Comparison of solutions resulted from direct problems formulated as FRE

Amin Ghodousiana,111 Corresponding authorEmail address<EMAIL_ADDRESS>(Amin Ghodousian)., Sara Zalb

a,Faculty of Engineering Science, College of Engineering, University of
Tehran, P.O.Box 11365-4563, Tehran, Iran.

bDepartment of Engineering Science, College of Engineering, University of
Tehran, Tehran, Iran.

Abstract

In this paper, we investigate direct solution of FRE and compare their results
with expected real consequences. We give an applied example formulated by a
FRE problem and show that FRE defined by maximum t-conorm and an arbitrary
t-norm can yield different interpretations for our example. A necessary
condition and a sufficient condition are presented that guarantee FRE defined
by maximum t-conorm and minimum t-norm attain the same solutions as human mind
does. Also, we present a t-conorm and use it instead of maximum t-conorm in
FRE to obtain solutions with highest similarity with real ones. Finally, we
show that under some conditions, FRE defined by any t-conorm and any t-norm
may find a solution which is not reasonable.

Keywords: Fuzzy relation, composition of fuzzy relations, Fuzzy relational
equations, Fuzzy operators.

1\. Introduction

Fuzzy relational equations (FRE) are a generalization of Boolean relation
equations. At first, Sanchez [39] investigated and developed this theory and
its applications. Generally, fuzzy set theory has a number of properties that
make it suitable for formulizing the uncertain information upon which many
applied concepts such as medical diagnosis and treatment are usually based.
Because of this fact the theory of FRE was originally applied in problems of
the medical diagnosis [39]. Later, The concept of FRE is used in many problems
such as system analysis[38], decision making[1], fuzzy controller [10], fuzzy
modeling[42], fuzzy analysis, especially fuzzy arithmetic[10]. In [21] it has
been established that the majority of fuzzy inference systems can be
implemented using FRE. Pedrycz [30] categorized and extended two ways of the
generalizations of FRE in terms of sets under discussion and various
operations which are taken into account. Since then , many theoretical
improvements have been investigated and many applications have been presented.
For instance, we can refer to [3,5,7,11-14,16,17,22,25,31-33,35,37,41,43].
Klement et al. [18-20] presented the basic analytical and algebraic properties
of triangular norms and important classes of fuzzy operators generalization
such as Archimedean, strict and nilpotent t-norms. Pedrycz and Vasilakos [34]
converted a highly dimensional relational equation into a serious of single
input FRE. In [36] author demonstrate how problems of interpolation and
approximation of fuzzy functions are connected with solvability of systems of
FRE. He focused on the problem of approximate solvability of a system of FRE.
In [40], FRE has been extended to the setting of interval-valued FRE with a
max-t-norm composition and three types of solution sets have been proposed.
Markovskii showed that solving max-product FRE is closely related to the
covering problem which is an NP-hard problem [24]. In [2] Chen and Wang
designed an algorithm for obtaining the logical representation of all minimal
solutions. They showed that a polynomial time algorithm to find all minimal
solutions of FRE with max-min composition may not exist. An interesting
application of the fuzzy relations theory is in the field of image processing
[9] and [6]. Also, the various types of FRE with continuous triangular norms
were used for compression / decompression of images [4,15,28,23]. In [4] the
image was divided in blocks and then the FRE with a t-norm were utilized for
compression each block where results were obtained using the Lukasiewicz
t-norm . In [26] any monochromatic image is interpreted as a fuzzy relation in
which the entries are the normalized values of the pixels. This method is
based essentially on the fact that the reconstructed images is obtained as the
greatest or the smallest solution of a system of fuzzy equations. Also,
authors showed processes for coding/decoding color images in the RGB and YUV
spaces by using FRE of max-t type where t is the Yager t-norm. In [27] authors
used particular FRE of compression / decompression of color images in the RGB
and YUV spaces .Nobuhara et al. [29] formulated and solved a problem of image
reconstruction using eigen fuzzy sets. They proposed two algorithms of
generating eigen fuzzy sets used in the reconstruction process.Di Nola and
Russo [8] focused on the algebraic structures of such these problems by
considering the Lukasiewicz transform as a residuated map. In this paper, we
firstly focus on a direct problem defined by the most familiar form of fuzzy
relational equations which is formulated as follows:

$xo\,A=b$ (1)

Where$I\,=\,\\{\,1\,,\,2\,,\,...\,,\,n\,\\}$,
and$J\,=\,\\{\,1\,,\,2\,,\,...\,,\,m\,\\}$.$A\,=\,(\,a_{ij}\,)_{n\times m}$ is
a fuzzy matrix in which $0\,\,\leq\,\,a_{ij}\,\,\leq\,\,1$,$\forall\,i\in\,I$
and $\forall j\in J\,$. $x$ is a given $n$-dimensional fuzzy vector, and $b$
is an unknown $m$-dimensional fuzzy vector in which $0\leq x_{i}\leq 1$
,$\forall i\in I$, and $0\leq b_{j}\leq 1$, $\forall j\in J$, respectively.
Also, ”$o$” is an arbitrary t-norm, and $a_{j}$ is the $j$th column of
matrix$A$. Equivalently, Problem(1) can be written with more details as
follows:

$xo\,a_{j}\,=\,\mathop{\max}\limits_{i=1}^{n}\\{\,t(\,x_{i},a_{ij})\,\\}\,=\,b_{i},\forall
j\in J$ (2)

We present an applied example that is formulated by (2) as a direct problem in
which $x$ and $A$ are given and $b$ is unknown. We present a necessary
condition and a sufficient condition in which direct solution of Problem (2)
defined by minimum t-norm leads to a solution coinciding with consequence
resulted from human mind. Also, we show that direct Problem (2) defined by
each t-norm cannot yield a rational vector$b$ for our example. In latter case,
provided with some assumptions hold, we present a t-conorm (instead of maximum
operator in (2)) by which we can solve direct Problem (2) (in which
$t(a,b)=\min(a,b)$) with acceptable results and we find a vector $b$ that is
exactly the same as rational consequences which are obtained by human mind.
Also, we investigate conditions in which direct Problem (2) defined by each
t-norm and each t-conorm cannot yield a reasonable solution.

