SCIENCE
Vol. TRANSPORTATION 39, No. 4, November 2005, pp. 446450 issn 0041-1655 eissn 1526-5447 05 3904 0446
informs
doi 10.1287/trsc.1050.0127 2005 INFORMS
On a Paradox of Trafc Planning
Faculty of Mathematics, Ruhr-University Bochum, 44780 Bochum, Germany, [e-mail] Department of Finance and Operations Management, Isenberg School of Management, University of Massachusetts, Amherst, Massachusetts 01003 {[e-mail], [e-mail]}
Dietrich Braess
Anna Nagurney, Tina Wakolbinger
F
or each point of a road network, let there be given the number of cars starting from it, and the destination of the cars. Under these conditions one wishes to estimate the distribution of trafc ow. Whether one street is preferable to another depends not only on the quality of the road, but also on the density of the ow. If every driver takes the path that looks most favorable to him, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the trafc that results in longer individual running times. Key words: trafc network planning; paradox; equilibrium; critical ows; optimal ows; existence theorem History: Received: April 2005; revision received: June 2005; accepted: July 2005. Translated from the original German: Braess, Dietrich. 1968. ber ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung 12 258268.
1.
The distribution of trafc ow on the roads of a trafc network is of interest to trafc planners and trafc controllers. We assume that the number of vehicles per unit time is known for all origin-destination pairs. The expected distribution of vehicles is based on the assumption that the most favorable routes are chosen among all possible ones. How favorable a route is depends on its travel cost. The basis for the evaluation of cost is travel time. The road network is modeled by a directed graph for the mathematical treatment. A (travel) time is associated with each link. The computation of the most favorable distribution can be considered solved if the travel time for each link is constant, i.e., if the time is independent of the number of vehicles on the link. In this case, it is equivalent to computing the shortest distance between two points of a graph and determining the corresponding critical (here meaning shortest) path. See Bellman (1958), von Falkenhausen (1963), and Pollack and Wiebenson (1960). In more realistic models, however, one has to take into account that the travel time on the links will strongly depend on the trafc ow. Our investigations will show that we will encounter new effects compared to the model with ow-independent costs. Specically, a more precise formulation of the problem will be required. We have to distinguish between ow that will be optimal for all vehicles and ow
446
Introduction
that is achieved if each user attempts to optimize his own route. Referring to a simple model network with only four nodes, we will discuss typical features that contradict facts that seem to be plausible. Central control of trafc can be advantageous even for those drivers who think that they will discover more protable routes for themselves. Moreover, there exists the possibility of the paradox that an extension of the road network by an additional road can cause a redistribution of the ow in such a way that increased travel time is the result.
2. Graph and Road Network
Directed graphs are used for modeling road maps, and the links, the connections between the nodes, have an orientation (Berge 1958, von Falkenhausen 1966). Two links that differ only by their direction are depicted in the gures by one line without an arrowhead. In general, the nodes are associated with street intersections. Whenever a more detailed description is necessary, an intersection may be divided into (four) nodes with each one corresponding to an adjacent road; see Figure 2 (Pollack and Wiebenson 1960). We will use the following notation for the nodes, links, and ows. The indices belong to nite sets. Because we use each index only in connection with one variable, we do not write the range of the indices.
