## 7 Conclusions  
In this paper we propose an optimization model to solve one of the major concerns of WSN, the sensor placement problem, and to answer one the most popular research questions: What is the minimum number of sensor nodes to achieve full coverage? The proposed approach is based on the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Method’s Properties** & **Map1 UL-A** & **Map2 UL-B** \\ \hline \multirow{3}{*}{The proposed algorithm} & Number of sensors & 9 & 78 \\  & Average & 100\% & 89.48\% \\  & CPU Time & 6 m 42 s & 5.2 h \\ \hline \multirow{3}{*}{GD Algorithm} & Number of sensors & 12 & 60 \\  & Average & 91.38\% & 83.17\% \\  & CPU Time & 72 s & 22.2 m \\ \hline \multirow{3}{*}{ CMA-ES Algorithm} & Number of sensors & 12 & 60 \\  & Average & 90.44\% & 87.63\% \\ \cline{1-1}  & CPU Time & 15.9 m & 11.1 h \\ \hline \hline \end{tabular}
\end{table}
Table 2: Coverage percentage, number of sensors, and processing time of the different methods. (Time is reported in seconds (s), minutes (m), and hours (h)).  
Figure 23: Illustration of the UL-B network connection graph.  
Figure 22: Optimal sensor deployment on area “UL-B”.  
combination of the gradient method and the SA algorithm, and can solve both the area coverage problem and the barrier coverage problem, for applications with coverage and connectivity degree equal to one.  
The simulation results of the two scenarios (area coverage and barrier coverage problem tested in Sections 6.1 and 6.2 on small maps with 100 \(\times\) 100 pixels) show that the model selects the optimal deployment design from all the generated models. The selected deployment design meets the application requirements in the two cases, in terms of coverage and connectivity, while defining the optimum number of sensor nodes. Another scenario was implemented in Section 6.3 in order to compare the proposed method to other optimization methods implemented in [8]. The CMA-ES and the GD algorithms were proposed by authors in [8] for the sensor placement optimization problem on the Laval Campus University. The same map specifications were used. The first sub-part, UL-A, is a small map with 100 \(\times\) 100 pixels and the second sub-part, UL-B, is larger with 500 \(\times\) 500 pixels. In comparison with results in [8], the proposed model shows better performance on the smaller map in terms of obtaining the maximum coverage with the optimum number of sensor nodes, while the processing time is acceptable, 5 times superior to the GD method and two times inferior to the CMA-ES method. However, on the larger map, it performs slightly better than average by reaching a slightly better coverage but with more sensor nodes than those needed by the two other methods, while the processing time is 13 times superior to the GD algorithm and 2 times inferior to the CMA-ES method.  
Future work could involve using another optimization method for nodes on the boundary of the region of interest in order to minimize the processing time. We will continue our research on WSN deployment strategies in order to provide solutions for k-coverage and k-connectivity problems while minimizing the network cost and maximizing the network lifetime.  
The last two authors were partially funded by the Spanish Ministry of Economy and Competitiveness (National Program for Research, Development and Innovation), project DArDOS TIN2015-65845-C3-3-R, with funds of the European Union (European Regional Development Fund - ERDF).  
Y.K., [PERSON] and [PERSON] conceived the research and conducted the simulations; [PERSON] designed and implemented the algorithm; [PERSON] and [PERSON] analyzed the data, results and verified the theory; All authors participated in the writing of the manuscript.  
The authors declare no conflict of interest.