Table 1 Notations used in formulations

V

= set of nodes in the multi-modal network.

E

= set of arcs in the multi-modal network.

M

= set of the transport modes, M={p,b,m,t}.

p

= the private mode.

b

= the bus mode.

m

= the metro mode.

t

= the transfer mode.

i,j

= index of nodes, i,j∈N.

(i,j,k)

= connections between the adjacent nodes of mode k.

ξ _ijk

= travel time/transfer time on arc (i,j,k).

c _ijk

= travel cost on arc (i,j,k), k∈M∖{t}.

d _k

= fixed cost when entering mode k.

x _ijk

= 1, if arc (i,j,k) is on the path, 0 otherwise, (i,j,k)∈E.

y _k

= 1, if x _ijk =1 for a certain (i,j,k)∈E, 0 otherwise, k∈M∖{t}.