## 4 Formulation for Optimal Solution

Table 1: _Cont._the sensor nodes' performance in that zone would not be influenced by environmental factor \(f\), which means the environmental effect metric \(EI_{i}^{f}(t)\) is set to 1. Suppose zone \(i\)'s environmental data are beyond the typical range. In that case, environmental factor \(f\) will detrimentally affect the sensor nodes in that zone, causing them to deteriorate and increasing the risk of malfunctions. Therefore, \(EI_{i}^{f}(t)\) is decreased. Figure 1 depicts the trend of the given function in Equation (1), which reflects the environment single factor impact metric. Actually, many environmental factors influence the performance of WSNs. So, the environmental impact metric should be determined for multiple environmental factors as follows [15]:  
\[MEI_{i}(t)=\min\Bigl{\{}EI_{i}^{f_{1}}(t),EI_{i}^{f_{2}}(t)\Bigr{\}} \tag{2}\]  
where \(EI_{i}^{f_{1}}(t)\) and \(EI_{i}^{f_{2}}\) denotes to the single environmental effect metrics of zone \(i\) at time \(t\) in relation to the environmental factors \(f_{1}\) and \(f_{2}\). Indeed, suppose the sensor nodes' performance in a certain zone would be severely affected by one of the many environmental factors. In that case, the sink node shouldn't cross that zone. Additionally, its sensor nodes should be prevented from being selected as a next-hop through the routing process of urgent messages. Therefore, among the environmental metrics values of different factors, the multiple environmental impact metric is represented by the minimum value [18]. On the other hand, if such a wildfire occurs in a particular zone, it is expected to spread rapidly to neighboring zones. So, neighboring zones are still considered hazard zones even though this incident did not spread to them. For urgent data, to keep the mobile sink path and routing paths from environmental risks as early as possible, it is important to link each zone's environmental effect metric with its neighbors. A zone's environmental impact on zone \(j\) is defined as follows [15]:  
\[NEI_{ij}(t)=K_{ij}MEI_{j}(t) \tag{3}\]  
The attenuation coefficient, \(K_{ij}\) was calculated as follows:  
\[K_{ij}=1+\frac{ED_{ij}-ED_{ij\min}}{ED_{ij\max}-ED_{ij\min}} \tag{4}\]  
where \(ED_{ij\min}\) and \(ED_{ij\max}\) represent the minimum and maximum distance between the center of zone \(i\) and \(j\). For square/circular zone with side length/diameter, a, \(ED_{ij\min}\) and \(ED_{ij\max}\) could be calculated as follows:  
\[ED_{ij\min}=a \tag{5}\]  
\[ED_{ij\max}=2\sqrt{2}a \tag{6}\]  
Figure 1: Single factor environmental impact metric curve function. For rectangular shaped zone with side length, \(L\), and side width, \(w\), \(ED_{ij\text{min}}\) and \(ED_{ij\text{max}}\) could be calculated as follows:  
\[ED_{ij\text{min}}=L \tag{7}\]  
\[ED_{ij\text{max}}=2\sqrt{L^{2}+w^{2}} \tag{8}\]  
Equations (3) and (4) show that the attenuation coefficient is proportional to zone \(i\) and \(j\) distance. This is because each zone is more sensitive to environmental threats from its neighbors when they are closer. As numerous bordering zones surround each zone, each has its own neighboring environmental effect metric. As the final neighbouring environmental effect is the least value across all adjacent metrics is chosen as follows:  
\[NEI_{i}(t)=\min\{NEI_{ij}(t)\big{|}_{j\in NZ_{i}}\} \tag{9}\]  
Finally, the final environmental effect of zone \(i\) is computed by combining the multiple environmental impacts \(MEI_{i}(t)\) and the environmental impact of the neighbouring zone \(NEI_{i}(t)\), as follows:  
\[FEI_{i}(t)= \begin{cases}MEI_{i}(t)&if\ NEI_{i}(t)=1\\ \frac{MEI_{i}(t)+NEI_{i}(t)}{2+\exp(-K_{ij})}&otherwise\end{cases}  \tag{10}\]  
According to Equation (10), the proposed final environmental effect metric is based on the insignificant nearby environmental impact during normal operation periods. If a neighbour's environmental effect is one, it is appropriate to ignore it. As a zone's neighbour's attenuation coefficient drops, so should its final environmental metric.