# Dynamic Bottleneck Identification of Manufacturing Resources in Complex Manufacturing System
## 3 Material and Methods
### Complex Manufacturing System and Its Decoupling
#### 3.1.2 Decoupling the CMS  
Setting up buffers on the production line can improve system stability and output [40]. However, the decoupling effect of the buffer, taking the production line with the buffer as a whole to analyze the bottleneck, tends to cause confusion and misjudgment of the bottlenecks.  
It should be noted that only effective buffers have the decoupling effect on the production system, so we must first find all effective buffer locations in the system. We define the effective buffer as the buffer that continues to act on the production system within the project time window [17]. Because when the buffer content reaches the limit value, the buffer loses its elasticity and cannot reduce the impact of the disturbance on the system. We define the buffers that arrive at the limit value within the project time window as invalid buffers. This article only considers the limited buffer, which shows that the buffer's capacity is not unlimited.  
Definition 1: Effective buffer is a buffer that will never be full or empty within the project time window.  
Assume the set of each buffer content records in different times for the manufacturing system can be denoted as:  
\[\mathbf{V}_{j}=\big{\{}v_{1,j},v_{2,j},\cdots v_{m,j}\big{\}} \tag{9}\]Assume the matrix of buffers' content records for the manufacturing system can be denoted as:  
\[\mathbf{V}_{i,j}=\left[\begin{array}{cccc}v_{1,1}&v_{1,2}&\cdots&v_{1,N}\\ v_{2,1}&v_{2,2}&\cdots&v_{2,N}\\ \vdots&\vdots&\ddots&\vdots\\ v_{M,1}&v_{M,2}&\cdots&v_{M,N}\end{array}\right] \tag{10}\]  
where \(m\) represents the number of time instances, \(n\) represents the number of buffers, and \(V\) represents the buffer matrix (10) of size \(m\times n\);  
**Definition 2**.: _Buffer j is the effective buffer in a system with \(n\) machines and \(m\) buffers during a period if:_  
\[\begin{array}{l}\forall v_{ij}\in\mathbf{V}_{j}\\ 0<v_{ij}<C_{\max},i=1,2,\cdots,m\end{array} \tag{11}\]  
Only the finite buffer is considered in this paper, which means the buffer capacity cannot be infinite. The effective buffering determination algorithm is proposed to find out the effective buffers of the system, as shown in Figures 2 and 3. Then, the bottleneck state of each part should be identified by analyzing the fine-grained machine state of each part after decoupling. The overall bottleneck state of the production system should be obtained without confusion.  
To explain the effective buffer, we present an example of a six-machine-five-buffer (6M5B) tandem line during one shift (shown in Figure 4). The buffer content information over the time interval is shown in Figure 5.  
According to the buffer content information combined with the definition of an effective buffer, it can be known that \(B2\) and \(B4\), which never reached their limit, are the effective buffers of the system. The system is decoupled into three stages as shown in Figure 4: \(M1\) and \(M2\); \(M3\) and \(M4\); and \(M5\) and \(M6\), and each stage has its bottleneck state.  
Figure 2: The effective buffering determination algorithm.  
Assume the set of effective buffers for the manufacturing system can be denoted as:  
\[B=\{b_{1i},b_{2i},\cdots,b_{in}\} \tag{12}\]  
In addition, assume the set of machines for the \(i\)-th stage can be denoted as:  
\[M_{i}=\{M_{i1},M_{i2},\cdots M_{in}\} \tag{13}\]  
Figure 4: The six-machine-five-buffer (6M5B) tandem line.  
Figure 3: The effective buffering determination flowchart.