## 3 Multimetric Map-Aware Routing Protocol (MMMR)
### Forwarding Decision  
MMMR makes forwarding decisions hop-by-hop based on geographic information. When a node wants to send a packet, it has first to find the optimal next forwarding node from its list of neighbors.  
When a sender node receives hello messages from its neighbors in transmission range, the node updates its neighbors list with all those nodes that sent their packets with enough power to be considered as a neighbor. After that, the node evaluates and assigns a total multimetric qualification to each neighbor. We assign the same weights (\(w_{1},w_{2},w_{3},w_{4}\)) to each metric (\(u_{1},u_{2},u_{3},u_{4}\)) in the qualification of each neighbor, \(i\). We propose a weighted geometric average metric with Equation (26). A global geometric score is used, because it is less sensitive than the arithmetic metric in the extreme values of the metric components. Due to the fact that all the metrics are exponential functions with the same base, the next forwarding node will be the one that has the greatest sum of all the exponents.  
\[\bar{u}_{i} = \prod_{j=1}^{4}u_{j}^{w_{j}}=u_{1}^{w_{1}}\cdot u_{2}^{w_{2}} \cdot u_{3}^{w_{3}}\cdot u_{4}^{w_{4}} \tag{26}\] \[= \prod_{j=1}^{4}u_{j}^{w_{j}}=e^{-\left(\frac{d_{i}}{d_{ref}} \right)^{\alpha\,w_{1}}}\cdot e^{-\left(\frac{\Delta d(t)}{d_{ref}}\right)^{ \alpha\,w_{2}}}\cdot e^{-\frac{1}{\mu_{i}(t)}\,w_{3}}\cdot e^{-\frac{1}{\Delta \bar{u}_{i}}\,w_{4}}\]  
Because of this, to simplify the computation of the final value, the logarithmic of each metric is used as shown in Equation (27), which is a monotonic transformation of the original score function.  
\[\ln\bar{u_{i}} = \sum_{j=1}^{4}w_{j}\cdot\ln u_{j}=w_{1}\cdot\ln e^{-\left(\frac{d_{i }}{d_{ref}}\right)^{\alpha}}+w_{2}\cdot\ln e^{-\left(\frac{\Delta d(t)}{d_{ref}} \right)^{\alpha}}+w_{3}\cdot\ln e^{-\frac{1}{\rho_{i}(t)}}+w_{4}\cdot\ln e^{- \frac{1}{\Delta\text{BE}_{i}}}\] \[= -w_{1}\cdot\left(\frac{d_{i}}{d_{ref}}\right)^{\alpha}-w_{2}\cdot \left(\frac{\Delta d(t)}{d_{ref}}\right)^{\alpha}-w_{3}\cdot\frac{1}{\rho_{i}( t)}-w_{4}\cdot\frac{1}{\text{ABE}_{i}}\] \[-\ln\bar{u_{i}} = w_{1}\cdot\left(\frac{d_{i}}{d_{ref}}\right)^{\alpha}+w_{2}\cdot \left(\frac{\alpha\cdot d^{\alpha-1}\cdot\parallel\vec{v}\parallel\cdot\cos \theta}{d_{ref}^{\alpha}}\right)+w_{3}\cdot\frac{1}{\rho_{i}(t)}+w_{4}\cdot \frac{1}{\text{ABE}_{i}} \tag{27}\]  
We give a score to each node using the negative of the logarithm in the final calculation of the metric. Therefore, the best next forwarding node is the neighbor with the lowest multimetric value obtained applying the MMMR algorithm. Notice that the exponent of the metrics distance and trajectory include also the exponent alpha, which could be merged with the weight of the metric (see Equation (26)) in a unique exponent as a new compound weight. Nevertheless, we preferred not to do that in order to emphasize the difference between the metric itself and the correspondent weight and to not distort their role in the equation. As previously said, we give the same degree of importance to all the metrics, _i.e._, we set \(w_{j}=1/4\), as the starting point of our work.