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

4-2 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

4-3 A Product Mix Example Problem 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

4-4 A Product Mix Example (2 of 8) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-5 A Product Mix Example Data (3 of 8) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-6 Decision Variables: x 1 = sweatshirts, front printing (in boxes) x 2 = sweatshirts, back and front printing (in boxes) x 3 = T-shirts, front printing (in boxes) x 4 = T-shirts, back and front printing (in boxes) 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 8) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-7 A Product Mix Example Computer Solution with Excel (5 of 8) 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 8) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-9 Exhibit 4.3 A Product Mix Example Solution with QM for Windows (7 of 8) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-10 Exhibit 4.4 A Product Mix Example Solution with QM for Windows (8 of 8) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-11 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

4-12 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

4-13 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

4-14 Exhibit 4.5 A Diet Example Computer Solution with Excel (4 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-15 Exhibit 4.6 A Diet Example Solution with Excel Solver Window (5 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-16 $70,000 to divide between several investments Municipal bonds – 8.5% annual return Certificates of deposit – 5% Treasury bills – 6.5% Growth stock fund – 13% -No more than 20% of investment should be in Municipal Bonds - The amount invested in deposit cannot exceed the amount invested in the other three alternatives. - At least 30% in treasury bills and deposit - More should be invested in deposits and treasury bills than in bonds and funds; the ratio should be at least 1.2 to 1. where x 1 = amount ($) 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

4-17 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 (1 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-18 An Investment Example Computer Solution with Excel (2 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.7

4-19 Exhibit 4.8 An Investment Example Solution with Excel Solver Window (3 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-20 An Investment Example Sensitivity Report (4 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.9

4-21  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 Data and Problem Definition (1 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-22 Maximize Z = 20,000x ,000x 2 + 9,000x 3 subject to: 15,000x 1 + 6,000x 2 + 4,000x 3  100,000 x 1  4 x 2  10 x 3  7 x 1 + x 2 + x 3  15 x 1, x 2, x 3  0 where x 1 = number of television commercials x 2 = number of radio commercials x 3 = number of newspaper ads A Marketing Example Model Summary (2 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-23 Exhibit 4.10 A Marketing Example Solution with Excel (3 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-24 Exhibit 4.11 A Marketing Example Solution with Excel Solver Window (4 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-25 Warehouse supply ofRetail store demand Television Sets:for television sets: 1 - Cincinnati 300A - New York Atlanta 200B - Dallas Pittsburgh 200C - Detroit200 Total 700Total600 A Transportation Example Problem Definition and Data (1 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-26 Minimize Z = $16x 1A + 18x 1B + 11x 1C + 14x 2A + 12x 2B + 13x 2C + 13x 3A + 15x 3B + 17x 3C subject to: x 1A + x 1B + x 1C  300 x 2A + x 2B + x 2C  200 x 3A + x 3B + x 3C  200 x 1A + x 2A + x 3A = 150 x 1B + x 2B + x 3B = 250 x 1C + x 2C + x 3C = 200 All x ij  0 A Transportation Example Model Summary (2 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Double-subscripted variable: x ij : Number of products shipped from warehouse i to store j, for i=1,2,3 and j=A,B,C.

4-27 Exhibit 4.15 A Transportation Example Solution with Excel (3 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-28 Exhibit 4.16 A Transportation Example Solution with Solver Window (4 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-29 A Blend Example Problem Definition and Data (1 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall -We should produce at least 3000 barrels/day from each oil grade. (Super, Premium, Extra)

