## 3 Problem Definition and Solution Method
### Mathematical Model  
The following mathematical model is suggested to find the solution the company is looking for in periodic meetings. Here, two separate objective functions are given at once. The company should look for the objective function solution it wants according to the strategy it has determined.  
Data sets and indices are as follows:  
\(N\): Work Set (1, 2, \(n\));  
\(K\): Row Set (1, 2, \(n\));  
\(M\): Machine Set (1, 2, \(m\));  
\(J\): work index \(j\) = 1, 2, \(n\);  
\(K\): row index \(k\) = 1, 2, \(n\);  
\(I\): machine index \(i\) = 1, 2, \(m\).  
Parameters are as follows:  
\(W_{j}\) = weight of job \(j\);  
\(R_{j}\) = release date of job \(j\);  
\(P_{j}\) = processing time of job \(j\);  
\(D_{j}\) = due date of job \(j\).  
Decision variables are as follows:  
\(X_{\text{jki}}\) = {1: if job \(j\) is assigned to row \(k\) of machine \(i\); 0: o.w.};  
\(CT_{ki}\): completion time of job in row \(k\) on machine \(i\);  
\(ST_{ki}\): starting time of job in row \(k\) on machine \(i\);  
\(T_{j}\) = max (\(CT_{j}-d_{j}\); 0);  
\(U\) = \(\sum_{k=\ 0}^{n}\delta(T_{I})\)\(\delta(x)\) = \(\left\{\begin{array}{ll}1,&x>0;\\ 0,&o.w\end{array}\right.\)  
Objective Functions are as follows:  
Min \(\sum_{j=1}^{n}w_{j}T_{j}\): minimizing total weighted tardiness;Min \(U\): minimizing the number of tardy jobs.  
Constraints are as follows:  
\[\sum_{i\ =\ 1}^{m}\sum_{k\ =\ 1}^{n}x_{Jki}\ =\ 1\qquad\forall j\ \in N \tag{1}\]  
\[\sum_{k\ =\ 1}^{n}x_{Jki}\leq 1\qquad\qquad\forall k\ \in K\ \ \forall i\ \in M \tag{2}\]  
\[ST_{ki}\geq\sum_{j\ =\ 1}^{n}r_{Jki}\qquad\qquad\forall k\ \in K\ \ \forall i\ \in M \tag{3}\]  
\[ST_{(k+1)i}\geq ST_{ki}+\sum_{j\ =\ 1}^{n}p_{J}x_{Jki}\qquad\forall k\ =\ 1,\ldots n-1\ \ \ \forall i\ \in M \tag{4}\]  
\[CT_{(k+1)i}\geq ST_{ki}+\sum_{j\ =\ 1}^{n}p_{J}x_{Jki}\ \ \forall k\ \in K\ \ \forall i\ \in M \tag{5}\]  
\[x_{ji}\ =\ \{0,1\}\qquad\forall k\ \in K\ \ \forall i\ \in M\ \forall j\ \in N \tag{6}\]  
\[ST_{ki},\ CT_{ki}\geq 0\ \forall k\ \in K\ \ \forall i\ \in M \tag{7}\]  
**Assignment Constraint (1):** Each job must be assigned to exactly one position in the schedule. This ensures that all jobs are scheduled without duplication or omission.  
**Uniqueness Constraint (2):** Each position can be occupied by at most one job. This prevents multiple jobs from being assigned to the same position.  
**Release Time Constraint (3):** A job cannot start before its predefined release time. This ensures that production does not begin prematurely.  
**Sequencing Constraint (4):** Ensures that when a job is scheduled in a sequence, the next job can only start after the completion of the previous one. This maintains the order of execution.  
**Completion Time Constraint (5):** Defines the completion time of a job as the sum of its start time and processing duration. This ensures that the model correctly accounts for processing times.  
**Variable Definitions (6):**\(x_{ji}\) is a binary variable ensuring that job assignments follow the given constraints.  
**Non-Negativity Constraints (7):**\(ST_{ki}\) and \(CT_{ki}\) are non-negative, ensuring a feasible scheduling timeline.