A Zadeh ′ s max-min composition operator for 3-dimensional triangular fuzzy number

Abstract

We generalized triangular fuzzy numbers from $ℝ$ to $ℝ^{2}$ . By defining parametric operations between two α-cuts, which are regions, we obtained parametric operations for two triangular fuzzy numbers defined on $ℝ^{2}$ . We also generalized triangular fuzzy numbers from $ℝ^{2}$ to $ℝ^{3}$ . By defining parametric operations between two α-cuts, which are subsets of $ℝ^{3}$ , we derived parametric operations for two triangular fuzzy numbers defined on $ℝ^{3}$ . For the calculation of Zadeh’s principle operators, the definition of parametric operations between two α-cuts, which are subsets of $ℝ^{3}$ , is critical.

Keywords

Zadeh’s max-min composition operator 3-dimensional triangular fuzzy number 47N99

1 Introduction

Many results exist for Zadeh’s principle operator. The results for Zadeh’s principle operator or Zadeh’s max-min compositions have been utilized in principle operators for two fuzzy sets, fuzzy logics, and fuzzy engineering. We have previously proven the operators for various types of fuzzy sets (see [6, 11]). We generalized the results on 1-dimensional fuzzy set to 2-dimensional fuzzy set in [4 , 10]. We also generalized triangular fuzzy numbers from $ℝ$ to $ℝ^{2}$ . By defining parametric operations between two region valued α-cuts, we obtained parametric operations for two triangular fuzzy numbers defined on $ℝ^{2}$ in [2]. We also demonstrated that 2-dimensional Zadeh’s max-min operator constitutes the generalization of 1-dimensional Zadeh’s max-min operator in [5]. In this paper, we generalized triangular fuzzy numbers from $ℝ^{2}$ to $ℝ^{3}$ . By defining parametric operations between two α-cuts, which are subsets of $ℝ^{3}$ , we determined parametric operations for two triangular fuzzy numbers defined on $ℝ^{3}$ . For the calculation of Zadeh’s principle operators, the definition of parametric operations between two α-cuts, which are subsets of $ℝ^{3}$ , is essential. This result can be utilized in the proof that the 3-dimensional case is the generalization of the 2-dimensional case.

2 Preliminaries

We define α-cut and α-set of the fuzzy set A on $ℝ$ with the membership function μ_A (x).

Definition 2.1. An α-cut of the fuzzy number A is defined by $A_{α} = {x \in ℝ ∣ μ_{A} (x) \geq α}$ if α ∈ (0, 1] and $A_{0} = cl {x \in ℝ ∣ μ_{A} (x) > α}$ . For α ∈ (0, 1), the set A^α = {x ∈ X ∣ μ_A (x) = α} is said to be the α-set of the fuzzy set A, and A⁰ is the boundary of ${x \in ℝ ∣ μ_{A} (x) > α}$ and A¹ = A₁.

Following Zadeh, Dubois, and Prade, the extension principle is defined as follows:

Definition 2.2. [13] The extended addition A (+) B, extended subtraction A (-) B, extended multiplication A (·) B, and extended division A (/) B are fuzzy sets with membership functions as follows. For all x ∈ A and y ∈ B,

$μ_{A (*) B} (z) = sup_{z = x * y} min {μ_{A} (x), μ_{B} (y)}, * = +, -, \cdot, /$

We defined the parametric operations for two fuzzy numbers defined on $ℝ$ , and showed that the results for parametric operations are the same as those for extended operations in [2]. For this, we proved that, for all fuzzy numbers A and all α ∈ [0, 1], there exists a piecewise continuous function f_α (t) defined on [0, 1], such that A_α = {f_α (t) |t ∈ [0, 1]}. If A is continuous, then the corresponding function f_α (t) is also continuous. The corresponding function f_α (t) is stated to be the parametric α-function of A. The parametric α-function of A is denoted by f_α (t) or f_A (t).

