# Operator-Based Adaptive Tracking Capacity Control in Complex Manufacturing Processes
## 4 Numerical Simulation
### Robustness Analysis  
Stochastic customer demands and different types of products were considered when measuring and analyzing the robustness of the capacity control systems. In this part, robustness was defined as the mean and standard deviation of the output variables. The dis  
Figure 10: Dynamic performance of the products.  
tribution of the error between the planned and current WIP at each workstation is given in Figure 11. The blue and red error bars are the distributions of errors in the PI-RRCF and ISPI-RRCF capacity control systems. These showed that the distributions in the red bars of all workstations were highly concentrated, with smaller mean values and standard deviations than those of the blue bars, which indicated that the WIP values of the workstations were distributed closer to the desired levels in the ISPI-RRCF system, with a higher robustness. Additionally, the red error bars were distributed over zero, which meant that the WIP levels of the workstations were lower than the planned level in most cases. This illustrated the lower possibility of bottleneck occurrence in this control system.  
Figure 12 shows the distribution of products while considering the input and output rates. The black error bars present the distribution of order input rates. The blue and red error bars in the PI-RRCF and ISPI-RRCF were relatively broad, but the latter distribution was still relatively concentrated. This showed that the robustness when considering the order output rate of products in the ISPI-RRCF system was still higher than that in the PI-RRCF system.  
Figure 11: Error distributions between the planned and current WIP of the workstations.  
Figure 12: Input and output rate distributions of the products.  
The overall statistics of the transient and robustness performance in both the ISPI-RRCF and PI-RRCF control systems are summarized in Table 3. The mean number of RMTs (MRMT) and standard deviation of RMTs (SDRMT) at each workstation were close in the ISPI-RRCF and PI-RRCF control systems. Nonetheless, the mean value of the absolute error (MAE) between the desired and actual WIP, as well as the standard deviation of the absolute error (SDAE), were smaller in the ISPI-RRCF system, which further indicated the higher robustness of this capacity control system. The transient performance when considering the overshoot and settling times of these two control systems was also collected. This further showed that the dynamic performance in the ISPI-RRCF system was better than that in the PI-RRCF system, with smaller overshoots and shorter settling times.  
In summary, for the capacity control of the complex job shop system with transportation delays, reconfiguration delays, and stochastic demand, the key system performance indexes and control performance indexes are discussed. The system performance indexes were represented by the WIP of the workstations and the output rate of the products, while the control performance indexes were represented by the transients and robustness. The performance of the proposed adaptive ISPI-RRCF and PI-RRCF control systems is compared and analyzed in this section. The system performance indexes showed that both control systems could keep the WIP and product output rate at the desired value to ensure productivity and satisfy customer demands. However, ISPI-RRCF had better transient and robustness performance in terms of its lower overshoots, shorter settling times, and smaller means and standard deviations. This indicated that the proposed algorithm could greatly improve the performance to avoid serious bottlenecks and balance the capacity distribution for high productivity, low costs, and high customer satisfaction.