## 4 Closed-Loop Supply-Chain Decision Making and Model Analysis  
### Profit Function Construction  
The power battery manufacturer is the entrusting party in the closed-loop supply chain system, and entrusts third-party recyclers to recycle used power storage batteries in the commercial market and to carry out dismantling and use processes to extract available elements as remanufactured raw materials. When a battery manufacturer signs a contract with a recycler that has a disassembly capacity of \(r_{i}\), the battery manufacturer expects the profit to be:  
\[\Pi_{M_{i}}=\left(e_{\mathrm{ij}}r_{i}+\theta+\xi\right)\left[f(v)\left(p_{n} -c_{r}\right)-f(v)\beta_{j}-p_{t}\right]-T_{j} \tag{1}\]  
Since the battery manufacturer is risk-neutral, its expected profit is equal to its expected utility, and the battery manufacturer expects the utility to be:  
\[E(\Pi_{M_{i}})=\left(e_{\mathrm{ij}}r_{i}+\theta\right)\left[f(v)\left(p_{n} -c_{r}\right)-f(v)\beta_{j}-p_{t}\right]-T_{j} \tag{2}\]  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Decision Variables** & \\ \hline \(T_{j}\) & The manufacturer pays the recycler a fixed payment \\ \(\beta_{j}\) & Proportion of revenue sharing between manufacturers and recyclers \\ \hline
**Model Parameter** & \\ \hline \(p_{t}\) & Manufacturer gives recycler maximum recycling price \\ \(p_{m}\) & Recyclers pay the largest recycling price for auto companies, \(p_{t}>p_{m}\) \\ \(p_{n}\) & Power battery sales price \\ \(c_{r}\) & Dismantling raw material cost after use \\ \(Q\) & Recycling function, \(Q=r_{i}e_{ij}+\theta+\xi\) \\ \(r_{i}\) & Recycler’s real disassembly use level \\ \(e_{ij}\) & Recycler recovery effort level, i indicates the real information of the recycler, and j indicates the information reported by the recycler to the manufacturer \\ \(\theta\) & Recovery amount when recovery effort is 0 \\ \(\xi\) & Market random factor \\ \(\mu\) & Recycler recycling effort costs, \(\mu=e_{ij}^{2}/2\) \\ \(\delta\) & Proportion of high recycling and recycling capacity recyclers in the market, \(\delta\in[0,1]\) \\ \(f(v)\) & Battery remanufacturing rate, \(f(v)=v^{r}\in[0,1]\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Decision variables and model parameters.  
The third-party recycler is based on a risk aversion feature, and its utility function uses the negative exponential function \(U_{R}(\Pi_{T})=-e^{\rho\Pi_{T}}\), where \(\rho\) is the degree of risk aversion, and \(\rho\in\{\rho>0,\rho=0,\rho<0\}\), indicating cases of being risk adverse, risk neutral, and risk tolerant, respectively. When the real dismantling ability is ri and the reporting ability to the client is ri, the third-party recycler expects the profit to be:  
\[\Pi_{T_{ij}}=\left(e_{\bar{\mathrm{i}}}r_{i}+\theta+\xi\right)\left[f(v)\beta_ {j}-p_{m}+p_{t}\right]-\frac{1}{2}ke_{\bar{\mathrm{i}}}^{2}+T_{j} \tag{3}\]  
According to the deterministic equivalent income method, the solution yields the expected utility of the recycler \(E(\Pi_{T_{ij}})\):  
\[E(\Pi_{T_{ij}})=\left(e_{\bar{\mathrm{i}}}r_{i}+\theta\right)\left[f(v)\beta_ {j}-p_{m}+p_{t}\right]-\frac{1}{2}ke_{\bar{\mathrm{i}}}^{2}-\frac{1}{2}\rho \sigma^{2}\left[f(v)\beta_{j}\right]^{2}+T_{j} \tag{4}\]