In section 2, we give an example to prove that direct Problem (2) may result
in a vector $b$ which is too farther from the expected rational conclusions.
In section 3, we present a t-conorm which is constructed by the convex
combination of drastic sum and maximum operators. Then, we solve direct
Problem (2) defined by minimum t-norm in which foregoing t-conorm takes the
place of maximum operator, and we show that vector $b$obtained by the new
problem has the highest similarity with the expected rational conclusions.
Finally, conditions are presented in which any kind of Problem (2) (defined by
arbitrary t-conorm and t-norm) does not result in a rational solution.

2\. An applied example: pea or melon?

Consider a basket including several peas, strawberries, bananas, and melons.
Also, consider a man who is aware of those four types of contents deciding to
distinguish those objects only by touching them with closed eyes. By this
assumption that the contents in the basket are just those four foregoing
kinds, one will able to evaluate them in his mind by considering some
properties(for instance, their weights). When one touches an object with
sufficient large value of weight (for example he assigns 1 to the weight of
that object from the interval [0,1]) he may argue that what is in his hands is
melon. On the other hand, when one touches a melon, he can feel that what is
in his hands has large value of weight among other contents included in the
basket. Then, we can define a fuzzy relation in which, membership value of
being melon and simultaneously having large value of weight, is too high. By
similar reasons, we may make a table based on four objects; pea, strawberry,
banana, melon, and three properties; long, Heavy, and voluminous as follows:

|  Pea |  Strawberry |  Banana |  Melon
---|---|---|---|---
Long |  0 |  0.3 |  0.98 |  0.7
Heavy |  0.001 |  0.01 |  0.1 |  0.99
Voluminous |  0.004 |  0.042 |  0.3 |  1

Table 1. Fuzzy relation between four objects; pea, strawberry, banana, melon,
and three properties; long, heavy, voluminous.

By our assumption, an object selected from basket is melon iff its length =
0.7 or its weight = 0.99 or its volume = 1. Similar statements are true for
being banana, being strawberry, and being pea. Furthermore, table 1 can be
shown by a fuzzy relational matrix as follows:

$\begin{array}[]{c}{}\hfil\\\ {Length}\\\ {Weight}\\\
{Volume}\end{array}\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,\,\,\,\,\,\,\,\,S\,\,\,\,\,\,\,\,\,\,\,B\,\,\,\,\,\,\,\,\,\,M}\\\
{\,\,\,\,\,\left[\,\begin{array}[]{cccc}{0}&{0.3}&{0.98}&{0.7}\\\
{0.001}&{0.01}&{0.1}&{0.99}\\\
{0.004}&{0.042}&{0.3}&{1}\end{array}\,\right]=A}\end{array}$

where$P,S,B,$and$M$denote pea, strawberry, banana, and melon, respectively.

Now, suppose we select an object $x$ from the basket (with closed eyes) and we
evaluate its properties as; $length(x)=x_{1}$, $weight(x)=x_{2}$, and
$volume(x)=x_{3}$, where $x_{1},x_{2},x_{3}\in[0,1]$. We show these pieces of
information by a 3-dimensional vector as shown below:

$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,L\,\,\,\,\,\,\,\,W\,\,\,\,\,\,\,V}\\\
{x=\left[\begin{array}[]{ccc}{x_{1}}&{x_{2}}&{x_{3}}\end{array}\right]}\end{array}$

where $L,W,$and $V$ denote length, weight, and volume, respectively.

Then, by the composition of fuzzy relations, we can find the kind of this
object by observing vector $b$ which is attained by solving direct Problem (2)
as follows:

$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,\,\,\,\,\,\,\,\,S\,\,\,\,\,\,\,\,\,\,\,B\,\,\,\,\,\,\,\,\,\,M\,\,\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,\,\,S\,\,\,\,\,\,B\,\,\,\,\,\,\,M}\\\
{\,\,\left[\begin{array}[]{ccc}{x_{1}}&{x_{2}}&{x_{3}}\end{array}\right]\,\,o\,\left[\,\begin{array}[]{cccc}{0}&{0.3}&{0.98}&{0.7}\\\
{0.001}&{0.01}&{0.1}&{0.99}\\\
{0.004}&{0.042}&{0.3}&{1}\end{array}\,\right]=\left[\begin{array}[]{cccc}{b_{1}}&{b_{2}}&{b_{3}}&{b_{4}}\end{array}\right]}\end{array}$

Vector $b$determines the membership value of each contents in the basket to
which object $x$ belongs. Precisely,$b_{1}$ is the degree of which $x$ is pea,
$b_{2}$ is the degree of which $x$ is strawberry, $b_{3}$is the degree of
which $x$ is banana, and $b_{4}$ is the degree of which $x$ is melon.

