数理解析研究所講究録
1068 巻 1998 年 142-149

142
Some Pseudo-Order of Fuzzy Sets on
千葉大学教育学四千葉大学理学部

蔵野正美

千葉大学理学部

中神潤–

北九州大学経済学部

吉田祐治

安田正實

$\mathbb{R}^{n}$(Masami KURANO)(Masami YASUDA)(Jun-ichi NAKAGAMI)(Yuji YOSHIDA)

Abstract
The aim of this paper is to define an order on a class of fuzzy sets which isextending a pseudo-order for fuzzy numbers, and its characterization and several
relations of the previous results are discussed. The idea comes from a set-relationin -dimensional Euclid space given by Kuroiwa, Tanaka and Ha (1997). We induce
the order of a class of fuzzy sets by a closed convex cone and characterize it by using
the projection into the dual cone. Especially, a structure of the lattice is described
for the class of rectangle-type fuzzy sets.
$n$: Pseudo-order, fuzzy $\max$ order, multidimensional fuzzy sets, rectangle-type
fuzzy sets.
$I_{1^{J}e}rywo\Gamma ds$

1. Introduction and notations
In the theory of optimization it is a quite important problem how to induce a naturala simple problem
definition of order on the class of considering systems. Since it
about a order on the fuzzy set theory, many auther tried to consider its natural extension.
Ram\’ik and \v{R}im\’anek [8] has introduced a partial order on the set of fuzzy numbers,
called the fuzzy $\max$ order. The present authors also tried to optimize the dynamic fuzzy
system [4]. Also there are various types of order relations on the class of fuzzy numbers.
See [3], [11] and their references. Congxin and Cong [1] have descrived the fuzzy number
lattice.
This paper is to extend the fuzzy $\max$ order of fuzzy numbers to a class of fuzzy sets
defined on . The pseudo order for fuzzy sets is induced by a closed convex cone $K$ in. Also, the structure of a latticeand characterized by the projection in the dual cone
is discussed for the class of rectangle-type fuzzy sets. By our works we can imagine themuch wider application to the fuzzy optimization problem. Our idea of the motivation
given by Kuroiwa, Tanaka and Ha [5] and Kuroiwa [6],originates from a set-relation inare used in set-valued optimizations.
in which various types of set-relations inIn the remainder of this section, we will give some notations and review a vector
an nbe the set of all real numbers andby a convex cone. Let
ordering ofby their membership functions
dimensional Euclidean space. We write fuzzy sets on(see Nov\’ak [7] and Zadeh [10]). The -cut $(\alpha\in[0,1])$ of the fuzzy set:
is defined ason$\mathrm{i}\mathrm{s}\mathrm{n}’ \mathrm{t}$$\mathbb{R}^{n}$$\mathbb{R}^{n}$$K^{+}$$\mathbb{R}^{n}$$\mathbb{R}^{n}$$\mathbb{R}^{n}$$\mathbb{R}^{n}$$\mathbb{R}$$\mathbb{R}^{n}$$\tilde{s}$$\mathbb{R}^{n}arrow[0,1]$$\tilde{s}$$\alpha$$\mathbb{R}^{n}$$\tilde{s}_{\alpha}:=\{x\in \mathbb{R}^{n}|\tilde{s}(x)\geq\alpha\}(\alpha>0)$and$\tilde{s}_{0}:=\mathrm{c}1\{x\in \mathbb{R}^{n}|\tilde{s}(x)>0\}$

,

143

where cl denotes the closure of the set. A fuzzy set$\tilde{s}(\lambda x+(1-\lambda)y)\geq\tilde{s}(x)$where

A $b= \min\{a, b\}$ . Note thatA

is called convex if$\tilde{s}$$x,$ $y\in \mathbb{R}^{n},$$\tilde{s}(y)$$\lambda\in[0,1]$,

is convex iff the -cut
is a convex set for all$\alpha\in[0,1]$ .
Let
be the set of all convex fuzzy sets whose membership functions
are upper-semicontinuous and normal:
and have acompact support. When the one-dimensional case $n=1$ , the fuzzy
are called fuzzy
denotes the set of all fuzzy numbers.
numbers and.
Letbe the set of all compact convex subsets of , andbe the set of all. For
rectangles in, we have. We write a rectangle
by
in
$a$

$\tilde{s}$$\tilde{s}_{\alpha}$$\alpha$$\mathcal{F}(\mathbb{R}^{n})$$\mathbb{R}^{n}arrow[0,1]$$\tilde{s}$$( \sup_{x\in \mathbb{R}^{n}}\tilde{s}(X)= 1)$

se.t.

