84
A. Kerber and R. Bruggemann
This evaluation associates, e.g. with the refrigerant ref 4 , the truth value tv(ref 4
has (ODP;GWP;ALT)) = (0.00431373;0.00513514;0.00040594).
As the entries of this table are in L = [0,1] 3 , this evaluation is a mapping
E :
ref 1 , ref 2 , . . .
× {(ODP ; GW P ; ALT )} → [0, 1]
3 ,
with, e.g., the value
E
ref 4 , (ODP ; GW P ; ALT )
= (0.00431373; 0.00513514; 0.00040594) .
These values are elements of the lattice L = [0,1] 3 , and hence we may consider
such an evaluation as an L-subset of the set of refrigerants. This way of analysis
implies that we have no more crisp sets (an element is a member of a set: “yes” or
“no”), but fuzzy sets, where the membership can be any number between 0 and 1.
The general case reads as follows:
1.1 Definition
An evaluation E of objects o i ∈ O w.r.t. attributes a k ∈ A and over L is a mapping
E : O × A → L : (o i , a k ) → E ((o i , a k )) = tv (o i has a k ) ,
i.e. we consider it as an L-subset E of O × A, containing (o i ,a k ) with the truth value
tv(o i has a k ) ∈ L.
1.2 Basic theory
Evaluations of objects o i w.r.t. attributes a k .
– Consider
L
O×A
:= {E | E : O × A → L} ,
the set of all L–subsets of O × A, for a given lattice L. In case L = [0,1] 3 , an L-subset
of O × A is an association of triples of parameter values to the pairs (o,a) ∈ O × A.
By this generalization of the evaluation we can choose a set theory and its logic
over L and this allows problem–orientation. Hereby we adopt the fuzzy notion
of membership functions (Pollandt 1997) as the set theoretical operations such as
subset-set relation, inclusion, set differences or union are not necessarily related to
crisp sets.
A. Kerber and R. Bruggemann
This evaluation associates, e.g. with the refrigerant ref 4 , the truth value tv(ref 4
has (ODP;GWP;ALT)) = (0.00431373;0.00513514;0.00040594).
As the entries of this table are in L = [0,1] 3 , this evaluation is a mapping
E :
ref 1 , ref 2 , . . .
× {(ODP ; GW P ; ALT )} → [0, 1]
3 ,
with, e.g., the value
E
ref 4 , (ODP ; GW P ; ALT )
= (0.00431373; 0.00513514; 0.00040594) .
These values are elements of the lattice L = [0,1] 3 , and hence we may consider
such an evaluation as an L-subset of the set of refrigerants. This way of analysis
implies that we have no more crisp sets (an element is a member of a set: “yes” or
“no”), but fuzzy sets, where the membership can be any number between 0 and 1.
The general case reads as follows:
1.1 Definition
An evaluation E of objects o i ∈ O w.r.t. attributes a k ∈ A and over L is a mapping
E : O × A → L : (o i , a k ) → E ((o i , a k )) = tv (o i has a k ) ,
i.e. we consider it as an L-subset E of O × A, containing (o i ,a k ) with the truth value
tv(o i has a k ) ∈ L.
1.2 Basic theory
Evaluations of objects o i w.r.t. attributes a k .
– Consider
L
O×A
:= {E | E : O × A → L} ,
the set of all L–subsets of O × A, for a given lattice L. In case L = [0,1] 3 , an L-subset
of O × A is an association of triples of parameter values to the pairs (o,a) ∈ O × A.
By this generalization of the evaluation we can choose a set theory and its logic
over L and this allows problem–orientation. Hereby we adopt the fuzzy notion
of membership functions (Pollandt 1997) as the set theoretical operations such as
subset-set relation, inclusion, set differences or union are not necessarily related to
crisp sets.