## 3 Mathematical Modeling
### Multi-Objective Model for FJSP-LS
#### 3.2.6. Maximum and Total Sublot Finish-Time Separation (\(Z_{6}\) and \(Z_{7}\))  
In lot streaming, subplots are treated independently. As a result, one sublot of a job may be finished much sooner than the other sublot of the same job. This may increase work-in-process inventory as the entire job can not be made available for shipment or assembly within a reasonable time window. Hence, in this research, we introduce an objective function to minimize the gap between the earliest and the latest finish-times of subplots of the same job. In doing so, first we defined a variable \(\hat{g}_{j}\) that assumes the earliest finish-time among all the subplots of a job, \(\min_{\forall(s,j)\mid\gamma_{sj}=1}\{d_{s,j}\}\), as enforced by the constraint in Equation (39) and the objective function. The latest finish-time of the subplots of a job is its departure time \(\hat{d}_{j}\), which was discussed previously. With these variables defined, the sublot finish separation time of a job, \(\hat{h}_{j}\), is the difference \(\hat{d}_{j}-\hat{g}_{j}\), enforced by the constraint in Equation (40). The constraint in Equation (41) and the objective function will enforce \(\hat{h}_{max}\) to assume the value \(\max_{\forall j}\{\hat{h}_{j}\}\). The total sublot finish time separation \(\hat{h}_{total}\) is evaluated using the constraint in Equation (42). The objective function terms \(Z_{6}\) and \(Z_{7}\) correspond to the values of \(\hat{h}_{max}\) and \(\hat{h}_{total}\), respectively, as enforced by the constraints in Equations (43) and (44). \[\hat{g}_{j}\leq d_{s,j}+\Omega(1-\gamma_{s,j});\ \ \forall(s,j) \tag{39}\]  
\[\hat{h}_{j}=\hat{d}_{j}-\hat{g}_{j};\ \ \forall(j) \tag{40}\]\[\hat{h}_{max}\geq\hat{h}_{j};\ \ \forall(j) \tag{41}\]  
\[\hat{h}_{total}=\sum_{j=1}^{I}\hat{h}_{j} \tag{42}\]  
\[Z_{6}=\hat{h}_{max} \tag{43}\]  
\[Z_{7}=\hat{h}_{total} \tag{44}\]  
2.7. Maximum Workload, Total Workload and Maximum Workload-Difference (\(Z_{8}\), \(Z_{9}\) and \(Z_{10}\))  
In addition to makespan and flowtime, two other objectives commonly considered in FJSP scheduling are the minimization of maximum and total machine workload. They represent the intention of protecting machines from overuse [43]. Moreover, in this research, we noted that, in the presence of alternative routing and sequence-dependent setup time, these objectives could result in a substantially reduced overall system workload with a moderate increase in makespan. This can significantly free up machine operators for other activities, such as quality improvement, development, and maintenance. The necessary variables and constraints to impose these categories of objective functions are discussed below. The workload on machine \(m\) (i.e., \(l_{m,\rho,s,j}\)) because of an assigned operation \(o\) of sublot \(s\) of job \(j\) comprises the setup and the actual processing of the operation. The value of this variable is assigned by the constraints in Equations (45)-(48). The overall workload on machine \(m\), \((\hat{l}_{m})\), comprises the workloads because of all the operations assigned to it from the current schedule and its release date \(D_{m}\) as shown in Equation (49). The release date may represent the amount of work that spills into the current planning and scheduling period from the previous one. The objective function along with the constraints in Equations (50) and (51) set the values of \(\hat{l}_{max}=\max_{vm}\{l_{m}\}\) and \(\hat{l}_{min}=\min_{vm}\{l_{m}\}\), respectively. The workload difference between the maximally and the least loaded machines (maximum workload difference, \(\hat{l}_{diff}\)) is calculated using the constraint in Equation (52). The total workload on the system \(\hat{l}_{total}\) is evaluated by the constraint in Equation (53). The objective function terms \(Z_{8}\), \(Z_{9}\), and \(Z_{10}\) represent the values of \(\hat{l}_{max}\), \(\hat{l}_{total}\), and \(\hat{l}_{diff}\) as enforced by the constraints in Equations (54)-(56), respectively.