1. Introduction to Management Science
8th Edition by
Bernard W. Taylor III
Chapter 4
Linear Programming: Modeling
Examples
Chapter 4 - Linear Programming: Modeling Examples

2. 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
Chapter 4 - Linear Programming: Modeling Examples

3. Problem Definition (1 of 7)
A Product Mix Example
Problem 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?
Chapter 4 - Linear Programming: Modeling Examples

4. A Product Mix Example Data (2 of 7)
Chapter 4 - Linear Programming: Modeling Examples

5. Model Construction (3 of 7)
A Product Mix Example
Model Construction (3 of 7)
Decision Variables:
Objective Function:
Model Constraints:
Chapter 4 - Linear Programming: Modeling Examples

6. Computer Solution with Excel (4 of 7)
A Product Mix Example
Computer Solution with Excel (4 of 7)
Exhibit 4.1
Chapter 4 - Linear Programming: Modeling Examples

7. Solution with Excel Solver Window (5 of 7)
A Product Mix Example
Solution with Excel Solver Window (5 of 7)
Exhibit 4.2
Chapter 4 - Linear Programming: Modeling Examples

8. Solution with QM for Windows (6 of 7)
A Product Mix Example
Solution with QM for Windows (6 of 7)
Exhibit 4.3
Chapter 4 - Linear Programming: Modeling Examples

9. Solution with QM for Windows (7 of 7)
A Product Mix Example
Solution with QM for Windows (7 of 7)
Exhibit 4.4
Chapter 4 - Linear Programming: Modeling Examples

10. Data and Problem Definition (1 of 5)
A Diet Example
Data and Problem Definition (1 of 5)
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.
Chapter 4 - Linear Programming: Modeling Examples

11. Model Construction – Decision Variables (2 of 5)
A Diet Example
Model Construction – Decision Variables (2 of 5)
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
Chapter 4 - Linear Programming: Modeling Examples

12. A Diet Example Model Summary (3 of 5) Minimize Z = subject to:
Chapter 4 - Linear Programming: Modeling Examples

13. Computer Solution with Excel (4 of 5)
A Diet Example
Computer Solution with Excel (4 of 5)
Chapter 4 - Linear Programming: Modeling Examples
Exhibit 4.5

14. Solution with Excel Solver Window (5 of 5)
A Diet Example
Solution with Excel Solver Window (5 of 5)
Exhibit 4.6
Chapter 4 - Linear Programming: Modeling Examples

15. An Investment Example Model 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
Chapter 4 - Linear Programming: Modeling Examples

16. An Investment Example Model Summary (2 of 5) Decision Variables:
Maximize Z =
subject to:
Chapter 4 - Linear Programming: Modeling Examples

17. Computer Solution with Excel (2 of 5)
An Investment Example
Computer Solution with Excel (2 of 5)
Exhibit 4.7
Chapter 4 - Linear Programming: Modeling Examples

18. Solution with Excel Solver Window (3 of 5)
An Investment Example
Solution with Excel Solver Window (3 of 5)
Exhibit 4.8
Chapter 4 - Linear Programming: Modeling Examples

19. Sensitivity Report (5 of 5)
An Investment Example
Sensitivity Report (5 of 5)
Exhibit 4.9
Chapter 4 - Linear Programming: Modeling Examples

20. Data and Problem Definition (1 of 6)
A Marketing Example
Data and Problem Definition (1 of 6)
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
Chapter 4 - Linear Programming: Modeling Examples

21. A Marking Example Model Summary (2 of 6) Decision variables:
Maximize Z =
subject to:
Chapter 4 - Linear Programming: Modeling Examples

22. Solution with Excel (3 of 6)
A Marking Example
Solution with Excel (3 of 6)
Exhibit 4.10
Chapter 4 - Linear Programming: Modeling Examples

23. Solution with Excel Solver Window (4 of 6)
A Marking Example
Solution with Excel Solver Window (4 of 6)
Exhibit 4.11
Chapter 4 - Linear Programming: Modeling Examples

24. Integer Solution with Excel (5 of 6)
A Marking Example
Integer Solution with Excel (5 of 6)
Exhibit 4.12
Exhibit 4.13
Chapter 4 - Linear Programming: Modeling Examples

