Introduction to management science 10e by bernard taylor chapter 04

Linear
Programming:
Modeling Examples
Chapter 4

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4-1


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

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4-2


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?

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4-3


A Product Mix Example

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(2 of 8)

4-4


A Product Mix Example
Data (3 of 8)

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4-5


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
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sweatshirts


A Product Mix Example
Computer Solution with Excel (5 of
8)

Exhibit
4.1
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4-7


A Product Mix Example
Solution with Excel Solver Window
(6 of 8)

Exhibit
4.2

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4-8


A Product Mix Example
Solution with QM for Windows (7 of
8)

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Exhibit 4.3
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A Product Mix Example
Solution with QM for Windows (8 of
8)

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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
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Hallmilligrams 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
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x

= slices of wheat toast

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A Diet Example
Model Summary (3 of 5)
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
+ 3x + 26x10 ≥

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400 mg of calcium

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A Diet Example
Computer Solution with Excel (4 of
5)

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Exhibit 4.5
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A Diet Example
Solution with Excel Solver Window
(5 of 5)

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Exhibit 4.6

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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 - x 1 - x 3 - x 4 ≤ 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 ($) invested in certificates of
deposit
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x = amount ($) invested in treasury bills 4-16


An Investment Example
Computer Solution with Excel (2 of
4)

Exhibit 4.7
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4-17


An Investment Example
Solution with Excel Solver Window

(3 of 4)

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Exhibit 4.8
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An Investment Example
Sensitivity Report (4 of 4)

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Exhibit 4.9
4-19


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 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
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A Marketing Example
Solution with Excel (3 of 6)
Exhibit
4.10

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4-22


A Marketing Example
Solution with Excel Solver Window
(4 of 6)

Exhibit
4.11

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4-23


A Marketing Example
Integer Solution with Excel (5 of 6)

Exhibit 4.12

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Exhibit 4.13

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A Marketing Example
Integer Solution with Excel (6 of 6)

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Exhibit

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