Cost Accounting (14th Edition)

APPENDIX 417
The three solid lines on the graph in Exhibit 11-14 show the existing constraints for assembly and testing and the
materials-shortage constraint. 4 The feasible or technically possible alternatives are those combinations of quantities of
snowmobile engines and boat engines that satisfy all the constraining resources or factors. The shaded “area of feasible
solutions” in Exhibit 11-14 shows the boundaries of those product combinations that are feasible.
Step 3: Compute the optimal solution. Linear programming (LP) is an optimization technique used to maximize the
objective function when there are multiple constraints. We present two approaches for finding the optimal solution
using LP: trial-and-error approach and graphic approach. These approaches are easy to use in our example because
there are only two variables in the objective function and a small number of constraints. Understanding these
approaches provides insight into LP. In most real-world LP applications, managers use computer software packages to
calculate the optimal solution. 5
Trial-and-Error Approach
The optimal solution can be found by trial and error, by working with coordinates of the corners of the area of feasible
solutions.
First, select any set of corner points and compute the total contribution margin. Five corner points appear in
Exhibit 11-14. It is helpful to use simultaneous equations to obtain the exact coordinates in the graph. To illustrate, the corner
point (S = 75, B = 90) can be derived by solving the two pertinent constraint inequalities as simultaneous equations:
2S + 5B = 600 (1)
1S + 0.5B = 120 (2)
Multiplying (2) by 2: 2S + B = 240 (3)
Subtracting (3) from (1): 4B = 360
Therefore, B = 360 , 4 = 90
Substituting for B in (2): 1S + 0.5(90) = 120
S = 120 - 45 = 75
Given S = 75 snowmobile engines and B = 90 boat engines, TCM = ($240 per snowmobile engine
engines) + ($375 per boat engine * 90 boat engines) = $51,750.
*
75 snowmobile
250
200
Testing
department
constraint
Exhibit 11-14
Linear Programming:
Graphic Solution for
Power Recreation
Boat Engines (Units)
150
100
50
Area
of feasible
solutions
Materials-shortage constraint
for boat engines
Optimal corner
(75, 90)
Equal
contribution
margin lines
Assembly
department
constraint
0
0
50
100 150 200 250 300
Snowmobile Engines (Units)
4 As an example of how the lines are plotted in Exhibit 11-14, use equal signs instead of inequality signs and assume for the
assembly department that B = 0; then S = 300 (600 machine-hours ÷ 2 machine-hours per snowmobile engine). Assume that
S = 0; then B = 120 (600 machine-hours ÷ 5 machine-hours per boat engine). Connect those two points with a straight line.
5 Standard computer software packages rely on the simplex method. The simplex method is an iterative step-by-step procedure
for determining the optimal solution to an LP problem. It starts with a specific feasible solution and then tests it by substitution
to see whether the result can be improved. These substitutions continue until no further improvement is possible and the
optimal solution is obtained.