# Entropy Weighted TOPSIS Based Cluster Head Selection in Wireless Sensor Networks under Uncertainty
## 4 Cluster Heads Formation Method for WSN  
The nodes designated to serve as CHs should inform the rest of the nodes in the network that they have been selected for such a function. To achieve this, each of the chosen CHs transmits an appropriate signal throughout the network announcing their selection as cluster heads. This brief message provides the network node identification and a header, characterizing it as an update message. Each non-cluster head node selects the CH that is closest to it and uses the least amount of transmission energy to form its cluster. The link is connected to the CH with the shortest distance and received signal amplitude. The node and base station (BS) will respond directly if the distance between it and the CH is longer than its distance from the BS. Otherwise, it connects the cluster using the shortest distance. Here, we have considered the shortest distance measure as Euclidean distance. If \(db\) is the distance between CH and a designated node, then \(db\) can be measured using the formula as follows:  
\[db=\sqrt{(X-W_{x})^{2}+(Y-W_{y})^{2}} \tag{5}\]  
where \((X,Y)\) and \((W_{x},W_{y})\) are the position of cluster head (CH) and node location. The BS, which is situated outside of the network area, is a node with improved processing abilities and no limited battery life. In order to obtain an appropriate signal-to-noise ratio (SNR), a \(k\) bit of information is transmitted over a distance \(db\) using a conventional radio energy dissipation model [23]. During data communication, the data transmission energy consumption (\(ET_{e}\)) and energy consumption due to data reception (\(ED_{rx}\)) are approximated using Equations (6) and (7). \[ET_{e}=\left\{ \begin{array}{ll}k\times EL_{e}\,+\,k\times\,e_{fs}\times db^{ 2}&if\,db\,\leq\,db_{0}\\ k\times\,EL_{e}\,+\,k\times e_{mp}\,\times\,db^{4}&if\,db\,\geq\,db_{0}\\ \end{array} \right. \tag{6}\]  
\[ED_{rx}=k\times EL_{e} \tag{7}\]  
It is to be noted that the electronic energy \(EL_{e}\) depends on various features, viz. coding of digital devices, modulation, filtering, and bandwidth of the signal, whereas \(e_{fs}db^{2}\) and \(e_{mp}db^{4}\) are dependent on the distance to the receiver and acceptable bit-error rate. Following the creation of the clusters, the CH, after receiving all CH connect signals from every node, assigns a timeframe for each designated node. The responsibility of gathering information from all cluster nodes belongs to each cluster head. The CH transmits the message to the BS after applying data aggregation when a packet of data from all the individuals is received. It has been observed that several protocols have been used for re-clustering strategy and choosing CHs using a probabilistic approach rather than a deterministic approach. Data transmission and re-clustering proceed for several cycles until all nodes are still alive. As dead nodes begin to appear, the number of active nodes in the cluster diminishes, and the smaller clusters that have lesser power than the predetermined threshold are combined with the larger ones. As a result, the cluster size starts to slow down anytime the number of active nodes diminishes. Determining the number of CHs in \begin{table}
\begin{tabular}{l l} \hline
**Symbol** & **Description** \\ \hline \(e_{fs}\) & Amplification of energy to overcome open space \\ \(e_{mp}\) & Amplification of energy to navigate the multi-path \\ \(ED_{rx}\) & The usage of energy during data receipt \\ \(O_{c}\) & The optimal number of cluster heads \\ \(Z\) & The dimensions of the square area \\ \(N_{node}\) & The total number of nodes in the network \\ \(N_{c}\) & The number of nodes in a cluster \\ \(R\) & The reliability of a cluster \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._each cycle is crucial for increasing the WSN's lifetime and energy efficiency.