Kurano, Masami

Nakagami, Jun-ichi

Yasuda, Masami

Yoshida, Yuji

Kyoto University Research Information Repository

Title Sequences of Fuzzy Sets on $\mathbb{R}^n$ (Decision Theoryin Mathematical Modelling)Author(s)Kurano, Masami; Yasuda, Masami; Nakagami, Jun-ichi;Yoshida, YujiCitation数理解析研究所講究録 (1999), 1079: 233-237Issue Date1999-02URL http://hdl.handle.net/2433/62675RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversitySequences of Fuzzy Sets on $\mathbb{R}^{n}$(Masami KURANO)(Masami YASUDA)– (Jun-ichi NAKAGAMI)(Yuji YOSHIDA)AbstractIn this paper, we study the convergence of a sequence of fuzzy sets on $\mathbb{R}^{n}$ whichin monotone w.r.t. a pseudo order $\neg K\prec$ induced by a closed convex cone $K$ . Ourstudy is carried out by restricting the class of fuzzy sets into the subclass in which$\neg\prec K$ becomes a partial order and a monotone convergence theorem is approved.This restricted subclass of fuzzy sets is created and characterized in the concept ofa determining class.Keywords: Pseudo-order, fuzzy $\max$ order, multidimensional fuzzy sets, monotone con-vergence theorem, determining class, rectangle-type fuzzy sets.1. Introduction and notationsIn our previous paper [3], we have introduced a pseudo order, $\neg K\prec$ , in the class of fuzzysets, which is natural extension of fuzzy $\max$ order (cf. [2], [6]) in fuzzy numbers on $\mathbb{R}$ andinduced by a closed convex cone $K$ in $\mathbb{R}^{n}$ . For a lattice-structure of the fuzzy $\max$ order,see [1], [10]. Here, we study the convergence of a sequence of fuzzy sets on $\mathbb{R}^{n}$ which ismonotone w.r.t. a pseudo order $\neg K\prec$ . Our study is done by restricting the class of fuzzysets into the subclass in which $\neg K\prec$ becomes a partial order and a monotone convergencetheorem is approved. This restricted subclass of fuzzy sets is created and characterizedin the concept of a determining class.In the remainder of this section, we will give some notations and review a vectorordering of $\mathbb{R}^{n}$ by a convex cone. In Section 2, a pseudo order of fuzzy sets on $\mathbb{R}^{n}$is reviewed referring to our previous paper [3]. In Section 3, we introduce a conceptdetermining class and give a convergence theorem for convex compact subclass $\mathbb{R}^{n}$ . Theseresults are applied to obtain a monotone convergence theorem for fuzzy sets on $\mathbb{R}^{n}$ inSection 4.Let $\mathbb{R}$ be the set of all real numbers and $\mathbb{R}^{n}$ an $n$-dimensional Euclidean space. Wewrite fuzzy sets on $\mathbb{R}^{n}$ by their membership functions $\tilde{s}$ : $\mathbb{R}^{n}arrow[0,1]$ (see Nov\’ak [5] andZadeh [9] $)$ . The $\alpha$-cut $(\alpha\in[0,1])$ of the fuzzy set $\tilde{s}$ on $\mathbb{R}^{n}$ is defined as$\tilde{s}_{\alpha}:=\{x\in \mathbb{R}^{n}|\tilde{s}(x)\geq\alpha\}(\alpha>0)$ and $\tilde{s}_{0}:=\mathrm{c}1\{x\in \mathbb{R}^{n}|\tilde{s}(x)>0\}$ ,where cl denotes the closure of the set. A fuzzy set $\tilde{s}$ is called convex if$\tilde{s}(\lambda x+(1-\lambda)y)\geq\tilde{s}(x)$ A $\tilde{s}(y)$ $x,$ $y\in \mathbb{R}^{n},$ $\lambda\in[0,1]$ ,1079 1999 233-237 233where $a$ $\wedge b=\min\{a, b\}$ . Note that $\tilde{s}$ is convex iff the $\alpha$-cut $\tilde{s}_{\alpha}$ is a convex set for all$\alpha\in[0,1]$ . Let $\mathcal{F}(\mathbb{R}^{n})$ be the set of all convex fuzzy sets whose membership functions$\tilde{s}$ : $\mathbb{R}^{n}arrow[0,1]$ are upper-semicontinuous and normal $( \sup_{x\in \mathbb{R}^{n}}\tilde{S}(X)=1)$ and have acompact support. When the one-dimensional case $n=1$ , the fuzzy sets are called fuzzynumbers and $\mathcal{F}(\mathbb{R})$ denotes the set of all fuzzy numbers.Let $C(\mathbb{R}^{n})$ be the set of all compact convex subsets of $\mathbb{R}^{n}$ , and $C_{r}(\mathbb{R}^{n})$ be the set of allrectangles in $\mathbb{R}^{n}$ . For $\tilde{s}\in \mathcal{F}(\mathbb{R}^{n})$ , we have $\tilde{s}_{\alpha}\in C(\mathbb{R}^{n})(\alpha\in[0,1])$ . We write a rectanglein $C_{r}(\mathbb{R}^{n})$ by$[x, y]=[x_{1}, y_{1}]\cross[x_{2}, y_{2}]\cross\cdots\cross[x_{n}, y_{n}]$for $x=(x_{1}, x_{2}, \cdots, x_{n}),$ $y=(y_{1}, y_{2}, \cdots, y_{n})\in \mathbb{R}^{n}$ with $x_{i}\leq y_{i}(i=1,2, \cdots , n)$ . For thecase of $n=1,$ $C(\mathbb{R})=C_{r}(\mathbb{R})$ and it denotes the set of all bounded closed intervals. When$\tilde{s}\in \mathcal{F}(\mathbb{R}^{n})$ satisfies $\tilde{s}_{\alpha}\in C_{r}(\mathbb{R}^{n})$ for all $\alpha\in[0,1],\tilde{s}$ is called a rectangle-type. We denoteby $\mathcal{F}_{r}(\mathbb{R}^{n})$ the set of all rectangle-type fuzzy sets on $\mathbb{R}^{n}$ . Obviously $\mathcal{F}_{r}(\mathbb{R})=\mathcal{F}(\mathbb{R})$ .The definitions of addition and scalar multiplication on $F(\mathbb{R}^{n})$ are as follows: For$\tilde{m},\tilde{n}\in \mathcal{F}(\mathbb{R}^{n})$ and $\lambda\geq 0$ ,(1.1) $( \tilde{m}+\tilde{n})(x):=x_{1},x_{2\in \mathbb{R}^{n}};\sup_{x_{2x}x1+=}\{\tilde{m}(x_{1})\wedge\tilde{n}(x_{2})\}$ ,(1.2) $(\lambda\tilde{m})(x):=\{$$\tilde{m}(x/\lambda)$ if $\lambda>0$$I_{\{0\}(x})$ if $\lambda=0$$(x\in \mathbb{R}n)$ ,where $I_{\{\cdot\}}(\cdot)$ 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,$ $B\subset \mathbb{R}^{n}$ , the following holds immediately.(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])$ .Let $K$ be a non-empty cone of $\mathbb{R}^{n}$ . Using this $K$ , we can define a pseudo-order relation$\neg K\prec$ on $\mathbb{R}$ by $x\neg\prec_{K}y$ iff $y-x\in K$ . Let $\mathbb{R}_{+}^{n}$ be the subset of entrywise non-negative elementsin $\mathbb{R}^{n}$ . When $K=\mathbb{R}_{+}^{n}$ , the order $\neg K\prec$ will be denoted $\mathrm{b}\mathrm{y}\prec_{\neg n}$ and $x\neg\prec_{n}y$ means that $x_{i}\leq y_{i}$for all $i=1,2,$ $\cdots$ , $n$ , where $x=(x_{1}, x_{2}, \cdots, x_{n})$ and $y=(y_{1}, y_{2}, \cdots , y_{n})\in \mathbb{R}^{n}$ .2. A pseudo-order on $F(\mathbb{R}^{n})$In this section, we review a pseudo order introduced by [3]. Henceforth we assume thatthe convex cone $K\subset \mathbb{R}^{n}$ is given. A pseudo order $\neg K\prec$ on $C(\mathbb{R}^{n})$ is defined, whose idea isbased a set-relation treated in [4], as follows.For $A,$ $B\in C(\mathbb{R}^{n}),$ $A\neg\prec KB$ means the following (C.a) and (C.b) :(C.a) For any $x\in A$ , there exists $y\in B$ such that $x\prec_{\neg Ky}$ .(C.b) For any $y\in B$ , there exists $x\in A$ such that $x\neg K\prec y$ .When $K=\mathbb{R}_{+}^{n}$ , the relation $\neg K\prec$ on $C(\mathbb{R}^{n})$ will be written simply by $\neg n\prec$ and for$[x, y],$ $[x’, y’]\in C_{r}(\mathbb{R}^{n}),$ $[x, y]\neg n\prec[x’, y’]$ means $x\neg\prec_{n^{X’}}$ and $y\neg n\prec y’$ .Using a pseudo order $\neg K\prec$ on $C(\mathbb{R}^{n})$ , a pseudo order $\neg K\prec$ on $\mathcal{F}(\mathbb{R}^{n})$ is defined as follows.For $\tilde{s},\tilde{r}\in \mathcal{F}(\mathbb{R}^{n}),\tilde{s}\neg\prec K\tilde{r}$ means the following (F.a) and (F.b):234(F.a) For any $x\in \mathbb{R}^{n}$ , there exists $y\in \mathbb{R}^{n}$ such that $x\neg K\prec y$ and $\tilde{s}(x)\leq\tilde{r}(y)$ .(F.b) For any $y\in \mathbb{R}^{n}$ , there exists $x\in \mathbb{R}^{n}$ such that $x\neg\prec_{K}y$ and $\tilde{s}(x)\geq\tilde{r}(y)$ .In [3], for $\tilde{s},\tilde{r}\in \mathcal{F}(\mathbb{R}^{n})$ , it is shown that $\tilde{s}\prec_{\neg K}\tilde{r}$ if and only if $\tilde{s}_{\alpha}\neg\prec_{K}\tilde{r}_{\alpha}$ on $C(\mathbb{R}^{n})$ forall $\alpha\in[0,1]$ . Define the dual cone of a cone $K$ by$K^{+}:=$ { $a\in \mathbb{R}^{n}|a\cdot x\geq 0$ for all $x\in K$ },where $x\cdot y$ denotes the inner product on $\mathbb{R}^{n}$ for $x,$ $y\in \mathbb{R}^{n}$ . For a subset $A\subset \mathbb{R}^{n}$ and$a\in \mathbb{R}^{n}$ , we define(2.1) $a\cdot A:=\{a\cdot x|x\in A\}(\subset \mathbb{R})$ .The equation (2.1) means the projection of A on the extended line of the vector $a$ if$a\cdot a=1$ . It is trivial that $a\cdot A\in C(\mathbb{R})$ if $A\in C(\mathbb{R}^{n})$ and $a\in \mathbb{R}^{n}$ .Lemma 2.1 ([3]). Let $A,$ $B\in C(\mathbb{R}^{n})$ . $A\neg K\prec B$ on $C(\mathbb{R}^{n})$ if and only if $a\cdot A\neg 1\prec a\cdot B$on $C(\mathbb{R})$ for all $a\in K^{+}$ , where $\neg 1\prec$ is the natural $ord$er on $C(\mathbb{R})$ .For $a\in \mathbb{R}^{n}$ and $\tilde{s}\in \mathcal{F}(\mathbb{R}^{n})$ , we define a fuzzy number $a\cdot\tilde{s}\in \mathcal{F}(\mathbb{R})$ by(2.2) $a\cdot\tilde{s}(x)$ $:= \sup_{\alpha\in[0,1]}\min\{\alpha, 1a\cdot\overline{s}_{\alpha}(x)\}$ , $x\in \mathbb{R}$ .where $1_{D}(\cdot)$ is the classical indicator function of a closed interval $D\in C(\mathbb{R})$ .We define a partial relation $\neg M\prec$ on $\mathcal{F}(\mathbb{R})$ as follows ([6]): For $\tilde{s},\tilde{r}\in \mathcal{F}(\mathbb{R}),\tilde{s}\neg