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An alternative formulation of the simultaneous route and departure-time choice equilibrium problem
Transpn Rex-C. Vol. 4. No. 6, pp. 339-357, 1996 Copyright Q 1996Elsevier Science Ltd Printed in Great Britain. All rights reserved 096&090X/96 $ I5.I0 + 0.00
Pergamon
PII: so%8-o%x(%)ooo1&6
AN ALTERNATIVE FORMULATION OF THE SIMULTANEOUS ROUTE AND DEPARTURE-TIME CHOICE EQUILIBRIUM PROBLEM FRANCISCO
J. JAUFFRED’
and DAVID BERNSTEIN2
‘Center for Transportation Studies, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. *Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ 08544, U.S.A. Abstract-Traditional (static) network equilibrium models have always been formulated in a route-based fashion rather than a vehicle-based fashion. That is, the decision variables have been the number of vehicles using each route rather than the route choices of each vehicle. Given the success of this approach, it is not surprising that recent “dynamic” network equilibrium models have been formulated in a similar way. That is. the decision variables in these models are usually the route-specific departure rates over time. In this paper, we develop a vehicle-based equilibrium model of simultaneous route and departure-time choice and discuss the possible advantages of this approach. We then describe a heuristic for solving this model and demonstrate its effectiveness on several small examples. Copyright c 1996 Elsevier Science Ltd
I. INTRODUCTION
In order to accurately evaluate many of the transportation policy questions of interest today (e.g. the impact of telecommunications, route guidance and driver information systems, and time-varying congestion pricing) we need higher resolution models than we have needed in the past. That is, things that in the past have been treated as “black boxes” must now be modeled in some detail. To that end, a great deal of attention has been devoted to incorporating traffic dynamics into models of route and departure-time choice. Traditional models of route choice, in which travel times are a function of the average usage of a facility over some (fairly long) period, are no longer considered adequate. Instead, models are being developed which explicitly account for the “timing” of vehicle interactions, how these interactions affect travel times, and how these travel times ultimately affect behavior. Much of the effort in this area has been devoted to the development of equilibrium models that include these kinds of time-dependent phenomena. Some of these models simply attempt to expand Wardrop’s (Wardrop, 1952) model of route choice to include traffic dynamics more explicitly. Others also attempt to include the departure-time choices of the commuters. In both cases, the models have been enormously influenced by the way in which Wardrop (1952) originally posed the problem. Recall that his original criterion was stated as follows: “The journey times on all the routes actually used are equal, and less than those which would be experienced... on any unused route” (p. 345). What is interesting about this definition of equilibrium is that it is stated in route-based as opposed to vehicle-bused terms. That is, equilibrium is defined in terms of attributes of the routes rather than attributes of the vehicles, hence the number of decision variables is related to the number of routes and not the number of vehicles. This is true of Wardrop’s original definition as well as many other related definitions of equilibrium that have been developed subsequently [see, for example, Dafermos and Sparrow, 1969; Dafermos, 1971; Smith, 1984; Heydecker, 1986 and Bernstein and Smith 19941. In many respects, it is not at all surprising that all of these definitions are route-based given the origins of the problem (i.e. traffic flow theory). In particular, observe that traffic volumes and flows have historically been assumed to be real-valued (rather than
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Francisco J. Jauffred and David Bernstein
integer-valued). Combining this assumption of real-valued volumes with a vehicle-based model one is left with a formulation in which there are an infinite number of decision variables. Specifically, letting D denote the total number of vehicles, it is necessary to solve for the routes choices of the infinite number of “vehicles” in the interval [O,D]. On the other hand, combining the assumption of real-valued volumes with a route-based model leads to a formulation in which there are only a finite number of decision variables. That is, letting /denote the finite number of (loopless) paths in the network, it is only necessary to solve for the number of vehicles on each path in /i Hence, in the traditional setting, the mathematics of route-based models is much simpler than the mathematics of vehicle-based models. What may seem surprising is that even when route volumes are assumed to be integervalued the equilibrium problem has been presented in route-based terms [see, for example, Rosenthal, 1973 and Bernstein, 19901. The resulting models are generally formulated as finite-dimensional integer programs in which the decision variables are the (integervalued) number of vehicles on each route. One could, just as easily, define the problem in such a way that the decision variables are the route choices of the finite number of vehicles. However, since these integer-valued models have principally been developed in order to gain insight into the real-valued models, it is not really surprising that they have been route-based. What is much more surprising, at least to us, is that almost all equilibrium models that include time-dependent phenomena have been formulated in a route/time-based fashion. That is, these models have been formulated in terms of route-specific departure rates (or volumes) over time [see, for example, Friesz et al., 1993; Bernstein et al., 1993; Ran et al., 1996; Wie et af., 1996; and Zhu and Marcotte, 19961.’ We think this is surprising because, when tra@ic dynamics are incorporated, such route/ time-based formulations are not of lower dimensionality than vehicle-based formulations.
The decision variables in these route/time-based models are the route-specific departure rates over time. Hence they are naturally infinite dimensional unless time is discretized (which, as discussed in Section 5, introduces other difficulties and inaccuracies). On the other hand, there do appear to be some advantages to using a vehicle-based formulation. Specifically, using discrete vehicles it may be possible to develop a finite dimensional vehicle-based formulation in which the decision variables are the route and departure-time choices of each of the (finite number of) vehicles. In addition, even infinite dimensional vehicle-based models (i.e. vehicle-based models developed assuming that vehicles are real-valued) seem to be easier to motivate understanding than their route/time-based counterparts since we can actually discuss the problem in a game-theoretic setting that involves players and their actions. Hence, in this paper we take the first steps towards developing a vehicle-based formulation of the simultaneous route and departure-time choice (SRD) equilibrium problem. We begin in Section 2 by defining such equilibria in a vehicle-based context and formulating the equilibrium problem as a collection of minimization problems. In Section 3 we develop a heuristic for solving this problem and then, in Section 4, we solve some illustrative examples. Finally, in Section 5 we discuss how these results might be extended. 2. THE SETTING
As discussed above, our purpose is to develop a model of rush-hour (automobile) traffic in which commuters choose both the time they are going to depart from their homes and the roads they are going to use to get from home to work. To formalize this choice process, let R (Z),R +(Z+), and R + +(Z+ +) denote the real numbers (integers), nonnegative reals (nonnegative integers), and positive reals (positive integers), respectively ‘Though our interest is in developing a model set on a general network that can be solved numerically, it is worth pointing out that even those models set on simple networks and solved analytically. See, for example, Vickrey (1969). Fargier (1981), Hendrickson and Kocur (1981) Mahmassani and Herman (1984) Daganzo (l985), Arnott ef al. (1990), and Kuwahara (1990), have been formulated in a route/time-based fashion.
Vehicle-based
equilibrium
model
341
and let the relevant departure time horizon be given by [0,7’j~R + . Further, let 6 = ( I, , ) denote an abstract transportation network comprised of a finite set of nodes, I, with cardinality N= 1 I 1, and a finite set of links (or arcs), /, with cardinality A = 1 , I. The number of commuters traveling between any pair of nodes in I x I is assumed to be known. For simplicity, we will only consider those origin-destination pairs with a positive number of commuters and will denote the set of all such pairs by ,Y (with cardinality W). The relevant demand vector is then denoted by D = (D,.:wEY )ER~+ with the total demand being given by Q = CwEWD ,,,. Commuters are assumed to be identical except, perhaps, for their origin and destination. We will refer to commuters traveling between O-D pair w as w-commuters. Each commuter travels from a specific origin node to a specific destination node on a route, p, which is simply an acyclic chain of links.* The set of all possible routes for w-commuter is denoted by ./;1,and has (finite) cardinality P&O. Further, the set of all relevant routes is denoted by / = U,,, P,., and has cardinality P = C,,, P,.. As mentioned in the Introduction, what makes our formulation of the SRD equilibrium problem different from earlier formulations is the decision variables. To date, most SRD equilibrium models have been formulated in such a way that the decision variables are the departure rates over time (i.e. using a route/time-based approach). However, we think complicated models are easier to understand when they are formulated in terms of the actual decisions being made by the actual decision-makers. In this case, the decisionmakers are the commuters, and each w-commuter chooses a departure-time, t~[o,T], and a route, pi /;y. Hence, in the remainder of this section we will develop a formulation of the SRD equilibrium problem in terms of these players and actions (i.e. using a vehicle-based approach). Specifically, we now let t,,. and rw denote the departure-time choice function and route choice function for O-D pair WEI. In other words, tJ9(.) denotes the departure-time choice of w-commuter 9 and r,,,(qI.) denotes the route choice of w-commuter 9. Further, we let t=(t, ,...,t& and r = (r ,,...,rW). At this point, we have not specified how commuters make these choices. In general, these choices can be based on a variety of different factors. As this discussion progresses, we will assume that these choices are based on some notion of travel cost. Now, observe that since we have implicitly ordered each set of w-commuters, we must restrict the set of feasible departure-time patterns to: /L~ = (t,,, : [tw(91.)- t,(kl.)](q - k) > 0, for all 9, k E [0, D,.] with 9 # k). That is, given any two w-commuters AA,=(r,,.:r,,, maps into 1) it follows terns is given by the pair ( /-, A))= Now, each feasible route and
(1)
9 and k, if 9 > k then 9 must depart after k. So, letting that the set of feasible route and departure-time pat(L&,,
1.,,.,u,~~,, n,,.).
departure-time pattern, (t,r)E( /, n’), gives rise to an arrival-time pattern, t,: [O,D,J x ( 1; AJ)-R + + , for w-commuters. We will denote the arrival-time for w-commuter 9 under route and departure-time pattern (t,r) by t,V (q[t,r) and hence the travel-time for this same w-commuter is given by r,.(qlt,r)-t,,.(qlt-{t,,.},r). We are not concerned here with the exact nature of these arrival-time operators, they can be developed in a variety of different ways (e.g. systems of differential or integral equations). We do, however, assume that they are regular in the sense of Friesz ef al. (1993) (Theorem 1 and footnote 3). That is, these arrival-time operators are assumed to be of a form which makes them monotonically increasing in 9. In Section 4 and Appendix A we present a specific example of the types of arrival-time operators that can be used. Further, following de Palma et al. (1983) we assume that the cost a commuter incurs is a function of both the travel time and the arrival time (as it compares to some desired arrival time). In particular, if T*ER + + denotes the (common) desired arrival-time for all 2We use p to denote a physical choice.
route rather
than r because we will later use the notation
r to denote a route
Francisco J. Jauffred and David Bernstein
342
commuters and &R+ denotes the actual arrival-time for a particular commuter, then the schedule delay cost, I&R + + +R + , incurred by that commuter is given by:
if T* > CY ifT*=a if T’ < ~7
(2)
where a! denotes the (common) value of early arrival time and /I denotes the (common) value of late arrival time. Thus, the travel cost for w-commuters, c,:[O,&] x( 7;~ )+R + + , is given by: c&It,
r) = y. [tw(qlt, r) - tdqlt - IhI, 41 + dM4lt~
4.
