# An Improved Genetic Algorithm with Adaptive Variable Neighborhood Search for FJSP
## 1 Introduction  
Along with the rapid development of modern manufacturing, science, and technology, consumers' demands are becoming more and more personalized and customized. The production mode of the manufacturing companies is gradually shifting from continuous, large quantity production to multi-variety and small quantity production. The flexible job-shop scheduling problem (FJSP) embodies the flexibility of production, that is, the different operations of each job in FJSP can be processed on different machines with different processing times. The job can select the machine according to the actual situation, such as workload and the idle state of the machine. It is more suitable for actual production practice activities and has certain theoretical and practical value for its research. The FJSP problem was first proposed by [PERSON] and [PERSON] in 1990. In recent years, many scholars have achieved certain results in solving the FJSP problem. [PERSON] et al. [1] proposed the method of combining multi-objective particle swarm optimization and variable neighborhood search to solve the FJSP problem effectively. [PERSON] et al. [2] proposed an improved quantum genetic algorithm based on the earliness/tardiness penalty coefficient. The double penalty coefficient is designed in the algorithm, and a double-chain coding method was proposed. At last, the standard example demonstrates the effectiveness of the algorithm. [PERSON] et al. [3] used a combination of tabu search (TS) and genetic algorithm (GA) to optimize the two goals of maximum completion time and cost reduction. A hybrid non-dominated sorting simulated annealing algorithm for FJSP was proposed by [PERSON] et al. [4], in which critical or incapable machines were eliminated by non-dominated sorting for all operations and simulated annealing was used to search for an optimal solution. [PERSON] et al. [5] solve the flexible job-shop problem with a multi-agent model based on mixed-element heuristics. [PERSON] et al. [6] proposed a bilevel neighborhood search hybrid algorithm to solve FJSP, which achieved remarkable results. [PERSON] et al. [7] proposed a new genetic algorithm (NGA) for FJSP, which used a new chromosome representation and different crossover and mutation strategies and obtained better experimental results. [PERSON] et al. [8] proposed a hybrid genetic algorithm (HGA), which used the Taguchi method to optimize the parameters of the genetic algorithm, and in the paper, a novel coding mechanism was proposed to solve the problem of invalid work assignment. [PERSON] et al. [9] proposed an effective memetic algorithm (MA), which combines the local search ability of TS with the global search capability of GA to provide an effective way to \"explore\" and \"exploit.\" [PERSON] et al. [10] developed a hybrid evolutionary algorithm (HEA), which proposed a new operator based on the average Hamming distance to maintain population diversity, adopted different crossover and mutation strategies, and introduced the critical path theory and local search process of the preferred strategy to improve the convergence speed. [PERSON] et al. [11] proposed an efficient hybrid algorithm (HA), which combines the genetic algorithm (GA) and tabu search (TS), and this algorithm uses TS with a good local search ability to mine, which balances the intensity and diversity of the scheduling. In view of these, this paper proposes an improved genetic algorithm with an adaptive variable neighborhood search (IGA-AVNS) for solving FJSP. Genetic algorithm (GA) is characterized by parallelism, adaptiveness, and self-learning, whose principle is simple, robust, versatile, has good global search performance, and can quickly find most solutions in the solution space. However, a single GA has a poor local search capability, so the entire search process is time-consuming and the efficiency of the algorithm is not high. However, variable neighborhood search (VNS) [12; 13; 14] can make up for the above defects and has a strong local search ability.