Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 4
Linear Programming: Modeling
Examples
Chapter 4 - Linear Programming: Modeling Examples
1
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
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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.
$25,000 available for a production run.
500 dozen blank T-shirts and sweatshirts in stock.
How many dozens (boxes) of each type of shirt to produce?
Chapter 4 - Linear Programming: Modeling Examples
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A Product Mix Example
Data (2 of 7)
Processing
Time (hr)
Per dozen
Cost
($)
per dozen
Profit
($)
per dozen
Sweatshirt - F
0.10
36
90
Sweatshirt – B/F
0.25
48
125
T-shirt - F
0.08
25
45
T-shirt - B/F
0.21
35
65
Chapter 4 - Linear Programming: Modeling Examples
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A Product Mix Example
Model Construction (3 of 7)
Decision Variables:
Objective Function:
Model Constraints:
Chapter 4 - Linear Programming: Modeling Examples
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A Product Mix Example
Computer Solution with Excel (4 of 7)
Exhibit 4.1
Chapter 4 - Linear Programming: Modeling Examples
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A Product Mix Example
Solution with Excel Solver Window (5 of 7)
Exhibit 4.2
Chapter 4 - Linear Programming: Modeling Examples
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A Product Mix Example
Solution with QM for Windows (6 of 7)
Exhibit 4.3
Chapter 4 - Linear Programming: Modeling Examples
8
A Product Mix Example
Solution with QM for Windows (7 of 7)
Exhibit 4.4
Chapter 4 - Linear Programming: Modeling Examples
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A Diet Example
Data and Problem Definition (1 of 5)
Breakfast Food
Cal
1. Bran cereal (cup)
90
2. Dry cereal (cup)
110
3. Oatmeal (cup)
100
4. Oat bran (cup)
90
5. Egg
75
6. Bacon (slice)
35
7. Orange
65
8. Milk-2% (cup)
100
9. Orange juice (cup) 120
10. Wheat toast (slice)
65
Fat Cholesterol Iron Calcium Protein Fiber Cost
(g)
(mg)
(mg)
(mg)
(g)
(g)
($)
0
0
6
20
3
5
0.18
2
0
4
48
4
2
0.22
2
0
2
12
5
3
0.10
2
0
3
8
6
4
0.12
5
270
1
30
7
0
0.10
3
8
0
0
2
0
0.09
0
0
1
52
1
1
0.40
4
12
0
250
9
0
0.16
0
0
0
3
1
0
0.50
1
0
1
26
3
3
0.07
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
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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
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A Diet Example
Model Summary (3 of 5)
Minimize Z =
subject to:
Chapter 4 - Linear Programming: Modeling Examples
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A Diet Example
Computer Solution with Excel (4 of 5)
Exhibit 4.5
Chapter 4 - Linear Programming: Modeling Examples
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A Diet Example
Solution with Excel Solver Window (5 of 5)
Exhibit 4.6
Chapter 4 - Linear Programming: Modeling Examples
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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
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An Investment Example
Model Summary (2 of 5)
Decision Variables:
Maximize Z =
subject to:
Chapter 4 - Linear Programming: Modeling Examples
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An Investment Example
Computer Solution with Excel (2 of 5)
Exhibit 4.7
Chapter 4 - Linear Programming: Modeling Examples
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An Investment Example
Solution with Excel Solver Window (3 of 5)
Exhibit 4.8
Chapter 4 - Linear Programming: Modeling Examples
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An Investment Example
Sensitivity Report (5 of 5)
Exhibit 4.9
Chapter 4 - Linear Programming: Modeling Examples
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A Marketing Example
Data and Problem Definition (1 of 6)
Television
Commercial
Radio
Commercial
Newspaper Ad
Exposure
(people/ad or
commercial)
20,000
Cost
$15,000
12,000
6,000
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
Chapter 4 - Linear Programming: Modeling Examples
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A Marking Example
Model Summary (2 of 6)
Decision variables:
Maximize Z =
subject to:
Chapter 4 - Linear Programming: Modeling Examples
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A Marking Example
Solution with Excel (3 of 6)
Exhibit 4.10
Chapter 4 - Linear Programming: Modeling Examples
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A Marking Example
Solution with Excel Solver Window (4 of 6)
Exhibit 4.11
Chapter 4 - Linear Programming: Modeling Examples
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A Marking Example
Integer Solution with Excel (5 of 6)
Exhibit 4.12
Exhibit 4.13
Chapter 4 - Linear Programming: Modeling Examples
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A Marking Example
Integer Solution with Excel (6 of 6)
Exhibit 4.14
Chapter 4 - Linear Programming: Modeling Examples
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A Transportation Example
Problem Definition and Data (1 of 3)
Warehouse supply of
Television Sets:
Retail store demand
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
1
A
$16
B
$18
C
$11
2
14
12
13
3
13
15
17
Chapter 4 - Linear Programming: Modeling Examples
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A Transportation Example
Model Summary (2 of 4)
Decision variables:
Minimize Z =
subject to:
Chapter 4 - Linear Programming: Modeling Examples
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A Transportation Example
Solution with Excel (3 of 4)
Exhibit 4.15
Chapter 4 - Linear Programming: Modeling Examples
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A Transportation Example
Solution with Solver Window (4 of 4)
Exhibit 4.16
Chapter 4 - Linear Programming: Modeling Examples
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A Blend Example
Problem Definition and Data (1 of 6)
Maximum Barrels
Available/day
Cost/barrel
1
4,500
$12
2
2,700
10
3
3,500
14
Component
Grade
Component Specifications
Selling Price ($/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
Extra
At least 60% of 1
At least 10% of 2
Chapter 4 - Linear Programming: Modeling Examples
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18
30
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
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A Blend Example
Model Summary (3 of 6)
Maximize Z =
subject to:
Chapter 4 - Linear Programming: Modeling Examples
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A Blend Example
Solution with Excel (4 of 6)
Exhibit 4.17
Chapter 4 - Linear Programming: Modeling Examples
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A Blend Example
Solution with Solver Window (5 of 6)
Exhibit 4.18
Chapter 4 - Linear Programming: Modeling Examples
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A Blend Example
Sensitivity Report (6 of 6)
Exhibit 4.19
Chapter 4 - Linear Programming: Modeling Examples
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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
1
2
3
4
5
6
Chapter 4 - Linear Programming: Modeling Examples
Computer Orders
105
170
230
180
150
250
36
A Multi-Period Scheduling Example
Decision Variables (2 of 5)
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)
Chapter 4 - Linear Programming: Modeling Examples
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A Multi-Period Scheduling Example
Model Summary (3 of 5)
Model summary:
Minimize Z =
subject to:
Chapter 4 - Linear Programming: Modeling Examples
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A Multi-Period Scheduling Example
Solution with Excel (4 of 5)
Exhibit 4.20
Chapter 4 - Linear Programming: Modeling Examples
39
A Multi-Period Scheduling Example
Solution with Solver Window (5 of 5)
Exhibit 4.21
Chapter 4 - Linear Programming: Modeling Examples
40
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
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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
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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
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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
Chapter 4 - Linear Programming: Modeling Examples
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Example Problem Solution
Solution with QM for Windows (5 of 5)
Exhibit 4.24
Chapter 4 - Linear Programming: Modeling Examples
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