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

4-2 A Product Mix Example Problem Definition (1 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-3 A Product Mix Example Problem Definition (1 of 6) 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? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-4 A Product Mix Example (2 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-5 A Product Mix Example Data (3 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-6 Decision Variables: x 1 = number of dozens of sweatshirts, front printing x 2 = number of dozens of sweatshirts, back and front printing x 3 = number of dozens of T-shirts, front printing x 4 = number of dozens of T-shirts, back and front printing Objective Function: Maximize Z = $90x 1 + $125x 2 + $45x 3 + $65x 4 Model Constraints: 0.10x x x x 4  72 hr 3x 1 + 3x 2 + x 3 + x 4  1,200 boxes $36x 1 + $48x 2 + $25x 3 + $35x 4  $25,000 x 1 + x 2  500 dozen sweatshirts x 3 + x 4  500 dozen T-shirts A Product Mix Example Model Construction (4 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-7 A Product Mix Example Computer Solution with Excel (5 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.1

4-8 Exhibit 4.2 A Product Mix Example Solution with Excel Solver Window (6 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-9 A Diet Example Data and Problem Definition (1 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-10 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. A Diet Example Data and Problem Definition (1 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-11 x 1 = cups of bran cereal x 2 = cups of dry cereal x 3 = cups of oatmeal x 4 = cups of oat bran x 5 = eggs x 6 = slices of bacon x 7 = oranges x 8 = cups of milk x 9 = cups of orange juice x 10 = slices of wheat toast A Diet Example Model Construction – Decision Variables (2 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-12 MinimizeZ = 0.18x x x x x x x x x x 10 subject to: 90x x x x x x x x x x 10  420 calories 2x 2 + 2x 3 + 2x 4 + 5x 5 + 3x 6 + 4x 8 + x 10  20 g fat 270x 5 + 8x x 8  30 mg cholesterol 6x 1 + 4x 2 + 2x 3 + 3x 4 + x 5 + x 7 + x 10  5 mg iron 20x x x 3 + 8x x x x 8 + 3x x 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 5x 1 + 2x 2 + 3x 3 + 4x 4 + x 7 + 3x 10  12 x i  0, for all j A Diet Example Model Summary (3 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-13 Exhibit 4.5 A Diet Example Computer Solution with Excel (4 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-14 Exhibit 4.6 A Diet Example Solution with Excel Solver Window (5 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-15 Formulate a linear programming model for the problem An Investment Example Model Summary (1 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-16 Maximize Z = $0.085x x x x 4 subject to: x 1  $14,000 x 2 - x 1 - x 3 - x 4  0 x 2 + x 3  $21, x 1 + x 2 + x x 4  0 x 1 + x 2 + x 3 + x 4 = $70,000 x 1, x 2, x 3, x 4  0 where x 1 = amount ($) 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 fund An Investment Example Model Summary (2 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-17 An Investment Example Computer Solution with Excel (3 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.7

4-18 Exhibit 4.8 An Investment Example Solution with Excel Solver Window (4 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-19 An Investment Example Sensitivity Report (5 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.9

4-20 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall