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A path-based traffic assignment algorithm based on the TRANSYT traffic model
Transportation Research Part B 35 (2001) 163±181
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A path-based trac assignment algorithm based on the TRANSYT trac model S.C. Wong a,*, Chao Yang b, Hong K. Lo c a
b
Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, People's Republic of China Department of Road and Trac Engineering, Tongji University, Shanghai 200092, People's Republic of China c Department of Civil Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, People's Republic of China Received 8 October 1998; received in revised form 5 July 1999; accepted 14 July 1999
Abstract This paper presents a path-based trac assignment formulation and its solution algorithm for solving an asymmetric trac assignment problem based on the TRANSYT trac model, a well-known procedure to determine the queues and delays in a signal-controlled network with explicit considerations of the signal coordination eects and platoon dispersion on the streets. The solution algorithm employs a Frank±Wolfe method to identify the descent direction at each iteration, which requires the input of the derivatives information. A post-simulation sensitivity analysis is developed to estimate the derivatives in the TRANSYT trac model. Good agreement of results with the values determined by numerical dierentiation is obtained. Using these derivatives information, the Frank±Wolfe method shows a good convergence behavior to the equilibrium solution. Comparison with other methods is also discussed in a numerical example to demonstrate the eectiveness of the proposed methodology. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Trac signals have been used for many years to resolve con¯icting trac movements, both vehicular and pedestrian, especially at intersections. In urban streets where the spacing between adjacent intersections is comparatively short, better operational performance of the signal-controlled intersections can often be obtained by taking into account the interaction between adjacent intersections in the determination of signal settings. Such co-ordination among intersections over *
Corresponding author. Tel.: +852-2859-2668; fax: +852-2559-5337. E-mail address: [email protected] (S.C. Wong).
0191-2615/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 1 - 2 6 1 5 ( 9 9 ) 0 0 0 4 4 - 2
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an area is called area trac control. To model the trac behavior in a signal-controlled network taking into account the signal co-ordination eects and platoon dispersion, the TRANSYT trac model (Robertson, 1969; Vincent et al., 1980) has been well accepted as a useful tool in evaluating the queues and delays on the links in the network, and is now widely used worldwide. Recently, studies has been undertaken to improve the modeling capabilities of the TRANSYT trac model by using a more ¯exible speci®cation of signal timings, the group-based method (Wong, 1995, 1996, 1997), for the case that the link ¯ows are ®xed. However, it has been observed that the equilibrium ¯ow pattern of a network is strongly related to the trac signal settings. By changing the control strategies, the trac will arrange itself in a user optimal manner (Wardrop, 1952), for which users traveling from any origin to any destination react to the new control strategy by choosing paths such that their individual cost is minimized. This redistribution eect on equilibrium trac ¯ow pattern will aect the performance of the network. Several authors have pointed out the importance of considering signal control in conjunction with trac equilibrium and studied the general characteristics of this non-convex optimization problem. Among them, Allsop (1974) was one of the ®rst who suggested that the signal control can be explored to aect the distribution and assignment of trac on an equilibrium network, and provided a rigorous mathematical framework for the problem. This problem has been known as equilibrium trac signal settings in the literature. In this paper, attempt is made to determine the equilibrium trac pattern based on the TRANSYT trac model. The diculties associated with detailed junction modeling in road trac assignment have been documented in Heydecker (1983), in which it was shown that for most signal control policies the conditions for good convergence behavior in trac assignment are violated. Despite the fact that it is assumed in this paper that the signal settings are ®xed, the problem is still dicult to solve as the cost functions, in contrast to the conventional network assignment problems with separable cost functions (i.e. the travel time on a link is unaected by the ¯ows on all the other links in the network), are generally asymmetric and non-convex. For this class of asymmetric trac assignment problems with explicit treatment of the eect of intersections, Meneguzzer (1995) and Berka and Boyce (1996) have developed a practical diagonalization algorithm and demonstrated its eectiveness in solving large-scale real problems. However, the signal co-ordination eects were not modeled in their papers. When signal co-ordination is considered in the asymmetric trac assignment problems, Hall et al. (1980) developed a simulation-assignment model, SATURN, for the evaluation of trac management schemes. The simulation module was based on a TRANSYT like trac model employing the concept of cyclic ¯ow pro®les and the assignment module was basically a conventional link-based assignment method (LeBlanc et al., 1975). The problem was also solved by a diagonalization algorithm in which at each iteration the set of separable cost functions, approximated by polynomial curves whose parameters were estimated from the last simulation results, was generated for use in the assignment module. It was however mentioned in Hall et al. (1980) that (even though the assignment module guaranteed to converge in itself) the simulation/ assignment loop does not necessarily converge to the equilibrium solution where neither ¯ows nor delays change signi®cantly. Oscillation of results have been found in certain networks, especially when the interaction of trac at adjacent signal-controlled intersections is signi®cant. The major advantage of the SATURN model is that the very dicult problem is, at each simulation/assignment iteration, transformed into a more manageable link-based trac
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assignment problem with separable cost functions that can be solved by the ecient algorithm (LeBlanc et al., 1975). Over the last twenty years, the SATURN model has been developed into a widely used commercial package, in which a number of extensions including the stochastic assignment, multi-class assignment and various signal control policies, etc. were also incorporated. Notwithstanding the continuous eort on the advancement of the SATURN package, there is not much research on the improvement of solution algorithm for solving the resulting dicult asymmetric trac assignment problem, albeit certain shortcomings are observed as mentioned in the previous paragraph. The objective of this paper is to investigate the application of a pathbased algorithm to solve this strongly asymmetric trac assignment problem based on the TRANSYT trac model, given the fact that due to the constantly increasing power of computers which helps in relaxing the computational constraints, the path-based assignment algorithms have received much attention recently (Jayakrishnan et al., 1994; Cascetta et al., 1997; Bell et al., 1997; Lo, 1998). Another motivation on the development of this path-based assignment module is to enable a sensitivity analysis on the network performance index with respect to the signal timing variables to be established in further studies; that can be used in the optimization of signal settings taking into account of the signal co-ordination and re-routing eects of trac in the network. Such sensitivity analysis, however, demands a relatively harsh requirement on the convergence criteria of the assignment algorithm, that is to adhere tightly to the user-optimal conditions. Existing approaches to this asymmetric problem, while producing reasonable results for most planning and engineering purposes, are not very satisfactory for the present requirements. Patriksson (1994) has provided a very comprehensive discussion on the various approaches to asymmetric trac assignment problems. For practical applications, two approaches are generally adopted: the diagonalization algorithms and descent algorithms. The proposed path-based method belongs to the class of descent algorithms. For comparison purposes, two other algorithms for asymmetric assignment are employed. The ®rst is the diagonalization algorithm from SATURN as it also uses the TRANSYT trac model for evaluation of the queues and delays. The other is the method of successive averages (MSA), a descent algorithm developed in this study also based on the TRANSYT trac model (She, 1985). In Section 2, a path-based trac assignment formulation due to Lo (1998) is given, and then a Frank±Wolfe solution algorithm for this formulation is derived in Section 3. For the solution algorithm, it requires the input of derivatives information of the TRANSYT trac model brie¯y described in Section 4. A post-simulation sensitivity analysis for these derivatives information is discussed in Section 5. A numerical example comparing three solution algorithms, the diagonalization method used in SATURN, the method of successive averages and the Frank±Wolfe algorithm in Section 3, is given in Section 6 to demonstrate the eectiveness of the proposed methodology. 2. A path-based trac assignment formulation Consider a road network consisting of a set of links, A, and a set of nodes, N. Let K N N be the set of OD pairs. Denote qk as the demand between OD pair k, k 2 K. These demands, when assigned onto the network, give rise to link ¯ows va ; a 2 A. The travel time on a link depends on
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the trac pattern in the network and is stated as ta v, where v va ; a 2 A is the set of link ¯ows in the network. For a general case, the travel time functions may be asymmetric, i.e. ota v ota0 v 6 ova0 ova
1
for some pairs of links. It is generally dicult to solve this class of S assignment problems (Patriksson, 1994). Let Pk be the set of paths between OD pair k, and P k2K Pk be the set of all paths in the network. Denote fkp as the path ¯ow on path p between OD pair k. We have the ¯ow conversation equations as X fkp qk 8k 2 K: 2 p2Pk
The corresponding link ¯ows are given by XX va dkpa fkp 8a 2 A;
3
k2K p2Pk
where
dkpa
1 0
if link a belongs to path p between OD pair k; otherwise;
is the link-path incidence variable. The travel time along a path can be expressed as X hkp dkpa ta 8p 2 Pk ; k 2 K;
4
5
a2A
where hkp is the travel time on path p between OD pair k. Let uk be the least travel time between OD pair k. We have uk min hkp :
6
p2Pk
A network equilibrium that satis®es WardropÕs user optimal principle for path ¯ows is achieved when for each OD pair, the travel times incurred by travelers on all used paths are equal and minimum, whereas the travel times on all unused paths are greater than or equal to that on the used paths. That is if fkp > 0 then hkp uk hkp P uk fkp P 0
8p 2 Pk ; 8p 2 Pk ;
8p 2 Pk ; k 2 K;
k 2 K; k 2 K:
7a 7b 7c
This network equilibrium problem has been formulated as a non-linear complementarity problem (NCP) in the literature: X Xÿ Z f hkp f ÿ uk fkp 0; 8a k2K p2Pk
X p2Pk
fkp ÿ qk 0
8k 2 K;
8b
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hkp f ÿ uk P 0 fkp P 0
8p 2 Pk ; k 2 K;
8p 2 Pk ; k 2 K;
167
8c 8d
where f fkp ; p 2 Pk ; k 2 K and uk is directly determined from Eq. (6). The link-based version of the symmetric NCP can be solved by a number of solution algorithms (Florian and Hearn, 1995). For an asymmetric NCP, it is very dicult to derive an ecient algorithm for the problem. In Lo (1998), a weak formulation to approximate the solution of the NCP was suggested, in which the problem was written as an optimization problem that is more easy to deal with. The term ``weak'' is used to describe LoÕs formulation because the solution obtained is only a numerical approximation to the solution from NCP. The NCP in Eq. (8a) is modi®ed as the following non-linear mathematical program (NMP): X Xÿ hkp f ÿ uk fkp 9a min Z f; u f;u
k2K p2Pk
subject to X fkp ÿ qk 0
8k 2 K;
9b
8p 2 Pk ; k 2 K;
9c
p2Pk
hkp f ÿ uk P 0 uk P 0
8k 2 K;
9d
fkp P 0
8p 2 Pk ; k 2 K;
9e
where u uk ; k 2 K and Z is the objective function of the NMP which is generally non-convex in nature. The non-linear constraints (9c) ensure that the path travel times between OD pair k are greater than or equal to the variables uk which at optimality are equal to the least travel time associated with the OD pair (see Proposition 1 below). The constraints in NMP form a closed feasible region for the determination of optimal path ¯ows. The existence of a solution for the NMP can be easily shown by setting u 0 and assigning all demands onto the respective shortest paths in the empty network. Since the travel times must be positive, the resulting ¯ow pattern satis®es all constraints Eq. (9b)±(9e) and hence is a candidate for the solution of NMP. The NMP is a non-linear mathematical program with non-convex objective function and a set of linear and non-linear constraints. Therefore, the solution is not unique. The major dierences between the NCP in Eq. (8a)±(8d) and the NMP in Eq. (9a)±(9e) are twofold. Firstly, the variables u is considered as endogenous (state) variables in the NCP, based on Eq. (6), but they are treated as exogenous (control) variables in the NMP. However, it is interesting to note that the optimal solution of NMP automatically satis®es Eq. (6). Proposition 1. At the optimal solution of the NMP in Eq. (9a)±(9e), the optimal values, u uk ; k 2 K, are the least travel times between the respective OD pairs.
