## 4 Numerical Results and Sensitivity Analysis
### Sensitivity Analyses According to the Customer Preference \(\rho\)  
In this subsection, a detailed analysis of the impact of the changes in customer preferences toward purchasing a product from one channel or the other on the proposed optimal solutions is presented. In reality, customer preferences towards a particular channel, a specific product or proposed services represent a crucial parameter in attracting more customers and eventually increase the demand and the profit. Therefore, the value of the customer preference for purchasing the product from the online platform \(\rho\) was increased progressively and the evolution of the obtained optimal solutions was examined. The following results were obtained by using the parameter values illustrated in Table 1. Figure 4 displays the mean of the optimal unit prices for the manufacturer, the pharmacy and the online platform, the mean of the obtained optimal demands in both distribution channels, while Figure 5 below presents the optimal total profits generated by the pharmacy and the online platform for the different values of customer preferences. As illustrated in the figure above, the progressive increase in the value of the customer preference for online purchasing, \(\rho\), enhanced the attractiveness of the online channel to customers, leading to an expansion in the online market share \(\rho\)-\(A(t)\). This resulted in an increase in the optimal online demand \(D_{o}\)\({}^{*}(t)\) for each period \(t\) and consequently, an increase in the mean online demand during the planning horizon \(H\). As a matter of fact, the total demand of the online platform grew from 7 568 \(P\) for \(\rho\) = 0.2 to 16,119 \(P\) for \(\rho\) = 0.8, which represents a rise of 113%. Furthermore, as the value of the customer online preference \(\rho\) increased, both the mean of the obtained optimal unit online selling prices \(P_{o}\)\({}^{*}(t)\) and the mean optimal wholesale prices \(W_{o}\)\({}^{*}(t)\) proposed by the manufacturer to the online platform increased. This rise in the pricing decisions related to the online channel was a direct outcome of the increase in the online market share and demand. In fact, by quadrupling the online market share, the online platform could afford an increase in its proposed prices without greatly impacting its demand. Thus, the mean of the optimal unit online sale prices \(P_{o}\)\({}^{*}(t)\) witnessed a rise of 34%, and the mean of the optimal wholesale prices \(W_{o}\)\({}^{*}(t)\) increased by 22% due to the increase in the customer preference for online purchasing. Nonetheless, the customer preference towards purchasing the products from the pharmacy (\(1-\rho\)) dropped whenever \(\rho\) increased. Hence, for each period \(t\), the pharmacy's market  
Figure 4: Optimal demands and pricing strategies during the study according to the customer online preference \(\rho\). Figure 5: Optimal total profits of the pharmacy and the online channel generated during the planning horizon according to the customer online preference \(\rho\). share (\((1-\rho)\cdot A(t)\)), the obtained optimal demand \(D_{p}\)*(\(t\)) and the mean of the demands of the pharmacy during the planning horizon \(H\) decreased, as a consequence of the increase in the customer online preference \(\rho\). Consequently, the optimal results showed that the total demand of the pharmacy during the planning horizon dropped by 57% from 16,282 \(P\) for \(\rho\) = 0.2 to 7 051 \(P\) for \(\rho\) = 0.8. In addition, as a consequence of the increase in value of the customer online preference \(\rho\), the mean of the obtained pharmacy's optimal unit selling prices \(P_{p}\)*(\(t\)) and the mean of the optimal wholesale prices \(W_{p}\)*(\(t\)) proposed by the manufacturer to the pharmacy decreased by 31% and 26%, respectively, due to a decrease in the pharmacy's optimal demand \(D_{p}\)*(\(t\)) for all the periods.