# Energy Efficient Routing Protocol in Novel Schemes for Performance Evaluation
## 4 Delay (D(n))
### Derivation of Optimal Relay Selection Fitness Function  
During the consistent state time frame, information is communicated. When multi-bounce pathways are utilized, the organization incorporates both intra- and between bunch correspondence. The ideal hand-off hub is picked in view of factors such as the leftover energy of hubs, start to finish delay, and closeness among CH and part hubs.  
(a) Distance between cluster member and CH:  
The distance between the cluster member and Ch is the average length between the sensor node, as well as the CHs with which it is related. The maximum length shows maximum hops, which may raise the energy consumption of the network. Equation (12) can be used to minimize the first parameter, '\(r_{1}\)' \[r_{1}\ =\sum_{j=1}^{m}(\sum_{i=1}^{ij}\ \textit{{dis}}(\textbf{{s}}_{i},\textbf{{CH}}))\] (12) Here, \(d_{i}\) (\(S_{i}\), cluster head) is distance between cluster member and respective CH \[\text{Dis }(\text{Si },\ \text{CH})=\sqrt[2]{(xpos_{CH}-xpos_{node})^{2}}+(ypos_{CH}- ypos_{node}).^{2}\] where: \(xpos_{CH}\) and \(xpos_{node}\) are \(x\) position of the cluster head and node; \(ypos_{CH}\) and \(ypos_{node}\) are \(y\) position of the cluster head and node.  
Residual energy (\(\text{E}_{\text{SI}}\)): through relay nodes, total cluster members provide data to respective CHs. Due to insufficient energy or exhaustion; the minimal energy node can expire or stop operating during data transmission. Equation (13) can be used to calculate the minimizing of the second parameter, 'r\({}_{2}\)'  
\[r_{2}\ =\sum_{j=1}^{m}\cdot\frac{1}{E_{si}} \tag{13}\]  
Residual energy \(=E_{Si}=E_{initial}-E_{consumed}\);  
\(E_{initial}\) denotes initial energy of sensor nodes;  
\(E_{consumed}\) represents consumed energy of sensor nodes.  
There is the time required to communicate between sensor nodes.  
As delay period grows, the network's energy usage grows, as well. For efficient data transmission, the relay nodes should have the shortest possible time delay.  
Network's communication delay, the node's ETC and Propagation delay influences the delay. This delay function 'r\({}_{3}\)' can all be minimized as Equation (14)  
\[r_{3}=\sum_{i=1}^{m}\mathbf{min}(D_{si}) \tag{14}\]  
The delay can be determined as Equation (15)  
\[\text{Delay}\ \text{Dsi}=\sum_{i=1}^{m}ETC_{I}\ (TD+PD_{i}) \tag{15}\]  
where  
\(ETC_{I}\rightarrow\) denotes the ith node expected transmission count;  
\(\text{TD}\rightarrow\) denotes complete transmissions of the entire network;  
\(PD_{i}\rightarrow\) represents the ith node propagation delay.  
The \(ETC_{I}\) of node is determined by the RPDR of sensor node to FPDR at time 't'., and Equation (16) is used to compute it.  
\[\text{ETC}_{I}=\frac{1}{f_{i}\left(t\right)+R_{i}\left(t\right)} \tag{16}\]  
where:  
\(R_{i}(t)\) denotes the ith node's RPDR (received packet delivery ratio);  
\(F_{i}(t)\) denotes to the ith node's FPDR (forward packet delivery ratio) at time 't'.  
Therefore, FF is computed from Equation (17)  
\[\text{fitness}=\ \beta_{1}*f_{1}+\ \beta_{2}*f_{2}+\beta_{3}*f_{3} \tag{17}\]  
Here, \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\) are weighted numbers allocated to every parameter.  
\[\sum_{i=1}^{3}\beta_{i}=1\ \ and\ \ \beta\ \in\ (0,1).\]  
From the below Algorithm 4 which states about the optimal relay node of Cluster heads and cluster members along with no of nodes and no of sensors. We also look for response of optimal and relay nodes and fitness