## 2 Process Quality Control Demand and Construction of Control Network
The process quality control model of the three-cylinder engine balance shaft system at the stage of production and operation can be expressed as:  
\[APMSCOP=MIN(SAO(e_{41}),PPS(e_{42}),PPS(e_{43}),PPS(e_{44}),PPS(e_{45})) \tag{3}\]  
2.2 Benefit and Cost Analysis of Process Quality Control Model of the Three-Cylinder Engine Balance Shaft System for Manufacturing Supply Chain  
(1) Stage of preparation before product processing  
According to the basic definition in 4.1, the demand function for three-cylinder engine assembly suppliers is:  
\[D(p,\theta,d)= \begin{bmatrix}a&0&0&0\\ 0&-p&0&0\\ 0&0&\theta&0\\ 0&0&0&d\end{bmatrix}\begin{bmatrix}1\\ b\\ r\\ t\end{bmatrix}  \tag{4}\]  
In the equation, \(a\) is the engine assembly supplier's demand order capacity for this product, \(\theta\) is the production quality level of the product provided by the balance shaft system small assembly supplier, \(p\) is the sales price of the balance shaft system, and \(d\) is the testing level of the engine assembly supplier. \(b\), \(r\), and \(t\) are, respectively, the sensitivity of the three-cylinder engine assembly supplier demand to production quality, price, and the testing level. Assume \(c_{s}(\theta)=k\theta^{2}\) and \(c_{m}(d)=\lambda d^{2}\), where \(k\) is the impact coefficient of the cost invested by the balance shaft system supplier for production quality, \(\lambda\) is the impact coefficient of the cost invested by the engine assembly supplier for testing, \(c_{s}(\theta)\) is the machining cost invested by the balance shaft system supplier to ensure the production quality level, and \(c_{m}(d)\) is the testing cost invested by the three-cylinder engine assembly supplier. The overall profit maximization of the manufacturing supply chain is used as the indicator to determine the quality level and propose appropriate prices to the balance shaft system small assembly suppliers. At the stage of preparation before product processing, the product process quality contract coordination model for the three-cylinder engine assembly supplier, balance shaft system small assembly supplier, and manufacturing supply chain can be shown in Formula (5). \[ \begin{array}{ll}\max_{p,\theta,d}&\prod_{T}(p,\theta,d)=\ (p-c)\times \left[\begin{array}{cccc}a&0&0&0\\ 0&-p&0&0\\ 0&0&\theta&0\\ 0&0&0&d\end{array}\right]\left[\begin{array}{c}1\\ r\\ b\\ t\end{array}\right]-c_{s}(\theta)-c_{m}(d)\\ &\\ &\prod_{S}(\theta)=(w-c)\times\left[\begin{array}{cccc}a&0&0&0\\ 0&-p&0&0\\ 0&0&\theta&0\\ 0&0&0&d\end{array}\right]\left[\begin{array}{c}1\\ r\\ b\\ t\end{array}\right]-c_{s}(\theta)>0\\ &\\ &\prod_{M}(p,d)=(p-w)\times\left[\begin{array}{cccc}a&0&0&0\\ 0&-p&0&0\\ 0&0&\theta&0\\ 0&0&0&d\end{array}\right]\left[\begin{array}{c}1\\ r\\ b\\ t\end{array}\right]-c_{m}(d)>0\\ &\\ &p>w>c\end{array}  \tag{5}\]  
In Formula (5), \(w\) is the expected wholesale price proposed by the engine assembly supplier to the balance shaft system supplier, and \(c\) is the machining cost of the balance shaft system supplier's products. \(\Pi_{M}(p,d)\), \(\Pi_{S}(\theta)\), and \(\Pi_{T}(p,\theta,d)\), respectively, represent the revenue functions of the three-cylinder engine assembly supplier, the small assembly supplier of the balance shaft system, and the manufacturing supply chain. (2) Stage of sample product performance evaluation At this stage, the three-cylinder engine balance shaft assembly supplier and the small assembly supplier have established a clear cooperative relationship. How to better ensure the design quality of the balance shaft system is the main task at this stage. Therefore, different from [15], this paper proposes to take the balance shaft system supplier as the main entity for process quality control to construct the demand function of the three-cylinder engine assembly supplier, as shown in Formula (6). \[D(p,\theta,e)= \begin{bmatrix}a&0&0&0\\ 0&-p&0&0\\ 0&0&\theta&0\\ 0&0&0&e\end{bmatrix}\begin{bmatrix}1\\ b\\ r\\ \beta\end{bmatrix}