## 3 System Modeling
### Energy Model  
We considered three different types of sensor node in terms of energy levels--i.e., normal, intermediate and advanced. If \(E_{n}\) is the initial energy of normal sensor nodes, then the initial energy of the intermediate and advanced sensor nodes should be higher than the normal sensor nodes by a factor of \(\mu=\frac{(E_{int}-E_{n})}{E_{n}}\) and \(\tau=\frac{(E_{ab}-E_{n})}{E_{n}}\), respectively. Using this information, we can determine the total energy of the sensor nodes in Equation (1) as:  
\[E_{total}=\ N\ (E_{n}+R_{int}(E_{int}-E_{n})\ +R_{adv}(E_{adv}-E_{n}))=\ NE_{n}(1+R_{int} \mu+\ R_{adv}\tau) \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Notation** & **Meaning** & **Notation** & **Meaning** \\ \hline \(E_{n}\) & Initial energy of normal sensor nodes & \(R_{L}\) & Long-radio range \\ \(E_{att}\) & Initial energy of intermediate sensor nodes & \(W_{n}/S_{n}\) & Working/Sleeping schedule \\ \(E_{adv}\) & Initial energy of advanced sensor nodes & \(F_{ST}\) & Status transition frequency \\ \(\mu\) & Energy weight factor of intermediate sensor nodes & \(TF_{V,V_{j}}\) & Time-Frequency parameter \\ \(\tau\) & Energy weight factor of advanced sensor nodes & \(R_{E}\) & Residual Energy of a sensor \\ \(R_{att}\) & Ratio of intermediate sensor nodes & \(LQR\) & Link quality factor(S/N) \\ \(R_{adv}\) & Ratio of advanced sensor nodes & \(D_{nRS}\) & Distance to Base Station \\ \(E_{adv}\) & Energy spent for running radio electronics & \(N_{appret}\) & Number of supported sensors \\ \(l\) & Number of bits & \(E_{A}(a)\) & Hesitant Fuzzy Entropy \\ \(d_{V,V_{j}}\) & Euclidean distance b/w transmitter and receiver & \(C_{A}\big{(}\alpha,\alpha^{C}\big{)}\) & Cross Entropy \\ \(\varepsilon_{fs}\) & Energy dissipation in free space fading & \(\eta_{V}\) & Entropy coefficient weight \\ \(\varepsilon_{app}\) & Energy dissipation in multi-path fading & \(D_{Ti}\) & Final data attribute value \\ \(N\) & Number of sensor nodes & \(C_{E}\) & Communication round \\ \(N_{C}\) & Number of clusters & \(\alpha(x)_{x_{i}}\) & ith value in fuzzy set \\ \(E_{aware}\) & Energy consumed during sensing & \(l_{a(x)}\) & Length of fuzzy set \\ \(E_{A}\) & Energy consumed during aggregation & \(\alpha,\beta,\gamma,\sigma\delta\) & Weights for \(TF_{V,V_{j}}\), \(R_{e}\), \(LQR\), \(D_{nRS}\) and \(N_{appret}\) \\ \(r\) & Compression ratio & \(L_{V_{C00}V_{C00}}\) & Link connectivity b/w CM(i) and CM(j) \\ \(R_{CC}\) & Communication-Computation ratio & \(L_{V_{C00}V_{C00}}\) & Link connectivity b/w CM and CM(j) \\ \(E_{CM}\) & Energy consumed by cluster head & \(T_{CP}\) & Data Collection period \\ \(E_{CM}\) & Energy consumed by cluster member & \(\zeta\) & CH Election procedure set \\ \(E_{cluster}\) & Energy consumption of a cluster & \(T\) & Time period for sending aggregated data to \\ \(R_{S}\) & Short-radio range & t & Time period for sending integrated data to \\  & & CH by CM \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation description. Figure 1: Network Model: heterogeneous wireless sensor networks (WSN) with three types of sensor nodes. where \(N\) is the total number of sensor nodes in the network, \(R_{int}\) is the ratio of intermediate sensor nodes and \(R_{adv}\) is the ratio of advanced sensor nodes in the network. Furthermore, we considered the simplified energy consumption model based on [26] for radio energy dissipation during transmission and reception. According to this model, the energy required to transmit \(l\) bits of data over distance \(d\) can be given in Equation (2) as:  
\[E_{T}\Big{(}V_{i},V_{j}\Big{)}=\ \left\{ \begin{array}{cc}E_{dec}l+\ \varepsilon_{ fg}ld_{V_{i}V_{j}}^{2}&d<d_{0}\\ E_{dec}l+\ \varepsilon_{mpl}ld_{V_{i}V_{j}}^{4}&d\geq d_{0}\end{array} \right.