# Multi-Location Assortment Optimization with Drone and Human Courier Joint Delivery
## 3 Model Description  
### Problem Definition  
We consider a multi-location assortment planning problem under the cooperative delivery by drone and human courier. We use \(S_{i}\) to denote the assortment list for online store \(i\), and the geographical regions are divided into \(M\) parts, \(i\in M,M=\{1,\ldots,m\}\). The ratio of customers visiting an online retailer \(i\) is denoted by \(\theta_{i}\) and satisfies \(\sum_{i\in M}\theta_{i}=1\). The products are denoted by \(N=\{1,\ldots,n\}\), and the online assortment consists of part of \(N\). Customers can order a product in any online shop if it can be delivered to the spot where the customer is. Customers from spot \(k,k\in M\) want to purchase a product in an online shop \(i,i\in M\) according to the MNL model. In online store \(i\) with assortment \(S_{i}\), the preference of product \(j\in S_{i}\) delivered to spot \(k\) is \(u^{j}_{ik}\), \(u^{0}_{i}\) for the no-purchase option. Customers from spot \(k\) who visit online store \(i\) only consider the products in \(S_{i}\) that can be delivered to \(k\), and buy product \(j\) with probability \(u^{j}_{ik}/(u^{0}_{i}+\sum_{l\in S^{k}_{i}}u^{l}_{ik})\) or leave without buy anything with probability \(u^{0}_{i}/(u^{0}_{i}+\sum_{l\in S^{k}_{i}}u^{l}_{ik})\), \(s^{i}_{i}\) meaning the products in \(S_{i}\) can be delivered to spot \(k\) as well. Once the customer buys the product, the order will be served by drone or human courier. A drone can deliver goods within \(D_{max}\) and more than \(D_{min}\), a human courier can deliver goods within \(D_{min}\). Figure 1 shows the distribution mode. We denote the marginal revenue of product \(j\) in online shop \(i\) is \(r^{j}_{i}\), \(j\in N\), the expected revenue of demand from spot \(k\) to online store \(i\) is \(\sum_{l\in S^{k}_{i}}r^{l}_{i}u^{l}_{ik}/(u^{0}_{i}+\sum_{l\in S^{k}_{i}}u^{l} _{ik})\). We define variable \(\gamma^{j}_{ik}\) to be \(1\) if product \(j\) can be delivered from spot \(i\) to \(k\), \(i\in M,k\in M,j\in N\), and \(0\) otherwise. Variable \(x^{j}_{i}=1\) when product \(i\) is provided in \(S_{i}\), and \(0\) otherwise. We define variable \(y^{j}_{ik}\) as equal to \(1\) if product \(j\) can be delivered from spot \(i\) to \(k\) using a drone, and \(0\) otherwise. Similarly, variable \(z^{j}_{ik}\) is equal to \(1\) if product \(j\) can be delivered from spot \(i\) to \(k\) by human courier, and \(0\) otherwise. By introducing these binary variables, our MADH problem can be formulated as (1)-(8). We summarize all the other symbols in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(x^{j}_{i}\) & \(1\) if products \(j\) is provided in the online store \(i\) and \(0\) otherwise \\ \(y^{j}_{ik}\) & \(1\) if product \(j\) can be delivered from spot \(i\) to \(k\) by drone and \(0\) otherwise \\ \(z^{j}_{ik}\) & \(1\) if product \(j\) can be delivered from spot \(i\) to \(k\) by human courier and \(0\) otherwise \\ \(\theta_{i}\) & Probability of demand for spot \(i\) \\ \(\gamma^{j}_{ik}\) & Decision variable \(\gamma^{j}_{ik}\) be \(1\) if product \(j\) can be delivered from spot \(i\) to \(k\) and \(0\) otherwise \\ \(u^{j}_{ik}\) & Preference associated with product \(j\) in assortment \(S_{i}\) for demand from spot \(k\) \\ \(u^{0}_{i}\) & No-purchase preference of spot \(i\) \\ \(r^{j}_{i}\) & Marginal revenue of product \(j\) in spot \(i\) \\ \(W\) & Drone load capacity \\ \(c_{i}\) & The cardinality capacity of products in spot \(i\) \\ \(w_{j}\) & The weight of product \(j\) \\ \(D_{max}\) & Maximum delivery distance of the drone \\ \(D_{min}\) & Maximum delivery distance of the human courier \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Nomenclature. \[\pi* = \max\sum_{i\in M}\sum_{k\in M}\sum_{j\in N}\theta_{i}r_{i}^{j}u_{ik}^ {j}q_{ik}^{j}\] (19) s.t.