60
R. Bruggemann and L. Carlsen
of Nardo (2008) the weights should not too much deviate from 1/m, m being the
number of indicators. In the case of the weighted sum, Eq. 4, the condition for
crucial weigths, namely CI(x) = CI(y), with x,y two objects of the object set, can
be relaxed as follows.
δ ’ (i1, i2) :=
j =1,...,m
g(j ) ∗ (x (i1, j) − x (i2, j))
(14)
In Eq. 14 i1 and i2 are labelling different objects of the object set, j is labelling
different indicators.
If the weights are taken the same value (g(j) = 1/m for all j) then the sum of
differences of the matrix entries alone is decisive:
δ (i1, i2) :=
j =1,...,m
(x (i1, j) − x (i2, j))
(15)
Scanning the whole object set and calculating the δ-values for all pairs, where
x(i1) x(i2) a distibution for δ is obtained. If then the distribution has high values
in the range g(j) = 1/m then the number of incomparabilities will relatively high
and thence the decision situation difficult. The acceptance of Eq. 15 implies that
the values of the data matrix x are considered as sharp. When noise may perturb
the entries of the data matrix x, then a probability scheme is to be developped
whose result leads to an expectation value for δ(i1,i2). However, this as well as
the following three points are future tasks.
Three further points are worth to be mentioned here:
1. Up to now the uncertainty degree s was the same for each weight. However in
reality the conceptual uncerteinty for each weight may be different. This aspects
needs still a lot of investigations.
2. The determination of a composite indicator due to Eq. 4 needs that the indicators
fulfill certain scaling levels. At least they must be metric in nature, in order to
let multiplication with a scalar (the weights) and summation a mathematically
meaningful. In the case of the child well-being, it is clear that a) a normalization
of ranks, as well as the subsequent algorithmic comination is at least mathematically questionable. Nevertheless the example is important enough, to consider
the indicators as if they are metric quantities.
3. It is completely clear that the weighted sum, with its high potential for compensation (Munda 2008) has advantages because of its transparency, but there are
other high sophisticated decision support systems (DSS). However an analysis
as in this and the former papers will be extremely difficult for those other DSS,
not only because of their more involved mathematical structure, but also, because
usually more parameters influence the final result, i.e. the final ranking.
Hence there is still much of work possible, to clarify the role of weights and other
parameters in other DSS.
R. Bruggemann and L. Carlsen
of Nardo (2008) the weights should not too much deviate from 1/m, m being the
number of indicators. In the case of the weighted sum, Eq. 4, the condition for
crucial weigths, namely CI(x) = CI(y), with x,y two objects of the object set, can
be relaxed as follows.
δ ’ (i1, i2) :=
j =1,...,m
g(j ) ∗ (x (i1, j) − x (i2, j))
(14)
In Eq. 14 i1 and i2 are labelling different objects of the object set, j is labelling
different indicators.
If the weights are taken the same value (g(j) = 1/m for all j) then the sum of
differences of the matrix entries alone is decisive:
δ (i1, i2) :=
j =1,...,m
(x (i1, j) − x (i2, j))
(15)
Scanning the whole object set and calculating the δ-values for all pairs, where
x(i1) x(i2) a distibution for δ is obtained. If then the distribution has high values
in the range g(j) = 1/m then the number of incomparabilities will relatively high
and thence the decision situation difficult. The acceptance of Eq. 15 implies that
the values of the data matrix x are considered as sharp. When noise may perturb
the entries of the data matrix x, then a probability scheme is to be developped
whose result leads to an expectation value for δ(i1,i2). However, this as well as
the following three points are future tasks.
Three further points are worth to be mentioned here:
1. Up to now the uncertainty degree s was the same for each weight. However in
reality the conceptual uncerteinty for each weight may be different. This aspects
needs still a lot of investigations.
2. The determination of a composite indicator due to Eq. 4 needs that the indicators
fulfill certain scaling levels. At least they must be metric in nature, in order to
let multiplication with a scalar (the weights) and summation a mathematically
meaningful. In the case of the child well-being, it is clear that a) a normalization
of ranks, as well as the subsequent algorithmic comination is at least mathematically questionable. Nevertheless the example is important enough, to consider
the indicators as if they are metric quantities.
3. It is completely clear that the weighted sum, with its high potential for compensation (Munda 2008) has advantages because of its transparency, but there are
other high sophisticated decision support systems (DSS). However an analysis
as in this and the former papers will be extremely difficult for those other DSS,
not only because of their more involved mathematical structure, but also, because
usually more parameters influence the final result, i.e. the final ranking.
Hence there is still much of work possible, to clarify the role of weights and other
parameters in other DSS.