## 3 Problem Formulation
### Modeling
\begin{table}
\begin{tabular}{l l} \hline \hline
**Sets** & **Description** \\ \hline \(N_{1}\) & the current set of orders that have been picked up but not delivered \\ \(N_{2}\) & the current order set that has not been picked up \\ \(N\) & set of orders involved in the current rolling period, \(N=N_{1}\cup N_{2}\), \(n\in N\) \\ \(I\) & set of restaurant nodes, \(I=\{1,2,\ldots,|I|\}\) \\ \(I^{virtual}\) & set of virtual restaurant nodes, \(I^{virtual}=\{|I|+1,|I|+2,\ldots,|N_{2}|\}\) \\ \(J\) & set of customer nodes, \(J=\{|N_{2}|+1,|N_{2}|+2,\ldots,|N_{2}|+|J|\}\) \\ \(J^{virtual}\) & set of virtual customer nodes, \(J^{virtual}=\{|N_{2}|+|J|+1,|N_{2}|+|J|+2,\ldots,|N_{1}|+2|N_{2}|\}\) \\ \(A\) & set of all nodes, \(A=I\cup J\) \\ \(It\) & set of restaurant nodes expansion, \(It=I\cup I^{virtual}\) \\ \(Jt\) & set of customer nodes expansion, \(Jt=J\cup I^{virtual}\) \\ \(At\) & set of all nodes’ expansion, \(At=It\cup Jt\) \\ \(map_{l}^{1}\) & virtual (actual) restaurant node i mapping of the actual restaurant node, \(i\in It\) \\ \(map_{l}^{2}\) & virtual (actual) customer node j mapping of the actual customer node, \(j\in J\) \\  & the restaurant node number corresponding to the order \(n\); if the restaurant node \\ \(\alpha_{n}\) & number corresponding to the order \(n\) is \(i\), then there is \(i=\alpha_{n}\), and there is an inverse mapping \(n=\text{arg}(i)\), \(i\in It\), \(n\in N\) \\  & the customer node number corresponding to the order \(n\); if the customer node \\ \(\beta_{n}\) & number corresponding to the order \(n\) is \(j\), then there is \(j=\beta_{n}\), and there is an inverse mapping \(n=\text{arg}\beta(i)\), \(j\in Jt\),\(n\in N\) \\ \hline \hline Parameters & \\ \hline \(V\) & maximum capacity of delivery vehicle \\ \(t_{ij}^{d}\) & travel time from node \(i\) to node \(j\), \(i\in It\), \(j\in Jt\) \\ \(s_{0}\) & number of the initial node \\ \(T_{0}\) & the ready time of the AGV \\ \(r_{n}\) & order time for order \(n\), \(n\in N\) \\ \(b_{n}\) & the latest delivery time for order \(n\), \(n\in N\) \\ \(v_{n}\) & the volume occupied by commodities for order \(n\), \(n\in N\) \\ \(t_{n}^{p}\) & meal preparation time for order \(n\); it follows the Gaussian distribution, \(t_{n}^{p}\sim(\mu_{i},\sigma_{i}^{2})\), \\  & \(n\in N\) \\ \hline \(t_{n}^{s}\) & customer service time for order \(n\), \(n\in N\) \\ \hline \multicolumn{2}{l}{Decision variable} \\ \hline \(x_{ij}\) & if the deliverer travels from node \(i\) to node \(j\), the decision variable is equal to 1, \\  & otherwise \(x_{ij}\) is equal to 0, \(i\in At\cup s_{0},j\in At\) \\ \hline \multicolumn{2}{l}{Other variables} \\ \hline \(s_{l}\) & the number of the \(l\)th node to be visited; if the sequence of the node \(i\) in the path is \(l\), \(i=s_{l}\), and there is an inverse mapping \(l=\text{args}(i)\), \(l\in\{1,2,\ldots,|A^{\prime}|\}\), \(i\in At\) \\ \(p_{l}^{1}\) & if the \(l\)th node and the next node merge to pick up food, \(p_{l}^{1}\) is equal to 1, otherwise \\  & \(p_{l}^{1}\) is equal to 0, \(l\in\{1,\ldots,|A^{\prime}|-1\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sets, parameters, and variables. Objective function 1 minimizes the expected order timeout rate; \(\omega\) is the timeout rate for all orders. \[\text{min}F_{1}=E(\omega) \tag{1}\]  
Objective function 2 minimizes the expected total customer waiting time. From the perspective of the customer, the customer needs the AGV to deliver the meals as soon as possible. The derivation of the completion time \(g_{n}\) of the order \(n\) is shown in (14). \[\text{min}F_{2}=E\Bigg{(}\sum_{n\in N}(g_{n}-r_{n})\Bigg{)} \tag{2}\]  
Objective function 3 maximizes the scheduling after-effect. A look-forward objective is introduced to adjust the impact of this decision on subsequent scheduling. See Section 3.1.5 for a detailed description. In terms of scheduling, orders are completed intensively in the early stage, while orders are completed relatively dispersed in the later stage. In the objective function, it is expressed as (3).