## 4 Proposed Work
### Unequal Cluster-Tree Topology Construction
\begin{table}
\begin{tabular}{c c c} \hline
**Notations** & **Description** & **Notations** & **Description** \\ \hline V & Denotes the sensors deployed & df & Energy required for CH to perform aggregation \\ \hline E & Connectivity of the nodes & \((CH-intra(en))\) & Energy required for CH in intra-cluster routing \\ \hline p. q & Two sensor nodes in the network & \((CH-R_{x}-intra(en))\) & Energy required for CH in reception of signals inside the cluster \\ \hline TR & Range of transmission & \((CH-df-intra(en))\) & Energy required for CH in aggregation of received signals from CM \\ \hline c & Cluster comprising of p number of nodes & \((CH-inter(en))\) & Energy required for CH in inter-cluster routing \\ \hline cp & Cluster center & \((CH-T_{x}-intra(en))\) & Energy required for CH in transmission of signals between the clusters \\ \hline \(X_{Nj}\) & Total number of nodes in the network for CH election & \((CH-CS-inter(en))\) & Energy required for CH in sensing of signals \\ \hline RE & Remaining energy & idle \((en)\) & Energy required in idle state \\ \hline \(Tx(en)\) & Energy required for transmission & a, b & Distance between two nodes in the cluster \\ \hline Rx(en) & Energy required for reception & C & Capacity of the node referred to in terms of buffer size \\ \hline k & Length of message bits & CR & Current ratio \\ \hline \(\epsilon_{apr}\)\(\epsilon_{apr}\) & Power consumption of amplifier in two-hop & PM & Profit margins \\ \hline \(ent_{dir}\) & Signal processing energy & P(A) & Probability of node being in active mode \\ \hline CS (en) & Sensing cost & PS(S) & Probability of node being in sleep mode \\ \hline CM & Cluster members & R & Sum of evidence of the values of residual energy \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations table. Once the clusters are formed, the cluster head selection takes place by implementing multi-objective black widow optimization (MoBWo). The main purpose of choosing this algorithm is an early convergence, and it obtains the optimized fitness values as compared to other algorithms such as PSO, GA and other traditional algorithms. In order to choose the proper CH for data collection and aggregation, various sets of parameters are biologically optimized in MoBWo that are follows,  
* Spider Size = 40
* Total Number of Generations = 100
* Reproduction Rate = 0.6
* Cannibalism Rate = 0.44
* Mutation Rate = 0.4  
The MoBWo is processed by random initialization of the black widow population. Two kinds of spiders' behavior are inherited in the CH selection, i.e., male black widow and female black widow. The initialization of black widow spiders is given in below,  
\[X_{N,i}=\left[ \begin{array}{ccc}x_{1,1}&x_{1,2}&x_{1,i}\\ x_{2,1}&x_{2,2}&x_{2,i}\\ x_{N,1}&x_{N,1}&x_{N,i}\end{array} \right] \tag{1}\]  
where \(X_{N,i}\) are the initial values of the population. In this paper, \(X_{N,i}\) is represented as the total number of nodes in the network for CH selection. The CH is selected based on multiple objectives like energy objective, distance objective and capacity objective. For optimum CH selection, these objectives must be defined as follows:  
* Energy objective: residual energy must be higher to elect that node for CH, since minimum energy causes packet retransmission or packet losses issues. * Distance objective: distance must be lower than the BS to choose it as the CH. Longer distance causes the huge end-to-end delay. * Capacity objective: a node must consist of lower capacity because a lower capacity node can carry the huge amount of packets from the CM. Based on the above objectives, the best set of nodes is selected as CH in the number of triangles. Each objective metric can be calculated by the following. Figure 2: Unequal Sierpinski cluster-tree topology. The remaining energy of the sensor node is expressed as,  
\[RE=\sum\{T_{x}(en)\ +\ R_{x}(en)\ +\ idle(en)\ +\ sleep(en)\}-IE \tag{2}\]  
\[T_{x}(en)\ =\ T_{x}(en_{dec})\ +\ T_{x}(en_{amp})=\ \left\{