1 ENGM 792

2 Prototype Example K-Log Lumber Mill Warehouse

3 Prototype Example K-Log Lumber Mill Warehouse

4 Prototype Example K-Log Lumber Mill Warehouse

5 Prototype Example K-Log Lumber Mill Warehouse

6 Prototype RC DO OC SF AL SP

7 Prototype RC DO OC SF AL SP Supply Demand

8 Prototype Supply Demand

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 = 10X X X X X X X X X 36

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 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 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 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 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 Cornerrequires 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 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 Transportation Algorithm Arbitrarily select u 2 = 0

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

37 Transportation Algorithm 10 = u u 1 = = u u 3 = 4

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

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

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

41 Transportation Algorithm d 23 = =

42 Transportation Algorithm d 31 = =

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

44 Transportation Algorithm Let non-basic cell with largest -d ij 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 v 1 = 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

73 Class Problem Arbitrarily select u 1 = 0.

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

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

76 Class Problem optimal All d ij > 0 Solution is optimal

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

78 Initialization (Vogel’s)

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 Initialization (Vogel’s) Table 8.17 H&L

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

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

85 Dummy Warehouse Suppose total supply exceeds total demand.

86 Dummy Warehouse Add dummy warehouse with 0 cost.

87 Dummy Supplier Suppose total demand exceeds total supply.

88 Dummy Supplier

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