## 4 Numerical Results and Sensitivity Analysis
### Numerical Example  
This subsection is dedicated to presenting the obtained optimal solutions and the different optimal pricing and remanufacturing strategies proposed by the developed model in this paper. This numerical example aimed to present the different optimal pricing and remanufacturing decisions along the supply chain and to examine their evolution over the planning horizon \(H.\) For that, the results discussed in this subsection were obtained according to the data presented in Table 1 and for a customer preference for purchasing the products from the online platform \(\rho=0.5\). Figures 2 and 3 below describe the obtained optimal unit wholesale and selling prices, the optimal pharmacy and online demands and the manufacturer's optimal remanufacturing strategy during the planning horizon \(H.\)  
Figure 2 presented above illustrates the changes and the evolution, during the planning horizon \(H,\) of the obtained optimal unit wholesale prices proposed by the manufacturer \begin{table}
\begin{tabular}{l l l l l} \hline \(H=12\) & \(A(1)=6000\ P\) & \(A(2)=5000\ P\) & \(A(3)=7000\ P\) & \(A(4)=8000\ P\) \\ \hline \(A(5)=6000\ P\) & \(A(6)=5000\ P\) & \(A(7)=10\ 000\ P\) & \(A(8)=9000\ P\) & \(A(9)=6000\ P\) \\ \hline \(A(10)=4000\ P\) & \(A(11)=5000\ P\) & \(A(12)=6000\ P\) & \(Mc=100\ mu/P\) & \(\delta=40\ mu/P\) \\ \hline \(\alpha_{p}=12\ P/mu\) & \(\alpha_{o}=14\ P/mu\) & \(\beta=8\ P/mu\) & \(R_{p}=0.2\) & \(R_{o}=0.2\) \\ \hline \multicolumn{4}{l}{\(\underline{U}as=20\ mu/P\)} & \(C_{o}=5\%\) & \(B=500,000\ mu\) & \\ \hline \end{tabular}
\end{table} 
Table 1: Key parameters. Figure 3: Optimal demands and remanufacturing strategy during the horizon. Figure 2: Optimal pricing decisions of the manufacturer, the online platform and the pharmacy during the horizon. \(W_{p}\)\({}^{*}\)(\(t\)) and \(W_{o}\)\({}^{*}\)(\(t\)) to the pharmacy and to the online platform, respectively, and the optimal unit selling prices to the customers proposed by the pharmacy \(P_{p}\)\({}^{*}\)(\(t\)) and by the online platform \(P_{o}\)\({}^{*}\)(\(t\)). As presented above, for all periods, the unit wholesale prices \(W_{p}\)\({}^{*}\)(\(t\)) and the unit selling prices \(P_{p}\)\({}^{*}\)(\(t\)) of the pharmacy were slightly higher than those proposed to and by the online selling platform, \(W_{o}\)\({}^{*}\)(\(t\)) and \(P_{o}\)\({}^{*}\)(\(t\)). The difference in the pricing decisions related to the pharmacy and the online platform was caused by the gap between the self-price sensitivities \(\alpha_{p}\) and \(\alpha_{o}\). According to assumption 2, since the online channel was more self-sensitive to the variation in its proposed prices (\(\alpha_{o}>\alpha_{p}\)), the obtained optimal online unit prices \(W_{o}\)\({}^{*}\)(\(t\)) and \(P_{o}\)\({}^{*}\)(\(t\)) were marginally lower than those proposed by the pharmacy \(W_{p}\)\({}^{*}\)(\(t\)) and \(P_{p}\)\({}^{*}\)(\(t\)). In addition, as presented in Figures 2 and 3, the obtained optimal pricing strategies and demands fluctuated during the planning horizon \(H\) following the variation in the market size \(A\)(\(t\)). For example, the highest demands, unit wholesale and selling prices in both distribution channels were offered during period 7 (largest market size \(A\)(7) = 10,000 \(P\)), while the lowest demands and pricing decisions were obtained for period 10 (smallest market size \(A\)(10) = 4000 \(P\)). Also, by studying Figure 3, the pharmacy's demand \(D_{p}\)\({}^{*}\)(\(t\)) and the online demand \(D_{o}\)\({}^{*}\)(\(t\)) were very similar for each period \(t\), since both distribution channels shared the market (\(\rho\) = 0.5) equally. In fact, the pharmacy sold a total of 11,666 \(P\) during the planning horizon while the online platform's total demand was equal to 11,843 \(P\). This slight difference between the pharmacy and the online platform in terms of total demand generated during the planning horizon is explained by the low gap in the pricing strategies of both channels.