Traffic Equilibria and Its Solution in Congested Road Networks

©

( :up\righl IF ,\( , ( :()lItll,J ill Tr.IIl'i>t1rt;l ti()I\ S\" ll'IlI ' . \ ' iellll ;1. ;\II,tr i;1. jq ;--i li

TRAFFIC EQUILIBRIA AND ITS SOLUTION IN CONGESTED ROAD NETWORKS H. Inouye [)1'/w rl tll t'1I1

11/ L h ·i/ FII ,!.!:iIlI'tTillp: . .\d/fw/

III

FlIgill t'f'rillg.

['lI i1. 'l' ni(y.

(Jh O,\'IIIJl fl

r ' II, lt illlfl ll fl lw. ()IWHllll fI . ) fI /HlI I

Abs tract . The traffic equi1ibria on a network with congested flows is dealt by considering queues at intersection s . The travel time on a link with congested flow is expressed by the sum of the travel time in non-congested flow regime and the imaginary waiting time at the end of the link. Then the traffic equi1ibria are formulated by adding capacity restraints explicitly to the usual equilibrium traffic assignment problem. The Lagrange multiplier as sociated with a capacity restraint is equivalent to the waiting time of the link. The solution of the problem provides equilibrium flows with waiting times on conges ted links. Two solution methods of the problem are proposed. One is the application of Lagrange multiplier method, and the other is the application of interior penalty function method. In view of the computational resu1 ts, the former is accurate. On the contrary, the latter is approximate but more practical for large scale networks. Keywords. Road traffic; graph theory; prediction; non1 inear prograr.llning; traffic control; traffic as s ignment. I NTRODUCTI Ol~ For the planning of a future road network and for the control of road traffic flows, it is indispensable to ana1yze the distribution of traffic flows on a road network. For this purpose, many traffic assignment methods have been developed up to now. The traffic equilibrium method which was first proposed by Wardrop (1952) and was systematized by Beckman, Mcguire and Winsten (1956) and Jtrgensen (1963) is the most basic and s ignificant method. Concerning to the fixed demand equilibrium problem, namely, the equilibrium traffic assignment problem, many researchers (Daffermos,1971; Leb1anc, Mor10k and Dierska11a, 1974; Leventha1, Nemhauser and Trotter, 1973; Nguyen, 1974; Ruiter, 1974; Sasaki and Inouye, 1974; and others) developed various kinds of solution. In these methods, the solution using Frank-Holfe algorithm developed by Leb1anc, Mor10k and Dierska11a (1974) i s understood to be the most practical. This solution has been improved by F10rian (1977) and others. In the latest researches (Daffermos, 1980; Smith,1979; and others ), the asymmetric equilibrium traffic assignment problems are treated.

sis may be extended to the network equilibrium problems. In a street network, the intersection capacity on a link i s less than the roadway capacity. So that, inter sections become to bottleneck and queues with the head at intersection tend to generate. In this paper, we show how to deal with the traffic equi1ibria in a road network with congested flows. He decompose the travel ti me on a congested link into travel time in non-congested regime and delay at the end of the link. The traffic equi1ibria in these situations can be formulated by adding capacity restraints explicitly to the usual equilibrium traffic assignment problem. The solution of this problem provides not only equilibrium flows but also the length or delays of traffic congestion. We propose two solution methods of this problem. One is the application of Lagrange multiplier method. The other is the application of interior penalty function method. The characteri s tics of these solution methods are investigated through computational examples.

By the way, it is demanded in practical traffic problems to estimate the length or waiting time of traffic congestions. but in usual traffic assignment methods, it is incapable. Because it is assumed in usual methods that the flow is uniform on a link and that the link travel time is a monotone increasing function of flow. In reality, non-congested flow and congested flow coexist in a saturated link. Besides , the travel time is monotone increasing in non-congested regime, but it is monotone decreasing in congested regime. Recently, Okutani (1984) proposed an equilibrium traffic assignment model on a congested network with two valued travel time functions. But in this r,10de1, the definition of traffic equi1ibria is different from the original one by \Jardrop (1952). Besides the equilibrium state is not unique, so that it is difficult to request the most likely solution. ~Iewell (1977) once ana1yzed the effect of queues generated at a bottleneck of a freeway on the route selection between freeway and street. This ana1y-

TRAVEL TIt~E ON A CONGESTED LI i~K Now, a road network is given as a set of nodes and links. Traffic demands between each centroids are also given. To begin with, let us assume the traffic flow is approximately stationary. Fig. 1 shows an arbitrary 1ink ij with flow q. -- q

conge s ted fl ow

non- conge s ted flow

I. Fig. 1.

267

I.

