Location-routing problem with multi-product cross-dock centers considering heterogeneous capacitated vehicles and split delivery and pickup in a multi-echelon supply chain

Cross-dock is a logistic strategy with the aim of reducing inventory and increasing customer satisfaction. It is described as process of moving goods from suppliers to customers through cross-dock centers without or with little storage, and products must be collected to cross-dock centers before being shipped to customers and after consolidation, freights are delivered to customer with outbound vehicles. This research considers location-rounting problem with cross-dock throughout a multi-echelon supply chain by integrating suppliers, cross-dock centers and costumers. In this problem, cross-dock center, vehicle, supplier, costumer, and product are multiple. Also, cross-dock centers and vehicles have limited capacity, split pickup and delivery are allowed, and vehicles are heterogeneous and have constraint for product type shipping. The aim of the model is to minimize the sum of the opening costs of cross-dock centers and the distribution and operational costs of vehicles. This problem arises from the perishable food products distribution industry. For the problem, an integrated mixed integer nonlinear programming model is proposed. Since the problem under study is NP-hard, a particle swarm optimization algorithm is presented to tackle larg-sized problems. Also, a lower bound is developed for the problem. In order to show the performance of the introduced algorithm, different test problems with small, medium and large sizes are provided randomly and solved by the proposed approaches. Computational results show the high performance of the proposed algorithm.