In a certain experiment, we select an object $\dot{x}$ from the basket and
find
that$\dot{x}=\left[\begin{array}[]{ccc}{0.004}&{0.002}&{0.003}\end{array}\right]$.
By comparing three properties of object$\dot{x}$ with that of pea, strawberry,
banana, and melon, it is easy to see that $\dot{x}$ should be a pea.
Therefore, we must have the same conclusion by finding vector$b$ through
solving Direct problem (2), too. At first, we solve problem $\dot{x}o\,A=b$ in
which ”$o$” is minimum t-norm and

$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,\,\,\,\,\,\,\,\,S\,\,\,\,\,\,\,\,\,\,\,B\,\,\,\,\,\,\,\,\,\,M}\\\
{\,\,\,\,A=\,\left[\,\begin{array}[]{cccc}{0}&{0.3}&{0.98}&{0.7}\\\
{0.001}&{0.01}&{0.1}&{0.99}\\\
{0.004}&{0.042}&{0.3}&{1}\end{array}\,\right]}\end{array}$

Easy calculations show that the result is the vector below:

$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,\,\,\,\,\,\,\,\,\,\,S\,\,\,\,\,\,\,\,\,\,\,\,\,B\,\,\,\,\,\,\,\,\,\,\,\,\,M}\\\
{\,b=\,\,\left[\begin{array}[]{cccc}{0.003}&{0.004}&{0.004}&{0.004}\end{array}\right]}\end{array}$

According to the vector $b$ above, $\dot{x}$ is a pea with membership value
0.003, and it is a melon with membership value 0.004 (?!). If we solve problem
above with product t-norm, we will attain solutions as objectionable as
previous one:

$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,B\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,M}\\\
{\,b=\,\,\left[\begin{array}[]{cccc}{0.000012}&{0.0012}&{0.00392}&{0.003}\end{array}\right]}\end{array}$

The result will be too wonderful if we solve problem above with Lukasiewicz
t-norm or drastic product. In both cases, we have:

$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,S\,\,\,\,B\,\,\,\,\,\,\,\,M}\\\
{\,b=\,\,\left[\begin{array}[]{cccc}{0}&{0}&{0}&{0.003}\end{array}\right]}\end{array}$

More wonderful result is attained when we try to distinguish an object
$\ddot{x}=\left[\begin{array}[]{ccc}{0}&{0.001}&{0.004}\end{array}\right]$ via
direct solution of Problem (2). By considering the components of table 1, it
is readily attained that $\ddot{x}$ is exactly a pea. But, when Problem (2) is
defined by minimum, product, Lukasiewicz, and drastic product, yields
vectors$b_{1}$, $b_{2}$, $b_{3}$, and $b_{4}$ below, respectively:

$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,\,\,\,\,\,\,\,\,\,\,S\,\,\,\,\,\,\,\,\,\,\,B\,\,\,\,\,\,\,\,\,\,\,\,\,M}\\\
{\,b_{1}=\,\,\left[\begin{array}[]{cccc}{0.004}&{0.004}&{0.004}&{0.004}\end{array}\right]}\end{array}$

$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,B\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,M}\\\
{\,b_{2}=\,\,\left[\begin{array}[]{cccc}{16\times 10^{-6}}&{168\times
10^{-6}}&{12\times 10^{-4}}&{4\times 10^{-3}}\end{array}\right]}\end{array}$

$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,S\,\,\,\,B\,\,\,\,\,\,M}\\\
{\,b_{3}=b_{4}=\,\,\left[\begin{array}[]{cccc}{0}&{0}&{0}&{0.004}\end{array}\right]}\end{array}$

Actually, we expect $b_{1}\geq b_{2}\geq b_{3}\geq b_{4}$ by a rational
justification in all cases above while results attained from Problem (2) have
inverse ordering in all cases. In a general case, suppose
$\stackrel{{\scriptstyle...}}{{x}}=\left[\begin{array}[]{ccc}{\stackrel{{\scriptstyle...}}{{x}}_{1}}&{\stackrel{{\scriptstyle...}}{{x}}_{2}}&{\stackrel{{\scriptstyle...}}{{x}}_{3}}\end{array}\right]$
is a selected object from the basket. Since $a_{i1}<a_{i4}$ for
each$i\in\\{\,1\,,\,2\,,\,3\,\\}$, then by the non-decreasing property of
t-norms we have $t(\stackrel{{\scriptstyle...}}{{x}}_{i}\,,\,a_{i1})\leq
t(\stackrel{{\scriptstyle...}}{{x}}_{i}\,,\,a_{i4})$ for
each$i\in\\{\,1\,,\,2\,,\,3\,\\}$and for each t-norm.Therefore,
$\mathop{\max}\limits_{i=1}^{3}\\{t(\stackrel{{\scriptstyle...}}{{x}}_{i}\,,\,a_{i1})\\}\leq\mathop{\max}\limits_{i=1}^{3}\\{t(\stackrel{{\scriptstyle...}}{{x}}_{i}\,,\,a_{i4})\\}$which
implies $b_{1}\leq b_{4}$. Since latter inequality is satisfied for each
$\stackrel{{\scriptstyle...}}{{x}}\in[0,1]^{3}$ and for each t-norm, then
problem (2) defined by each t-norm cannot truly distinguish between a pea and
a melon.

3\. Modification of Problem (2) In this section, we investigate how human mind
can solve example above with true solutions while Problem (2) does not. Also,
we try to modify Problem (2) in a way which leads us to the same consequences
as made by human mind. The process which is used in the mind for solving
example above is based on the comparison, and comparison process itself is
done by a difference operation. Actually, if we want to formulate the solution
process of mind for our example with highest similarity with direct problem
(2), we have:

$1-\mathop{\min}\limits_{i=1}^{n}\\{\,\left|\,x_{i}-a_{ij}\,\right|\,\\}=b_{j}\,\,\,\,,\forall
j\in J$ (3)

or equivalently;

$\mathop{\max}\limits_{i=1}^{n}\\{\,1-\left|\,x_{i}-a_{ij}\,\right|\,\\}=b_{j}\,\,\,\,,\forall
j\in J$ (4)

in which $x_{i}$ and $a_{ij}$ are given$\forall i\in I$ and $\forall j\in J$,
and $b_{j}$ is unknown $\forall j\in J$.

