# A Hybrid Crow Search Algorithm for Solving Permutation Flow Shop Scheduling Problems
## 1 Introduction  
Several optimization methods have been proposed in artificial intelligence to find interesting patterns or optimization results for larger NP-hard optimization problems within a reasonable amount of time. To date, several real-world problems have been investigated to address this issue, such as data mining [1; 2; 3; 4], DNA fragment assembly [5; 6], DNA compression [7], Internet of Things [8; 9], knapsack problem [10], networks of evolutionary processors [11], scheduling [12], and traveling salesman [13]. Among these, the permutation flow shop scheduling problem (PFSP) has attracted significant attention and has important roles in scheduling research. The PFSP was first proposed by [PERSON] [14] in 1953, and has attracted extensive attention since then. Several theoretical, algorithmic, and computational research studies have addressed this problem. The basic concept of the PFSP is that \(n\) jobs need to be processed on a sequence of \(m\) machines, such that each job needs to be processed on all machines in the same order. [PERSON] addressed it as a two-machine problem. [PERSON] proved that the makespan minimization of the PFSP is an NP-hard problem [15]. For large problem instances, both the traditional dynamic programming approach and the heuristic approach require excessivecomputational time. To date, the Nawaz-Enscore-Ham (NEH) heuristic [16] is considered to be one of the most effective heuristic approaches for solving the PFSP; numerous variants of the NEH algorithm have been proposed, such as those in References [17; 18; 19; 20; 21; 22]. Metaheuristic approaches are often applied to solve NP-hard combinatorial optimization problems. Various metaheuristic algorithms [23] continue to attract increasing interest in optimization research. This type of research is typically inspired by natural behavior, and has yielded algorithms such as the artificial bee colony (ABC) [24], ant colony optimization (ACO) [25], cuckoo search (CS) [26], differential evolution (DE) [27], firefly algorithm (FA) [28], gravitational search algorithm (GSA) [29], particle swarm optimization (PSO) [30], and whale optimization (WA) [31]. Recently, the combinatorial optimization community has attempted to solve the PFSP by using metaheuristic approaches, such as those in References [32; 33; 34; 35; 36; 37]. One of the recently proposed swarm-based intelligence algorithms is the crow search algorithm (CSA) [38]. The CSA is based on the intelligent behavior of crows, who tend to store excess food in hiding places and retrieve it when required. Crows may try to follow other crows to steal their hidden food. The CSA simulates crow strategies with awareness probability and flight length parameters. The CSA provides an efficient search strategy for solving optimization problems. Furthermore, it is considered to be quicker than and superior to the PSO algorithm. The performance of a simple CSA is controlled by two parameters: the flight length \(fL\) and the awareness probability \(AP\). Although the CSA is better than PSO, similar issues occur; namely, it is necessary to avoid premature convergence and enhance diverse abilities [39; 40]. In addition, the original CSA is more suitable for continuous optimization problems, whereas little research has been done on transferring the approach to combinatorial optimization problems. It is a challenge to map the CSA to PFSP problems. Thus, in this paper, we propose a hybrid CSA (HCSA) approach to minimize makespan in PFSP problems. First, to make the CSA suitable for solving the PFSP, the smallest position value rule is applied to convert continuous numbers into job sequences. Our HCSA applies an NEH technique to create a population with excellent quality and diversity. Finally, to enhance the quality of the solutions and avoid premature convergence, simulated annealing (SA) [41] is combined with the variable neighborhood search (VNS) [23] to produce SA-VNS, which provides a local search to enhance the exploitation and exploration of HCSA.