(3)
where y denotes the (common) value of travel-time.3 With these fundamental concepts in hand, we can now define what we mean by an equilibrium. Of course, a natural place to begin is with a Nash-like [see Nash, 19521 definition in which no individual commuter can reduce her/his cost by unilaterally changing either her/his route or departure-time. Unfortunately, this is not as easy as it seems in our current context since we do not really have individual vehicles (i.e., since q is assumed to be real-valued). What we might do instead is to think about changes in (t,,,,r,) for a particular q. For example, given a route and departure-time pattern, (t,,,,r,,J, we could represent a change in the behavior of the qth vehicles by creating a new pattern, (&, rW), in which ;,,&I.) =r,Jkj.) and ?,(k(.) = t,(kl.) for all k#q. However, it is relatively easy to see (at an intuitive level) that such an approach is virtually meaningless. In particular, if we adopt the fairly standard and reasonable assumption that the travel time on a route is some function of the number of vehicles using the links in that route, then it follows that the costs under (i,, F,,,) would be identical to those under (t,,,,r,J, regardless of the values of (t,,,,r,,,) and q. This is because the number of vehicles on a link is some function of an integral involving the route and departure-time functions. Hence, since (t,,,,r,,,) and (i,, rW> can only differ on a set of measure zero, the trajectories of all link volumes will be the same under both route and departure-time patterns. To define a notion of equilibrium in this context we instead make use of a convenient fiction first introduced by Devarajan (1981) for the static problem. In particular, we suppose that each origindestination pair assigns people to routes and departure-times in such a way that each commuter experiences equal cost [in the spirit of Wardrop, 19521and that the O-D pair experiences minimum cost. More formally, letting t-, = rw), and using the notation t_, + tw 0 I,... ,tw-~,tW+ I,..., td and r-,,, = (rl, .. .. rw-l, rw+l, . ... and r+ + rW to denote (tl, .... t,-1, t,, &,,+I, ... . tw) s t and (ri, .. .. r,,,-1, r,,,, r,,,+l, . . . . rw) E r respectively, we have the following preliminary definition:4 Dejinition I. (i) A route and departure-time pattern, (t,r)E( Y;M), is said to be vehicleneutral iff there exist constants, CwE R,,, such that:
c&It, r) = &
(4)
for all WEP and qE[O,D,]. (ii) The set of all vehicle-neutral route and departure-time patterns in ( 7;~) is denoted by 1 w. (iii) Given (t-w,r_w)E/’ w-‘, the set of all (t,,,,r,,,) such that (t,r)El w is denoted by r,(t_,,,r_,).With this, we now have the following definition of equilibrium: 3The assumption that all commuters have the same desired arrival-time, value of travel-time, value of earlytime, and value of late-time is for notational convenience only. This assumption could be relaxed without significantly changing the analysis that follows. 4We do not use measure-theoretic arguments in Definitions 1 and 2 because we prefer to simplify the exposition and because they are not necessary for the discussion that follows. Specifically, since we ultimately solve for equilibria by discretixing the demand, we do not need to concern ourselves with the behavior of the system on sets of measure zero. Friesz et al. (1993) provide a careful discussion of the issues involved.
Vehicle-based equilibrium model
DeJinition 2. A route and departure-time
343
pattern, (t,r)~( CM), is said to be an equili-
brium if
for all WEE and qE[O,D,,,].That is, a route and departure-time pattern is an equilibrium if no w-commuter wants to trade places with another w-commuter and, subject to this condition, they are minimizing their costs. When c, is continuously differentiable in q it follows from Definitions 1 and 2 that each SD pair, WEZ, is trying to solve the following infinite dimensional minimization problem:5 D, min c&It, r) dq/& &V.~,)E( ' [email protected] J
(6)
0
st .
.
k&l~~ dq
r>=
o
(7)
given the route and departure-time patterns of the W-l other SD pairs, (ti ,..., t,,t,+ I ,..., t&TW-’ and (rl,..., rW,rw+r,..., r&E x,“-‘. The equivalence between equation 7 and vehicle-neutrality can be seen by integrating equation 7. The equivalence between equation 6 and equation 5 can be seen by observing that when equation 7 is satisfied, the cost incurred by all w-commuters is the same. Hence the cost to each w-commuter is simply the total cost experienced by all w-commuters divided by the number of w-commuters. Since the total cost is simply the “sum” of the costs experienced by each commuter, the total cost in this case is given by 4 c,(qlt, r) dq and 0 D, the (common) cost to each commuter is given by J c,(qlt, r) dq/D,. It is important to 0
note that, although the objective function in this problem has a very similar structure to the objective function in math programming formulations of the so-called static equilibrium problem, it is actually unrelated. In this problem, the objective function represents the average cost experienced by w-commuters, whereas in the static problem the objective function is an artificial construct. Also, in this problem the upper limit of integration is known, whereas in the static problem the upper limit of integration is one of the decision variables. This is discussed more fully below. 3. A SOLUTION APPROACH
Though the assumption of continuous differentiability is unlikely to hold in practice, the W minimization problems in equations 6 and 7 do provide a useful way to motivate a solution method for the more general problem. Hence, we will now develop a heuristic for soIving the general equilibrium problem by proceeding as if the cost operators are continuously differentiable functions. 3.1. Overview of the approach Specifically, we will solve the equilibrium problem by iteratively solving the minimization problems in equations 6 and 7 for each O-D pair sequentially. That is, at each iteration, 1, we will solve the minimization problem in equations 6 and 7 for a particular (rD pair w, treating all other O-D pairs as given. 50f course, it is well known that such a collection of minimization problems can be formulated as a single variational inequality problem [see, for example, Harker and Pang, 19901.With appropriate assumptions and letting the subdifferential of e, be given by &Jr(O)] = {a : c&r (0), .... s”,, .... zw(O)]l c,&(O)] + a]s, - r,(O)] for all s.+, it is relatively easy to show that a route and departure-time pattern, (F. ?) E ( /-, H), with dc,(qjt,r)/dq=O, q~[O,D,,,] is an equilibrium iff C,,, gc,[7(0)][rw(O)- T,JO)] 2 0 for all (r,r)~( /,-,Y) with dc,(q] t,r)/dq = 0, qE[O,D,,,].However, as our concern here is primarily with developing a solution method we will not dwell on this variational inequality formulation.
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Francisco J. Jauffred and David Bernstein
Letting t,,,l denote the value of t, at iteration 1 and I-,,,~denote the value of r,,, at iteration 1, this process can be summarized as follows: Step 0. Set I= 1. Find an initial feasible solution (tl,o ,..., tw,o), (q. ,..., rw,& Step 1. Solve the necessary minimization problem for each O-D pair consecutively. Step 1.1. Set w= 1. Step 1.2. Solve equations 6 and 7 for O-D pair w given
(t1,r, *.., t,-l.I, ~,+I.I-I..., tw,r-l) and (~I,L . .. . rw-l,~, rw+l.~-l..., QV,I-I) Step 1.1 Step 2. If (~IJ, .... rw,i)=(ti, I- 1, .... fw,i - 1) and (Q.L ... . QVJ)=(~IJ-I, .... rw,r-1) then STOP. Otherwise set I= I+ 1 and GOT0 step 1. Step 1.3. Set w = w + 1. If w< W then GOT0
Many different convergence criteria can be used in Step 2 of this heuristic. We chose to stop iterating when w
Dw
U[
tw,dxl-)-
M-1
w
tw,,-~(Xl-)]2 dx 5 et
and
c W-1
0
Dw
J[r,&+) - rw,,-~(xl~)]z dx i
G.
0
Obviously, this method requires that we be able to solve the necessary minimization problem for a single O-D pair given some level of “background traffic” (i.e. traffic that is generated by the other O-D pairs). We describe a method for doing so below. First, we consider the case when there is a single route connecting a single origin and destination, then we generalize to the case of multiple routes connecting a single origin and destination. In both cases, we develop an optimization problem assuming that the cost operator and the departure-time function are continuously differentiable, relax the complicating constraints using a penalty function, and solve using finite differences. In order to simplify the notation in the following sections we will no longer include the subscripts indexing the O-D pair (since we will only be concerned with a single O-D pair at a time). Also for notational simplicity we will denote dc(ql+)/dq by c?(ql.) and dt(q(.)/dq by Q(11.). 3.2. Solving the departure-time choice problem for a single O-D pair On a network with a single O-D pair and a single route equations 6 and 7 reduce to the following departure-time choice problem: D
(8) s.t. t E 7-
(9)
c’(qlt) = 0
(10)
Now, observe from the definition of /in equation 1 that when t is continuously differentiable equation 9 can be ;&ten as /(q) > 0. In addition, observe that since D is D
constant, t is a solution of m;ln J’ c(qlt) dq/D iff it is also a solution of m,in J c(qlt) dq. n
n
Using these two observations w’e can re-write equations 8, 9 and 10 as thi following infinite dimensional minimization problem:
(11) s.t. c’(q(t) = 0
(12)
t’(ql*)’ 0
(13)
345
Vehicle-based equilibrium model
Though there are many potential ways to solve equations 11, 12 and 13 perhaps the most obvious is to relax equation 12 as follows: D
rn;n V =
c(qlt) + h@‘(qlt)l dq
J
(14)
0
s.t.
t’(ql.) > 0
(15)
where M is a penalty function and then solve the resulting Euler-Lagrange equations. In this case, this would require that we solve the following system of differential equations:
av at
d av=, dq ar
(16)
av --0 atw)
(17)
However, since in reality c and t may not be differentiable, this does not seem to be the best approach in practice. Instead, we make use of the method of finite differences. To do so we will discretize the closed interval, [O,D]cR+, into n~z+ +, subintervals in such a way that [O,D]= [O,qllU[cll,q2lU[qn-l,q,l and O
i= l,...,n
where q. = 0 by definition. Further, for any function,J[O,D]-+R,
(18) we will let:
Af(qi)=f(q;)-f(qi-l), i= l,...,n.
(19)
Then, it follows that we can approximate equations 14 and 15 in terms of these finite differences as follows: mm V = &c(q;lt)
i=o
+ M[Ac(qil~)l)Aq
s.t. At(qil.) > 0, i = 0, . . . , H.
Observe that it follows from equations 19, 3 and 2 that Ac(qi]t)= y.[As(qii.)-At(qjJ.)]
(20) (21) +
Adt(qiI.)I.
The easiest way to interpret this problem is to consider the case in which Aqi= 1 for i= 1,...,n (i.e. in which we have discrete vehicles). Equation 21 simply requires that vehicle i depart after vehicle i- 1. The two terms in the objective function represent the total cost of travel for the n( = D) vehicles and the penalty for violating vehicle neutrality. Given an appropriate choice of penalty function, equation 20 will be minimized when 5 M[Ac(qilt)] = 0, m ’ which case all n vehicles experience the same cost. When Aq, > 1 the i-0
problem can be interpreted in the same way, except that we must think about packets of vehicles. As Aqi+O for all i= 1,...,n the problem in equations 20 and 21 approaches the problem in equations 14 and 15. The only remaining difficulty in solving this problem is evaluating c(q;jt), the cost experienced by vehicle (or packet) i under departure-time pattern t. In general, it can be quite computationally expensive to evaluate c(qilt) since, as the notation indicates, it depends on the complete trajectory oft. We overcome this difficulty by using an iteratively updated estimate of this cost. We denote the estimate by R[.] and update it on a packetby-packet basis, using the cost calculated for packet i-l as the estimate for packet i.
