Treasury Management Linear Programming

1. LINEAR PROGRAMMING PROBLEMS-
FORMULATIONS - 1
Decision Sciences - 2018
Dr. Gunjan Malhotra
gmalhotra@imt.edu; mailforgunjan@gmail.com
7/23/2018 1IMT, Ghaziabad, INDIA

2. Question 1..
• A small magazine publisher wants to determine the best combination of two
possible magazines to print for the month of July. Backyard magazine, which
he has published for years, is a steady seller. The publisher wants to make
sure he prints at least 400 copies to meet his demand from the newsstands.
Porch is a new venture and has received the benefit of a great deal of
advance publicity. The publisher is hoping that by positioning it near
Backyard, he will pick up some spillover demand from his regular readers.
Also, he is hoping that the advertising campaign will bring in a new type of
reader from a potentially very lucrative market. He wants to print at least 300
copies of Porch. The cover price for Backyard is $3.50; he is pricing Porch at
$4.50 because other magazine in this market seem to be able to command
this type of higher price. The publisher has 36 hours of printing time available
for this production run. He also has 30 hours in the collation department,
where the magazines are actually assembled. Each Backyard requires 2.5
minutes per copy to print and 1.8 minutes per copy to collate. Each Porch
requires 2 minutes to print and 2 minutes to collate. How many of each
magazine should the publisher print to maximize his revenue?

3. Answer 1.
• Let X = number of copies of Backyard,
Y= number of copies of Porch.
• Objective: Maximize revenue = $3.50X + $4.50Y
• Subject to: 2.5X + 2Y  2,160 Print time, minutes
x ≥ 400 Min Backyard to print
Y ≥ 300 Min Porch to print
1.8X + 2Y  1,800 Collate time, minutes
X, Y  0 Non-negativity

4. Question 2..
• Children’s Woodcarving, Inc., manufactures two types of wooden
toys: soldiers and trains. A soldier sells for $27 and uses $10 worth
of raw materials. Each soldier that is manufactured increases
Children’s variable labor and overhead costs by $14. A train sells for
$21 and uses $9 worth of raw materials. Each train built increases
Children’s variable labor and overhead costs by $10. The
manufacturer of wooden soldiers and trains requires two types of
skilled labor: carpentry and finishing. A soldier requires 2 hours of
finishing labor and 1 hour of carpentry labor. A train requires 1 hour
of finishing and 1 hour of carpentry labor. Each week, Children can
obtain all the needed raw material but only 100 finishing hours and
80 carpentry hours. Demand for trains is unlimited, but at the most
40 soldiers are bought each week. Children’s wants to maximize
weekly profit (revenue – costs). Formulate a mathematical model of
Children’s situation that can be used to maximize the firm’s weekly
profit.

5. Answer 2.
• LPP formulation:
• Max. Z = 3 X1 + 2 X2
• Subject to
• 2 X1 + X 2 ≤ 100 (Finishing Constraint)
• X1 + X2 ≤ 80 (Carpentry Constraint)
• X1 ≤ 40 (Constraint on demand for
soldiers)
• X1 , X2 ≥ 0 (sign restriction/ non-negativity
condition)

6. Question 3.
• Truckco manufactures two types of trucks: 1 and 2. Each
truck must go through the painting shop and assembly
shop. If the painting shop were completely devoted to
painting Type 1 trucks, then 800 per day could be painted;
if the painting shop were completely devoted to painting
Type 2 trucks, then 700 per day could be painted. If the
assembly shop were completely devoted to assembling
truck 1 engines, then 1500 per day could be assembled; if
the assembly shop were completely devoted to assembling
truck 2 engines, then 1200 per day could be assembled.
Each type 1 truck contributes $300 to profit; each type 2
truck contributes $500. Formulate an LP that will maximize
Truckco’s profit.

