## 4 Numerical Results and Sensitivity Analysis
### Sensitivity Analyses According to the Customer Preference \(\rho\)
Thus, as presented in Figure 5, the increase in the value of the customer preference for purchasing the product via the online platform significantly impacted the total profits generated by both distribution channels during the planning horizon \(H\). Specifically, the shift in the customer preference \(\rho\) from 0.2 to 0.8 led to a remarkable increase in the optimal total profit generated by the online platform \(\pi_{O}\)* of 341% and a corresponding increase in the total commission paid by the online platform in return for the provided AS by the pharmacy of 183%. These results were directly linked to the rise in the optimal demands \(D_{o}\)*(\(t\)) and the optimal unit selling prices \(P_{o}\)*(\(t\)) of the online platform for all the periods of study. However, the pharmacy's optimal total profit \(\pi_{P}\)* experienced a 63% decrease due to the drop in both optimal pharmacy's demands \(D_{p}\)*(\(t\)) and optimal unit selling prices \(P_{p}\)*(\(t\)) proposed by the pharmacy. Notably, for \(\rho\) = 0.8, half of the total profit of the pharmacy was generated through the received commission from the online channel for the AS. Thus, the sensitivity analyses on the obtained solutions conducted in this subsection show that the changes in the value of the customer preference towards purchasing products online significantly affects the optimal pricing decisions and demands in both distribution channels, resulting in an important impact on the optimal total profits generated in each channel and the total sales commission between the online platform and the pharmacy. Hence, precisely studying the customer behavior and preferences is an essential activity for managers in order to adopt an adequate pricing strategy that optimizes their total profits. Sensitivity Analyses following the Variations in the Scaling Parameter of the Collection Cost \(B\)  
This subsection is dedicated to assessing and examining the impact of a variation in the value of the collection cost's scaling parameter \(B\) on the proposed optimal solutions, specifically the optimal remanufacturing strategy. In fact, designing and wisely managing the remanufacturing process rely heavily on the investments and the costs borne by the supply chain. However, in this paper, the costs related to the remanufacturing activities were only represented by the costs of collecting used products by the manufacturer. Thus, the evolution of several optimal decisions was studied according to the increase in the value of the scaling parameter \(B\) of the cost of collecting used products for the remanufacturing activities from \(B\) = 200,000 \(mu\) to \(B\) = 900,000 \(mu\). Also, the customer preference towards purchasing products online was fixed at \(\rho\) = 0.5, in order to examine the evolution of the optimal solution under a market equitably balanced between both channels in terms of demands. Hence, Table 2 below illustrates the variation in the obtained optimal remanufacturing rate for the different values of the scaling parameter \(B\) of the cost of collecting used products. As presented in Table 2, the obtained optimal remanufacturing rates \(\tau^{*}(t)\) dropped for each period \(t\) whenever the value of the collection cost's scaling parameter increased gradually from \(B\) = 200,000 \(mu\) to \(B\) = 900,000 \(mu\). Thus, by increasing the value \(B\), the mean during the planning horizon \(H\) of the obtained optimal remanufacturing rates \(\tau^{*}(t)\) gradually decreased from 0.4 for \(B\) = 200,000 \(mu\) to 0.09 for \(B\) = 900,000 \(mu\). Hence, as a consequence of the rise in the value of the scaling parameter \(B\), new products manufactured during the planning horizon contained less and less remanufactured components. As explained above, the manufacturer bore for each period \(t\) the costs related to the remanufacturing activities represented by the collection cost \(B\cdot\tau(t)^{2}/2\).