# A Scheduling Algorithm for Multi-Workshop Production Based on BOM and Process Route
## 3 Process Network Diagram Model Based on BOM and Process Route  
BOM is the product composition structure and its attribute description of enterprise production organization, and the process route is the relationship set of the product process and operation, which reflects the different process information of products. For the scheduling problem of complex products in a multi-workshop production environment, its research object has complex constraints, and its product and composition are complex. These components and their relationship can be expressed by the BOM structure of the product, and the constraint relationship between processes can be reflected by the product's process routes. If the processing and assembly process are regarded as the processing steps of the product, then a network diagram can be constructed by using BOM and process routes. The network diagram can describe the process relationship of product production. Through the process network diagram, the complex product scheduling problem across workshops can be described. The process network diagram is a directed diagram, in which the nodes represent the process of the product, the edges represent the constraints between the processes, the nodes without the succeeding ones represent the last process of the product, and the completion of the process indicates the completion of the product. The process network diagram of the complex products scheduling problem in multi-workshop production can be constructed as follows:  
1. For a single product, its BOM can be expressed as \(X=(P,A)\), in which \(P\) is the set of product nodes, \(A\) is the relationship matrix of the product nodes, and \(a_{ij}=1\) indicates that \(P_{i}\) is the child node of \(P_{j}\). \[A=\{a_{ij}\},i,j=1,...,n\] (1) \[a_{ij}=\left\{ \begin{array}{c}1,P_{i}\in P_{j}^{c}\\ 0\end{array} \right.\] (2)
2. The process route is the production process stage of products in different workshops--that is, the process flow between workshops; the process sequence is the internal process of products in each workshop. In the actual production environment, some processes of the product allow the selection of equipment in different workshops, so the problem of equipment balance caused by equipment selection in multi-workshop environment will arise. However, in this paper, we do not elaborate on the balance of multi-workshop equipment. The multi-workshop expression proposed in this paper is to facilitate the elaboration of the production process of the complex products. For the process route and process sequence of a product, it can be expressed as \(X^{\prime}\) = (\(G\), \(W\), \(Q\), \(B\), \(M\)), in which \(G\) is the set of product process routes, \(W\) is the set of workshops, \(Q\) is the set of product process nodes, \(B\) is the relation matrix of process nodes, and \(M\) is the set of equipment; \(g_{is}\) = 1 means that the product \(P_{i}\) is processed by workshop s, \(b_{isth}\) = 1 means that the process \(h\) of \(P_{i}\) in workshop \(s\) is the next operation of process \(k\), \(b_{iskn}\) = 1 means that the process \(k\) of \(P_{i}\) in workshop \(s\) is processed by equipment \(m\). For the process route and operation series of a product, it can be expressed as \(X^{\prime}\) = (\(Q\), \(B\)), in which \(Q\) is the process node set of a certain product, \(B\) is the relationship matrix of process nodes, \(b_{kh}\)=1 indicates that process \(h\) of \(P_{i}\) is the succeeding node of process \(k\). \[G=\left\{g_{is}\right\},i=1,...,n;s=1,...,w \tag{3}\]  
\[g_{is}=\left\{ \begin{array}{c}1,s\in G_{i}\\ 0\end{array} \right. \tag{4}\]  
\[B\)=\(\left\{b_{isth}\right\},k,h=1,...,l;Q_{isk},Q_{ish}\in Q_{is} \tag{5}\]  
\[l=L(i,s) \tag{6}\]  
\[b_{isth}=\left\{ \begin{array}{c}1,h\)=\(k+1\\ 0\end{array} \right. \tag{7}\]  
\[b_{iskm}=\left\{ \begin{array}{c}1,m\in M_{s}\\ 0\end{array} \right. \tag{8}\]  
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