As the manufacturing industry rapidly shifts toward high-mix, large-scale production and extreme personalization, scheduling optimization has become significantly more complex than before. Existing search-based methods such as mathematical optimization, metaheuristics, and constraint programming face fundamental limitations in handling the high variability and uncertainty that characterize real-world manufacturing environments. Deep neural network technologies are emerging at the forefront. When coupled with the human-centric manufacturing paradigm of Industry 5.0, there is growing anticipation that Large Language Models (LLMs), capable of interpreting constraints and objectives expressed in natural language, could lead to more flexible and sustainable manufacturing ecosystems. In this study, we propose a method for solving manufacturing system scheduling optimization by utilizing an LLM, a form of unsupervised deep neural networks. Specifically, we focus on the Job Shop Scheduling Problem (JSSP), known to be an NP-hard task, and introduce a process wherein the JSSP is represented in natural language and then trained with an LLM using Low-Rank Adaptation (LoRA) and an enhanced prompting technique. Our results show that this approach achieves scheduling performance comparable to existing neural network methods, thereby suggesting the potential for LLM-based scheduling optimization to serve as a key tool in human-centric manufacturing under Industry 5.0.