## 2 PPCFP Problem Formulation
### Optimization Modeling
#### 2.2.3 Modeling  
From the above analysis, in order to optimize both the yield (Q) and the production capacity utilization (C) of two tasks simultaneously, optimization objectives can be defined as follows [33,34].  
\[Task1:\left\{\begin{aligned} & Maximize\quad Q_{1,1}=\\ & k_{p}(\frac{f*k_{d1}}{k_{t}})^{\frac{1}{2}}(I_{1}*e^{-\frac{t}{ \text{th}}})^{\frac{1}{2}}M_{1}e^{-10N_{1}}\\ & Maximize\quad C_{1,2}=\\ &\frac{k}{e^{(w_{1}*M_{1}+w_{2}*I_{1}+w_{3}*N_{1})}e^{-(334.15-T_ {1})}}\\ & Task2:\left\{\begin{aligned} & Maximize\quad Q_{2,1}=\\ & k_{p}(\frac{f*k_{d2}}{k_{t}})^{\frac{1}{2}}(I_{2}*e^{-\frac{t}{ \text{th}}})^{\frac{1}{2}}M_{2}e^{-10N_{2}}\\ & Maximize\quad C_{2,2}=\\ &\frac{k}{e^{(w_{1}*M_{2}+w_{2}*I_{2}+w_{3}*N_{2})}e^{-(334.15-T_ {2})}}\\ \end{aligned}\right.\\ & s.t.\left\{\begin{aligned} & M_{min}\leq M_{i}\leq M_{max}, & i=1,2.\\ & I_{min}\leq I_{i}\leq I_{max},& i=1,2.\\ & N_{min}\leq N_{i}\leq N_{max},& i=1,2.\\ \end{aligned}\right.\\ \end{aligned}\right. \tag{2}\]  
As the Equations (3) and (5) show, the \(T_{i}\) require to be calculated by mathematical formulas. According to the heat conservation theorem, the change of heat in the polymerization reactor is equal to the sum of the input and output energy of the system. For simplicity,  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **t(S)** & \(T_{1}\)**(T)** & \(q_{M_{1}}\)**(g/S)** & \(T_{2}\)**(T)** & \(q_{M_{1}}\)**(g/S)** \\ \hline Conditon 1 & 1 & 333.96 & 1054 & 333.97 & 946 \\ Conditon 2 & 2 & 333.77 & 1028 & 333.78 & 972 \\ Conditon 3 & 3 & 333.57 & 1042 & 333.62 & 958 \\ Conditon 4 & 4 & 333.35 & 1084 & 333.47 & 916 \\ Conditon 5 & 5 & 333.23 & 1001 & 333.23 & 999 \\ Conditon 6 & 6 & 333.04 & 1012 & 333.06 & 988 \\ Conditon 7 & 7 & 332.78 & 1067 & 332.95 & 933 \\ Conditon 8 & 8 & 332.57 & 1083 & 332.80 & 917 \\ Conditon 9 & 9 & 332.48 & 1014 & 332.53 & 986 \\ Conditon 10 & 10 & 332.22 & 1058 & 332.42 & 942 \\ \hline \hline \end{tabular}
\end{table}
Table 4: 10 typical operational conditions in production process.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Name** & \(M_{1}\) & \(I_{1}\) & \(N_{1}\) & \(M_{2}\) & \(I_{2}\) & \(N_{2}\) \\ \hline Lower & 0.1 & 0.01 & 0.01 & 0.1 & 0.01 & 0.01 \\ bound & 1 & 0.1 & 0.1 & 1 & 0.1 & 0.1 \\ Unit & mol/L & mol/L & mol/L & mol/L & mol/L & mol/L \\ \hline \hline \end{tabular}
\end{table}
Table 3: Boundaries of decision variables.  
the changes in chemical reaction heat, kinetic energy, and potential energy of solution in the polymerization reactor are negligible. The formula is presented in Equation (9) [35].  
\[mH\frac{dT_{i}}{dt} =q_{M_{i}}H(T_{si}-T_{i})\] \[\quad+U_{tra}A_{tra}(\overline{T_{jac}}-T_{i}),i=1,2. \tag{9}\]  
Assume the temperature of the solution \(T_{s}\) during the time from the polymer reservoir to the polymerizer satisfies exponential distribution. The formulas are shown in Equation (10).  
\[T_{si}=\left\{\begin{array}{ll}\mu_{i}e^{-\lambda_{i}t}&\text{if }t\geq 0,i=1,2.\\ 0&\text{others}\end{array}\right. \tag{10}\]  
As shown in Equations (2) and (4), the \(k_{d}\) requires to be calculated, which is likely to vary with ambient temperature [36], and its relationship with temperature can be expressed by the widely used Arrhenius formula. The formula is presented in Equation (11).  
\[k_{di}=A_{d}e^{-(\frac{k_{d}}{K_{i}})^{\beta}},i=1,2. \tag{11}\]