4-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Linear Programming: Modeling Examples Chapter 4.

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

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

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A Product Mix ExampleProblem Definition (1 of 8)

A 4-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.

■ A 12-sweatshirts box is three times the size of a standard box.

■ $25,000 available for a production run.

■ There are 500 dozen blank T-shirts and sweatshirts in stock.

■ How many dozens (boxes) of each type of shirt to produce?

■ T-shirt with Front Print

■ T-shirt with Front and Back Print

■ Sweatshirt with Front Print

■ Sweatshirt with Front and Back Print

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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

72 25,000

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 hr 3x1 + 3x2 + x3 + x4 1,200 boxes

$36x1 + $48x2 + $25x3 + $35x4 $25,000 x1 + x2 500 dozen sweatshirts

x3 + x4 500 dozen T-shirts

A Product Mix ExampleModel Construction (4 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 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

+ 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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$70,000 to divide between several investments• Municipal bonds – 8.5% annual return• Certificates of deposit – 5%33• Treasury bills – 6.5%• Growth stock fund – 13%

-No more than 20% of investment should be in Municipal Bonds- The amount invested in deposit cannot exceed the amount invested in the other three alternatives.- At least 30% in treasury bills and deposit- More should be invested in deposits and treasury bills than in bonds and funds; the ratio should be at least 1.2 to 1.where x1 = amount ($) invested in municipal bonds x2 = amount ($) invested in certificates of deposit x3 = amount ($) invested in treasury bills x4 = amount ($) invested in growth stock fund

An Investment Example

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Maximize Z = $0.085x1 + 0.05x2 + 0.065 x3+ 0.130x4

subject to:x1 $14,000

x2 - x1 - x3- x4 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 ($) invested in certificates of deposit x3 = amount ($) invested in treasury bills x4 = amount ($) invested in growth stock fund

An Investment ExampleModel Summary (1 of 4)


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Exposure (people/ad or commercial)

Cost

Television Commercial 20,000 $15,000

Radio Commercial 12,000 6,000

Newspaper Ad 9,000 4,000

Budget limit $100,000

Television time available for 4 commercials

Radio time for 10 commercials

Newspaper space for 7 ads

Resources available for no more than 15 commercials and/or ads

A Marketing Example (The Biggs Dept. Store Chain hires an advertising firm))Data and Problem Definition (1 of 6)


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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

A Marketing ExampleModel Summary (2 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 Example (Zephyr TV Comp.)Problem 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)


Double-subscripted variable:

xij : Number of products shipped from warehouse i to store j, for i=1,2,3 and j=A,B,C.




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Component Maximum Barrels

Available/day Cost/barrel

1 4,500 $12

2 2,700 10

3 3,500 14

Oil Grade Component Needs 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

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A Blend Example (A petroleum company produces three grades of motor oil from three components)Problem 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 0 0.70x2s - 0.30x1s - 0.30x3s 0 0.60x1p - 0.40x2p - 0.40x3p 0 0.75x3p - 0.25x1p - 0.25x2p 0 0.40x1e- 0.60x2e- - 0.60x3e 0 0.90x2e - 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

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Solution for Blend Example

X1s = 1500 barrels

X2s = 600 barrels

X3s = 900 barrels

X1p = 1200 barrels

X2p = 1800 barrels

X1e = 1800 barrels

X2e = 1300 barrels

X3e = 900 barrels

Z = $76800Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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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 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 50 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 = 250week 6rj, oj, ij 0

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


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Slotion for Multi-Period Scheduling Example

r1 = 160 computers produced in regular time in week 1




r5= 160 computers produced in regular time in week 5


o3 = 25 computer produced with overtime in week 3

o4 = 20computer produced with overtime in week 4



i1 = 55 computers carried over in inventory in week 1




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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 units of 16-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)


4-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Linear Programming: Modeling Examples Chapter 4.

Documents

printing x

eggs x

oranges x

dozen sweatshirts x

cups of milk x

cups of oatmeal x

slices of bacon x

cups of bran cereal