A Dynamic User Equilibrium Model for Traffic Assignment in Urban Areas

Transcription

1 A Dynamic User Equilibrium Model for Traffic Assignment in Urban Areas Luce Brotcorne Lamih/roi, Université de Valenciennes, France Daniel De Wolf GREMARS, Université de Lille 3, France Michel Gendreau CRT and DIRO, Université de Montréal, Canada Martine Labbé ISRO and SMG, Université libre de Bruxelles, Belgium 1. Introduction The growing traffic congestion in urban areas creates an urgent need for a better transportation planning. In this context, traffic assignment models can be used to help public authorities to test the impact of various transportation policies such as enlarging areas where bottleneck occurs, work start time flexbility etc. The representation of traffic variations over time allows dynamic transportation models to tackle real situations. As departure time choice sharply influences traffic congestion, the simultaneous route and departure time models seem to be the more accurate dynamic models. These problems have been treated by using either simulation or analytical methods. The simulation based approaches lead to easiest implementations of traffic control strategies and allow studies on the influence of information system techniques in the context of Intelligent Trafic System. However, the number of attributes involved are normally quite large and the properties of the solution remain uncertain. In contrast, with analytical based methods, the optimality conditions can be derived from preset driver behaviour principle such as utility maximisation. Moreover, the sensitivity analysis for different scenarios is easier to perform since the procedure is usually less time consuming that with a simulation based approach. Many analytical based dynamic user optimal departure time and route choice models have been proposed: e.g. Ran et al. (1992,1994), Bernstein et al. (1993), Friesz et al. (1993), Wie et al. (1993), Ran et al. (1996), Gendreau and Bouzaiene-Ayari (1997), Janson (1993) and Jayakrishnan (1995). The presence of time-dependent congestion makes the dynamic equilibrium problem much more difficult than the static one. Thus in most existing models major simplifications are made to make the problem more tractable and the proposed algorithms are generally heuristics. In this paper, we develop a new dynamic user equilibrium model that includes route choices and departure time decisions on a multiple origin-destination network. The transportation demand is the number of users leaving their origin to reach their destination within a time window around the desired arrival time. The model determines for each user a choice of path and departure time such that he cannot decrease his disutility by unilaterally changing his departure time or/and his path. The disutility integrates travel times and schedule delays. Although the dynamic traffic assignment problem is a continuous phenomenon, one major drawback of continuous models is the c 2000 Kluwer Academic Publishers. Printed in the Netherlands. Dynamic3.tex; 2/02/2000; 10:24; p.1

2 2 necessary discretisation used to solve them. To provide a more realistic description, we consider a finite set of possible departure time variables and continuous traversal times. Users with same choice of departure time and first arc of path are uniformly loaded on the network during a time interval whose lower bound is the departure time. In order to reduce asymmetric interactions the path travel time of users of a choice is computed according to the number of users of the choice and the number of users preceding him on the arc. The originality of our model is based on these two models caracteristics. A finite dimensional variational inequality formulation of the problem is given and the existence of a dynamic user equilibrium is proved. A heuristic method that simulates daily changes of departure time and paths of users traveling repeatedly on the network is proposed. The paper is organized as follows. In Section 2, we describe the formulation of the model as a variational inequality problem. A proof of existence of a dynamic user equilibrium is given in Section 3. We present the main lines of the solution algorithm in Section 4. Section 5 is devoted to numerical results. Finally, some conclusions are drawn in Section Problem formulation The notations introduced in this paper are summarized in Table I. The transportation network is represented by a graph G = (N, A) where N is the node set and A the arc set. A node represents either an origin or a destination, or a street crossing. An arc represents a one way street. Let [0, T] be the time period of study. The length of this time interval is sufficient to allow all the users to complete their trip with some flexibility of departure times. Moreover, we denote by S = {0,1,2,...T } [0, T] the finite set of possible departure times for users. To simplify notations, we define a commodity k as the set of users traveling between the same origin-destination pair (o(k), d(k)) and having the same desired arrival time at(k) at destination. The set of all commodities k is denoted by K. The transportation demand is given by n k the total number of users associated with the same commodity, i.e., those with same origin o(k), same destination d(k) and same desired arrival time at(k). Note that users do not generally have to arrive at their destination at a precise moment, but they have some latitude to arrive slightly earlier or later. This leads us to define a time window of half-width around the desired arrival time at(k). Users of each commodity k K can be spread among several choices of departure time τ and path p linking origin o(k) to destination d(k). SP(k) is the set of these choices, P(k) is the set of all possible paths linking origin o(k) to destination d(k). The decision variables of the problem are thus the number of users n k τ,p of commodity k with the same choice (τ, p). There is a finite number of possible departure times τ S at any origin node. However, all users with same departure time τ and same first arc a on the path will have an effective departure time belonging to an interval with lower bound τ. These users will be uniformly loaded on the first arc of the path during the departure time interval. The rate of leaving users is chosen in such a way that all users with same first arc of path and same departure time lower bound τ will enter the arc during the time interval [τ, τ + 1[. Thus the departure rate varies with the number of users whereas the length of the departure time interval is fixed (See Figure 1). The network capacity is specified through travel time functions D a continuous in time. For all t [0, T], D a (t) represents the traversal time for a user entering on Dynamic3.tex; 2/02/2000; 10:24; p.2

