# Review of Traffic Assignment and Future Challenges
## 3 Traffic Assignment: Theoretical Background
### Static Traffic Assignment (STA)
#### 3.1.1 User Equilibrium (\(Ue\)): Definition and Formulation of the Problem  
In the illustrative example, it is evident that the fair assignment (depicted by the yellow circle in Figure 2) aligns with the preferences of rational road users, each striving to minimize their travel time. Indeed, we are dealing with independent users, where each user seeks their own interest. The first Wardrop principle describes the behavior of users within this assignment. This principle is given by the following definition:  
**Definition 2** (First Wardrop Equilibrium Principle). : _An assignment is said to be at equilibrium according to the first Wardrop principle if the following two conditions are satisfied:_  
* _For each origin-destination pair,_ \(o\)_-_\(d\) _, the generalized costs of each utilized route are less than or equal to those of alternative (unused) routes._
* _If multiple routes are used for an_ \(o\)_-_\(d\) _pair, their generalized costs are equal._  
According to the definition of the Wardrop principle, it is clear that no user has an interest in changing their route alone. Indeed, if they modify their route, they increases the cost of their journey. This principle is also known as a \(UE\). Several mathematical formulations of [PERSON]'s first principle can be found in the literature. In the following, we present the one given in [16]. Let there be a network of vertices that includes demand origins and destinations, denoted, respectively, as \(o\in O\) and \(d\in D\). Let a constant demand, \(q^{od}\), be assigned to a set of routes, \(K^{od}\). Noting the flow assigned to route \(k^{\prime}nK^{od}\) by \(r_{k}^{od}\), we have  
\[\sum_{k\in K^{od}}r_{k}^{od}=q^{od}. \tag{3}\]  
Equation (3) reflects the fact that all demand from \(o\) to \(d\) is affected. The Wardrop's first principle of \(UE\) is written as follows:  
\[ \begin{array}{rcl}c(K^{\prime})=\min_{k\in K^{od}}(c(k))&\Rightarrow&r_{k}^ {od}\geq 0\\ c(K^{\prime})>\min_{k\in K^{od}}(c(k))&\Rightarrow&r_{k}^{od}=0,\end{array}  \tag{4}\]  
where \(c(k)\) is the generalized cost of route \(k\). Let us note from Equation (4) that the first principle is not written in the classical form of an optimization problem, i.e., an objective function under constraints. Although the formulation given in Equation (4) is common, the contribution of [PERSON] et al. [17] has enabled us to reformulate the problem by distinguishing between each segment of the road network. Let us consider the road network represented as a graph, \(G=(S,A)\), where \(S\) denotes the set of vertices, \(s\in S\), and \(A\) represents the set of directed arcs, \(a\in A\). The vertices stand for points of interest and intersections. We have \(O\subset S\) and \(D\subset S\). Each road segment is modeled by a directed arc, \(a\), connecting two vertices, \(s_{i}\) and \(s_{j}\), where \(i\
eq j\). Each route, \(k^{od}\), consists of a set of successive arcs, \(a\in I_{k}^{od}\). Therefore, we have  
\[c(k^{od})=\sum_{a\in I_{k}^{od}}c(a), \tag{5}\]  
where \(c(a)\) denotes the generalized cost of arc \(a\) (route segment). This Equation (5) states that the generalized cost of a route is the sum of the costs of its constituent arcs. This is true in the case where the generalized cost is the travel time; see Section 2.2. For all \(o\)-\(d\) and all \(k^{od}\) routes, the calculation of \(r_{k}^{od}\) defines the flow through each arc, \(a\), of the graph, \(G\). Let us denote \(r_{a}\), the flow assigned to arc \(a\). Thus, we have  
\[r_{a}=\sum_{I_{k}^{od}\
i a}r_{k}^{od}. \tag{6}\]  
In other words, the flow assigned to arc \(a\) is equal to the sum of the flows of all routes that pass through this arc. The static traffic assignment at equilibrium involves solving the following optimization problem:\[ \begin{split}\min_{k_{a}}&\sum_{a\in A}\int_{0}^{r_{a}}c_{ a}(z)dz\\ \text{s.t.}&(3),(5),(6)\\ & r_{k}^{od}\geq 0,r_{a}\geq 0\end{split}  \tag{7}\]  
The problem (7) is a convex optimization problem as it involves minimizing an integral of an increasing function over a compact domain.