Pedestrian route-choice and activity scheduling theory and models

S.P. Hoogendoorn, P.H.L. Bovy / Transportation Research Part B 38 (2004) 169–190 181
Markov diffusion process in two dimensions with nearest-neighbor transitions that are determined
by the stochastic differential Eq. (4) (Fleming and Soner, 1993). Solving this (discrete) stochastic
dynamic programming problem is related to solving Eq. (24) by replacing the partial derivatives
with the appropriate finite differences. Let u i denote the unit vector in the ith dimension (i ¼ 1; 2).
The forward finite difference D þ x i
W and backward finite difference D xi
W are defined by
D x i
W :¼ d 1 ½W ðt; x du i Þ W ðt; xÞŠ for i ¼ 1; 2 ð31Þ
For the second-order terms, we then use the following approximations
D 2 x i
W :¼ d 2 ½W ðt; x þ du i Þ 2W ðt; xÞþW ðt; x du i ÞŠ for i ¼ 1; 2
D x i x j
W :¼ 1 2 d 2 ½W ðt; x þ dðu i u j ÞÞ þ 2W ðt; xÞþW ðt; x dðu i u j Þ i
ÞŠ ð32Þ
1 2 d 2 ½W ðt; x þ du i ÞþW ðt; x þ du j ÞþW ðt; x du i ÞþW ðt; x du j ÞŠ
In numerically approximating Eq. (24), the following solution approach is proposed (Fleming and
Soner, 1993):
W ðt h; xÞ ¼W ðt; xÞ hHðx; D x i
W ; D 2 x i
W ; D x i x j
W Þ ð33Þ
where the numerical Hamiltonian is defined by
(
Hðx; D x i
W ; D 2 x i
W ; D x i x j
W Þ¼ min Lðt; x; vÞ þ X
v2V a ðt;xÞ
i
v þ i
D þ x i
W
v i
D xi
W
þ 1 X
H ii ðx; vÞD 2 x
2
i
W þ 1 X
ðH þ ij
2
ðx; vÞDþ x i x j
W
i
i
H ij
ðx; vÞD xi x j
W Þ
ð34Þ
where a þ :¼ maxfa; 0g and a :¼ minfa; 0g. This numerical solution approach has been adopted
in the dedicated microscopic pedestrian flow simulation model NOMAD (Hoogendoorn, 2001).
Example 1 (Route choice and activity area choice for free-flow conditions in Schiphol Plaza). This
example considers Schiphol Plaza, which is a multi-purpose multi-modal transfer station. Fig. 1
shows a snapshot of the microscopic simulation model NOMAD described in (Hoogendoorn,
2001). In this figure, exits E1–E5 indicate exits from Schiphol Plaza; escalators E6 and E7 indicate
exits to the train platforms. V1 and V2 depict the locations of the newspaper vendors. The colors
reflect pedestrians having distinct activity schedules; the color intensity reflects gender (light: female,
dark: male).
Fig. 2a and b respectively show the expected minimum perceived disutility functions W and
example routes for pedestrians using the escalators E6 or E7 to get to the train platform and
pedestrians using either of the exits E1–E5 to get outside. The expected minimum perceived
disutility functions have been determined using the numerical solution approach described in
Section 5.5.3, with route weights c 1 ¼ 1, c 2 ¼ 10, c 3 ¼ 1:5, c 4 ¼ c 5 ¼ 0, and a m ¼ 1 and b m ¼ 0:1
(for all obstacles m). Eq. (29) shows that the optimal routes are perpendicular to the iso-expected
minimum perceived disutility function curves depicted in the figures. Fig. 2a shows three exemplar
)