Multi-Class Demand Matrix Adjustment

Transcription

1 Multi-Clss Dend Mtrix Adjustent Yolnd Norieg Michel Florin Noveber 2007 CIRRELT

2 Multi-Clss Dend Mtrix Adjustent Yolnd Norieg 1,*, Michel Florin 1 1 Interuniversity Reserch Centre on Enterprise Networks, Logistics nd Trnsporttion (CIRRELT) Université de Montrél, P.O. Box 6128, Sttion Centre-ville, Montrél, Cnd H3C 3J7 Abstrct. In this pper, the grdient ethod for djusting single clss origin-destintion trix by using observed flows (see Spiess, 1990) is extended to djusting siultneously the origin-destintion trices of severl clsses of trffic. The lgorith is developed in detil nd coputtionl tests deonstrte the efficiency of the ethod. A coprison is crried out with two other wys of sequentilly djusting origin-destintion trices of severl clsses by using the trnsporttion plnning network of the Montrel region. Keywords. Multi-clss equilibriu ssignent, dend djustent, grdient ethod. Acknowledgeents. This reserch ws supported in prt by n individul operting grnt fro the Nturl Sciences nd Engineering Reserch Council of Cnd (NSERC). We would like to thnk Mr. Pierre Trebly of the Ministry of Trnsporttion of Quebec for perission to use the dt of the MOTREM odel for the Montrel region. Results nd views expressed in this publiction re the sole responsibility of the uthors nd do not necessrily reflect those of CIRRELT. Les résultts et opinions contenus dns cette publiction ne reflètent ps nécessireent l position du CIRRELT et n'enggent ps s responsbilité. * Corresponding uthor: Yolnd.Norieg@cirrelt.c Dépôt légl Bibliothèque ntionle du Québec, Bibliothèque ntionle du Cnd, 2007 Copyright Norieg, Florin nd CIRRELT, 2007

3 Multi-clss Dend Mtrix Adjustent 1. Introduction The djustent of n origin-destintion (O-D) trix by using observed flows (counts) on the links nd turns of trnsporttion plnning network hs ttrcted the ttention of ny reserchers. The ethods proposed y be subdivided into two ctegories depending whether the network considered is ssigned constnt trvel ties or flowdependent trvel ties. Soe of the contributions de for O-D trix djustent on uncongested networks include those of Vn Zuylen nd Willusen (1980), Mher (1983), Cscett (1984), Bell (1984), Spiess (1987), Tin nd Willusen (1989), Willusen (1984), Bell (1991) nd Bierlire nd Toint (1994). When the network considered for the O-D trix djustent is subject to congestion the underlying route choice ethod is n equilibriu ssignent. Soe of the nuerous contributions de for this version of the proble re those of LeBlnc nd Frhngin (1982), Nguyen (1984), Fisk (1988, 1989), Spiess (1990), Kwki et l (1992), Florin nd Chen (1995), Yng et l (1992) nd Yng et l (1994). In this cse the O-D trix djustent ethod y be forulted s bi-level optiiztion proble or, s others denote such probles theticl progring proble with equilibriu constrints (MPEC). Severl of these ethods hve been ipleented in prctice (see Vn Vliet, 1982, nd Spiess, 1990), nd re used on regulr bsis for the djustent of n out-of-dte O-D trix for the evlution of conteplted supply chnges in short ter plnning horizon. The considertion of severl clsses of trffic, such s privte crs nd different types of trucks, hs led to the coon use of ulti-clss ssignents to predict the use of the trnsporttion infrstructure. In turn, this hs creted the need to djust siultneously the dend for severl clsses of trffic by using link nd turn counts for ech clss. Most of the pplictions resorted to the sequentil use of single clss O-D djustent ethod. The one exception is the pper by Wong et l (2005) which considers the djustent of O-D trices by severl clsses by using n entropy bsed ethod. In the context of freight flows, Crinic et l (2001) propose siultneous ulti-clss O-D djustent ethod but do not present n ipleenttion or coputtionl results. In this rticle the grdient ethod developed by Spiess (1990) for the djustent of single clss O-D trix is extended to ultiple clsses of trffic. The ethod is developed nd ipleented for coputtions in the Ee trnsporttion plnning softwre pckge (see INRO, 2007). Then coprisons re crried out with two wys of djusting O-D trices for severl clsses sequentilly by using the trnsporttion plnning network of the Montrel Region. CIRRELT

