# Development of Operator Theory in the Capacity Adjustment of Job Shop Manufacturing Systems
## 6 Conclusions and Outlook  
In this paper, an operator-based approach incorporating RMTs was developed to control the capacity of job shop systems. A mathematical model integrating the flexibility of RMTs was presented for the capacity control. The design of the RRCF controller was discussed in the capacity control of a general job shop system. To evaluate the effectiveness of the RRCF control approach, a PID control was applied as a benchmark in the simulation of a four-workstation job shop manufacturing system. Then, conclusions are summarized as follows:  
* The proposed RRCF capacity control method can deal with the constant demands and solve bottlenecks to ensure practical stability of the system in the WIP levels of all workstations and the output rates of all products. Additionally, compared to the PID method, the RRCF showed better transient and steady state performances, with shorter settling times and lower overshoots.
* For the stochastic demands, Monte Carlo simulation was utilized to evaluate the robustness of these two control systems. The simulation results illustrated that RRCF was more robust than PID with less mean and standard deviation of the absolute errors between planned and current WIP levels.
* The simulation results also supported the applicability and effectiveness of the proposed capacity control approach by integrating of RMTs with the RRCF method.
* As this capacity control approach is in a decentralized architecture, it also can be used in large-scale job shop systems. In future work, the following research directions can be considered:
* The mathematical model can be further extended and integrated with more performance indicators, e.g., backlog and inventory for more complex problems with various perspectives.  
Figure 13: Distribution of errors at each workstation for stochastic demands.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Controller** & \multicolumn{4}{c}{**PID**} & \multicolumn{4}{c}{**RRCF**} \\ \hline
**Workstation** & **1** & **2** & **3** & **4** & **1** & **2** & **3** & **4** \\ \hline \(E(X)\) & 2.20 & 1.10 & 1.10 & 2.19 & 2.20 & 1.10 & 1.11 & 2.19 \\ \(\sigma(X)\) & 0.56 & 0.54 & 0.63 & 0.97 & 1.12 & 0.76 & 0.86 & 1.96 \\ \(E(Y)\) & 17.58 & 17.00 & 21.26 & 32.76 & 13.11 & 17.33 & 19.48 & 24.58 \\ \(\sigma(Y)\) & 14.41 & 13.46 & 17.00 & 25.56 & 10.28 & 13.71 & 15.25 & 19.03 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Performances of system for stochastic demands.  
* The proposed capacity control approach is designed from the customer perspective. Another work can be to develop an effective reconfiguration rule to optimize the performance, at the same time satisfying the demands.  
Validation, P.L., Q.Z. and J.P.; Writing--Original Draft Preparation, P.L. and Q.Z.; Writing--Review & Editing, P.L., Q.Z. and J.P.; Supervision, J.P.  
This research was funded by the FUSION and gLINK programs of ERASMUS MUNDUS grant number 2013-2541/001-011-EM and 2014-0861/001-001-EM, respectively.  
The publication of this paper was supported by Staats- und Universitaetsbibliothek Bremen (SuUB), Germany.  
The authors declare no conflict of interest.