Post on 16-Dec-2015
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ENGM 792ENGM 792
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Prototype ExamplePrototype Example
K-Log Lumber Mill Warehouse
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Prototype ExamplePrototype Example
K-Log Lumber Mill Warehouse
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Prototype ExamplePrototype Example
K-Log Lumber Mill Warehouse
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Prototype ExamplePrototype Example
K-Log Lumber Mill Warehouse
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12137
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PrototypePrototype
RC
DO
OC
SF
AL
SP
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78
137
56
1112
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PrototypePrototype
RC
DO
OC
SF
AL
SP
10
78
137
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1112
150
80
120
130
100
120
Supply
Demand
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PrototypePrototype
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137
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1112
150
80
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130
100
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Supply
Demand
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PrototypePrototype
Min Z = Transportation Costss.t.
Total amount shipped from plant i = Capacity at i
Demand at each Warehouse is satisfied
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PrototypePrototype
Min Z = 10X14 + 7X15 + 8X16 + 13X24 + 7X25 + 5X26 + 6X34 + 11X35 + 12X36
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PrototypePrototype
Min Z = 10X14 + 7X15 + 8X16 + 13X24 + 7X25 + 5X26 + 6X34 + 11X35 + 12X36
s.t.X14 + X15 + X16 = 130
X24 + X25 + X26 = 100
X34 + X35 + X36 = 120
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PrototypePrototype
Min Z = 10X14 + 7X15 + 8X16 + 13X24 + 7X25 + 5X26 + 6X34 + 11X35 + 12X36
s.t.X14 + X15 + X16 = 130
X24 + X25 + X26 = 100
X34 + X35 + X36 = 120
X14 + X24 + X34 = 150
X15 + X25 + X35 = 80
X16 + X26 + X36= 120
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Prototype Prototype (re-index (re-index warehouse)warehouse)
Min Z = 10X11 + 7X12 + 8X13 + 13X21 + 7X22 + 5X23 + 6X31 + 11X32 + 12X33
s.t.X11 + X12 + X13 = 130
X21 + X22 + X23 = 100
X31 + X32 + X32 = 120
X11 + X21 + X31 = 150
X12 + X22 + X32 = 80
X13 + X23 + X33= 120
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General FormulationGeneral FormulationTransportation Problem Transportation Problem MinZ c X
s t
X s i m
X d j n
ijj
n
i
m
ij
ijj
n
i
iji
m
j
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1
1
1 2
1 2
. .
, , ,...,
, , ,...,
Also, requires that supply matches demand.
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General Format General Format Transportation Problem Transportation Problem
Z X14 X15 X16 X24 X25 X26 X34 X35 X36 RHS
1 10 7 8 13 7 5 6 11 12 00 1 1 1 1300 1 1 1 1000 1 1 1 1200 1 1 1 1500 1 1 1 800 1 1 1 120
Also, requires that supply matches demand.
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Suppliers Supply10 7 8
Rapid City 13013 7 5
Albert Lea 1006 11 12
Spokane 120 Demand
San Francisco Ok City DaytonWarehouses
150 80 120
Transportation Transportation TableauTableau
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Transportation Transportation TableauTableau
Total Demand = Total Supply
Suppliers Supply10 7 8
Rapid City 13013 7 5
Albert Lea 1006 11 12
Spokane 120 Demand 350150 80 120
San Francisco Ok City DaytonWarehouses
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Initial Feasible Initial Feasible SolutionSolution
Northwest Corner requires m+n-1 basic variablesVogel’s ApproximationRussel’s Approximation (Not done for class)
Suppliers Supply10 7 8
Rapid City 13013 7 5
Albert Lea 1006 11 12
Spokane 120 Demand
San Francisco Ok City DaytonWarehouses
150 80 120
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Suppliers Supply10 7 8
Rapid City 13013 7 5
Albert Lea 1006 11 12
Spokane 120 Demand
San Francisco Ok City DaytonWarehouses
150 80 120
Initial Feasible Initial Feasible SolutionSolution
Northwest Corner
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Initial Feasible Initial Feasible SolutionSolution
Northwest Corner
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 0 120 120 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
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Initial Feasible Initial Feasible SolutionSolution
Total Cost = 10(130) + 13(20) + 7(80) + 11(0) + 12(120) = $3,560
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 0 120 120 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
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Clever IdeaClever Idea
Suppose we can find a loop to move units around.
