SlideShare a Scribd company logo
1 of 51
Linear Programming: 
Modeling Examples 
Chapter 4 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 4-1
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 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice 
Hall
4-3 
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. 
■ $25,000 available for a production run. 
■ 500 dozen blank T-shirts and sweatshirts in stock. 
■ How many dozens (boxes) of each type of shirt to produce? 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-4 
A Product Mix Example (2 of 8) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-5 
A Product Mix Example 
Data (3 of 8) 
Processing 
Time (hr) 
Per dozen 
Cost 
($) 
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 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-6 
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: 
0.10x1 + 0.25x2+ 0.08x3 + 0.21x4  72 hr 
3x1 + 3x2 + x3 + x4  1,200 boxes 
$36x1 + $48x2 + $25x3 + $35x4  $25,000 
x1 + x2  500 dozen sweatshirts 
x3 + x4  500 dozen T-shirts 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-7 
A Product Mix Example 
Computer Solution with Excel (5 of 8) 
Exhibit 4.1 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-8 
A Product Mix Example 
Solution with Excel Solver Window (6 of 8) 
Exhibit 4.2 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-9 
A Product Mix Example 
Solution with QM for Windows (7 of 8) 
Exhibit 4.3 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-10 
A Product Mix Example 
Solution with QM for Windows (8 of 8) 
Exhibit 4.4 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Cost 
($) 
0.18 
0.22 
0.10 
0.12 
0.10 
0.09 
0.40 
0.16 
0.50 
0.07 
4-11 
A Diet Example 
Data and Problem Definition (1 of 5) 
Breakfast Food 
Cal 
Fat 
(g) 
Cholesterol 
(mg) 
Iron 
(mg) 
Calcium 
(mg) 
Protein 
(g) 
Fiber 
(g) 
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 
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. 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-12 
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 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-13 
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  12 
xi  0, for all j 
A Diet Example 
Model Summary (3 of 5) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-14 
Exhibit 4.5 
A Diet Example 
Computer Solution with Excel (4 of 5) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-15 
A Diet Example 
Solution with Excel Solver Window (5 of 5) 
Exhibit 4.6 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-16 
An Investment Example 
Model Summary (1 of 4) 
Maximize Z = $0.085x1 + 0.05x2 + 0.065 x3+ 0.130x4 
subject to: 
x1  $14,000 
x2 - x1 - x3- x4  0 
x2 + x3  $21,000 
-1.2x1 + x2 + x3 - 1.2 x4  0 
x1 + x2 + x3 + x4 = $70,000 
x1, x2, x3, x4  0 
where 
x1 = amount ($) invested in municipal bonds 
x2 = amount ($) invested in certificates of deposit 
x3 = amount ($) invested in treasury bills 
x4 = amount ($) invested in growth stock fund 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-17 
An Investment Example 
Computer Solution with Excel (2 of 4) 
Exhibit 4.7 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-18 
An Investment Example 
Solution with Excel Solver Window (3 of 4) 
Exhibit 4.8 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-19 
An Investment Example 
Sensitivity Report (4 of 4) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 
Exhibit 4.9
4-20 
A Marketing Example 
Data and Problem Definition (1 of 6) 
Exposure 
(people/ad or 
commercial) 
Cost 
Television Commercial 20,000 $15,000 
Radio Commercial 2,000 6,000 
Newspaper Ad 9,000 4,000 
 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 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-21 
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 = number of television commercials 
x2 = number of radio commercials 
x3 = number of newspaper ads 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-22 
Exhibit 4.10 
A Marketing Example 
Solution with Excel (3 of 6) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-23 
A Marketing Example 
Solution with Excel Solver Window (4 of 6) 
Exhibit 4.11 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-24 
A Marketing Example 
Integer Solution with Excel (5 of 6) 
Exhibit 4.13 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 
Exhibit 4.12
4-25 
A Marketing Example 
Integer Solution with Excel (6 of 6) 
Exhibit 4.14 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-26 
A Transportation Example 
Problem Definition and Data (1 of 3) 
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 
Unit Shipping Costs: 
From Warehouse 
To Store 
A B C 
1 $16 $18 $11 
2 14 12 13 
3 13 15 17 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-27 
A Transportation Example 
Model Summary (2 of 4) 
Minimize Z = $16x1A + 18x1B + 11x1C + 14x2A + 12x2B + 13x2C + 
13x3A + 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 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-28 
A Transportation Example 
Solution with Excel (3 of 4) 
Exhibit 4.15 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-29 
A Transportation Example 
Solution with Solver Window (4 of 4) 
Exhibit 4.16 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-30 
A Blend Example 
Problem Definition and Data (1 of 6) 
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 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
A Blend Example 
Problem Statement and Variables (2 of 6) 
■ Determine the optimal mix of the three components in each grade 
4-31 
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). 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-32 
A Blend Example 
Model Summary (3 of 6) 
Maximize Z = 11x1s + 13x2s + 9x3s + 8x1p + 10x2p + 6x3p + 6x1e 
+ 8x2e + 4x3e 
subject to: 
x1s + x1p + x1e  4,500 bbl. 
x2s + x2p + x2e  2,700 bbl. 
x3s + x3p + x3e  3,500 bbl. 
0.50x1s - 0.50x2s - 0.50x3s  0 
0.70x2s - 0.30x1s - 0.30x3s  0 
0.60x1p - 0.40x2p - 0.40x3p  0 
0.75x3p - 0.25x1p - 0.25x2p  0 
0.40x1e- 0.60x2e- - 0.60x3e  0 
0.90x2e - 0.10x1e - 0.10x3e  0 
x1s + x2s + x3s  3,000 bbl. 
x1p+ x2p + x3p  3,000 bbl. 
x1e+ x2e + x3e  3,000 bbl. 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 
all xij  0
4-33 
Exhibit 4.17 
A Blend Example 
Solution with Excel (4 of 6) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-34 
A Blend Example 
Solution with Solver Window (5 of 6) 
Exhibit 4.18 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-35 
Exhibit 4.19 
A Blend Example 
Sensitivity Report (6 of 6) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-36 
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 Holding Cost: $10/computer per week 
Order schedule: 
Week Computer Orders 
1 105 
2 170 
3 230 
4 180 
5 150 
6 250 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-37 
A Multi-Period Scheduling Example 
Decision Variables (2 of 5) 
Decision Variables: 
rj = regular production of computers in week j 
(j = 1, 2, …, 6) 
oj = overtime production of computers in week j 
(j = 1, 2, …, 6) 
ij = extra computers carried over as inventory in week j 
(j = 1, 2, …, 5) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-38 
A Multi-Period Scheduling Example 
Model Summary (3 of 5) 
Model summary: 
Minimize Z = $190(r1 + r2 + r3 + r4 + r5 + r6) + $260(o1+o2 
+o3 +o4+o5+o6) + 10(i1 + i2 + i3 + i4 + i5) 
subject to: 
rj  160 computers in week j (j = 1, 2, 3, 4, 5, 6) 
oj  150 computers in week j (j = 1, 2, 3, 4, 5, 6) 
r1 + o1 - i1 = 105 week 1 
r2 + o2 + i1 - i2 = 170 week 2 
r3 + o3 + i2 - i3 = 230 week 3 
r4 + o4 + i3 - i4 = 180 week 4 
r5 + o5 + i4 - i5 = 150 week 5 
r6 + o6 + i5 = 250 week 6 
rj, oj, ij  0 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-39 
A Multi-Period Scheduling Example 
Solution with Excel (4 of 5) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 
Exhibit 4.20
4-40 
A Multi-Period Scheduling Example 
Solution with Solver Window (5 of 5) 
Exhibit 4.21 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-41 
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 
Input 2 = supplementary funds/student 
Input 3 = average educational level of parents 
Output 1 = average reading SOL score 
Output 2 = average math SOL score 
Output 3 = average history SOL score 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-42 
A Data Envelopment Analysis (DEA) Example 
Problem Data Summary (2 of 5) 
Inputs Outputs 
School 1 2 3 1 2 3 
Alton .06 $260 11.3 86 75 71 
Beeks .05 320 10.5 82 72 67 
Carey 
.08 
340 
12.0 
81 
79 
80 
Delancey 
.06 
460 
13.1 
81 
73 
69 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-43 
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 y1 + 460y2 + 13.1y3 = 1 
86x1 + 75x2 + 71x3 .06y1 + 260y2 + 11.3y3 
82x1 + 72x2 + 67x3  .05y1 + 320y2 + 10.5y3 
81x1 + 79x2 + 80x3  .08y1 + 340y2 + 12.0y3 
81x1 + 73x2 + 69x3  .06y1 + 460y2 + 13.1y3 
xi, yi  0 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-44 
A Data Envelopment Analysis (DEA) Example 
Solution with Excel (4 of 5) 
Exhibit 4.22 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-45 
A Data Envelopment Analysis (DEA) Example 
Solution with Solver Window (5 of 5) 
Exhibit 4.23 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-46 
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? 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-47 
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) + 0.06(xhb + xfb + xcb) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-48 
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: xhm + xfm + xcm + xhb + xfb+ xcb  36,000 ounces 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-49 
Example Problem Solution 
Model Summary (4 of 5) 
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 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 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
4-50 
Example Problem Solution 
Solution with QM for Windows (5 of 5) 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 4-51