$\mathrm{s}$$\mathcal{F}(\mathbb{R})$$C(\mathbb{R}^{n})$$\mathbb{R}^{n}$$\mathbb{R}^{n}$$c_{r}(\mathbb{R}^{n})$$\tilde{s}_{\alpha}\in C(\mathbb{R}^{n})(\alpha\in[0,1])$$\tilde{s}\in \mathcal{F}(\mathbb{R}^{n})$$C_{r}(\mathbb{R}^{n}.)$$[x, y]=[X_{1,y_{1}]}\cross[x_{2}, y_{2}]\cross\cdots[x_{n},$ $y_{n}|$for $x=(x_{1}, x_{2}, \cdots, x_{n}),$with $x_{i}\leq y_{i}(i=1,2, \cdots, n)$ . For thecase of $n=1,$
and it denotes the set of all bounded closed intervals. Whensatisfies
for allis called a rectangle-type. We denoteby
the set of all rectangle-type fuzzy sets on. Obviously.
The definitions of addition and scalar multiplication on
are as follows: Forand,$y=(y_{1}, y_{2}, \cdots, y_{n})\in \mathbb{R}^{n}$$C(\mathbb{R})--c_{r}(\mathbb{R})$$\alpha\in[0,1],\tilde{s}$$\tilde{s}_{\alpha}\in C_{r}(\mathbb{R}^{n})$$\tilde{s}\in \mathcal{F}(\mathbb{R}^{n})$$\mathbb{R}^{n}$$\mathcal{F}_{r}(\mathbb{R}^{n})$$\mathcal{F}_{r}(\mathbb{R})=\mathcal{F}(\mathbb{R})$$\mathcal{F}(\mathbb{R})$$\tilde{m},\tilde{n}\in \mathcal{F}(\mathbb{R})$$\lambda\geq 1$

{

(1.1)

$( \tilde{m}+\tilde{n})(x):=x_{1},x_{2}\in \mathbb{R}\cdot,1+x_{2}=x\sup_{x}$$\tilde{m}(x/\lambda)$(1.2)$(\lambda\tilde{m})(x):=\{$$I_{\{0\}}(x)$ifif$\tilde{m}(x1)$A$\tilde{n}(x_{2})$$\lambda>0$$(x\in \mathbb{R})$$\lambda=0$},,

is an indicator. By using set operations $A+B:=\{x+y|x\in A, y\in B\}$ and$\lambda A:=\{\lambda x|x\in A\}$ for any non-empty sets $A,$
, the following holds immediately.

where$I_{\{\cdot\}}(\cdot)$$B\subset \mathbb{R}$(1.3)$(\tilde{m}+\tilde{n})_{\alpha}:=\tilde{m}_{\alpha}+\tilde{n}_{\alpha}$and$(\lambda\tilde{m})_{\alpha}=\lambda\tilde{m}_{\alpha}$$(\alpha\in[0,1])$.

be a non-empty cone of . Using this , we can define a pseudo-order relation
by
iff
. Let
be the subset of entrywise non-negative elementsin. When, thewill be denoted byandmeans thatfor all $i=1,2,$, where $x=(x_{1}, x_{2}, \cdots, x_{n})$ and.
In Section 2, we will introduce a pseudo-order relation onwhich is characterized
by the scaralization technique. In section 3, the lattice structure is discussed for the classof rectangle-type fuzzy sets.
$\backslash \prec K$Leton$\mathbb{R}$$IC$$\mathbb{R}^{n}$$I\mathrm{t}^{\nearrow}$$x\backslash \prec_{K}y$$y-x\in I\mathrm{f}$$\mathbb{R}^{n}$$I\acute{\mathrm{t}}=\mathbb{R}_{+}^{n}$$\mathbb{R}_{+}^{n}$$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\prec_{\backslash K}$$\backslash n\prec$$x_{i}\leq y_{i}$$x\backslash \prec_{n}y$$y=(y_{1}, y_{2}, \cdots, y_{n})\in \mathbb{R}^{n}$$\cdots,$ $n$$\mathcal{F}(\mathbb{R}^{n})$2. A pseudo-order on$\mathcal{F}(\mathbb{R}^{n})$First we introduce a binary relation on, by which a pseudo-order onHenceforth we assume that the convex cone
is given.
We define a binary relationon: For $A,$:
following (C.a) and (C.b)$C(\mathbb{R}^{n})$$\mathcal{F}(\mathbb{R}^{n})$is given.

$K\subset \mathbb{R}^{n}$$\backslash K\prec$$(\mathrm{c}.\mathrm{f}.

[5], [6])$$C(\mathbb{R}^{n})$$B\in C(\mathbb{R}^{n}),$$A\prec_{K}B\backslash$means the144(C.a) For any $x\in A$ , there exists $y\in B$ such that$x\backslash \prec Ky$(C.b) For any $y\in B$ , there exists $x\in A$ such that$x\backslash \prec Ky$

Lemma 2.1. The$\mathrm{r}\mathrm{e}l$a tion$\backslash \prec K$

is a pseudo-order on.
.

