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Chapter 8 Linear Programming: Modeling Examples

Introduction to Management Science 8th Edition by Bernard W. Taylor III. Chapter 8 Linear Programming: Modeling Examples. Chapter Topics. A Product Mix Example A Diet Example An Investment Example A Marketing Example A Transportation Example A Blend Example

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Chapter 8 Linear Programming: Modeling Examples

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  1. Introduction to Management Science 8th Edition by Bernard W. Taylor III Chapter 8 Linear Programming: Modeling Examples Chapter 8 - Linear Programming: Modeling Examples

  2. Chapter Topics • A Product Mix Example • A Diet Example • An Investment Example • A Marketing Example • A Transportation Example • A Blend Example • A Multi-Period Scheduling Example • A Data Envelopment Analysis Example Chapter 8 - Linear Programming: Modeling Examples

  3. A Product Mix Example Problem Definition (1 of 7) • 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? Chapter 8 - Linear Programming: Modeling Examples

  4. A Product Mix Example Data (2 of 7) Chapter 8 - Linear Programming: Modeling Examples

  5. A Product Mix Example Model Construction (3 of 7) Decision Variables: x1 = sweatshirts, front printing x2 = sweatshirts, back and front printing x3 = T-shirts, front printing x4 = 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 doz. sweatshirts x3 + x4 500 doz. T-shirts Chapter 8 - Linear Programming: Modeling Examples

  6. A Product Mix Example Computer Solution with Excel (4 of 7) Exhibit 8.1 Chapter 8 - Linear Programming: Modeling Examples

  7. A Product Mix Example Solution with Excel Solver Window (5 of 7) Exhibit 8.2 Chapter 8 - Linear Programming: Modeling Examples

  8. A Product Mix Example Solution with QM for Windows (6 of 7) Exhibit 8.3 Chapter 8 - Linear Programming: Modeling Examples

  9. A Product Mix Example Solution with QM for Windows (7 of 7) Exhibit 8.4 Chapter 8 - Linear Programming: Modeling Examples

  10. A Diet Example Data and Problem Definition (1 of 5) 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. Chapter 8 - Linear Programming: Modeling Examples

  11. A Diet Example Model Construction – Decision Variables (2 of 5) 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 Chapter 8 - Linear Programming: Modeling Examples

  12. A Diet Example Model Summary (3 of 5) 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 2x2 + 2x3 + 2x4 + 5x5 + 3x6 + 4x8 + x10 20 270x5 + 8x6 + 12x8 30 6x1 + 4x2 + 2x3 + 3x4+ x5 + x7 + x10 5 20x1 + 48x2 + 12x3 + 8x4+ 30x5 + 52x7 + 250x8 + 3x9 + 26x10 400 3x1 + 4x2 + 5x3 + 6x4 + 7x5 + 2x6 + x7+ 9x8+ x9 + 3x10 20 5x1 + 2x2 + 3x3 + 4x4+ x7 + 3x10 12 xi  0, i=1,…,10. Chapter 8 - Linear Programming: Modeling Examples

  13. A Diet Example Computer Solution with Excel (4 of 5) Exhibit 8.5 Chapter 8 - Linear Programming: Modeling Examples

  14. A Diet Example Solution with Excel Solver Window (5 of 5) Exhibit 8.6 Chapter 8 - Linear Programming: Modeling Examples

  15. An Investment Example Model Summary (1 of 4) Maximize Z = $0.085x1 + 0.05x2 + 0.065 x3+ 0.130x4 subject to: x1 14,000 – x1 + x2 – x3 – x4 0 x2 + x3 21,000 –1.2x1 + x2 + x3 – 1.2 x4 0 x1 + x2 + x3 + x4 = 70,000 x1, x2, x3, x4 0 where x1 = amount invested in municipal bonds ($) x2 = amount invested in certificates of deposit ($) x3 = amount invested in treasury bills ($) x4 = amount invested in growth stock fund($) Chapter 8 - Linear Programming: Modeling Examples

  16. An Investment Example Computer Solution with Excel (2 of 4) Exhibit 8.7 Chapter 8 - Linear Programming: Modeling Examples

  17. An Investment Example Solution with Excel Solver Window (3 of 4) Exhibit 8.8 Chapter 8 - Linear Programming: Modeling Examples

