# Unfair and Risky? Profit Allocation in Closed-Loop Supply Chains by Cooperative Game Approaches
## 5 Numerical Experiment  
In this section, we illustrate how L's risk aversion parameter \(\eta\) and F's fairness concern parameter \(\lambda\) affect profit distribution under the five models. To obtain the schemes of profit and utility in relation to parameters \(\eta\) and \(\lambda\) only, we set other variables and parameters at: the variance of demand uncertainty \(\sigma^{2}=0.64\), the correlation coefficient between new products and remanufactured products \(\rho=-0.5\), unit production cost for a new product \(c_{n}=1\), unit production cost for a remanufactured product \(c_{r}=0.5\), exogenous unit cost for recycling a preowned product \(A=0.1\), consumers' value deduction for a remanufactured product \(\delta=0.5\). By plugging these values into the profit and utility functions in Section 4, we obtain individual profit and utility functions with variables \(\eta\) and \(\lambda\). The effect of L's risk aversion parameter \(\eta\) and F's fairness concern parameter \(\lambda\) on profit utility is graphically illustrated in Figures 2 and 3. In the DC model, the utility of F is negative, as shown in Figure 2c, due to \(\frac{\pi_{\text{DC}}^{DC}}{\pi_{\text{M}}^{DC}}<\frac{\lambda}{\lambda+1}\). Figure 2 demonstrates that in the DC model, L's risk aversion parameter \(\eta\) affects all parties in different ways. [PERSON]'s utility decreases linearly as \(\eta\) increases. [PERSON]'s profit and [PERSON]'s utility have non-linear relations with \(\eta\) and \(\lambda\).  
Figure 3 visually shows how \(\eta\) and \(\lambda\) affect individual profit and utility under different models. Figure 2a illustrates that [PERSON]'s profit in the LF model is affected by both \(\eta\) and \(\lambda\). Figure 2b shows that [PERSON]'s profit utility in the MF model is in a negatively linear relation to its risk aversion parameter \(\eta\). Figure 2c shows that [PERSON]'s profit utility in the ML model is in a positive linear relation to its fairness concern parameter \(\lambda\). By comparing Figures 2a and 3a, it can be seen that [PERSON]'s profit is significantly higher in the LF model than in the DC model.  
Figure 3: Individual utilities under the LF, MF, and ML models are: (**a**) [PERSON]’s profit in the LF model; (**b**) [PERSON]’s utility in the MF model; (**c**) [PERSON]’s utility in the ML model.  
Figure 2: Individual utilities under the DC model are: (**a**) [PERSON]’s profit in the DC model; (**b**) [PERSON]’s utility in the DC model; (**c**) [PERSON]’s utility in the DC model.  
By comparing Figures 2b and 3b, it can be seen that [PERSON]'s utility is slightly higher in the MF model than in the DC model. By comparing Figures 2c and 3c, it can be seen that [PERSON]'s utility is higher in the ML model than in the DC model.  
These numerical studies confirm that the decentralized case is the worst for profit optimization among all models. [PERSON], [PERSON], and [PERSON] achieve higher optimal profit even when the other two parties form a coalition than when no coalition exists. In addition, it is further verified that the impact of risk aversion parameter \(\eta\) and fairness concern parameter \(\lambda\) is dynamic, not always positive or negative. The larger \(\lambda\) does not always lead to more profit being transferred to F. This phenomenon is different from what has been observed in supply chains [18] which consider only fairness concern but not risk aversion. The larger \(\eta\), the less profit is transferred to L. It is understandable that lower risk endurance brings lower profit return.