4-1 Copyright © 2013 Pearson Education Solving Examples of Linear Programming Models Chapter 4.

4-1Copyright © 2013 Pearson Education

Solving Examples of Linear Programming

Models

Chapter 4


Chapter Topics

1. A Product Mix Example

2. A Diet Example

3. An Investment Example

4. A Marketing Example

5. A Transportation Example

6. A Blend Example

7. A Multiperiod Scheduling Example

8. A Data Envelopment Analysis Example


A Product Mix Example

Quick-Screen is a clothing manufacturing company that specializes in producing commemorative shirts immediately following major sporting. The company has been contracted to produce a standard set of shirts for the winning team, either State University or Tech, following a college football game. The items produced include two sweatshirts, one with silk-screen printing on the front and one with print on both sides, and two T-shirts of the same configuration.



The company has to complete all production within 72 hours after the game, at which time a trailer truck will pick up the shirts. The company will work around the clock. The truck has enough capacity to accommodate 1,200 standard-size boxes. A standard size box holds 12 T-shirts, and a box of 12 sweatshirts is three times the size of a standard box. The company has budgeted $25,000 for the production run. It has 500 dozen blank sweatshirts and T-shirts each in stock, ready for production.


A Product Mix ExampleProblem Definition

Four-product T-shirt/sweatshirt manufacturing company.

■ Must complete production within 72 hours

■ Truck capacity = 1,200 standard sized boxes.

■ Standard size box holds12 T-shirts.

■ One-dozen sweatshirts box is three times size of standard box.

■ $25,000 available for a production run.

■ 500 dozen blank T-shirts and sweatshirts in stock.

■ How many dozens (boxes) of each type of shirt to produce?




Processing Time (hr) Per dozen

Cost ($)

per dozen

Profit ($)

per dozen

Sweatshirt - F 0.10 $36 $90

Sweatshirt – B/ F 0.25 48 125

T-shirt - F 0.08 25 45

T-shirt - B/ F 0.21 35 65

A Product Mix ExampleData

Resource requirements for the product mix example.


Decision Variables:x1 = sweatshirts, front printingx2 = sweatshirts, back and front printingx3 = T-shirts, front printingx4 = T-shirts, back and front printing

Objective Function: Maximize Z = $90x1 + $125x2 + $45x3 + $65x4

Model Constraints:

0.10x1 + 0.25x2+ 0.08x3 + 0.21x4 72 hr 3x1 + 3x2 + x3 + x4 1,200 boxes

$36x1 + $48x2 + $25x3 + $35x4 $25,000 x1 + x2 500 dozen sweatshirts

x3 + x4 500 dozen T-shirts

A Product Mix ExampleModel Construction


A Diet Example

Breathtakers, a health and fitness center, operates a morning fitness program for senior citizens. The program includes aerobic exercise, either swimming or step exercise, followed by a healthy breakfast in the dining room. Breathtakers’ dietitian wants to develop a breakfast that will be high in calories, calcium, protein, and fiber, which are especially important to senior citizens, but low in fat and cholesterol. She also wants to minimize cost. She has selected the following possible food items, whose individual nutrient contributions and cost from which to develop a standard breakfast menu are shown in the following table:


Breakfast to include at least 420 calories, 5 milligrams of iron, 400 milligrams of calcium, 20 grams of protein, 12 grams of fiber, and must have no more than 20 grams of fat and 30 milligrams of cholesterol.

Breakfast Food Cal

Fat (g)

Cholesterol (mg)

Iron (mg)

Calcium (mg)

Protein (g)

Fiber (g)

Cost ($)

1. Bran cereal (cup) 2. Dry cereal (cup) 3. Oatmeal (cup) 4. Oat bran (cup) 5. Egg 6. Bacon (slice) 7. Orange 8. Milk-2% (cup) 9. Orange juice (cup)

10. Wheat toast (slice)

90 110 100

90 75 35 65

100 120

65

0 2 2 2 5 3 0 4 0 1

0 0 0 0

270 8 0

12 0 0

6 4 2 3 1 0 1 0 0 1

20 48 12

8 30

0 52

250 3

26

3 4 5 6 7 2 1 9 1 3

5 2 3 4 0 0 1 0 0 3

0.18 0.22 0.10 0.12 0.10 0.09 0.40 0.16 0.50 0.07

A Diet ExampleData and Problem Definition


x1 = cups of bran cereal

x2 = cups of dry cereal

x3 = cups of oatmeal

x4 = cups of oat bran

x5 = eggs

x6 = slices of bacon

x7 = oranges

x8 = cups of milk

x9 = cups of orange juice

x10 = slices of wheat toast

A Diet ExampleModel Construction – Decision Variables


Minimize Z = 0.18x1 + 0.22x2 + 0.10x3 + 0.12x4 + 0.10x5 + 0.09x6 + 0.40x7 + 0.16x8 + 0.50x9 + 0.07x10

subject to:90x1 + 110x2 + 100x3 + 90x4 + 75x5 + 35x6 +

65x7 + 100x8 + 120x9 + 65x10 420 calories

2x2 + 2x3 + 2x4 + 5x5 + 3x6 + 4x8 + x10 20 g fat

270x5 + 8x6 + 12x8 30 mg cholesterol

6x1 + 4x2 + 2x3 + 3x4+ x5 + x7 + x10 5 mg iron

20x1 + 48x2 + 12x3 + 8x4+ 30x5 + 52x7 + 250x8

+ 3x9 + 26x10 400 mg of calcium

3x1 + 4x2 + 5x3 + 6x4 + 7x5 + 2x6 + x7

+ 9x8+ x9 + 3x10 20 g protein

5x1 + 2x2 + 3x3 + 4x4+ x7 + 3x10 12xi 0, for all j

A Diet ExampleModel Summary


Kathleen Allen, an individual investor, has $70,000 to divide among several investments. The alternative investments are municipal bonds with an 8.5% annual return, certificates of deposit with a 5% return, treasury bills with a 6.5% return, and a growth stock fund with a 13% annual return. The investments are all evaluated after 1 year. However, each investment alternative has a different perceived risk to the investor; thus, it is advisable to diversify. Kathleen wants to know how much to invest in each alternative in order to maximize the return.

An Investment Example


The following guidelines have been established:1.No more than 20% in municipal bonds2.Investment in CDs should not exceed the other three alternatives3.At least 30% invested in t-bills and CDs4.More should be invested in CDs and t-bills than in municipal bonds and growth stocks by a ratio of 1.2 to 15.All $70,000 should be invested.

An Investment ExampleModel Summary


An Investment ExampleModel


An Investment ExampleModel

4-1 Copyright © 2013 Pearson Education Solving Examples of Linear Programming Models Chapter 4.

Documents

printing x

dozen sweatshirts x

investment example

marketing example

transportation example

blend example

diet example breathtakers

dozen sweatshirts box