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Network Models with Excel. Simple Structure Intuition into solver Numerous applications Integral data means integral solutions. Netherlands. Amsterdam. 500. *. 800. The Hague. *. Germany. 500. Tilburg. *. 700. *. Antwerp. Leipzig. *. Belgium. 400. *. Liege. 200. Nancy. - PowerPoint PPT Presentation
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Network Models with Excel
Simple StructureIntuition into solverNumerous applicationsIntegral data means integral
solutions
PROTRAC Engine Distribution
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800 700
500
400
900
200
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*
*
*
*
*
*
Belgium
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
500
800
700
500
200
400
900
Transportation Costs
To DestinationFrom Origin Leipzig Nancy Liege TilburgAmsterdam 120 130 41 62Antwerp 61 40 100 110The Hague 102.5 90 122 42
Unit transportation costs from harbors to plants
Minimize the transportation costs involved in
moving the engines from the harbors to the
plants
A Transportation Model
PROTRAC Transportation ModelUnit Cost From/To Leipzig Nancy Liege Tilburg
Amsterdam 120.0$ 130.0$ 41.0$ 62.0$ Antwerp 61.0$ 40.0$ 100.0$ 110.0$ The Hague 102.5$ 90.0$ 122.0$ 42.0$
Shipments From/To Leipzig Nancy Liege Tilburg Total Available
Amsterdam - - - - - 500Antwerp - - - - - 700The Hague - - - - - 800Total - - - - - Required 400 900 200 500
Total Cost From/To Leipzig Nancy Liege Tilburg Total
Amsterdam -$ -$ -$ -$ -$ Antwerp -$ -$ -$ -$ -$ The Hague -$ -$ -$ -$ -$ Total -$ -$ -$ -$ -$
Model ComponentsAdjustables or Variables
By changing cells selection ranges separated by commas
Objective Target Cell Min or Max
Constraints LHS is a cell reference >=, <=, = (others for later) RHS is a cell reference or number.
How the Solver works
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Belgium
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
500
800
700
500
200
400
900
A Basic Feasible Solution
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800 700
500
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900
200
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*
Belgium
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
500
800
700
500
200
400
900
500
400
100
200
800
0
Finding an Entering Variable
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Belgium
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
500
800
700
500
200
400
900
Finding an Entering Variable
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800 700
500
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900
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Belgium
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
500
800
700
500
200
400
900
Computing Reduced Cost
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800 700
500
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900
200
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*
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Belgium
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
$122
Computing Reduced Cost
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800 700
500
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900
200
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*
*
*
*
*
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
$122
$100
Computing Reduced Cost
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800 700
500
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900
200
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*
*
*
*
*
*
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
$122
$100
$40
Computing Reduced Cost
500
800 700
500
400
900
200
*
*
*
*
*
*
*
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
$122
$100
$40
$90
Costs$122 $ 40 $162
Saves$100 $ 90 $190
Net $28
Finding a Leaving Variable
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Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
0
200
100
800
Red flows decrease.
Green flowsincrease.
Leaving variableis first to reach 0
New Basic Feasible Solution
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800 700
500
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200
