# Development of Operator Theory in the Capacity Adjustment of Job Shop Manufacturing Systems
## 3 Mathematical Model  
Job shop manufacturing systems have a high flexibility and can produce numerous types of products, as these may flow flexibly between all workstations. The variables in the model are given in Table 1. In general, for a job shop system with \(n\) workstations, the simplified model of the \(j\)th (\(j=1,\ldots,n\)) workstation is illustrated in Figure 5. The input rate of the workstation is the sum of output rate from all workstations, including the workstation itself and a possible initial stage. The output rate of the workstation is the current capacity. Figure 4: Equivalent of Figure 3.  
Figure 3: Nonlinear feedback system with disturbances. The workstation receives orders from the initial stage (\(k=0\)) and workstations \(k\in\{1,\ldots,n\}\) and delivers its products to a final stage (\(i=0\)) and workstation \(i\in\{1,\ldots,n\}\). Thus, for the \(j\)th workstation, considering the transportation delay \(\tau_{2}\) between each workstations, the orders input rate is:  
\[X_{j}(t)=\sum_{k=0}^{n}X_{kj}(t-\tau_{2})=X_{0j}(t-\tau_{2})+\sum_{k=1}^{n}W_{ kj}(t-\tau_{2}), \tag{3}\]  
where \(X_{0j}(t)\) is the order input from the initial stage. The order output rate of the \(j\)th workstation is given by:  
\[W_{j}(t)=\sum_{j=0}^{n}p_{ji}\cdot W_{j}(t)=\sum_{j=0}^{n}W_{ji}(t), \tag{4}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Variable** & **Description** \\ \hline \(X_{kj}(t)\) & Orders input rate from workstation \(k\) to workstation \(j\) for \(k,j\in\{0,\ldots,n\}\) \\ \(X_{j}(t)\) & Orders input rate of workstation \(j\in\{1,\ldots,n\}\) \\ \(W_{jk}(t)\) & Orders output rate of workstation \(k\) to workstation \(k\) for \(j,k\in\{1,\ldots,n\}\) \\ \(W_{j}(t)\) & Orders output rate of workstation \(j\in\{1,\ldots,n\}\) \\ \(u_{j}(t)\) & Input signal of workstation \(j\in\{1,\ldots,n\}\), which is equal to the number of RMTs \\ \(y_{j}(t)\) & Output signal of workstation \(j\in\{1,\ldots,n\}\), which is the WIP level of the workstation \\ \(c_{j}(t)\) & Current capacity of workstation \(j\in\{1,\ldots,n\}\) \\ \(c_{j}(t)\) & Maximum capacity of workstation \(j\in\{1,\ldots,n\}\) \\ \(p_{jk}\) & Flow probability that the output orders from workstation \(j\) to workstation \(k\) for \(j,k\in\{0,\ldots,n\}\) \\ \(n^{RMT}\) & Number of RMTs in the system \\ \(n^{RMT}_{l}\) & Number of RMTs in workstation \(j\in\{1,\ldots,n\}\) \\ \(n^{DMT}_{l}\) & Number of DMTs in workstation \(j\in\{1,\ldots,n\}\) \\ \(v^{DMT}_{l}\) & Production rate of DMTs in workstation \(j\in\{1,\ldots,n\}\) \\ \(v^{RMT}_{j}\) & Production rate of RMTs in workstation \(j\in\{1,\ldots,n\}\) \\ \(d_{j}(t)\) & Disturbances in workstation \(j\in\{1,\ldots,n\}\) \\ \(\tau_{1}\) & Reconfiguration delay \\ \(\tau_{2}\) & Transportation delay \\ \(n\) & Number of workstations in the system \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables within the workstation with RMTs. Figure 5: Model of \(j\)th workstation with reconfigurable machine tools (RMTs). where the flow probabilities satisfy \(\sum\limits_{i=0}^{n}p_{ji}=1\) for all \(j\in\{1,\ldots,n\}\). The current WIP as the output signal of the \(j\)th workstation is the integral difference between the order input and output rate plus disturbances over time:  
\[y_{j}(t)=y_{j}(0)+\int_{0}^{t}(X_{j}(\tau)+d_{j}(\tau)-W_{j}(\tau))d\tau. \tag{5}\]  
The latter variable is of particular importance as for a high level of WIP. The order output rate is equal to the capacity of the workstation, that is:  
\[W_{j}(t)=c_{j}(t)=\sum\limits_{i=0}^{n}p_{ji}c_{j}(t).\]  
In order to reflect the functionality of RMTs for capacity adjustment, in [31], an extended model of a job shop system with DMTs and RMTs was proposed, cf. Figure 5 for a sketch of workstation \(j\) with an assigned number of RMTs. Due to the high productivity of DMTs, this kind of machines will also be adopted in the system. The overall system includes \(n^{RMT}\) reconfigurable and \(n^{DMT}\) dedicated machine tools.