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1 ENGM 792. 2 Prototype Example K-Log Lumber Mill Warehouse.

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Presentation on theme: "1 ENGM 792. 2 Prototype Example K-Log Lumber Mill Warehouse."— Presentation transcript:

1 1 ENGM 792

2 2 Prototype Example K-Log Lumber Mill Warehouse

3 3 Prototype Example K-Log Lumber Mill Warehouse

4 4 Prototype Example K-Log Lumber Mill Warehouse

5 5 Prototype Example K-Log Lumber Mill Warehouse

6 6 Prototype RC DO OC SF AL SP

7 7 Prototype RC DO OC SF AL SP Supply Demand

8 8 Prototype 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 X X X X X X X X 36

11 11 Prototype Min Z = 10X X X X X X X X X 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 X X X X X X X X 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 X X X X X X X X 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          ,,,...,,,,..., 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 = 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, (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 u 1 = = u 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 = =

42 42 Transportation Algorithm d 31 = =

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

44 44 Transportation Algorithm Let non-basic cell with largest -d ij enter basis

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, (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 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, (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, (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

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

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)


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