  18. An Investment Example Sensitivity Report (4 of 4) Exhibit 8.9 Chapter 8 - Linear Programming: Modeling Examples

  19. A Marketing Example Data and Problem Definition (1 of 6) • 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 Chapter 8 - Linear Programming: Modeling Examples

  20. A Marketing Example Model Summary (2 of 6) 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 = Exposure from Television Commercial (people) x2 = Exposure from Radio Commercial (people) x3 = Exposure from Newspaper Ad (people) Chapter 8 - Linear Programming: Modeling Examples

  21. A Marketing Example Solution with Excel (3 of 6) Exhibit 8.10 Chapter 8 - Linear Programming: Modeling Examples

  22. A Marketing Example Solution with Excel Solver Window (4 of 6) Exhibit 8.11 Chapter 8 - Linear Programming: Modeling Examples

  23. A Marketing Example Integer Solution with Excel (5 of 6) Exhibit 8.12 Exhibit 8.13 Chapter 8 - Linear Programming: Modeling Examples

  24. A Marketing Example IntegerSolution with Excel (6 of 6) Exhibit 8.14 Chapter 8 - Linear Programming: Modeling Examples

  25. A Transportation Example Problem Definition and Data (1 of 3) 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 600 Chapter 8 - Linear Programming: Modeling Examples

  26. A Transportation Example Model Summary (2 of 4) 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 xij 0, i=1,2,3 and j= A,B,C. Chapter 8 - Linear Programming: Modeling Examples

  27. A Transportation Example Solution with Excel (3 of 4) Exhibit 8.15 Chapter 8 - Linear Programming: Modeling Examples

  28. A Transportation Example Solution with Solver Window (4 of 4) Exhibit 8.16 Chapter 8 - Linear Programming: Modeling Examples

  29. A Blend Example Problem Definition and Data (1 of 6) Chapter 8 - Linear Programming: Modeling Examples

  30. A Blend Example Problem Statement and Variables (2 of 6) • 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 motor oil (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). Chapter 8 - Linear Programming: Modeling Examples

  31. production requirement component availability blend specification; e.g.: x1s/(x1s+x2s+x3s) ≥ 0.50 x1s ≥ 0.50·(x1s+x2s+x3s) 0.50·(x1s–x2s–x3s) ≥ 0 A Blend Example Model Summary (3 of 6) Maximize Z = 11x1s + 13x2s + 9x3s + 8x1p + 10x2p + 6x3p + 6x1e + 8x2e + 4x3e subject to: x1s + x1p + x1e 4,500 x2s + x2p + x2e 2,700 x3s + x3p + x3e 3,500 0.50·(x1s – x2s – x3s)  0 0.70x2s – 0.30·(x1s + x3s)  0 0.60x1p – 0.40·(x2p + x3p)  0 0.25·(3·x3p – x1p – x2p)  0 0.20·(2·x1e – 3·x2e- – 3·x3e)  0 0.10·(9·x2e – x1e – x3e)  0 x1s + x2s + x3s 3,000 x1p+ x2p + x3p 3,000 x1e+ x2e + x3e 3,000 xij 0, i=1,2,3, and j=s,p,e. Chapter 8 - Linear Programming: Modeling Examples

  32. A Blend Example Solution with Excel (4 of 6) Exhibit 8.17 Chapter 8 - Linear Programming: Modeling Examples

  33. A Blend Example Solution with Solver Window (5 of 6) Exhibit 8.18 Chapter 8 - Linear Programming: Modeling Examples

  34. A Blend Example Sensitivity Report (6 of 6) Exhibit 8.19 Chapter 8 - Linear Programming: Modeling Examples

  35. A Multi-Period Scheduling Example Problem Definition and Data (1 of 5) Production Capacity: 160 computers per week 50 more computers with overtime Assembly Costs: $190 per computer regular time; $260 per computer overtime Inventory Cost: $10/comp. per week Order schedule: Week (j) Computer Orders (dj) 1 105 2 170 3 230 4 180 5 150 6 250 Chapter 8 - Linear Programming: Modeling Examples

