# An Improved Adaptive Genetic Algorithm for Two-Dimensional Rectangular Packing Problem
## 1 Introduction
However, if the fitness of the individual is close to or equal to the maximum fitness of the population, the crossover probability and mutation probability will be close or equal to zero, making the evolution of genetic algorithm stagnant. [PERSON] proposed an improved AGA [26] for this problem. However, when the average fitness is close to the maximum fitness of the contemporary population, it is easy to cause a large number of individuals to have a lower crossover probability and mutation probability, which will stagnate the evolution. This improved AGA has also been applied to many fields such as the three-dimensional container loading problem [27] and the laminate stacking sequence optimization [28]. In addition, the algorithm proposed by [PERSON] is only in view of individual fitness. This algorithm adaptively changes the crossover probability and mutation probability from the local perspective, but the dynamic adjustment of crossover and mutation are not considered whether they have a positive impact on the population evolution process from the overall perspective of population evolution. According to the evolution of later generations, the AGA [29] proposed by [PERSON] recorded the number of algebras in which the fitness of the optimal individual of the population did not change and dynamically adjusted the crossover and mutation probabilities. However, the algorithm has two shortcomings: (1) the incremental value and decrement value of crossover probability and mutation probability are greatly affected by fitness, and if the value range of the fitness function is not between 0 and 1, the adaptive algorithm will no longer be applicable; (2) although the crossover probability and mutation probability of each generation are dynamically adjusted, the crossover probability and mutation probability of all individuals in a certain generation are the same, and the performance is consistent for individuals with different fitness. In this paper, in order to overcome the shortcomings of the two kinds of adaptive genetic algorithms, an improved method called hybrid adaptive genetic algorithm (HAGA) is proposed to supplement [PERSON] and [PERSON]'s algorithms. It is the key for HAGA to work through the joint action of individual fitness from local perspective and the global perspective of population evolution. The presented algorithm not only can obtain differential performance for individuals but also deals with the impact of dynamic changes on population evolution to quickly find a further improved solution. First of all, through the research and analysis of the 2 DRP, a two-dimensional packing model is established to maximize the filling rate with relevant constraints. Then, the initial population is constructed by two-stage encoding, and the heuristic lowest horizontal line algorithm is used in the process of placement to decode the packing solution and calculate the individual fitness. After the termination rules of the design are satisfied in the iterative optimization process of HAGA, the packing scheme with the largest filling rate is selected in the feasible solutions. To verify the performance of the proposed algorithm, zero-waste and non-zero-waste instances are tested. The computational results show that the proposed HAGA is more efficient than the compared algorithms. The rest of this paper is organized as follows. In Section 2, the mathematical model of 2 DRP is built. The improved method, HAGA, for the 2 DRP problem is introduced in detail in Section 3. Then, Section 4 provides the computational results and comparisons. Finally, the conclusion of this paper is given in Section 5 with some closing remarks.