4 Multi-clss Dend Mtrix Adjustent The pper is orgnized s follows. The next section contins the forultion of the odel nd the developent of the grdient bsed lgorith. The ipleenttion of the ethod is presented in Section 3 nd the coputtionl results re presented in Section 4. A short conclusion ends this rticle. 2. The odel forultion nd the solution lgorith In this section, the nottion used is introduced in order to stte the theticl forultion for the proble. The nodes of the rod network re denoted n, n N nd the links re denoted, A, where N is the set of nodes nd A is the set of links. The set of O-D pirs is denoted by I nd it is convenient to refer to the O-D pir with index i = ( p, q), i I, where p, q N. Â A is the set of links where counts re vilble. The dend for trvel by user clss M for the origin-destintion pir i is denoted s gi where M is the set of clsses. These dends y use pths K i K where K is the set of ll routes K = nd k is pth index. The pth flow of clss on the route k is denoted h k K i () i nd gives rise to link flows link flow v on link is the su of the clss flows v of clss, = 1,.., M on link, A; the totl ˆ v = v, for = 1,..., M. The counts by clss re denoted s v. And finlly, s( v) is the trvel tie on link for totl link flow. v The copct forultion of the bi-level (or MPEC) ulti-clss O-D djustent proble is given by 1 2 Min Z( g) = ( v ˆ v ) (1) 2 M Aˆ Subject to v = ssign( g) (2) where ssign( g) is the nottion used to indicte tht the vector of flows v is the result of the ulti-clss equilibriu ssignent of dend g, This ssignent proble is: v Min F() v s () v dv Subject to = (3) A 0 CIRRELT

5 Multi-clss Dend Mtrix Adjustent v = M v A (4) k Ki v h k = δ h k k A, M (5) = g i I k Ki i, i I M (6) h 0 k K, M (7) k i δ k 1 if k for ode = (8) 0 otherwise In order to develop the solution lgorith it is necessry to derive the locl grdient of the objective function by ssuing tht the pth proportions p k re fixed. This is the siplifiction tht ws used in the grdient coputtion in Spiess (1990). Using the pth proportions p k : hk pk = k K, (9) i M gi v cn be rewritten s: v = δ k i I k Ki gi i I k Ki p k k g i = δ p A, M (10) k Considering tht the pth proportions re loclly constnts: v = δk pk,, g i k Ki A i I M (11) Hence the grdient y be coputed s: Z( g ) Z( g ) v * = gi v g i = ( v vˆ ) δ p Aˆ k k k Ki = p δ ( v vˆ ) (12) k Ki k k Aˆ CIRRELT

6 Multi-clss Dend Mtrix Adjustent * In order to obtin the optil step length by clss, λ, the following proble ust be solved: Z( g ) Min Z( g (1 ) i λ ) (13) λ g Subject to i Z( g ) λ 1 v The derivtive is coputed s: i I, M, g i 0 (14) dz( λ ) dv Z( λ ) = * (15) dλ dλ v Aˆ Since dv dg v v ' = = * = g ( p ( v vˆ ))( i i k δ k dλ i Idλ g i I i k K ˆ i A p ) k k k Ki δ (16) nd then Z( g ) 1 2 = ( ( ˆ ) v v v v 2 ) Aˆ 1 = v + v vˆ = v + v vˆ (17) v Aˆ 2 ( ( λ ' ) ) λ ' 2 Aˆ dz( λ ) = v ' ( ' ˆ v λv v ) + (18) dλ nd by nnulling (18) the optil step size is then: λ * = Aˆ v '( vˆ v ) Aˆ ( v ') 2 M (19) It is worthwhile to note tht the optil step size is different for ech clss of trffic. The stteent of the lgorith is given next. CIRRELT

