## 4 Case Study
### Approach to the Case Study  
The case presented is based on the process used in the study case of [10]: an assembly process of a 2D product composed of four parts which is assembled in a 3-stage manufacturing process and it is inspected at the end of the process, as illustrated in Figure 9.  
Three possible scenarios have been considered on this process depending on the type of control used:  
* Scenario_0: No correction actions are applied. The VFS_Basic_Monitor&Control block monitors some of the variables of each stage and performs a statistical analysis of the final inspection of products carried out in Stage 4. Measurements of this final inspection are used to calculate the mean and the standard deviation of the geometric deviations obtained at each inspected point. In addition, the maximum geometric deviation and its corresponding inspection point is identified, as well as the number of products that do not meet specifications. * Scenario_1: The in-process measuring capability of Stage_1 is used and, based on the measurement results, a VFS_FeedForward_Monitor&Control class calculates the optimal position of the locators in the subsequent stages for each assembly order to minimize the deviations obtained in the final inspection of each product. It must be also considered that the fixtures at Stages 2, 3, and 4 have actuators to adjust the position of each locator. For this reason, the setup times of these stages are modified to simulate the adjustment of the actuators. * Scenario_2: the VFS_FeedForward&Backward_Monitor&Control block is used. In addition to the control actions considered in Scenario_1, corrections are calculated in the locators of Stage_1 using a statistical analysis of the measurements detected in the previous processed assemblies in Stage_1. For this reason, the setup time of Stage_1 is also modified to simulate the adjustment of the actuators. Figure 10 graphically represents the two control alternatives considered. Figure 9: Case study process schema. However, both Scenario_1 and Scenario_2 require the use of actuators that adjust the position of the locators. In order to refine the productivity analysis of the proposed scenarios, it has been considered that the process can use automatic or manual actuators, with their respective consequences in setting times. Thus, a total of five alternative cases are obtained to study and compare:  
* Case 1: Simulation of Scenario_0. * Case 2: Simulation of Scenario_1 using manual actuators. * Case 3: Simulation of Scenario_1 using automatic actuators. * Case 4: Simulation of Scenario_2 using manual actuators. * Case 5: Simulation of Scenario_2 using automatic actuators. Some general parameters considered for the case study are presented below:  
* The processing time means for each stage in Case_1 are: 200 s, 220 s, 180 s, and 80 s, respectively. These time means are maintained in the rest of the cases, except in Stage_1 which includes an in-process measurement; its new time mean is 240 s. A 10 s standard deviation is defined for every process time. * The setup time for each stage with actuators is 80 s for manual actuators and 20 s for automatic actuators. * Initial part deviations are modelled as a normal distribution with 0 mean and 0.05 mm standard deviation at coordinates X and Z. In addition, another pair of random deviations at coordinates X and Z are generated to deduce a random orientation deviation. * Initial locator deviations are modelled as a normal distribution with 0 mean and 0.02 mm standard deviation at coordinates X and Z. However, a bias of the positioners has been modelled by introducing a progressive increase in the mean parameter of the locator deviation distribution, following a quadratic function. This mean parameter is restored to 0 every 3000 s to simulate their adjustment, repair, or replacement. * The measurement noise for the Stage_1 (assembly station with in-process measurement) is defined by a normal distribution with 0 mean and 0.03 mm standard deviation.