Ordering of convex fuzzy sets — A brief survey and new results

M. Kurano(Chiba Univ.) M. Yasuda(Chiba Univ.) J. Nakagami(Chiba Univ.) and Y. Yoshida(Kitakyushu Univ.)

Abstract Concerning with the topics of a fuzzy max order, a brief survey on ordering of fuzzy numbers is presented in this article, and we will consider an extension to that of fuzzy sets. An extension of the fuzzy max order as a pseudo order is investigated and defined on a class of fuzzy sets on ℝ^{n}(n ≥ 1). This order is developed by using a non-empty closed convex cone and characterized by the projection into its dual cone. Especially a structure of the lattice can be illustrated with the class of rectangle-type fuzzy sets.

Keywords : Fuzzy set, fuzzy max order, pseudo order, n-dimensional fuzzy set, rectangletype fuzzy set.

1. Introduction

Fuzzy set theory has made applications in many fields of management science, operations research and statistics (cf.[20, 25, 31]), in which ordering or ranking fuzzy sets is a fundamental problem in fuzzy optimization or fuzzy decision making. Many methods of ordering fuzzy numbers have been proposed in the literature (see, for example, the survey paper, Bortolan and Degani[2]). Also, in multiple criteria decision making several procedures for ranking fuzzy multicriteria alternatives are investigated (for example, see [5, 29]). Each method has its own advantages and disadvantages, so that the ordering method should be chosen to be suitable for the particular problem. Among various ordering methods, a partial order on the set of fuzzy numbers, called the fuzzy max order, introduced by Ramík and R̆imánek[27] is very interesting in the concerns of pure mathematics because it is a natural extension of the order over real numbers and includes many theoretical and applicable potentials.

In this paper, concentrating on the fuzzy max order, we present a brief survey on ordering fuzzy sets in a real line ℝ, which motivates our new attempt of ordering highdimensional fuzzy sets. The fuzzy max order of fuzzy numbers is extended to a pseudo order on a class of fuzzy sets defined on an n-dimensional Euclidean space ℝ^{n}.

The pseudo order for fuzzy sets is defined by a non-empty closed convex cone K in ℝ^{n} and characterized by the projection into its dual cone K^{+}. Also, the structure of a lattice is discussed for the class of rectangle-type fuzzy sets. So, we can imagine the much wider application to the fuzzy optimization problem. Our idea of the motivation originates from a set-relation in ℝ^{n} given by Kuroiwa[22], Kuroiwa, Tanaka and Ha[23], in which various types of set-relations in ℝ^{n} are used in set-valued optimizations.

The outline for this paper is as follows: The next section contains some notations and basic concepts of fuzzy set theory referring to the text books (cf. [10, 26]); several methods of ordering fuzzy sets of a real line are surveyed concentrating on the fuzzy max order and its related topics in the third section; a pseudo order on the class of fuzzy sets on ℝ^{n} is originally introduced as an extension of the fuzzy max order. Its characterization and the structure of a lattice is considered in the fourth section.

2. Notations and basic concepts

In this section we describe notations and basic concepts of fuzzy set theory (cf. [10, 20, 26, 30, 35]).

Let ℝ be the set of all real numbers and ℝ^{n} an n-dimensional Euclidean space. We write a fuzzy set on ℝ^{n} by its membership function s̃ : ℝ^{n} → [0, 1] (cf. [26, 35]). The α-cut (α ∈ [0, 1]) of the fuzzy set s̃ on ℝ^{n} is defined as

s̃_{α} := {x ∈ ℝ^{n} | s̃(x) ≥ α} (α > 0) and s̃_{0} := cl{x ∈ ℝ^{n} | s̃(x) > 0},

where cl denotes the closure of the set. A fuzzy set s̃ is called convex if

s̃(λx + (1 − λ)y) ≥ s̃(x) ∧ s̃(y) x, y ∈ ℝ^{n}, λ ∈ [0, 1],

where a ∧ b := min{a, b}. Note that s̃ is convex if and only if the α-cut s̃_{α} is a convex set for all α ∈ [0, 1]. Let F(ℝ^{n}) be the set of all convex fuzzy sets whose membership functions s̃ : ℝ^{n} → [0, 1] are upper-semicontinuous and normal (sup_{x∈ℝ^{n}} s̃(x) = 1) and have a compact support. When the one-dimensional case n = 1, the fuzzy sets are called fuzzy numbers and F(ℝ) denotes the set of all fuzzy numbers.

Let C(ℝ^{n}) be the set of all compact convex subsets of ℝ^{n}, and C_{r}(ℝ^{n}) be the set of all rectangles in ℝ^{n}. For s̃ ∈ F(ℝ^{n}), we have s̃_{α} ∈ C(ℝ^{n}) (α ∈ [0, 1]). We write a rectangle in C_{r}(ℝ^{n}) by

[x, y] = [x_{1}, y_{1}] × [x_{2}, y_{2}] × ⋯ × [x_{n}, y_{n}]

for x = (x_{1}, x_{2}, ⋯, x_{n}), y = (y_{1}, y_{2}, ⋯, y_{n}) ∈ ℝ^{n} with x_{i} ≤ y_{i} (i = 1, 2, ⋯, n). For the case of n = 1, C(ℝ) = C_{r}(ℝ) and it denotes the set of all bounded closed intervals. When s̃ ∈ F(ℝ^{n}) satisfies s̃_{α} ∈ C_{r}(ℝ^{n}) for all α ∈ [0, 1], s̃ is called a rectangle-type. We denote by F_{r}(ℝ^{n}) the set of all rectangle-type fuzzy sets on ℝ^{n}. Obviously F_{r}(ℝ) = F(ℝ).

