# Development of Operator Theory in the Capacity Adjustment of Job Shop Manufacturing Systems
## 3 Mathematical Model
Here, all RMTs can be used within all workstations but only perform one operation in the specific period. Moreover, each DMT can only perform one operation and is assigned to a specific workstation. Hence, each workstation consists of a fixed number of DMTs and a variable number of RMTs. Therefore, the maximal capacity of a workstation is given by:  
\[\varepsilon_{j}(t)=n_{j}^{DMT}\cdot v_{j}^{DMT}+n_{j}^{RMT}\cdot v_{j}^{RMT}. \tag{6}\]  
The number of RMTs in each workstation is considered to be our new degree of freedom. If the association of an RMT to a workstation is changed over time via \(u_{j}(t)\), this renders the maximal capacity time-variant. Assuming a high WIP, each workstation is operating at its maximal capacity, and its output rate is given by:  
\[W_{j}(t)=n_{j}^{DMT}\cdot v_{j}^{DMT}+u_{j}(t-\tau_{1})\cdot v_{j}^{RMT}, \tag{7}\]  
where \(\tau_{1}\) is the reconfiguration delay when RMTs change the operation between workstations. Note that if an RMT is reconfigured from workstation \(j\) to workstation \(k\), the capacity of workstation \(k\) increases only after a lag \(\delta\) (e.g., 2 h), while the capacity of workstation \(j\) decreases immediately. This reveals the description:  
\[\tau_{1}= \begin{cases}\delta,&u_{j}(t^{+})\geq u_{j}(t^{-}),\\ 0,&\text{else}.\end{cases}  \tag{8}\]  
Then, the WIP level can be controlled via the function \(u_{j}(\cdot)\) for all workstations in (5). The input-output model of the system, which is the plant operator \(P_{j}\) for \(j=1,\ldots,n\), is given by:  
\[ \begin{split} P_{j}:y_{j}(t)=& y_{j}(0)+\int_{0}^{t}X_{ 0j}(\tau-\tau_{2})+d_{j}(\tau)\\ &+\sum\limits_{k=1}^{n}p_{kj}\cdot(n_{k}^{DMT}\cdot v_{k}^{DMT}+u_{ k}(\tau-\tau_{1}-\tau_{2})\cdot v_{k}^{RMT})\\ &-(n_{j}^{DMT}\cdot v_{j}^{DMT}+u_{j}(\tau-\tau_{1})\cdot v_{j}^{RMT })d\tau,\end{split}  \tag{9}\]  
where \(u_{j}\subseteq U\) and \(y_{j}\subseteq Y\), for all \(j=1,\ldots,n\). Additionally, the number of RMTs in the system is assumed to be limited by \(n^{RMT}\), and each workstation contains at least 0 RMTs. This reveals the control constraints:  
\[0\leqslant u_{j}(t)\qquad\text{and}\qquad\sum\limits_{j=1}^{n}u_{j}(t)\leqslant n ^{RMT}. \tag{10}\]  
Note that similar to \(n_{j}^{PMT}\) but in contrast to the input and output rate values \(X_{j}(\cdot)\) and \(W_{j}(\cdot)\), our control \(u_{j}(\cdot)\) is integer instead of continuous. To compensate for the issue, the truncation \(\lfloor\cdot\rfloor\) is utilized. Moreover, for the constraint \(\sum_{j=1}^{n}u_{j}(t)\leq n^{RMT}\), the fractional approach [32] is utilized, which is given as:  
\[\hat{u}_{j}(t)=\left\{ \begin{aligned} &\left\lfloor u_{j}(t)\right\rfloor,&\text{if}\ \sum\limits_{j=1}^{n}u_{j}(t)\leq n^{RMT}\\ &\left\lfloor\frac{n^{RMT}}{\sum\limits_{k=1}^{n}u_{k}(t)}u_{j}(t )\right\rfloor,&\text{else.}\end{aligned} \right.\]  
Note that if the number of utilized RMTs does not exceed the total number of RMTs, then \(\hat{u}_{j}(t)\) corresponds to the truncated value of \(u_{j}\). Otherwise, the fractional discrete value of RMTs in the sum over all workstations is considered. Note that model (9) only applies if the system is working at a high WIP level. In this case, the order output rate equals the maximum capacity, i.e., the WIP level can be controlled via the assignment of RMTs \(u_{j}(\cdot)\) for all workstations.