# Review of Traffic Assignment and Future Challenges
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### Illustrative Example  
To illustrate the previously communicated scope of the studied traffic assignment problems, a concrete example is presented in the following. Let us consider the road network presented in Figure 1. This network represents the traffic supply. It has one origin and one destination, named \(o\) and \(d\), respectively. There are two itineraries to link \(d\) to \(o\). Each itinerary, \(k_{i}\), is a road segment, \(a_{i}\), with a given capacity, \(Q_{max}^{i}\), and a function relating the travel time, \(c_{i}(r_{i})\), to the flow, \(r_{i}\). We have for each road segment \(r_{i}\), the following characteristics:  
* In road segment \(a_{1}\), the capacity, \(Q_{max}^{1}=1800\) vph, and the travel time behave as follows: \[c_{1}(r_{1})=10\bigg{(}1+2\Big{(}\frac{r_{1}}{1800}\Big{)}^{2}\bigg{)}.\] (1)
* In road segment \(a_{2}\), the capacity, \(Q_{max}^{2}=3600\) vph, and the travel time behave as follows: \[c_{2}(r_{2})=20\bigg{(}1+2\Big{(}\frac{r_{2}}{3600}\Big{)}^{2}\bigg{)}.\] (2)  
The time in Equations (1) and (2) is given in minutes. The presented time functions for both roads are based on \(BPR\)[13] curves, which are widely used to model traffic behavior. One can notice from Equations (1) and (2) that when both roads are empty, the users spend 10 min and 20 min for traveling \(a_{1}\) and \(a_{2}\), respectively. Naturally, \(a_{1}\) is more attractive without considering the demand.  
Let us examine a scenario involving 3000 vehicles destined to travel from origin \(o\) to destination \(d\) within a one-hour time interval. We denote this demand as \(q^{od}\), and consequently, \(q^{od}=3000\) vph. Figure 2 illustrates all possible scenarios of traffic assignment. This figure gives \(c_{1}\) and \(c_{2}\), according to \(r_{1}\) and \(r_{2}\). Note that, from the problem statement, we have \(r_{1}+r_{2}=q^{od}\). From this figure, three assignments can be distinguished:  
* Shortest path (All-or-nothing): Without considering Equations (1) and (2) as though both roads are devoid of traffic, all vehicles follow road \(a_{1}\) because it is the shortest path. However, since travel time increases with traffic flow, an all-or-nothing assignment cannot be adopted to predict road use. In this figure, the solution of the all-or-nothing assignment is presented by a green circle. According to Equation (1), the users spend more than an hour to go from \(o\) to \(d\).
* Optimal assignment (Centralized): If we consider that energy consumption is proportional to travel time, the manager of a fleet of the 3000 vehicles has a strong incentive to minimize the average travel time. The red curve in Figure 2 presents the average travel time. The optimal solution is represented by the red circles. One can note that, with this solution, the vehicles taking road \(a_{1}\) spend less travel time than the ones using road \(a_{2}\).
* Fair assignment (Decentralized): In this configuration, each vehicle optimizes its itinerary, so that there is no better solution. The fair assignment is represented by the yellow circle.  
Figure 1: Example of a network of two alternative roads.  
This example clearly illustrates the difference between centralized and decentralized assignments. Assuming a cost proportional to travel time, the discrepancy in the sums of travel times between decentralized and centralized assignments is generally referred to as the cost of anarchy [14].