Linear Programs 713line BB.^2 Finally, the binding labor constraint is given by the equation 5S
5E 2,000 and is graphed as DD. The firm’s feasible region of production
consists of S and E combinations that simultaneously satisfy all three con-
straints: the shaded area bounded by OABCD.
The company still has to determine the output combination (the point in
the feasible region) that maximizes total contribution. To find this point, we
draw in contribution contours—lines indicating combinations of S and E that
yield a fixed value of contribution. For instance, we can graph the contour cor-
responding to a contribution of $75,000 by using the equation 500S 300E
75,000. This contour is shown in Figure 17.2. (Check this by noting that the
horizontal intercept is S 150, since the firm can earn $75,000 by producing
only standard models. In turn, the vertical intercept is E 250.) Figure 17.2
also depicts contours corresponding to contributions of $120,000, $140,000,
and $200,000. Note that increasing the contribution causes a parallel (north-
east) shift in the contour. Obviously, larger production quantities are neces-
sary to generate the greater contribution. Note, however, that the contour
slopes do not change because the ratio of contributions is always $500 to $300.
The optimal solution is found at the corner of the feasible region that touches the
highest contribution contour.In Figure 17.2, this occurs at point C. Here, the con-
tour corresponding to the $140,000 contribution just touches the feasible
region. As the figure shows, this is the best the firm can do. The firm can con-
sider other feasible production plans, but any such plan lies on a lower contri-
bution contour. For instance, point B’s plan (200 standard models and 100
economy models) produces only $130,000 in contribution (i.e., lies on a lower
contour). At point D the firm earns even less. However, “pie in the sky” pro-
duction plans are irrelevant. The firm cannot attain a higher contribution—say,
$200,000—because such a contour lies wholly outside the feasible region.
We can reinforce the visual solution to the LP problem by using marginal
analysis. Suppose the firm takes point D as a candidate for its optimal produc-
tion plan. Using marginal analysis, the firm asks whether it could increase its
contribution by moving to some other point on the edge of the feasible region.
Suppose it considers moving in the direction of C, producing more standard
models and fewer economy models. (Note that segment DC portrays the bind-
ing labor constraint.) To produce an extra standard model requires 5 addi-
tional hours of labor; with all labor utilized, this means producing one fewer
economy model (which frees up 5 labor-hours). Would such a move improve
the firm’s profit? It certainly would! The net increase in contribution is $200.
(The gain is $500 in contribution for the extra standard unit minus $300 in
lost contribution from the economy unit that is no longer produced.) Thus, the
firm should make the one-unit switch. But having switched one unit, it can(^2) The easiest way to graph any constraint line is to pinpoint its two intercepts; that is, set one of the
decision variables equal to zero and solve for the other. Doing this for the hard-disk equation, we
find E 0 with S 250 and, in turn, S 0 with E 500. Thus, the horizontal intercept is 250 (at
B) and the vertical intercept is 500 (at B).
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