Bi-level models for OD matrix estimation

Transcription

1 TNK084 Trffc Theory seres Vol.4, number. My 2008 B-level models for OD mtrx estmton Hn Zhng, Quyng Meng Abstrct- Ths pper ntroduces two types of O/D mtrx estmton model: ME2 nd Grdent. ME2 s mxmum-entropy bsed model, t s populr becuse t s esy to deploy. However, t cn not tke congeston nto ccount. An extenson of ME2 model cn solve ths problem. It cn lso be chrcterzed s B-level model. As ME2, Grdent model s lso formulted s n optmzton problem. A grdent wll be found, whch cn be chrcterzed s devton from the orgnl OD mtrx to the estmted OD mtrx. Keywords: OD mtrx estmton, ME2, B-level, Grdent, User Equlbrum I. Introducton Socety depends on the moblty provded by trnsportton networks n order to mke t possble for ts member to prtcpte n essentl ctvtes []. In ths cse, get to know the number of people n dfferent regon s qute mportnt. O/D mtrx represents the number of people from some orgns to some destntons, for exmple, from home to work plce. In ths cse, t s crucl nput dt for trffc plnnng. Trdtonlly, O/D mtrx s estmted by home ntervew, questonnre, etc. These methods not only cost lot of tme but lso expensve. Currently, some modern technology s used for O/D mtrx estmton, for exmple, GPS trckng. However, t lso needs expensve on-bord GPS equpment. In ths pper, two chep nd tme effectve O/D mtrx estmton models (ME2 nd Grdent) re ntroduced. In the theory prt, we wll ntroduce ME2, Modfcton ME2 nd Grdent Model seprtely. And n the mplementton prt, we wll focus on the Modfcton ME2 (B-level) model. II. Problem Anlyss In ths prt, theoretcl ntroducton nd nlyss bout ME2, Modfcton ME2 nd Grdent model wll be crred out. II.. Theory A populr model for O/D mtrx estmton s Mxmum entropy mxmum estmton (ME2). It s bsed on nformton-mnmzng prncples. It wll lwys reproduce the observed lnk flows to wthn gven tolernce provded the constrns defne fesble spce [2]. ME2 s proporton-fxed scheme. Ths mens t wll not tke the congeston on the lnk nto ccount. In ths model, wht we need for the nput s the trget O/D mtrx, observed lnk flow nd n ntl proporton p. ME2 s represented by the formul s: γ = γ 2 = The purpose of ths objectve functon s to decrese the dfference between the estmted nd trget O/D mtrx s smll s possble, nd to mke the estmted lnk flow s close to the observed lnk flow s possble. A b-level model for orgns destntons mtrx estmton cn be used to mprove the qulty of such ntl estmton [3]. A dstngushng chrcterstc of multlevel system s tht the decson mker t one level my be ble to nfluence the behvor of decson mker t nother level but not completely control hs ctons []. Modfcton ME2 (MME2) s n extenson of ME2 model, nd t s b-level model. Wherever congeston plys n mportnt role n route choce ths ssumpton becomes questonble s the route choce proportons nd the trp mtrx become nterdependent. Becuse of ts theoretcl nd prctcl dvntges, equlbrum ssgnment s the nturl frmework for extendng the ME2 model for the congested network cse [2]. One of the fetures of the extenson ME2 model s ts multplctve nture. Ths mens tht f cell n the pror mtrx s zero t wll remn zero n the soluton s well [2].The pproch of the modfcton of ME2 looks s Fg..

