## 1 Introduction  
The problem of two-dimensional irregular packing entails the placement of irregular shapes within a limited area with the objective of optimizing material utilization. The utilization of sheets is prevalent in various industry sectors, including mechanical manufacturing, metal processing, aerospace engineering, automobile manufacturing, and furniture manufacturing [1]. At the societal level, the enhancement of material utilization can have a substantial impact on environmental conservation. In addition, even a slight increase in packing density can result in significant economic gain. The primary focus of manufacturing industries is undeniably the optimization of material utilization, a task of utmost importance in their efforts to mitigate the rising expenses associated with raw materials. To exemplify the significant cost reduction possibilities, it is worth examining the leather industry in China, which, in the year 2021, achieved an astonishing production output of 597 million square meters of leather [2]. Even a slight increase of 0.01% in material utilization has the potential to save up to 60,000 square meters of leather. Additionally, during the period of January to December 2021, China recorded an import of 612,500 tons of cattle and horse leather, with a total value amounting to US52.096 billion [3]. The potential to increase leather utilization rates not only offers significant economic advantages, but also aligns with the urgent requirement for environmental sustainability, emphasizing the critical necessity for inventive approaches. The primary objective of this research is to enhance material utilization rates, acknowledging this as the fundamental focus of the study's purpose. While prioritizing material utilization, the ultimate objective is to achieve the best possible balance with respect to calculation time. Striking an optimal balance between these two critical elements is essential for the success of the study. Currently, there has been significant research conducted on the problem of two-dimensional irregular packing, resulting in the proposal of various solutions. These solutions are commonly classified into three distinct categories: mathematical programming methods, heuristic algorithms, and machine learning methods [4]. Mathematical programming methods can be used to handle complex constraints and obtain optimal solutions. For instance, [PERSON] et al. proposed a separation overlap polygon algorithm based on nonlinear programming, which finds a new position with the least overlap on a material board by exchanging the placement positions of the two polygons on the original material board [5]. [PERSON] et al. introduced a mixed integer model that establishes a correlation between discrete points, commonly represented as grids, on the material board and each specific type of workpiece using binary decision variables. The representation of the workpiece type in the model requires the use of binary variables to indicate whether a particular workpiece type is selected. Additionally, discrete points are employed to determine the placement position of the workpiece [6]. [PERSON] et al. proposed an innovative semi-continuous mixed integer programming model for the two-dimensional cutting and packing problem of irregularly shaped workpieces, and partially simplified the developed mathematical model [7]. [PERSON] et al., introduced a mixed integer quadratically constrained programming model to address the irregular strip packing problem, taking into account the continuous rotation of workpieces. The authors also conducted an analysis of the model's strengths and weaknesses [8]. [PERSON] et al. proposed a mathematical programming model that addresses the integration of irregular strip packing and cutting path determination problems [9]. Heuristic algorithms, including greedy algorithms, simulated annealing algorithms, and genetic algorithms, have the capability to yield satisfactory solutions within a limited timeframe.