2018 ICSEE International Conference on the Science of Electrical Engineering 978-1-5386-6378-3/18/$31.00 ©2018 IEEE Solving the Capacitated Open Vehicle Routing Problem Algorithm, Based on Probability Distribution Modeling of Saving Matrix Uri Lipowezky, Boris Korenfeld Global R&D, Gett Tel Aviv, Israel uril@gett.com , boris@gett.com Ianir Ideses R&D Amazon Herzliya, Israel ianir.ideses@gmail.com Abstract— Same day last-mile parcels delivery services are recently gaining a great deal of attention in the applied data science community. In the Tel Aviv metropolitan area, smart mobility companies provide delivery services for small parcels using scooter fleets. Scooter couriers deliver the parcels from the local post offices to the final customers. Since a scooter has a limited capacity and is paid with respect to the total traveling distance, the problem can be formulated as a Capacitated Open Vehicle Routing Problem (COVRP). The proposed approach is based on the high sparseness of the Clarke-Wright (CW) saving matrix for COVRP in case of parcels delivery from post offices. In this case the CW algorithm can be separated to construction of all feasible Hamiltonian paths at the first phase and its optimal combination by solving maximal weight clique problem (MWCP) at the second phase. Computational results show that the proposed algorithm is competitive, improves known benchmark results on some instances and saves 40-80% of delivery cost in the Tel Aviv metropolitan area. Keywords— capacitated open vehicle routing problem, Clarke- Wright algorithm, saving matrix values distribution, maximal weight clique problem I. INTRODUCTION Same day or instant last-mile delivery services, especially of parcels recently are getting a great deal of attention in the applied data science community. This is mainly attributed to the significant rise of e-commerce and insufficient supply of traditional delivery services [16]. Following [16], in the near future, autonomous agents will deliver 80% of parcels, mainly in the urban areas with high density. With relation to this, the development of relevant decisions of vehicle routing problem for business to client (B2C) service becomes extremely relevant. Recently available intensive big data exploration about existing paths and delivery density distribution allows effective modeling and evaluation of existing and proposed vehicle routing algorithms. In the Tel Aviv metropolitan area, which can be used as an example of very high dense area, the following last-mile dispatching model is used. Parcels are concentrated in local post offices (depots) by long distance suppliers such as the Israeli Post, UPS, etc. The big parcels are delivered directly to the customers’ home or the nearest parcel locker without going into the post office storage. For the small packages Gett provides a scooters and electrical bikes fleet for each depot that operates independently from any other service and dispatches all parcels collected in a post office once a day. Bike and scooter couriers are not required to return to the depot and are paid proportionally to the total traveling distance from depot to customers. Since both bikes and scooters have limited capacity, e.g. up to six parcels, in order to minimize the dispatching costs we propose to solve the Capacitated Open Vehicle Routing Problem (COVRP) [18]. Fig. 1. Depot point (D) and customer points (A and B) in square  on  lattice. COVRP is a natural extension of the well known Traveling Salesman Problem (TSP) [9] for more than one vehicle of limited capacity and no requirement to the vehicle to return to the depot after completing the service. These minor, at first sight, modifications make the classical closed solutions of vehicle routing problem (VRP), described e.g. in [27] inapplicable. Moreover, there are many other practical applications in which COVRP naturally arises. This happens, for example, every time when a company hires an external fleet or uses third party logistics providers [24]. In such situations, the cost of the delivery is usually proportional to the distance travelled while loaded. A. Related Works From a graph theoretical viewpoint, COVRP is a reduction from the Hamiltonian cycles finding problem required in classical VRP to set of Hamiltonian paths. However, as it has