## 1 Introduction
The validity of membership in these clusters is based on performance measures using the criteria of hypervolume and density [25]. In [26], the authors proposed a fuzzy-based controller for sink selection and clustering in wireless sensor networks. Their proposed method considers the capacity of sink nodes to optimize data transmission and prevent packet loss. Furthermore, a fuzzy-based controller is designed to select cluster heads by reducing the number of fuzzy rules using the R-implications method, which decreases the complexity and energy consumption of the system. Simulation results demonstrate that this approach enhances network lifetime, reduces packet loss, and improves energy efficiency compared to traditional clustering techniques. In [27], an anti-swing control method for overhead crane systems was developed by integrating the clustering salp swarm algorithm (CSSA) with a fuzzy PID controller. The proposed approach enhances population diversity and improves convergence accuracy by introducing clustering techniques into the original salp swarm algorithm (SSA). Similarly, a fuzzy control approach based on the extraction and reduction of fuzzy rules through the FCM clustering algorithm was presented in [28]. This method optimizes a self-tuning fuzzy proportional-derivative controller by reducing the number of fuzzy rules while maintaining control accuracy. The authors demonstrated the effectiveness of these approaches through simulation experiments, showing superior performance compared to conventional fuzzy and PID controllers in mitigating overhead crane oscillations along with improved control precision. A fuzzy tuning approach for the controller parameters of a parallel manipulator based on clustering analysis was proposed in [29]. The new technique improves tracking performance throughout the robot workspace by identifying specific configurations that represent different levels of load inertia. Using the FCM clustering algorithm, inertial data are classified and representative cluster centers are determined to enable efficient tuning of the controller parameters for arbitrary configurations. The results obtained from a prototype demonstrate the effectiveness of this methodology in improving control quality without requiring complex real-time computations. The authors of [30] presented a hybrid approach combining a fuzzy logic controller (FLC) and FCM to enhance the asset management of power transformers. Their proposed methodology provides diagnoses and prognoses of incipient power transformer faults using a system that incorporates critical parameters such as dissolved gas analysis, interfacial tension, etc. Their study used a dataset consisting of 200 transformer tests for verification, demonstrating that the fuzzy clustering approach can effectively classify insulation conditions into four clusters ranging from healthy insulation to insulation that is nearing the end of its useful life. In [31] the authors demonstrated target recognition capability in remote sensing images acquired by a small UAV using fuzzy clustering techniques. To facilitate improved object detection and localization, the fuzzy clustering algorithm was applied to construct an intelligent target recognition model that effectively segments and classifies remote sensing data. The performance of the proposed system was verified through simulation experiments, demonstrating its effectiveness and robustness. In [32], the authors proposed a fuzzy control framework for aircraft engines by integrating dynamic clustering modeling with robust control strategies. Their approach utilizes a hierarchical clustering technique to optimize the number of fuzzy rules, ensuring accurate representation of the engine's dynamic behavior while maintaining computational efficiency. Experimental verification through hardware-in-the-loop (HIL) tests demonstrated the effectiveness of the proposed model, achieving improved engine performance with reduced settling times and overshoots.