3 3 Table I. Notations N i A a or (i, j) p set of nodes a node set of arcs an arc a path [0, T] time period of study S (τ, p) k o(k) d(k) at(k) K P(k) SP(k) n k finite set of departure times a choice of departure time and path a commodity (i.e. a set of users associated to a triplet (o(k), d(k), at(k)) origin of commodity k destination of commodity k desired arrival time of commodity k set of commodities set of possible paths linking origin o(k) to destination d(k) set of choices of path and departure time for commodity k total number of users belonging to commodity k n k τ,p number of users of commodity k with choice (τ, p) f a(t) f a(t) x a(t) y a(t) D a(t) load on arc a at time t revised load on arc a at time t number of users entering arc a at time t number of users leaving arc a at time t traversal time of arc a for some users entering on the arc at time t fa,τ,p k (t) number of users of commodity k with choice (τ, p) present on arc a at time t x k a,τ,p (t) number of users of commodity k with choice (τ, p) entering arc a at time t y k a,τ,pt) number of users of commodity k with choice (τ, p) s k τ,p (i, t) leaving arc a at time t arrival time at node i for users of commodity k with choice (τ, p) leaving the origin at effective time t. C k (τ, p) disutility function corresponding to the choice (τ, p) α β γ associated to commodity k half-width of the time window around the desired arrival time travel time coefficient in the disutility function early arrival penalty coefficient in the disutility function late arrival penalty coefficient in the disutility function Dynamic3.tex; 2/02/2000; 10:24; p.3

4 4 Entry rate on the first arc Figure 1. Entry on the first arc of a path t arc a at time t. It is computed according to the number of users of the choice to which the entering user belongs and the number of users of other choices preceding him on the arc. Function D a is defined to satisfy the so called First In First Out (FIFO) property stating that a user cannot arrive earlier at his destination by leaving his origin later. This is a natural assumption in urban congested network. Let g a be a non-decreasing piecewise linear function from IR + into [0, T] that represents the classical arc performance function linking the arc traversal time with the number of users present on arc a. According to Branston (1975), a piecewise linear function with two breakpoints seems reasonable to model the arc performance function. The arc traversal time function D a is then defined as follows: { ga ( f a (0)) if t = 0 D a (t) = { ( ) } max g a fa (t) ; D a (t ǫ) ǫ, ǫ [0, t] otherwise (1) For t > 0, g a ( f a (t)) represents the arc performance function evaluated at the revised load value on an arc at time t. The definition of f a (t) will be precised hereafter. As illustrated in Figure 2, a traversal time function given by that term may be unrealistic and may violate the FIFO assumption. Suppose that a great number of users leave the arc a in a small interval starting at time t, then if the traversal time of an arc is only defined as D a (t) = g a ( f a (t)), it will suddenly decrease (represented in Figure 2 by the dash line). Furthermore, it would be possible that a user entering the arc just after t will pass users entered before t. This justifies the presence of the second term in the maximum of (1). With such a definition, the arc traversal time of a user who entered just after t will be determined by a line segment of slope 1 starting at point (t, D a (t)). The following proposition asserts that the arc traversal time function respects the FIFO queue discipline. PROPOSITION 1. Let g a (.) be a nondecreasing function from IR + into [0, T]. We have that: t 2 > t 1, t 2 + D a (t 2 ) t 1 + D a (t 1 ). (2) Proof: Without loss of generality suppose that t 1 = t 2 ǫ for ǫ > 0. By definition (1), it immediately follows that: D a (t 2 ) D a (t 2 ǫ) ǫ Dynamic3.tex; 2/02/2000; 10:24; p.4

