## 4 Numerical Results
### Small Network  
The small network consisted of 16 oriented links (see Figure 2), whose link performances (i.e., the link travel times) were expressed by means of a BPR function [26], that is:  
\[t_{l}=(L_{l}/v_{0,l})\cdot(1+\alpha\cdot(f_{l}/Cap_{l})^{\beta}) \tag{10}\]  
where \(t_{l}\) is the travel time associated to link \(l\); \(L_{l}\) is the length of link \(l\); \(v_{0,l}\) is the free-flow speed associated to link \(l\); \(f_{l}\) is the link flow associated to link \(l\); \(Cap_{l}\) is the capacity of link \(l\); \(\alpha\) and \(\beta\) are function parameters. The main features of the considered network are summarised in Table 1.  
Figure 1: Values of weights \(w_{\text{SUN}}\).  
Figure 2: Small network.  
Bold links have higher capacity and free-flow speed with respect to the others. We assume that nodes 1, 2, 3 and 4 are also the origins and destinations of the trips, according to the OD vector, \(\mathbf{d}\), reported in Table 2. In the tests, we adopted Multinomial Logit as the stochastic path choice model and assumed that the cost of each link was equal to the running time expressed in minutes with parameter \(\theta\) equal to 0.5 (coefficient of variation about 0.2). We implemented the proposed algorithm in Visual Basic. We tested the proposed algorithm with the following values of parameter \(\eta\): 1 (classic MSA), 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, 0.05, 0.01, 0. For each value, we tested the OD vector reported in Table 2 multiplied by the following coefficients: 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0. The use of different OD vectors, as well as different multipliers, is to analyse the algorithm's performance for different congestion levels. Indeed, the vectors thus obtained produce average saturation degrees (flow/capacity ratio) on the network between 0.249 and 0.838, as reported in Table 3. The convergence threshold was assumed to be equal to 1% for all tests.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Link** & **Length (km)** & **Capacity (veh/h)** & **Free-Flow Speed (km/h)** & **BPR Param. \(\mathbf{\alpha}\)** & **BPR Param. \(\mathbf{\beta}\)** \\ \hline
1–5 & 2 & 1800 & 60 & 2.0 & 4.0 \\
5–1 & 2 & 1800 & 60 & 2.0 & 4.0 \\
1–3 & 2 & 1800 & 50 & 2.0 & 4.0 \\
3–1 & 2 & 1800 & 50 & 2.0 & 4.0 \\
3–6 & 3 & 3600 & 90 & 0.2 & 4.0 \\
6–3 & 3 & 3600 & 90 & 0.2 & 4.0 \\
5–6 & 2 & 3600 & 80 & 0.2 & 4.0 \\
6–5 & 2 & 3600 & 80 & 0.2 & 4.0 \\
5–2 & 4 & 1800 & 50 & 2.0 & 4.0 \\
2–5 & 4 & 1800 & 50 & 2.0 & 4.0 \\
6–2 & 5 & 1800 & 70 & 2.0 & 4.0 \\
2–6 & 5 & 1800 & 70 & 2.0 & 4.0 \\
6–4 & 5 & 3600 & 90 & 0.2 & 4.0 \\
4–6 & 5 & 3600 & 90 & 0.2 & 4.0 \\
4–2 & 3 & 1800 & 60 & 2.0 & 4.0 \\
2–4 & 3 & 1800 & 60 & 2.0 & 4.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Features of the small network.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Origin** & **Destination** & **OD Flow (veh/h)** \\ \hline
1 & 2 & 1700 \\
2 & 1 & 600 \\
1 & 3 & 500 \\
3 & 1 & 700 \\
1 & 4 & 650 \\
4 & 1 & 100 \\
2 & 3 & 800 \\
3 & 2 & 450 \\
2 & 4 & 750 \\
4 & 2 & 450 \\
3 & 4 & 1250 \\
4 & 3 & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Origin-destination vector of the small network.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Origin** & **Destination** & **OD Flow (veh/h)** \\ \hline
1 & 2 & 1700 \\
2 & 1 & 600 \\
1 & 3 & 500 \\
3 & 1 & 700 \\
1 & 4 & 650 \\
4 & 1 & 100 \\
2 & 3 & 800 \\
3 & 2 & 450 \\
2 & 4 & 750 \\
4 & 2 & 450 \\
3 & 4 & 1250 \\
4 & 3 & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Average saturation degrees on the small network.  
Note that the reduction in the number of iterations, and hence, in computing time, provided by the proposed algorithm can be very significant for medium-high congested networks, i.e., up to \(-61\%\) compared with the traditional version of the MSA algorithm.