Here we give the extension principle introduced by Zadeh which provides a general method for fuzzification of non-fuzzy mathematical concepts.

The extension principle (cf. [10]): Let f be a map from ℝ^{n} to ℝ such that y = f(x_{1}, ⋯, x_{n}). It allows us to induce a map from n fuzzy sets s̃_{i} on ℝ to a fuzzy set s̃ on ℝ through f such that

s̃(y) := sup_{x_{1},··· ,x_{n}∈ℝ_{y=f(x_{1},··· ,xn)}}{min(s̃_{1}(x_{1}), ⋯, s̃_{n}(x_{n}))}

where s̃(y) := 0 if f^{−1}(y) = ϕ.

Applying the extension principle, the addition and the scalar multiplication on ℝ are extended to those on F(ℝ) as follows:

For s̃, r̃ ∈ F(ℝ) and λ ≥ 0,

(2.1)  (s̃+ r̃)(x) := sup_{x_{1},x_{2}∈ℝ_{x_{1}+x_{2}=x}}{s̃(x_{1}) ∧ r̃(x_{2})},

(2.2)  (λs̃)(x) := \{ \begin{matrix}s̃(x/λ) & if λ > 0 \\ I_{\{0\}}(x) & if λ = 0 \end{matrix} (x ∈ ℝ),

where I_{{·}}(·) is an indicator. By using set operations A + B := {x + y | x ∈ A, y ∈ B} and λA := {λx | x ∈ A} for any non-empty sets A, B ⊂ ℝ, the following holds immediately:

(2.3)  (s̃+ r̃)_{α} = s̃_{α} + r̃_{α} and (λs̃)_{α} = λs̃_{α} (α ∈ [0, 1]).

Also, for s̃, r̃ ∈ F(ℝ), m̃ax(s̃, r̃) and m̃in(s̃, r̃) are defined by

(2.4)  m̃ax(s̃, r̃)(y) := sup_{x_{1},x_{2}∈ℝ_{y=max(x_{1},x_{2})}}{s̃(x_{1}) ∧ r̃(x_{2})}

and

(2.5)  m̃in(s̃, r̃)(y) := sup_{x_{1},x_{2}∈ℝ_{y=min(x_{1},x_{2})}}{s̃(x_{1}) ∧ r̃(x_{2})}

respectively. The images of m̃ax(s̃, r̃) and m̃in(s̃, r̃) are illustrated as follows:

1

s̃  r̃

0

Fig.1: s̃ and r̃

1

m̃ax(s̃, r̃)
m̃in(s̃, r̃)

0

Fig.2: m̃ax(s̃, r̃) and m̃in(s̃, r̃)

We need a representative theorem (cf. [10, 26]) which is a basic tool for the fuzzy interval analysis.

The representative theorem:

(i) For any s̃ ∈ F(ℝ^{n}), s̃(x) = sup_{α∈[0,1]}{α ∧ 1s̃_{α}(x)}, x ∈ ℝ.

(ii) Conversely, for a family of subsets {D_{α} ∈ C(ℝ^{n}) | 0 ≤ α ≤ 1} with D_{α} ⊂ D_{α′}

for α′ ≤ α and ∩_{α′<α}D_{α′} = D_{α}, we set s̃(x) := sup_{α∈[0,1]}{α ∧ 1_{D_{α}}(x)}, x ∈ ℝ,

then s̃ belongs to F(ℝ^{n}) and satisfies s̃_{α} = D_{α}, α ∈ [0, 1].

3. A brief survey on ordering of fuzzy numbers

In this section, we give a brief survey on method for ordering fuzzy numbers which is mainly concerning to the fuzzy max order.

3.1. Fuzzy max order

The following binary relation ≼ has been formulated first by Ramík and R̆imánek[27]. Let s̃ and r̃ be two fuzzy numbers. Then s̃ ≼ r̃ if and only if sup s̃_{α} ≤ sup r̃_{α} and inf s̃_{α} ≤ inf r̃_{α} for each α ∈ [0, 1], where s̃_{α} and r̃_{α} are α-cuts of s̃ and r̃ respectively. Without any confusion we put s̃_{α} := [inf s̃_{α}, sup s̃_{α}] and r̃_{α} := [inf r̃_{α}, sup r̃_{α}]. Obviously the binary relation ≼ satisfies the axioms of a partial order relation on F(ℝ) and is called the fuzzy max order.

Some properties of the relation ≼ are investigated in Ramík and R̆imánek[27].

Proposition 1. Let s̃, r̃ be fuzzy numbers.

(i) The inequality s̃ ≼ r̃ if and only if there are m, n, t^{∗} ∈ ℝ with m ≤ t^{∗} ≤ n, s̃(m) = r̃(n) = 1 and s̃(t) ≥ r̃(t) for any t ≤ t^{∗} and s̃(t) ≤ r̃(t) for any t > t^{∗}.

(ii) The following three conditions (a) to (c) are equivalent:

(a) s̃ ≼ r̃, (b) m̃ax{s̃, r̃} = r̃, (c) m̃in{s̃, r̃} = s̃, where m̃in and m̃ax are defined in (2.4) and (2.5).

