# Reliability Analysis of Multi-Process Machining Based on Information Entropy
## 3 Graphical Representation of Machining Process  
The machining process of mechanical parts is complicated, so it is essential to visualize the complex machining process. Graphs in Graph Theory can visually express the complex coupling relationship between different things, and it is very suitable to use the graph to present the complex machining process [15]. A graph is composed of a set of vertices and edges, an edge connecting two vertices. A vertex is an abstraction of something, and an edge connecting two vertices is used to represent a certain relationship between two things [23]. During the machining of mechanical parts, we can abstract the processes as the vertices in the graph and regard the relationship between the two processes as the edge between the two vertices [24]. In Figure 1, this type of graph representing the machining process is called a directed-weighted process graph (abbreviated as DWPG). The DWPG consists of vertices, edges and transfer coefficients, and has the following related definitions:  
1. Vertex: Representing the process in the machining, i.e., a vertex is equivalent to the corresponding process. All the vertices constitute a set V.
2. Edge: Representing the connection between two vertices, and from the previous process to the following process. All the edges constitute a set E.
3. Transfer coefficient: Representing the transmission of machining error between two processes. All the transfer coefficients constitute a set T.
4. In-degree: The number of edges pointing to a vertex. 5. Out-degree: The number of edges that a vertex points to other vertices. 6. Weighted in-degree: The sum of the transfer coefficients on the edges pointing to a vertex, which measures the influence of the machining errors of the previous vertices on the current vertex. 7. Weighted out-degree: The sum of the transfer coefficients on the edges of a vertex pointing to other vertices, which measures the extrapolation of vertex uncertainty. 8. Process quality random variable: The vertex's interval of process quality distribution is divided into multiple sub-intervals with width k. The values of the process quality that fall into the same sub-interval are regarded as one class. In this situation, the process quality of the vertex can be viewed as a discrete random variable, called the process quality random variable (abbreviated as PQRV). It is denoted by a capital letter. 9. Inherent process quality random variable: In the absence of interference from other vertices, the process quality distribution is determined by the intrinsic attributes of the vertex. At this time, the vertex's interval of process quality distribution is divided into multiple sub-intervals with width k. The values of the process quality that fall into the same sub-interval are regarded as one class. In this situation, the inherent process quality of the vertex can be viewed as a discrete random variable, called the inherent process quality random variable (abbreviated as IPQRV). It is denoted by a capital letter with an \"int\" subscript. The IPQRV of a vertex is an intrinsic property of the vertex (not shown in the DWPG). 10. Vertex entropy: Measuring the uncertainty of the PQRV. The larger the value, the higher the uncertainty. 11. Vertex inherent entropy: Measuring the uncertainty of the IPQRV. The larger the value, the higher the uncertainty. A DWPG contains multiple process subgraphs. A vertex with an out-degree but no in-degree represents the first process of machining a feature, i.e., machining directly on the blank. A vertex with an in-degree but no out-degree represents the last process of machining a feature, and other processes do not use the geometric elements of the feature as process benchmarks. This situation means that the feature does not affect other machining features. The DWPG can be viewed as composed of single-chain process subgraphs (called single-chain subgraphs) and multi-chain process subgraphs (called multi-chain subgraphs).