legal. For example in set_scaling, a choice must be
made between SCALE_EXTREME, SCALE_RANGE, SCALE_MEAN, SCALE_GEOMETRIC or
SCALE_CURTISREID. They may not be combined with each other. This is also tested:</p>
<pre>>eulpsolve("set_scaling", lp, "SCALE_MEAN|SCALE_RANGE");
eulpsolve returned an error:
SCALE_RANGE cannot be combined with SCALE_MEAN</pre>
<p>Everywhere constants must be provided, numeric or string values may be provided.
The routine automatically interpretes them. </p>
<p>Returning constants is a different
story. The user must let lp_solve know how to return it. Numerical or as string.
The default is numerical:</p>
<pre>>eulpsolve("get_scaling", lp)
131</pre>
<p>To let lp_solve return a constant as string, a call to a new function must be
made: return_constants</p>
<pre>>eulpsolve("return_constants", 1);</pre>
<p>From now on, all returned constants are returned as string:</p>
<pre>>eulpsolve("get_scaling", lp)
SCALE_MEAN|SCALE_INTEGERS</pre>
<p>This for all routines until return_constants is again called with 0:</p>
<pre>>eulpsolve("return_constants", 0);</pre>
<p>The (new) current setting of return_constants is always returned by the call.
Even when set:</p>
<pre>>eulpsolve("return_constants", 1)
1</pre>
<p>To get the value without setting it, don't provide the second argument:</p>
<pre>>eulpsolve("return_constants")
1</pre>
<p>In the next part of this documentation, return_constants is the default, 0, so all
constants are returned numerical and provided constants are also numerical. This
to keep the documentation as compatible as possible with older versions. But
don't let you hold that back to use string constants in your code.</p>
<a name="notebooks"></a>
<h3>Notebooks and Euler files</h3>
<p>Euler can execute a sequence of statements stored in files.
There are two types. Notebooks and Euler files.</p>
<p>Notebooks contains a number of Euler statements that are stored in a file.
They should have the extension *.en and are first loaded in the text window of Euler and then executed.
There are some notebook examples with the distribution of lpsolve.
</p>
<p>Euler files. If you want to develop longer and more complicated programs, it becomes useful to put all
function definitions and all commands into external Euler files. These files should have the
extension *.e, and can be loaded into Euler with the load command. Files in the current
directory will be found using the name alone. The current directory is the directory, where the
current notebook is loaded from or saved to. Otherwise, use the full path the file, or include the
directory of the file into the Euler path. Euler files often contain support functions (subroutines) that can be used
in your code.
</p>
<p>You can also edit these files with your favourite text editor (or notepad).</p>
<h4>loadlpsolve.en</h4>
<p>Loads the eulpsolve driver in Euler. Also loads Euler files lpsolve.e and lpmaker.e. See further for their usage.
</p>
<p>To execute and also see which commands are executed in the debug window, use following commands:</p>
<pre>File, Open notebook...
Open loadlpsolve.en
File, Run all Commands in this Notebook
</pre>
<h4>example1.en</h4>
<p>Contains the commands as shown in the first example of this article.</p>
<h4>example2.en</h4>
<p>Contains the commands as shown in the second example of this article.</p>
<h4>example3.en</h4>
<p>Contains the commands of a practical example. See further in this article.</p>
<h4>example4.en</h4>
<p>Contains the commands of a practical example. See further in this article.</p>
<h4>example5.en</h4>
<p>Contains the commands of a practical example. See further in this article.</p>
<h4>example6.en</h4>
<p>Contains the commands of a practical example. See further in this article.</p>
<h4>lpsolve.e</h4>
<p>This Euler file uses the API to create a higher-level function called lpsolve.
This function accepts as arguments some matrices and options to create and solve an lp model.
See the beginning of the file or enter help lpsolve to see its usage:</p>
<pre>>load "lpsolve"
>help lpsolve</pre>
<pre> LPSOLVE Solves mixed integer linear programming problems.
SYNOPSIS: {obj,x,duals} = lpsolve(f,a,b,e,vlb,vub,xint,scalemode,keep)
solves the MILP problem
max v = f'.x
a.x <> b
vlb <= x <= vub
x(int) are integer
ARGUMENTS: The first four arguments are required:
f: n vector of coefficients for a linear objective function.
a: m by n matrix representing linear constraints.
b: m vector of right sides for the inequality constraints.
e: m vector that determines the sense of the inequalities:
e(i) = -1 ==> Less Than
e(i) = 0 ==> Equals
e(i) = 1 ==> Greater Than
vlb: n vector of lower bounds. If empty or omitted,
then the lower bounds are set to zero.
vub: n vector of upper bounds. May be omitted or empty.
xint: vector of integer variables. May be omitted or empty.
scalemode: scale flag. Off when 0 or omitted.
keep: Flag for keeping the lp problem after it's been solved.
