Decision aid methodologies in transportation

Transcription

1 Decson ad methodologes n transportaton Lecture 7: More Applcatons Prem Kumar prem.vswanathan@epfl.ch Transport and Moblty Laboratory

2 Summary We learnt about the dfferent schedulng models We also learnt about demand-supply nteractons n the form of revenue management concepts We learnt to mmc expectatons and use solver wth Spreadsheets We have learnt about OR applcatons n martme and learnt to use mathematcal solvers Today, we wll see some more applcatons We wll work more wth MATHPROG n the lab

3 Applcatons n Ralways: Locomotve Assgnment Basc Inputs Tran Schedule over a perod of plannng horzon A set of locomotves, ther current locatons and propertes Output Assgnment schedule of locomotves to trans Constrants Locomotve mantenance Tonnage and HP requrement of tran Several other constrants Objectve Cost mnmzaton

4 Locomotve Assgnment: Some Features A tran s typcally assgned a group of multple locomotves called a consst that usually travels together Each tran has a dfferent HP and Tonnage requrement that depends on the number of cars attached Locomotves can ether pull trans actvely or deadhead on them. Locomotves can also lght travel. Trans need not have the same daly schedule.

5 Locomotve Assgnment: Mathematcal Model Decson Varables Locomotve-Tran assgnment schedule Actve locomotves Deadhead locomotves Lght travel locomotves Parameters Locomotve avalablty, mantenance schedule and features Tran schedule / tme-table and tran features Infrastructure features for sectons and yards

6 Locomotve Assgnment: Hard Constrants Horsepower requrements Tonnage requrements Fleet sze lmtatons Consstency of the assgnments Locomotve avalablty at yards and sectons Repeatablty of the soluton Soluton robustness and recoverablty

7 Locomotve Assgnment: Lterature Reference: Ahuja et al (2003, 2005) Number of trans per week: over 3,500 Number of locomotves: over 2,000 Number of locomotve types: 5 Sze of the nteger programmng problem: Number of nteger varables: 200,000 Number of constrants: 67,000

8 Locomotve Assgnment: Soluton Methodology Input Data Shorter perod Schedulng Problem Complete tme horzon Schedulng Problem Soluton Two-stage optmzaton allows us to reduce the problem sze substantally whle gvng an opportunty to mantan consstency

9 Locomotve Assgnment: Soluton Methodology Input Data Form trantran connectons that can be served by the same locomotve Determne locomotves for lght travel and deadheadng dependng on locomotve mbalances Determne mnmal cost assgnment of locomotves Soluton Determne the three sets of decson varables usng a sequental process.

10 Locomotve Assgnment: Model Results Increase n effcency by about 15%. Ralroad company felt that they could save about locomotves by the use of ths model.

11 Ralroad Blockng Problem Problem: Orgn-Destnaton of shpments gven Each shpment contans dfferent number of cars Tran routes and tme table known Capacty of the network and trans known Magntude: Thousands of trans per month 50, ,000 shpments wth an average of 10 cars (Ahuja et al) Desgn the network on whch commodtes flow

12 Comparson wth Arlne Schedule Desgn Hongkong Delh Frankfurt Chcago Zürch Prague Pars Geneva Lyon Lsbon Munch Mlan Venna

13 Ralroad Blockng Problem Blockng Arcs Orgns Yards Destnatons Reference: Ahuja et al: Ralroad Blockng Problems

14 Ralroad Blockng Problem: Model Decson Varables: Blockng arcs to a yard wth orgn (or destnaton) selected, or not Route followed by the shpments along the blockng arcs Constrants: Number of blockng arcs at each node Volume of cars passng through each node Capacty of the network and tran schedule Objectve Functon: Mnmze the number of ntermedate handlng and the sum of dstance travelled (dfferent objectves can be weghted)

15 Ralroad Blockng Problem: Problem Scale Network sze: 1,000 orgns 2,000 destnatons 300 yards Number of network desgn varables: 1,000x x x2,000 1 mllon Number of flow varables: 50,000 commodtes flowng over 1 mllon potental arcs Reference: Ahuja et al: Ralroad Blockng Problems

16 Ralroad Blockng Problem: Complexty Network desgn problems are complex for many reasons. Apart from the large number of varables, there can be several competng solutons wth the same value of the objectve functon Problems wth only a few hundred network desgn varables can be solved to optmalty Ralroads want a near-optmal and mplementable soluton wthn a few hours of computatonal tme. Reference: Ahuja et al: Ralroad Blockng Problems

17 Ralroad Blockng Problem: Soluton Approach Integer Programmng Based Methods Slow and mpractcal for large scale nstances Network Optmzaton Methods Start wth a feasble solutons Gradually mprove the soluton one node at a tme Reference: Ahuja et al: Ralroad Blockng Problems

18 Ralroad Blockng Problem: Soluton Approach Start wth a feasble soluton of the blockng problem Optmze the blockng soluton at only one node (leavng the soluton at other nodes unchanged) and reroute shpments Repeat as long as there are mprovements. Reference: Ahuja et al: Ralroad Blockng Problems

19 Ralroad Blockng Problem: Soluton Approach Out of about 3,000 arcs emanatng from a node, select 50 arcs and redrect up to 50,000 shpments to mnmze the cost of flow. Problem nstance could be solved for one node usng CPLEX n one hour. Reference: Ahuja et al: Ralroad Blockng Problems

