Geoscience Reference
In-Depth Information
X
X
X
X
y
j
, for all stories
s
, sectors
m
and regions
j
x
sm
ij
a
smn
j
x
jk
þ
¼
:
si
sn
k
ð
17
:
2
Þ
where
a
j
smn
is the quantity of inputs from sector
m
required to produce one unit of
output of sector
n
at
s-intensity
of land use (i.e.
s
-story building) in zone
j
, the
quantities being measured in value terms, and
y
j
m
is the final demand for the output
of sector
m
in zone
j
. The materials balance constraint represents the conservation
of shipments between origin and destination zones. Zones are assumed to be
indifferent regarding the source of the supply of commodities.
3. The sum of shipments of sector
m
from zone
i
to zone
j
by all modes equals the
total shipment of sector
m
from zone
i
to zone
j
.
X
x
smw
ij
x
ij
, for all sectors
m
, and regions
i
,
j
¼
:
ð
17
:
3
Þ
sm
4. The total flow (tons) of sector
m
on all routes
r
of mode
w
between regions
i
and
j
equals the total shipment ($) of sector m from
i
to
j
by mode
w
divided by an
exogenous ratio
gm
($/ton) that converts commodity shipments from value of
commodity shipments to weight of network flows.
X
x
mw
ij
g
m
, for all sectors
m
, all modes
w
, and regions
i
,
j
h
mw
ijr
¼
ð
17
:
4
Þ
r
The ratio is derived from the 1993 Commodity Flow Survey by dividing the
total value of shipments ($) by total weight of the shipments (tons) throughout
the U.S.
24
5. Finally, route flows must be nonnegative, which implies that links flows are also
nonnegative:
h
mw
ijr
0, for all sectors
m
, and all routes
r
of mode
w
and zones
i
,
j
ð
17
:
5
Þ
Having defined the constraints on interregional and route flows, the objective
function is as follows. On the assumption that shippers collectively desire to
minimize their total shipment distances and thus the dispersion of shipments across
origins and destinations and modes can be represented by interzonal and modal
dispersion (entropy) functions, the objective function is defined as follows:
Z
X
a
d
a
min
h
,
x
ðÞ¼
h
;
x
ðÞ
d
ˉ
0
aw
"
#
þ
"
!#
X
g
m
X
ij
X
m
g
m
X
ijws
x
ij
X
i
x
smw
ij
x
ij
1
1
x
ij
ln
x
smw
ij
þ
ln
ʱ
m
m
ð
17
:
6
Þ
24
The values of these ratios are shown in Table 3 of Ham et al. (
2005
).
