## 2 Evaluating of Machine Criticality for an Automatic Production Line
\begin{array}{l}Ey_{j}=na+b_{1}E\lambda_{j1}+b_{2}E\lambda_{j2}+ \cdots+b_{K}E\lambda_{jK}\\ E\lambda_{j1}y_{j}=aE\lambda_{j1}+b_{1}E\lambda_{j1}{}^{2}+b_{2}E\lambda_{j1} \lambda_{j2}+\cdots+b_{K}E\lambda_{j1}\lambda_{jK}\\ E\lambda_{j2}y_{j}=aE\lambda_{j2}+b_{1}E\lambda_{j1}\lambda_{j2}+b_{2}E\lambda_ {j2}{}^{2}+\cdots+b_{K}E\lambda_{j2}\lambda_{jK}\\ \cdots\cdots\\ E\lambda_{jK}y_{j}=aE\lambda_{jK}+b_{1}E\lambda_{j1}\lambda_{jK}+b_{2}E\lambda _{j2}\lambda_{jK}+\cdots+b_{K}E\lambda_{jK}{}^{2}\end{array} \right., \tag{3}\]  
where the values of \(a\), \(b_{1}\), \(b_{2}\),..., \(b_{K}\) can be gained by solving Equations (3) above. The value of \(b_{i}\) indicates the effect of the machine failure rate \(\lambda_{i}\) on the operating rate of production line \(Y\), and the larger the \(b_{i}\), the greater the effect of \(\lambda_{i}\) on the operating rate \(Y\). Figure 1: The typical layout and structure of automatic production line, where \(B_{i}\) (\(i=1\), 2,..., \(K\)) is the buffer area, \(L_{i}\) (\(i=1\), 2,..., \(K\)) is the corresponding buffer capacity, \(\lambda_{i}\) (\(i=1\), 2,..., \(K\)) is the machine failure rate, and it is assumed that the time between failure (TBF) of each machine is Weibull distribution. Finally, the criticality of machine in the production line is evaluated according to the parameter \(b_{i}\) in Equation (3), and so the machines can be classified into three groups on the basis of the ABC (Activity Based Classification, ABC) analysis method [20; 21; 22]. The principle of ABC classification analysis method comes from [PERSON]'s principle, which classifies the top 15-20% goods occupied 65-80% value of whole system into A group, the following 30-40% goods occupied 15-20% value of whole system into B group, and the other 40-55% goods occupied 5-15% value of the whole system into C group. For the automatic production line, the most critical machines have an important effect on the operating rate of production line, so the top 20% (Upper limit of A group ratio in ABC classification method) machines were evaluated as the most critical machines (A group). The secondary critical machines have a lower influence on the operating rate of the automatic production line compared with that of the most critical machines and the production enterprises pay more attention to saving maintenance costs. Thus, the following 30% (Lower limit of B group ratio in the ABC classification method) of machines were evaluated as the secondary critical machines (B group). The other 50% of the machines were evaluated as the general machines (C group). Namely, the machines with the parameter \(b_{i}\) at the top 20% are classified into the most critical machines and that the following 20-50% are classified into the secondary critical machines; thus, the rest of machines are classified into the general machines. The failure of most and secondary critical machines all have a great influence on the operating rate of production line; therefore, the preventive maintenance should be carried out for them. The failure of general machines has a smaller influence on the operating rate, and the breakdown maintenance is suitable for the general machines.