Download presentation
Presentation is loading. Please wait.
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 Algorithm
3 d12 = 7 -(-3) - 7 = +3
40
Transportation Algorithm
3 3 d13 = 8 -(-3) - 8 = +3
41
Transportation Algorithm
3 3 3 d23 = = -3
42
Transportation Algorithm
3 3 3 11 d31 = = -11
43
Transportation Algorithm
3 3 3 11 Note: -3 is the same thing we got earlier by finding a loop.
44
Transportation Algorithm
3 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)
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.