More Related Content

What's hot

Supply Chain activities at Tapal Tea
Supply Chain activities at Tapal TeaSupply Chain activities at Tapal Tea
Supply Chain activities at Tapal TeaMohammedEmad1987
 
Final eb ch p supply chain management of kfc
Final eb ch p supply chain management of kfcFinal eb ch p supply chain management of kfc
Final eb ch p supply chain management of kfcazmatmengal
 
Task 5 - 5 porter model (KFC)
Task 5 - 5 porter model (KFC)Task 5 - 5 porter model (KFC)
Task 5 - 5 porter model (KFC)ilyanaismarau90
 
VoIP Project Request For Proposal Presentation
VoIP Project Request For Proposal PresentationVoIP Project Request For Proposal Presentation
VoIP Project Request For Proposal Presentationbrianbish10795
 
Expansion Strategies of Starbucks
Expansion Strategies of StarbucksExpansion Strategies of Starbucks
Expansion Strategies of StarbucksVishesh Srivastava
 
Extended Pro Forma Statement
Extended Pro Forma StatementExtended Pro Forma Statement
Extended Pro Forma StatementCarrine Aulia
 
Import Export Trading Company Powerpoint Presentation Slides
Import Export Trading Company Powerpoint Presentation SlidesImport Export Trading Company Powerpoint Presentation Slides
Import Export Trading Company Powerpoint Presentation SlidesSlideTeam
 
