# A Traffic Equilibrium Model for Multi-Modal Networks with Uncertain Demands
## 2 Notations and Network Representation
### Multi-Modal Transport Network  
Consider a multi-modal transportation network, \(G=(N,A)\), where \(N\), \(A\), respectively, are the sets of physical nodes and physical links. As shown in Figure 1, the network includes two \(OD\) pairs and four transport modes. The origin is set as Node 1, and the destinations as Node 9 and Node 12, respectively. The transport modes in the multi-modal network are, respectively, bicycle (transport mode 1), bus (mode 2), subway (mode 3), and auto (mode 4). Specifically, there are three lines of transport mode buses with corresponding operating frequencies and bus vehicles with a specific number of seats and capacity. The physical node in the network is also a transfer node; that is, travellers can complete a probable transfer on any physical node. In this multi-modal transportation network, travellers can choose one or more modes of transportation.There will be these travel modes: bicycle, bus, subway, car, bicycle+bus, bicycle+subway, bus+subway. In practice, if the traveller chooses car (mode 4), it is impossible for transfer behavior to occur most of the time. Therefore, the combination of cars and other modes of transportation is not considered. Traffic assignment in a multi-modal network is more complicated than the assignment of pure vehicular or bus trips. In multi-modal networks, it involves pure-mode trips versus combined-mode trips. Compared to a pure-mode trip, in a combined-mode trip, travellers choose not only the routes, but also the transport modes and the kinds and locations of transfers. To flexibly present combined- and pure-mode trips, the supernetwork concept is adopted, in which the probable transfer rules are automatically captured completely. This  
Figure 1: Multi-modal transportation network. is accomplished by constructing the sub-network layer for each transport mode and connecting the origin to the sub-network of each transport mode by the access link, which represent online travel behavior and off-road behavior. And nodes of the sub-network of each transport mode are connected to the corresponding (same-named) nodes of other single-transport-mode networks by transfer links, connecting the destinations to the sub-network of each transport mode, which represent the transfer behavior in the physical node. According to the actual travel behavior of most travellers and the convenience of research, we assume that a traveller cannot transfer more than two times, and the car mode does not involve any transfers. In Figure 2, the super network \(G^{\prime}=(V,L)\) is illustrated, where \(V,L\), respectively, are the sets of nodes and links. It is constructed so that each single-transport-mode network is represented individually on separate layers interconnected using transfer links. The links can be divided into transfer links, physical links, network access links, and network departure links. The nodes can be divided into the origin, the destination, and physical nodes (which are also transfer nodes). After a multi-modal network has been transformed into a supernetwork, it can be considered as a simple network without the need to consider transfer feasibility and related issues. On the other hand, a route in the supernetwork automatically combines the mode transfer choices, which can be decoded for the specific modes used and transfer locations selected. Then, the travellers' choices in the supernetwork can be broadly categorized into two groups: travel mode choice, which involves the selection of a pure or a combined transport mode, and travel route choice. In addition, care must be exercised in defining the maximum number of transfers and formulating the set of probable transfers so as to strike a balance between capturing realistic transfer behaviors and the proliferation of nodes in the supernetwork. In this study, it is assumed that a traveller cannot transfer more than two times, and the car mode does not involve any transfers.