## 3 Problem Formulation
\begin{table}
\begin{tabular}{l l} \hline \hline
**Nomenclature** & **Description** \\ \hline \(SM\)\(x\)\(y\) & Coordinates, longitude and latitude for SMs \\ \(BS\)\(x\)\(y\) & Coordinates, longitude and latitude for the base station \\ \(C_{off}\)\(x\)\(y\) & Coordinates, longitude and latitude for the central office \\ \(\gamma_{ij}\) & Vector of pairs of adjacent nodes \\ \(\beta\) & Vector preliminary computation of end to end delay and FSPL \\ \(a\) & Result vector of end to end delay and FSPL calculated with the ODB algorithm \\ \(n\) & Number of smart meters \\ \(A\) & Georeferenced area \\ \(Z\) & Universal data aggregation point \\ \(X\) & Smart meters \\ \(G\) & Adjacency matrix \\ \(m\) & Capacity restriction \\ \(s\) & Length cluster \\ \(k\) & Number of clusters \\ \(w\) & Maximum number of hops allowed \\ \(h\) & Hop number counter \\ \(C_{1}\), \(C_{2}\), \(C_{3}\) & Unit costs, cellular, WiFi and optical fiber \\ \(C_{orf}\), \(C_{all}\), \(C_{fop}\) & Total costs, WiFi, cellular and optical fiber \\ \(r_{ds}\), \(r_{db}\) & WiFi and cellular coverage restriction, respectively \\ \(r_{ni}\), \(r_{ns}\) & Haversine distance (m) of the intra- and inter-cluster \\ \(dist_{i,j}\) & Haversine distance matrix \(n\times n\) \\ \(d_{fop}\) & Distance (m) of optical fiber \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Variables used. SM, Smart Meter; FSPL, Free Space Path Loss; ODB, Optimal Delay Balancing. In Equations (1)-(3), the total costs of each technology are expressed: WiFi, cellular and optical fiber:  
\[C_{wf}=C_{2}*\sum_{j=1}^{k}(s_{j}-1) \tag{1}\]  
\[C_{cell}=C_{1}*k \tag{2}\]  
\[C_{fop}=C_{3}*d_{fop} \tag{3}\]  
where \(s_{j}\) represents the length of each cluster, \(k\) is the maximum number of clusters to be deployed in the network and \(d_{fop}\) is the required distance of optical fiber to be used in the FiWi network. In this way, the optimization problem can be expressed as follows:  
\[min\ C_{wf}+C_{cell}+C_{fop} \tag{4}\]  
subject to:  
\[C_{i}\in\Re^{+},\forall\ i=1,2,3. \tag{5}\]  
\[\sum_{s,k\in\ n}(s-1)+k=n,\ \forall\ s,k\in n;\forall\ n\in A(n) \tag{6}\]  
\[\sum_{SM\in A(n)}SM=Z_{i,j},\ \forall\ Z\in A(n) \tag{7}\]  
\[\sum_{SM\in A(n)}SM=X_{i,j},\ \forall\ X\in A(n) \tag{8}\]  
\[\sum_{s\in S}S\leq\ m,\ \forall\ S\in A(n);\forall\ m>1 \tag{9}\]  
\[C_{wf}=C_{2}*\sum_{j=1}^{k}(s_{j}-1) \tag{1}\]0  
\[C_{wf}=C_{2}*\sum_{j=1}^{k}(s_{j}-1) \tag{1}\]1  
\[C_{wf}=C_{2}*\sum_{j=1}^{k}(s_{j}-1) \tag{1}\]2  
Equation (4) corresponds to the objective function, which consists of minimizing the costs of implementation on a FiWi network. Equation (5) necessarily asserts that there are three types of costs. Equation (6) presents a restriction of verification, in which it must be satisfied that the sum of WiFi links and the sum of cellular links does not exceed the total number of SMs deployed at A. This ensures that there are no loops within the wireless network. Equations (7) and (8) enable any SM belonging to A to be able to be a UDAP. The restriction of capacity, of Equation (9), limits the number of intra-cluster SMs that will be able to bring together each cluster. In Equation (10), the maximum radio coverage allowed is restricted to give way to the existence of an intra-cluster link. It is very important to mention that the referential distances that are restricted are those from point to point, which are given between an SM and its respective UDAP; in such a way that all the nodes that comply with the restriction would be able to form part of a cluster by a single jump or multiple jumps. Finally, the model verifies the capacity of the conglomerate and the maximum radio coverage. Therefore, if a node needs more than one jump to transmit the information to the UDAP and if the referential distance allows it, the resulting length would be the sum from the initial node, passing through each transition node, until reaching the UDAP. In Equation (11), the maximum radio coverage allowed is restricted to make way for the existence of inter-cluster links.