7. Answer 3.
• Decision Variables:
• x1 = Number of Type 1 Trucks produced daily
• x2 = Number of Type 2 Trucks produced daily
• Expressing profit in hundreds of dollars we obtain the
following formulation:
• LPP:
• Max z = 3x1 + 5x2
• s.t. x1/800 + x2/700  1 (Paint Shop Const.)
• x1/1500 + x2/1200  1 (Engine Shop Const.)
• x10, x20

8. Question 4..
• Quitmeyer Electronics Incorporated manufacturers six
microcomputer peripheral devices: internal modems (I), external
modems(E) , graphics circuit boards (C), floppy disk drive (F),
hard disk drives (H), and memory expansion boards (M). Each of
these technical products requires time, in minutes, on three
types of electronic testing equipment, as shown in the table.
Time requirements I E C F H M
Test Device 1 7 3 12 6 18 17
Test Device 2 2 5 3 2 15 17
Test Device 3 5 1 3 2 9 2

9. Question 4…cont..
• The first two test devices are available 120 hours per week. Test
device 3 requires more preventive maintenance and may be used
only 100 hours each week. The market for all six computer
components is vast, and Quitmeyer Electronics believes that it can
sell as many units of each product as it can manufacture. The
following table summarizes the revenues and material costs for
each product:
Device Revenue per unit
($)
Material cost per
unit ($)
Internal modems (I) 200 35
External modems(E) 120 25
graphics circuit boards (C) 180 40
floppy disk drive (F) 130 45
hard disk drives (H) 430 170
memory expansion boards (M) 260 60

10. Question 4…cont..
• In addition, variable labor costs are $15 per
hour or test device1, $12 per hour for test
device 2, and $18 per hour for test device 3.
Quitmeyer wants to determine the product
mix that maximizes profits. Formulate the
problems an LP model solve it by using excel.

11. Answer 4.
• Maximize Profits =
$161.35 I + $92.95 E + $135.50 C + $82.50 F +
$249.80 H + $191.75 M
Subject to
7 I + 3 E + 12 C + 6 F + 18 H + 17 M ≤ 720
2 I + 5 E + 3 C + 2 F + 15 H + 17 M ≤ 720
5 I + E + 3 C + 2 F + 9 H + 2 M ≤ 600
All variables ≥ 0

12. Question 5..
One of the clients has just instructed the Heinlein and Krampf brokerage firm
to invest $250,000 that she obtained recently through the sale of land
holdings in Ohio. The client has a good deal of trust in the firm, but she also
has her own ideas about the distribution of the funds being invested. In
particular, she requests that the firm select whatever stocks and bonds it
believes are well rated, but within the following guidelines:
• Municipal bonds, nursing home stock, and drug company stock should
constitute at least 20%, 10%, and 10%, respectively, of the total amount
invested.
• At least 40% of the funds should be placed in a combination of electronics
and aerospace firms, with each accounting for at least 15%.
• No more than 50% of the total amount invested in electronics and
aerospace firms should be placed in a combination of nursing home and
drug company stock, both of which carry high risk.
Subject to these restraints, the client’s goal is to maximize projected return on
investments. The analysts at Heinlein and Krampf, aware of these guidelines,
prepare a list of high-quality stocks and bonds and their corresponding rates
of return:

13. Question 5…cont..
• Formulate this portfolio selection problem
using LP and solve it by using Excel.
Investment Projected Rate of Return
Los Angeles municipal bonds 5.3%
Thompson Electronics, Inc. 6.8%
United Aerospace Corp. 4.9%
Palmer Drugs 8.4%
Happy days Nursing Homes 11.8%

14. Answer 5.
• Maximize Returns:
0.053 L + 0.068 T + 0.049 U + 0.084 P + 0.118 H
Subject to
L + T + U + P + H ≤ 250000
L ≥ 20% (L + T + U + P + H )
H ≥ 10% (L + T + U + P + H )
P ≥ 10% (L + T + U + P + H )
T + U ≥ 40% (L + T + U + P + H )
T ≥ 15% (L + T + U + P + H )
U ≥ 15% (L + T + U + P + H )
P + H ≤ 50% (T + U)
All variables ≥ 0

