A small backpack manufacturer carries four different

A small backpack manufacturer carries four different models of backpacks, made of canvas, plastic, nylon, and leather. The bookstore, which will exclusively sell the backpacks sell the backpacks, expects to be able to sell between 15 and 40 of each model. The store has agreed to pay $35.50 for each canvas backpack, $39.50 for each plastic backpack, $42.50 for each nylon backpack, and $69.50 for each leather backpack that can be delivered by the end of the following week.

One worker can work on either canvas or plastic, can complete a backpack in 1.5 hours, and will charge $7.00 per hour to do the work. This worker can work a maximum of 90 hours during the next week. Another worker can sew backpacks made of nylon fabric. He can complete a bag in 1.7 hours, will charge $8.00 per hour to work and can work 42.5 hours in the next week. A third worker has the ability to sew leather. He can complete a book bag in 1.9 hours, will charge $9.00 per hour to work, and can work 80 hours during the next week. The following table provides additional info about each backpack. What is the best combination of backpacks to provide the store to maximize the profit?

BACKPACK MODELS

MATERIALS REQUIRED (square yards)

MATERIAL AVAILABLE

(square yards)

COST/SQUARE YARD

CANVAS

2.25

200

$4.50

PLASTIC

2.40

350

$4.25

NYLON

2.10

700

$7.65

LEATHER

2.60

550

$9.45

3-2

A manufacturer of travel pillows must determine the production plan for the next production cycle. He wishes to make at least 300 of each of the three models that his firm offers and no more than 1200 of any one model. The specifics for each model are shown in the following table. How many pillows of each type should be manufactured in order to maximize total profit?

PILLOW MODEL

SELLING PRICE

CUTTING

SEWING

FINISHING

PACKING

Junior Travel

$5.75

0.10

0.05

0.18

0.20

Travel Pillow

$6.95

0.15

0.12

0.24

0.20

Deluxe Travel

$7.50

0.20

0.18

0.20

0.20

Available Hours

450

550

600

450

Cost Per Hour

$7.00

$9.00

$8.50

$7.25

3-8

A gear manufacturer is planning next week’s production run for four types of gears. If necessary, it is possible to outsource any type of gear from another gear company located nearby. The following table and the table below show next week’s demand, revenue per unit, outsource cost per unit, time (in hours) required per unit in each production process, and the availability and cost of these processes. The nearby company can supply a maximum of 300 units of each type of gear next week. What should be the production and/or outsource plan for the next week to maximize profit?

GEAR TYPE

GEAR A

GEAR B

GEAR C

GEAR D

Demand

400

500

450

600

Revenue

$12.50

$15.60

$17.40

$19.30

Outsource

$7.10

$8.10

$8.40

$9.00

PROCESS

GEAR A

GEAR B

GEAR C

GEAR D

HOURS AVAILABLE

COST PER HOUR

Forming

0.30

0.36

0.38

0.45

500

$9.00

Hardening

0.20

0.30

0.24

0.33

300

$8.00

Deburring

0.30

0.30

0.35

0.25

310

$7.50

3-10

The advertising director of a large retail store in Columbus, Ohio, is considering three advertising media possibilities (1) ads in the Sunday Columbus Dispatch newspaper, (2) ads in a local trade magazine that is distributed free to all houses in the city and northwest suburbs, and (3) ads on Columbus’ WCC-TV station. She wishes to obtain a new-customer exposure level of at least 50% within the city and 60% in the northwest suburbs. Each TV ad has a new customer exposure level of 5% in the city and 3% in the northwest suburbs. The Dispatch ads have corresponding exposure levels per ad of 3.5% and 3%, respectively, while the trade magazine has exposure levels per ad of 0.5% and 1%, respectively. The relevant costs are $1,000 per Dispatch ad, $300 per trade magazine ad, and $2,000 per TV ad. The advertising policy is that no single media type should consume more than 45% of the total amount spent. Find the advertising strategy that will meet the store’s objective at minimum cost.

