Fuzzy Reasoning Techniques for GDSS ∗
Robert Fullér†
Department of Computer Science, Eötvös Loránd University,
P.O.Box 157, H-1502 Budapest 112, Hungary
Luisa Mich
Dep. of Computer and Management Sci., University of Trento,
via Inama 5, I-38100 Trento, Italy
Abstract
Fedrizzi and Mich [Fed91] presented a new Group Decision Support
System (GDSS) logic architecture in which linguistic variables and fuzzy
production rules were used for reaching consensus. In [Fed92] it was shown
that when all the fuzzy numbers representing the performance levels have
continuous membership function, then the consensus degrees (defined by a
certain similarity measure) relative to each alternative are stable under small
changes of the experts’ performance levels.
Generalizing the method of [Fed91], we represent the knowledge via
fuzzy production rules of more complex form, which makes it possible to
determine the actual (and overall) group performance level in one step by
using Zadeh’s compositional rule of inference in the consensus management
module.
∗
in: Proceedings of EUFIT’93 Conference, September 7-10, 1993 Aachen, Germany, Verlag
der Augustinus Buchhandlung, Aachen, 1993 937-940.
†
Partially supported by the Hungarian Research Foundation OTKA under contracts T 4281,
I/3-2152 and T 7598.
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1
Introduction
Fedrizzi and Mich [Fed91] presented a new GDSS logic architecture in which
linguistic variables and fuzzy production rules have been used for reaching consensus.
In this model the experts’ preferences are formalized in a matrix of linguistic
variables (see also [Fed86]), the ranking of alternatives for each expert is performed by using one of the many algorithms proposed in the literature (see e.g.
[Bor85]), the knowledge representation is performed via fuzzy production rules
of the form
If Pi1 is Fi1 and . . . and Pik is Fik then Pi is Fi
where, Pij is the performance valued by the expert Ej and relative to the alternative Ai , Fij is the fuzzy number that represents the semantics of Pij , Pi is the
performance that the group of experts assign to alternative Ai , Fi is the fuzzy
number that represents the semantics of Pi (i=1,. . . ,m, j=1,. . . ,k), the subsystem
for consensus management contains the following fuzzy inference schemes
If Pi1 is Fi1 and . . . and Pik is Fik then Pi is Fi
Pi1 is Fi1∗ and . . . and Pik is Fik∗
Pi is Fi∗
where the fuzzy number Fij∗ represents the actual performance level given by the
expert Ej , the consequence (i.e. the group performance level) Fi∗ is calculated by
Zadeh’s compositional rule of inference as
Fi∗ = (Fi1∗ × · · · × Fik∗ ) ◦ (Fi1 × · · · × Fik → Fi ),
the degree of consensus, αi , relative to alternative Ai is calculated in the following
manner,
k
σ(Fij∗ , Fi∗ )
αi = j=1
,
k
where the measure of similarity between Fij∗ and Fi∗ , σ(Fij∗ , Fi∗ ), is defined by
σ(Fij∗ , Fi∗ ) = T (1/2 + min{N (Fij∗ |Fi∗ ), 1/2}, Π(Fij∗ |Fi∗ )),
T is a continuous triangular norm, Π and N denote conditional possibility and
necessity, respectively.
We recall now the definition of conditional possibility, necessity and Cartesian
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product of fuzzy sets.
Definition 1. Let A and B be two fuzzy predicates defined on the real line R.
Knowing that ’X is B’ is true,
(i) the degree of possibility that the proposition ’X is A’ is true, Π(A|B), is given
by
Π(A|B) = sup min{A(t), B(t)},
t∈R
(ii) the degree of necessity that the proposition ’X is A’ is true, N (A|B), is given
by
N (A|B) = 1 − Π(¬A|B),
where A and B are the possibility distributions (for simplicity we write A instead
of µA ) defined by the predicates A and B, respectively.
Definition 2. Let Fi , i = 1, . . . , k, be fuzzy sets of the real line. Their Cartesian
product, F1 × . . . × Fk , is defined by
(F1 × . . . × Fk )(u1 , . . . , uk ) = F1 (u1 ) ∧ . . . ∧ Fk (uk ).
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The one-step approach
We represent the knowledge via fuzzy production rules of more complex form,
If F11 × · · · × F1m and . . . and Fk1 × · · · × Fkm then F ,
where Fij is the performance level valued by the i-th expert and relative to the j-th
alternative, and F is the group performance level, and the subsystem for consensus
management contains the following fuzzy inference scheme
If F11 × · · · × F1m and . . . and Fk1 × · · · × Fkm then F
∗
∗
∗
∗
F11
× · · · × F1m
and . . . and Fk1
× · · · × Fkm
F∗
where the fuzzy number Fij∗ represents the actual performance level given by the
expert Ej and relative to the alternative Ai , the consequence (i.e. the group performance level) F ∗ is calculated by Zadeh’s compositional rule of inference as
∗
∗
∗
∗
× · · · × F1m
& . . . & Fk1
× · · · × Fkm
◦
F ∗ = F11
(F11 × · · · × F1m & . . . & Fk1 × · · · × Fkm → F )
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which allows to determine the actual (and overall) group performance level in one
step by using Zadeh’s compositional rule of inference in the consensus management module.
Our model can be interpreted as follows: The rules represent earlier situations, in
which we were able to reach consensus. Then we try to reach consensus in the
present situation by using max −T convolution of the actual performance levels
and the past situations.
3
Acknowledgement
We are thankful to Prof.Hannu Nurmi of University of Turku for his helpful comments.
References
[Bor85]
G.Bortolan and R.Degani, A review of some methods for ranking fuzzy
subsets, Fuzzy Sets and Systems, 1985, Vol.1. 1-19.
[Fed86]
M.Fedrizzi, Group decision making and consensus: a fuzzy approach,
AMSES review, No.9, Pitagora Press Bologna, 1986 (in Italian).
[Fed91]
M. Fedrizzi and L. Mich, Consensus reaching in group decisions using
production rules, In: Proc. of Annual Conference of the Operational
Research Society of Italy, Riva del Garda, Italy, 118-121 (1991).
[Fed92]
M.Fedrizzi and R.Fullér, On stability in group decision support systems under fuzzy production rules, in: R.Trappl ed., Proceedings of
the Eleventh European Meeting on Cybernetics and Systems Research,
World Scientific Publisher, London, 1992, Vol.1. 471-478.
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