## 4 Closed-Loop Supply-Chain Decision Making and Model Analysis
### Incentive Mechanism Design
#### 4.2.1 Information Screening Contract Model
The sequence of the recycling channel decision is shown in Table 2.  
When the power battery manufacturer (the entrusting party) faces the recycler (the agent) with different types of recycling and dismantling capabilities in the market, in order to enable the agent to truthfully report its true dismantling use level \(r_{i}\), the entrusting party must satisfy the agent's Participation Constraint (IR). At the same time, to induce the agent to adopt the optimal recovery effort level, the Principal Information Screening Contract must satisfy the agent's incentive compatibility constraint (IC), that is, the recycler expects profit maximization. The client provides different types of contracts \(\{(T_{H},\beta_{H}),(T_{L},\beta_{L})\}\), enabling high-capacity recyclers and low-capacity recyclers to choose the type of contract, allowing them to avoid untruthful reporting. This achieves the goal of information screening of recycling capability and proposed efforts. The contract can be obtained by solving the following mode:  
\[\max_{T_{L}\beta_{L}T_{H},\beta_{H}}E(\Pi_{M}) =\delta\left[\left(e_{\text{HH}}r_{H}+\theta\right)\left(f(v)\left( p_{n}-c_{r}\right)-f(v)\beta_{H}-p_{t}\right)-T_{H}\right]+ \tag{5}\] \[\left(1-\delta\right)\left[\left(e_{\text{LL}}r_{L}+\theta\right) \left(f(v)\left(p_{n}-c_{r}\right)-f(v)\beta_{L}-p_{t}\right)-T_{L}\right]\] \[s.t =\left\{ \begin{array}{l}\max_{\epsilon_{HH}}E(\Pi_{T_{H}r_{H}^{ \prime}})\geq\max_{\epsilon_{HL}}E(\Pi_{T_{T_{H}r_{L}}})\\ \max_{\epsilon_{LL}}E(\Pi_{T_{T_{L}r_{L}}})\geq\max_{\epsilon_{LH}}E(\Pi_{T_{ T_{L}r_{H}}})\\ E(\Pi_{T_{H}r_{H}})\geq\pi\\ E(\Pi_{T_{T_{L}r_{L}}})\geq\pi\end{array} \right.\] (6) Among them: \[\max_{\epsilon_{HH}}E(\Pi_{T_{H}r_{H}}) =\left(e_{\text{HH}}r_{H}+\theta\right)\left[f(v)\beta_{H}-p_{m}+ p_{t}\right]-\frac{1}{2}ke_{\text{HH}}^{2}-\frac{1}{2}\rho\sigma^{2}\left(f(v) \beta_{H}\right){}^{2}+T_{H}\] (10) \[\max_{\epsilon_{HL}}E(\Pi_{T_{H}r_{L}}) =\left(e_{\text{HL}}r_{H}+\theta\right)\left[f(v)\beta_{L}-p_{m}+ p_{t}\right]-\frac{1}{2}ke_{\text{HL}}^{2}-\frac{1}{2}\rho\sigma^{2}\left(f(v) \beta_{L}\right){}^{2}+T_{L}\] (11) \[\max_{\epsilon_{LL}}E(\Pi_{T_{L}r_{L}}) =\left(e_{\text{LL}}r_{L}+\theta\right)\left[f(v)\beta_{L}-p_{m}+ p_{t}\right]-\frac{1}{2}ke_{\text{LL}}^{2}-\frac{1}{2}\rho\sigma^{2}\left(f(v) \beta_{L}\right){}^{2}+T_{L}\] (12) \[\max_{\epsilon_{LH}}E(\Pi_{T_{L}r_{H}}) =\left(e_{\text{LH}}r_{L}+\theta\right)\left[f(v)\beta_{H}-p_{m}+ p_{t}\right]-\frac{1}{2}ke_{\text{LH}}^{2}-\frac{1}{2}\rho\sigma^{2}\left(f(v) \beta_{H}\right){}^{2}+T_{H} \tag{13}\]  
In the formula, constraints (6) and (7) are the incentive compatibility constraints (IC), and Formula (6) indicates that the high dismantling use ability recycler faithfully chooses the contract \((T_{H},\beta_{H})\). At the given level of concealment, Formula (7) indicates that recyclers with low disassembly capability truthfully selects the contract \((T_{L},\beta_{L})\) whose profit is greater than or equal to the level at which the ability is disguised. Formulae (8) and (9) represent the participation constraint (IR), where \(\pi\) retains the profit for the agent, indicating that the recycler accepts the contract on the condition that its guaranteed equivalent income is greater than or equal to \(\pi\).