# Reliability Analysis of Multi-Process Machining Based on Information Entropy
## 4 The Reliability Analysis of the Machining Process
### Analysis of Error Transmission  
The machining quality of mechanical parts is determined by multiple processes [26]. The machining quality of each process is not constant and has certain fluctuations that often follow a normal distribution. The fluctuation of the machining quality of mechanical parts is also the result of each process's combined action of the machining quality fluctuations. Therefore, it is necessary to analyze the error transmission.  
There is a complex coupling relationship between processes. The machining quality of each process is not only affected by the 5M1E of the current process but the machining error of the previous processes [19]. It is specifically reflected in that the machining error of the previous processes will affect the machining quality distribution of the current process, as shown in Figure 4. It is difficult to model so many variables simultaneously in the actual analysis, and even if a model is established, the accuracy of the model is not very satisfactory. In addition, many contingent factors affect each process's machining quality [27], but these factors cannot be predicted beforehand. Therefore, an error transmission analysis method based on process quality distribution is proposed in this paper. The main idea is to determine the influence of the previous vertex on the current vertex by calculating the mutual information between the PQRV of two vertices. This method enables the quantification of error transmission and quantitatively evaluates the degree of correlation between two processes.  
Figure 4: Influence of machining errors in previous processes on the distribution of machining quality in the current process.  
In the single-chain case, the influence between two vertices is defined as the transfer coefficient \(T(B;C)\), where \(B\) is the PQRV of the previous vertex and \(C\) denotes the PQRV of the current vertex.  
\[T(B;C)=I(B;C) \tag{1}\]  
There are two necessary explanations for the error transmission analysis in the case of a single chain.  
1. The PQRV of the vertices is usually regarded as a continuous distribution, but it is necessary to discretize the continuous distribution into a discrete probability distribution for introducing concepts such as information entropy and mutual information. The specific measure divides the continuous distribution into intervals and treats each interval as a discrete value.
2. In information theory, mutual information is bidirectional [21], i.e., \(I(B;C)=I(C;B)\), but in the single-chain case, the transfer coefficient only represents the error transfer from the previous vertices to the current vertex, because the machining process is moving in one direction. Therefore, there is no \(T(B;C)=T(C;B)\).  
In the case of the multi-chain, the influence between vertices is defined as a transfer coefficient \(T\big{(}X^{i};Y\big{)}\), where \(Y=\big{(}Y^{1},\ldots,Y^{n}\big{)}\), \(T\big{(}X^{i};Y\big{)}\) denotes the sum of the influence of vertex \(V_{X^{i}}\) on vertex \(V_{Y}\), and satisfies the following equation:  
\[T(X^{i};Y)=I(X^{i},Y^{i})-\frac{1}{n}\sum_{i=1}^{n-1}I(Y^{i+1};Y^{i},Y^{i-1}, \ldots,Y^{1}) \tag{2}\]