# An Assessment Scheme for Road Network Capacity under Demand Uncertainty
## 2 Network Capacity Problem under Demand Uncertainty
### Road Network Capacity Model with Uncertain Demand  
In this subsection, we extend the classical reserve capacity model by treating the initial/current O-D demand as an uncertain parameter. The following two reasons are given for using the reserve capacity model to assess capacity of the network with uncertain demand:  
1. Network capacity is represented by a uniformly increased or decreased multiplier; therefore, the reserve capacity model is straightforward to solve.
2. Given that the initial/current O-D matrix is regarded as an uncertain parameter in the reserve capacity model, the demand distribution is no longer fixed but variable within a given range.  
Firstly, the deterministic reserve capacity model is given as follows.  
\[max\;\mu, \tag{2}\]  
\[s.t.\;v_{a}(\mu)\leq C_{a},\;\forall a\in A, \tag{3}\]  
where \(v_{a}(\mu)\) represents the flows on link \(a\), which is an implicit function of the multiplier \(\mu\), the relationships of which are described in the PSL model [25].  
\[min_{f}\sum_{a}\int_{0}^{v_{a}}t_{a}(x)dx-\frac{1}{\theta_{k}}\sum_{ij}\sum_{ k}f_{k}^{ij}ln\varpi_{k}^{ij}+\frac{1}{\theta_{k}}\sum_{ij}\sum_{k}f_{k}^{ij} \ln f_{k}^{ij}, \tag{4}\]  
\[s.t.\sum_{k\in K_{ij}}f_{k}^{ij}=\mu q_{ij},\;\forall i\in I,\;j\in J, \tag{5}\]  
\[v_{a}=\sum_{i}\sum_{j}\sum_{k}f_{k}^{ij}\cdot\delta_{a,k}^{ij},\;\forall a\in A, \tag{6}\]  
\[f_{k}^{ij}\geq 0,\;\forall i\in I,\;j\in J,\;k\in K_{ij}, \tag{7}\]  
where \(C_{a}\) represents the capacity of road link \(a\). \(f_{k}^{ij}\) represents the flow between O-D pairs (\(i\), \(j\)) on route \(k\). \(\theta_{k}\) represents the dispersion parameter related to route selection. \(\delta_{a,k}^{ij}\) represents the link/route incidence parameter and equals 1 if route \(k\) (from \(i\) to \(j\)) travels through link \(a\) and 0 otherwise. \(t_{a}(v_{a})\) represents the travel time function of the link \(a\). \(\varpi_{k}^{ij}\in(0,1]\) represents the path-size factor, which aims to deal with the route-overlapping problem. The typical form is given by [PERSON] and [PERSON] [25]:  
\[\varpi_{k}^{ij}=\sum_{a\in A_{k}}\frac{L_{a}}{L_{k}^{ij}}\frac{1}{\sum_{l\in K _{ij}}\delta_{al}^{ij}},\;\forall k\in K_{ij},\;i\in I,\;j\in J, \tag{8}\]where \(L_{a}\) represents the length of link \(a\). \(L_{k}^{ij}\) represents the length of route k connecting O-D pairs \((i,j)\), and \(A_{k}\) represents the collection of all links of route k between O-D pairs \((i,j)\).  
In the above network capacity model, the objective function Equation (2) for the upper-level problem aims to maximize the multiplier of O-D demand. Constraint (3) indicates that the flow of each link cannot exceed its physical volume constraints. The objective function Equation (4) for the traffic assignment considers both route choice and route overlap problem, in which the first term is the classical Beckmann transform term, the second term considers the path-size factor for dealing with route overlap, and the third term is connected to the entropy maximization for stochastic equilibrium. Equations (5) and (6) are defined as demand conservation and flow conservation, respectively. Equation (7) is the nonnegative constraint.  
Further, according to the above reserve capacity network model, we introduce the uncertain demand variable \(\mathbf{q}\in\mathcal{Q}\), where \(Q=\left\{q_{ij}|q_{ij}^{L}\leq q_{ij}\leq q_{ij}^{U}\right\}\) is an uncertain demand set based on existing demand. Figure 4 compares the exact solution approach and the approximate approach. Clearly, it is very expensive to solve the network capacity with all possible O-D demand patterns because solving the traffic assignment model requires a large number of iterations. Hence, an approximate solution algorithm is proposed in the next section aiming to address the inefficiency of exact solution approach in solving network capacity with uncertain travel demand.