# Optimization of Multimodal Discrete Network Design Problems Based on Super Networks
### The Multimodal Network Design Problem  
There are a few studies about the multimodal DNDP, which mainly focus on two aspects: one is how to optimize the bus network considering congestion interaction with other modes. The other one is how to optimize both the car network and the bus network. For the first aspect, [PERSON] and [PERSON] studied transit network design problem in the multimodal network and presented a systematic phase-wise methodology that considering both the effect of congestion and integration of modal transfers [30]. Moreover, they modeled inter-route and inter-modal transfers through the state augmented multimodal (SAM) network approach. [PERSON] et al. established a multimodal network design problem model to solve the problem of optimal frequency design for bus services. They applied sensitivity analysis to define linear approximation functions between the Probit SUE link flows and the design parameters, which are then used as constraints in the sub-problem of the NDP instead of the original stochastic user equilibrium (SUE) condition [31]. [PERSON] et al. considered the impact of bike-sharing on transfer, optimized the design problem of the bimodal transit network, and proposed a continuum model to optimize the bimodal transit system and shared bikes in a grid network at the same time [32]. For the second aspect, [PERSON] et al. optimized both car and bus networks by increasing the capacity of existing roads and the service frequency of bus routes, but the network topology remained unchanged and the added optimization was not considered [33]. [PERSON] et al. simultaneously optimized the auto network and bus network by establishing a single-level mathematical program with complementarity constraints and solved the model with an active-set algorithm, however, they only considered two transportation modes without considering the modal interactions [34]. [PERSON] et al. reviewed the research status of cycling network planning and design in cold-climate cities in China and worldwide and summarized the relevant methods that can be used in China [35]. [PERSON] et al. investigated the optimal network design problem of bike paths, which are on or adjacent to roadways but are physically separated from motorized traffic within the existing urban network. They proposed a mixed-integer nonlinear nonconvex model for the problem and solved the proposed model with a global optimization method and a matheuristic [36]. In addition, in this paper, based on the optimal scheme of the bike network design problem, we use the known bike routes and walking routes to connect car, bus, rail transit networks as a multimodal transportation network, then study the multimodal transportation DNDP. In summary, we consider the interactions among various transportation modes and comprehensively optimize the multimodal DNDP of the car, bus, and rail transit, which is meaningful and innovative. The main contributions are listed as follows:  
Firstly, the slow travel modes in the multimodal transportation network are considered in this paper, which is usually neglected in existing studies. Apart from walking, bike-sharing has become another popular transfer mode. Therefore, we take the bicycle and walking as transfer modes to describe the modal interactions of car, bus, and rail transit. Secondly, the car, bus, and rail transit network are simultaneously studied for the optimization of multimodal DNDP. Most studies only considered the car network and bus network. However, owing to the large capacity, fast speed, and high comfort of rail transit, it occupies a very important position in multimodal travel behavior. Therefore,it is necessary to consider rail transit into multimodal DNDP. Moreover, a multimodal super network topology is presented, which clearly expresses inter-modal transfers of car, bus, and rail transit networks, as well as inter-route transfers of the bus and rail transit network.