$C(\mathbb{R}^{n})$.

such that. Let $A,$ $B,$
for
Proof. It is trivial thatby two cases (c.a) and (C.b). Case(C.a): Since. We will checkandand there exists $z\in C$ such
, for any $x\in A$ there exists $y\in B$ such thatand
. Therefore it holds that
, we haveis a pseudo-order on. Sincethat,
and. Case(C.b):Since
for any $x\in A$ there exists $z\in C$ such that.
and there exists $x\in A$ such thatfor any $z\in C$ there exists $y\in B$ such that
. Therefore it holds that for any $z\in C$, we haveis a pseudo-order on
Since.
there exists $x\in A$ such that.D.
. Thus the lemma holds.
From the above (a) and (b), we obtain$C\in C(\mathbb{R}^{n})$$A\in C(\mathbb{R}^{n})$$A\backslash \prec_{K}A$$A\backslash \prec_{K}B$$A\backslash \prec_{K}B$$A\backslash \prec KC$$B\backslash \prec_{K}C$$x\backslash \prec Ky$$B\backslash \prec KC$$\mathbb{R}^{n}$$x\backslash \prec_{K}z$$\backslash K\prec$$y\backslash \prec_{K}z$$B\backslash \prec_{K}C$$A\backslash \prec_{K}B$$x\backslash \prec_{K}z$$x\backslash \prec Ky$$y\backslash \prec_{K}z$$\mathbb{R}^{n}$$x\backslash \prec_{K}z$$\backslash K\prec$$x\backslash \prec_{K}z$$\mathrm{Q}.\mathrm{E}$$A\backslash \prec I1^{\prime c}$on$[x, y],$means
Next, we introduce a binary relationmeans the following (F.a) and (F.b):

When$K=\mathbb{R}_{+}^{n}$, the relation$[x’, y/]\in C_{r}(\mathbb{R}^{n}),$and forwill be written simply by.
and. The relation
: Let
on

$C(\mathbb{R}^{n})$$\backslash K\prec$$[x, y]_{\backslash n}\prec[x^{\prime/}, y]$$\prec_{\backslash n}$$y\backslash \prec_{n}y’$$x\prec_{n}x^{J}\backslash$$\backslash \prec K$$\tilde{s},\tilde{r}\in \mathcal{F}(\mathbb{R}^{n})$$\mathcal{F}(\mathbb{R}^{n})$$\tilde{s}\backslash \prec_{K}\tilde{r}$(F.a) For any$x\in \mathbb{R}^{n}$, there exists$y\in \mathbb{R}^{n}$such that$x\prec\backslash Ky$(F.b) For any$y\in \mathbb{R}^{n}$, there exists$x\in \mathbb{R}^{n}$such that$x\backslash \prec_{K}y$Lemma 2.2. The relation$\prec_{\backslash K}$

is a pseudo-order onand$\tilde{s}(x)\leq\tilde{r}(y)$.

and$\tilde{s}(x)\geq\tilde{r}(y)$.

$\mathcal{F}(\mathbb{R}^{n})$.

andsuch that. Let
for
Proof. It is trivial that
andby two cases (F.a) and (F.b). Case(F.a):Since. We will check, and there
andsuch that
there exists, for any
, weis a pseudo-order on. Sinceandsuch thatexiststhere exists. Therefore it holds that for any
and
have
there
, for anyand. Case(F.b) Since
andsuch thatsuch that, and there exists
andsuch thatexists. Therefore it holds, we haveis a pseudo-order on. Sinceand.
andsuch that
there existsthat for any.D.
. Thus the lemma holds.
From the above (a) and (b), we obtain$\tilde{s}\backslash \prec_{K}\tilde{s}$$\tilde{s}\in \mathcal{F}(\mathbb{R}^{n})$$\tilde{s},\tilde{r},\tilde{p}\in \mathcal{F}(\mathbb{R}^{n})$$\tilde{s}\backslash \prec_{K}\tilde{r}$$\tilde{s}\prec_{K}\tilde{r}\backslash$$\tilde{s}\backslash \prec K\tilde{p}$$\tilde{r}\backslash \prec_{K}\tilde{p}$$x\in \mathbb{R}^{n}$$\tilde{r}\backslash \prec K\tilde{p}$$z\in \mathbb{R}^{n}$$y\in \mathbb{R}^{n}$$y\backslash \prec_{K}z$$\mathbb{R}^{n}$$\tilde{r}(y)\leq\tilde{p}(z)$$\backslash K\prec$$x\backslash \prec_{K}z$$y\in \mathbb{R}^{n}$$\tilde{s}(x)\geq\tilde{r}(y)$$z\in \mathbb{R}^{n}$$\tilde{s}(x)\leq\tilde{p}(z)$$y\backslash \prec_{K}z$$z\in \mathbb{R}^{n}$$x\in \mathbb{R}^{n}$$\tilde{s}(x)\leq\tilde{p}(z)$$x\backslash \prec_{K}z$$\tilde{s}(x)\leq\tilde{r}(y)$$x\backslash \prec_{K}y$$\tilde{s}\backslash \prec K\tilde{r}$$x\in \mathbb{R}^{n}$$\tilde{s}(x)\geq\tilde{r}(y)$$\mathbb{R}^{n}$$x\backslash \prec Ky$$x\backslash \prec Kz$$\backslash K\prec$$x\in \mathbb{R}^{n}$$z\in \mathbb{R}^{n}$$\tilde{r}\backslash \prec K\tilde{p}$$x\backslash \prec_{K}z$$\tilde{s}(x)\geq\tilde{p}(z)$$\mathrm{Q}.\mathrm{E}$$\tilde{s}\backslash \prec_{K}\tilde{p}$The following lemma implies the correspondence between the pseudo-order on
for the -cuts.
for fuzzy sets and the pseudo-order on$C(\mathbb{R}^{n})$Lemma 2.3. Let$\alpha\in(0,1]$ .

