April 1999
By Michael Florian
Network equilibrium models are commonly used for the prediction of traffic patterns in transportation networks that are subject to congestion phenomena. Their application, in various transportation planning contexts, has increased dramatically over the past 25 years due to the development of efficient solution algorithms and the continually increasing power of personal computers and other computing platforms. However, they are based on concepts that were stated some 75 years ago. The idea of traffic equilibrium originated as early as 1924, when F.M. Knight gave a simple and intuitive description of a postulate of route choice under congested conditions, as follows:
"Suppose that between two points there are two highways, one of which is broad enough to accommodate without crowding all the traffic which may care to use it, but is poorly graded and surfaced; while the other is a much better road, but narrow and quite limited in capacity. If a large number of trucks operate between the two termini and are free to choose either of the two routes, they will tend to distribute themselves between the roads in such proportions that the cost per unit of transportation, or effective returns per unit of investment, will be the same for every truck on both routes. As more trucks use the narrower and better road, congestion develops, until at a certain point it becomes equally profitable to use the broader but poorer highway."
Twenty-eight years later, in a seminal contribution, Wardrop [1952] stated two principles that formalize this notion of equilibrium and introduced the alternative behavior postulate of the minimization of the total travel costs. His first principle states that "the journey times in all routes actually used are equal and less than those which would be experienced by a single vehicle on any unused route." Under certain assumptions, another interpretation of this principle is that the routes actually used are the shortest in time (or cost) under the prevailing traffic conditions and their perception by the travelers. Wardrop's first principle of route choice, which is identical to the notion postulated by Knight, became accepted as a sound and simple behavioral principle to describe the spreading of trips over alternate routes due to congested conditions. The traffic flows that satisfy this principle are usually referred to as "user optimal" flows, since each user chooses the route that is perceived to be the best. On the other hand, the "system optimal" is characterized by Wardrop's second principle.
The first mathematical model of network equilibrium was formulated by Beckmann, McGuire and Winsten in 1956. Their contribution was the starting point for intense research and then application of this route choice model. The integration of network equilibrium models in the field of transportation planning began in the 1970s and became more common as the algorithms used to solve various model variations were embedded in user-friendly software packages. Urban and regional planners could then attain easy access to these models without the need to have extensive mathematical programming background. One such software package is EMME/2, which is used by more than 700 organizations in countries on five continents. Applications carried out with EMME/2 will be used to illustrate this article.
Transportation Planning:
Demand, Supply and Performance
The quantitative approaches to transportation planning construct models that predict the demand for travel for an urban area or region based on the socio-economic characteristics of the resident population and the levels of service from origins to destinations provided by the existing transport infrastructure. The aim of travel demand models is to predict when trips originate, their destination and which modes they use. The theory and practice of travel demand models are very rich — and a great variety of econometric models and methods are used to formulate and calibrate such models by using survey data. The result of a calibration exercise is one or more demand functions which share the property that as the level of service deteriorates for an element in the set of choices, the demand for that element decreases. For instance, if the road network becomes congested, and heavy tolls are imposed, a shift may occur from the car mode to a rail mode which is unaffected by congestion. Thus, travel demand models serve to predict the demand for travel by time of day, by destination and by mode, for origin-destination pairs. This prediction may be in the form of an origin-destination matrix of trips or an origin-destination matrix of demand functions.
In order to derive a prediction of the flows on the links of the network, a network model is used to represent the supply of transportation infrastructure and services, and the demand is "assigned" on it.
The network models that are most commonly used are steady-state models, in spite of the fact that all traffic phenomena are temporal. One considers a given period of time for which the demand for travel has been quantified, as described above. Then one seeks to determine the flow pattern that results from the interaction of the demand and the performance of the transport infrastructure available.
A deterministic network equilibrium model of route choice uses a representation of the transportation network with a set of nodes and arcs. The nodes represent origins and destinations of traffic and intersection of links, while arcs represent the road network and, to represent transit lines, they are suitably concatenated. Figure 1 is part of the road network of the city of Portland, Ore. The costs on the arcs of the network are usually monotone, increasing and separable in order to model congestion. The resulting models may be formulated as nonlinear cost network optimization models. When the travel time on an arc depends on the flow on other arcs, and this dependence is not symmetric, the resulting models are variational inequality problems with an embedded network structure. The arc cost functions are also referred to as performance functions, since they describe how the link performs when it is subject to an increasing amount of traffic.
Figure 1: A portion of the road network of Portland, Ore.
The interaction between the travel demand models, the network infrastructure supplied and its performance determine the way in which the routes are chosen. The resulting link flows are the solution of deterministic network equilibrium models with fixed or variable demand. In the planning context, one defines scenarios where one of these corresponds to an existing situation. The others correspond to future situations that may differ in the travel demand or in the transport infrastructure. By predicting the flows and demands for each of the future scenarios and comparing them to the existing situation, conclusions may be drawn as to which contemplated actions are likely to be most beneficial.
The transportation planning process uses descriptive models that are used to evaluate future scenarios against a base scenario and obtain indications of the best courses of action to adopt. The OR contribution is to be found in the formulation of the models, their analysis and the algorithms used for solving the convex cost network optimization models or the monotone cost variational inequalities which result. (See Florian and Hearn, [1995].)
