# Memetic Chains for Improving the Local Wireless Sensor Networks Localization in Urban Scenarios +
## 2 Node Location Problem in Wireless Sensor Networks  
The NLP aims to find the most appropriate location for the sensors in space in order to reduce the system uncertainties [55], maximize the coverage [56], deal with potential sensor failures [29] or reduce the energy consumption of the network [57]. Mathematically, the NLP entails the definition of the three-dimensional Cartesian coordinates of each of the architecture sensors \((\langle s_{i}\rangle=\langle x_{i},\;y_{i},\;z_{i}\rangle)\). for achieving a particular set of n sensors \(\{S_{i}=(\langle s_{1}\rangle,\ldots,\;\langle s_{n}\rangle)\}\) within the possible combinations of sensors (S) which maximizes the system properties that are enhanced during the optimization process. The definition of the Cartesian coordinates during the optimization must follow a discrete characterization of the LPS coverage space (i.e., Target Location Environment (TLE)) and the space in which the sensor nodes can be located (i.e., Node Location Environment (NLE)) due to the impossibility for dealing with a direct optimization of the NLP without a discretization [35; 58]. This makes the NLP to be a combinatorial problem which has been characterized as NP-Hard for guaranteeing the target coverage in wireless sensor networks [32]. In addition, the localization NLP has been also assigned as NP-Hard when a reduction of the system uncertainties is addressed [31]. Therefore, the consideration of the NLP for localization is  
Figure 1: Memetic Chains optimization proposed for the Node Location Problem (NLP). doubly NP-Hard since the sensor network must provide the least uncertainty in the target location determination but also attaining the stable coverage in the entire TLE. Consequently, the mathematical model proposed in this paper for the localization NLP is as follows [26]:  
\[\textit{Maximize}\ Z=ff(CRB(S_{i})) \tag{1}\]  
\[\textit{Subject to}:\]  
\[x_{lim_{1}}\ \leq\ x_{i}\ \leq x_{lim_{2}};\ \forall x_{i}\ \in s_{i};\ s_{i}\ \in S_{i};\ s_{i}\
otin U \tag{2}\]  
\[y_{lim_{1}}\ \leq\ y_{i}\ \leq y_{lim_{2}};\ \forall y_{i}\ \in s_{i};\ s_{i}\ \in S_{i};\ s_{i}\
otin U \tag{3}\]  
\[z_{lim_{1}}\ \leq\ z_{i}\ \leq z_{lim_{2}};\ \forall z_{i}\ \in s_{i}\ ;s_{i}\ \in S_{i};\ s_{i}\
otin U \tag{4}\]  
\[\textit{Cov}_{TLE_{k}}\ \geq n_{min_{TDOA}};\ \forall\ k\in n_{TLE} \tag{5}\]  
\[\textit{Cov}_{TLE_{k}}\ =\ \sum_{i=1}^{n}\textit{Cov}_{TLE_{ki}} \tag{6}\]  
\[\textit{Cov}_{TLE_{ki}}\ =\ \left\{\begin{array}{ll}1\ \textit{if}\ \textit{SNR}_{TLE_{ki}}\ \geq \textit{SNR}_{threshold}\\ 0\ \methodology. It comprises a zone of coverage of the LPS, defined as TLE, the zone where the architecture sensors can be located (NLE) and an Obstacle Area (OA) where the sensors and the target cannot be located (i.e., the interior of the buildings) [26]. It supposes a challenging benchmark for the optimization since the LPS is covering two different streets which generates potential difficult links of the positioning signal among the sensors which are particularly covering each of these two regions. The TLE and the NLE are discretized for preserving the representativity of the space of solutions achieving time-effective results [35]. The TLE is discretized under a spatial resolution of 2 m in the Cartesian coordinates maintaining the representativeness of the scenario since the selection of this value is based on a pre-optimization study from which slightly differences in the statistical parameters involved in the process are achieved considering lower resolutions. The NLE allows an elevation of the sensors of 2 to 10 m from the base surface (i.e., the ground and the buildings) for avoiding or reducing the multipath effects and other disruptive phenomena on signals produced near the reference surface. The resolution of the NLE varies depending on the dynamic codification of the individuals through a sedation technique that we first introduced in [35].