PAZ351 MATHEMATICAL MODELING FOR BUSINESS APPLICATIONS
LINEAR PROGRAMMING
EXAMPLES
March 27, 2007
1. (Production Mix) A clothier makes coats and slacks. The two resources
required are wool cloth and labor. The clothier has 150 square yards of
wool and 200 hours of labor available. Each coat requires 3 square yards
of wool and 10 hours of labor, whereas each pair of slacks requires 5 square
yards of wool and 4 hours of labor. The profit for a coat is $50, and the
profit for slacks is $40. The clothier wants to determine the number of
coats and pairs of slacks to make so that profit will be maximized.
(a) Formulate a linear programming model for this problem.
(b) Solve this model by using graphical analysis.[Tay07]
2. (Production Mix) A jewelry store makes necklaces and bracelets from gold
and platinum. The store has 18 ounces of gold and 20 ounces of platinum.
Each necklace requires 3 ounces of gold and 2 ounces of platinum, whereas
each bracelet requires 2 ounces of gold and 4 ounces of platinum. The demand for bracelet is no more than four. A necklace earns $300 in profit and
a bracelet, $400. The store wants to determine the number of necklaces
and bracelets to make in order to maximize profit.
(a) Formulate a linear programming model for this problem.
(b) Solve this model by using graphical analysis.[Tay07]
3. (Blending) My diet requires that all the food I eat come from the one of the
four basic food groups (chocolate cake, ice cream, soda and cheesecake).
At present, the following four foods are available for consumption: brownies, chocolate ice cream, cola and pineapple cheesecake. Each brownie
costs 50¢, each scoop of chocolate ice cream costs 20¢, each bottle of cola
costs 30¢, and each piece of pineapple cheesecake costs 80. Each day, I
must ingest at least 500 calories, 6 oz of chocolate, and 10 oz of sugar
and 8 oz of fat. The nutritional content per unit of each food is shown
in Table1 . Formulate a linear programming model that can be used to
satisfy my daily nutritional requirements at minimum cost[Win04].
4. (Production Mix) Giapetto’s Woodcarving, Inc., manufactures two types
of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10
1
Calories
Brownie
Chocolate Ice cream (1 Scoop)
Cola (1 Bottle)
Pineapple Cheesecake (1 Piece)
400
200
150
500
Chocolate
(ounces)
3
2
0
0
Sugar
(ounces)
2
2
4
4
Fat
(ounces)
2
4
1
5
Table 1: Diet Problem
worth of raw materials. Each soldier that is manufactured increases Giapettos variable labor and overhead costs by $14. A train sells for $21 and
uses $9 worth of raw materials. Each train built increases Giapettos variable labor and overhead costs by $10. The manufacture of wooden soldiers
and trains requires two types of skilled labor: carpentry and finishing. A
soldier requires 2 hours of finishing and 1 hour of carpentry labor. A train
requires 1 hour of finishing and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw material but only 100 finishing hours
and 80 carpentry hours. Demand for train is unlimited, but at most 40
soldiers are bought each week. Giapetto wants to maximize weekly profit
(revenues costs). Formulate a mathematical model of Giapettos situation
that can be used to maximize Giapettos weekly profit[Win04].
Price
Raw Materials
Labor Cost
Carpentry
Finishing
Demand
Soldier
$27
$10
$14
1 hour
2 hours
40
Train
$21
$9
$10
1 hour
1 hour
Unlimited
Capacity
80 hours
100 hours
-
Table 2: Giapetto’s Woodcarving, Inc.
5. (Production Mix) A knitting machine can produce 1, 000 pants or 3, 000
shirts (or a combination of the two) each day. The finishing department
can handle either 1, 500 pants or 2, 000 shirts (or a combination of the
two) each day. The marketing department requires that at least 400
pants be produced each day. The company’s stated objective is profit
maximization[TM93].
(a) If the profit from a pair of pants is $4 and that derived from a shirt is
$1.50, how many of each type should be produced? Solve graphically.
(b) If the profit from a shirt is $2, what should be the minimum profit
derived from selling a pair of pants that will justify production of
pants?
(c) Examine the solutions to (a) and (b) and interpret the difference
between them.
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6. (Employee Assignment) A store sells men’s and ladies’ tennis shoes. It
makes a profit of $1 a pair on the men’s shoes and $1.20 a pair on ladies’.
It takes two minutes of a salesperson’s time and two minutes of a cashier’s
time to sell a pair of men’s shoes. It takes three minutes of a salesperson’s
time and one minute of a cashier’s time per pair of women shoes. The store
is open eight hours per day, during which time there are two salespersons
and one cashier on duty. How much of the salespersons’ and the cashiers’
time should be allocated to the men’s and ladies’ shoes should the store
sell in order to maximize profit each day? What is the profit[TM93]?
