# Hybrid Flow Shop Scheduling Problems Using Improved Fireworks Algorithm for Permutation
## 2 Literature Review  
Methods for solving FSPs are divided into three categories: exact algorithms, construction algorithms, and intelligent algorithms. Exact algorithms accurately obtain the optimal schedule, while they are greatly affected by the scale of the problem and the amount of calculation. Exact algorithms mainly include integer programming, the branch and bound method, and other methods. [PERSON] et al. [16] used the branch and bound algorithm to study a class of two-stage HFSPs below 20 jobs. However, the applicability of the method is limited as the number of jobs increases. [PERSON] et al. [17] proposed different mixed integer programming equations to solve an FSP with up to 100 jobs. In a sense, these methods belong to the enumeration method and unnecessary calculations are excluded based on the operation results. However, the computational cost of these algorithms increases exponentially with the increase of jobs. In general, exact algorithms are suitable for small-scale scheduling problems, while they are insufficient to solve large-scale scheduling problems. Heuristic algorithms for obtaining a feasible or the suboptimal solution have been rapidly developed, considering the huge cost of getting an optimal solution. A constructive heuristic algorithm can construct the solution of the problem through certain rules and quickly obtain the solution of the problem. However, the construction rules are usually complicated, and the solution obtained is not satisfactory. Construction algorithms mainly include the Palmer method, the Nawaz-Enscore-Ham method, the Gupta method, etc. [18; 19; 20]. The Johnson rule obtained good results when dealing with an FSP involving two machines [21]. It laid a good foundation for solving FSPs with multiple machines by using heuristic rules later. The Campbell-Dudek-Smith rule is an extended form of the Johnson rule, which is often used to solve scheduling problems involving multiple machines [22]. The Nawaz-Enscore-Ham method is regarded to be the most effective construction algorithm. [PERSON] and [PERSON] [23] proved that other heuristic algorithms do not perform as well as the Nawaz-Enscore-Ham algorithm in dealing with PFSPs. [PERSON] and [PERSON] [24] proposed a new constructive heuristic algorithm named the modified Johnson's algorithm, which applied the Johnson rule and a pair-splitting strategy. The performance of the modified [PERSON]'s algorithm was better than those of three other classic heuristics ([PERSON], Campbell-Dudek-Smith, and Gupta), and its computational cost is made much lower by verifying the benchmark function. A main method of solving FSPs is the use of intelligent optimization algorithms, which can obtain optimal solutions with a high probability in a reasonable time. Several intelligent algorithms, such as the evolutionary algorithm [25], particle swarm optimization (PSO), grey wolf optimizer, firefly algorithm, and ant colony optimization algorithm have been extensively used [26; 27; 28; 29]. [PERSON] and [PERSON] [30] introduced a heuristic rule to convert the continuous position values into discrete arrangement values in the optimization problem to solve the FSP based on the Cuckoo algorithm. The application of the algorithm can effectively reduce the completion time and average processing time. [PERSON] et al. [31] introduced the scheduling rules and heuristic methods in the monkey search algorithm. The method has been proven suitable to FSP compared with other approaches. [PERSON] et al. [32] adopted the Hill function to redefine the inertia coefficient of the velocity update equation in the PSO, which can avoid the premature convergence. The application of the method can effectively improve energy efficiency in HFSPs. [PERSON] et al. [33] put forward a new crossover and mutation strategy on the traditional backtracking search algorithm, which can improve the diversity of solutions and improve the ability to solve discrete problems.