5 5 D a (t) -1 Figure 2. Arc traversal time function t t which is equivalent to (2). The dependence of the arc travel time function value D a (t) on the revised load charge f a (t) is first introduced with an example. It will later be defined for the general case. Consider a single arc a and a single commodity of n users distributed on 2 choices whose lower bounds of departure time intervals are (τ 1 ) and (τ 2 ). We assume that τ 1 < τ τ 2 < τ Our goal is to study the influence of the number of users present on an arc at time t on the travel time value at that time. Thus we make the assumption that the slope of traversal time function will be greater or equal to 1 for all times. More precisely, we assume that D a (t) = g a ( f a (t)) t [0, T]. Without loss of generality, we also assume that function D a has no breakpoints in the interval ]τ 2, τ 2 + 1[. Indeed, if it is not the case, the same reasoning can be done on a subinterval. Let t [τ 2, τ 2 + 1[ be the departure time of user x of choice (τ 2, a). Suppose that all users with choice (τ 1, a) have not left the arc a at time t, more precisely, assume that t < τ 1 +1+D a (τ 1 +1). We note fa(t), 1 respectively fa(t), 2 the number of users with choices (τ 1, a) and respectively (τ 2, a) present on arc a at time t. Classically the evaluation of the traversal time value D a (t) through function g a depends on the number of users present on arc a at time t : fa(t) 1 + fa(t). 2 More precisely, we assume that fa(t) 1 + fa(t) 2 users of choices (τ 1, a) and (τ 2, a) will be present on arc a during the presence time interval of the user of choice (τ 2, a) entering at time t. However, this can lead to overestimate D a (t) since the users of choice (τ 1, a) leave the arc long before user x. Then, we only need to consider the average number of users of choice (τ 1, a) (computed from time t to the last leaving time of users of choice (τ 1, a)) that are present on the arc to compute D a (t). This average value is given by: 1 τ D a (τ 1 + 1) t τ1+1+d a(τ 1+1) t f 1 a(t) dt. (3) Since D a (t) depends on the number of users of choice (τ 2, a), we could consider by analogy that only the average number of users of the choice (τ 2, a) (computed from time t to the leaving time of the user x of choice (τ 2, a) entering at t) will be present on the arc: 1 t + D a (t) t {f2 a(t){τ 2 + D a (τ 2 ) t} f2 a(t){t + D a (t) τ 2 D a (τ 2 )}}, Dynamic3.tex; 2/02/2000; 10:24; p.5

6 6 or, 1 2 f2 a(t) τ 2 + D a (τ 2 ) t. (4) D a (t) However, this value require the knowledge of the travel time value D a (t) to compute g a ( f a (t)) and thus D a (t). Therefore, we will approximate the average value (4) by f 2 a(t). This is reasonable since (4) is upper-bounded by f 2 a(t) due to the FIFO hypothesis: τ 2 + D a (τ 2 ) t + D a (t). Finally, to compute D a (t) we take into account the time interval by which the user entering at time t will be delayed by users of other choices entered before him on the arc. More precisely, the delay induced by users f 1 a(t) on the travel time of user x of choice (τ 2, a) entering the arc at time t [τ 2, τ 2 + 1[ will depend on their position with respect to x. If the users f 1 a(t) are far from user x, they will slightly delay him. However, if they are just preceding him, the induced delay will be more significant. To take into account these observations, the load charge (3) will be weighted by a ratio between the interval where users f 1 a(t) and x are both present and the travel time of user x entering at time t. This ratio is given by: τ D a (τ 1 + 1) t. D a (t) The farther the users of choice (τ 1, a) are from the users x of choice (τ 2, a), the smaller is the numerator τ 1 +1+D a (τ 1 +1) t and thus the smaller will be the impact of the presence of the users of choice (τ 1, a) on the computation of the traversal time for x. In conclusion, the contribution of the users of choices (τ 1, a) and (τ 2, a) in the computation of the travel time D a (t) is given for t [τ 2, τ 2 + 1] by or: f a (t) = τ D a (τ 1 + 1) t D a (t){τ D a (τ 1 + 1) t} f a (t) = 1 D a (t) τ1+1+d a(τ 1+1) t τ1+1+d a(τ 1+1) t f 1 a(t) dt + f 2 a(t). (5) f 1 a(t) dt + f 2 a(t). (6) Next, two cases have to be considered to evaluate the contribution of the users of choice (τ 1, a). First if the users of the choice (τ 1, a) are still present on the arc at time t (τ 1 + D a (τ 1 ) t ) (cf. Figure 3) then f a (t) = 1 D a (t) {n τ 1 (τ 1 t) (7) + n } τ 1 2 (D a(τ 1 + 1) D a (τ 1 )) + fa(t). 2 Second, if the users of the choice (τ 1, a) have started to leave the arc at time t (τ 1 + D a (τ 1 ) < t ) (cf. Figure 4) then f a (t) = τ D a (τ 1 + 1) t D a (t) f 1 a(t) 2 + f 2 a(t). (8) Thus f a (.) is linear t [τ 2, τ 1 + D a (τ 1 )] (see (7)) and nonlinear t [τ 1 + D a (τ 1 ), τ D a (τ 1 + 1)] (see (8)). As a result functions g a ( f a (t)) and D a (t) are respectively linear and nonlinear. We approximate the nonlinear part of function Dynamic3.tex; 2/02/2000; 10:24; p.6

