Multi-stage resource leveling problem in sharing logistics network is a multi-objective optimization problem, which is strongly non-deterministic polynomial hard in open loop environment. In this paper, we attempt to develop a logistics task-resource allocation model for this problem, which not only considers the total cost and duration for sharing logistics network, but also refers to the resource efficiency intra-stage and stability inter-stage for resource providers. As the defects of slow convergence, weak local search and easy-to-precocious in traditional algorithms, an improved multi-objective artificial bee colony algorithm is developed with adaptive neighborhood rules. The process of algorithm improvement involves: (I) an adaptive moving step size in population update strategy instead of random step size and (II) an adaptive weigh updating method with multiple neighborhood search rules in local optimum. The results show that the improved algorithm can effectively solve multi-stage resource leveling problem proposed in this paper, compared with traditional artificial bee colony algorithm, non-dominated sorting genetic algorithm-II and multi-objective particle swarm optimization, and can obtain a better non-dominated solution set with multiple metrics for algorithm.