Definition 2.3. [2] Let A and B be two continuous fuzzy numbers defined on $ℝ$ and f_A (t) , f_B (t) be the parametric α-functions of A and B, respectively. The parametric addition, parametric subtraction, parametric multiplication, and parametric division are fuzzy numbers that have their α-cuts as follows:

(1) parametric addition A (+) _pB: (A (+) _pB) _α = {f_A (t) + f_B (t) | t ∈ [0, 1]}

(2) parametric subtraction A (-) _pB: (A (-) _pB) _α = {f_A (t) - f_B (1 - t) | t ∈ [0, 1]}

(3) parametric multiplication A (·) _pB: (A (·) _pB) _α = {f_A (t) · f_B (t) | t ∈ [0, 1]}

(4) parametric division A (/) _pB: (A (/) _pB) _α = {f_A (t)/f_B (1 - t) | t ∈ [0, 1]}

Theorem 2.4. [2] Let A and B be two continuous fuzzy numbers defined on $ℝ$ . Then, we have A (+) _pB = A (+) B, A (-) _pB = A (-) B, A (·) _pB = A (·) B, and A (/) _pB = A (/) B.

Remark 2.5. Zadeh defined max-min operators in different ways for positive and negative fuzzy numbers. When defined using parametric operators as above, it can be defined in one way without distinguishing between positive and negative cases, and the result is the same as that calculated by Zadeh’s positive case.

We defined the 2-dimensional triangular fuzzy numbers on $ℝ^{2}$ as a generalization of triangular fuzzy numbers on $ℝ$ . We then defined the parametric operations between two 2-dimensional triangular fuzzy numbers. For that, we have to calculate operations between α-cuts in $ℝ$ . The α-cuts are intervals in $ℝ$ , but in $ℝ^{2}$ the α-cuts are regions, which makes the existing method of calculations between α-cuts unusable. We interpret the existing method from a different perspective, and apply the method to the region valued α-cuts on $ℝ^{2}$ .

Definition 2.6. A fuzzy set A with a membership function

$μ_{A} (x, y) = {\begin{matrix} 1 - \sqrt{\frac{(x - x_{1})^{2}}{a^{2}} + \frac{(y - y_{1})^{2}}{b^{2}}}, & b^{2} (x - x_{1})^{2} + a^{2} (y - y_{1})^{2} \leq a^{2} b^{2}, \\ 0, & otherwise, \end{matrix}$

where a, b > 0 is called the 2-dimensional triangular fuzzy number and denoted by (a, x₁, b, y₁) ².

Note that μ_A (x, y) is a cone. The intersections of μ_A (x, y) and the horizontal planes z = α (0 < α < 1) are ellipses. Moreover, the intersections of μ_A (x, y) and the vertical planes $y - y_{1} = k (x - x_{1}) (k \in ℝ)$ are symmetric triangular fuzzy numbers in those planes. If a = b, ellipses become circles. The α-cut A_α of a 2-dimensional triangular fuzzy number A = (a, x₁, b, y₁) ² is an interior of ellipse in an xy-plane, including the boundary

$A_{α} = {(x, y) \in ℝ^{2} | b^{2} (x - x_{1})^{2} + a^{2} (y - y_{1})^{2} \leq a^{2} b^{2} (1 - α)^{2}} = {(x, y) \in ℝ^{2} | (\frac{x - x_{1}}{a (1 - α)})^{2} + (\frac{y - y_{1}}{b (1 - α)})^{2} \leq 1} .$

Definition 2.7. A 2-dimensional fuzzy number A defined on $ℝ^{2}$ is called a convex fuzzy number if, for all α ∈ (0, 1), the α-cuts

$A_{α} = {(x, y) \in ℝ^{2} | μ_{A} (x, y) \geq α}$

are convex subsets in $ℝ^{2}$ .

