Real-World Applications
3.11 Multiperiod Financial Models
during the month in which it is purchased). At the beginning of month 1, 100 trucks and 200 cars are in inventory. At the end of each month, a holding cost of $150 per vehicle is assessed. Each car gets 20 mpg, and each truck gets 10 mpg. During each month, the vehicles produced by the company must average at least 16 mpg. Formulate an LP to meet the demand and mileage requirements at minimum cost (include steel costs and holding costs).
6 Gandhi Clothing Company produces shirts and pants.
Each shirt requires 2 sq yd of cloth, each pair of pants, 3.
During the next two months, the following demands for shirts and pants must be met (on time): month 1—10 shirts, 15 pairs of pants; month 2—12 shirts, 14 pairs of pants.
During each month, the following resources are available:
month 1—90 sq yd of cloth; month 2—60 sq yd. (Cloth that is available during month 1 may, if unused during month 1, be used during month 2.)
During each month, it costs $4 to make an article of clothing with regular-time labor and $8 with overtime labor.
During each month, a total of at most 25 articles of cloth-ing may be produced with regular-time labor, and an un-limited number of articles of clothing may be produced with overtime labor. At the end of each month, a holding cost of
$3 per article of clothing is assessed. Formulate an LP that can be used to meet demands for the next two months (on time) at minimum cost. Assume that at the beginning of month 1, 1 shirt and 2 pairs of pants are available.
7 Each year, Paynothing Shoes faces demands (which must be met on time) for pairs of shoes as shown in Table 37. Workers work three consecutive quarters and then receive one quarter off. For example, a worker may work during quarters 3 and 4 of one year and quarter 1 of the next year.
During a quarter in which a worker works, he or she can produce up to 50 pairs of shoes. Each worker is paid $500 per quarter. At the end of each quarter, a holding cost of $50 per pair of shoes is assessed. Formulate an LP that can be used to minimize the cost per year (labor holding) of meeting the demands for shoes. To simplify matters, assume
that at the end of each year, the ending inventory is zero.
(Hint: It is allowable to assume that a given worker will get the same quarter off during each year.)
8 A company must meet (on time) the following demands:
quarter 1—30 units; quarter 2—20 units; quarter 3—40 units. Each quarter, up to 27 units can be produced with regular-time labor, at a cost of $40 per unit. During each quarter, an unlimited number of units can be produced with overtime labor, at a cost of $60 per unit. Of all units produced, 20% are unsuitable and cannot be used to meet demand. Also, at the end of each quarter, 10% of all units on hand spoil and cannot be used to meet any future demands. After each quarter’s demand is satisfied and spoilage is accounted for, a cost of $15 per unit is assessed against the quarter’s ending inventory. Formulate an LP that can be used to minimize the total cost of meeting the next three quarters’ demands. Assume that 20 usable units are available at the beginning of quarter 1.
9 Donovan Enterprises produces electric mixers. During the next four quarters, the following demands for mixers must be met on time: quarter 1—4,000; quarter 2—2,000;
quarter 3—3,000; quarter 4—10,000. Each of Donovan’s workers works three quarters of the year and gets one quarter off. Thus, a worker may work during quarters 1, 2, and 4 and get quarter 3 off. Each worker is paid $30,000 per year and (if working) can produce up to 500 mixers during a quarter. At the end of each quarter, Donovan incurs a holding cost of $30 per mixer on each mixer in inventory. Formulate an LP to help Donovan minimize the cost (labor and inventory) of meeting the next year’s demand (on time). At the beginning of quarter 1, 600 mixers are available.
T A B L E 37
Quarter 1 Quarter 2 Quarter 3 Quarter 4
600 300 800 100
money market funds. Returns from investments may be immediately reinvested. For ex-ample, the positive cash flow received from investment C at time 1 may immediately be reinvested in investment B. Finco cannot borrow funds, so the cash available for invest-ment at any time is limited to cash on hand. Formulate an LP that will maximize cash on hand at time 3.