2 Hn Zhng, Quyng Meng P Fg.. Approch of MME2 MME2 s formulted s: (g, g ˆ )+ (f, ) fˆ g 0 where f s found by solvng, = I hr=f, A hr 0, =,,I, r R At the sme tme, ME2 model cn ccommodte other sources of dt provded they cn be ncorported s lner constrnts. Such s the fctor of trp length dstrbuton, we cn trnslte t to constrn s followng: k ( / T T j = p ) j j k Where T s the totl number of trps, p k s the k proporton of trps n cost (length) rng, j s f trps between nd j, nd 0 otherwse [2]. ME2 hs been wdely used nd mplemented, prtculrly n UK. The model hs, however, some known lmttons, one of the lmttons rses when trffc hs grown mrkedly between the pror trp mtrx nd the present. As the model estmtes the mtrx closest to the pror, when loded on the network, reproduces the trffc counts, ths my led to dstortons. Another lmtton s the fct tht t consders the trffc counts s error free observtons on non-stochstc vrbles [2]. Grdent model s lso formulted s n optmzton problem: I A F (g) = 2 ˆ 2 ( g ˆ g ) + ( f f ) 2 = 2 = The grdent s gven by: A F f = ( g gˆ ) + ( f fˆ ) g g = ME2 model Equlbrum ssgnment Here the objectve functon to be mnmzed s mesure of dstnce between observed nd ssgned volumes. For the trdtonl model whch mesure of dstnce f ( ĝ, g) nd mpose the equlty between observed nd ssgnment vlues s constrnts, whle ths pproch ndeed provdes men to choose the best demnd mtrx mong those stsfyng the condtons on g the observed volumes, t lso ncrese the complexty of the problem to be solved, thus, theses models re very dffcult to pply to lrge scle problem network. However, the grdent method whch lwys follows the drecton of the lrgest yeld wth respect to mnmzng the objectve functon nd, thus, t lso ssures us not to devte from the strtng soluton more thn necessry [4]. The grdent method cn be mplemented usng the stndrd verson of the wdely used EMME/2 trnsportton plnnng softwre. The result of reserch shows tht the EMME/2 whch computton ech terton by usng the grdent method cn be hndled the lrgest network (up to 600 trffc zone, 2,560,000 O/D prs nd lnks) [4]. So, the grdent method s more sutble for solve the lrge network problems. The procedure for grdent model looks lke ths:. Strt wth some OD-mtrx (0) g, for exmple, the mtrx to be updted ĝ. Let k=0 2. Compute the trffc equlbrum for the gven trvel demnd. Let the lnk flow soluton be (k ) gven by f. 3. Compute how (n whch drecton) the OD-mtrx should be chnges, tht s, compute (k ) the grdent to F w.r.t g n g nd denote the (k ) drecton d. 4. Check some termnton crter. 5. Compute the step length by performng lne serch n the gven drecton. Fnd the step length k such tht ( k ) ( k ) ( k ) F ( g + k d ) < F ( g ) ( k + ) ( k ) ( k ) 6. Let g = g + d nd go to Step. II.2. k Relzton For the relzton, we wll just focus on the modfcton ME2 model (b-level model). And we wll crry out ths model by Mtlb lter on. Durng ths project, we wll focus more on how to mke t come true, tht s mplementton, but not the theory. For the mplementton, we wll gve fctve network, whch contns 0 nodes nd 20 rcs The volume dely (cost) functon s defned s: The prmeter α,β,γ, nd the cpcty C re ndvdul for ech rc nd depend on the chrcterstcs of the correspondng rod segment. The b-level procedure wll be ntroduced below, nd our project wll lso follow these pproches, our workng tme wll be ssgned. However, the tme wll probbly be re-ssgned ccordng to our progress. k. ME2 -Clculte g wth respect to the gven trget OD mtrx gˆ, observed lnk flow vˆ nd n ntl 2

