# Evolutionary Algorithms for Optimization Sequence of Cut in the Laser Cutting Path Problem
## 2 The Laser Cutting Path Problem (LCPP)  
A laser cutting machine is an automated device for precise cutting and design projects in various industries. It offers high cutting speed, narrow kerf, excellent cutting quality, and versatility. The laser cutting path is crucial in determining the cutting quality, processing efficiency, and energy consumption, directly impacting production costs. The required time is divided into cutting and air time (head movement between different edges/patterns) in the laser cutting process. Therefore, the specific CPDP applied to laser cutting machines is known as laser cutting path planning (LCPP).  
The primary objective of the LCPP problem is to optimize the cutting time, which depends on two key parameters that simulate the laser cutting device: cutting velocity (\(V_{c}\)) and air-moving (or sliding) velocity (\(V_{m}\)) of the cutting head. The first parameter is influenced by the machine's hardware, the shape of the pieces in the input layout, and the characteristics of the material being cut. The second parameter controls the speed at which the cutting head moves without cutting. In this study, we aim to address a generalized type of CPDP [6] considering the earlier constraints, distinguishing it from the research conducted by [PERSON] et al. [10].  
Furthermore, LCPP involves optimizing the distances between cutting points where no cutting is performed, reducing air time. Therefore, strategies must minimize the length of unnecessary laser head movements. This situation can also be considered an \"empty trip\" problem in laser cutting [11].  
Figure 1 illustrates an optimal solution obtained by input layout instance. Note that, in the LCPP problem, the choice process for every solution considers the source (\(S\)) coordinates (\(x=0,\ y=0\)). Usually, the top-left corner of the laser cut machines. The main question is: starting from the source point, how can a laser cutting machine (a computational method) select the best sequence of edges that minimizes the complete cutting process time?  
Furthermore, in Figure 1, the LCPP considers the initial and the final air time (represented by a dashed line, a sequence with the numbers 1 and 18). In every cutting process, the head of the device will move without cutting from the origin to the first point of the input layout (sequence number 1). The same happens when all the edges are cut, and the head must return to the origin (sequence number 18). Thus, it only moves the head when it does not perform a cut. The cost function uses a \(V_{m}\) parameter and the Chebyshev distance between points to compute the time required for that movement. All the rest of the head machine moves (sequences 2-17) increase the final process time to cut the demand of an input layout. For this case, each cut uses the parameter \(V_{c}\) and the size of the edge. Note that the sequence numbers \(11\) and \(12\) represent the same edge. Nonetheless, we divide common points for our proposal to increase path possibilities. For example, the node of the square that touches the triangle allows one edge division between sequence numbers \(11\) and \(12\).  
Figure 1: An optimal solution example for laser cutting path planning.