Geoscience Reference
In-Depth Information
TOAW
=
exp(
−
β
d
)
(7.4)
ij
ii j
ij
where
T
ij
is the simulated flow between origin
i
and destination
j
O
i
is the known origin mass (i.e. the pupil population in this case study)
A
i
is the balancing factor to constrain the model, ensuring all origin masses are consistent with
those observed
W
j
is the destination attractiveness (i.e. the average point score for the school in this case)
Parameter β is calibrated and
A
i
is calculated using the following equation:
1
exp(
A
=
∑
i
(7. 5)
W
−
β
d
)
j
ij
j
This origin-constrained model equation can be reordered into Equations 7.6 and 7.7
(Openshaw 1998):
*
TW d
ij
=
exp
(
−
β
)
(7.6)
j
ij
O
*
i
T
=
T
∑
ij
ij
(7.7)
T
ij
j
This resulting simplification of the model calculation separates the constraint from the model equa-
tion. Therefore, the constraint can be applied to different model forms without altering the con-
straint calculation. Here, beta is a parameter to be optimised controlling the relative flow ()
T
ij
*
between
i
and
j
as a function of distance. It does not depend at all on the absolute values of the flow
and hence is independent of the constraints
O
i
.
7.8.3 r
ePreSenting
the
e
quationS
As with the previous case study, a crucial step is to formalise the model (or parameters) to be opti-
mised in a way that is consistent with the structure of a GA. Within this example, each equation (or
layer representing different groups of pupils) can be represented as a chromosome that is made up
of different genes. Here, genes are composed of four separate components: a data item, a parameter,
a function and an operator. Following the advice of Openshaw (1998), the component parts of the
gene can be encoded into strings that can be manipulated and decoded to form valid equations. To
encode and decode the genes, the lookup tables shown together as Table 7.6 were used.
If we were to take a typical representation of an SI model,
T OD fd
ij
=
()
(7.8)
i
j
ij
and encode the different components, the codes in Table 7.7 would be produced. For valid
equations to be built, each gene must contain one of the four component parts discussed earlier.
However, the final gene in the SI model used in this example has no operator in the equation.
Therefore, the same operator, that is, 3, is encoded for this component of each gene, and when
the equation is resolved, the final operator in the equation is ignored. Concatenating these three
