1: Linear Programming

Appendix 7.1: Linear Programming

Factors such as factory capacity, personnel time, floor space, and so forth constrain most man- agerial decisions. If the firm has enough time before implementing a decision, it can relax constraints by increasing capacity. In the short run, however, decision makers face a constrained amount of resources available to them. Linear programming solves problems of this type. We refer to linear programming as a constrained optimization technique because it solves for the optimal use of scarce (that is, constrained) resources.

Two simple examples demonstrate how linear programming works. We solve these using graphs and simple algebra. More complex problems require some systematic procedure like the simplex method, described in textbooks on operations research and quantitative methods. Most linear programming problem solutions result from computer implementation of the simplex method or variations of it.

P R O F I T M A X I M I Z AT I O N Example Moline Company produces two products, 1 and 2. The contribution margins per unit of the two

products follow:

Product

Contribution Margin per Unit

Fixed costs are the same regardless of the combination of Products 1 and 2 the firm produces; therefore, the firm wants to maximize the total contribution per period of these two products.

Both products have a positive contribution margin. If Moline Company faces no constraints, it should make (and sell) both products, eliminating our problem. When production of a unit of each product consumes the same quantity of a scarce resource, managers solve the problem by making and selling only the highest contribution item. For our example, if Product 1 and Product 2 each require one hour of machine time and the quantity of machine hours is finite, Moline will choose Product 2, all else being equal. Products usually do not consume equal amounts of scarce resources, however, so the problem is to find the optimal mix of products given the amount of a scarce resource each product consumes.

244 Chapter 7 Differential Cost Analysis for Operating Decisions

Moline Company uses two scarce resources to make the two products, labor time and machine time. Twenty-four hours of labor time and 20 hours of machine time are available each day. The amount of time required to make each product follows:

Product 1 2

Labor Time ..................................................................................................

1 hour per unit 2 hours per unit

Machine Time .............................................................................................

1 hour per unit 1 hour per unit

This problem formulation follows. (X 1 and X 2 refer to the quantity of Products 1 and 2 produced and sold.)

1. Maximize: $3X 1 þ $4X 2 ¼ Total Contribution

2. Subject to:

X 1 þ 2X 2 ¼ 24 Labor Hours

3. X 1 þX 2 ¼ 20 Machine Hours

The first line, the objective function, states the objective of our problem as a linear equation. Here the objective is to maximize total contribution where each unit of Product 1 contributes $3 and each unit of Product 2 contributes $4. The lines that follow specify the parameters of the constraints. Line (2) is the labor time constraint, which states that each unit of Product 1 requires

1 labor hour and each unit of Product 2 requires 2 labor hours. Total labor hours cannot exceed 24 per period (that is, one day). Line (3) is the machine time constraint, which states that Product 1 and Product 2 each use 1 machine hour per unit, and total machine hours cannot exceed 20.

Exhibit 7.15 graphs the constraints. The shaded area shows feasible production; production does not use more scarce resources than are available. The lowercase letters show the corner points. We find the optimal solution by deriving the total contribution margin at each point, using the following steps.

Linear Programming. Graphics Solution: Comparison of Corner and Noncorner Points

EXHIBIT 7.15

24 Labor Time

ab

Machine Time

0 12 20 X 2

Appendix 7.1: Linear Programming 245

Step 1 Find the production level of Product 1 and Product 2 at each point. Points a and c are straight-

forward. At a, X 1 ¼ 20 and X 2 ¼ 0; at c, X 2 ¼ 12 and X 1 ¼ 0. Point b requires solving for two unknowns using the two constraint formulas:

Labor Time:

X 1 þ 2X 2 ¼ 24

Machine Time:

X 1 þX 2 ¼ 20

Setting these two equations equal, we have

24 2 2 4 ¼X 2 :

If X 2 ¼ 4, then

At point b, Moline produces 16 units of Product 1 and 4 units of Product 2. Step 2

Find the total contribution margin at each point. (Recall that the unit contribution margins of Products 1 and 2 are $3 and $4.) Exhibit 7.16 shows the solution. It is optimal to produce at point b,

where X 1 ¼ 16 and X 2 ¼ 4. Why must the optimal solution be at a corner? If production moves away from the corner at point b in any feasible direction, total contribution will not increase and generally will be lower. Exhibit 7.17 shows a movement away from point b in four feasible directions.

