1. Linear Programming Modeling
Decision Science
Chapter 4
Linear Programming Modeling
and computer solution

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
A Data Envelopment Analysis Example

3. Problem Definition (1 of 8)
A Product Mix Example
Problem Definition (1 of 8)
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?

4. A Product Mix Example (2 of 8)
Chapter 4 - Linear Programming: Modeling Examples

5. A Product Mix Example
Data (3 of 8)

6. Model Construction (4 of 8)
A Product Mix Example
Model Construction (4 of 8)
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

7. Computer Solution with Excel (5 of 8)
A Product Mix Example
Computer Solution with Excel (5 of 8)
Chapter 4 - Linear Programming: Modeling Examples
Exhibit 4.1

8. Solution with Excel Solver Window (6 of 8)
A Product Mix Example
Solution with Excel Solver Window (6 of 8)
Chapter 4 - Linear Programming: Modeling Examples
Exhibit 4.2

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

10. Solution with QM for Windows (8 of 8)
A Product Mix Example
Solution with QM for Windows (8 of 8)
Chapter 4 - Linear Programming: Modeling Examples
Exhibit 4.4

11. 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.

12. 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

13. A Diet Example Model Summary (3 of 5)
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
2x x3 + 2x4 + 5x5 + 3x6 + 4x8 + x10  20
270x5 + 8x6 + 12x8  30
6x1 + 4x2 + 2x3 + 3x4+ x5 + x7 + x10  5
20x1 + 48x x3 + 8x4+ 30x5 + 52x x8
+ 3x9 + 26x10  400
3x1 + 4x2 + 5x3 + 6x4 + 7x5 + 2x6 + x7+ 9x8+ x9 + 3x10  20
5x1 + 2x2 + 3x3 + 4x4+ x7 + 3x10  xi  0, for all j

14. Computer Solution with Excel (4 of 5)
A Diet Example
Computer Solution with Excel (4 of 5)
Exhibit 4.5

15. A Diet Example Solution with Excel Solver Window (5 of 5)
Exhibit 4.6

16. 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

17. A Marketing Example Model Summary (2 of 6)
Maximize Z = 20,000x1 + 12,000x2 + 9,000x3
subject to:
15,000x1 + 6,000x 2+ 4,000x3  100,000
x1  4
x2  10
x3  7
x1 + x2 + x3  15
x1, x2, x3  0
where
x1 = Exposure from Television Commercial (people)
x2 = Exposure from Radio Commercial (people)
x3 = Exposure from Newspaper Ad (people)

18. A Marketing Example Solution with Excel (3 of 6)

19. Solution with Excel Solver Window (4 of 6)
A Marketing Example
Solution with Excel Solver Window (4 of 6)
Chapter 4 - Linear Programming: Modeling Examples

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

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

22. 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

23. A Transportation Example Model Summary (2 of 4)
Chapter 4 - Linear Programming: Modeling Examples
Minimize Z = $16x1A + 18x1B + 11x1C + 14x2A + 12x2B + 13x2C x3A + 15x3B + 17x3C
subject to:
x1A + x1B+ x1C  300
x2A+ x2B + x2C  200
x3A+ x3B + x3C  200
x1A + x2A + x3A = 150
x1B + x2B + x3B = 250
x1C + x2C + x3C = 200
All xij  0

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

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

26. Problem Statement and Variables (2 of 6)
A Blend Example
Problem Statement and Variables (2 of 6)
Chapter 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).

27. A Blend Example Model Summary (3 of 6)
Chapter 4 - Linear Programming: Modeling Examples
Maximize Z = 11x1s + 13x2s + 9x3s + 8x1p + 10x2p + 6x3p + 6x1e x2e + 4x3e
subject to:
x1s + x1p + x1e  4,500
x2s + x2p + x2e  2,700
x3s + x3p + x3e  3,500
0.50x1s x2s x3s  0
0.70x2s x1s x3s  0
0.60x1p x2p x3p  0
0.75x3p x1p x2p  0
0.40x1e- 0.60x2e x3e  0
0.90x2e x1e x3e  0
x1s + x2s + x3s  3,000
x1p+ x2p + x3p  3,000
x1e+ x2e + x3e  3,000
xij  0

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

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

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

31. 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 Holding Cost: $10/computer per week
Order schedule: Week Computer Orders

32. A Multi-Period Scheduling Example Decision Variables (2 of 5)
rj = regular production of computers in week j
(j = 1, 2, …, 6)
oj = overtime production of computers in week j
ij = extra computers carried over as inventory in week j
(j = 1, 2, …, 5)

33. Model summary: subject to: rj  160 (j = 1, 2, 3, 4, 5, 6)
A Multi-Period Scheduling Example Model Summary (3 of 5)
Model summary:
Minimize Z = $190(r1 + r2 + r3 + r4 + r5 + r6) + $260(o1 + o2 + o o4 + o5 +o6) + 10(i1 + i2 + i3 + i4 + i5)
subject to:
rj  160 (j = 1, 2, 3, 4, 5, 6)
oj  150 (j = 1, 2, 3, 4, 5, 6)
r1 + o1 - i1 = 105
r2 + o2 + i1 - i2 = 170
r3 + o3 + i2 - i3 = 230
r4 + o4 + i3 - i4 = 180
r5 + o5 + i4 - i5 = 150
r6 + o6 + i5 = 250
rj, oj, ij  0

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

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

36. A Data Envelopment Analysis (DEA) Example Problem Definition (1 of 5)
DEA compares a number of service units of the same type based on their inputs (resources) and outputs. The result indicates if a particular unit is less productive, or efficient, than other units.
Elementary school comparison:
Input 1 = teacher to student ratio
Output 1 = average reading SOL score
Input 2 = supplementary $/student
Output 2 = average math SOL score
Input 3 = parent education level
Output 3 = average history SOL score

37. A Data Envelopment Analysis (DEA) Example
Problem Data Summary (2 of 5)
Chapter 4 - Linear Programming: Modeling Examples

38. A Data Envelopment Analysis (DEA) Example
Decision Variables and Model Summary (3 of 5)
Decision Variables:
xi = a price per unit of each output where i = 1, 2, 3
yi = a price per unit of each input where i = 1, 2, 3
Model Summary:
Maximize Z = 81x1 + 73x2 + 69x3
subject to:
.06 y y y3 = 1
86x x2 + 71x3 .06y y y3
82x x2 + 67x3  .05y y y3
81x x2 + 80x3  .08y y y3
81x x2 + 69x3  .06y y y3
xi, yi  0

39. A Data Envelopment Analysis (DEA) Example
Solution with Excel (4 of 5)

40. A Data Envelopment Analysis (DEA) Example
Solution with Solver Window (5 of 5)
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?

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 = $0.05(xhm + xfm + xcm) (xhb + xfb + xcb)

43. Example Problem Solution Model Formulation (3 of 5)
Step 3: Formulate the Model Constraints
Amount of each ingredient available each week:
xhm + xhb  9,600 ounces of horse meat
xfm + xfb  12,800 ounces of fish
xcm + xcb  16,000 ounces of cereal additive
Recipe requirements:
Meow Chow
xfm/(xhm + xfm + xcm)  1/2 or - xhm + xfm- xcm  0
Bow Chow
xhb/(xhb + xfb + xcb)  1/2 or xhb- xfb - xcb  0
Can Content Constraint
xhm + xfm + xcm + xhb + xfb+ xcb  36,000 ounces

44. Example Problem Solution Model Summary (4 of 5)
Chapter 4 - Linear Programming: Modeling Examples
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

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