Theorem 2.8. [2] Let A be a continuous convex fuzzy number defined on $ℝ^{2}$ , and $A^{α} = {(x, y) \in ℝ^{2} | μ_{A} (x, y) = α}$ be the α-set of A. Then, for all α ∈ (0, 1), there exist continuous functions $f_{1}^{α} (t)$ and $f_{2}^{α} (t)$ defined on [0, 2π] such that $A^{α} = {(f_{1}^{α} (t), f_{2}^{α} (t)) \in ℝ^{2} | 0 \leq t \leq 2 π} .$

If A is a continuous convex fuzzy number defined on $ℝ^{2}$ , then the α-set A^α is a closed circular convex subset in $ℝ^{2}$ .

Definition 2.9. Let A and B be convex fuzzy numbers defined on $ℝ^{2}$ , and

$A^{α} = {(f_{1}^{α} (t), f_{2}^{α} (t)) \in ℝ^{2} | 0 \leq t \leq 2 π}, B^{α} = {(g_{1}^{α} (t), g_{2}^{α} (t)) \in ℝ^{2} | 0 \leq t \leq 2 π}$

be the α-sets of A and B, respectively. For α ∈ (0, 1), we define that the parametric addition, parametric subtraction, parametric multiplication, and parametric division of two fuzzy numbers A and B are fuzzy numbers that have their α-sets as follows:

(1) parametric addition A (+) _pB:

$\begin{matrix} (A (+)_{p} B)^{α} = {(f_{1}^{α} (t) + g_{1}^{α} (t), f_{2}^{α} (t) + g_{2}^{α} (t)) \\ \in ℝ^{2} | 0 \leq t \leq 2 π} \end{matrix}$

(2) parametric subtraction A (-) _pB:

$(A (-)_{p} B)^{α} = {(x_{α} (t), y_{α} (t)) \in ℝ^{2} | 0 \leq t \leq 2 π},$

where

$x_{α} (t) = {\begin{matrix} f_{1}^{α} (t) - g_{1}^{α} (t + π), & if 0 \leq t \leq π \\ f_{1}^{α} (t) - g_{1}^{α} (t - π), & if π \leq t \leq 2 π \end{matrix}$

and

$y_{α} (t) = {\begin{matrix} f_{2}^{α} (t) - g_{2}^{α} (t + π), & if 0 \leq t \leq π \\ f_{2}^{α} (t) - g_{2}^{α} (t - π), & if π \leq t \leq 2 π \end{matrix}$

(3) parametric multiplication A (·) _pB:

$\begin{matrix} (A (\cdot)_{p} B)^{α} = {(f_{1}^{α} (t) \cdot g_{1}^{α} (t), f_{2}^{α} (t) \cdot g_{2}^{α} (t)) \\ \in ℝ^{2} | 0 \leq t \leq 2 π} \end{matrix}$

(4) parametric division A (/) _pB: $(A (/)_{p} B)^{α} = {(x_{α} (t), y_{α} (t)) \in ℝ^{2} | 0 \leq t \leq 2 π},$

where

$\begin{matrix} x_{α} (t) = \frac{f_{1}^{α} (t)}{g_{1}^{α} (t + π)} (0 \leq t \leq π), \\ x_{α} (t) = \frac{f_{1}^{α} (t)}{g_{1}^{α} (t - π)} (π \leq t \leq 2 π) \end{matrix}$

and

$\begin{matrix} y_{α} (t) = \frac{f_{2}^{α} (t)}{g_{2}^{α} (t + π)} (0 \leq t \leq π), \\ y_{α} (t) = \frac{f_{2}^{α} (t)}{g_{2}^{α} (t - π)} (π \leq t \leq 2 π) \end{matrix}$

For α = 0 and α = 1, $(A (*)_{p} B)^{0} = lim_{α \to 0^{+}} (A (*)_{p} B)^{α}$ and $(A (*)_{p} B)^{1} = lim_{α \to 1^{-}} (A (*)_{p} B)^{α}$ , where * = + , - , · , /.

