1. ENGM 631 Optimization
Transportation Problems

2. Prototype Example
K-Log Lumber Mill
Warehouse

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

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

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

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

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

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

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

10. Prototype
Min Z = 10X14 + 7X15 + 8X X24 + 7X25 + 5X X X X36

11. Prototype
Min Z = 10X14 + 7X15 + 8X X24 + 7X25 + 5X X X X36
s.t.
X14 + X15 + X = 130
X24 + X25 + X26 = 100
X34 + X35 + X36 = 120

12. Prototype
Min Z = 10X14 + 7X15 + 8X X24 + 7X25 + 5X X X X36
s.t.
X14 + X15 + X = 130
X24 + X25 + X26 = 100
X34 + X35 + X36 = 120
X X X34 = 150
X X X35 = 80
X X X36 = 120

13. Prototype (re-index warehouse)
Min Z = 10X11 + 7X12 + 8X X21 + 7X22 + 5X X X X33
s.t.
X11 + X12 + X = 130
X21 + X22 + X23 = 100
X31 + X32 + X32 = 120
X X X31 = 150
X X X32 = 80
X X X33 = 120

14. General Formulation Transportation Problem
Min Z c X s t i m d j n ij = å 1 2 . ,
Also, requires that supply matches demand.

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

16. Excel Solver Setup

17. Excel Solver Setup

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. Transportation Tableau

20. Transportation Tableau
Total Demand = Total Supply

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

22. Initial Feasible Solution
Northwest Corner

23. Initial Feasible Solution
Northwest Corner

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

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

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

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

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

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

30. Clever Idea For each unit I can move around the loop, I can save
= 3 per unit of flow

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

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

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

34. Finding the Best Loop Basic Cell cij = ui + vj
Nonbasic Cell dij = cij - ui – vj
Note: book doesn’t use d’s page 321

35. Transportation Algorithm
Arbitrarily select u2 = 0

36. Transportation Algorithm
13 = 0 + v v1 = 13
7 = 0 + v v2 = 7

37. Transportation Algorithm
10 = u u1 = -3
11 = u u3 = 4

38. Transportation Algorithm
12 = 4 + v v3 = 8

39. Transportation Algorithm3
d12 = 7 -(-3) - 7 = +3

40. Transportation Algorithm3 3
d13 = 8 -(-3) - 8 = +3

41. Transportation Algorithm3 3 3
d23 = = -3

42. Transportation Algorithm3 3 3 11
d31 = = -11

43. Transportation Algorithm3 3 3 11
Note: -3 is the same thing we got earlier by finding a loop.

44. Transportation Algorithm3 3 3 11
Let non-basic cell with largest -dij enter basis.

45. Transportation Algorithm
Find a feasible loop.

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

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

48. Transportation Algorithm
Note that ui and vj must now be recomputed from new basis.
Arbitrarily select v1 = 0

49. Class Problem
Find u1, u2, u3, v2, v3
dij for non-basic cells

50. Class Problem 8 14
Find u1, u2, u3, v2, v3 and dij for non-basic cells

51. Class Problem 14
Find most -dij. Find feasible loop for transfer.

52. Class Problem
Find most -dij. Find feasible loop for transfer.

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

54. Class Problem
Arbitrarily select u2 = 0. Find other multiplier values.

55. Class Problem
Arbitrarily select u2 = 0. Find other multiplier values.

56. Class Problem
Arbitrarily select u2 = 0. Find other multiplier values.

57. Class Problem
Arbitrarily select u2 = 0. Find other multiplier values.

58. Class Problem Find all dij values. Select largest –dij to leave basis.11 8 3
Find all dij values. Select largest –dij to leave basis.

59. Class Problem
Find largest -dij. Find feasible loop for transfer.

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

61. Class Problem
Arbitrarily select u1 = 0. Find other multiplier values.

62. Class Problem
Arbitrarily select u1 = 0. Find other multiplier values.

63. Class Problem
Arbitrarily select u1 = 0. Find other multiplier values.

64. Class Problem
Arbitrarily select u1 = 0. Find other multiplier values.

65. Class Problem
Arbitrarily select u1 = 0. Find other multiplier values.

66. Class Problem Find all dij values. Select largest –dij to leave basis.8
Find all dij values. Select largest –dij to leave basis.

67. Class Problem 8
Find largest -dij. Find feasible loop.

68. Class Problem
Find largest -dij. Find feasible loop.

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

70. Class Problem
Arbitrarily select u1 = 0.

71. Class Problem
Arbitrarily select u1 = 0. Find other multipliers.

72. Class Problem
Arbitrarily select u1 = 0. Find other multipliers.

73. Class Problem 6 3 8 8
All dij > Solution is optimal.

74. Class Problem Z = 10(30) + 7(80) + 8(20) + 5(100) + 6(120) = 2,240 6 3

75. Initialization (Vogel’s)

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

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

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

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

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

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

82. Dummy Warehouse
Suppose total supply exceeds total demand.

83. Dummy Warehouse
Add dummy warehouse with 0 cost.

84. Dummy Supplier
Suppose total demand exceeds total supply.

85. Dummy Supplier

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