## Appendix B Algorithm Performance Comparison  
Most of the MSA-based algorithms proposed in the literature for solving the traffic assignment problem are based on a modification of the weight formulation (i.e., term \(1/k\) in the traditional MSA algorithm) by adopting a \(1/\xi(k)\) function, that is:  
\[f^{k}=f^{k-1}+1/\xi(k)\left(f_{\text{SUN}}{}^{k}-f^{k-1}\right) \tag{15}\]  
The differences in formulation among these algorithms are related to the adopted \(\xi(k)\) function. Indeed, the traditional MSA algorithm [2; 3] assumes:  
\[\xi(k)=k \tag{16}\]  
Likewise, our proposal described in Section 3 adopts:  
\[\xi(k)=1+(k\text{-}1)\cdot\eta \tag{17}\]  
Obviously, in the case of \(\eta=1\), our proposal provides the traditional MSA formulation, that is (17) degenerates into (16).  
In order to provide some comparisons in terms of convergence speed of the proposed algorithm with some other MSA-based algorithms, we have considered:* the _Refresh Memory method_ (RM), also indicated as _Restarting MSA method_ (RMSA), proposed by Reference [18] and applied by Reference [12], whose \(\xi(k)\) function may be formulated as follows: \[\xi(k)=1,2,\ldots,\zeta,2,3,\ldots,2\zeta,4,5,\ldots,4\zeta,\ldots,x,x+1,\ldots,x \zeta,\ldots\] (12) with: \[x\text{ integer }x=1,2,4,8,\ldots\] (13) where we have assumed \(\zeta=10\) (method acronym 'RM 10') as proposed by Reference [18];
* the _Polyak method_ (POL) [9], whose \(\xi(k)\) function is formulated as follows: \[\xi(k)=k^{2/3}\] (14)
* the _Nagurney and Zhang method_ (NAZ) [10], whose \(\xi(k)\) function is formulated as follows: \[\xi(k)=1,2,2,3,3,3,4,4,4,4,4,\ldots,x,x,\ldots,x\text{ ($x$ times), $\ldots$}\] (15)
* the _Bar-Gera and Boyce method_ (BGB) [11], whose \(\xi(k)\) function is formulated as follows: \[\xi(k)=\zeta\] (16) where we have assumed \(\zeta\) equal to 5 (method acronym 'BGB 5') and equal to 10 (method acronym 'BGB 10')  
Our proposal has been classified as _muffled method_ by adopting the acronym 'M \(\eta\)', so that 'M 1' represents the traditional MSA formulation.  
The comparison among algorithm formulations has been implemented in the case of the urban network of Benevento (described in Section 4.2) by adopting a \(Cv\) value equal to 0.05. Numerical results are shown in Table 1 and Figure 11.  
Numerical results have shown that the proposed approach (i.e., the muffled method), at least in the case under consideration (the urban network of Benevento with a \(Cv\) coefficient equal to 0.05) almost always provides the best results, i.e., it requires the minimum number of iterations to reach convergence.  
It is worth noting that, although the convergence can be demonstrated theoretically (as already shown in Appendix A), a theoretical proof cannot be performed in the case of the convergence speed property which represents an empirical element to be verified by means of numerous and appropriate tests.  
However, although it is not possible to generalise the optimal performance of the proposed methodology in an absolute way, the results obtained give hope that at least the proposed method can be included among the ones providing optimal performances. Obviously, the confirmation of this statement can only be obtained through a large number of tests in the case of different networks both in terms of context (urban vs. rural), network dimension (urban, provincial or regional network), demand and/or congestion levels.