# Evolutionary Algorithms for Optimization Sequence of Cut in the Laser Cutting Path Problem
## 1 Introduction  
Researchers frequently aspire to reduce production costs by considering the context of cutting materials. Integrating CAD (computer-aided design) and CAM (computer-aided manufacturing) systems has significantly enhanced the production capacity of device tools for generating NC (numerical control) programs. Consequently, developers have created numerous commercial computer system packages to automate the NC programming process for diverse cutting applications. This advancement has resulted in the extensive adoption of automated devices across manufacturing industries, enabling the precise cutting of various materials such as clothes, papers, glasses, and sheet metals. For example, cutting clothes involves working with polygon-shaped pieces that must be carefully positioned on a \"strip\" to minimize waste. This process, known as cutting and packing (C&P), is classified as an optimization problem. It aims to efficiently arrange items within a given space, maintaining the same dimension [1, 2]. Employing heuristics and exact techniques for C&P can provide a significant commercial advantage [3]. For comprehensive surveys on C&P problems, refer to [4, 5]. In the nesting process, which is the initial stage, the objective is to minimize the amount of raw material required. This process can also involve additional constraints specific to the application at hand. Laser cutting, for instance, is a widely used technique for separating sheet metal items, making it one of the primary methods employed once an optimal layout has been determined. Once a suitable arrangement of polygons (layout) has been found, the next step is to compute the optimal cutting path the cutting device should follow to minimize the processing time. This problem is known as the cutting path determination problem (CPDP) [6]. The CPDP focuses on determining the most efficient cutting path for a tool or machine to manufacture a specific part or product. Cutting can be categorized into two main categories: complete cutting, where each polygon is cut entirely before moving on to the next polygon, and partial cutting, where an item can be cut in portions, allowing for switching between partially cut pieces during the cutting process. The CPDP plays a crucial role in the manufacturing industry, and extensive research has been conducted to develop efficient algorithms and approaches to solve this problem across various manufacturing processes. The aim is to optimize the cutting path and minimize processing time, leading to improved efficiency and productivity in the manufacturing industry. Various studies have explored the impact of laser cutting materials, specifically by examining the effects of processing heat and different speed parameters on cutting quality. However, our research specifically focuses on reducing the empty laser cutting path, mainly when dealing with instances involving connected and separate nested pieces. We incorporate two parameters to represent the air movement (\(V_{m}\)) and cutting speeds (\(V_{c}\)) of a laser cutting machine. Thus, we are able to simulate the elapsed time to cut the pieces in different paths. It is worth reinforcing that such processes can be time consuming in most applications and gains, however tiny, represent a large improvement overall. We encounter the laser cutting path planning (LCPP) problem in this context. LCPP is a specific type of CPDP that aims to quickly determine a cutting path that minimizes the overall time required to cut all parts from the given layout. This optimization considers two crucial parameters: \(V_{m}\) and \(V_{c}\). The total time involved in the laser cutting operation consists of the actual processing time and the movement time, during which the laser head moves without cutting between different nodes and edges (representing the items). Consequently, the actual processing time can be determined once the machine and material cut speed limitations are correctly defined.