## 3 Problem Definition and Solution Method
### Proposed Methodology
\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 \multirow{6}{*}{Apparent Tardiness Cost (ATC)} & Prioritizes jobs based on a calculated cost of apparent tardiness, which considers the difference between the job’s current completion time and its due date. & Helps in minimizing overall tardiness and meeting due dates. & Requires accurate estimation of apparent tardiness costs and may not consider other factors such as job priorities or processing times. \\ \cline{2-4}  & Prioritizes jobs based on their due dates, with jobs having earlier due dates processed first. & Effective for minimizing the maximum lateness of jobs. & May lead to suboptimal utilization of machines if jobs with later due dates could be processed earlier. \\ \hline \multirow{6}{*}{Shortest Processing Time (SPT)} & Prioritizes jobs based on the shortest processing time, with shorter jobs processed first. & Reduces the average completion time of jobs. & Can lead to starvation of longer jobs, especially if there are many short jobs. \\ \cline{2-4}  & Prioritizes jobs based on time, calculated as processing time divided by job weight. & Useful in balancing the load across machines when job lengths vary significantly. & May increase the average completion time of jobs and can delay short jobs. \\ \hline \multirow{6}{*}{Weighted Shortest Processing Time (WSPT)} & Prioritizes jobs based on their weighted processing time, calculated as processing time divided by job weight. & Minimizes the weighted sum of completion times. & Requires accurate weights for each job and can be computationally intensive. \\ \cline{2-4}  & Prioritizes jobs based on their slack time, calculated as due date minus (current time plus processing time). & Reduces the risk of job lateness by prioritizing jobs close to their due dates. & May deprioritize already late jobs, making them close to their due dates. & May deprioritize already late jobs, making them even later. \\ \hline \multirow{6}{*}{Minimum Slack (MS)} & Prioritizes jobs based on their slack time, calculated as the maximum of zero or due due plus (current time plus processing time). & Ensures that late jobs are still given priority, reducing maximum lateness. & Similar to FSTLP but may not always optimize other performance metrics. \\ \cline{2-4}  & Prioritizes jobs based on their flexibility, calculated as due date minus release date. & Effective in environments where job due dates are critical. & Jobs with more flexibility may be unnecessarily delayed. \\ \cline{2-4}  & Processes jobs in the order they arrive, based on release date. & Simple and fair, easy to implement. & May not optimize any specific performance metric and can lead to inefficients. \\ \cline{2-4}  & Processes the most recently arrived job first. & Can be useful in specific scenarios where recent jobs are more urgent. & Rarely used due to potential for high lateness and job starvation. \\ \hline \multirow{2}{*}{Random Scheduling} & Jobs are processed in a random order. & Simple to implement, useful for testing. & Inefficient and may lead to suboptimal scheduling performance. \\ \cline{2-4}  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: 13 Dispatching rules used in MIU. Each dispatching rule has its own trade-offs and is suited to different production environments and objectives. Choosing the right rule depends on factors such as job characteristics, machine capabilities, and production goals. Using multiple dispatching rules at the same time will cover each other's disadvantages and ensure that the most advantageous one is selected. In the literature, various resolution methods have been employed to address predictive-reactive rescheduling problems in parallel-machine environments, including Linear Mathematical Models [55], Mixed Integer Programming (MIP) [26], LP-Relaxation-Based Two-Stage Algorithms, the Time-Indexed LP Model [4], and Simulated Annealing Algorithms [2].