1. 1 ENGM 792

2. 2 Prototype Example K-Log Lumber Mill Warehouse

3. 3 Prototype Example K-Log Lumber Mill Warehouse 10 7 8

4. 4 Prototype Example K-Log Lumber Mill Warehouse 10 7 8 6 11 12

5. 5 Prototype Example K-Log Lumber Mill Warehouse 10 7 8 6 11 12 13 7 5

6. 6 Prototype RC DO OC SF AL SP 10 7 8 13 7 5 6 11 12

7. 7 Prototype RC DO OC SF AL SP 10 7 8 13 7 5 6 11 12 150 80 120 130 100 120 Supply Demand

8. 8 Prototype 1 6 5 4 2 3 10 7 8 13 7 5 6 11 12 150 80 120 130 100 120 Supply Demand

9. 9 Prototype Min Z = Transportation Costs s.t. Total amount shipped from plant i = Capacity at i Demand at each Warehouse is satisfied

10. 10 Prototype Min Z = 10X 14 + 7X 15 + 8X 16 + 13X 24 + 7X 25 + 5X 26 + 6X 34 + 11X 35 + 12X 36

11. 11 Prototype Min Z = 10X 14 + 7X 15 + 8X 16 + 13X 24 + 7X 25 + 5X 26 + 6X 34 + 11X 35 + 12X 36 s.t. X 14 + X 15 + X 16 = 130 X 24 + X 25 + X 26 = 100 X 34 + X 35 + X 36 = 120

12. 12 Prototype Min Z = 10X 14 + 7X 15 + 8X 16 + 13X 24 + 7X 25 + 5X 26 + 6X 34 + 11X 35 + 12X 36 s.t. X 14 + X 15 + X 16 = 130 X 24 + X 25 + X 26 = 100 X 34 + X 35 + X 36 = 120 X 14 + X 24 + X 34 = 150 X 15 + X 25 + X 35 = 80 X 16 + X 26 + X 36 = 120

13. 13 Prototype (re-index warehouse) Min Z = 10X 11 + 7X 12 + 8X 13 + 13X 21 + 7X 22 + 5X 23 + 6X 31 + 11X 32 + 12X 33 s.t. X 11 + X 12 + X 13 = 130 X 21 + X 22 + X 23 = 100 X 31 + X 32 + X 32 = 120 X 11 + X 21 + X 31 = 150 X 12 + X 22 + X 32 = 80 X 13 + X 23 + X 33 = 120

14. 14 General Formulation Transportation Problem MinZcX st Xsim Xdjn ij j n i m j n i i m j          11 1 1 12 12..,,,...,,,,..., Also, requires that supply matches demand.

15. 15 General Format Transportation Problem Also, requires that supply matches demand.

16. 16 Excel Solver Setup

17. 17 Excel Solver Setup

18. 18 Excel Solver Setup  Note Excel Solver does not use a special transportation problem method.  It just solves the problem with the usual LP software.  For larger problems Excel Solver will be considerably slower than software designed to for transportation problems

19. 19 Transportation Tableau

20. 20 Transportation Tableau Total Demand = Total Supply

21. 21 Initial Feasible Solution Northwest Cornerrequires m+n-1 basic variables Vogel’s Approximation Russel’s Approximation (Not done for class)

22. 22 Initial Feasible Solution Northwest Corner

23. 23 Initial Feasible Solution Northwest Corner

24. 24 Initial Feasible Solution Total Cost = 10(130) + 13(20) + 7(80) + 11(0) + 12(120) = $3,560

25. 25 Clever Idea Suppose we can find a loop to move units around.

26. 26 Clever Idea Suppose we can find a loop to move units around.

27. 27 Clever Idea Suppose we can find a loop to move units around.

28. 28 Clever Idea Suppose we can find a loop to move units around.

29. 29 Clever Idea Suppose we can find a loop to move units around.

30. 30 Clever Idea For each unit I can move around the loop, I can save -5 + 12 - 11 + 7 = 3 per unit of flow

31. 31 Clever Idea I can move at most 80 units around this loop

32. 32 Clever Idea I can move at most 80 units around this loop

33. 33 Clever Idea Total Cost = 10(130) + 13(20) + 11(80) + 5(80) + 12(40) = $3,320 = $3,560 - 3(80)

34. 34 Finding the Best Loop Basic Cell c ij = u i + v j Nonbasic Celld ij = c ij - u i – v j Note: book doesn’t use d’s page 321

35. 35 Transportation Algorithm Arbitrarily select u 2 = 0

36. 36 Transportation Algorithm 13 = 0 + v 1 v 1 = 13 7 = 0 + v 2 v 2 = 7

37. 37 Transportation Algorithm 10 = u 1 + 13 u 1 = -3 11 = u 3 + 7 u 3 = 4

38. 38 Transportation Algorithm 12 = 4 + v 3 v 3 = 8

39. 39 Transportation Algorithm d 12 = 7 -(-3) - 7 = +3 3

40. 40 Transportation Algorithm d 13 = 8 -(-3) - 8 = +3 33

41. 41 Transportation Algorithm d 23 = 5 -0 - 8 = -3 33 3

42. 42 Transportation Algorithm d 31 = 6 -4 - 13 = -11 33 3 11

43. 43 Transportation Algorithm Note: -3 is the same thing we got earlier by finding a loop. 33 3 11

44. 44 Transportation Algorithm Let non-basic cell with largest -d ij enter basis. 33 3 11

45. 45 Transportation Algorithm Find a feasible loop.

