# Hybrid Genetic and Spotted Hyena Optimizer for Flow Shop Scheduling Problem
## 3 Flow Shop Scheduling Problems (FSSPs)  
Flow Shop Scheduling Problems (FSSPs) is a class of scheduling problems that arise in manufacturing and production systems, where a set of jobs or tasks must be processed through a series of machines in a specific order. FSSPs is a well-studied optimization problem in the field of operations research and has numerous applications in various industries, such as automotive, semiconductor, food processing and textile production, among others. In a typical flow shop environment, there are multiple machines or workstations, and each job must be processed on every machine in a predetermined sequence. The main objective of the FSSP is to determine the optimal scheduling of jobs on machines to achieve certain performance criteria, such as minimizing makespan, total completion time, total tardiness or a combination of these objectives. The importance of flow shop scheduling problems lies in their ability to optimize the utilization of resources, reduce production costs and improve overall efficiency in manufacturing systems. Effective flow shop scheduling can lead to:  
* Reduced production lead time: By optimizing the job sequence and minimizing the idle time of machines, flow shop scheduling can significantly reduce the total production time. * Improved resource utilization: Efficient scheduling ensures that machines are utilized optimally, reducing idle time and maximizing production throughput. * Enhanced customer satisfaction: Timely delivery of products and shorter lead times can increase customer satisfaction and help to maintain a competitive edge in the market. * Lower inventory costs: By reducing work-in-process inventory and minimizing production time, flow shop scheduling can help lower inventory holding costs. * Increased competitiveness: Effective flow shop scheduling allows companies to be more agile and responsive to market demands, thus enhancing their competitive position. Despite its importance, FSSP is a challenging combinatorial optimization problem known to be NP-hard, meaning that finding an optimal solution becomes increasingly difficult as the problem size increases. As a result, researchers have developed various heuristic and metaheuristic algorithms to find near-optimal solutions for the FSSP in a reasonable amount of time, such as particle swarm optimization, genetic algorithms, simulated annealing and ant colony optimization, among others. These methods have been successfully applied to tackle real-world flow shop scheduling problems, resulting in significant improvements in manufacturing efficiency and cost reduction. The Fob-Shop Scheduling Problem (FSSP) involves assigning a set of n jobs, each consisting of multiple operations, to a set of m machines. The primary objective is to find a schedule that minimizes the makespan (\(Cmax\)), which is the total completion time of all jobs. A solution can be represented as an n \(\times\) m vector of operation sequences that optimizes the completion time. \[C_{\max}=\max(t_{ij}+p_{ij}) \tag{1}\] \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Algorithm** & **Approach** & **Operators** & **Population Size** & **Selection** \\ \hline GA-SHOA & Hybrid & Uniform Crossover, Mutation & 50–100 & Roulette Wheel \\ SSO & Nature-inspired & Randomization, Clustering & 20–50 & Roulette Wheel \\ SCE-OB & Evolutionary & Crossover, Mutation & 50–200 & Rank-based \\ CLS-BFO & Metaheuristic & Local search, randomization & 50–100 & Roulette Wheel \\ ACGA & Hybrid & Adaptive Crossover, Mutation & 50–100 & Rank-based \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of optimization algorithms for the FSSP. \[min(C_{nm}+1)\] (2) where \[C_{kl}\leq\ C_{ji}-d_{kl};\ j=1,\ldots,n;\ i=1,\ldots,m;\ kl\ \in\ P_{ji}\] (3) \[\sum_{jieo(t)}^{n}r_{ji}\leq\ 1;\ i\ \in\ M;\ t\ \geq\ 0\] (4) \[C\_{-ji}\geq\ 0;\ j=1,\ldots,n;\ i=1,\ldots,m\] (5) The constraints are as follows: * Constraint (2) minimizes the finish time of the operation on machine \(m+1\) (the makespan).