A second approach to solving LP problems employs the corner point method. This technique is simpler conceptually than the isoprofit line approach, but it involves looking at the profit at every corner point of the feasible region.

The mathematical theory behind LP states that an optimal solution to any problem (that is, the values of T and C that yield the maximum profit) will lie at a corner point, or extreme point, of the feasible region. Hence, it is necessary to find the values of the variables only at each corner; an optimal solution will lie at one (or more) of them.

The first step in the corner point method is to graph the constraints and find the feasible region. This was also the first step in the isoprofit method, and the feasible region is shown again in Figure 7.9. The second step is to find the corner points of the feasible region. For the Flair Furniture example, the coordinates of three of the corner points are obvious from observing the graph. These are (0, 0), (50, 0), and (0, 80). The fourth corner point is where the two constraint lines intersect, and the coordinates must be found algebraically by solving the two equations simultaneously for two variables.

There are a number of ways to solve equations simultaneously, and any of these may be used. We will illustrate the elimination method here. To begin the elimination method, select a variable to be eliminated. We will select T in this example. Then multiply or divide one equation by a number so that the coefficient of that variable (T) in one equation will be the negative of the coefficient of that variable in the other equation. The two constraint equations are

To eliminate T, we multiply the second equation by $\mathrm{-}2:$

and then add it to the first equation:

or

Doing this has enabled us to eliminate one variable, T, and to solve for C. We can now substitute 40 for C in either of the original equations and solve for T. Let’s use the first equation. When $C=40,$ then

or

Thus, the last corner point is (30, 40).

The next step is to calculate the value of the objective function at each of the corner points. The final step is to select the corner with the best value, which would be the highest profit in this example. Table 7.3 lists these corners points with their profits. The highest profit is found to be $4,100, which is obtained when 30 tables and 40 chairs are produced. This is exactly what was obtained using the isoprofit method.

Table 7.4 provides a summary of both the isoprofit method and the corner point method. Either of these can be used when there are two decision variables. If a problem has more than two decision variables, we must rely on computer software or use the simplex algorithm discussed in Module 7.