3 Hn Zhng, Quyng Meng proporton 0 p. In order to solve ths problem, ME2 model s mplemented here. The objectve functon we wnt to mxmze s: whch wll subject to: nd Here we pln to use the optmzton functon n Excel to solve ths optmzton problem. 2. Network lodng -Snce we hve got new g, we wll mke ths g s n nput n ths step, nd solve the network lodng problem bsed on User Equlbrum problem to clculte new proporton p, nd lnk flow v. Here we wll use Mtlb to crry out ths problem. The bsc theory for ths equlbrum problem s: hr > 0 cr = π pq,( p, q) C, r Rpq h = 0 c π,( p, q) C, r R r r pq Here, h represents the route flow, represents the route cost, route cost s summry of lnk cost, π denote the mnmum cost n ech OD pr. It pq mens f the cost on specfc route s hgher thn the mnmum level; the route flow wll be zero. In other words, the drvers wll lwys choose the chepest route. Under ths condton, shortest pth problem wll be solved. 3. Iterton After one terte fnshed. Return to step. It wll go nto loop. But, when to stop s problem. Actully there s no ccurte stop pont. The m of terton s to mke the dfference between the referenced OD mtrx nd estmted OD mtrx smller nd smller. In Mtlb, we cn set the terton by ourselves. II.3. Gol The gol of ths project s to study the theory of dfferent methods on OD mtrx estmton, whch re ME2 nd Grdent. Furthermore, we wll py more ttenton to the b-level model-modfcton ME2. At lst, we wll gve fctve network wth 0 nodes nd 20 rcs, nd mplement modfcton ME2 on ths model by Mtlb. Becuse the tme s lmted, we wll not go nto the theory deeply; we wll focus more on the mplementton prt. pq c r III. Implementton Followng tht we wll ntroduce the mplementton of our project n detl. The progress beng ntroduced s the sme s wht we hve ntroduced bove. However, t wll nvolve n more detled nformton, such s the network, dt, Mtlb code, etc.. Intlzton As we tlked bove, the network we wll ntroduce here s smple fctve network wth 9 OD prs, 0 nodes nd 20rcs [Fg. 2]. The trget OD mtrx s shown n Tble. Fg.2. Fctve Network Tble. OD Flow Then we llocte ll of the OD flow onto the network. The flow s llocted mnly bsed on the cost functon, whch we wll ntroduce n the followng prt. In ths cse, the flow on the network cn be close to the relty. The detled lnk flow wll be shown n Appendx. When the flow s llocted onto the network, the proporton s esy to clculte. Ths cn be done by Excel. Followng tht we should gve the observed lnk flow. Here we gve lnk 3-8 wth the flow 65, nd lnk 7-9 wth the flow 68. Now, we hve got the nput for ME2,.e. trget OD mtrx- ĝ, proporton-p nd observed lnk flow- vˆ, we wll contnue to updte the current trget OD mtrx ĝ. 2. Updte g by ME2 model Here we use the problem solver n Excel to solve the optmzton problem. As we tlked bove, we should mnmze the objectve functon wth respect to three constrnts. The result s shown n Tble

4 Hn Zhng, Quyng Meng 3. Implement the network lodng problem by Mtlb to updte the proporton. Here we wll tlk bout network lodng problem together wth the computer code n detl. The code s ttched n Appendx 2. The network lodng s mnly bsed on the theory of User Equlbrum, whch should be strted by defnng trvel dely functon or cost functon. The dely functon s defned s: d (α * + ) χ t ( f )= f β * f d f The trvel dely functon represents the cost on ech lnk, nd t s relted wth lnk flow. Tht s to sy, when the cost s hgh on ths lnk, the flow should be lght ccordng to the rel condton, vce vers. The prmetersα, β nd χ re ndvdul for ech lnk, nd dependent on the chrcterstcs of the correspondng rod segment, such s length, etc. In Mtlb code, A s to defne the cost functon. The formt s: [StrtNode EndNode ALPHA BETA GRT]. These prmeters cn be defned by us ccordng to our fctve network. Once we hve defned the dely functon, t wll not be chnged n the followng pproch. The mn de of User Equlbrum s to fnd out the grdent of the orgnl lnk flow bsed on shortest pth theory, tht s, serches drecton,.e. to whch lnk the flow should be llocted. Ths lnk should be the chepest one comprng to other lnks. Followng tht, clculte the step length to mke sure how much flow should be llocted to the trget lnk. These steps should be terted untl none of the lnk hs lower cost. At ths tme, we cn get fnl resonble soluton, whch s seres of updte lnk flow. The number of terton s dependent on the complexty of network,.e. number of OD pr (NOC), number of nodes (NON), number of rcs (NOA), nd lso dependent on wht knd of ccurcy we wnt to get. For the Mtlb code, the nput s g, whch we hve just got from the prevous step. The output s lnk flow- v. If we run the code for proporton, we cn get proporton of ech lnk. Ths s wht we re relly nterested n. And lso, the output of ths pproch s the nput for next terton to clculte ME2 by Excel. The regulton we should set frstly looks lke: NoA = 20; NoN = 0; NoC = 9; And then, set the terton s 50 round,.e. mxt = 50; set the ccurcy, cc = e-6. Durng our mplementton, we cn not get soluton wth ths ccurcy sometmes. In ths cse, f we wden the restrcton, for exmple, set cc= e-5, we cn get result. After tht, set the cold strt, cold =. Ths ndctes tht the procedure s ntlzed by defult strtng soluton. 4. Return to pproch 2. Once we hve got n updte proporton, we return bck to ME2 model to re-clculte the optmzton problem nd get n updte g. These steps should be repeted gn nd gn untl we get resonble soluton. IV. Result Here we wll lst our result generted ech terton of B-level Model. And the necessry nlyss nd explnton wll lso be crred out. Snce wht we re nterested n s OD mtrx-g, we wll just lst g n ech terton. For the proporton nd lnk flow clculted by Mtlb, we wll not lst here. Orgnl g: g0: g: g2: g3: g4:

