# Novel Approach Sizing and Routing of Wireless Sensor Networks for Applications in Smart Cities
## 3 Problem Formulation
### Wireless Sensor Network Routing
#### 3.2.3 Multiple Flow Routing
Table 4 identifies the innovations of this research in contrast to those of other proposals that address the problem of the sizing and routing of wireless sensors. The present work focuses on scenarios such as the smart metering of electrical energy; in fact, it focuses on relaxing the complexity of the sizing and routing problem by independently and consecutively solving each stage. In this way, the present work was compared with other proposals to highlight the contributions regarding the problem addressed. Figure 7: Routing with the capacity of the links—Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Source Node**} & \multirow{2}{*}{\begin{tabular}{c} **Destination** \\ **Node** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Requirement** \\ **\# of** \\ **Flows** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Link Cost** \\ **(Kbps)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Link** \\ **Capacity**\#** \\ **of Flows** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **MILP**–**\# of \\ **Flows** \\ \end{tabular} } \\ \hline
49 & 40 & 1 & 230 & 10 & 1 \\ \hline
40 & 39 & 1 & 240 & 10 & 2 \\ \hline
39 & 2 & 1 & 247 & 10 & 3 \\ \hline
2 & 26 & 1 & 250 & 10 & 4 \\ \hline
26 & 42 & 1 & 248 & 10 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: ZigBee simulation (250 kbps)—flows were generated by using LPSolve. ## 5 Conclusions

The model for the resolution of the problem of flow capacity achieved the optimal coordination of data routing in the network layer and allocated resources in the physical layer by fixing the costs in the capacities of the DAPs, thus providing a cross-layer solution. The novel contribution of this model was the establishment of a multi-hop WSN that used cross-layer information to determine the routing of the network layers while involving decision aspects, such as wireless link capacity and traffic flow demands. The model did not use clustering methods to relax the NP-complete problem. This paper focused on the sizing and routing of wireless sensor networks deployed for the provision of new IoT services and applications for smart cities by considering the locations of DAPs in a neighborhood area network. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Goal** & **Proposal** & **A1**[26] & **A2**[23] & **A3**[25] & **A4**[24] & **A5**[27] \\ \hline
**Sizing** & & & & & & \\ \hline DAP location & \begin{tabular}{c} Candidate \\ sites \\ \end{tabular}  & Random & Random & Random & Random & Random \\ Haversine distance & ✓ & ✓ & ✗ & ✓ & ✓ & ✗ \\ Euclidean distance & ✗ & ✗ & ✓ & ✗ & ✗ & ✗ \\ Optimization MILP & ✓ & ✗ & ✗ & ✗ & ✗ & ✗ \\ K-means clustering & ✗ & ✓ & ✓ & ✓ & ✓ & ✓ \\ Coverage & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ DAP capacity & ✓ & ✗ & ✗ & ✗ & ✗ & ✗ \\ \hline

Figure 8: Coverage radius vs. total path cost. Therefore, to achieve this objective, a problem was formulated in three stages (graph theory, multi-casting, and multiple flows) that were sequentially simulated, but had the same goal: to minimize the cost of the minimum number of DAPs and the minimum distance. This paper presented an actual urban area that was geo-referenced with the latitude and longitude of each technological resource, and the performance of the optimization models was evaluated. Simulation results verified that the proposed solution could reduce the number of DAPs and generate random answers based on clustering methods. Additionally, a minimum-cost tree was initially created by considering the weight of the cost per distance for the backup wireless network. Finally, the binding capacity restriction was added. Inter-DAP interference was not considered in this work, as it can be solved by assigning different channels to each DAP. E.I. : Conceptualization, Methodology, Validation, Writing--review and editing. J.I., A.O. : Conceptualization, Methodology, Software, Writing--original draft. J.I. and A.O. : Data curation, Formal analysis. E.I. : Supervision. E.I. : Writing--review and editing.