# Enhancing E-Grocery-Delivery-Network Resilience with Autonomous Delivery Robots
## 3 Problem Definition  
### Notion  
In this stochastic E-grocery delivery network, we assumed that customers cannot be visited directly by vans. In order to minimize the unmet demand and the waiting time, the allocations of customers' orders to micro hubs and micro hubs to the depots are a crucial step, which is determined according to the information of orders, including the coordination of the customers, the deadline of the orders, the weight of the orders, the coordination of the micro hubs, and the capacity of the micro hubs. We illustrate this two-echelon E-grocery network in (Figure 1), where vans connect depots (\(d_{1}\)) to micro hubs (\(s_{1}\),\(s_{2}\), \(s_{3}\), and \(s_{4}\)), and ADRs connect micro hubs to customers (\(f_{1}\),\(f_{2}\), \(f_{3}\), \(f_{4}\), \(f_{5}\), and \(f_{6}\)).  
To mathematically formulate the two-echelon delivery network, we represented the two-echelon vehicle routing problem with the stochastic service time and demand (2E  
Figure 1: A schematic graph showing the potential solution of the 2E-VRP-SSSD.  
VRP-SSSD) as an undirected graph \(W=(N,\,A)\) consisting of a node set \(N\) and arc set \(A\), and a vehicle set \(K\). Furthermore, the node set \(N\) was partitioned into four subsets, which are denoted as depots \((O)\), micro hubs \((S)\), customers \((F)\), and delayed customers \((F^{\prime})\). We arranged it as follows: \(N=O\ \cup S\cup F\ \cup F^{\prime}\), \(N_{1}=O\ \cup\ S\), \(N_{2}=S\ \cup\ F\cup F^{\prime}\). And the arc set \(A\) was partitioned into two subsets; set \(A_{1}=\{(i,j)|i\in N_{1},\ j\in S,i\
eq j\}\) consisted of the arcs that can be traversed by vans in the first echelon. Similarly, set \(A_{2}=\{(i,j)|i\in N_{2},\ j\in F\cup F^{\prime},i\
eq j\}\) consisted of the arcs that can be travelled by ADRs in the second echelon. For each arc \((i,j)\in A\), let \(\rho_{ijk}^{an}\) be the travel time of van \(k\) from \(i\) to \(j\), \((i,j)\in A_{1}\), and \(\tau_{ijk}^{ADR}\) be the travel time of the ADRs \(k\) from \(i\) to \(j\), where \(i\in N_{2},\ j\in F\). Given that some demand \(j\in F^{\prime}\) cannot be satisfied, we still assigned \(j\) a traveling time \(\tau_{ijk}^{ADR}\geq L\), where \(i\in N_{1},\ j\in F\), \(L\) is a sufficiently large value. Accordingly, the 2 EVRP-SSSD was identified: A limited fleet of vans \(k\in K_{van}\) with capacity \(Q^{can}\) start from a depot \(o\) in \(O\) in the first echelon, visit all of the assigned micro hubs, and finally return to the origin depot \(o\) with capacity \(Q_{o}\), \(o\in O\). In the second echelon, a limited fleet of robots \(k\in K_{ADR}\) with capacity \(Q^{ADR}\) start from a micro hub \(s\) in \(S\), visit all of the assigned customers, and finally return to the origin micro hub \(s\) with capacity \(Q_{s}\), \(s\in S\). We made the following assumptions before the model was developed:  
* Each route is served by one vehicle.
* Each trip must start and terminate at the same depot or micro hub.
* The orders assigned to a micro hub cannot be more than the maximum capacity of the micro hub.
* Without the loss of generality, travel times and preparation times are integrated into the service times.
* A limited number of vans and ADRs is available at each depot and micro hub. Building upon the notations introduced in [45; 46; 47], we present the notations specific to this study.