$\tilde{s},\tilde{r}\in \mathcal{F}(\mathbb{R}^{n}).\tilde{s}\backslash \prec_{K}\tilde{r}$on$\mathcal{F}(\mathbb{R}^{n})$$\mathcal{F}(\mathbb{R}^{n})$$\alpha$if and on $ly$ if$\tilde{s}_{\alpha\backslash K}\prec\tilde{r}_{\alpha}$on$C(\mathbb{R}^{n})$for all145. Then, Two cases (a)onand $\alpha\in(0,1]$ . Suppose
Proof. Let, there exists
such that. Sinceand (b) are considered. Case(a): Let. Since, there. Case(b): Let. Namelyand.
. Namelyandsuch thatexistsfor all $\alpha\in(0,1]$ from the above (a) and (b).
on
Therefore we getonfor all $\alpha\in(0,1]$ . Then, Two cases
On the other hand, suppose. If $\alpha=0$ , then
and
. Put
are considered. Case(a’): Let
and, there exists
such. Since. While, if $\alpha>0$ , then
. If
. Put
. Case(b’): Let. And we havethat$\alpha--0$ ,,. Since. While, if $\alpha>0$ , then
and
then
.
. And we havesuch that
there exists.
we obtain
andfrom the above Caseon
Therefore we getthis lemma. Q.E.D.
$\tilde{s},\tilde{r}\in \mathcal{F}(\mathbb{R}^{n})$$x\in\tilde{s}_{\alpha}$$\alpha\leq\tilde{s}(x)\leq\tilde{r}(y)$$x\backslash \prec I\{y$$x\in \mathbb{R}^{n}$$\mathcal{F}(\mathbb{R}^{n})$$\tilde{s}\backslash \prec K\tilde{\Gamma}$$y\in \mathbb{R}^{n}$$\tilde{s}\backslash \prec K\tilde{r}$$y\in\tilde{r}_{\alpha}$$y\in\tilde{r}_{\alpha}$$\tilde{s}(x)\geq\tilde{r}(y)\geq\alpha$$x\backslash \prec Ky$$x\in\tilde{s}_{\alpha}$$C(\mathbb{R}^{n})$$\tilde{s}_{\alpha\backslash K}\prec\tilde{r}_{\alpha}$$C(\mathbb{R}^{n})$$\tilde{s}_{\alpha\backslash K}\prec\tilde{r}_{\alpha}$$(\mathrm{a}’)$$\alpha=\tilde{s}(x)$$x\in \mathbb{R}^{n}$$(\mathrm{b}’)$$\tilde{s}(x)=0\leq\tilde{r}(x)$$x\in\tilde{s}_{\alpha}$$x\backslash \prec Kx$$y\in\tilde{r}_{\alpha}$$\tilde{s}_{\alpha\backslash }\prec_{K^{\tilde{\Gamma}_{\alpha}}}$$\tilde{s}(x)=\alpha\leq\tilde{r}(y)$$x\backslash \prec_{K}y$$x\in\tilde{s}_{\alpha}$$y\in\tilde{r}_{\alpha}$.’$(\mathrm{a}’)$$\mathcal{F}(\mathbb{R}^{n})$$(\mathrm{b}$$\tilde{s}\backslash \prec K.\tilde{r}$$I\mathrm{t}^{\nearrow+}:=$

where$a\in \mathbb{R}^{n}${$I\mathrm{t}’$$a\in \mathbb{R}^{n}|a\cdot x\geq 0$

denotes the inner product on, we define$\mathbb{R}^{n}$for allfor$x\in I\mathrm{t}^{\nearrow}$$x,$ $y\in \mathbb{R}^{n}$$a\cdot A:=\{a\cdot x|x\in A\}(\subset \mathbb{R})$The equation (2.1) means the projection of$a\cdot a=1$ . It is trivial thatif$a\cdot A\in C(\mathbb{R})$

Lemma 2.4. Let $A,$is theall $a\in K^{+}$ , where$B\in C(\mathbb{R}^{n})$.

