network models ii

Network Models II

Shortest Path Cross Docking

Enhance Modeling Skills Modeling with AMPL

15057 Spring 03 Vande Vate

The Shortest Path Model

Find the shortest path from Home to 5


Direction

Two-way street

One-way street


03ShortestPathModelxls


Challenge

Build a Solver model


A Solver Model

The Objective Minimize $U$21

The Variables $C$13$J$20

The Constraints

Only Travel on existing edges

Number From ndash Number To = Net Required


1048673 Number From -Number To = Net Required

Number of times - Number of =

we leave times we enter +1 at Home (we leave once)

-1 at Site 5 (we arrive once)

0 everywhere else

each time we arrive (if ever) we leave


Flow Conservation

Assignment Model Sum across each row = 1 Sum down each column = 1 Each variable appears in 2 constraints

Shortest Path Model Sum across a row -Sum down the column = 0 Trips out of a site -Trips into the site Each variable appears in constraints

Compare with Assignment Model


Each variable appears in at most two constraints At most one constraint as -the variable At most one constraint at + the variable

Assignment Sum across each row = 1 Sum down each column = 1

Shortest Path Sum across the a row -sum down the col =

Network Flow Problems


Bounds

Variables can also have bounds

eg in the Shortest Path Model

Number of times we use each variable Lower bound gt=0

Upper bound lt=1 if it is an edge 0 otherwise


If the bounds and RHS are integral the solution will be integral It the costs are integral the reduced costs and marginal values will be integral

Can be solved very quickly Limited demands on memory

Properties of Network Flows


Crossdocking

3 plants 1048673 2 distribution centers 2 customers Minimize shipping costs


A Network ModelMinimum Cost Network Flow Problem

Challenge



Objective Minimize $K$28 Variables $C$17$D$19 $C$23$D$24 Constraints

Do not exceed supply at the plants

Meet customer demand

Do not exceed shipping capacity


A Solver Model

Flow conservation at the DCs $C$28$D$28 = 0

Supply and Demand like Autopower 1048673 Flow conservation at DCs like Shortest Path

And


Lots of applications1048673 Simple Models1048673 Optimal Solutions Quickly1048673 Integral Data Integral Answers

Network Flows Good News


Underlying Assumptions Single Homogenous Product Linear Costs No conversions or losses


Network Flows Bad News

Homogenous Product

Must be able to interchange positions of product anywhere

No Fixed Charges 1048673 No Volume Discounts 1048673 No Economies of Scale

Linear Costs


Network Flows Simple Formulation Flow Out (sum across a row) lt= Capacity Flow In (sum down a column) gt= Demand Flow In -Flow Out = Constant Limited by Homogenous Product Linear Costs etc

Integer Data give Integral Solutions

Summary



Problems with Excel Solver Integration of ldquoModelrdquo and Data Example Change the time horizon of our Inventory Model Excel is a limited database tool Algebraic Modeling Languages Separate the ldquoModelrdquo from the Data Keep the data in databases

Modeling with AMPL


How they work


Established in US 1048673 Very good book 1048673 Lower barrier to entry 1048673 Free ldquostudentrdquo version 1048673 Industrial strength tool

Why AMPL


Pseudo AMPL to discuss models In class In exams

Need to be precise about Whatrsquos a parameter variable hellip

Indexing relationships between variables data constraints Challenges and Project

Our Use of AMPL


AMPL is very detailed Expect 1 or 2 per team to master Rest to read and understand Brings out the real issues Practical implementation --you can oversee Data issues --the real challenge

Valuable tool

Is this necessaryvaluable


set ORIG1048673 set DEST1048673 param supply ORIG1048673 param demand DEST1048673 param cost ORIG DEST1048673 var Trans ORIG DEST gt= 0

The Transportation Model


minimize Total_Cost sumo in ORIG d in DEST cost[od]Trans[od] st Supply o in ORIG sumd in DEST Trans[od] lt= supply[o] st Demand d in DEST sumo in ORIG Trans[od] gt= demand[d]

Transportation Model


An Access Database called TransportationDatamdb Tables in the database Origins Supply information

Destinations Demand information

The Data


Cost Unit transportation costs

The Costs

AMPL reads the model and the data combines the two and produces (in human readable form) hellip


Produced by the command expand gtAMPLOutputtxt

AMPLrsquos Output


table OriginTable IN ODBC DPersonal15057TransportationDatamdbOriginsORIG lt-[Origin] supply~SupplyExplanation lsquotablersquo is a keyword that says we will read or write data lsquoOriginTablersquo is a name we made up No other AMPL model entity can have this name lsquoINrsquo is a key word that says we are reading data

ldquoODBCrdquo says we are using ODBC to read the data

Reading Data


DPersonal15057TransportationDatamdbrdquo is the path to the database Alternatively you can create a DSN (data source name) for this file say TransportData and use the command ldquoDSN=TransportDatardquo

Originsrdquo is the name of the table in the database Alternatively we can use an SQL command like ldquoSQL=SELECT FROM Originsrdquo

The is syntax What follows is the mapping of the data we read to AMPL objects that will hold it

The brackets [] around Origin mean that this field in the database indexes the data eg 500 is the supply for Amsterdam