4-30 ■ Determine the optimal mix of the three components in each grade of motor oil that will maximize profit. ■ Decision variables: The quantity of each of the three components used in each grade of gasoline (9 decision variables)  x ij = 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 Example Problem Statement and Variables (2 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-31 x ij = 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). X1S+X2S+X3S will give you the total amount of Super. So that; MAX Z = PROFIT = TOTAL REVENUE – TOTAL COST TOT.REVENUE= 23 (X1S+X2S+X3S) + 20 (X1P+X2P+X3P) + 18 (X1E+X2E+X3E) TOTAL COST= 12 (X1S+X1P+X1E) + 14 (X2S+X2P+X2E) + 10 (X3S+X3P+X3E) MAX Z = 23 (X1S+X2S+X3S) + 20 (X1P+X2P+X3P) + 18 (X1E+X2E+X3E) - 12 (X1S+X1P+X1E) - 14 (X2S+X2P+X2E) - 10 (X3S+X3P+X3E) A Blend Example Problem Statement and Variables Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-32 x ij = 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). For example; X1S+X2S+X3S will give you the total amount of Super. So that; X1S / (X1S+X2S+X3S) >= 0,5 X2S / (X1S+X2S+X3S) =< 0,3 X1P / (X1P+X2P+X3P) >= 0,4 X3P / (X1P+X2P+X3P) =< 0,25 X1E / (X1E+X2E+X3E) >= 0,6 X2E / (X1E+X2E+X3E) >= 0,1 A Blend Example Problem Statement and Variables Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-33 ■ Company wants to produce at least 3,000 barrels of each grade of motor oil. x 1s + x 2s + x 3s  3,000 bbl. x 1p + x 2p + x 3p  3,000 bbl. x 1e + x 2e + x 3e  3,000 bbl. A Blend Example Problem Statement and Variables (2 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-34 Maximize Z = 11x 1s + 13x 2s + 9x 3s + 8x 1p + 10x 2p + 6x 3p + 6x 1e + 8x 2e + 4x 3e subject to: x 1s + x 1p + x 1e  4,500 bbl. x 2s + x 2p + x 2e  2,700 bbl. x 3s + x 3p + x 3e  3,500 bbl. 0.50x 1s x 2s x 3s  x 2s x 1s x 3s  x 1p x 2p x 3p  x 3p x 1p x 2p  x 1e x 2e x 3e  x 2e x 1e x 3e  0 x 1s + x 2s + x 3s  3,000 bbl. x 1p + x 2p + x 3p  3,000 bbl. x 1e + x 2e + x 3e  3,000 bbl. A Blend Example Model Summary (3 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall all x ij  0

4-35 Exhibit 4.17 A Blend Example Solution with Excel (4 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-36 Exhibit 4.18 A Blend Example Solution with Solver Window (5 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-37 Exhibit 4.19 A Blend Example Sensitivity Report (6 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-38 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 Problem Definition and Data (1 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-39 Decision Variables: r j = regular production of computers in week j (j = 1, 2, …, 6) o j = overtime production of computers in week j (j = 1, 2, …, 6) i j = extra computers carried over as inventory in week j (j = 1, 2, …, 5) A Multi-Period Scheduling Example Decision Variables (2 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-40 Model summary: Minimize Z = $190(r 1 + r 2 + r 3 + r 4 + r 5 + r 6 ) + $260(o 1 +o 2 +o 3 +o 4 +o 5 +o 6 ) + 10(i 1 + i 2 + i 3 + i 4 + i 5 ) subject to: r j  160 computers in week j (j = 1, 2, 3, 4, 5, 6) o j  50 computers in week j (j = 1, 2, 3, 4, 5, 6) r 1 + o 1 - i 1 = 105week 1 r 2 + o 2 + i 1 - i 2 = 170week 2 r 3 + o 3 + i 2 - i 3 = 230 week 3 r 4 + o 4 + i 3 - i 4 = 180week 4 r 5 + o 5 + i 4 - i 5 = 150week 5 r 6 + o 6 + i 5 = 250week 6 r j, o j, i j  0 A Multi-Period Scheduling Example Model Summary (3 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-41 A Multi-Period Scheduling Example Solution with Excel (4 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.20

4-42 Exhibit 4.21 A Multi-Period Scheduling Example Solution with Solver Window (5 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-43 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. ■ 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? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-44 Step 1: Define the Decision Variables x ij = 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(x hm + x fm + x cm ) (x hb + x fb + x cb ) Example Problem Solution Model Formulation (2 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-45 Step 3: Formulate the Model Constraints Amount of each ingredient available each week: x hm + x hb  9,600 ounces of horse meat x fm + x fb  12,800 ounces of fish x cm + x cb  16,000 ounces of cereal additive Recipe requirements: Meow Chow: x fm /(x hm + x fm + x cm )  1/2 or - x hm + x fm - x cm  0 Bow Chow: x hb /(x hb + x fb + x cb )  1/2 or x hb - x fb - x cb  0 Can Content: x hm + x fm + x cm + x hb + x fb + x cb  36,000 ounces Example Problem Solution Model Formulation (3 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-46 Step 4: Model Summary Maximize Z = $0.05x hm + $0.05x fm + $0.05x cm + $0.06x hb x fb x cb subject to: x hm + x hb  9,600 ounces of horse meat x fm + x fb  12,800 ounces of fish x cm + x cb  16,000 ounces of cereal additive - x hm + x fm - x cm  0 x hb - x fb - x cb  0 x hm + x fm + x cm + x hb + x fb + x cb  36,000 ounces x ij  0 Example Problem Solution Model Summary (4 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-47 Example Problem Solution Solution with QM for Windows (5 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall