# Multi-Location Assortment Optimization with Drone and Human Courier Joint Delivery
## 5 Numerical Study
### The Influence of Marginal Revenue Varying by Weight  
In this subsection, we explore the influence of marginal revenue that varies according to the weight of the product. We assume there are eight spots (online stores) ranging from A to H, and 20 products. The capacity of each spot varies from \([6,8]\) in uniform distribution, \(u_{i}^{0}=20\), \(u_{ik}^{j}\) is of \([0,1]\) uniform distribution, \(d_{ik}=d_{ki},i\in M,k\in M\) is of \([1,20]\) uniform distribution and \(d_{ii}=0,i\in M\). Table 1 lists the details of these distances and the capacity of each spot. The \(D_{max},D_{min}\), and \(W\) are 6, 3, and 3, respectively. We set \(w_{j}\) to be of \([1,5]\) uniform distribution. We consider four scenarios where the \(r_{j}(r_{i}^{j}=r_{j},i\in M,j\in N)\) and \(w_{j},j\in N\) conform to the relation of linear-increasing, linear-decreasing, increasing-decreasing, and random. When the weight of product \(w_{j}\) increases, we set \(r_{j}=w_{j}\) as the linear-increasing \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**M**} & \multirow{2}{*}{**N**} & \multirow{2}{*}{\(c_{i}\)} & \multirow{2}{*}{\(u_{i}^{0}\)} & \multirow{2}{*}{\(u_{ik}^{j}\)} & \multicolumn{2}{c}{**MILP**} & \multicolumn{2}{c}{**Conic + MC**} \\ \cline{5-8}  & & & & & **Value** & **CPU/s** & **Gap** & **Value** & **CPU/s** \\ \hline
10 & & & & & 2.416 & 600 & \(5.57\times 10^{-6}\) & 2.416 & 13.086 \\
15 & & & & & 2.758 & 600 & \(3.36\times 10^{-5}\) & 2.758 & 57.993 \\
20 & 200 & [10, 12] & 20 & [0, 1] & 2.968 & 600 & \(6.78\times 10^{-6}\) & 2.968 & 161.322 \\
25 & & & & & 3.199 & 600 & \(2.57\times 10^{-5}\) & 3.199 & 310.179 \\
30 & & & & & 3.352 & 600 & \(5.03\times 10^{-6}\) & 3.352 & 372.597 \\ \hline
20 & & & & & 2.869 & 600 & \(5.08\times 10^{-7}\) & 2.869 & 42.040 \\  & & & & & 2.837 & 600 & \(1.92\times 10^{-5}\) & 2.837 & 65.766 \\  & & & & & 2.939 & 600 & \(7.33\times 10^{-8}\) & 2.939 & 80.716 \\  & & & & & 2.929 & 600 & \(3.77\times 10^{-5}\) & 2.929 & 88.233 \\  & & & & & 2.956 & 600 & \(3.21\times 10^{-5}\) & 2.956 & 113.929 \\  & & & & & 2.969 & 600 & \(1.72\times 10^{-6}\) & 2.969 & 127.364 \\ \hline
20 & & & [4, 6] & & & 2.272 & 600 & \(6.10\times 10^{-8}\) & 2.272 & 29.100 \\  & & [6, 8] & & & 2.507 & 600 & \(6.32\times 10^{-6}\) & 2.507 & 52.958 \\  & & [8, 10] & & & 2.800 & 600 & \(9.89\times 10^{-6}\) & 2.800 & 64.435 \\  & & [10, 12] & & & 3.014 & 600 & \(5.82\times 10^{-7}\) & 3.014 & 140.901 \\  & & [12, 14] & & & 3.137 & 600 & \(2.91\times 10^{-5}\) & 3.137 & 259.235 \\  & & [14, 16] & & & 3.254 & 600 & \(4.20\times 10^{-6}\) & 3.254 & 343.928 \\ \hline
20 & & & & 10 & & 3.611 & 600 & \(1.18\times 10^{-5}\) & 3.611 & 220.315 \\  & & & 20 & & 2.804 & 600 & \(6.77\times 10^{-6}\) & 2.804 & 108.650 \\  & & & 30 & [0, 1] & 2.499 & 600 & \(7.72\times 10^{-7}\) & 2.499 & 83.416 \\  & & & 40 & & 2.274 & 600 & \(2.02\times 10^{-6}\) & 2.274 & 74.115 \\  & & & 50 & & 1.962 & 600 & \(9.75\times 10^{-8}\) & 1.962 & 65.058 \\ \hline
20 & & & & [0, 1] & 2.820 & 600 & \(5.84\times 10^{-7}\) & 2.820 & 116.823 \\  & & & & [1, 2] & 3.885 & 600 & \(2.69\times 10^{-6}\) & 3.885 & 209.405 \\  & & & [2, 3] & 4.294 & 600 & \(5.33\times 10^{-6}\) & 4.294 & 170.582 \\  & & & & [3, 4] & 4.408 & 600 & \(1.31\times 10^{-5}\) & 4.408 & 95.440 \\  & & & & [4, 5] & 4.492 & 600 & \(8.11\times 10^{-8}\) & 4.492 & 178.419 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Effectiveness of MILP and Conic + MC. scenario, set \(r_{j}=-w_{j}+5\) as the linear-decreasing scenario, set \(r_{j}=\sin(\pi*w_{j}/5)*5\) as the increasing-decreasing scenario, and \(r_{j}\) is of \([1,5]\) uniform distribution in the random scenario. Table A2 lists the results of marginal revenues in different scenarios. For the convenience of expression, we denote them as L-In, L-De, In-De, and Ran. Figure 2 shows the results of our tests. It shows the weight distribution of products from A to H in four scenarios. The number in the middle of the rectangle represents the number of the product. Table A2 lists the product number and its weight.