:1

A Link with Congested flow

H.

268

Let qmax be the roadway capacity of the link and C be the intersection capacity of the link. ~lormally the intersection capacity is less than the roadway capacity. So that if the flow is temporarily larger than C, they are stored in the link and the traffic congestion generates from the end of the link. At an equilibrium state, the length of traffic congestion will be steady. Let L' be the length of traffic congestion, and L be the length of the link. As is illustrated in Fig. 2, we denote the speed-flow relationship of the link by V1(q) in non-congested regime and V2(q) in congested regime.

Inolly(~

The function f(q) is monotone increasing and from l ' ~ L, the waiting time w should be H V1 (C)-V2(C) } (4) w ~ V1(C),V 2(C) The waiting time w means a traffic congestion with the length of

L'

wV 1(C)V 2(C) ( 5) V (C)-V (C) 2 1 Fig. 3 shows the relationship between travel time and flow. If the flow is in non-congested regime, (t,q) is on POP]' and in congested regime (t,q) is on P Pr =

1

v

TRAFFIC EQUILIBRIA non-congested regime

Now, we formulate the traffic equi1ibria on a network with congested flows on the basis of the hypothesis that any users select the minimum time route . First, we consider a simple case.

--- -------- --',- --, --rt I I

I

conges ted . reglme , in k '

q

--o

Fig. 2 Speed-flow Relationship The function V1(q) is monotone decreasing, and V2(q) is monotone increasing. Then the travel time on the link is given by t

L- L'

L'

= Vi1CT + V2(C) =

L

L'

V]TC) + { V2(CT -

Fig . 4. L'

Vi1CT }

(1 )

The first term of the right side of Eq. (1) is equivalent to the ordinary travel time in non-congested regime. Therefore, the second term represents the delay due to traffic congestion. We deal the latter as the imaginary waiting time at the end of the link. Then the travel time t on any link can be expressed as the sum of the travel time f(q) in non-congested regime and the waiting time w at the end of link. That is (2) t = f(q) + w where, w= 0 (O 0

A Simple Network

Fig. 4 illustrates a simple network with two links 1 and 2 between node i and j. Let 0 be the traffic demand from i to j, q, and q2 be the flow, f 1 (q) and f 2(q) be the link travel time function, and Cl and C2 be the intersection capacity, where suffix 1 and 2 indicate link number. We assume link 1 is saturated and traffic congestion with waiting time w generates. Then, (6)

is true from the hypothesis stated previously. With respect to flows, the equation q1 + q2 = 0 (7)

q,

Cl

(8)

should hold. The equil ibrium flows q1' q2 and waiting time w can be decided from above equations. Tha t is

P2

q1

C,

q2

0 - Cl

w = f 2(0- Cl) - f 1 (C 1) Provided that this solution is true when 0
Fig. 3.

Travel time-flow Relationship

Next we consider the general case. Let 0i k be the traffic demand from i to k, xij k be the flow on 1ink ij with destination k, Xij be the total flow

2(,9

Cungestcd Ru,ld :\c{\mrks

A. k

on link ij, wij be the waiting time on link ij, and Cijbe the intersection capacity of link ij respectively. Then the traffic equi1ibria on a congested network may be expressed as follows. 1: xij k , k k 1: x .. 1:1 xl'1 k j 1J

x1J.. =

lIij

(9) 11 i,k

1

x .. k > 0 ,

11 ij, k

(12)

=

, f ij (X ij ) + wij = t.k 1

(x i /

>0)

(x .. k =0 )

•• > t.k f ij (X ij ) + W1J = 1 W

ij = 0 ,

(Xij < Cij )

wij ,;: 0 ,

(X ij = Cij )

1J

(13)

} 11 i j

(14)

By the way, these equilibrium conditions are equivalent to the following mathematical programming problem (P). (P)

