1.225J J (ESD 205) Transportation Flow Systems
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1 .5J J (ESD 5 Trasportatio Flow Systems Lecture 6 Itroductio to Optimizatio Pro. Ismail Chabii ad Pro. Amedeo R. Odoi Lecture 6 Outlie Mathematical programs (MPs Formulatio o shortest path problems as MPs Formulatio o U.O. traic assigmet as a MP Relatioship betwee U.O. ad S.O. traic assigmet Solvig S.O. traic assigmet by had Lecture summary.5, /9/ Lecture 6, Page
2 Optimizatio: Mathematical Programs Geeral ormulatio ( variables, m costraits: mi z (,,..., Subject to (s.t.: g (,,..., b g (,,..., b g (,,..., g j (,,..., Notes: m (,,..., b M m : decisio variables : A costrait g (,,..., = b g(,,..., b ad g(,,..., b b j Objective uctio Feasible set M a (,,..., = Mi z( = (,,..., g (,,..., b g(,,..., b.5, /9/ Lecture 6, Page Types o Mathematical Programs (MPs Liear programs (LPs: objective uctio is liear, ad costraits are liear No-liear programs (NLPs: objective uctio is liear. (costraits are usually liear. Otherwise, there might be more tha oe optimal solutio (idig such a solutio ca be a very time cosumig task I decisio variables are urther costraied to take iteger values, a liear program is a iteger program I decisio variables are costraied to take / values: a iteger program is a / iteger program I some, but ot all, variables are costraied to take iteger values: a liear program is called a mied iteger program.5, /9/ Lecture 6, Page 4
3 MPs Are Tools or Trasportatio Aalysts Various models ad quatitative aalysis questios i trasportatio ca be ormulated as miimizatio (or maimizatio problems: Shortest path problems Traic assigmet models i cogested etworks Sigal settig problems Ramp-meterig optimizatio Eamples o questios related to modelig: Formulate a model as a mathematical program (MP (there might be more tha oe model or the same modelig questio Study the properties o the model (i.e. does the model possess oe or multiple solutios? Is it easy to id a solutio? Fid a solutio to the model (Oe may settle or a approimate reasoable solutio, as it is ot always possible (desirable to id a optimal solutio i a reasoable amout o time.5, /9/ Lecture 6, Page 5 How to Solve Mathematical Programs? Graphically: This gives you a eel o what is happeig By had usig a systematic aalytical method Use your sotware such as Xpress-MP, GAMES, LINDO, CPLEX, Ecel: Sotware tools are computer implemetatios o systematic methods There are also specialized sotware or some trasportatio applicatios Eamples: TRANSCAD ad EMME/ or static traic assigmet (We have liceses o these sotware systems i the CTS Computig lab.5, /9/ Lecture 6, Page 6
4 Shortest Path Problems As A LP: Eample We wat to id shortest paths rom all odes to Node 5 Decisio variables:, i ij =, c ij, ( i, j A arc( i, j is the cost o eh arc is used otherwise , /9/ Lecture 6, Page 7 All-to to-oe Shortest Path Problem As A LP Formulatio: mi = = = = = 4, ( i, j A ij Note: costraits the decisio variables must be itegers should have bee added. However, it is kow theoretically that the above LP possesses a iteger solutio, ad tools eist to it..5, /9/ Lecture 6, Page 8 4
5 Oe-to to-oe Shortest Path Problem As A LP Formulatio: mi = = = = =, ( i, j A ij Note: we could have added the t that the decisio variables are either or. However, it is kow theoretically that the above LP possesses a - solutio, ad tools eist that provide such solutio.5, /9/ Lecture 6, Page 9 SO Static Traic Assigmet as a MP: Eample mi t ( + t ( + t ( + Demad,, No - egativity mi t where = = = + Deiitio o lik lows ( + t( + t( = mi m ( d + m ( d + m ( m ( d = ( t ( d, m ( d = ( t ( d, m ( d = ( t ( d d.5, /9/ Lecture 6, Page 5
6 Objective Fuctio For U.O. Traic Assigmet 5 Lik, O D 5 Lik, t ( = + t ( = + t t t ( t ( Miimum o ( t ( w dw + t ( w dw , /9/ Lecture 6, Page UO & SO Traic Assigmet As MPs: R7 Eample mi t ( d + t ( d + t ( d mi m ( d + m ( d + m ( + + d,,,, = = = + U.O.: All used paths have, betwee ay O-D pair, equal ad miimum travel time = = = + S.O.: All used paths have, betwee ay O-D pair, equal ad miimum margial travel times.5, /9/ Lecture 6, Page 6
7 Solvig SO Static Traic Assigmet: Eamples 5 Lik, O D 5 Lik, t ( = + t ( = + a b c t ( = + t ( = 9 + t ( = ( q, = ( 8, q.5, /9/ Lecture 6, Page Solvig SO Static Traic Assigmet: Eamples q Lik, O D q Lik, t ( = + t ( = + UO solutios or three values o q: q=/8: =, =q q=/: =, =q q=5: =, = SO solutios or three values o q: m(=+; m(=+4 q=/8: =, =q q=/4: =, =q q=5: =(-+4q/6, =(+q/6.5, /9/ Lecture 6, Page 4 7
8 SO Static Traic Assigmet: Eample mi s t. t ( + t ( + t (. Demad +,, No - egativity = = = + Deiitio o lik lows * * * S.O. solutio: (,, = (,, mi t ( + t ( + t ( 8 ( = mi m ( d + m ( d + m ( d t t d ( d, m ( ( = d t where m ( ( =, m ( ( = ( d d d.5, /9/ Lecture 6, Page 5 Lecture 6 Outlie Mathematical programs (MPs Formulatio o shortest path problems as MPs Formulatio o U.O. traic assigmet as a MP Relatioship betwee U.O. ad S.O. traic assigmet Solvig S.O. traic assigmet by had.5, /9/ Lecture 6, Page 6 8
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