1. MT 2351 Chapter 2 An Introduction to Linear Programming

2. MT 2352 Courtesy of NPR: The Mathematician Who Solved Major Problems http://www.npr.org/dmg/dmg.php?prgCode=WESAT&showDate=21-May- 2005&segNum=14& George Dantzig

3. MT 2353 General Form of an LP Model

4. MT 2354 General Form of an LP Model where the cs, as and bs are constants determined from the problem and the xs are the decision variables

5. MT 2355 Components of Linear Programming An objective Decision variables Constraints Parameters

6. MT 2356 Assumptions of the LP Model Divisibility - basic units of xs are divisible Proportionality - as and cs are strictly proportional to the xs Additivity - each term in the objective function and constraints contains only one variable Deterministic - all cs, as and bs are known and measured without error Non-Negativity (caveat)

7. MT 2357 Sherwood Furniture Company Recently, Sherwood Furniture Company has been interested in developing a new line of stereo speaker cabinets. In the coming month, Sherwood expects to have excess capacity in its Assembly and Finishing departments and would like to experiment with two new models. One model is the Standard, a large, high-quality cabinet in a traditional design that can be sold in virtually unlimited quantities to several manufacturers of audio equipment. The other model is the Custom, a small, inexpensive cabinet in a novel design that a single buyer will purchase on an exclusive basis. Under the tentative terms of this agreement, the buyer will purchase as many Customs as Sherwood produces, up to 32 units. The Standard requires 4 hours in the Assembly Department and 8 hours in the Finishing Department, and each unit contributes $20 to profit. The Custom requires 3 hours in Assembly and 2 hours in Finishing, and each unit contributes $10 to profit. Current plans call for 120 hours to be available next month in Assembly and 160 hours in Finishing for cabinet production, and Sherwood desires to allocate this capacity in the most economical way.

8. MT 2358 Sherwood Furniture Company – Linear Equations

9. MT 2359 Sherwood Furniture Company – Graph Solution

10. MT 23510 Sherwood Furniture Company – Graph Solution Constraint 1

11. MT 23511 Sherwood Furniture Company – Graph Solution Constraint 1

12. MT 23512 Sherwood Furniture Company – Graph Solution Constraint 2

13. MT 23513 Sherwood Furniture Company – Graph Solution Constraint 1 & 2

14. MT 23514 Sherwood Furniture Company – Graph Solution Constraint 3

15. MT 23515 Sherwood Furniture Company – Graph Solution Constraint 1, 2 & 3

16. MT 23516 Sherwood Furniture Company – Graph Solution

17. MT 23517 Sherwood Furniture Company – Graph Solution

18. MT 23518 Sherwood Furniture Company – Solve Linear Equations

19. MT 23519 Sherwood Furniture Company – Solve Linear Equations

20. MT 23520 Sherwood Furniture Company – Solve Linear Equations

21. MT 23521 Sherwood Furniture Company – Graph Solution Optimal Point (15, 20)

22. MT 23522 Sherwood Furniture Company – Slack Calculation

23. MT 23523 Sherwood Furniture Company - Slack Variables Max 20x 1 + 10x 2 + 0S 1 + 0S 2 + 0S 3 s.t. 4x 1 + 3x 2 + 1S 1 = 120 8x 1 + 2x 2 + 1S 2 = 160 x 2 + 1S 3 = 32 x 1, x 2, S 1,S 2,S 3 >= 0

24. MT 23524 Sherwood Furniture Company – Graph Solution 2 3 1

25. MT 23525 Sherwood Furniture Company – Slack Calculation Point 1 Point 1

26. MT 23526 Sherwood Furniture Company – Graph Solution 2 3 1

27. MT 23527 Sherwood Furniture Company – Slack Calculation Point 2 Point 2

28. MT 23528 Sherwood Furniture Company – Graph Solution 2 3 1

29. MT 23529 Sherwood Furniture Company – Slack Calculation Point 3 Point 3

30. MT 23530 Sherwood Furniture Company – Slack Calculation Points 1, 2 & 3 Point 1Point 2Point 3

31. MT 23531 Sherwood Furniture Company – Slack Variables For each constraint the difference between the RHS and LHS (RHS-LHS). It is the amount of resource left over. Constraint 1; S 1 = 0 hrs. Constraint 2; S 2 = 0 hrs. Constraint 3; S 3 = 12 Custom

