1. Solving Examples of Linear Programming Models
Chapter 4

2. Chapter Topics A Product Mix Example A Diet Example
An Investment Example
A Marketing Example
A Transportation Example
A Blend Example
A Multiperiod Scheduling Example
A Data Envelopment Analysis Example

3. 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.

4. A Product Mix Example
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.

5. A Product Mix Example Problem 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?

6. A Product Mix Example

7. A Product Mix Example Data
Resource requirements for the product mix example.

8. A Product Mix Example Model Construction Decision Variables:
x1 = sweatshirts, front printing
x2 = sweatshirts, back and front printing
x3 = T-shirts, front printing
x4 = T-shirts, back and front printing
Objective Function:
Maximize Z = $90x1 + $125x2 + $45x3 + $65x4
Model Constraints:
0.10x x x x4  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

9. 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:

10. Data and Problem Definition
A Diet Example
Data and Problem Definition
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.

11. Model Construction – Decision Variables
A Diet Example
Model Construction – Decision Variables
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

12. A Diet Example Model Summary
Minimize Z = 0.18x x x x x x x x x x10
subject to:
90x x x3 + 90x4 + 75x5 + 35x6 + 65x7
+ 100x x9 + 65x10  420 calories
2x x3 + 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 + 48x x3 + 8x4+ 30x5 + 52x x8
+ 3x9 + 26x10  mg of calcium
3x1 + 4x2 + 5x3 + 6x4 + 7x5 + 2x6 + x x8+ x9 + 3x10  20 g protein
5x1 + 2x2 + 3x3 + 4x4+ x7 + 3x10  12
xi  0, for all j

13. An Investment Example
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.

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

15. An Investment Example
Model

16. An Investment Example
Model