7 Multi-clss Dend Mtrix Adjustent The Multi-Clss O-D Adjustent Algorith Step 0. Initiliztion. Itertion l = 0, Step 1. Multi-clss ssignent. Multi-clss ssignent of dend g l ( M ) to, obtin link volues v l for A, M Step 2. Link derivtives nd objective function. Coputtion of the link derivtives l, l, 1 ( v ˆ v ) for A ˆ, M l, l, 2 nd the objective function ( v ˆ v ) 2 M Aˆ If the xiu nuber of itertions L is reched go to Step 7. Step 3. Assignent to copute the grdient trix. Multi-clss ssignent with pth nlysis to copute the grdient trices: Z( g ),, Z( g) = = ( ) l l, l, l, l l ˆ l, pk δk v v g l i k K ˆ i A Step 4. Assignent to obtin the derivtives. Multi-clss ssignent with pth nlysis to obtin the derivtives: v g Z g δ l, l, l, l, l, ' = i ( ) k p k i I l k Ki Step 5. Updte of the dend trices. For ech clss M : Coputtion of the xil grdient s l, l, l, x g = x( Z( g) / g ) i Coputtion of the optil step length s λ Updte of the dend trix: g + = g + in( λ,1)* Z( g) / x g l, * l, 1 l, l, * l, l, i i = Aˆ v ' ( vˆ v ) l, l, l, Aˆ ( v ') l, 2 Step 6. Itertion counter. Updte the itertion counter l = l+ 1 nd return to Step 1. Step 7. End. CIRRELT

8 Multi-clss Dend Mtrix Adjustent 3. Ipleenttion of the Algorith The ipleenttion of the lgorith ws done in the Ee 3 (INRO, 2007) trnsporttion plnning softwre. Three pproches were tested. The first pproch is the ulti-clss djustent where the dend of ll clsses is djusted siultneously. The second pproch djusts the dend for one clss t tie, but letting the flows of ll the other clsses vrible during the ssignents. In the third pproch, the dend of one clss is djusted t tie, but in this vrint the volues of the other clsses re fixed. For the first pproch, the lredy existing EMME/2 cro dedj22.c (by Heinz Spiess) ws odified to consider ultiple clsses of vehicles. The new cro, dedjc.c, uses the «generlized cost ulti-clss ssignent with pth nlysis» option, which hs becoe vilble recently. The use of this option kes possible to solve the proble crrying out only M + 2 ulti-clss ssignents per itertion. In fct, this kind of ssignent llows the user to nlyse the pths for ll the clsses t the se tie nd to sve the clss specific volues. However, M ssignents re required in order to specify the prticulr clss link ttribute for the pth nlysis (Step 3). The totl nuber of ulti-clss ssignents becoes L*( M + 2), where L is the xiu nuber of itertions. As the cro dedjc.c llows to select the clsses for which the dend ust be djusted, this cro ws odified slightly to ipleent the second pproch. The new cro, clled dedj-seq.c, hs n dditionl externl loop which llows it to djust sequentilly the dend for one clss of vehicles t the tie, leving the other dends fixed but the flows vrible. The procedure is s follows: For i = 1 to M Cll dedjc.c (clss i) The nuber of ulti-clss ssignents by itertion decreses to 3, but the totl nuber of itertions increses M ties, so one cn expect tht this cro will be ore tie consuing. The totl nuber of ulti-clss ssignents goes up to 3* L* M. Note tht this ount considers tht the se xiu nuber of itertions is chosen for ll the clsses. For the third pproch, new cro ws developed. This cro, clled dedj-fix.c, ipleents n itertive sequence of clls to the single clss dend djustent cro dedj22.c. A ulti-clss ssignent is crried out before clling dedj22.c in order to clculte the volues of the clsses tht re not djusted. These volues re fixed s bckground flows. The totl nuber of ulti-clss ssignents is M, nd the totl nuber of single clss ssignents is 3* L* M. CIRRELT

9 Multi-clss Dend Mtrix Adjustent All the cros were dpted for the Montrel Region dt set described below but could be esily custoized for ny other ppliction. dedjc.c ws lredy generlized to work with ny dt set. 4. The Coputtionl Tests The coputtionl tests were crried out by using one network dt set originting fro the Region of Metropolitn Montrel. The corresponding network is displyed in Figure 1. The considered Montrel network uses 3 clsses of trffic: privte cr, regulr trucks nd hevy trucks. Figure 1. The Montrel Network The network chrcteristics re given in Tbles 1 to 3. The dend is given in Tble 4. AM pek PM pek Off pek Zones (centroids) Regulr nodes Links Clsses Tble 1. The network preters Clss Regulr truck Hevy truck Chrcteristics One unit, 2 or 3 xles One unit, 4 xles, or ore thn one unit Tble 2. The Truck chrcteristics CIRRELT