Ice Cream Industry in Pakistan
Ice Cream Industry in PakistanIce Cream Industry in Pakistan
Ice Cream Industry in PakistanJunaid Inam
 
Starbucks Mission Social Responsibility & Brand Strength Case Study Sample
Starbucks Mission Social Responsibility & Brand Strength Case Study SampleStarbucks Mission Social Responsibility & Brand Strength Case Study Sample
Starbucks Mission Social Responsibility & Brand Strength Case Study SampleStudents Assignment Help
 
Final paper starbucks a strategic plan.edited
Final paper starbucks a strategic plan.editedFinal paper starbucks a strategic plan.edited
Final paper starbucks a strategic plan.editedTee Dashner
 
Cotton On: Internationalisation Strategy
Cotton On: Internationalisation StrategyCotton On: Internationalisation Strategy
Cotton On: Internationalisation StrategyPooja Das
 

What's hot (20)

Final report
Final reportFinal report
Final report
 
Supply Chain activities at Tapal Tea
Supply Chain activities at Tapal TeaSupply Chain activities at Tapal Tea
Supply Chain activities at Tapal Tea
 
Final eb ch p supply chain management of kfc
Final eb ch p supply chain management of kfcFinal eb ch p supply chain management of kfc
Final eb ch p supply chain management of kfc
 
Task 5 - 5 porter model (KFC)
Task 5 - 5 porter model (KFC)Task 5 - 5 porter model (KFC)
Task 5 - 5 porter model (KFC)
 
VoIP Project Request For Proposal Presentation
VoIP Project Request For Proposal PresentationVoIP Project Request For Proposal Presentation
VoIP Project Request For Proposal Presentation
 
Expansion Strategies of Starbucks
Expansion Strategies of StarbucksExpansion Strategies of Starbucks
Expansion Strategies of Starbucks
 
Slides for ch08
Slides for ch08Slides for ch08
Slides for ch08
 
Marketing Plan for Nespresso
Marketing Plan for Nespresso Marketing Plan for Nespresso
Marketing Plan for Nespresso
 
Extended Pro Forma Statement
Extended Pro Forma StatementExtended Pro Forma Statement
Extended Pro Forma Statement
 
Import Export Trading Company Powerpoint Presentation Slides
Import Export Trading Company Powerpoint Presentation SlidesImport Export Trading Company Powerpoint Presentation Slides
Import Export Trading Company Powerpoint Presentation Slides
 
Ice Cream Industry in Pakistan
Ice Cream Industry in PakistanIce Cream Industry in Pakistan
Ice Cream Industry in Pakistan
 
Starbucks Mission Social Responsibility & Brand Strength Case Study Sample
Starbucks Mission Social Responsibility & Brand Strength Case Study SampleStarbucks Mission Social Responsibility & Brand Strength Case Study Sample
Starbucks Mission Social Responsibility & Brand Strength Case Study Sample
 
CocaCola Audit
CocaCola AuditCocaCola Audit
CocaCola Audit
 
The Coca Cola company
The Coca Cola companyThe Coca Cola company
The Coca Cola company
 
Final paper starbucks a strategic plan.edited
Final paper starbucks a strategic plan.editedFinal paper starbucks a strategic plan.edited
Final paper starbucks a strategic plan.edited
 
Pepsi
PepsiPepsi
Pepsi
 
Rsh qam11 ch10 ge
Rsh qam11 ch10 geRsh qam11 ch10 ge
Rsh qam11 ch10 ge
 
Ufone report
Ufone reportUfone report
Ufone report
 
Cotton On: Internationalisation Strategy
Cotton On: Internationalisation StrategyCotton On: Internationalisation Strategy
Cotton On: Internationalisation Strategy
 
KFC Analysis
KFC Analysis KFC Analysis
KFC Analysis
 

Similar to Taylor introms10 ppt_04

Taylor introms10 ppt_04
Taylor introms10 ppt_04Taylor introms10 ppt_04
Taylor introms10 ppt_04QA Cmu
 
Taylor introms10 ppt_03
Taylor introms10 ppt_03Taylor introms10 ppt_03
Taylor introms10 ppt_03QA Cmu
 
Taylor introms10 ppt_02
Taylor introms10 ppt_02Taylor introms10 ppt_02
Taylor introms10 ppt_02QA Cmu
 
Chapter 3 linear programming
Chapter 3   linear programmingChapter 3   linear programming
Chapter 3 linear programmingsarkissk
 
256 slide for management faculty science for excel
256 slide for management faculty science for excel256 slide for management faculty science for excel
256 slide for management faculty science for excelimamaliazizov1
 
chapter2-2-_to_be_sent.ppt................
chapter2-2-_to_be_sent.ppt................chapter2-2-_to_be_sent.ppt................
chapter2-2-_to_be_sent.ppt................ssuser486a8b
 
Algebra 2 Section 0-2
Algebra 2 Section 0-2Algebra 2 Section 0-2
Algebra 2 Section 0-2Jimbo Lamb
 
Operations Research
Operations ResearchOperations Research
Operations Researchajithsrc
 
Linear Programming Free Power Point Slide Presentation.ppt
Linear Programming Free Power Point Slide Presentation.pptLinear Programming Free Power Point Slide Presentation.ppt
Linear Programming Free Power Point Slide Presentation.pptmarkgilvinson
 
