A linear programming (L.P.) problem consists of a linear objective function to be optimized and a set of linear constraints (inequalities and/or equations). The basic idea behind the graphical method of solving L.P. problem with two variables is that each pair of values [x1, x2] can be represented as a point in the two-dimensional coordinate system. With such representation, we will be able to visualize the set of all feasible solutions as a graphical region, called the feasible region or the feasible set, and then identify the optimal solution (assuming it exists).

GRULP II is a learning program for solving L.P. problems with
two variables by graphical method. The name of the program **
GRULP II **
is a shortcut of the following Slovak words:

**G**raficke=graphical

**R**iesenie=solution
of

**U**loh=problems
of

**L**inearneho=linear

**P**rogramovania=programming.

Once you have selected the number of constraints, select the
**OK **button to display the input form. Fill the input form with the
coefficients of each constraint and the objective function. Also select the type
of problem (minimization or maximization) and the type (<=, >=, =) and the right
hand side constant of each constraint.

Program will represent the constraints in the coordinate system with x and x2 axes by the appropriate halfplane. Before this, the user has to enter the coordinates of points, which determine the border line of the constraint and also coordinate of the point, which belongs to the halfplane.

As a next step, check the answer, what is the feasible set and click to the intersection of the graphical solution of the constraints.

The program will display feasible set in the next figure. The user has to enter the number for which the line of the objective function will be depicted. GRULP II allows moving of the objective function’s line and displays the changed value of the objective function.

The last step is correct checking of the optimal solution’s set.

Finally, the program will show the arithmetic result of the solving this L.P. problem: optimal values of x1, x2 and the optimal objective-function value.

The program GRULP II assists user while solving L.P. problems. The program controls all user steps and alerts if the case of some inaccuracy. The solving of problem is accompanies by detail descriptions.

The main advantages of GRULP II are that program speeds up computation, is interactive and visual. This freeware program is available on the net (http://www.mathe-online.at/materialien/Danka.Timkova or Slovak version http://www.science.upjs.sk/diplom/grulp).

**REMARKS**

- Because of the graphical solution in the two-dimensional coordinate system, the number of variables is fixed to 2.
- The number of constraints is limited to 20.
- The decimal notation of the number should be enter by ‘.’ (not by ‘,’).
- After entering the mathematical model of the L.P. problem, is it not possible to change it.
- Classically, the applet assumes the variables to be non negative.
- Because the applet assumes that all variables are non-negative, the problem is solved only in the first quadrant of coordinate system.
- Because of the specific representation of number by some softvers, sometimes only approximate values of result are shown.