# Hybrid Flow Shop Scheduling Problems Using Improved Fireworks Algorithm for Permutation
## 1 Introduction  
Effectively solving scheduling problems is a topic in the optimization field [1]. There are scheduling problems in all walks of life, such as medical resource allocation, power system scheduling, wireless network optimization, and electric vehicle scheduling [2; 3; 4; 5; 6]. Production scheduling is a common scheduling type and an important part of industry optimization scheduling. A good scheduling strategy contributes to production efficiency and enables the entire production process to be executed successfully. Production scheduling has been widely studied to consider the increasing scale of production and intense market competition [7].  
In the process of production scheduling, the producers arrange a set of available production resources to complete the production plan. During the process, the jobs are assigned to the correspondingmachines. The job sequence and operation time on each machine are determined to achieve one or more performance indexes [8]. Flow shop scheduling is a common type of production scheduling that has been adopted in many large-scale industrial productions. The permutation flow shop scheduling problems (PFSP) is a type of typical combinatorial optimization problem that exists widely in the field of industrial automation [9]. The permutation flow shop is regarded as a simplified model of numerous actual flow production lines, and it is suitable for mass production. A hybrid flow shop consists of multiple parallel machines at each stage, where the processing situation is more complicated than the permutation flow shop [10]. As the extension of the PFSP, the hybrid flow shop scheduling problem (HFSP) can flexibly adapt to the actual production conditions [11]. The solution methods for these two types of problems help production firms find the optimal scheduling strategy. In view of this, two kinds of problems were studied.  
The flow shop scheduling problem (FSP) was developed for focusing on mathematical programming, multi-objective optimization, etc. [12]. These methods are often used to find the optimal solution, while it is insufficient to handle large-scale issues [13]. It is shown that the FSPs belong to a class of NP-hard problems when the scale is more than three and the optimization algorithms with polynomial time have not yet been found [14]. For these typical scheduling problems, it is difficult to obtain satisfactory results by the traditional optimization methods. Finding efficient methods for solving FSPs has always been an important topic in academia and industry. The intelligent algorithm obtains the optimal schedule in a short time by searching the neighborhood of the solution and improving the current solution continuously. At present, the trend of study is to solve FSPs with intelligent algorithms. The fireworks algorithm (FWA) is an intelligent algorithm inspired by the rule of fireworks explosion and has shown good performance in solving complex optimization problems [15]. Through the explosion and mutation of fireworks, the FWA has strong global search ability, but also has disadvantages such as premature convergence and cumbersome calculation. In this study, an improved fireworks algorithm (IFWA) is put forward for enhancing the performance of FWA and solving the two scheduling problems.  
The improvement involves three processes, including explosion, mutation, and selection, compared with the original algorithm. The results confirm the performance of the improved algorithm by solving two instances from the FSP. The rest of the study is structured as follows: In Section 2, the relevant approaches for FSP and improvement strategies for the FWA are introduced. Section 3 introduces the FWA and IFWA. In Section 4, the FSP and HFSP are described in detail. In Section 5, two instances are given to test the IFWA's performance in solving the FSP and the results are discussed. Finally, the conclusions and future studies are discussed.