# Multi-Location Assortment Optimization with Drone and Human Courier Joint Delivery
## 5 Numerical Study
### The Influence of Delivery Distance on Revenue  
In this subsection, we consider the impact of delivery distance on the whole revenue, where the marginal revenue of a product varies from the four scenarios in Section 5.2. We use the same parameter settings as Section 5.2, and vary the \(D_{max}\) and \(D_{min}\) from \(\{3,4,5,6\}\) and \(\{1,2,3,4,5\}\). The results are shown in Table 4. There are also four weight-marginal revenue relations, namely L-In, L-De, In-De, and Ran, and Figure 3 is divided into four parts according to them. In each part, the abscissa represents \(D_{max}\), the vertical above represents the \(Gap^{*}\), and below is the revenue in each \(D_{min}\) for the same \(D_{max}\). If we assume \(D_{max}=D_{a},D_{min}=D_{\beta}\), the revenue can be expressed as \(R_{a-\beta}\), then the \(Gap^{*}\) is equal to \((D_{a-a^{\prime}\{a^{\prime}=-1\}}-D_{a-\beta})/D_{a-a^{\prime}\{a^{\prime}= -1\}}\) which is shown as a circle in the top half of each part and is numbered as \(a^{\prime}-\beta\). The size of the circle indicates the gap size. For the rectangle shown below, in the middle, the number above represents \(D_{min}\), below is the retailers' optimal revenue under this delivery setting, and we only show the maximization and minimization of it. Moreover, the width of the rectangle represents the revenue in each \(D_{min}\) under this \(D_{max}\). In addition, we use the depth of green to represent the revenue level under different \(D_{max}\) and \(D_{min}\) as shown at the bottom of the Figure. From Figure 3, when \(D_{max}\) is fixed, we can easily see that L-In and Ran offer obvious improvements to revenue if we improve the \(D_{min}\), as the green grows darker and the \(Gap^{*}\) decreases quickly. Therefore, retailers can have a significant increase in revenue when they improve the \(D_{min}\) in these two situations. In addition, the rate of gap reduction is not \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \(D_{max}\) & \multicolumn{3}{c}{3} & \multicolumn{3}{c}{4} & \multicolumn{3}{c}{5} & \multicolumn{3}{c}{6} \\ \hline \(D_{min}\) & **1** & **2** & **1** & **2** & **3** & **1** & **2** & **3** & **4** & **1** & **2** & **3** & **4** & **5** \\ \hline L-In & 0.887 & 1.037 & 0.923 & 1.068 & 1.248 & 0.944 & 1.083 & 1.258 & 1.370 & 0.962 & 1.092 & 1.263 & 1.373 & 1.424 \\ L-De & 0.818 & 0.823 & 0.911 & 0.916 & 0.919 & 0.953 & 0.958 & 0.961 & 0.963 & 0.986 & 0.991 & 0.994 & 0.997 & 0.997 \\ In-De & 1.387 & 1.424 & 1.528 & 1.564 & 1.600 & 1.591 & 1.623 & 1.653 & 1.675 & 1.648 & 1.676 & 1.704 & 1.725 & 1.736 \\ Ran & 1.019 & 1.110 & 1.102 & 1.190 & 1.295 & 1.148 & 1.234 & 1.335 & 1.387 & 1.181 & 1.262 & 1.359 & 1.408 & 1.427 \\ \hline \end{tabular}
\end{table} 
Table 4: The revenue under different delivery distances and marginal revenues. Figure 3: Impacts of delivery distance. slowing down, which means the increase of \(D_{min}\) is effective in obtaining a higher revenue for retailers. However, when In-De improves slightly, L-De is even less. Moreover, we can find when the \(D_{max}\) grows bigger, the \(Gap^{*}\) increases fast in 3-1, and then the trends flatten out in all scenarios. This reflects that when \(D_{max}\) is fixed, no matter what \(D_{min}\) is, the \(Gap^{*}\) between \(D_{max}-D_{max-1}\) and \(D_{max}-1\) is within 40% in the L-In situation, 5% in L-De, 10% in In-De and 20% in Ran under this parameter setting. As the rate of gap reduction is not slowing down, one can speculate the \(Gap^{*}\) when the \(D_{max}\) is given. Moreover, we can also conclude that in the L-In and Ran scenarios, \(R_{\alpha-\beta}\) is larger than \(R_{\alpha-\beta^{\prime}\{\beta^{\prime}<\beta\}}\), \(R_{\alpha^{\prime}\{\alpha^{\prime}<\alpha\}-\beta}\) and \(R_{\alpha^{\prime}\{\alpha^{\prime}>\alpha\}-\beta^{\prime}\{\beta^{\prime}< \beta\}}\) as the green is darker in \(R_{\alpha-\beta}\) obviously. The L-De and In-De are also consistent with this conclusion, but the difference is relatively small.