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cation with cutting points determined by the reduction
percent c will place alternative A in the class C
i
for that i minimizing the absolute
value of the difference A
i
+
A benevolent classi
c)A
i
−
.
−
(1
−
cation for the same reduction percent will place
alternative A in that class Ci
i
for the i determined by the hostile procedure modi
Analogously, a hostile classi
ed
by replacing the difference A
i
+
A
i
−
.
Another way to generate more benevolent and more hostile extremes without the
need to determine a rate of reduction may be applied if the combination of the
criteria is by joint probabilities. In that case, to obtain the benevolent classi
A
i
−
with (1
c)Ai
+
−
−
−
cation,
the joint probability of being above all the pro
les of the class may be replaced by
the probability of being above at least one of them. On the other end, in the
computation of the hostile extreme, the joint probability of being below all the
pro
les of the class will be replaced by the probability of being below at least one
of them.
That means, assuming maximal dependence, to get the benevolent extreme in the
computation of A
i
+
, substituting max
j
A
i
+
for min
j
A
i
+
and, to get the hostile extreme,
substituting max
j
A
i
−
for min
j
A
i
−
. In the case of independence, instead of max
j
A
i
+
and max
j
A
i
−
, respectively, 1
A
i
−
) enter here.
To conclude this section a practical hint on the generation of pro
A
i
+
) and 1
− π
j
(1
−
− π
j
(1
−
les may be
useful. In practice, we have to classify not a unique alternative, but a set of different
alternatives, as in the ranking problem. This may be used in the process of gen-
erating pro
les for r classes would be formed with
coordinates given by the quantiles of order 1/2r, 3/2r,
…
, (2r
−
1)/2r, for every
criterion.
les. For instance, central pro
8.5 Classification of Car Models
The classi
cation setup may be applied to the data of 20 car models studied in the
previous chapters. Suppose that there are
five classes where they might be classi
ed
each identi
les, given in Table
8.1
.
The probabilities of each alternative being above and below all the four pro
ed by four pro
les
for each of the 5 classes are presented in Table
8.2
, assuming triangular distribu-
tions with extremes 0 and 1 and composing the criteria by a weighted average with
the weights derived in Chap.
2
from pairwise comparison: 0.03 for beauty and for
power, 0.05 for price, 0.08 for comfort, 0.15 for gas consumption and 0.33 for
reliability and for safety.
The last column of this table presents the classi
cation obtained. Only Car3,
Car16 and Car17 are classi
ed in the highest class and no alternative is classi
ed in
the lowest.
If instead of triangular distributions, normal distributions with standard devia-
tions of 0.4
—
an approximate value for the observed standard deviations in the
samples of all the different criteria
—
are employed, the classi
cation obtained is
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