If we solve example above for
$\dot{x}=\left[\begin{array}[]{ccc}{0.004}&{0.002}&{0.003}\end{array}\right]$
and $\ddot{x}=\left[\begin{array}[]{ccc}{0}&{0.001}&{0.004}\end{array}\right]$
by direct problem (3) or (4), we will have reasonable vectors $\dot{b}$, and
$\ddot{b}$, respectively as follows:

$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,\,\,\,\,\,\,\,\,S\,\,\,\,\,\,\,\,\,\,\,\,\,\,B\,\,\,\,\,\,\,\,\,\,\,\,M}\\\
{\,\dot{b}=\,\,\left[\begin{array}[]{cccc}{0.999}&{0.992}&{0.902}&{0.304}\end{array}\right]}\end{array}$
$\begin{array}[]{l}{\,\,\,\,\,\,\,\,\,\,\,P\,\,\,\,\,\,\,\,\,S\,\,\,\,\,\,\,\,\,\,\,\,B\,\,\,\,\,\,\,\,\,\,M}\\\
{\,\ddot{b}=\,\,\left[\begin{array}[]{cccc}{1}&{0.991}&{0.974}&{0.3}\end{array}\right]}\end{array}$

Here the question is whether there exists a t-norm by which vector $b$resulted
from Problem (2) is the same as one attained from Problem (4), or else by
which t-norm, Problem (2) yields a vector $b$ with the most accurate
approximation to vector $b$ in (4). Lemma 1 and its corollary below show that
not only vector $b$ resulted from (2) defined by an arbitrary t-norm is not
generally equal to its corresponding vector in (4), but also under some
conditions, the amount of the accuracy for vector $b$ in (2) as an
approximation solution for vector $b$ in (4) is bounded above. However,
Problem (2) is not generally equal to Problem (4), and also under some
conditions, the accuracy of Problem (2) (defined by each t-norm) to
approximate Problem (4) can not be desirably increased.

Lemma 1. Given Problem (2).
Then,$\mathop{\max}\limits_{i=1}^{n}\\{\,t(\,x_{i},a_{ij})\\}\leq$$\mathop{\max}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij})\,\\}\leq$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$$,\forall
j\in J$, where $t$ is an arbitrary t-norm. Proof. The first inequality is well
known. For the second, fix a $j_{0}\in J$ and suppose$x_{i}\leq a_{ij_{0}}$for
some$i\in I$(otherwise, if$x_{i}>a_{ij_{0}}$, result is similarly attained).
In this case, if$x_{i}\leq 2a_{ij_{0}}-1$, then$x_{i}\leq
1-(a_{ij_{0}}-x_{i})\leq a_{ij_{0}}$. Otherwise, if$x_{i}>2a_{ij_{0}}-1$,
then$x_{i}\leq a_{ij_{0}}<1-(a_{ij_{0}}-x_{i})$. Therefore, in both cases, we
have:

$\,\min(x_{i},a_{ij_{0}})\,\leq 1-\left|x_{i}-a_{ij_{0}}\right|\,$ (5)

Now, since last inequality holds for each$i\in I$, we have
$\mathop{\max}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij_{0}})\,\\}\leq$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij_{0}}\right|\,\\}$.
Since $j_{0}\in J$ was arbitrary, then latter inequality is satisfied for
each$j\in J$.

Corollary 1. Given Problem (2)defined by minimum t-norm. Consider a fixed
$j_{0}\in J$ and suppose $\min(x_{i},a_{ij_{0}})>2\max(x_{i},a_{ij_{0}})-1$,
$\forall i\in I$. Then,
$\mathop{\max}\limits_{i=1}^{n}\\{\,t(\,x_{i},a_{ij_{0}})\\}\leq$
$\mathop{\max}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij_{0}})\,\\}<$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij_{0}}\right|\,\\}$.

Corollary 1 shows that under some conditions, Problem (2) defined by each
t-norm may not even a good approximation of real solution resulted from
Problem (4). Lemma 2 part (a) below gives a sufficient condition by which
Problem (2) defined by minimum t-norm finds rational vector $b$, exactly.
Also, a necessary condition has been given in part (b) when Problem (2)
defined by minimum t-norm provides a reasonable vector $b$ the same as we
expect. Moreover,

Lemma 2. Given Problem (2).

a) If $\max(x_{i},a_{ij})=1$,$\forall i\in I$ and $\forall j\in J$, then
$\mathop{\max}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij})\,\\}=$
$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$,$\forall
j\in J$ .

b) Suppose
$\mathop{\max}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij})\,\\}=$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$,$\forall
j\in J$. Then, for each $j\in J$ there exist at least one$i_{j}\in I$ such
that $\max(x_{i_{j}},a_{i_{j}j})=1$. Proof. a) If $\max(x_{i},a_{ij})=1$, then
$\min(x_{i},a_{ij})=$$1-\left|x_{i}-a_{ij}\right|\,$. Therefore, if
$\max(x_{i},a_{ij})=1$, $\forall i\in I$, we have
$\min(x_{i},a_{ij})=$$1-\left|x_{i}-a_{ij}\right|\,$, $\forall i\in I$ which
implies part (a). (b) Consider a fixed $j_{0}\in J$ and suppose
$\mathop{\max}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij_{0}})\,\\}=$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij_{0}}\right|\,\\}$.Let

$\min(x_{i_{1}},a_{i_{1}j_{0}})=\mathop{\max}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij_{0}})\,\\}$and$1-\left|x_{i_{2}}-a_{i_{2}j_{0}}\right|$$=\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij_{0}}\right|\,\\}$

Then, by assumption of part (b), we have:

$\min(x_{i_{1}},a_{i_{1}j_{0}})=1-\left|x_{i_{2}}-a_{i_{2}j_{0}}\right|\qquad(*)$

Now, by (5), we have:

$\min(x_{i_{1}},a_{i_{1}j_{0}})\leq
1-\left|x_{i_{1}}-a_{i_{1}j_{0}}\right|\leq\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij_{0}}\right|\,\\}=1-\left|x_{i_{2}}-a_{i_{2}j_{0}}\right|$

Which together with (*) imply
$\min(x_{i_{1}},a_{i_{1}j_{0}})=$$1-\left|x_{i_{1}}-a_{i_{1}j_{0}}\right|$.
Since $j_{0}\in J$ was arbitrary, last equality completes the proof.

