Are521 Transportation11.Ppt

8/14/2019 Are521 Transportation11.Ppt

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The Transportation Problem

For ARE 521 QUANTITATIVETECHNIQUES

Offered by Prof. G. DSouzaPrepared by Tanya Borisova


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Transportation Problem This problem involves the shipment of a homogeneous

product from a number of supply locations to a number ofdemand locations.

Problem: given needs at the demand locations, how shouldwe take limited supply at supply locations and move thegoods. Further suppose we wish to minimize cost.

Based on McCarl 2005


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Basic Concept Objective:Minimize cost

Variables:Quantity of goods shipped from eachsupply point to each demand point

Restrictions:

Non negative shipmentsSupply availability at a supply pointDemand need at a demand pointBased on McCarl 2005


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Formulating the Problem:Basic notation and the decision variable We denote the supply locations as i

We denote the demand locations asj Let us define our fundamental decision

variable as the set of individual shipmentquantities from each supply location to

each demand location and denote thisvariable as Movesupplyi,demandj



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Formulating the Problem:The objective functionWe want to minimize total shipping cost so we need an expression for

shipping costLet us define a data item per unit cost of shipments from each supplylocation to each demand location as costsupplyi,demandjOur objective then becomes to minimize the sum of the shipment costsover all supplyi, demandj pairsMinimize

supplyi demandj

costsupplyi,demandj Movesupplyi,demandj

hich is the per unit cost of moving from each supply location to eachdemand location times the amount shipped summed over all possibleshipment routesBased on McCarl 2005


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Formulating the Problem: Constraintsthe sum of outgoing shipments from the supplyi supply point to all possible

destinations (demandj) to not exceed supplysupplyi (resource availabilityconstraints)

Movesupplyi,demandj supplysupplyidemandj

shipments into the demandj demand point should be greater than or equal

to demand at that point. Incoming shipments include shipments from allpossible supply points supplyi to the demandj demand point (minimumrequirement constraints).supplyi

Movesupplyi,demandj demanddemandj

nonnegative shipments Movesupplyi,demandj 0Based on McCarl 2005


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Standard FormMin c ij xiji =1j =1mn

= 1, .., ni = 1, .., m

xj =1mnij

ai , bj,

xi =1ij

xij 0

here

i supply locations,j demand locations,cij unit shipping cost, xij shipment volumes,bj demand levels,ai supply limits


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Exampletrucking company has contracted to supply weekly threesupermarkets located in Modesto, Sonoma, and Fresno,California, with potatoes kept at two warehouses in

Sacramento and Oakland, also in California. The weeklyavailability (supply) of potatoes at the warehouses is 400and 600 tons, respectively. The supermarkets weeklyneeds (demands) are 200, 500, and 300 tones of potatoes,respectively. (the transportation unit costs are given below.)

The objective is to find supply routs that minimizes the totaltransportation cost of the trucking firm. The contractstipulates that all supermarkets must receive at least thequantity of potatoes specified above.

Based on Paris 1991


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Example (cont.) Two warehouses: Sacramento and Oakland Three supermarkets: Modesto, Sonoma, and

Fresno Quantity availability: Supply availableSacramento 400 tones

Oakland600 tones Demand requiredModesto200Sonoma500Fresno300Based on Paris 1991


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Example (cont.):Transportation Unit Cost (dollars per ton)

arehouse

ModestoSacramentoOaklandBased on Paris 1991

Supermarkets

Sonoma404050

30Fresno8090


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Network Representationc11 (cost11)1) Sacramento

400c12 (cost12)

c13 (cost13)x12 (move12)

2. Sonoma500

x22 (move22)x11 (move11) 1. Modesto

200

x21 (move21)c21 (cost21)2) Oakland

600c22 (cost22)c23 (cost23)x13 (move13)

x23 (move23)3. Fresno

300Based on Paris 1991


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Example:Problem Formulation Objective function:

minimize total cost of shipmentMinTC =

Min cost11move11 + cost12move12+ cost13move13 + cost21move21 +cost22move22 + cost23move23 Demand constraint move11 + move21 demand1 move12 + move22 demand2 move13 + move23 demand3

cost moveiji

ij

moveiij

demand


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demand j ,for allj

E l


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Example:Problem Formulation (cont.) Supply constraints move11 + move12 + move13 supply1 move21 + move22 + move23 supply2 moveij

supplyi ,for all i

moveij supplyi ,for all i

E l T bl


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Example:Tableaumove11(x11)

move12(x12)

move13(x13)

move21(x21)

move22(x22)

move23(x23)

4015080

4013090

min


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min200500300

-400-60011

-1-1-1-111

-1

1


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-1

E l Optimal Shipping Pattern


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Example: Optimal Shipping PatternDestinationModestoSonomaFresnoUnits

ariableUnits

ariableUnits

ariable

Sacramento100move11(x11)

