Linear Programming: Modeling

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

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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?

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A Product Mix Example Data (2 of 7)
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A Product Mix Example Model Construction (3 of 7)
Decision Variables Objective Function
Model Constraints
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A Product Mix Example Computer Solution with
Excel (4 of 7)
Exhibit 4.1
7
A Product Mix Example Solution with Excel Solver
Window (5 of 7)
Exhibit 4.2
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A Product Mix Example Solution with QM for
Windows (6 of 7)
Exhibit 4.3
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A Product Mix Example Solution with QM for
Windows (7 of 7)
Exhibit 4.4
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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.
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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
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A Diet Example Model Summary (3 of 5)
Minimize Z subject to
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A Diet Example Computer Solution with Excel (4 of
5)
Exhibit 4.5
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A Diet Example Solution with Excel Solver Window
(5 of 5)
Exhibit 4.6
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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 lt in other 3 alternatives
• At 30 in treasury bills and CD
• Ratio in (TB and CD) to (MB and GSF) gt 1.2 to 1

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An Investment Example Model Summary (2 of 5)
Decision Variables Maximize Z subject
to
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An Investment Example Computer Solution with
Excel (2 of 5)
Exhibit 4.7
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An Investment Example Solution with Excel Solver
Window (3 of 5)
Exhibit 4.8
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An Investment Example Sensitivity Report (5 of 5)
Exhibit 4.9
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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

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A Marking Example Model Summary (2 of 6)
Decision variables Maximize Z subject to
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A Marking Example Solution with Excel (3 of 6)
Exhibit 4.10
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A Marking Example Solution with Excel Solver
Window (4 of 6)
Exhibit 4.11
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A Marking Example Integer Solution with Excel (5
of 6)
Exhibit 4.12
Exhibit 4.13
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A Marking Example Integer Solution with Excel (6
of 6)
Exhibit 4.14
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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
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A Transportation Example Model Summary (2 of 4)
Decision variables Minimize Z subject to
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A Transportation Example Solution with Excel (3
of 4)
Exhibit 4.15
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A Transportation Example Solution with Solver
Window (4 of 4)
Exhibit 4.16
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A Blend Example Problem Definition and Data (1 of
6)
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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).

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A Blend Example Model Summary (3 of 6)
Maximize Z subject to
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A Blend Example Solution with Excel (4 of 6)
Exhibit 4.17
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A Blend Example Solution with Solver Window (5 of
6)
Exhibit 4.18
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A Blend Example Sensitivity Report (6 of 6)
Exhibit 4.19
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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 Computer
Orders 1
105 2
170 3 230
4 180
5 150 6
250
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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)
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A Multi-Period Scheduling Example Model Summary
(3 of 5)
Model summary Minimize Z subject to
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A Multi-Period Scheduling Example Solution with
Excel (4 of 5)
Exhibit 4.20
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A Multi-Period Scheduling Example Solution with
Solver Window (5 of 5)
Exhibit 4.21
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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?

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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
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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
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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
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Example Problem Solution Solution with QM for
Windows (5 of 5)
Exhibit 4.24