Chapter 14 Decision Making: Relevant Costs and Benefits 645

APPENDIXTOCHAPTER

Linear Programming
When a firm produces multiple products, management must decide how much of each output to pro-
duce. In most cases, the firm is limited in the total amount it can produce, due to constraints on resources Learning Objective 14-8
such as machine time, direct labor, or raw materials. This situation is known as a product-mix problem. Formulate a linear program to
To illustrate, we will use International Chocolate Company’s Phoenix plant, which produces Chew- solve a product-mix problem
ies and Chompo Bars. Exhibit 14–21 provides data pertinent to the problem. with multiple constraints.
Linear programming is a powerful mathematical tool, well suited to solving International Choco-
late Company’s product-mix problem. The steps in constructing the linear program are as follows:
1. Identify the decision variables, which are the variables about which a decision must be made.
International Chocolate’s decision variables are as follows:
Decision X = Number of cases of Chewies to produce each month
variables Y = Number of cases of Chompo Bars to produce each month

2. Write the objective function, which is an algebraic expression of the firm’s goal. International
Chocolate’s goal is to maximize its total contribution margin. Since Chewies bring a contribu-
tion margin of $1 per case, and Chompos result in a contribution margin of $2 per case, the
firm’s objective function is the following:
Objective function Maximize Z = X + 2Y

3. Write the constraints, which are algebraic expressions of the limitations faced by the firm,
such as those limiting its productive resources. International Chocolate has a constraint for
machine time and a constraint for direct labor.
Machine-time constraint .02X + .05Y ≤ 700
Labor-time constraint .20X + .25Y ≤ 5,000

Suppose, for example, that management decided to produce 20,000 cases of Chewies and 6,000
cases of Chompos. The machine-time constraint would appear as follows:
(.02)(20,000) + (.05)(6,000) = 700

Thus, at these production levels, the machine-time constraint would just be satisfied, with no
machine hours to spare.

Graphical Solution
To understand how the linear program described above will help International Chocolate’s management
solve its product-mix problem, examine the graphs in Exhibit 14–22. The two colored lines in panel A
represent the constraints. The colored arrows indicate that the production quantities, X and Y, must lie on
or below these lines. Since the production quantities must be nonnegative, colored arrows also appear on
the graphs’ axes. Together, the axes and constraints form an area called the feasible region, in which the
solution to the linear program must lie.
The purple slanted line in panel A represents the objective function. Rearrange the objective func-
tion equation as follows:
Z − __
Z = X + 2Y → Y = __ 1X
2 2

Chewies Chompo Bars

Exhibit 14–21
Data for Product-Mix Prob-
Contribution margin per case ............................................................................................... $1.00 $2.00 lem: International Chocolate
Machine hours per case ........................................................................................................ .02 .05 Company
Direct-labor hours per case .................................................................................................. .20 .25

Machine Direct-Labor
Hours Hours

Limited resources: hours available per month ..................................................................... 700 5,000

646 Chapter 14 Decision Making: Relevant Costs and Benefits

Exhibit 14–22 A. Constraints, Feasible Region, and Objective Function

Product-Mix Problem Y (cases of Chompos)
Expressed as Linear
Program: International
Chocolate Company Labor-time constraint
.20X + .25Y ≤ 5,000
20,000

15,000

Machine-time constraint
.02X + .05Y ≤ 700
10,000

Feasible
region
5,000

X (cases of Chewies)
5,000 10,000 15,000 20,000 25,000 30,000 35,000

Objective function:
Z = X + 2Y

B. Solution of Linear Program

Y (cases of Chompos)

20,000

15,000

Optimum: X = 15,000; Y = 8,000

10,000
8,000

5,000 Feasible
region

X (cases of Chewies)
5,000 10,000 15,000 20,000 25,000 30,000 35,000

This form of the objective function shows that the slope of the equation is −½, which is the slope of
the objective-function line in the exhibit. Management’s goal is to maximize total contribution margin,
denoted by Z. To achieve the maximum, the objective-function line must be moved as far outward and
upward in the feasible region as possible, while maintaining the same slope. This goal is represented in
panel A by the arrow that points outward from the objective-function line.