Solution Finco must decide how much money should be placed in each investment (including money market funds). Thus, we define the following decision variables:
A dollars invested in investment A B dollars invested in investment B C dollars invested in investment C D dollars invested in investment D E dollars invested in investment E
St dollars invested in money market funds at time t (t 0, 1, 2)
Finco wants to maximize cash on hand at time 3. At time 3, Finco’s cash on hand will be the sum of all cash inflows at time 3. From the description of investments A–E and the fact that from time 2 to time 3, S2will increase to 1.08S2,
Time 3 cash on hand B 1.9D 1.5E 1.08S2
Thus, Finco’s objective function is
max z B 1.9D 1.5E 1.08S2 (62)
In multiperiod financial models, the following type of constraint is usually used to relate decision variables from different periods:
Cash available at time t cash invested at time t
uninvested cash at time t that is carried over to time t 1 If we classify money market funds as investments, we see that
Cash available at time t cash invested at time t (63) Because investments A, C, D, and S0are available at time 0, and $100,000 is available at time 0, (63) for time 0 becomes
100,000 A C D S0 (64)
At time 1, 0.5A 1.2C 1.08S0is available for investment, and investments B and S1 are available. Then for t 1, (63) becomes
T A B L E 38
Cash Flow ($) at Time*
0 1 2 3
A 1 0.50 1 0
B 0 1 0.50 1
C 1 1.2 0 0
D 1 0 0 1.9
E 0 0 1 1.5
*Note: Time 0 present; time 1 1 year from now; time 2 2 years from now; time 3 3 years from now.
0.5A 1.2C 1.08S0 B S1 (65) At time 2, A 0.5B 1.08S1is available for investment, and investments E and S2are available. Thus, for t 2, (63) reduces to
A 0.5B 1.08S1 E S2 (66)
Let’s not forget that at most $75,000 can be placed in any of investments A–E. To take care of this, we add the constraints
A 75,000 (67)
B 75,000 (68)
C 75,000 (69)
D 75,000 (70)
E 75,000 (71)
Combining (62) and (64)–(71) with the sign restrictions (all variables 0) yields the fol-lowing LP:
max z B 1.9D 1.5E 1.08S2
s.t. A C D S0 100,000 0.5A 1.2C 1.08S0 B S1
A 0.5B 1.08S1 E S2
A 75,000 B 75,000 C 75,000 D 75,000 E 75,000 A, B, C, D, E, S0, S1, S2 0
We find the optimal solution to be z 218,500, A 60,000, B 30,000, D 40,000, E 75,000, C S0 S1 S2 0. Thus, Finco should not invest in money market funds. At time 0, Finco should invest $60,000 in A and $40,000 in D. Then, at time 1, the
$30,000 cash inflow from A should be invested in B. Finally, at time 2, the $60,000 cash inflow from A and the $15,000 cash inflow from B should be invested in E. At time 3, Finco’s $100,000 will have grown to $218,500.
You might wonder how our formulation ensures that Finco never invests more money at any time than the firm has available. This is ensured by the fact that each variable Si
must be nonnegative. For example, S0 0 is equivalent to 100,000 A C D 0, which ensures that at most $100,000 will be invested at time 0.
Real-World Application
Using LP to Optimize Bond Portfolios
Many Wall Street firms buy and sell bonds. Rohn (1987) discusses a bond selection model that maximizes profit from bond purchases and sales subject to constraints that minimize the firm’s risk exposure. See Problem 4 for a simplified version of this model.
P R O B L E M S
Group A
1 A consultant to Finco claims that Finco’s cash on hand at time 3 is the sum of the cash inflows from all investments, not just those investments yielding a cash inflow at time 3.
Thus, the consultant claims that Finco’s objective function should be
max z 1.5A 1.5B 1.2C 1.9D 1.5E
1.08S0 1.08S1 1.08S2
Explain why the consultant is incorrect.
2 Show that Finco’s objective function may also be written as max z 100,000 0.5A 0.5B 0.2C 0.9D 0.5E
0.08S0 0.08S1 0.08S2
3 At time 0, we have $10,000. Investments A and B are available; their cash flows are shown in Table 39. Assume that any money not invested in A or B earns no interest.
Formulate an LP that will maximize cash on hand at time 3. Can you guess the optimal solution to this problem?
Group B
4† Broker Steve Johnson is currently trying to maximize his profit in the bond market. Four bonds are available for purchase and sale, with the bid and ask price of each bond as shown in Table 40. Steve can buy up to 1,000 units of each bond at the ask price or sell up to 1,000 units of each bond at the bid price. During each of the next three years, the person who sells a bond will pay the owner of the bond the cash payments shown in Table 41.
Steve’s goal is to maximize his revenue from selling bonds less his payment for buying bonds, subject to the constraint that after each year’s payments are received, his current cash position (due only to cash payments from bonds and not purchases or sale of bonds) is nonnegative. Assume
that cash payments are discounted, with a payment of $1 one year from now being equivalent to a payment of 90¢
now. Formulate an LP to maximize net profit from buying and selling bonds, subject to the arbitrage constraints previ-ously described. Why do you think we limit the number of units of each bond that can be bought or sold?
