Nowadays, an increasing number of vehicle routing problem with stochastic demands (VRPSD) models have been studied to meet realistic needs in the field of logistics. In this paper, a bi-objective vehicle routing problem with stochastic demands (BO-VRPSD) was investigated, which aims to minimize total cost and customer dissatisfaction. Different from traditional vehicle routing problem (VRP) models, both the uncertainty in customer demands and the nature of multiple objectives make the problem more challenging. To cope with BO-VRPSD, a membrane-inspired multi-objective algorithm (MIMOA) was proposed, which is characterized by a parallel distributed framework with two operation subsystems and one control subsystem, respectively. In particular, the operation subsystems leverage a multi-objective evolutionary algorithm with clustering strategy to reduce the chance of inferior solutions. Meanwhile, the control subsystem exploits a guiding strategy as the communication rule to adjust the searching directions of the operation subsystems. Experimental results based on the ten 120-node instances with real geographic locations in Beijing show that, MIMOA is more superior in solving BO-VRPSD to other classical multi-objective evolutionary algorithms.