Braess, Nagurney, and Wakolbinger: On a Paradox of Trafc Planning
Transportation Science 39(4), pp. 446450, 2005 INFORMS
447 In the general case with more than one origindestination pair we introduce index sets B such that the groups U B contain the paths with the same origin and destination nodes. Here, species the origin-destination pair. Analogously to (2.3), we have =
B
h
j
k
e
d a
Figure 1
f
g
b
c
(2 3 )
nodes of the graph. oriented links of the graph. ow on u (vehicles/time). We consider trafc networks with stationary ows. It is useful to regard the total ow as the sum of threads with each thread being associated with an origin-destination pair. Each thread corresponds to a path in the graph. We need to consider only paths without cycles. paths that do not contain links more than U once. ow along U (vehicles/time). . the vector with the components The ows on paths and links are related by the arcpath incidence matrix C whose coefcients c assume only the values 0 or 1 because cycles are excluded: 1 if link u is contained in path U c= (2.1) 0 otherwise. Obviously, = c (2.2)
ai u
Which paths are considered optimal is determined by their respective travel costs. The costs depend on the length of the road, travel time, and other costs (cf. von Falkenhausen 1966, p. 23). The dominating feature is travel time, and for reasons of clarity we identify costs with travel time. In this way, it is also clear that costs depend on the volume of trafc. Moreover, we regard the model as deterministic, and stochastic arguments are deliberately ignored.1 The following denitions are to be understood in this framework: t travel time required on u if u carries ow =. T ik travel time for getting to ai from ak on the path U . (If this is impossible, the function value is .) The superscripts i and k will be suppressed if ai is the destination and ak is the origin of . Travel time depends on , in particular on the ow on U . Because travel time for a path is the sum of the times for its links, we have T = ct (2.4)
Of course, all ow variables in a trafc network are nonnegative. For simplicity, a part of the general considerations will be done only for the special case in which the total ow has a common origin node and a common destination node. Those nodes will be denoted as a0 and a , respectively. The total ow is given by = (2.3)
= is Here, the functional relationship given by (2.2). Moreover, we dene the most unfavorable time by T ik and T = max T
ik
= max T ik
=0
(2.5)
B
=0
(2.6)
where ai and ak are the nodes of the destination of the paths U B and of the origins, respectively. The functions t are assumed to have the following properties: I. t 0. II. t is a nondecreasing function.
1 For the trafc planner, deviations in trafc densities are not interesting for the design as long as the impact on travel time is small. Therefore, we do not obtain serious errors if individual (stochastic) quantities are replaced by their mean values. It is known that problems arise in models with ow-independent costs because one needs a criterion as to how to assign a portion of ow to nearly optimal (suboptimal) paths (von Falkenhausen 1966).
Figure 2
448 III. t is semicontinuous, i.e., lim 0 < 0 t = t 0. The rst two assumptions are natural in view of the problem setting. Assumption 3 simplies the mathematical treatment. In this case, the functions t are lower semicontinuous (Natanson 1961); i.e., we have lim t
0
Braess, Nagurney, and Wakolbinger: On a Paradox of Trafc Planning
Transportation Science 39(4), pp. 446450, 2005 INFORMS
z
4 5
3
b
c
t
0
(2.7)
1
2
In Sections 4 and 5 we will assume continuity for even further simplication.
a
Figure 3
3.
We will discuss which ow distributions admit travel times for all drivers to be as short as possible when the trafc networks have only one origin-destination pair. The time that is needed to reach the destination in the most unfavorable case measures how well the ows are distributed. This time is given by Equation (2.6). The total ow = is considered xed, with given. Denition. The ow is optimal if the relation T holds for all with = (3.2) T (3.1)
Optimality
Proof. To prove the theorem, we consider a minin mal sequence n , i.e., a sequence with = and n limn T = inf T = . Because the n paths do not contain cycles, it follows that 0 n . Due to the boundedness, a subseand 0 n quence n with convergent values for can be selected. Let n lim =
It follows from (2.4), (2.7), and lower semicontinuity that n T lim T (3.4)
Let U be a path with = 0. Then, we have n = 0 for all sufciently large , and consequently for the minimal sequence lim T
n
It is essential in this denition that the value of a ow distribution is guided by the travel time of all drivers (in contrast to the setting in the next section). The concept and the results do not differ substantially if the mean value of the travel time 1 T (3.3)
lim T
n
= inf T
=
(3.5)
and not the maximal time, T , determines the quality. It cannot be decided by mathematical arguments which specication is more appropriate. This decision must be left to the trafc planners.2 We will only postulate the following consistency property: It should be impossible to redistribute optimal ows so that each driver achieves a reduction of the costs. Now we turn to the ows that are optimal according to (3.1). Theorem. Assume that t is from semicontinuous below whenever 0 . Then, an optimal ow exists with =.
2 Optimality could be easily dened for trafc with several origins and destinations on the basis of mean values of travel times. On the other hand, can one require that someone be content with a long drive to reduce the mean travel time? Our denition has an advantage that will become clear later. The optimal solution is at least as advantageous for each driver as the equilibrium.
