## 3 Problem Definition and Solution Method
### Proposed Methodology
While these approaches provide structured optimization techniques, they also present significant limitations, particularly in large-scale and dynamic production environments. Mathematical programming models, such as MIP and the Time-Indexed LP Model, can theoretically provide optimal solutions but often become impractical for real-world applications due to their excessive computational requirements. As problem size increases, the solution time grows exponentially, making them unsuitable for large-scale industrial scheduling. The LP-Relaxation-Based Two-Stage Algorithm attempts to mitigate this issue by simplifying the problem space, but this comes at the cost of deviating from true optimality. Simulated Annealing and other heuristic-based approaches, while effective in certain conditions, may suffer from inconsistent performance and require extensive parameter tuning to achieve satisfactory results. In contrast, our approach employs Python-based dispatching rules, which offer significant advantages in terms of computational efficiency, adaptability, and scalability. Unlike complex mathematical models that demand high processing power, dispatching rules provide fast, heuristic-based scheduling decisions. Python's (version 3.11.5) robust ecosystem, including libraries such as NumPy, Pandas, and SciPy, allows for efficient data handling and rapid simulations, making it particularly well suited for large-scale scheduling problems. One of the key strengths of our method lies in its computational speed. Dispatching rules operate with minimal processing overhead, enabling real-time decision-making in production environments where quick responsiveness is crucial. Additionally, our approach implements 13 different dispatching rules, ensuring adaptability to various production scenarios without requiring extensive modifications. Moreover, Python's modular structure allows for seamless integration with existing production planning systems, facilitating a smooth transition from theoretical models to practical implementation. Despite these advantages, our approach is not without limitations. This study does not use the model to calculate LB-UB (lower and upper bounds), which could present an opportunity for integration with researchers working on exact models. Despite the \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Dispatching Rule** & **Definition** & **Advantages** & **Disadvantages** \\ \hline Weighted Shortest Processing Time with Due Date (WSPTDD) & Prioritizes jobs based on combination of processing time, weight, and due date, calculated as processing time multiplied by (due date/weight). & Balances due dates and processing times, optimizing a weighted objective. & Requires accurate weights and due dates for each job, computationally intensive. \\ \hline Critical Ratio (CR) & Prioritizes jobs based on the critical ratio, calculated as time until due date divided by processing time. & Helps in managing jobs close to their due dates and balances processing times and due dates. & May not prioritize jobs effectively if processing times and due dates vary widely. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._advantages of using dispatching rules, the absence of LB-UB calculations means that global optimality cannot be guaranteed. Furthermore, handling complex constraints--such as machine-specific priorities, maintenance schedules, and intricate sequencing requirements--is more straightforward in mathematical programming approaches than in dispatching-rule-based heuristics. In conclusion, our Python-based dispatching rule approach offers significant advantages in terms of speed, adaptability, and real-world applicability while avoiding the computational burdens associated with traditional optimization models. However, to further enhance its effectiveness, a hybrid approach that integrates heuristics with advanced optimization techniques could be explored to address its limitations, particularly in terms of optimality and complex constraint handling.