346
Francisco J. Jauffred and David Bernstein
Using this iterative estimation process, it is relatively easy to solve equations 20 and 21. The complete heuristic for the departure-time choice problem with a single route and CD pair can be summarized as follows: Step 0. Set R[t(qo)] = 0. Set i= 0. Step 1. Solve the following problem:
$~,“1 = (Nt(qiI.)l+ MAc(qilt)l). Let At*(qil*) and q denote the solution and the optimal value of the objective function respectively. Step 2. Set R[t(qi+ ,I.)] = Z;. Set i= i+ 1. If i
7;~ C(C[~ilt(O), At(qil-)I
+ R[t(q~l.)l)Aq
r=O
The particular penalty function we choose to use is M[x] =K(x/e)” where mEZ+ +, KEZ + + is a large number, and EEZ + + is a small number. As illustrated in the example in Section 4.1, this approach seems to work reasonably well. 3.3. Solving the equilibrium problem for a single O-D pair In principle, it is not difficult to expand the single (TD pair problem to include route choice as well as departure-time choice. In particular, it follows from the same arguments used in developing equations 11, 12 and 13 that we need to solve the following problem: D y”
c(qlt, ’
4
dq
(22)
J 0
s.t. c’(qlt, r) = 0
t’(d)
’
rE R
0
(23) (24) (25)
The difficulty is that this is an infinite dimensional, nonlinear, mixed integer programming problem. Thus, in this form, this problem is very to difficult to solve. However, as it turns out, it is possible to transform this problem into one that is much easier to deal with. To do so we treat each route as if it has its own one-to-one, strictly monotone, and differentiable departure-time function, $(q). This allows us to develop a formulation without the route choice function, r. To see why, first observe that the O-D departure-time function and the route-specific departure-time functions are related as follows: (26) Intuitively, equation 26 says time) equals the sum of the pair. (Recall that since we are specific indexes). Now, observe that given all p&f, and hence that:
that the O-D departure rate (measured in vehicles per unit path-specific departure rates for all paths serving the O-D only considering a single O-D pair we have dropped the O-D our assumption about tp, the inverse function, r;*, exists for
(27)
341
Vehicle-based equilibrium model So,
it follows that
dqp=
$J+ dq P
for all pc.2 Letting h,eR+ further follows that:
(28)
p
denote the number of commuters
h
that choose route p, it
(29)
P
for all PG./“,and hence that: (30) Finally, letting h = (hp : p E P) E Rr denote the complete vector of route choices, it follows that: rc
&wchp=D.
(31)
PC ’
Substituting equation 30 into equation 22 and equation 3 1 into equation 25 it follows that we can alternatively solve the following minimization problem:
(32)
s.t. c'(qplt) = 0, p E /
(33) (34)
$h?pl.) >
0, p
h,zO,
PE
6 ’
(35)
/’
(36)
where c’(qplt) denotes dc(qP(t)/dqp and tk(qp(.) denotes dtp(qp)l.)/dqP. Even with this transformation, this problem can be very dificult to solve as a result of equation 33. Hence, as in the single-route case, we relax this problem using a penalty function to obtain: 4
min th
(c(qp
It)
+
Mc’(qp
s.t. chp
IOI)$$
dqp
(37)
P
= D
(38)
RE/
(39)
()(qpl.) > 0, p E .” h,zO,
PE
(40)
/.
Further, letting: cp(qlt)
=
(c(qpIt)
+
~4qpM)~
;
* P.
(41)
348
Francisco J. Jauffred and David Bernstein
we can re-write this problem as:
(42)
s.t.
c
hp =
D
(43)
PE *
$(qpl*) > 0, p E .4 hp 10,
p E 4.
(45)
It is interesting to note that, given a specific departure-time function (found, for example, using the heuristic described above), we can solve for the partial equilibrium route-choices by solving the following problem: (46)
s.t. chp
= D
(47)
hp > 0, p E .Y.
(48)
PE ’
This problem is clearly very similar in structure to the original programming formulation developed by Beckmann et al. (1956). However, it is important to observe that the nature of this problem is quite different from that of Beckmann et al. (1956). In particular, observe that in their model, the objective function is artificial in the sense that it is used simply because it leads to Karush-Kuhn-Tucker conditions which are equivalent to the equilibrium conditions. That is, the integral of the costs has no (immediately obvious) behavioral interpretation. In our model, the “cost function” used in the objective already includes a term that is designed to result in vehicle-neutral costs. Hence, the integral of this “cost function” can be interpreted as the total cost [see the derivation above] and this is being minimized in order to choose amongst all possible vehicle-neutral patterns [see the discussion of Definition 2 above]. Returning to the complete problem given in equations 42-45, it is again convenient to use a discrete approximation of this problem and solve using the method of finite differences. In particular, assuming that the total number of commuters has been divided into subintervals of size Aq, we have the following discrete approximation:
s.t. pp PE’
=$
Atpo’l~)10, p E ./, j = 1,. . . np n,LO,
pc.;P
(50) (51)
(52)
where cPo’l.)denotes the cost incurred by thejth packet of size Aq that chooses route p, n,, is the number of packets that choose route p, and Af,,~l~) is the (finite) difference in departure-times between packet i and packet i- 1 on route p.
349
Vehicle-based equilibrium model
Fig. I. A three-link network.
We can solve this problem in much the same way as we solved the departure-time choice problem. Given an initial departure-time, we can calculate the costs by sequentially assigning each packet to the route with the minimum value of c,(jl.) + M[Ac,ol.)]. In the case of a tie, the route with the earlier departure-time is chosen. This process continues until all of the packets are assigned. A simple search procedure is then used to determine the optimal initial departure-time. 4. NUMERICAL EXAMPLES
We now illustrate the above heuristic using two simple examples. In both of these examples we assume that the travel-time on link a is a function only of the service rate of the queue at the end of that link, so, and the free-flow travel-time, d,. Thus, for the case of a single link, u:
r(q + 41.) = max
d, + t(q + A&), t(&) + $
I
.
(53)
When there is no queue, the arrival time is simply the departure-time plus the free-flow travel-time. However, when there is a queue, the arrival-time of the (q+ Aq)th commuter is just the arrival time of the qth commuter plus the queuing time.6 4.1. An example of the departure-time choice problem for a single O-D pair Consider the three-link network in Fig. 1 in which there are 50 people traveling from node a to node d. The delay function parameters on these links are given in Table 1. All commuters are assumed to have a desired arrival time of 9. The early arrival cost is given by IY= 0.5, the late arrival cost is given by fi = 2, and the travel cost is given by y = 1. In order to make this problem somewhat more interesting, we assume that there is exogenous traffic, fO, entering the tail node of each of these three links as follows: f*(t) = 20, 0 5 t < 2
(54)
h(t) = 60,
(55)
0 I t5 4
fs(t) = 50, 5 5 t 5 12.
(56)
Table 1. Delay function parameters for the network in Fig. 1
Link 1 Link 2 Link 3
Free-flow time (I)
Service rate (3,)
t
40 70 50
1
6With this structure, it is actually possible to obtain analytic solutions to some problems. This is discussed further in Appendix A. lil(C) 4/6-B
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Francisco J. Jauffred and David Bernstein
The equilibrium solution departure-time function obtained using the heuristic described above is shown in Fig. 2. The first departure occurs at time 3.7995. The slope of the departure-time function is approximately 0.0225. That is, the headways between vehicles when they depart is 0.0225. As shown in Fig. 3, this is, in fact, an equilibrium. 4.2. A complete example We now consider a complete example of route and departure-time choice with multiple SD pairs. The network for this example is shown in Fig. 4 and the delay function parameters are given in Table 2. Now, there are two O-D pairs, ad and b-d with two routes connecting each. Route 1 for a-d uses links 1 and 3, route 2 for a-d uses links 1 and 4, route 1 for b-d uses links 2
s-
6_
I
I
I
I
10
20
30
40
*4
Fig. 2. The equilibrium departure-time function.
I?-
4
I 10
I 20
I
I
30
40
Fig. 3. The cost function in equilibrium.
w
4
Fig. 4. A four-link network with two O-D pairs.
)4
Vehicle-based equilibrium model
351
Table 2. Delay function parameters for the network in Fig. 4
Link Link Link Link
Free-flow time (I&)
Service rate (s,)
1
50
I 2 3 4
1
15
1
60
1
40
and 3, and route 2 for b-d uses links 2 and 4. Again, all commuters have a desired arrival time of 9 but now they have different early and late costs. In particular, for a-d commuters the early arrival cost is given by (Y,d=2, the late arrival cost is given by Bad= 2, and the travel cost is given by yad=4, whereas for b-d commuters the early arrival cost is given by c&j = 1, the late arrival cost is given by jIbd= 2 and the travel cost is given ybd = 4. The total number of commuters traveling from a to d is given by Dad= 1033 and the total number of commuters traveling from b to d is given by Du = 283. The equilibrium departure-time function for each of the O-D pairs is shown in Figs 5 and 6. [Note that negative departure-times are perfectly meaningful in this context, they simply represent times before 0.1 As you can see, the solution has two distinct regimes. Initially, only a-d commuters depart. Then commuters between both C&D pairs depart.
w A 15 -
10 -
s-
0
Fig. 5. The departure-time function for a-d.
9-
6_
3-
I so
I 100
I 150
I zoo
Fig. 6. The departure-time function for b-d.
I 250
*q
352
Francisco J. Jauffred and David Bernstein w 4 9_
3-
I 50
I 100
I 150
I 2m
I 250
)4
Fig. 7. The departure-time function for bd route 1.
As it turns out, ad commuters always choose their route 1 (though at three different rates; the first 366.67 vehicles have a departing headway of l/100, the next 133.33 have a departing headway of l/80, and the remaining vehicles have a departing headway of 3/100). On the other hand, bd commuters do split between the two routes, and their route choice functions are shown in Figs 7 and 8. Approximately 80.55 bd commuters will choose route 1, and the remainder will choose route 2. For the route 1 commuters, the departing headway between the first 66.67 commuters is 3/80 and the departing headway for the remainder is 9/50. For the route 2 commuters, the departing headway between the first 133.33 commuters is 3/160 and for the remainder is 91250. In this solution a-d commuters experience higher costs than bd commuters, and this is not surprising given that they have a higher early penalty. The resulting costs are illustrated in Figs 9 and 10. The cost for ad commuters is approximately 29.33 and the cost for a-d commuters is approximately 11.33.
5. CONCLUSIONS AND FUTURE RESEARCH
The primary objective of this paper was to demonstrate that there is an alternative to route/time-based formulations of the SRD equilibrium problem and that this alternative may be more attractive. Specifically, we have shown that it is possible to develop a vehicle-based
I
I
so
loo
I
150
I
200
I 250
Fig. 8. The departure-time function for b-d route 2.
)4
353
Vehicle-based equilibrium model
m-
10 -
I
I
I
I
200
400
600
800
*4
Fig. 9. The cost function in equilibrium for a-d.
formulation of the SRD equilibrium problem that can be motivated in a Nash-like fashion and has no more decision variables than existing route/time-based formulations (which are infinite dimensional). We have also shown that it is possible, at least for small networks, to solve a discrete approximation using a relatively simple heuristic. Though much work still needs to be done to demonstrate that this technique will work on large networks, our initial tests are encouraging. Hence, we intend to pursue this heuristic further. This will principally involve developing methods to evaluate the cost operators more efficiently (which was the principle bottleneck in solving these small problems). However, there is an interesting aspect of the discrete approximation that we used to solve for an equilibrium that we have not yet fully explored. In particular, observe that when Aqi= 1, the discrete approximation is tantamount to a model with integer-valued vehicles and real-valued time. This is interesting because no model of this kind has, to our knowledge, been developed as yet. As shown in Table 3, a variety of equilibrium models have been developed with realvalued vehicles and time, real-valued vehicles and integer-valued time, and integer-valued vehicles and integer-valued time. In general, models that would go in the upper left-hand comer of this table use infinite-dimensional variational inequality or optimal control formulation, while models that would go in the upper right-hand comer of this table use finite-dimensional variational inequality, optimal control, and math programming formulations. Finally, models that would go in the lower right-hand comer of this table typically use some form of time-based simulation in place of closed-form cost operators.