Lemma 3. Consider Problem (2) and Suppose $i_{1},i_{2}\in I$ and $j\in J$. a)
If we have $\min(x_{i_{1}},a_{i_{1}j})\,=\,\min(x_{i_{2}},a_{i_{2}j})=0$ or,
if $\min(x_{i_{1}},a_{i_{1}j})\,=0$ and$\,\min(x_{i_{2}},a_{i_{2}j})\neq
0$,then:

$S_{D}\\{\min(x_{i_{1}},a_{i_{1}j})\,,\,\min(x_{i_{2}},a_{i_{2}j})\\}\leq\max\\{1-\left|x_{i_{1}}-a_{i_{1}j}\right|\,,\,1-\left|x_{i_{2}}-a_{i_{2}j}\right|\,\\}$

b) If $\min(x_{i_{1}},a_{i_{1}j})\,\neq 0$ and
$\,\min(x_{i_{2}},a_{i_{2}j})\neq 0$, then:

$\max\\{1-\left|x_{i_{1}}-a_{i_{1}j}\right|\,,\,1-\left|x_{i_{2}}-a_{i_{2}j}\right|\,\\}\leq
S_{D}\\{\min(x_{i_{1}},a_{i_{1}j})\,,\,\min(x_{i_{2}},a_{i_{2}j})\\}$

Proof. a) Suppose
$\min(x_{i_{1}},a_{i_{1}j})\,=\,\min(x_{i_{2}},a_{i_{2}j})=0$. In this case,
we have
$S_{D}\\{\min(x_{i_{1}},a_{i_{1}j})\,,\,\min(x_{i_{2}},a_{i_{2}j})\\}=0$. Now,
by noting that $0\leq 1-\left|x_{i}-a_{ij}\right|\leq 1$, $\forall i\in
I$and$\forall j\in J$, the result is attained. In another case, suppose
$\min(x_{i_{1}},a_{i_{1}j})\,=0$ and $\,\min(x_{i_{2}},a_{i_{2}j})\neq 0$.
Then, by the definition of drastic sum and Lemma 1, we have:

$S_{D}\\{\min(x_{i_{1}},a_{i_{1}j})\,,\,\min(x_{i_{2}},a_{i_{2}j})\\}=\,\min(x_{i_{2}},a_{i_{2}j})\leq\,1-\left|x_{i_{2}}-a_{i_{2}j}\right|\,\leq\max\\{1-\left|x_{i_{1}}-a_{i_{1}j}\right|\,,\,1-\left|x_{i_{2}}-a_{i_{2}j}\right|\,\\}$

that implies statement. (b) Suppose $\min(x_{i_{1}},a_{i_{1}j})\,\neq 0$ and
$\,\min(x_{i_{2}},a_{i_{2}j})\neq 0$. Then, by the definition of drastic sum,
we have
$S_{D}\\{\min(x_{i_{1}},a_{i_{1}j})\,,\,\min(x_{i_{2}},a_{i_{2}j})\\}=1$. This
fact together with
$0\leq\max\\{1-\left|x_{i_{1}}-a_{i_{1}j}\right|\,,\,1-\left|x_{i_{2}}-a_{i_{2}j}\right|\,\\}\leq
1$ prove part b.

Lemmas 1 and 3 lead us to a theorem below by which we will able to introduce
and use a new t-conorm instead of maximum operator in (2), for compensating
the difference between solution attained from (2) and real solution resulted
from (4).

Theorem 1. Consider Problem (2) and suppose for each $j\in J$ there exist at
least two $i_{1},i_{2}\in I$ such that $\min(x_{i_{1}},a_{i_{1}j})\,\neq
0$and$\,\min(x_{i_{2}},a_{i_{2}j})\neq 0$. Then,
$\mathop{\max}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij})\,\\}\leq$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}\leq$$\mathop{S_{D}}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij})\,\\}$.
Proof. First inequality is the same which was proved in Lemma 1. Also, second
inequality is obtained by repeating Lemma 3.

Now, let $i_{1},i_{2}\in I$ such that $\min(x_{i_{1}},a_{i_{1}j})\,\neq
0$and$\,\min(x_{i_{2}},a_{i_{2}j})\neq 0$. By Theorem 1,
$\mathop{\max}\limits_{i=i_{1}}^{i_{2}}\\{\,\min(x_{i},a_{ij})\,\\}\leq$$\mathop{\max}\limits_{i=i_{1}}^{i_{2}}\\{1-\left|x_{i}-a_{ij}\right|\,\\}\leq$$\mathop{S_{D}}\limits_{i=i_{1}}^{i_{2}}\\{\,\min(x_{i},a_{ij})\,\\}$.
Then, for some $\lambda\in[0,1]$, we have:
$\lambda\mathop{\max}\limits_{i=i_{1}}^{i_{2}}\\{\,\min(x_{i},a_{ij})\,\\}+(1-\lambda)$$\mathop{S_{D}}\limits_{i=i_{1}}^{i_{2}}\\{\,\min(x_{i},a_{ij})\,\\}$=$\mathop{\max}\limits_{i=i_{1}}^{i_{2}}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$in
which :