Ejercicios y problemas sobre maximización y minimización por el método gráfico
Ejercicios y problemas sobre maximización y minimización por el método gráfico Ejercicios y problemas sobre maximización y minimización por el método gráfico
Ejercicios y problemas sobre maximización y minimización por el método gráfico osvaldo bacile lambrechts
 
Multiplication ppt
Multiplication pptMultiplication ppt
Multiplication pptAbha Arora
 
Duality in Linear Programming
Duality in Linear ProgrammingDuality in Linear Programming
Duality in Linear Programmingjyothimonc
 
Formulation Lpp
Formulation  LppFormulation  Lpp
Formulation LppSachin MK
 
Bài Giảng: Quy hoạch tuyến tính (Linear Programming)
Bài Giảng: Quy hoạch tuyến tính (Linear Programming)Bài Giảng: Quy hoạch tuyến tính (Linear Programming)
Bài Giảng: Quy hoạch tuyến tính (Linear Programming)Hiền Tôn Nguyễn Trọng
 
Tutorials--Polynomial Expansion
Tutorials--Polynomial ExpansionTutorials--Polynomial Expansion
Tutorials--Polynomial ExpansionMedia4math
 

Similar to Taylor introms10 ppt_04 (20)

Taylor introms10 ppt_04
Taylor introms10 ppt_04Taylor introms10 ppt_04
Taylor introms10 ppt_04
 
Taylor introms10 ppt_03
Taylor introms10 ppt_03Taylor introms10 ppt_03
Taylor introms10 ppt_03
 
Taylor introms10 ppt_02
Taylor introms10 ppt_02Taylor introms10 ppt_02
Taylor introms10 ppt_02
 
Chapter 3 linear programming
Chapter 3   linear programmingChapter 3   linear programming
Chapter 3 linear programming
 
256 slide for management faculty science for excel
256 slide for management faculty science for excel256 slide for management faculty science for excel
256 slide for management faculty science for excel
 
chapter2-2-_to_be_sent.ppt................
chapter2-2-_to_be_sent.ppt................chapter2-2-_to_be_sent.ppt................
chapter2-2-_to_be_sent.ppt................
 
Rsh qam11 ch08 ge
Rsh qam11 ch08 geRsh qam11 ch08 ge
Rsh qam11 ch08 ge
 
Algebra 2 Section 0-2
Algebra 2 Section 0-2Algebra 2 Section 0-2
Algebra 2 Section 0-2
 
Operations Research
Operations ResearchOperations Research
Operations Research
 
Chapter 3
Chapter 3Chapter 3
Chapter 3
 
Linear Programming Free Power Point Slide Presentation.ppt
Linear Programming Free Power Point Slide Presentation.pptLinear Programming Free Power Point Slide Presentation.ppt
Linear Programming Free Power Point Slide Presentation.ppt
 
Algebra 3
Algebra 3Algebra 3
Algebra 3
 
Ejercicios y problemas sobre maximización y minimización por el método gráfico
Ejercicios y problemas sobre maximización y minimización por el método gráfico Ejercicios y problemas sobre maximización y minimización por el método gráfico
Ejercicios y problemas sobre maximización y minimización por el método gráfico
 
Multiplication ppt
Multiplication pptMultiplication ppt
Multiplication ppt
 
Duality in Linear Programming
Duality in Linear ProgrammingDuality in Linear Programming
Duality in Linear Programming
 
Formulation Lpp
Formulation  LppFormulation  Lpp
Formulation Lpp
 
Bài Giảng: Quy hoạch tuyến tính (Linear Programming)
Bài Giảng: Quy hoạch tuyến tính (Linear Programming)Bài Giảng: Quy hoạch tuyến tính (Linear Programming)
Bài Giảng: Quy hoạch tuyến tính (Linear Programming)
 
Quicker maths
Quicker mathsQuicker maths
Quicker maths
 
Tutorials--Polynomial Expansion
Tutorials--Polynomial ExpansionTutorials--Polynomial Expansion
Tutorials--Polynomial Expansion
 
Intro week3 excel vba_114e
Intro week3 excel vba_114eIntro week3 excel vba_114e
Intro week3 excel vba_114e
 

Recently uploaded

Falcon Invoice Discounting: The best investment platform in india for investors
Falcon Invoice Discounting: The best investment platform in india for investorsFalcon Invoice Discounting: The best investment platform in india for investors
Falcon Invoice Discounting: The best investment platform in india for investorsFalcon Invoice Discounting
 
Marel Q1 2024 Investor Presentation from May 8, 2024
Marel Q1 2024 Investor Presentation from May 8, 2024Marel Q1 2024 Investor Presentation from May 8, 2024
Marel Q1 2024 Investor Presentation from May 8, 2024Marel
 
Abortion pills in Jeddah ! +27737758557, cytotec pill riyadh. Saudi Arabia" A...
Abortion pills in Jeddah ! +27737758557, cytotec pill riyadh. Saudi Arabia" A...Abortion pills in Jeddah ! +27737758557, cytotec pill riyadh. Saudi Arabia" A...
Abortion pills in Jeddah ! +27737758557, cytotec pill riyadh. Saudi Arabia" A...bleessingsbender
 
