## 2 Problem Formulation
### Constraints  
In this study, the constraints of the FJSP-cSH design are shown as follows according to the actual production situation:  
\[C_{i}=\max TP_{ijk}^{E},\ \ \ i\in[1,\ n],j\in[1,\ w],k\in[1,\ m] \tag{4}\]  
\[TS_{pqk}^{B}\geq TP_{ijk}^{E}-(1-y_{ijpqk})\times L,\ i,p\in[1,\ n],j,q\in[1, \ w],k\in[1,\ m] \tag{5}\]  
\[TP_{ijk}^{B}=TS_{ijk}^{E},\ i\in[1,\ n],j\in[1,\ w],k\in[1,\ m] \tag{6}\]  
\[TC_{ij}^{B}\geq TP_{ijk}^{E},\ i\in[1,\ n],j\in[1,\ w-1],k\in[1,\ m] \tag{7}\]  
\[TP_{ij(j+1)k2}^{B}\geq TC_{ij}^{E}-(1-z_{jik12})\times L,\ i\in[1,n],j\in[1, w-1],k1,k2\in[1,\ m] \tag{8}\]  
\[\sum_{k=1}^{m}x_{ijk}=1,\ i\in[1,\ n],j\in[1,\ w] \tag{9}\]  
\[\sum_{h=1}^{NOH}t_{ijh}\leq 1,\ i\in[1,\ n],j\in[1,\ w-1] \tag{10}\]  
\[t_{ijh}\times T_{h}^{B}=t_{ijh}\times TC_{ij}^{B}\leq t_{ijh}\times TC_{ij}^{ E}=t_{ijh}\times T_{h}^{E},\ \ \ i\in[1,\ n],j\in[1,\ w-1],h\]  
\[\in[1,NOC] \tag{11}\]  
\[x_{ijk}\times TP_{ijk}^{E}=x_{ijk}\times(TP_{ijk}^{B}+TP_{ijk}),\ i\in[1,\ n],j \in[1,\ w],k\in[1,\ m[ \tag{12}\]  
\[x_{ijk}\times TS_{ijk}^{E}=x_{ijk}\times(TS_{ijk}^{B}+TS_{ijk}),\ i\in[1,\ n],j \in[1,\ w],k\in[1,\ m] \tag{13}\]  
\[z_{ijk1k2}\times TC_{ij}^{E}=z_{ijk1k2}\times(TC_{ij}^{B}+TC_{k1k2}),\ i\in[1,\ n],\ j\in[1,\ w-1],\ k1,k2\]  
\[\in[1,\ m] \tag{14}\]  
\[TP_{ijk}^{B},\ TS_{ijk}^{B},\ TC_{ij}^{B},\ T_{h}^{B}\geq 0,\ i\in[1,\ n],j\in[1,\ w ],k\in[1,\ m],h\in[1,NOC] \tag{15}\]  
Equation (4) states that the completion time of the job \(J_{i}\) is the completion time of the last operation for the job \(J_{i}\). Equation (5) states that only one operation can be processed by one machine at the same time, and the start setup time of the next operation of the same machine is not less than the completion processing time of the previous operation. Equation (6) states that when a machine processes an operation of a job, the start processing time is equal to the completion setup time of the operation. Equation (7) states that the starting handling time after a process of a job is not less than the completion processing time of the operation. Equation (8) states that a job can start the next operation only after the previous operation of the job is completed, and the start processing time of the operation is not less than the completion handling time of the job after the previous operation. Equation (9) states that each operation will be allocated to only one machine. Equation (10) states that each operation will be allocated to at most one handling task. Equation (11) states that the operation is the same as the beginning time and the completion time of the corresponding handling task. Equation (12) is the processing time constraint: the completion processing time of operation \(O_{ij}\) on machine \(M_{k}\) is equal to the sum of the start processing time and the processing time on machine \(M_{k}\). Equation (13) is the setup time constraint: the completion setup time of operation \(O_{ij}\) on the machine \(M_{k}\) is equal to the sum of the start setup time and the setup time on the machine \(M_{k}\). Equation (14) is the handling time constraint: when the handling occurs between operation \(O_{ij}\) and operation \(O_{i(j+1)}\), it is handled from the work center of machine \(M_{k1}\) to the work center of machine \(M_{k2}\), and the completion handling time is equal to the sum of the start handling time after operation \(O_{ij}\) and the handling time from the work center of machine \(M_{k1}\) to the work center of machine \(M_{k2}\). Equation (15) is the working time constraint: the start time of processing, preparing, and handling shall not be less than 0.