Linnear programming system of equations and restrictions

Question

While doing a linnear programming problem, i came with this system of equations of 10 variables, and 7 restrictions (7 equations and 10 inequalities).

The objtective is to minimize the function:

$$Z=3x_{14}+5x_{15}+8x_{25}+9x_{23}+2x_{45}+4x_{35}+2x_{46}+5x_{56}+7x_{57}+2x_{37}$$

Subject to:

$437.5=x_{14}+x_{15}$

$437.5=x_{25}+x_{23}$

$x_{23}=x_{35}+x_{37}$

$x_{14}=x_{45}+x_{46}$

$x_{15}+x_{25}=x_{56}+x_{57}$

$x_{46}+x_{56}=600$

$x_{57}+x_{37}=275$

And

$x14≤300$

$x15≤300$

$x25≤300$

$x23≤200$

$x45≤200$

$x35≤200$

$x46≤300$

$x56≤300$

$x57≤300$

$x37≤200$

There are programms wich are suposed to solve this, but no one recives equations as an innput. What can i do?

Answer 1

I have good experiences with LINGO from LINDO Systems. A free version (6 months) can be downloaded here.

$\color{blue}{\texttt{Input example}}$

$MAX \ \ 5 X1 + 7 X2 \\ ST \\ 3 X1 + 4 X2 < 650\\ 2 X1 + 3 X2 < 500$

The default setting for the variables is, that they are non-negative.

$X1, \ X2 \geq 0$

$\color{blue}{\texttt{Output example}}$

 Global optimal solution found.
  Objective value:                              1137.500
  Infeasibilities:                              0.000000
  Total solver iterations:                             1
  Elapsed runtime seconds:                         17.99

  Model Class:                                        LP

  Total variables:                      2
  Nonlinear variables:                  0
  Integer variables:                    0

  Total constraints:                    3
  Nonlinear constraints:                0

  Total nonzeros:                       6
  Nonlinear nonzeros:                   0



                                Variable           Value        Reduced Cost
                                      X1        0.000000           0.2500000
                                      X2        162.5000            0.000000

                                     Row    Slack or Surplus      Dual Price
                                       1        1137.500            1.000000
                                       2        0.000000            1.750000

                                   3        12.50000            0.000000

Answer 2

There are many linear programming solvers available. Look at Wikipedia, for example. I would think that nearly all of these would accept equations as well as inequalities. But if by some chance you're stuck with a program that only accepts inequalities, you can replace each equation $a = b$ by the pair of inequalities $a \le b$ and $a \ge b$