15. Question 6..
• The famous Y.S.Chang Restaurant is open 24 hours a day.
Waiters and busboys report for duty at 3 A.M., 7A.M., 11A.M.,
3P.M., 7P.M., or 11P.M., and each works an 8-hour shift. The
following table shows the minimum number of workers
needed during the six periods into which the day is divided:
• How should Chang schedule his workers so that the total staff
required for one day’s operation is minimized?
Period Time Workers Required
1 3A.M. – 7A.M. 3
2 7A.M. – 11A.M. 12
3 11A.M. – 3P.M. 16
4 3P.M. – 7P.M. 9
5 7P.M. – 11P.M. 11
6 11P.M. – 3A.M. 4

16. Answer 6.
• Minimize the Staff Salary:
X1 + X2 + X3 + X4 + X5 + X6
• Subject to:
X1 + X2 ≥ 12
X2 + X3 ≥ 16
X3 + X4 ≥ 9
X4 + X5 ≥ 11
X5 + X6 ≥ 4
X6+ X1 ≥ 3
All variables ≥ 0

17. Question 7..
• South Central Utilities has just announced the August 1
opening of its second nuclear generator at its Baton Rouge,
Louisiana, nuclear power plant. Its personnel department has
been directed to determine how many nuclear technicians
need to be hired and trained over the remainder of the year.
The plant currently employs 350 fully trained technicians and
projects the following personnel needs:
Month Personnel Hours Needed
August 40,000
September 45,000
October 35,000
November 50,000
December 45,000

18. Question 7..cont…
• By Louisiana law, a reactor employee can actually work no more than 130
hours per month. (Slightly over 1 hour per day is used for check-in and
check-out, record keeping, and daily radiation health scans.) Policy at
South Central Utilities also dictates that layoffs are not acceptable in those
months when the nuclear plant is overstaffed. So, if more trained
employees are available than are needed in any month, each worker is still
fully paid, even though he or she is not required to work the 130 hours.
• Training new employees is an important and costly procedure. It takes one
month of one-on-one classroom instruction before a new technician is
permitted to work alone in the reactor facility. Therefore, South Central
must hire trainees one month before they are actually needed. Each
trainee teams up with a skilled nuclear technicians and requires 90 hours
of that employee’s time, meaning that 90 hours less of the technician’s
time are available that month for actual reactor work.
• Personnel department records indicate a turnover rate of trained
technicians at 2% per month. In other words, 2% of the skilled technicians
at the start of any month resign by the end of that month. A trained
technician earns a monthly salary of $4,500, while trainees are paid $2000
during their one month of instruction.
• Formulate this staffing problem using LP and solve it by using Excel.

19. Answer 7.
Let X = no. of trained technicians available at start of the month
Y = no. of trainees beginning in the month
• Minimize total salaries paid
$4500 (X1 + X2 + X3 + X4 + X5) + $2000 (Y1 + Y2 + Y3 + Y4 + Y5)
• Subject to
130 X1 – 90 Y1 ≥ 40000 (Aug. need, hours)
130 X2 – 90 Y2 ≥ 45000 (Sep. need, hours)
130 X3 – 90 Y3 ≥ 35000 (Oct. need, hours)
130 X4 – 90 Y4 ≥ 50000 (Nov. need, hours)
130 X5 – 90 Y5 ≥ 45000 (Dec. need, hours)
X1 = 350 (Starting staff on Aug 1)
X2 = X1 + Y1 -0.02 X1 (Staff on Sep. 1)
X3 = X2 + Y2 -0.02 X2 (Staff on Oct. 1)
X4 = X3 + Y3 -0.02 X3 (Staff on Nov. 1)
X5 = X4 + Y4 -0.02 X4 (Staff on Dec. 1)
All Variables ≥ 0