7 7 1 f a (t) τ 2 t τ 2 +1 τ 1 +D a (τ 1 ) Figure 3. Presence of choice τ 1, entry of choice τ 2. fa 1 (t) τ 1 +1+D a ( τ 1 +1) t τ 1 + D a (τ ) τ 1 2 t τ 2 +1 τ Da ( τ 1 +1) t Figure 4. Exit of choice τ 1, entry of choice τ 2. g a ( f a (t)) on the interval [τ 1 +D a (τ 1 ), τ 1 +1+D a (τ 1 +1)] by a piecewise linear function with breakpoints defined by τ 1 + D a (τ 1 ), τ D a (τ 1 + 1), corresponding to times associated with the entry or leaving of users and abscissa of function g a () breakpoints. The remaining difficulty with expression (6) is that it requires the knowledge of D a (t) to compute f a (t) used to compute D a (t). To remedy to this difficulty the denominator value D a (t) is approximated by the value of function D a at the last breakpoint preceding t. On the basis of the reasoning developed in the previous example, we can deduce the value of the arc performance function g a ( f a (t)) at any time. We note tep k τ,p and ted k τ,p the entry times of the first and last user of the group n k τ,p on the arc a and tsp k τ,p and tsd k τ,p the exit times of the first and last user of the same group on the arc a. Let t 0 the last breakpoint before t of the travel time function D a (t). The computation of the value g a ( f a (t)) is described hereafter. Step 1: (Initialization). Initialize f a (t) = 0 t T. Step 2: (Computation of f a (t)). For all commodities k K, For all choices (τ, p) SP(k) such that n k τ,p 0 and a p: Dynamic3.tex; 2/02/2000; 10:24; p.7

8 8 1. Determine the times: tep k τ,p, ted k τ,p, tsp k τ,p and tsd k τ,p. 2. If t [tep k τ,p, ted k τ,p], where [tep k τ,p, ted k τ,p] is the entry time interval for the users with choice (τ, p) : Evaluate the number of users f k a,τ,p(t) of the choice (τ, p) present on arc a at time t. f = f + f k a,τ,p (t). 3. If t [ted k τ,p, tsp k τ,p], where [ted k τ,p, tsp k τ,p] is the time interval for which all the users of choice (τ, p) are present: Determine f = n k τ,p(tep t) nk τ,p{d a (ted) D a (tep)}. f = f + f D a(t 0). 4. If t [tsp k τ,p, tsl k τ,p], where [tsp k τ,p, tsl k τ,p] is the exit time interval for the users with choice (τ, p): Determine the number of users f k a,τ,p(t) of the choice (τ, p) present on arc a at time t. Compute f = (tsd t) f k a,τ,p (t) 2. f = f + f D a(t 0). Step 3: (Computation of g a ( f a (t))). Evaluate the piecewise linear function g a at the load value f a (t). Given the transportation demand and the network capacity, we now model the problem of the determination of the path and departure time choices (τ, p) of users to reach a user equilibrium. We first describe the constraints of the model. As mentioned previously, the decision variables of the problem are the number of users n k τ,p of each commodity k with choice (τ, p) SP(k) where p can be defined by the node sequence p = {i 0, i 1,..., i m } or the arc sequence p = {a 1, a 2,..., a m } The variables used are denoted by f a (t), x a (t) and y a (t) for a A. Specifically, f a (t) represents the total number of users present on arc a at time t [0, T]. x a (t) and y a (t) are the rates at which users, respectively, enter or leave arc a at time t. Further, they will be referred as the inflow and exit rate on arc a at time t. These variables are the sum of partial variables for choices (τ, p) associated to commodities k. For all t [0, T]: fa,τ,p(t) k = f a (t), (τ,p) SP(k) k K (τ,p) SP(k) k K (τ,p) SP(k) k K (τ,p) SP(k) x k a,τ,p(t) = x a (t), y k a,τ,p(t) = y a (t). n k τ,p = n k k K, (9) x k a,τ,p a 1=a (t) = nk τ,p a 1=a k K, (τ, p) SP(k), t [τ, τ + 1], (10) df a (t) dt = x a (t) y a (t) a A, t [0, T], (11) Dynamic3.tex; 2/02/2000; 10:24; p.8

