## 1 Introduction  
Today, people are changing their shopping behavior. Recent advances in information and communication technologies have introduced new shopping channels and models, which make possible the expansion of e-commerce and, consequently, the emergence of new challenges in operational research, transportation, and logistics areas. Specifically, regarding modern e-commerce business models, new decision variables and constraints have been incorporated in them, leading to emerging variants of distribution problems in supply chain management. Unlike brick-and-mortar stores, where salespeople are available to support and help customers to make their purchases, the popularization of mobile devices with access to the Internet has promoted the use of different shopping channels. The online channel is an example that has emerged as a competitive marketing channel to that of traditional retail centers, transforming e-commerce into a global trend and an important tool for every business worldwide [1]. With e-commerce, customers are immersed in an environment of a plethora of information, opinions, and access to a vast combined supply of stock, which together allows them to browse through different stores in an online environment. According to some experts, the online shopping channels were predicted to kill off the physical ones. However, they co-exist and have completely transformed the way customers shop nowadays [2]. The use of a variety of shopping channels is referred to as 'omnichannel retailing', where, instead of having only the single option of physically visiting a store to buy products, consumers can also buy them via onlineshopping to be delivered at their homes. Hence, the company also becomes responsible for delivering goods ordered online to consumers. Therefore, this new retailing environment has introduced the need to design integrated distribution systems for both serving online customers and for replenishing the stocks of retailer stores [3]. This paper focuses on the emerging omnichannel vehicle routing problem (OM-VRP) concept, which is an extension of the classical vehicle routing problem (VRP) in which a two-echelon distribution network is considered. The VRP aims to design cargo vehicle routes with minimum transportation costs, and sometimes the problem also considers facility location costs [4]. The generated routes are designed to distribute goods between depots and a set of final customers during a planning period [5]. Since the OM-VRP tackles simultaneously the replenishment of a set of retailer stores and the direct shipment of the products from those retailer stores to customers in an omnichannel retailing environment, the OM-VRP can be seen as an integrated problem that combines both the VRP and the pick-up and delivery problem (PDP). In the OM-VRP, which was first introduced by [PERSON] et al. [1], a single distribution center supplies a group of retailer stores-i.e., the first-echelon. In turn, this set of retail centers serve a set of online customers-the second-echelon-by using a single fleet of vehicles in both delivery levels in order to reduce transportation costs. Apart from reducing operating costs and improving supply chain competitiveness, the optimization of this two-echelon problem holds the potential to improve customer service levels and to enhance the on-time delivery of customer orders [6]. VRPs and PDPs are frequently considered in the literature as deterministic problems, where customers' demands and travel times are constant values. In real-life, however, it is frequent that both demands and travel times are exposed to some degree of uncertainty. In these scenarios, it is more accurate to model these constant variables as random variables. In this paper, we extend the previous OM-VRP formulation by replacing the deterministic travel times by stochastic ones. To the best of our knowledge, it is the first time that a stochastic version of the OM-VRP has been considered.