32. MT 23532

33. MT 23533

34. MT 23534

35. MT 23535

36. MT 23536

37. MT 23537

38. MT 23538 Pet Food Company A pet food company wants to find the optimal mix of ingredients, which will minimize the cost of a batch of food, subject to constraints on nutritional content. There are two ingredients, P1 and P2. P1 costs $5/lb. and P2 costs $8/lb. A batch of food must contain no more than 400 lbs. of P1 and must contain at least 200 lbs. of P2. A batch must contain a total of at least 500 lbs. What is the optimal (minimal cost) mix for a single batch of food?

39. MT 23539 Pet Food Company – Linear Equations

40. MT 23540 Pet Food Company – Graph Solution

41. MT 23541 Pet Food Company – Graph Solution Constraint 1

42. MT 23542 Pet Food Company – Graph Solution Constraint 1

43. MT 23543 Pet Food Company – Graph Solution Constraint 2

44. MT 23544 Pet Food Company – Graph Solution Constraint 1 & 2

45. MT 23545 Pet Food Company – Graph Solution Constraint 3

46. MT 23546 Pet Food Company – Graph Solution Constraint 1, 2 & 3

47. MT 23547 Pet Food Company – Solve Linear Equations

48. MT 23548 Pet Food Company – Graph Solution

49. MT 23549 Pet Food Company – Solve Linear Equations

50. MT 23550 Pet Food Company – Solve Linear Equations

51. MT 23551 Pet Food Company – Graph Solution Optimal Point (300, 200)

52. MT 23552 Pet Food Company – Slack/ Surplus Calculation

53. MT 23553 Pet Food Co. – Linear Equations Slack/ Surplus Variables Min 5P 1 + 8P 2 + 0S 1 + 0S 2 + 0S 3 s.t. 1P 1 + 1S 1 = 400 1P 2 - 1S 2 = 200 1P 1 + 1P 2 - 1S 3 = 500 P 1, P 2, S 1,S 2,S 3 >= 0

54. MT 23554 Pet Food Co. – Slack Variables For each constraint the difference between the RHS and LHS (RHS-LHS). It is the amount of resource left over. Constraint 1; S 1 = 100 lbs.

55. MT 23555 Pet Food Co. – Surplus Variables For each constraint the difference between the LHS and RHS (LHS-RHS). It is the amount bt which a minimum requirement is exceeded. Constraint 2; S 2 = 0 lbs. Constraint 3; S 3 = 0 lbs.

56. MT 23556

57. MT 23557

58. MT 23558

59. MT 23559

60. MT 23560

61. MT 23561 Special Cases Alternate Optimal Solutions No Feasible Solution Unbounded Solutions

62. MT 23562 Alternate Optimal Solutions

63. MT 23563 Alternate Optimal Solutions

64. MT 23564 Alternate Optimal Solutions

65. MT 23565 Alternate Optimal Solutions

66. MT 23566 Alternate Optimal Solutions

67. MT 23567 Alternate Optimal Solutions

68. MT 23568 Alternate Optimal Solutions

69. MT 23569 Alternate Optimal Solutions

70. MT 23570 Alternate Optimal Solutions

71. MT 23571 Alternate Optimal Solutions A B

72. MT 23572 Alternate Optimal Solutions

73. MT 23573 Alternate Optimal Solutions

74. MT 23574

75. MT 23575

76. MT 23576

77. MT 23577

78. MT 23578

79. MT 23579 Special Cases Alternate Optimal Solutions No Feasible Solution Unbounded Solutions

80. MT 23580 No Feasible Solution

81. MT 23581 No Feasible Solution

82. MT 23582 No Feasible Solution

83. MT 23583

84. MT 23584

85. MT 23585

86. MT 23586 Special Cases Alternate Optimal Solutions No Feasible Solution Unbounded Solutions

87. MT 23587 Unbounded Solutions

88. MT 23588 Unbounded Solutions

89. MT 23589 Unbounded Solutions

90. MT 23590 Unbounded Solutions

91. MT 23591

92. MT 23592

93. MT 23593

94. MT 23594

95. MT 23595