## 1 Introduction
Originating from the field of the design of experiments, RSM efficiently navigates complex parameter spaces, elucidating the intricate relationships between input variables and responses [26; 27]. Widely adopted in diverse disciplines, RSM empowers researchers and engineers to systematically explore, model, and optimize processes, ensuring efficiency and precision. By providing a structured approach to experimentation, RSM significantly contributes to advancing scientific understanding and technological innovation, making it an indispensable tool for optimizing complex systems and processes. RSM facilitates a comprehensive understanding of the intricate relationships between FDM [28] and EDM [29; 30] parameters, such as raster angle, air gap, raster width, current, pulse on time, and pulse off time, and critical performance measures like compressive stress, percentage deformation, breaking stress, material removal rate, tool wear rate, and surface roughness. The utilization of RSM not only aids researchers in uncovering the optimal combinations of parameters for enhanced outcomes but also contributes to advancing the efficiency and effectiveness of these advanced manufacturing techniques [31; 32]. In this study, the authors employed response surface methodology (RSM), a well-established Design of Experiments (DOE) technique, to explore the impact of three input variables--laser power (LP), focal plane position (FPP), and cutting speed (CS)--on the -kerr wall's geometry and quality in 5 mm thick polymethyl methacrylate (PMMA) sheets. Utilizing the Design Expert v17 software, the study extensively measured and statistically analyzed top and bottom kerr width, the ratio of the upper kerr to the lower kerr, and the cut angle of the kerr wall. An optimization process was undertaken to identify optimal laser cutting parameters that yield the desired geometrical characteristics of the cut kerr. Seventeen cutting experiments were conducted at the optimal settings to validate the optimization results, comparing the outcomes with the software optimization results. The primary objectives were directed at enhancing kerr quality by determining the most effective combination of the input parameters--laser power, cutting speed, and focal plane position. The optimization method employed in this study was centered around the desirability approach, a well-established technique in the field of Design of Experiments (DOE). Desirability functions are utilized for the systematic and comprehensive evaluation of multiple responses simultaneously, enabling the determination of optimal process conditions. By assigning desirability values to different levels of each response, the approach facilitates the identification of parameter combinations that collectively meet the desired criteria [33]. This methodology is particularly advantageous in laser cutting processes, where multiple factors influence the quality and geometry of the kerr. Through the desirability approach, this research aims to not only enhance specific aspects of the kerr, such as width and angle, but also find a balance that maximizes the overall cutting performance. This investigation leveraged Design of Experiments (DOE) and response surface methodology (RSM) to model the intricate relationship between the input parameters and the kerr's quality. The optimization process, guided by a desirability function, sought parameter values maximizing the desirable outcomes, such as a minimized kerr width, an improved cut angle, and a specific ratio of upper to lower kerr dimensions.