M\prec\tilde{r}$means that $\tilde{s}_{\alpha}\neg\prec 1\tilde{r}_{\alpha}$ for all $\alpha\in[0,1]$ .The following theorem gives the correspondence between the pseudo-order $\neg K\prec$ on$\mathcal{F}(\mathbb{R}^{n})$ and the fuzzy $\max$ order $\neg M\prec$ on $\mathcal{F}(\mathbb{R})$ .Lemma 2.2 ([3]). For $\tilde{s},\tilde{r}\in \mathcal{F}(\mathbb{R}^{n}),\tilde{s}\neg\prec_{K}\tilde{r}$ if and on$ly$ if $a\cdot\tilde{s}\neg\prec Ma\cdot\tilde{T}$ for all $a\in K^{+}$ .Let $\rho_{n}$ be the Hausdorff metric on $C(\mathbb{R}^{n})$ , that is, for $A,$ $B\in C(\mathbb{R}^{n}),$ $\rho_{n}(A, B)=$$\max_{a\in A}d(a, B)\vee\max d(b, A)b\in B$ ’ where $d$ is a metric in $\mathbb{R}^{n}$ and $d(x, Y)= \min_{y\in Y}d(x, y)$ for $x\in \mathbb{R}^{n}$and $Y\in C(\mathbb{R}^{n})$ . It is well-known that $(C(\mathbb{R}n), \rho n)$ is a complete separable metric space. Asequence $\{D_{f}\}_{\ell}\infty=1\subset C(\mathbb{R}^{n})$ converges to $D\in C(\mathbb{R}^{n})$ w.r.t. $\rho_{n}$ if $\rho_{n}(D_{\ell}, D)arrow \mathrm{O}$ as $parrow\infty$ .Definition( $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}$ of fuzzy set, [8]).For $\{\tilde{s}_{\ell}\}_{p_{=}1}\infty\subset \mathcal{F}(\mathbb{R}^{n})$ and $\tilde{r}\in \mathcal{F}(\mathbb{R}^{n}),\tilde{S}\ell$ converges to $\tilde{r}\mathrm{w}.\mathrm{r}.t$ . $\rho_{n}$ if $\rho_{n}(\tilde{s}_{f\alpha},,\tilde{r}_{\alpha}1\text{ }arrow 0$ as$parrow\infty$ except at $\mathrm{m}ostCO$untable $\alpha\in[0,1]$ .In the sequel, the monotone convergence theorems for fuzzy sets are given under theconcept of the above convergence.2353. Sequences in $C(\mathbb{R}^{n})$In this section, restricting $C(\mathbb{R}^{n})$ into the subclass by use of the concept of determiningclass, we prove the monotone convergence theorem for $C(\mathbb{R}^{n})$ .Let $L\subset C(\mathbb{R}^{n})$ and $A\subset \mathbb{R}^{n}$ . Then we say that $L$ is determined by $A$ if $a\cdot D=a\cdot F$for $a\in A$ and $D,$ $F\in \mathrm{J}3$ implies $D=F$ .Note that $\mathcal{L}$ determined by some $A\subset \mathbb{R}^{n}$ is closed w.r.t. $\rho_{n}$ . Obviously, $C_{r}(\mathbb{R}^{n})$ isdetermined by $\{\mathrm{e}_{1}, \mathrm{e}_{2}, \cdots, \mathrm{e}_{n}\}$ . Also, by the separation theorem, $C(\mathbb{R}^{n})$ is determined$\mathbb{R}^{n}$ .Theorem 3.1. Let $K$ be a closed $co\mathrm{n}$vex cone of $\mathbb{R}^{n}$ . Suppose that $L\subset C(\mathbb{R}^{n})$ isdetermin$ed$ by $K^{+}$ . Then, the pseudo order $\neg K\prec$ is a partial one in the restricted class L.Proof. It suffices to show that $\neg K\prec$ is antisymmetric in $L$ . Let $D,$ $F\in L$ satisfy that$D\prec_{K}F\neg$ and $F\prec_{k}D\neg$ . By Lemma 2.1, $aD\neg\prec_{1}aF$ and $\dot{a}F\neg\prec_{1}aD$ for all $a\in K^{+}$ . Since $\neg 1\prec$ia a partial order, $aF=aD$ for all $a\in K^{+}$ , which implies $F=D$ from the determiningproperty of $K^{+}$ . Q.E.D.The