## 3 Materials and Methods
### Mathematical Formulation of the Proposed Model
\[\text{T}_{\text{hl}}+\text{ps}_{\text{ij}}\leq\text{T}_{\text{h(l+1)}}+\left( 1-\text{Y}_{\text{jjhl}}\right)\text{.}\text{B }\text{ }\text{\forall}\text{i}\text{,j,h,l}=1,\ldots,\text{l}_{\text{h}}-1 \tag{11}\]  
The beginning of each operation on the time axis must be simultaneous with the beginning of the operation on the machine on which the operation is performed. \[\text{e}_{\text{jih}}-\text{t}_{\text{jih}}=\sum_{\text{k}\text{O}_{\text{ij} }\in\text{M}_{\text{ij}}}\left(\text{P}_{\text{jih}}\text{.}\text{X}_{\text{ jih}}\right)\text{ }\text{ }\text{\forall}\text{i}\text{,j} \tag{12}\]  
The difference between the end time and the start time of the operation \(\text{O}_{\text{ij}}\) must be equal to the execution time of the operation \(\text{O}_{\text{ij}}\) (process time) on the machine \(\text{M}_{\text{h}}\) from the subset \(\text{M}_{\text{ij}}\). \[\text{T}_{\text{hl}}\leq\text{t}_{\text{ij}}+\left(1-\text{Y}_{\text{jjhl}} \right)\text{.}\text{B }\text{ }\text{\Delta T}_{\text{hl}}+\left(1-\text{Y}_{\text{jjhl}}\right)\text{.} \text{B }\geq\text{t}_{\text{ij}}\text{ }\text{\forall}\text{i}\text{,j,h,l} \tag{13}\]At maximum, one operation \(\mathrm{O_{ij}}\) can be performed on the lth place on the machine \(\mathrm{M_{h}}\) from the subset \(\mathrm{M_{ij}}\). \[\sum_{i=1}^{n}\sum_{j=1}^{k_{i}}Y_{ij\mathrm{hl}}\leq 1\ \forall h,1 \tag{14}\]  
One operation \(\mathrm{O_{ij}}\) can be assigned for execution-only on one machine \(\mathrm{M_{h}}\) from the subset \(\mathrm{M_{ij}}\). \[\sum_{h=1}^{\left|\mathrm{M_{hi}}\right|}X_{ij\mathrm{h}}=1\ \forall i,j \tag{15}\]  
One operation \(\mathrm{O_{ij}}\) can be performed only once on an assigned machine \(\mathrm{M_{h}}\) from the subset \(\mathrm{M_{ij}}\). \[\sum_{i=1}^{h_{h}}Y_{ij\mathrm{hl}}=X_{ij\mathrm{h}}\ \forall i,j,h \tag{16}\]  
Determining the subset of machines \(\mathrm{M_{ij}}\) on which operation \(\mathrm{O_{ij}}\) can be performed  
\[\text{e}_{\text{jh}}\text{--completion time of operation }\text{O}_{\text{ij}}\text{ on hth machine} \tag{8}\]0  
For each variable parameter, the following constraints are applied:  
\[\text{e}_{\text{jh}}\text{--completion time of operation }\text{O}_{\text{ij}}\text{ on hth machine} \tag{8}\]1  
The objective function of the mathematical model embodies the optimization criteria. In the instance discussed in this study, the objective is to simultaneously minimize makespan and maximize revenues. In the current literature, such cases are not discussed enough. This goal consists of two opposite goals for the observed company model. One company in the production shop has, as mentioned, m machines. The operational costs of the machines are not the same. From costing analysis or analytical bookkeeping, the exact operational costs of every machine can be calculated. For example, if the execution time of one operation is 30 min, and suppose that on every machine that time is the same, the cost of execution on a machine \(\mathrm{M_{1}}\) will be \(30\cdot\mathrm{K_{1}}=0.5\ \mathrm{hour}\cdot 20\frac{\epsilon}{\mathrm{hour}}=10\ \mathrm{\upxi}\), and on machine \(\mathrm{M_{3}}\) it will be \(30\cdot\mathrm{K_{3}}=0.5\ \mathrm{hour}\cdot 15\frac{\epsilon}{\mathrm{hour}}=7.5\ \mathrm{\upxi}\). In real systems, the execution time for the same operation on a different machine is not the same, because even if the machine belongs to the same subset of machines \(\mathrm{M_{ij}}\), their technical and technological characteristics are different. With this in mind, it must be considered that one operation can be performed on several appropriate machines that belong to a subset of machines \(\mathrm{M_{ij}}\) and the time for execution on every machine from the subset must be calculated as the input value of the model, as mentioned previously. \(\mathrm{K_{h}}\) constants (used in the above example), representing the operational cost of machine/per time unit, are defined. This problem could be considered as multi-criteria optimization, with two criteria: first to minimize the makespan and then to maximize earnings.