## 4 Proposed Scheme
### Relay-Node Selection and Rotation  
To improve the network lifetime while maintaining balanced energy consumption among network nodes, relay nodes in addition to cluster heads are utilised in the proposed method. The scheme is adaptable to varying network scales, determining the most energy  
Figure 7: Multi-hop inter-cluster routing to transmit data to BS. efficient location for relay nodes. Given that the \(ith\) cluster head \(CH_{i}\) needs to transmit data to the destination (i.e., base station \(BS\)), direct transmission becomes energy efficient if the condition \(d\leq d_{0}^{min}\) is satisfied. The value of \(d_{0}^{min}\) can be calculated as:  
\[d_{0}^{min}=\sqrt[2]{\frac{\varepsilon_{o}(\gamma)}{1-2^{-\frac{\gamma}{2}}}}\]  
Figure 8 illustrates the search region for relay nodes in the network. The choice of the relay node is determined by considering the distance between the cluster head \(CH_{i}\) and the destination node \(D\), aiming to reduce energy consumption while maintaining efficient network performance. In the diagram, \(d_{CH-O}\) represents the distance between the cluster head \(CH_{i}\) and the central point \(O\), which is a potential relay position. Distances \(d_{CH-A}\) and \(d_{A-D}\) denote the distances between the cluster head and a potential relay node \(A\), and between the relay node and the destination \(D\), respectively. The relay-node search area, represented by the dashed circle, is governed by the energy dynamics of the network, with the goal of balancing the load across relay nodes to optimise energy efficiency. Parameter \(\gamma\) can take values of 2 or 4, based on the first-order radio model parameters \(E_{ELEC}\), \(\varepsilon_{fs}\), and \(\varepsilon_{mp}\), which influence the calculation of the threshold distances. The parameters are defined as follows:  
* For \(\gamma=2\), the energy threshold is given by \(\varepsilon_{o}(\gamma)=\frac{E_{ELEC}}{\varepsilon_{fs}}\), reflecting the free space model. * For \(\gamma=4\), the energy threshold is \(\varepsilon_{o}(\gamma)=\frac{E_{ELEC}}{\varepsilon_{mp}}\), representing the multi-path fading model. These energy thresholds dictate the conditions under which direct or multi-hop communication is more energy efficient. When \(\gamma=2\), the network operates under the free-space propagation model, and when \(\gamma=4\), the multi-path fading model is applied. The selection of relay nodes is thus guided by these parameters, ensuring minimal energy consumption and maximising the network's lifetime. Figure 9 presents the flow chart detailing the stepwise operation of the proposed balanced energy and maximum network lifetime inter-cluster multi-hop communication method. The process begins with the farthest cluster head from the base station and calculates the number of hops required for energy-efficient transmission. This approach continues progressively towards relay nodes closer to the base station. Figure 8: Relay-node search region based on energy change. The goal is to identify the optimal relay-node positions to ensure that direct transmission from the closer relay nodes is more energy efficient than multi-hop routing from farther nodes. This strategy extends the scalability of the network and allows the proposed method to operate adaptively in networks of varying scale, which is essential for supporting IoT infrastructure. By continuously adjusting based on the energy consumption model, the method helps balance the energy usage across the network and prolong its overall lifetime. The pseudo-code in the Algorithm 3 ensures the energy efficient relay-node selection and rotation of relay role among nodes within the set of candidate relay nodes, all executed at an appropriate frequency to optimise energy usage and network stability. Figure 9: Flow chart of the proposed relay selection and rotation method. ```
0:\(n^{\prime}\) number nodes, their locations \((x_{i},y_{i},z_{i})\), heterogeneous fixed data rate \(T_{i}\), and initial energies \(\mathbb{E}_{i}\) of each sensor node, sensing region and its boundaries.