## 3 Preliminaries
### FCM Clustering Scheme in Routing Layer  
Fuzzy c-means (FCM) clustering is utilized in the forwarder selection process to identify low-density regions for optimal packet forwarding. As an unsupervised clustering technique based on fuzzy set theory, FCM allows nodes to belong to multiple clusters with varying membership degrees. Unlike traditional hard clustering, this method assigns a participation cost to each node, proportional to its distance from the cluster centroid, enabling dynamic adaptation to network density. The number of clusters is determined locally within the sender node's transmission range, ensuring flexibility in dynamic network conditions [11; 54]. Previous research has explored FCM at the global network level for WSN optimization [11; 54]. However, this study is the first to integrate FCM locally within an OR protocol for forwarder selection, significantly reducing control overhead by eliminating the need for network-wide clustering updates. Instead, clustering calculations are performed locally, enhancing efficiency while minimizing communication overhead. Given these advantages, FCM is adopted as the clustering mechanism in the proposed OR protocol. However, the protocol is kept lightweight by localizing all FCM and FR computations within the one-hop neighborhood and avoiding them completely if the neighbor count is small. The following part of this section outlines the detailed procedure for FCM calculation. When the neighborhood density or the number of potential forwarders surrounding a sensor node exceeds a defined threshold, the sender node initiates the clustering algorithm using the fuzzy c-means (FCM) method. As an unsupervised clustering technique, FCM does not require predefined convergence values for cluster formation. Instead, the process begins with the initialization of a membership matrix. The membership matrix (\(U\)) is a two-dimensional array that assigns each node a degree of belonging to each cluster. The sum of membership values for any given node across all clusters equals 1. Once the number of clusters is determined, the membership matrix is initialized, and subsequent calculations are performed iteratively until the optimal clusters are identified. To initialize the membership matrix, the sender node first determines the optimal number of clusters  
Figure 3: Data transmission procedure in the MAC layer (BoX-MAC) in the proposed FCM-OR protocol. based on the number of neighboring nodes (primary forwarder set) within its transmission range. The following formula is used for calculating the optimal number of clusters:  
\[C_{opt}=\left(\sqrt{\frac{n}{2\pi}}\right)*\left(\sqrt{\frac{e_{fs}}{e_{mp}}} \right)*\left(\frac{\delta}{d(s,f)^{2}}\right). \tag{1}\]  
Here, \(n\) represents the number of nodes in the primary forwarder set, which also represents all known neighbor nodes under the sender node's transmission range. Additionally, \(e_{fs}\) and \(e_{mp}\) represent the energy required for transmitting a single bit of data to obtain the minimum bit error rate; they are dependent on the transmission distance in a free-space model or a multi-path fading model, respectively [55]. \(\delta\) denotes the transmission range of a sensor node, which is the same for all the nodes in the network. Additionally, \(d(s,f)\) represents the average Euclidean distance between the calculating sender node and the forwarder nodes. Initially, cluster centroids are randomly assigned based on the number of clusters and the communication range of the  
a node receives a transmission request, it executes the FCM clustering algorithm if the number of potential forwarders exceeds a predefined threshold, segmenting forwarders into clusters. Nodes from the smallest cluster are selected for FR computation, while those farther from the sink are excluded to prevent unnecessary forwarding. The FR value is attached to outgoing packets, enabling receiving nodes to assess their eligibility as forwarders.