## 4 Case Application and Benefit-Cost Analysis
### Application and Benefit-Cost Analysis of the Stage of Preparation before Product Processing  
In the practical case, two suppliers of the balance shaft system--SS1 and SS2--offered intention of cooperation. The biggest difference between the two proposals is the design of the driven gear. The key performance parameters of the driven gears provided by the existing supplier and the other two suppliers are shown in Table 3.  
Based on the content in Section 3, \(EMC(e_{ii})\)(\(i\) = 1, 2, 3, 4, 5), \(EMC(e_{2i})\)(\(i\) = 1, 2, 3, 4, 5), and \(EMC(e_{3i})\)(\(i\) = 1, 2, 3, 4, 5) represent the manufacturing capacity of the existing supplier, supplier \(SS1\), and supplier \(SS2\) based on mass, static unbalance, rotational inertia, processing cost, and damping rate. The comprehensive manufacturing capacity of the existing supplier, supplier \(SS_{1}\), and supplier \(SS_{2}\) can be represented by \(MSCMP_{1}\), \(MSCMP_{2}\), and \(MSCMP_{3}\). Based on model (1), the process quality control model for the three-cylinder engine balance shaft system at the stage of preparation before product processing can be expressed as:  
\[\left\{ \begin{array}{c}MSCMP_{1}=\sum\limits_{i=1}^{5}{(EMC(e_{1i}))}\\ MSCMP_{2}=\sum\limits_{i=1}^{5}{(EMC(e_{2i}))}\\ MSCMP_{3}=\sum\limits_{i=1}^{5}{(EMC(e_{3i}))}\\ MAX(MSCMP_{1},MSCMP_{2},MSCMP_{3})\end{array} \right. \tag{8}\]  
With SPSSAU 21.0 and the analytic hierarchy process (AHP), the weights and eigenvectors in the criteria hierarchy are calculated, and the final score of each supplier is obtained. The following results are obtained through calculation: \(MSCMP_{1}\) = 0.3226, \(MSCMP_{2}\) = 0.2146, \(MSCMP_{3}\) = 0.3956, thus \(MAX(MSCMP_{1},\)\(MSCMP_{2},\)\(MSCMP_{3})\) = 0.3956. The results indicate that supplier \(SS_{2}\) is the optimal supplier among the three. The product provided by supplier \(SS_{2}\) satisfies all the requirements in terms of performance indicators, and its cost is 27% lower than that of the existing supplier. Although the cost of supplier \(SS_{1}\) is 27% lower than that of the existing supplier, the damping rate of its product is only 55% of that of the existing supplier, leading to the lowest score. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parts**} & \multicolumn{4}{c}{**Performance Indicators**} & \multirow{2}{*}{**Processing Cost**} & \multirow{2}{*}{**Damping Effect**} \\ \cline{2-2} \cline{4-6}  & **Mass** & & & & & \\ \hline \multirow{2}{*}{Balance Shaft} & \multirow{2}{*}{558 g} & \(m_{xu}\) = \(\pm\)5 g\(\cdot\)cm & \multirow{2}{*}{1939 g\(\cdot\)cm\({}^{2}\)} & \multirow{2}{*}{48} & \multirow{2}{*}{Connector} \\  & & & & & \\ \hline \multirow{2}{*}{Drive Gear} & \multirow{2}{*}{290 g} & \(\pm\)0.1 g\(\cdot\)cm & 4450–5000g\(\cdot\)cm\({}^{2}\) & 112 & \
\begin{tabular}{c} Power \\ Transmission \\ \end{tabular} \\ \hline \multirow{2}{*}{Driven Gear} & \multirow{2}{*}{449 g} & \(\pm\)0.15 g\(\cdot\)cm & 4450–5000g\(\cdot\)cm\({}^{2}\) & 120 &
\begin{tabular}{c} Power \\ Transmission \\ \end{tabular} \\ \hline \multirow{2}{*}{Counterweight A} & \multirow{2}{*}{327 g} & \(m_{gg}\) = \(\pm\)5 g\(\cdot\)cm & \multirow{2}{*}{3483 g\(\cdot\)cm\({}^{2}\)} & \multirow{2}{*}{57} & \multirow{2}{*}{Balancing Torque} \\  & & \(m_{gg}\) = \(\pm\)5 g\(\cdot\)cm & & & \\ \hline \multirow{2}{*}{Counterweight B} & \multirow{2}{*}{273 g} & \(m_{xu}\) = \(\pm\)5 g\(\cdot\)cm & \multirow{2}{*}{1630 g\(\cdot\)cm\({}^{2}\)} & \multirow{2}{*}{45} & \multirow{2}{*}{Balancing Torque} \\  & & \(m_{gu}\) = 475 \(\pm\) 12 g\(\cdot\)cm & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Requirements for key performance indicators of the three-cylinder engine balance shaft system.