$\lambda=\frac{\mathop{\max}\limits_{i=i_{1}}^{i_{2}}\\{1-\left|x_{i}-a_{ij}\right|\,\,\\}-\mathop{\max}\limits_{i=i_{1}}^{i_{2}}\\{\min(x_{i},a_{ij})\\}}{\mathop{S_{D}}\limits_{i=i_{1}}^{i_{2}}\\{\min(x_{i},a_{ij})\\}-\mathop{\max}\limits_{i=i_{1}}^{i_{2}}\\{\min(x_{i},a_{ij})\\}}\qquad(*)$

Now, by statements above, we define $S_{a,b}(a,b):\,[0,1]^{2}\to[0,1]$ such
that $S_{a,b}(a,b)=$$\lambda\max\\{\min(a,b)\\}+(1-\lambda)S_{D}(a,b)$in
which$\lambda$ is found by (*). It is easy to verify that $S_{a,b}(a,b)$ is
really a t-conorm. Also, by extension of $S_{a,b}(a,b)$for $n$ arguments and
usage of it instead of maximum operator in Problem (2), we have:

$\mathop{S_{M}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}=\lambda\mathop{\max}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij})\,\\}+(1-\lambda)\mathop{S_{D}}\limits_{i=1}^{n}\\{\,\min(x_{i},a_{ij})\,\\}$

in which$M=\\{\min(x_{i},a_{ij})\,,\,\forall i\in I\,\\}$and $\lambda$ is
found by equality below:

$\lambda=\frac{\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\,\\}-\mathop{\max}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}}{\mathop{S_{D}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}-\mathop{\max}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}}\qquad(**)$

However, if assumption in Theorem 1 hold, statements above show that there
exists t-conorm $S_{a,b}(a,b)$ such that
$\mathop{S_{M}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}=$
$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$, $\forall
j\in J$. Now, we have proved theorem below.

Theorem 2. Consider Problem (2) and suppose for each $j\in J$ there exist at
least two $i_{1},i_{2}\in I$ such that $\min(x_{i_{1}},a_{i_{1}j})\,\neq
0$and$\,\min(x_{i_{2}},a_{i_{2}j})\neq 0$. Define
$S_{a,b}(a,b):\,[0,1]^{2}\to[0,1]$ such that
$S_{a,b}(a,b)=$$\lambda\max\\{\min(a,b)\\}+$ $(1-\lambda)S_{D}(a,b)$ and

$\lambda=\frac{\mathop{\max}\limits_{i=i_{1}}^{i_{2}}\\{1-\left|x_{i}-a_{ij}\right|\,\,\\}-\mathop{\max}\limits_{i=i_{1}}^{i_{2}}\\{\min(x_{i},a_{ij})\\}}{\mathop{S_{D}}\limits_{i=i_{1}}^{i_{2}}\\{\min(x_{i},a_{ij})\\}-\mathop{\max}\limits_{i=i_{1}}^{i_{2}}\\{\min(x_{i},a_{ij})\\}}$

Then, $S_{a,b}(a,b)$ is a t-conorm and
$\mathop{S_{M}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}=$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$,
$\forall j\in J$, in which
$M=\\{\min(x_{i},a_{ij})\,,\,\forall i\in I\,\\}$and $\lambda$ is found by

$\lambda=\frac{\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\,\\}-\mathop{\max}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}}{\mathop{S_{D}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}-\mathop{\max}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}}.$

Theorem 3. Consider Problem (2) and Suppose for each $j\in J$ there exists at
most one $i_{j}\in I$ such that $\min(x_{i_{j}},a_{i_{j}j})\neq 0$. Then, for
each t-conorm $S$ and each t-norm T, we have
$\mathop{S}\limits_{i=1}^{n}\\{t(x_{i},a_{ij})\\}\leq$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$,
$\forall j\in J$.

Proof. Since for each $a,b\in[0,1]$ , we have $T(a,b)\leq\min(a,b)$ and
$S(a,b)\leq S_{D}(a,b)$for each t-norm T and each t-conorm $S$, it is
sufficient to prove
$\mathop{S_{D}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}<$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$.
At first, suppose that for a fixed $j\in J$, we have
$\min(x_{i},a_{ij})\\}=0$, $\forall i\in I$. Then,
$\mathop{S_{D}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}=0$ which implies
inequality. In another case, suppose for a fixed $j\in J$ there exists exactly
one $i_{j}\in I$ such that $\min(x_{i_{j}},a_{i_{j}j})\neq 0$. In this case,
$\mathop{S_{D}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}=\min(x_{i_{j}},a_{i_{j}j})$.
Now, by (5) we have:

$\mathop{S_{D}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}=\min(x_{i_{j}},a_{i_{j}j})\leq\,1-\left|x_{i_{j}}-a_{i_{j}j}\right|\,\leq\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$

that requires the inequality.

Corollary 2. Consider Problem (2) and consider a fixed $j\in J$. If two
following conditions hold, then
$\mathop{S}\limits_{i=1}^{n}\\{T(x_{i},a_{ij})\\}<$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$.

(a) For each $j\in J$ there exists $i\in I$ such that $\max(x_{i},a_{ij})<1$.