9 y k a,τ,p(t + D a (t)) = x k a,τ,p(t) k K, a A, t [0, T], (12) y(i k l 1,i l ),τ,p (t) = xk (i l,i l+1 ),τ,p(t) k K, (τ, p) SP(k), (13) (i l 1, i l, i l+1 ) p, t [0, T]. The final constraints are the non-negativity constraints: n k τ,p 0, k K, (τ, p) SP(k), f a (t) 0, a A, t [0, T], x k a,τ,p(t) 0, k K, (τ, p) SP(k), a p, t [0, T], y k a,τ,p(t) 0, k K, (τ, p) SP(k), a p, t [0, T]. The first group of constraints (9) ensures that all the users of each commodity k will choose a departure time and a path to reach their destination. The second group of constraints (10) guarantees a uniform distribution of the users entering the same first arc of path during the same departure time interval. The third group (11) specifies that the variation of the number of users present on an arc at a given time is the difference between the inflow and exit rates on this arc at that time. The fourth group (12) of constraints guarantees that the exit rate on arc a at time t of users of commodity k with choice (τ, p) is equal to the entry rate of these users on the arcs at time t where t = t D a (t). Finally, the fifth group of constraints (13) is composed of load conservation equation at intermediate nodes. Contraints (9) define allocation of users to choice of path and departure time although contraints (10), (11), (12), and (13) model the users propagation on the network. In order to define the dynamic user equilibrium conditions for the dynamic traffic assignment, we define the disutility functions associated with choices (τ, p) for each commodity k. As the users are linearly loaded on the first arc of the path and the traversal time functions are continuous in time, the arrivals at destination are also continuous. Moreover, each user endures a disutility value corresponding to its effective departure time. The disutility function for a user of a commodity k is a weighted sum of travel time and penalties for arrival outside the time window of half-width around at(k). The total path travel time is defined as the summation of arc traversal times determined at the time users enter the arc. Thus, the model described in this paper is based on the notion of actual path times computed as follows. Let s k τ,p(i, t) be the arrival time at node i for the users of n k τ,p leaving the first node of path p at effective time t; and let {i 1, i 2,..., i l } denote the sequence of nodes belonging to the path p P(k). The variables s k τ,p(i, t) are computed recursively as follows. s k τ,p(i 2, t) = t + D i1,i 2 (t) s k τ,p(i m+1, t) = s k τ,p(i m, t) + D imi m+1 (s k τ,p(i m, t)), m = 1,..., l 1 and this for all t [τ, τ + 1[. As we suppose a uniform distribution of the users, the disutility value of the users of a commodity k with choice (τ, p) will be defined as the average disutility value of the users computed during the arrival time interval. If the discretisation step for departure times is sufficiently small, we can hope that the disutility values of the users with same choice will not be too spread out around this average value. 9 Dynamic3.tex; 2/02/2000; 10:24; p.9

10 10 Then, the disutility function corresponding to the choice (τ, p) associated to the commodity k is written as: where C k (τ, p) = α +β +γ τ+1 τ τ+1 τ τ+1 τ [x] + = [s k τ,p(d(k), t) t] dt { [at(k) s k τ,p(d(k), t)] + dt [s k τ,p(d(k), t) at(k) ] + dt, x if x 0, 0 otherwise. We assume that the disutility function penalty coefficient associated with the path travel time is greater than the penalty coefficient associated with an early arrival at destination, i.e.: α β. (14) This assumption means that the users prefer to wait at their destination rather than in their car. Furthermore, empirical studies show that this inequality holds (see for example Small (1982)). The dynamic user equilibrium conditions can then be defined as follows. We denote by N A is the set of feasible assignments and n the assignments n k τ,p vector. More precisely, N A = n = (..., nk τ,p,...), k K,(τ, p) SP(k) (τ,p) SP(k) n k τ,p = n k, n k τ,p 0. DEFINITION 1. A traffic assignment n N A is said to be a dynamic user equilibrium if the two following conditions are satisfied for all commodities k K: 1. All the choices (τ, p) SP(k) which are used have the same disutility function value. 2. All the choices (τ, p) SP(k) which are unused, produce a disutility function value greater or equal to those of the used choices. There exists a finite number of user equilibrium conditions since the number of path and departure time choices is finite and the disutility value of a set of users with same choice (τ, p) is the average disutility value of the users of the choice. Mathematically the user equilibrium conditions can be written as follows: DEFINITION 2. n N A is a dynamic user equilibrium if the following conditions are satisfied: n k [ τ,p C k (τ, p) C min k ] = 0 k K, (τ, p) SP(k), C k (τ, p) C k min 0 k K, (τ, p) SP(k), n k τ,p 0 k K, (τ, p) SP(k),. (15) Dynamic3.tex; 2/02/2000; 10:24; p.10

