The XNPV Function
3.7 Short-Term Financial Planning ‡
LP models can often be used to aid a firm in short- or long-term financial planning (also see Section 3.11). Here we consider a simple example that illustrates how linear pro-gramming can be used to aid a corporation’s short-term financial planning.§
Semicond is a small electronics company that manufactures tape recorders and radios. The per-unit labor costs, raw material costs, and selling price of each product are given in Table 10. On December 1, 2002, Semicond has available raw material that is sufficient to manufacture 100 tape recorders and 100 radios. On the same date, the company’s balance sheet is as shown in Table 11, and Semicond’s asset–liability ratio (called the current ra-tio) is 20,000/10,000 2.
Semicond must determine how many tape recorders and radios should be produced during December. Demand is large enough to ensure that all goods produced will be sold.
All sales are on credit, however, and payment for goods produced in December will not Short-Term Financial Planning
E X A M P L E 1 1
(Assume that any fraction of an investment may be purchased.)
3 Suppose that r, the annual interest rate, is 0.20, and that all money in the bank earns 20% interest each year (that is, after being in the bank for one year, $1 will increase to
$1.20). If we place $100 in the bank for one year, what is the NPV of this transaction?
4 A company has nine projects under consideration. The NPV added by each project and the capital required by each project during the next two years is given in Table 9. All
figures are in millions. For example, Project 1 will add $14 million in NPV and require expenditures of $12 million during year 1 and $3 million during year 2. Fifty million is available for projects during year 1 and $20 million is available during year 2. Assuming we may undertake a fraction of each project, how can we maximize NPV?
Group B
5† Finco must determine how much investment and debt to undertake during the next year. Each dollar invested reduces the NPV of the company by 10¢, and each dollar of debt increases the NPV by 50¢ (due to deductibility of interest payments). Finco can invest at most $1 million during the coming year. Debt can be at most 40% of investment. Finco now has $800,000 in cash available. All investment must be paid for from current cash or borrowed money. Set up an LP whose solution will tell Finco how to maximize its NPV. Then graphically solve the LP.
T A B L E 8
Cash Flow (in $ Thousands) at Time
Investment 0 1 2 3
1 6 5 7 9
2 8 3 9 7
T A B L E 9
Project
1 2 3 4 5 6 7 8 9
Year 1 Outflow 12 54 6 6 30 6 48 36 18
Year 2 Outflow 3 7 6 2 35 6 4 3 3
NPV 14 17 17 15 40 12 14 10 12
†Based on Myers and Pogue (1974).
‡This section covers material that may be omitted with no loss of continuity.
§This section is based on an example in Neave and Wiginton (1981).
be received until February 1, 2003. During December, Semicond will collect $2,000 in accounts receivable, and Semicond must pay off $1,000 of the outstanding loan and a monthly rent of $1,000. On January 1, 2003, Semicond will receive a shipment of raw material worth $2,000, which will be paid for on February 1, 2003. Semicond’s manage-ment has decided that the cash balance on January 1, 2003, must be at least $4,000. Also, Semicond’s bank requires that the current ratio at the beginning of January be at least 2.
To maximize the contribution to profit from December production, (revenues to be re-ceived) (variable production costs), what should Semicond produce during December?
Solution Semicond must determine how many tape recorders and radios should be produced dur-ing December. Thus, we define
x1 number of tape recorders produced during December x2 number of radios produced during December To express Semicond’s objective function, note that
100 50 30 $20
90 35 40 $15 As in the Giapetto example, this leads to the objective function
max z 20x1 15x2 (33)
Semicond faces the following constraints:
Constraint 1 Because of limited availability of raw material, at most 100 tape recorders can be produced during December.
Constraint 2 Because of limited availability of raw material, at most 100 radios can be produced during December.
Contribution to profit
Contribution to profit
T A B L E 10 Cost Information for Semicond
Tape Recorder Radio
Selling price $100 $90
Labor cost $ 50 $35
Raw material cost $ 30 $40
T A B L E 11 Balance Sheet for Semicond
Assets Liabilities
Cash $10,000
Accounts receivable§ $ 3,000 Inventory outstanding¶ $ 7,000
Bank loan $10,000
§Accounts receivable is money owed to Semicond by customers who have previously purchased Semicond products.
¶Value of December 1, 2002, inventory 30(100) 40(100)
$7,000.
Constraint 3 Cash on hand on January 1, 2002, must be at least $4,000.
Constraint 4 (January 1 assets)/(January 1 liabilities) 2 must hold.
Constraint 1 is described by
x1 100 (34)
Constraint 2 is described by
x2 100 (35)
To express Constraint 3, note that
January 1 cash on hand December 1 cash on hand
accounts receivable collected during December
portion of loan repaid during December
December rent December labor costs
10,000 2,000 1,000 1,000 50x1 35x2
10,000 50x1 35x2
Now Constraint 3 may be written as
10,000 50x1 35x2 4,000 (36 )
Most computer codes require each LP constraint to be expressed in a form in which all variables are on the left-hand side and the constant is on the right-hand side. Thus, for computer solution, we should write (36 ) as
50x1 35x2 6,000 (36)
To express Constraint 4, we need to determine Semicond’s January 1 cash position, ac-counts receivable, inventory position, and liabilities in terms of x1and x2. We have already shown that
January 1 cash position 10,000 50x1 35x2
Then
January 1 accounts receivable December 1 accounts receivable
accounts receivable from December sales
accounts receivable collected during December
3,000 100x1 90x2 2000
1,000 100x1 90x2
It now follows that
Value of January 1 inventory value of December 1 inventory
value of inventory used in December
value of inventory received on January 1
7,000 (30x1 40x2) 2,000
9,000 30x1 40x2
We can now compute the January 1 asset position:
January 1 asset position January 1 cash position January 1 accounts receivable
January 1 inventory position
(10,000 50x1 35x2) (1,000 100x1 90x2)
(9,000 30x1 40x2)
20,000 20x1 15x2
Finally,
January 1 liabilities December 1 liabilities December loan payment
amount due on January 1 inventory shipment
10,000 1,000 2,000
$11,000 Constraint 4 may now be written as
2 Multiplying both sides of this inequality by 11,000 yields
20,000 20x1 15x2 22,000 Putting this in a form appropriate for computer input, we obtain
20x1 15x2 2,000 (37)
Combining (33)–(37) with the sign restrictions x1 0 and x2 0 yields the follow-ing LP:
max z 20x1 15x2
s.t. 20x1 15x2 100 (Tape recorder constraint) s.t. 20x1 15x2 100 (Radio constraint) s.t. 50x1 35x2 6,000 (Cash position constraint) s.t. 20x1 15x2 2,000 (Current ratio constraint)
x1, x2 0 (Sign restrictions)
When solved graphically (or by computer), the following optimal solution is obtained:
z 2,500, x1 50, x2 100. Thus, Semicond can maximize the contribution of De-cember’s production to profits by manufacturing 50 tape recorders and 100 radios. This will contribute 20(50) 15(100) $2,500 to profits.
P R O B L E M S
Group A
20,000 20x1 15x2
11,000
1 Graphically solve the Semicond problem. 2 Suppose that the January 1 inventory shipment had been valued at $7,000. Show that Semicond’s LP is now infeasible.