1Chapter 4 - Linear Programming: Modeling Examples

Chapter 4

Linear Programming: ModelingExamples

Introduction to Management Science

8th Edition

by

Bernard W. Taylor III

2Chapter 4 - Linear Programming: Modeling Examples

Chapter Topics

A Product Mix Example

A Diet Example

An Investment Example

A Marketing Example

A Transportation Example

A Blend Example

A Multi-Period Scheduling Example

3Chapter 4 - Linear Programming: Modeling Examples

A Product Mix ExampleProblem Definition (1 of 7)

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?

4Chapter 4 - Linear Programming: Modeling Examples

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 (2 of 7)

5Chapter 4 - Linear Programming: Modeling Examples

Decision Variables:

Objective Function:

Model Constraints:

A Product Mix ExampleModel Construction (3 of 7)

6Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.1

A Product Mix ExampleComputer Solution with Excel (4 of 7)

7Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.2

A Product Mix ExampleSolution with Excel Solver Window (5 of 7)

8Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.3

A Product Mix ExampleSolution with QM for Windows (6 of 7)

9Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.4

A Product Mix ExampleSolution with QM for Windows (7 of 7)

10Chapter 4 - Linear Programming: Modeling Examples

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 (1 of 5)

11Chapter 4 - Linear Programming: Modeling Examples

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 (2 of 5)

12Chapter 4 - Linear Programming: Modeling Examples

Minimize Z =

subject to:

A Diet ExampleModel Summary (3 of 5)

13Chapter 4 - Linear Programming: Modeling ExamplesExhibit 4.5

A Diet ExampleComputer Solution with Excel (4 of 5)

14Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.6

A Diet ExampleSolution with Excel Solver Window (5 of 5)

15Chapter 4 - Linear Programming: Modeling Examples

An Investment ExampleModel Summary (1 of 5)

Kathleen has $70,000 to invest.

Municipal bonds (MB): 8.5%

Certificates of deposit (CD): 5%

Treasury bills (TB): 6.5%

Growth stock fund (GSF): 13%

No more than 20% in municipal bonds

Amount in CD < in other 3 alternatives

At 30% in treasury bills and CD

Ratio in (TB and CD) to (MB and GSF) > 1.2 to 1

16Chapter 4 - Linear Programming: Modeling Examples

Decision Variables:

Maximize Z =

subject to:

An Investment ExampleModel Summary (2 of 5)

17Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.7

An Investment ExampleComputer Solution with Excel (2 of 5)

18Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.8

An Investment ExampleSolution with Excel Solver Window (3 of 5)

19Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.9

An Investment ExampleSensitivity Report (5 of 5)

20Chapter 4 - Linear Programming: Modeling Examples

Budget limit $100,000

Television time for four commercials

Radio time for 10 commercials

Newspaper space for 7 ads

Resources for no more than 15 commercials and/or ads

Exposure (people/ad or commercial)

Cost

Television Commercial

20,000 $15,000

Radio Commercial

12,000 6,000

Newspaper Ad 9,000 4,000

A Marketing ExampleData and Problem Definition (1 of 6)

21Chapter 4 - Linear Programming: Modeling Examples

Decision variables:

Maximize Z =

subject to:

A Marking ExampleModel Summary (2 of 6)

22Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.10

A Marking ExampleSolution with Excel (3 of 6)

23Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.11

A Marking ExampleSolution with Excel Solver Window (4 of 6)

24Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.12

Exhibit 4.13

A Marking ExampleInteger Solution with Excel (5 of 6)

25Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.14

A Marking ExampleInteger Solution with Excel (6 of 6)

26Chapter 4 - Linear Programming: Modeling Examples

Warehouse supply of Retail store demand Television Sets: for television sets:

1 - Cincinnati 300 A - New York 150

2 - Atlanta 200 B - Dallas 250

3 - Pittsburgh 200 C - Detroit 200

Total 700 Total 600

To Store From Warehouse

A B C

1 $16 $18 $11

2 14 12 13

3 13 15 17

A Transportation ExampleProblem Definition and Data (1 of 3)

27Chapter 4 - Linear Programming: Modeling Examples

Decision variables:

Minimize Z =

subject to:

A Transportation ExampleModel Summary (2 of 4)

28Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.15

A Transportation ExampleSolution with Excel (3 of 4)

29Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.16

A Transportation ExampleSolution with Solver Window (4 of 4)

30Chapter 4 - Linear Programming: Modeling Examples

Component Maximum Barrels

Available/day Cost/barrel

1 4,500 $12

2 2,700 10

3 3,500 14

Grade Component Specifications Selling Price ($/bbl)

Super At least 50% of 1

Not more than 30% of 2 $23

Premium At least 40% of 1

Not more than 25% of 3

20

Extra At least 60% of 1 At least 10% of 2

18

A Blend ExampleProblem Definition and Data (1 of 6)

31Chapter 4 - Linear Programming: Modeling Examples

Determine the optimal mix of the three components in each grade of motor oil that will maximize profit. Company wants to produce at least 3,000 barrels of each grade of motor oil.

