This paper presents a new bi-objective multi-modal hub location problem with multiple assignment and capacity considerations for the design of an urban public transportation network under uncertainty. Because of the high construction costs of hub links in an urban public transportation network, it is not economic to create a complete hub network. Moreover, the demand is assumed to be dependent on the utility proposed by each hub. Thus, the elasticity of the demand is considered in this paper. The presented model also has the ability to compute the number of each type of transportation vehicles between every two hubs. The objectives of this model are to maximize the benefits of transportation by establishing hub facilities and to minimize the total transportation time. Since exact values of some parameters are not known in advance, a fuzzy multi-objective programming based approach is proposed to optimally solve small-sized problems. For medium and large-sized problems, a meta-heuristic algorithm, namely multi-objective particle swarm optimization is applied and its performance is compared with results from the non-dominated sorting genetic algorithm. Our experimental results demonstrated the validity of our developed model and approaches. Moreover, an intensive sensitivity analyze study is carried out on a real-case application related to the monorail project of the holy city of Qom.