Fig. 10. The cost function in equilibrium for b-d.
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Francisco J. Jauffred and David Bernstein
What makes this intriguing is that the most realistic of the four categories seems to be the one that is ignored. On6 possible explanation for this is that, in general, the most realistic case is also the most complex. However, for (so-called) dynamic traffic assignment problems, this may not be true. To see this observe that, in the past, there have been two obstacles that needed to be overcome in the development of such models. The first is the problem of overtaking (or violation of the first-in, first-out discipline). The second is the problem of solving infinite-dimensional problems. Yet, models that would fall in the lower left-hand corner of Table 3 may be able to easily overcome both of these obstacles. In particular, it follows from Friesz et al. (1993), (Theorem 1 and footnote 3) that, in continuous time, overtaking can be avoided with appropriately defined delay functions. In addition, using integer-valued vehicles actually decreases the number of decision variables (i.e. making an infinite-dimensional problem finite-dimensional). Hence, in a future paper we will develop a vehicle-based formulation in which vehicles are explicitly integer-valued and time is real-valued. In this context we will first explore appropriate definitions of SRD equilibrium. Using a slight modification of the notation introduced above, it is easy to see what such a definition might look like. First, let tW= ( t ;, ...1 t&j E [0, TID” and rw = (,,...,r&) rw
E jtw
denote the vectors of departure-time choices and route choices for the D, w-commuters respectively. Further, let t = (t’,...,tW)~[O,i’JQ and r= (r’,...,?‘)~ /Q denote the complete vectors of departure-time and route choices and let c,” :[O, TIQ x YQ + R+ denote the cost experienced by commuter n of O-D pair w. Then, it seems quite natural to say that a route and departure-time pattern, (t,r), is a simultaneous route and departuretime choice (SRD) equilibrium if and only if (ifQ: (57) for all WET and n= 1,...,D,. That is, a route and departure-time pattern, (t r), is an SRD equilibrium if no commuter can reduce his or her travel cost given the behavior of all other commuters. However, it turns out that this definition may not be appropriate. This will be discussed in a future paper. In addition, we will explore methods of evaluating the cost operator cF(t, r) within this context. As it turns out, the problem of evaluating these cost operators has proven to be the most difficult aspect of incorporating the dynamics of traffic flow into network equilibrium models. In particular, even if a link’s performance (i.e. travel-time) at a particular time is a function of the number of vehicles on that link at that time, it is very difficult to model link performance as a function of departure-time and route choices. This is because the number of vehicles on a link at a particular time is itself determined by the travel-time on (potentially) all of the links in the network. To date, two approaches have been proposed to model these interactions. One involves time-based simulation and the other involves systems of integral and/or differential equations. However, it may also be possible to use a fixed-point formulation based on event-based simulation when vehicles are integer-valued and time is real-valued. This may enable us both to understand the properties of these operators and to evaluate them quickly and efficiently. Table 3. A categorization of some recent research
Real-valued vehicles
Integer-valued vehicles
Real-valued time
Integer-valued time
Bernstein et al. (1993) Friesz et al. (1993) Ran ef al. (I 996)
Smith (1993) Drissi-Kai’touni and Hameda-Benchekroun (1992) Janson (1991) Wie ef al. (1991) Mahmassani and Peeta (1992) Leonard ef al. (I 978)’
‘Some people have argued that the model developed by Leonard ef al. (1978) uses real-valued time. However, our understanding of their model is that it uses discrete time slices (i.e. it assumes that flow rates are constant over intervals of time). Hence, we still consider it a discrete time model.
Vehicle-based equilibrium model
355
Finally, we will explore questions related to the existence of such equilibria. In this regard, it may be necessary to add some “noise” to the model since discrete vehicles can give rise to discontinuities in the travel delay functions. These discontinuities can, in some cases, mean that no deterministic equilibrium exists. Hence, it may be necessary to consider a stochastic generalization of this model. Acknowledgements-This CONACYT.
research was supported in part by the Mexican Institute of Transportation
and
REFERENCES Amott, R., de Palma, A. and Lindsey, R. (1990) Departure time and route choice for the morning commute. Transportation Research -B, 7.4209228.
Bernstein, D. (1990) Programmable network equilibria. Ph.D. dissertation, University of Pennsylvania. Bernstein, D. and Smith, T. E. (1994) Equilibria for networks with lower semicontinuous costs: with an application to congestion pricing. Transportation Science, 28, 221-235. Bernstein, D., Friesz, T. L., Tobin, R. L. and Wie, B.-W. (1993) Variational control models of dynamic network traffic equilibrium. In Proceedings of the 12th International Symposium on Transportation and Trajic Theory, pp. 107-126. Elsevier, Amsterdam. Dafermos, S. C. (1971) An extended traffic assignment model with applications to two-way traffic. Transportation Science, 5, 366389.
Dafermos, S. C. and Sparrow, F. T. (1969) The traffic assignment problem for a general network. Journal of Research of National Bureau Stand, 73B, 91-118.
Dagatuo, C. F. (1985) The uniqueness of a time-dependent equilibrium distribution of arrivals at a single bottleneck. Transportation Science, 19, 29-37. Devarajan, S. (1981) A note on network equilibria and noncooperative games. Transportation Research -B, 15, 421426. Drissi-Kaitouni, 0. and Hameda-Benchekroun, A. (1992) A dynamic traffic assignment model and solution algorithm. Transportation Science, 26, 119-128. Fargier, P. H. (1981) Effects of the choice of departure time on road traffic congestion: theoretical approach. In Proceedings of the Eight International Symposium on Transportation and Tra@c Theory, pp. 223-263. University of Toronto Press. 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. Operations Research, 41, 379-191. Harker, P. and Pang, J. S. (1990) Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory. Algorithms and Applied Mathematical Programming, 48, 161-220. Hendrickson, C. and Kocur, G. (1981) Schedule delay and departure time decisions in a deterministic model. Transportation Science, 15, 62-77. Heydecker, B. G. (1986) On the definition of traffic equilibrium. Transportation Research -B, 20,435440. Janson, B. N. (1991) Dynamic traffic assignment for urban road networks. Transportation Research -B, 25, 143-161.
Kuwahara, M. (1990) Equilibrium queuing patterns at a two-tandem bottleneck during the morning peak. Transportation Science, 24, 2 17-229.
Leonard, D. R., Tough, J. B. and Baguley, P. C. (1978) CONTRAM: a traffic assignment model for predicting flows and queues during peak periods. Research Report 841, Transport and Road Research Laboratory, Crowthome, Berkshire, U.K. Mahmassani, H. S. and Herman, R. (1984) Dynamic user equilibrium departure time and route choice on idealized traffic arterials. Transportation Science, 18, 362-384. Mahmassani, H. S. and Peeta, S. (1992) Dynamic assignment for route guidance decisions. Presented at the 39th North American Meetings of the Regional Science Association fnternational.
Nash, J. F. (1952) Non-cooperative games. Annals of Mathematics, 54, 286295. de Palma, A., Ben-Akiva, M., Lefevre, C. and Litinas, N. (1983) Stochastic equilibrium mode1 of peak period traffic congestion. Transportation Science, 17,430-453. Ran, B., Hall, R. W. and Boyce, D. E. (1996) A link-based variational inequality model for dynamic departure time/route choice. Transportation Research -B, 30, 31-46. Rosenthal, R. W. (1973) The network equilibrium problem in integers. Networks, 3, 53-59. Smith, M. J. (1984) Two alternative definitions of traffic equilibrium. Transportation Research -B, 18, 63-65. Smith, M. J. (1993) A new dynamic traffic model and the existence and calculation of dynamic user equilibria on congested capacity-constrained road networks. Transportation Research -B, 27,49-63. Vickrey, W. S. (1969) Congestion theory and transport investment. Papers and Proceedings of the American Economic Review, Vol. 59, pp. 251-261. Wardrop, J. G. (1952) Some theoretical aspects of road traffic research. In Proceedings of the Institution of Civil Engineers, Part II, pp. 32>378. Wie, B.-W., Friesz, T. L., Tobin, R. L. and Bernstein, D. (1991) A discrete time, nested cost operator approach to the dynamic network user equilibrium problem. Presented at TRISTAN. Wie, B.-W., Tobin, R. L., Bernstein, D. and Friesz, T. L. (1996) A comparison of system optimum and user equilibrium dynamic traffic assignment with schedule delays. Transportation Research -C, 3, 38941 I. Zhu, D. L. and Marcotte, P. (1996) On the existence of solutions to the dynamic user equilibrium problem. Manuscript, Universite de Montreal, Canada.
356
Francisco J. Jauffred and David Bernstein APPENDIX A: OBTAINING ANALYTIC SOLUTIONS
When travel times are modeled as deterministic queues it is sometimes possible to obtain analytic solutions to the SRD equilibrium problem. Some such results have already been presented by Vickrey (1969); Fargier (1981); Daganzo (1985); Arnott et 01. (1990); and Kuwahara (1990). In this Appendix A we demonstrate how the results above can also be used to obtain analytic solutions in some cases. For sake of simplicity, we only consider the case of departure time choice. That is, we consider networks with one &D pair and one route connecting that origin and destination. However, these results can be generalized to include route choices. First, suppose that the route of interest consists of a single link. Then, expressing equation 53 in terms of finite differences, it follows that d, + f(ql.)
-
t(ql.)
+
At(ql.). $
I
Then, allowing the differences to become arbitrarily small, we have the following: d, + t(q/.)
-
s(ql,)
= 0 +
Wql.) ~ dq
Wql.) = dq
and d, + t(q1.)
-
dr(ql.) c 0 =+ dq
r(ql.)
1 = -. $0
More generally, we can include exogenous traffic departing at a ratefO(f) by defining the as follows:
equivalenr
service
rare,
s;,
that is, by assuming that the equivalent service rate is proportional to the fraction of stream vehicles that enter the link in that particular period of time. Alternatively,
Then, substituting into equation Al yields
4, + W) - 441.) + At(&),
Aq +h(r)Ar s,
cw
and allowing the differences to become arbitrarily small da + f(qj.)
-
r(ql.)
dr(ql.)
< 0 =+ -
1+h@ &I
= ~.
dq
(A7)
Now, observe from equation I2 that:
1
d&It) ----=ooj~ dq
+ &(ql.)]
dr(4.) = 0 drl
(‘48)
and hence that (A9) Hence, it follows that in equilibrium:
(AlO) In general, when routes contain multiple links, the maximization in equation Al must be taken over more terms. In particular, a route with m links will have m+ I terms on the right-hand-side of equation 12. Following through, one is left with 2”’ dynamic equations of the type shown in equations A2 and A3. Therefore, this is a fairly tedious process for long routes. As an example, consider a single link network with d= 0, s = 50, and Q = 100. The early arrival cost is OS/min, the late arrival cost is Z/min and the travel cost is I/min. Further, suppose there is an exogenous traffic source of 50/h and f = 9. For early arrivals:
df(ql.) -=dq
w
1 _yj.!+5=j$
I
(Al 1)
Vehicle-based equilibrium model
351
whereas for late arrivals: !g
WI.) -=dq
3
1-so.!#=-~
(Al21
Hence, there are no late arrivals and the equilibrium departure-time functions must satisfy:
$.
6413)
$ = (01.) + &.
C414)
r(q1.)= r(O1.)+ Further, the arrival-time function is given by
?(q(.)= (01.) + Therefore, the average cost is: &I.)=
[r(Ocol., +$] - [@I.) +$] + 0.5[9 - (m9 +$)I
= 4.5 - 0.5 t(ol.).