$n\mathrm{a}t\mathrm{u}r\mathrm{a}l$$A$$\mathrm{T}\mathrm{h}.\mathrm{u}\mathrm{s}\backslash$onorder on},

. For a subset$A\subset \mathbb{R}^{n}$and.

on the extended line of the vector.
and$C(\mathbb{R}^{n})$$C(\mathbb{R})$

$a$

if

$a\in \mathbb{R}^{n}$$A\in C(\mathbb{R}^{n})$$A\backslash \prec_{K}B$$\backslash 1\prec$

$)$

by

$x\cdot y$(2.1)$\tilde{s}_{\alpha\backslash K\alpha}\prec\tilde{r}$$\tilde{s}(x)\geq\alpha=\tilde{r}(y)$$x\backslash \prec Ky$

Define the dual cone of a cone$\alpha--\tilde{r}(y)$$y\in \mathbb{R}^{n}$$\tilde{s}(x)\geq 0=\tilde{r}(y)$$x\backslash \prec KX$$\tilde{s}\backslash \prec K\tilde{r}$if and only if$a\cdot A\backslash \prec_{1}a\cdot B$

on$C(\mathbb{R})$for.

. Consider the two cases (a) and (b). Case(a): For anyon
Proof. Suppose$a\cdot x\in a\cdot A$ , there exists $y\in B$ such that, then. Then $y-x\in I\mathrm{f}.$ If, there exists $x\in A$ such
. Case(b): For any$a\cdot(y.-x)\geq 0$ and i.e.
. From. Then $y-x\in K$ . If $a\in K^{+}$ , then $a\cdot(y-x)\geq 0$ and i.e.
that.
the above cases (a) and (b), we have thaton
On the other hand, to prove the inverse statement, we assume that
. Case(i): There exists
does not hold. Then we have the following two cases (i)and$x\in A$ such thatfor all $y\in B$ . Then $B\cap(x+K)=\emptyset$ . Sincesuch that$x+K$ are closed convex, by the separation theorem there exists. Hence we suppose that there exists$a\cdot y<a\cdot x+a\cdot z$ for all $y\in B$ and all
is a cone, and so we havefor allsuch that $a\cdot z\geq 0$ . Then
. This contradicts $a\cdot y<a\cdot x+a\cdot z$ . Therefore
asfor all $z\in K$ . Especially taking $z=0\in K$ , we get $a\cdot y<a\cdot x$ for all
we obtain$y\in B$ . This contradicts. Case(ii): There exists $y\in B$ such that $y-x\not\in K$
for all $x\in A$ . Then we derive the contradiction in a similar way to the case (i).
$A\backslash \prec_{K}B$$C(\mathbb{R}^{n})$$a$$x\backslash \prec Ky$$\in K^{+}$$a\cdot y\in a\cdot B$$a\cdot x\leq a\cdot y$$a\cdot x\leq a\cdot y$$x\backslash \prec_{K}..y$$a\cdot A\backslash \prec_{1}a\cdot B$$A\backslash \prec_{K}B$$C(\mathbb{R}^{n})$$\mathrm{a}\mathrm{n}\dot{\mathrm{d}}(\mathrm{i}\mathrm{i})$$B$$y-x\not\in I\mathrm{t}^{\nearrow}$$a\in \mathbb{R}(a\neq 0)$$z\in I\{^{F}$$\lambda z\in I\backslash ^{\nearrow}$$z\in I\mathrm{f}$$a\cdot x+a\cdot\lambda_{Z}=a\cdot x+\lambda a\cdot zarrow-\infty$$\lambdaarrow\infty$$a\cdot z\geq 0$$a\cdot A\backslash \prec_{1}a\cdot B$$\lambda\geq 0.\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{C}\mathrm{e}I1^{\Gamma}$

146
Therefore the inverse statement holds from the results of the above (i) and (ii). The
proof of this lemma is completed. Q.E.D.
For

and

$a\in \mathbb{R}^{n}$$\tilde{s}\in \mathcal{F}(\mathbb{R}^{n})$(2.2), we define a fuzzy number,$a\cdot\tilde{s}\in \mathcal{F}(\mathbb{R})$$x\in \mathbb{R}$$a \cdot\tilde{s}(x):=\sup_{\alpha\in[0,1]}\min\{\alpha, 1_{a\cdot\overline{S}_{\mathrm{O}}}(x)\}$

by

.

