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R. Bruggemann and L. Carlsen
2. even minor numerical differences (possibly being considered as scientifically
irrelevant) are evaluated and lead to comparabilities/incomparabilities, although
the objects should better be considered as equivalent.
These two aspects have led to many activities, such as fuzzy concepts that
were introduced (Wieland and Bruggemann 2013; Bruggemann et al. 2011). The
role of incomparabilities were analyzed in details (cf. e.g. Bartel and Mucha
2014; Bruggemann and Carlsen 2014a, b, 2015, 2017). Two objects x,y mutually
incomparable are denoted as x y. When the orientation x ≤ y or x ≥ y is of minor
interest then the mere fact of comparability is denoted by x ⊥ y.
2.2 Modelling a Decision Support System
As already mentioned, for the sake of public acceptance, the model for decision
support should be simple and the weighted sum of indicator values of an object
seems to be the best starting point. The synthetic indicator, also known as composite
indicator CI of an object x will be calculated, according to Eq. 4.
CI (x) = Σ (g (j) ∗ x (i, j)) i = 1, . . . , n; j
1, . . . , m
( 4 )
The entries of a data matrix x(i,j) must be metric in order to combine them by
multiplication with a scalar g(j) and subsequent additions. Furthermore, the entries
x(i,j) and the weights g(j) should conveniently be normalized, i.e., being elements
of the range [0,1]. The boundary condition for the weights g(j) is:
Σ g (j) = 1 j =, 1, . . . , m
( 5 )
and
0 ≤ g (j) ≤ 1
( 6 )
The crucial point is that (as already mentioned) most often a sharp value for the
weights g(j) cannot be given. Therefore the theoretical concept (already published
and discussed in more detail in Bruggemann and Carlsen 2017) can be described by
the following six items:
1. g(j) is taken from an interval [g(j) min , g(j) max ], for the sake of simplicity it is
written: g(j) min = gjmin and g(j) max = gjmax.
2. s(j): = gjmax – gjmin is called the degree of uncertainty with respect to the
weights.
3. it is assumed that s(j) = s for all j.
4. the evolution concept
5. identification of the number of incomparabilities, U, as a leading quantity and
6. in refinement of (5), the concept of Us and other quantities derived from U.
R. Bruggemann and L. Carlsen
2. even minor numerical differences (possibly being considered as scientifically
irrelevant) are evaluated and lead to comparabilities/incomparabilities, although
the objects should better be considered as equivalent.
These two aspects have led to many activities, such as fuzzy concepts that
were introduced (Wieland and Bruggemann 2013; Bruggemann et al. 2011). The
role of incomparabilities were analyzed in details (cf. e.g. Bartel and Mucha
2014; Bruggemann and Carlsen 2014a, b, 2015, 2017). Two objects x,y mutually
incomparable are denoted as x y. When the orientation x ≤ y or x ≥ y is of minor
interest then the mere fact of comparability is denoted by x ⊥ y.
2.2 Modelling a Decision Support System
As already mentioned, for the sake of public acceptance, the model for decision
support should be simple and the weighted sum of indicator values of an object
seems to be the best starting point. The synthetic indicator, also known as composite
indicator CI of an object x will be calculated, according to Eq. 4.
CI (x) = Σ (g (j) ∗ x (i, j)) i = 1, . . . , n; j
1, . . . , m
( 4 )
The entries of a data matrix x(i,j) must be metric in order to combine them by
multiplication with a scalar g(j) and subsequent additions. Furthermore, the entries
x(i,j) and the weights g(j) should conveniently be normalized, i.e., being elements
of the range [0,1]. The boundary condition for the weights g(j) is:
Σ g (j) = 1 j =, 1, . . . , m
( 5 )
and
0 ≤ g (j) ≤ 1
( 6 )
The crucial point is that (as already mentioned) most often a sharp value for the
weights g(j) cannot be given. Therefore the theoretical concept (already published
and discussed in more detail in Bruggemann and Carlsen 2017) can be described by
the following six items:
1. g(j) is taken from an interval [g(j) min , g(j) max ], for the sake of simplicity it is
written: g(j) min = gjmin and g(j) max = gjmax.
2. s(j): = gjmax – gjmin is called the degree of uncertainty with respect to the
weights.
3. it is assumed that s(j) = s for all j.
4. the evolution concept
5. identification of the number of incomparabilities, U, as a leading quantity and
6. in refinement of (5), the concept of Us and other quantities derived from U.