1: fi ij o

j

f .. (X)dX

( 15)

lIij

(16)

1J

subject to 1: x .. k , k 1J 1: x .. k 1: xl' k j 1J 1 1 Xij

X.. < C.. , 1J =

xi /

,;: 0,

D.k

11 i ,k

1

(17)

lIij

(18)

lIij,k

(19)

1J

Because, the Kuhn-Tucker conditions for problem (P) are equivalent to conditions (9)-(14). Let denote Lagrange multipliers associated with (17) and (18) by Ai k and ~ ij respectively, then the Lagrangian function of problem (P) may be formulated as 1: x .. k ~(x,l,p) = 1: f k 1J f . . (X)dX ij 0 1J +

H Ai k (3

xi /

-

Yx1i k -

Di k )

+1:p .. (1:x .. k -C .. ) ij 1J k 1J 1J

(20)

The necessary and sufficient conditions for x* to be the optimal solution of problem (P) are given from Kuhn-Tucker theorem as follows. 1: x.. k* - 1: Xl' k* -- 0 . k ,

j

1

1J

x .. k* > 0 , 1J

=

=0

,

(25)

1

1

11 i ,k

(26)

lIij

These are equivalent to (9)-(14), if Xij *, Xijk*, Ai k and ~ ij are replaced by X , xij k , tik and wij 1J

} 11 i ,k

Eq. (9) and (10) represent the continuity and the demand and supply of flows. Eq. (11) shows the capacity restraint on each links. Eq. (12) shows non-negativity of flows. Eq. (13) imp1 ies the hypothesis that any users select the minimum time route. Eq. (14) means that the waiting time is produced when the flow is saturated.

min F(x)

~ ij

vJhere, (11 )

1J

1J

11 i ,k (24)

( 10)

11 ij

1J

(x .. k*= 0)

~ i j ,;: 0 ,

Ok

X.. < C.. , 1J =

(Xi/* > 0) }

1

(21 )

11 i j

(22)

11 ij ,k

(23)

respectively. Whence, it i s concluded that the optimal solution of (P) yields the traffic equi1ibria on a congested network, and the Lagrange multipliers associated with capacity restraints are equivalent to the waiting times on each link. Then we can obtain the equilibrium flows and waiting times on congested links by solving problem (P) with respect to flows and Lagrange multipliers. In usual traffic equilibrium methods, capacity restraints (18) are not dea1ed explicitly. For example, Daganzo (1977a, 1977b) showed solutions with capacity restraint algorithm, in which the travel time function is suggested such that 1im f(X) = (27) X-+C This character acts as capacity restraint, but we can not get the length or waiting time of traffic congestions. 00

In problem (P), with respect to total link flows Xij (lIij), the feasible region is clearly convex and the objective function is strictly convex. So that, the equilibrium solution is unique. With res~ect to link flows with a certain destination xij (lIij,k), the optimal solution is not unique. But the problem has a convexity, so we can use various mathematical programming methods to request the optimal solution numerically. By the way, condition (4) which represents the suppremum of waiting time is not considered in problem (P). If a link is saturated in whole length and the traffic congestion extend to upper links, then condition (4) is violated. In this case, as Newe11 (1977) showed, the equilibrium state is not unique. It seems that other conditions influence to the extension of queues. So, in this study we do not object such a case that condition (4) is violated. SOLUTION BY MULTIPLIER METHOD It is a particular feature of problem (P) that we can estimate the equilibrium flows on a network with the waiting time or length of traffic congestions. Therefore, it is desirable to use such a solution method that can provide the optimal solution with Lagrange multipliers. The Lagrange multiplier method for non-linear programming with constraints is one of them. In this method, the augmented Lagrangian function is formulated by adding penalty functions associated with constraint to Lagrangian function. This function is matua11y optimized with respect to variables and Lagrange multipliers. When penalty parameters are magnified , the exploratory point approaches gradually to the optimal point. Hot only equality constraints but also inequality constraints can be dea1ed in this method. Now, multiplier method is applied. The constraints (17)-(19) are transformed into penalty functions. Then we get following augmented Lagrangian function.