Solution The result of moving the objective-function line as far as possible in the indicated direction
is shown in panel B of the exhibit. The objective-function line intersects the feasible region at exactly
one point, where X equals 15,000 and Y equals 8,000. Thus, International Chocolate’s optimal product
mix is 15,000 cases of Chewies and 8,000 cases of Chompos per month. The total contribution margin
is calculated as shown below.
Total contribution margin = (15,000)($1) + (8,000)($2) = $31,000

Simplex Method and Sensitivity Analysis Although the graphical method is instructive, it is a
cumbersome technique for solving a linear program. Fortunately, mathematicians have developed a more
 Chapter 14 Decision Making: Relevant Costs and Benefits 647

efficient solution method called the simplex algorithm. A computer can apply the algorithm to a complex
linear program and determine the solution in seconds. In addition, most linear programming computer
packages provide a sensitivity analysis of the problem. This analysis shows the decision maker the extent to
which the estimates used in the objective function and constraints can change without changing the solution.

Managerial Accountant’s Role

What is the managerial accountant’s role in International Chocolate’s product-mix decision? The production
manager in the company’s Phoenix plant makes this decision, with the help of a linear program. However,
the linear program uses information supplied by the managerial accountant. The coefficients of X and Y in
the objective function are unit contribution margins. Exhibit 14–14 shows that calculating these contribu-
tion margins requires estimates of direct-material, direct-labor, variable-overhead, and variable selling and
administrative costs. These estimates were provided by a managerial accountant, along with estimates of the
machine time and direct-labor time required to produce a case of Chewies or Chompos. All of these estimates
were obtained from the standard-costing system, upon which the Phoenix plant’s product costs are based.
Thus, the managerial accountant makes the product-mix decision possible by providing the relevant cost data.
Linear programming is widely used in business decision making. Among the applications are
blending in the petroleum and chemical industries; scheduling of personnel, railroad cars, and aircraft;
and the mixing of ingredients in the food industry. In all of these applications, managerial accountants
provide information crucial to the analysis.6

Review Questions
14–1. List the seven steps in the decision-making process. 14–17. What is a joint production process? Describe a spe-
14–2. Describe the managerial accountant’s role in the cial decision that commonly arises in the context of a
decision-making process. joint production process. Briefly describe the proper
14–3. Distinguish between qualitative and quantitative deci- approach for making this type of decision.
sion analyses. 14–18. Are allocated joint processing costs relevant when
14–4. Explain what is meant by the term decision model. making a decision to sell a joint product at the split-off
point or process it further? Why?
14–5. A quantitative analysis enables a decision maker to put
a “price” on the sum total of the qualitative characteris- 14–19. Briefly describe the proper approach to making a pro-
tics in a decision situation. Explain this statement, and duction decision when limited resources are involved.
give an example. 14–20. What is meant by the term contribution margin per unit
14–6. What is meant by each of the following potential charac- of scarce resource?
teristics of information: relevant, accurate, and timely? Is 14–21. How is sensitivity analysis used to cope with uncer-
objective information always relevant? Accurate? tainty in decision making?
14–7. List and explain two important criteria that must be sat- 14–22. There is an important link between decision making and
isfied in order for information to be relevant. managerial performance evaluation. Explain.
14–8. Explain why the book value of equipment is not a rel- 14–23. List four potential pitfalls in decision making, which
evant cost. represent common errors.
14–9. Is the book value of inventory on hand a relevant cost? Why? 14–24. Why can unitized fixed costs cause errors in decision
14–10. Why might a manager exhibit a behavioral tendency to making?
inappropriately consider sunk costs in making a decision? 14–25. Give two examples of sunk costs, and explain why they
14–11. Give an example of an irrelevant future cost. Why is it are irrelevant in decision making.
irrelevant? 14–26. “Accounting systems should produce only relevant data
14–12. Define the term opportunity cost, and give an example and forget about the irrelevant data. Then I’d know
of one. what was relevant and what wasn’t!” Comment on this
remark by a company president.
14–13. What behavioral tendency do people often exhibit with
regard to opportunity costs? 14–27. Are the concepts underlying a relevant-cost analysis
still valid in an advanced manufacturing environment?
14–14. How does the existence of excess production capacity
Are these concepts valid when activity-based costing is
affect the decision to accept or reject a special order?
used? Explain.
14–15. What is meant by the term differential cost analysis?
14–28. List five ways that management can seek to relax a
14–16. Briefly describe the proper approach for making a deci- constraint by expanding the capacity of a bottleneck
sion about adding or dropping a product line. operation.

6
For other linear programming applications, see J. D. Dotson, “Five Areas of Application for Linear Programming,”
www.sciencing.com, May 21, 2018.