1. Overview

Contemporary transportation networks are intensively used and congested at various degrees, notably road transportation systems in urban areas. What is less known is the spatial logic behind the generation, attraction and distribution of traffic on a network. There are two important concepts related with understanding traffic in transport systems:

• The transport demand between places must either be known or estimated. For instance, the gravity model offers a methodology to estimate potential flows between locations if a set of attributes are known, such as respective distances and emission and attraction variables.
• The transport supply between places must also either be known or estimated. This involves establishing a set of paths between places that are generating and attracting movements. This includes the geometric definition of transport networks with the graph theory.

However, a fundamental concept is absent, that is how traffic is distributed in a transport network when we know its structure, capacity and the spatial demand.

An assignment problem is the distribution of traffic in a network considering a demand between locations and the transport supply of the network. Assignment methods are looking for a way to model the distribution of traffic in a network according to a set of constraints, notably related to transport capacity, time and cost.

The purchase of a an airplane ticket is a classic example of traffic assignment. For instance, a potential traveler wishes to go from city A to city B at a specific date and around a specific time. A query to a reservation system (either through a travel agent or, increasingly, online) will offer a set of choices (paths) along with a price quote for each path. The traveler will likely chose the least expensive path, which may not necessarily be a direct path and may involve a transfer at city C. When tens of thousands of travelers make these decision each day, the assignment of passengers to paths (air service) become a very complex task for airline companies and their reservation (traffic assignment) system. On the other hand, airline companies use these decisions to adjust their transport supply (mainly planes) to match the demand as closely as possible. This type of problem can be solved using optimization methods.

2. Traffic and its Properties

Traffic is the number of units passing on a link in a given period of time and it is commonly represented by Q(a,b), that is the amount of traffic passing on the a,b link (between a and b). Units can be vehicles, passengers, tons of freight, etc. Because of the characteristics of transportation networks, there are two major types of traffic flows:

• Uninterrupted traffic. Traffic regulated by vehicle-vehicle interactions and interactions between vehicles and the transport infrastructure. The most common example of uninterrupted traffic is an highway.
• Interrupted traffic. Traffic regulated by an external means, such as a traffic signal, which often create a queuing. Under interrupted flow conditions, vehicle-vehicle interactions and vehicle-infrastructure interactions play a less important part. The most common example of interrupted traffic in the urban circulation regulated by traffic signals such as lights and stop signs.

Traffic is not a spatial interaction as an interaction represents movements between locations (origins and destinations), while traffic represents movements on links of a network. Traffic could be similar to an interaction when the transport network is equal to the set of Origin / Destination (O/D) pairs, but this is very unlikely.

• Traffic is represented in a graph (network) by its value; the number of any units flowing (cars, people, tons etc.). The intensity of the traffic is proportional to the load of the network.
• Traffic is also represented in a graph by its assignment; how the traffic is distributed on a graph according to supply and demand.

Traffic is assigned on a network according to a sequence of links where every link has its own value and direction where several conditions that must be satisfied:

• There must exists nodes in the graph where traffic can be generated and attracted. These nodes are generally associated to centroids in an O-D matrix.
• The minimal (l(a,b)) and maximal (k(a,b)) capacities of every link must be respected. k(a,b) is the transport supply on the link (a,b).
• Transport demand must respected. The O/D matrix has equal inputs and outputs (closed system).
• There is conservation of the traffic at every node that is not an origin or a destination.

There are also two general measures of traffic in a network:

• Maximal Load (ML): Number of units of traffic that a network can support at a point in time. The maximal load is the summation of the capacity of all links.

• Load (L): Number of units of traffic that a network supports while fulfilling a transport demand. Load is the summation of the traffic of all links.

When the load of a network reaches the maximal load, congestion is reached.

3. Traffic Maximization and Costs Minimization

Traffic in a transportation network can be represented from two perspectives, traffic maximization and costs minimization. Traffic maximization involves the determination of the maximal transport demand that a network or a section of network can support between its nodes.

It involves maximizing traffic for all links, where the traffic on links must be equal or lower to the capacity of the link. The heuristic method is the easiest way to solve this equation for simple networks. Cost minimization involves the determination of the minimal transport costs considering a known demand. Transport costs on a link are expressed by g(Q(a,b)) and the minimization function by:

The goal of this equation is to minimize the summation of transport costs (global cost) of each link subject to capacity constraints. Once again, the heuristic method is the easiest way to solve this equation for simple networks. Several types of costs are involved in the minimization procedure.

• Click here to download some traffic assignment problems (PowerPoint).