## 3 System Flexibility Digital Representation
### Data Modelling and Data Analysis
#### 3.3.1 Static Functional Model (SFM)  
The static function model (SFM) mainly describes and analyses the system static characteristic parameters, so as to evaluate the ability of the PS to meet the enterprise's requirements. The model focuses on the system feature analysis network (SFAN) described in Figure 3, which integrates and analyzes the static characteristic parameters according to the different interaction relationships of the product, process, and machine dimensions, and combines the proposed flexibility static evaluation metrics to form a multidimensional digital twin flexibility evaluation model. The multidimensional digital twin flexibility evaluation model covers a product-dimension flexibility evaluation model, process-dimension flexibility evaluation model, and machine-dimension flexibility evaluation model. With analyzing the relationships and properties of the components described in the descriptive metamodel, the characteristic matrices are defined in Equations (1)-(8), they are: product family judgment matrix \(VV\); product process correlation matrix \(VP\); product machine correlation matrix \(PM\); product process analysis matrix \(PP\); process sequence changeability matrix \(PT\); process insertability matrix \(MM\); product conversion time analysis matrix \(VQ\); conversion time analysis matrix \(VT\). \[VV=\left(vv_{il}\right)_{I\times I}\quad\forall i,l\in[1,I] \tag{1}\]  
\[VP=\left(vp_{ij}\right)_{I\times I}\quad\forall j\in[1,J] \tag{2}\]  
\[PM=\left(pm_{jk}\right)_{J\times K}\quad\forall k\in[1,K] \tag{3}\]  
\[PP=\left(pp_{il}/pp_{il}\right)_{I\times I} \tag{4}\]  
\[PT=\left(pt_{ij}\right)_{I\times I}\quad\forall s,j\in[1,J] \tag{5}\]  
\[MM=\left(mm_{ko}\right)_{K\times K}\quad\forall k,o\in[1,K] \tag{6}\]  
\[VQ=\left(vc_{i}\right)_{I\times I} \tag{7}\]  
\[VT=VT_{I\times I}=\left(tr_{il}\right)_{I\times I} \tag{8}\]  
Figure 3: System feature analysis network. Product dimension flexibility aims at analysing the similarity and the number of product varieties that a PS can produce. First, the structural and process similarity analysis matrix \(Si_{v}\) of the product is obtained by analyzing the product homology relationship and processing technology of all products. \[Si_{v}=0.5(PP+VV)=\left(Si_{v_{i},v_{h}}\right)_{n\times n} \tag{9}\]  
\[VP=\left(vp_{ij}\right)_{I\times I}\quad\forall j\in[1,J] \tag{2}\]0  
Combined with the quantity of product varieties that can be produced by the production system and the product similarity analysis matrix, the product similarity value \(Si\) is obtained, and \(Si\) can be given by  
\[VP=\left(vp_{ij}\right)_{I\times I}\quad\forall j\in[1,J] \tag{2}\]1  
The flexible evaluation of product dimension can be analysed from two aspects. For one thing, the more product families the production system can produce, the more flexibility the production system will be. For another, the lower the similarity between product varieties, the higher the flexibility of the production system. Therefore, the product dimension flexibility evaluation model \(F_{Pd}\) can be represented as:  
\[VP=\left(vp_{ij}\right)_{I\times I}\quad\forall j\in[1,J] \tag{2}\]2  
Process dimension flexibility is mainly determined by the adjustment degree of the processing technology for the existing production system structure, including the changing ability \(R_{o}\) of the processing path for each product and the production efficiency \(R_{q}\) of the production system to transform production between different product varieties. The calculation of \(R_{o}\) for the changeability of the product processing path includes the house of quality model of the process path variability (\(Rv\)), which is represented in \[VP=\left(vp_{ij}\right)_{I\times I}\quad\forall j\in[1,J] \tag{2}\]3  
Analysing the rank relationship of the process path variability quality house model, the quantity of alternative paths for product variety \(r_{v_{i}}\) is obtained. Thus the process path  
\[VP=\left(vp_{ij}\right)_{I\times I}\quad\forall j\in[1,J] \tag{2}\]4 The house of the process path variability.