(a) Formulate as a linear programming problem.
(b) Solve graphically
7. (Production Mix) The Flair Furniture Company produces inexpensive tables and chairs. The production process for each is similar in that both
require a certain number of hours of carpentry work and a certain number of labor hours in the painting and vanishing department. Each table
takes 4 hours of carpentry and 2 hours in the painting and varnishing shop.
Each chair requires 3 hours of carpentry and 1 hour in the painting and
varnishing. During the current production period, 240 hours of carpentry
and 100 hours in the painting and varnishing time are available. Each
table sold yields a profit of $7; each chair produced is sold for a $5 profit.
Flair Furnitures problem is to determine the best possible combination of
tables and chairs to manufacture in order to reach the maximum profit.
(a) Formulate this production mix situation as an LP problem[RSH06].
8. (Media Selection) Dorian Auto manufactures luxury cars and trucks. The
company believes that its most likely customers are high-income women
and men. To reach these groups, Dorian Auto has embarked on an ambitious TV advertising campaign and has decided to purchase 1-minute
commercial spots on two type of programs: comedy shows and football
games. Each comedy commercial is seen by 7 million high-income women
and 2 million high-income men. Each football commercial is seen by 2
million high-income women and 12 million high-income men. A 1-minute
comedy ad costs $50,000, and a 1-minute football ad costs $100,000. Dorian would like the commercials to be seen by at least 28 million highincome women and 24 million high-income men. Use linear programming
to determine how Dorian Auto can meet its advertising requirements at
minimum cost[Win04].
9. (Production Mix) An auto company manufactures cars and trucks. Each
vehicle must be processed in the paint shop and body assembly shop. If
the paint shop were only painting the trucks, then 40 per day could be
painted. If the paint shop were only painting the cars, then 60 per day
could be painted. If the body shop were only producing cars, it could
process 50 per day. If the body shop were only producing trucks, it could
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process 50 per day. Each truck contributes $300 to profit, and each car
contributes $200 to profit.
(a) Use linear programming to determine a daily production schedule
that will maximize the companys profits.
(b) Suppose that auto dealers require that the auto company produce at
least 30 trucks and 20 cars. Find the optimal solution to the new
LP[Win04].
10. (Production Mix) Reddy Mikks produces both interior and exterior paints
from two raw materials, M1 and M2. Table3 provides the basic data of the
problem A market surveys restricts the maximum daily demand of interior
Raw material, M1
Raw material, M2
Profit per ton($1000)
Tons of raw material per ton of
Exterior Paint Interior Paint
6
4
1
2
5
4
Maximum daily
availability (tons)
24
6
Table 3: Reddy Mikks
paint to 2 tons. Additionally, the daily demand for interior demand cannot
exceed that of exterior paint by more than one ton. Reddy Mikks wants to
determine the optimum (best) product mix of interior and exterior paints
that maximizes the total daily profit[Tah97].
11. (Blending) The Holiday Meal Turkey Ranch is considering buying two
different brands of turkey feed and blending them to provide a good, lowcost diet for its turkeys. Each feed contains, in varying proportions, some
or all of the three nutritional ingredients essential for fattening turkeys.
Each pound of Brand 1 purchased contains 5 ounces of ingredients A, 4
ounces of ingredient B, and 1/2 ounce of ingredient C. Each pound of
Brand 2 contains 10 ounces of ingredient A, 3 ounces of ingredient C. The
Brand 1 feed costs the ranch 2 cents a pound , while the Brand 2 feed costs
3 cents a pound. Table4 shows the minimum monthly intake requirement
for each nutritional ingredient. The owner of the ranch would like to use
LP to determine the lowest- cost diet that meets the minimum monthly
intake requirement for each nutritional ingredient[RSH06].
Ingredient
A
B
C
Minimum monthly
requirement per turkey
90oz.
48oz.
1.5oz.
Table 4: Holiday Meal Turkey Ranch
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12. (Labor Scheduling) A post office requires different numbers of full-time
employees on different days of the week. The number of full-time employees required on each day is given in Table5. Union rules state that each
full-time employee must work five consecutive days and then receive two
days off. For example, an employee who works Monday to Friday must
be off on Saturday and Sunday. The post office wants to meet its daily
requirements using only full-time employees. Formulate an LP that the
post office can use to minimize the number of full-time employees that
must be hired[Win04].
