55 luce accuracy (sidt2009)

8/14/2019 55 LUCE Accuracy (SIDT2009)

1/6

SIDT 2009 International Conference 1

Guido Gentile

On the accurate convergence of deterministic

assignment when comparing scenarios for large

networks: investigating the LUCE algorithm

Guido Gentile1

1Dipartimento di Idraulica Trasporti e Strade, Sapienza Universit di Roma

[email protected]

1. IntroductionIn this paper we investigate a new algorithm (Gentile, 2009) to solve the static

assignment problem to multimodal transport networks with deterministic route

choice, called Linear User Cost Equilibrium (LUCE), that has been recently adopted

in the software VISUM. To find a descent direction, local shifts of flows that satisfy

the total cost lowering rule are determined exploiting the inexpensive information

provided by the derivatives of the link costs with respect to link flows.

LUCE achieves a very high convergence speed that compares favourably to the other

methods, while it assigns the demand flow of each o-d couple on several paths at

once. This algorithm is therefore particularly suitable when the aim is to simulate the

effects of supply or demand changes on large networks, since in this case a relatively

small inaccuracy of the solution and a poor set of loaded paths can lead to relevant

design mistakes; indeed, very often in practice the stop criterion is based on a

run-time budget instead that on predefined convergence goals.

Although traffic assignment is a rather mature issue in transport modelling, to find

an accurate equilibrium on real networks is still a difficult problem to be solved. In

practice, the algorithms currently available are not truly satisfactory for many

applications: they simply dont converge in reasonable time fine enough to allow

consistent comparisons between design scenarios. Indeed, apparently small errors in

the iterative procedure do not allow to appreciate the real differences among


2/6

2 Milan 29-30 June 2009

Guido Gentile

equilibria and may lead to false conclusions in relevant projects, thus vanishing any

modelling effort.

This is a well recognized problem by the recent literature (see, for example, Boyce et

al., 2004), but is not really acceptable by practitioners, who are asking researchers

and developers to enhance the algorithm convergence. To satisfy this necessity and

overcome such a drawback, the main software producers in the world are in these

days changing their traffic assignment procedures (e.g. VISUM, TRANSCAD,

EMME), thus animating the international debate and the competition on this topic.

In this context, LUCE gives a noticeable contribution: it converges between 10 and

100 times faster than other recent algorithms and allows on large networks to reach

in few minutes and iterations the relative gap of 10E-8, that is considered enough for

any application. This result conveys a considerable advance in modelling practice,

more than in algorithm theory.

The LUCE algorithm is here applied to several networks (elementary, toy and real)

with different levels of congestion, to show the importance of an accurate

convergence when comparing different simulated scenarios.

2. Main aspects of the algorithmSpecularly to Origin-Based methods, the assignment problem is partitioned by

destinations. The main idea is to seek at every node a deterministic equilibrium

(Wardrop, 1952) for the local route choice of users directed toward a same

destination among the links of its forward star that belong to the current bush a

bush is an acyclic sub-graph that connects each origin to the destination at hand. The

cost function associated to each one of these travel alternatives expresses the average

impedance to reach the destination by continuing the trip with that link, linearized at

the current flow pattern. It can be proved that the solution to such linear program in

terms of destination flows, recursively applied for each node of the bush in

topological order, provides a descent direction with respect to the classicalsum-integral objective function (Beckmann et. al., 1956). The network loading is

then performed through the corresponding splitting rates, thus avoiding explicit path

enumeration.

To grasp immediately the concept underlying LUCE, we can refer to the simplest


3/6


Guido Gentile

network where one o-d pair with demand D is connected by two links with cost

function c1(f1) and c2(f2), respectively. At the current flow pattern f = (D/2,D/2), it

is c1 < c2 (see Figure 1, below), so that an all or nothing approach would lead to a

descent direction (D, 0), which is far away from the equilibrium f* (black circle in

the Figure). Our approach, instead, is to consider the first order approximations of

the cost functions at the current flow pattern, i.e. ca + ca( fa)/fa ( fa - fa), and

determine a user equilibrium e among these lines (white circle in the Figure): this

descent direction efficiently approaches the equilibrium f*, and in most cases can be

taken as the new iterate with a step one.

Figure 1. Linear Cost User Equilibrium between two paths.

Contrary to the classical All Or Nothing assignment to shortest paths, the network

loading map resulting from the application of the LUCE algorithm a) assigns

(implicitly) the demand flow of each o-d couple on several paths at once and b) is a

one-to-one function that combined with the link cost function yields a well-defined

fixed point operator, thus offering both computational and theoretical advantages.

For each iteration, the proposed algorithm requires no shortest path calculations and

two visits of the bush links for each destination, that is equal to the complexity of the

STOCH single pass procedure (Dial, 1971) for the Logit network loading.

c1

c2

f1 f2

c1(f1)

c2(f2)

f e f*

D / 2

c2

D


4/6

4 Milan 29-30 June 2009

Guido Gentile

3. Comparison with other algorithmsLUCE presents a higher convergence speed, both in terms of computing time and

iterations, than the most advanced Origin-Based methods recently proposed.

