Proceedings of the 2007 IEEE International Conference on MonE04Networking, Sensing and Control, London, UK, 15-17 April 2007

A Study on Multiple user-classes Dynamic traffic assignmentRunmei Li Sharron Tang

Abstract-This paper studies the multiclass dynamic traffic In this paper, a discrete time multiclass dynamic userassignment problem. A multiple user-classes dynamic user optimal traffic assignment model is proposed. Based on theoptimal traffic assignment model is proposed. The phenomenon overtaking analysis, two methods of classification are given.of overtaking existing in traffic network is considered. The The distinction between user-classes is typically based ondistinction between user-classes is typically based on vehicle vehicle travel speed. For the same class user, thetravel speed. According to it, vehicles are classified into different First-in-first-out (FIFO) principle will hold. The multipleclasses. Asymmetric interactions between the user-classessharing the same road infrastructure are taken into account. A user-classes link travel time functions are proposed whereset of multiple user-classes link travel time functions are asymmetries interactions between the user-classes sharing theproposed. For the same class user, the First-in-first-out same road infrastructure are taken into account. A simple caseprinciple holds. A case study illustrates the model. study illustrates how the model works numerically.

In saturated traffic networks, overtaking is less, so trafficI. INTRODUCTION flows studied here are undersaturated.

T HERE are many types of vehicles traveling in realisticl traffic networks which have different characteristics, e.g. II. THE VEHICLES CLASSIFICATION BASED ON OVERTAKING

cars and trucks. By introducing multiple user-classes in DTA ANALYSISmodels, more refined estimates of travel times and networkflows can be obtained, especially user-class specific times In 1992, Carey[8] proposed the FIFO discipline in DTAand flows. Hence, better solutions can be found concerning problem firstly. The FIFO is observed implicitly in staticinfrastructure investments and better decisions can be made assignment, because it is assumed that travel times arein traffic management, especially in relation to distinct target identical for all travelers on the same route, and all travelersgroups[l]. have the same free flow speed. This discipline should be

Multiclass traffic had been introduced in a general observed in a dynamic assignment, since we do not generallymathematical model for static models by Dafermos[2-3]. expect that travelers arrive at the destination earlier than thoseExtensions and improvements in the theory of multiple who departed the origin before them. Denote the link traveluser-classes in static models can be found in Wynter[4]. time for flows entering link a at time t as Ca(t), the travel timeAmong the first to use an analytical approach to modeling for flows entering link a at time t+A t as Ca(t+ A t). The FIFOmulticlass DTA were Lo et al. [5] and Ran et al. [6]. In these discipline can be written as follows for a link a:papers, travelers were classified into those who follow t+c,(t) < t +At+ca(t+ At) (1)predetermined routes, those who follow a stochastic dynamic +C t t+A a( t 1Uredeasmignmenutes,a those who follow a dynamic Overtaking violates the FIFO rule for traffic propagationE assignment, and those who follow a dynamic UE on links, although it might happen on two or more lane links.assignment. The link travel time functions used are not

t

class-specific but depend on the total flow. Ran and Boyce[7] t + Ca (0 > t + At + Ca (t + At) (2)stratify travelers on route diversion willingness, income, age, When only one user class is considered, overtaking can beand driving behavior. Again, only route choice is considered avoided by restricting the time interval A t, link travel time orto be different between user-classes. The link travel time the length of link [7]. However, the phenomenon offunction is identical for all travelers. Michiel [1] proposed a overtaking exists in realistic traffic networks widelymulticlass DTA model, in which the distinction between including the overtaking between different types of vehiclesuser-classes is typically based on vehicle characteristics, e.g. such as cars and trucks and the same type of vehicles thatcars and trucks. have different free flow speed. The models ignoring the

phenomenon are inappropriate.Overtaking complicated the study of DTA. Firstly, the

Manuscript received January 21, 2007. This work is partly funded by a . . . . . . T..the

National Basic Research Program of China (973 Program, 2006CB705500), violation ofFIFO during vehicles is in contradiction with thethe Key Program of the National Natural Science Foundation of China concept of equilibrium flows. On the other hand, when(60334020,60504025), a project of The Sci-Tech Foundation of Beijing overtaking existing, there may be asymmetric interactionsJiaotong University(2005RC016),a project of the Key Sci-Tech Project ofShandongProvince(2004GG1104001),akeyprojectofShandongAcademy diffeen vehiclessharingethetsame roadtinfrastructure.of Sciences (2004029). different vehicles sharing the same road infTastructure.

