# Advanced Optimization Algorithm Combining a Fuzzy Inference System for Vehicular Communications
## 2 System Models
### Sugeno Fuzzy Inference Systems
This flexibility is particularly useful in complex scenarios such as clustering algorithms in vehicular networks, where factors like residual energy, node degree, and distance to the sink need to be considered simultaneously [27]. By leveraging these characteristics and advantages, Sugeno fuzzy systems can significantly enhance the performance, reliability, and efficiency of routing protocols and other decision-making processes in vehicular communication networks. Their ability to handle uncertainty, adapt to changing conditions, and make complex decisions based on multiple inputs makes them an invaluable tool in the development of robust and efficient vehicular communication systems [28]. Figure 2 illustrates a Sugeno-type FIS architecture with two input variables and one output variable. The first input, labeled \"Distance\", operates within a range of [3; 150] and employs four membership functions; the Gaussian combination membership function was chosen for the first input. The second input, \"CSI\", spans the range [0, 1] and utilizes five membership functions; the Gaussian membership function was used for the second input. These inputs feed into a central processing block containing 20 fuzzy rules, which serves as the system's inference engine. The output variable, marked as \"\(P_{rk}\)\", is displayed in a cyan-highlighted box with a range of [0.4, 0.8], and comprises five membership functions; the Gaussian membership function was also used for the output. The entire system is connected through directional lines, demonstrating the flow of information from inputs through the fuzzy rule base to generate the final output. Figure 3 illustrates fuzzy membership functions depicting four linguistic variables (VN, N, F, and VF) represented across a distance domain ranging from 0 to 140 units. Each linguistic variable is characterized by a trapezoidal membership function that overlaps  
Figure 2: Sugeno-FIS diagram. with adjacent functions, creating smooth transitions between categories. The leftmost function, _Very Near_ (VN), peaks at the lower distance values, followed by _Near_ (N), _Far_ (F), and _Very Far_ (VF), which progress towards higher distance values. The functions exhibit symmetrical crossover points at approximately 0.5 degrees of membership, where adjacent functions intersect, ensuring a continuous and well-distributed fuzzy classification system. The trapezoidal shapes maintain full membership (a value of 1) over certain intervals before gradually decreasing, providing stable regions of complete belonging to each linguistic category. Figure 4 shows a fuzzy membership function plot depicting five linguistic variables represented by overlapping curves across a CSI range from 0 to 1. The membership functions are labeled as _very loose_ (VL), _loose_ (L), _moderate_ (M), _tight_ (T), and _very tight_ (VT), each represented by distinct colored curves that peak at different CSI values. Each curve exhibits a triangular or trapezoidal shape with smooth transitions, where the membership value reaches a maximum of 1 at its peak and gradually decreases to 0 at its boundaries. The curves are systematically distributed across the domain with significant overlap between adjacent membership functions, ensuring continuous coverage and smooth linguistic transitions. The VL function peaks at 0, L at approximately 0.25, M at 0.5, T at 0.75, and VT at 1.0, creating a symmetrical distribution of linguistic variables across the CSI range. Figure 5 illustrates a fuzzy membership function plot displaying the Gaussian membership function of the \(P_{rk}\) domain ranging from 0.4 to 0.8. The membership functions are labeled as _very low_ (VL), _low_ (L), _medium_ (M), _high_ (H), and _very high_ (VH), with each function reaching a maximum degree of membership of 1.0 at their respective peaks. The VL function starts at full membership at 0.4 and decreases linearly, while the VH function increases to full membership as it approaches 0.8.