## 4 Proposed Decoupling Dynamic Point  
The proposed model involves dynamically assigning the decoupling point within the system. The primary objective of adjusting the decoupling point is to minimize total costs associated with tardiness and work in process within the manufacturing system. This dynamic adjustment is based on evaluating the earliness or tardiness of products delivered to customers.  
When the generic customer places an order, it defines the maximum waiting time for the product, following Equation (1):  
\[waiting\ time=Tnow+processing\ time\ last*UNIF[min,max] \tag{1}\]  
Equation (1) defines the waiting time of the customer considering the time when the customer places the order (_Tnow_), the processing time of the last production stage (processing time last), and an uncertain parameter by a uniform distribution. This approach follows the total work content (TWK) rule, with an allowance factor extracted by a uniform distribution [20]. If the order is delivered over this waiting time, the customer can request a penalty proportional to the tardiness.  
The delivery time is the time when the order is given to the customer. Then, for each order delivered to the customer, two parameters are computed: _tardiness_ and _earliness_.  
Figure 1: MTS, MTO, and hybrid FFS.  
If the delivery time is lower than the waiting time, the earliness parameter is updated following Equation (2):  
\[earliness=earliness+(delivery\;time-waiting\;time) \tag{2}\]  
If the delivery time is greater than the waiting time, the earliness parameter is updated following Equation (3):  
\[tadiness=tardiness+(waiting\;time-delivery\;time) \tag{3}\]  
The decision on the decoupling point change can follow two review approaches: periodic and continuous review.  
The periodic policy, at fixed period \(Tp\), evaluates the _earliness_ and _tardiness_ during the last period \(Tp\).  
If the earliness is greater than the tardiness, the decoupling point is moved one stage towards the end of the flow line. This means that the flow line is closer to an MTO because the orders are delivered early compared to the waiting time expected by the customers.  
Otherwise, if the tardiness is greater than the earliness, then the decoupling point is moved one stage towards the start of the flow line. This means that the flow line tends to an MTS to reduce the tardiness of the orders. After that, the earliness and tardiness are set to zero.  
The continuous policy starts the movement of the decoupling point when certain events occur. The model applies the following steps.  
The continuous model monitors the number of products delivered consecutively in either an early or late state. If the number of consecutive orders exceeds a threshold, the model adjusts the decoupling point: moving it one stage towards the end of the production line for consecutive tardy orders, or one stage towards the beginning of the flow line for consecutive early orders.  
The main assumptions of the system are as follows: each machine can process only one item at a time; customer orders are fulfilled according to the earliest due date (EDD) rule; and the transportation time of items between machines is negligible.