Runmei Li is with School of Electronics and Information Engineering, When FIFO can be guaranteed for each user-class. TheBeijing Jiaotong University, Beijing 100044, China (telephone:86-10-63525876, e-mail: lrm-99041@sohu.com). Sharron Tang iS with rao foetkn sta ifrn srcasshvInstitute of Automation Shandong Academy of Sciences, Jinan 250014, China, different travel speed on the same link at the same time. So,(telephone: 86-531-8260-5491, e-mail: sharron@ieee. org ).

1-4244-1076-2/07/$25.00 ©2007 IEEE 513

based on the phenomenon of overtaking, two methods of horizon (k E K c( [0 T]). Denote origin-destination (O-D)classification are used to get the user-classes. For the same pairs as (r,s). h - (k) is departure flow rate on route p fromuser-class, the FIFO should hold. The main goal of this paper l Pis to propose an analytical multiple user-classes DTA model origin r toward destination s during interval k for class i users.that can explain the interactions during different user-classes. Denote uj a (k) is inflow into link a during interval k for class

A. Classification based on the types ofvehicles i user, V, a (k) is exit flow, x/ia (k) is number of vehicles onVehicles can be classified into three types simply: 1) Car. 2) link a at the beginning of interval k for class i user. Let the

Passenger car. 3) Truck. multiple user-class time-dependent travel demand rates beThese vehicles are common in traffic networks. Each type defined by q rS (k), is the flow rate of class i users departing at

vehicle has its own characteristics, such as maximum speed, time instant k from origin r to destination s. Here the demandvehicle size and acceleration. is assumed to be inelastic and given. Denote the route travel

The capability of trucks to accelerate is worse. The time cost function for class i users over route p E prs from origin rthat trucks used in start process is longer than that of cars. Sofor multiple user-class traffic flows, vehicles that speed t' (k) ,

slowly will have more influence on other vehicles. The where Prs is the set of feasible routes from r to s for user classvehicles that have better performance of acceleration will i. Let us further denote the minimal route travel function forhave less influence on other vehicles [9]. users of class i corresponding to that O-D pair and departingTo the index ofvelocity, the highest speed ofcars can be up at the same time instant by )r- (k). Denote the link travel time

to 120kilometres, and for passenger cars, trucks or for class i user on link a during interval k as Z,a (k)motorcycles, the highest speed is under 90kilometers. Sothose vehicles will have different behavior when they travel Definitionl: Multiple user-classes dynamic user optimalon the same link at the same time. In some conditions, (DUO) state, For each user-class and for each O-D pair, theovertaking during different types ofvehicles is possible. route travel costs for all users traveling between a specific

O-D pair and departing at the same time are equal, and lessB. Classifcation based on characteristics ofdrivers than (or equal to) the route travel costs which would beObserving the traffic flows on links, we can see that even experienced by a single user on any unused feasible route for

for the same type vehicles, overtaking also occurs often that user-class[1].because of the different characteristics of drivers. A typical Assume that time is discretized using intervals duringexample is that the driver speed of jackaroo is often slower which the flow rate is stationary. The average class i travelthan that of veteran. So, it is possible for the same type of demand rate q (k) is given by the number ofusers departingvehicles to travel at different speeds on the same link at the d t i k d bsame time. Overtaking during the same type of vehicles will d interval.l an dr f ies aen no define

occur especiallyin freeways. interval. The link and route flow variables are now defined asoccur especially in freeways.Ran and Boyce[6] stratify travelers into m classes for the average flow rates during a time interval.

