## 4 Case Application and Benefit-Cost Analysis
### Application and Benefit-Cost Analysis of the Stage of Sample Product Performance Evaluation  
Through the above analysis, \(SS_{2}\) is the optimal supplier for the balance shaft system. The task of \(SS_{2}\) is to cooperate with downstream suppliers to process and assemble products according to the proposed parts production techniques, and then deliver the balance shaft system to the engine assembly supplier for final assembly. The assembly process of the balance shaft system is: 1 press-fit the bearing on the cylinder block; 2 insert the balance shaft units (including counterweight block B) into the cylinder block; 3 set in the elastic gear; and 4 set in counterweight block A. During the assembly, it is required to ensure the dimensions and tolerances of the bearing, counterweights blocks (A and B), and the center of the elastic gear to the center of the balance shaft. L1, the dimension of counterweight block B to the center of the balance shaft, is taken as an example to verify the feasibility of the process quality control model at this stage. The composition loop of the dimension chain for the balance shaft system assembly is shown in Figure 9. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{3}{*}{**Key Parameter**} & \multicolumn{1}{c}{**Decentralized Decision-Making/**} & \multicolumn{1}{c}{**Centralized Decision-Making/**} & \multicolumn{1}{c}{**Decentralized Decision-Making/**} \\  & **without Considering** & **without Considering** & **Considering** & **Considering** \\  & **Inspection** & **Inspection** & **Inspection** & **Inspection** & **Inspection** \\ \cline{2-5}  & \multicolumn{1}{c}{**Existing Research Methods**[16; 17]} & \multicolumn{1}{c}{**Method Proposed in this Paper**} \\ \hline Maximum Profit for Engine Assembly Supplier (RMB) & 2500 & 1500 & 4210 & 4863 \\ \hline Maximum Profit for Balance Shaft System Small Assembly Supplier (RMB) & 3700 & 3600 & 6104 & 4268 \\ \hline Maximum Profit for Manufacturing Supply Chain (RMB) & 5300 & 4800 & 10,315 & 9130 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Decision results of manufacturing supply chain contract coordination in the preparation stage before product processing. Figure 9: Counterweight block B assembles dimension chain. In Figure 9, \(L_{11}=11\pm 0.1\), \(L_{12}=56.4\pm 0.2\), \(L_{13}=19.2\), \(L_{14}=52\pm 0.2\), and \(L_{1}=L_{11}+L_{12}+L_{13}+L_{14}+L_{15}=138^{+0.5}_{-0.5}+L_{15}\). Counterweight block \(B\) and the balance shaft are obtained through rough processing by parts suppliers, and then fine finished by the balance shaft system supplier. During the assembly, the dimension and tolerance of counterweight block \(B\) to the center of the coordinate system shall be ensured, which is \(L_{1}\) in Figure 9. The keys for the control accuracy of \(L_{1}\) are the dimensional accuracies of counterweight block \(B\) and the end face of the balance shaft, which require two processes: turning and grinding. Combined with the analysis in Section 3, the manufacturing precision assemblies of the balance shaft and counterweight block \(B\) are represented by \(V_{I31}\) and \(V_{I32}\), respectively. The manufacturing accuracies of turning them are \(w_{I31}\) and \(w_{I32}\), and the manufacturing accuracies of grinding the balance shaft system supplier are represented by \(w_{I33}\) and \(w_{I34}\). By adopting the cost function and quality loss function of turning and grinding presented in [30], the process quality control model at this stage is constructed:  
\[\left\{ \begin{aligned} & SPC(e_{31})=NPC(e_{31})+NPC(e_{32})\\ & SPC(e_{32})=+NPC(e_{33})+NPC(e_{34})\\ & M(c_{m})=C_{m}(w_{I31})+C_{m}(w_{I32})+C_{m}(w_{I33})+C_{m}(w_{I34 })\\ & M(c_{l})=C_{l}(w_{I31})+C_{l}(w_{I32})+C_{l}(w_{I33})+C_{l}(w_{I34 })\\ & MAX(SCPC),MIN(M(c_{m}),MIN(M(c_{l}))\end{aligned} \right.