Geoscience Reference
In-Depth Information
For a 2-additive capacity
µ
, the Shapley value of an isolated criterion i is given by
X
S l f
ðÞ ¼
i
½
ð
#
ð
S
n
K
Þ
1
Þ!#
ð
ðÞ
K
Þ!=#
ðÞ!
S
l ð
K
[
fgÞ l
i
ðÞ
K
K S n fg
or
2 X
S l f
ðÞ ¼l f
i
ðÞþ
i
1
=
I
l ij
j 2 S n fg
for
l ij ¼ l
ð
fg
;
Þ l
if l
fð :
I
i
j
i
j
I
µ ij represents an interaction between i and j, in the sense that
I
µ ij = 0 corresponds to independence between i and j;
µ ij > 0 means some complementarity between i and j, i.e., for the decision
maker, both criteria have to be satisfactory in order to get a satisfactory alternative;
and
I
I
µ ij < 0 means some substitutability or redundancy between i and j, i.e., for the
decision maker, the satisfaction of one of the two criteria is suf
cient to have a
satisfactory alternative.
With this notation, for any x = (x 1 ,
,x m ), the Choquet integral of x with
respect to the 2-additive capacity
µ
is given by:
m
2 X
i ; j 2 S
C
l ð
x 1 ; ...;
x m Þ ¼
S l ð i Þ
x i
1
=
Il ij j
x i
x j j:
i¼1
2.3.2 Example of Application of the Choquet Integral
An example of application of the concept of capacity and the Choquet integral may
be constructed by revising the car models choice problem presented to show how to
use the AHP approach.
Essentially, a car has no value if it does not move. Thus a model evaluated as
unsatisfactory from the point of view of power suffers from a basic limitation. In
that sense, the value of the presence of any of the other attributes depends on the
presence of power. To take this into account, a capacity might be employed to
improve that study. It would derive from the weights there employed, which
assumed additivity, the capacity of any set of criteria that includes power, but would
assign a null capacity to any set that does not include power. This would result in a
final score of zero for those cars which were assigned a value of zero with respect to
power.
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