## 3 Modeling of Circular Quality Responses  
In this section, in order to define functional relations between inputs, process parameters that were varied through experimental trials and measured plasma jet circular cut quality responses modeling will be performed by the application of fuzzy logic. Fuzzy logic is a form of artificial intelligence that has proven particularly useful for describing complex manufacturing processes and systems. The plasma jet manufacturing process can be characterized as one of these. Usually, due to ambiguous and imprecise information regarding analyzed responses, it is not possible to implement conventional mathematical modeling approaches such as regression analysis [25]. In these scenarios, fuzzy logic stands out as an effective solution for establishing relations between process inputs and outputs to better understand the process behavior and process parameters effects, as well as to reach optimal solutions regarding desirable response values [15]. Each fuzzy logic system consists of a few elements: a fuzzification module, fuzzy inference module, and defuzzification module. The fuzzification module assigns a fuzzy linguistic value to each input by determining its degree of membership within the range [0, 1]. The degree of membership is defined by membership functions. There are a few membership functions that can be applied in this step, such as: triangular, Gaussian, trapezoidal, sigmoid, linear, generalized bell, etc. The fuzzy inference module uses a knowledge base of fuzzy IF-THEN rules and membership functions to establish relationships between the inputs and outputs. Two of the most famous fuzzy inference modules are Mamdani and Sugeno. Mamdani is applied more usually than Sugeno due to its intuitiveness and ease of use. The fuzzy inference module is in this way in charge for defining fuzzy linguistic values of system outputs. Finally, the defuzzification module reshapes aggregated fuzzy outputs into exact non-fuzzy values. There are also a few defuzzification methods that can be applied in this step. Some of them are centroid, the middle of the maximum, the smallest of the maximum, the largest of the maximum, bisector, etc. [15; 26]. In this paper, the Mamdani fuzzy inference system was applied to model the relationship between the inputs-plasma jet cutting process parameters--the cutting speed, arc current, and cutting height--and the outputs-circular cut quality responses--the circular  
Figure 3: Experimental responses. kerr width, circular level angle, circularity error--top, and circularity error--bottom. The settings of this system are as followed: method: min or method: max; implication method: min; aggregation method: max; and defuzzification method: centroid. The architecture of the developed fuzzy logic system is presented in Figure 4.  
The input and outputs of the fuzzy logic system were described by Gaussian membership functions. For each input, three Gaussian membership functions were defined linguistically as follows: L (low), M (medium), and H (high). Additionally, outputs from the fuzzy logic system were defined by five (the circular kerr width and circularity error--top) and seven (circular bevel angle and circularity error--bottom) Gaussian membership functions linguistically described as: VL (very low), L (low), ML (medium low), M (medium), MH (medium high), H (high), and VH (very high). These defined membership functions for the inputs and outputs are presented in Figures 5 and 6. In the next step, in order to establish functional relations between such defined fuzzy logic system inputs and outputs, based on the experimental results and preliminary and empirical findings, a set of 27 fuzzy IF-THEN rules was generated. These rules are listed in Table 4. Finally, the defuzzification module from the MATLAB R2023a Fuzzy logic toolbox converted aggregated fuzzy logic outputs data into exactly expressed numerical non-fuzzy values of a circular kerr width, circular bevel angle, circularity error--top, and circularity error--bottom responses.