The discrete-time dynamic multiple user-classes optimalsame type vehicles according to the socio-economic sae scharacteristics of each traveler. For example, we can classify sta rs (k)asr: (k)] hZ5 (k) 0travelers based on route diversion willingness, income, age XW X XWand driving behavior. IrC- (k)-;T- (k) . 0 Vie m,k,p,r,s (3)

Throughout this study, we assume that the O-D demandsare given, and drivers have sufficient and perfect knowledge h,'§ (k) .0of traffic network and make routing decisions in a user For each class i, the route travel time i7sI,(k) depends onoptimal manner. not only flow of itself but also the other types of flow. So the

route travel time of class i is the function of all classes vehicleIIIUSTIME-DOPTISCLTRE MULTIPLENUSENFR-CLASSESLNAMIC flows. And for each user-class, the route travel time functionUSER OPTIMAL TRAFFIC ASSIGNMENT FORMULATION. may be different.

Since solving continuous-time model for practical We can rewrite the equation (3) as follows:problems is not feasible, in this section time will be S[* (k) -)rrs (k)]- [hs (k)hrs* (k)] > 0 (4)discretized in a finite number of time intervals, yielding a l'P "P "P "Pfinite-dimensional variational inequality (VI) problem. By adding up the summation over r,s,p,k, it yields:Furthermore, we will convert the route-based model E i (k)[h7rs (k) - h7* (k)]formulation proposed into a link-based model formulation by k r s p (5)assuming additive costs. The resulting model is a - F r (k) [h (k) -hJ*s(k)].0finite-dimensional VI problem. k r s p

Inputs of the model are a network G=(N, A), consisting of By making the substitution of:nodes N and directed links A, and certain characteristics for Z h7 W (k) = EhrS* (k) (6)each link. i(i=l,. .....,m)is class designation. T denotes the time p . ...............p

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The second term vanishes and the remaining term results in E h,- (k) = qrs (k) (14)the following variational inequality equation: pE=-P

Z Z Z Z r1W (k)Eh1 (k)- h[1 (k)] >0 (7) Assumed that mid-node don't produce flow. Then fork r s p mid-nodej, exit flow at interval k equal to inflow:

Equation (7), accompanied with the relevant constraints, is ,ui,a (t) = VY,a (t) (15)in fact a route-based multiple user-classes DUO route choice aeA(j) aeB(j)model. Since any route-based network model needs to B. Flow propagation constraints based on FIFOprincipleperform route enumeration during the solution procedure,only small sized problems can be solved. This route-based The consistent movements oftravelers forward in time andDUO route choice model converted into a link based DUO space are described by the flow propagation constraints. Theroute choice model by using the following two definitional flow propagation satisfied FIFO for class i users can beconstraints [10]. written as:

Denote Ursa (t, k) is part of inflow rate for link a during uj a (k)= Via (k + ia (k)) Va, k (16)i,ap ~~~~~~~~~Fordifferent user classes, equation (I16) cannot beinterval k that is departing origin r over path p toward Fora di

destination s during interval t for class i users ( i e m ). g675, (t,k) , dynamic effective flow rate factors, equal 1, if C. Nonnegativity constraintsinflow rate on link a during interval k departs origin r over Xi a(0) = 0, Vi,a(0) = 0 (17)pathp toward destination s during interval t; otherwise, equal Xia (k) 2 0, via (k) > 0 (18)0. Then:

u (La-(k) ZE E E E h, rs (Axrs'5,p (t,k)(8) UiXa (k) >O° h,' (k) >0 (19)r s t p

s (t) = ZTa (k)4rsp (t, k) (9) V. DYNAMIC MULTIPLE USER-CLASSES LINK TRAVEL TIME

k a FUNCTIONBy applying equation (8) and (9) into equation (7), one In references [2>7], only route choice is considered to be

obtains: different between user-classes. The link travel time functions

E E EZ [Z Z <ia (k)trsap (t, k)] depend on the total link flows. The link travel time function is

r s t p k aeA (10) identical for all travelers. In this paper, the distinction[hrs (t)- h rs*(t)] > 0 between user-classes is typically based on vehicle travel

