A Two-Phase Gradient Projection Algorithm for Solving the Combined Modal Split and Traffic Assignment Problem with Nested Logit Function

Table 1

List of variables.

Notation	Definition
Set
R	Set of origins, RN
S	Set of destinations, SN
RS	Set of O-D pairs rs
	Set of routes connecting O-D pair rs
	Set of upper nests connecting O-D pair rs
	Set of lower nests connecting O-D pair rs in upper nest u
	Set of transit modes connecting O-D pair rs
Parameters and inputs
	Given exogenous utility of mode m in nest u connecting O-D pair rs
	Expected perceived utility of mode m in nest u connecting O-D pair rs
	O-D specific parameter
	Upper nest parameter of each O-D pair rs (i.e., )
	Conditional probability
	Marginal probability
	Convergence error in the inner loop
	Convergence error in the outer loop
,	Dwelling times on link a of bus and metro
, ,	Equal to 1 for link a with a bus line, metro line, and bus-metro line and 0 otherwise
	Length on link a
Ws	Walking speed (i.e., 4 km/h)
	Weighting time for the transit mode (bus and metro) on link a
	Transit fare connecting O-D pair rs
VOT	Inverse value of time for converting transit fare into equivalent time unit
	Total travel demand connecting O-D pair rs
	Capacity on link a
Decision variables
	Route flow on the mode m of route k connecting O-D pair rs
	Route cost on the mode m of route k connecting O-D pair rs
	Minimum cost on the mode m connecting O-D pair rs
	Route flow of the private transport mode on route k connecting O-D pair rs
	Route cost of the private transport mode route k connecting O-D pair rs
	Route cost of the private transport mode shortest route connecting O-D pair rs
	Positive-definite scaling factor between route k and the shortest route of O-D pair rs
	Link flow of private transport mode on link a
	Travel time of private transport mode on link a
	Route-link indicator is 1 if link a is on route k between O-D pair rs and 0 otherwise
	Total demand of public transport mode connecting O-D pair rs
	Demand from public transport mode m in the lower nest connecting O-D pair rs
	Generalized cost of public transport mode m connecting O-D pair rs
,	Bus and metro route-link indicators: 1 if link a is on bus route B or metro route M connecting O-D pair rs and 0 otherwise
Abbreviations
GP	Gradient projection
CMSTA	Combined modal split and traffic assignment
NL	Nested logit
UE	User equilibrium
MSA	Method of successive averages
CDA	Combined distribution and assignment
MP	Mathematical program
NCP	Nonlinear complementarity problem
VI	Variational inequality
FP	Fixed point
MNL	Multinomial logit
CNL	Cross-nested logit
PCL	Paired combinatorial logit
KKT	Karush–Kuhn–Tucker
MAE	Mean absolute error
RMSE	Root-mean-square error
ATT	Average travel time
IGP	Improved gradient projection