## 3 Mathematical Model
### Objective Functions  
In this paper, the maximum completion time of batch jobs, the total number of machine adjustments, the total number of workpiece handlings, and the total machine load are \begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Definitions** \\ \hline \(i\) & job type number \\ \(j\) & job number \\ \(k,k^{\prime}\) & job process number \\ \(WC_{m}\) & \(m\)-th work center \\ \(N_{WC}\) & total number of work centers \\ \(M_{ml}\) & the \(l\)-th processing equipment in the \(m\)-th work center, \(WC_{m}\) \\ \(g\) & total number of job types \\ \(n_{i}\) & total number of batch jobs of type \(i\) jobs \\ \(n_{WC_{m}}\) & total number of machines and equipment in the \(WC_{m}\) of the work center \\ \(n_{ij}\) & total number of operations of the \(j\)-th job in the \(i\)-th job type \\ \(t^{P}_{(i,j,k,m,l)}\) & \(k\)-th operation of the \(j\)-th job in the \(i\)-th job type \\ \(t^{P}_{(i,j,k,m,l)}\) & processing time of the process, \(O_{ijk}\), on the equipment, \(M_{ml}\) \\ \(t^{P}_{(i,j,k,m,l)}\) & start time and end time of the process, \(O_{ijk}\), on the equipment, \(M_{ml}\) \\ \(t^{P}_{ijkml}\) & setup time of process, \(O_{ijk}\), before processing on equipment \(M_{ml}\) \\ \(t^{S}_{(i,j,k,m,l)}\) & starting and ending setup time of the process, \(O_{ijk}\), on the equipment, \(M_{ml}\) \\ \(t^{P}_{ijkml}\) & handling time after the process, \(O_{ijk}\), is processed on the equipment, \(M_{ml}\) \\ \(t^{h}_{(i,j,k,m,l)}\) & start time and end time of job handling after the completion of the process, \(O_{ijk}\) \\ \(t^{P}_{(i,j,k,m,l)}\) & an infinite positive number \\ \(C_{ij}\) & completion time of the \(j\)-th job in the \(i\)-th job type \\ \(C_{mxx}\) & completion time of the \(k\)-th operation of the \(j\)-th job in the \(i\)-th job type \\ \(F_{r}\) & maximum completion time \\ \(T_{ml}\) & function of the \(r\)-th decision objective \\ \(T_{ml}\) & total processing time of the \(l\)-th processing equipment in the \(m\)-th work center, \(WC_{m}\) \\ \(T_{load}\) & total machine load \\ \(NOM_{ml}\) & amount of downtime of the \(l\)-th processing equipment in the \(m\)-th work center, \(WC_{m}\) \\ \(NOC_{ijk}\) & number of handling tasks of the job after the \(k\) operation of the \(j\)-th job of type \(i\) is completed \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter variable. considered objectives through which to optimize the optimal production scheduling scheme. The following are the designed decision-making objective functions:  
(1) Minimize the maximum completion time. \[min\ F_{1}=min\ C_{max}=min\ max\{C_{ij}|1\leq i\leq g,1\leq j\leq n_{i}\}\] (1)  
Shortening the maximum completion time of an order batch task and ensuring punctual delivery comprise the most fundamental optimization index in scheduling research. (2) Minimize the total number of machine adjustments. \[min\ F_{2}=min{\sum_{m=1}^{N_{WC}}{\sum_{l=1}^{n_{WCm}}{NOM}}}\ \\] (2)  
Minimizing the total number of adjustments to the processing equipment can reduce the number of equipment interruptions in production, decrease the likelihood of human intervention and enhance the quality stability of continuous processing operations, and the utilization rate of the machines. Additionally, this also helps to reduce the energy consumption of the equipment. In flexible job shop scheduling, one machine may process different workpieces, so adjustments need to be made. In accordance with the final production scheduling result, the total number of machine adjustments to the processing equipment is calculated. (3) Minimize the total number of workpiece handlings. \[min\ F_{3}=min{\sum_{i=1}^{g}{\sum_{j=1}^{n_{i}}{\sum_{k=1}^{n_{j}-1}{NOC_{ijk }}}}}\] (3)  
Minimizing the total number of workpiece handling tasks can reduce the number of handling equipment inputs and the number of handling workpieces, and improve the utilization rate of handling equipment, which can also indirectly affect the cost related to workpiece handling tasks.