The scheme of the developed framework is depicted in Figure 5.4. The main challenge of the studied modular multi-product production line is that the station times are uncertain and depend on which products, produced at a given station, are handled by the model predictive controller that utilises the state-space model presented in Section 5.2. The denitions of the sequenced modular products π(k) are stored in the MES.
In the ideal case, there is a degree of freedom to optimize the sequence of the production by minimising the risk of conveyor stoppage and maximising the total utility/productivity of the production line. The proposed MPC has been developed for the same purpose, so it can maximise the benets of the sequencing as it will be demonstrated in the case study (in Section 5.5) where the sequencing algorithm proposed in [254] was incorporated into the proposed framework.
Based on this information, the model calculates which elementary activities should be performed at a given station. As the activity times are represented by fuzzy sets, the optimistic/shortest expected activity time is the left-hand side supreme of the fuzzy set, ti,L = min ([Ai]α), while the pessimistic/longest expected activity time
400 600 800 1000 1200
t
400 600 800 1000 1200
t
Figure 5.3: An example of how two fuzzy variables can be totaled based on their α-cuts. The parameters of the fuzzy set with dashed lines are a1 = 110, b1 = c1 = 50 and d1 = 450, while the parameters of the other fuzzy variables
area2 = 90,b2=c2= 40and d2 = 650.
is the right-hand side supreme of the fuzzy set, ti,R = max ([Ai]α). Based on this concept, optimistic and pessimistic estimates of the activity times,ta,{L,R}(k+j|k), nishing times,tf,{L,R}(k+1|k), and the delays,td,{L,R}(k+1|k) = tf,{L,R}(k+1|k)−
tc(k+ 1), wheretd,L(k)is the lower andtd,R(k)the upper bound of the duration of the delay at the start of the kth cycle. (The {L, R}notation in lowercase denotes the left, or right-hand supreme of the fuzzy sets). tc(k) is the sum of elapsed cycle times of the kth cycle. According to this denition, this value is positive during delays and negative when the operators work ahead of schedule. As is shown in Figure 5.4, based on the extracted (defuzzied) activity times related to a given condence (α-cut), the model predictive controller calculates the optimal cycle time as control signal, u(k). The gure also illustrates that data collected concerning the elementary activity times can be used to update the parameters of the fuzzy sets based on the method that was presented in the previous section.
The following subsections will illustrate how the information extracted from the fuzzy activity-time models can be incorporated into model-based control schemes.
As will be presented in the following subsection, when the control signal is calcu-lated to prevent stoppage of the conveyor in the following cycle time, a one-step-ahead predictive controller is dened. Based on the constrained minimisation of the delay, in a prediction horizonHp, a more sophisticated optimal control solution will also be proposed in the remaining part of this section.
Figure 5.4: The scheme of the proposed fuzzy activity time-based model pre-dictive controller (MPC). The fuzzy activity times are identied based on his-torical data collected from the conveyor. The models of the MPC are updated
based on the sequence of the produced products.
5.4.1 One-step-ahead predictive control
Using the ta,{L,R}(k+ 1|k)prediction of the lower or upper bound of the activity times (for a givenα) at the beginning of thek-th cycle, thetf,L(k+ 1|k)lower and tf,R(k+ 1|k) upper boundaries of the completion times can be calculated. Based on the tc(k+ 1)start time of the (k+ 1)th cycle, the delay at every(k+j)th cycle can be predicted according to Eq.(5.10).
td,{L,R}(k+ 1|k) = tf,{L,R}(k+ 1|k)−tc(k+ 1) =tf,{L,R}(k+ 1|k)−(tc(k) +u(k)) (5.10) whereu(k)denotes the cycle time set at the beginning of the kthcycle. Therefore, the kth cycle starts at tc(k) and nishes at tc(k+ 1) =tc(k) +u(k).
The upper bound of the delay, td,R(k), cannot exceed a critical limit, therefore, a ccrit value can be dened which is equal to this critical limit or less than it.
The cycle cannot be started should it be impossible to nish the tasks before this limit. Therefore, the criteria for not stopping the conveyor can be formulated as Eq. (5.11).
max (tf,R(k+ 1|k))−(tc(k) +u(k))< ccrit (5.11) wheremax (tf,R(k+ 1|k))denotes the delay of the slowest operator (workstation).
By expanding the expression (tf,R(k+ 1|k)), the Eq. (5.12) can be derived.
max ((tf,R(k) +ta,R(k+ 1|k)))−(tc(k) +u(k))< ccrit (5.12) It should be noticed that tf,R(k) is a measured value of the completion of thekth cycle, whileta,R(k+ 1|k)is a predicted one, therefore, making it possible to derive a one-step-ahead predictive controller.
The proposed algorithm continuously sets the cycle timeu(k)for everykthcycle to avoid any stoppages, so the control signal/the cycle time should exceed the upper
bound of the nishing time. The Eq. (5.13) shows how to u(k) is dened.
max(td,R(k) +ta,R(k+ 1|k))−ccrit < u(k) (5.13)
The Eq. (5.14) shows that stoppages can be prevented by setting the u(k) to ensure that the expected maximum delay should be less than u(k).
u(k) = max(td(k) +ta,R(k+ 1|k))−ccrit (5.14) If an unpaced conveyor is used during the production, the speed of the line can be dened as the output of the controller by using thes(k) = u(k)l transformation, where l stands for the length of the conveyor line.
5.4.2 Constrained fuzzy model predictive control
In addition to the one-step-ahead predictive control, a much more eective model predictive control scheme that minimizes the eect of tuning for horizon of longer duration, Hp, by determining a control sequence of lengthHc u∗(k) =
[u(k), u(k+ 1), . . . , u(k+Hc)] where Hc denotes the control horizon was formu-lated.
As the control horizon cannot exceed the prediction horizon (Hp ≥ Hc), it is assumed that the control variable remains constant after the control horizon has ended until the end of the prediction horizon u(k +Hc + 1), . . . , u(k +Hp) = u(k+Hc).
In a similar manner to the cost functions of simple assembly line balancing prob-lems [255], several types of cost functions can be dened, e.g., the cost function can be formalized to minimize the cycle time which in turn minimizes any delay to the expected nishing times in Eq. (5.15), which also optimizes the utilities of the operators and attempts to ensure a well-balanced workload.
min
u∗(k)u∗(k)TRu∗(k) (5.15) The type of model predictive control can primarily be determined from the den-ition of the control constraints as the formulation of the control sequence seeks
to avoid stoppages to the conveyor belt due to the accumulation of a delay. The constraints that ensure this leads to the formulation of a quadratic optimisation problem in Eq. (5.16).
ARu∗(k)<bR (5.16)
whereAR denotes a lower triangular matrix andbR =tc(k) +ccrit−tf,R(k+j|k). Therefore, the constraint applied to the cycle time should be rearranged in the form of Eq. (5.16). By rearranging Eq. (5.11):
tf,R(k+j|k)−tc(k)−ccrit<
Hp
X
j=1
u(k+j−1) (5.17)
−
Hp
X
i=j
u(k+j −1)< tc(k) +ccrit−tf,R(k+j|k) (5.18)