## 2 Related Works  
Numerous routing methods have been developed with the goal of increasing energy efficiency and lengthening network lifetime in order to address the problem of limited energy in WSNs. LEACH [18] is one of the most well-known clustering routing methods for WSNs. Each iteration of the LEACH protocol is divided into two stages: clustering and stabilization. During the clustering phase, each node generates a random integer between 0 and 1, which it subsequently compares to the value obtained from a threshold calculation. The node is not elected as CH in the following round if the random number is less than the threshold and it was not elected in the preceding \(1/p\) (\(p\) is the proportion of nodes that became CH in the current round) rounds. During the stabilization phase, the sensor nodes transmit the information they have collected to the CH. After completing data fusion on the gathered data, CH sends the information in a single hop to the aggregate node. The CH needs more energy to run because it has a higher mission load than other nodes, which is a major drawback of the LEACH algorithm. The energy-efficient multi-hop routing protocol L-LEACH (Energy Balanced) [19] solely takes the distance component into account. Like LEACH, L-LEACH is divided into two phases. Every round, the energy load is split equally among the network nodes and new CHs are chosen. Nonetheless, a noteworthy constraint of the algorithm is its disregard for the algorithm's intricacy and the corresponding network overhead. This oversight could be considered a negative in assessing the overall performance of L-LEACH. In order to send data from orphan nodes in the network that cannot be fully covered to the BS, [PERSON] et al. [20] designed the O-LEACH (orphan-LEACH) protocol. The protocol proposes two solutions. One method is to have a cluster member serve as a gateway, enabling non-clustered nodes to join the cluster by designating that gateway node as the CH, so that the data of the isolated nodes can be sent through the gateway node aggregation to the CH of the cluster where the gateway node is located. Another solution is that cluster members can form subclusters with orphan nodes, and the first orphan node to join will serve as the cluster head for that subcluster. It is important to remember that this approach works only in situations when there is a finite number of cluster members. The Multi-Hop LEACH (MH-LEACH) protocol was created by [PERSON] et al. [21] to find the best pathways to the BS via several intermediate CHs. According to this protocol, each CH broadcasts an advertisement message in order to build its routing table according to the strength of the signal it receives. However, they are forwarded to the CH to be checked for possible looping or an opposite direction before employing these routes. [PERSON] et al. [22] and [PERSON] et al. [23] propose LEACH-ANT and LEACH-IACA protocols, respectively, both of which are based on LEACH for cluster head selection and utilize an ant colony algorithm to find the optimal path in the inter-cluster transfer phase. However, the algorithm suffers from local optimization and slow convergence speeds. Swarm intelligence has a lot of potential for WSN routing optimization. In order to extend the network life cycle, [PERSON] et al. [24] proposed a clustering algorithm combining Mayfly and Aquila called MFA-AOA. The algorithm uses MFA for CH selection using variables like energy and distance, among others, to build the fitness function and choose CHs. The algorithm uses AOA to select the optimal route between CH and BS. But the algorithm does not adequately consider the complexity. [PERSON] et al. [25] presented the IAOAR routing protocol, which is based on the Archimedean optimization technique and can reduce energy efficiency. The algorithm selects the optimal CH based on AOA.