300move13

(x13)Oakland100move21(x21)

500move22(x22)

E l Optimal Solution


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Example: Optimal Solution Objective value $47,000VariableMove11(x11)

Value1000300100

5000

Reducedcost

020

00

EquationSupply1Supply2Demand1

Demand2

Slack


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Slack0000

0Shadowprice

Eps040

3080Move12(x12)

Move13(x13)

Move21(x21)Move22(x22)

010Demand3

Move23(x23)

Example: Solution


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Example: Solution shadow price represents marginal values

of the resources, e.g.

marginal value of additional units at supplylocations are (close to) zero

marginal value of unit decrease in demand atdemand locations = unit transportation cost to

that location reduced cost represents marginal costs offorcing nonbasic variable into the solution


Example: Reduced Cost


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Example: Reduced Cost Assume the trucking company is required to ship 1tone of potatoes from Sacramento to Sonoma

Shipment from Sacramento to Sonoma cost $50 The trucking company also decreases supply from Oakland to Sonoma (saving of $30) To stay within supply limits, the company decreases

supply from Sacramento to Modesto (saving of $40) To meet demand constraints, the company increases

supply from Oakland to Modesto (cost $40) Resulting change in transportation cost = reduced cost = $50 - $30 + $40 - $40 = $20


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Transportation Model:


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Transportation Model:Important Extensions Transshipment

Transshipment through intermediatewarehouses is permitted

Spatial equilibrium Quantities supplied and demanded depend on

prices

GAMS IDE


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GAMS-IDE GAMSIDE GAMS Generalized Algebraic Modeling System IDE Integrated Development Environment, A Windows graphical interface to run GAMS

Using GAMSIDE Open the IDE through the icon Create a project Create or open an existing file of GAMS instructions

GAMSIDE Files


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GAMSIDE FilesprojectBased on McCarl and Gillig

What is a


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What is aProject? The project location

determines where allsaved files are placed(to place files elsewhereuse the save as dialogue) The project saves file names and program options

associated with the effort. It is recommended to define a new project every time youwish to change the file storage directory.

GAMSIDE


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GAMSIDEProject

Define a project name

and location. Put it in adirectory you want to use. All files associatedith this project will be saved in that directory.

In the File name area type in a name for the project fileyou wish to use. Save the file.

File name extension gprstands for GAMS project.

Slide is taken from McCarl and Gillig


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Opening or Creating


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a File with GAMSInstructions

You can

Create a new file Open an existing file Open a model library file

File name extension .gms GAMSIDE recognizes codes (and adds color

differentiation) only in the files saved as .gms files.

GAMS Model Library


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GAMS Model Library collected models to be

used as of examples Standard textbook examples

to illustrate problem formulation / GAMS features models that have been used in policy or sector analysis and

are interesting for both the methods and the data they use

Output and Log files


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Output and Log files Log file Located in the same directory as the project file File extension - .log

Contain whatever GAMS-IDE shows during the run (content of Process window) Clicking on any of the black-colored line will activate the .lst output file Clicking Reading Solution for model open the .lst file and position

the window at the SOLVE SUMMARY

Clicking on the red-colored line will cause the cursor to jump in

the .gms file to the line with error Output (listing) file Located in the same directory as the project file File name extension - .lst Stores results of the model run

Running GAMS programs


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g p g click on the run button, or use File menu, or

press the F9 key Note Option Compile performs

initial check of the program

Option Run check the program, and (if there isno error) run optimization and generate output

GAMS Help


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GAMS Help GAMS Users Guide McCarl Guide

Solver Manual (discussion of various GAMSSolvers) GAMS documents additional materials onGAMS, IDE, Solvers

Useful GAMS Websites


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Download GAMS student versionhttp://www.gams.com/download/

GAMS development corporationhttp://www.gams.com/ Bruce McCarls web-sitehttp://agecon2.tamu.edu/people/faculty/mccarl-

bruce/

References


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McCarl, B. Course Materials from GAMS 2 Class.http://www.gams.com/mccarl/useide.pdfMcCarl B. 2005. Basic LP problem formulationshttp://agecon2.tamu.edu/people/faculty/mccarl-bruce/622class/overhead05basiclpformulate.pdf

McCarls and Spreens book athttp://agecon2.tamu.edu/people/faculty/mccarl-bruce/books.htmMcCarl B & D Gillig. Introduction to GAMS IDE.http://agecon2.tamu.edu/people/faculty/mccarl-bruce/641clas/02_641_intro_gamsIDE_.pdfParis 1991. An Economic Interpretation of Linear Programming. Iowa

State University Press / Ames

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