Exhibit 7.18 compares contributions at those noncorner points with the contribution at corner point b. Although these examples show intuitively that the contribution margin declines away from the corner point, we can prove mathematically our assertion that the optimal solution always lies on a corner point.

S E N S I T I V I T Y A N A LY S I S The contribution margins and costs in the objective functions are estimates, subject to error.

Decision makers frequently need to know how much the estimates can change before the decision changes.

To demonstrate our point, we use our earlier profit-maximization problem for Moline Company, which we formulated as follows:

Maximize:

$3X 1 þ $4X 2 ¼ Total Contribution

Subject to:

X 1 þ 2X 2 ¼ 24 Labor Hours X 1 þX 2 ¼ 20 Machine Hours:

EXHIBIT 7.16

Optimal Product Mix

X 1 X 2 1 2 Total

a .................................................................................

b .................................................................................

c .................................................................................

246 Chapter 7 Differential Cost Analysis for Operating Decisions

Linear Programming. Graphics Solution: Comparison of Corner and Noncorner Points

Labor Time

Machine Time

3.0 4.5 12 20 X 2 4.0

EXHIBIT 7.18

Comparison of Corner Point and Noncorner Points

Production Contribution

Point

X 1 X 2 1 2 Total

b3 a ...........................................................................................................................

b4 b ...........................................................................................................................

a Let X 1 ¼ 15 and find X 2 as follows: X 1 2

15 2 2X 2 ¼9 X 2 ¼ 4:5:

b Let X 2 ¼ 3 and find X 1 as follows: X 1 2

Appendix 7.1: Linear Programming 247

EXHIBIT 7.19

Optimal Product Mix: Revised Cost Estimates

Production

Contribution

Point a

X 1 X 2 1 2 Total

b The graph in Exhibit 7.15 presents these points. Four units

Linear Programming. Graphics Solution: Increase in

EXHIBIT 7.20

Machine Time from 20 to 21 Hours

Machine Time

Labor c Time

0 12 21 X 2

Suppose that the variable cost estimate for Product 2 was $0.50 per unit too low, so Product 2’s unit contribution margin should have been $3.50 instead of $4.00. What effect would this have? We have calculated the new contributions in Exhibit 7.19. If you compare this exhibit with Exhibit 7.16, you will see that the contribution for Product 2 changes; thus the total contribution changes. The optimal decision to produce 16 units of Product 1 and 4 units of Product 2 does not change, however. In spite of the change in costs and thus in contributions, the decision does not change. In this example, the unit contribution margin of Product 2 would have to drop to less than $3 per unit before the optimal decision would change, assuming that all other things remained constant.

Most linear programming computer programs can provide this type of sensitivity analysis. With it, managers and accountants can ascertain how much a cost or contribution margin can change before the optimal decision will change.

OPPORTUNITYCOSTS Any constrained resource has an opportunity cost, which is the profit forgone by not having an

additional unit of the resource. For example, suppose that Moline Company in our previous example could obtain one additional hour of machine time. With one more hour of machine time,

248 Chapter 7 Differential Cost Analysis for Operating Decisions

level at point b as follows:

The new total contribution at point b would be $3(18) þ $4(3) ¼ $66, compared to $64 when machine time was constrained to 20 hours per day, as shown for point b in Exhibit 7.18. Thus the opportunity cost of not having an extra hour of machine time is $2 ( ¼ $66 – $64).

Linear programming computer programs regularly provide for opportunity costs, called shadow prices. Opportunity cost data indicate the benefits of acquiring more units of a scarce resource. For example, if Moline Company could rent one more machine hour for less than $2 per hour, the company would profit by doing so, all other things being equal.