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Proof. Consider a particular OD pair k. It is clear from Eq. (9c) that uk 6 hkp ; 8p 2 Pk , where hkp is the corresponding travel time on path p at the optimal solution of the NMP. Now assume that uk is not the minimum value of hkp ; 8p 2 Pk , i.e. uk < hkp ; 8p 2 Pk . There exists a positive quantity j minp2Pk hkp ÿ uk , such that the objective value in Eq. (9a) can be further reduced by replacing uk as uk j for the OD pair k, whereas all the constraints in Eq. (9b)±(9e) are still satis®ed. This is a contradiction that uk is already the optimal solution to the NMP. Therefore, uk minp2Pk hkp , and is the least travel time between OD pair k at the optimal solution of NMP. This completes the proof. Another major dierence between the NCP and NMP is that instead of equating the function Z to zero in NCP, the problem becomes of minimizing the objective function Z in the NMP. Proposition 2. If the solution of the NCP exists, this solution is a local minimum of the NMP. Proof. Since the objective value of the NMP is non-negative, i.e. Z P 0, from Eq. (9c) and (9e), and the solution of the NCP satis®es all the constraints in the NMP if hkp P 0 (this is always true as the travel time must be positive), the solution of the NCP must be a local minimum of the NMP. From Proposition 2, the network equilibrium problem is solved when we can ®nd a local minimum of the NMP for which the objective value is suciently close to zero. For asymmetric network equilibrium problems, the objective function Z is generally non-convex. Therefore, multiple solutions exist. However, for this class of problems, even the NCP and other methodologies such as the variational inequality (VI) are of the similar order of diculty in solving the network equilibrium problem. Moreover, computational experience has shown that it is generally not too dicult to obtain a good local minimum of the NMP which is suciently close to the equilibrium solution. 3. Solution algorithm To solve the NMP in Eq. (9b)±(9e), the Frank±Wolfe method is employed. Firstly, the auxiliary solution of the problem is obtained by solving the following linearized subproblem (Vajda, 1961): ! XX X X oh~mn X qk yk 10a h~kp ÿ u~k f~mn xkp ÿ min Z~ x;y ofkp m2K n2Pm k2K p2Pk k2K subject to X xkp qk
8k 2 K;
10b
p2Pk
XX m2K n2Pm
! ! X X oh~kp oh~kp xmn ÿ yk P f~ ÿ h~kp ofmn ofmn mn m2K n2Pm
8p 2 Pk ; k 2 K;
10c
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yk P 0 8k 2 K;
169
10d
10e xkp P 0 8p 2 Pk ; k 2 K; where Z~ is the objective function of the linear program in Eq. (10a)±(10e) which is derived by linearizing the objective function of the original NMP in Eq. (9a). Constraints (10c) are the linearization of the non-linear constraints (9c) in the original problem. In Eq. (10a)±(10e) x xkp ; p 2 Pk ; k 2 K and y yk ; k 2 K are, respectively the auxiliary variables for path ¯ows and least travel times, u~k is the current least travel time, and h~kp and f~kp are respectively the current path travel time and path ¯ow on path p, between OD pair k obtained from the previous iteration, and oh~kp ohkp 11 ofmn ofmn fmn f~mn denotes the derivative of path travel time with respect to path ¯ow evaluated at the current trac pattern obtained from the previous iteration. Note that the Jacobian matrix " # oh~kp M 8p 2 Pk ; k 2 K; n 2 Pm ; m 2 K 12 ofmn is in general asymmetric for the problem concerned in this paper. After solving the linear program, a line search is carried out as follows: u a x; y ÿ ~f; ~ u 13a min Z ~f; ~ a
subject to 1 P a P 0; hkp ~f a x ÿ ~f ÿ u~k ÿ a yk ÿ u~k P 0;
13b p 2 Pk ; k 2 K;
13c
where ~f f~kp ; p 2 Pk ; k 2 K and ~ u ~ uk ; k 2 K denote the collections of current values from the previous iteration, and x xkp ; p 2 Pk ; k 2 K and y yk ; k 2 K are the auxiliary solutions obtained from the linear program in Eq. (10a)±(10e) . Constraint (13c) ensures the feasibility of the solution during the line search. Since both ~f and x satisfy the ¯ow conservation Eqs. (9b) and (10b), any point in the line search with 1 P a P 0 also satis®es the ¯ow conservation equation. During the line search, each point is evaluated by the TRANSYT trac model to determine the delays at intersections. In Lo (1998), a set of pre-de®ned paths are used for the calculation. In the algorithm shown below, the method of column generation is employed in which paths are generated during the solution process (Patriksson, 1994). The solution algorithm is summarized below: Step 1: Find the shortest paths pk ; 8k 2 K in the empty network. Assign the demands onto the respective shortest paths to determine the travel times in the network and the Jacobian matrix 0 0 0 M 0 ; Set Pk pk , fkpk qk , uk hkpk , 8k 2 K; Set I 0. Step 2: Solve the linear program in Eq. (10a)±(10e) using the Jacobian matrix M I to obtain ÿ I x ; y I .
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Step 3: Carry out the line search in Eq. (13a)±(13c) to determine for the minimum objective I I value Zmin and optimal step size amin . I ÿ Step 4: Updateÿ the trac ¯ows as f I1 f I amin x I ÿ f I and the least travel times as I u I1 u I amin y I ÿ u I . Assign the ¯ows onto the network to determine the updated travel times in the network and Jacobian matrix M I1 . Step 5: Find the shortest paths pk ; 8k 2 K in the network based on the trac pattern obtained I from Step 4. If there exists any OD pair k such that pk \ Pk ;, an empty set (i.e. a new path is generated), the travel time on this new path h~kpk may be less than the current value of the variable u~k obtained so far. This may violate the constraint in Eq. (9c) and make the solution infeasible if this path is to be added to the path set. In such a case, the variable of least travel time has to be uk ; h~kpk to ensure feasibility of the solution. This may lead to a jump in the adjusted by u~k min ~ I1 I I1 I pk [ Pk , fkp fkp if p 6 pk and objective value during the solution process. Further set Pk I1 fkp 0 if p pk ; update the Jacobian matrix M I1 to include the new path as well; and go to Step 2. I I Step 6: If pk \ Pk 6 ;; 8k 2 K (i.e. no new path is generated) and Zmin 6 e, an acceptable error, then stop; otherwise, set I I 1 and go to Step 2. When the algorithm quits from Step 6, it ensures that the objective value (or gap) Z is suciently close to the equilibrium solution by an acceptable tolerance and under this approximated equilibrium solution, all paths that were not generated from the algorithm would have travel times greater than the respective least travel times. This makes sure that at the equilibrium solution there is no path (used or unused) that has a travel time less than the respective least travel time variables; otherwise the solution algorithm would return to Step 2 for further computation from Step 5, before leaving the algorithm from Step 6.