25. Integer Solution with Excel (6 of 6)
A Marking Example
Integer Solution with Excel (6 of 6)
Exhibit 4.14
Chapter 4 - Linear Programming: Modeling Examples

26. A Transportation Example Problem Definition and Data (1 of 3)
Warehouse supply of Retail store demand Television Sets: for television sets:
1 - Cincinnati A - New York 150
2 - Atlanta B - Dallas 250
3 - Pittsburgh C - Detroit 200
Total Total 600
Chapter 4 - Linear Programming: Modeling Examples

27. A Transportation Example Model Summary (2 of 4)
Decision variables:
Minimize Z =
subject to:
Chapter 4 - Linear Programming: Modeling Examples

28. A Transportation Example Solution with Excel (3 of 4)
Exhibit 4.15
Chapter 4 - Linear Programming: Modeling Examples

29. A Transportation Example Solution with Solver Window (4 of 4)
Exhibit 4.16
Chapter 4 - Linear Programming: Modeling Examples

30. Problem Definition and Data (1 of 6)
A Blend Example
Problem Definition and Data (1 of 6)
Chapter 4 - Linear Programming: Modeling Examples

31. Problem Statement and Variables (2 of 6)
A Blend Example
Problem Statement and Variables (2 of 6)
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).
Chapter 4 - Linear Programming: Modeling Examples

32. A Blend Example Model Summary (3 of 6) Maximize Z = subject to:
Chapter 4 - Linear Programming: Modeling Examples

33. Solution with Excel (4 of 6)
A Blend Example
Solution with Excel (4 of 6)
Exhibit 4.17
Chapter 4 - Linear Programming: Modeling Examples

34. Solution with Solver Window (5 of 6)
A Blend Example
Solution with Solver Window (5 of 6)
Exhibit 4.18
Chapter 4 - Linear Programming: Modeling Examples

35. Sensitivity Report (6 of 6)
A Blend Example
Sensitivity Report (6 of 6)
Exhibit 4.19
Chapter 4 - Linear Programming: Modeling Examples

36. A Multi-Period Scheduling Example Problem Definition and Data (1 of 5)
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
Chapter 4 - Linear Programming: Modeling Examples

37. A Multi-Period Scheduling Example Decision Variables (2 of 5)
rj = regular production of computers per week j
(j = 1 - 6)
oj = overtime production of computers per week j
ij = extra computers carried over as inventory in week j
(j = 1 - 5)
Chapter 4 - Linear Programming: Modeling Examples

38. A Multi-Period Scheduling Example Model Summary (3 of 5)
Minimize Z =
subject to:
Chapter 4 - Linear Programming: Modeling Examples

39. A Multi-Period Scheduling Example Solution with Excel (4 of 5)
Exhibit 4.20
Chapter 4 - Linear Programming: Modeling Examples

40. A Multi-Period Scheduling Example Solution with Solver Window (5 of 5)
Exhibit 4.21
Chapter 4 - Linear Programming: Modeling Examples

41. Example Problem Solution Problem 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?
Chapter 4 - Linear Programming: Modeling Examples

42. Example Problem Solution Model Formulation (2 of 5)
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 =
Chapter 4 - Linear Programming: Modeling Examples

43. Example Problem Solution Model Formulation (3 of 5)
Step 3: Formulate the Model Constraints
Amount of each ingredient available each week:
Recipe requirements:
Meow Chow
Bow Chow
Can Content Constraint
Chapter 4 - Linear Programming: Modeling Examples

44. Example Problem Solution Model Summary (4 of 5)
Step 4: Model Summary
Maximize Z = $0.05xhm + $0.05xfm + $0.05xcm + $0.06xhb xfb xcb
subject to:
xhm + xhb  9,600 ounces of horse meat
xfm + xfb  12,800 ounces of fish
xcm + xcb  16,000 ounces of cereal additive
- xhm + xfm- xcm  0
xhb- xfb - xcb  0
xhm + xfm + xcm + xhb + xfb+ xcb  36,000 ounces xij  0
Chapter 4 - Linear Programming: Modeling Examples

45. Example Problem Solution Solution with QM for Windows (5 of 5)
Exhibit 4.24
Chapter 4 - Linear Programming: Modeling Examples