5 A small toy store, Toyco projects the monthly cash flows (in thousands of dollars) in Table 42 during the year 2003.
A negative cash flow means that cash outflows exceed cash inflows to the business. To pay its bills, Toyco will need to borrow money early in the year. Money can be borrowed in two ways:
a Taking out a long-term one-year loan in January. In-terest of 1% is charged each month, and the loan must be paid back at the end of December.
b Each month money can be borrowed from a short-term bank line of credit. Here, a monthly interest rate of 1.5% is charged. All short-term loans must be paid off at the end of December.
At the end of each month, excess cash earns 0.4% in-terest. Formulate an LP whose solution will help Toyco maximize its cash position at the beginning of January, 2004.
6 Consider Problem 5 with the following modification:
Each month Toyco can delay payments on some or all of the cash owed for the current month. This is called “stretching payments.” Payments may be stretched for only one month, and a 1% penalty is charged on the amount stretched. Thus, if it stretches payments on $10,000 cash owed in January, then it must pay 10,000(1.01) $10,100 in February. With this modification, formulate an LP that would help Toyco maximize its cash on hand at the beginning of January 1, 2004.
†Based on Rohn (1987).
T A B L E 39
Time A B
0 $1 $0
1 $0.2 $1
2 $1.5 $0
3 $0 $1.0
T A B L E 40
Bond Bid Price Ask Price
1 980 990
2 970 985
3 960 972
4 940 954
T A B L E 41
Year Bond 1 Bond 2 Bond 3 Bond 4
1 100 80 70 60
2 110 90 80 50
3 1,100 1,120 1,090 1,110
T A B L E 42
Month Cash Flow Month Cash Flow
January 12 July 7
February 10 August 2
March 8 September 15
April 10 October 12
May 4 November 7
June 5 December 45
7 Suppose we are borrowing $1,000 at 12% annual interest with 60 monthly payments. Assume equal payments are made at the end of month 1, month 2, . . . month 60. We know that entering into Excel the function
PMT(.01, 60, 1,000) would yield the monthly payment ($22.24).
It is instructive to use LP to determine the montly pay-ment. Let p be the (unknown) monthly paypay-ment. Each month we owe .01 (our current unpaid balance) in interest. The remainder of our monthly payment is used to reduce the un-paid balance. For example, suppose we un-paid $30 each month.
At the beginning of month 1, our unpaid balance is $1,000.
Of our month 1 payment, $10 goes to interest and $20 to paying off the unpaid balance. Then we would begin month 2 with an unpaid balance of $980. The trick is to use LP to determine the monthly payment that will pay off the loan at the end of month 60.
8 You are a CFA (chartered financial analyst). Madonna has come to you because she needs help paying off her credit card bills. She owes the amounts on her credit cards shown in Table 43. Madonna is willing to allocate up to $5,000 per month to pay off these credit cards. All cards must be paid off within 36 months. Madonna’s goal is to minimize the total of all her payments. To solve this problem, you must understand how interest on a loan works. To illustrate, suppose Madonna pays $5,000 on Saks during month 1.
Then her Saks balance at the beginning of month 2 is 20,000 (5,000 .005(20,000))
This follows because during month 1 Madonna incurs .005(20,000) in interest charges on her Saks card. Help Madonna solve her problems!
9 Winstonco is considering investing in three projects. If we fully invest in a project, the realized cash flows (in millions of dollars) will be as shown in Table 44. For example, project 1 requires cash outflow of $3 million today
and returns $5.5 million 3 years from now. Today we have
$2 million in cash. At each time point (0, .5, 1, 1.5, 2, and 2.5 years from today) we may, if desired, borrow up to $2 million at 3.5% (per 6 months) interest. Leftover cash earns 3% (per 6 months) interest. For example, if after borrowing and investing at time 0 we have $1 million we would receive
$30,000 in interest at time .5 years. Winstonco’s goal is to maximize cash on hand after it accounts for time 3 cash flows. What investment and borrowing strategy should be used? Remember that we may invest in a fraction of a project. For example, if we invest in .5 of project 3, then we have cash outflows of $1 million at time 0 and .5.
T A B L E 43
Card Balance ($) Monthly Rate (%)
Saks Fifth Avenue 20,000 .5
Bloomingdale’s 50,000 1
Macys 40,000 1.5
T A B L E 44
Cash Flow
Time (Years) Project 1 Project 2 Project 3
0 3.0 2 2.0
0.5 1.0 .5 2.0
1 1.8 1.5 1.8
1.5 1.4 1.5 1
2 1.8 1.5 1
2.5 1.8 1.2 1
3 5.5 1 6