5 Hn Zhng, Quyng Meng g5: g6: g7: g8: n 2 ( x, x2, ) = MSD ( x, x2 ) = n In our exmple, the nput s. In ths cse, x = g ; x2 g + =. The result looks s Tble2: Devt on g G0 G G2 G3 G4 G5 G6 G7 G G2 G3 G4 G5 G6 G7 G Tble2. Men Squre Devton From Tble2, we cn see tht the devton between g8 nd g7 s huge, whch s If we mke n nlyss curve, we cn see the Men Squre Devton between ech g clerly. From Fg.3, we cn see tht from pont to pont3, the curve descends grdully. At the pont4, the curve goes up, nd then, descends gn. From pont5 to pont7, the curve s stble. At pont8, t scends suddenly. Under ths condton, the fnl result s g7. Snce OD mtrx s stble fctor, n other words, when the updte OD mtrx does not chnge lot comprng to the prevous mtrx. We cn sy tht ths s the fnl result. So, from our exmple, fter eght tertons, the OD mtrx g seems dffers lot wth the prevous one. We cn check from the tble of g8. From pont to pont4, we got flow ; from pont2 to pont6, we got flow ; from pont3 to pont5, the flow s And f we compre the prevous OD mtrx g (go-g7) wth the lst one (g8), the prevous g re more stble. We cn judge from our ntutve, t seems s the fnl terton. We cn stop here. In ths cse, the fnl OD mtrx s g7: V. Vldton So fr, we hve got our result for B-level OD mtrx estmton. However, the judgment bove s just our ntuton. We need vldton tool to check our result nd to confrm our judgment. Here we use Men Squre Devton s vldton tool. It s lso crred out by Excel. The mn de of Men Squre Devton s to clculte the dfference between smplng rry nd the referenced rry. The smller dfference t s, the better result we wll get. The formul s: Fg.3. Men Squre Devton VI. Improvement The vldton tool- Men Squre Devton shows clerly tht the devton between ech of the two g dffers lot. Even t becomes stble t g7, the mnmum devton s Cn we fnd better method to solve optmzton problem, nd get some more resonble nd ccurte soluton? Then we turned to Mtlb gn. Fortuntely, thnks to Cls- our supervsor. He helped us to wrte pece of code, whch cn solve the ME2 by Mtlb. We wll ttch ths code n Appendx 3. Ths pece of code s mnly bsed on ME2 model. It cn solve the optmzton problem by the powerful engne of Mtlb. In ths cse, the result wll be more resonble nd ccurte. In ddton to tht, we do not need to nput the lrge mount of dt nto Excel. Ths s very mportnt ssue n relty. Snce the network n relty s 5

