## 4 Numerical Results and Sensitivity Analysis
### Sensitivity Analyses According to the Customer Preference \(\rho\)
Therefore, for the same values of remanufacturing decisions \(\tau(t)\), the augmentation in the value of \(B\) resulted in a rise in the collection cost, leading to a drop in the total profits generated by the manufacturer. Accordingly, by increasing the scaling parameter \(B\), the optimal remanufacturing rate \(\tau^{*}(t)\) decreased during all the periods in order to reduce the costs related to collecting used products. Thus, the costs generated by the remanufacturing process dictated the remanufacturing strategy adopted by the manufacturer. Also, as mentioned in Section 4.1, the obtained optimal remanufacturing rate \(\tau^{*}(t)\) fluctuated from one period to another following the demands and the market size \(A(t)\). Hence, for all the values of \(B\) considered in this study, the highest and the lowest remanufacturing efforts were achieved during the 7 th and the 10 th periods, respectively. Figure 6 below describes the evolutions of key financial indicators according to the different values of the scaling parameter \(B\) of the cost of collecting used products. As already elaborated, as a consequence of the increase in the scaling parameter \(B\), the optimal remanufacturing rate \(\tau^{*}(t)\) obtained by the proposed model in this study significantly decreased for each period \(t\). Therefore, the manufacturing process of new products required less remanufactured components causing a rise in the optimal unit manufacturing cost (\(Mc-\delta\cdot\tau^{*}(t)\)). Essentially, this increase was caused by the decrease in the optimal unit saving related to the remanufacturing activities \(\delta\cdot\tau^{*}(t)\), due to the drop in the remanufacturing rates. Hence, the total savings generated by the remanufacturing process cumulated by the manufacturer during the planning horizon \(H\) significantly decreased whenever the scaling parameter of the collection cost increased from \(B\) = 200,000 \(mu\) to \(B\) = 900,000 \(mu\). As a matter of fact, the optimal total savings related to the remanufacturing process dropped by almost 79%, as presented in Figure 6. The increase in the unit optimal \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \(B\) & \(\tau^{*}(t)\) & \(\tau^{*}(2)\) & \(\tau^{*}(3)\) & \(\tau^{*}(4)\) & \(\tau^{*}(5)\) & \(\tau^{*}(6)\) & \(\tau^{*}(7)\) & \(\tau^{*}(8)\) & \(\tau^{*}(9)\) & \(\tau^{*}(10)\) & \(\tau^{*}(11)\) & \(\tau^{*}(12)\) & \(mean\) \(\tau^{*}(t)\) \\ \hline
200,000 & 0.36 & 0.29 & 0.44 & 0.52 & 0.37 & 0.29 & 0.67 & 0.59 & 0.37 & 0.22 & 0.29 & 0.37 & 0.4 \\ \hline
300,000 & 0.24 & 0.19 & 0.29 & 0.34 & 0.24 & 0.19 & 0.44 & 0.39 & 0.24 & 0.14 & 0.19 & 0.24 & 0.26 \\ \hline
400,000 & 0.18 & 0.14 & 0.22 & 0.25 & 0.18 & 0.14 & 0.33 & 0.29 & 0.18 & 0.11 & 0.14 & 0.18 & 0.2 \\ \hline
500,000 & 0.14 & 0.11 & 0.17 & 0.2 & 0.14 & 0.12 & 0.26 & 0.23 & 0.14 & 0.09 & 0.12 & 0.14 & 0.15 \\ \hline
600,000 & 0.12 & 0.1 & 0.14 & 0.17 & 0.12 & 0.1 & 0.22 & 0.19 & 0.12 & 0.07 & 0.1 & 0.12 & 0.13 \\ \hline
700,000 & 0.1 & 0.08 & 0.12 & 0.14 & 0.1 & 0.08 & 0.19 & 0.17 & 0.1 & 0.06 & 0.08 & 0.1 & 0.11 \\ \hline
800,000 & 0.09 & 0.07 & 0.11 & 0.13 & 0.09 & 0.07 & 0.16 & 0.14 & 0.09 & 0.05 & 0.07 & 0.09 & 0.1 \\ \hline
900,000 & 0.08 & 0.06 & 0.1 & 0.11 & 0.08 & 0.06 & 0.14 & 0.13 & 0.08 & 0.05 & 0.06 & 0.08 & 0.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimal remanufacturing strategy \(\tau^{*}(t)\) according to the scaling parameter \(B\). Figure 6: Optimal total savings and collection cost according to the scaling parameter \(B\). manufacturing cost due to the decrease in the unit savings led to a drop in the optimal total profit of the supply chain of 2.8%. On the other hand, despite the rise in the scaling parameter \(B\), the collection cost \(B\cdot\tau^{\prime}(t)^{2}/2\) decreased for each \(t\) due to the heavy drop in the obtained optimal remanufacturing effort. Therefore, the total collection cost of used products during the planning horizon \(H\) experienced a drop of 79% in response to the rise of the scaling parameter \(B\).