3-11

A grocery chain wants to promote the sale of a new flavor of ice cream by issuing up to 15,000 coupons by mail to preferred customers. The budget for this promotion has been limited to $12,000. The following table shows the expected increased sales per coupon and the probability of coupon usage for the various coupon amounts under consideration.

COUPON AMOUNT

INCREASED SALES PER COUPON (CARTONS)

PROBABILITY COUPON WILL BE USED

$1.00

1.50

0.80

$0.85

1.40

0.75

$0.70

1.25

0.60

$0.55

1.00

0.50

$0.40

0.90

0.42

For example, every $1-off coupon issued will stimulate sales of 1.5 additional cartons. However since the probability that a $1-off coupon will actually be used is only 0.80, the expected increased sales per coupon issued is 1.2 (= 0.8 x 1.5) cartons. The selling price per carton of ice cream is $3.50 before the coupon value is applied. The chain wants at least 20% of the coupons issued to be of the $1-off variety and at least 10% of the coupons issued to be of each of the other four varieties. What is the optimal combination of coupons to be issued, and what is the expected net increased revenue from this promotion?

3-15

A finance major has inherited $200,000 and wants to invest it in a diversified portfolio. Some of the investments she is considering are somewhat risky. These include international mutual funds, which should earn 12.25% over the next year, and U.S. stocks, which should earn 11.5% over the next year. She has therefore decided that she will put no more than 30% of her money in either of these investments and no more than a total of 50% in both investments. She also wants to keep some of her investment in what is considered a liquid state, so that she can divest quickly if she so chooses. She believes school bonds, which returns 5% interest, short-term certificate of deposit, which return 6.25% interest, and tax-free municipal bonds, which return 8.75%, to be reasonably liquid. She will keep no more than 40% of her money in these investments and no more than 15% in any one of these investments. She believes that T-bills are also considered liquid and less risky and that they will return 7.5%. However, she has decided to invest no more than 25% of her investment in T-bills. She wishes to have experience investing in different types of instruments, so she will invest at least 10% of her money in each of the six types of investment choices. What is the optimal investment strategy for her to follow?

3-19

A hospital is moving from 8-hour shifts for its lab techs to 12-hour shifts. Instead of working five 8-hour days, the lab tech would work three days on and four days off in the first week followed by four days on and three days off in the second week, for a total of 84 hours every two weeks. Because the peak demand times in the hospital appear to be between 5 a.m. and 7 a.m. and between 5 p.m. and 7 p.m., four 12-hour shifts will be arranged according to the table below.

SHIFTS

WORK TIMES

PAY RATE/WEEK

A and A (alt)

5 a.m. – 5 p.m.

$756

B and B (alt)

7 a.m. – 7 p.m.

$840

C and C (alt)

5 p.m. – 5 a.m.

$882

D and D (alt)

7 p.m. – 7 a.m.

$924

The shift pay differentials are based on the most and least desirable times to begin and end work. In any one week, techs on shift A might work Sunday through Tuesday, while techs on shift A (alt) would work at the same times but on Wednesday through Saturday. In the following week, techs on shift A would work Sunday through Wednesday, while techs on shift A (alt) would work the corresponding Thursday through Saturday. Therefore, the same number of techs would be scheduled for shift A as for shift A (alt). The requirement for lab techs during the 24-hour day are shown in the following table. What is the most economical schedule for the lab techs?

5 a.m. – 7 a.m.

7 a.m. – 5 p.m.

5 p.m. – 7 p.m.

7 p.m. – 5 a.m.

Lab Tech needed

12

8

14

10

3-34

A steel company is producing steel for a new contract. The contract specifies the information in the following table for the steel.

MATERIAL

MINIMUM

MAXIMUM

Maganese

2.10%

3.10%

Silicon

4.30%

6.30%

Carbon

1.05%

2.05%

The steel company mixes batches of eight different available materials to produce each ton of steel according to the specification. The table at the bottom of this page 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 the company can meet the specifications under the contract while minimizing costs.

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.%

18.0%

50

$0.10

Carbide 2

0%

25.0%

20.0%

200

$0.12

Carbide 3

0%

23.%

25.0%

100

$0.09