Decision variables: The quantity of each of the three components used in each grade of gasoline (9 decision variables); xij = barrels of component i used in motor oil grade j per day, where i = 1, 2, 3 and j = s (super), p (premium), and e (extra).

A Blend ExampleProblem Statement and Variables (2 of 6)

32Chapter 4 - Linear Programming: Modeling Examples

Maximize Z =

subject to:

A Blend ExampleModel Summary (3 of 6)

33Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.17

A Blend ExampleSolution with Excel (4 of 6)

34Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.18

A Blend ExampleSolution with Solver Window (5 of 6)

35Chapter 4 - Linear Programming: Modeling ExamplesExhibit 4.19

A Blend ExampleSensitivity Report (6 of 6)

36Chapter 4 - Linear Programming: Modeling Examples

Production Capacity: 160 computers per week 50 more computers with overtime

Assembly Costs: $190 per computer regular time; $260 per computer overtime

Inventory Cost: $10/comp. per week

Order schedule: Week Computer Orders 1 105 2 170 3 230 4 180 5 150 6 250

A Multi-Period Scheduling ExampleProblem Definition and Data (1 of 5)

37Chapter 4 - Linear Programming: Modeling Examples

Decision Variables:

rj = regular production of computers per week j(j = 1 - 6)

oj = overtime production of computers per week j(j = 1 - 6)

ij = extra computers carried over as inventory in week j(j = 1 - 5)

A Multi-Period Scheduling ExampleDecision Variables (2 of 5)

38Chapter 4 - Linear Programming: Modeling Examples

Model summary:

Minimize Z =

subject to:

A Multi-Period Scheduling ExampleModel Summary (3 of 5)

39Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.20

A Multi-Period Scheduling ExampleSolution with Excel (4 of 5)

40Chapter 4 - Linear Programming: Modeling Examples

Exhibit 4.21

A Multi-Period Scheduling ExampleSolution with Solver Window (5 of 5)

41Chapter 4 - Linear Programming: Modeling Examples

Example Problem SolutionProblem Statement and Data (1 of 5)

Canned cat food, Meow Chow; dog food, Bow Chow.

Ingredients/week: 600 lb horse meat; 800 lb fish; 1000 lb cereal.

Recipe requirement: Meow Chow at least half fish; Bow Chow at least half horse meat.

2,250 sixteen-ounce cans available each week.

Profit /can: Meow Chow $0.80; Bow Chow $0.96.

How many cans of Bow Chow and Meow Chow should be produced each week in order to maximize profit?

42Chapter 4 - Linear Programming: Modeling Examples

Step 1: Define the Decision Variables

xij = ounces of ingredient i in pet food j per week, where i = h (horse meat), f (fish) and c (cereal), and j = m (Meow chow) and b (Bow Chow).

Step 2: Formulate the Objective Function

Maximize Z =

Example Problem SolutionModel Formulation (2 of 5)

43Chapter 4 - Linear Programming: Modeling Examples

Step 3: Formulate the Model Constraints

Amount of each ingredient available each week:

Recipe requirements:Meow Chow

Bow Chow

Can Content Constraint

Example Problem SolutionModel Formulation (3 of 5)

44Chapter 4 - Linear Programming: Modeling Examples

Step 4: Model Summary

Maximize Z = $0.05xhm + $0.05xfm + $0.05xcm + $0.06xhb + 0.06xfb + 0.06xcb

subject to:xhm + xhb 9,600 ounces of horse meatxfm + xfb 12,800 ounces of fishxcm + xcb 16,000 ounces of cereal additive- xhm + xfm- xcm 0 xhb- xfb - xcb 0xhm + xfm + xcm + xhb + xfb+ xcb 36,000 ounces

xij 0

Example Problem SolutionModel Summary (4 of 5)

45Chapter 4 - Linear Programming: Modeling Examples

Example Problem SolutionSolution with QM for Windows (5 of 5)

Exhibit 4.24