4. The TRANSYT trac model 4.1. The trac model The TRANSYT trac model (Robertson, 1969; Vincent et al., 1980) simulates the movement of trac through a network and takes into account the eect of platoon dispersion. It is a widely used procedure to determine the queues and delays in a signal-controlled network with explicit consideration of the signal co-ordination eects. However, the trac model does not consider the re-routing of trac in the network in response to the trac conditions. In this paper, the TRANSYT trac model is employed for the evaluation of delays in the network, which forms the basic module of the assignment problem discussed in the paper. In our model, the cruise time on a link is ®xed, but the delays at the end of the link depend on the ¯ows of many other links in the network. The travel time on a link, therefore, consists of two components: the cruise time and the delay at the end of the link. It can be expressed as ta v tac da v
8a 2 A;
14
where da v is the delay at the end of a link a, which is determined by the TRANSYT trac model. This travel time function is generally asymmetric with respect to the link ¯ows, and
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171
therefore the problem falls into the class of asymmetric network equilibrium problems discussed in Section 2. The solution algorithm given in Section 3 can be used to solve the problem. In the trac model, the delay consists of two components, i.e. da v dau v dar v;
15
where dau v and dar v are the uniform and random-and-oversaturation delays, respectively. The uniform delay represents the delay incurred with an identical pattern of trac arrives during every cycle, and the random-and-oversaturation delay takes into account, respectively, the variations in trac arrivals from cycle to cycle and the steady increase in queues on oversaturation links. Before discussing these delays, the following trac patterns as functions of time during a cycle are de®ned to describe how the vehicles arrive at and depart from a link: (i) IN pattern: the pattern of trac that would arrive at the stop line at the end of the link if the trac were not impeded by the signals at the stop line. (ii) OUT pattern: the pattern of trac leaving a link. (iii) GO pattern: the pattern of trac that would leave the stop line if there was enough trac to saturate the green. These de®nitions were employed in Vincent et al. (1980). To facilitate discussions in the subsequent sections, we also de®ne a trac pattern in this paper. (iv) EN pattern: the pattern of trac entering a link. The uniform components of delays and stops were obtained through simulation of two cycles of the IN, OUT, EN and GO patterns to obtain the queue formation patterns of all links, which were then used to calculate the uniform delays. For the random-and-oversaturation delays, approximate delay formulae were employed to estimate their values. 4.2. Uniform delay Let ia , oa , ga and ea be, respectively, the IN, OUT, GO and EN patterns on a link a. The GO pattern depends on the link characteristics such as saturation ¯ow and the signal settings. Since in this paper we are only concerned about the assignment problem, the signal settings are ®xed and unchanged. The optimization of signal settings taking into account the assignment capability will be studied elsewhere. From a simulation process, the OUT pattern from the link is determined from the IN and GO patterns oa G ia ; ga :
16
In a network, the EN pattern is aected by all the OUT patterns from the upstream links. Therefore, we can write ea H ob ;
b 2 Ba ;
17
where Ba is the set of upstream links of link a. The trac entering into a link will travel along the link in accordance with a linear recursive platoon dispersion function to determine the IN pattern as ia J ea :
18
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This linear recursive platoon dispersion function is additive, i.e. J A B J A J B:
19
This property is very useful in the determination of the Jacobian matrix that will be discussed in the subsequent section. For a network with a tree structure, the trac model only needs to iterate once using Eqs. (16)± (18). However, for a general network structure, the trac model has to iterate until the trac patterns stabilize. Then the uniform delays can be calculated from the stabilized (or converged) trac patterns as dau F ia ; ga ;
20
where ia is the stabilized IN pattern on link a. 4.3. Random-and-oversaturation delays The random-and-oversaturation is evaluated based on a sheared delay formulae used in Wong (1995), taking into account the initial random-and-oversaturation queue length for a prescribed interval. The formula was derived by applying the co-ordinate transformation method (Kimber and Hollis, 1979). The random-and-oversaturation rate of delay for a link is ®rst estimated by the following: o 1 nq r 2 21a Ua V a ÿ Ua ; Da 2 where Ua
1 ÿ qa la T 2 2 2Cqa ÿ L0a la T 8CL0a 2 la T ÿ 2C
21b
Va
2C 2L0a qa la T 2 ; la T ÿ 2C
21c
and
where la is the capacity of the link a which depends on the saturation ¯ow and signal settings, L0a the initial queue length at the start of the duration of analysis for link a, qa va =la the trac intensity (or degree of saturation) for link a, T the duration of the analysis, and C is the constant in the Pollaczek±Khintchine formula (Kendall, 1951) for which a value between 0.5 and 0.6 was suggested by Branston (1978). For application in area trac control, further research is required to calibrate the value of C so that a more accurate prediction of random-and-oversaturation delay can be obtained. The formula given in Eqs. (21a)±(21c) calculates the rate of delay on the link a. For assignment application, it is required to determine the average delay per vehicle on a link rather than the rate of delay. The average delay can be obtained as dar
Dra : va
22
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173
5. The Jacobian matrix For the solution algorithm in Section 3, it is required to input the Jacobian matrix M. However, since the travel times are evaluated using the TRANSYT trac model, the determination of the derivatives is not straight forward. In a recent study (Wong, 1995), the analytical derivatives of the delays and stops with respect to the signal timing variables were derived. In this section, a post-simulation sensitivity analysis is given to estimate the elements in the Jacobian matrix M, i.e. the derivative of the path travel time with respect to the path ¯ow. Combining Eqs. (5), (14) and (15), we have X ÿ dkpa tac dau dar 8p 2 Pk ; k 2 K: 23 hkp a2A
Taking dierentiation, u ohkp X oda odar dkpa ofmn ofmn ofmn a2A
8p 2 Pk ; k 2 K;
n 2 Pm ;
m 2 K:
24
The derivative can therefore be divided into two components: uniform and random-and-oversaturation. Denote Mu and Mr as the uniform and random-and-oversaturation components of the Jacobian matrix, respectively. We have M Mu Mr :
25
5.1. Uniform component The following sensitivity analysis is a ®rst order approximation to the derivatives based on the stabilized trac patterns obtained from the TRANSYT trac model. We ®rst consider the diagonal elements of the Jacobian matrix. Consider a path in the network composed by a sequence of links, a1 ; a2 ; . . . ; aw . Usually, the ®rst link connected to an origin is also a boundary link in the TRANSYT network. The IN pattern on the ®rst link is assumed to increase by a small ¯ow Df . Since this link is on the boundary of the network, the increased ¯ow is assumed to be uniformly distributed. Therefore, the change in IN pattern becomes Dia1 Df over the whole cycle. The derivative of the uniform delay on the ®rst link with respect to the path ¯ow concerned can then determined by the following: u u oda1 F ia1 Dia1 ; ga1 ÿ da1 ; ofmn Df
26
u where ia1 is the stabilized IN pattern on link a1 , and da1 is the uniform delay on the ®rst link a1 evaluated at the stabilized trac patterns of the TRANSYT trac model. The change in EN trac pattern on the second link a2 can be approximated by
Dea2 G ia1 Dia1 ; ga1 ÿ oa1 ;
27
oa1
is the stabilized OUT pattern on link a1 . Now applying the platoon dispersion, the where change in IN pattern on link a2 becomes Dia2 J Dea2
28
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since the platoon dispersion function is linearly addictive. The derivative for the second link can then be evaluated by u u oda2 F ia2 Dia2 ; ga2 ÿ da2 : ofmn Df
29
The process continues for all the subsequent links in a similar fashion. The uniform component of the derivative on the diagonal of the matrix corresponding to the path n 2 Pm is then obtained by X od u ohumn a ; ofmn aa1 ;a2;... ;aw ofmn
30
where odau =ofmn are obtained from Eqs. (26)±(29) if dmna 6 0, and odau =ofmn 0 if dmna 0. This serves as a ®rst approximation to the derivative. For the o-diagonal elements of the Jacobian matrix, their values can be determined by ohukp X od u dkpa a ofmn ofmn a2A
8p 2 Pk ; k 2 K:
31
The above procedure determines a column of the Jacobian matrix. All other columns can be evaluated by perturbing the respective path ¯ows. 5.2. Random-and-oversaturation component The derivative of the random-and-oversaturation delay on a link with respect to the path ¯ow variables can be determined by direct dierentiation of Eqs. (21a)±(22): i 8 hÿ 9 ÿ 2 ÿ1=2 1 < r v U oU =ov oV =ov U V ÿ oU =ov ÿ 2Dra = a a a a a a a a a a 2 oda d ; 32a ; mna ofmn : 2v2a where oUa ÿla T 2 4CT 2 la T ÿ 2C ova
32b
oVa 4CT 2L0a qa la T : la T ÿ 2C ova
32c
and
Note that the derivative vanishes if the link a does not lie on the path n 2 Pm . The elements in the Jacobian matrix are obtained as ohrkp X od r dkpa a ofmn ofmn a2A
8p 2 Pk ; k 2 K:
33
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6. Numerical example Consider a network shown in Fig. 1 consisting of 9 signal-controlled intersections, 60 links and 4 origin/destination nodes. The OD matrix is given as 2 3 0 270 270 230 6 250 0 290 310 7 7 veh=h: Q6 4 330 250 0 270 5 220 270 300 0 The derivatives of path travel times with respect to path ¯ows are determined by the post-simulation sensitivity analysis in Section 5. To demonstrate the eectiveness of the sensitivity analysis in estimating the derivatives, the results are compared with those calculated by numerical differentiation in which the ¯ow on a particular path is perturbed by a small amount and the TRANSYT trac model is used to evaluate travel times on all paths. The dierences of these values from the original path travel times divided by the perturbed ¯ow generate one column of the Jacobian matrix associated with the perturbed path. The procedure is repeated for all the paths to produce the full Jacobian matrix. The comparison of results is shown in Fig. 2 in which good agreement of results from sensitivity analysis and numerical dierentiation is found. However, since the sensitivity analysis is a post-simulation calculation on the stabilized trac
Fig. 1. The example network.
