## 4 Simulation Results
### Comparative Analysis  
In this section, thirteen MADM algorithms are simulated with various weightings, normalization techniques, and criteria associated with positioning accuracy employing experimental NMEA data time series. A comparative analysis is provided through an evaluation of the MADM algorithms in terms of ranking performance and robustness. An investigation of the importance and criticality of the criteria is also presented (sensitivity analysis), accompanied by the necessary ranking diagrams and simulation results. The investigated methods are the SAW, CODAS, COPRAS, GRA, MABAC, DIDEAL, TOPL1, TOPL2, TOPL3, VIKOR, ORESTE, PROMETHEE, and ELECTRE, where TOPL1 is the TOPSIS using the L1 norm or taxicab/Manhattan distance metric, TOPL2 employs the L2 norm, which is the Euclidean distance of each alternative from the best and worst ideal points, TOPL3 employs the infinite norm distance metric, while four distinct preference functions were employed for PROMETHEE. The five alternative vehicles move in simultaneous but separate trajectories that span 2000 epochs. For each trajectory epoch, a decision matrix is formed with criteria values extracted from the NMEA sentences of the five aiding vehicles. Hence, a decision matrix associated with a specific timestamp (epoch) yields as many ranking results as the number of MADM algorithms and normalization techniques. In summary, in our MADM simulations, a trajectory of \(T=2000\) timestamps subject to \(Q=6\) criteria is examined employing \(P=5\) alternative vehicles (veh. #02-veh. #06) while the ranking results of 13 different MADM algorithms and five normalization techniques are reported. Table 5 exhibits the decision matrix with a of size (\(5\times 6\)) of a random timestamp filled with the required criteria values. The Amb Stat takes values of 1.0 for a fixed solution and 0.5 for a float solution. \begin{table}
\begin{tabular}{c c c c c} \hline a/a & **Criteria Name** & **NMEA Sentence** & **NMEA Field No** & **Type** \\ \hline
1 & Number of satellites (NS) & GNS & 7 & Benefit \\
2 & Range RMS (L1 RMS) & GST & 2 & Benefit \\
3 & Horizontal std (Hz std) & GST & 6,7 & Expense \\
4 & Vertical std (V std) & GST & 6,7 & Expense \\
5 & Integer ambiguity status (Amb Stat) & GNS & 6 & Benefit \\
6 & Horizontal dilution of precision (HDOP) & GNS & 8 & Expense \\ \hline \end{tabular}
\end{table} 
Table 4: Criteria used for implementing MADM technique in C-DGNS positioning. Table 6-9 present the simulations' ranking results after applying the MADM methods SAW, TOPL1, CODAS, and PROMETHEE, indicatively, with the criteria input data of Table 5. The index P\({}_{i}\) represents the performance score of the \(i_{\text{th}}\) alternative, and A\({}_{i}\) expresses the ID of the \(i_{\text{th}}\) alternative with \(i\) = 2, 3, 4, 5, and 6. It is observed that SAW, TOPL1, and CODAS produce identical rankings while PROMETHEE yields a slightly different top-three ranking. Moreover, the five normalization techniques are in good agreement with each other and with their corresponding ranking results. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Rank** & **MAX** & \multicolumn{2}{c}{**SUM**} & \multicolumn{2}{c}{**VEC**} & \multicolumn{2}{c}{**MAX-MIN**} & \multicolumn{2}{c}{**DEA**} \\ \hline a/a & P\({}_{\text{i}}\) & A\({}_{\text{i}}\) & P\({}_{\text{i}}\) & A\({}_{\text{i}}\) & P\({}_{\text{i}}\) & A\({}_{\text{i}}\) & P\({}_{\text{i}}\) & A\({}_{\text{i}}\) \\
**I** & 0.948 & 3 & 0.432 & 3 & 0.728 & 3 & 0.908 & 3 & 0.947 & 3 \\
**II** & 0.797 & 6 & 0.368 & 6 & 0.605 & 6 & 0.679 & 6 & 0.913 & 6 \\
**III** & 0.786 & 2 & 0.363 & 2 & 0.595 & 2 & 0.664 & 2 & 0.909 & 2 \\
**IV** & 0.589 & 5 & 0.257 & 5 & 0.441 & 5 & 0.401 & 5 & 0.731 & 5 \\
**V** & 0.289 & 4 & 0.094 & 4 & 0.194 & 4 & 0.016 & 4 & 0.499 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: SAW.