## 4 MMS-DSR Mathematical Formalization
### Numerical Illustration  
To provide a concrete understanding of the MMS-DSR scoring mechanism, we present a numerical example with network topology, as illustrated in Figure 2. This hypothetical scenario considers several candidate routes, each with different metrics for latency, bandwidth, reliability, and beamforming efficacy (BE), typical of diverse urban paths.  
The routes and their respective metrics are summarized in Table 3.  
Using the scoring function we assume the following parameters: \(\delta=0.2\), \(\lambda=0.3\), \(\beta=0\), Experience \(E=10\), dynamic weights at time \(t\): \(w_{1}(t)=0.35\), \(w_{2}(t)=0.4\), \(w_{3}(t)=0.2\), \(w_{4}(t)=0.05\).  
To ensure a fair comparison, the metrics are normalized to a \([0,1]\) scale using the formula:  
\[x_{\text{norm}}=\frac{x-x_{\text{min}}}{x_{\text{max}}-x_{\text{min}}} \tag{6}\]  
The routes and their respective normalized metrics are summarized in Table 4.  
The scoring function \(S(r,t)\) for each route is calculated using the normalized values. For example for \(r1\) we have the following:  
\[S(r1,t)=0.6125\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Route** & **Latency** & **Bandwidth** & **Reliability** & **Beamforming** & **Pprediction(r,t)** & **BF(r,t)** \\  & **(ms)** & **(Mbps)** & **Reliability** & **Efficacy (BE)** & **Pprediction(r,t)** & **BF(r,t)** \\ \hline \(r_{1}\) & 10 & 50 & 0.9 & 15 & 0.92 & 20 \\ \(r_{2}\) & 8 & 40 & 0.95 & 18 & 0.87 & 22.5 \\ \(r_{3}\) & 12 & 60 & 0.85 & 12 & 0.90 & 15 \\ \(r_{4}\) & 15 & 45 & 0.92 & 17 & 0.91 & 21 \\ \(r_{5}\) & 9 & 55 & 0.88 & 14 & 0.89 & 19 \\ \(r_{6}\) & 10 & 50 & 0.9 & 15 & 0.92 & 20 \\ \(r_{7}\) & 8 & 40 & 0.95 & 18 & 0.87 & 22.5 \\ \(r_{8}\) & 12 & 60 & 0.85 & 12 & 0.90 & 20 \\ \(r_{9}\) & 15 & 45 & 0.92 & 17 & 0.91 & 22.5 \\ \(r_{10}\) & 9 & 55 & 0.88 & 14 & 0.90 & 15 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Example routes with associated metrics.  
Figure 2: Network topology diagram illustrating the routes from Node A to Node J with respective metrics.  
Repeating the same calculations for each route we will obtain the values showed in Table 5:  
Even if by applying the scoring function, the path r9 emerges as the optimal route with the highest score of 0.775833, we must remember that in the classic DSR protocol, the primary metric for selecting the optimal path is typically the number of hops. Carrying out the route analysis of the graph illustrated in Figure 3, the possible paths from source to destination and the related Scoring Function (SF) values are shown in Table 6:  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Route** & **Latency** & **Bandwidth** & **Reliability** & **BE Norm** & **P prediction** & **BF** \\  & **Norm** & **Norm** & **Norm** & **Norm** & **Norm(r,t)** & **Norm(r,t)** \\ \hline r1 & 0.286 & 0.50 & 0.5 & 0.500 & 1.0 & 0.667 \\ r2 & 0.000 & 0.00 & 1.0 & 1.000 & 0.0 & 1.000 \\ r3 & 0.571 & 1.00 & 0.0 & 0.000 & 0.6 & 0.000 \\ r4 & 1.000 & 0.25 & 0.7 & 0.833 & 0.8 & 0.800 \\ r5 & 0.143 & 0.75 & 0.3 & 0.333 & 0.4 & 0.533 \\ r6 & 0.286 & 0.50 & 0.5 & 0.500 & 1.0 & 0.667 \\ r7 & 0.000 & 0.00 & 1.0 & 1.000 & 0.0 & 1.000 \\ r8 & 0.571 & 1.00 & 0.0 & 0.000 & 0.6 & 0.667 \\ r9 & 1.000 & 0.25 & 0.7 & 0.833 & 0.8 & 1.000 \\ r10 & 0.143 & 0.75 & 0.3 & 0.333 & 0.6 & 0.000 \\ \hline \end{tabular}
\end{table}
Table 4: Normalized metrics for each route.  
Figure 3: Network topology diagram illustrating the routes from Node A to Node J with respective metrics.  
\begin{table}
\begin{tabular}{c c} \hline
**Route** & **Scoring Function (SF)** \\ \hline r1 & 0.612500 \\ r2 & 0.425000 \\ r3 & 0.420000 \\ r4 & 0.715833 \\ r5 & 0.453333 \\ r6 & 0.612500 \\ r7 & 0.425000 \\ r8 & 0.620000 \\ r9 & 0.775833 \\ r10 & 0.333333 \\ \hline \end{tabular}
\end{table}
Table 5: Scoring Function values for each route after normalization.  
Considering that MMS-DSR prioritizes the route with the highest cumulative scoring function (SF) and minimum number of hops, the chosen optimal path would be the following:  
\(A->B->J\) with a total \(SF\) of 1.225.