## 4 Proposed Protocol
### CH Selection
#### 4.1.2 Step 2: Selection of CHs  
In this step, an FIS is adopted to compute the probabilities of becoming a CH for all candidate CHs stored in \(S_{CH}\). We employed four fuzzy inputs including residual energy, number of neighboring nodes, distance to the BS, and node centrality to maximize the rationalization of CH selection. The parameters considered as FIS input characteristics are detailed as follows:  
* Residual energy (\(f_{CH_{1}}\)): This parameter indicates the remaining energy of node \(i\) denoted by \(E(N_{i})\) in the current round. The CH serves as a critical node for data transmission, requiring a large amount of energy. This characteristic becomes more critical in the later stages of operation. In contrast to existing protocols that use a linearly declining weight constraint on this parameter, our approach determines the energy threshold at step 4.1.1 and selects nodes with the highest residual energy as candidate CHs. * Number of neighboring nodes (\(f_{CH_{2}}\)): This parameter refers to the number of direct communication connections established between each node within its communication range and its surrounding nodes, representing each cluster's compactness. A larger value of this parameter is considered a more satisfactory feature because it indicates a shorter intra-cluster distance. * Distance to the BS (\(f_{CH_{3}}\)): This parameter concerns the Euclidean distance between the candidate node \(i\) and the BS which is expressed by \(Dis(i,BS)\). The nodes with a closer distance to the BS have a higher probability of being selected as CHs because they consume less communication energy. Since the nodes are static, the BS has collected the distance information of all nodes based on RSSI in the first iteration. * Node centrality (\(f_{CH_{4}}\)): Our protocol in this paper selects CHs before cluster formation, so we use the average distance between a candidate CH and its set of neighboring nodes (\(Neighbor(N_{i})\)) [35] to represent node centrality, which can be computed by \[f_{CH_{4}}=\frac{\sum_{i\in Neighbor(N_{i})}d(CH,i)}{Neighbor(N_{i})}\] (6) where \(d(CH,i)\) is the Euclidean distance between candidate CH and one of its neighbors. The smaller the value of \(f_{CH_{4}}\), the closer the candidate CH is to the center of the cluster. Nodes selected as CHs that are close to the cluster centroid can minimize the energy consumption of data transmission. Using the above-mentioned inputs, the FLS outputs a chance to indicate the priority of nodes becoming CHs. The whole process involves normalization, fuzzification, a fuzzy logic system, and defuzzification, as shown in Figure 2.  
Where normalization aims to standardize input variables from different ranges into a uniform range between 0 and 1, this step maps membership values to a standardized range that allows us to make efficient comparisons and calculations. Each normalized variable is mapped to a corresponding fuzzy linguistic variable via the membership functions. In our proposed method, each selected input variable of each fuzzy system has different membership functions represented as trapezoidal, triangular-trapezoidal, and triangular. The fuzzy outputs for all the FISs are derived from the set of selected features obtained by the optimization process. We employed seven linguistic variables to express the outputs, providing enhanced flexibility in formulating the fuzzy rules. The detailed linguistic variables and parameters can be found in Table 1.  
The membership functions of the inputs and output can be formulated based on previous works [13; 17] and experimental experiences. These are illustrated in Figure 3 and Figure 4, respectively.