On fuzzy screening systems ∗
Christer Carlsson
IAMSR, Åbo Akademi University,
DataCity A 3210, SF-20520 Åbo, Finland
e-mail: ccarlsso@abo.fi
Robert Fullér †
Comp. Sci. Dept., Eötvös Loránd University,
Muzeum krt. 6-8, H-1088 Budapest, Hungary
e-mail: rfuller@abo.fi
Abstract
Yager introduced a fuzzy screening procedure to select, from a large class
of alternatives, a small subset to be further investigated. In his methodology
the Kleene-Dienes implication operator was employed to find the unit score
of each alternative by each expert. Generalizing Yager’s principle we suggest
the use of S- and R-implication operators to find the unit score.
1
Introduction
The fuzzy screening system described in [2] is a two stage process. In the first
stage, individual experts are asked to provide an evaluation of the alternatives. This
evaluation consists of a rating for each alternative on each of the criteria, where
each of the criteria may have a different level of importance. The values to be used
for the evaluation of the ratings and importances are drawn from a linguistic scale
which makes it easier for the evaluator to provide a single value rating for each
evaluator for each alternative. This rating is again a linguistic value from the same
simple linguistic scale.
∗
in: Proceedings of EUFIT’95 Conference, August 28-31, 1995 Aachen, Germany, Verlag Mainz,
Aachen, 1995 1261-1264.
†
Presently visiting post-doc researcher at Turku Centre for Computer Science and IAMSR. Partially supported by OTKA T 4281, T 14144 and T 7598
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In the second stage, a methodology introduced in [1] is used to aggregate the individual experts’ evaluations to obtain an overall linguistic value for each object.
This overall evaluation can then be used by the decision maker as an aid in the
selection process. The problem in question consists of three components: (i) a collection of alternative solutions from amongst which we desire to select some subset to be investigated further; (ii) a group of experts whose opinions are solicited
in screening the alternatives; (iii) is a collection of criteria which are considered
relevant in the choice of the alternatives to be further considered.
For each alternative each expert is required to provide his opinion in terms of elements from the following scale S
{OU (S7 ), VH (S6 ), H (S5 ), M (S4 ), L (S3 ), VL (S2 ), N (S1 )}
where OU stands for Outstanding, VH for Very High, H for High, M for Medium,
L for Low, VL for Very Low, and N for None. Assume that for an alternative an
expert provides a collection of n values
{P1 , . . . , Pn }
where Pj is the rating of the alternative on the j-th criteria by the expert. Each Pj
is an element in the set of allowable scores S.
Then the unit score of each alternative by each expert, denoted by U, is calculated
as follows
U = min{N eg(Ij ) ∨ Pj )}
(1)
j
where Ij denotes the importance of the j-th critera. The operations max, min and
N eg are defined by
max(Si , Sj ) = Si if Si ≥ Sj , min(Si , Sj ) = Sj if Sj ≤ Si , N eg(Si ) = S8−i
Yager notes that the formula (1) can be seen as a measure of the degree to which
an alternative satisfies the following proposition:
All important criteria are satisfied
Example 1 Consider some alternative with the following scores on five criteria
Criteria:
C1
C2
C3
C4
C5
Importance:
VH
VH
M
L
VL
Score:
M
L
OU
VH
OU
2
In this case
U = min{N eg(V H) ∨ M, N eg(V H) ∨ L,
N eg(M ) ∨ OU, N eg(L) ∨ V H, N eg(V L) ∨ OU }
= min{V L ∨ M, V L ∨ L, M ∨ OU, H ∨ V H, V H ∨ OU }
= min{M, L, OU, V H, OU } = L
The essential reason for the low performance of this alternative is that it performed
low on the second criterion which has a very high importance.
There are three important classes of fuzzy implication operators:
• S-implications: defined by
x → y = S(n(x), y)
where S is a t-conorm and n is a negation on [0, 1]. These implications
arise from the Boolean formalism p → q = ¬p ∨ q. We shall use the
following S-implications: x → y = min{1 − x + y, 1} (Łukasiewitz) and
x → y = max{1 − x, y} (Kleene-Dienes).
