## 1 Introduction
Although researchers have developed many algorithms to solve FSSPs, there still exist some shortcomings in these algorithms, such as local optimization and high computational cost, especially when solving large-scale problems. Table 1 summarizes some of the algorithms proposed for FSSPs in the literature. Most of these studies demonstrate that combinational heuristic and metaheuristic approaches which include advantages of more than one algorithm are very useful for solving flow shop problems. For instance, [PERSON] et al. [18] studied a hybrid genetic algorithm for a no-wait flow shop problem. [PERSON] et al. [19] presented an improved iterated greedy algorithm with a tabu-based reconstruction strategy. An iterated greedy algorithm in [20], a high-performing memetic algorithm in [21], and a discrete self-organizing migrating algorithm in [22] have been used for flow shop problems with the minimum makespan criterion and obtained better results. [PERSON] et.al [23] proposed a hybrid combinatorial particle swarm optimization (HCPSO) algorithm as a resolution technique for solving this problem. [PERSON] and [PERSON] [24] proposed a new multiparent operator which could greatly improve the performance of a GA search. The GA and the simulated annealing (SA) algorithm are well-known metaheuristics that have been successfully used in flow shop problems. The genetic algorithm has the advantages of strong global optimization ability, fast speed, strong versatility, and easy implementation. However, it has the drawback of poor local search ability, which decreases search efficiency, especially in the late period of optimization. On the other hand, the simulated annealing algorithm has strong local search capabilities to make up the shortcomings of the genetic algorithm. GA + SA has been used to solve job shop [25], open shop [26], and flexible flow shop problems [27]. However, to the best of our knowledge, there are no reports on its application in flow shop scheduling, especially with improved crossover and mutation operators and adaptive simulated annealing. Therefore, we propose a hybrid genetic simulated annealing (HGSA) algorithm to solve flow shop scheduling problems. The main contributions of this paper include the following:  
1. We propose the novel HGSA algorithm to solve FSSPs. It is characterized by an operational coding in GA and a hormone regulation mechanism in the simulated annealing part of the algorithm. The MME [28] algorithm, combined with the NEH [5] and MinMax (MM) [29] heuristic algorithms, is used for initialization of the population. We use location-based intersection and two-point intersection for crossover operations with either one of these two being randomly selected. In the mutation process, twors mutation or inversion mutation [30] is randomly employed to mutate the population. After the crossover and mutation operation is completed, the best individuals are retained, and the simulated annealing operation is performed on these solutions. 2. Using the widely adopted Taillard benchmark FSSPs, we conducted extensive experiments and showed that our HGSA algorithm achieved better results than the baseline algorithms to which it was compared in our study. We showed that our HGSA algorithm's high performance for FSSPs is based on its hybrid search strategy, twors/inversion mutation, location-based/two-point crossovers, and their combination with the MME heuristic algorithm for population initialization. The rest of this paper is organized as follows: Section 2 introduces the definition of the problem and mathematical model of the flow shop scheduling problem, Section 3 describes the HGSA, Section 4 shows the experimental results of HGSA, and Section 5 provides conclusions and future works.