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 0 120 120 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
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Clever IdeaClever Idea
Suppose we can find a loop to move units around.
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 0 120 120 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
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Clever IdeaClever Idea
Suppose we can find a loop to move units around.
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 0 120 120 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
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Clever IdeaClever Idea
Suppose we can find a loop to move units around.
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 0 120 120 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
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Clever IdeaClever Idea
Suppose we can find a loop to move units around.
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 0 120 120 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
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Clever IdeaClever Idea
For each unit I can move around the loop, I can save -5 + 12 - 11 + 7 = 3 per unit of flow
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 0 120 120 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
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Clever IdeaClever Idea
I can move at most 80 units around this loop
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 0 120 120 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
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Clever IdeaClever Idea
I can move at most 80 units around this loop
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 80 40 120 Demand
San Francisco Ok City DaytonWarehouses
150 80 120
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Clever IdeaClever Idea
Total Cost = 10(130) + 13(20) + 11(80) + 5(80) + 12(40) = $3,320 = $3,560 - 3(80)
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 80 40 120 Demand
San Francisco Ok City DaytonWarehouses
150 80 120
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Finding the Best Finding the Best LoopLoop
Basic Cell cij = ui + vj
Nonbasic Cell dij = cij - ui – vj
Note: book doesn’t use d’s page 321
Suppliers Supply10 7 8
Rapid City 130 13013 7 5
Albert Lea 20 80 1006 11 12
Spokane 0 120 120 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 Demand
Vj
150 80 120
San Francisco Ok City Dayton
Warehouses
Arbitrarily select u2 = 0
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
13 7
13 = 0 + v1 v1 = 13
7 = 0 + v2 v2 = 7
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130 -3
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 4 Demand
Vj
150 80 120
13 7
San Francisco Ok City Dayton
Warehouses
10 = u1 + 13 u1 = -3
11 = u3 + 7 u3 = 4
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130 -3
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
13 7 8
12 = 4 + v3 v3 = 8
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130 -3
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
13 7 8
d12 = 7 -(-3) - 7 = +3
3
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130 -3
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
13 7 8
d13 = 8 -(-3) - 8 = +3
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130 -3
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
13 7 8
d23 = 5 -0 - 8 = -3
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3
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130 -3
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
13 7 8
d31 = 6 -4 - 13 = -11
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3
11
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130 -3
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
13 7 8
Note: -3 is the same thing we got earlier by finding a loop.
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3
11
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130 -3
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
13 7 8
Let non-basic cell with largest -dij enter basis.
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3
11
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130 -3
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
13 7 8
Find a feasible loop.
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130 -3
13 7 5Albert Lea 20 80 100 0
6 11 12Spokane 0 120 120 4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
13 7 8
Move the maximim unit flow around the loop.
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 20 80 100
6 11 12Spokane 0 120 120 Demand
Vj
150 80 120
San Francisco Ok City Dayton
Warehouses
Move the maximim unit flow around the loop.Total Cost = 10(130) + 13(20) + 7(80) + 12(120) = $3,560
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Transportation Transportation AlgorithmAlgorithm
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 20 80 100
6 11 12Spokane 0 120 120 Demand
Vj
150 80 120
0
San Francisco Ok City Dayton
Warehouses
Note that ui and vj must now be recomputed from new basis.Arbitrarily select v1 = 0
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 20 80 100
6 11 12Spokane 0 120 120 Demand
Vj
150 80 120
0
San Francisco Ok City Dayton
Warehouses
Find u1, u2, u3, v2, v3
dij for non-basic cells
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130 10
13 7 5Albert Lea 20 80 100 13
6 11 12Spokane 0 120 120 6 Demand
Vj
150 80 120
0 -6 6
San Francisco Ok City Dayton
Warehouses
Find u1, u2, u3, v2, v3 and dij for non-basic cells
8
14
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 20 80 100
6 11 12Spokane 0 120 120 Demand
Vj
150 80 120
San Francisco Ok City Dayton
Warehouses
Find most -dij. Find feasible loop for transfer.
