# Optimization of the One-Size-Fits-All Layout Problem Based on Preparing Material for Steel Bridges
## 3 Layout Strategy  
Since the parts for bridge materials are cut from steel plates, the layout needs to facilitate the subsequent cutting processes. Therefore, in this study, the \"one size fits all\" layout strategy was applied. The basic idea of this algorithm is to cut through the entire plate along the layout path from one end of the plate to the other during each cutting process. This method is called \"one size fits all\" because it uses the same direction at each cutting stage to ensure that the entire plate is cut while meeting the process requirements. We call cutting in the same direction a stage. In the actual production process, in order to improve the efficiency, the cutting process is usually divided into three or four stages. To achieve this, the three-stage precise nesting and the three-stage non-precise nesting methods are commonly used as cutting strategies. Figures 1 and 2 illustrate these two nesting methods. In these diagrams, rectangles with numbers represent parts, where the same number indicates parts of the same size. The vertical lines outside the rectangles represent cuts, with different numbers denoting different cutting stages. The arrows of different colors represent different stages. During cutting, we refer to cuts in the same direction as one stage. In the three-stage precise layout method, square pieces can be cut to an accurate size within three stages. This method ensures a high production efficiency while meeting specification requirements as closely as possible. In the non-precise method, an additional fourth stage of cutting may be required for some square pieces to meet the specification requirements. The \"one size fits all\" layout strategy not only considers the process requirements and production efficiency but also focuses on meeting product specifications. When preparing materials for steel bridges, manufacturing companies choose an appropriate cutting stage and layout method according to specific requirements to achieve the best production results. This integrated approach to process and specification provides greater flexibility in the production process to accommodate different types of products and their requirements. Due to the difference in the number of stages, different authors use different names for each stage. Figure 3 shows specific details of the key stage modules. During the actual cutting process, the initial cut can be made along either the long or short side. Figure 3 illustrates an example perpendicular to one of the sides. Figure 1: Three-stage non-precise layout. Figure 2: Three-stage precise layout. In taking the three stages of the cutting process as an example (see Figure 3), the first stage involves horizontal cutting to generate modules that we call stripes. The steel plate is divided into Stripe1 and Stripe2 along the red lines. The second stage involves vertical cutting to generate modules called stacks. For example, Stripe1 is further cut along the blue lines to form Stack1, Stack2, and Stack3. The third stage involves horizontal cutting to generate modules we call items. For instance, Stack1 is further cut along the black lines to form Item1, Item2, and Item3. The cutting process in these three stages facilitates the subsequent cutting of parts for preparing material for steel bridges. The layout used in this study was a three-stage non-precision layout, which can be described as follows: \(n\) rectangular parts \(\{\text{Item}_{1},\text{Item}_{2},\cdots,\text{Item}_{n}\}\) of different sizes, with a quantity \(\{m_{1},m_{2},\cdots,m_{n}\}\), are placed without overlapping onto a rectangular motherboard \(S\) with width \(W\) and an unlimited height. Under the condition that the \"one size fits all\" constraint is met, the height \(H\) of the board is minimized, thereby maximizing the utilization rate of the motherboard. The following constraints must be met during the placement of rectangular parts onto the motherboard:  
1. Different rectangular parts \(\text{Item}_{i}\) and \(\text{Item}_{j}\) must not overlap each other;
2.