Geoscience Reference
In-Depth Information
Appendix: Integrated Three-Dimensional Urban Activity Model:
Formulation, Optimality Conditions and Solution Algorithm
22
Model Formulation
This model formulation may be regarded as integrating a Leontief-Strout-Wilson
(Leontief and Strout
1963
; Wilson
1970
) interregional commodity shipment model
with a transportation network model. Exports and imports of commodities in an
urban area are given by the urban input-output models for the period of analysis
taken as 1 year. Factor input includes labor so that commuters are included in the
model as well as commodity flows among traffic zones. Then the problem is to
predict the shipments by sector/commodity between each pair of zones and the
transportation network flows resulting from these shipments.
Let
f
a
w
denote the total flow (tons) on link
a
of mode
w
over an effective distance
(miles) of
d
a
w
(
f
a
w
). For this prototype model, link distances are either considered to
be fixed, or to be effectively longer than the actual length in the case of congested
road links, representing the additional operating costs of congestion. The tradi-
tional Bureau of Public Roads (BPR) link performance function is utilized for roads
to represent this effective distance, which increases as a function of the flow to
capacity ratio raised to a positive power. Sensitivity analysis results are reported for
powers of 0 (no congestion), 4 and 8. For mass transit modes including railways,
link distances are fixed, corresponding to the assumption of unlimited capacity.
Let
h
ijr
mw
denote the flow (tons) of output of sector
m
from region
i
to region
j
by
mode
w
on route
r
, and
X
s
x
smw
denote the shipment ($) of output of sector
m
(including housing outputs assuming one per each labor
23
) produced in
s
story
building from zone
i
to zone
j
by mode
w
. To define the relationships among these
variables and other quantities, consider the following definitions:
1. Link flow equals the sum of the route flows on all routes between all zones using
link
a
of mode
w
:
X
X
h
mw
ijr
aw
ijr
f
a
, for all links
a
and modes
w
φ
¼
:
ð
17
:
1
Þ
m
ijr
φ
ijr
aw
is 1, if route
r
from
i
to
j
uses link
a
of mode
w
, and 0, otherwise.
2. The materials balance constraint states the total shipment of commodity
m
into
zone
j
from all zones
i
is equal to the use of that commodity for producing all
commodities (intermediate demand) in
s
story of production facilities plus zonal
final demand.
where
22
The original paper title is “Implementation and estimation of a combined model of interregional,
multimodal commodity shipments and transportation network flows”, as described in Footnote 24.
23
Assuming each household requires a house, m denotes labor in input sector, but housing outputs
in output sector.
