## 3 The Proposed Fuzzy Model and WSN Clustering Technique
### Linguistic Variables' Measurement and Normalization
We are interested in the nodes located within a predefined range as long as they are not closer in distance to any other pre-selected CH because they will be probably members of the current node if this current node wins CH election. Since we use the average of the consumed energies for the current node, we use the term _average of local consumed energy_ and shorten it as \(AVG\_Energy\) to point out the _location suitability_. Note that we measure the average of the consumed energy rather than the total consumed energy to guarantee that the size of the group does not have an effect on the normalized value. Some consider, mistakenly, that if a node has a lower sum of distances to its future probable members, then is situated at a more suitable location to become CH. This stems from the assumption that the nodes closer to the centroid of the group will consume less total energy. However, since the energy is exponentially proportional to the distance, it becomes necessary to consider members with extreme distances. This means that the location with the minimum average distances to other surrounding nodes in a predefined range is not always the most suitable location for a node to become a CH since it may not always result in minimum total consumed energy for its surrounding group of nodes. To illustrate this idea, consider the scenario depicted in Figure 2. Here, node G has the minimum average distance of 4.62488 m from its surrounding nodes, and is the closest to the centroid. If selected as a CH, it will result in an average consumed energy of 2.535168 \(\times\) 10\({}^{-7}\) J, for one byte of data from each neighboring node. On the other hand, node \(F\) has an average distance of 5.12655 m, but results in an average consumed energy of 2.529408 \(\times\) 10\({}^{-7}\) J. For that reason, we base the computation of location suitability for a node to become a CH on the consumed energy rather than the total distances. 4. **Density of surrounding nodes:** Selecting CHs surrounded by dense nodes over CHs surrounded by sparse nodes improves the energy consumption by increasing the opportunity for nodes with more neighbors in their vicinity to become CHs. Thereby, the local consumed energy for the group members is decreased. _Density_ is calculated by the number of surrounding nodes within a predefined range normalized using Equation (1), where \(Value(var)\) is the density of the current node, \(Max(var)\) is the maximum density value through all candidate nodes, and \(Min(var)\) is the minimum density value through all candidate nodes. 5. **Compaction of surrounding nodes:** Group compaction is a measure of how the surrounding nodes are distributed around the current node. A node surrounded by more neighbors in a closer vicinity is considered of higher _compaction_degree. Selecting a node with a higher compactiondegree minimizes the total energy consumption. This criterion is important to assign different priorities for CH candidates with the same number of surrounding nodes within a predefined range. In other words, it is beneficial for distinguishing candidates surrounded by the same density of sensor nodes. _Compaction_ is calculated as the ratio of the number of nodes located within the first vicinity region to those located within the second vicinity region. The first vicinity region is that region that is within half of a predefined radius distance, whereas the second vicinity region is the one that is within the nodes' radius. As shown in Figure 3, the small circles surrounding A and B are considered as the first vicinities, and the larger ones are considered as the second vicinities for nodes A and B. The distance to which to extend the first vicinity is a design parameter and may be any fraction of the radius. Figure 3: First and second vicinities of nodes A and B.  
Figure 2: Simple WSN scenario for location suitability computation.