Comparison of Selected Procedures for Generating Activated Carbon. . .
171
2011; Bruggemann et al. 2014), the latter gives an insight in the relative importance
of the included indicators (Bruggemann and Patil 2011; Bruggemann et al. 2014)
and antichain analyses (Bruggemann and Voigt 2011).
2.3.1 Average Ranks
The average ranking is expressed as average height from bottom (min. Height = 1)
to the top (max height = n, i.e., the maximum number of objects, here n = 21)
(Bruggemann and Annoni 2014). The average rank is generated by calculating all
linear order preserving sequences (set LE), the linear extensions of the original
partial order. From LE_0 the statistical characterization for each object is obtained.
For example the characterization is calculated as the average value an object has,
taken all positions of this object within LE_0, the averaged heights. It is clear that
this procedure is computationally extremely difficult. Hence, approximations were
developed (Bruggemann et al. 2004, Bruggemann and Carlsen 2011).
2.3.2 Sensitivity Analysis
For the sentivity analysis (Bruggemann et al. 2001; Bruggemann and Patil 2011;
Bruggemann et al. 2014), let Q be the set of all indicators, then taken all indicators
of Q leads to a partial order, which is called PO_0. The corresponding set of linear
extensions is denoted by LE_0. Leaving out one indicator of Q, say r j , then another
partial order results, which is denoted as PO_j.
Both partial orders can be described by an adjacent matrix, say A_0 for PO_0
and A_j for PO_j.
Taken the Euclidian Distance (squared) quantifies the role of indicator qj in
PO_0. This is a sensitivity measure for the indicators of set Q, describing the
structural changes of the partial order leaving one indicator out. This is not
immediately a measure of the sensitivity of the indicators for a ranking, because
the ranking is per definition a linear order and here derived over many interim steps.
If a linear order is obtained by all orders in LE_0, the set of linear extensions
taken from PO_0, then any PO_j will also lead to a corresponding set LE_j. And
this set is the more differing from LE_0 the larger the sensitivity is. Therefore the
ranking due to averaged heights is as more affected by indicator r j as larger its
sensitivity is.
2.3.3 Indicator Conflicts – Tripartite Graphs
In order visually to display and thus better understand the role of individual
indicators for incomparisons, the concept of tripartite graph was introduced by
Bruggemann and Voigt (2011). Here an intuitive approach is presented again assuming a case with three indicators. Imagine that Objx has better values (i.e. higher
171
2011; Bruggemann et al. 2014), the latter gives an insight in the relative importance
of the included indicators (Bruggemann and Patil 2011; Bruggemann et al. 2014)
and antichain analyses (Bruggemann and Voigt 2011).
2.3.1 Average Ranks
The average ranking is expressed as average height from bottom (min. Height = 1)
to the top (max height = n, i.e., the maximum number of objects, here n = 21)
(Bruggemann and Annoni 2014). The average rank is generated by calculating all
linear order preserving sequences (set LE), the linear extensions of the original
partial order. From LE_0 the statistical characterization for each object is obtained.
For example the characterization is calculated as the average value an object has,
taken all positions of this object within LE_0, the averaged heights. It is clear that
this procedure is computationally extremely difficult. Hence, approximations were
developed (Bruggemann et al. 2004, Bruggemann and Carlsen 2011).
2.3.2 Sensitivity Analysis
For the sentivity analysis (Bruggemann et al. 2001; Bruggemann and Patil 2011;
Bruggemann et al. 2014), let Q be the set of all indicators, then taken all indicators
of Q leads to a partial order, which is called PO_0. The corresponding set of linear
extensions is denoted by LE_0. Leaving out one indicator of Q, say r j , then another
partial order results, which is denoted as PO_j.
Both partial orders can be described by an adjacent matrix, say A_0 for PO_0
and A_j for PO_j.
Taken the Euclidian Distance (squared) quantifies the role of indicator qj in
PO_0. This is a sensitivity measure for the indicators of set Q, describing the
structural changes of the partial order leaving one indicator out. This is not
immediately a measure of the sensitivity of the indicators for a ranking, because
the ranking is per definition a linear order and here derived over many interim steps.
If a linear order is obtained by all orders in LE_0, the set of linear extensions
taken from PO_0, then any PO_j will also lead to a corresponding set LE_j. And
this set is the more differing from LE_0 the larger the sensitivity is. Therefore the
ranking due to averaged heights is as more affected by indicator r j as larger its
sensitivity is.
2.3.3 Indicator Conflicts – Tripartite Graphs
In order visually to display and thus better understand the role of individual
indicators for incomparisons, the concept of tripartite graph was introduced by
Bruggemann and Voigt (2011). Here an intuitive approach is presented again assuming a case with three indicators. Imagine that Objx has better values (i.e. higher