H,

270

Lr s t(X;A,IJ,V) = l: , , ij

J

IIlOll\('

The above function can be rewrite as follows.

l:x, ,k k 1J f, ,(X)dX

y,,'(g,,(X)) }dX 1J 1J 1J f 01J {f,,(X)+tn -- l: f Xij U,,(X)+t n -e 1_X 1• dX (33)

+ l: l: A' k h ' k(X) + ~ r L L:{ h, k(x )} 2 ik 1 ik 1 1

ij

+ l: t-Cmax {O, IJ ' ,+sg , ,( x ) } 2_ ~ , ,2 J ij S 1J 1J 1J + L l:

ij k

1 C max { 0, v" k - tx" k ,, 2 - { v' , k } 2J 2t

1J

1J

1J

X, '

F(x,t n ) = l: ij

1J

(28)

0

ij

1J

Fi g. 5 illustrates the travel time function and derivative of penalty functions.

\~here,

vi / (\lij,k) are Lagrange multipliers associated with Eq, (19), r,s and t are the penalty parameters, and - l: x"k+ l:X lk+D, k j 1J 1 1 1

h/(X)

\I

i ,k

t 1, ' t Z.' \ .

(29)

f (X)

gij (x) = ~ xi/-C ij

11 i j

(30)

The solution algorithm i s as follows.

Step

Set initial values of X,A,IJ,V,r,s and t. set n=O. Minimize Lr,s,t(X;A,IJ,V) with respect to x. Write x(n) for the optima l solution.

Step 2

If

Step 0

h,k(x(n»= 0,

lIi,k

gij(x(n» ~ O,

lIij

1

Step 3

Step 4

e Fig. 5.

x' ,k(n »O, lIij,k 1J ~ then stop. Refresh Lagrange multipliers by A, k(n+ 1) =A ' k(n) +r (n) h , k(x (n) ), 11' k 1 1 1 1, IJ" (n+l)=IJ" (n)+s(n)g" (x(n», 11 ij 1J 1J 1J v' ,k (n+ 1 )=v, k (n) _t (n) x, ,k (n), 11 i j , k 1J 1J 1J Magnify penalty parameters. r(n+l )=ar(n) s ( n+ 1 )=Ss (n ) t (n+ 1) =yt (n ) Set n=n+l and return to Step 1.

In this algorithm, r(O), s(O) and t(O) are 1 to 100 and a, S and y are 2 to 10. Lagrange multipl ier s Akk (Ilk) are put zeros. In the minimizing step of augmented Lagrangian function, various kinds of optimization methods without constraints can be used. For example, Newton's method is powerfull for a small network. When the network is large, variable metric method will be efficient. SOLUTION BY PENALTY FUNCTI ON METHOD In this solution, only capacity restraints (18) are transformed into inter10r penalty function, and the transformed problem is solved by usual methods. As a interior penalty function, we use ljJij (g) = - log (-g/C ij ) (31) n Let { t } , n= 0,1,2,··· denote penalty parameters, then we obtain following augmented objectitive function F(x, t n ) = F(x) + t n L ljJ, ,( g, ,(x) ) ij 1J 1J X, ,

= l: 1J fij(X)dX _ t n l: {log(e, ,- X, ,)-loge , , } ij,r 0 ij 1J 1J 1J (32)

x

Travel Time Function and Derivative of Penalty functions.

Then, the following transformed problems (pn) are formulated. (pn) subject to k

lIij

Xij = ~ Xij , l: Xi j

k

- l: Xl i 1

k

= Di

k

11 i,k

lIij,k The ser ies of the optimal so lution of these problems , { x(n) }, converges to the optimal solution of the original problem (P), when t n approaches O. to/hi 1e, IJ iJ, = tn i/liJ· ' (gl'J' (x»

= tn. - C \ ' 11 ij (34) ij - ij converges to the Lagrange multipliers associated with capacity restraints (18). Accordingly, they are equivalent to the waiting ti me of each links. On the other hand, (pn) i s a usual equilibrium traffic assig nment problem with the cost function fij(X)+tn ljJ i/(gij(X ». Therefore, we can use existing many met hod s to require equilibrium flows. The solution method using Frank-Wo1fe algorithm developed by Leb1anc, Morlok and Dierskal1a (1973) is one of the most efficient and practical method. But some modification should be necessary to apply this algorithm, because the exploratory point may exceed the interior feasible region. So that, the step size in one dimensional search is restrained such that the total link flow does not exceed the capacity in every 1inks. The solution algorithm is as follows. Algorithm (A) Step 0 Select a set of interior feasible solution xij k( 1 ) , (11 ij ,k). Set n=1. Step Assign every traffic demands to their minimum ti me routes under link times

