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R. Bruggemann et al.
• reflexivity: the object can be compared with itself
• antisymmetry: if x ≤ y and y ≤ x ⇒ x = y
• transitivity: if x ≤ y and y ≤ z ⇒ x ≤ z
A special realization of order relations is given by Eqs. (1, 2, and 3):
x(i) = (x (i, 1) , x (i, 2) , . . . ., x (i, m))
(1)
The quantity x(i, j) is the value of the i th object (i = 1, . . . ,n) (here the i th metal),
the j th indicator (j = 1,..,m) (here the jth site) and m the number of indicators used
(here m = 10).
Equation 1 describes a mapping X ➔ IR m , wherein X is the set of objects (the
metals) and IR m is the set of tuples of real numbers with m components. Note that
the tuples are also denoted as data profiles.
According to m = 10 sites, we will have a system of 10 indicators.
x(i1) ≤ x(i2) : ⇐⇒ (x (i1, 1) , . . . , x (i1, m)) ≤
x (i2, 1) , . . . ., x
i2, m
(2)
Equation 2 needs clarification, as it is not yet clear under which conditions one
tuple (that of x(i1) is to be considered less or equal to that of x(i2). The way how
Eq. 2 can be given a meaning, opens the door to many variants. By Eq. 3
(x (i1, 1) , . . . , x (i1, m)) ≤ (x (i2, 1) , . . . .x (i2, m)) : ⇐⇒ x (i1, j) ≤x (i2, j)
for all j = 1, .., m
(3)
a special partial order is defined. Two objects, following Eq. 3 are called
“comparable”, otherwise “incomparable”.
The immediate relation to the data and the corresponding indicators has two
consequences:
1. Any order relation x ≤ y is a direct reflection of the data values of x and y. This
is in contrast to many decision support systems, where an order relation cannot
easily be traced back to the original data, i.e., to the data matrix.
2. The partial order methodology, based on Eq. 3 is applicable wherever a data
matrix is available and where a ranking aim can be defined.
In the literature the method, based on Eq. 3 together with appropriate supporting
software, is often denoted Hasse diagram technique (HDT) (Galassi et al. 1996;
Grisoni et al. 2015; Halfon and Reggiani 1986; Bruggemann et al. 2001, 2008; Patil
and Taillie 2004; Klein and Ivanciuc 2006; Simon et al. 2004, 2006; Helm 2003;
Bruggemann and Voigt 2008, 2011, 2012; Carlsen and Bruggemann 2011, 2014a, b;
Carlsen 2008a, b, 2013, 2018; Newlin and Patil 2010; Annoni et al. 2014; Sørensen
R. Bruggemann et al.
• reflexivity: the object can be compared with itself
• antisymmetry: if x ≤ y and y ≤ x ⇒ x = y
• transitivity: if x ≤ y and y ≤ z ⇒ x ≤ z
A special realization of order relations is given by Eqs. (1, 2, and 3):
x(i) = (x (i, 1) , x (i, 2) , . . . ., x (i, m))
(1)
The quantity x(i, j) is the value of the i th object (i = 1, . . . ,n) (here the i th metal),
the j th indicator (j = 1,..,m) (here the jth site) and m the number of indicators used
(here m = 10).
Equation 1 describes a mapping X ➔ IR m , wherein X is the set of objects (the
metals) and IR m is the set of tuples of real numbers with m components. Note that
the tuples are also denoted as data profiles.
According to m = 10 sites, we will have a system of 10 indicators.
x(i1) ≤ x(i2) : ⇐⇒ (x (i1, 1) , . . . , x (i1, m)) ≤
x (i2, 1) , . . . ., x
i2, m
(2)
Equation 2 needs clarification, as it is not yet clear under which conditions one
tuple (that of x(i1) is to be considered less or equal to that of x(i2). The way how
Eq. 2 can be given a meaning, opens the door to many variants. By Eq. 3
(x (i1, 1) , . . . , x (i1, m)) ≤ (x (i2, 1) , . . . .x (i2, m)) : ⇐⇒ x (i1, j) ≤x (i2, j)
for all j = 1, .., m
(3)
a special partial order is defined. Two objects, following Eq. 3 are called
“comparable”, otherwise “incomparable”.
The immediate relation to the data and the corresponding indicators has two
consequences:
1. Any order relation x ≤ y is a direct reflection of the data values of x and y. This
is in contrast to many decision support systems, where an order relation cannot
easily be traced back to the original data, i.e., to the data matrix.
2. The partial order methodology, based on Eq. 3 is applicable wherever a data
matrix is available and where a ranking aim can be defined.
In the literature the method, based on Eq. 3 together with appropriate supporting
software, is often denoted Hasse diagram technique (HDT) (Galassi et al. 1996;
Grisoni et al. 2015; Halfon and Reggiani 1986; Bruggemann et al. 2001, 2008; Patil
and Taillie 2004; Klein and Ivanciuc 2006; Simon et al. 2004, 2006; Helm 2003;
Bruggemann and Voigt 2008, 2011, 2012; Carlsen and Bruggemann 2011, 2014a, b;
Carlsen 2008a, b, 2013, 2018; Newlin and Patil 2010; Annoni et al. 2014; Sørensen