where
is the classical indicator function of a closed interval.
We define a partial relationon
as follows ([8]): Formeans thatfor.
The following theorem gives the correspondence between the pseudo-orderand the fuzzy $\max$ orderon.
$1_{D}(\cdot)$$D\in C(\mathbb{R})$$\mathcal{F}(\mathbb{R})$$\backslash \prec M$$\mathrm{a}\mathrm{l}1\alpha\in[0,1]$$\tilde{s}_{\alpha\backslash 1}\prec\tilde{r}_{\alpha}$$\mathcal{F}(\mathbb{R}^{n})$$\tilde{s},\tilde{r}\in \mathcal{F}(\mathbb{R}^{n}),\tilde{S}\prec\backslash K\tilde{r}$Proof. From Lemmas 2.3 and 2.4,$\alpha\in(0,1]$ . Is equivalent to
and

iff

$a\cdot\tilde{s}\prec_{M}\backslash a\cdot\tilde{r}$iff
for all$\tilde{s}\prec_{K}\tilde{r}\backslash$$a\cdot\tilde{s}\backslash \prec Ma\cdot\tilde{\Gamma}$$\{\tilde{s}_{k}\}_{k=1}\infty\subset \mathcal{F}(\mathbb{R}^{n})$$\backslash K\prec$on$\mathcal{F}(\mathbb{R})$$\backslash M\prec$

Theorem 2.1. For

For$\tilde{s},\tilde{r}\in \mathcal{F}(\mathbb{R}^{n}),\tilde{s}\backslash \prec_{M}\tilde{r}$for all $a\in K^{+}$ .
for all for all
Q.E.D.

$a\cdot\tilde{s}_{\alpha\backslash 1}\prec a\cdot\tilde{\Gamma}_{\alpha}$$a\in I\mathrm{t}^{\nearrow+}$.

$\tilde{s}\in \mathcal{F}(\mathbb{R}^{n}),\lim_{karrow\infty}\tilde{s}_{k}=\tilde{s}$means that$a$$\in I\iota^{\nearrow+}$and$\sup\rho(\tilde{s}_{k,\alpha},\tilde{s}_{\alpha})arrow 0$$\alpha\in[0,1]$$(karrow\infty)$, whereLemma 2.5.
$\lim_{karrow\infty^{\tilde{S}}k}=\tilde{s}$$\tilde{s}_{k,\alpha}$

is the -cut of$\alpha$Let. ThenProof. Trivial.

$\{\tilde{s}_{k}\}_{k=1}^{\infty}\subset \mathcal{F}(\mathbb{R})$$\tilde{s}_{1\backslash M}\prec\tilde{s}$$\mathrm{Q}.\mathrm{E}$

Theorem 2.2. Let. Then$\lim_{karrow\infty^{\tilde{s}_{k}}}=\tilde{s}$$\tilde{s}_{k}$

and
and

$\rho$

is the Hausdorff metric on$\tilde{s}\in \mathcal{F}(\mathbb{R})$such th at$C(\mathbb{R}^{n})$.

$\tilde{s}k\backslash \prec M^{\tilde{S}}k+1(k\geq 1)$

an

$d$

an

$d$

.

.D.

$\{\tilde{s}_{k}\}_{k=1}^{\infty}\subset \mathcal{F}(\mathbb{R}^{n})$$\tilde{s}_{1\backslash K}\prec\tilde{s}$$\tilde{s}\in \mathcal{F}(\mathbb{R}^{n})$such th at$\tilde{s}_{k\backslash }\prec_{K}\tilde{s}_{k+1}(k\geq 1)$.

Proof. From Theorem 2.1, for all. By Lemma 2.3,$\tilde{s}_{k}=a\cdot\tilde{\mathit{8}}$

and

$a$

we haveall. From Theorem 2.1,$\in K^{+}$$a\cdot\tilde{s}_{1}\prec\backslash Ma\cdot\tilde{S}$$a\cdot\tilde{s}_{k\backslash M}\prec a\cdot\tilde{s}_{k+1}(k\geq 1)$$a\in I\iota^{\nearrow+}$and$\lim_{karrow\infty^{a}}$$\tilde{s}_{1}\prec_{K}\tilde{s}\backslash \cdot$.

Q.E.D.

Remark. Let the map
be continuous. A pointis said to be efficient
if $x_{0}\in[0,1]$ andfor some $x\in[0,1]$ implies $g(x)=g(X_{0})$ . Then, by applying
the same idea as in Lemma 3.2 of Furukawa [2], we observe that there exists at least one
efficient point in $[0,1]$ . In fact, considering, if necessary, a partial orderon the class ofthe quotient sets with respect to the equivalence
defined byiff, we can assume thatandis a partial order on. By theorem 2.2 and thecontinuity of $\{g(x)|x\in[0,1]\}$ can be proved to be an inductively ordered set. So, by
Zorn’s lemma $\{g(x)|x\in[0,1]\}$ has an efficient element.
$g:[0,1]arrow \mathcal{F}(\mathbb{R}^{n})$$x_{0}$$g(x_{0})\backslash \prec Kg(x)$$\backslash \prec K$$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\sim_{K}$$\tilde{r}\backslash \prec_{K}\tilde{s}$$\tilde{s}\sim_{K}\tilde{r}$$\tilde{s}\backslash \prec K\tilde{r}$$\mathcal{F}(\mathbb{R}^{n})$$\backslash K\prec$$g,$

