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 x 2 = sweatshirts, back and front printing x 3 = T-shirts, front printing x 4 = 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 8) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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

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

4-9 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

4-10 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

4-11 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

4-12 $70,000 to divide between several investments Municipal bonds – 8.5% annual return Certificates of deposit – 5%33 Treasury bills – 6.5% Growth stock fund – 13% -No more than 20% of investment should be in Municipal Bonds - The amount invested in deposit cannot exceed the amount invested in the other three alternatives. - At least 30% in treasury bills and deposit - More should be invested in deposits and treasury bills than in bonds and funds; the ratio should be at least 1.2 to 1. where x 1 = amount ($) 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

4-13 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

4-14  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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-15 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

4-16 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-17 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-18 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 (Zephyr TV Comp.) Problem Definition and Data (1 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-19 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.

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

4-21 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-22 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-23 A Blend Example (A petroleum company produces three grades of motor oil from three components) Problem Definition and Data (1 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-24 ■ 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)  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

4-25 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

4-26 Solution for Blend Example X 1s = 1500 barrels X 2s = 600 barrels X 3s = 900 barrels X 1p = 1200 barrels X 2p = 1800 barrels X 1e = 1800 barrels X 2e = 1300 barrels X 3e = 900 barrels Z = $76800 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-27 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

4-28 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

4-29 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

4-30 Slotion for Multi-Period Scheduling Example r 1 = 160 computers produced in regular time in week 1 r 2 = 160 computers produced in regular time in week 2 r 3 = 160 computers produced in regular time in week 3 r 4 = 160 computers produced in regular time in week 4 r 5 = 160 computers produced in regular time in week 5 r 6 = 160 computers produced in regular time in week 6 o 3 = 25 computer produced with overtime in week 3 o 4 = 20computer produced with overtime in week 4 o 5 = 30 computer produced with overtime in week 5 o 6 = 50 computer produced with overtime in week 6 i 1 = 55 computers carried over in inventory in week 1 i 2 = 45 computers carried over in inventory in week 2 i 5 = 40 computers carried over in inventory in week 5 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

4-31 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

4-32 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

4-33 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

4-34 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

4-35 Example Problem Solution Solution with QM for Windows (5 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall