Linear Programming: Modeling Examples - Winonacourse1.winona.edu/mwolfmeyer/BA340/Chapt_04.pdf0.10x 1 + 0.25x 2 + 0.08x 3 + 0.21x 4 ≤72 hr 3x 1 + 3x 2 + x 3 + x 4 ≤1,200 boxes

4-1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Linear Programming: Modeling Examples

Chapter 4

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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 Multiperiod Scheduling Example

A Data Envelopment Analysis Example

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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A Product Mix ExampleProblem 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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A Product Mix Example (2 of 8)


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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

A Product Mix ExampleData (3 of 8)


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Decision Variables:x1 = sweatshirts, front printingx2 = sweatshirts, back and front printingx3 = T-shirts, front printingx4 = 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 hr3x1 + 3x2 + x3 + x4 ≤ 1,200 boxes

$36x1 + $48x2 + $25x3 + $35x4 ≤ $25,000x1 + x2 ≤ 500 dozen sweatshirtsx3 + x4 ≤ 500 dozen T-shirts

A Product Mix ExampleModel Construction (4 of 8)


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A Product Mix ExampleComputer Solution with Excel (5 of 8)


Exhibit 4.1

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Exhibit 4.2

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


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Exhibit 4.3

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


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Exhibit 4.4

A Product Mix ExampleSolution 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) 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

0.18 0.22 0.10 0.12 0.10 0.09 0.40 0.16 0.50 0.07

A Diet ExampleData and Problem Definition (1 of 5)


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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

A Diet ExampleModel Construction – Decision Variables (2 of 5)


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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 calories2x2 + 2x3 + 2x4 + 5x5 + 3x6 + 4x8 + x10 ≤ 20 g fat270x5 + 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 ≥ 12xi ≥ 0, for all j

A Diet ExampleModel Summary (3 of 5)


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Exhibit 4.5

A Diet ExampleComputer Solution with Excel (4 of 5)


4-15Exhibit 4.6

A Diet ExampleSolution with Excel Solver Window (5 of 5)


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Maximize Z = $0.085x1 + 0.05x2 + 0.065 x3+ 0.130x4subject to:

x1 ≤ $14,000x2 - x1 - x3- x4 ≤ 0x2 + x3 ≥ $21,000-1.2x1 + x2 + x3 - 1.2 x4 ≥ 0x1 + x2 + x3 + x4 = $70,000x1, x2, x3, x4 ≥ 0

wherex1 = amount ($) invested in municipal bondsx2 = amount ($) invested in certificates of deposit x3 = amount ($) invested in treasury billsx4 = amount ($) invested in growth stock fund

An Investment ExampleModel Summary (1 of 4)


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An Investment ExampleComputer Solution with Excel (2 of 4)


Exhibit 4.7

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Exhibit 4.8

An Investment ExampleSolution with Excel Solver Window (3 of 4)


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An Investment ExampleSensitivity Report (4 of 4)


Exhibit 4.9

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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

Resources for no more than 15 commercials and/or ads

A Marketing ExampleData and Problem Definition (1 of 6)


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Maximize Z = 20,000x1 + 12,000x2 + 9,000x3subject to:

15,000x1 + 6,000x 2+ 4,000x3 ≤ 100,000x1 ≤ 4x2 ≤ 10x3 ≤ 7x1 + x2 + x3 ≤ 15x1, x2, x3 ≥ 0

wherex1 = number of television commercialsx2 = number of radio commercialsx3 = number of newspaper ads

A Marketing ExampleModel Summary (2 of 6)


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Exhibit 4.10

A Marketing ExampleSolution with Excel (3 of 6)


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Exhibit 4.11

A Marketing ExampleSolution with Excel Solver Window (4 of 6)


4-24Exhibit 4.13

A Marketing ExampleInteger Solution with Excel (5 of 6)


Exhibit 4.12

4-25Exhibit 4.14

A Marketing ExampleInteger Solution with Excel (6 of 6)


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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 600Unit Shipping Costs:

From Warehouse To Store A B C 1 $16 $18 $11 2 14 12 13 3 13 15 17

A Transportation ExampleProblem Definition and Data (1 of 3)


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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

A Transportation ExampleModel Summary (2 of 4)


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Exhibit 4.15

A Transportation ExampleSolution with Excel (3 of 4)


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Exhibit 4.16

A Transportation ExampleSolution with Solver Window (4 of 4)


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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

A Blend ExampleProblem Definition and Data (1 of 6)


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■ 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).

A Blend ExampleProblem Statement and Variables (2 of 6)


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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 ≥ 00.70x2s - 0.30x1s - 0.30x3s ≤ 00.60x1p - 0.40x2p - 0.40x3p ≥ 00.75x3p - 0.25x1p - 0.25x2p ≤ 00.40x1e- 0.60x2e- - 0.60x3e ≥ 00.90x2e - 0.10x1e - 0.10x3e ≥ 0

x1s + x2s + x3s ≥ 3,000 bbl.x1p+ x2p + x3p ≥ 3,000 bbl.x1e+ x2e + x3e ≥ 3,000 bbl.

A Blend ExampleModel Summary (3 of 6)


all xij ≥ 0

4-33Exhibit 4.17

A Blend ExampleSolution with Excel (4 of 6)


4-34Exhibit 4.18

A Blend ExampleSolution with Solver Window (5 of 6)


4-35Exhibit 4.19

A Blend ExampleSensitivity Report (6 of 6)


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Production Capacity: 160 computers per week50 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 ExampleProblem Definition and Data (1 of 5)


Week Computer Orders1 1052 1703 2304 1805 1506 250

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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)

A Multi-Period Scheduling ExampleDecision Variables (2 of 5)


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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 1r2 + o2 + i1 - i2 = 170 week 2r3 + o3 + i2 - i3 = 230 week 3r4 + o4 + i3 - i4 = 180 week 4r5 + o5 + i4 - i5 = 150 week 5r6 + o6 + i5 = 250 week 6rj, oj, ij ≥ 0

A Multi-Period Scheduling ExampleModel Summary (3 of 5)


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A Multi-Period Scheduling ExampleSolution with Excel (4 of 5)


Exhibit 4.20

4-40Exhibit 4.21

A Multi-Period Scheduling ExampleSolution with Solver Window (5 of 5)


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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) ExampleProblem Definition (1 of 5)


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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

A Data Envelopment Analysis (DEA) ExampleProblem Data Summary (2 of 5)


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Decision Variables:

xi = a price per unit of each output where i = 1, 2, 3yi = a price per unit of each input where i = 1, 2, 3

Model Summary:

Maximize Z = 81x1 + 73x2 + 69x3subject to:

.06 y1 + 460y2 + 13.1y3 = 186x1 + 75x2 + 71x3 ≤.06y1 + 260y2 + 11.3y382x1 + 72x2 + 67x3 ≤ .05y1 + 320y2 + 10.5y381x1 + 79x2 + 80x3 ≤ .08y1 + 340y2 + 12.0y381x1 + 73x2 + 69x3 ≤ .06y1 + 460y2 + 13.1y3

xi, yi ≥ 0

A Data Envelopment Analysis (DEA) ExampleDecision Variables and Model Summary (3 of 5)


4-44Exhibit 4.22

A Data Envelopment Analysis (DEA) ExampleSolution with Excel (4 of 5)


4-45

Exhibit 4.23

A Data Envelopment Analysis (DEA) ExampleSolution with Solver Window (5 of 5)


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Example Problem SolutionProblem 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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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)

Example Problem SolutionModel Formulation (2 of 5)


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Step 3: Formulate the Model Constraints

Amount of each ingredient available each week:xhm + xhb ≤ 9,600 ounces of horse meatxfm + xfb ≤ 12,800 ounces of fishxcm + 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

Example Problem SolutionModel Formulation (3 of 5)


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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 meatxfm + xfb ≤ 12,800 ounces of fishxcm + xcb ≤ 16,000 ounces of cereal additive- xhm + xfm- xcm ≥ 0 xhb- xfb - xcb ≥ 0xhm + xfm + xcm + xhb + xfb+ xcb ≤ 36,000 ounces xij ≥ 0

Example Problem SolutionModel Summary (4 of 5)


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Example Problem SolutionSolution with QM for Windows (5 of 5)


4-51Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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Linear Programming: Modeling Examples - Winonacourse1.winona.edu/mwolfmeyer/BA340/Chapt_04.pdf0.10x 1 + 0.25x 2 + 0.08x 3 + 0.21x 4 ≤72 hr 3x 1 + 3x 2 + x 3 + x 4 ≤1,200 boxes