## 2 Modeling and Formulation
### Air-R  
Air-R has been adopted in the military field over the last decade and achieved great success [24]. Air-R currently has a wide range of nonmilitary applications, from remote sensing [25] to autonomous aerial water sampling [28] to, or in marine environments (such as in a mobile data collector) [29]. Researchers have shown that they can significantly replace sensors using a mobile robot [25]. For instance, using Air-R to prolong the sensor network lifetime based on recharging or replacing is a promising solution, especially in locations where they are normally inaccessible to humans. Figure 3 shows the Air-R model and the unloading of a fully functional R-WSN in the monitoring field. Figure 2: Replacement prose flowchart. The major mobile robot problems are long travel distance and the exhaustive searching process to discover the failure (i.e., increase mobile robot effort)--especially in unsafe places for humans to reach the failed nodes, such as volcanoes, etc. The benefits and characteristics of the \(BS\) are utilized to meet various requirements that any application demands. Our study utilizes the \(BS\) to expect any network failure, solve problems before expanding, and find the failed nodes that need to be replaced in the network. Air-R successfully reduces work pressure and the travel distance; the efforts of the Air-R are also decreased. Sensor nodes are usually distributed over a large area, and Air-R can cover a large area because of its fast-moving speed and freeways. Air-R can also replace sensor nodes at locations that are ordinarily unreachable. Therefore, the replacement process is easy because no complicated mechanical mechanism is required to operate the sensor node. Figure 3 shows a moving example of Air-R to discover failure and replace failed nodes. In this example shown in Figure 4c, the Air-R starts from the \(BS\) to sector \(SN_{1}^{1}\), then to \(SN_{3}^{2}\) and \(SN_{2}^{2}\), etc. We have experienced several monitoring fields to show good results. If many sensor nodes are distributed widely, deciding which nodes should be replaced when having a node failure in one or more sector is challenging. The continuous synchronized information exchange between \(BS\) and Air-R allows the detection and replacement of the failed nodes in the R-WSN as a long-term monitoring system (by regularly checking the recharging request from the sensor node on the field to the \(BS\)). In this study, we answer the following question: Given the \(BS\), Air-R, and a set of sensor networks \(N\), what is the most optimal strategy for the Air-R to replace the failed nodes in sectors to make the network more sustainable, and therefore prolong the lifetime of the R-WSN as much as possible? The Air-R is allowed to fly to the next failed node at all times after receiving the replacement mission from the \(BS\) in \(RT\). At this point, we can distinguish two cases: either the Air-R is at the \(BS\), or it is anywhere else in the field. If the first case occurs, then \(RT\) can be considered; if the second case occurs, then the possible situations should be checked (as in the schedule replacement algorithm (SRA) in Algorithm 2). The situations are as follows:  
Figure 4: Air-R scenario in detecting and replacing failed sensors and unloading new ones. (**a**) normal R-WSN; (**b**) some failed nodes detected; (**c**) Air-R replacing failed nodes. Figure 3: Air-R unloading fully functional R-WSN. **Situation 1. ** The replacement time (\(T_{\textit{replace}}\)) to replace all failed nodes in \(NR\)s may be larger than the round-trip time \(T\); or, the energy needed to cover the complete \(RT\) (\(E_{RT}^{\max}\)) distance may also be more than that available in the Air-R (\(E_{Air-R}\), expressed in terms of time needed as a condition), as shown in (11):  
\[(T_{\textit{replace}}(NRs)\geq T)\vee(E_{Air-R}<E_{RT}^{\max}) \tag{11}\]  
**Situation 2.