For the fuzzy max order on fuzzy numbers, Congxin and Cong[6] proved that the bounded set of fuzzy numbers must have supremum and infimum. The basic proof is as follows. For any sequence of fuzzy numbers \{s̃_{n}\}_{n=1}^{∞}, let \underline{D}_{α} := lim_{α′↑α} sup_{n≥1} inf s̃_{nα′} and

\overline{D}_{α} := lim_{α′↑α} sup_{n≥1} sup s̃_{nα′}, where s̃_{nα} is the α-cut of s̃_{n}. Then, the family of closed subsets

\{D_{α} := [\underline{D}_{α}, \overline{D}_{α}] | α ∈ [0, 1]\} satisfies condition (ii) of the representation theorem, so that s̃ defined by

s̃(x) := sup_{α∈[0,1]}{α ∧ 1_{D_{α}}(x)} (x ∈ ℝ)

belongs to F(ℝ) and s̃ = sup_{n≥1} s̃_{n}.

Similarly, the infimum of {s̃_{n}}_{n=1}^{∞} is constructed. These results derived the interesting mathematical fact that the continuous fuzzy-valued function on a closed interval has a maximum and minimum. Also, the structure of lattice for fuzzy numbers is discussed in Zhang and Hirota[37].

To be suitable for computations and treatments, a class of fuzzy numbers, called an L-R-fuzzy number, is introduced in many text books.

Let L, R : [0, ∞) → [0, 1] be two non-increasing and not constant functions with L(0) = R(0) = 1 and L(x_{0}) = R(x_{0}) = 0 for some x_{0} > 0. A fuzzy number s̃ is called an L-R-fuzzy number if there exist real numbers m, n(m ≤ n), α, β(α, β > 0) such that

s̃(x) = \{ \begin{matrix}L(\frac{m−x}{α}) & for x ≤ m, \\ 1 & for m ≤ x ≤ n, \\ R(\frac{x−n}{β}) & for n ≤ x. \end{matrix}

Given functions L, R with the properties in the above definition, the L-R-fuzzy number specified by m, n, α, β is denoted by an ordered tetradic (m, n, α, β)_{L-R}, which includes the triangular and trapezoidal fuzzy numbers. Then, the fuzzy max order on the set of L-R-fuzzy numbers is characterized by inequalities of the elements. (cf. [27]).

In particular, the symmetric fuzzy number s̃ = (m, m, α, α)_{L-L} is called an L-fuzzy number and denoted by (m, α)_{L}. Furukawa[14] extended the L-fuzzy number (m, α)_{L} with α > 0 to the case of α ∈ ℝ and proved that for αβ ≥ 0 the fuzzy max order (m, α)_{L} ≼ (n, β)_{L} if and only if x_{0}|α − β| ≤ n − m where x_{0} is the zero point of L. Moreover Furukawa[14] introduced the linear operations on the set of extended L-fuzzy numbers by

(m, α)_{L} ⊕ (n, β)_{L} = (m + n, α + β)_{L},
λ(m, α)_{L} = (λm, λα)_{L} for any scalar λ ∈ ℝ.

The fuzzy max order is proved to be adapted to the above operations. Also Furukawa [15] introduced a parametric order relation on L-fuzzy numbers which is an extension of the fuzzy max order and its total order relation. The fuzzy optimization problems related to the fuzzy max order are dealed with many authors; e.g., Furukawa[15], Kurano et al.[21], Yoshida[34] and others.

3.2. Other ordering methods

Besides the fuzzy max order, a large collection of methods for ordering fuzzy sets of linear line has been developed in the literature. A simple method for ordering fuzzy numbers consists of the definition of ordering or ranking function. Let f : F(ℝ) → ℝ, where a natural ranking relation ≤ on ℝ is defined. Then, using this ranking function, we can define the order relation ≼ on F(ℝ) as follows:

s̃ ≼ r̃ implies f(s̃) ≤ f(r̃) for s̃, r̃ ∈ F(ℝ).

If, for all s̃, r̃ ∈ F(ℝ), f(s̃ + r̃) = f(s̃) + f(r̃) and f(λs̃) = λf(s̃) for λ ≥ 0 are satisfied, it is called a linear ranking function. As a simple ranking function, we can construct a family of ranking procedures as follows:

f_{p}(s̃) := ∫_{0}^{1}(p_{α} inf s̃_{α} + q_{α} sup s̃_{α}) dα, where ∫_{0}^{1}(p_{α} + q_{α}) dα = 1.

Several variants of this family are discussed by the authors; see, e.g., Adamo[1], Campos, Bonzalez and Vila[4], Fortemps and Roubens[12] and Yager[33]. Gonzalez and Vila[17] defined the dominance relations by means of a ranking function evaluated into ℝ^{n}. For a ℝ^{n}-valued ranking function f : F(ℝ) → ℝ^{n}, the order relation ≼ on F(ℝ) is defined using an order relation ≼_{n} on ℝ^{n}(defined in the next section) as follows:

s̃ ≼ r̃ implies f(s̃) ≼_{n} f(r̃).

From flexibility of order relations on ℝ^{n}, this seems to be useful in the suitable optimization problem of fuzzy decision making.

Another approach to ordering fuzzy numbers is discussed by using the fuzzy ordering (cf. [36]). From the point of view of possibility theory, Dubois and Prade[8] defined four fuzzy relations on F(ℝ): Pos(s̃ ≼ r̃), Pos(s̃ ≺ r̃), Nes(s̃ ≼ r̃) and Nes(s̃ ≺ r̃).

