## 4 Simulation Results
### Sensitivity Analysis  
In this sub-section, a sensitivity analysis is carried out to determine the most critical position-related criteria and to evaluate the robustness of the investigated MADM algorithms [24; 25]. A sensitivity analysis can be performed for all the available methods, but in our case, only \(n=3\) MADM methods are compared: SAW, CODAS, and TOPL1, while \(m=6\) alternative vehicles are examined for a trajectory of \(k=2000\) epochs. Each epoch consists of data from \(i=6\) criteria, respectively: Number of satellites, L1 RMS double-difference phase residual, Hz std, V std, Amb Status, and HDOP indicator. Seven weight matrices are constructed: at first, a matrix with \(1/6\) equal weight for every criterion is formulated, and then six different weight matrices where each criterion weight is set to have a weight of \(2/6\) and the remaining criteria have equal weights. Thus, the weight matrices are presented analytically in Table 11. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**\% Simulations** & **I** & **II** & **III** & **IV** & **V** \\ \hline
**veh. \#02** & 14.3 & 3.2 & 82.5 & 0 & 0 \\
**veh. \#03** & 85.7 & 1.6 & 12.7 & 0 & 0 \\
**veh. \#04** & 0 & 0 & 0 & 0 & 100 \\
**veh. \#05** & 0 & 0 & 0 & 100 & 0 \\
**veh. \#06** & 0 & 95.2 & 4.8 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: All MADM simulations. Figure 3: Mean ranking of I/II/III positions of alternative vehicles for the entire trajectory using all normalizations and 13 MADM methods. (**Top**) Equal weights; (**Bottom**) Unequal weights. Every timestamp corresponds to a column that includes the ranking of alternative vehicles for each MADM method (\(rank_{m/n}\)), and thus the dimensions of the ranking matrix for every timestamp are \((mxn)\). The subsequent step is the correlation coefficient matrix calculation using [PERSON]'s correlation method between the ranking matrix obtained from the equal weight matrix \(w_{0}\) with the results of the rest weight matrices (\(w_{1}\), \(w_{2}\), \(w_{3}\), \(w_{4}\), \(w_{5}\), and \(w_{6}\)), where the result is a \((nxi)\) matrix at every timestamp. In order to calculate the total correlation coefficient matrix from the complete trajectory, a sum along the third dimension is performed, and then it is divided by the total number of timestamps; this is the total correlation index (TCI). Conclusively, the criterion with the smaller TCI is the most sensitive one per each MADM method. The [PERSON]'s correlation coefficient _rho_ between the two ranking vectors \(X\), \(Y\) is given by:  
\[rho(X,Y)=1-\frac{6\sum\limits_{i=1}^{n}d_{i}^{2}}{n(n^{2}-1)} \tag{2}\]  
where, \(d_{i}\) is the difference in the \(i_{\text{th}}\) rank between two ranking vectors (i.e., two MADM methods) and \(n\) is the length of each vector (number of alternatives). If the input is a ranking matrix \(X\) with multiple columns, \(rho(X)\) returns a matrix with the correlations between each pair of its columns. The [PERSON]'s correlation coefficient \(\rho\) is a metric of the linear correlation between two random variables, \(X\) and \(Y\), and it can be alternatively employed using the real-valued performance scores of the MADM methods instead of the integer-valued ranking vectors. It is defined as follows:  
\[\rho(X,Y)=1-\frac{\sum\limits_{i=1}^{n}\left(X_{i}-\overline{X} \right)\left(Y_{i}-\overline{Y}\right)}{\left\{\sum\limits_{i=1}^{n}\left(X_{ i}-\overline{X}\right)^{2}\sum\limits_{j=1}^{n}\left(Y_{j}-\overline{Y} \right)^{2}\right\}^{1/2}} \tag{3}\]  
where \(\overline{X}\), \(\overline{Y}\) are the mean values and \(n\) is the length of each vector. Table 12 summarizes Spearman's correlation results along the complete trajectory, where clearly, the smaller TCI values are observed in criteria L1 RMS double-difference phase residuals, HDOP, and Hz std, while the other criteria are less sensitive. Table 13 includes the Pearson's correlation results where the smaller TCI values are also spotted in criteria L1 RMS, HDOP, and V std, whereas the other criteria are slightly less sensitive.