3. Further results
In this section, we investigate a pseudo-orderon. To this end, we need the following lemma.
$\backslash I\prec\iota’$$K^{+}\subset \mathbb{R}^{n}$$\mathcal{F}_{r}(\mathbb{R}^{n})$for a polyhedral cone $K$ with

147

Lemma 3.1. Let$a,$ $b\in \mathbb{R}_{+}^{n}$

(3.1)

and

$A\in C_{r}(\mathbb{R}^{n})$. Then for any scalars$(\lambda_{1}a+\lambda_{2}b)\cdot A=\lambda_{1}(a\cdot A)+\lambda_{2}(b\cdot A)$$\lambda_{2}\geq 0$$\lambda_{1},$, it holds,

where the arithmetic in (3.1) is defined in (2.1).

Proof. Letwith$\lambda_{1}a\cdot x+\lambda_{2}b\cdot y\in\lambda_{1}(a\cdot A)+\lambda_{2}(b\cdot B)$$x,$ $y\in

A$. It suffices to show that. Define $z=(z_{1}, z_{2}, \cdots, z_{n})$ by$\lambda_{1}a\cdot x+\lambda_{2}b\cdot b\in(\lambda_{1}a+\lambda_{2}b)\cdot A$$z_{i}=(\lambda_{1}a_{ii}x+\lambda_{2}b_{iy_{i}})/(\lambda_{1}a_{i}+\lambda_{2}b_{i})$if$(\lambda_{1}a_{i}+\lambda_{2}b_{i})=0$$=x_{i}$Then, clearly.

$(\lambda_{1}a_{i}+\lambda_{2}b_{i})>0$$(i=1,2, \cdots, n)$$(\lambda_{1}a+\lambda_{2}b)\cdot z=\lambda_{1}a\cdot x+\lambda_{2}b\cdot y$$\lambda_{1}a\cdot x+\lambda_{2}b\cdot y=(\lambda_{1}a+\lambda_{2}b)\cdot A$

if. Since$A\in C_{r}(\mathbb{R}^{n}),$$Z\in A$ ,so that

Q.E.D.

Henceforth, we assume that $K$ is a polyhedral convex cone withexist vectors
such that$K^{+}\subset \mathbb{R}^{n}$, i.e., there$b^{i}\in \mathbb{R}_{+}^{n}(i=1,2, \cdots, m)$$I\zeta=$Then, it is well-known{$x\in \mathbb{R}^{n}|b^{i}\cdot x\leq 0$$(\mathrm{c}.\mathrm{f}.

[9])$

that

$I\iota^{\nearrow+}$

for all $i=1,2,$$\cdots,$ $m$}.

is expressed as$K^{+}= \{x\in \mathbb{R}^{n}|x=\sum_{i=1}^{m}\lambda_{i}bi, \lambda_{i}\geq 0(i=1,2, \cdots, m)\}$The above dual cone

$I\mathrm{t}^{\nearrow+}$

is denoted simply by$I\mathrm{t}^{\nearrow+}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{V}\{b1, b2, \cdots, b^{m}\}$The pseudo-order$A\backslash \prec_{k}B$.

is characterized in the following.

$\backslash \prec K\mathrm{o}\mathrm{n}.C_{r}(\mathbb{R}n)$Corollary 3.1. Let
if and only if.

.

$I\iota^{\nearrow+}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{V}\{b^{1}, b^{2}, \cdots, b^{m}\}$$b^{i}\cdot A\backslash \prec_{1}b^{i}\cdot B$

withfor all $i=1,2,$$\cdots,$,. Then, for $A,$
in, where is a pseudo-order on$B\in C_{r}(\mathbb{R}^{n})$$b^{i}\in \mathbb{R}_{+}^{n}$$\backslash 1\prec$$C_{r}(\mathbb{R})$Proof. We assume thatexistswith$\lambda_{i}\geq 0$, $m$ . For any
for all $i=1,2,$. From Lemma 3.1 we have:$b^{i}\cdot A\backslash \prec_{1}b^{i}\cdot B$$a= \sum_{i=1}^{m}\lambda_{i}bi$$\cdots$$a$$\in K^{+}$, there.

$a \cdot A=\sum_{i=1}^{m}\lambda_{i}(b^{i}\cdot A)\backslash 1\prec=\sum_{i=1}^{m}\lambda_{i}(b^{i}\cdot B)=a\cdot B$

Thus, by Lemma 2.4,
Q.E.D.
of Corollary holds.