Theorem 2.10. [2] Let A = (a₁, x₁, b₁, y₁) ² and B = (a₂, x₂, b₂, y₂) ² be two 2-dimensional triangular fuzzy numbers. We then have the following:

(1) A (+) _pB = (a₁ + a₂, x₁ + x₂, b₁ + b₂, y₁ + y₂) ²

(2) A (-) _pB = (a₁ + a₂, x₁ - x₂, b₁ + b₂, y₁ - y₂ ) ²

(3) (A (·) _pB) ^α = {(x_α (t) , y_α (t)) | 0 ≤ t ≤ 2π} , where $\begin{matrix} x_{α} (t) = x_{1} x_{2} + (x_{1} a_{2} + x_{2} a_{1}) (1 - α) cos t \\ + a_{1} a_{2} (1 - α)^{2} {cos}^{2} t \end{matrix}$

and $\begin{matrix} y_{α} (t) = y_{1} y_{2} + (y_{1} b_{2} + y_{2} b_{1}) (1 - α) sin t \\ + b_{1} b_{2} (1 - α)^{2} {sin}^{2} t . \end{matrix}$

(4) (A (/) _pB) ^α = {(x_α (t) , y_α (t)) | 0 ≤ t ≤ 2π} , where $\begin{matrix} x_{α} (t) = \frac{x_{1} + a_{1} (1 - α) cos t}{x_{2} - a_{2} (1 - α) cos t} and \\ y_{α} (t) = \frac{y_{1} + b_{1} (1 - α) sin t}{y_{2} - b_{2} (1 - α) sin t} . \end{matrix}$

Therefore, A (+) _pB and A (-) _pB become 2-dimensional triangular fuzzy numbers, but A (·) _pB and A (/) _pB are not 2-dimensional triangular fuzzy numbers.

Remark 2.11. The 2-dimensional case is an extended concept of the 1-dimensional case. As in 1-dimension, fuzzy numbers in 2-dimension are defined in all xy-planes without distinguishing positive and negative regions. Cutting a 2-dimensional fuzzy number at its vertices into a plane parallel to the xz-plane or the yz-plane produces a 1-dimensional fuzzy number on the cut plane. As a consequence, if 2-dimensional results are cut into planes and reduced to one dimension, they are consistent with the 1-dimensional results. For the sake of understanding, cuts were made in a plane parallel to the xz-plane or yz-plane, but cutting at its vertex into any vertical plane is consistent with the 1-dimensional results. The above results are proven in [5], and a corresponding graph is presented below for clear illustration.

3 3-dimensional triangular fuzzy numbers

In this section, we define the 3-dimensional triangular fuzzy numbers on $ℝ^{3}$ as a generalization of triangular fuzzy numbers on $ℝ^{2}$ . We then want to define parametric operations between two 3-dimensional triangular fuzzy numbers. To achieve this, we have to calculate operations between α-cuts in $ℝ^{3}$ . The α-cuts are regions in $ℝ^{2}$ , but in $ℝ^{3}$ the α-cuts are some subsets of $ℝ^{3}$ , which makes the existing method of calculations between α-cuts inapplicable. We interpret the existing method from a different perspective, and apply the method to the subset valued α-cuts on $ℝ^{3}$ .

Definition 3.1. A fuzzy set A with a membership function

$\begin{matrix} μ_{A} (x, y, z) \\ = {\begin{matrix} 1 - \sqrt{\frac{(x - x_{1})^{2}}{a^{2}} + \frac{(y - y_{1})^{2}}{b^{2}} + \frac{(z - z_{1})^{2}}{c^{2}}}, if b^{2} c^{2} (x - x_{1})^{2} \\ + c^{2} a^{2} (y - y_{1})^{2} + a^{2} b^{2} (z - z_{1})^{2} \leq a^{2} b^{2} c^{2}, \\ 0, otherwise, \end{matrix} \end{matrix}$

where a, b, c > 0 is called the 3-dimensional triangular fuzzy number and denoted by (a, x₁, b, y₁, c, z₁) ³.