4. On an extension to a pseudo order on F(ℝ^{n})

In this section, we extend the fuzzy max order on F(ℝ) to a pseudo order on F(ℝ^{n}).

4.1. A pseudo order on F(ℝ^{n})

We will review a vector ordering on ℝ^{n} by a non-empty convex cone K ⊂ ℝ^{n}. Using this K, we can define a pseudo order relation ≼_{K} on ℝ by x ≼_{K} y if and only if y − x ∈ K. Let ℝ_{+}^{n} be the subset of entrywise non-negative elements in ℝ^{n}. When K = ℝ_{+}^{n}, the order ≼_{K} will be denoted by ≼_{n} and x ≼_{n} y means that x_{i} ≤ y_{i} for all i = 1, 2, ⋯, n, where x = (x_{1}, x_{2}, ⋯, x_{n}) and y = (y_{1}, y_{2}, ⋯, y_{n}) ∈ ℝ^{n}.

First we introduce a binary relation on C(ℝ^{n}), by which a pseudo order on F(ℝ^{n}) is given. Henceforth we assume that the convex cone K ⊂ ℝ^{n} is given. We define a binary relation ≼_{K} on C(ℝ^{n}) by abuse of notation. For A, B ∈ C(ℝ^{n}), A ≼_{K} B means the following (C.a) and (C.b) (cf. [22, 23]):

(C.a) For any x ∈ A, there exists y ∈ B such that x ≼_{K} y.

(C.b) For any y ∈ B, there exists x ∈ A such that x ≼_{K} y.

Lemma 4.1. The binary relation ≼_{K} is a pseudo order on C(ℝ^{n}).

Proof. It is trivial that A ≼_{K} A for A ∈ C(ℝ^{n}). Let A, B, C ∈ C(ℝ^{n}) such that A ≼_{K} B and B ≼_{K} C. We will check A ≼_{K} C by two cases (C.a) and (C.b). Case(C.a): Since A ≼_{K} B and B ≼_{K} C, for any x ∈ A there exists y ∈ B such that x ≼_{K} y and there exists z ∈ C such that y ≼_{K} z. Since ≼_{K} is a pseudo order on ℝ^{n}, we have x ≼_{K} z.

Therefore it holds that for any x ∈ A there exists z ∈ C such that x ≼_{K} z. Case(C.b): Since A ≼_{K} B and B ≼_{K} C, for any z ∈ C there exists y ∈ B such that y ≼_{K} z and there exists x ∈ A such that x ≼_{K} y. Since ≼_{K} is a pseudo order on ℝ^{n}, we have x ≼_{K} z. Therefore it holds that for any z ∈ C there exists x ∈ A such that x ≼_{K} z. From the above (C.a) and (C.b), we obtain A ≼_{K} C. Thus the lemma holds. □

When K = ℝ_{+}^{n}, the binary relation ≼_{K} on C(ℝ^{n}) will be written simply by ≼_{n} and for [x, y], [x′, y′] ∈ C_{r}(ℝ^{n}), [x, y] ≼_{n} [x′, y′] means x ≼_{n} x′ and y ≼_{n} y′.

Next, we introduce a binary relation ≼_{K} on F(ℝ^{n}): Let s̃, r̃ ∈ F(ℝ^{n}). The relation s̃ ≼_{K} r̃ means the following (F.a) and (F.b):

(F.a) For any x ∈ ℝ^{n}, there exists y ∈ ℝ^{n} such that x ≼_{K} y and s̃(x) ≤ r̃(y).

(F.b) For any y ∈ ℝ^{n}, there exists x ∈ ℝ^{n} such that x ≼_{K} y and s̃(x) ≥ r̃(y).

Note that the notation ≼_{K} denotes the binary relation on ℝ^{n}, C(ℝ^{n}), F(ℝ^{n}) with some abuse of notation.

Lemma 4.2. The binary relation ≼_{K} is a pseudo order on F(ℝ^{n}).

Proof. It is trivial that s̃ ≼_{K} s̃ for s̃ ∈ F(ℝ^{n}). Let s̃, r̃, p̃ ∈ F(ℝ^{n}) such that s̃ ≼_{K} r̃ and r̃ ≼_{K} p̃. We will check s̃ ≼_{K} p̃ by following two cases (F.a) and (F.b). Case(F.a): Since s̃ ≼_{K} r̃ and r̃ ≼_{K} p̃, for any x ∈ ℝ^{n} there exists y ∈ ℝ^{n} such that x ≼_{K} y and s̃(x) ≤ r̃(y), and there exists z ∈ ℝ^{n} such that y ≼_{K} z and r̃(y) ≤ p̃(z). Since ≼_{K} is a pseudo-order on ℝ^{n}, we have x ≼_{K} z and s̃(x) ≤ p̃(z). Therefore it holds that for any x ∈ ℝ^{n} there exists z ∈ ℝ^{n} such that x ≼_{K} z and s̃(x) ≤ p̃(z). Case(F.b): Since s̃ ≼_{K} r̃ and r̃ ≼_{K} p̃, for any z ∈ ℝ^{n} there exists y ∈ ℝ^{n} such that y ≼_{K} z and r̃(y) ≤ p̃(z), and there exists x ∈ ℝ^{n} such that x ≼_{K} y and s̃(x) ≤ r̃(y). Since ≼_{K} is a pseudo-order on ℝ^{n}, we have x ≼_{K} z. Therefore it holds that for any z ∈ ℝ^{n} there exists x ∈ ℝ^{n} such that x ≼_{K} z and s̃(x) ≤ p̃(z). From the above (F.a) and (F.b), we obtain s̃ ≼_{K} p̃. Thus the lemma holds. □

The following lemma implies the correspondence between the pseudo order on F(ℝ^{n}) for fuzzy sets and the pseudo order on C(ℝ^{n}) for the α-cuts.