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Fig. 2. Comparison of derivatives obtained from sensitivity analysis and numerical dierentiation.
patterns in the TRANSYT trac model, no re-run of the full simulation procedure is required as in the case of numerical dierentiation, the computational eort is greatly reduced. Three solution schemes are employed to solve the problem and their results are compared. The ®rst scheme is the diagonalization algorithm used in SATURN (Hall et al., 1980), in which the SATURN package is employed to solve the network equilibrium problem. The second scheme is the method of successive averages (MSA) and the third scheme is the Frank±Wolfe (FW) method given in Section 3. To demonstrate the eectiveness of these schemes under dierent levels of congestion in the network, a multiplying factor is applied to the OD matrix to scale the demands. To illustrate the convergence characteristics of the three schemes, a relative gap is employed as follows: ÿ P P k2K p2Pk hkp ÿ uk fkp P P 100%; Relative gap k2K p2Pk uk fkp where the denominator term represents the system travel time in the network when all travelers are taking the least time routes. This relative gap function is also used in SATURN as one of the convergence criteria. Generally, the smaller the relative gap, the closer is the solution to the user equilibrium pattern. In the example calculations, two cases are considered. The ®rst case uses the OD matrix shown above, whereas the second case doubles the demands in the matrix. The convergence characteristics of the three schemes are shown in Fig. 3, with a tolerance of 0.1% for the relative gap. From the ®gure, it can be seen that the MSA is subject to the slowest convergence rate among the
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Fig. 3. The convergence characteristics of dierent solution schemes.
schemes, particularly for the second case (the congested network). The SATURN shows oscillations even with a large number of iterations, especially for the congested network. However, it has to be emphasized that although the SATURN results ¯uctuate after certain number of iterations, the results are actually quite close to the user optimal pattern (good enough for most engineering and planning purposes); but further reduction of the relative gap cannot be obtained which is inherent from the general property of diagonization algorithms. The FW gives a very rapid convergence for both uncongested and congested networks, which performs the best among all schemes. Although the convergence characteristics of all schemes are very dierent, they can all produce reasonable equilibrium trac patterns. This is illustrated in Fig. 4. However, as illustrated in Fig. 4(a) for MSA, once an inecient path is generated during the solution process, it is very hard and needs many iterations to remove or divert these assigned trac o the inecient path. Therefore, a number of elements are deviated from the ideal line, but the ¯ows associated with these odiagonal elements are only very small. For the SATURN model, there are also quite a number of o-diagonal elements as shown in Fig. 4(b), which is due to the unsatisfactory convergence of the algorithm (i.e. ¯uctuates around the equilibrium solution), especially for the congested network. For the FW algorithm, all used and unused paths generated during the solution process are kept, and from Fig. 4(c), it can be seen that all used paths are subject to the respective minimum path
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Fig. 4. (a) Path travel times against the respective minimum values for MSA. (b) Path travel times against the respective minimum values for SATURN. (c) Path travel times against the respective minimum values for FW.
travel time whereas all unused paths (generated and stored but with zero path ¯ow) have travel times greater than or equal to the respective minimum values. This demonstrates that the useroptimal conditions are fully satis®ed for the solution obtained. To illustrate the performance of these schemes under dierent congestion levels, the OD matrix is multiplied by a multiplying factor to scale the demands in the network. The number of iterations needed to compute the equilibrium solution by these schemes are plotted in Fig. 5(a) with an acceptable tolerance set as 0.1% for the relative gap. It can be seen that the number of iterations generally increases with the congestion level (i.e. a larger multiplying factor) for MSA. However, the SATURN while normally requires only small number of iterations, may not always converge to the desire tolerance. This may be due to the relatively harsh requirement on the relative gap. Nevertheless, the FW method can always achieve this tight convergence criteria, but it requires only more or less similar number of iterations as with the SATURN algorithm (if converged). Since the FW evaluates the TRANSYT trac model (the most time-consuming part of calculation) several times in one iteration during the line search while the other two methods only evaluate once in one iteration, a fairer comparison is made by plotting the number of TRANSYT evaluations (whereas in SATURN it counts the number of simulations) against the values of the
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Fig. 5. (a) Number of iterations against the multiplying factor on the OD matrix. (b) Number of TRANSYT evaluations against the multiplying factor on the OD matrix.
multiplying factors, as shown in Fig. 5(b). The use of a number of TRANSYT evaluations, instead of actual computing time, provides a better picture about the computing time requirement among the schemes, because the overheads associated with the SATURN package and the implementation of the proposed method are dierent; and for practicality concerns, it is expected that the computing eort is roughly proportional to the number of TRANSYT runs. From Fig. 5(b), it can be seen that a similar trend as with Fig. 5(a) is also observed that the number of TRANSYT evaluations increase with the congestion level. The FW again requires only small number of evaluations for most cases and hence is found to be the most ecient for this example problem. 7. Conclusions A path-based trac assignment formulation together with a Frank±Wolfe (FW) solution algorithm for solving the asymmetric trac assignment problem based on the TRANSYT trac model has been presented. The method determines an equilibrium trac pattern taking into account the signal co-ordination eects and platoon dispersion on the streets. A post-simulation sensitivity analysis has been developed to evaluate the derivatives information of the TRANSYT trac model, which forms a major input to the FW solution algorithms to determine the descent direction for line search. A numerical example has been given to demonstrate the eectiveness of the proposed methodology, in which comparison with the SATURN diagonalization method and the method of successive averages has also been made. It has been shown that the methodology presented in this paper provides a promising approach of solving the dicult asymmetric trac assignment problem based on TRANSYT trac model. The next step of this research is to study the optimization of signal timings for this signal-controlled network (Wong and Yang, 1999).
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Acknowledgements This research was supported by a research grant HKU 7015/97E from the Hong Kong Research Grant Council. Transport Research Laboratory is acknowledged for supplying the source code of the TRANSYT program for this research. We would like to thank two anonymous referees for their helpful suggestions and comments on an earlier version of the paper. References Allsop, R.E., 1974. Some possibilities for using trac control to in¯uence trip distribution and route choice. In: Proceedings of the Sixth International Symposium on Transportation and Trac Theory. Elsevier, Amsterdam, pp. 345±374. Bell, M.G.H., Cassir, C., Grosso, S., Clement, S., 1997. Path ¯ow estimation in trac system management. In: Proceedings of International Federation of Automatic Control Transportation Systems, Chania, Greece, pp. 1316± 1321. Berka, S., Boyce, D.E., 1996. Generating highway travel times with a large-scale, asymmetric user equilibrium assignment model. In: Bianco, L., Toth, P. (Eds.), Advanced Methods in Transportation Analysis. Springer, Berlin, pp. 29±61. Branston, D., 1978. A comparison of observed and estimated queue lengths at oversaturated trac signals. Trac Engineering and Control 19, 322±327. Cascetta, E., Russo, F., Vitetta, A., 1997. Stochastic user equilibrium assignment with explicit path enumeration comparison of models and algorithms. In: Proceedings of International Federation of Automatic Control Transportation Systems, Chania, Greece, pp. 1078±1084. Florian, M., Hearn, D., 1995. Network equilibrium models and algorithms. In: Ball, M.O. (Ed.), Handbooks in Operations Research and Management Science, vol. 8, Network Routing. Elsevier, Netherlands. Hall, M.D., Van Vliet, D., Willumsen, L.G., 1980. SATURN ± a simulation-assignment model for the evaluation of trac management schemes. Trac Engineering and Control 21, 168±176. Heydecker, B.G., 1983. Some consequences of detailed junction modelling in road trac assignment. Transportation Science 17, 263±281. Jayakrishnan, R., Tsai, W.K., Prashker, J.N., 1994. Faster path-based algorithm for trac assignment. Transportation Research Record 1443, 75±83. Kendall, D.G., 1951. Some problems in the theory of queues. Journal of Royal Statistical Society B 13, 151±185. Kimber, R.M., Hollis, E.M., 1979. Trac queues and delays at road junctions. TRRL Report LR909, Transport and Road Research Laboratory, Crowthorne. LeBlanc, L.J., Morlok, E.K., Pierskalla, W., 1975. An ecient approach to solving the road network equilibrium trac assignment problem. Transportation Research 9, 309±318. Lo, H.K., 1998. A path-based trac assignment formulation. Paper presented at the 77th TRB Annual Meeting, 11±15 January, Washington, DC. Meneguzzer, C., 1995. An equilibrium route choice model with explicit treatment of the eect of intersections. Transportation Research B 29, 329±356. Patriksson, M., 1994. The Trac Assignment Problem: Models and Methods, VSP, Utrecht, The Netherlands. Robertson, D.I., 1969. TRANSYT: a trac network study tool. RRL Report LR 253, Road Research Laboratory, Crowthorne. She, Y., 1985. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall, Englewood Clis, NJ. Vajda, S., 1961. Mathematical Programming. Addison-Wesley, London. Vincent, R.A., Mitchell, A.I., Robertson, D.I., 1980. User guide to TRANSYT version 8. TRRL Report LR 888, Transport and Road Research Laboratory, Crowthorne.