The inequalities (3.4) and (3.5) imply that is an optimal ow distribution. We now turn to a model example with four nodes. For convenience, the link travel times t are linear functions. (Moreover, the graph does not contain cycles.) t1 t2 t5 = t3 = t4 = 10 = 50 + = 10 + (3.6)
(a) If a total ow of = 2 is to be guided from a to z, the optimal solution is
abcz
=2
abz
=
acz
=0
T
= 52
(b) If a total ow of = 6 is to be guided from a to z, the optimal solution is
abcz
=0
abz
=
acz
=3
T
= 83
(c) If a total ow of = 20 is to be guided from a to z, the optimal solution is
abcz
=0
abz
=
acz
= 10
T
= 160
Obviously, all the solutions are unique.
Braess, Nagurney, and Wakolbinger: On a Paradox of Trafc Planning
Transportation Science 39(4), pp. 446450, 2005 INFORMS
449 An indirect proof is easy. Set
i
Each driver attempts to nd for himself the most favorable path. It is assumed that he obtains the information that is necessary for determining the route. Therefore, our approach differs signicantly from the approach in a game-theoretic consideration; see also the footnote in Section 2. We consider once more the model example from the last section. If the volume of trafc is as in cases (a) or (c), then it is most protable to move in accordance with the optimal ow. This is different in case (b). The optimal ow moves along paths abz and acz . There exists, however, a path for which travel time . Supis lower. Specically, Tabcz = 70 < 83 = T pose that the vehicles are distributed as the optimal ow. Those drivers to which the link travel times are known would move to path abcz and destroy the optimality. If the drivers of two vehicles possess perfect information from experience, they will not choose paths with different travel times. Therefore, we consider the hypothesis as realistic that the trafc ow will be distributed in a manner that will be called critical. Denition. The ow is a critical ow3 if for all paths U T T =T T if if =0 =0 (4.1)
4.
Critical Flow
= min T i0
If the inequalities in the theorem would not hold, then there would exist a path from a0 to a with T < T . Before we deduce the existence of critical ows and additional properties, we provide a numerical result for the model example above. The unique critical ow in case (b) with total ow = 6 is
abcz
=
abz
=
acz
=2
and
a
=0
b
= 40
c
= 52
z
= 92
Obviously, criticality has the following meaning. The destination will be reached on all paths with nonvanishing ow at the same time.4 Travel time on paths with no ow is the same or even larger. The analogous property holds for all nodes in between. Theorem. Let be a critical ow. Then, a number i exists for each node ai such that for all paths that pass ai T T
i i
Hence, we have already obtained the result that the critical ow does not always coincide with the optimal ow. This happens not only for the optimality criterion introduced in the last section, but for each consistent denition because there obviously exist ows such that travel time of all vehicles is smaller than 92. Although each driver chooses the most favorable path for himself, none of them achieves the value that each one of them could achieve at optimal ow. In this framework, we also recognize a paradoxical fact. If the link u5 is eliminated in the road network, the critical ow coincides with the optimal ow; the distribution of the trafc ow is improved in this case. This means that for real-life trafc practice: In unfavorable situations an extension of the road network may lead to increased travel times.
= T ik
i i
if if
i
=0 =0
k
Moreover, we have for all paths U that run from ak to ai =T and, moreover,
The existence of critical ows for a given total ow can be shown for continuous and nondecreasing func. This will be done by the reduction to a tions t is nondecreasing, the convex program. Because t associated function f =
0
5. An Existence Theorem
t
d
(5.1)
We restrict ourselves to continuous functions t . If only semicontinuity is assumed, relations (4.1) have to be replaced by T T
+
3
T T
if if
=0 =0
is convex. We do not restrict our attention to trafc with only one origin and one destination and choose a slightly more general denition. Denition. The ow is a critical ow if for all paths U with B T T =T T if if =0 =0 (5.2) are contin. Then, the
Here, T + and T denote the upper and lower limits, respectively, where jumps occur. The values coincide with T at points of continuity. The existence theorem in Section 5 also holds in this more general case.
