## 5 Summary of Conclusions and Future Research  
This study examined a manufacturing scenario with multiple product types, each with a different production rate and a predetermined, cyclical demand. All the products can be produced using identical machines operated by several operators. Each machine can produce any product, and changing between product types requires an operator's fixed time for a setup process. The goal is to determine the minimum number of machines and operators needed to meet production requirements while ensuring that all product types are produced in every cycle. The problem is classified as NP-hard because it is analogous to the BPP with conflicts.  
To tackle this challenge, we developed a two-stage algorithm. In the first stage, the algorithm identifies the minimum number of machines needed to meet production requirements and creates a production schedule for each machine. This process includes determining the order in which different product types will be produced on each machine, utilizing a well-known heuristic algorithm. In the second stage, the algorithm calculates the number of operators required to carry out the setup processes defined in the first stage. This stage is also addressed using heuristic methods. We examined both the upper and lower bounds to evaluate the quality of the solution and assess its potential optimality.  
The algorithm was validated using a real-world test case. In the first stage, the heuristic algorithm identified the optimal number of machines needed. The second stage presented some complexity, as it is a special case of the BPP problem with constraints. We illustrate the solution to the second-stage problem using rings to demonstrate that overlapping is prevented.  
The model provides practical tools for industrial managers seeking to reduce costs and optimize production under resource constraints. Its adaptability to various manufacturing contexts, including industries facing high product variety and cost pressures, highlights its strategic value. Although this study focuses on a manufacturing system, the model can also be applied to other contexts.  
Limitations and Future Research Although the proposed algorithm yields promising results and demonstrates practical applicability, it has several limitations. It is a heuristic approach that does not guarantee optimal solutions in all instances. Additionally, it assumes a fixed production sequence and does not allow for the reallocation of idle time, which may reduce its flexibility in more dynamic or uncertain production environments.  
Future research should aim to enhance the algorithm's adaptability and performance by exploring hybrid metaheuristic approaches or integrating dynamic scheduling capabilities. Another direction involves applying the model in scenarios where production quantities are optimized based on trade-offs among inventory holding costs, setup costs, and limited storage capacity. Moreover, extending the model to non-manufacturing domains--such as healthcare, logistics, and services--could further validate its versatility and broaden its impact. These efforts would contribute to developing a more robust and generalizable framework for resource optimization in cyclic systems.  
Conceptualization, Y.H. and B.K.; methodology, Y.H. and B.K.; software, B.K.; validation, Y.H. and B.K.; formal analysis, B.K.; data curation, Y.H. and B.K.; writing--original draft preparation, Y.H.; writing--review and editing, Y.H. and B.K.; visualization, Y.H.; supervision, Y.H.; project administration, Y.H. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.