## 1 Introduction  
Optimizing machine allocation and workforce scheduling is critical for minimizing operational costs in industries such as automotive, textile, and electronics. However, traditional scheduling models often overlook key factors such as machine interference and the cyclical nature of production demands. This study proposes a novel bin-packing-based approach to determine the optimal number of machines and operators in cyclic manufacturing environments. By systematically incorporating production constraints, the model addresses limitations in existing heuristic methods and significantly improves resource allocation efficiency. Many industries, including food, textiles, plastics, electronics, and rubber, face significant manufacturing challenges, such as determining the number of production machines, operators, and production scheduling. A paint factory may produce multiple types of paint, each considered a distinct product; a food processing facility produces items with different packaging, colors, and shapes; and a textile factory may produce clothing in multiple colors and sizes. These challenges highlight the need for an optimal manufacturing strategy in production environments where multiple product types are manufactured using several nearly identical machines with limited capacity; efficient workforce management and cyclic production planning are essential. The primary goals in such settings include optimizing the number of machines and operators, establishing a production schedule that aligns with these objectives, and ensuring the timely fulfillment of customer orders. These considerations become particularly important when skilled labor is scarce to operate costly machinery and equipment. A key feature of such production systems is that a single operator manages multiple, nearly identical machines. Assessing the efficiency of these systems requires analyzing the average cycle time, which consists of machine runtime, service time, and downtime. Downtime can occur when one or more machines require setup or service, but the operator is busy servicing another machine, leaving the machines requiring service idle until the operator becomes available. This phenomenon is referred to as machine interference [1]. Integrating machines and human labor is crucial in modern manufacturing, directly impacting efficiency, precision, and production speed. As technology evolves, the interaction between machines and human operators becomes increasingly sophisticated, emphasizing the need for a well-balanced approach that enhances productivity while supporting workforce development. The optimization of machine scheduling is one of the most critical issues in modern manufacturing, and many studies have been conducted on optimizing production schedules [2; 3]. In this context, determining the optimal number of machines required to meet the total demand for various products and the number of operators needed for those machines is a challenging problem. This paper analyzes the problem of minimum machines and operators as a combinatorial optimization closely related to the bin packing problem (BPP), a well-known NP-hard problem involving efficiently allocating items to a finite number of bins. In our proposed model, the machines are bins containing (produce) several product types, and the operators are bins containing (operate) several machines. The bin packing problem is one of the most widely studied problems in combinatorial optimization [4] and the references there. In this study, repetitive (cyclic) production refers to a planning framework where all product types are manufactured in repeating cycles within a fixed time horizon, such as a week or a month. Each product is produced in batches that exactly match its demand for the cycle, and the production sequence repeats identically in each cycle. This cyclic structure simplifies scheduling and capacity planning and reflects real-world industrial practices in environments with stable and recurring customer orders.