## 4 Case Study
### Flexibility Evaluation  
(1) System Flexibility Characteristics  
To realize the digital expression of multidimensional system flexibility, the first step is to extract the data of flexibility feature in the system design scheme. According to the analysis of the two design schemes, the basic static characteristic parameters used to describe the flexibility in production system design schemes are extracted, part of parameters are shown in Table 3. Scheme A can produce four types of products, which belong to the same product family. Scheme A has 10 kinds of process capacity, and the number of device types is 9. Among them, the number of types of machine with processing capacity is seven (feeding machine and discharging machine are not counted). Scheme B can produce five types of products, which belong to two product families. Scheme B also has 10 kinds of process capacity, and the number of device types is 9. Among them, the number of types of machine with processing capacity is eight (AGV is not counted). For the following flexibility analysis, a random failure rate of 0-20% was set for each processing machine, with a failure rate increase of 0.5. The hourly output of each product in scheme A and B is 1028 pieces, and different production layouts and timing are given in the design documents of the two schemes. (2) Data modelling and data analysis  
The characteristic parameters extracted from the production system design documents are converted according to the matrix (1)-(8) can obtain interaction relation matrix between the scheme A and B. Then, according to the requirements of flexibility static evaluation model, quantitative indexes of evaluation models of product dimension, process dimension and equipment dimension can be obtained, as shown in Table 4. According to the requirements of Equations (22) and (23), the dynamic prediction of system flexibility can be realized based on the equipment layout, processing technology, product types, processing time sequence, and other information extracted from the design document of the production system scheme. \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \(Q_{V}\) & \(Q_{P}\) & \(Q_{M}\) & \(Q_{V\Psi y}\) \\ \hline Plan A & 4 & 10 & 9 & 1 \\ Plan B & 5 & 10 & 9 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Scheme A and B: basic flexibility static characteristic parameters. After feature extraction and modelling of the production system design scheme, data analysis can be carried out to evaluate the static flexibility of the system based on AHP and the dynamic flexibility of the system based on Plant Simulation platform. The dynamic flexibility evaluation model of the system design schemes are shown in Figure 6.  
The flexibility static evaluation of the PSDS combines the subjective demand of the enterprise for flexibility production. Therefore, a questionnaire survey was conducted on 15 production personnel (including a production manager, process analyst, and product quality inspector) in the demand enterprise to obtain their evaluation on the importance of different dimensions of flexibility in system design. The evaluation results are summarized and analyzed according to Equations (24)-(26), which helps to obtain the flexibility weights of different dimension. The weights are 0.26102, 0.21505, and 0.52393, respectively. Additionally, the static flexibility evaluation index in Table 5 can be obtained with the multi-dimensional flexibility evaluation index of the PSDS. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{3}{c}{**Product-Dimension**} & \multicolumn{3}{c}{**Process-Dimension**} & \multicolumn{3}{c}{**Machine-Dimension**} \\ \hline \multicolumn{3}{c}{\(Si\)} & \(F_{Pd}\) & \(R_{o}\) & \(R_{q}\) & \(F_{Pc}\) & \(MT\) & \(Mex\) & \(F_{M}\) \\ \hline A & 1 & 0.125 & 0 & 0.875 & 0.4375 & 0.0123 & 0 & 0.062 \\ B & 0.79 & 0.33 & 0 & 0.875 & 0.4375 & 0.0123 & 1 & 0.062 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Flexibility static evaluation index.