## Appendix C Network Design Problem (NDP)  
The NDP tackled in the test reported in Section 4.4 is a network capacity expansion problem where there are limited resources for improving the performances of a rural road network. The problem is the same tackled in Reference [28], slightly simplified, considering only the demand in the morning peak-hour. The problem is formulated as follows:  
\[\mathbf{y^{opt}}=\text{Arg}\ y\ \text{min}\ (\beta_{1}\cdot(\Sigma_{l}\ c_{l}(\mathbf{y}, f^{*}(\mathbf{y}))\cdot f^{*}(\mathbf{y}))+\beta_{2}\cdot(\Sigma_{i}\ y\ c_{i})+\beta_{3}\cdot(\Sigma_{l} \ (Rec_{km}\cdot\Sigma_{l}\ f^{*}(\mathbf{y})\cdot L_{l}))) \tag{11}\]  
subject to:  
\[y_{i}=0/1\quad\forall\ i\in I \tag{12}\]  
\[\Sigma_{i}\ y_{i}\ c_{i}\leq B \tag{13}\]  
\[f^{*}(\mathbf{y})=\mathbf{A}\ \mathbf{P}(\mathbf{A}^{\text{T}}\mathbf{\chi}(\mathbf{f^{*}},\mathbf{y}))\ \mathbf{d} \tag{14}\]  
where \(y_{i}\) is the decision variable related to road \(i\); \(\mathbf{y}\) is the decision variable vector; \(\beta_{1}\), \(\beta_{2}\) and \(\beta_{3}\) are the weights of the objective function terms; \(c_{l}(.)\) is the cost function on road link \(l;f^{*}\) is the equilibrium flow on road link \(l;f^{*}\) is the equilibrium flow vector; \(cr_{i}\) represents the cost of the improving intervention on road \(i\) (on an hourly basis, assuming a useful life of the intervention of 20 years); \(Rec_{km}\) is the average environmental cost produced by a car travelling 1 km (\(\xi\)/km); \(L_{l}\) is the length of road link \(l\); \(B\) is the total available budget (on an hourly basis); \(A\) is the link-path incidence matrix; \(\mathbf{P}\) is the path choice probability matrix; \(\mathbf{\chi}(\ )\) is the cost function vector; \(\mathbf{d}\) is the demand vector.  
The objective function to be minimised, shown in (11), is given by the sum of user costs, improvement costs and environmental costs in terms of air pollution produced by cars. Other objective function formulation could also have been used (for instance, which would have considered the impact of the design solutions on road accident phenomena or on the severity of accidents) which would obviously have produced different optimal solutions. But the purpose of this application is to show the benefits of adopting the proposed assignment algorithm in providing the same solution as traditional algorithms, but in significantly shorter calculation times, rather than providing an analysis of the best objective function to be used in network design problems.  
However, the decision variable \(y_{i}\) is equal to 1 if the road \(i\) is improved, 0 otherwise. Equation (12) identifies the constraint on the binary nature of decision variables, Equation (13) is the budget constraint and Equation (14) represents the equilibrium assignment constraint.  
In order to apply the proposed model to test network described in Section 4.3, we have to identify the roads, \(i\), that can be improved, and for each of them, the corresponding improvement costs. Each road, \(i\), is composed by several road links, \(l\). In this test network, we identify 102 roads, and therefore, 102 decision variables, \(y_{i}\). In Figure 11, the 102 roads are reported. In Table 11 the possible interventions are summarised, in terms of performance improvements and costs.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Current**} & \multicolumn{2}{c}{**Improved Configuration**} & \multicolumn{3}{c}{**Current**} & \multicolumn{3}{c}{**Improved Configuration**} \\ \cline{2-11}
**Road** & \multicolumn{2}{c}{**Configuration**} & \multicolumn{2}{c}{**Improved Configuration**} & \multicolumn{3}{c}{**Road**} & \multicolumn{2}{c}{**Configuration**} & \multicolumn{2}{c}{**Improved Configuration**} \\ \cline{2-11}  & \(v_{\mathbf{0}}\) & \(Cap\) & \(v_{\mathbf{0}}\) & \(Cap\) & \(c_{\mathbf{r_{1}}}\) **(f/