In contemporary manufacturing systems, the extension of the traditional job-shop scheduling problem (JSP) to the flexible job-shop scheduling problem (FJSP) is of significant importance. FJSP introduces more complexities compared to the traditional JSP by allowing each operation the flexibility to be executed on any of the machines from a designated group, and is acknowledged as NP-hard. In order to effectively solve FJSP and minimize the maximum completion time, a new method called Competitive domain Genetic Algorithm (CNGA) is proposed. CNGA merges the broad-reaching search features of genetic algorithms (GA) with the focused search strengths of domain search, enhanced by a competitive mechanism, to effectively balance exploration and exploitation. CNGA’s design includes efficient coding strategies, innovative genetic operators, and dynamic neighborhood structures that are specifically tailored to address the complexity and needs of FJSPS. The algorithm introduces a competition mechanism to enhance the diversity and quality of solutions. Through this mechanism, multiple solution candidates can compete with each other in the iterative process, so as to improve the possibility of finding the global optimal solution. To verify the performance of CNGA, this study tested the algorithm using two well-known FJSP benchmark examples, including 16 open questions, and compared it with some algorithms. The results show that CNGA is excellent in solving FJSP problems, and its superiority is better in solving quality, which provides a new solution for complex scheduling problems in manufacturing field.