(b) $\min(x_{i},a_{ij})=0$, $\forall i\in I$. Proof. Since
$\mathop{S}\limits_{i=1}^{n}\\{T(x_{i},a_{ij})\\}\leq\mathop{S_{D}}\limits_{i=1}^{n}(\min(x_{i},a_{ij})\\}$for
each t-norm $T$and each t-conorm $S$, it is sufficient to prove
$\mathop{S_{D}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}<$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$.
By assumption (b),
$\mathop{S_{D}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}=0$. Also, from
Theorem 3, we have
$\mathop{S}\limits_{i=1}^{n}\\{t(x_{i},a_{ij})\\}\leq$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$.
Then, to prove the corollary, it is sufficient to show
$\mathop{S}\limits_{i=1}^{n}\\{t(x_{i},a_{ij})\\}\neq$$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$.
By contrary, suppose
$\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\}$$=\mathop{S}\limits_{i=1}^{n}\\{t(x_{i},a_{ij})\\}=0$.
Therefore, $1-\left|x_{i}-a_{ij}\right|=0$, $\forall i\in I$, which implies
$\min(x_{i},a_{ij})=$$1-\left|x_{i}-a_{ij}\right|$, $\forall i\in I$. Thus,
$\max(x_{i},a_{ij})=1$, $\forall i\in I$, which contradicts assumption (a).
Then, the proof is complete.

Briefly, the amount of the accuracy of vector $b$resulted from the direct
solution of FRI problem $xo\,A=b$ modeling an applied topic can be categorized
as follows:

  * •

If $\max\\{x_{i},a_{ij}\\}=1$, $\forall i\in I$and $\forall j\in J$, Problem
(2) defined by minimum t-norm yields exact solution(see Lemma 2 part (b)). In
this case, Problem (2) is formulated as
$xo\,a_{j}\,=\,\mathop{\max}\limits_{i=1}^{n}\\{\,\min(\,x_{i},a_{ij})\,\\}\,=\,b_{i}$$,\forall
j\in J$.

  * •

If for each $j\in J$ there exist at least two $i_{1},i_{2}\in I$ such that
$\min(x_{i_{1}},a_{i_{1}j})\,\neq 0$and$\,\min(x_{i_{2}},a_{i_{2}j})\neq 0$,
then Problem (2) defined by minimum t-norm and $S_{a,b}$ t-conorm given in
Theorem 2, yields exact solution(See Theorems 1 and 2). Here, Problem (2) is
formulated as:
$xo\,a_{j}\,=\,\mathop{S_{M}}\limits_{i=1}^{n}\\{\,\min(\,x_{i},a_{ij})\,\\}\,=\,b_{i}$$,\forall
j\in J$.

  * •

However, there are conditions in which we cannot obtain reasonable solutions;

    * –

If necessary condition stated in Lemma 2 part (b) does not hold, then Problem
(2) defined by minimum t-norm fails to provide a rational solution for vector
$b$.

    * –

If conditions expressed in Corollary 1 hold, we have:

$\mathop{\max}\limits_{i=1}^{n}\\{T(x_{i},a_{ij})\\}<\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\},\forall
j\in J$

which implies that Problem (2) defined by any t-norm does not yield a
justifiable solution $b$.

    * –

If conditions in Corollary 2 hold, we have:

$\mathop{S}\limits_{i=1}^{n}\\{T(x_{i},a_{ij})\\}<\mathop{\max}\limits_{i=1}^{n}\\{1-\left|x_{i}-a_{ij}\right|\,\\},\forall
j\in J$

which requires Problem (2) defined by each t-norm and each t-conorm will
attain a solution farther than one we expect. Also, in this case, the best
solution of modified Problem (2) is attained by problem below:

$\mathop{S_{D}}\limits_{i=1}^{n}\\{\min(x_{i},a_{ij})\\}=b_{i},\forall j\in J$

Then, the accuracy of each modification of Problem (2) is bounded above.

Therefore, Problem (2) may not guarantee that vector $b$ is the same as we
expect. Moreover, in a worse case, it may exists an example for which the
results of Problem (2) in direct solution is not even a good approximation for
expected rational consequences.

Conclusion  In this paper, we investigated direct solution of FRE and compared
their results with expected real consequences. It was shown that FRE defined
by maximum t-conorm and an arbitrary t-norm yields different interpretations.
A necessary condition and a sufficient condition were presented that guarantee
FRE defined by maximum t-conorm and minimum t-norm attain the same solutions
as human mind does. Also, we presented a t-conorm and use it instead of
maximum t-conorm in FRE to obtain solutions with highest similarity with real
ones. Moreover, we showed that under some conditions, FRE defined by any
t-conorm and any t-norm may find a solution which is not reasonable.

References  [1] Bellman R.E., Zadeh L.A., Decision-making in fuzzy
environment, Management Sci. 17(1970).141-164.

[2] Chen L., Wang P.P., Fuzzy relation equations(I): the general and
specialized solving algorithms, Soft Computing,6(2002)428-435.

[3] Di Martino F., Sessa S., Digital watermarking in coding/decoding processes
with fuzzy relation equations, Soft Computting,10(2006)238-243.

[4] Di Martino F., Loia V., Sessa S., A method in the compression
/decompression of images using fuzzy equations and fuzzy similarities. In:
Proceedings of Conference IFSA 2003 (29/6-2/7/2003, Istanbul, Turkey),
pp.524-527(2003).

[5] Di Nola A., Pedrycz W., Sessa S., Some theoretical aspects of fuzzy
relation equations describing fuzzy systems, Inform.Sci.34(1984)241-264.

[6] Di Nola A., Pedrycz W., Sessa S., Fuzzy relation equations and algorithms
of inference mechanism in expert systems, in: M.M. Gupta, A. Kandel, W.
Bandler and J.B. Kiszka, Eds., Approximate Reasoning in Expert System
(Elsevier Science Publishers (North Holland), Amsterdam, 1985)355-367.

[7] Di Nola A., Pedrycz W., Sessa S., Sanchez E., Fuzzy relation equations
theory as a basis of fuzzy modeling: an overview, Fuzzy Sets and Systems
40(1991)415-429.

[8] Di Nola A., Russo C., Lukasiewicz transform and its application to
compression and reconstruction of digital images, Information Sciences
177(2007)1481-1498.

[9] Di Nola A., Sessa S., Pedrycz W., Sanchez E., Fuzzy relation equations and
their applications to knowledge engineering, Kluwer, Dordrecht, (1989).