11 Finally we can state the dynamic user equilibrium conditions as a variational inequality problem. Let C(n) be the vector of disutility functions C k (τ, p), k K, (τ, p) SP(k). We mention here the dependency of this global disutility function on the decision variables n k τ,p. THEOREM 1. The simultaneous route-departure equilibrium problem defined in (15) is equivalent to the following variational inequality problem: Find a point ñ N A such that: C(ñ) t (n ñ) 0 n N A. (16) Existence of a User Equilibrium THEOREM 2. The simultaneous route and departure time dynamic user equilibrium in Definition (2) has a solution. Proof: First observe that the feasible set N A is non empty, compact and convex. Second note that the functions C k (τ, p) are continuous functions in variables n = (...; n k τ,p;...). Indeed, the functions C k (τ, p) are the integrals of the destination arrival time functions s k τ,p(d(k), t). These are continuous functions of the load variables f a (t) which are continuous in the decision variables n k τ,p. As C(n) is a continuous function on a convex, compact, non empty set N A, it results (see Harker and Pang (1990)) there exists a solution for the variational inequality problem (16). 4. Solution algorithm The algorithm is based on an iterative scheme that attemps to capture the day-today changes of path and departure time that the users of each commodity make. One iteration of the algorithm can be described as follows. The current state of the network (i.e. the arc traversal times) resulting from an assignment of users of each commodity to several choices of path and departure time is first evaluated (Loading Step). Then, on the basis of the new arc traversal times, a new choice of path and departure time is computed for each commodity (Computation of a choice Step). Finally, the users of each commodity are distributed among the new choices of path and departure time computed in this iteration, and those used in preceding iterations in order that the used choices have the same disutility value (Reallocation Step). This process is repeated until each user has found a satisfying departure time and path. The algorithm is illustrated in Figure 5. The First Step concerns the initialization of the process. For each commodity k, we determine the best choice of path and departure time in the static network in which each arc traversal time corresponds to a free flow situation. We denote these paths and times by p and τ respectively. We then assume that all users of each commodity make the same choice (τ, p), namely n k τ,p = n k k K. The Second Step is the loading. The number of users present on each arc (f a ()), the revised loads ( f a ()) and the corresponding traversal times (D a ()) are determined according to the current assignment of the users of each commodity k to their choices Dynamic3.tex; 2/02/2000; 10:24; p.11

12 12 Initialization Loading New Choices Disutility Values Evaluation Stopping Criterium false Reallocation true Stop Figure 5. Solution Algorithm. of departure time and path. The loading is defined to satisfy the users propagation constraints on the network (10), (11), (12), and (13). The passing of time is handled by an event drive technique. A detailed description of the loading step is given in Brotcorne (1998)). On the basis of the traversal times evaluated in the loading, the Third Step computes a new choice of path and departure time (τ, p) for each commodity k. Moreover, it evaluates the disutility value C k (τ, p) for each used choice associated to each commodity k (i.e for all (τ, p) for which n k τ,p > 0 and for the new choices). The algorithm to compute a new choice for a given commodity k searches for the best departure time and path of a single user which is added to the network without altering the link loads. In other words, the arc traversal times are uniquely determined by the revised loads value before adding the new user. To simplify notations, the index k referring to a given commodity is omitted for all variables and the disutility function. The algorithm relies upon the following decomposition principle: ( ) min C(τ, p) = min τ,p τ Given a departure time, the problem min C(p τ) p min C(p τ) p = min C (τ). τ determines the path linking the origin to the destination minimizing the disutility function. The optimal departure time is obtained by minimizing a unidimensional function C (τ). Its analytical expression is unknown but its value can be computed for any fixed departure time τ by solving the optimal departure time problem. Dynamic3.tex; 2/02/2000; 10:24; p.12

13 Let us first consider the optimal path problem. For a given departure time τ from the origin, the marginal user will enter the first arc of his path during the time interval [τ, τ + 1[ and will encounter the average disutility of the path for departure inside this interval. Similarly to de Palma et al.(1990b), it can be proved that the path with minimal average traversal time will be the one with minimal average disutility value. However, the determination of the path with minimal average traversal time value cannot be solved with classical shortest path algorithms since a sub-path of an optimal one may not be optimal (c.f. Brotcorne (1998)). To avoid this difficulty, we modify the assumption according to which the marginal user chooses the path with minimal average path travel time. More precisely, we consider that he chooses the optimal path p for a departure from the origin at τ Then, we define the value C(p τ) as the average disutility of this path p for a departure inside [τ, τ + 1[. It follows from de Palma et al.(1990b) that the optimal path p for the marginal user leaving the origin at τ can be determined by Dijkstra s algorithm. The optimal departure time problem is solved by evaluating the function C (τ) on a subset of departure time S S. S can for example correspond to the morning rush hour and be determined from lower and upper bounds on travel times of users. Note that the choice determined with this procedure may not be the minimal average disutility value choice. Moreover, it is possible that its disutility value is not lower that the disutility values of the used choices. In this case, the new choice is not considered. The Fourth step is the convergence test. Let Cmin k be the smaller disutility value associated with each commodity k. If the following condition is reached for each commodity, the algorithm stops. max k K max (τ,p) n k τ,p >0 13 { C k (τ, p) Cmin k } ǫ. (17) C k min The Fifth Step is a departure time path-swapping process. More precisely, for each commodity k, we swap users from some used choices of path and departure time to the least costly choices. The distribution attempts to equalize the disutility function values of these choices. For each commodity k, the variation of the number of users with choice (τ, p) is given by: n k τ,p(θ k ) = n k τ,p + θ k dn k τ,p. The direction of movement dn k τ,p and the step θ k are determined as follows. Let Nmin k be the number of choices for which the minimal value Ck min is reached. The number of users with non-optimal choice will be decreased proportionally to the difference between the disutility function values C k (τ, p) of the considered choice, and Cmin k. The users taken from the non-minimal choices are then transferred to the least costly ones. The direction of movement for the allocation to the non-optimal choices (τ, p) and the optimal ones (τ, p) are thus respectively given by: dn k τ,p = C k min C k (τ, p). dn k τ,p = (τ,p) (τ,p) N k min dn k τ,p. (18) Dynamic3.tex; 2/02/2000; 10:24; p.13

