## 4 System Model Definition and Formulation  
One base station (BS) and a large number of sensor nodes comprise the system infrastructure. Two categories are used to categorize all sensor nodes. Common nodes and cluster head nodes are the two kinds of nodes. Common nodes' duties include keeping track of environmental data and sending sensing data to the cluster head node. The cluster head node is systematically chosen from the common nodes. The cluster head receive the data from common nodes, aggregates the data and transmits the data to the BS.  
The energy model is the first-order radio model. We just take into account energy use during the communication phase. The energy dissipation caused by data transmission, reception and aggregation makes up the total energy usage. In this paradigm, a common node and a cluster head node exchange \(L\)-bit data, and Equation (5) can be used to determine the data transmission energy consumption, whereas Equation (6) is used to determine energy consumption during receiving the data.  
\[E_{TX}(L,d)=(E_{elec}+\varepsilon_{amp})L \tag{5}\]  
\[E_{RX}(L)=E_{elec}L \tag{6}\]  
\(E_{TX}(L,d)\) is the energy consumption during the transmission of \(L\)-bit of data, and \(E_{RX}(L)\) is the energy consumption during receiving the data. Equation (7) can be used to  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(d\) & Distance to base station \\ \(d_{0}\) & Fixed measuring distance to base station \\ \((\mathbb{C}_{x},\mathbb{C}_{y})\) & Coordinate of cluster head in a WSNs \\ \((N_{x},N_{y})\) & Coordinate of node in a WSNs \\ \(E_{initial}\) & Initial energy \\ \(E_{elec}\) & Electronics energy \\ \(E_{TX}\) & Data transmission energy consumption \\ \(\varepsilon_{fs}\) & Energy amplification to overcome open area \\ \(\varepsilon_{mp}\) & Energy amplification in order to navigate the multi-path \\ \(\varepsilon_{amp}\) & The amplifier energy consumption during transmission phase \\ \(E_{RX}\) & Energy consumption while data reception \\ \(K_{opt}\) & Number of cluster heads that is optimal \\ \(L\) & Length of data \\ \(N\) & Number of nodes in the network as a whole \\ \(H\) & Distance between the special node and the common node \\ \(N_{c}\) & Number of clusters \\ \(X-Axis\) & Direction of length of the base station \\ \(Y-Axis\) & Direction of breadth of the base station \\ \(m\) & Number of alternatives \\ \(n\) & Number of criteria \\ \hline \hline \end{tabular}
\end{table}
Table 3: Table of symbols.  
determine the amplifier's energy usage during the transmission phase, where \(\varepsilon_{amp}\) is the amplifier energy consumption during the transmission phase.  
\[\varepsilon_{amp}=\left\{\begin{array}{ll}\varepsilon_{fs}d^{2}\text{ if }d\leq d_{0}\\ \varepsilon_{mp}d^{4}\text{ if }d>d_{0}\end{array}\right. \tag{7}\]  
If the value of \(d\) less than or equal to \(d_{0}\), then the sensor node will use the free-space propagation model. On the other hand, if the system uses a multipath fading channel which uses an \(\varepsilon_{fs}\) and \(\varepsilon_{mp}\) communication energy parameter, it can be used to calculate the value of \(d_{0}\) by Equation (8).  
\[d_{0}=\sqrt{\frac{\varepsilon_{fs}}{\varepsilon_{mp}}} \tag{8}\]  
Determining the number of cluster heads in each cycle is crucial for increasing the WSN's lifetime and energy efficiency. We determine the optimal cluster size \(k_{opt}\)[33] using Equation (9).  
\[k_{opt}=\sqrt{\frac{\varepsilon_{fs}}{\pi(\varepsilon_{mp}d_{toBS}^{4}-E_{ dec})}}M\sqrt{N} \tag{9}\]  
where \(M\), \(N\) are represented as an area covered and the number of nodes in the system. BS defines the base location.