# A Traffic Equilibrium Model for Multi-Modal Networks with Uncertain Demands
## 2 Notations and Network Representation  
### Notation  
* \(G\) multi-modal transport network: \(G=(N,A)\);
* \(N\) set of physical nodes: \(N=\{n\}\);
* \(A\) set of physical links: \(A=\{a\}\);
* \(I\) set of transport modes;\(i\) individual transport mode, \(i\in I:1(bike),2(bus),3(subway),4(car)\);
* \(J\) set of bus lines of transport mode bus;
* \(j\) individual bus line: \(j\in J\);
* \(M\) set of travel modes;
* \(m\) individual travel modes: \(m\in M\);
* \(P_{w}\) set of routes between the origin and destination \((OD)\) node pair \(w\): \(p\in P_{w}\);
* \(p_{w}^{m}\) set of routes in travel mode \(m\) between \(OD\) pair \(w\): \(P_{w}=\sum\limits_{m}P_{w}^{m}\);
* \(q_{w}\) travel demands between \(OD\) pair \(w\);
* \(q_{w}^{m}\) the proportion of passengers for travel mode \(m\) in \(OD\) pair \(w\): \(q_{w}=\sum\limits_{m}q_{w}^{m}\);
* \(G^{\prime}\) super network: \(G^{\prime}=(V,L)\);
* \(V\) set of nodes: \(V=\{v\}\);
* \(L\) set of links: \(L=\{l\}\);
* \(v_{l}\) passenger flow on link \(l\);
* \(v_{l}^{\prime}\) passenger flow of transport mode \(i\) on physical link \(l\);
* \(f_{w}^{p}\) passenger flow of route \(p\) between \(OD\) pair \(w\);
* \(c_{l}\) travel cost of link \(l\);
* \(c_{w}^{p}\) travel cost of route \(p\) between \(OD\) pair \(w\);
* \(\sigma_{lp}\) incidence relationship between link and route; if link is on route, \(\sigma_{lp}=1\), and otherwise it is \(0\);
* \(t_{l}\) travel time of link \(l\);
* \(t_{l}^{i}\) travel time of transport mode \(i\) on physical link \(l\);
* \(t_{l}^{h}\) travel time on transfer link \(l\);
* \(t_{l}^{s}\) travel time on network access link \(l\);
* \(t_{l}^{x}\) travel time on network decorture link \(l\);
* \(u_{l}\) crowd discomfort of link \(l\);
* \(u_{l}^{i}\) crowd discomfort of transport mode \(i\) on physical link \(l\);
* \(u_{l}^{i}\) crowd discomfort on transfer link \(l\);
* \(u_{l}^{i}\) crowd discomfort on network access link \(l\);
* \(u_{l}^{x}\) crowd discomfort on network departure link \(l\);
* \(m_{l}\) travel face of link \(l\);
* \(m_{l}^{i}\) travel face of transport mode \(i\) on physical link \(l\);
* \(m_{l}^{h}\) travel face on transfer link \(l\);
* \(m_{l}^{s}\) travel face on network access link \(l\);
* \(m_{l}^{x}\) travel face on network departure link \(l\);
* \(\lambda_{t}\) coefficient for travel time;
* \(\lambda_{u}\) coefficient for crowd discomfort;
* \(\lambda_{m}\) coefficient for travel face;
* \(t_{l}^{h^{0}}\) free-flow travel time for transport mode \(i\) on physical link \(l\);
* \(t_{l}^{h^{i}}\) walking time for transfer to transport mode \(i\) on transfer link \(l\);
* \(t_{l}^{di}\) waiting time for transfer to transport mode \(i\) on transfer link \(l\);
* \(u^{i^{0}}\) crowd discomfort of transport mode \(i\) per time unit;
* \(u^{h^{0}}\) crowd discomfort for transfer per time unit;
* \(m^{i^{0}}\) travel face of mode \(i\) per distance unit;
* \(m^{h^{0}}\) travel face for transfer;
* \(h_{l}\) length of link \(l\);
* \(B_{l}\) road capacity for transport mode \(1\) (non-motor vehicles) on physical link \(l\);
* \(C_{l}\) road capacity for transport modes \(2\) and \(4\) (motor vehicles) on physical link \(l\);
* \(\alpha,\beta\) modal parameters of travel time function;
* \(\alpha^{\prime},\beta^{\prime}\) modal parameters of crowd discomfort function;
* \(\gamma\) crowd discomfort-time conversion coefficient;
* \(\pi\) travel fare-time conversion coefficient;