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

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ENGM 792ENGM 792

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Prototype ExamplePrototype Example

K-Log Lumber Mill Warehouse

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PrototypePrototype

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1112

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PrototypePrototype

RC

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SP

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1112

150

80

120

130

100

120

Supply

Demand

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PrototypePrototype

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150

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120

130

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120

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




Spokane 120 Demand 350150 80 120


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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)




Spokane 120 Demand


150 80 120

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Spokane 120 Demand


150 80 120


Northwest Corner

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Northwest Corner


Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 0 120 120 Demand 150 80 120


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


Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 0 120 120 Demand 150 80 120


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Clever IdeaClever Idea

Suppose we can find a loop to move units around.


Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 0 120 120 Demand 150 80 120


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Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 0 120 120 Demand 150 80 120


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Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 0 120 120 Demand 150 80 120


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Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 0 120 120 Demand 150 80 120


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Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 0 120 120 Demand 150 80 120


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For each unit I can move around the loop, I can save -5 + 12 - 11 + 7 = 3 per unit of flow


Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 0 120 120 Demand 150 80 120


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I can move at most 80 units around this loop


Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 0 120 120 Demand 150 80 120


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I can move at most 80 units around this loop


Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 80 40 120 Demand


150 80 120

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Total Cost = 10(130) + 13(20) + 11(80) + 5(80) + 12(40) = $3,320 = $3,560 - 3(80)


Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 80 40 120 Demand


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


Rapid City 130 13013 7 5

Albert Lea 20 80 1006 11 12

Spokane 0 120 120 Demand 150 80 120


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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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Suppliers Supply Ui

10 7 8Rapid City 130 130

13 7 5Albert Lea 20 80 100 0


Vj


Warehouses

150 80 120

13 7

13 = 0 + v1 v1 = 13

7 = 0 + v2 v2 = 7

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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


Warehouses

10 = u1 + 13 u1 = -3

11 = u3 + 7 u3 = 4

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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


Warehouses

150 80 120

13 7 8

12 = 4 + v3 v3 = 8

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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


Warehouses

150 80 120

13 7 8

d12 = 7 -(-3) - 7 = +3

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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


Warehouses

150 80 120

13 7 8

d13 = 8 -(-3) - 8 = +3

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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


Warehouses

150 80 120

13 7 8

d23 = 5 -0 - 8 = -3

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3

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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


Warehouses

150 80 120

13 7 8

d31 = 6 -4 - 13 = -11

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3

11

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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


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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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


Warehouses

150 80 120

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Let non-basic cell with largest -dij enter basis.

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3

11

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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


Warehouses

150 80 120

13 7 8

Find a feasible loop.

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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


Warehouses

150 80 120

13 7 8

Move the maximim unit flow around the loop.

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Suppliers Supply Ui

10 7 8Rapid City 130 130

13 7 5Albert Lea 20 80 100


Vj

150 80 120


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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Suppliers Supply Ui

10 7 8Rapid City 130 130

13 7 5Albert Lea 20 80 100


Vj

150 80 120

0


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


Vj

150 80 120

0


Warehouses

Find u1, u2, u3, v2, v3

dij for non-basic cells

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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


Warehouses

Find u1, u2, u3, v2, v3 and dij for non-basic cells

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14

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Suppliers Supply Ui

10 7 8Rapid City 130 130

13 7 5Albert Lea 20 80 100


Vj

150 80 120


Warehouses

Find most -dij. Find feasible loop for transfer.

14

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Suppliers Supply Ui

10 7 8Rapid City 130 130

13 7 5Albert Lea 20 80 100


Vj

150 80 120


Warehouses

Find most -dij. Find feasible loop for transfer.

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Suppliers Supply Ui

10 7 8Rapid City 130 130

13 7 5Albert Lea 80 20 100


Vj


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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Suppliers Supply Ui

10 7 8Rapid City 130 130

13 7 5Albert Lea 80 20 100 0


Vj

150 80 120


Warehouses

Arbitrarily select u2 = 0. Find other multiplier values.

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Suppliers Supply Ui

10 7 8Rapid City 130 130

13 7 5Albert Lea 80 20 100 0


Vj


Warehouses

150 80 120

7 5


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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


Warehouses


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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


Warehouses


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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


Warehouses

150 80 120

-1 7 5

11 8

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Find all dij values. Select largest –dij to leave basis.

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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


Warehouses

150 80 120

-1 7 5

Find largest -dij. Find feasible loop for transfer.

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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


Vj

150 80 120


Warehouses

Total Cost = 10(50) + 7(80) + 5(100) + 6(100) + 12(20)

= $2,400 = 3,280 - 11(80)

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Suppliers Supply Ui

10 7 8Rapid City 50 80 130 0

13 7 5Albert Lea 100 100


Vj

150 80 120


Warehouses


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Suppliers Supply Ui

10 7 8Rapid City 50 80 130 0

13 7 5Albert Lea 100 100


Vj


Warehouses

150 80 120

10 7


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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


Warehouses


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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


Warehouses

150 80 120

10 7 16


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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


Warehouses


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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


Warehouses

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Find all dij values. Select largest –dij to leave basis.

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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


Warehouses

Find largest -dij. Find feasible loop.

8

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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


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


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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Suppliers Supply Ui

10 7 8Rapid City 30 80 20 130 0

13 7 5Albert Lea 100 100


Vj


Warehouses

150 80 120

Arbitrarily select u1 = 0.

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Suppliers Supply Ui

10 7 8Rapid City 30 80 20 130 0

13 7 5Albert Lea 100 100


Vj


Warehouses

150 80 120

10 7 8

Arbitrarily select u1 = 0. Find other multipliers.

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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


Warehouses

150 80 120

10 7 8

Arbitrarily select u1 = 0. Find other multipliers.

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Suppliers Supply Ui

10 7 8Rapid City 30 80 20 130 0

13 7 5Albert Lea 100 100 -3


Vj


Warehouses

150 80 120

10 7 8

All dij > 0 Solution is optimaloptimal.

6 3

8 8

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Suppliers Supply Ui

10 7 8Rapid City 30 80 20 130 0

13 7 5Albert Lea 100 100 -3


Vj


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&LTable 8.17 H&L

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82


83


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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


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

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San Francisco

Add dummy warehouse with 0 cost.

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Dummy SupplierDummy SupplierSuppliers Supply



6 11 12Spokane 120 Demand 150 120 150


Suppose total demand exceeds total supply.

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Dummy SupplierDummy Supplier

Suppliers Supply



6 11 12Spokane 150

0 0 0Dummy 70 Demand

Warehouses

150 120 150


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Final slideFinal slide

Transportation Problem Northwest corner Method Transportation Tableau Method Vogler’s approximation (Initialization)

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

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