Cost Accounting (14th Edition)

418 CHAPTER 11 DECISION MAKING AND RELEVANT INFORMATION
Second, move from corner point to corner point and compute the total contribution margin at each corner point.
Trial Corner Point (S, B) Snowmobile Engines (S) Boat Engines (B) Total Contribution Margin
1 (0, 0) 0 0 $240(0) + $375(0) = $0
2 (0, 110) 0 110 $240(0) + $375(110) = $41,250
3 (25,110) 25 110 $240(25) + $375(110) = $47,250
4 (75, 90) 75 90 $240(75) + $375(90) = $51,750 a
5 (120, 0) 120 0 $240(120) + $375(0) = $28,800
a The optimal solution.
The optimal product mix is the mix that yields the highest total contribution: 75 snowmobile engines and 90 boat
engines. To understand the solution, consider what happens when moving from the point (25,110) to (75,90). Power
Recreation gives up $7,500 [$375 * (110 – 90)] in contribution margin from boat engines while gaining $12,000
[$240 * (75 – 25)] in contribution margin from snowmobile engines. This results in a net increase in contribution
margin of $4,500 ($12,000 – $7,500), from $47,250 to $51,750.
Graphic Approach
Consider all possible combinations that will produce the same total contribution margin of, say, $12,000. That is,
This set of $12,000 contribution margins is a straight dashed line through [S = 50 ($12,000 ÷ $240); B = 0)] and
[S = 0, B = 32 ($12,000 ÷ $375)] in Exhibit 11-14. Other equal total contribution margins can be represented by
lines parallel to this one. In Exhibit 11-14, we show three dashed lines. Lines drawn farther from the origin represent
more sales of both products and higher amounts of equal contribution margins.
The optimal line is the one farthest from the origin but still passing through a point in the area of feasible solutions.
This line represents the highest total contribution margin. The optimal solution—the number of snowmobile
engines and boat engines that will maximize the objective function, total contribution margin—is the corner point
(S = 75, B = 90). This solution will become apparent if you put a straight-edge ruler on the graph and move it outward
from the origin and parallel with the $12,000 contribution margin line. Move the ruler as far away from the origin
as possible—that is, increase the total contribution margin—without leaving the area of feasible solutions. In
general, the optimal solution in a maximization problem lies at the corner where the dashed line intersects an extreme
point of the area of feasible solutions. Moving the ruler out any farther puts it outside the area of feasible solutions.
Sensitivity Analysis
$240S + $375B = $12,000
What are the implications of uncertainty about the accounting or technical coefficients used in the objective function
(such as the contribution margin per unit of snowmobile engines or boat engines) or the constraints (such as the number
of machine-hours it takes to make a snowmobile engine or a boat engine)? Consider how a change in the contribution
margin of snowmobile engines from $240 to $300 per unit would affect the optimal solution. Assume the contribution
margin for boat engines remains unchanged at $375 per unit. The revised objective function will be as follows:
TCM = $300S + $375B
Using the trial-and-error approach to calculate the total contribution margin for each of the five corner points
described in the previous table, the optimal solution is still (S = 75, B = 90). What if the contribution margin of snowmobile
engines falls to $160 per unit? The optimal solution remains the same (S = 75, B = 90). Thus, big changes in
the contribution margin per unit of snowmobile engines have no effect on the optimal solution in this case. That’s
because, although the slopes of the equal contribution margin lines in Exhibit 11-14 change as the contribution margin
of snowmobile engines changes from $240 to $300 to $160 per unit, the farthest point at which the equal contribution
margin lines intersect the area of feasible solutions is still (S = 75, B = 90).
Terms to Learn
This chapter and the Glossary at the end of the book contain definitions of the following important terms:
book value (p. 410)
business function costs (p. 395)
constraint (p. 416)
decision model (p. 391)
differential cost (p. 399)
differential revenue (p. 399)
full costs of the product (p. 395)
incremental cost (p. 399)
incremental revenue (p. 399)