## Figure

Graphic Solution of the Linear Programming Problem

Points along the isoprofit line represent all possible combinations of X and Y that result in the same profit level. Point M is on the highest isoprofit curve that intersects the feasible space. Thus, it represents the output combination that will maximize total profit given input constraints.

Quantity of Y (per time period)

Total profit contribution = \$12 QX + \$9Qy

Quantity of Y (per time period)

Total profit contribution = \$12 QX + \$9Qy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Quantity of X (per time period)

approach, point M in the figure is indicated as the optimal solution. At point M, the firm produces 6 units of X and 4 units of Y, and the total profit is \$108 [=(\$12 X 6) + (\$9 X 4)], which is the maximum available under the conditions stated in the problem. No other point within the feasible spaces touches so high an isoprofit curve.

Using the combined graphic and analytical method introduced in the preceding section, M can be identified as the point where QX = 6 and QY = 4. At M, constraints on inputs A and B are binding. At M, 32 units of input A and 10 units of input B are being completely used to produce X and Y. Thus, Equations 9.4 and 9.5 can be written as equalities and solved simultaneously for QX and QY. Subtracting two times Equation 9.5 from Equation 9.4 gives

Substituting 6 for QX in Equation 9.5 results in corner point

Spot in the feasible space where the X-axis, Y-axis, or constraint conditions intersect

Notice that the optimal solution to the linear programming problem occurs at a corner point of the feasible space. A corner point is a spot in the feasible space where the X-axis, Y-axis, or constraint conditions intersect. The optimal solution to any linear programming problem always lies at a corner point. Because all of the relations in a linear programming problem must be linear by definition, every boundary of the feasible space is linear. Furthermore, the objective function is linear. Thus, the constrained optimization of the objective function takes place either at a corner of the feasible space or at one boundary face, as is illustrated by Figure 9.8.

In Figure 9.8, the linear programming example has been modified by assuming that each unit of either X or Y yields a profit of \$5. In this case, the optimal solution to the problem includes any of the combinations of X and Y found along line LM. All of these combinations are feasible and result in a total profit of \$50. If all points along line LM provide optimal combinations of output, the combinations found at corners L and M are also optimal. Because the firm is indifferent about producing at point L or at point M, or at any point in between, any such location provides an optimal solution to the production problem.

The search for an optimal solution can be limited to just the corners of each linear programming problem's feasible space. This greatly reduces the number of necessary computations. 