Note that μ_A (x, y) is a cone in $ℝ^{2}$ , but it is not possible to know the shape of μ_A (x, y, z) in $ℝ^{3}$ . The α-cut A_α of a 3-dimensional triangular fuzzy number A = (a, x₁, b, y₁, c, z₁) ³ is the following set

$A_{α} = {(x, y, z) \in ℝ^{3} | \frac{(x - x_{1})^{2}}{a^{2}} + \frac{(y - y_{1})^{2}}{b^{2}} + \frac{(z - z_{1})^{2}}{c^{2}} \leq (1 - α)^{2}} = {(x, y, z) \in ℝ^{3} | (\frac{x - x_{1}}{a (1 - α)})^{2} + (\frac{y - y_{1}}{b (1 - α)})^{2} + (\frac{z - z_{1}}{c (1 - α)})^{2} \leq 1} .$

Definition 3.2. A 3-dimensional fuzzy number A defined on $ℝ^{3}$ is called a convex fuzzy number if for all α ∈ (0, 1), the α-cuts

$A_{α} = {(x, y, z) \in ℝ^{3} | μ_{A} (x, y, z) \geq α}$

are convex subsets in $ℝ^{3}$ .

Theorem 3.3. Let A be a continuous convex fuzzy number defined on $ℝ^{3}$ and $A^{α} = {(x, y, z) \in ℝ^{3} | μ_{A} (x, y, z) = α}$ be the α-set of A. Then, for all α ∈ (0, 1), there exist continuous functions $f_{1}^{α} (s), f_{2}^{α} (s, t)$ , and $f_{3}^{α} (s, t) (0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2})$ , such that $\begin{matrix} A^{α} = {(f_{1}^{α} (s), f_{2}^{α} (s, t), f_{3}^{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, \\ - \frac{π}{2} \leq t \leq \frac{π}{2}} . \end{matrix}$

Proof. Let α ∈ (0, 1) be fixed. Since A is a convex fuzzy number defined on $ℝ^{3}$ , the α-cut A_α is a convex subset in $ℝ^{3}$ . Therefore, the set $A^{α} = {(x, y, z) \in ℝ^{3} | μ_{A} (x, y, z) = α}$

is a subset in $ℝ^{3}$ . Let $A_{2}^{α} = {(x, y) \in ℝ^{2} | (x, y, z) \in A_{α}}$ , and $\bar{A_{2}^{α}}, \bar{A^{α}}$ are the boundaries of $A_{2}^{α}$ and A^α, respectively. The upper surface of A^α is the graph of a continuous concave function h₁ (x, y), and the lower surface of A^α is also the graph of a continuous convex function h₂ (x, y) defined on $A_{2}^{α}$ . Let

$l = inf {x | (x, y) \in \bar{A_{2}^{α}}} and m = sup {x | (x, y) \in \bar{A_{2}^{α}}} .$

The upper boundary of $\bar{A_{2}^{α}}$ is the graph of some continuous concave function g₁ (x) defined on [l, m], and the lower boundary of $\bar{A_{2}^{α}}$ is also the graph of some continuous convex function g₂ (x) defined on [l, m] (see [2]). Define

$f_{1}^{α} (s) = \frac{1}{2} (m - l) (cos s - 1) + m, if s \in [0, 2 π] .$

Then, $f_{1}^{α} (s)$ moves from m to l if 0 ≤ s ≤ π, and from l to m if π ≤ s ≤ 2π. Let

$P (s) = {\begin{matrix} \frac{1}{2} (g_{1} (f_{1}^{α} (0) - g_{1} (f_{1}^{α} (π))) (cos s - 1) + g_{1} (f_{1}^{α} (0)), & 0 \leq s \leq π, \\ \frac{1}{2} (g_{2} (f_{1}^{α} (π) - g_{2} (f_{1}^{α} (2 π))) (cos s - 1) + g_{2} (f_{1}^{α} (π)), & π \leq s \leq 2 π . \end{matrix}$

Define

$f_{2}^{α} (s, t) = {\begin{matrix} \frac{1}{2} (P (s) - g_{1} (f_{1}^{α} (s))) (sin t - 1) + P (s), & 0 \leq s \leq π, - \frac{π}{2} \leq t \leq \frac{π}{2}, \\ \frac{1}{2} (P (s) - g_{2} (f_{1}^{α} (s))) (sin t - 1) + P (s), & π \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2} . \end{matrix}$

