9781118041581

Sensitivity Analysis and Shadow Prices 719

not only the numerical solution to the problem in question but also a wide vari-

ety of sensitivity analyses. Thus, a solid understanding of sensitivity analysis is

essential in order to take full advantage of the power of linear programming.

Changes in the Objective Function

It is natural to ask how changes in the coefficients of the objective function

affect the optimal decision. In the computer firm’s production problem, for

instance, the current contributions are $500 and $300 per unit of each model

type. Obviously, if market prices or variable unit costs change, so will the con-

tributions. How would such changes affect the firm’s optimal production mix?

As a concrete example, suppose the firm anticipates that the current indus-

try price for an economy model, $1,000, will fall to $900 in the coming months.

Thus the firm expects that contribution per unit of the economy model will fall

to 900  700 $200 as a result. Assuming an unchanged contribution for

model S, the firm’s new objective function is

.

With the sizable drop in E’s contribution per unit, intuition suggests that the firm

should reduce the output of E and increase the output of S. Figure 17.4 indicates

that this is indeed the case. It shows the same feasible region as Figure 17.2. But

the slopes of the contribution contours change. In Figure 17.4, the highest con-

tribution contour touches the feasible region at point B, where the hard disk

and DVD constraints are binding. (The slope of the new contribution contour is

500/200    2.5, and this falls between the slopes of these two constraint lines,

2 and .) We find the values of the decision variables at optimal corner B by

solving the equations S 200 and 2S E 500. The resulting values are S 

200 and E 100, and maximum contribution is $120,000. In contrast, if the firm

were to maintain its old production mix, S 100 and E 300 (at point C), it

would earn a contribution of only $110,000. To sum up, the firm should respond

to the fall in economy model contribution by shifting to a greater quantity of

standard models.

A general conclusion emerges from this example: The optimal production

plan depends on the relativecontributions of the two models. To see this, write

the objective function in the form SS EE, where Sand Edenote the

contribution per unit for the respective models. The slope of the contribution

contour is E/ S    S/E. Depending on the ratio of model contributions,

one of the following three plans is optimal:

Point B (S 200, E 100) provided   S/E   2

Point C (S 100, E 300) provided   2  S/E   1

Point D (S 0, E 400) provided     1  S/E0.

500S200E

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