## 3 Details on the Stochastic OM-VRP
The objective of this extended problem is to minimize the total travel cost of the vehicle routes such that: (i) every route starts and ends at the depot; (ii) the routes do not violate the maximum tour length; (iii) every node \(i\in N\) is visited by only one vehicle and only once; (iv) the total load of the vehicle does not exceed the vehicle capacity; (v) the total consumer demand to be fulfilled by a vehicle for a specific product does not exceed the inventory picked up by the vehicle; (vi) the retail store determined to satisfy the consumer's demand must be visited before the consumer and by the same vehicle; and (vii) the routes must have a reliability level greater than a user-set parameter, \(R_{min}\). The reliability level of a set of routes is defined as the probability that all routes are completed within the maximum time limit of \(T_{max}\). Two alternative mathematical formulations of the deterministic OM-VRP can be found in [PERSON] et al. (2018), [PERSON] et al. (2019). The differences in this case are a stochastic objective  
Figure 1: A general schema of the OM-VRP. function and probabilistic constraint regarding route completion reliability. Let \(x_{fkij}\) be a binary decision variable indicating whether or not vehicle \(f\) in the fleet \(F\) traverses edge \(ij\) in the \(kth\) node visit in its route. The stochastic objective function is given in Equation (1), where \(D_{ij}\) is the stochastic delay associated with traversing edge \(ij\). \[min\ E\left(\sum_{f\in F}\sum_{k\in K}\sum_{i\in V}\sum_{j\in V}x_{fkij}\left(t _{ij}+D_{ij}\right)\right). \tag{1}\]  
The probabilistic route completion reliability constraint is given in Equation (2). \[R_{min}\leq P\left(\sum_{i\in V}\sum_{j\in N}\sum_{k\in K}\left(t_{ij}+\tau+D_{ ij}\right)x_{fkij}+\sum_{i\in V}\sum_{k\in K}\left(t_{i0}+D_{i0}\right)x_{fk0} \leq T_{max},\ \forall f\in F\right). \tag{2}\]  
Equation (2) also accounts for the drop times (\(\tau\)) that are required when performing deliveries and pickups. In this work we employ a simheuristic approach for addressing the stochastic aspects of the OM-VRP. The main novelty of the OM-VRP, in comparison to other VRPs, is the precedence constraints regarding the collection of customer orders from retailers and subsequent delivery to customers. The vehicle capacity constraint regarding the retailer demands that can be satisfied are expressed by Equation (3), where \(OP_{j}\) denotes the number of ordered product units of node \(j\) (which is zero for customer nodes) and \(H\) denotes the vehicle capacity. \[\sum_{i\in N}\sum_{j\in V}\sum_{l=1}^{k}x_{fllj}OP_{j}\leq H,\ \forall f\in F,\ \forall k\in K. \tag{3}\]  
The customer order precedence constraints are expressed by Equation (4), where \(OD_{jq}\) denotes the number of items of type \(q\in Q\) ordered by node \(j\) (which is zero for retailer nodes) and \(P_{jq}\) denotes the number of items picked up of type \(q\in Q\) at node \(j\) (which is zero for customer nodes). \[\sum_{i\in V}\sum_{j\in V}\sum_{l=1}^{k}x_{fllj}\left(P_{jq}-OD_{jq}\right) \geq 0,\ \forall f\in F,\ \forall k\in K,\ \forall q\in Q. \tag{4}\]  
Table 1 provides the details for a small-sized instance, which is composed of one central warehouse, four retail stores, and nine online customers. For this instance, Table 1 provides: the geographic coordinates (\(X\) and \(Y\)) of each node, the demand required from the depot by each retail center \(i\in R\), the demands for each customer \(j\) for each product type \(q\) (\(OD_{jq}\)), and the inventory of each available product \(P_{iq}\), \(q\in\{1,2,3\}\) at the retail centers. Since the products are negligible in terms of capacity, they do not affect the vehicle load capacity. Accordingly, Figure 2 presents the optimal solution for the provided example. The routes are performed with a total cost of 455.91 distance units, and a fleet of vehicles each with a capacity of 100 weight units is employed for performing the routes.