# Dynamic Bottleneck Identification of Manufacturing Resources in Complex Manufacturing System
## 3 Material and Methods
### Complex Manufacturing System and Its Decoupling  
#### 3.1.1 Time Series Flow Modelling of CMS  
CMS is an organizational system that is constructed to achieve a predetermined manufacturing purpose. It is an organic whole with specific functions composed of manufacturing processes, hardware, software, and related personnel. The CMS can be described as the workshop arranging to process \(j\) parts with a total of \(N_{j}\). The complex system must complete the processing of the workpiece within the delivery period. For the convenience of discussion, the description of the symbols involved in the model are shown in Table 2.  
The manufacturing process needs to meet the following constraints:  
(1) Each process needs to be processed in order, according to the requirements of the process route of the workpiece;  
\[C_{ij}\leq S_{i(j+1)},\forall i\in[1,N_{i}],\forall j\in[1,N_{j}-1] \tag{1}\]  
(2) A workpiece can only be processed on one machine at the same time;  
\[\sum_{k=1}^{m}x_{ijk}=1,\forall i\in[1,N_{i}],\forall j\in[1,N_{j}],\forall k \in[1,m] \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(j\), \(i\), \(k\) & sequence numbers of workpiece, process, and machine \\ \(N_{j}\), \(N_{i}\), \(m\) & number of workpieces, processes, and machines \\ \(S_{ij}\), \(C_{ij}\) & the start and completion time of the \(i\)-th process of the \(j\)-th workpiece \\ \(t_{\beta\ast}\), \(t_{\alpha\ast}\) & the processing start time and end time of the active state \\ \(P_{\beta\ast}\) & the processing time of the \(i\)-th process of the \(j\)-th workpiece on the machine \(h\) \\ \(S_{all}\) & the processing start time of the \(l\)-th process of the \(u\)-th workpiece; \\ \(x_{ijk}\) & the decision variable for the machine selection of the process \\ \(y_{ijkl}\) & the decision variable is selected for the procedure \\ \(S_{ij}\) & the shifting bottleneck degree of station \(i\) in time window \(j\) \\ \(\upsilon(a_{nm})\) & the value of the \(m\)-th attribute under scene \(c\) at time \(n\) \\ \(c_{n}\) & the scene at time \(n\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of bottleneck identification.  
where \(x_{ijk}\) is one when the process \(O_{ij}\) selects the machine \(M_{k}\) and \(x_{ijk}\) is zero when the process \(O_{ij}\) does not select the machine \(M_{k}\).  
(3) One machine can only process one workpiece at the same time;  
\[S_{ij}+P_{ijh}\leq S_{ul}+L(1-y_{ijhul}) \tag{3}\]  
\[C_{ij}\leq S_{j(i+1)}+L(1-y_{ijhul(i+1)}) \tag{4}\]  
where \(s_{j(i+1)}\) represents the processing completion time of the \(i\) procedure of the \(j\)-th workpiece, and it is also the processing start time of the \(i+1\)-th procedure of the \(j\)-th workpiece; \(L\) presents a sufficiently large positive number.  
In addition, there is a buffer with a specific capacity in front of each piece of equipment, and the equipment is unreliable. The processing time of the equipment to the workpiece is variable.  
Assume the total time in an effective state of machine \(i\) can be expressed as:  
\[T_{i}=\sum_{q=1}^{n}\big{(}t_{qe}-t_{qs}\big{)} \tag{5}\]  
Assume the set of bottleneck machines can be denoted as:  
\[B=\big{\{}\,m|\text{max}(g_{i1},g_{i2}\cdots g_{ij}),m\in(M_{1},M_{2}\cdots M _{n})\big{\}} \tag{6}\]  
Abstractly represent the production activities involved in the manufacturing process as a manufacturing process time series flow model (MP-DataModel). MP-DataModel consists of multiple object data models (Ob-DataModel), and its description method is as follows:  
\[Ob\text{-}DataModel=\{Object,DSet\} \tag{7}\]  
where _object_ represents the manufacturing process object and _DSet_ represents the collected data generated by the object during the manufacturing process. _DSet_ is time-series data, and its description method is:  
\[DSet=\{<v(a_{1}\cdots a_{1m}),ct_{1}>,<v(a_{2}\cdots a_{2m}),ct_{2}>,\cdots,<v( a_{n1}\cdots a_{nm}),ct_{n}>\} \tag{8}\]