## 2 Evaluating of Machine Criticality for an Automatic Production Line  
The operating rate is one of the most important performance indexes to evaluate the automatic production line, which is defined by the ratio of actual output to theoretical output in unit time. The main factors that influence the operating rate of production line are machine failure rate, mean time to repair (MTTR), buffer capacity and production line balancing rate. Because the automatic production line is often a complicated manufacturing system, it is very difficult to establish a definite analytical model between all influence factors and the operating rate. Considering the maintainability, the optimal maintenance strategy directly influenced the failure rate and repair rate of the machines; therefore, the machine failure rate was considered as the key influence factor of the operating rate in this study. The buffer capacity and balancing rate of the production line mostly depended on the design performance of production line and they were taken as the quantitative design parameters into the preventive maintenance model. For an automatic production line with \(K\) machines (\(M_{1}\), \(M_{2}\),..., \(M_{K}\)), the typical layout and structure are shown in Figure 1.  
For the automatic production line, the preventive maintenance is to inspect the machines, and replace parts and lubrication in a certain maintenance period. The preventive maintenance will make the machine return to the state of completely new or good, as long as the parts and lubrication are correctly replaced according to the criteria. The effect of preventive maintenance on the different machine is the same. Therefore, it can be assumed that preventive maintenance has the same effect on the failure rate of each machine, and the operating rates of production line are solved by simulation for the different combinations of the machines' failure rate. In order to ensure the simulation accuracy, many groups of combinations of the failure rate are needed, especially while the production line is comprised of a high number of machines, the simulation calculation will be very difficult and take a long time. To improve the calculation efficiency, the simulation analysis of the production line is carried out in the present study based on the uniform design method [18; 19]. First of all, the range of \(\lambda_{i}\) value of each machine can be determined according to the machines' failure data. The \(N\) levels were taken for each \(\lambda_{i}\) (\(N\) is 3-5 times the \(K\) of the machines), and then the uniform design table \(U_{N}\) (\(N^{5}\)) is established to combine the failure rate for each machine. Finally, the different machines' failure rate combinations are put into the simulation model to calculate the corresponding operating rates of production line \(Y_{j}\), and \(N\) groups of data samples (\(Y_{j}\), \(\lambda_{j1}\), \(\lambda_{j2}\),..., \(\lambda_{jK}\)) (\(j=1\), 2,..., \(K\)) are obtained. The failure rate of each machine in the production line doesn't affect each other, which means that the variables \(\lambda_{j}\) are independent of each other. Therefore, the regression equation of operating rate \(Y\) can be expressed as:  
\[\overset{\wedge}{Y}=a+b_{1}\lambda_{1}+b_{2}\lambda_{2}+\cdots+b_{K}\lambda_{ K}, \tag{1}\]  
where \(a\), \(b_{1}\), \(b_{2}\),..., \(b_{K}\) are the unknown parameters in Equation (1). The fitting regression equation is performed by the minimal mean square error method for the \(N\) groups of samples (\(Y_{j}\), \(\lambda_{j1}\), \(\lambda_{j2}\),..., \(\lambda_{jK}\)) (\(j=1\), 2,..., \(N\)) and expressed as:  
\[\min M=E\Big{(}Y_{j}-\overset{\wedge}{Y_{j}}\Big{)}^{2}=E\big{(}Y_{j}-a-b_{1 }\lambda_{j1}-b_{2}\lambda_{j2}-\cdots-b_{K}\lambda_{jK}\big{)}^{2}. \tag{2}\]  
The partial derivation of each parameter in Equation (2) is taken, and the partial derivation is set to be zero. The following equations are gained:  
\[\left\{