# Optimization of Multimodal Discrete Network Design Problems Based on Super Networks
## 6 Conclusions  
In this paper, we comprehensively optimize a multimodal transportation DNDP that is composed of car, bus, rail transit, bicycle, and walking. Firstly, a multimodal super network topology that includes four types of links (entering links, leaving links, driving links, transfer links) is proposed, which clearly expresses inter-modal transfers among car, bus, and rail transit networks, as well as inter-route transfers in the bus or rail transit network. And to complete the above transfers, walking and bike-sharing are considered as transfer modes. Specifically, in Section 2 we present a simple multimodal transportation network that meets two types of the above transfer relationships and we also discuss the process of establishing a multimodal transportation super network topology. Secondly, the generalized cost formulas of each type of links in the super network topology are defined. Among them, the generalized cost is mainly composed of four types, namely travel time, monetary cost, and comfort loss, risk reserve time. Based on the above formulas, a bi-objective programming model to minimize the network operation cost and construction cost is proposed to decide whether to expand or add links for the car, bus, and rail network respectively. Moreover, based on the minimum cost flow algorithm and simulated annealing algorithm, we develop a hybrid heuristic algorithm to solve the proposed model, which can efficiently and effectively get the near-optimal solution. Finally, two numerical tests of a simple test network (designed in Section 2) and an actual network (an area from Xizhimen to Dongdan in Beijing, China) are implemented:  
1. For the simple network, to test the feasibility of the developed heuristic algorithm, an exact algorithm is first used to solve the model. And the results show that the proposed algorithm is effective and feasible within the allowable error range of 1.5%. 2. For the actual network, 12 groups of numerical experiments are carried out to analyze the optimal schemes under different weight coefficients. The results indicate that the choice of the near-optimal scheme is affected by the weight coefficients, and the results are consistent with the actual situation, Consequently, the proposed model and developed algorithm can promote the planning of multimodal transportation systems in the actual transportation network. In this paper, the proposed model and developed algorithm are flexible, and the objective function and constraints can be modified. In our future works, we can consider the network accessibility, environmental pollution, and other aspects, and modify the objective function and constraints correspondingly. In addition, the algorithm can be combined with other heuristic algorithms, such as tabu search algorithm, genetic algorithm, and so on, to get the near-optimal solution more quickly. The study of multimodal transportation DNDP is conductive to guide passengers' multimodal travel behavior and relieve congestion. The practical implications are as follows: (1) By studying the multimodal transportation DNDP, the optimal investment scheme of