Explanation


ORIG lt-[Origin] says that the values of the field Origin will define the set ORIG of origins supply~Supply says that the values of the parameter supply should hold the values read from the field Supply in the database

read table OriginTable reads the data

Explanation Continued


table DestinationTable IN ODBC DPersonal15057TransportationDatamdbldquoDestinationsDEST lt-[Destination] demand~DemandExplanation lsquotablersquo is a keyword that says we will read or write data DestinationTablersquo is a name we made up No other AMPL model entity can have this name lsquoINrsquo is a key word that says we are reading data


Reading Data



rdquoDestinationsrdquo is the name of the table in the database Alternatively we can use an SQL command like ldquoSQL=SELECT FROM Destinationsrdquo


The brackets [] around Destination mean that this field in the database indexes the data eg 400 is the demand for Leipzig

Explanation


DEST lt-[Destination] says that the values of the field Destination will define the set DEST of destinations demand~Demand says that the values of the parameter demand should hold the values read from the field Demand in the database



table CostTable IN ODBCldquoDPersonal15057TransportationDatamdbldquoCost[origin destination] costExplanation lsquotablersquo is a keyword that says we will read or write data

lsquoCostTablersquo is a name we made up No other AMPL model entity can have this name

lsquoINrsquo is a key word that says we are reading data


Reading Cost


DPersonal15057TransportationDatamdbrdquo is the path to the database Alternatively you can create a DSN (data source name) for this file say TransportData and use the command ldquoDSN=TransportDatardquo rdquoCostrdquo is the name of the table in the database Alternatively we can use an SQL command like ldquoSQL=SELECT FROM Costrdquo


The brackets [] around origin and destination mean that these two fields in the database index the data eg 120 is the unit transportation cost from Amsterdam to Leipzig

Explanation


We donrsquot have an lt-here because we are not defining the members of a set

We read the values of the field cost in the database into the parameter cost Note that since these two names are identical we donrsquot need the ~

read table CostTable reads the data


model d15057TransportationModelmod1048673 option solver cplex use cplex to solve1048673 solve1048673 display Trans


Running AMPL


table TransOutTable OUT ODBC DPersonal15057TransportationDatamdbldquoTransOutorigin in ORIG destination in DEST

Trans[origin destination] gt 0 -gt [origin destination] Trans[origindestination]~Trans write table TransOutTable

Explanation lsquotablersquo is a keyword that says we will read or write data

lsquoTransOutTablersquo is a name we made up No other AMPL model entity can have this name

Writing Output


lsquoOUTrsquo is a key word that says we are writing data ldquoODBCrdquo says we are using ODBC to write the data DPersonal15057TransportationDatamdbrdquo is the path to the database Or you can use ldquoDSN=helliprdquo TransOutrdquo is the name of the table to create AMPL drops and writes this table Any data currently in the table is lost is syntax It separates the description of the destination from the definition of the data and the mapping of the columns

Explanation


origin in ORIG destination in DEST Trans[origin destination] gt 0 defines the index set that will control the data to write out This says to only report on origin-destination pairs where we actually send a positive flow -gt is syntax It separates the indexing from the data definition and mapping to fields of the output table

[origin destination] indicates that the records of the output table are indexed by the origin-destination pairs AMPL will write a new record for each pair

Trans[origindestination]~Trans says to create a field called Trans in the table and to populate it with the values of the Trans variable

More Explanation


write table TransOutTable actually writes the data 1048673 The output is

More details available at 1048673

httpwwwamplcomcmcswhatamplNEWtableshtml

Explanation Completed

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The Shortest Path Model

Find the shortest path from Home to 5


Direction

Two-way street

One-way street




Challenge



A Solver Model



The Constraints








0 everywhere else



Flow Conservation










Bounds










Crossdocking




Challenge








A Solver Model



And








Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Slide 44

Slide 45

Direction

Two-way street

One-way street




Challenge



A Solver Model



The Constraints








0 everywhere else



Flow Conservation










Bounds










Crossdocking




Challenge








A Solver Model



And








Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Challenge



A Solver Model



The Constraints








0 everywhere else



Flow Conservation










Bounds










Crossdocking




Challenge








A Solver Model



And








Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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A Solver Model



The Constraints








0 everywhere else



Flow Conservation










Bounds










Crossdocking




Challenge








A Solver Model



And








Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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0 everywhere else



Flow Conservation










Bounds










Crossdocking




Challenge








A Solver Model



And








Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Bounds










Crossdocking




Challenge








A Solver Model



And








Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Crossdocking




Challenge








A Solver Model



And








Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Challenge








A Solver Model



And








Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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A Solver Model



And








Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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And








Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Homogenous Product



Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Linear Costs




Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Summary




Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Slide 45



Modeling with AMPL


How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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How they work



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Slide 43

Slide 44

Slide 45



Why AMPL





Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Our Use of AMPL



Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Valuable tool











The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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The Data



The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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The Costs




AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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AMPLrsquos Output




Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Reading Data






Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Explanation








Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Reading Data






Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Explanation









Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Reading Cost





Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Explanation








Running AMPL






Writing Output



Explanation





More Explanation






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Running AMPL






Writing Output



Explanation





More Explanation






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Writing Output



Explanation





More Explanation






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Explanation





More Explanation






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More Explanation






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network models ii

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