## 4 Relay Scheme based on FAHP
### Measurement of Relative Importance
#### 4.2.3 Criterion 3: Number of Neighbors  
As done with the two previous criteria, distance and number of hops, the numbers of neighbors between two candidate nodes are compared, and a final value should be obtained within the Saaty scale [1; 9] to build the corresponding pairwise matrix. In a random network, is impossible to know a priori what the maximum number of neighbors of a node is. However, we can approximate the problem to be in an intermediate situation bounded by the best and the worst topologies. The best topology we can imagine is a dense and regular mesh of nodes, where all of them are at the distance \(d_{min}\), imposed by the deployment conditions for the network. When more nodes are included, a node in the center of a sphere of radius \(d_{TX}\) (transmission range) will have a greater number of neighbors. Due to the problem being 3D, it is necessary to think of filling this sphere (radius \(d_{TX}\)) with a regular structure of nodes. The solution is to use regular polyhedrons, which can be inscribed in a sphere and keep the regularity. That is, the nodes (vertices) will keep the distance \(d_{min}\) between adjacent vertices. Among the five regular polyhedrons known (tetrahedron, cube, octahedron, dodecahedron, and icosahedron), the polyhedron with the greatest number of vertices is the dodecahedron with 20 vertices and 12 faces, shown in Figure 7.  
The best topology for having the maximum number of neighbors is to fill the big sphere of radius \(d_{TX}\) with equal dodecahedrons of side length \(d_{min}\). This way, between two adjacent nodes (vertices), a distance \(d_{min}\) is conserved and the mesh is a 3D structure. In spite of existing holes in the space when we fill a sphere with dodecahedra (the rhombic dodecahedron is the structure that tessellates the space without holes), we follow with this reasoning (simpler), and by using the volumes of the sphere and dodecahedron, the expression to obtain the total number of neighbors (nodes) \(N_{n,\ max}\) can be approximated as  
\[V_{sphere}=4\ \pi\ d_{TX}^{3} \tag{26}\]  
\[V_{dodecahedron}=\frac{1}{4}\Big{(}15+7\sqrt{5}\Big{)}d_{min}^{3} \tag{27}\]  
\[N_{n,\ max}\ \sim\frac{V_{sphere}}{V_{dodecahedron}}\times 20-N_{common} \tag{28}\]  
where \(N_{common}\) is the number of common vertices when a set of dodecahedra are joined together. This number is easy to obtain in a practical way, and a possible empirical solution is presented in Table 2 (it is not the only one, depending on the position selected for the new dodecahedron with respect to others). It is assumed that every added dodecahedron is set to share the greatest number of faces possible, for its nodes are nearest to the rest of the vertices already present. So, we can approximate that the new one added always shares three common faces. Figure 7: Dodecahedron: sided (**left**) and polyhedral (**right**) shape. are evaluated on the Saaty scale, as shown in Table 4. These values include the reciprocal numbers calculated as their inverse values (e.g., if Distance vs. No. Hops value is 6, then No. Hops/Distance will be its reciprocal with the value 1/6). An importance value of 6, in the middle of the Saaty scale, is used for stability, although other values are possible, and the method works fine. The next step is to apply the fuzzy technique to the relative importance of the criteria in Table 4 by introducing TFNs (4), yielding values presented in Table 5.  
According to (11), we can calculate the geometric mean \(\widetilde{g}\) from every criterion from Table 5, yielding values in Table 6. As an example, for the distance criterion, its TFN \(\widetilde{g}\) extended over every criterion of the three considered will be calculated as follows:  
\[\widetilde{g}=(l,\ m,u)=\left((\text{1-5-5})^{\frac{1}{3}},\ (\text{1-6-6})^{\frac{1}{3}},\ (\text{1-7-7})^{\frac{1}{3}}\right)=(\text{2.924, 3.3019, 3.6593}) \tag{30}\]  
Once the mean of every criterion has been obtained, is time to calculate the fuzzy weights from (12).