A multi Operator Genetic Algorithm For Solving The Capacitated Vehicle Routing Problem with Cross-Docking Problem Alaa Touihri Institut Sup´ erieur de Gestion, University of Tunis, Tunisia Email: alaa.touihri2011@gmail.com Olfa Dridi Institut Sup´ erieur de Gestion, University of Tunis, Tunisia Email: dridi o@yahoo.fr Saoussen Krichen Institut Sup´ erieur de Gestion, University of Tunis, Tunisia Email: krichen s@yahoo.fr Abstract—This study will discuss the capacitated vehicle rout- ing problem with simultaneous pick-up and delivery in a cross– docking environment. The transportation system includes three levels in the supply chain management:(1)Suppliers, (2)Retailers, (3)Customers. This paper addresses the CVRPCD, where a set of homogeneous vehicles are used to transport products from the suppliers to the corresponding customers via a cross-dock. The objective of the CVRPCD is to minimize the total traveled distance while respecting time window constraints at the nodes and a time horizon for the whole transportation operation. In this paper, a mixed integer programming formulation for the CVRPCD is proposed. I. I NTRODUCTION The advantages of the cross-docking technique have been increasingly appreciated in literature and in practice. This appreciation, coupled with the advances of numerous applica- tions in the vehicle routing problem (VRP) across numerous practical contexts, presents an opportunity to explore the Capacitated VRP with cross-docking (CVRPCD). This study considers a single product and single cross-dock wherein capacity-homogeneous vehicles start at cross-dock and finish in the cross-dock after serving all customers(Delivery process) and all suppliers (different pickup points). The vehicles are scheduled to route in the network synchronously to arrive at the cross-dock center simultaneously. In the delivery opera- tions, all customers must be served at most once and deliveries should be finished at a predetermined duration. We model CVRPCD as a mixed-integer linear program that minimizes the total cost (vehicle hiring cost and transportation cost). A genetic algorithm (GA) is proposed to solve the problem. GA is first verified by solving the CVRPCD benchmark instances; We have tested our algorithm on a set of VRPCD benchmarks with some modification of the benchmark to be adapted with our approach, the results are compared with those obtained by Augerat et al (2012) instances. Computational results show that GA can obtain optimal solutions to more than 80% of instances. The remainder of this paper is organized as follows. A detailed description of the CVRPCD is given in the next section. A mixed integer formulation of the problem is then presented, followed by a hybrid heuristic approach to solve the problem. Computational results are presented and conclusion follow. II. RELATED WORK Cross-docking is a new warehousing strategy in logistics. It is defined as the consolidation of products from incoming shipments so that they can be easily sorted at a distribution center for outgoing shipments. The distribution center in this case is referred to as a cross-dock. It essentially eliminates the inventory holding function of a traditional warehouse while still allowing consolidation. The advantages of the cross-docking technique have been increasingly appreciated in literature and in practice. This appreciation, coupled with the advances of numerous ap- plications in the vehicle routing problem (VRP) across nu- merous practical contexts, presents an opportunity to explore the Capacitated VRP with cross-docking (CVRPCD). Many companies are trying to develop efficient strategies to control the physical flow of their supply chain. The important aspects in finding new strategies is minimizing the total cost and achieving a high level of agility, flexibility, and reliability for various demands. Cross-docking is one innovative strategy to minimize unnecessary cost, particularly in terms of inventory and customer service level (Apte et Viswanathan, 2000). The shipments arriving from disparate sources are regrouped and dispatched directly by the outgoing trailers without being stored. Shipments typically spend less than 24 hours at the cross-dock, sometimes less than an hour. This way, cross- docking not only provides good customer service but also yields substantial advantages over traditional warehousing: reduction in inventory investment, storage space, handling cost and order-cycle time, as well as faster inventory turnover and accelerated cash flow (Cook, Gibson and MacCurdy, 2005; Apte and Viswanathan, 2000). Agustina, Lee, and Piplani (2010) noted that cross-docking is important for the efficient operation of a distribution network because it reduces or eliminates the storage activities that belong to the warehousing system. In general, the concept of cross-docking does not allow products to be stored at the cross-docking center but may occur whenever the inventory cost incurred is lower than