By changing the order of the summation, it follows: speed. The essential distinction between these two(1 1) classification is that for the latter, different users maybe have

z Z<,a(k)[Ui,a(k) iU,a(k) . 0 different link travel time on the same link, and users of ak acA

Then for all class users: certain class may restrict the movements of other user-classes~ E T<a (k)[ui,a6,(k) - Uiea(k)] . 0 (12) present on the same road at the same time. In most instances,

k m aeA this interaction is asymmetric.Denote w-rax is free flow travel speed of class i user, and:

IV. MULTIPLE USER-CLASSES CONSTRAINTS SET Wrax > wimaxEQUIVALENCE ANALYSIS , +]),a

The travel speed for class i user on link a during interval kTo guarantee that link flows describe a feasible pattern, de pend

several constraints to the flows need to be imposed including Wla (k) son three factors when the link shape given: 1)flow conservation, flow propagation, nonnegativity and FIFO The free travel speed of class i user; 2) The flow density onconstraints. These constraints are part ofmost analytical DTA link a during interval k; 3) The flow density of every class imodels. Additional constraints can be proposed in order to (i=1,2, ... ,m) on link a during interval k. Wia (k) can be writtenmake the flows empirically more realistic and to make typical as:interactions between user-classes more apparent. The w (k) = fp (k) pi (k) p () (20)following is a straightforward extension of these constraints W la (k) pa(k),Pi,a

=1,2,n m,a fk)) (20)

from homogeneous single user-class case to heterogeneous Where Pa (k), Pia (k), i 1,2, , m are flow density andmultiple user-class case considered in this paper. In saturated the flow density of class i user on link a during interval k.traffic networks, overtaking is less, and there is no overtaking Consider that travel speeds of class 1,2,... ,i- 1 users arein queue, so traffic flows studied here are undersaturated. quicker than that of class i user. And the vehicles that have

A. State equation and conservation constraint lower travel speed will have more impedance than the quickerThe link state function is: on class i user. So we can write the travel speed function of

Xia ( + 1)Xia() + [ja (k - Vi k~. (13)class i user as follows:

The flow conservation constraints can be written as Wia (k) f (Pa (k),Pi+W a(k)< DPma (k)) (21)follows: We can describe the travel speed models of every class user

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on link a during interval k based on modified Greenshields 0.3km 4.9kmspeed-density model: Ik / (3) 4.9km 1.6km

wa, (k) wmaxa[1]JJ T r ax )a,, ] (22P (4) 5.2kmi (6) 5Pa j=j+l Pa (5)

Where, [1jap (k) .i8i is the interaction factor of class Fig. 1. Traffic network for numerical examplempmax We only consider one direction-from south to north. For

j user on cassuse. isgive parmetes. p-isthe simplicity, only one O-D pair (1,5) is examined. Denote routej user on class i user. cx, /Aj is given parameters. prnaxis the 1, 2 and 3 are: (1)-(2)-(3)-(6), (1)-(2)-(4)-(6) and (1)-(5)-(6)

maximal flow density on link a. Let: respectively. Link (4) can only be used by truck. Threey=[1_m(Pa(k))a ]3 user-classes are presented on the road network: fast car (class

P7max 1), truck (class 2) and slow car (class 3). They have different

(Pj,a k/3)[1j(PJa())c travel speed and interact with one another. The parametersrj (Pi.a (k), aj;;,1 ) = max that will be used in the multiple user-classes speed functions

