This section deals with geometric representation of a goal programming
problem. The graphical method of solving goal programming problem is
quite similar to the graphical method of linear programming.

The graphical method is not appropriate for solving real large scale problems; however, the pictorial interpretation will help you in developing intuition about the workings of a goal programming model.

## Graphical Method Example: Goal Programming

Minimize z = P_{1}d_{1}^{−} + 2P_{2}d_{2}^{−} + P_{2}d_{3}^{−} + P_{3}d_{1}^{+}

subject to

x_{1} + x_{2} + d_{1}^{−} - d_{1}^{+} = 450

x_{1} + d_{2}^{−} = 250

x_{2} + d_{3}^{−} = 350

x_{1}, x_{2}, d_{1}^{−},
d_{2}^{−}, d_{3}^{−},
d_{1}^{+} ≥ 0

Solution.

The problem is graphed in the following figure.

The region bounded by OPQR represents the solution set.

The optimal solution to the problem is given below:

x_{1 }= 250, x_{2} = 350, d_{1}^{−} = 0, d_{2}^{−} = 0, d_{3}^{−} = 0, d_{1}^{+} = 150