10 Multi-clss Dend Mtrix Adjustent Code Period Hours Counts NI Night 0 :00 to 6 : AM AM Pek 6 :00 to 9 : OPD Off Pek Dy 9 :00 to 15 : PM PM Pek 15 :30 to 18 : OPN Off Pek Night 18 :30 to 24 : Tble 3. The Tie Periods nd the Nuber of Link Counts Period Auto Regulr Truck Hevy truck NI 246,212 7,048 7,542 AM 976,715 26,631 15,367 OPD 1'905,037 84,091 42,157 PM 1'259,606 24,305 14,804 OPN 1'007,921 47,703 27,111 Tble 4. The Dend A set of logistic volue dely functions hve been clibrted for ll the links of the network 1. The logistic functions re continuous positive non decresing functions of the flow. Their functionl for is: 0 η s ( v ) *(1 ) = t + A 1 + α /(( v + θ )/ c )^β 0 where s( v) is the trvel tie on link ; t is the trvel tie in the link t free flow speed; η, α, θ nd β re the preters for the link ; c is the cpcity of the link nd v = v s previously entioned. The originl dend ws perturbed to test the perfornce of the three solution ethods. For every clss, the totl dend ws decresed pproxitely 20 percent nd only the lrgest O-D pir vlues were chnged. The totl dend To be djusted by type of vehicle nd period of the dy is listed in Tble 5. Period Auto Regulr Truck Hevy truck NI 205,673 5,785 6,086 AM 805,512 21,619 12,486 OPD 1'566,806 68,808 33,445 PM 1'021,659 19,812 11,986 OPN 822,865 38,798 22,303 Tble 5. The Dend To be djusted 1 Service de l odélistion des systèes de trnsport. Ministry of Trnsporttion of Quebec. CIRRELT

11 Multi-clss Dend Mtrix Adjustent For the second nd the third djustent pproches the vehicle clsses were considered in the order of decresing totl dend: uto, regulr truck nd then hevy truck. The stopping criterion for the ssignent routines is shown on Tble 6. The nuber of itertions for the djustent process ws fixed to 5 for ll the tests nd ll the clsses. Period Nuber of itertions Reltive Gp Norlized Gp NI AM OPD PM OPN Tble 6. The Assignent Preters All three ethods worked s expected. The results re very stisfctory. In ll the cses 2 the objective function nd the R coefficient (fro the regression between the observed nd the siulted link flows) were iproved. Moreover, the slope of the regression, B, is lso iproved, especilly for the trucks; showing tht the issing dend is prtilly recovered. In the following, the first pproch, in which ll the clsses re djusted t the se tie is referred s MC Adj.; the second pproch, where the djustent is done sequentilly clss by clss, leving ll the flows vrible, is identified s SEQ Adj.; nd the third pproch, in which one clss is djusted t the tie considering the flows of the rest of the clsses s fixed is clled FIX Adj. The regression coefficients R 2 nd B re listed in Tbles 7 nd 8, respectively. The objective function vlues re listed in Tble 9 nd grphiclly presented in Figures 2 to 4 for the three congested periods of the dy. Soe explntions re needed to understnd Figures 2 to 4. In the cse of the ulti-clss djustent (MC Adj.) even if 5 itertions were requested, 7 vlues re coputed; the first one corresponds to the initiliztion itertion nd the lst one is the vlue t the end of the djustent. The se procedure is used in the sequentil djustent (SEQ Adj.), but in this cse the lst vlue of the objective function for clss corresponds to the initil vlue for the next clss. In the fixed flows djustent cse (FIX Adj.) the globl objective function is coputed only four ties; before ech clss dend djustent nd t the end of the djustent process. CIRRELT

12 Multi-clss Dend Mtrix Adjustent NI AM OPD PM OPN To be djusted MC Adj. SEQ Adj. FIX Adj. Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Tble 7. The R 2 regression coefficient NI AM OPD PM OPN To be djusted MC Adj. SEQ Adj. FIX Adj. Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Tble 8. The B regression coefficient To be djusted MC Adj. SEQ Adj. FIX Adj. NI AM OPD PM OPN Tble 9. The Objective function CIRRELT

13 Multi-clss Dend Mtrix Adjustent Objective function. AM MC Adj SEQ Adj FIX Adj Vlue All clsses / Auto Regulr truck Hevy truck Itertion Figure 2. Iproveent of the objective function vlue. AM Objective function. Off Pek Dy MC Adj SEQ Adj FIX Adj Vlue All clsses / Auto Regulr truck Hevy truck Itertion Figure 3. Iproveent of the objective function vlue. OPD CIRRELT

