Research Article
A Divide-and-Conquer Approach for Solving Fuzzy
Max-Archimedean -Norm Relational Equations
Jun-Lin Lin, Hung-Chjh Chuan, and Laksamee Khomnotai
Department of Information Management and Innovation Center for Big Data and Digital Convergence, Yuan Ze University,
Taoyuan 32003, Taiwan
Correspondence should be addressed to Jun-Lin Lin; jun@saturn.yzu.edu.tw
Received 2 January 2014; Accepted 22 April 2014; Published 11 May 2014
Academic Editor: Juan Carlos Cort´ es
Copyright © 2014 Jun-Lin Lin et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A system of fuzzy relational equations with the max-Archimedean t-norm composition was considered. Te relevant literature
indicated that this problem can be reduced to the problem of fnding all the irredundant coverings of a binary matrix. A divide-
and-conquer approach is proposed to solve this problem and, subsequently, to solve the original problem. Tis approach was
used to analyze the binary matrix and then decompose the matrix into several submatrices such that the irredundant coverings
of the original matrix could be constructed using the irredundant coverings of each of these submatrices. Tis step was performed
recursively for each of these submatrices to obtain the irredundant coverings. Finally, once all the irredundant coverings of the
original matrix were found, they were easily converted into the minimal solutions of the fuzzy relational equations. Experiments
on binary matrices, with the number of irredundant coverings ranging from 24 to 9680, were also performed. Te results indicated
that, for test matrices that could initially be partitioned into more than one submatrix, this approach reduced the execution time by
more than three orders of magnitude. For the other test matrices, this approach was still useful because certain submatrices could
be partitioned into more than one submatrix.
1. Introduction
Solving a system of fuzzy relational equations is a subject of
great scientifc interest [1, 2]. Tis work considers a system of
fuzzy relational equations of the form
max {(
1
,
11
),(
2
,
21
),...,(
,
1
)}=
1
max {(
1
,
12
),(
2
,
22
),...,(
,
2
)}=
2
...,
max {(
1
,
1
),(
2
,
2
),...,(
,
)}=
,
(1)
where
,
,
∈[0,1] for each , 1⩽⩽ and for each ,
1⩽⩽ and represents a continuous Archimedean -norm
function. System (1) can be succinctly written in the following
equivalent matrix form:
∘=, (2)
where =(
)
1×
is the matrix of unknowns, =(
)
×
is
the matrix of coefcients, =(
)
1×
is the right-hand side of
the system, and the symb “∘” represents a max-Archimedean
-norm composition.
Di Nola et al. [3] indicated that, given a continuous -
norm for in system (1) and assuming the existence of solu-
tions, the solution set of system (1) can be fully determined
by the greatest solution and a fnite number of minimal
solutions. It is well-known that the greatest solution can be
easily computed, but fnding all minimal solutions is difcult.
Li and Fang [4] demonstrated that the systems of max--norm
equations can be divided into two categories, depending on
the function in the system. When is continuous and
Archimedean, the minimal solutions correspond one-to-one
to the irredundant coverings of a set covering problem. When
is continuous and non-Archimedean, the minimal solutions
correspond to a subset of constrained irredundant coverings
of a set covering problem. Li and Fang [5] discussed the
necessary and sufcient conditions for solving max--norm
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 315290, 10 pages
http://dx.doi.org/10.1155/2014/315290