## 5 Conclusions and Research Prospects  
The results obtained on the small and on the real-scale networks show that the proposed algorithm is able to reduce the number of iterations for convergence, and hence, computing times compared with the classic MSA approach; in particular, the reductions arrive in percentage up to -79% with respect to the classic MSA for the best value of the parameter. On all networks the results are similar, showing that the benefits of the proposed algorithm are significant for medium-high congested demand levels (average saturation degrees over about 0.4); less significant, but not negligible, are the benefits for lower demand levels.  
In all tests, the best values of parameter \(\eta\) lie between 0.3 and 0.6. An important characteristic of the proposed algorithm is that there is a parameter that can be opportunely chosen, so to optimise the performances before starting with network design procedure; moreover, examining Figures 3, 5, 6 and 8 it is evident that, especially for medium-high levels of demand, a minimum can be found. In our test, we adopt a 0.1 step, but the parameter \(\eta\) can assume any value between 1 (MSA) and 0 (0 excluded), and therefore, other values can be tested in order to find the one that minimises the computing times.  
The reduction in computing times can be not important if the assignment procedure is performed only once or a few times: with current computing capabilities, the savings are lower than a minute on the real-scale network. By contrast, if the assignment is a subroutine of the Network Design Problem, that requires the calculation of the equilibrium traffic flows many thousands of times, the reduction in  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **MSA (\(\eta\) = 1.0)** & **Muffled (\(\eta\) = 0.5)** & **\% Difference** \\ \hline Examined solutions & 6019 & 6223 & 3.39\% \\ Examined neighbourhoods & 59 & 61 & 3.39\% \\ \hline SNLs & 72,893 & 39,103 & \(-\)46.36\% \\ Computing time [h] & 5,83 & 3.25 & \(-\)44.18\% \\ Starting objective function value [€] & 38,010,674,174 & 37,971,126,906 & \\ Final objective function value [€] & 29,363,169,248 & 29,305,075,507 & \\ \% variation of objective function value & \(-\)22.75\% & \(-\)22.82\% & \\ \hline \hline \end{tabular}
\end{table}
Table 10: Comparison between algorithms.  
Figure 9: Comparison between algorithms in terms of computing times vs. objective function values.  
computing time of an assignment is transferred to the computing time of the network design algorithm, allowing as much as several days to be saved in the calculation. In our test on the rural road network, the proposed algorithm is able to save about 44% of computing time, equivalent to about 2.5 h for a single NSA procedure.  
Future research will focus on testing other real-scale networks, testing the algorithm on multimodal networks and under the assumption of elastic demand and proposing a similar algorithm for solving the combined assignment-control problem. Moreover, testing the proposed algorithm within other real-scale network design procedures will be the subject of future studies.  
Conceptualisation, M.G. and L.D.; Data curation, M.B., M.G., M.M., L.D.; Formal analysis, M.G. and L.D.; Investigation, M.G. and L.D.; Methodology, M.G. and L.D.; Validation, M.B. and M.M.; Writing--original draft, M.B., M.G., M.M., L.D.; Writing--review and editing, M.B., M.G., M.M., L.D. All authors have read and agreed to the published version of the manuscript.  
This research was partially funded by Federico II University of Naples (Italy), grant number 000009-ALTRO-R-2020-LD \"Department research DICEA\" and by University of Sannio, University Research Fund FRA_2018.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.