sequence $\{D_{f}\}^{\infty}\ell_{=}1\subseteq C(\mathbb{R}^{n})$ is said to be bounded w.r.t. $\prec_{\neg K}$ if there exists$F,$ $D\in C(\mathbb{R}^{n})$ such that $F\neg\prec_{K}D_{\ell\neg}\prec_{K}D$ for all $\ell\geq 1$ and said to be monotone w.r.t. $\neg K\prec$if $D_{1}\neg\prec KD_{2}\neg K\prec\ldots$ . Then, as an application of Theorem 3.1, we have the following,whose proof is omitted.Theorem 3.2. Let $K$ be a closed convex cone $of\mathbb{R}^{n}$ with $K^{+}\cap(\mathbb{R}_{+}^{n})^{o}\neq\emptyset$ . $S\mathrm{u}$ppose that$,\mathrm{C}\subset C(\mathbb{R}^{n})$ is determin$ed$ by $K^{+}$ . Then, any sequence $\{D_{\mathit{1}}\}_{f}^{\infty}=1\subset L$ which is $\mathrm{m}$onoton$e$and bounded $w.\mathrm{r}.t$ . $\neg K\prec$ converges $\mathrm{w}.\mathrm{r}.\mathrm{t}$ . $\rho_{n}$ , where $A^{o}$ is a set of inner points in $A$ .The following results are concerned with the scalarization method.Corollary 3.1. Let $K=\{\lambda a|\lambda\geq 0\}$ for some $a\in\overline{\mathbb{R}_{+}^{n}}$ . Then, any sequence in $C(\mathbb{R}^{n})$with monotonicity and boundedness $w.\mathrm{r}.t$ . $\neg K\prec$ converges $w.r.t$ . $\rho_{n}$ .Corollary 3.2. Any sequence in $C_{r}(\mathbb{R}^{n})$ with monotonicity and $bo$undedness $w.r.t$ . $\neg n\prec$converges $\mathrm{w}.\mathrm{r}.t$ . $\rho_{n}$ .4. Sequences in $F(\mathbb{R}^{n})$In this section, applying the results in Section 3, we give the monotone convergencetheorem in $\mathcal{F}(\mathbb{R}^{n})$ . Let $\tilde{\epsilon}\subseteq \mathcal{F}(\mathbb{R}^{n})$ and $A\subset \mathbb{R}^{n}$ . Then we say that ,$\tilde{\mathrm{C}}$ is determined by $A$if $a\tilde{s}=a\tilde{r}$ for all $a\in \mathrm{A}$ and $\tilde{s},\tilde{r}\in\tilde{L}$ implies $\tilde{s}=\tilde{r}$ .Note that $\tilde{L}$ determined by some $A\subset \mathbb{R}^{n}$ is closed in the convergence given in Defi-nition 1. Applying Lemma 2.2, the same proof as Theorem 3.1 is useful in proving thefollowing.Theorem 4.1. Let $K$ be a closed convex cone of $\mathbb{R}^{n}$ . Suppose that $\tilde{L}\subset \mathcal{F}(\mathbb{R}^{n})$ is236determined by $K^{+}$ . Then, a pseudo $or\mathrm{d}er\prec_{\neg K}$ is a partial order in the restricted class$\overline{L}$ .In order to get the convergence theorem, we need the concept of directionality givenin [8]. Put the surface of the unit ball by $U:=\{x\in \mathbb{R}^{n}|||x||=1\}$ . Let$V\subseteq U$ . Then,for $D,$ $D’\in C(\mathbb{R}^{n})$ with $D\subset D’$ , we call $D’V$-directional to $D$ (written by $D’\supset vD$ ) ifthere exists a real $\lambda>0,$ $y\in D$ and $z\in D’$ such that(i) $d(z, y)=\rho_{n}(D’, D)$ and (ii) $z-y=\lambda v$ for some $v\in V$ .Definition 2. Let $V\subset U\subseteq \mathbb{R}^{n}$ . For $\tilde{s}\in \mathcal{F}(\mathbb{R}^{n}),\tilde{s}$ is called $V$ -directional if$\tilde{s}_{\alpha}\supset_{V^{\tilde{S}}\alpha}$’for $0\leq\alpha\leq\alpha’\leq 1$ .Theorem 4.2. Let $K$ be a closed convex cone of $\mathbb{R}^{n}$ with $K^{+}\cap(\mathbb{R}_{+}^{n})^{O}\neq\emptyset$ . Supposethat $\tilde{L}\subset \mathcal{F}(\mathbb{R}^{n})$ is determined by $K^{+}$ . Then, any monotone and bounded sequence$\{\tilde{s}_{l}\}_{\ell}\infty=1\subset\tilde{L}$ with $\tilde{s}_{l}(\ell\geq 1)$ is $V$ -directional for a finite set $V$ converges $w.r.t$ . $\rho_{n}$ .Corollary 4.1. Let $K=\{\lambda a|\lambda\geq 0\}$ for some $a\in\overline{\mathbb{R}_{+}^{n}}$ . Then, any $se\mathrm{q}\mathrm{u}ence\{\tilde{S}\ell\}\subset$$\mathcal{F}(\mathbb{R}^{n})$ satisfying that it is $\mathrm{m}$onotone and bounded $\mathrm{w}.\mathrm{r}.t$ . $\neg K\prec$ and $\tilde{s}_{f}(l\geq 1)$ is V-direction $\mathrm{a}l$ for a finite set $V$ converges $w.r.\mathrm{t}$ . $\rho_{n}$ .Corollary 4.2. A$n\mathrm{y}$ sequence in $\mathcal{F}_{r}(\mathbb{R}^{n})w\mathrm{i}$th $\mathrm{m}$onotonicity and boundedn$ess\mathrm{w}.r.\mathrm{t}$ .$\neg n\prec$ converges $w.r.t$ . $\rho_{n}$ .References[1] W.Congxin and C.Cong, “The supremum and infimum of the set of fuzzy numbersand its application,” J. Math. Anal. Appl. 210 (1997) 499-511.[2] N.Furukawa, $‘(\mathrm{p}_{\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}}\mathrm{t}\mathrm{r}\mathrm{i}_{\mathrm{C}}$ orders on fuzzy numbers and their roles in fuzzy optimiza-tion problems,” Optimization 40 (1997) 171-192.[3] M.Kurano, M.Yasuda, J.Nakagami and Y.Yoshida, ((Some pseudo order offuzzy setson $\mathbb{R}^{n},$” (Preprint).[4] D.Kuroiwa, T.Tanaka and T.X.D.Ha,$‘\zeta \mathrm{O}\mathrm{n}$ cone convexity of set-valued maps,” Non-linear $Analysis_{2}$ Theory $\not\in y$ Applications 30 (1997) 1487-1496.[5] V.Nov\’ak, Fuzzy Sets and Their Applications (Adam Hilder,Bristol-Boston, 1989).[6] J.Ram\’ik and J.\v{R}im\’anek, (‘Inequality relation between fuzzy numbers and its useinfuzzy $\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{i}_{\mathrm{Z}}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n},$ ” Fuzzy Sets and Systems 16 (1985) 123-138.[7] J.Stoer and C.Witzgall, Convexity and Optimization in FiniteDimensions $I$(Springer-Verlag, Berlin and New York, 1970).[8] Y.Yoshida, M.Yasuda, J.Nakagami and ,M, .Kurano, “A limit theorem in dynamicfuzzy systems with a monotone property, Fuzzy sets and Systems 94(1998) 109-119.[9] L.A.Zadeh, “Fuzzy sets,” Inform. and Control 8 (1965) 338-353.[10] K.Zhang and K.Hirota, $‘(\mathrm{O}\mathrm{n}$ fuzzy number lattice$(\tilde{R}, \neg\prec),$” Fuzzy sets and Systems92 (1997) 113-122.237

Sequences of Fuzzy Sets on $\mathbb{R}^n$ (Decision Theory in Mathematical Modelling)

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Sequences of Fuzzy Sets on $\mathbb{R}^n$ (Decision Theory in Mathematical Modelling)

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Sequences of Fuzzy Sets on $\mathbb{R}^n$ (Decision Theory in Mathematical Modelling)