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Wardrop, J.G., 1952. Some theoretical aspects of road trac research. Proceedings of Institution of Civil Engineers 2, 325±378. Wong, S.C., 1995. Derivatives of the performance index for the trac model from TRANSYT. Transportation Research B 29, 303±327. Wong, S.C., 1996. Group-based optimisation of signal timings using the TRANSYT trac model. Transportation Research B 30, 217±244. Wong, S.C., 1997. Group-based optimisation of signal timings using parallel computing. Transportation Research C 5, 123±139. Wong, S.C., Yang, C., 1999. An iterative group-based signal optimization scheme for trac equilibrium networks. Journal of Advanced Transportation, in press.
www.elsevier.com/locate/trb
A path-based trac assignment algorithm based on the TRANSYT trac model S.C. Wong a,*, Chao Yang b, Hong K. Lo c a
b
Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, People's Republic of China Department of Road and Trac Engineering, Tongji University, Shanghai 200092, People's Republic of China c Department of Civil Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, People's Republic of China Received 8 October 1998; received in revised form 5 July 1999; accepted 14 July 1999
Abstract This paper presents a path-based trac assignment formulation and its solution algorithm for solving an asymmetric trac assignment problem based on the TRANSYT trac model, a well-known procedure to determine the queues and delays in a signal-controlled network with explicit considerations of the signal coordination eects and platoon dispersion on the streets. The solution algorithm employs a Frank±Wolfe method to identify the descent direction at each iteration, which requires the input of the derivatives information. A post-simulation sensitivity analysis is developed to estimate the derivatives in the TRANSYT trac model. Good agreement of results with the values determined by numerical dierentiation is obtained. Using these derivatives information, the Frank±Wolfe method shows a good convergence behavior to the equilibrium solution. Comparison with other methods is also discussed in a numerical example to demonstrate the eectiveness of the proposed methodology. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Trac signals have been used for many years to resolve con¯icting trac movements, both vehicular and pedestrian, especially at intersections. In urban streets where the spacing between adjacent intersections is comparatively short, better operational performance of the signal-controlled intersections can often be obtained by taking into account the interaction between adjacent intersections in the determination of signal settings. Such co-ordination among intersections over *
Corresponding author. Tel.: +852-2859-2668; fax: +852-2559-5337. E-mail address: [email protected] (S.C. Wong).
0191-2615/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 1 - 2 6 1 5 ( 9 9 ) 0 0 0 4 4 - 2
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an area is called area trac control. To model the trac behavior in a signal-controlled network taking into account the signal co-ordination eects and platoon dispersion, the TRANSYT trac model (Robertson, 1969; Vincent et al., 1980) has been well accepted as a useful tool in evaluating the queues and delays on the links in the network, and is now widely used worldwide. Recently, studies has been undertaken to improve the modeling capabilities of the TRANSYT trac model by using a more ¯exible speci®cation of signal timings, the group-based method (Wong, 1995, 1996, 1997), for the case that the link ¯ows are ®xed. However, it has been observed that the equilibrium ¯ow pattern of a network is strongly related to the trac signal settings. By changing the control strategies, the trac will arrange itself in a user optimal manner (Wardrop, 1952), for which users traveling from any origin to any destination react to the new control strategy by choosing paths such that their individual cost is minimized. This redistribution eect on equilibrium trac ¯ow pattern will aect the performance of the network. Several authors have pointed out the importance of considering signal control in conjunction with trac equilibrium and studied the general characteristics of this non-convex optimization problem. Among them, Allsop (1974) was one of the ®rst who suggested that the signal control can be explored to aect the distribution and assignment of trac on an equilibrium network, and provided a rigorous mathematical framework for the problem. This problem has been known as equilibrium trac signal settings in the literature. In this paper, attempt is made to determine the equilibrium trac pattern based on the TRANSYT trac model. The diculties associated with detailed junction modeling in road trac assignment have been documented in Heydecker (1983), in which it was shown that for most signal control policies the conditions for good convergence behavior in trac assignment are violated. Despite the fact that it is assumed in this paper that the signal settings are ®xed, the problem is still dicult to solve as the cost functions, in contrast to the conventional network assignment problems with separable cost functions (i.e. the travel time on a link is unaected by the ¯ows on all the other links in the network), are generally asymmetric and non-convex. For this class of asymmetric trac assignment problems with explicit treatment of the eect of intersections, Meneguzzer (1995) and Berka and Boyce (1996) have developed a practical diagonalization algorithm and demonstrated its eectiveness in solving large-scale real problems. However, the signal co-ordination eects were not modeled in their papers. When signal co-ordination is considered in the asymmetric trac assignment problems, Hall et al. (1980) developed a simulation-assignment model, SATURN, for the evaluation of trac management schemes. The simulation module was based on a TRANSYT like trac model employing the concept of cyclic ¯ow pro®les and the assignment module was basically a conventional link-based assignment method (LeBlanc et al., 1975). The problem was also solved by a diagonalization algorithm in which at each iteration the set of separable cost functions, approximated by polynomial curves whose parameters were estimated from the last simulation results, was generated for use in the assignment module. It was however mentioned in Hall et al. (1980) that (even though the assignment module guaranteed to converge in itself) the simulation/ assignment loop does not necessarily converge to the equilibrium solution where neither ¯ows nor delays change signi®cantly. Oscillation of results have been found in certain networks, especially when the interaction of trac at adjacent signal-controlled intersections is signi®cant. The major advantage of the SATURN model is that the very dicult problem is, at each simulation/assignment iteration, transformed into a more manageable link-based trac
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assignment problem with separable cost functions that can be solved by the ecient algorithm (LeBlanc et al., 1975). Over the last twenty years, the SATURN model has been developed into a widely used commercial package, in which a number of extensions including the stochastic assignment, multi-class assignment and various signal control policies, etc. were also incorporated. Notwithstanding the continuous eort on the advancement of the SATURN package, there is not much research on the improvement of solution algorithm for solving the resulting dicult asymmetric trac assignment problem, albeit certain shortcomings are observed as mentioned in the previous paragraph. The objective of this paper is to investigate the application of a pathbased algorithm to solve this strongly asymmetric trac assignment problem based on the TRANSYT trac model, given the fact that due to the constantly increasing power of computers which helps in relaxing the computational constraints, the path-based assignment algorithms have received much attention recently (Jayakrishnan et al., 1994; Cascetta et al., 1997; Bell et al., 1997; Lo, 1998). Another motivation on the development of this path-based assignment module is to enable a sensitivity analysis on the network performance index with respect to the signal timing variables to be established in further studies; that can be used in the optimization of signal settings taking into account of the signal co-ordination and re-routing eects of trac in the network. Such sensitivity analysis, however, demands a relatively harsh requirement on the convergence criteria of the assignment algorithm, that is to adhere tightly to the user-optimal conditions. Existing approaches to this asymmetric problem, while producing reasonable results for most planning and engineering purposes, are not very satisfactory for the present requirements. Patriksson (1994) has provided a very comprehensive discussion on the various approaches to asymmetric trac assignment problems. For practical applications, two approaches are generally adopted: the diagonalization algorithms and descent algorithms. The proposed path-based method belongs to the class of descent algorithms. For comparison purposes, two other algorithms for asymmetric assignment are employed. The ®rst is the diagonalization algorithm from SATURN as it also uses the TRANSYT trac model for evaluation of the queues and delays. The other is the method of successive averages (MSA), a descent algorithm developed in this study also based on the TRANSYT trac model (She, 1985). In Section 2, a path-based trac assignment formulation due to Lo (1998) is given, and then a Frank±Wolfe solution algorithm for this formulation is derived in Section 3. For the solution algorithm, it requires the input of derivatives information of the TRANSYT trac model brie¯y described in Section 4. A post-simulation sensitivity analysis for these derivatives information is discussed in Section 5. A numerical example comparing three solution algorithms, the diagonalization method used in SATURN, the method of successive averages and the Frank±Wolfe algorithm in Section 3, is given in Section 6 to demonstrate the eectiveness of the proposed methodology. 2. A path-based trac assignment formulation Consider a road network consisting of a set of links, A, and a set of nodes, N. Let K N N be the set of OD pairs. Denote qk as the demand between OD pair k, k 2 K. These demands, when assigned onto the network, give rise to link ¯ows va ; a 2 A. The travel time on a link depends on
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the trac pattern in the network and is stated as ta v, where v va ; a 2 A is the set of link ¯ows in the network. For a general case, the travel time functions may be asymmetric, i.e. ota v ota0 v 6 ova0 ova
1
for some pairs of links. It is generally dicult to solve this class of S assignment problems (Patriksson, 1994). Let Pk be the set of paths between OD pair k, and P k2K Pk be the set of all paths in the network. Denote fkp as the path ¯ow on path p between OD pair k. We have the ¯ow conversation equations as X fkp qk 8k 2 K: 2 p2Pk
The corresponding link ¯ows are given by XX va dkpa fkp 8a 2 A;
3
k2K p2Pk
where
dkpa
1 0
if link a belongs to path p between OD pair k; otherwise;
is the link-path incidence variable. The travel time along a path can be expressed as X hkp dkpa ta 8p 2 Pk ; k 2 K;
4
5
a2A
where hkp is the travel time on path p between OD pair k. Let uk be the least travel time between OD pair k. We have uk min hkp :
6
p2Pk
A network equilibrium that satis®es WardropÕs user optimal principle for path ¯ows is achieved when for each OD pair, the travel times incurred by travelers on all used paths are equal and minimum, whereas the travel times on all unused paths are greater than or equal to that on the used paths. That is if fkp > 0 then hkp uk hkp P uk fkp P 0
8p 2 Pk ; 8p 2 Pk ;
8p 2 Pk ; k 2 K;
k 2 K; k 2 K:
7a 7b 7c
This network equilibrium problem has been formulated as a non-linear complementarity problem (NCP) in the literature: X Xÿ Z f hkp f ÿ uk fkp 0; 8a k2K p2Pk
X p2Pk
fkp ÿ qk 0
8k 2 K;
8b
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hkp f ÿ uk P 0 fkp P 0
8p 2 Pk ; k 2 K;
8p 2 Pk ; k 2 K;
167
8c 8d
where f fkp ; p 2 Pk ; k 2 K and uk is directly determined from Eq. (6). The link-based version of the symmetric NCP can be solved by a number of solution algorithms (Florian and Hearn, 1995). For an asymmetric NCP, it is very dicult to derive an ecient algorithm for the problem. In Lo (1998), a weak formulation to approximate the solution of the NCP was suggested, in which the problem was written as an optimization problem that is more easy to deal with. The term ``weak'' is used to describe LoÕs formulation because the solution obtained is only a numerical approximation to the solution from NCP. The NCP in Eq. (8a) is modi®ed as the following non-linear mathematical program (NMP): X Xÿ hkp f ÿ uk fkp 9a min Z f; u f;u
k2K p2Pk
subject to X fkp ÿ qk 0
8k 2 K;
9b
8p 2 Pk ; k 2 K;
9c
p2Pk
hkp f ÿ uk P 0 uk P 0
8k 2 K;
9d
fkp P 0
8p 2 Pk ; k 2 K;
9e
where u uk ; k 2 K and Z is the objective function of the NMP which is generally non-convex in nature. The non-linear constraints (9c) ensure that the path travel times between OD pair k are greater than or equal to the variables uk which at optimality are equal to the least travel time associated with the OD pair (see Proposition 1 below). The constraints in NMP form a closed feasible region for the determination of optimal path ¯ows. The existence of a solution for the NMP can be easily shown by setting u 0 and assigning all demands onto the respective shortest paths in the empty network. Since the travel times must be positive, the resulting ¯ow pattern satis®es all constraints Eq. (9b)±(9e) and hence is a candidate for the solution of NMP. The NMP is a non-linear mathematical program with non-convex objective function and a set of linear and non-linear constraints. Therefore, the solution is not unique. The major dierences between the NCP in Eq. (8a)±(8d) and the NMP in Eq. (9a)±(9e) are twofold. Firstly, the variables u is considered as endogenous (state) variables in the NCP, based on Eq. (6), but they are treated as exogenous (control) variables in the NMP. However, it is interesting to note that the optimal solution of NMP automatically satis®es Eq. (6). Proposition 1. At the optimal solution of the NMP in Eq. (9a)±(9e), the optimal values, u uk ; k 2 K, are the least travel times between the respective OD pairs.
