# A Traffic Equilibrium Model for Multi-Modal Networks with Uncertain Demands
## 4 Effective Travel Cost (ETC) and Traffic Equilibrium Model
### Variational Inequality Formulation  
The equilibrium conditions are given by the intersection of the following two subsets of conditions:  
The choice of travel mode: For each \(OD\) pair \(w\in W\), the proportion of users in every travel mode is given as the logit model function, which is defined as follows:  
\[q_{w}^{m}=q_{w}\dfrac{\exp(-\theta\mu_{w}^{m})}{\sum_{m\in M}\exp(-\theta\mu_{ w}^{m})}\quad m\in M,w\in W \tag{27}\]  
where \(q_{w}^{m}\) is the proportion of passengers using travel mode \(m\) for \(OD\) pair \(w\); \(q_{w}\) is the travel demands for \(OD\) pair \(w\); \(\mu_{w}^{m}\) is the ETC of the minimum route in travel mode \(m\) for \(OD\) pair \(w\); \(\theta\) is a parameter of travel utility perception variation. Figure 3: ETC definition cases. The choice of route: It is assumed that each hyper-path on the multi-modal network has an ETC that models these implicit choices. The users' behaviors are modelled through a version of [PERSON]'s user-optimal principle. Therefore, in the equilibrium state, no traveller can unilaterally change the travel path to reduce the ETC; that is, all the used paths have the same ETC, which is less than or equal to the ETC of the unused paths. This condition can be formulated as follows:  
\[f_{w}^{p}\left(c_{w}^{p}-\mu_{w}^{m}\right)=0,c_{w}^{p}-\mu_{w}^{m}\geq 0, \forall p\in P_{w}^{m},\quad m\in M,w\in W \tag{28}\]  
Then, the feasible set can be described as follows:  
\[\sum_{m\in M}q_{w}^{m}=q_{w},\quad w\in W \tag{29}\]  
\[\sum_{p\in P_{w}^{m}}f_{w}^{p}=q_{w}^{m},\quad m\in M,w\in W \tag{30}\]  
\[v_{l}=\sum_{w\in W}\sum_{p\in P_{w}^{m}}f_{w}^{p}\sigma_{w}^{lp},\quad l\in L \tag{31}\]  
\[f_{w}^{p}\geq 0,\quad p\in P_{w},w\in W \tag{32}\]  
\[q_{w}^{m}>0,\quad m\in M,w\in W \tag{33}\]  
where (29) and (30) are the travel demand conservation constraints, (31) is a definitional constraint that summarizes all the route flows that pass through a given link, and (32) and (33) are non-negativity constraints on the route flows. Then, the equilibrium model can be formulated as a variational inequality problem \(VI(f,\Omega)\), as follows:  
\[\sum_{w\in W}\sum_{m\in M}\sum_{p\in P_{w}^{m}}c_{w}^{p}\left(f_{w}^{p^{*}} \right)\left(f_{w}^{p}-f_{w}^{p^{*}}\right)+\sum_{w\in W}\sum_{m\in M}\frac{1 }{\theta}\ln\frac{q_{w}^{m^{*}}}{q_{w}}\left(q_{w}^{m}-q_{w}^{m^{*}}\right)\geq 0 \tag{34}\]  
where \(\Omega\) represents constraints (29)-(33). The following two propositions state the equivalence of the VI formulation and the equilibrium model, as well as the existence of an equilibrium solution. Proposition 1: _If the effective route travel time function \(c_{w}^{p}\) is positive, and the solution of the VI problem (34) is equivalent to the equilibrium solution of the ETC model._  
Proof. : By considering the Karush-Kuhn-Tucker conditions of model (34), we have  
\[\left[\sum_{p\in P_{w}^{m}}c_{w}^{p}\left(f_{w}^{p}\right)-\mu_{w}^{m}\right] f_{w}^{p}=0,\sum_{p\in P_{w}^{m}}c_{w}^{p}\left(f_{w}^{p}\right)-\mu_{w}^{m} \geq 0,\quad\forall p\in P_{w}^{m},w\in W \tag{35}\]  
Figure 3:0  
Then, from (33), we obtain  
Figure 3:1  
which can be written in the following form:  
Figure 3:2Combined with (29), this yields  
Figure 3:3  
When (38) is substituted into (37), we obtain  
Figure 3:4  
The final equation is the logit model (34). If the route ETC function \(\xi_{w}^{p}\) is positive and continuous, the ETC model has at least one solution. Based on Proposition 1, solving the equivalent VI formulation is sufficient. Note that the feasible set is nonempty and convex. Furthermore, according to the assumption, the route ETC is continuous. Thus, the VI problem (21) has at least one solution.