8
J. Wittmann
2.2.3 Level 3: Aggregation
Aggregating is the usage mainly applied: not only one but several aspects of a
system are
(a) measured (see level 1)
(b) classified (see level 2),
(c) weighted to each other, and
(d) functionally combined to one resulting single value.
As can already be seen from the example in 2.2.2, the task of judging is in most
cases not a one-dimensional problem but a multidimensional one. In the context of
indicators and system optimisation, it is better to speak of a multi-criteria problem.
Different aspects should be considered when assessing the state of the system
measured by the indicator. As in Level 1 and Level 2, a scale must be introduced for
each of these individual aspects and an evaluation or definition of threshold values
must be carried out.
At this point the problem arises that the evaluation of sub-criteria is contradictory
and therefore no simple and unambiguous decision can be made. Ultimately, the
Hasse diagrams, which are the subject of many contributions in this volume,
represent an alternative solution to this decision problem by defining a partial order
for the subcriteria.
The second fundamental alternative is to combine the individual criteria into an
aggregated value. This can be done by arbitrary mathematical operations. Usually,
the values of the subcriteria are added, but multiplication, exponentiation and any
other connections are also conceivable. In order to compensate for imbalances
with regard to the dimensions of the criteria but also to realize an applicationspecific weighting of the subcriteria, the values are usually weighted before they
are subjected to the aggregation function. The aim of this aggregation is always to
determine a one-dimensional indicator value with only one scale, on the basis of
which a clear ranking or a clear decision can then be made.
In the muesli example, the quantity available for each type of fruit must be determined, a weighting factor must be assigned, and the individual values determined
in this way must be aggregated (for example, by forming totals) to determine the
final indicator value for the “quality” of the muesli. Obviously, the problem of the
weighting of the individual aspects (“Can 2 slices of mango compensate for the
lack of 20 pieces of apple?”) and the decision for the aggregation operation (sum
formation? product? ...) come to light. The advantage of this alternative, however,
is that there is a single indicator value at the end and no incomparability has to be
discussed, as occurs with the use of partial orders.
2.2.4 Hierarchy of the Levels
The hierarchy of the levels is obvious: level 1 describes the access to a quantity
under observation, level 2 deals with the range of the values of the quantity observed,
J. Wittmann
2.2.3 Level 3: Aggregation
Aggregating is the usage mainly applied: not only one but several aspects of a
system are
(a) measured (see level 1)
(b) classified (see level 2),
(c) weighted to each other, and
(d) functionally combined to one resulting single value.
As can already be seen from the example in 2.2.2, the task of judging is in most
cases not a one-dimensional problem but a multidimensional one. In the context of
indicators and system optimisation, it is better to speak of a multi-criteria problem.
Different aspects should be considered when assessing the state of the system
measured by the indicator. As in Level 1 and Level 2, a scale must be introduced for
each of these individual aspects and an evaluation or definition of threshold values
must be carried out.
At this point the problem arises that the evaluation of sub-criteria is contradictory
and therefore no simple and unambiguous decision can be made. Ultimately, the
Hasse diagrams, which are the subject of many contributions in this volume,
represent an alternative solution to this decision problem by defining a partial order
for the subcriteria.
The second fundamental alternative is to combine the individual criteria into an
aggregated value. This can be done by arbitrary mathematical operations. Usually,
the values of the subcriteria are added, but multiplication, exponentiation and any
other connections are also conceivable. In order to compensate for imbalances
with regard to the dimensions of the criteria but also to realize an applicationspecific weighting of the subcriteria, the values are usually weighted before they
are subjected to the aggregation function. The aim of this aggregation is always to
determine a one-dimensional indicator value with only one scale, on the basis of
which a clear ranking or a clear decision can then be made.
In the muesli example, the quantity available for each type of fruit must be determined, a weighting factor must be assigned, and the individual values determined
in this way must be aggregated (for example, by forming totals) to determine the
final indicator value for the “quality” of the muesli. Obviously, the problem of the
weighting of the individual aspects (“Can 2 slices of mango compensate for the
lack of 20 pieces of apple?”) and the decision for the aggregation operation (sum
formation? product? ...) come to light. The advantage of this alternative, however,
is that there is a single indicator value at the end and no incomparability has to be
discussed, as occurs with the use of partial orders.
2.2.4 Hierarchy of the Levels
The hierarchy of the levels is obvious: level 1 describes the access to a quantity
under observation, level 2 deals with the range of the values of the quantity observed,