14
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 20 80 100
6 11 12Spokane 0 120 120 Demand
Vj
150 80 120
San Francisco Ok City Dayton
Warehouses
Find most -dij. Find feasible loop for transfer.
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 80 20 100
6 11 12Spokane 20 100 120 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
Total Cost = 10(130) + 7(80) + 5(20) + 6(20) + 12(120) = $3,280 = 3,560 - 20(14)
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 80 20 100 0
6 11 12Spokane 20 100 120 Demand
Vj
150 80 120
San Francisco Ok City Dayton
Warehouses
Arbitrarily select u2 = 0. Find other multiplier values.
57
Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 80 20 100 0
6 11 12Spokane 20 100 120 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
7 5
Arbitrarily select u2 = 0. Find other multiplier values.
58
Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 80 20 100 0
6 11 12Spokane 20 100 120 7 Demand
Vj
150 80 120
7 5
San Francisco Ok City Dayton
Warehouses
Arbitrarily select u2 = 0. Find other multiplier values.
59
Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130
13 7 5Albert Lea 80 20 100 0
6 11 12Spokane 20 100 120 7 Demand
Vj
150 80 120
-1 7 5
San Francisco Ok City Dayton
Warehouses
Arbitrarily select u2 = 0. Find other multiplier values.
61
Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130 11
13 7 5Albert Lea 80 20 100 0
6 11 12Spokane 20 100 120 7 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
-1 7 5
11 8
3
Find all dij values. Select largest –dij to leave basis.
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 130 130 11
13 7 5Albert Lea 80 20 100 0
6 11 12Spokane 20 100 120 7 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
-1 7 5
Find largest -dij. Find feasible loop for transfer.
63
Class Problem Class Problem Suppliers Supply Ui
10 7 8Rapid City 50 80 130 0
13 7 5Albert Lea 100 100
6 11 12Spokane 100 20 120 Demand
Vj
150 80 120
San Francisco Ok City Dayton
Warehouses
Total Cost = 10(50) + 7(80) + 5(100) + 6(100) + 12(20)
= $2,400 = 3,280 - 11(80)
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 50 80 130 0
13 7 5Albert Lea 100 100
6 11 12Spokane 100 20 120 Demand
Vj
150 80 120
San Francisco Ok City Dayton
Warehouses
Arbitrarily select u1 = 0. Find other multiplier values.
65
Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 50 80 130 0
13 7 5Albert Lea 100 100
6 11 12Spokane 100 20 120 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
10 7
Arbitrarily select u1 = 0. Find other multiplier values.
66
Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 50 80 130 0
13 7 5Albert Lea 100 100
6 11 12Spokane 100 20 120 -4 Demand
Vj
150 80 120
10 7
San Francisco Ok City Dayton
Warehouses
Arbitrarily select u1 = 0. Find other multiplier values.
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 50 80 130 0
13 7 5Albert Lea 100 100
6 11 12Spokane 100 20 120 -4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
10 7 16
Arbitrarily select u1 = 0. Find other multiplier values.
68
Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 50 80 130 0
13 7 5Albert Lea 100 100 -11
6 11 12Spokane 100 20 120 -4 Demand
Vj
150 80 120
10 7 16
San Francisco Ok City Dayton
Warehouses
Arbitrarily select u1 = 0. Find other multiplier values.
69
Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 50 80 130 0
13 7 5Albert Lea 100 100 -11
6 11 12Spokane 100 20 120 -4 Demand
Vj
150 80 120
10 7 16
San Francisco Ok City Dayton
Warehouses
8
Find all dij values. Select largest –dij to leave basis.
