## 4 Numerical Results and Sensitivity Analysis
### Sensitivity Analyses According to the Sales Commission Co  
A detailed sensitivity analysis examining the impact of the unit sales commission \(Co\) paid by the online platform to the pharmacy in return of the provided AS on the optimal obtained solutions is illustrated in this subsection. In this work, the obtained optimal total profits of both distribution channels were studied in order to determine the sales commission \(Co\)\({}^{*}\) that represented the break-even point for the profits of both channels. Therefore, \(Co\)\({}^{*}\) denotes to the break-even point of the unit sales commission that allows the pharmacy and the online platform to generate equal total profits. Hence, the variation in the optimal total profits of both distribution channels was examined for different values of unit sales commission \(Co\) and customer preference for online purchasing \(\rho\). Also, the following results were conducted on the proposed model according to the data presented in Table 1. Figure 7 below illustrates the evolution of the optimal total profit of the pharmacy \(\pi_{P}\)\({}^{*}\) and the online optimal total profit \(\pi_{O}\)\({}^{*}\) generated during the planning horizon \(H\) for different unit sales commissions \(Co\) and different customer preferences \(\rho\). As presented above, the variation in the values of the unit sales commission \(Co\) impacted the optimal total profit generated in each distribution channel. In particular, for each customer preference towards purchasing products via the online channel \(\rho\), the increase in the value of the unit sales commission \(C_{0}\) led to an increase in the optimal total profit \(\pi_{P}\)\({}^{*}\) of the pharmacy and a decrease in the optimal online total profit \(\pi_{O}\)\({}^{*}\) generated during the planning horizon \(H\). In fact, following the rise in the value of the unit sales commission \(C_{o}\), the total commission paid by the online platform to the pharmacy in return for the provided AS for products sold online (i.e., \(C_{o}\)\(\cdot\)\(P_{o}\)\({}^{*}(t)\)) increased for each period \(t\). Hence, the total paid commission during the planning horizon \(H\) increased for each customer preference for online purchasing \(\rho\). For instance, for \(\rho=0.6\), the total paid commission during the planning horizon \(H\) quadrupled from 128,468.5 \(mu\) for \(C_{0}=2\%\) (representing 14% of the pharmacy's total profit and 11% of the online total profit) to 513,382 \(mu\) for \(C_{o}=8\%\) (which corresponds to 35% of the pharmacy's total profit and 48% of the online total profit). Therefore, for each value of the customer preference towards purchasing products from the online platform \(\rho\), the increase in the total commission during the planning horizon \(H\) due to the rise in the value of the unit sales commission \(C_{o}\) generatedmore revenues for the pharmacy and less profits for the online channel. These outcomes explain the variations in the optimal total profit \(\pi_{P}{{}^{*}}\) of the pharmacy and the optimal online total profit \(\pi_{O}{{}^{*}}\). On the other hand, as illustrated in Figure 7, the value of the break-even point according to the unit sales commission \(\text{C}_{o}{{}^{*}}\) increased whenever the customer online preference \(\rho\) increased. Furthermore, the break-even point \(\text{C}_{o}{{}^{*}}\) indicated the value of the unit sales commission \(\text{C}_{o}\) allowing the pharmacy and the online platform to generate equal total profits during the planning horizon \(H\). For \(\rho=0.5\), since both distribution channels shared the market and generated similar demands, the pharmacy's optimal total profit \(\pi_{P}{{}^{*}}\) was slightly higher than the online optimal total profit \(\pi_{O}{{}^{*}}\) due to the gap between the self-price sensitivities \(\alpha_{p}\) and \(\alpha_{o}\). In addition, no unit sales commission \(\text{C}_{o}\) could achieve the break-even point condition related to the total profits for \(\rho=0.5\).