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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

Instituto Superior Técnico

REGIÕES E REDES(REGIONS AND NETWORKS)

Lecture 2: Transport networks design and evaluation

Xpress presentation - Laboratory work

Eng. Luis Martínez

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

OUTLINE Xpress presentation

Xpress structure Create input files for Xpress Xpress-IVE overview Xpress-IVE key features Mosel programming language key words Declare and initialize variables in Mosel Declare an objective function and constraints in Mosel Declare and run the optimization in Mosel Output the results (library “mmive”)

Xpress Laboratory work with two network design examples A minimum spanning tree problem example A Min-cost-flow problem example

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS PRESENTATION - XPRESS STRUCTURE

Xpress-MP is a suite of mathematical modeling and optimization tools used to solve linear, integer, quadratic, non-linear, and stochastic programming problems.

Solver engines: Xpress-Optimizer (LP, MIP, QP, MIQP, QCQP, NLP) Xpress-SLP (NLP, MINLP) Xpress-SP is a Stochastic Programming tool for solving optimization problems

involving uncertainty Xpress-Kalis is Constraint Programming software (discrete combinatorial problems)

Model building and development tools: Xpress-Mosel programming language Xpress-BCL is an object-oriented library Xpress-IVE is a complete visual development environment for Xpress-Mosel under

Windows Xpress-Application Developer (XAD) extends Xpress-Mosel with an API for

graphical user interface development

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS PRESENTATION - CREATE INPUT FILES FOR XPRESS

The input files are created in .dat files format These files should contain the vectors and matrices with the input

variables identified for reading from the Xpress engine.

Arcs: [(1 1) "Lisboa" "Odivelas"(2 1) "Lisboa" "Loures"(3 1) "Lisboa" "Amadora"(4 1) "Lisboa" "Oeiras"(5 1) "Lisboa" "Sintra"(6 1) "Lisboa" "Cascais"(7 1) "Lisboa" "Mafra"(8 1) "Lisboa" "Vila Franca de Xira“]

x: [("Lisboa")111461 ("Odivelas") 108843 ("Loures") 110098 ("Amadora") 105211]

flow: [0 15900 26500 26500 2385015900 0 9540 13780 1590026500 9540 0 14840 1908026500 13780 14840 0 4240023850 15900 19080 42400 0]

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS PRESENTATION - XPRESS-IVE OVERVIEW

Optimization processOptimization resultsMosel editor

Variables and constraints activity and output

Run and debug control

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS PRESENTATION - XPRESS-IVE KEY FEATURES

The code file should have the following structure: model – model name uses – libraries to be used ("mmxprs","mmive“) parameters – define parameters of the model – filename of the input data file

(DATAFILE= ……dat) end-parameters declarations – declare variables and ranges of variables of the input file end-declarations initializations from DATAFILE – initialize variables from file (insert variables

names) end-initializations declarations – declare decision variables end-declarations end-model

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS PRESENTATION - MOSEL PROGRAMMING LANGUAGE KEY WORDS

Key words: Variables types

string – text integer – integer number real – real number mpvar – decision variable array(range) of type of variable

forall(range) – iterator forall(range) do end-do – cycle with actions if end-if – conditional action sum – somation := – assign value; =, >=,<= – equality and inequality operators

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS PRESENTATION - DECLARE AND INITIALIZE VARIABLES IN MOSEL

Variables declaration and initialization: declarations

NODES: set of stringx: array(NODES) of realy: array(NODES) of realA: array(ARCS:set of integer,1..2) of stringDIST: array (ARCS) of realdemand:array(NODES,NODES) of integer

end-declarations initializations from DATAFILE

A x y demandend-initializations

Declare range

Declare input variables (range size defined by the input data file)

Variables read in the file

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS PRESENTATION - DECLARE AN OBJECTIVE FUNCTION AND CONSTRAINTS

Objective function declaration Cost:=sum(i in ARCS)(a*flow(i) + b*Exist(i))*DIST(i)

Constraints declaration forall(a in NODES)

Total(a):= sum(i in ARCS|A(i,1)=a) flow(i)>= sum(b in NODES) demand(a,b)

FlowT:=sum(c in ARCS) flow(c) = sum(i,j in NODES) demand(i,j) forall(c in ARCS) flow(c) is_integer forall(c in ARCS) Exist(c) is_binary

Decision variablesConstraint variable (linctr)

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS PRESENTATION - DECLARE AND RUN THE OPTIMIZATION IN MOSEL

After the declaration of the objective function and all the constraints we need to define the optimization method that will apply (minimization, maximization) minimize (Cost) maximize (utility)

After the definition of the optimization method (if we don’t define any output results – text or graphical), we need to close the model with the key word end-model

Having all the complete code needed we can now run the model in the Run button.

Optimize the identified constraint variable

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS PRESENTATION - OUTPUT THE RESULTS

Write output results in text: getsol(mpvar) – get the solution values of the variable getobjval – get final value of the objective function writeln(string) – write in a string line write(string) – write a string strfmt(number) – write as string

Draw graphs: CnctGraph:= IVEaddplot("Road", IVE_YELLOW) TermGraph:= IVEaddplot("Cities", IVE_GREEN) IVEdrawpoint(TermGraph, x(i), y(i)) IVEdrawline(CnctGraph, x(A(i)), y(A(i)), x(A(j)), y(A(j))) IVEdrawlabel(CnctGraph, (x(A(i))+x(A(j)))/2, (y(A(i))+y(A(j)))/2,

string(getsol(flow(a,j))))

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS LABORATORY - A MINIMUM SPANNING TREE PROBLEM EXAMPLE (I)

Goal: Design a network that connects all the nodes of the graph at minimum cost Decision Variables:X: array (NODES, NODES) – binaryLevel: array (NODES) - integer

Objective function: Constraints:Number of connections:

Avoid Subcycle:

Direct all connections towards the root:

∑=

⋅Nodes

jiijLength

1,α

1/

1,−=∑

<>

=

NxjiNodes

jiij

ijijji xNNlevellevel ×+−+≥∀ 1,

11/

1=∀ ∑

<>

=

jN

iijj x

Road length: 2333.06Connections: A Coruña-Pontevedra Lugo-A Coruña Asturias-Lugo Cantabria-León Vizcaya-Cantabria Guipúzcoa-Vizcaya León-Asturias Burgos-Vizcaya Navarra-Guipúzcoa Madrid-Salamanca Barcelona-Zaragoza Valencia-Zaragoza Sevilla-Badajoz Salamanca-León Zaragoza-Navarra Badajoz-Salamanca

Spain Network Example

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Instituto Superior Técnico / MPOT and MUOT – Regions and Networks 2011/2012

XPRESS LABORATORY - A MIN-COST-FLOW MULTI-TERMINAL PROBLEM EXAMPLE (I)

Goal: Assign demand flow the an existing network from a several sources to a several sink nodes at minimum cost considering capacity constraints on the links of the graph

Decision Variables:Flow: array (ARCS,ODpairs) – integer

Objective function:

Constraints:Total source flow:

Node equilibrium:

Capacity constraint:

Positive flow:

ioMaximumRatCapacityFlow a

ODPairs

iaiarcs ⋅≤∀ ∑

=1

0, ≥∀ ajjODPairsarcs Flow

∑=

=

≥∀ji OOArcs

ijijODPairsj DemandFlow

/

1,

∑∑=

=

=

=

=∀nOArcs

kjk

nDArcs

iijjODPairsnodes

ki

FlowFlow/

1,

/

1,

i

ODPairsArcs

jiij LengthFlow ⋅⋅∑

==

,

1,1α

Spain Network Example