$A\backslash K\prec B$

Lemma 3.2. Let(3.2)$a,$ $b\in \mathbb{R}_{+}^{n}$follows. By applying Lemma 2.4 again, the ‘only if’ partand$\tilde{s}\in \mathcal{F}_{r}(\mathbb{R}^{n})$. Then, for any$\lambda_{1},$$(\lambda_{1}a+\lambda_{2}b)\cdot\tilde{s}=\lambda 1(a\cdot\tilde{S})+\lambda_{2}(b\cdot\tilde{s})$,$\lambda_{2}\geq 0$

,

148

where the arithmetic in (3.2) are given in (1.1), (1.2) and (2.2).

Proof. For any $\alpha\in[0,1]$ , it follows from the definition and Lemma 3.1 that$[(\lambda_{1}a+\lambda_{2}b)\cdot\tilde{s}]_{\alpha}=(\lambda_{1}a+\lambda 2b)\cdot\tilde{S}_{\alpha}=\lambda 1(a\cdot\tilde{s}\alpha)+\lambda 2(b\cdot\tilde{s}_{\alpha})$.

$=\lambda_{1}(a\cdot\tilde{s})_{\alpha}+\lambda_{2}(b\cdot\tilde{S})\alpha=[\lambda_{1}(a\cdot\tilde{\mathit{8}})+\lambda 2(b\cdot\tilde{s})]_{\alpha}$

The last equality follows from (1.3). The above shows that (3.3) holds.

$\mathrm{Q}.\mathrm{E}$.D.

The main results in this section are given in the following.

Theorem 3.1. Let$\tilde{s}\prec_{\backslash K}\tilde{r}$$I\mathrm{t}^{\nearrow+}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{V}\{b^{1}, b^{2}, \cdots, b^{m}\}$if and only iff

with$b^{i}\in \mathbb{R}^{n}.$$b^{i}\cdot\tilde{s}\backslash M\prec b^{i}\cdot\tilde{r}$The.n, for

for $i=1,2,$

Proof. It suffices to prove the ‘if’ part of Theorem 3.1. For any. Applying Lemma 3.2, we havewith$\tilde{s},\tilde{r}\in \mathcal{F}_{r}(\mathbb{R}^{n})$$\cdots,$ $m$,.

$a\in I\mathrm{t}^{\prime+}$, there exists$a= \sum_{i=1}^{m}\lambda_{i}bi$$\lambda_{i}\geq 0$$a \cdot\tilde{s}=\sum_{i=1}^{m}\lambda_{i}(b^{i}\cdot\tilde{S})\backslash \prec M\sum_{i=1}^{m}\lambda i(b^{i}\cdot\tilde{r})=a\cdot\tilde{r}$From Theorem 2.1,$\tilde{s}\backslash k\prec\tilde{r}$follows.

$\mathrm{Q}.\mathrm{E}$Figure 1:Whenandously,$I\mathrm{f}=\mathbb{R}^{n}$$\backslash 1\prec$$\backslash \prec M$, the pseudo-order
are the same.

$\backslash I^{\mathrm{g}’}\prec$,.D.

$\tilde{v}=\sup(\tilde{s},\tilde{r})$on$\mathcal{F}_{r}(\mathbb{R}^{n})$will be simply written by$\backslash n\prec$. Obvi-

149.
Congxin and Cong [1] described the structure of the fuzzy number lattice. So that, by Theorem 3.1,
andWhen
. Therefore, bymeans
for all $i=1,2,$
we see that for
applying the same method as [1], we can describe the structure of the fuzzy set lattice.
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numbers and its application,” J. Math. Anal. Appl. 210 (1997) 499-511.
[2] N.Furukawa, “Characterization of optimal policies in vector-valued Markovian
decision processes,” Math. Oper. Research. 5 (1980) 271-279.
[3] N.Furukawa, “Parametric orders on fuzzy numbers and their roles in fuzzy
optimization problems,” Optimization 40 (1997) 171-192.

-type fuzzy de[4] M.Kurano, M.Yasuda, J.Nakagami and Y.Yoshida, “cision processes with a discounted reward on a closed interval,” European
Journal of Operational Research 92 (1996) 649-662.
$\mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}_{\mathrm{o}\mathrm{v}}$[5] D.Kuroiwa, T.Tanaka and T.X.D.Ha, “On cone convexity of set-valued maps,”
Nonlinear $Analysi_{S}$ Theory Applications 30 (1997) 1487-1496.
$\not\in y$)[6] D.Kuroiwa, . “The natural criteria in set-valued optimization,”
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RIMS[7] V.Nov\’ak, Fuzzy Sets and Their Applications (Adam Hilder, Bristol-Boston,1989).

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$(\tilde{R}, \backslash \prec),$”

Fuzzy sets and