Managerial Accounting 5th Edition by Stacey Whitecotton test bank.docx
Managerial Accounting 5th Edition by Stacey Whitecotton test bank.docxManagerial Accounting 5th Edition by Stacey Whitecotton test bank.docx
Managerial Accounting 5th Edition by Stacey Whitecotton test bank.docxssuserf63bd7
 
Unveiling Falcon Invoice Discounting: Leading the Way as India's Premier Bill...
Unveiling Falcon Invoice Discounting: Leading the Way as India's Premier Bill...Unveiling Falcon Invoice Discounting: Leading the Way as India's Premier Bill...
Unveiling Falcon Invoice Discounting: Leading the Way as India's Premier Bill...Falcon Invoice Discounting
 
Horngren’s Cost Accounting A Managerial Emphasis, Canadian 9th edition soluti...
Horngren’s Cost Accounting A Managerial Emphasis, Canadian 9th edition soluti...Horngren’s Cost Accounting A Managerial Emphasis, Canadian 9th edition soluti...
Horngren’s Cost Accounting A Managerial Emphasis, Canadian 9th edition soluti...ssuserf63bd7
 
Putting the SPARK into Virtual Training.pptx
Putting the SPARK into Virtual Training.pptxPutting the SPARK into Virtual Training.pptx
Putting the SPARK into Virtual Training.pptxCynthia Clay
 
Ital Liptz - all about Itai Liptz. news.
Ital Liptz - all about Itai Liptz. news.Ital Liptz - all about Itai Liptz. news.
Ital Liptz - all about Itai Liptz. news.htj82vpw
 
Understanding Financial Accounting 3rd Canadian Edition by Christopher D. Bur...
Understanding Financial Accounting 3rd Canadian Edition by Christopher D. Bur...Understanding Financial Accounting 3rd Canadian Edition by Christopher D. Bur...
Understanding Financial Accounting 3rd Canadian Edition by Christopher D. Bur...ssuserf63bd7
 
Pimpri Chinchwad Call Girl Just Call♥️ 8084732287 ♥️Top Class Call Girl Servi...
Pimpri Chinchwad Call Girl Just Call♥️ 8084732287 ♥️Top Class Call Girl Servi...Pimpri Chinchwad Call Girl Just Call♥️ 8084732287 ♥️Top Class Call Girl Servi...
Pimpri Chinchwad Call Girl Just Call♥️ 8084732287 ♥️Top Class Call Girl Servi...pr788182
 
Getting Real with AI - Columbus DAW - May 2024 - Nick Woo from AlignAI
Getting Real with AI - Columbus DAW - May 2024 - Nick Woo from AlignAIGetting Real with AI - Columbus DAW - May 2024 - Nick Woo from AlignAI
Getting Real with AI - Columbus DAW - May 2024 - Nick Woo from AlignAITim Wilson
 
DUNGARPUR 💋 Call Girl 9827461493 Call Girls in Escort service book now
DUNGARPUR 💋 Call Girl 9827461493 Call Girls in  Escort service book nowDUNGARPUR 💋 Call Girl 9827461493 Call Girls in  Escort service book now
DUNGARPUR 💋 Call Girl 9827461493 Call Girls in Escort service book nowkapoorjyoti4444
 
GURGAON CALL GIRL ❤ 8272964427❤ CALL GIRLS IN GURGAON ESCORTS SERVICE PROVIDE
GURGAON CALL GIRL ❤ 8272964427❤ CALL GIRLS IN GURGAON  ESCORTS SERVICE PROVIDEGURGAON CALL GIRL ❤ 8272964427❤ CALL GIRLS IN GURGAON  ESCORTS SERVICE PROVIDE
GURGAON CALL GIRL ❤ 8272964427❤ CALL GIRLS IN GURGAON ESCORTS SERVICE PROVIDEkajalroy875762
 
10 Influential Leaders Defining the Future of Digital Banking in 2024.pdf
10 Influential Leaders Defining the Future of Digital Banking in 2024.pdf10 Influential Leaders Defining the Future of Digital Banking in 2024.pdf
10 Influential Leaders Defining the Future of Digital Banking in 2024.pdfciolook1
 
How does a bike-share company navigate speedy success? - Cyclistic
How does a bike-share company navigate speedy success? - CyclisticHow does a bike-share company navigate speedy success? - Cyclistic
How does a bike-share company navigate speedy success? - CyclisticChristofer Vizcaino
 
Pre Engineered Building Manufacturers Hyderabad.pptx
Pre Engineered  Building Manufacturers Hyderabad.pptxPre Engineered  Building Manufacturers Hyderabad.pptx
Pre Engineered Building Manufacturers Hyderabad.pptxRoofing Contractor
 
PALWAL CALL GIRL ❤ 82729*64427❤ CALL GIRLS IN PALWAL ESCORTS
PALWAL CALL GIRL ❤ 82729*64427❤ CALL GIRLS IN PALWAL ESCORTSPALWAL CALL GIRL ❤ 82729*64427❤ CALL GIRLS IN PALWAL ESCORTS
PALWAL CALL GIRL ❤ 82729*64427❤ CALL GIRLS IN PALWAL ESCORTSkajalroy875762
 