271

Congested Road :\ ct ",orb

f i j ( Xi j (n ) ) +t nljJ ij (g i . (x ( n) ) ) , (\I i j ) . Write zi/' (\fij ,k) for assigned 1ink flows. I

Step 2

r~inimize F(x(n)+ad, t n) with respect to Cl , in a range of

0" Cl < mi n (1, mi n Dij >O where, k k k(n) d ij = Zij -x ij , DiJ = ~di/' Step 3

Step 4 Step 5

\lij,k

\lij

Write Cl* for the optimal value. Refresh flows by x .. k(n) :=x .. k(n)+Cl*d .. k(n), \lij,k lJ 1J lJ _ (n) * Xij (n)..-X +a D \lij ij ij , Repeat Step 1 to Step 3 given times. If flows converge, then stop. Otherwi se, set t n+1 =yt n and n=n+1, then return to Step 1.

In the above algorithm, y=O .l is suitable. While the penalty part of the transformed objective function t n L 1jJ • • (g .. (x(n))) (35) i j 1 J lJ n converges to 0 when penalty parameter t approaches O. Whence, we may terminate the computation when P(x,t n ) is full y reduced in comparison with the value of the objective function. An interior feasible solution should be given in this algorithm. That is a rather difficult problem if demands are close to the network capacity. But we can get it by use of interior penalty function method itself. That is an algorithm to surround flows gradually within given capacities. When the boundary of the feasible region approaches to an exploratory point, the value of the objective function increases sharply. So that, the exploratory point moves leaving the boundary by an minimizing step of the objective function. Then we may enclose flows in a more small region. The algorithm is as follows. Algorithm (AD) Step 0 Set temporary capacities infynity, namely Cij * = 00 , \lij Step 1

Practice Step 1 to Step 3 of algorithm (A) , provided that F(x,t) = .L. lJ

Step 2 Step 3

X. .

1

- } dX f 0lJ {f lJ.. (X) + t· Ci/-X

If X. < C.. , then se t C.. *=C .. , Otherwise lJ = lJ lJ lJ set Ci{=Xi/t X (\liJ). If C =C ij for any ij, then stop. iJ Otherwise go to Step 1.

The value of ~ X is about 0.1, and as a value of t the initial value of penalty parameter is su itable. Cor~PUTATI

ONAl

EXAr~PlES

let us take some computational examples, to see the characteristics of solution algorithms. Fig. 6 illustrates a road network used in the computation. Ilode 1 to 4 are centroids and node 5 to 8 are intermediate nodes. The travel time function and capacity of every links are showed in TABLE 1. TABLE 2

Fig. 6.

Network Used in the Computation.

TABLE 1 Travel Time Function and Capacities

f(X ) 1 ink No. 5x10- 6X2+5 1,2,4,5 5xlO- 6X2+5 3,6 25xlO- 6X2+20 7,8 15xlO- 6X2+10 9,1 Ci , 11 ,1 2 13,14,15,16,17 , lOxlO- 6X2+7 19, 20,22,23,24 ' 10x lO- 6X2+7 18,21

C 1000 800 600 600 600 500

TABLE 2 Traffic Demand Matrix

~

1

2

1 2 3 4

-

300

300 900 400

-

3 900 500

500 600

-

4 400 600 400

400

-

indicates the traffic demand matrix. Fir st, we show the computational result by multiplier method. The initial value of flows were created by assigning traffic demands adequately over the network and those of Lagrange multipliers were set zeros. In the optimization of augmented lagrangian function, Ilewton's method was used. The computational result is showed in TABLE 3, provided that r(1)=s(1)=t(1)=10 2 . By three times iteration, flows and multipliers fully converged to the optimal so lution. The result implies that the traffic congestion arises in six links. Especially, in node 7 capacities are somewhat poor, so that long congestions arise. In view of the computational result, this method gives a very accurate solution, but there exists a problem that lagrange multipliers diverge if the minimization of augmented lagrangian function is not accurate. In actual traffic assignment problems, the network i s very big. So that, the accurate minimization of augmented lagrangian function i s a hard question. Whence, the application of this me thod to an actual large scale problem may be rather difficult. Next, we show the computational result by interior penalty function method. An interior feasible solution was obtained by 12 times repetition using Algorithm (AD) , provided that t=103. Then equilibrium flows and waiting times were requested by Algorithm (A). TABLE 4 shows the computational