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*
*
*
*
*
*
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
500
800
700
500
200
400
900
Tilburg
New Basic Feasible Solution
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800 700
500
400
900
200
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*
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*
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*
*
Germany
Netherlands
The Hague
Amsterdam
Antwerp
Nancy
Liege
Tilburg
Leipzig
Miles
100500
500
800
700
500
200
400
900
Tilburg
400
500
200300
600
Quantity Discounts
Minimize Cost
Shipment Size
Tot
al C
ost
$4
$3
Crossdocks and Warehouses
PlantSupply
(Tons/Year) CustomerDemand (Tons/Year)
Plant 1 200 Customer 1 400Plant 2 300 Customer 2 180Plant 3 100
Inter Plant Plant 1 Plant 2 Plant 3Plant to DC DC 1 DC 2
Plant 1 -$ 5.0$ 3.0$ Plant 1 5.0$ 5.0$ Plant 2 9.0$ -$ 9.0$ Plant 2 1.0$ 1.0$ Plant 3 4.0$ 8.0$ -$ Plant 3 1.0$ 0.5$
Plant to Customer Customer 1 Customer 2
DC to DC DC 1 DC 2
Plant 1 20.0$ 20.0$ DC 1 -$ 1.2$ Plant 2 8.0$ 15.0$ DC 2 0.8$ -$ Plant 3 10.0$ 12.0$
Customer to
Customer Customer 1 Customer 2Customer 1 -$ 1.0$ Customer 2 7.0$ -$
The RedBrand Company Example 15.4
A Network Flow Model with Transshipments
Transportation Costs ($ 000/Ton)
Flow Balance
At the DCs Flow into the DC - Flow out of the DC = 0
At the Plants Flow out of Plant - Flow into the Plant Supply
At the Customers Flow into the Cust. - Flow out of the Cust. Demand
A Solver ModelRedBrand Minimum Cost Network Flow Problem
Unit Shipping Costs
Inter Plant Plant 1 Plant 2 Plant 3Plant to
DC DC 1 DC 2Plant 1 -$ 5.0$ 3.0$ Plant 1 5.0$ 5.0$ Plant 2 9.0$ -$ 9.0$ Plant 2 1.0$ 1.0$ Plant 3 0.4$ 8.0$ -$ Plant 3 1.0$ 0.5$
Plant to Customer Customer 1 Customer 2 DC to DC DC 1 DC 2
Plant 1 20.0$ 20.0$ DC 1 -$ 1.2$ Plant 2 8.0$ 15.0$ DC 2 0.8$ -$ Plant 3 10.0$ 12.0$
Customer to
Customer Customer 1 Customer 2DC to
Customer Customer 1 Customer 2Customer 1 -$ 1.0$ DC 1 2.0$ 12.0$ Customer 2 7.0$ -$ DC 2 2.0$ 12.0$
Shipments
Inter Plant Plant 1 Plant 2 Plant 3 Total OutPlant to
DC DC 1 DC 2 Total OutPlant 1 - - - - Plant 1 - - - Plant 2 - - - - Plant 2 - - - Plant 3 - - - - Plant 3 - - -
Total In - - - Total In - -
Plant to Customer Customer 1 Customer 2 Total Out DC to DC DC 1 DC 2 Total Out
Plant 1 - - - DC 1 - - - Plant 2 - - - DC 2 - - - Plant 3 - - - Total In - -
Total In - -
Customer to
Customer Customer 1 Customer 2 Total OutDC to
Customer Customer 1 Customer 2 Total OutCustomer 1 - - - DC 1 - - - Customer 2 - - - DC 2 - - - Total In - - Total In - -
Net FlowsNet Flow
Out Supply Net Flow In Demand Net FlowPlant 1 - 200 Customer 1 - 400 DC 1 - Plant 2 - 300 Customer 2 - 180 DC 2 - Plant 3 - 100
Transportation Costs ($ 000/Ton)
Arc Capacities
Inter Plant Plant 1 Plant 2 Plant 3Plant to
DC DC 1 DC 2Plant 1 200 200 200 Plant 1 200 200Plant 2 200 200 200 Plant 2 200 200Plant 3 200 200 200 Plant 3 200 200
Plant to Customer Customer 1 Customer 2 DC to DC DC 1 DC 2
Plant 1 200 200 DC 1 200 200Plant 2 200 200 DC 2 200 200Plant 3 200 200
Customer to
Customer Customer 1 Customer 2DC to
Customer Customer 1 Customer 2Customer 1 200 200 DC 1 200 200Customer 2 200 200 DC 2 200 200
Incurred Costs
Inter Plant Plant 1 Plant 2 Plant 3 Total OutPlant to
DC DC 1 DC 2 Total OutPlant 1 -$ -$ -$ -$ Plant 1 -$ -$ -$ Plant 2 -$ -$ -$ -$ Plant 2 -$ -$ -$ Plant 3 -$ -$ -$ -$ Plant 3 -$ -$ -$
Total In -$ -$ -$ -$ Total In -$ -$ -$
Plant to Customer Customer 1 Customer 2 Total Out DC to DC DC 1 DC 2 Total Out
Plant 1 -$ -$ -$ DC 1 -$ -$ -$ Plant 2 -$ -$ -$ DC 2 -$ -$ -$ Plant 3 -$ -$ -$ Total In -$ -$ -$
Total In -$ -$ -$
Customer to
Customer Customer 1 Customer 2 Total Out DC to
Customer Customer 1 Customer 2 Total OutCustomer 1 -$ -$ -$ DC 1 -$ -$ -$ Customer 2 -$ -$ -$ DC 2 -$ -$ -$ Total In -$ -$ -$ Total In -$ -$ -$
Total Shipping Cost -$
Transportation Capacities (Tons)
Network Flow Models
Variables are flows of a single homogenous commodity
Constraints are Net flow Supply/Demand Lower Bound Flow on arc Upper Bound
Theorem: If the data are integral, any solution solver finds will be integral as well.
An Important Special Case
One unit available at one plantOne unit required at one customerMinimizing the cost of shipping is....