Then, $f_{2}^{α} (s, t)$ moves from P (s) to $g_{1} (f_{1}^{α} (s))$ if $0 \leq s \leq π, - \frac{π}{2} \leq t \leq \frac{π}{2}$ , and $f_{2}^{α} (s, t)$ moves from $f_{1}^{α} (s)$ to $g_{2} (f_{1}^{α} (s))$ if $π \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}$ . We then have

$A_{2}^{α} = {(f_{1}^{α} (s), f_{2}^{α} (s, t)) \in ℝ^{2} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}} .$

If we define $\begin{matrix} f_{3}^{α} (s, t) \\ = {\begin{matrix} h_{1} (f_{1}^{α} (s), f_{2}^{α} (s, t)), & 0 \leq s \leq π, - \frac{π}{2} \leq t \leq \frac{π}{2}, \\ h_{2} (f_{1}^{α} (s), f_{2}^{α} (s, t)), & π \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}, \end{matrix} \end{matrix}$

we have $\begin{matrix} A^{α} = {(f_{1}^{α} (s), f_{2}^{α} (s, t), f_{3}^{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, \\ - \frac{π}{2} \leq t \leq \frac{π}{2}} . \end{matrix}$

The proof is now complete.

Remark 3.4. We proved that Theorem 3.3 is satisfied in the case that A is a continuous convex fuzzy number. If A is a piecewise continuous convex fuzzy number, we can prove similarly (see [2]).

The membership function of the 3-dimensional triangular fuzzy number is a function defined on $ℝ^{3}$ with values in [0, 1]. In case of $A = (\sqrt{6}, 3, 2 \sqrt{2}, 5, 2, 7)^{3}$ , we represent the values of membership function with colors as shown in Figure 1. Cutting the graph of $A = (\sqrt{6}, 3, 2 \sqrt{2}, 5, 2, 7)^{3}$ with plane z = 7, we get Figure 2. In Figure 3, we restrict the domain to the cutting plane, and then represent the membership function as the graph on $ℝ^{2}$ . Then we know that the 3-dimensional triangular fuzzy number is an extension of the 2-dimensional triangular fuzzy number. In 3-dimensional triangular fuzzy number, the α-cut must be an ellipsoid including interior and the α-set must be an ellipsoidal surface. The $\frac{1}{2}$ -set for $A = (\sqrt{6}, 3, 2 \sqrt{2}, 5, 2, 7)^{3}$ is given in Figure 4 for 0 ≤ s ≤ 2π and $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . In case of 0 ≤ s ≤ π and $0 \leq t \leq \frac{π}{2}$ , the graphs are given in Figure 5 and Figure 6, respectively. Therefore, we can confirm that the range of s and t is 0 ≤ s ≤ 2π and $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , respectively.

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6
Definition 3.5. Let A and B be two continuous convex fuzzy numbers defined on $ℝ^{3}$ , and

$A^{α} = {(x, y, z) \in ℝ^{3} | μ_{A} (x, y, z) = α} = {(f_{1}^{α} (s), f_{2}^{α} (s, t), f_{3}^{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}}, B^{α} = {(x, y, z) \in ℝ^{3} | μ_{B} (x, y, z) = α} = {(g_{1}^{α} (s), g_{2}^{α} (s, t), g_{3}^{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}}$

be the α-sets of A and B, respectively. For α ∈ (0, 1), we define that the parametric addition, parametric subtraction, parametric multiplication, and parametric division of two fuzzy numbers A and B are fuzzy numbers that have their α-sets as follows:

(1) parametric addition A (+) _pB:

$(A (+)_{p} B)^{α} = {(f_{1}^{α} (s) + g_{1}^{α} (s), f_{2}^{α} (s, t) + g_{2}^{α} (s, t), f_{3}^{α} (s, t) + g_{3}^{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}}$