In the following Figure 2 we compare the convergence performances of five

different algorithms on the Chicago network, which is one of the best known

benchmark with 1768 zones and 39018 links.

Convergence of different algorithms for Chicago

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

0 50 100 150 200 250 300 350 400 450 500run time on a 2.0GHz CPU [min]

relativeg

ap

LUCE

FW

OBA

B

PG

Figure 2 Performance comparison of LUCE to other four assignment algorithms on the

Chicago network, whose convergence patterns are retrieved from papers and presentations.

Frank Wolfe by LeBlanc (1973) and Nguyen (1973) is here reported just to recall

that an accurate solution of the assignment problem requires in practice more

specialized procedures.

Bar-Geras OBA (2002) takes into account link and node cost derivates with the aim

of calculating Newton-type flow shifts among the paths of the bush, thus obtaining a

descent direction in the space of link flows.

LUCE exploits the same information, but instead of seeking some parallelism to

non-linear optimization techniques like the OBA, it adopts an ad-hoc approach

founded on the intuitive ideas of local equilibrium and first-order approximation.


5/6


Guido Gentile

Algorithm B proposed by Dial (2006) is a link based procedure that shifts trips from

max- to min-paths to make their cost difference minima.

The Projected Gradient algorithm, recently revised by Florian (2009), is a path based

method that regards their costs as the gradient of the sum integral objective function.

4. ConclusionsWe summarize below the main properties of the proposed algorithm to solve the

classical equilibrium problem with fixed demand, deterministic route choice andseparable arc cost functions.

LUCE is formally proved to converge at equilibrium. We tested that it is actually

capable of solving numerically the traffic assignment problem with any wanted

accuracy, under the condition that the computer arithmetic precision is twice as

much the aimed relative gap. The convergence shows to be linear, but its rate

decreases with the level of congestion and with the sharpness of cost functions. To

alleviate this problem, that affects any algorithm, a hyperbolic shape and a piecewise

linear shape of the arc cost function can replace the power based BPR.

LUCE appears to be at least 10 times faster than the most advanced methods recently

proposed in the literature. Few iterations, say 20, are usually enough to reach aconvergence suitable for most applications.

LUCE has a linear complexity in terms of links and zones, unlike many other

methods that are path based or require shortest path searches on cyclic graphs. In

particular, the runtime is not sensitive to the number of o-d couples but only to the

number of destinations, given the implicit path enumeration.

LUCE does not need to store in memory single o-d paths but only destination bushes,

that however offer a much richer finite set of used alternatives, whose probabilities

can be easily obtained ex-post.

LUCE allows for a fixed point formulation of the equilibrium problem also in the

deterministic case without introducing one to many maps.LUCE is based on a simple idea that can be understood with a single chart and

exploits the concept of derivatives that is familiar to any engineer.

LUCE is implemented in VISUM, one of the most widespread software for transport

planning.


6/6

6 Milan 29-30 June 2009

Guido Gentile

References

Beckmann M., McGuire C., Winston C. (1956) Studies in the economics of transportation. Yale

University Press, New Haven, CT.

Bar-Gera H. (2002) Origin-based algorithm for the transportation assignment problem. Transportation

Science 36, 398-417.

Boyce D., Ralevic-Dekic B., Bar-Gera H. (2004) Convergence of traffic assignments: how much is

enough? ASCE Journal of Transportation Engineering 130, 49-55.

Dial R. (1971) A probabilistic multipath assignment model than obviates path enumeration. Transport

Research 5, 83-111.

Dial R. (2006) A path-based user-equilibrium traffic assignment algorithm that obviates path storage

and enumeration. Transport Research B 40, 917-936.

Florian M. (2009) New look at projected gradient method for equilibrium assignment. Transportation

Research Board Annual Meeting 2009, Paper #09-0852.

Gentile G. (2009) Linear User Cost Equilibrium: a new algorithm for traffic assignment, submitted to

Transportation Research B.

LeBlanc L. (1973) Mathematical programming algorithms for large scale network equilibrium and

network design problems. Ph.D. Dissertation. Department of Industrial Engineering and Management

Sciences, Northwestern University.

Nguyen S. (1973) A mathematical programming approach to equilibrium methods of traffic assignment

with fixed demands. Publication 138, Dpartement dInformatique et de Recherche Oprationnelle,

Universit de Montral, Qubec.

Nguyen S., Pallottino S. (1988) Equilibrium traffic assignment for large scale transit networks.

European Journal of Operational Research 37, 176-186.

Wardrop J. (1952) Some theoretical aspects of road traffic research. Proceedings of the Institution of

Civil Engineers, Part 2, 325-378.

55 luce accuracy (sidt2009)

Documents