Introduction to management science 10e by bernard taylor chapter 04

Decision Variables: x1 = sweatshirts, front printing x2 = sweatshirts, back and front printing x3 = T-shirts, front printing x4 = T-shirts, back and front printing T-A Product Mix Exampl

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Linear Programming:

Modeling Examples

Chapter 4

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■ 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

produce?

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A Product Mix Example (2 of 8)

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Processing Time (hr) Per dozen

Cost ($) per dozen

Profit ($) per dozen

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Decision Variables:

x1 = sweatshirts, front printing

x2 = sweatshirts, back and front printing

x3 = T-shirts, front printing

x4 = T-shirts, back and front printing

T-A Product Mix Example

Model Construction (4 of 8)

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A Product Mix Example

Computer Solution with Excel (5 of 8)

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Exhibit

4.1

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Exhibit 4.2

Solution with Excel Solver Window

(6 of 8)

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Exhibit 4.3

Solution with QM for Windows (7 of 8)

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Exhibit 4.4

Solution with QM for Windows (8 of

8)

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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.

Breakfast Food

Cal

Fat (g)

Cholesterol (mg)

Iron (mg)

Calcium (mg)

Protein (g)

Fiber (g)

Cost ($)

1 Bran cereal (cup)

2 Dry cereal (cup)

9 Orange juice (cup)

10 Wheat toast (slice)

A Diet Example

Data and Problem Definition (1 of 5)

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x 1 = cups of bran cereal

x 2 = cups of dry cereal

x 9 = cups of orange juice

x 10 = slices of wheat toast

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Minimize Z = 0.18x 1 + 0.22x 2 + 0.10x 3 + 0.12x 4 +

0.10x 5 + 0.09x 6 + 0.40x 7 + 0.16x 8 + 0.50x 9 + 0.07x 10

subject to:

90x 1 + 110x 2 + 100x 3 + 90x 4 + 75x 5 + 35x 6 + 65x 7

+ 100x 8 + 120x 9 + 65x 10  420 calories

2x 2 + 2x 3 + 2x 4 + 5x 5 + 3x 6 + 4x 8 + x 10  20 g fat

270x 5 + 8x 6 + 12x 8  30 mg cholesterol 6x 1 + 4x 2 + 2x 3 + 3x 4 + x 5 + x 7 + x 10  5 mg iron 20x 1 + 48x 2 + 12x 3 + 8x 4 + 30x 5 + 52x 7 + 250x 8 + 3x 9 + 26x 10  400 mg of calcium

3x 1 + 4x 2 + 5x 3 + 6x 4 + 7x 5 + 2x 6 + x 7

+ 9x 8 + x 9 + 3x 10  20 g protein

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x 1 = amount ($) invested in municipal bonds

x 2 = amount ($) invested in certificates of deposit

x 3 = amount ($) invested in treasury bills

x 4 = amount ($) invested in growth stock

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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

ads

A Marketing Example

Data and Problem Definition (1 of 6)

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x 1 = number of television commercials

x 2 = number of radio commercials

x 3 = number of newspaper ads

Model Summary (2 of 6)

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Exhibit 4.10

Solution with Excel (3 of 6)

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Exhibit 4.11

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Exhibit 4.13

Integer Solution with Excel (5 of 6)

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Exhibit 4.12

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Exhibit 4.14

Integer Solution with Excel (6 of 6)

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Television Sets: for television sets:

Total 700Unit Shipping Costs: Total 600

From Warehouse To Store

Problem Definition and Data (1 of 3)

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Minimize Z = $16x 1A + 18x 1B + 11x 1C + 14x 2A + 12x 2B + 13x 2C + 13x 3A + 15x 3B + 17x 3C

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Exhibit 4.15

A Transportation Example

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Exhibit 4.16

A Transportation Example

Solution with Solver Window (4 of 4)

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A Blend Example

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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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Exhibit 4.17

A Blend Example

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Exhibit 4.18

A Blend Example

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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:

A Multi-Period Scheduling Example

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oj  150 computers in week j (j = 1, 2, 3, 4, 5, 6)

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Exhibit 4.20

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Exhibit 4.21

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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 ratioInput 2 = supplementary funds/studentInput 3 = average educational level of parents

Output 1 = average reading SOL scoreOutput 2 = average math SOL scoreOutput 3 = average history SOL score

A Data Envelopment Analysis (DEA)

Example

Problem Definition (1 of 5)

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Problem Data Summary (2 of 5)

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.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

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Exhibit 4.22

Example

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Exhibit 4.23

Example

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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

Bow Chow at least half horse meat

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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

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Step 3: Formulate the Model Constraints

Amount of each ingredient available each week:

xhm + xhb  9,600 ounces of horse meat

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Example Problem Solution

Solution with QM for Windows (5 of

5)

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