Pa are given in Table 1. The travel demand for each ofthe first 3Then we can rewrite equation (22) as follows: time intervals is given by Table 2, traveling from origin node

a (I to destination node 5. The length of time interval iswia (k) = WXi,a . jy. y a(p (k) ,ai/ (23) A =30sec, T=600sec. So K=20. Table 3 gives the free travel

j.=i+lWhen 1m+a](k)Pa(k), we have: time on each link of each class user. We use the traffic

When PN1,a (k) = Pa (k), we have: demand of first time interval to initialize the network. Table 4W1,a (k) = W2,a (k) = = Wm,a (k) (24) and Table 5 summarize the outcomes ofthe model. In Table 4,

So the boundary conditions of equation (23) are: the results of dynamic optimization based on model (12) and(13-19) are shown. Table 5 gives the actual route flow and

Wlmax. [I (P_ 1k )a.X ]f18i = route travel time for the test network.Pa °a =2.5, ,21 = 32 = 031 A = a21 =6, a32 3, 31 =1.5,

,max (Pa(k)y m2 813m2 = ,max pax =0. 125veh/m.L2, pmax1Pa TABLE 1 PARAMETER VALUES

Length of vehicle Free travel speed Vehiclewnax . [f _-( p'1 (k)J)agll ]/11 (25) (m) (kilometer/h) proportionpax Class 1 4 120 65%

Class 2 8 80 15%wmax ri - (Pa V"FnZ 1,2 ]fm1,2 = mx=max Class 3 4 65 20%2, max1 Wm-_1

TABLE 2 O-D DEMAND (VEHICLE/UNIT TIME INTERVAL)Time interval 1 2 3

p, (k) Class 1 32.5 39 26[fmax (faVV)x2l ]/521 max Class 2 7.5 9 6

Pa Class 3 10 12 8

Let La is the length of link a, the travel time for class i user TABLE 3 FREE TRAVEL TIME (SECOND)on link a during interval k is:

Link 1 2 3 4 5 6

z, (k)= La Va,i,k (26) Class 1 29.88 9 147 156 48

W,(aW(k) Class 2 45 13.5 220.5 220.5 234 72

Class 3 55.38 16.62 271.38 -- 288 88.62VI. CASE STUDY link not available for cars.

The network used is from reference [1 ] (Fig. 1).TABLE 4 DYNAMIC OPTIMIZATION RESULT FOR EACH CLASS USER OF THE TEST NETWORK

Class 1 Class 2 Class 3Entering

Link time Number Link travel Number Link travel Number Link travelinterval Inflow of Inflow of t Inflow of

vehicles time(s) vehicles time(s) vehicles time(s)

1 32.5 32.5 43.9876 7.5 15 58.4930 10 20 71.69772 39 32.5 43.9876 9 15 58.4930 12 20 71.6977

1 3 26 39 49.6994 6 16.5 64.9975 8 22 79.58104 ---- 65 62.4380 ---- 15 83.8935 ---- 20 102.89675 ---- 26 37.1302 ---- 6 50.1327 ---- 20 61.50536 ---- 0 ---- ---- 6 45.3492 ---- 8 55.7737

2 1 16.8 0 9 7.5 0 13.5 5.1 0 16.622 16.8 0 9 7.5 0 13.5 5.1 0 16.62

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3 25.0 0 9 7.5 0 13.5 6.0 0 16.624 0.0 0 9 7.8 0 13.5 6.0 0 16.625 3.2 0 9 6.0 0 13.5 0.0 0 16.626 0.0 0 9 0.0 0 13.5 6.6 0 16.622 16.8 84.0 153.5957 ---- ---- ---- 5.1 45.9 275.34843 25.0 84.0 153.5957 5.1 45.9 275.34844 0.0 92.2 153.9696 ---- ---- ---- 6.0 45.9 275.99875 3.2 75.4 153.4087 ---- ---- ---- 6.0 46.8 274.75586 0.0 61.8 153.0789 ---- ---- ---- 0 47.7 273.92607 0.0 45.0 151.7191 ---- ---- ---- 6.6 42.6 272.80708 ---- 28.2 151.6136 ---- ---- ---- ---- 44.1 272.2382

