# Solution to Solid Wood Board Cutting Stock Problem
## 1 Introduction  
In recent years, with the rapid development of the social economy, the desire for, and expectation of, a high quality of life is becoming more common. Solid wood furniture is increasingly valued and favored by people because of its unique aesthetic feeling and superior material characteristics; the market demand for solid wood furniture is showing an upward trend year on year [1, 2, 3]. However, some countries have low forestation; in China, the forest coverage rate is only three-fifths of the world average, and the per capita forest area is less than one-quarter of the world average [4]. Solid wood board resources are scarce; this, together with the implementation of a forest cutting quota policy, has led to the supply of solid wood board in China being highly dependent on imports [5, 6, 7, 8]. In addition, the production technology of most furniture enterprises remains outdated, which results in solid wood boards being only partially used. This waste results in increased production costs and the need to manage waste [9]. Therefore, measures must be taken to alleviate the divergence between supply and demand of solid wood boards by saving raw materials [10]. In 2020, 322 million pieces of wooden furniture were produced in China, with the average board consumption per piece of furniture being 0.6 m\({}^{3}\). If the existing board utilization rate is increased by 1%, 193 million m\({}^{3}\) boards can be saved; this shows that a slight increase in the utilization rate of raw materials can cause a significant decrease in the total consumption of raw materials. As the first step of the furniture manufacturing process, layout directly determines the utilization rate of raw materials [11, 12, 13]. Therefore, areasonable layout is an important measure to reduce material costs and improve economic benefits and is thus of great significance in promoting the sustainable development of the furniture manufacturing industry. As shown in Figure 1, the solid wood board cutting stock problem (CSP) refers to placing the required standard boards of different lengths on the original boards of different lengths. Under the condition of meeting the requirements for the demand and size of standard boards, the used original boards are fully utilized to maximize the board utilization rate. In this paper, we only consider length, which thus belongs to the one-dimensional cutting stock problem. The one-dimensional cutting stock problem is a typical optimization problem. In terms of computational complexity, it is an NP-hard problem with the highest complexity; the solution time increases exponentially with the problem scale, and there is no accurate optimal solution in the polynomial time [14; 15; 16]. At present, most studies use heuristic methods to solve optimization problems, such as polar bear optimization [17; 18; 19; 20], grey wolf optimizer [21; 22; 23], and genetic algorithm [24; 25; 26], which can find good approximate solutions in a limited time. The one-dimensional cutting stock problem has been studied by scholars for years. Common layout methods include column generation, greedy algorithms, and intelligent optimization algorithms. [PERSON] [27] proposed a priority-based goal programming approach to solve the one-dimensional cutting stock problem with a random demand mix. This methodology can evaluate the effects of various raw material order levels to account for demand randomness at the time of order. [PERSON] [28] presented an integer programming model for the 1 DCSPUL (one-dimensional cutting stock problem with usable leftovers) with limited leftover types, and described a heuristic algorithm based on a column-generation procedure for its solution. [PERSON] [29] proposed a modified greedy heuristic (MGH), which first orders the pair items, or the odd ones in the case that there are more items in the problem. The experimental results showed that the MGH is more effective than the  
Figure 1: Solid wood board CSP. greedy heuristic.