(2) parametric subtraction A (-) _pB:

$(A (-)_{p} B)^{α} = {(f_{1}^{α} (s) - g_{1}^{α} (s + π), f_{2}^{α} (s, t) - g_{2}^{α} (s + π, t), f_{3}^{α} (s, t) - g_{3}^{α} (s + π, t)) \in ℝ^{3} | 0 \leq s \leq π, - \frac{π}{2} \leq t \leq \frac{π}{2}},$ $(A (-)_{p} B)^{α} = {(f_{1}^{α} (s) - g_{1}^{α} (s - π), f_{2}^{α} (s, t) - g_{2}^{α} (s - π, t), f_{3}^{α} (s, t) - g_{3}^{α} (s - π, t)) \in ℝ^{3} | π \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}}$

(3) parametric multiplication A (·) _pB:

$(A (\cdot)_{p} B)^{α} = {(f_{1}^{α} (s) \cdot g_{1}^{α} (s), f_{2}^{α} (s, t) \cdot g_{2}^{α} (s, t), f_{3}^{α} (s, t) \cdot g_{3}^{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}}$

(4) parametric division A (/) _pB:

$(A (/)_{p} B)^{α} = {(\frac{f_{1}^{α} (s)}{g_{1}^{α} (s + π)}, \frac{f_{2}^{α} (s, t)}{g_{2}^{α} (s + π, t)}, \frac{f_{3}^{α} (s, t)}{g_{3}^{α} (s + π, t)}) \in ℝ^{3} | 0 \leq s \leq π, - \frac{π}{2} \leq t \leq \frac{π}{2}}, (A (/)_{p} B)^{α} = {(\frac{f_{1}^{α} (s)}{g_{1}^{α} (s - π)}, \frac{f_{2}^{α} (s, t)}{g_{2}^{α} (s - π, t)}, \frac{f_{3}^{α} (s, t)}{g_{3}^{α} (s - π, t)}) \in ℝ^{3} | π \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}}$

For α = 0 and α = 1, $(A ()_{p} B)^{0} = lim_{α \to 0^{+}} (A ()_{p} B)^{α}$ and $(A ()_{p} B)^{1} = lim_{α \to 1^{-}} (A ()_{p} B)^{α}$ , where * = + , - , · , /.

Theorem 3.6. Let A = (a₁, x₁, b₁, y₁, c₁, z₁) ³ and B = (a₂, x₂, b₂, y₂, c₂, z₂) ³ be two 3-dimensional triangular fuzzy numbers. Then, we have the following:

(1) A (+) _pB = (a₁ + a₂, x₁ + x₂, b₁ + b₂, y₁ + y₂, c₁ + c₂, z₁ + z₂) ³ .

(2) A (-) _pB = (a₁ + a₂, x₁ - x₂, b₁ + b₂, y₁ - y₂, c₁ + c₂, z₁ - z₂) ³ .

(3) $(A (\cdot)_{p} B)^{α} = {(x_{α} (s), y_{α} (s, t), z_{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}},$ where

$\begin{matrix} x_{α} (s) = x_{1} x_{2} + (x_{1} a_{2} + x_{2} a_{1}) (1 - α) cos s + a_{1} a_{2} \\ (1 - α)^{2} {cos}^{2} s, \end{matrix}$

$\begin{matrix} y_{α} (s, t) = y_{1} y_{2} + (y_{1} b_{2} + y_{2} b_{1}) (1 - α) sin s cos t + \\ b_{1} b_{2} (1 - α)^{2} {sin}^{2} s {cos}^{2} t \end{matrix}$

and

$\begin{matrix} z_{α} (s, t) = z_{1} z_{2} + (z_{1} c_{2} + z_{2} c_{1}) (1 - α) sin s sin t \\ + c_{1} c_{2} (1 - α)^{2} {sin}^{2} s {sin}^{2} t . \end{matrix}$