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Proof. Consider a particular OD pair k. It is clear from Eq. (9c) that uk 6 hkp ; 8p 2 Pk , where hkp is the corresponding travel time on path p at the optimal solution of the NMP. Now assume that uk is not the minimum value of hkp ; 8p 2 Pk , i.e. uk < hkp ; 8p 2 Pk . There exists a positive quantity j minp2Pk hkp ÿ uk , such that the objective value in Eq. (9a) can be further reduced by replacing uk as uk j for the OD pair k, whereas all the constraints in Eq. (9b)±(9e) are still satis®ed. This is a contradiction that uk is already the optimal solution to the NMP. Therefore, uk minp2Pk hkp , and is the least travel time between OD pair k at the optimal solution of NMP. This completes the proof. Another major dierence between the NCP and NMP is that instead of equating the function Z to zero in NCP, the problem becomes of minimizing the objective function Z in the NMP. Proposition 2. If the solution of the NCP exists, this solution is a local minimum of the NMP. Proof. Since the objective value of the NMP is non-negative, i.e. Z P 0, from Eq. (9c) and (9e), and the solution of the NCP satis®es all the constraints in the NMP if hkp P 0 (this is always true as the travel time must be positive), the solution of the NCP must be a local minimum of the NMP. From Proposition 2, the network equilibrium problem is solved when we can ®nd a local minimum of the NMP for which the objective value is suciently close to zero. For asymmetric network equilibrium problems, the objective function Z is generally non-convex. Therefore, multiple solutions exist. However, for this class of problems, even the NCP and other methodologies such as the variational inequality (VI) are of the similar order of diculty in solving the network equilibrium problem. Moreover, computational experience has shown that it is generally not too dicult to obtain a good local minimum of the NMP which is suciently close to the equilibrium solution. 3. Solution algorithm To solve the NMP in Eq. (9b)±(9e), the Frank±Wolfe method is employed. Firstly, the auxiliary solution of the problem is obtained by solving the following linearized subproblem (Vajda, 1961): ! XX X X oh~mn X qk yk 10a h~kp ÿ u~k f~mn xkp ÿ min Z~ x;y ofkp m2K n2Pm k2K p2Pk k2K subject to X xkp qk
8k 2 K;
10b
p2Pk
XX m2K n2Pm
! ! X X oh~kp oh~kp xmn ÿ yk P f~ ÿ h~kp ofmn ofmn mn m2K n2Pm
8p 2 Pk ; k 2 K;
10c
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yk P 0 8k 2 K;
169
10d
10e xkp P 0 8p 2 Pk ; k 2 K; where Z~ is the objective function of the linear program in Eq. (10a)±(10e) which is derived by linearizing the objective function of the original NMP in Eq. (9a). Constraints (10c) are the linearization of the non-linear constraints (9c) in the original problem. In Eq. (10a)±(10e) x xkp ; p 2 Pk ; k 2 K and y yk ; k 2 K are, respectively the auxiliary variables for path ¯ows and least travel times, u~k is the current least travel time, and h~kp and f~kp are respectively the current path travel time and path ¯ow on path p, between OD pair k obtained from the previous iteration, and oh~kp ohkp 11 ofmn ofmn fmn f~mn denotes the derivative of path travel time with respect to path ¯ow evaluated at the current trac pattern obtained from the previous iteration. Note that the Jacobian matrix " # oh~kp M 8p 2 Pk ; k 2 K; n 2 Pm ; m 2 K 12 ofmn is in general asymmetric for the problem concerned in this paper. After solving the linear program, a line search is carried out as follows: u a x; y ÿ ~f; ~ u 13a min Z ~f; ~ a
subject to 1 P a P 0; hkp ~f a x ÿ ~f ÿ u~k ÿ a yk ÿ u~k P 0;
13b p 2 Pk ; k 2 K;
13c
where ~f f~kp ; p 2 Pk ; k 2 K and ~ u ~ uk ; k 2 K denote the collections of current values from the previous iteration, and x xkp ; p 2 Pk ; k 2 K and y yk ; k 2 K are the auxiliary solutions obtained from the linear program in Eq. (10a)±(10e) . Constraint (13c) ensures the feasibility of the solution during the line search. Since both ~f and x satisfy the ¯ow conservation Eqs. (9b) and (10b), any point in the line search with 1 P a P 0 also satis®es the ¯ow conservation equation. During the line search, each point is evaluated by the TRANSYT trac model to determine the delays at intersections. In Lo (1998), a set of pre-de®ned paths are used for the calculation. In the algorithm shown below, the method of column generation is employed in which paths are generated during the solution process (Patriksson, 1994). The solution algorithm is summarized below: Step 1: Find the shortest paths pk ; 8k 2 K in the empty network. Assign the demands onto the respective shortest paths to determine the travel times in the network and the Jacobian matrix 0 0 0 M 0 ; Set Pk pk , fkpk qk , uk hkpk , 8k 2 K; Set I 0. Step 2: Solve the linear program in Eq. (10a)±(10e) using the Jacobian matrix M I to obtain ÿ I x ; y I .
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Step 3: Carry out the line search in Eq. (13a)±(13c) to determine for the minimum objective I I value Zmin and optimal step size amin . I ÿ Step 4: Updateÿ the trac ¯ows as f I1 f I amin x I ÿ f I and the least travel times as I u I1 u I amin y I ÿ u I . Assign the ¯ows onto the network to determine the updated travel times in the network and Jacobian matrix M I1 . Step 5: Find the shortest paths pk ; 8k 2 K in the network based on the trac pattern obtained I from Step 4. If there exists any OD pair k such that pk \ Pk ;, an empty set (i.e. a new path is generated), the travel time on this new path h~kpk may be less than the current value of the variable u~k obtained so far. This may violate the constraint in Eq. (9c) and make the solution infeasible if this path is to be added to the path set. In such a case, the variable of least travel time has to be uk ; h~kpk to ensure feasibility of the solution. This may lead to a jump in the adjusted by u~k min ~ I1 I I1 I pk [ Pk , fkp fkp if p 6 pk and objective value during the solution process. Further set Pk I1 fkp 0 if p pk ; update the Jacobian matrix M I1 to include the new path as well; and go to Step 2. I I Step 6: If pk \ Pk 6 ;; 8k 2 K (i.e. no new path is generated) and Zmin 6 e, an acceptable error, then stop; otherwise, set I I 1 and go to Step 2. When the algorithm quits from Step 6, it ensures that the objective value (or gap) Z is suciently close to the equilibrium solution by an acceptable tolerance and under this approximated equilibrium solution, all paths that were not generated from the algorithm would have travel times greater than the respective least travel times. This makes sure that at the equilibrium solution there is no path (used or unused) that has a travel time less than the respective least travel time variables; otherwise the solution algorithm would return to Step 2 for further computation from Step 5, before leaving the algorithm from Step 6.