14 Multi-clss Dend Mtrix Adjustent Objective function. PM MC Adj SEQ Adj FIX Adj Vlue All clsses / Auto Regulr truck Hevy truck Itertion Figure 4. Iproveent of the objective function vlue. PM Soe conclusions result fro the study of Figures 2 to 4. The iproveent of the objective function with the second nd the third pproches is significnt for the utos (note tht the dend of trucks is lwys less thn 10% of the uto dend). In the cse of the SEQ Adjustent for trucks, fter the first itertion, the iproveent is negligible; however, if only 1 itertion is done for these clsses of vehicles, the fit between the siulted nd the observed flows will not be good enough. The djusted dend is shown in Tble 10 nd sury of the coputtion ties is given in Tble 11. The tests were crried out on n Intel Core (TM) 2 CPU 2.13GHz, 3.25 Gb of RAM. NI AM OPD PM OPN Originl To be djusted MC Adj. SEQ Adj. FIX Adj. Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Auto Regulr truck Hevy truck Tble 10. The Adjusted Dend CIRRELT

15 Multi-clss Dend Mtrix Adjustent Period MC Adj. SEQ Adj. FIX Adj. NI AM OPD PM OPN Tble 11. The Coputtion Ties (in.) Bsed on the coputtion ties, the first choice would be the FIX Adjustent, then the MC Adjustent. The SEQ Adjustent is very tie consuing nd the results re lost the se s the others. Four scttergrs which copre the observed vs. the siulted flows re presented in Figures 5 to 8. They correspond to the Off Pek Dy period of the dy, for crs nd regulr trucks. The first two scttergrs show the fit of the dend To be djusted. The two lst scttergrs show the fit fter the MC Adj. ws crried out. As one cn see, the fit is lwys significntly iproved fter the djustents. To be Adjusted. Autos. Off Pek Dy Period Figure 5. Link flow scttergr. Observed vs. siulted utos before the djustent CIRRELT

16 Multi-clss Dend Mtrix Adjustent To be Adjusted. Regulr trucks. Off Pek Dy Period Figure 6. Link flow scttergr. Observed vs. siulted regulr trucks before the djustent MC Adjustent. Autos. Off Pek Dy Period Figure 7. Link flow scttergr. Observed vs. siulted utos fter the djustent CIRRELT

17 Multi-clss Dend Mtrix Adjustent MC Adjustent. Regulr trucks. Off Pek Dy Period Figure 8. Link flow scttergr. Observed vs. siulted regulr trucks fter the djustent Results indicte tht using the MC Adjustent is the best option. The SEQ Adjustent tkes uch ore tie, nd the results re not uch better. The FIX Adjustent is the fstest but the fit is not the best one cn get. With this ethod there is not continuous feedbck ong clsses. Conclusions The siultneous djustent of severl origin-destintion trices, for severl clsses of trffic, is fesible by using the ethod presented in this rticle. The siultneous ethod is preferble to the sequentil djustent of the origin-destintion trix of ech clss. The generliztion of the grdient ethod of Spiess (1990) for severl clsses is n efficient lgorith. Acknowledgeents This reserch ws supported in prt by n individul Operting Grnt fro NSERC. CIRRELT