Day
Day
Day
Day
Day
Day
Day
Day
1
2
3
4
5
6
7
=
=
=
=
=
=
=
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
Number of Full-time Employees Required
17
13
15
19
14
16
11
Table 5: Post Office
13. (Production Mix) The Dakota Furniture Company manufactures desks,
tables and chairs. The manufacture of each type of furniture requires lumber and two types of skilled labor: finishing and carpentry. The amount
of each resource needed to make each type of furniture is given in Table6.
Currently, 48 board feet of lumber, 20 finishing hours and 8 carpentry
Resource
Lumber (board ft)
Finishing hours
Carpentry hours
Desk
8
4
2
Table
6
2
1.5
Chair
1
1.5
0.5
Table 6: Dakota Furniture Company
hours are available. A desk sells for $60, a table for $30, and a chair for
$20. Dakota believes that demand for desks and chairs is unlimited, but
at most 5 tables can be sold. Because the available resources have already
been purchased, Dakota wants to maximize total revenue.
14. (Production) A baker has 30 oz of flour and 5 packages of yeast. Baking
a loaf of bread requires 5 oz of flour and 1 package of yeast. Each loaf
of bread can be sold for 30¢. The baker may purchase additional flour at
4¢/oz or sell leftover flour at the same price. Formulate and solve an LP
to help the baker maximize profits (revenues-costs).
15. (Financial) Semicond is a small electronics company that manufactures
tape recorders and radios. The per-unit labor costs, raw material costs,
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Selling Price
Labor Cost
Raw Material Cost
Tape Recorder
$100
$50
$30
Radio
$90
$35
$40
Table 7: Semicond Price-Costs
and selling price of each product are given in Table7. On December 1,
1998, Semicond has a available raw material that is sufficient to manufacture 100 tape recorders and 100 radios. On the same date, the companys
balance sheet is as shown in Table8, and Semiconds asset / liability ratio
(called the current ratio) is 20000/10000 = 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
Cash
Accounts receivable1
Inventory Outstanding
Bank Loan
2
Assets
$10,000
$3,000
$7,000
Liabilities
$10,000
Table 8: Semicond Price-Costs
sold. All sales are on credit, however, and payment for goods produced in
December will not be received until February 1, 1999. 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, 1999, Semicond will receive a shipment of raw material worth
$2, 000, which will be paid for the January 1, 1999. Semiconds management has decided that the cash balance on January 1, 1999 must be at
least $4, 000. also, Semiconds bank requires that the current ratio at the
beginning of the January be at least 2. To maximize the contribution
to profit from December production, (revenues to be received) (variable
production costs), what should Semicond produce during December?
16. (Media Selection) The Win Big Gambling Club promotes gambling junkets from a large Midwestern city to casinos in Bahamas. The club has
budgeted up to $8, 000 per week for local advertising. The money is to be
allocated among four promotional media: TV spots, newspaper ads, and
two types of radio advertisements. Win Big’s goal is to reach the largest
possible high-potential audience through various media. Table9 presents
the number of potential gamblers reached by making use of advertisement
in each of four media. It also provides the cost per advertisement placed
and the maximum number of ads that can be purchased per week.
Win Big’s contractual arrangements require that at least five radio spots
be placed each week. To ensure a broad-scoped promotional campaign,
management also insists that no more than $1, 800 can be spent on radio
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Medium
TV Spot(1 minute)
Daily Newspaper(full-page ad)
Radio Spot(30 seconds, prime time)
Radio Spot(1 minute, afternoon)
Audience
Reached per ad
5,000
8,500
2,400
2,800
Cost
ad($)
800
925
290
380
Maximum
ads per week
12
5
25
20
Table 9: Win Big Gambling Club
advertising each week. Formulate this media selection problem by linear
programming [RSH06].
17. (Marketing Research) Management Sciences Associates (MSA) is a marketing and computer research firm based in Washington D.C., that handles
consumer surveys. One of its clients is a national press service that periodically conducts political polls on issues of several requirements in order
to draw statistically valid conclusions on the sensitive issue of new U.S.
immigration laws:
(a) Survey at least 2, 300 U.S. households in total.
(b) Survey at least 1, 000 households whose heads are 30 years of age or
younger.
(c) Survey at least 600 households whose heads are between 31 and 50
years of age.
(d) Ensure that at least 15% of those surveyed live in state that borders
on Mexico.
(e) Ensure that no more than 20% of those surveyed who are 51 years of
age or over live in a state that borders on Mexico.
MSA decides that all surveys should be conducted in person. It estimates
that the costs of reaching people in each age and region category are as
in Table10. MSA’s goal is to meet the five sampling requirements at the
Region
State bordering Mexico
State not bordering Mexico
Cost per Person Surveyed ($)
Age≤30 Age 31-50 Age≥51
$7.50
$6.80
$5.50
$6.90
$7.25
$6.10
Table 10: Management Sciences Associates
least possible cost. Formulate an LP to solve MSA’s problem [RSH06].