14 14 The reallocation step θ k associated with commodity k is determined by the successive average method (Powell and Sheffi (1982)). More precisely, θ k = min{b k, 1 l mod L }. where l is the iteration counter, L is a fixed number of iterations and B k the bound on step θ k to keep non negativity of allocations of users to choices. More precisely, B k = min (τ,p) (τ,p) dn k τ,p n k. τ,p Note that the ratio 1 l mod L aims to slow the decrease of step θk since it is reduced only after a block of L iterations. 5. Numerical results We consider a network with 3 links and 2 commodities. These are described in Figure 6 and in Tables II, III. Two types of roads are present: roads with high capacity and roads with low capacity. 0 1 Figure 6. A simple network. 2 Table II. Links Description. Origin Destination Length Congestion Free-flow Minimal density speed speed (km) (veh/km) (km/h) (km/h) The arc performance function used in the computation of the arc traversal time is a piecewise linear approximation with 2 breakpoints and increased by the free flow travel time of a function proposed by Jayakrishnan et al. (1995). This is a density based link cost function that is monotonically non-decreasing and convex. The parameters used in the disutility function are those calibrated by Small (1982): α = 6.4 ( travel time coefficient), β = 3.9 ( penalty for early arrivals coefficient), γ = 15.2 ( penalty for late arrivals coefficient). Dynamic3.tex; 2/02/2000; 10:24; p.14

15 15 Table III. Commodities Description. Commodity Origin Destination Desired arrival time Table IV. Solutions obtained for commodities with 500 users. Commodity Departure Path Number Average Average Maximum time of users time disutility relatif value criteria 1 (0, 1, 54) 44 0,2, , ,2, , ,2, , ,2, , ,2, (0, 2, 54) 50 0, , (0, 1, 54) 45 0, ,2, , ,2, , ,2, , , ,2, (0, 2, 54) 49 0, , , , , All times (departure time and travel time) used in the numerical results are given according to the following time units. One time unit corresponds to 10 minutes. Thus, Dynamic3.tex; 2/02/2000; 10:24; p.15

16 16 Table V. Solutions obtained for commodities with 600 users. Commodity Departure Path Number Average Average Maximum time of users time disutility relatif value criteria 1 (0, 1, 54) 44 0, ,2, , ,2, , ,2, , ,2, , ,2, (0, 2, 54) 49 0, , , (0, 1, 54) 44 0, ,2, , ,2, , , ,2, , , ,2, (0, 2, 54) 49 0, , , , , the desired arrival time which is 9 a.m. is 54 in the time unit system. The solutions for desired time windows with half-width = 1 or 2 time units are described in Tables IV for commodities with 500 users (n 1 = n 2 = 500) and in Tables V for commodities with 600 users (n 1 = n 2 = 600). The successive average reallocation steps are reduced of 1 unit after a block of L = 50 iterations. The solutions in Tables IV and V are the best ones obtained after 6000 iterations, i.e., the ones with the minimal relative criterion (17). Their values are given in the last column of Tables IV and V. Dynamic3.tex; 2/02/2000; 10:24; p.16