(Al51
To minimize the cost we must make the first departure-time as close to 9 without causing late arrivals. Hence, the equilibrium departure-time function is given by: r(ql.) = 5 + $ with equilibrium cost of 2.
Pergamon
PII: so%8-o%x(%)ooo1&6
AN ALTERNATIVE FORMULATION OF THE SIMULTANEOUS ROUTE AND DEPARTURE-TIME CHOICE EQUILIBRIUM PROBLEM FRANCISCO
J. JAUFFRED’
and DAVID BERNSTEIN2
‘Center for Transportation Studies, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. *Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ 08544, U.S.A. Abstract-Traditional (static) network equilibrium models have always been formulated in a route-based fashion rather than a vehicle-based fashion. That is, the decision variables have been the number of vehicles using each route rather than the route choices of each vehicle. Given the success of this approach, it is not surprising that recent “dynamic” network equilibrium models have been formulated in a similar way. That is. the decision variables in these models are usually the route-specific departure rates over time. In this paper, we develop a vehicle-based equilibrium model of simultaneous route and departure-time choice and discuss the possible advantages of this approach. We then describe a heuristic for solving this model and demonstrate its effectiveness on several small examples. Copyright c 1996 Elsevier Science Ltd
I. INTRODUCTION
In order to accurately evaluate many of the transportation policy questions of interest today (e.g. the impact of telecommunications, route guidance and driver information systems, and time-varying congestion pricing) we need higher resolution models than we have needed in the past. That is, things that in the past have been treated as “black boxes” must now be modeled in some detail. To that end, a great deal of attention has been devoted to incorporating traffic dynamics into models of route and departure-time choice. Traditional models of route choice, in which travel times are a function of the average usage of a facility over some (fairly long) period, are no longer considered adequate. Instead, models are being developed which explicitly account for the “timing” of vehicle interactions, how these interactions affect travel times, and how these travel times ultimately affect behavior. Much of the effort in this area has been devoted to the development of equilibrium models that include these kinds of time-dependent phenomena. Some of these models simply attempt to expand Wardrop’s (Wardrop, 1952) model of route choice to include traffic dynamics more explicitly. Others also attempt to include the departure-time choices of the commuters. In both cases, the models have been enormously influenced by the way in which Wardrop (1952) originally posed the problem. Recall that his original criterion was stated as follows: “The journey times on all the routes actually used are equal, and less than those which would be experienced... on any unused route” (p. 345). What is interesting about this definition of equilibrium is that it is stated in route-based as opposed to vehicle-bused terms. That is, equilibrium is defined in terms of attributes of the routes rather than attributes of the vehicles, hence the number of decision variables is related to the number of routes and not the number of vehicles. This is true of Wardrop’s original definition as well as many other related definitions of equilibrium that have been developed subsequently [see, for example, Dafermos and Sparrow, 1969; Dafermos, 1971; Smith, 1984; Heydecker, 1986 and Bernstein and Smith 19941. In many respects, it is not at all surprising that all of these definitions are route-based given the origins of the problem (i.e. traffic flow theory). In particular, observe that traffic volumes and flows have historically been assumed to be real-valued (rather than
340
Francisco J. Jauffred and David Bernstein
integer-valued). Combining this assumption of real-valued volumes with a vehicle-based model one is left with a formulation in which there are an infinite number of decision variables. Specifically, letting D denote the total number of vehicles, it is necessary to solve for the routes choices of the infinite number of “vehicles” in the interval [O,D]. On the other hand, combining the assumption of real-valued volumes with a route-based model leads to a formulation in which there are only a finite number of decision variables. That is, letting /denote the finite number of (loopless) paths in the network, it is only necessary to solve for the number of vehicles on each path in /i Hence, in the traditional setting, the mathematics of route-based models is much simpler than the mathematics of vehicle-based models. What may seem surprising is that even when route volumes are assumed to be integervalued the equilibrium problem has been presented in route-based terms [see, for example, Rosenthal, 1973 and Bernstein, 19901. The resulting models are generally formulated as finite-dimensional integer programs in which the decision variables are the (integervalued) number of vehicles on each route. One could, just as easily, define the problem in such a way that the decision variables are the route choices of the finite number of vehicles. However, since these integer-valued models have principally been developed in order to gain insight into the real-valued models, it is not really surprising that they have been route-based. What is much more surprising, at least to us, is that almost all equilibrium models that include time-dependent phenomena have been formulated in a route/time-based fashion. That is, these models have been formulated in terms of route-specific departure rates (or volumes) over time [see, for example, Friesz et al., 1993; Bernstein et al., 1993; Ran et al., 1996; Wie et af., 1996; and Zhu and Marcotte, 19961.’ We think this is surprising because, when tra@ic dynamics are incorporated, such route/ time-based formulations are not of lower dimensionality than vehicle-based formulations.
The decision variables in these route/time-based models are the route-specific departure rates over time. Hence they are naturally infinite dimensional unless time is discretized (which, as discussed in Section 5, introduces other difficulties and inaccuracies). On the other hand, there do appear to be some advantages to using a vehicle-based formulation. Specifically, using discrete vehicles it may be possible to develop a finite dimensional vehicle-based formulation in which the decision variables are the route and departure-time choices of each of the (finite number of) vehicles. In addition, even infinite dimensional vehicle-based models (i.e. vehicle-based models developed assuming that vehicles are real-valued) seem to be easier to motivate understanding than their route/time-based counterparts since we can actually discuss the problem in a game-theoretic setting that involves players and their actions. Hence, in this paper we take the first steps towards developing a vehicle-based formulation of the simultaneous route and departure-time choice (SRD) equilibrium problem. We begin in Section 2 by defining such equilibria in a vehicle-based context and formulating the equilibrium problem as a collection of minimization problems. In Section 3 we develop a heuristic for solving this problem and then, in Section 4, we solve some illustrative examples. Finally, in Section 5 we discuss how these results might be extended. 2. THE SETTING
As discussed above, our purpose is to develop a model of rush-hour (automobile) traffic in which commuters choose both the time they are going to depart from their homes and the roads they are going to use to get from home to work. To formalize this choice process, let R (Z),R +(Z+), and R + +(Z+ +) denote the real numbers (integers), nonnegative reals (nonnegative integers), and positive reals (positive integers), respectively ‘Though our interest is in developing a model set on a general network that can be solved numerically, it is worth pointing out that even those models set on simple networks and solved analytically. See, for example, Vickrey (1969). Fargier (1981), Hendrickson and Kocur (1981) Mahmassani and Herman (1984) Daganzo (l985), Arnott ef al. (1990), and Kuwahara (1990), have been formulated in a route/time-based fashion.
Vehicle-based
equilibrium
model
341
and let the relevant departure time horizon be given by [0,7’j~R + . Further, let 6 = ( I, , ) denote an abstract transportation network comprised of a finite set of nodes, I, with cardinality N= 1 I 1, and a finite set of links (or arcs), /, with cardinality A = 1 , I. The number of commuters traveling between any pair of nodes in I x I is assumed to be known. For simplicity, we will only consider those origin-destination pairs with a positive number of commuters and will denote the set of all such pairs by ,Y (with cardinality W). The relevant demand vector is then denoted by D = (D,.:wEY )ER~+ with the total demand being given by Q = CwEWD ,,,. Commuters are assumed to be identical except, perhaps, for their origin and destination. We will refer to commuters traveling between O-D pair w as w-commuters. Each commuter travels from a specific origin node to a specific destination node on a route, p, which is simply an acyclic chain of links.* The set of all possible routes for w-commuter is denoted by ./;1,and has (finite) cardinality P&O. Further, the set of all relevant routes is denoted by / = U,,, P,., and has cardinality P = C,,, P,.. As mentioned in the Introduction, what makes our formulation of the SRD equilibrium problem different from earlier formulations is the decision variables. To date, most SRD equilibrium models have been formulated in such a way that the decision variables are the departure rates over time (i.e. using a route/time-based approach). However, we think complicated models are easier to understand when they are formulated in terms of the actual decisions being made by the actual decision-makers. In this case, the decisionmakers are the commuters, and each w-commuter chooses a departure-time, t~[o,T], and a route, pi /;y. Hence, in the remainder of this section we will develop a formulation of the SRD equilibrium problem in terms of these players and actions (i.e. using a vehicle-based approach). Specifically, we now let t,,. and rw denote the departure-time choice function and route choice function for O-D pair WEI. In other words, tJ9(.) denotes the departure-time choice of w-commuter 9 and r,,,(qI.) denotes the route choice of w-commuter 9. Further, we let t=(t, ,...,t& and r = (r ,,...,rW). At this point, we have not specified how commuters make these choices. In general, these choices can be based on a variety of different factors. As this discussion progresses, we will assume that these choices are based on some notion of travel cost. Now, observe that since we have implicitly ordered each set of w-commuters, we must restrict the set of feasible departure-time patterns to: /L~ = (t,,, : [tw(91.)- t,(kl.)](q - k) > 0, for all 9, k E [0, D,.] with 9 # k). That is, given any two w-commuters AA,=(r,,.:r,,, maps into 1) it follows terns is given by the pair ( /-, A))= Now, each feasible route and
(1)
9 and k, if 9 > k then 9 must depart after k. So, letting that the set of feasible route and departure-time pat(L&,,
1.,,.,u,~~,, n,,.).
departure-time pattern, (t,r)E( /, n’), gives rise to an arrival-time pattern, t,: [O,D,J x ( 1; AJ)-R + + , for w-commuters. We will denote the arrival-time for w-commuter 9 under route and departure-time pattern (t,r) by t,V (q[t,r) and hence the travel-time for this same w-commuter is given by r,.(qlt,r)-t,,.(qlt-{t,,.},r). We are not concerned here with the exact nature of these arrival-time operators, they can be developed in a variety of different ways (e.g. systems of differential or integral equations). We do, however, assume that they are regular in the sense of Friesz ef al. (1993) (Theorem 1 and footnote 3). That is, these arrival-time operators are assumed to be of a form which makes them monotonically increasing in 9. In Section 4 and Appendix A we present a specific example of the types of arrival-time operators that can be used. Further, following de Palma et al. (1983) we assume that the cost a commuter incurs is a function of both the travel time and the arrival time (as it compares to some desired arrival time). In particular, if T*ER + + denotes the (common) desired arrival-time for all 2We use p to denote a physical choice.
route rather
than r because we will later use the notation
r to denote a route
Francisco J. Jauffred and David Bernstein
342
commuters and &R+ denotes the actual arrival-time for a particular commuter, then the schedule delay cost, I&R + + +R + , incurred by that commuter is given by:
if T* > CY ifT*=a if T’ < ~7
(2)
where a! denotes the (common) value of early arrival time and /I denotes the (common) value of late arrival time. Thus, the travel cost for w-commuters, c,:[O,&] x( 7;~ )+R + + , is given by: c&It,
r) = y. [tw(qlt, r) - tdqlt - IhI, 41 + dM4lt~
4.
