# Hybrid Genetic and Spotted Hyena Optimizer for Flow Shop Scheduling Problem
## 1 Introduction  
Combinatorial problems [1] involve finding the optimal arrangement, selection or sequencing of a finite set of elements based on specific constraints and objectives. They are prevalent in various fields, such as mathematics, computer science, engineering, operations research, economics and biology. The importance of solving combinatorial problems stems from their practical applications in decision-making, resource allocation and optimization tasks critical for the efficiency and effectiveness of real-world systems. Examples of combinatorial problems include the traveling salesman problem [2], which seeks the shortest route for a salesman visiting a set of cities once before returning to the starting point; the knapsack problem, which aims to select items with different weights and values to maximize the total value without exceeding a given weight limit; and the graph coloring problem, which assigns colors to graph vertices so that no two adjacent vertices share the same color. These problems, although seemingly different, share common challenges in terms of computational complexity, as they often require exploring an extensive solution space to identify the optimal solution. One specific combinatorial problem of significant practical importance is the Flow Shop Scheduling Problem (FSSP) [3]. In this problem, a set of jobs must be processed on a collection of machines in a specific order, with each job consisting of multiple operations executed sequentially. The objective is to find a schedule that minimizes the makespan or the total time required to complete all jobs. Due to the combinatorial nature of this problem, finding an optimal solution becomes increasingly difficult as the number of jobs and machines increases. Swarm intelligence algorithms [4] have emerged as effective approaches for tackling the optimization of makespan in the FSSP. These algorithms are inspired by the collective behavior of social organisms such as birds, fish, and insects, and their ability to solve complex problems through decentralized and self-organizing processes. Some widely used swarm intelligence algorithms for solving the FSSP include Particle Swarm Optimization (PSO) [5], Ant Colony Optimization (ACO) [6], Whale Swarm Algorithm (WOA) [7] and Rat Swarm Optimization [8]. In the context of the FSSP [9], these algorithms encode potential solutions as particles, ants or bees and iteratively explore the solution space by updating the position or path of these agents based on their own experiences and the collective knowledge of the swarm. The agents collaborate and compete, allowing the swarm to converge towards an optimal or near-optimal solution for the makespan. The effectiveness of swarm intelligence algorithms in solving the FSSP has been demonstrated in various industrial applications, such as manufacturing, transportation and logistics, where efficient job scheduling is critical for reducing production costs, improving resource utilization and enhancing overall performance. In this article, we propose a novel hybrid algorithm that combines the Spotted Hyena Optimizer (SHO) [10] and Genetic Algorithm (GA) [11] to solve the Flow Shop Scheduling Problem (FSSP). The FSSP is a particular case of the Flow Shop Scheduling Problem (FSSP), in which jobs are processed on a set of machines in a specific order, and all jobs follow the same sequence of operations on the machines. The primary objective is to minimize the makespan, which is the total time required to complete all jobs. The Spotted Hyena Optimizer (SHO) is a nature-inspired optimization algorithm based on the unique hunting behavior of spotted hyenas. It incorporates four main mechanisms: searching, encircling, attacking and mobbing. These mechanisms help explore the solution space efficiently and exploit the best solutions found during the search process. The Genetic Algorithm (GA) is a widely-used, population-based optimization method inspired by the principles of natural selection and genetics.