Looking for Alternatives? Split-Shots as an Exemplary Case
155
2.1 The Basic Equation of Hasse Diagram Technique
In its basis partial ordering appears pretty simple as the only mathematical relation
among the objects is “≤” (Bruggemann and Carlsen 2006a, b; Bruggemann and
Patil 2011). As the basis for a comparison of objects, here split-shot alternatives,
characterization by the group of indicators is used (vide infra). This series of
indicators, r i , characterizes the single split-shot alternatives. Thus, if one of the
alternatives (x) is characterized by the set of indicators r i (x), i = 1,...,m, where m is
the number of indicators, it can be compared to another alternative (y), characterized
by the indicators r i (y). Thus, y < x iff
r i (y) ≤ r i (x) for all i = 1, . . . , m
( 1 )
Equation 1 is a very hard and strict requirement for establishing a comparison.
It demands that all indicators of x should be better (or at least equal) than those
of y. Further, let X be a set of alternatives included in the analysis, i.e., X = {Pb,
Bi, cer, ste, Sn, W}, 1 x will be ordered higher (better) than y, i.e., x > y, if at least
one of the indicator values for x is higher than the corresponding indicator value
for y and no indicator for x is lower than the corresponding indicator value for y.
On the other hand, if r i (x) > r i (y) for some indicator r i and r j (x) < r j (y) for some
other indicator r j , x and y will be called incomparable (notation: x || y) expressing
the mathematical contradiction due to conflicting indicator values. A set of mutual
incomparable objects is called an antichain. When all indicator values for x are
equal to the corresponding indicator values for y, i.e., r i (x) = r i (y) for all r i , the two
compared elements will have identical rank and will be considered as equivalent,
i.e., x ~ y. The analysis of Eq. 1 can be visualized by a Hasse diagram.
2.2 The Hasse Diagram
The Eq. 1 is the basic for the Hasse diagram technique (HDT) (Bruggemann
and Carlsen 2006a, b; Bruggemann and Patil 2011). Hasse diagrams are visual
representations of the partial order. In the Hasse diagram comparable objects are
connected by a sequence of lines (Bruggemann and Carlsen 2006a, b; Bruggemann
and Patil 2011; Bruggemann and Munzer 1993; Bruggemann and Voigt 1995, 2008).
1 Pb: Lead, Bi: Bismut, cer: Ceramics, ste: Steel, Sn: Tin, W: Wolfram.
155
2.1 The Basic Equation of Hasse Diagram Technique
In its basis partial ordering appears pretty simple as the only mathematical relation
among the objects is “≤” (Bruggemann and Carlsen 2006a, b; Bruggemann and
Patil 2011). As the basis for a comparison of objects, here split-shot alternatives,
characterization by the group of indicators is used (vide infra). This series of
indicators, r i , characterizes the single split-shot alternatives. Thus, if one of the
alternatives (x) is characterized by the set of indicators r i (x), i = 1,...,m, where m is
the number of indicators, it can be compared to another alternative (y), characterized
by the indicators r i (y). Thus, y < x iff
r i (y) ≤ r i (x) for all i = 1, . . . , m
( 1 )
Equation 1 is a very hard and strict requirement for establishing a comparison.
It demands that all indicators of x should be better (or at least equal) than those
of y. Further, let X be a set of alternatives included in the analysis, i.e., X = {Pb,
Bi, cer, ste, Sn, W}, 1 x will be ordered higher (better) than y, i.e., x > y, if at least
one of the indicator values for x is higher than the corresponding indicator value
for y and no indicator for x is lower than the corresponding indicator value for y.
On the other hand, if r i (x) > r i (y) for some indicator r i and r j (x) < r j (y) for some
other indicator r j , x and y will be called incomparable (notation: x || y) expressing
the mathematical contradiction due to conflicting indicator values. A set of mutual
incomparable objects is called an antichain. When all indicator values for x are
equal to the corresponding indicator values for y, i.e., r i (x) = r i (y) for all r i , the two
compared elements will have identical rank and will be considered as equivalent,
i.e., x ~ y. The analysis of Eq. 1 can be visualized by a Hasse diagram.
2.2 The Hasse Diagram
The Eq. 1 is the basic for the Hasse diagram technique (HDT) (Bruggemann
and Carlsen 2006a, b; Bruggemann and Patil 2011). Hasse diagrams are visual
representations of the partial order. In the Hasse diagram comparable objects are
connected by a sequence of lines (Bruggemann and Carlsen 2006a, b; Bruggemann
and Patil 2011; Bruggemann and Munzer 1993; Bruggemann and Voigt 1995, 2008).
1 Pb: Lead, Bi: Bismut, cer: Ceramics, ste: Steel, Sn: Tin, W: Wolfram.