## 2 Process Quality Control Demand and Construction of Control Network
The weight of the capacity on the manufacturing accuracy \(w_{li}\) of each single node is recorded as \(NPC\)(\(e_{3i}\)); the overall manufacturing accuracy of the supply chain and the manufacturing accuracy of a single node are represented by \(SCPC\)(\(e_{3i}\)) and \(NPC\)(\(e_{3i}\)), respectively. Here, \(SCPC\)(\(e_{3i}\)) is not the linear superposition of \(NPC\)(\(e_{3i}\)), because the manufacturing accuracy of the previous node directly affects the manufacturing accuracy and cost of the next node, and the influencing coefficient \(\lambda=\{\lambda_{i}\ |\ i=1,\ 2,\ \ldots,\ 7\}\) is different. Meanwhile, the manufacturing cost of each node is represented by \(M\)(\(c\)) = \(\{mc_{i}\ |\ i=1,\ 2,\ \ldots,\ 7\}\). Considering the constraint of the manufacturing cost, the sum of the manufacturing accuracy on all arcs is the overall manufacturing capability of the three-cylinder engine balance shaft system at this stage, and the optimal combination of manufacturing accuracy of the node represents the maximum overall manufacturing accuracy. Therefore, the product process quality control model for the three-cylinder engine balance shaft system at the stage of sample product performance evaluation can be expressed as:  
\[\left\{ \begin{aligned} SCPC&=\sum\limits_{i=2}^{7} \left(NPC(e_{31})+\varphi_{i}NPC(e_{3i})+NPC(e_{38})\right)\\ & M(c)=\sum\limits_{i=1}^{7}mc_{i}\\ & MAX(SCPC),MIN(M(c))\end{aligned} \right. \tag{2}\]  
Note: since arc \(e_{38}\) represents the assembling process of the balance shaft system to the engine assembly, we assume that the assembling method and level are certain, thus the value of \(NPC\)(\(e_{38}\)) is fixed. (3) Stage of production and operation  
The stage of production and operation is the period after the three-cylinder engine balance shaft system comes onto the market. At this stage, the control relationship between the automobile manufacturer and suppliers over the operation state of the system, and the tracking and analysis of the operation state information are the theoretical basis for formulating an optimal maintenance scheme or implementing the emergency plan for recall events. Therefore, the operation performance states obtained by the automobile manufacturer (or the engine assembly supplier), by the node enterprises of the balance shaft system suppliers, and by the parts suppliers for the balance shaft, the gear, etc., are taken as the node element in the metagraph. The constrained relationship of operation performance states among parts, components, systems, and assemblies are taken as the arcs in the path. The metagraph for the stage of production and operation is shown in Figure 8.  
At the stage of production and operation, it is assumed that the operation performance information of the three-cylinder engine balance shaft system can be tracked and collected in real time. The arcs of the paths are represented by \(e_{4i}=<V_{14i},w_{14i}>\), in which \(V_{14i}\)(i = 1, 2,..., 5) is the overall operation performance of the balance shaft system. The weight of the capacity on \(e_{4i}\) is recorded as \(SAO\)(\(e_{4i}\)), and the weight of the capacity on  
Figure 8: Metagraph for the stage of production and operation. the manufacturing accuracy \(w_{li}\) of each single node is recorded as \(PPS(e_{4i})\). When the performance of one part in the manufacturing supply chain cannot meet the requirements, an alarm message will be generated. Therefore, the minimum operation performance of all arcs is the overall operating performance of the entire supply chain.