JIND CALL GIRL ❤ 8272964427❤ CALL GIRLS IN JIND ESCORTS SERVICE PROVIDE
JIND CALL GIRL ❤ 8272964427❤ CALL GIRLS IN JIND ESCORTS SERVICE PROVIDEJIND CALL GIRL ❤ 8272964427❤ CALL GIRLS IN JIND ESCORTS SERVICE PROVIDE
JIND CALL GIRL ❤ 8272964427❤ CALL GIRLS IN JIND ESCORTS SERVICE PROVIDEkajalroy875762
 

Recently uploaded (20)

Falcon Invoice Discounting: The best investment platform in india for investors
Falcon Invoice Discounting: The best investment platform in india for investorsFalcon Invoice Discounting: The best investment platform in india for investors
Falcon Invoice Discounting: The best investment platform in india for investors
 
Marel Q1 2024 Investor Presentation from May 8, 2024
Marel Q1 2024 Investor Presentation from May 8, 2024Marel Q1 2024 Investor Presentation from May 8, 2024
Marel Q1 2024 Investor Presentation from May 8, 2024
 
Abortion pills in Jeddah ! +27737758557, cytotec pill riyadh. Saudi Arabia" A...
Abortion pills in Jeddah ! +27737758557, cytotec pill riyadh. Saudi Arabia" A...Abortion pills in Jeddah ! +27737758557, cytotec pill riyadh. Saudi Arabia" A...
Abortion pills in Jeddah ! +27737758557, cytotec pill riyadh. Saudi Arabia" A...
 
Managerial Accounting 5th Edition by Stacey Whitecotton test bank.docx
Managerial Accounting 5th Edition by Stacey Whitecotton test bank.docxManagerial Accounting 5th Edition by Stacey Whitecotton test bank.docx
Managerial Accounting 5th Edition by Stacey Whitecotton test bank.docx
 
Unveiling Falcon Invoice Discounting: Leading the Way as India's Premier Bill...
Unveiling Falcon Invoice Discounting: Leading the Way as India's Premier Bill...Unveiling Falcon Invoice Discounting: Leading the Way as India's Premier Bill...
Unveiling Falcon Invoice Discounting: Leading the Way as India's Premier Bill...
 
Horngren’s Cost Accounting A Managerial Emphasis, Canadian 9th edition soluti...
Horngren’s Cost Accounting A Managerial Emphasis, Canadian 9th edition soluti...Horngren’s Cost Accounting A Managerial Emphasis, Canadian 9th edition soluti...
Horngren’s Cost Accounting A Managerial Emphasis, Canadian 9th edition soluti...
 
Putting the SPARK into Virtual Training.pptx
Putting the SPARK into Virtual Training.pptxPutting the SPARK into Virtual Training.pptx
Putting the SPARK into Virtual Training.pptx
 
Ital Liptz - all about Itai Liptz. news.
Ital Liptz - all about Itai Liptz. news.Ital Liptz - all about Itai Liptz. news.
Ital Liptz - all about Itai Liptz. news.
 
Understanding Financial Accounting 3rd Canadian Edition by Christopher D. Bur...
Understanding Financial Accounting 3rd Canadian Edition by Christopher D. Bur...Understanding Financial Accounting 3rd Canadian Edition by Christopher D. Bur...
Understanding Financial Accounting 3rd Canadian Edition by Christopher D. Bur...
 
Pimpri Chinchwad Call Girl Just Call♥️ 8084732287 ♥️Top Class Call Girl Servi...
Pimpri Chinchwad Call Girl Just Call♥️ 8084732287 ♥️Top Class Call Girl Servi...Pimpri Chinchwad Call Girl Just Call♥️ 8084732287 ♥️Top Class Call Girl Servi...
Pimpri Chinchwad Call Girl Just Call♥️ 8084732287 ♥️Top Class Call Girl Servi...
 
Home Furnishings Ecommerce Platform Short Pitch 2024
Home Furnishings Ecommerce Platform Short Pitch 2024Home Furnishings Ecommerce Platform Short Pitch 2024
Home Furnishings Ecommerce Platform Short Pitch 2024
 
Getting Real with AI - Columbus DAW - May 2024 - Nick Woo from AlignAI
Getting Real with AI - Columbus DAW - May 2024 - Nick Woo from AlignAIGetting Real with AI - Columbus DAW - May 2024 - Nick Woo from AlignAI
Getting Real with AI - Columbus DAW - May 2024 - Nick Woo from AlignAI
 
DUNGARPUR 💋 Call Girl 9827461493 Call Girls in Escort service book now
DUNGARPUR 💋 Call Girl 9827461493 Call Girls in  Escort service book nowDUNGARPUR 💋 Call Girl 9827461493 Call Girls in  Escort service book now
DUNGARPUR 💋 Call Girl 9827461493 Call Girls in Escort service book now
 
GURGAON CALL GIRL ❤ 8272964427❤ CALL GIRLS IN GURGAON ESCORTS SERVICE PROVIDE
GURGAON CALL GIRL ❤ 8272964427❤ CALL GIRLS IN GURGAON  ESCORTS SERVICE PROVIDEGURGAON CALL GIRL ❤ 8272964427❤ CALL GIRLS IN GURGAON  ESCORTS SERVICE PROVIDE
GURGAON CALL GIRL ❤ 8272964427❤ CALL GIRLS IN GURGAON ESCORTS SERVICE PROVIDE
 
10 Influential Leaders Defining the Future of Digital Banking in 2024.pdf
10 Influential Leaders Defining the Future of Digital Banking in 2024.pdf10 Influential Leaders Defining the Future of Digital Banking in 2024.pdf
10 Influential Leaders Defining the Future of Digital Banking in 2024.pdf
 
