# Swarm Intelligence Techniques for Mobile Wireless Charging
## 4 Recharge Optimization Problem with Multiple Mobile Chargers (ROPMMCs)  
This section presents our multiple charger recharge optimization problem (MCROP) and considers some practical challenges for real sensing applications. The first challenge is the palpable increase in MC's _\"movement energy\"_ consumption, due to the improper coordination procedures for energy provisioning to the sensors. The _\"movement energy\"_ of the MC is the amount of energy spent in traversing the network to recharge the sensors. In this instance, the trajectories of the MCs should be of paramount importance to minimize the movement energy consumption and travelling costs. Thus, the recharge route should be carefully structured to reflect the MC's current energy position and traveling costs. The second constraint has to do with the non-uniformity in energy consumption occasioned by data transmissions. We note that some nodes assume higher energy consumption rates than others and should be given higher priority recharge-request considerations (i.e., they should be allocated a higher recharge frequency) to maintain the network's functionality. This is particularly the case with nodes close to the CHs with higher data transmission activities compared to other nodes. We adopted an adaptive recharge threshold, which is proportional to the energy consumption rates at various cluster-regions. Hence, nodes close to the CHs are given a higher charging frequency than others. The recharge routes also take into consideration all the concerns stated earlier. Figure 4: Network partition based on clustering. Next, we formulate the wireless recharging problem as a multiple charger recharge optimization problem (MCROP) and show that it is \(\mathcal{NP}\)-complete from the geometric traveling salesman problem (G-TSP). Our aim is to minimize the MCs' total cost of traveling when they traverse the network to charge the sensors. The TSP with profits [44] presents a reward by visiting a city with the aim of maximizing profits. The reward, in our problem, denotes the amount of replenishable energy by a sensor-node, while the cost is a measurement of the energy cost used to traverse the entire network. The MCROP can be formulated as follows:  
MCROP: Given a set of MCs, M = {1, 2,... m}, and a set of rechargeable sensor-nodes, S\({}_{r}\) = {1, 2,... s\({}_{r}\)}, where each sensor-node is capable of storing E units of energy for each s\({}_{r}\)\(\in\) S\({}_{r}\) in a list L\({}_{s}\) of pairs \(\left(t^{r}_{s_{r}},\,e^{s}_{s_{r}}\right),\ r\geq 1\), where \(t^{r}_{s_{r}}\) represents the time taken to generate the \(r\)th message of \(s_{r}\), and \(e^{s}_{s_{r}}\) is the energy expended by the sensor-node in its message transmission. We consider a graph, G = (U, V), where \(U_{j}(j\in S_{r})\) is the location of the sensor-nodes \(j\) to be visited, and V is the set of edges. Each edge \(V_{j,k}\) has a traveling energy cost \(e_{j,k}\) which varies directly with the distance between the nodes \(j\) and \(k\), and a latency cost \(c_{jk}=t_{j}+t_{jk}\), where \(t_{j}\) is the recharge time of node \(j\) from its current energy status to its full capacity, and \(t_{jk}\) is the traveling time from node \(j\) to node \(k\). The recharge capacity \(R_{c}\) of an MC is used to determine the maximum number of sensor-nodes to be recharged before the MC's energy is depleted. We note that different MCs could have varying recharge capacities during the run. We also consider an \(|S_{r}|\times|S_{r}|\) matrix \(R\), where \(R_{j,k}\) is the distance between sensors \(j\) and \(k\) and an MC, M, which can recharge a sensor-node to its initial energy in a timely manner. If we assume that the MC moves at a constant speed v, then the time of travel between \(j\) and \(k\) can be calculated as \(\frac{R_{jk}}{\
u}\). The MCROP is to determine the feasibility of scheduling an MC for node-energy replenishment to minimize/avoid packet loss arising from insufficient energy. We adapt the method in [22] to prove that MCROP is \(\mathcal{NP}\)-complete.