(3)
where y denotes the (common) value of travel-time.3 With these fundamental concepts in hand, we can now define what we mean by an equilibrium. Of course, a natural place to begin is with a Nash-like [see Nash, 19521 definition in which no individual commuter can reduce her/his cost by unilaterally changing either her/his route or departure-time. Unfortunately, this is not as easy as it seems in our current context since we do not really have individual vehicles (i.e., since q is assumed to be real-valued). What we might do instead is to think about changes in (t,,,,r,) for a particular q. For example, given a route and departure-time pattern, (t,,,,r,,J, we could represent a change in the behavior of the qth vehicles by creating a new pattern, (&, rW), in which ;,,&I.) =r,Jkj.) and ?,(k(.) = t,(kl.) for all k#q. However, it is relatively easy to see (at an intuitive level) that such an approach is virtually meaningless. In particular, if we adopt the fairly standard and reasonable assumption that the travel time on a route is some function of the number of vehicles using the links in that route, then it follows that the costs under (i,, F,,,) would be identical to those under (t,,,,r,J, regardless of the values of (t,,,,r,,,) and q. This is because the number of vehicles on a link is some function of an integral involving the route and departure-time functions. Hence, since (t,,,,r,,,) and (i,, rW> can only differ on a set of measure zero, the trajectories of all link volumes will be the same under both route and departure-time patterns. To define a notion of equilibrium in this context we instead make use of a convenient fiction first introduced by Devarajan (1981) for the static problem. In particular, we suppose that each origindestination pair assigns people to routes and departure-times in such a way that each commuter experiences equal cost [in the spirit of Wardrop, 19521and that the O-D pair experiences minimum cost. More formally, letting t-, = rw), and using the notation t_, + tw 0 I,... ,tw-~,tW+ I,..., td and r-,,, = (rl, .. .. rw-l, rw+l, . ... and r+ + rW to denote (tl, .... t,-1, t,, &,,+I, ... . tw) s t and (ri, .. .. r,,,-1, r,,,, r,,,+l, . . . . rw) E r respectively, we have the following preliminary definition:4 Dejinition I. (i) A route and departure-time pattern, (t,r)E( Y;M), is said to be vehicleneutral iff there exist constants, CwE R,,, such that:
c&It, r) = &
(4)
for all WEP and qE[O,D,]. (ii) The set of all vehicle-neutral route and departure-time patterns in ( 7;~) is denoted by 1 w. (iii) Given (t-w,r_w)E/’ w-‘, the set of all (t,,,,r,,,) such that (t,r)El w is denoted by r,(t_,,,r_,).With this, we now have the following definition of equilibrium: 3The assumption that all commuters have the same desired arrival-time, value of travel-time, value of earlytime, and value of late-time is for notational convenience only. This assumption could be relaxed without significantly changing the analysis that follows. 4We do not use measure-theoretic arguments in Definitions 1 and 2 because we prefer to simplify the exposition and because they are not necessary for the discussion that follows. Specifically, since we ultimately solve for equilibria by discretixing the demand, we do not need to concern ourselves with the behavior of the system on sets of measure zero. Friesz et al. (1993) provide a careful discussion of the issues involved.
Vehicle-based equilibrium model
DeJinition 2. A route and departure-time
343
pattern, (t,r)~( CM), is said to be an equili-
brium if
for all WEE and qE[O,D,,,].That is, a route and departure-time pattern is an equilibrium if no w-commuter wants to trade places with another w-commuter and, subject to this condition, they are minimizing their costs. When c, is continuously differentiable in q it follows from Definitions 1 and 2 that each SD pair, WEZ, is trying to solve the following infinite dimensional minimization problem:5 D, min c&It, r) dq/& &V.~,)E( ' [email protected] J
(6)
0
st .
.
k&l~~ dq
r>=
o
(7)
given the route and departure-time patterns of the W-l other SD pairs, (ti ,..., t,,t,+ I ,..., t&TW-’ and (rl,..., rW,rw+r,..., r&E x,“-‘. The equivalence between equation 7 and vehicle-neutrality can be seen by integrating equation 7. The equivalence between equation 6 and equation 5 can be seen by observing that when equation 7 is satisfied, the cost incurred by all w-commuters is the same. Hence the cost to each w-commuter is simply the total cost experienced by all w-commuters divided by the number of w-commuters. Since the total cost is simply the “sum” of the costs experienced by each commuter, the total cost in this case is given by 4 c,(qlt, r) dq and 0 D, the (common) cost to each commuter is given by J c,(qlt, r) dq/D,. It is important to 0
note that, although the objective function in this problem has a very similar structure to the objective function in math programming formulations of the so-called static equilibrium problem, it is actually unrelated. In this problem, the objective function represents the average cost experienced by w-commuters, whereas in the static problem the objective function is an artificial construct. Also, in this problem the upper limit of integration is known, whereas in the static problem the upper limit of integration is one of the decision variables. This is discussed more fully below. 3. A SOLUTION APPROACH
Though the assumption of continuous differentiability is unlikely to hold in practice, the W minimization problems in equations 6 and 7 do provide a useful way to motivate a solution method for the more general problem. Hence, we will now develop a heuristic for soIving the general equilibrium problem by proceeding as if the cost operators are continuously differentiable functions. 3.1. Overview of the approach Specifically, we will solve the equilibrium problem by iteratively solving the minimization problems in equations 6 and 7 for each O-D pair sequentially. That is, at each iteration, 1, we will solve the minimization problem in equations 6 and 7 for a particular (rD pair w, treating all other O-D pairs as given. 50f course, it is well known that such a collection of minimization problems can be formulated as a single variational inequality problem [see, for example, Harker and Pang, 19901.With appropriate assumptions and letting the subdifferential of e, be given by &Jr(O)] = {a : c&r (0), .... s”,, .... zw(O)]l c,&(O)] + a]s, - r,(O)] for all s.+, it is relatively easy to show that a route and departure-time pattern, (F. ?) E ( /-, H), with dc,(qjt,r)/dq=O, q~[O,D,,,] is an equilibrium iff C,,, gc,[7(0)][rw(O)- T,JO)] 2 0 for all (r,r)~( /,-,Y) with dc,(q] t,r)/dq = 0, qE[O,D,,,].However, as our concern here is primarily with developing a solution method we will not dwell on this variational inequality formulation.
344
Francisco J. Jauffred and David Bernstein
Letting t,,,l denote the value of t, at iteration 1 and I-,,,~denote the value of r,,, at iteration 1, this process can be summarized as follows: Step 0. Set I= 1. Find an initial feasible solution (tl,o ,..., tw,o), (q. ,..., rw,& Step 1. Solve the necessary minimization problem for each O-D pair consecutively. Step 1.1. Set w= 1. Step 1.2. Solve equations 6 and 7 for O-D pair w given
(t1,r, *.., t,-l.I, ~,+I.I-I..., tw,r-l) and (~I,L . .. . rw-l,~, rw+l.~-l..., QV,I-I) Step 1.1 Step 2. If (~IJ, .... rw,i)=(ti, I- 1, .... fw,i - 1) and (Q.L ... . QVJ)=(~IJ-I, .... rw,r-1) then STOP. Otherwise set I= I+ 1 and GOT0 step 1. Step 1.3. Set w = w + 1. If w< W then GOT0
Many different convergence criteria can be used in Step 2 of this heuristic. We chose to stop iterating when w
Dw
U[
tw,dxl-)-
M-1
w
tw,,-~(Xl-)]2 dx 5 et
and
c W-1
0
Dw
J[r,&+) - rw,,-~(xl~)]z dx i
G.
0
Obviously, this method requires that we be able to solve the necessary minimization problem for a single O-D pair given some level of “background traffic” (i.e. traffic that is generated by the other O-D pairs). We describe a method for doing so below. First, we consider the case when there is a single route connecting a single origin and destination, then we generalize to the case of multiple routes connecting a single origin and destination. In both cases, we develop an optimization problem assuming that the cost operator and the departure-time function are continuously differentiable, relax the complicating constraints using a penalty function, and solve using finite differences. In order to simplify the notation in the following sections we will no longer include the subscripts indexing the O-D pair (since we will only be concerned with a single O-D pair at a time). Also for notational simplicity we will denote dc(ql+)/dq by c?(ql.) and dt(q(.)/dq by Q(11.). 3.2. Solving the departure-time choice problem for a single O-D pair On a network with a single O-D pair and a single route equations 6 and 7 reduce to the following departure-time choice problem: D
(8) s.t. t E 7-
(9)
c’(qlt) = 0
(10)
Now, observe from the definition of /in equation 1 that when t is continuously differentiable equation 9 can be ;&ten as /(q) > 0. In addition, observe that since D is D
constant, t is a solution of m;ln J’ c(qlt) dq/D iff it is also a solution of m,in J c(qlt) dq. n
n
Using these two observations w’e can re-write equations 8, 9 and 10 as thi following infinite dimensional minimization problem:
(11) s.t. c’(q(t) = 0
(12)
t’(ql*)’ 0
(13)
345
Vehicle-based equilibrium model
Though there are many potential ways to solve equations 11, 12 and 13 perhaps the most obvious is to relax equation 12 as follows: D
rn;n V =
c(qlt) + h@‘(qlt)l dq
J
(14)
0
s.t.
t’(ql.) > 0
(15)
where M is a penalty function and then solve the resulting Euler-Lagrange equations. In this case, this would require that we solve the following system of differential equations:
av at
d av=, dq ar
(16)
av --0 atw)
(17)
However, since in reality c and t may not be differentiable, this does not seem to be the best approach in practice. Instead, we make use of the method of finite differences. To do so we will discretize the closed interval, [O,D]cR+, into n~z+ +, subintervals in such a way that [O,D]= [O,qllU[cll,q2lU[qn-l,q,l and O
i= l,...,n
where q. = 0 by definition. Further, for any function,J[O,D]-+R,
(18) we will let:
Af(qi)=f(q;)-f(qi-l), i= l,...,n.
(19)
Then, it follows that we can approximate equations 14 and 15 in terms of these finite differences as follows: mm V = &c(q;lt)
i=o
+ M[Ac(qil~)l)Aq
s.t. At(qil.) > 0, i = 0, . . . , H.
Observe that it follows from equations 19, 3 and 2 that Ac(qi]t)= y.[As(qii.)-At(qjJ.)]
(20) (21) +
Adt(qiI.)I.
The easiest way to interpret this problem is to consider the case in which Aqi= 1 for i= 1,...,n (i.e. in which we have discrete vehicles). Equation 21 simply requires that vehicle i depart after vehicle i- 1. The two terms in the objective function represent the total cost of travel for the n( = D) vehicles and the penalty for violating vehicle neutrality. Given an appropriate choice of penalty function, equation 20 will be minimized when 5 M[Ac(qilt)] = 0, m ’ which case all n vehicles experience the same cost. When Aq, > 1 the i-0
problem can be interpreted in the same way, except that we must think about packets of vehicles. As Aqi+O for all i= 1,...,n the problem in equations 20 and 21 approaches the problem in equations 14 and 15. The only remaining difficulty in solving this problem is evaluating c(q;jt), the cost experienced by vehicle (or packet) i under departure-time pattern t. In general, it can be quite computationally expensive to evaluate c(qilt) since, as the notation indicates, it depends on the complete trajectory oft. We overcome this difficulty by using an iteratively updated estimate of this cost. We denote the estimate by R[.] and update it on a packetby-packet basis, using the cost calculated for packet i-l as the estimate for packet i.