[10] Dubois D., Prade H., Fuzzy sets and systems: Theory and Applications
(Academic Press, New York,1980).

[11] Fernandez M.J., Gil P., Some specific types of fuzzy relation equations,
Information sciences 164(2004)189-195.

[12] Guo F.F., Xia Z.Q., An algorithm for solving optimization problems with
one linear objective function and finitely many constraints of fuzzy relation
inequalities, Fuzzy Optimization and Decision Making, 5 (2006)33-47.

[13] Han S.C., Li H.X., Wang J.Y., Resolution of finite fuzzy relation
Equations based on strong pseudo-t-norms, Applied Mathematics Letters
19(2006)752-757.

[14] Han S.C., Li H.X., Nots on “pseudo-t-norms and implication operators on a
complete brouwerian lattice” and “Pseudo-t-norms and implication operators:
direct products and direct product decompositions ” ,Fuzzy Sets and Systems
153(2005)289-294.

[15] Hirota K., Pedrycz W., Fuzzy relational compression, IEEE Transactions on
Systems, Man and Cybernetics, Part B 29(3)(1999)407-415.

[16] Hirota K., Pedrycz W., Solving Fuzzy relational equations through logical
filtering, Fuzzy Sets Syst. 81(1996)355-364.

[17] Hirota K., Pedrycz W., Specificity shift in solving ,Fuzzy Sets Syst. 106
(1999)211-220.

[18] Klement E.P., Mesiar R., Pap E.,Triangular norms. Position paper I: basic
analytical and algebraic properties,Fuzzy Sets and Systems 143 (2004) 5-26.

[19] Klement E.P., Mesiar R., Pap E., Triangular norms. Position paper II,
Fuzzy Sets and Systems 145(2004)411-438.

[20] Klement E.P., Mesiar R., Pap E., Triangular norms. Position paper III,
Fuzzy Sets and Systems 145(2004)439-454.

[21] Klir G.J., Yuan B., Fuzzy sets and fuzzy logic: Theory and Applications,
Prentice-Hall,PTR,USA,1995.

[22] Loia V., Sessa S., Fuzzy relation equations for coding/decoding processes
of images and videos, Information sciences 171(2005)145-172.

[23] Loia V., Pedrycz W., Sessa S., Fuzzy relation calculus in the compression
and decompression of fuzzy relations.Inter J Image Graphics,2(2002)1-15

[24] Markovskii A.V., On the relation between equations with max-product
composition and the covering problem, Fuzzy Sets and Systems 153
(2005)261-273.

[25] Nobuhara H., Pedrycz W., Hirota K., A digital watermaking algorithm using
image compression method based on fuzzy relational equations. In: Proceedings
of FUZZ-IEEE 2002.IEEE Press, vol 2,pp.1568-1573.

[26] Nobuhara H., Hirota K., Pedrycz W., Relational image compression:
optimizations through the design of fuzzy coders and YUV colour space, Soft
Computing 9(2005)471-479.

[27] Nobuhara H., Hirota K., Di Martino F., Pedrycz W., Sessa S., fuzzy
relation equations for compression/decompression processes of colour images in
the RGB and YUV colour spaces(2005).

[28] Nobuhara H., Pedrycz W., Hirota K., Fast solving method of fuzzy
relational equation and its application to lossy image compression. IEEE Trans
Fuzzy Sys,8(3)(2000)325-334.

[29] Nobuhara H., Bede B., Hirota K., On various eigen fuzzy sets and their
application to image reconstruction,Information Sciences 176 (2006) 2988-3010.

[30] Pedrycz W., Fuzzy relational equations with generalized connectives and
their applications, Fuzzy Sets and Systems 5(1983)185-201.

[31] Pedrycz W., On generalized fuzzy relational equations and their
applications, Journal of Mathematical Analysis and Applications 107
(1985)520-536.

[32] Pedrycz W., s-t fuzzy relational equations, Fuzzy Sets and Systems 59
(1993)189-195.

[33] Pedrycz W., Inverse problem in fuzzy relational equations, Fuzzy Sets and
Systems 36(1990)277-291.

[34] Pedrycz W., Vasilakos A.V., Modularization of fuzzy relational equations,
Soft Computing 6(2002).33-37.

[35] Peeva K., Kyosev Y., Fuzzy relational calculus, theory, applications and
software, World Scientific,2004.

[36] Perfilieva I., Fuzzy function as an approximate solution to a system of
fuzzy relation equations, Fuzzy Sets and Systems 147(2004)363-383.

[37] Pedrycz W., An approach to the analysis of fuzzy systems, Int. J. Control
34(1981)403-421.

[38] Sanchez E., Solution in composite fuzzy relation equations: Application
to medical diagnosis in Brouwerian logic, In Fuzzy Automata and Decision
Processes,(Edited by M. M. Gupta , G . N . Saridis and B R
Games),pp.221-234,North-Holland,New York , (1977).

[39] Wang S., Fang S.C., Nuttle H.L.W., Solution sets of interval-valued fuzzy
relational equations, Fuzzy Optimization and Decision Making 2 (1)
(2003)41-60.

[40] Wang G.J., On the logic foundation of fuzzy reasoning, Inform.Sci. 117
(1) (1999)47-88.

[41] Wenstop F., Deductive verbal models of organizations, Int. J. Man-Machine
Studies 8 (1976).293-311.

[42] Wu Y.K., Guu S.M., A note on fuzzy relation programming problems with
max-strict-t-norm composition, Fuzzy Optimization and Decision
Making,3(2004)271-278.

[43] Xiong Q.Q., Wang X.P., Some properties of sup-min fuzzy relational
equations on infinite domains, Fuzzy Sets and Systems 151(2005)393-402.