## 3 Details on the Stochastic OM-VRP  
Usually, retail stores must be supplied from the depot with a large number of products, which are packed and frequently measured in terms of the number of pallets. In turn, and in contrast to retail stores' demand, online customer orders are of negligible size but require processing at a physical retail store before their delivery. Since orders must be processed at retail stores before delivery, products ordered online cannot be shipped directly from the central warehouse to online customers. In the OM-VRP, a single fleet of vehicles is employed to simultaneously perform three different operations: (i) bulk deliveries to retail stores from a main central depot; (ii) the pick-up of online customer orders from these retail stores; and finally, (iii) the deliver of online customers' orders. Different product types are available at each retail store. Likewise, each retail store has a limited inventory of processed products that can be delivered to online customers. Therefore, the vehicle responsible for a particular online order must first visit a retail store with that product in stock. The available processed inventory and bulk demand for each retail store are known in advance. It is assumed that bulk inventory does not contribute to a retailer's available inventory since bulk deliveries cannot be processed within the drop time of delivery. As a simplifying assumption, each online customer orders a single product. When more than one item is ordered, the consumer is replicated in the same location as a 'virtual' consumer, and each product is delivered separately. According to [PERSON] et al. [1], this strategy of considering single-item orders by consumers guarantees the solution feasibility and also minimizes the distribution cost. Figure 1 depicts how the processes of visiting retail centers and online customers are performed by the same fleet of vehicles. A route starts from the single and central depot, and the same cargo vehicle visits a set of retail centers and customers. Deliveries to customers cannot be directly performed from the depot. Hence, drivers must meet precedence constraints in order to pick-up customer orders from the retail centers before they can be delivered. Since retailers are supplied by the depot, these nodes are both delivery and pick-up points. On the other hand, customers require only the delivery of items from these retail centers, making them, delivery nodes. Apart from precedence and inventory constraints, the service of each route is limited by a maximum tour length and vehicle capacity limit. We can define the distribution network of the OM-VRP as directed graph \(G=(V,A)\). The set of vertices \(V=\{0,1,2,\ldots,r+c\}\) is composed of a single depot (0), \(r\) retail centers, and \(c\) online customers. In other words, the problem is represented by a set \(R\) of retail stores, which are supplied by the depot 0, and a set \(C\) of online customers, which are posteriorly supplied by the retail centers. The set \(N=R\cup C\) comprises both sets \(R\) and \(C\), in which \(R=\{1,2,\ldots,r\}\) and \(C=\{r+1,r+2,\ldots,r+c\}\). The same fleet of capacitated vehicles, initially stationed at the central depot at the start time \(t_{0}\), is employed for serving both the retail centers and consumers. For the complete mathematical formulation of this problem, readers are referred to [PERSON] et al. (2018). In this work, we extend previous work on this topic by considering the case in which edge-traversal times are stochastic. We define the time to traverse edge \((i,j)\) as \(T_{ij}=t_{ij}+D_{ij}\) where \(t_{ij}\) is the deterministic edge-traversal time (which represents the edge-traversal time under 'ideal' traffic conditions), \(D_{ij}\) is a log-normally distributed delay term, and \(T_{ij}\) denotes the distribution of edge-traversal times.