346
Francisco J. Jauffred and David Bernstein
Using this iterative estimation process, it is relatively easy to solve equations 20 and 21. The complete heuristic for the departure-time choice problem with a single route and CD pair can be summarized as follows: Step 0. Set R[t(qo)] = 0. Set i= 0. Step 1. Solve the following problem:
$~,“1 = (Nt(qiI.)l+ MAc(qilt)l). Let At*(qil*) and q denote the solution and the optimal value of the objective function respectively. Step 2. Set R[t(qi+ ,I.)] = Z;. Set i= i+ 1. If i
7;~ C(C[~ilt(O), At(qil-)I
+ R[t(q~l.)l)Aq
r=O
The particular penalty function we choose to use is M[x] =K(x/e)” where mEZ+ +, KEZ + + is a large number, and EEZ + + is a small number. As illustrated in the example in Section 4.1, this approach seems to work reasonably well. 3.3. Solving the equilibrium problem for a single O-D pair In principle, it is not difficult to expand the single (TD pair problem to include route choice as well as departure-time choice. In particular, it follows from the same arguments used in developing equations 11, 12 and 13 that we need to solve the following problem: D y”
c(qlt, ’
4
dq
(22)
J 0
s.t. c’(qlt, r) = 0
t’(d)
’
rE R
0
(23) (24) (25)
The difficulty is that this is an infinite dimensional, nonlinear, mixed integer programming problem. Thus, in this form, this problem is very to difficult to solve. However, as it turns out, it is possible to transform this problem into one that is much easier to deal with. To do so we treat each route as if it has its own one-to-one, strictly monotone, and differentiable departure-time function, $(q). This allows us to develop a formulation without the route choice function, r. To see why, first observe that the O-D departure-time function and the route-specific departure-time functions are related as follows: (26) Intuitively, equation 26 says time) equals the sum of the pair. (Recall that since we are specific indexes). Now, observe that given all p&f, and hence that:
that the O-D departure rate (measured in vehicles per unit path-specific departure rates for all paths serving the O-D only considering a single O-D pair we have dropped the O-D our assumption about tp, the inverse function, r;*, exists for
(27)
341
Vehicle-based equilibrium model So,
it follows that
dqp=
$J+ dq P
for all pc.2 Letting h,eR+ further follows that:
(28)
p
denote the number of commuters
h
that choose route p, it
(29)
P
for all PG./“,and hence that: (30) Finally, letting h = (hp : p E P) E Rr denote the complete vector of route choices, it follows that: rc
&wchp=D.
(31)
PC ’
Substituting equation 30 into equation 22 and equation 3 1 into equation 25 it follows that we can alternatively solve the following minimization problem:
(32)
s.t. c'(qplt) = 0, p E /
(33) (34)
$h?pl.) >
0, p
h,zO,
PE
6 ’
(35)
/’
(36)
where c’(qplt) denotes dc(qP(t)/dqp and tk(qp(.) denotes dtp(qp)l.)/dqP. Even with this transformation, this problem can be very dificult to solve as a result of equation 33. Hence, as in the single-route case, we relax this problem using a penalty function to obtain: 4
min th
(c(qp
It)
+
Mc’(qp
s.t. chp
IOI)$$
dqp
(37)
P
= D
(38)
RE/
(39)
()(qpl.) > 0, p E .” h,zO,
PE
(40)
/.
Further, letting: cp(qlt)
=
(c(qpIt)
+
~4qpM)~
;
* P.
(41)
348
Francisco J. Jauffred and David Bernstein
we can re-write this problem as:
(42)
s.t.
c
hp =
D
(43)
PE *
$(qpl*) > 0, p E .4 hp 10,
p E 4.
(45)
It is interesting to note that, given a specific departure-time function (found, for example, using the heuristic described above), we can solve for the partial equilibrium route-choices by solving the following problem: (46)
s.t. chp
= D
(47)
hp > 0, p E .Y.
(48)
PE ’
This problem is clearly very similar in structure to the original programming formulation developed by Beckmann et al. (1956). However, it is important to observe that the nature of this problem is quite different from that of Beckmann et al. (1956). In particular, observe that in their model, the objective function is artificial in the sense that it is used simply because it leads to Karush-Kuhn-Tucker conditions which are equivalent to the equilibrium conditions. That is, the integral of the costs has no (immediately obvious) behavioral interpretation. In our model, the “cost function” used in the objective already includes a term that is designed to result in vehicle-neutral costs. Hence, the integral of this “cost function” can be interpreted as the total cost [see the derivation above] and this is being minimized in order to choose amongst all possible vehicle-neutral patterns [see the discussion of Definition 2 above]. Returning to the complete problem given in equations 42-45, it is again convenient to use a discrete approximation of this problem and solve using the method of finite differences. In particular, assuming that the total number of commuters has been divided into subintervals of size Aq, we have the following discrete approximation:
s.t. pp PE’
=$
Atpo’l~)10, p E ./, j = 1,. . . np n,LO,
pc.;P
(50) (51)
(52)
where cPo’l.)denotes the cost incurred by thejth packet of size Aq that chooses route p, n,, is the number of packets that choose route p, and Af,,~l~) is the (finite) difference in departure-times between packet i and packet i- 1 on route p.
349
Vehicle-based equilibrium model
Fig. I. A three-link network.
We can solve this problem in much the same way as we solved the departure-time choice problem. Given an initial departure-time, we can calculate the costs by sequentially assigning each packet to the route with the minimum value of c,(jl.) + M[Ac,ol.)]. In the case of a tie, the route with the earlier departure-time is chosen. This process continues until all of the packets are assigned. A simple search procedure is then used to determine the optimal initial departure-time. 4. NUMERICAL EXAMPLES
We now illustrate the above heuristic using two simple examples. In both of these examples we assume that the travel-time on link a is a function only of the service rate of the queue at the end of that link, so, and the free-flow travel-time, d,. Thus, for the case of a single link, u:
r(q + 41.) = max
d, + t(q + A&), t(&) + $
I
.
(53)
When there is no queue, the arrival time is simply the departure-time plus the free-flow travel-time. However, when there is a queue, the arrival-time of the (q+ Aq)th commuter is just the arrival time of the qth commuter plus the queuing time.6 4.1. An example of the departure-time choice problem for a single O-D pair Consider the three-link network in Fig. 1 in which there are 50 people traveling from node a to node d. The delay function parameters on these links are given in Table 1. All commuters are assumed to have a desired arrival time of 9. The early arrival cost is given by IY= 0.5, the late arrival cost is given by fi = 2, and the travel cost is given by y = 1. In order to make this problem somewhat more interesting, we assume that there is exogenous traffic, fO, entering the tail node of each of these three links as follows: f*(t) = 20, 0 5 t < 2
(54)
h(t) = 60,
(55)
0 I t5 4
fs(t) = 50, 5 5 t 5 12.
(56)
Table 1. Delay function parameters for the network in Fig. 1
Link 1 Link 2 Link 3
Free-flow time (I)
Service rate (3,)
t
40 70 50
1
6With this structure, it is actually possible to obtain analytic solutions to some problems. This is discussed further in Appendix A. lil(C) 4/6-B
350
Francisco J. Jauffred and David Bernstein
The equilibrium solution departure-time function obtained using the heuristic described above is shown in Fig. 2. The first departure occurs at time 3.7995. The slope of the departure-time function is approximately 0.0225. That is, the headways between vehicles when they depart is 0.0225. As shown in Fig. 3, this is, in fact, an equilibrium. 4.2. A complete example We now consider a complete example of route and departure-time choice with multiple SD pairs. The network for this example is shown in Fig. 4 and the delay function parameters are given in Table 2. Now, there are two O-D pairs, ad and b-d with two routes connecting each. Route 1 for a-d uses links 1 and 3, route 2 for a-d uses links 1 and 4, route 1 for b-d uses links 2
s-
6_
I
I
I
I
10
20
30
40
*4
Fig. 2. The equilibrium departure-time function.
I?-
4
I 10
I 20
I
I
30
40
Fig. 3. The cost function in equilibrium.
w
4
Fig. 4. A four-link network with two O-D pairs.
)4
Vehicle-based equilibrium model
351
Table 2. Delay function parameters for the network in Fig. 4
Link Link Link Link
Free-flow time (I&)
Service rate (s,)
1
50
I 2 3 4
1
15
1
60
1
40
and 3, and route 2 for b-d uses links 2 and 4. Again, all commuters have a desired arrival time of 9 but now they have different early and late costs. In particular, for a-d commuters the early arrival cost is given by (Y,d=2, the late arrival cost is given by Bad= 2, and the travel cost is given by yad=4, whereas for b-d commuters the early arrival cost is given by c&j = 1, the late arrival cost is given by jIbd= 2 and the travel cost is given ybd = 4. The total number of commuters traveling from a to d is given by Dad= 1033 and the total number of commuters traveling from b to d is given by Du = 283. The equilibrium departure-time function for each of the O-D pairs is shown in Figs 5 and 6. [Note that negative departure-times are perfectly meaningful in this context, they simply represent times before 0.1 As you can see, the solution has two distinct regimes. Initially, only a-d commuters depart. Then commuters between both C&D pairs depart.
w A 15 -
10 -
s-
0
Fig. 5. The departure-time function for a-d.
9-
6_
3-
I so
I 100
I 150
I zoo
Fig. 6. The departure-time function for b-d.
I 250
*q
352
Francisco J. Jauffred and David Bernstein w 4 9_
3-
I 50
I 100
I 150
I 2m
I 250
)4
Fig. 7. The departure-time function for bd route 1.
As it turns out, ad commuters always choose their route 1 (though at three different rates; the first 366.67 vehicles have a departing headway of l/100, the next 133.33 have a departing headway of l/80, and the remaining vehicles have a departing headway of 3/100). On the other hand, bd commuters do split between the two routes, and their route choice functions are shown in Figs 7 and 8. Approximately 80.55 bd commuters will choose route 1, and the remainder will choose route 2. For the route 1 commuters, the departing headway between the first 66.67 commuters is 3/80 and the departing headway for the remainder is 9/50. For the route 2 commuters, the departing headway between the first 133.33 commuters is 3/160 and for the remainder is 91250. In this solution a-d commuters experience higher costs than bd commuters, and this is not surprising given that they have a higher early penalty. The resulting costs are illustrated in Figs 9 and 10. The cost for ad commuters is approximately 29.33 and the cost for a-d commuters is approximately 11.33.
5. CONCLUSIONS AND FUTURE RESEARCH
The primary objective of this paper was to demonstrate that there is an alternative to route/time-based formulations of the SRD equilibrium problem and that this alternative may be more attractive. Specifically, we have shown that it is possible to develop a vehicle-based
I
I
so
loo
I
150
I
200
I 250
Fig. 8. The departure-time function for b-d route 2.
)4
353
Vehicle-based equilibrium model
m-
10 -
I
I
I
I
200
400
600
800
*4
Fig. 9. The cost function in equilibrium for a-d.
formulation of the SRD equilibrium problem that can be motivated in a Nash-like fashion and has no more decision variables than existing route/time-based formulations (which are infinite dimensional). We have also shown that it is possible, at least for small networks, to solve a discrete approximation using a relatively simple heuristic. Though much work still needs to be done to demonstrate that this technique will work on large networks, our initial tests are encouraging. Hence, we intend to pursue this heuristic further. This will principally involve developing methods to evaluate the cost operators more efficiently (which was the principle bottleneck in solving these small problems). However, there is an interesting aspect of the discrete approximation that we used to solve for an equilibrium that we have not yet fully explored. In particular, observe that when Aqi= 1, the discrete approximation is tantamount to a model with integer-valued vehicles and real-valued time. This is interesting because no model of this kind has, to our knowledge, been developed as yet. As shown in Table 3, a variety of equilibrium models have been developed with realvalued vehicles and time, real-valued vehicles and integer-valued time, and integer-valued vehicles and integer-valued time. In general, models that would go in the upper left-hand comer of this table use infinite-dimensional variational inequality or optimal control formulation, while models that would go in the upper right-hand comer of this table use finite-dimensional variational inequality, optimal control, and math programming formulations. Finally, models that would go in the lower right-hand comer of this table typically use some form of time-based simulation in place of closed-form cost operators.
Fig. 10. The cost function in equilibrium for b-d.