  36. A Multi-Period Scheduling Example Decision Variables (2 of 5) Decision Variables: rj = regular production of computers per week j (j = 1,…,6) -- decision variable oj = overtime production of computers per week j (j = 1,…,6) -- decision variable ij = extra computers carried over as inventory in week j (j = 1,…,5) — computed as ij = rj + oj + ij–1 – dj, where dj is the demand (orders) as tabulated on p.35. Note: the ij variables are being calculated, so that they are not independent from the rj and oj decision variables: the ij’s are functions of rj and oj. Chapter 8 - Linear Programming: Modeling Examples

  37. Overall constraints Individual constraints Nontrivial for the ij! A Multi-Period Scheduling Example Model Summary (3 of 5) 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 (j = 1, …, 6) oj 150 (j = 1, …, 6) r1 + o1 - i1 105 r2 + o2 + i1 - i2 170 r3 + o3 + i2 - i3 230 r4 + o4 + i3 - i4 180 r5 + o5 + i4 - i5 150 r6 + o6 + i5 250 rj, oj, ij 0 Chapter 8 - Linear Programming: Modeling Examples

  38. A Multi-Period Scheduling Example Solution with Excel (4 of 5) Exhibit 8.20 Chapter 8 - Linear Programming: Modeling Examples

  39. A Multi-Period Scheduling Example Solution with Solver Window (5 of 5) Exhibit 8.21 Chapter 8 - Linear Programming: Modeling Examples

  40. A Data Envelopment Analysis (DEA) Example Problem Definition (1 of 5) 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 ratio Output 1 = average reading SOL score Input 2 = supplementary $/student Output 2 = average math SOL score Input 3 = parent education level Output 3 = average history SOL score Chapter 8 - Linear Programming: Modeling Examples

  41. A Data Envelopment Analysis (DEA) Example Problem Data Summary (2 of 5) Chapter 8 - Linear Programming: Modeling Examples

  42. A Data Envelopment Analysis (DEA) Example Decision Variables and Model Summary (3 of 5) Decision Variables: xi = a price per unit of each output where i = 1, 2, 3 yi = a price per unit of each input where i = 1, 2, 3 Model Summary: Maximize Z = 81·x1 + 73·x2 + 69·x3 subject to: .06y1 + 460y2 + 13.1y3 = 1 86x1 + 75x2 + 71x3.06y1 + 260y2 + 11.3y3 82x1 + 72x2 + 67x3 .05y1 + 320y2 + 10.5y3 81x1 + 79x2 + 80x3 .08y1 + 340y2 + 12.0y3 81x1 + 73x2 + 69x3 .06y1 + 460y2 + 13.1y3 xi, yi 0, i=1,2,3. Chapter 8 - Linear Programming: Modeling Examples

  43. A Data Envelopment Analysis (DEA) Example Solution with Excel (4 of 5) Exhibit 8.22 Chapter 8 - Linear Programming: Modeling Examples

  44. A Data Envelopment Analysis (DEA) Example Solution with Solver Window (5 of 5) Exhibit 8.23 Chapter 8 - Linear Programming: Modeling Examples

  45. 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 (1/2) fish;Bow Chow at least half (1/2) 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? Chapter 8 - Linear Programming: Modeling Examples

  46. Example Problem Solution Model Formulation (2 of 5) 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) Chapter 8 - Linear Programming: Modeling Examples

  47. Example Problem Solution Model Formulation (3 of 5) Step 3: Formulate the Model Constraints Amount of each ingredient available each week: xhm + xhb 9,600 ounces of horse meat xfm + xfb  12,800 ounces of fish xcm + 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 Constraint xhm + xfm + xcm + xhb + xfb+ xcb 36,000 ounces Chapter 8 - Linear Programming: Modeling Examples

  48. Example Problem Solution Model Summary (4 of 5) 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 meat xfm + xfb  12,800 ounces of fish xcm + xcb 16,000 ounces of cereal additive - xhm + xfm- xcm 0 xhb- xfb - xcb 0 xhm + xfm + xcm + xhb + xfb+ xcb 36,000 ounces xij 0, i=h,f,m, and j=m,b. Chapter 8 - Linear Programming: Modeling Examples

  49. Example Problem Solution Solution with QM for Windows (5 of 5) Exhibit 8.24 Chapter 8 - Linear Programming: Modeling Examples

  50. Chapter 8 - Linear Programming: Modeling Examples

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