Example 3.  Consider the region in the plane defined by the inequalities:  

          

3 (b).  Find the maximum of      over the region .  

Solution 3 (b).

Graph the domain defined by the constraints.

Enter the objective function.

Use the Mathematica subroutine LinearProgramming to find the solution (note carefully the "-" signs in front of the matrix and vector in the second and third arguments).

Use the simplex algorithm to find this solution.  

Enter the coefficients of the decision variables and right side of the tableau.  

Mathematica will fill in the columns for the slack variables and append a column of zeros which will be use to calculate the ratios for determining the pivot rows.

Run the subroutine  SimplexMethod.

There are no negative coefficients in the bottom row, so we are done.

From column 1 and row 2 we see that .  From column 2 and row 1 we see that .  The final point    is the optimal feasible solution.  The simplex method has moved from the origin along the edges to the point .  The value of the objective function at the solution point is located in the bottom row of the column "",  i.e.  .

We are done!

Aside.  The slack variables could be computed from (4) using the initial coefficients of the tableau:

       for rows    in the tableau.    

The bottom line of the tableau corresponds to the augmented objective function.

Graph the domain defined by the constraints and the level curves of the objective function.

(c) John H. Mathews 2005