46. 46 Transportation Algorithm Move the maximim unit flow around the loop.

47. 47 Transportation Algorithm Move the maximim unit flow around the loop. Total Cost = 10(130) + 13(20) + 7(80) + 12(120) = $3,560

48. 48 Transportation Algorithm Note that ui and vj must now be recomputed from new basis. Arbitrarily select v 1 = 0

49. 49 Class Problem Find u 1, u 2, u 3, v 2, v 3 d ij for non-basic cells

50. 50 Class Problem Find u 1, u 2, u 3, v 2, v 3 d ij for non-basic cells

51. 51 Class Problem Find u 1, u 2, u 3, v 2, v 3 d ij for non-basic cells

52. 52 Class Problem Find u 1, u 2, u 3, v 2, v 3 and d ij for non-basic cells 8 14

53. 53 Class Problem Find most -d ij. Find feasible loop for transfer. 14

54. 54 Class Problem Find most -d ij. Find feasible loop for transfer.

55. 55 Class Problem Total Cost = 10(130) + 7(80) + 5(20) + 6(20) + 12(120) = $3,280 = 3,560 - 20(14)

56. 56 Class Problem Arbitrarily select u 2 = 0. Find other multiplier values.

57. 57 Class Problem Arbitrarily select u 2 = 0. Find other multiplier values.

58. 58 Class Problem Arbitrarily select u 2 = 0. Find other multiplier values.

59. 59 Class Problem Arbitrarily select u 2 = 0. Find other multiplier values.

60. 60 Class Problem Arbitrarily select u 2 = 0. Find other multiplier values.

61. 61 Class Problem 118 3 Find all d ij values. Select largest –d ij to leave basis.

62. 62 Class Problem Find largest -d ij. Find feasible loop for transfer.

63. 63 Class Problem Total Cost = 10(50) + 7(80) + 5(100) + 6(100) + 12(20) = $2,400 = 3,280 - 11(80)

64. 64 Class Problem Arbitrarily select u 1 = 0. Find other multiplier values.

65. 65 Class Problem Arbitrarily select u 1 = 0. Find other multiplier values.

66. 66 Class Problem Arbitrarily select u 1 = 0. Find other multiplier values.

67. 67 Class Problem Arbitrarily select u 1 = 0. Find other multiplier values.

68. 68 Class Problem Arbitrarily select u 1 = 0. Find other multiplier values.

69. 69 Class Problem 8 Find all d ij values. Select largest –d ij to leave basis.

70. 70 Class Problem Find largest -d ij. Find feasible loop. 8

71. 71 Class Problem Find largest -d ij. Find feasible loop.

72. 72 Class Problem Total Cost = 10(30) + 7(80) + 8(20) + 5(100) + 6(120) = $2,240 = 2,400 - 8(20)

73. 73 Class Problem Arbitrarily select u 1 = 0.

74. 74 Class Problem Arbitrarily select u 1 = 0. Find other multipliers.

75. 75 Class Problem Arbitrarily select u 1 = 0. Find other multipliers.

76. 76 Class Problem optimal All d ij > 0 Solution is optimal. 63 88

77. 77 Class Problem Z = 10(30) + 7(80) + 8(20) + 5(100) + 6(120) = 2,240 63 88

78. 78 Initialization (Vogel’s)

79. 79 Initialization (Vogel’s) Table 8.17 H&L

80. 80 Initialization (Vogel’s) Table 8.17 H&L

81. 81 Initialization (Vogel’s) Table 8.17 H&L

82. 82 Initialization (Vogel’s) Table 8.17 H&L

83. 83 Initialization (Vogel’s) Table 8.17 H&L

84. 84 Initialization (Vogel’s) Table 8.17 H&L

85. 85 Dummy Warehouse Suppose total supply exceeds total demand.

86. 86 Dummy Warehouse Add dummy warehouse with 0 cost.

87. 87 Dummy Supplier Suppose total demand exceeds total supply.

88. 88 Dummy Supplier

89. 89 Final slide  Transportation Problem  Northwest corner Method  Transportation Tableau Method  Vogler’s approximation (Initialization)