Lemma 4.3. Let s̃, r̃ ∈ F(ℝ^{n}). s̃ ≼_{K} r̃ on F(ℝ^{n}) if and only if s̃_{α} ≼_{K} r̃_{α} on C(ℝ^{n}) for all α ∈ (0, 1].

Proof. Let s̃, r̃ ∈ F(ℝ^{n}) and α ∈ (0, 1]. Suppose s̃ ≼_{K} r̃ on F(ℝ^{n}). Then, two cases (a) and (b) are considered. Case(a): Let x ∈ s̃_{α}. Since s̃ ≼_{K} r̃, there exists y ∈ ℝ^{n} such that x ≼_{K} y and α ≤ s̃(x) ≤ r̃(y). Namely y ∈ r̃_{α}. Case(b): Let y ∈ r̃_{α}. Since s̃ ≼_{K} r̃, there exists x ∈ ℝ^{n} such that x ≼_{K} y and s̃(x) ≥ r̃(y) ≥ α. Namely x ∈ s̃_{α}. Therefore we get s̃_{α} ≼_{K} r̃_{α} on C(ℝ^{n}) for all α ∈ (0, 1] from the above (a) and (b).

On the other hand, suppose s̃_{α} ≼_{K} r̃_{α} on C(ℝ^{n}) for all α ∈ (0, 1]. Then, two cases (a’) and (b’) are considered. Case(a’): Let x ∈ ℝ^{n}. Put α = s̃(x). If α = 0, then x ≼_{K} x and s̃(x) = 0 ≤ r̃(x). While, if α > 0, then x ∈ s̃_{α}. Since s̃_{α} ≼_{K} r̃_{α}, there exists y ∈ r̃_{α} such that x ≼_{K} y. And we have s̃(x) = α ≤ r̃(y). Case(b’): Let y ∈ ℝ^{n}. Put α = r̃(y). If α = 0, then x ≼_{K} x and s̃(x) ≥ 0 = r̃(y). While, if α > 0, then y ∈ r̃_{α}. Since s̃_{α} ≼_{K} r̃_{α}, there exists x ∈ s̃_{α} such that x ≼_{K} y. And we have s̃(x) ≥ α = r̃(y).

Therefore we get s̃ ≼_{K} r̃ on F(ℝ^{n}) from the above Case (a’) and (b’). Thus we obtain this lemma. □

For the case of K = ℝ_{+}, Lemma 4.3 says that the order relation ≼_{1} on F(ℝ) (That is, n = 1) is the fuzzy max order mentioned in Section 3.

Define the dual cone of a cone K by

K^{+} := {a ∈ ℝ^{n} | a · x ≥ 0 for all x ∈ K},

where x · y denotes the inner product on ℝ^{n} for x, y ∈ ℝ^{n}. For a subset A ⊂ ℝ^{n} and a ∈ ℝ^{n}, we define

(4.1)  a · A := {a · x | x ∈ A} (⊂ ℝ).

The equation (4.1) means that a · A is the projection of A on the parallel line with the vector a if a · a = 1. It is trivial that a · A ∈ C(ℝ) if A ∈ C(ℝ^{n}) and a ∈ ℝ^{n}.

Lemma 4.4. Let A, B ∈ C(ℝ^{n}). A ≼_{K} B on C(ℝ^{n}) if and only if a · A ≼_{1} a · B on C(ℝ) for all a ∈ K^{+}.

Proof. Suppose A ≼_{K} B on C(ℝ^{n}). Consider the two cases (a) and (b). Case(a): For any x ∈ A, there exists y ∈ B such that x ≼_{K} y. Then y − x ∈ K. If a ∈ K^{+}, then a · (y − x) ≥ 0 and i.e. a · x ≤ a · y. Case(b): For any y ∈ B, there exists x ∈ A such that x ≼_{K} y. Then y − x ∈ K. If a ∈ K^{+}, then a · (y − x) ≥ 0 and i.e. a · x ≤ a · y. From the above cases (a) and (b), we have that a · A ≼_{1} a · B.

On the other hand, to prove the inverse statement, we assume that A ≼_{K} B on C(ℝ^{n}) does not hold. Then we have the following two cases (i) and (ii). Case(i): There exists x ∈ A such that y − x ∉ K for all y ∈ B. Then B ∩ (x + K) = ∅. Since B and x + K are closed convex, by the separation theorem there exists a ∈ ℝ^{n} (a ≠ 0) such that a · y < a · x + a · z for all y ∈ B and all z ∈ K. Now, we suppose that there exists z ∈ K such that a · z < 0. Then λz ∈ K for all λ ≥ 0 since K is a cone, and so we have a · x + a · λz = a · x + λa · z → −∞ as λ → ∞. This contradicts a · y < a · x + a · z. Therefore we obtain a · z ≥ 0 for all z ∈ K, which implies a ∈ K^{+}. Especially taking z = 0 ∈ K, we get a · y < a · x for all y ∈ B. This contradicts a · A ≼_{1} a · B. Case(ii): There exists y ∈ B such that y − x ∉ K for all x ∈ A. Then we derive a contradiction in the similar way to the case (i). Therefore the inverse statement holds from the results of the above (i) and (ii). The proof of this lemma is completed. □

For a ∈ ℝ^{n} and s̃ ∈ F(ℝ^{n}), applying the representation theorem we define a fuzzy number a · s̃ ∈ F(ℝ) by

(4.2)  a · s̃(x) := sup_{α∈[0,1]}{α ∧ 1_{a·s̃_{α}}(x)}, x ∈ ℝ.

