Framework for Spatial Models for Future Cities - Regional Science Matters - page 361

m
j

exp g m

m
j

ʵ ¼ ʼ 8 m , j ð 17 : 22 Þ

A simpler form for x ij m is:

8

d ij

x ij ¼ X i ʲ

m
i

m
j exp

ʵ mij , i 6¼ j ð 17 : 23 Þ

By substituting Eq. ( 17.23 ) into Eq. ( 17.19 ), an interregional commodity
shipment function by mode can be obtained:

2
4

3
5 8

exp

a m

mw
ij

ʴ

d ij

x mw
ij

X i ʲ

m
i

m
j

¼ ʵ exp mwij , i 6¼ j ð 17 : 24 Þ

X

a m

mw
ij

exp ʴ

w

The amount of interzonal flows including passengers and freight in Equation
( 17.24 ) depends on the magnitude of the Lagrange multiplier associated with zone
i ( ʵ i m ), and the probability of choosing mode for shipments of

passengers and freight. A comparable result is found in the auto occupancy factors
of multiclass travel forecasting models in Boyce and Bar-Gera ( 2003 ).

ʲ i m ) and zone j (

Solution Algorithm

A generalization of the algorithm proposed by Evans ( 1976 ) was used to solve the
optimization problem. In this case each iteration of the algorithm uses Wilson's
( 1970 ) iterative balancing method to generate the subproblem interregional com-
modity shipments, and the all-or-nothing assignment method to find the subproblem
network link flows.
The convergence of Wilson's iterative balancing method can be judged by the
relative error between the observed and calculated final demands for each sector
and each region, or the relative change of balancing factors. 25 Unlike the experience
of Rho et al. ( 1989 ), convergence was satisfactory.

References

Batty M (1976) Urban modelling: algorithms, calibrations, predictions. Cambridge University
Press, Cambridge, MA
Ben-Ayed O, Blair CE, Boyce DE, LeBlanc LJ (1992) Construction of a real-world bilevel linear
programming model of the highway network design problem. Ann Oper Res 34:1-4

25 The former convergence criterion was used successfully with 0.001 as the stopping value in
Ham et al. ( 2005 ).