6 Hn Zhng, Quyng Meng fr more complex thn our exmple. Followng we wll lst our result clculted by ths code. Orgnl g: g0: g: g2: g3: g8: From these results clculted by Mtlb, we cn see tht the results seems more resonble thn whch clculted by Excel. Followng let s check the Men Squre Devton between ech g n Tble 3. Unlke the devton shown s bove, the devton here s qute smll. In ths cse, the result s more stble thn the result clculted by Excel. From Fg.4, the devton decreses dstnctly from strt pont. And then, becomes stble. At pont8, the devton s lmost 0. At ths tme, the terton should stop here. The fnl result s g8 [Tble 4]. G0 G G2 G3 G4 G5 G6 G7 G G2 G3 G4 G5 G6 G7 G8 Devton Tble3. Men Squre Devton Tble4 Fnl OD Mtrx g4: g5: g6: g7: Fg.4. Men Squre Devton VII. Concluson Our B-level OD Mtrx Estmton s crred out mnly bsed on the combnton of Excel nd Mtlb. We use Excel to solve the optmzton problem, nd use Mtlb to solve network lodng problem. Once durng our mplementton of ME2 model, we cn not get fesble soluton by Excel, no mtter wht the constrnts we set. At lst, we hd to modfy our trget OD mtrx. And then, we got result fnlly. In ths cse, Excel seems not smrt tool to solve the optmzton problem. In relty, the network nd trget OD mtrx s fxed. We cn not modfy them rbtrrly just to ft for solver n Excel. Furthermore, n rel condton, the network s more complex, nd the number of dt s huge. We cn 6

7 Hn Zhng, Quyng Meng not nput ths huge number of dt by hnd. Even f we cn do tht mnully, the Excel cn not mnge such complex network. Under ths condton, our orgnl method cn just ft for smll network wth lght trffc. However, our mproved method seems cn del wth these problems. There s no need to nput the dt by hnd gn nd gn. Wht mtter most s the mproved method cn get more ccurte nd resonble result. Summry Followng s the tme we spent on dfferent prt of our project: Problem nlyss: 30 hours Implementton nd result (ME2 nd Network lodng): 60 hours Vldton: 5 hours Improvement: 20 hours 7

8 TNK084 Trffc Theory seres Vol.4, number. My 2008 Appendx Intl Lnk Flow OD

9 TNK084 Trffc Theory seres Vol.4, number. My 2008 Appendx2 Mtlb Code for Network Lodng Problem % MATLAB. % % Prmeter specfctons of ll 3 prmeters for runnng % the mydsd functon (wth COLD START). % The prmeter nmes re stted for cll of the form: % % [v,h,r,r2,r3,r4,r5] = % mydsd(noa,non,noc,g,a,mxt,cc,cold,r,r2,r3,r4,r5); % NoA = 20; NoN = 0; NoC = 9; g = [ ]; A = [ e e e e e e e e e e e e e e e e e e e e-5 5.0]; mxt = 50; cc = e-6; cold = ; r = 0; r2 = 0; r3 = 0; r4 = 0; 9

10 Hn Zhng, Quyng Meng r5 = 0; [v,h,r,r2,r3,r4,r5] = mydsd(noa,non,noc,g,a,mxt,cc,cold,r,r2,r3,r4,r5); %Compute proportons p = zeros(noc,noa); for = :NoC for = :NoA %for ech route n the OD-pr for r = :r() %for ech route r chech every lnk the n route for l = r2(,r):(r2(,r)+r3(,r)-) %when route r uses lnk f (r4(l) == ) %...dd the OD flow p(,) = p(,) + h(,r); end end end %dvde the sum by the totl OD-demnd p(,) = p(,)/g(,3); end end 0

11 TNK084 Trffc Theory seres Vol.4, number. My 2008 cler g=[ ]'; g_ht=[ ]'; g_low=zeros(9,); Appendx3 Mtlb Code for ME2 Model p=[ ]; f_ndex=[8 4]; f_obs=[35 50]; [g_sol,entvl] = fmncon(@(g) sum(g.*log(g./g_ht)-g),g_ht,[],[],p(:,f_ndex)',f_obs,g_low)

12 TNK084 Trffc Theory seres Vol.4, number. My 2008 Acknowledgements Specl thnks to our supervsor Cls Rydergren. References [] Athnsos M., Blevel progrmmng n trffc plnnng: models, methods nd chllenge (995) [2] Ortuzr nd Wllumsen, Modelng Trnsport (990, p408, p47) [3] Cls Rydergren, Trffc theory lecture, 2008 [4] Spess, A grdent pproch for the O-D mtrx problem (990, p.3) 2