If omitted, the lp will be deleted when solved.
OUTPUT: A nonempty output is returned if a solution is found:
obj: Optimal value of the objective function.
x: Optimal value of the decision variables.
duals: solution of the dual problem.
</pre>
<p>Example of usage. To create and solve following lp-model:</p>
<pre>max: -x1 + 2 x2;
C1: 2x1 + x2 < 5;
-4 x1 + 4 x2 <5;
int x2,x1;
</pre>
<p>The following command can be used:</p>
<pre>>load "lpsolve"
>{obj, x}=lpsolve([-1, 2], [2, 1; -4, 4], [5, 5], [-1, -1], [], [], [1, 2]);
>obj
3
>x
1
2
</pre>
<h4>lpmaker.e</h4>
<p>This Euler file is analog to the lpsolve Euler file and also uses the API to create a higher-level function called lpmaker.
This function accepts as arguments some matrices and options to create an lp model. Note that this Euler file only
creates a model and returns a handle.
See the beginning of the file or enter help lpmaker to see its usage:</p>
<pre>>load "lpmaker"
>help lpmaker</pre>
<pre> LPMAKER Makes mixed integer linear programming problems.
SYNOPSIS: lp_handle = lpmaker(f,a,b,e,vlb,vub,xint,scalemode,setminim)
make the MILP problem
max v = f'.x
a.x <> b
vlb <= x <= vub
x(int) are integer
ARGUMENTS: The first four arguments are required:
f: n vector of coefficients for a linear objective function.
a: m by n matrix representing linear constraints.
b: m vector of right sides for the inequality constraints.
e: m vector that determines the sense of the inequalities:
e(i) < 0 ==> Less Than
e(i) = 0 ==> Equals
e(i) > 0 ==> Greater Than
vlb: n vector of non-negative lower bounds. If empty or omitted,
then the lower bounds are set to zero.
vub: n vector of upper bounds. May be omitted or empty.
xint: vector of integer variables. May be omitted or empty.
scalemode: scale flag. Off when 0 or omitted.
setminim: Set maximum lp when this flag equals 0 or omitted.
OUTPUT: lp_handle is an integer handle to the lp created.
</pre>
<p>Example of usage. To create following lp-model:</p>
<pre>max: -x1 + 2 x2;
C1: 2x1 + x2 < 5;
-4 x1 + 4 x2 <5;
int x2,x1;
</pre>
<p>The following command can be used:</p>
<pre>>load "lpmaker"
>lp=lpmaker([-1, 2], [2, 1; -4, 4], [5, 5], [-1, -1], [], [], [1, 2])
0
</pre>
<p>To solve the model and get the solution:</p>
<pre>>eulpsolve("solve", lp)
0
>eulpsolve("get_objective", lp)
3
>eulpsolve("get_variables", lp)
1
2
</pre>
<p>Don't forget to free the handle and its associated memory when you are done:</p>
<pre>>eulpsolve("delete_lp", lp);</pre>
<h4>lpdemo.en</h4>
<p>Contains several examples to build and solve lp models.</p>
<h4>ex.en</h4>
<p>Contains several examples to build and solve lp models.
Also solves the lp_examples from the lp_solve distribution.</p>
<a name="A_practical_example"></a>
<h3>A practical example</h3>
<p>We shall illustrate the method of linear programming by means of a simple example,
giving a combination graphical/numerical solution, and then solve both a slightly as well as a substantially
more complicated problem.</p>
<p>Suppose a farmer has 75 acres on which to plant two crops: wheat and barley.
To produce these crops, it costs the farmer (for seed, fertilizer, etc.) $120 per acre for the
wheat and $210 per acre for the barley. The farmer has $15000 available for expenses.
But after the harvest, the farmer must store the crops while awaiting favourable market conditions.
The farmer has storage space for 4000 bushels. Each acre yields an average of 110 bushels of wheat
or 30 bushels of barley. If the net profit per bushel of wheat (after all expenses have been subtracted)
is $1.30 and for barley is $2.00, how should the farmer plant the 75 acres to maximize profit?</p>
<p>We begin by formulating the problem mathematically.
First we express the objective, that is the profit, and the constraints
algebraically, then we graph them, and lastly we arrive at the solution
by graphical inspection and a minor arithmetic calculation.</p>
<p>Let x denote the number of acres allotted to wheat and y the number of acres allotted to barley.
Then the expression to be maximized, that is the profit, is clearly</p>
<p align="center">P = (110)(1.30)x + (30)(2.00)y = 143x + 60y.</p>
<p>There are three constraint inequalities, specified by the limits on expenses, storage and acreage.