18 Multi-clss Dend Mtrix Adjustent We would like to thnk Mr. Pierre Trebly of the Ministry of Trnsporttion of Quebec for perission to use the dt of the MOTREM odel for the Montrel region. References Bell M. G. H., (1984), Log-liner odels for the estition of origin-destintion trices fro trffic counts, In Proc. of the Ninth Interntionl Syposiu on Trnsporttion nd Trffic Theory, Delft, The Netherlnds. Bell M. G. H., (1991), The estition of origin-destintion trices by constrined generlized lest squres, Trnspn. Res. Vol. 25B. Bierlire M., Toint P. H. L., (1994), Meuse: An origin-destintion trix estitor tht exploits structure, Trnspn. Res. Vol. 29B. Cscett E., (1984), Estition of trip trices fro trffic counts nd survey dt: A generlized lest squres estitor, Trnspn. Res. Vol. 18B. Cscett E., Nguyen S., (1988), A unified frework for estiting or updting origin/destintion trices fro trffic counts, Trnspn. Res. Vol. 22B. Chen Y., (1994), Bilevel progring probles: nlysis, lgoriths nd pplictions, Publiction 984, Centre for reserch on trnsporttion, University of Montrel, Cnd. Crinic T. G., Dufour G., Florin M., Lrin D., Leve Z., (2001), Dend Mtrix Adjustent for Multiodl Freight Networks, Trnspn. Res. Rec. 1771, Pper Fisk C. S., Boyce D. E., (1983), A note on trip trix estition fro link trffic count dt, Trnspn. Res. Vol. 17B. Fisk C. S., (1988), On cobining xiu entropy trip trix estition with user optil ssignent, Trnspn. Res. Vol. 22B. Fisk C. S., (1989), Trip trix estition fro link trffic counts: the congested network cse, Trnspn. Res. Vol. 23B. Florin M., Chen Y., (1995), A coordinte descent ethod for the bi-level OD trix djustent proble, Int. Trns. Opl. Res. Vol. 2, No. 2. Gur Y. J., Turnquist M., Schneider M., LeBlnc L., Kurth D., (1980), Estition of n origin-destintion trip tble bsed on observed link volues nd turning oveents, Technicl report RD , FHWA, U.S. Deprtent of Trnsporttion, Wshington D.C. CIRRELT

19 Multi-clss Dend Mtrix Adjustent INRO Consultnts Inc., (2007), Ee On-line Docuenttion. Jornsten K., Nguyen S., (1979), On the estition of trip trix fro network dt, Technicl Report LiTH-MAT-R-79-36, Deprtent of Mthetics, University of Linköping, Linköing, Sweden, (revised, April 1983). Jornsten K., Nguyen S., (1983), Estition of n OD trip trix fro network dt: dul pproches, Technicl report LiTH-MAT-R , Deprtent of Mthetics, University of Linköping, Sweden. Kwki S., Lu H., Hirobt Y., (1992), Estition of origin-destintion trices fro link trffic counts considering the interction of the trffic odes, Ppers in Regionl Science, 71. LeBlnc L. J., Frhngin K., (1982), Selection of trip tble which reproduces observed link flows, Trnspn. Res. Vol. 16B. Mher M., (1983), Inferences on trip trices fro observtions on link volues: A Byesin sttisticl pproch, Trnspn. Res. Vol. 17B. Nguyen S., (1984), Estiting origin-destintion trices fro observed flows, Trnspn. Res., Vol. 17B. Sherli H. D., Sivnndn R., Hobeik A. G., (1994), A liner progring pproch for synthesizing origin-destintion trip tbles fro link trffic volues, Trnspn. Res. Vol. 28B. Spiess H., (1987), A xiu-likelihood odel for estiting origin-destintion trices, Trnspn. Res. Vol. 21B. Spiess H., (1990), A Grdient Approch for the O-D Mtrix Adjustent proble, Publiction 693, Centre for reserch on trnsporttion, University of Montrel, Cnd. Tin O.Z., Willusen L.G., (1989), Trnsport dend odel estition fro trffic counts, Trnsporttion 16. Vn Vliet D., (1982), SATURN - A odern ssignent odel, Trffic Eng. Control 23. Vn Zuylen H., Willusen L.G., (1980), The ost likely trip trix estited fro trffic counts, Trnspn. Res. Vol. 14B. Willusen, L. G., (1984), Estiting tie-dependent trip trices fro trffic counts, in Proceedings of the Ninth Interntionl Syposiu on Trnsporttion nd Trffic Theory, J. Volüller nd R. Herslg (eds.), VNU Science Press, Utrecht, The Netherlnds. CIRRELT

20 Multi-clss Dend Mtrix Adjustent Wong S. C., Tong C. O., Wong K. I., L W. H. K., Lo H. K., Yng H., Lo H. P., (2005), Estition of Multiclss Origin-Destintion Mtrices fro Trffic Counts, Journl of Urbn Plnning nd Developent, Vol. 131, No. 1. Yng H., Sski T., Iid Y., Askur Y., (1992), Estition of origin-destintion trices fro link trffic counts on congested networks, Trnspn. Res. Vol. 26B. Yng H., Iid Y., Sski T., (1994), The equilibriu-bsed Origin-Destintion trix estition proble, Trnspn. Res. Vol. 28B. CIRRELT