5.5 Examples of applications
5.5.1 Illustrative example
TheNw = 5workstations of the studied illustrative assembly line produceN = 500 modular products during every shift in 10-piece batches of the same product type.
TheNm = 5modules determine the elementary activities that should be performed at each workstation. The nominal time values of these activities are presented in matrix θ, where each row represents a workstation and each column provides the activity time of the related module at the given station:
θ = denote the type of products and the columns the modules. The nominal values ta of the station times can be calculated as ta=θTpp, as is depicted in Fig. 5.5 for the products of the lowest degree of complexity p1 = [1,0,0,0,0] (base module) to the highest degree of complexity p6 = [1,1,1,0,0]:
As is depicted in Fig. 5.5, the process is not perfectly balanced and a signicant dierence between the maxima of the activity times is present. Instead of the described deterministic station times, the proposed more realistic L-R fuzzy sets were used to represent the activity times. According to our expertise, the same strategy was followed with regard to the fuzzication of the activity times as the parameters of the fuzzy set (Eq. 5.4) were set atb=c(therefore, a single maximum was dened) and set to the nominal values (the average cycle times), namely a = b/15 and d = b/10, respectively. These fuzzy sets were applied to analyse
1 2 3 4 5 Station times of product type 1 200
300 400
Assembly time [s] 1 2 3 4 5
Station times of product type 2 200
300 400
Assembly time [s]
1 2 3 4 5
Station times of product type 3 200
300 400
Assembly time [s] 1 2 3 4 5
Station times of product type 4 200
300 400
Assembly time [s]
1 2 3 4 5
Station times of product type 5 200
300 400
Assembly time [s] 1 2 3 4 5
Station times of product type 6 200
300 400
Assembly time [s]
Figure 5.5: Station times (ta,w(k+j|k)) of dierent types of products calcu-lated according to the parameter matrices Pand θof the presented illustrative example. The bars represent dierent workstations. The blue segments illustrate the station times, while the yellow parts highlight the dierence from the max-imum time of the bottleneck. As can be seen, the production line is not perfectly
balanced and there are signicant dierence between the station times.
the stochastic simulation by Monte Carlo simulation-based random generation to ensure that the distribution of the generated random variables approximates the membership functions.
The model permits the operators to work ahead of schedule for a certain duration of time. However, since the operators must not disturb each other, none of the elements of the vector td(k) can exceed a critical value cah, which is usually half of the average cycle time. In that case, the delay time (td) is negative, therefore, the operators work ahead of schedule, but according to this dened constraint cannot leave their assigned workstation. When a constant setting time is applied, the cycle time is set to the maximum of the station times calculated using the α= 0.1, ccrit = 120 and cah = 120.
Fig. 5.6 depicts the results when the cycle time was constant. In this case, every operator can complete their designated tasks, but the eciency of production is low compared to the controlled cycle time set in Fig. 5.7. As can be seen in the subplot at the top of both gures, the same sequences of products were produced in both cases. The subplots in the middle depict show the controlled cycle times, which was constant in the rst case, while the subplots at the bottom present how the delays vary in the case of dierent cycletimes and work stations.
The negative delays show that the operators worked ahead of shedule at given workstation and when the cycle times were constant the system was excessively optimized to prevent stoppages. However, by applying a model predictive control scheme, the system is optimized to prevent stoppages and maintain a high level of productivity.
Figure 5.6: Production of N = 500 products in batches with constant cycle time,u= 348. The bottom plot shows the time delay(td(k)) at every
worksta-tion where the colors represent the operators.
50 100 150 200 250 300 350 400 450
Figure 5.7: Production ofN = 500products in batches with model predictive control of the cycle time according to the following parameters: Hp = 2,Hc= 1 and α = 0.1. Control of the cycle time maximises the productivity, so the improvement in performance compared to the xed cycle time is 11%. The bottom plot shows the time delay(td(k))at every workstation where the colors
represent the operators.
The numerical results are presented in Table 5.1. The performance of the produc-tion line is measured as the average producproduc-tion time calculated by dividing the total time required to produce N products by the number of products tf(N)/N, the value of which is sensitive to the number of stoppages since when production has to be stopped, one cycle time is required to restart the conveyor belt.
As is illustrated by the results, when the cycle time is controlled, the productivity of the production line is enhanced by 11%. By decreasing α, the robustness of the controller is increased, thanks to a reduction in the number of stoppages. An increase in the prediction horizon also enhances the degree of robustness. A larger prediction horizon usually results in a slightly slower, balanced response and robust performance.
Table 5.1: A comparison between the average production times (tf(N)/N) and number of stoppages when the cycle time is constant and dierent settings
are applied to the controllers.
Scenario Prod. time
[min]
# of stop-pages
Constant cycle time, u= 348 348.6 0 One-step-ahead predictive
con-trol, α= 0.1
312.4 6
Model predictive control, Hp = 5,Hc= 3, α= 0.1
309.4 0
Model predictive control, Hp = 2,Hc= 1, α= 0.1
309.2 0
One-step-ahead predictive con-trol, α= 0.05
309.9 2
Model predictive control, Hp = 5,Hc= 3, α= 0.05
309.8 0
Model predictive control, Hp = 2,Hc= 1, α= 0.05
309.5 0