They are respectively:</p>
<p align="center">
120x + 210y <= 15000<br>
110x + 30y <= 4000<br>
x + y <= 75
</p>
<p>Strictly speaking there are two more constraint inequalities forced by the fact that the farmer cannot plant
a negative number of acres, namely:</p>
<p align="center">x >= 0, y >= 0.</p>
<p>Next we graph the regions specified by the constraints. The last two say that we only need to consider
the first quadrant in the x-y plane. Here's a graph delineating the triangular region in the first quadrant determined
by the first inequality.</p>
<pre>
>X = 0.1:0.05:125;
>Y1 = (15000. - 120*X)/210;
>plot2d(X,Y1,bar=1);
>insimg;
</pre>
<p><IMG alt="Source" src="Euler1.jpg" border="0"></p>
<p>Now let's put in the other two constraint inequalities.</p>
<pre>
>X = 0.1:0.05:38;
>Y1 = (15000. - 120*X)/210;
>Y2 = max((4000 - 110.*X)./30, 0);
>Y3 = max(75 - X, 0.);
>Ytop = min(min(Y1, Y2), Y3);
>plot2d(X,Ytop,bar=1);
>insimg;
</pre>
<p><IMG alt="Source" src="Euler2.jpg" border="0"></p>
<p>The black area is the solution space that holds valid solutions. This means that any point in this area fulfils the
constraints.
</p>
<p>Now let's superimpose on top of this picture a contour plot of the objective function P.</p>
<pre>
>X = 0.1:0.05:38;
>Y1 = (15000. - 120*X)/210;
>Y2 = max((4000 - 110.*X)./30, 0);
>Y3 = max(75 - X, 0.);
>Ytop = min(min(Y1, Y2), Y3);
>plot2d(X,Ytop,bar=1);
>n=(1000:1000:9000)';
>plot2d("(n-143*x)/60", add=1, color=2);
>insimg;
</pre>
<p><IMG alt="Source" src="Euler3.jpg" border="0"></p>
<p>The red lines give a picture of the objective function.
All solutions that intersect with the black area are valid solutions, meaning that this result also fulfils
the set constraints. The more the lines go to the right, the higher the objective value is. The optimal solution
or best objective is a line that is still in the black area, but with an as large as possible value.
</p>
<p>It seems apparent that the maximum value of P will occur on the level curve (that is, level
line) that passes through the vertex of the polygon that lies near (22,53).<br>
It is the intersection of x + y = 75 and 110*x + 30*y = 4000<br>
This is a corner point of the diagram. This is not a coincidence. The simplex algorithm, which is used
by lp_solve, starts from a theorem that the optimal solution is such a corner point.<br>
In fact we can compute the result:</p>
<pre>>x = [1 1; 110 30] \ [75; 4000]
21.875
53.125
</pre>
<p>The acreage that results in the maximum profit is 21.875 for wheat and 53.125 for barley.
In that case the profit is:</p>
<pre>
>P = [143 60] . x
6315.625
</pre>
<p>That is, $6315.625.</p>
<p>Note that these command are in notebook example3.en</p>
<p>Now, lp_solve comes into the picture to solve this linear programming problem more generally.
After that we will use it to solve two more complicated problems involving more variables
and constraints.</p>
<p>For this example, we use the higher-level Euler file lpmaker to build the model and then some lp_solve API calls
to retrieve the solution. Here is again the usage of lpmaker:</p>
<pre> LPMAKER Makes mixed integer linear programming problems.
SYNOPSIS: lp_handle = lpmaker(f,a,b,e,vlb,vub,xint,scalemode,setminim)
make the MILP problem
max v = f'.x
a.x <> b
vlb <= x <= vub
x(int) are integer
ARGUMENTS: The first four arguments are required:
f: n vector of coefficients for a linear objective function.
a: m by n matrix representing linear constraints.
b: m vector of right sides for the inequality constraints.
e: m vector that determines the sense of the inequalities:
e(i) < 0 ==> Less Than
e(i) = 0 ==> Equals
e(i) > 0 ==> Greater Than
vlb: n vector of non-negative lower bounds. If empty or omitted,
then the lower bounds are set to zero.
vub: n vector of upper bounds. May be omitted or empty.
xint: vector of integer variables. May be omitted or empty.
scalemode: scale flag. Off when 0 or omitted.
setminim: Set maximum lp when this flag equals 0 or omitted.
OUTPUT: lp_handle is an integer handle to the lp created.
</pre>
<p>Now let's formulate this model with lp_solve:</p>
<pre>
>f = [143, 60];
>A = [120, 210; 110, 30; 1, 1];
>b = [15000; 4000; 75];
>lp = lpmaker(f, A, b, [-1, -1, -1], [], [], [], 1, 0);
>solvestat = eulpsolve("solve", lp);
>eulpsolve("get_objective", lp)
6315.625
>eulpsolve("get_variables", lp)