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Francisco J. Jauffred and David Bernstein
What makes this intriguing is that the most realistic of the four categories seems to be the one that is ignored. On6 possible explanation for this is that, in general, the most realistic case is also the most complex. However, for (so-called) dynamic traffic assignment problems, this may not be true. To see this observe that, in the past, there have been two obstacles that needed to be overcome in the development of such models. The first is the problem of overtaking (or violation of the first-in, first-out discipline). The second is the problem of solving infinite-dimensional problems. Yet, models that would fall in the lower left-hand corner of Table 3 may be able to easily overcome both of these obstacles. In particular, it follows from Friesz et al. (1993), (Theorem 1 and footnote 3) that, in continuous time, overtaking can be avoided with appropriately defined delay functions. In addition, using integer-valued vehicles actually decreases the number of decision variables (i.e. making an infinite-dimensional problem finite-dimensional). Hence, in a future paper we will develop a vehicle-based formulation in which vehicles are explicitly integer-valued and time is real-valued. In this context we will first explore appropriate definitions of SRD equilibrium. Using a slight modification of the notation introduced above, it is easy to see what such a definition might look like. First, let tW= ( t ;, ...1 t&j E [0, TID” and rw = (,,...,r&) rw
E jtw
denote the vectors of departure-time choices and route choices for the D, w-commuters respectively. Further, let t = (t’,...,tW)~[O,i’JQ and r= (r’,...,?‘)~ /Q denote the complete vectors of departure-time and route choices and let c,” :[O, TIQ x YQ + R+ denote the cost experienced by commuter n of O-D pair w. Then, it seems quite natural to say that a route and departure-time pattern, (t,r), is a simultaneous route and departuretime choice (SRD) equilibrium if and only if (ifQ: (57) for all WET and n= 1,...,D,. That is, a route and departure-time pattern, (t r), is an SRD equilibrium if no commuter can reduce his or her travel cost given the behavior of all other commuters. However, it turns out that this definition may not be appropriate. This will be discussed in a future paper. In addition, we will explore methods of evaluating the cost operator cF(t, r) within this context. As it turns out, the problem of evaluating these cost operators has proven to be the most difficult aspect of incorporating the dynamics of traffic flow into network equilibrium models. In particular, even if a link’s performance (i.e. travel-time) at a particular time is a function of the number of vehicles on that link at that time, it is very difficult to model link performance as a function of departure-time and route choices. This is because the number of vehicles on a link at a particular time is itself determined by the travel-time on (potentially) all of the links in the network. To date, two approaches have been proposed to model these interactions. One involves time-based simulation and the other involves systems of integral and/or differential equations. However, it may also be possible to use a fixed-point formulation based on event-based simulation when vehicles are integer-valued and time is real-valued. This may enable us both to understand the properties of these operators and to evaluate them quickly and efficiently. Table 3. A categorization of some recent research
Real-valued vehicles
Integer-valued vehicles
Real-valued time
Integer-valued time
Bernstein et al. (1993) Friesz et al. (1993) Ran ef al. (I 996)
Smith (1993) Drissi-Kai’touni and Hameda-Benchekroun (1992) Janson (1991) Wie ef al. (1991) Mahmassani and Peeta (1992) Leonard ef al. (I 978)’
‘Some people have argued that the model developed by Leonard ef al. (1978) uses real-valued time. However, our understanding of their model is that it uses discrete time slices (i.e. it assumes that flow rates are constant over intervals of time). Hence, we still consider it a discrete time model.
Vehicle-based equilibrium model
355
Finally, we will explore questions related to the existence of such equilibria. In this regard, it may be necessary to add some “noise” to the model since discrete vehicles can give rise to discontinuities in the travel delay functions. These discontinuities can, in some cases, mean that no deterministic equilibrium exists. Hence, it may be necessary to consider a stochastic generalization of this model. Acknowledgements-This CONACYT.
research was supported in part by the Mexican Institute of Transportation
and
REFERENCES Amott, R., de Palma, A. and Lindsey, R. (1990) Departure time and route choice for the morning commute. Transportation Research -B, 7.4209228.
Bernstein, D. (1990) Programmable network equilibria. Ph.D. dissertation, University of Pennsylvania. Bernstein, D. and Smith, T. E. (1994) Equilibria for networks with lower semicontinuous costs: with an application to congestion pricing. Transportation Science, 28, 221-235. Bernstein, D., Friesz, T. L., Tobin, R. L. and Wie, B.-W. (1993) Variational control models of dynamic network traffic equilibrium. In Proceedings of the 12th International Symposium on Transportation and Trajic Theory, pp. 107-126. Elsevier, Amsterdam. Dafermos, S. C. (1971) An extended traffic assignment model with applications to two-way traffic. Transportation Science, 5, 366389.
Dafermos, S. C. and Sparrow, F. T. (1969) The traffic assignment problem for a general network. Journal of Research of National Bureau Stand, 73B, 91-118.
Dagatuo, C. F. (1985) The uniqueness of a time-dependent equilibrium distribution of arrivals at a single bottleneck. Transportation Science, 19, 29-37. Devarajan, S. (1981) A note on network equilibria and noncooperative games. Transportation Research -B, 15, 421426. Drissi-Kaitouni, 0. and Hameda-Benchekroun, A. (1992) A dynamic traffic assignment model and solution algorithm. Transportation Science, 26, 119-128. Fargier, P. H. (1981) Effects of the choice of departure time on road traffic congestion: theoretical approach. In Proceedings of the Eight International Symposium on Transportation and Tra@c Theory, pp. 223-263. University of Toronto Press. 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. Operations Research, 41, 379-191. Harker, P. and Pang, J. S. (1990) Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory. Algorithms and Applied Mathematical Programming, 48, 161-220. Hendrickson, C. and Kocur, G. (1981) Schedule delay and departure time decisions in a deterministic model. Transportation Science, 15, 62-77. Heydecker, B. G. (1986) On the definition of traffic equilibrium. Transportation Research -B, 20,435440. Janson, B. N. (1991) Dynamic traffic assignment for urban road networks. Transportation Research -B, 25, 143-161.
Kuwahara, M. (1990) Equilibrium queuing patterns at a two-tandem bottleneck during the morning peak. Transportation Science, 24, 2 17-229.
Leonard, D. R., Tough, J. B. and Baguley, P. C. (1978) CONTRAM: a traffic assignment model for predicting flows and queues during peak periods. Research Report 841, Transport and Road Research Laboratory, Crowthome, Berkshire, U.K. Mahmassani, H. S. and Herman, R. (1984) Dynamic user equilibrium departure time and route choice on idealized traffic arterials. Transportation Science, 18, 362-384. Mahmassani, H. S. and Peeta, S. (1992) Dynamic assignment for route guidance decisions. Presented at the 39th North American Meetings of the Regional Science Association fnternational.
Nash, J. F. (1952) Non-cooperative games. Annals of Mathematics, 54, 286295. de Palma, A., Ben-Akiva, M., Lefevre, C. and Litinas, N. (1983) Stochastic equilibrium mode1 of peak period traffic congestion. Transportation Science, 17,430-453. Ran, B., Hall, R. W. and Boyce, D. E. (1996) A link-based variational inequality model for dynamic departure time/route choice. Transportation Research -B, 30, 31-46. Rosenthal, R. W. (1973) The network equilibrium problem in integers. Networks, 3, 53-59. Smith, M. J. (1984) Two alternative definitions of traffic equilibrium. Transportation Research -B, 18, 63-65. Smith, M. J. (1993) A new dynamic traffic model and the existence and calculation of dynamic user equilibria on congested capacity-constrained road networks. Transportation Research -B, 27,49-63. Vickrey, W. S. (1969) Congestion theory and transport investment. Papers and Proceedings of the American Economic Review, Vol. 59, pp. 251-261. Wardrop, J. G. (1952) Some theoretical aspects of road traffic research. In Proceedings of the Institution of Civil Engineers, Part II, pp. 32>378. Wie, B.-W., Friesz, T. L., Tobin, R. L. and Bernstein, D. (1991) A discrete time, nested cost operator approach to the dynamic network user equilibrium problem. Presented at TRISTAN. Wie, B.-W., Tobin, R. L., Bernstein, D. and Friesz, T. L. (1996) A comparison of system optimum and user equilibrium dynamic traffic assignment with schedule delays. Transportation Research -C, 3, 38941 I. Zhu, D. L. and Marcotte, P. (1996) On the existence of solutions to the dynamic user equilibrium problem. Manuscript, Universite de Montreal, Canada.
356
Francisco J. Jauffred and David Bernstein APPENDIX A: OBTAINING ANALYTIC SOLUTIONS
When travel times are modeled as deterministic queues it is sometimes possible to obtain analytic solutions to the SRD equilibrium problem. Some such results have already been presented by Vickrey (1969); Fargier (1981); Daganzo (1985); Arnott et 01. (1990); and Kuwahara (1990). In this Appendix A we demonstrate how the results above can also be used to obtain analytic solutions in some cases. For sake of simplicity, we only consider the case of departure time choice. That is, we consider networks with one &D pair and one route connecting that origin and destination. However, these results can be generalized to include route choices. First, suppose that the route of interest consists of a single link. Then, expressing equation 53 in terms of finite differences, it follows that d, + f(ql.)
-
t(ql.)
+
At(ql.). $
I
Then, allowing the differences to become arbitrarily small, we have the following: d, + t(q/.)
-
s(ql,)
= 0 +
Wql.) ~ dq
Wql.) = dq
and d, + t(q1.)
-
dr(ql.) c 0 =+ dq
r(ql.)
1 = -. $0
More generally, we can include exogenous traffic departing at a ratefO(f) by defining the as follows:
equivalenr
service
rare,
s;,
that is, by assuming that the equivalent service rate is proportional to the fraction of stream vehicles that enter the link in that particular period of time. Alternatively,
Then, substituting into equation Al yields
4, + W) - 441.) + At(&),
Aq +h(r)Ar s,
cw
and allowing the differences to become arbitrarily small da + f(qj.)
-
r(ql.)
dr(ql.)
< 0 =+ -
1+h@ &I
= ~.
dq
(A7)
Now, observe from equation I2 that:
1
d&It) ----=ooj~ dq
+ &(ql.)]
dr(4.) = 0 drl
(‘48)
and hence that (A9) Hence, it follows that in equilibrium:
(AlO) In general, when routes contain multiple links, the maximization in equation Al must be taken over more terms. In particular, a route with m links will have m+ I terms on the right-hand-side of equation 12. Following through, one is left with 2”’ dynamic equations of the type shown in equations A2 and A3. Therefore, this is a fairly tedious process for long routes. As an example, consider a single link network with d= 0, s = 50, and Q = 100. The early arrival cost is OS/min, the late arrival cost is Z/min and the travel cost is I/min. Further, suppose there is an exogenous traffic source of 50/h and f = 9. For early arrivals:
df(ql.) -=dq
w
1 _yj.!+5=j$
I
(Al 1)
Vehicle-based equilibrium model
351
whereas for late arrivals: !g
WI.) -=dq
3
1-so.!#=-~
(Al21
Hence, there are no late arrivals and the equilibrium departure-time functions must satisfy:
$.
6413)
$ = (01.) + &.
C414)
r(q1.)= r(O1.)+ Further, the arrival-time function is given by
?(q(.)= (01.) + Therefore, the average cost is: &I.)=
[r(Ocol., +$] - [@I.) +$] + 0.5[9 - (m9 +$)I
= 4.5 - 0.5 t(ol.).
(Al51
To minimize the cost we must make the first departure-time as close to 9 without causing late arrivals. Hence, the equilibrium departure-time function is given by: r(ql.) = 5 + $ with equilibrium cost of 2.