4. The TRANSYT trac model 4.1. The trac model The TRANSYT trac model (Robertson, 1969; Vincent et al., 1980) simulates the movement of trac through a network and takes into account the eect of platoon dispersion. It is a widely used procedure to determine the queues and delays in a signal-controlled network with explicit consideration of the signal co-ordination eects. However, the trac model does not consider the re-routing of trac in the network in response to the trac conditions. In this paper, the TRANSYT trac model is employed for the evaluation of delays in the network, which forms the basic module of the assignment problem discussed in the paper. In our model, the cruise time on a link is ®xed, but the delays at the end of the link depend on the ¯ows of many other links in the network. The travel time on a link, therefore, consists of two components: the cruise time and the delay at the end of the link. It can be expressed as ta v tac da v
8a 2 A;
14
where da v is the delay at the end of a link a, which is determined by the TRANSYT trac model. This travel time function is generally asymmetric with respect to the link ¯ows, and
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therefore the problem falls into the class of asymmetric network equilibrium problems discussed in Section 2. The solution algorithm given in Section 3 can be used to solve the problem. In the trac model, the delay consists of two components, i.e. da v dau v dar v;
15
where dau v and dar v are the uniform and random-and-oversaturation delays, respectively. The uniform delay represents the delay incurred with an identical pattern of trac arrives during every cycle, and the random-and-oversaturation delay takes into account, respectively, the variations in trac arrivals from cycle to cycle and the steady increase in queues on oversaturation links. Before discussing these delays, the following trac patterns as functions of time during a cycle are de®ned to describe how the vehicles arrive at and depart from a link: (i) IN pattern: the pattern of trac that would arrive at the stop line at the end of the link if the trac were not impeded by the signals at the stop line. (ii) OUT pattern: the pattern of trac leaving a link. (iii) GO pattern: the pattern of trac that would leave the stop line if there was enough trac to saturate the green. These de®nitions were employed in Vincent et al. (1980). To facilitate discussions in the subsequent sections, we also de®ne a trac pattern in this paper. (iv) EN pattern: the pattern of trac entering a link. The uniform components of delays and stops were obtained through simulation of two cycles of the IN, OUT, EN and GO patterns to obtain the queue formation patterns of all links, which were then used to calculate the uniform delays. For the random-and-oversaturation delays, approximate delay formulae were employed to estimate their values. 4.2. Uniform delay Let ia , oa , ga and ea be, respectively, the IN, OUT, GO and EN patterns on a link a. The GO pattern depends on the link characteristics such as saturation ¯ow and the signal settings. Since in this paper we are only concerned about the assignment problem, the signal settings are ®xed and unchanged. The optimization of signal settings taking into account the assignment capability will be studied elsewhere. From a simulation process, the OUT pattern from the link is determined from the IN and GO patterns oa G ia ; ga :
16
In a network, the EN pattern is aected by all the OUT patterns from the upstream links. Therefore, we can write ea H ob ;
b 2 Ba ;
17
where Ba is the set of upstream links of link a. The trac entering into a link will travel along the link in accordance with a linear recursive platoon dispersion function to determine the IN pattern as ia J ea :
18
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This linear recursive platoon dispersion function is additive, i.e. J A B J A J B:
19
This property is very useful in the determination of the Jacobian matrix that will be discussed in the subsequent section. For a network with a tree structure, the trac model only needs to iterate once using Eqs. (16)± (18). However, for a general network structure, the trac model has to iterate until the trac patterns stabilize. Then the uniform delays can be calculated from the stabilized (or converged) trac patterns as dau F ia ; ga ;
20
where ia is the stabilized IN pattern on link a. 4.3. Random-and-oversaturation delays The random-and-oversaturation is evaluated based on a sheared delay formulae used in Wong (1995), taking into account the initial random-and-oversaturation queue length for a prescribed interval. The formula was derived by applying the co-ordinate transformation method (Kimber and Hollis, 1979). The random-and-oversaturation rate of delay for a link is ®rst estimated by the following: o 1 nq r 2 21a Ua V a ÿ Ua ; Da 2 where Ua
1 ÿ qa la T 2 2 2Cqa ÿ L0a la T 8CL0a 2 la T ÿ 2C
21b
Va
2C 2L0a qa la T 2 ; la T ÿ 2C
21c
and
where la is the capacity of the link a which depends on the saturation ¯ow and signal settings, L0a the initial queue length at the start of the duration of analysis for link a, qa va =la the trac intensity (or degree of saturation) for link a, T the duration of the analysis, and C is the constant in the Pollaczek±Khintchine formula (Kendall, 1951) for which a value between 0.5 and 0.6 was suggested by Branston (1978). For application in area trac control, further research is required to calibrate the value of C so that a more accurate prediction of random-and-oversaturation delay can be obtained. The formula given in Eqs. (21a)±(21c) calculates the rate of delay on the link a. For assignment application, it is required to determine the average delay per vehicle on a link rather than the rate of delay. The average delay can be obtained as dar
Dra : va
22
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5. The Jacobian matrix For the solution algorithm in Section 3, it is required to input the Jacobian matrix M. However, since the travel times are evaluated using the TRANSYT trac model, the determination of the derivatives is not straight forward. In a recent study (Wong, 1995), the analytical derivatives of the delays and stops with respect to the signal timing variables were derived. In this section, a post-simulation sensitivity analysis is given to estimate the elements in the Jacobian matrix M, i.e. the derivative of the path travel time with respect to the path ¯ow. Combining Eqs. (5), (14) and (15), we have X ÿ dkpa tac dau dar 8p 2 Pk ; k 2 K: 23 hkp a2A
Taking dierentiation, u ohkp X oda odar dkpa ofmn ofmn ofmn a2A
8p 2 Pk ; k 2 K;
n 2 Pm ;
m 2 K:
24
The derivative can therefore be divided into two components: uniform and random-and-oversaturation. Denote Mu and Mr as the uniform and random-and-oversaturation components of the Jacobian matrix, respectively. We have M Mu Mr :
25
5.1. Uniform component The following sensitivity analysis is a ®rst order approximation to the derivatives based on the stabilized trac patterns obtained from the TRANSYT trac model. We ®rst consider the diagonal elements of the Jacobian matrix. Consider a path in the network composed by a sequence of links, a1 ; a2 ; . . . ; aw . Usually, the ®rst link connected to an origin is also a boundary link in the TRANSYT network. The IN pattern on the ®rst link is assumed to increase by a small ¯ow Df . Since this link is on the boundary of the network, the increased ¯ow is assumed to be uniformly distributed. Therefore, the change in IN pattern becomes Dia1 Df over the whole cycle. The derivative of the uniform delay on the ®rst link with respect to the path ¯ow concerned can then determined by the following: u u oda1 F ia1 Dia1 ; ga1 ÿ da1 ; ofmn Df
26
u where ia1 is the stabilized IN pattern on link a1 , and da1 is the uniform delay on the ®rst link a1 evaluated at the stabilized trac patterns of the TRANSYT trac model. The change in EN trac pattern on the second link a2 can be approximated by
Dea2 G ia1 Dia1 ; ga1 ÿ oa1 ;
27
oa1
is the stabilized OUT pattern on link a1 . Now applying the platoon dispersion, the where change in IN pattern on link a2 becomes Dia2 J Dea2
28
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since the platoon dispersion function is linearly addictive. The derivative for the second link can then be evaluated by u u oda2 F ia2 Dia2 ; ga2 ÿ da2 : ofmn Df
29
The process continues for all the subsequent links in a similar fashion. The uniform component of the derivative on the diagonal of the matrix corresponding to the path n 2 Pm is then obtained by X od u ohumn a ; ofmn aa1 ;a2;... ;aw ofmn
30
where odau =ofmn are obtained from Eqs. (26)±(29) if dmna 6 0, and odau =ofmn 0 if dmna 0. This serves as a ®rst approximation to the derivative. For the o-diagonal elements of the Jacobian matrix, their values can be determined by ohukp X od u dkpa a ofmn ofmn a2A
8p 2 Pk ; k 2 K:
31
The above procedure determines a column of the Jacobian matrix. All other columns can be evaluated by perturbing the respective path ¯ows. 5.2. Random-and-oversaturation component The derivative of the random-and-oversaturation delay on a link with respect to the path ¯ow variables can be determined by direct dierentiation of Eqs. (21a)±(22): i 8 hÿ 9 ÿ 2 ÿ1=2 1 < r v U oU =ov oV =ov U V ÿ oU =ov ÿ 2Dra = a a a a a a a a a a 2 oda d ; 32a ; mna ofmn : 2v2a where oUa ÿla T 2 4CT 2 la T ÿ 2C ova
32b
oVa 4CT 2L0a qa la T : la T ÿ 2C ova
32c
and
Note that the derivative vanishes if the link a does not lie on the path n 2 Pm . The elements in the Jacobian matrix are obtained as ohrkp X od r dkpa a ofmn ofmn a2A
8p 2 Pk ; k 2 K:
33
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6. Numerical example Consider a network shown in Fig. 1 consisting of 9 signal-controlled intersections, 60 links and 4 origin/destination nodes. The OD matrix is given as 2 3 0 270 270 230 6 250 0 290 310 7 7 veh=h: Q6 4 330 250 0 270 5 220 270 300 0 The derivatives of path travel times with respect to path ¯ows are determined by the post-simulation sensitivity analysis in Section 5. To demonstrate the eectiveness of the sensitivity analysis in estimating the derivatives, the results are compared with those calculated by numerical differentiation in which the ¯ow on a particular path is perturbed by a small amount and the TRANSYT trac model is used to evaluate travel times on all paths. The dierences of these values from the original path travel times divided by the perturbed ¯ow generate one column of the Jacobian matrix associated with the perturbed path. The procedure is repeated for all the paths to produce the full Jacobian matrix. The comparison of results is shown in Fig. 2 in which good agreement of results from sensitivity analysis and numerical dierentiation is found. However, since the sensitivity analysis is a post-simulation calculation on the stabilized trac
Fig. 1. The example network.
