## 3 Problem Definition and Solution Method
### Proposed Methodology  
The problem has two stages. The first stage is to predict, and the second stage is to decide what will happen after the failure. In the literature, there are studies that use the same algorithm for both parts, as well as studies that apply different solution methods for both stages. In the first stage, an initial schedule is created, and in the second stage, rescheduling activities are carried out after the failure. A schedule will be created for the first stage, and periodic updates to the schedule will be allowed. The mathematical model created can be used at the beginning of production, without taking into account any fluctuations in processing times or rework, by using only the expected processing times values when creating the initial scheduling. Even in this case, our problem is NP-hard. The minimizing total weighted tardiness (TWT) problem involves scheduling a set of jobs, each with a given processing time, weight, and due date, such that the total weighted tardiness is minimized. The tardiness of each job is defined as the difference between its completion time and its due date, if the job is completed late. The Job-Shop Scheduling Problem (JSSP), the well-known NP-hard problem of scheduling, can be reduced to TWT. In the JSSP, a set of jobs must be scheduled on a set of machines, with each job having a predefined sequence of operations. The objective in the JSSP is to minimize the makespan,the total time required to complete all jobs. To reduce the JSSP to TWT, one conducts the following:  
* Each operation in the JSSP corresponds to a job in TWT. * The processing time of each operation in the JSSP is mapped directly to the processing time of the corresponding job in TWT. * The weight of all jobs in TWT is set to one (i.e., no job has a higher priority than another). * The due date for each job in TWT is set to the completion time of the corresponding operation in the JSSP. By this reduction, solving the TWT problem minimizes the total weighted tardiness, and the tardiness of a job in TWT corresponds to the delay in completing a job beyond its due date, which is determined by the JSSP instance. Thus, an optimal solution to TWT also provides an optimal schedule for the JSSP, proving that TWT is NP-hard. Similarly, the minimizing number of tardy jobs (NTJ) problem involves scheduling jobs so that the number of tardy jobs is minimized, which can also be reduced from the JSSP:  
* Each operation in the JSSP corresponds to a job in the NTJ. * The processing time of each operation in the JSSP is mapped to the corresponding job's processing time in the NTJ. * The due date for each job in the NTJ is set to the completion time of the corresponding operation in the JSSP. The goal of the NTJ is to minimize the number of tardy jobs, which is equivalent to minimizing the number of jobs whose completion time exceeds their due date. This objective directly relates to minimizing the number of operations that are completed after their due dates in the JSSP instance. Therefore, an optimal solution to the NTJ will correspond to an optimal solution for the JSSP, proving that the NTJ is NP-hard. When actual production times deviate from expected values or rework operations become necessary, the continuous optimization model must be re-executed using updated data. This adjustment is essential, as machine availability changes dynamically with each completed job, requiring real-time recalibration of the schedule. This makes the structure of the problem even more difficult. The fact that the structure of the problem is NP-hard and the amount of disturbance is very high has caused the need for a new methodology that is practical and gives good results. The methodology we developed is called Multi-Purpose Intelligent Utility (MIU). This methodology provides fast and good results to the targeted strategy based on a large number of distribution rules. Each dispatching rule has its own advantages and disadvantages.