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Grand Canyon BUS 660 Week 6 Assignment Problems

LINEAR PROGRAMMING FORMULATION PROBLEMS AND SOLUTIONS

7-14 The Electrocomp Corporation manufactures two
electrical products: air conditioners and large fans. The assembly process for
each is similar in that both require a certain amount of wiring and drilling.
Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan
must go through 2 hours of wiring and 1 hour of drilling. During the next
production period, 240 hours of wiring time are available and up to 140 hours
of drilling time maybe used. Each air conditioner sold yields a profit of $25.
Each fan assembled may be sold for a $15 profit. Formulate and solve this LP
production mix situation to find the best combination of air conditioners and
fans that yields the highest profit. Use the corner point graphical approach.

7-15 Electrocomp’s management realizes that it
forgot to include two critical constraints (see Problem 7-14). In particular,
management decides that to ensure an adequate supply of air conditioners for a
contract, at least 20 air conditioners should be manufactured. Because
Electrocomp incurred an oversupply of fans in the preceding period, management
also insists that no more than 80 fans be produced during this production
period. Resolve this product mix problem to find the new optimal solution.

7-16 A candidate for mayor in a small town has
allocated $40,000 for last-minute advertising in the days preceding the
election. Two types of ads will be used: radio and television. Each radio ad
costs $200 and reaches an estimated 3,000 people. Each television ad costs $500
and reaches an estimated 7,000 people. In planning the advertising campaign,
the campaign manager would like to reach as many people as possible, but she
has stipulated that at least 10 ads of each type must be used. Also, the number
of radio ads must be at least as great as the number of television ads. How
many ads of each type should be used? How many people will this reach?

7-17 The Outdoor Furniture Corporation
manufactures two products, benches and picnic tables, for use in yards and
parks. The firm has two main resources: its carpenters (labor force) and a
supply of redwood for use in the furniture. During the next production cycle,
1,200 hours of labor are available under a union agreement. The firm also has a
stock of 3500 feet of good-quality redwood. Each bench that Outdoor Furniture
produces requires 4 labor hours and 10 feet of redwood; each picnic table takes
6 labor hours and 35 feet of redwood. Completed benches will yield a profit of
$9 each, and tables will result in a profit of $20 each. How many benches and
tables should Outdoor Furniture produce to obtain the largest possible profit?
Use the graphical LP approach.

7-18 The dean of the Western College of Business
must plan the school’s course offerings for the fall semester. Student demands
make it necessary to offer at least 30 undergraduate and 20 graduate courses in
the term. Faculty contracts also dictate that at least 60 courses be offered in
total. Each undergraduate course taught costs the college an average of $2,500
in faculty wages, and each graduate course costs $3,000. How many undergraduate
and graduate courses should be taught in the fall so that total faculty
salaries are kept to a minimum?

7-19 MSA Computer Corporation manufactures two
models of minicomputers, the Alpha 4 and the Beta 5. The firm employs five
technicians, working 160 hours each per month, on its assembly line. Get research paper samples and course-specific study resources under   homework for you course hero writing service – Manage ment
insists that full employment (i.e., all 160 hours of time) be maintained for
each worker during next month’s operations. It requires 20 labor hours to assemble
each Alpha 4 computer and 25 labor hours to assemble each Beta 5 model. MSA
wants to see at least 10 Alpha 4s and at least 15 Beta 5s produced during the
production period. Alpha 4s generate $1,200 profit per unit, and Beta 5s yield
$1,800 each. Determine the most profitable number of each model of minicomputer
to produce during the coming month.

7-20 A winner of the Texas Lotto has decided to
invest $50,000 per year in the stock market. Under consideration are stocks for
a petrochemical firm and a public utility. Although a long-range goal is to get
the highest possible return, some consideration is given to the risk involved
with the stocks. A risk index on a scale of 1–10 (with 10 being the most risky)
is assigned to each of the two stocks. The total risk of the portfolio is found
by multiplying the risk of each stock by the dollars invested in that stock.
The following table provides a summary of the return and risk:

Stock

Estimated
Return

Risk
Index

Petrochemical

12%

9

Utility

6%

4

The investor would like to maximize the return on the
investment, but the average risk index of the investment should not be higher
than 6. How much should be invested in each stock? What is the average risk for
this investment? What is the estimated return for this investment?

7-21 Referring to the Texas Lotto situation in
Problem 7-20, suppose the investor has changed his attitude about the
investment and wishes to give greater emphasis to the risk of the investment.
Now the investor wishes to minimize the risk of the investment as long as a
return of at least 8% is generated. Formulate this as an LP problem and find
the optimal solution. How much should be invested in each stock? What is the
average risk for this investment? What is the estimated return for this
investment?