18. (Pricing and Marketing Strategy) The I. Kruger Paint and Wallpaper
Store is a large retail distributor of the Supertrex brand of vinyl wallcoverings. Kruger will enhance its citywide image in Miami if it can outsell
other retail stores in total number of rolls of Supertrex next year. It is
able to estimate the demand function as follows:
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Number of rolls of Supertrex sold = 20 x dollars spent on advertising + 6.8 x dollars spent on in-store displays + 12 x dollars
invested in on-hand wallpaper inventory - 65,000 x percentage
markup taken above wholesale cost of a roll
The store budgets a total of $17, 000 for advertising, in-store displays, and
on-hand inventory of Supertrex for next year. It decides it must spend
at least $3, 000 on advertising; in addition, at least 54% of the amount
invested in on-hand inventory should be devoted to displays. Markups
on Supertrex seen at other local stores range from 20% to 45%. Kruger
decides that its markup had the best be in this range as well.
(a) Formulate as an LP problem.
(b) Solve the problem.
(c) What is the difficulty with the answer?
(d) What constraint would you add[Tay07]?
19. (Employee Scheduling) Hong Kong Bank of Commerce and Industry is a
busy bank that has requirements for between 10 and 18 tellers, depending
on the time of the day. The lunch time, from noon to 2P.M., is usually the
heaviest. Table11 indicates the workers needed at various hours that the
bank is open.
Time Period
9 A.M.-10 A.M.
10 A.M.-11 A.M.
11 A.M.-Noon
Noon-1 P.M.
1 P.M.-2 P.M.
2 P.M.-3 P.M.
3 P.M.-4 P.M.
4 P.M.-5 P.M.
Number of Tellers Required
10
12
14
16
18
17
15
10
Table 11: Hong Kong Bank of Commerce and Industry
The bank now employs 12 full-time tellers, but many people are on its
roster of available part-time employees. A part-time employee must put
in exactly four hours per day but can start any time between 9A.M. and
1P.M. Part-timers are a fairly inexpensive labor pool, since no retirement
or lunch benefits are provided for them. Full-timers, on the other hand,
work from 9A.M. to 5P.M. but are allowed 1 hour for lunch.(Half of the full
timers eat at 11A.M., the other half at noon.) Full-timers thus provide 35
hours per week of productive labor time.
By corporate policy, the bank limits part-time hours to a maximum of
50% of the day’s total requirement. Part-timers earn $8 per hour (or $32
per day) on average, and full timers can earn $100 per day in salary and
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benefits, on average. The bank would like to set a schedule that would
minimize its total personnel cost. It will release one or more of its full-time
tellers if it is possible to do so[RSH06].
20. (Shipping) The Top Speed Bicycle Co. manufactures and markets a line
of 10-speed bicycles nationwide. The firm has final assembly plants in two
cities in which labor costs are low, New Orleans and Omaha. Its three
major warehouses are located near the large market areas of New York,
Chicago, and Los Angeles.
The sales requirements for the next year at the New York warehouse are
10,000 bicycles, at the Chicago warehouse 8,000 bicycles, and at the London warehouse 15,000 bicycles. The factory capacity at each location is
limited. New Orleans can assemble and ship 20,000 bicycles; the Omaha
plant can produce 15,000 bicycles per year. The cost of shipping one bicycle from each factory to each warehouse differs, and these unit shipping
costs are as in Table12 The company wishes to develop a shipping schedule
New Orleans
Omaha
New York
$2
$3
Chicago
$3
$1
Los Angeles
$5
$4
Table 12: Top Speed Bicycle Co
that will minimize its total annual transportation costs[RSH06].
21.
References
[RSH06] B. Render, R.M. Stair, and M. Hanna. Quantitative Analysis for Management. Prentice Hall, New Jersey, 9 edition, 2006. ITU Library
Number: T56.R46 2006.
[Tah97] H. Taha. Operations Research: an Introduction. Prentice Hall Inc.,
USA, 6 edition, 1997.
[Tay07] B.W. Taylor. Introduction to Management Science. Prentice Hall, New
Jersey, 9 edition, 2007. ITU Library Number: T56.T39.1990/T56.T39
1986.
[TM93] E. Turban and J.R. Meredith. Fundamentals of Management Science.
Plano, Tex. : Business Publications, 6 edition, 1993. ITU Library
Number: HD30.23.T87 1981.
[Win04] W.L. Winston. Operations Research, Applications and Algorithms.
Thomson Learning Inc. , Canada, 4 edition, 2004.
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