# Research on Process Quality Prediction and Control of Spindle Housings in Flexible Production Lines
## 4 Optimization of Critical Process Quality Prediction Control Based on GM(1,1)-BP
### Process Quality Eigenvalue Prediction Model  
The GM(1,1) prediction model is a standard grey system prediction model, which is based on the principle of first-order accumulation of quality data of crucial process characteristics collected through a spindle box flexible production line to generate sequences, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Indicator Layer A** & **Indicator Weights** & **Sub-Indicator Layer B** & **Sub-Indicator Weights** & **Total Weights** \\ \hline Machinery and & & Maintenance & 0.333 & 0.075 \\ Equipment & 0.225 & Tool management & 0.296 & 0.067 \\  & & Operation calibration & 0.371 & 0.083 \\ \hline \multirow{3}{*}{Materials} & \multirow{3}{*}{0.181} & Raw material quality & 0.371 & 0.067 \\  & & Process inspection & 0.333 & 0.060 \\  & & Material economy & 0.296 & 0.054 \\ \hline \multirow{3}{*}{Methods} & \multirow{3}{*}{0.203} & Reasonable tooling solutions & 0.450 & 0.091 \\  & & Reasonable process planning & 0.550 & 0.112 \\ \hline \multirow{3}{*}{Measurement} & \multirow{3}{*}{0.203} & Measurement accuracy requirements & 0.296 & 0.060 \\  & & Maintenance or replacement & 0.333 & 0.068 \\  & & Measurement process specification & 0.371 & 0.075 \\ \hline Environment & 0.188 & Dust contamination & 0.261 & 0.049 \\  & & Safety & 0.278 & 0.053 \\  & & Recyclability & 0.244 & 0.046 \\  & & Vibration noise & 0.217 & 0.041 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Total ranking of indicators at each level. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Main Influencing Factors** & **Control Methods and Measures** \\ \hline \multirow{3}{*}{1} & Reasonable Tooling Solutions & 1. Regularly check whether the horizontal machining center fixture is damaged \\  & & 2. Check that the fixture is reasonable and correct before machining \\  & & 3. Check that the jig adjustment top position is in place \\ \hline \multirow{3}{*}{2} & Reasonable process planning & 1. Regularly analyze the quality data of the spindle box for process improvement \\  & & 2. Stop production to repair abnormal process problems \\  & & 3. Repeatedly determine the process for different specifications of customized \\  & & spindle cases \\ \hline \multirow{3}{*}{3} & \multirow{3}{*}{Equipment operation verification} & 1. Horizontal machining center repair and calibration \\  & & 2. Calibration of fixed-point positioning of the AGV trolley operation \\ \cline{1-1}  & & 3. Robot gripping and placement position verification \\ \cline{1-1}  & & 4. Calibration of the collaboration between the three-dimensional exchange device \\ \cline{1-1}  & & and the transportation line \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Process quality control scheme for the flexible production line of a spindle case. establish differential equations, to generate accumulative prediction sequences by using a least squares transpose matrix, and finally to obtain the original sequence prediction values by accumulative reduction [20]. Usually, the original data sequence \(X^{0}\) is input in a grey system prediction model, the input original sequence values must all be positive, and the original sequence data are processed non-negatively before the feasibility analysis. The input data are shown in Equation (5):  
\[X^{0}=\left(x^{0}_{(1)},x^{0}_{(2)},\ldots,x^{0}_{(i)},\ldots,x^{0}_{(n)}\right) \tag{5}\]  
where n represents that there are n sample observations, and for each data in the original sequence \(X^{0}\) for the level ratio validation, all the level ratio values \(\sigma\) in the sequence should be within the specified range [21].