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# Fast Method of Solving Fuzzy Relational Equations

IP.com Disclosure Number: IPCOM000109595D
Original Publication Date: 1992-Sep-01
Included in the Prior Art Database: 2005-Mar-24
Document File: 6 page(s) / 229K

IBM

## Related People

Fujita, M: AUTHOR

## Abstract

Disclosed is a fast method of solving fuzzy relational equations strictly. Fuzzy relational equations can be applied to diagnosis systems in which the knowledge-related to symptoms and/or diagnoses contains some ambiguity.

This text was extracted from an ASCII text file.
This is the abbreviated version, containing approximately 29% of the total text.

Fast Method of Solving Fuzzy Relational Equations

Disclosed is a fast method of solving fuzzy relational
equations strictly.  Fuzzy relational equations can be applied to
diagnosis systems in which the knowledge-related to symptoms and/or
diagnoses contains some ambiguity.

In the conventional method, broadest range of solutions of
fuzzy relational equations can be obtained easily, but all the
possible combinations of symptoms and relations must be processed in
order to obtain strict solutions, which include much useful
information for reaching a diagnosis.  As this requires a long
processing time, fuzzy relational equations have not been commonly
used in practical diagnosis systems, where the amount of knowledge
related to symptoms and diagnoses is comparatively large.

In the disclosed method, symptoms and relations are evaluated
before they are combined, so as not to generate unnecessary
combinations.  If there are many symptoms and relations of the same
certainties, the amount of calculation will be greatly reduced.  This
method may reduce the restriction imposed by the processing time in
practical fuzzy diagnosis systems.
A. Conventional method

A diagnosis is reached by surmizing the causes of malfunctions
from observed symptoms, using diagnostic knowledge. However
diagnostic knowledge is often given as a cause of a symptom.  That
is, it is predicted that certain symptoms will occur if certain
causes are present.

If the certainties that causes(1-n) occur are given by the
vector x = (x1, x2, ..., xn), the certainties that symptoms(1-m) are
observed by the vector y = (y1,y2, ..., yn), and the certainty of the
relationship between symptom j and cause i by R = (rij), relation
between cause x and symptom y is described by the following fuzzy
relation:
y = x * R,
that is,
yj = max (xi min rij);  j = 1, 2, ..., m
i

Clearly, symptoms (= y) can be easily obtained from the given
causes (= x) and relations (= R), but this equation must be solved as
a fuzzy relational equation in order to find the certainties of
causes (= x) from given symptoms (= y) and relations (= R), which is
a common requirement in diagnosis.

The solutions of fuzzy relational equations are represented as
sections between the minimum and maximum certainty of cause xi.  The
broadest sections of each xi can be easily obtained by the
conventional method.

However, strict solutions of a fuzzy relational equation are
obtained as multiple sections of certainties of cause xi, where each
section shows a probable solution of the equation.

The following method of solving fuzzy equations is known:
1. Given y and R, generate matrix U and U_:
U = (uij) = (rij @ yj)
U_ = (u_ij) = (rij @_ yj)
where
rij @ yj = (yj, yj)  if rij > yj
rij @ yj = (yj, 1)   if rij = yj
rij @ yj = null      if rij < yj
rij @_ yj = (0, yj)  if rij > yj
...