Modeling and Operations Research

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TAYLMC04_0131961381.QXD

4/14/09

8:33 AM

Page 42

Chapter Four: Linear Programming: Modeling Examples
PROBLEM SUMMARY
1. “Product mix” example 2. “Diet” example 3. “Investment” example 4. “Marketing” example 5. “Transportation” example 6. “Blend” example 7. Product mix (maximization) 8. Sensitivity analysis (4–7) 9. Diet (minimization) 10. Product mix (minimization) 11. Product mix (maximization) 12. Product mix (maximization) 13. Product mix (maximization) 14. Ingredients mix (minimization) 15. Transportation (maximization) 16. Product mix (maximization) 17. Ingredients mix blend (minimization) 18. Crop distribution (maximization) 19. Monetary allocation (maximization) 20. Diet (minimization), sensitivity analysis 21. Transportation (maximization) 22. Transportation (minimization) 23. Warehouse scheduling (minimization) 24. School busing (minimization) 25. Sensitivity analysis (4–24) 26. Ingredients mixture (minimization) 27. Interview scheduling (maximization) 28. Multiperiod investments mixture (maximization) 29. Insurance policy mix (maximization) 30. Product mix (maximization) 31. Advertising mix (minimization), sensitivity analysis 32. Blend (maximization) 33. Multiperiod borrowing (minimization) 34. Multiperiod production scheduling (minimization) 35. Blend (maximization), sensitivity analysis 36. Assignment (minimization), sensitivity analysis 37. Transportation (minimization) 38. Scheduling (minimization) 39. Production line scheduling (maximization) 40. College admissions (maximization) 41. Network flow (minimization) 42. Blend (maximization) 43. Trim loss (minimization) 44. Multiperiod investment (maximization) 45. Multiperiod sales and inventory (maximization) 46. Multiperiod production and inventory (minimization) 47. Employee assignment (maximization) 48. Data envelopment analysis 49. Data envelopment analysis 50. Network flow (maximization) 51. Multiperiod workforce planning (minimization) 52. Integer solution (4–51) 53. Machine scheduling (maximization), sensitivity analysis 54. Cargo storage (maximization) 55. Broadcast scheduling (maximization) 56. Product mix (maximization) 57. Product mix/advertising (maximization) 58. Scheduling (minimization) 59. Consultant project assignment (minimization)

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60. Multiperiod workforce planning (minimization) 61. Multiperiod workforce (4–60) 62. Coal transportation (minimization) 63. Soccer field assignment (minimization) 64. Data envelopment analysis 65. Airline crew scheduling (maximization) 66. Product flow/scheduling (minimization) 67. Transshipment (minimization) 68. Assignment (minimization)

2.

With a minimum of 500 calories, the three food items remain the same; however, the amount of each and the total cost increases: x3 = 2.995 cups of oatmeal, x8 = 1.352 cups of milk, x10 = 1.005 slices of toast, and Z = $0.586. With a minimum of 600 calories, the food items change to x3 = 4.622 cups of oatmeal and 1.378 cups of milk and Z = $0.683. A change of variables would be expected given that 600 calories is greater than the upper limit of the sensitivity range for calories. Many different combinations of maximum servings of each of the 10 food items could be used. As an example, limiting the four hot and cold cereals, x1, x2, x3, and x4, to four cups, eggs to three, bacon to three slices, oranges to two, milk to two cups, orange juice to four cups, and wheat toast to four slices results in the following solution: x3 = 2 cups of oatmeal x4 = 1.464 cups of oat bran x5 = .065 eggs x8 = 1.033 cups of milk x10 = 4 slices of wheat toast Z = $0.828 Further limiting the servings of the four hot and cold cereals to two cups, x1 + x2 + x3 + x4 ≤ 2, results in the following solution: x3 = 2 cups of oatmeal x6 = .750 slices of bacon x8 = 2 cups of milk x9 = .115 cups of orange juice x10 = 4 slices of wheat toast Z = $0.925

PROBLEM SOLUTIONS
1. Since the profit values would change, the shadow...
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