7-24 The stock brokerage firm of Blank, Leibowitz, and Weinberger has
analyzed and recommended two stocks to an investors’ club of college
professors. The professors were interested in factors such as short term
growth, intermediate growth, and dividend rates. These data on each stock are
as follows:

Stock

Factor

Louisiana
Gas and Power

Trimex Insulation Company

Short term growth potential, per dollar invested

.36

.24

Intermediate growth potential (over next three years), per
dollar invested

1.67

1.5

Dividend rate potential

4%

8%

Each member of the club has an investment goal of (1) an
appreciation of no less than $720 in the short term, (2) an appreciation of at
least $5,000 in the next three years, and (3) a dividend income of at least
$200 per year. What is the smallest investment that a professor can make to
meet these three goals?

7-25 Woofer Pet Foods produces a low-calorie dog
food for overweight dogs. This product is made from beef products and grain.
Each pound of beef costs $0.90, and each pound of grain costs $0.60. A pound of
the dog food must contain at least 9 units of Vitamin 1 and 10 units of Vitamin
2. A pound of beef contains 10 units of Vitamin 1 and 12 units of Vitamin 2. A
pound of grain contains 6 units of Vitamin 1 and 9 units of Vitamin 2.
Formulate this as an LP problem to minimize the cost of the dog food. How many
pounds of beef and grain should be included in each pound of dog food? What is
the cost and vitamin content of the final product?

SOLUTIONS TO SELECT
PROBLEMS FROM CHAPTER 8

8-1 (Production problem) Winkler Furniture
manufactures two different types of china cabinets: a French Provincial model
and a Danish Modern model. Each cabinet produced must go through three
departments: carpentry, painting, and finishing. The table below contains all
relevant information concerning production times per cabinet produced and
production capacities for each operation per day, along with net revenue per
unit produced. The firm has a contract with anIndiana distributor to produce a minimum of
300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would
like to determine a product mix to maximize his daily revenue.
(a) Formulate as an LP problem.
(b) Solve using an LP software program or spreadsheet.

Cabinet Style

Carpentry (Hours/Cabinet)

Painting
(Hours/Cabinet)

Finishing
(Hours/Cabinet)

Net Revenue per Cabinet ($)

French Provincial

3

1.5

.75

28

Danish Modern

2

1

.75

25

Dept. capacity
(hrs)

360

200

125

8-2 (Investment decision problem) The Heinlein
and Krarnpf Brokerage firm has just been instructed by one of its clients to
invest $250,000 for her money obtained recently through the sale of land
holdings in Ohio. The client has a good deal of trust in the investment house,
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 they believe are well rated, but within the following guidelines:
(a)Municipal
bonds should constitute at least 20% of the investment.
(b)
At least 40% of the funds should be
placed in a combination of electronic firms, aerospace firms, and drug
manufacturers.
(c)
No more than 50% of the amount invested
in municipal bonds should be placed in a high-risk, high-yield nursing home
stock.
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.

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

(a) Formulate this portfolio selection problem using LP. (b)
Solve this problem.

8-3 (Restaurant work scheduling problem). The
famous Y. S. Chang Restaurant is open 24 hours a day. Waiters and busboys
report for duty at 3AM., 7 AM., 11 AM., 3 P.M., 7 P.M., or 11 P.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. Chang’s scheduling
problem is to determine how many waiters and busboys should report for work at
the start of each time period to minimize the total staff required for one
day’s operation. (Hint: Let Xi
equal the number of waiters and busboys beginning work in time period i, where
i = 1, 2,3,4,5,6.)

Period

Time

Number of Waiters
and Busboys Required

1

3 A.M–7 A.M.

3

2

7 A.M–11 A.M.

12

3

11 A.M–3 P.M.

16

4

3 P.M–7 P.M.

9

5

7 P.M–11 P.M.

11

6

11 P.M–3 A.M.

4

8-4 (Animal feed mix problem) The Battery Park
Stable feeds and houses the horses used to pull tourist-filled carriages
through the streets of Charleston’s historic waterfront area. The stable owner,
an ex-racehorse trainer, recognizes the need to set a nutritional diet for the
horses in his care. At the same time, he would like to keep the overall daily
cost of feed to a minimum.
The feed mixes available
for the horses’ diet are an oat product, a highly enriched grain, and a mineral
product. Each of these mixes contains a certain amount of five ingredients
needed daily to keep the average horse healthy. The table below shows these
minimum requirements, units of each ingredient per pound of feed mix, and costs
for the three mixes.
In
addition, the stable owner is aware that an overfed horse is a sluggish worker.
Consequently, he determines that 6 pounds of feed per day are the most that any
horse needs to function properly. Formulate this problem and solve for the
optimal daily mix of the three feeds.

