# Development of Operator Theory in the Capacity Adjustment of Job Shop Manufacturing Systems
## 5 Case Study
### Simulations for Constant Demands  
Considering constant customer demands, the resulting performances of all workstations with delays are shown in Figures 10-12. The allocation of respective RMTs at each workstation is illustrated in Figure 10. The error between the planned and current WIP level of each workstation is given in Figure 11. The red and blue lines represent the variables with PID and RRCF control, respectively. The performance of WIP levels and distribution of RMTs of each workstation in this dynamical capacity control process is analyzed as follows. At workstation 1, due to rush orders from the initial stage, the WIP level in workstation 1 increased and induced bottlenecks in the first 5 h, so the error in Figure 11 was highly decreased. Then both PID and RRCF controllers started assigning RMTs to this workstation, cf. Figure 10. Because of the reconfiguration delay, RMTs did not work for the first two hours, so the WIP was still increasing. However, after 2 h, PID and RRCF assigned 5 and 7 RMTs to this workstation, respectively, so the WIP in the RRCF case decreased more than in the PID one. After 10 h, the WIP of workstation 1 was lower than the planned level, so the number of RMTs was decreased. After around 50 h, the WIP in the RRCF case was practically stabilized (cf. [35], Chapter 2 for details), and the respective number of RMTs was also controlled between 2 to 3. The number of RMTs as the input signal is an integer, so the value of the WIP showed scattering but bounded behavior. For the PID case, the WIP level was practically stabilized after 130 h with higher overshoot. At workstation 2, the initial WIP was lower than the planned level, so the error between planned and current WIP at the beginning was positive. In the first two hours, the error increased a bit due to fewer input orders from Workstation 1. After 2 h, owing to more orders flowing in from Workstation 1 but limited capacity, the WIP level was quickly increased and the error decreased quickly in the first 10 h in Figure 11. Therefore, bottleneck also moved to this workstation. After 10 h, both controllers started assigning RMTs to workstation 2 to solve the bottleneck in Figure 10. However, the RRCF had a quicker response with fewer than 2 RMTs to practically stabilize the WIP levels after 30 h, while the PID showed a slower response and took 4 RMTs to resolve the bottleneck. Additionally, the respective WIP level displayed higher overshoot and longer settling time than the RRCF control. At workstations 3 and 4, the initial WIPs were less than the planned levels, so the initial error was positive in Figure 11. In the first two hours, as there was less order input and more output rate, the WIP level decreased, so the errors increased a bit. Then, due to increasing order inputs but limited capacity, the WIPs increased quickly over the planned levels and induced bottlenecks. After around 30 and 40 h at Workstations 3 and 4, respectively, both PID and RRCF controllers assigned RMTs to these two workstations. Then, the capacities increased, while the WIP levels decreased. However, the WIPs in RRCF were quickly stabilized and close to the planned levels. Same as for workstation 2, the RRCF control system showed a quicker response and used fewer RMTs than the PID control system to solve the bottlenecks. Moreover, the WIP levels of these two workstations within the RRCF control tended to be practically stable after 50 h, while the PID control system took around 150 h for the practical stabilization of the WIP levels. Additionally, the PID control showed higher overshoots and more oscillations. In Figures 10 and 11, both the PID and RRCF control approaches could practically