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Fig. 2. Comparison of derivatives obtained from sensitivity analysis and numerical dierentiation.
patterns in the TRANSYT trac model, no re-run of the full simulation procedure is required as in the case of numerical dierentiation, the computational eort is greatly reduced. Three solution schemes are employed to solve the problem and their results are compared. The ®rst scheme is the diagonalization algorithm used in SATURN (Hall et al., 1980), in which the SATURN package is employed to solve the network equilibrium problem. The second scheme is the method of successive averages (MSA) and the third scheme is the Frank±Wolfe (FW) method given in Section 3. To demonstrate the eectiveness of these schemes under dierent levels of congestion in the network, a multiplying factor is applied to the OD matrix to scale the demands. To illustrate the convergence characteristics of the three schemes, a relative gap is employed as follows: ÿ P P k2K p2Pk hkp ÿ uk fkp P P 100%; Relative gap k2K p2Pk uk fkp where the denominator term represents the system travel time in the network when all travelers are taking the least time routes. This relative gap function is also used in SATURN as one of the convergence criteria. Generally, the smaller the relative gap, the closer is the solution to the user equilibrium pattern. In the example calculations, two cases are considered. The ®rst case uses the OD matrix shown above, whereas the second case doubles the demands in the matrix. The convergence characteristics of the three schemes are shown in Fig. 3, with a tolerance of 0.1% for the relative gap. From the ®gure, it can be seen that the MSA is subject to the slowest convergence rate among the
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Fig. 3. The convergence characteristics of dierent solution schemes.
schemes, particularly for the second case (the congested network). The SATURN shows oscillations even with a large number of iterations, especially for the congested network. However, it has to be emphasized that although the SATURN results ¯uctuate after certain number of iterations, the results are actually quite close to the user optimal pattern (good enough for most engineering and planning purposes); but further reduction of the relative gap cannot be obtained which is inherent from the general property of diagonization algorithms. The FW gives a very rapid convergence for both uncongested and congested networks, which performs the best among all schemes. Although the convergence characteristics of all schemes are very dierent, they can all produce reasonable equilibrium trac patterns. This is illustrated in Fig. 4. However, as illustrated in Fig. 4(a) for MSA, once an inecient path is generated during the solution process, it is very hard and needs many iterations to remove or divert these assigned trac o the inecient path. Therefore, a number of elements are deviated from the ideal line, but the ¯ows associated with these odiagonal elements are only very small. For the SATURN model, there are also quite a number of o-diagonal elements as shown in Fig. 4(b), which is due to the unsatisfactory convergence of the algorithm (i.e. ¯uctuates around the equilibrium solution), especially for the congested network. For the FW algorithm, all used and unused paths generated during the solution process are kept, and from Fig. 4(c), it can be seen that all used paths are subject to the respective minimum path
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Fig. 4. (a) Path travel times against the respective minimum values for MSA. (b) Path travel times against the respective minimum values for SATURN. (c) Path travel times against the respective minimum values for FW.
travel time whereas all unused paths (generated and stored but with zero path ¯ow) have travel times greater than or equal to the respective minimum values. This demonstrates that the useroptimal conditions are fully satis®ed for the solution obtained. To illustrate the performance of these schemes under dierent congestion levels, the OD matrix is multiplied by a multiplying factor to scale the demands in the network. The number of iterations needed to compute the equilibrium solution by these schemes are plotted in Fig. 5(a) with an acceptable tolerance set as 0.1% for the relative gap. It can be seen that the number of iterations generally increases with the congestion level (i.e. a larger multiplying factor) for MSA. However, the SATURN while normally requires only small number of iterations, may not always converge to the desire tolerance. This may be due to the relatively harsh requirement on the relative gap. Nevertheless, the FW method can always achieve this tight convergence criteria, but it requires only more or less similar number of iterations as with the SATURN algorithm (if converged). Since the FW evaluates the TRANSYT trac model (the most time-consuming part of calculation) several times in one iteration during the line search while the other two methods only evaluate once in one iteration, a fairer comparison is made by plotting the number of TRANSYT evaluations (whereas in SATURN it counts the number of simulations) against the values of the
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Fig. 5. (a) Number of iterations against the multiplying factor on the OD matrix. (b) Number of TRANSYT evaluations against the multiplying factor on the OD matrix.
multiplying factors, as shown in Fig. 5(b). The use of a number of TRANSYT evaluations, instead of actual computing time, provides a better picture about the computing time requirement among the schemes, because the overheads associated with the SATURN package and the implementation of the proposed method are dierent; and for practicality concerns, it is expected that the computing eort is roughly proportional to the number of TRANSYT runs. From Fig. 5(b), it can be seen that a similar trend as with Fig. 5(a) is also observed that the number of TRANSYT evaluations increase with the congestion level. The FW again requires only small number of evaluations for most cases and hence is found to be the most ecient for this example problem. 7. Conclusions A path-based trac assignment formulation together with a Frank±Wolfe (FW) solution algorithm for solving the asymmetric trac assignment problem based on the TRANSYT trac model has been presented. The method determines an equilibrium trac pattern taking into account the signal co-ordination eects and platoon dispersion on the streets. A post-simulation sensitivity analysis has been developed to evaluate the derivatives information of the TRANSYT trac model, which forms a major input to the FW solution algorithms to determine the descent direction for line search. A numerical example has been given to demonstrate the eectiveness of the proposed methodology, in which comparison with the SATURN diagonalization method and the method of successive averages has also been made. It has been shown that the methodology presented in this paper provides a promising approach of solving the dicult asymmetric trac assignment problem based on TRANSYT trac model. The next step of this research is to study the optimization of signal timings for this signal-controlled network (Wong and Yang, 1999).
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Acknowledgements This research was supported by a research grant HKU 7015/97E from the Hong Kong Research Grant Council. Transport Research Laboratory is acknowledged for supplying the source code of the TRANSYT program for this research. We would like to thank two anonymous referees for their helpful suggestions and comments on an earlier version of the paper. References Allsop, R.E., 1974. Some possibilities for using trac control to in¯uence trip distribution and route choice. In: Proceedings of the Sixth International Symposium on Transportation and Trac Theory. Elsevier, Amsterdam, pp. 345±374. Bell, M.G.H., Cassir, C., Grosso, S., Clement, S., 1997. Path ¯ow estimation in trac system management. In: Proceedings of International Federation of Automatic Control Transportation Systems, Chania, Greece, pp. 1316± 1321. Berka, S., Boyce, D.E., 1996. Generating highway travel times with a large-scale, asymmetric user equilibrium assignment model. In: Bianco, L., Toth, P. (Eds.), Advanced Methods in Transportation Analysis. Springer, Berlin, pp. 29±61. Branston, D., 1978. A comparison of observed and estimated queue lengths at oversaturated trac signals. Trac Engineering and Control 19, 322±327. Cascetta, E., Russo, F., Vitetta, A., 1997. Stochastic user equilibrium assignment with explicit path enumeration comparison of models and algorithms. In: Proceedings of International Federation of Automatic Control Transportation Systems, Chania, Greece, pp. 1078±1084. Florian, M., Hearn, D., 1995. Network equilibrium models and algorithms. In: Ball, M.O. (Ed.), Handbooks in Operations Research and Management Science, vol. 8, Network Routing. Elsevier, Netherlands. Hall, M.D., Van Vliet, D., Willumsen, L.G., 1980. SATURN ± a simulation-assignment model for the evaluation of trac management schemes. Trac Engineering and Control 21, 168±176. Heydecker, B.G., 1983. Some consequences of detailed junction modelling in road trac assignment. Transportation Science 17, 263±281. Jayakrishnan, R., Tsai, W.K., Prashker, J.N., 1994. Faster path-based algorithm for trac assignment. Transportation Research Record 1443, 75±83. Kendall, D.G., 1951. Some problems in the theory of queues. Journal of Royal Statistical Society B 13, 151±185. Kimber, R.M., Hollis, E.M., 1979. Trac queues and delays at road junctions. TRRL Report LR909, Transport and Road Research Laboratory, Crowthorne. LeBlanc, L.J., Morlok, E.K., Pierskalla, W., 1975. An ecient approach to solving the road network equilibrium trac assignment problem. Transportation Research 9, 309±318. Lo, H.K., 1998. A path-based trac assignment formulation. Paper presented at the 77th TRB Annual Meeting, 11±15 January, Washington, DC. Meneguzzer, C., 1995. An equilibrium route choice model with explicit treatment of the eect of intersections. Transportation Research B 29, 329±356. Patriksson, M., 1994. The Trac Assignment Problem: Models and Methods, VSP, Utrecht, The Netherlands. Robertson, D.I., 1969. TRANSYT: a trac network study tool. RRL Report LR 253, Road Research Laboratory, Crowthorne. She, Y., 1985. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall, Englewood Clis, NJ. Vajda, S., 1961. Mathematical Programming. Addison-Wesley, London. Vincent, R.A., Mitchell, A.I., Robertson, D.I., 1980. User guide to TRANSYT version 8. TRRL Report LR 888, Transport and Road Research Laboratory, Crowthorne.
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Wardrop, J.G., 1952. Some theoretical aspects of road trac research. Proceedings of Institution of Civil Engineers 2, 325±378. Wong, S.C., 1995. Derivatives of the performance index for the trac model from TRANSYT. Transportation Research B 29, 303±327. Wong, S.C., 1996. Group-based optimisation of signal timings using the TRANSYT trac model. Transportation Research B 30, 217±244. Wong, S.C., 1997. Group-based optimisation of signal timings using parallel computing. Transportation Research C 5, 123±139. Wong, S.C., Yang, C., 1999. An iterative group-based signal optimization scheme for trac equilibrium networks. Journal of Advanced Transportation, in press.