How does a bike-share company navigate speedy success? - Cyclistic
How does a bike-share company navigate speedy success? - CyclisticHow does a bike-share company navigate speedy success? - Cyclistic
How does a bike-share company navigate speedy success? - Cyclistic
 
Pre Engineered Building Manufacturers Hyderabad.pptx
Pre Engineered  Building Manufacturers Hyderabad.pptxPre Engineered  Building Manufacturers Hyderabad.pptx
Pre Engineered Building Manufacturers Hyderabad.pptx
 
PALWAL CALL GIRL ❤ 82729*64427❤ CALL GIRLS IN PALWAL ESCORTS
PALWAL CALL GIRL ❤ 82729*64427❤ CALL GIRLS IN PALWAL ESCORTSPALWAL CALL GIRL ❤ 82729*64427❤ CALL GIRLS IN PALWAL ESCORTS
PALWAL CALL GIRL ❤ 82729*64427❤ CALL GIRLS IN PALWAL ESCORTS
 
HomeRoots Pitch Deck | Investor Insights | April 2024
HomeRoots Pitch Deck | Investor Insights | April 2024HomeRoots Pitch Deck | Investor Insights | April 2024
HomeRoots Pitch Deck | Investor Insights | April 2024
 
JIND CALL GIRL ❤ 8272964427❤ CALL GIRLS IN JIND ESCORTS SERVICE PROVIDE
JIND CALL GIRL ❤ 8272964427❤ CALL GIRLS IN JIND ESCORTS SERVICE PROVIDEJIND CALL GIRL ❤ 8272964427❤ CALL GIRLS IN JIND ESCORTS SERVICE PROVIDE
JIND CALL GIRL ❤ 8272964427❤ CALL GIRLS IN JIND ESCORTS SERVICE PROVIDE
 

