Network Models with Excel Simple Structure Intuition into solver Numerous applications Integral data means integral solutions

PROTRAC Engine Distribution 500 800 700 400 900 200 * Belgium Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50 500 800 500 400 700 200 900

Transportation Costs Minimize 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

Model Components Adjustables or Variables Objective Constraints 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 Belgium Netherlands Germany 500 800 700 400 200 900 200 * Belgium Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50

A Basic Feasible Solution 500 800 700 400 900 200 * Belgium Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50

Finding an Entering Variable 500 800 700 400 900 200 * Belgium Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50 500 800 500 400 700 200 900

Finding an Entering Variable 500 800 700 400 900 200 * Belgium Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50

Computing Reduced Cost 500 800 700 400 900 200 * Belgium Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50 $122

Computing Reduced Cost 500 800 700 400 900 200 * Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50 $122 $100

Computing Reduced Cost 500 800 700 400 900 200 * Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50 $122 $100 $40

Computing Reduced Cost 500 800 700 400 900 200 * Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50 $122 $100 $40 $90 Costs$122 $ 40 $162 Saves$100 $ 90 $190 Net $28

Finding a Leaving Variable 500 800 700 400 900 200 * Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50 Red flows decrease. Green flows increase. Leaving variable is first to reach 0

New Basic Feasible Solution 500 800 700 400 900 200 * Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50

New Basic Feasible Solution 500 800 700 400 900 200 * Germany Netherlands The Hague Amsterdam Antwerp Nancy Liege Tilburg Leipzig Miles 100 50 300 600

Quantity Discounts Minimize Cost Total Cost $3 $4 Shipment Size

Crossdocks and Warehouses

Flow Balance At the DCs At the Plants At the Customers 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 Model

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 plant One unit required at one customer Minimizing the cost of shipping is....