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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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		<TITLE>Free variables</TITLE>
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					<h1 align="left"><u>free variables</u></h1>
					<P align="left">Free variables are variables that have no lower or upper bound.
                                        By default, variables don't have an upper bound, but they do have a lower bound of zero.
                                        So they can only take positive values. Free variables can also become negative until -infinite.
                                        </P>
					<P align="left">
						lp_solve supports free variables since a long time.
                                                Internally, these variables are split in a positive and negative part. So the result of using free
                                                variables is that the number of columns increases. However this is transparent to the user.
                                                The API call <A HREF="set_unbounded.htm">set_unbounded</A> can be used to define a variable as free.</P>
					<P align="left">In the mps format, free variables can be specified in the
						BOUNDS section. See <a href="mps-format.htm">mps-format</a>.
						<br>
						<br>
						Example:
					</P>
					<pre>
NAME
ROWS
 N  R0
 L  R1
 G  R2
 G  R3
 G  R4
COLUMNS
    x1        R0        -1.000000000   R1        1.0000000000
    x1        R2        2.0000000000   R3        -1.000000000
    x2        R0        -2.000000000   R1        1.0000000000
    x2        R2        -1.000000000   R3        3.0000000000
    x3        R0        4.0000000000   R4        1.0000000000
    x4        R0        3.0000000000   R4        1.0000000000
RHS
    RHS       R1        5.0000000000   R4        0.5000000000
BOUNDS
<FONT color=red> FR BND       x2</FONT>
 UP BND       x3        10.000000000
 LO BND       x3        1.1000000000
<FONT color=red> FR BND       x4</FONT>
ENDATA
</pre>
					The red lines specify that variables x2 and x4 are free variables.

                                        <P align="left">In the lp format, free variables can be specified in the
						free section. See <a href="lp-format.htm">lp-format</a>.
                                                <br>
                                                <br>
						Example:
					</P>
					<pre>
max: x1 + 2x2 - 4x3 -3x4;
x1 + x2 &lt;= 5;
2x1 - x2 &gt;= 0;
-x1 + 3x2 &gt;= 0;
x3 + x4 &gt;= .5;
x3 &gt;= 1.1;
x3 &lt;= 10;
<FONT color=red>
free x2, x4;
</FONT>
</pre>
					<PRE>The red line specifies that variables x2 and x4 are free variables. The solution of this model is:</PRE>
					<pre>
Value of objective function: 5.73333

Actual values of the variables:
x1                        1.66667
x2                        3.33333
x3                            1.1
x4                           -0.6
</pre>
					<P align="left">
					As can be seen, the value of x4 is -0.6, a negative value.
                                        If the variable would not be set as free, it would not be negative.
                                        </P>
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