3 9 ---- 3.2 150.5358 ---- ---- ---- ---- 39.0 271.557010 ---- 3.2 149.8490 ---- ---- ---- ---- 33.9 271.4911

11---- ---- ---- ---- ---- ---- ---- 28.8 271.409812 ---- ---- ---- ---- ---- ---- ---- 23.7 271.374313 ---- ---- ---- ---- ---- ---- ---- 18.6 271.348714 ---- ---- ---- ---- ---- ---- ---- 12.6 271.329515 ---- ---- ---- ---- ---- ---- ---- 6.6 271.320116 ---- ---- ---- ---- ---- ---- ---- 6.6 271.3201

3---- ---- ---- 7.5 52.5 222.4296 ---- ---- ----

4---- ---- ---- 7.8 52.5 222.4296 ---- ---- ----

4 5---- ---- ---- 6.0 52.8 222.4572 ---- ---- ----

6---- ---- ---- 51.3 222.3219 ---- ---- ----

7---- ---- ---- 43.8 221.7324 ---- ---- ----

2 15.7 78.5 162.62 0.0 0.0 4.9 49 291.423 14.0 78.5 162.62 0.0 0.0 4.0 49 291.424 0.0 76.8 162.39 1.2 0.0 6.0 48.1 291.255 22.8 61.1 162.20 0.0 1.2 236.16 0.0 49.2 290.496 0.0 68.2 161.56 0.0 1.2 236.25 1.4 44.3 290.627 ---- 52.5 160.54 ---- 1.2 235.44 ---- 40.8 289.638 ---- 36.8 159.50 ---- 1.2 234.80 ---- 35.9 288.85

5 9 ---- 22.8 158.66 ---- 1.2 234.40 ---- 31.0 288.3810 ---- 22.8 158.06 ---- 1.2 234.32 ---- 26.1 288.2911 ---- ---- ---- ---- 1.2 234.07 ---- 21.2 287.9812 ---- ---- ---- ---- 1.2 234.05 ---- 16.3 287.9613 ---- ---- ---- ---- ---- ---- ---- 11.4 287.9414 ---- ---- ---- ---- ---- ---- ---- 7.4 287.9315 ---- ---- ---- ---- ---- ---- ---- 1.4 287.9316 ---- ---- ---- ---- ---- ---- ---- 1.4 287.937 32.5 97.5 81.1648 7.5 22.5 103.9060 10.0 40.0 127.00948 39.0 97.5 102.9820 7.5 22.5 132.7971 10.0 40.0 162.41849 0.0 104.0 109.2555 7.5 22.5 141.0373 10.0 40.0 172.511910 26.0 71.5 84.5059 7.5 22.5 108.3903 10.0 40.0 132.509511 ---- 65.0 81.1468 9.0 22.5 103.9060 10.0 40.0 127.0094

6 12 ---- 26.0 74.5982 6.0 31.5 92.7180 10.0 40.0 113.021813 ---- 26.0 82.7623 0.0 30.0 96.7801 10.0 50.0 117.120114 ---- ---- ---- 0.0 30.0 86.4003 12.0 50.0 104.159515 ---- ---- ---- 0.0 6.0 75.9262 0.0 52.0 92.317216 ---- ---- ---- ---- ---- ---- 8.0 32.0 89.238017 ---- ---- ---- ---- ---- ---- ---- 30.0 89.1416

----:link not available for cars

TABLE 5 ACTUAL ROUTE TRAVEL TIME DURING EVERY INTERVAL FOR EACH CLASS USER OF THE TEST NETWORK

Route Class 1 Class 2 Class 31 2 3 1 2 3 1 2 3

Route flow 16.8 25.0 3.2 0.0 0.0 0.0 6.0 6.0 6.61 RoutetRavelutime 309.5653 321.9245 285.2558 ---- ---- ---- 467.2129 462.4242 446.2234travel time