Feed Mix

Diet Requirement
(Ingredients)

Oat
Product
(units/lb)

Enriched
Grain
(units/lb)

Mineral
Product
(units/lb)

Minimum Daily Requirement
(units)

A

2

3

1

6

B

.5

1

.5

2

C

3

5

6

9

D

1

1.5

2

8

E

.5

.5

1.5

5

Cost/lb

$0.09

$0.14

$0.17

8-6 (Media selection problem) The advertising
director for Diversey Paint and Supply, a chain of four retail stores onChicago’s North Side, is
considering two media possibilities. One plan is for a series of half- page ads
in the Sunday Chicago Tribune newspaper, and the other is for advertising time
on Chicago TV. The stores are expanding their lines of do-it-yourself tools,
and the advertising director is interested in an exposure level of at least 40%
within the city’s neighborhoods and 60% in northwest suburban areas.
The
TV viewing time under consideration has an exposure rating per spot of 5% in
city homes and 3% in the northwest suburbs. The Sunday newspaper has
corresponding exposure rates of 4% and 3% per ad. The cost of a half-page
Tribune advertisement is $925; a television spot costs $2,000.
Diversey
Paint would like to select the least costly advertising strategy that would
meet desired exposure levels.
(a) Formulate using LP.
(b) Solve the problem.

8-11 (College meal selection problem) Kathy
Roniger, campus dietician for a small Idaho college, is responsible for
formulating a nutritious meal plan for students. For an evening meal, she feels
that the following five meal-content requirements should be met: (1) between
900 and 1,500 calories; (2) at least 4 milligrams of iron; (3) no more than 50
grams of fat; (4) at least 26 grams of protein; and (5) no more than 50 grams
of carbohydrates. On a particular day, Roniger’s food stock includes seven
items that can be prepared and served for supper to meet these requirements.
The cost per pound for each food item and the contribution to each of the five
nutritional requirements are given in the accompanying table:

Table
of Food Values and Costs

Food Item

Calories/
Pound

Iron
(mg/lb)

Fat
(gm/lb)

Protein
(gm/lb)

Carbs.
(gm/lb)

Cost/
Pound ($)

Milk

295

0.2

16

16

22

0.60

Ground Meat

1216

0.2

96

81

0

2.35

Chicken

394

4.3

9

74

0

1.15

Fish

358

3.2

0.5

83

0

2.25

Beans

128

3.2

0.8

7

28

0.58

Spinach

118

14.1

1.4

14

19

1.17

Potatoes

279

2.2

0.5

8

63

0.33

What combination and amounts of
food items will provide the nutrition Roniger requires at the least total food
cost?

8-12 (High tech production problem) Quitmeyer
Electronics Incorporated manufactures the following six microcomputer
peripheral devices: internal modems, external modems, graphics circuit boards,
CD drives, hard disk drives, and memory expansion boards. Each of these
technical products requires time, in minutes, on three types of electronic
testing equipment, as shown in the table the following table:

Internal
Modem

External
Modem

Circuit
Board

CD
Drive

Hard
Drive

Memory
Board

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

The first two test devices are available 120 hours per week.
The third (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 table that follows summarizes the revenues and
material costs for each product:

Device

Revenue Per Unit Sold ($)

Material Cost Per Unit ($)

Internal modem

200

35

External modem

120

25

Graphics circuit board

180

40

CD drive

130

45

Hard disk drive

430

170

Memory expansion board

260

60

In addition, variable labor costs are $15 per hour for test
device 1, $12 per hour for test device 2. and $18 per hour for test device 3.
Quitmeyer Electronics wants to maximize its profits.
(a) Formulate this problem as an LP model.
(b) Solve the problem by computer. What is the best product
mix?
(c) What is the value of an additional minute of time per
week on test device 1? Test device 2? Test device 3? Should Quitmeyer
Electronics add more test device time? If so, on which equipment?

8-15 (Material blending problem) Amalgamated Products has just received a
contract to construct steel body frames for automobiles that are to be produced
at the new Japanese factory in Tennessee. The Japanese auto manufacturer has
strict quality control standards for all of its component subcontractors and
has informed Amalgamated that each frame must have the following steel content:

Material

Minimum Percent

Maximum Percent

Manganese

2.1

2.3

Silicon

4.3

4.6

Carbon

5.05

5.35

Amalgamated mixes batches of eight
different available materials to produce one ton of steel used in the body
frames. The table below details these materials. Formulate and solve the LP
model that will indicate how much of each of the eight materials should be
blended into a 1-ton load of steel so that Amalgamated meets its requirements
while minimizing cost.

Material
Available

Manganese
(%)

Silicon
(%)

Carbon
(%)

Pounds
Available

Cost
Per Pound ($)

Alloy 1

70.0

15.0

3.0

No
limit

0.12

Alloy 2

55.0

30.0

1.0

300

0.13

Alloy 3

12.0

26.0

0

No
limit

0.15

Iron 1

1.0

10.0

3.0

No
limit

0.09

Iron 2

5.0

2.5

0

No
limit

0.07

Carbide 1

0

24.0

18.0

50

0.10

Carbide 2

0

25.0

20.0

200

0.12

Carbide 3

0

23.00

25.0

100

0.09

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