The following theorem gives the correspondence between the pseudo order ≼_{K} on F(ℝ^{n}) and the fuzzy max order ≼_{1} on F(ℝ).

Theorem 4.1. For s̃, r̃ ∈ F(ℝ^{n}), s̃ ≼_{K} r̃ if and only if a · s̃ ≼_{1} a · r̃ for all a ∈ K^{+}.

Proof. By (4.2) and the representative theorem, we have (a·s̃)_{α} = a·s̃_{α} for all α ∈ [0, 1]. On the other hand, from Lemmas 4.3 and 4.4, s̃ ≼_{K} r̃ if and only if a · s̃_{α} ≼_{1} a · r̃_{α} for all a ∈ K^{+}. Thus, noting the definition of the max order ≼_{1} on F(ℝ), Theorem 4.1 follows. □

For {s̃_{k}}_{k=1}^{∞} ⊂ F(ℝ^{n}) and s̃ ∈ F(ℝ^{n}), lim_{k→∞} s̃_{k} = s̃ means that sup_{α∈[0,1]} ρ_{n}(s̃_{kα}, s̃_{α}) → 0

(k → ∞), where s̃_{kα} is the α-cut of s̃_{k} and ρ_{n} is the Hausdorff metric on C(ℝ^{n}).

Lemma 4.5. Let {s̃_{k}}_{k=1}^{∞} ⊂ F(ℝ) and s̃ ∈ F(ℝ) such that s̃_{k} ≼_{1} s̃_{k+1} (k ≥ 1) and lim_{k→∞} s̃_{k} = s̃. Then we have s̃_{1} ≼_{1} s̃.

Proof. Trivial. □

Theorem 4.2. Let {s̃_{k}}_{k=1}^{∞} ⊂ F(ℝ^{n}) and s̃ ∈ F(ℝ^{n}) such that s̃_{k} ≼_{K} s̃_{k+1} (k ≥ 1) and lim_{k→∞} s̃_{k} = s̃. Then we have s̃_{1} ≼_{K} s̃.

Proof. From Theorem 4.1, for all a ∈ K^{+} it holds that a · s̃_{k} ≼_{1} a · s̃_{k+1} (k ≥ 1). Also, since (a · s̃_{k})_{α} = a · s̃_{kα} from (4.2) and ρ_{1}(a · s̃_{kα}, a · s̃_{α}) ≤ ∥a∥ρ_{n}(s̃_{kα}, s̃_{α}) for all k ≥ 1, we get lim_{k→∞} a·s̃_{k} = a·s̃ where ∥a∥ is a norm of a. By Lemma 4.5, it holds that a·s̃_{1} ≼_{1} a·s̃ for all a ∈ K^{+}. From Theorem 4.1, we have s̃_{1} ≼_{K} s̃. □

Remark. Let consider a continuous map g : [0, 1] → F(ℝ^{n}). A point x_{0} is said to be efficient if x_{0} ∈ [0, 1] and g(x_{0}) ≼_{K} g(x) for some x ∈ [0, 1] implies g(x) = g(x_{0}). Then, by applying the same idea as in Lemma 3.2 of Furukawa[13], we observe that there exists at least one efficient point in [0, 1]. In fact, considering, if necessary, a partial order ≼_{K} on the class of the quotient sets with respect to the equivalence relation ∼_{K} defined by s̃ ∼_{K} r̃ if and only if s̃ ≼_{K} r̃ and r̃ ≼_{K} s̃, we can assume that ≼_{K} is a partial order on F(ℝ^{n}). By Theorem 4.2 and the continuity of g, {g(x) | x ∈ [0, 1]} can be proved to be an inductively ordered set. So, by Zorn’s lemma {g(x) | x ∈ [0, 1]} has an efficient element.

4.2. Further results

In this section, we shall investigate the above pseudo order ≼_{K} on F_{r}(ℝ^{n}) for a polyhedral cone K with K^{+} ⊂ ℝ^{n}. To this end, we need the following lemma.

Lemma 4.6. Let a, b ∈ ℝ_{+}^{n} and A ∈ C_{r}(ℝ^{n}). Then for any scalars λ_{1}, λ_{2} ≥ 0, it holds

(4.3)  (λ_{1}a + λ_{2}b) · A = λ_{1}(a · A) + λ_{2}(b · A),

where the arithmetic in (4.3) is defined in (4.1).

Proof. Let λ_{1}a · x + λ_{2}b · y ∈ λ_{1}(a · A) + λ_{2}(b · B) with x, y ∈ A. It suffices to show that λ_{1}a · x + λ_{2}b · y ∈ (λ_{1}a + λ_{2}b) · A. Define z = (z_{1}, z_{2}, ⋯, z_{n}) by

z_{i} := \{ \begin{matrix}(λ_{1}a_{i}x_{i} + λ_{2}b_{i}y_{i})/(λ_{1}a_{i} + λ_{2}b_{i}) & if (λ_{1}a_{i} + λ_{2}b_{i}) > 0 \\ x_{i} & if (λ_{1}a_{i} + λ_{2}b_{i}) = 0 \end{matrix} (i = 1, 2, ⋯, n).