Route flow ---- ---- ---- 7.5 7.8 6.0 ---- ---- ----

2 Route 398.3286 393.6727 388.2987travel timeRoute flow 15.7 14.0 22.8 0.0 1.2 0.0 4.0 6.0 1.4

3 Route 309.5874 321.3427 284.7404 ---- 393.8748 ---- 467.1053 462.3901 447.0254travel time

As can be observed in Table 5, the same class user drawn as shown in Fig. 2. The time-space network essentiallydeparting from the same origin during the same interval also contains two dimensions; the horizontal axis denotes distance,experience approximately the same route travel time. and the vertical axis represents time.The corresponding time-space network of route 1 can be

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1: , [3] S. C. Dafermos, The traffic assignment problem for multiclass-user__.___.__.___._ transportation networks. Transportation Science. 1972,6, 73-87.

2 < X[4] L.M.Wynter, Advances in the theory and application of the multiclass3 9 _ *-* i \ traffic assignment problem. Ph.D. thesis, Ecole National des Ponts et

Chaussees, France. 1995.4 [5] H.K. Lo, A dynamic Traffic Assignment Formulation That

\x xX\* * *________ _ Encapsulates the Cell-Transmission Model. In: Ceder, A., Editor,1999. Transportation and Traffic Theory, Pergamon, Oxford, 327-350.

6 v >.\\ \ \ \ [6] B. Ran, D.E. Boyce, Modeling dynamic transportation networks: anZ.\.\>; x\ \ intelligent transportation system oriented approach. (Second ed.),

7 1996,Springer-Verlag, Berlin.8

__ _ v__ \ \ \ [7] B. Ran, H. K. Lo, and D. E. Boyce, A formulation and solution8 ' . X algorithm for a multi-class dynamic traffic assignment model. In:9 x \ x \\. \ Lesort, J.-B., Editor,1996. Transportation and Traffic Theory,10 _____& uX\ \ Pergamon, Oxford, 195-216.

10 . . \ \ -\%~ ~\ z\[8] M. Carey, Nonconvexity of the dynamic traffic assignment problem,\_ \ \ X, . K\. TransportationResearchB: 1992,26,127-133.

11 - --\̂\- x -- [9] J. Li, X. G. Fu, Z. M. Li, Advanced road traffic management. 2000,12 China Communications Press, China.

*\\ \ - \ sX * \. [10] Huey-kuo chen,Che-fu Hsueh, A model and an algorithm for the13 ... dynamic user-optimal route choice problem, Transportation Research14 _____

s \ s B:1998,32(3),219-234.14

15-00*00_181 \

16

17 N

18

19.

20Fig. 2. Time-space extension network

We can see from Fig. 2 that overtaking occurs amongdifferent class users, and for the same user-class, the FIFOprinciple holds.

VIII CONCLUSIONIn this paper, the multiple user-classes DUO route choice

problem is formulated as a discrete-time link-based VI modelthat deals with the multiple user-classes equilibrium problemsuccessfully when overtaking exists widely. The multiclasslink travel time functions are proposed which play animportant role in the model. The numerical example confirmsthat the equilibrated flow pattern is coincident with themulticlass dynamic user-optimal equilibrium condition.Some issues that need further study are summarized as

follows:1) Further research on realistic multiple user-classes travel

time model is required;2) The signal timing plan for each signalized intersection is

to be incorporated into the multiclass DUO assignmentmodel;

3) The solution algorithm should be studied further, such asthe genetic method which has a better chance to attain theglobal solutions for nonconvex problems.

REFERENCES[1] Michiel C J Bliemer, Piet H L Bovy, Quasi-variational inequality

formulation of the multiclass dynamic traffic assignment problem.Transportation Research: B, 2003, 37(6), 501-519.

[2] S. C. Dafermos, An extended traffic assignment model withapplications to two-way traffic. Transportation Science. 1971, 5,366-3 89.

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