H. I nOll\ 'c

272

result, povided that the number of repetitions in Frank-Wolfe algorithm is 1000. In this computation , flows converged by three time s iteration. But obtained flows and waiting times are somewhat different from those of the optimal solution. Even if more iterations are practiced, the solution is scarcely improved. This is because Frank-Wolfe algorithm is slow in convergence, and it provides merely an approximate solution under limited times repetition. But this solution method is practical, because the computer program does not require a great memory capacity for a large scale network. CONCLUSION In this study, we s howed how to deal with the traffic equilibria in a congested network on the extension of usual traffic equilibrium methods. It was enabled by decomposing link travel time into travel time in non-conge sted regime and waiting time. The so lution of the problem provides not only equilibrium flows but also waiting times or length s of traffic congestion on a congested network. These informations will be usefull for traffic control such as inflow control of expressway. But the problem how to deal with a case that traffic congestions expand to upper links remains. That is the subject for a future study. REFERENCES Beckman, M.J., C.B . McGuire, and C.B. Winsten(1956) . Studies in the Economic s of Transportation, Yale University Press. Dafermos, S.C. (1971). An extended traffic assign ment model with application to two-way traffic, Transpn. Science, Vol.5, 4, 366-389. Dafermos, S.C. (1980). Traffic equilibrium and variational inequal iti es, Transpn. Science, Vol.14 , 42-54. Daganzo, C.F. (1977a). On the traffic assignment problem with flow dependent costs-I, Transpn. Res., Vol.ll, 433-437.

TABLE 3 Computational Result by initial value link r~o

. 1 2 3 4

X

930. 0 930.0 790. 0 790. 0 790. 0 6 790.0 7 330 .0 8 330. 0 9 580.0 10 580 .0 11 400. 0 12 I 400. 0 13 ! 540 .0 14 540.0 15 590.0 16 590. 0 17 490. 0 18 490 .0 19 140 .0 20 140 . 0 21 490.0 22 490. 0 23 590.0 24 590 . 0 0.0 Ilh ( x )~ 1.1 219 F{x) xl05 ~

n=l (r=s=t=1 02) X

' .'

801. 3 0.0 0.0 800 . 3 800 .2 19.37 799 . 7 0.0 0.0 800 .1 800 .2 19.33 302 . 7 0.0 303.9 0.0 600 .1 6. 29 600. 1 6.26 492.4 0.0 491 .4 0.0 492.5 0. 0 492 .8 0.0 589 .2 0.0 0.0 589 . 1 0. 0 500 . 4 500 . 2 23 .61 0 .1 0.0 0.0 - 0.3 500 .2 23 .57 500 . 2 0. 0 588 .7 D.G 0. 0 589.5 2.033x l 0- l 1. 09 51xl05

~ultiplier

Method

n=2 (r=s=t= 103 ) X

800 . 0 0. 0 300 . 0 0.0 19.85 800.0 800.0 0 .0 300 . 0 0. 0 81JO . 0 19. 85 0. 0 1I 300.0 300 .0 0.0 II 600.0 6.80 .0 6 .80 ! 600 500.0 0 .0 SOO.O 0.0 500.0 0 .0 I I 500 .0 0.0 I ~ 600.0 0.0 i I 600.0 0 .0 ! 500. 0 0.0 I 24 .40 500.0 0.0 0.0 0.0 0 .0 24 . 40 500.0 500 . 0 0.0 600.0 0 .0 600 .0 0.0 2.776x l0- 4 l.l 016xl0 5