Then, clearly (λ_{1}a + λ_{2}b) · z = λ_{1}a · x + λ_{2}b · y. Since A ∈ C_{r}(ℝ^{n}), z ∈ A, so that λ_{1}a · x + λ_{2}b · y ∈ (λ_{1}a + λ_{2}b) · A. □

Henceforth, we assume that K is a polyhedral convex cone with K^{+} ⊂ ℝ_{+}^{n}, i.e., there exist vectors b^{i} ∈ ℝ_{+}^{n}(i = 1, 2, ⋯, m) such that

K = {x ∈ ℝ^{n} | b^{i} · x ≥ 0 for all i = 1, 2, ⋯, m}.

Then, it is well-known (cf. [30]) that K^{+} is expressed as

K^{+} = {x ∈ ℝ^{n} | x = ∑_{i=1}^{m} λ_{i}b^{i}, λ_{i} ≥ 0, i = 1, 2, ⋯, m}.

The above dual cone K^{+} is denoted simply by

K^{+} = cone convex{b^{1}, b^{2}, ⋯, b^{m}}.

The pseudo order ≼_{K} on C_{r}(ℝ^{n}) is characterized by the pseudo order ≼_{1} on C_{r}(ℝ).

Corollary 4.1. Let K^{+} = cone convex{b^{1}, b^{2}, ⋯, b^{m}} with b^{i} ∈ ℝ_{+}^{n}. Then, for A, B ∈ C_{r}(ℝ^{n}), A ≼_{K} B if and only if b^{i} · A ≼_{1} b^{i} · B for all i = 1, 2, ⋯, m.

Proof. We assume that b^{i} · A ≼_{1} b^{i} · B for all i = 1, 2, ⋯, m. For any a ∈ K^{+}, there exist λ_{i} ≥ 0 with a = ∑_{i=1}^{m} λ_{i}b^{i}. From Lemma 4.1, we have:

a · A = ∑_{i=1}^{m} λ_{i}(b^{i} · A) ≼_{1} ∑_{i=1}^{m} λ_{i}(b^{i} · B) = a · B.

Thus, by Lemma 4.4, A ≼_{K} B follows. By applying Lemma 4.4 again, the ‘only if’ part of Corollary holds. □

Lemma 4.7. Let a, b ∈ ℝ_{+}^{n} and s̃ ∈ F_{r}(ℝ^{n}). Then, for any λ_{1}, λ_{2} ≥ 0,

(4.4)  (λ_{1}a + λ_{2}b) · s̃ = λ_{1}(a · s̃) + λ_{2}(b · s̃),

where the arithmetic in (4.4) is given in (2.1), (2.2) and (4.2).

Proof. For any α ∈ [0, 1], it follows from the definition and Lemma 4.6 that

((λ_{1}a + λ_{2}b) · s̃)_{α} = (λ_{1}a + λ_{2}b) · s̃_{α} = λ_{1}(a · s̃_{α}) + λ_{2}(b · s̃_{α})
= λ_{1}(a · s̃)_{α} + λ_{2}(b · s̃)_{α} = (λ_{1}(a · s̃) + λ_{2}(b · s̃))_{α}.

The last equality follows from (2.3). The above shows that (4.4) holds. □

The main result in this section is given in the following.

Theorem 4.3. Let K^{+} = cone convex{b^{1}, b^{2}, ⋯, b^{m}} with b^{i} ∈ ℝ^{n}. Then, for s̃, r̃ ∈ F_{r}(ℝ^{n}),

s̃ ≼_{K} r̃ if and only if b^{i} · s̃ ≼_{1} b^{i} · r̃ for i = 1, 2, ⋯, m.

Proof. It suffices to prove the ‘if’ part of Theorem 4.3. For any a ∈ K^{+}, there exist λ_{i} ≥ 0 with a = ∑_{i=1}^{m} λ_{i}b^{i}. Applying Lemma 4.7, we have

a · s̃ = ∑_{i=1}^{m} λ_{i}(b^{i} · s̃) ≼_{1} ∑_{i=1}^{m} λ_{i}(b^{i} · r̃) = a · r̃,

From Theorem 4.1, s̃ ≼_{K} r̃ follows. □

m̃ax(s̃, r̃)
r̃

s̃

Fig 3: m̃ax(s̃, r̃)

Zhang and Hirota[37] described the structure of the fuzzy number lattice (F_{r}(ℝ), ≼_{1}). When K = ℝ^{n}, K^{+} = ℝ^{n} and K^{+} = cone convex{e^{1}, e^{2}, ⋯, e^{n}}. So that, by Theorem 4.3, we see that for s̃, r̃ ∈ F_{r}(ℝ^{n}), s̃ ≼_{n} r̃ means e^{i} · s̃ ≼_{1} e^{i} · r̃ for all i = 1, 2, ⋯, n. Therefore, by applying the same method as Zhang and Hirota[37], we can describe the structure of the fuzzy set lattice (F_{r}(ℝ^{n}), ≼_{n}). Figure 3 illustrates m̃ax(s̃, r̃) for s̃, r̃ ∈ F_{r}(ℝ^{2}).

Acknowledgement

Authors would express their thanks to the referees for several comments and corrections which led to a considerable improvement of the original version. Also, they are grateful to Associate Professor H. Kuwano of Kanazawa Gakuin University for useful discussions related to this paper.