20 Ralroad Blockng Problem: Future Ths s one of the ongong research open problems that s currently beng tackled by the ralroad ndustry Of course there are many such nterestng problems n ralways and we could gve example of only two n ths lecture

21 Arport Gate Assgnment: Objectves Gven a set of flght arrvals and departures at a major hub arport, what s the *best* assgnment of these ncomng flghts to arport gates so that all flghts are gated? Gatng constrants such as adjacent gate, LIFO gates, gate rest tme, towng, push back tme and PS gates are applcable

22 Arport Gate Assgnment: Problem Instance One of the largest n the world Over 1200 flghts daly Over 25 dfferent fleet types handled 60 gates and several landng bays Around 50,000 connectng passengers

23 Termnology Adjacent Gates: Two physcally adjacent gates such that when one gate has a wde boded arcraft parked on t, the other gate cannot accommodate another wde body Gate #1 Gate #2

24 Termnology LIFO: Last-In Frst-Out Gates These gates are one behnd the other makng t physcally mpossble for the arcraft n the nner gate to leave before the arcraft at outer gate departs Gate #2 Gate #1

25 Termnology Towng: At tmes, a turn occupes gate for a long tme because of the long gap between an ncomng flght arrval tme and outgong flght departure tme. Arcraft n such cases s towed away to a remote bay so that subsequent arrvals can be gated. Arcraft s brought back to the gate closer to ts departure tme. Orgnal Turn Tme Plane Arrval Plane Departure Tow Away Tow Back Tme Deplane to Tme Boardng for tme

26 Termnology Market: An orgn-destnaton par Turns: A par of ncomng and outgong flghts wth the same arcraft or equpment Gate Rest: Idle tme between a flght departure and next flght arrval to the gate. Longer gate rest helps pad any mnor schedule delays, though at the cost of schedule feasblty PS Gates: Premum Servce gates are a set of gates that get assgned to premum markets typcally where VIPs travel

27 Mathematcal Model Parameters a : scheduled arrval tme of turn b : scheduled departure tme of turn (k,l): two gates restrcted n the adjacent par E k E 1 1 1, : sets of equpment types such that when an arcraft of a type n E l k s 1 occupyng k, no arcraft of any type n may use l; and vce versa. Decson varables {0,1}: 1 f turn s assgned to gate k; 0 otherwse {0,1}: 1 f turn s not assgned to any gate; 0 otherwse {0,1}: 1 f long turn t s towed; 0 otherwse E l

28 Mathematcal Model Maxmze T k Ckxk C1 wt C2 K t L T y subject to: k K x k y 1 T y k y jk 1, j T; k K : a bj, a j b, j y k y jl y k y jl 1 1 F R, j T; k, l K; klfo, llfo LF : a j a bj, F R j, e E, e E LFo k j LFo l F R, j T; k, l K; klfo, llfo LF : a j b bj F R j, e E LFo, e k j E LFo l,

29 Mathematcal Model t k k w y y 2 1 t p p K k T t T L 2 1, :,,, t k k w y y : ; ;,, K k T t T L,, a b a a t p t p 2 1, t k k w y y ,, : ;, ; ;,, L b b a a J j K k k T t T,, 2 1 t p t p ,,, j j j j E e E e q k q k

30 Addtonal Objectves Mnmze the Gatng Costs Even the arport gates are nfrastructure nvestments, t costs the arport to manage them operatonally by provdng pushbacks, tow vehcles, etc. There are also penalzaton Maxmze Connecton Revenues Ths gatng objectve dentfes connectons at rsk for a hub staton and gates the turns nvolved such that connecton revenue s maxmzed Maxmze Robustness Flghts must be gated based on the past pattern of flght delays to provde adequate gate rest between a departng flght and the next arrvng flght

31 Assumptons Connecton revenue s realzed only f the passenger s able to deplane, walk between the gates and board before the connectng flght departs Deplanng tme, walkng tme and boardng tmes are provded as pont estmate nputs Connecton revenue s provded as pont estmate nputs For schedule robustness, gate rest accounts for mnmum gate rest and average delay of the turns

32 Input, Model and Output Inputs Methodology Outputs Flght Schedule LIFO Gate Rest Adjacency Push Tme Zones Used Connecton Revenue Turns Data Zone Informaton Gates Data Gate Adjacency Gate Rest Last In Frst Out Gates Push Back Tme Towng PS Gates Walkng Tmes

33 Revenue n Run Tme n Mn Results Lft n revenue observed for ths objectve when evaluated separately as well as n conjuncton wth other objectves Run tme ncreases as number of objectves ncrease mnmum gate rest ncreases Revenue vs Run Tme Comparson for Dfferent Gate Rests Base Model Optmze Connecton Revenue Optmze Connecton Revenue and Robustness 0

34 Results Desred gate rest s obtaned based on hstorcal pattern of delays; volaton s only 0.9% Run tme ncreases wth the complexty of objectves and ncreasng gate rest Objectve % Volaton Gatng wthout Robustness Objectve 4.0% Gatng wth Robustness Objectve 0.9% Gatng wth Optmzng Robustness and Connecton Revenue 0.9%