An M/M/c queue model for vehicle routing problem in multi-door cross-docking environments

Cross-docking is a strategy to facilitate a persistent process from suppliers to the consumer points, without longterm storage of products at a distribution warehouse. Products are collected from various origins by inbound trucks, unloaded to the cross-dock, reconsolidated with other products, and finally loaded onto outbound trucks within the same or next day. Because of the limited number of dock-doors as the main resources and the uncertain arrival time of trucks at the cross-dock, queue problems in such environments are unavoidable. This study considers a vehicle routing problem (VRP) for a multi-door cross-docking system with a queuing approach. A real application of the proposed model can be found in congestion conditions at the cross-docking yard, when the queuing time of the vehicles (i.e., the queuing delay) may reduce the quality of service. Moreover, in some cases, improper queue management at a facility such as a cross-dock may incur an economic cost associated with the waiting time in the queue. In this study, we focus on the receiving doors of a cross-dock and assume that the rate of truck arrivals at the cross-dock is a random variable. Moreover, the cross-dock is not able to provide service to all vehicles simultaneously; it has some limitations such as capacity constraints and service time restrictions. Thus, an M/M/c queuing formulation is proposed to model this cross-docking environment, in which the vehicles' dispatch plan for starting the pickup process would also be determined. In the proposed multi-channel queuing system, the arrival flow of trucks to the cross-docking terminal can be deemed as a Poisson process, resulting in a nonlinear mathematical formulation to optimize the problem. The model is then linearized. To handle its computational complexity, we develop a new Genetic Algorithm (GA) to obtain near-optimal solutions to the problem and compare them with those of the optimization software GAMS. Then, a sensitivity analysis is done on different parameters of the model, and their effect on transportation cost and waiting cost of vehicles in the queue is investigated. The results show that considering a queuing approach in this problem, even for a smallscale problem derived from a real case study, can improve the average waiting time of trucks in the queue considerably compared with the situation in which all parameters are assumed deterministic.