Instituto Superior Técnico Masters in Civil Engineering. Lecture 2: Transport networks design and evaluation Xpress presentation  Laboratory work


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1 Instituto Superior Técnico Masters in Civil Engineering REGIÕES E REDES () Lecture 2: Transport networks design and evaluation Xpress presentation  Laboratory work Eng. Luis Martínez
2 OUTLINE Xpress presentation Xpress structure Create input files for Xpress XpressIVE overview XpressIVE 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 Mincostflow problem example 2
3 XPRESS PRESENTATION  XPRESS STRUCTURE XpressMP is a suite of mathematical modeling and optimization tools used to solve linear, integer, quadratic, nonlinear, and stochastic programming problems. Solver engines: XpressOptimizer (LP, MIP, QP, MIQP, QCQP, NLP) XpressSLP (NLP, MINLP) XpressSP is a Stochastic Programming tool for solving optimization problems involving uncertainty XpressKalis is Constraint Programming software (discrete combinatorial problems) Model building and development tools: XpressMosel programming language XpressBCL is an objectoriented library XpressIVE is a complete visual development environment for XpressMosel under Windows XpressApplication Developer (XAD) extends XpressMosel with an API for graphical user interface development 3
4 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: [( ) "Lisboa" "Odivelas" (2 ) "Lisboa" "Loures" (3 ) "Lisboa" "Amadora" (4 ) "Lisboa" "Oeiras" (5 ) "Lisboa" "Sintra" (6 ) "Lisboa" "Cascais" (7 ) "Lisboa" "Mafra" (8 ) "Lisboa" "Vila Franca de Xira ] x: [("Lisboa")46 ("Odivelas") ("Loures") 0098 ("Amadora") 052] flow: [ ] 4
5 XPRESS PRESENTATION  XPRESSIVE OVERVIEW Run and debug control Variables and constraints activity and output Mosel editor Optimization results Optimization process 5
6 XPRESS PRESENTATION  XPRESSIVE 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) endparameters declarations declare variables and ranges of variables of the input file enddeclarations initializations from DATAFILE initialize variables from file (insert variables names) endinitializations declarations declare decision variables enddeclarations endmodel 6
7 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 enddo cycle with actions if endif conditional action sum somation := assign value; =, >=,<= equality and inequality operators 7
8 XPRESS PRESENTATION  DECLARE AND INITIALIZE VARIABLES IN MOSEL Variables declaration and initialization: declarations NODES: set of string x: array(nodes) of real y: array(nodes) of real A: array(arcs:set of integer,..2) of string DIST: array (ARCS) of real demand:array(nodes,nodes) of integer enddeclarations initializations from DATAFILE A x y demand endinitializations Declare range Declare input variables (range size defined by the input data file) Variables read in the file 8
9 XPRESS PRESENTATION  DECLARE AN OBJECTIVE FUNCTION AND CONSTRAINTS Objective function declaration Cost:=sum(i in ARCS)(a*flow(i) + b*exist(i))*dist(i) Constraint variable (linctr) Constraints declaration forall(a in NODES) Total(a):= sum(i in ARCS A(i,)=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 variables 9
10 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) Optimize the identified constraint variable 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 endmodel Having all the complete code needed we can now run the model in the Run button. 0
11 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))))
12 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) binary Level: array (NODES)  integer Objective function: Constraints: Number of connections: Nodes i, j= Nodes / i <> j xij i, j= α Length ij Avoid Subcycle: level level + N + N x i, j j i Direct all connections towards the root: Road length: = N ij N / j <> j x ij i= = Spain Network Example Connections: A CoruñaPontevedra LugoA Coruña AsturiasLugo CantabriaLeón VizcayaCantabria GuipúzcoaVizcaya LeónAsturias BurgosVizcaya NavarraGuipúzcoa MadridSalamanca BarcelonaZaragoza ValenciaZaragoza Sevilla Badajoz SalamancaLeón ZaragozaNavarra BadajozSalamanca 2
13 XPRESS LABORATORY  A MINCOST COSTFLOW MULTITERMINAL 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: nodes, jodpairs arcs ODPairs i= Flow ai Positive flow: i Capacity arcs Arcs / D = n i= Arcs / O = i O j j, ODPairs i= Flow Arcs, ODPairs ij i=, j= = a k k=, jodpairs Flowaj 0 α Flow Length Flow Arcs / O = n MaximumRatio ij ij Flow Demand k, j i j Spain Network Example 3
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