References

[1] Adamo, J. M., Fuzzy decision trees, Fuzzy Sets and Systems, 4 (1980), 207-219.

[2] Bortolan, G. and Degani, R. A., A review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems, 15 (1985), 1-20.

[3] Campos, L. and Verdegay, J. L., Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems, 32 (1989), 1-11.

[4] Campos, L, Gonzalez, A. and Vila, M.-A., On the use of the ranking function approach to solve fuzzy matrix game in a direct way, Fuzzy Sets and Systems, 49 (1992), 193-203.

[5] Choobineh, F. and Li, H., Ranking fuzzy multicriteria alternatives with respect to a decision maker’s fuzzy goal, Information Sci., 72 (1993), 143-155.

[6] Congxin, W. and Cong, C., The supremum and infimum of the set of fuzzy numbers and its application, J. Math. and Appl., 210 (1997), 499-511.

[7] Dubois, D., Systems of linear fuzzy constraints, Fuzzy Sets and Systems, 3 (1980), 37-48.

[8] Dubois, D. and Prade, H., Ranking fuzzy numbers in the setting of possibility theory, Information Sci., 30(1983), 183-224.

[9] Dubois, D. and Prade, H., Operations on fuzzy numbers, Int. J. Systems Sci., 9 (1978), 612-623.

[10] Dubois, D. and Prade, H., Fuzzy Sets and Systems: Theory and Applications, Academic Press, 1980.

[11] Fishburn, P. C., Nonlinear Preference and Utility Theory, John Hopkins Univ. Press, 1988.

[12] Fortempts, P. and Roubens, M., Ranking and defuzzification methods based on area compensation, Fuzzy Sets and Systems, 82 (1996), 319-330.

[13] Furukawa, N., Characterization of optimal policies in vector-valued Markovian decision processes, Math. Oper. Research, 5 (1980) , 271-279.

[14] Furukawa, N., An order relation and fundamental operations on fuzzy numbers and their applications to fuzzy linear programming, Bulletin Infor. Cybernetics, 25 (1993), 161-179.

[15] Furukawa, N., A parametric total order on fuzzy numbers and a fuzzy shortest route problem, Optimization 30 (1994), 367-377.

[16] Gen, M. and Cheng, R., Genetic Algorithms and Engineering Design, John Wiley & Sons, 1996.

[17] Gonzalez, A. and Vila, M.-A., Dominance relations on fuzzy numbers, Information Sci., 64 (1992), 1-16.

[18] Grzegorzewski, P., Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems 97 (1998), 83-94.

[19] Iamil, H. N. and Morsi, N. N., Fuzzy rank of a fuzzy matrix, Fuzzy Sets and Systems 41 (1991), 243-249.

[20] Kaufmann, A. and Gupta, M. M., Fuzzy mathematical Models in Engineering and Management Science, North-Holland 1988.

[21] Kurano, M.,Yasuda, M., Nakagami, J. and Yoshida, Y., Markov-type fuzzy decision processes with a discounted reward on a closed interval, European J. of Oper. Res., 92 (1996), 649-662.

[22] Kuroiwa, D., The natural criteria in set-valued optimization, RIMS kokyuroku, 1031 (1998), 85-90.

[23] Kuroiwa, D., Tanaka, T. and Ha, T. X. D., On cone convexity of set-valued maps, Nonlinear Analysis Theory & Applications, 30 (1997), 1487-1496.

[24] Liou, T.S. and Wang, M. J., Ranking fuzzy numbers with integral value, Fuzzy Sets and Systems 50 (1992), 247-255 .

[25] Li, H. X. and Yen, V. C., Fuzzy Sets and Fuzzy Decision-making, CRC Press, 1995.

[26] Novák, V., Fuzzy Sets and their Applications, Adam Hilder, Bristol-Boston, 1989.

[27] Ramík, J. and R̆imánek, J., Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy Sets and Systems 16 (1985),123-138.

[28] Ramík, J. and Rommelfanger, H., A single- and a multi-valued order on fuzzy numbers and its use in linear programming with fuzzy coefficients, Fuzzy Sets and Systems 57 (1993), 203-208.

[29] Sakawa, M., Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, 1993.

[30] Stoer, J. and Witzgall, C., Convexity and Optimization in Finite Dimensions I, Springer-Verlag, Berlin and New York, 1970.

[31] Stowinski, R.(editor), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, 1998.

[32] Tanaka, H., Ichihashi, H. and Asai, K., A formulation of fuzzy linear Programming problem based on comparison of fuzzy numbers, Control and Cybernetics 3 (1984),185-194.

[33] Yager, R. R., A procedure for ordering fuzzy subsets of the unit interval, Information. Sci., 24 (1981), 143-161.

[34] Yoshida, Y., A time-average fuzzy reward criterion in fuzzy decision processes, Information Sci., 110 (1998),103-112.

[35] Zadeh, L. A., Fuzzy sets, Inform. and Control, 8 (1965), 338-353.

[36] Zadeh, L. A., Similarity relations and fuzzy ordering, Information Sci., 3 (1971), 177-206.

[37] Zhang, K. and Hirota, K., On fuzzy number lattice (R, ≼), Fuzzy Sets and Systems 92 (1997), 113-122.

Masami Kurano

Department of Mathematics

Faculty of Education Chiba University Yayoi-cho, Inage-ku Chiba 263-8522, Japan