Transportation Planning Models in Practice
The need to plan transportation networks is pervasive. All cities in developed countries carry out quantitative transportation planning activities. Developing countries face the same need, as cities develop at high rates of growth and generate congestion on the existing transportation facilities. The challenge, then, is to provide a flexible set of tools, embedded in a software package, that is equally adaptable to developed and developing nations, and considers all the modes available for transportation. Such tools may be used in cities in North America, where the private car traffic usually carries the majority of the trips and transit services are relatively scarce; or in cities in South America, where the majority of the trips occur in transit modes and the private car serves fewer trips; or to some cities in the People's Republic of China, where a good part of the trips are taken by bicycle. The remarkable fact is that network equilibrium models may be adapted to most situations by ensuring that they consider multiple modes and multiple classes of traffic. One needs a flexible modeling framework that is adaptable to different needs. Some examples originating from applications carried out with EMME/2 in different countries illustrate the variety of situations that may be handled with the same framework.
The first example application originates in Italy, where the Comuna di Roma undertook a conventional transportation planning exercise that considered all the modes available in the Rome region. The aim was to plan for future developments of the road and transit networks in anticipation of the changes in the socio-economic characteristics of the region. The results of the study are summarized in Figures 2 and 3, which present the main traffic flows on the road and transit networks, respectively, with an indication of the levels of congestion (levels of service) on the road network and the different transit operators, respectively.
Figure 2 (above) and Figure 3 (below):
Traffic flows on the road and transit networks, respectively, in the Rome region.
Another example application originates from Portugal. In this study, which is more regional in nature, the aim was to determine if the southern part of the country had sufficient capacity to accommodate the intercity traffic. Figures 4 and 5 show the predicted traffic flows on the national road network south of Lisbon and the resulting levels of service.
Figure 4 (above) and Figure 5 (below):
The predicted traffic flows on the national road network south of Lisbon.
Other types of applications are more detailed in nature and focus on particular areas of a city where congestion at certain intersections is analyzed in detail. An example of the output of such an analysis is shown in Figure 6, where the flows on the road network are augmented with the turning movements at intersections. This application originates from the city of Winnipeg, Canada.
Figure 6: A detailed application focusing on a particular intersection in Winnipeg, Canada.
Another typical application of transportation planning models is the evaluation of the impact of a new major facility, such as the construction of a bridge. Figure 7 is an example of the comparison of two scenarios: one with the new bridge and one without the new bridge. The differences between the resulting flows obtained by applying the network equilibrium model on the two scenarios are shown in red and green. Red indicates a positive difference and green a negative difference. Thus the impact of the bridge may be easily seen. Some of the trips are shorter due to the presence of the bridge, but the links downstream of the bridge are more congested.
Figure 7: Impact of a new major facility: comparing two scenarios.
The development of toll roads or the restriction of traffic to certain parts of the city require specialized analysis techniques. In the case of toll roads, it has become common to carry out "stated preference" analysis of demand, where "panels" composed of potential users of such facilities are provided with several options, each involving a different travel time and travel cost. Based on the choices indicated, demand models for "toll" options vs. "non-toll" options are developed for several socio-economic factors. These demand models are then used to determine the "value of time" for these "classes" of travelers, and a forecast of the usage of toll roads and the corresponding revenues are obtained by employing multiclass network equilibrium models with generalized costs.
Projects related to toll roads are carried out in countries of both the developed and developing worlds. The multiclass network equilibrium models may be equally used for the analysis of truck traffic, the analysis of zones that have traffic restrictions and the analysis of HOV (high occupancy vehicles) lanes.
Transportation impacts the environment of a city. It is no longer sufficient in a transportation planning study to simply provide a forecast of the traffic flows on contemplated future scenarios. The noise and pollution impacts of urban traffic, as well as the effects of congestion on fuel consumption, must be quantified and displayed. In order to do so, one carries out computations based on the predicted link flows and speeds, which are the output of a network equilibrium model, by using formulae which correlate the traffic intensity, noise, emissions and fuel consumption.
Another major development during the past few years has been the growing popularity of geographic information systems (GIS). The advent of the digital maps, which are easily produced with GIS software packages, has provided transportation planners with new data sources for the development of traffic models and has also opened the possibility of enhancing graphical output of scenario simulations by adding annotations which correspond to the land use of the area studied. An example of these new possibilities is illustrated in Figure 8, which shows the major traffic flows in the city of Sulmona, Italy. The GIS databases are likely to be developed further in the near future, which implies that linkages between GIS systems and transportation planning databases will also be moved forward.
Figure 8: GIS-enhanced view of traffic flows in Sulmona, Italy.
Conclusion
The new challenges posed by the current issues in urban transportation planning may be met by adapting a flexible modeling approach which may be adapted for particular needs. This implies that many of the accepted practices must be modified and new analysis tools developed.
References
- Beckmann, M., C.B. McGuire and C.B. Winsten, (1956), "Studies in the Economics of Transportation," Yale University Press, New Haven, Conn.
- Knight, F.M., (1924), "Some Fallacies in the Interpretation of Social Costs," Q.J. Econ. , Vol. 38, p. 582-626.
- Wardrop, J.G., (1952), "Some Theoretical aspects of Road Traffic Research," Proc. Institute of Civil Engineers, Part II, Vol. 1, p. 325-378.
- Florian, M. and D. Hearn, (1995), "Network Equilibrium Models and Algorithms," chapter 6 in M.O. Ball et al. Eds., Handbook in OR & MS, Vol. 8, p. 485-550.
- INRO Consultants Inc., (1998), EMME/2 User's Manual, Release 9.0, 944 pages.
Michael Florian is president of INRO Consultants Inc. and professor in the Department of Computer Science and Operations Research and the Center for Research on Transportation at the Université de Montréal.
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