17 For = 1, 2, the users of the first commodity are split on the two possible paths (0, 1) and (0,2,1) linking 0 to 1. Even if there is only one path linking 0 to 1, the users of the second commodity are split among different choices of departure times in order to avoid congestion. This emphasizes the importance of the departure time choice in dynamic models. For both arrival time windows, penalties for early arrivals or slightly late arrivals at destination are observed for both commodities. For = 1 and n 1 = n 2 = 500, the users of commodity 1 with departure times 44 et 45 time units arrive sharply early at destination, although the users with departure times 46 and 48 arrive respectively slightly early or late. The users of the second commodity experience small early or late penalties for departure times 50 and 52 from the origin. For = 2 and n 1 = n 2 = 500, the commodities 1 and 2 users with departure time 45, 46 and 49 arrive early at destination, whereas the users of commodities 1 and 2 with respective departure times 50 and 53 arrive late. For = 1 and n 1 = n 2 = 600, the commodity 1 users arrive sharply early for departure times 44 and 45, although the ones with departure times 46 et 48 are respectively slightly early or late. The users of the second commodity arrive early for departure at 49 and slightly early or late for departure times equal to 50 and 52. For = 2 and n 1 = n 2 = 600, the users of the first commodity with departure time 44 arrive sharply early, whereas those with departure time 49 arrive slightly late. Concerning the commodity 2, the users with departure times 49 and 53 arrive slightly early or late. An increase of endured penalties occurs when the width of the arrival time window decreases or when the number of commodities increases. Moreover, for a larger arrival time window width, the users are split on a larger number of departure time and path choices. This leads to a decrease of each disutility values. Finally note that the described heuristic converges slowly to good solutions. Solutions described in Tables IV and V are the best ones reached after 6000 iterations, except for the problem defined with 500 commodites users and = 1 for which a solution of quality is determined after 1704 iterations. Note that after 3000 iterations, the maximum relative criterion value for the best known solution was already small for = 1 and reasonably small for = 2. Thus, for commodities with 500 users and = 2, the maximum relative criterion is equal to For commodities with 600 users and = 1, respectively = 2, it was equal to and Computation times on a Sun Ultra Sparc Station is on average equal to 385 seconds for 6000 iterations Conclusions In this paper, we have described a dynamic user equilibrium model for traffic assignment in urban transportation networks. Given the number of users traveling from the same origin to the same destination and willing to reach it at the same desired arrival time, the model determines both path and departure time of the users so that none of them can decrease his disutility by unilaterally changing his departure time and his path. The disutility measure is given by a weighted sum of the total path travel time and penalties for arrivals outside a desired time window. For the users of a given choice of path and departure time, the total path travel time is computed according to the number of users of the choice and the number of users preceding them. Moreove, it is defined to preserve the FIFO queue discipline. The users with the same choice of departure time and same first Dynamic3.tex; 2/02/2000; 10:24; p.17

18 18 arc of selected path are uniformly loaded on the first arc of their path during a time interval whose lower bound is the departure time. A finite dimensional variational inequality formulation of the problem is given and the existence of a dynamic user equilibrium is proved. A heuristic method is proposed to compute a dynamic user equilibrium defined by the model. Numerical results on small networks are promising even if more realistic models make good solutions more difficult to achieve. The results emphasizes the importance of the departure time choice for users. References Bernstein D., Friesz T.L., Tobin R.L. and Wie B.W. (1993). A variational control formulation of the simultaneous route and departure-time choice equilibrium problem, Transportation and Traffic Theory, Branston D. (1975). Link Capacity Functions : A Review, Transportation Research, 10, Brotcorne L. (1998). Approches opérationnelles et stratégiques des problèmes de trafic routier, Thèse de Doctorat, Université Libre de Bruxelles. de Palma A. and Hansen P. (1990). Optimum departure times for commuters in congested networks, Annals of Operations Research, 25, de Palma A., Hansen P. and Labbe M. (1990b). Commuters paths with penalties for early or later arrival time, Transportation Science 24, Friesz T.L., Bernstein D., Smith T.E., Tobin R.L. and Wie B.W. (1993). A variational inequality formulation of the dynamic network user equilibrium problem, Operational Research, 41, Gendreau M. and Bouzaiene-Ayari B. (1997). A discrete-time macroscopic dynamic traffic assignment model, Publication CRT 96-31, Center For Research on Transportation, University of Montreal. Janson B.N. (1993). Dynamic traffic assignment with schedule delay, presented at: The 72nd Annual Meeting of the Transportation Research Board, Washington DC. Jayakrishnan R., Wei K. Tsai and Chen A.(1995). A dynamic traffic assignment model with trafficflow relationships, Transportation Research, 3C, Leblanc L.J., Hegalson R.V.and Boyce D.E. (1985). Improved efficiency of the Franck-Wolfe algorithm for convex network programs, Transportation Science, 19, Harker P.T. and PANG J.-S. (1990). Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms and applications, Mathematical Programming, 48, Powell W. and Sheffi Y. (1982). The convergence of equilibrium algorithms with predetermined step sizes, Transportation Science, 16, Ran B., Boyce D.E and Leblanc L.J. (1992). Dynamic user optimal departure time and route choice model: a bilevel, optimal-control formulation, Advance Working Paper 12, University of l Illinois Chicago. Ran B. and Boyce D.E. (1994). Combined departure time/route choice models, in Dynamic urban transportation network models, Lecture Notes in Economics and Mathematical Systems 417, Springer Verlag. Ran B., Hall R.W. and Boyce D.E. (1996). A link-based variational inequality model for dynamic departure time/route choice, Transportation Research, 30B, Small K. (1982). The scheduling of consumer activities: work trips, Am. Econ. Rev. 82, Smith M.J. (1979). The existence, uniqueness and stability of traffic equilibria, Transportation Research, 13B, Wie B.W., Tobin R.L., Friesz T.L. and Bernstein D. (1995). A discrete nested cost operator approach to the dynamic network user equilibrium problem, Transportation Science, 29, Dynamic3.tex; 2/02/2000; 10:24; p.18