• R-implications: obtained by residuation of continuous t-norm T , i.e.
x → y = sup{z ∈ [0, 1] | T (x, z) ≤ y}
These implications arise from the Intutionistic Logic formalism. We shall
use the following R-implication: x → y = 1 if x ≤ y and x → y = y if
x > y (Gödel), x → y = min{1 − x + y, 1} (Łukasiewitz)
• t-norm implications: if T is a t-norm then
x → y = T (x, y)
Although these implications do not verify the properties of material implication they are used as model of implication in many applications of fuzzy
logic. We shall use the minimum-norm as t-norm implication (Mamdani).
2
Extensions
Let’s us rewrite equation (1) in the form
U = min{N eg(Ij ) ∨ Pj )} = N (P|I) = min{Ij → Pj }
j
j
3
(2)
where P = (P1 , . . . , Pn ) is the vector of performances, I = (I1 , . . . , In ) is the
vector the importances, N denotes the minimum-norm-based necessity, and → is
the Kleene-Dienes implication operator.
Generalizing (2) we define a large family of scoring functions by
U = min{Ij → Pj }
(3)
j
where → denotes a fuzzy implication operator.
Example 2 Consider Example 1 with Gödel implication (R-implication)
Criteria:
C1
C2
C3
C4
C5
Importance:
VH
VH
M
L
VL
Score:
M
L
OU
VH
OU
In this case
U = min{M, L, OU, OU, OU } = L
Similarly to Example 1 the essential reason for the low performance of this alternative is that it performed low on the second criterion which has a very high
importance.
We note that if → is an R-implication then the formula (3) can be seen as a measure
of the degree to which an alternative satisfies the following proposition:
All scores are bigger or equal than the importances
Really, if → is an R-implication and Ij ≤ Pj holds for all j then
U = OU.
And if Ij ≤ Pj holds for some j then the j-th criterion does not affect the unit
score.
Example 3 Consider Example 1 with Łukasiewicz implication
Criteria:
C1
C2
C3
C4
C5
Importance:
VH
VH
M
L
VL
Score:
M
L
OU
VH
OU
4
In this case
U = min{H, H, M, OU, OU, OU } = M
where we used the operation
OU − Sj + Si = S7−j+i for j ≥ i.
It means that the Łukasiewicz implication operator (which belongs to the intersection of the families of S− and R-implications) provides a better score (bigger
positive compensation) than the Kleene-Dienes implication in (2). This follows
from the relationship
max{1 − x, y} ≤ min{1 − x + y, 1}, x, y ∈ [0, 1].
It should be noted that t-norms are not appropriate for modeling the implication
operator in (3) because from the relationship
min{Ij → Pj } = min T (Ij → Pj ) ≤ min{Ij ∧ Pj } ≤ min Ij
j
j
j
j
it follows that the unit score could never exceed the score of the least important
criterion.
Summary 2.1 Generalizing Yager’s fuzzy screening method we have suggested
the use of S- and R-implication eperators to find the unit scores. If there are many
alternatives and only a few of them could be selected for further consideration then
we can use two closely related principles
All important criteria are satisfied
(S-implications), or,
All scores are bigger or equal than the importances
(R-implications).
It should be noted, however, that even we use a more compensative implication
operator in (3) it will select only a slightly larger subset for further consideration,
because the criteria satisfactions are aggregated by the most pessimistic min operator. A typical example for this case is to select the top 6 six from 300 submitted
project proposals. Then the unit scores can be determined by Yager’s original
method (1). Assume now the sponsors increase their support and 9 projects can be
selected. Then it is more appropriate to choose the Łukasiewicz implication in (3).
If an essentially bigger subset of projects can be supported (for example 30 from
300) then in (3) we should replace the pessimistic min aggregation by a Linguistic
OWA operator allowing much more compensation.
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References
[1] R.R. Yager, On ordered weighted averaging aggregation operators in
multi-criteria decision making, IEEE Transactions on Systems, Man and
Cybernetics 18(1988) 183-190.
[2] R.R.Yager, Fuzzy Screaning Systems, in: R.Lowen and M.Roubens eds.,
Fuzzy Logic: State of the Art, Kluwer, Dordrecht, 1993 251-261.
[3] R.R.Yager, Aggregation operators and fuzzy systems modeling, Fuzzy
Sets and Systems, 67(1994) 129-145.
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