4 Each path with nonvanishing ows is therefore a critical value in the spirit of standard optimization problems on graphs (Berge 1958).
Theorem. Assume that the functions t uous and nondecreasing for 0 =
450 solutions of the convex program f =
B
Braess, Nagurney, and Wakolbinger: On a Paradox of Trafc Planning
Transportation Science 39(4), pp. 446450, 2005 INFORMS
= Min! c = 0 (5.3)
is more than one group, the equations can no longer be related to a variational principle.5
6. Violation of Monotonicity
The critical ows in the model example of Section 3 are optimal in cases (a) and (c). It is obvious from the numerical solutions that we do not have implies the relations for critical ows (and consequently also not for optimal ones), although one might expect this. There are consequences for the numerical treatment of the problem, in particular, for an approximation procedure that was recently suggested. Let n be a given natural number. Determine the shortest paths for all origin-destination pairs of interest when there is no trafc on the roads. It is assumed that the nth portion of the ow chooses those routes. Now, the shortest paths for the new situation are evaluated, and another nth portion of the ow is determined. By proceeding in the same way, the total ow will be distributed on the trafc network. The result is considered to be an approximate solution. If this method is applied to the model example above with = 20, then path abcz is chosen in the rst steps of the procedure although this path carries zero ow in the solution. When n is increased, the approximate solutions do not converge to the correct solution. However, one can use the convex program (5.3) for the computation of the critical ows, and here well-known algorithms are available (Collatz and Wetterling 1966). We did not study whether the evaluation of the shortest paths helps to accelerate the codes.
The author thanks Prof. Dr. Helmut Werner for the promotion of this research and Prof. Dr. W. Kndel for valuable stimulating remarks.
are critical ows. Proof. The variables may be temporarily eliminated by substitution. The Kuhn-Tucker conditions (Collatz and Wetterling 1966) (with Lagrange multipliers and for the remaining constraints) are ct
B
=
=0
B (5.4)
=0 0 =0 =0
0 T T = 0 if 0 if
The equations in the rst line imply, due to (2.4), B
Hence, =T , and is a critical ow. The existence of a critical ow is immediate. The set of critical ows is convex. A solution is also the unique solution if the function t is strictly monotone at = for at least one link u . Therefore, we can guarantee uniqueness of the solution if each path contains at least one link on which t is strictly monotone on the whole domain. The possibility of characterizing critical ows as the solutions of a minimization problem is connected with a symmetry in the model. Roughly speaking, we can say: Each driver induces the same delay for the other drivers as the other one does for him. This symmetry no longer holds in a more general model. In particular, the travel time is not equal for all vehicles on a street with several lanes; it depends on the type (class) of the vehicle. The most signicant differences are between passenger cars and trucks. This can be partially incorporated into the theory by dividing the ows into groups: =
g g
Acknowledgment
References
Bellman, R. 1958. On a routing problem. Quart. Appl. Math. 16 8790. Berge, C. 1958. La Theorie des Graphes. Paris, France. Collatz, L., W. Wetterling. 1966. Optimierungsaufgaben. Springer, Berlin-Heidelberg-New York. Natanson, I. P. 1961. Theorie der Funktionen einer reellen Vernderlichen. Berlin, Germany. Pollack, M., W. Wiebenson. 1960. Solutions of the shortest route problemA review. Oper. Res. 8 224230. von Falkenhausen, H. 1963. Ein Verfahren zur Prognose der Verkehrsverteilung in einem geplanten Straennetz. Unternehmensforschung 7 7588. von Falkenhausen, H. 1966. Ein stochastisches Modell zur Verkehrsumlegung. Dissertation, Darmstadt, Germany.
5 We note that there is an analogous situation with the diffusion equations in reactor physics. The equations for models with one group of neutrons are self-adjoint while the multigroup equations are not.
=
g
g
The travel time of each group also depends on the amount of ow of the other groups on the corresponding link tg = tg
1 2
The arguments that led us to introduce the concept of critical ows can be directly extended. We do not provide the dening relationships because they differ from (5.2) merely by indices referring to groups. Nevertheless, there is an essential difference. If there