Taylor introms10 ppt_04

  • 1. Linear Programming: Modeling Examples Chapter 4 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 4-1
  • 2. 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 3. 4-3 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. ■ $25,000 available for a production run. ■ 500 dozen blank T-shirts and sweatshirts in stock. ■ How many dozens (boxes) of each type of shirt to produce? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 4. 4-4 A Product Mix Example (2 of 8) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 5. 4-5 A Product Mix Example Data (3 of 8) Processing Time (hr) Per dozen Cost ($) 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 6. 4-6 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: 0.10x1 + 0.25x2+ 0.08x3 + 0.21x4  72 hr 3x1 + 3x2 + x3 + x4  1,200 boxes $36x1 + $48x2 + $25x3 + $35x4  $25,000 x1 + x2  500 dozen sweatshirts x3 + x4  500 dozen T-shirts Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 7. 4-7 A Product Mix Example Computer Solution with Excel (5 of 8) Exhibit 4.1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 8. 4-8 A Product Mix Example Solution with Excel Solver Window (6 of 8) Exhibit 4.2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 9. 4-9 A Product Mix Example Solution with QM for Windows (7 of 8) Exhibit 4.3 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 10. 4-10 A Product Mix Example Solution with QM for Windows (8 of 8) Exhibit 4.4 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 11. Cost ($) 0.18 0.22 0.10 0.12 0.10 0.09 0.40 0.16 0.50 0.07 4-11 A Diet Example Data and Problem Definition (1 of 5) Breakfast Food Cal Fat (g) Cholesterol (mg) Iron (mg) Calcium (mg) Protein (g) Fiber (g) 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 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. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 12. 4-12 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 13. 4-13 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  12 xi  0, for all j A Diet Example Model Summary (3 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 14. 4-14 Exhibit 4.5 A Diet Example Computer Solution with Excel (4 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 15. 4-15 A Diet Example Solution with Excel Solver Window (5 of 5) Exhibit 4.6 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 16. 4-16 An Investment Example Model Summary (1 of 4) Maximize Z = $0.085x1 + 0.05x2 + 0.065 x3+ 0.130x4 subject to: x1  $14,000 x2 - x1 - x3- x4  0 x2 + x3  $21,000 -1.2x1 + x2 + x3 - 1.2 x4  0 x1 + x2 + x3 + x4 = $70,000 x1, x2, x3, x4  0 where x1 = amount ($) invested in municipal bonds x2 = amount ($) invested in certificates of deposit x3 = amount ($) invested in treasury bills x4 = amount ($) invested in growth stock fund Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 17. 4-17 An Investment Example Computer Solution with Excel (2 of 4) Exhibit 4.7 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 18. 4-18 An Investment Example Solution with Excel Solver Window (3 of 4) Exhibit 4.8 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 19. 4-19 An Investment Example Sensitivity Report (4 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.9
  • 20. 4-20 A Marketing Example Data and Problem Definition (1 of 6) Exposure (people/ad or commercial) Cost Television Commercial 20,000 $15,000 Radio Commercial 2,000 6,000 Newspaper Ad 9,000 4,000  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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 21. 4-21 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 = number of television commercials x2 = number of radio commercials x3 = number of newspaper ads Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 22. 4-22 Exhibit 4.10 A Marketing Example Solution with Excel (3 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 23. 4-23 A Marketing Example Solution with Excel Solver Window (4 of 6) Exhibit 4.11 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 24. 4-24 A Marketing Example Integer Solution with Excel (5 of 6) Exhibit 4.13 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.12
  • 25. 4-25 A Marketing Example Integer Solution with Excel (6 of 6) Exhibit 4.14 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 26. 4-26 A Transportation Example Problem Definition and Data (1 of 3) 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 Unit Shipping Costs: From Warehouse To Store A B C 1 $16 $18 $11 2 14 12 13 3 13 15 17 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 27. 4-27 A Transportation Example Model Summary (2 of 4) Minimize Z = $16x1A + 18x1B + 11x1C + 14x2A + 12x2B + 13x2C + 13x3A + 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 28. 4-28 A Transportation Example Solution with Excel (3 of 4) Exhibit 4.15 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 29. 4-29 A Transportation Example Solution with Solver Window (4 of 4) Exhibit 4.16 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 30. 4-30 A Blend Example Problem Definition and Data (1 of 6) 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 31. A Blend Example Problem Statement and Variables (2 of 6) ■ Determine the optimal mix of the three components in each grade 4-31 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). Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 32. 4-32 A Blend Example Model Summary (3 of 6) Maximize Z = 11x1s + 13x2s + 9x3s + 8x1p + 10x2p + 6x3p + 6x1e + 8x2e + 4x3e subject to: x1s + x1p + x1e  4,500 bbl. x2s + x2p + x2e  2,700 bbl. x3s + x3p + x3e  3,500 bbl. 0.50x1s - 0.50x2s - 0.50x3s  0 0.70x2s - 0.30x1s - 0.30x3s  0 0.60x1p - 0.40x2p - 0.40x3p  0 0.75x3p - 0.25x1p - 0.25x2p  0 0.40x1e- 0.60x2e- - 0.60x3e  0 0.90x2e - 0.10x1e - 0.10x3e  0 x1s + x2s + x3s  3,000 bbl. x1p+ x2p + x3p  3,000 bbl. x1e+ x2e + x3e  3,000 bbl. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall all xij  0
  • 33. 4-33 Exhibit 4.17 A Blend Example Solution with Excel (4 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 34. 4-34 A Blend Example Solution with Solver Window (5 of 6) Exhibit 4.18 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 35. 4-35 Exhibit 4.19 A Blend Example Sensitivity Report (6 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 36. 4-36 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 Holding Cost: $10/computer per week Order schedule: Week Computer Orders 1 105 2 170 3 230 4 180 5 150 6 250 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 37. 4-37 A Multi-Period Scheduling Example Decision Variables (2 of 5) Decision Variables: rj = regular production of computers in week j (j = 1, 2, …, 6) oj = overtime production of computers in week j (j = 1, 2, …, 6) ij = extra computers carried over as inventory in week j (j = 1, 2, …, 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 38. 4-38 A Multi-Period Scheduling Example Model Summary (3 of 5) Model summary: Minimize Z = $190(r1 + r2 + r3 + r4 + r5 + r6) + $260(o1+o2 +o3 +o4+o5+o6) + 10(i1 + i2 + i3 + i4 + i5) subject to: rj  160 computers in week j (j = 1, 2, 3, 4, 5, 6) oj  150 computers in week j (j = 1, 2, 3, 4, 5, 6) r1 + o1 - i1 = 105 week 1 r2 + o2 + i1 - i2 = 170 week 2 r3 + o3 + i2 - i3 = 230 week 3 r4 + o4 + i3 - i4 = 180 week 4 r5 + o5 + i4 - i5 = 150 week 5 r6 + o6 + i5 = 250 week 6 rj, oj, ij  0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 39. 4-39 A Multi-Period Scheduling Example Solution with Excel (4 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.20
  • 40. 4-40 A Multi-Period Scheduling Example Solution with Solver Window (5 of 5) Exhibit 4.21 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 41. 4-41 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 Input 2 = supplementary funds/student Input 3 = average educational level of parents Output 1 = average reading SOL score Output 2 = average math SOL score Output 3 = average history SOL score Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 42. 4-42 A Data Envelopment Analysis (DEA) Example Problem Data Summary (2 of 5) Inputs Outputs School 1 2 3 1 2 3 Alton .06 $260 11.3 86 75 71 Beeks .05 320 10.5 82 72 67 Carey .08 340 12.0 81 79 80 Delancey .06 460 13.1 81 73 69 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 43. 4-43 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 y1 + 460y2 + 13.1y3 = 1 86x1 + 75x2 + 71x3 .06y1 + 260y2 + 11.3y3 82x1 + 72x2 + 67x3  .05y1 + 320y2 + 10.5y3 81x1 + 79x2 + 80x3  .08y1 + 340y2 + 12.0y3 81x1 + 73x2 + 69x3  .06y1 + 460y2 + 13.1y3 xi, yi  0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 44. 4-44 A Data Envelopment Analysis (DEA) Example Solution with Excel (4 of 5) Exhibit 4.22 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 45. 4-45 A Data Envelopment Analysis (DEA) Example Solution with Solver Window (5 of 5) Exhibit 4.23 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 46. 4-46 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? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 47. 4-47 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) + 0.06(xhb + xfb + xcb) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 48. 4-48 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: xhm + xfm + xcm + xhb + xfb+ xcb  36,000 ounces Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 49. 4-49 Example Problem Solution Model Summary (4 of 5) 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 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 50. 4-50 Example Problem Solution Solution with QM for Windows (5 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
  • 51. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 4-51