# Operator-Based Adaptive Tracking Capacity Control in Complex Manufacturing Processes
## 4 Numerical Simulation
### Dynamic Performance Analysis  
The dynamic performance of each workstation and product in both the PI-RRCF and ISPI-RRCF capacity control systems while facing stochastic demands are shown in Figures 7-10. The dynamic optimization values of both the PI and ISPI tracking control parameters with the single neuron are given in Figure 9.  
possible bottlenecks and assign enough RMTs to balance the capacity distribution by using the customized flexibility of RMTs. This further illustrates the quicker response in the ISPI-RRCF control system for effectively deducing high overshoots and avoiding bottleneck problems, which highly decreased the production costs and improved the dynamic performance. Figure 8: Distribution of RMTs for stochastic demands. Figure 7: WIP levels of workstations for stochastic demands. For the tracking control parameter values, Figure 9 shows the dynamic optimization results obtained by using the single neuron in both the PI-RRCF and ISPI-RRCF capacity control processes. With the decoupling control, each workstation was an independent SISO system. Therefore, the parameter optimization was relatively independent in both the PI-RRCF and ISPI-RRCF systems. However, compared with Figure 8, the curves for the KP (i.e., \(C_{1}^{\text{q}}\)) and KI (i.e., \(C_{1}^{\text{t}}\)) in ISPI-RRCF showed relatively stable behaviors after 10 h, while the PI-RRCF took a longer time. The KI values in all workstations went toward 0 to avoid high overshoots when the error was over set threshold in the ISPI-RRCF control system. However, the KI values were not zeros in the PI-RRCF system, which explained one of the reasons for the high overshoots. From the overall settings, these values had relatively quicker updating in ISPI-RRCF in order to solve bottlenecks, and they had relatively stable settings when facing bounded stochastic input rates. This illustrated the effectiveness of the proposed adaptive algorithm based on a single neuron with supervised Hebb learning. On the other hand, this could also imply the better dynamic and robust performances in the ISPI-RRCF control system. Unlike the performance analysis in the workstations, the input and output rates of each product were not independent, which caused dynamic changes in the workstations' productivity. When facing stochastic demands, the input rates of each product were the same in both control systems, as shown with the black lines in Figure 10. The blue and red lines represent the product output rates in the PI-RRCF and ISPI-RRCF control systems. Products 1, 2, and 3 were produced starting from the first, second, and first workstations and were finished at the third, fourth, and fourth workstations, respectively. The output rate of each product was produced by the DMTs at the beginning. With the additional RMTs assigned to the later two workstations, the output rates were quickly increased around the input rate in 20 h in the ISPI-RRCF system. However, the PI-RRCF system took a longer time. Additionally, the output rates in ISPI-RRCF were around the input  
Figure 9: Value of the tracking controller’s parameters. rate for most of the time, which showed that there was greater customer satisfaction in this system. In summary, when facing stochastic customer demands, both control systems were able to provide the required output products, while the proposed ISPI-RRCF system still showed better dynamic performance with a quicker response for satisfying customers' stochastic demands. Considering the transient performance in the above figures, it was concluded that both control systems could deal with stochastic demands to guarantee that the WIP of workstations was kept in the vicinity of the desired level and that the output rate of the products was kept close to the input rates. However, the proposed adaptive ISPI-RRCF tracking capacity control system showed better behaviors with a quicker response and lower overshoot.