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 50 80 130 0
13 7 5Albert Lea 100 100 -11
6 11 12Spokane 100 20 120 -4 Demand
Vj
150 80 120
10 7 16
San Francisco Ok City Dayton
Warehouses
Find largest -dij. Find feasible loop.
8
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 50 80 130 0
13 7 5Albert Lea 100 100 -11
6 11 12Spokane 100 20 120 -4 Demand
Vj
150 80 120
10 7 16
San Francisco Ok City Dayton
Warehouses
Find largest -dij. Find feasible loop.
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Class Problem Class Problem Suppliers Supply Ui
10 7 8Rapid City 30 80 20 130 0
13 7 5Albert Lea 100 100
6 11 12Spokane 120 120 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
Total Cost = 10(30) + 7(80) + 8(20) + 5(100) + 6(120)
= $2,240 = 2,400 - 8(20)
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 30 80 20 130 0
13 7 5Albert Lea 100 100
6 11 12Spokane 120 120 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
Arbitrarily select u1 = 0.
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 30 80 20 130 0
13 7 5Albert Lea 100 100
6 11 12Spokane 120 120 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
10 7 8
Arbitrarily select u1 = 0. Find other multipliers.
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 30 80 20 130 0
13 7 5Albert Lea 100 100 -3
6 11 12Spokane 120 120 -4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
10 7 8
Arbitrarily select u1 = 0. Find other multipliers.
76
Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 30 80 20 130 0
13 7 5Albert Lea 100 100 -3
6 11 12Spokane 120 120 -4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
10 7 8
All dij > 0 Solution is optimaloptimal.
6 3
8 8
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Class Problem Class Problem
Suppliers Supply Ui
10 7 8Rapid City 30 80 20 130 0
13 7 5Albert Lea 100 100 -3
6 11 12Spokane 120 120 -4 Demand
Vj
San Francisco Ok City Dayton
Warehouses
150 80 120
10 7 8
Z = 10(30) + 7(80) + 8(20) + 5(100) + 6(120) = 2,240
6 3
8 8
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Initialization Initialization (Vogel’s) (Vogel’s)
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Initialization (Vogel’s) Initialization (Vogel’s) Table 8.17 H<able 8.17 H&L
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Initialization (Vogel’s) Initialization (Vogel’s) Table 8.17 H<able 8.17 H&L
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Initialization (Vogel’s) Initialization (Vogel’s) Table 8.17 H<able 8.17 H&L
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Initialization (Vogel’s) Initialization (Vogel’s) Table 8.17 H<able 8.17 H&L
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Initialization (Vogel’s) Initialization (Vogel’s) Table 8.17 H<able 8.17 H&L
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Initialization (Vogel’s) Initialization (Vogel’s) Table 8.17 H<able 8.17 H&L
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Dummy WarehouseDummy WarehouseSuppliers Supply
10 7 8Rapid City 150
13 7 5Albert Lea 120
6 11 12Spokane 150 Demand 150 80 120
San Francisco Ok City DaytonWarehouses
Suppose total supply exceeds total demand.
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Dummy WarehouseDummy Warehouse
Suppliers Supply10 7 8 0
Rapid City 15013 7 5 0
Albert Lea 1206 11 12 0
Spokane 150 Demand
Ok City Dayton
150 80 120
DummyWarehouses
70
San Francisco
Add dummy warehouse with 0 cost.
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Dummy SupplierDummy SupplierSuppliers Supply
10 7 8Rapid City 130
13 7 5Albert Lea 100
6 11 12Spokane 120 Demand 150 120 150
San Francisco Ok City DaytonWarehouses
Suppose total demand exceeds total supply.
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Dummy SupplierDummy Supplier
Suppliers Supply
10 7 8Rapid City 150
13 7 5Albert Lea 120
6 11 12Spokane 150
0 0 0Dummy 70 Demand
Warehouses
150 120 150
San Francisco Ok City Dayton
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Final slideFinal slide
Transportation Problem Northwest corner Method Transportation Tableau Method Vogler’s approximation (Initialization)