I

Daganzo, C.F. (1977b). On the traffic assignment problem with flow dependent cost-IT, Transpn. Res., Vol. 11, 439-441. FlorJan, 1'1. (1977). An improved linear approximation algorithm for the network equilibrium ( packet switching) problem, Proc. of the 1977 I . E.E.E. Conf. on Decision &Control, 812-818. J~rgensen, N.O. (1963). Some aspects of the urban traffic assignment problem, I.T.T . E. Graduate Report, University of California, Berkley. Leblanc, L.J., E.K. Morlok and W.P. Dierska11a (1974). An accurate and efficient approach to equilibrium traffic assignment on congested networks, Transpn. Res. Record, i2l, 12-23. Leventhal, T., G. Nemhauser, and L. Trotter Jr. (1973). A column generation algorithm for optimal traffic assignment, Transpn. Sci. ,Vol.7 ,2, 168-176. Newell, G.F. (1977). The effect of queues on the traffic assignment to freeways, Proc. of the 7th Int. Symp. on Transportation and Traffic Theory, 311-340. Nguyen, S. (1974). An algorithm for the traffic assignment problem, Transpn. Sci., Vol .8 , 3, 203-216. Okutani, 1. (1984). Equilibrium flows in a network with congested links, Proc. of the 9th Int . Symp. on Transportation and Traffic Theory, 253-271. Ruiter, E.R. (1974) . Implementation of operational network equilibrium procedures, Transpn. Res. Record, 491, 40-51. Sasa~ ana-H. Inouye (1974). Traffic Assignment by analogy to electric circuit, Proc. of the 6th Int . Symp. on Transportation and Traffic Theory, 495-518. Smith, M.J. (1979). Existence, uniqueness and stabil ity of traffic equil ibria, Transpn . Res., 13B, 295 -304. Wardrop, J.G. (1952). Some theoretical aspects of road traffic research, Proc. Inst. Civil Engrs. , 1, 325-378.

TABLE 4 Computational Result by Penalty Function I·lethod i nit i a 1 n=l value 3) (t= 10 (t=102) 1ink iJo. X X ~ 1 869 .6 809 .6 0. 53 2 847 .4 804.7 0.51 3 785 . 8 795.9 24.23 4 763.6 791. 0 0.48 5 791.4 796.8 0.49 6 788 .7 795.9 24.54 7 428.4 362.6 0.42 8 442.9 365.0 0.43 9 591.2 S9 1 . 1 11 .19 10 589 . 2 59 1.1 11. 20 11 500.8 511 .1 1. 23 12 508.4 513.6 1. 16 13 584.6 524. 8 1. 33 14 592.2 527.3 1. 38 15 491.2 545.3 1. 83 16 495 .9 , 546 .1 1. 86 17 505.7 i 498.9 0.99 18 489.1 I 496.5 28.80 19 83 .8 13.7 0 .1 7 20 83.8 13.7 0. 17 21 497 . 5 496.7 30.34 22 500.4 498.4 0.98 23 491.2 545.3 1. 83 24 495.9 546.2 1.86 15 1.1145xl 05 37 F{x ) ( 5.8517xl0 3 P{x) 5:~U1

n=2 (t= l O) X

n=3 (t= 1 ) lJ

X

~

809.7 0.05 809.8 0.01 803 .9 0.05 803 .9 0.01 799.5 18 .93 799.9 17. 83 793.7 0.05 794.1 0.01 800.2 0.05 800 .9 0.01 799.5 19 . 46 799.9 17. 83 353 . 9 0.04 353. 8 0.00 355.6 0 .04 355. 1 0.00 598.4 6.43 599.8 4.98 598 . 5 6.64 599.8 5.42 49 8. 5 0.10 496.4 0.01 502.6 0.10 501.0 0.01 508.7 0.11 506.3 0.01 512.8 0 .12 510 .8 0.01 547 .5 0. 19 546.7 0.02 548.1 0.02 0. 191547.6 501. 2 0.10 . 501.2 0.01 499.5 21. 31 I 500.0 22.30 10.2 0 .02 9.9 0.00 10.2 0 .02 9.9 0.00 499.5 22. 11 500.0 22 .13 503.0 0.10 503.6 0.01 547.5 0. 19 546.7 0.02 0. 19 ' 547.6 548.1 0.02 1.1 089x l0 5 1.1083x l0 5 6.969 1xl0 2 8.2824xlO