## 2 Problem Statement  
In the realm of manufacturing, specifically in the field of sheet cutting, it is a prevalent necessity to arrange and cut various shapes and quantities of workpieces on a predetermined size of raw material sheet. The objective is to optimize the utilization of each sheet by either maximizing its usage or minimizing the number of sheets required. This study exclusively addresses the issue of irregular packing in rectangular raw material areas. Mathematically, this problem can be described as follows:  
Given a rectangular raw material board \(\Omega\) with a length of \(L\) and \(p\) pieces of workpieces to be arranged, the \(i-th\) and \(j-th\) workpieces to be arranged are \(\Omega_{i}\) and \(\Omega_{j}\), respectively, with areas of \(S_{i}\) and \(S_{j}\). The initial position of each workpiece to be arranged is represented by a coordinate list composed of all its vertices coordinates. The initial position of the \(i-th\) workpiece to be arranged is \([[x_{1},y_{1}],[x_{i2},y_{2}],\cdots,[x_{in},y_{m}]]\) (\(n\) is the number of its vertices), that is, the coordinate list of all workpieces to be arranged is \([[[x_{1},y_{1}],[x_{12},y_{12}],\cdots,[x_{1n},y_{1n}]],[[x_{21},y_{21}],\)\([x_{22},y_{22}],\cdots,[x_{2n},y_{2n}]],\cdots,[[x_{p1},y_{p1}],[x_{p2},y_{p2}],\cdots,[x_{ pn},y_{m}]]]\). The angle that the workpiece to be arranged can be rotated relative to the initial posture is \(\theta\) (i.e., the workpiece to be arranged can be placed in different directions by a certain angle, typically \(90^{\circ}\) or its multiples, to increase the position variation of the workpiece, improve the search efficiency of the algorithm, and help the algorithm find more possible solutions and increase the probability of obtaining higher-quality optimal solutions). The maximum contour height of the packing result is \(H_{\text{max}}\), and the utilization rate of the board material is \(U\). A schematic diagram with some symbol meanings is shown in Figure 1.  
The objective is to minimize the maximum contour height of the packing result, denoted as \(H_{\text{max}}\), while ensuring that each workpiece is fully contained within the rectangular sheet (\(\Omega_{i}\in\Omega\)), that workpieces do not overlap (\(\Omega_{i}\cap\Omega_{j}=\varnothing\)), and that the allowable rotation angle \(\theta_{i}\) meets the specific process requirements. Moreover, the utilization rate of the board material \(U\) is constrained to fall within the range [0, 1]. Therefore, we define the two-dimensional irregular packing problem as follows:  
\[\text{minimize}\ H_{\text{max}} \tag{1}\]  
\[\text{subject to}\ \Omega_{i}\in\Omega,\ 1\leq i\leq p \tag{2}\]  
\[\Omega_{i}\cap\Omega_{j}=\varnothing,\ 1\leq i<j\leq p \tag{3}\]  
\[\theta_{i}\in\theta,\ 1\leq i\leq p,\ 0\leq\theta\leq 2\pi \tag{4}\]  
\[U\in[0,1] \tag{5}\]  
\[H_{\text{max}}\in\mathbb{R} \tag{6}\]  
Formula (2) indicates that all the workpieces to be arranged must be completely contained within the rectangular sheet, Formula (3) indicates that the workpieces to be arranged cannot overlap, and Formula (4) indicates that the allowable rotation angle of the workpieces to be arranged must meet the process requirements, for example, some leather workpieces need to be cut in the same direction to maintain the consistency of their surface texture.  
This mathematical model provides a foundation for the subsequent algorithmic approach that will effectively solve the irregular packing problem, with the primary objective being to minimize \(H_{\text{max}}\) and optimize material utilization.  
Figure 1: Schematic diagram of some symbol meanings.