(4) $(A (/)_{p} B)^{α} = {(x_{α} (s), y_{α} (s, t), z_{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}},$ where $\begin{matrix} x_{α} (s) = \frac{x_{1} + a_{1} (1 - α) cos s}{x_{2} - a_{2} (1 - α) cos s}, \\ y_{α} (s, t) = \frac{y_{1} + b_{1} (1 - α) sin s cos t}{y_{2} - b_{2} (1 - α) sin s cos t} \end{matrix}$

and

$z_{α} (s, t) = \frac{z_{1} + c_{1} (1 - α) sin s sin t}{z_{2} - c_{2} (1 - α) sin s sin t} .$

Therefore, A (+) _pB and A (-) _pB become 3-dimensional triangular fuzzy numbers, but A (·) _pB and A (/) _pB are not 3-dimensional triangular fuzzy numbers.

Proof. Since A and B are continuous convex fuzzy numbers defined on $ℝ^{3}$ , by Theorem 3.3, there exists $f_{1}^{α} (s), g_{1}^{α} (s), f_{i}^{α} (s, t),$ and $g_{i}^{α} (s, t) (i = 2, 3)$ , such that $\begin{matrix} A^{α} = {(f_{1}^{α} (s), f_{2}^{α} (s, t), f_{3}^{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, \\ - \frac{π}{2} \leq t \leq \frac{π}{2}}, \end{matrix}$

and $\begin{matrix} B^{α} = {(g_{1}^{α} (s), g_{2}^{α} (s, t), g_{3}^{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, \\ - \frac{π}{2} \leq t \leq \frac{π}{2}} . \end{matrix}$

Since A = (a₁, x₁, b₁, y₁, c₁, z₁) ³ and B = (a₂, x₂, b₂, y₂, c₂, z₂) ³, we have

$f_{1}^{α} (s) = x_{1} + a_{1} (1 - α) cos s, f_{2}^{α} (s, t) = y_{1} + b_{1} (1 - α) sin s cos t, f_{3}^{α} (s, t) = z_{1} + c_{1} (1 - α) sin s sin t,$

and $g_{1}^{α} (s) = x_{2} + a_{2} (1 - α) cos s, g_{2}^{α} (s, t) = y_{2} + b_{2} (1 - α) sin s cos t, g_{3}^{α} (s, t) = z_{2} + c_{2} (1 - α) sin s sin t .$

(1) Since

$f_{1}^{α} (s) + g_{1}^{α} (s) = x_{1} + x_{2} + (a_{1} + a_{2}) (1 - α) cos s,$ $\begin{matrix} f_{2}^{α} (s, t) + g_{2}^{α} (s, t) = y_{1} + y_{2} + (b_{1} + b_{2}) (1 - α) \\ sin s cos t, \end{matrix}$ and $\begin{matrix} f_{3}^{α} (s, t) + g_{3}^{α} (s, t) = z_{1} + z_{2} + (c_{1} + c_{2}) (1 - α) \\ sin s sin t, \end{matrix}$

we have

$(A (+)_{p} B)^{α} = {(x, y, z) \in ℝ^{3} | (\frac{x - x_{1} - x_{2}}{(a_{1} + a_{2}) (1 - α)})^{2} + (\frac{y - y_{1} - y_{2}}{(b_{1} + b_{2}) (1 - α)})^{2} + (\frac{z - z_{1} - z_{2}}{(c_{1} + c_{2}) (1 - α)})^{2} = 1} .$

Thus $\begin{matrix} A (+)_{p} B = (a_{1} + a_{2}, x_{1} + x_{2}, b_{1} + b_{2}, y_{1} + y_{2}, \\ c_{1} + c_{2}, z_{1} + z_{2})^{3} . \end{matrix}$

(2) If 0 ≤ s ≤ π,

$f_{1}^{α} (s) - g_{1}^{α} (s + π) = x_{1} - x_{2} + (a_{1} + a_{2}) (1 - α) cos s$

$\begin{matrix} f_{2}^{α} (s, t) - g_{2}^{α} (s + π, t) = y_{1} - y_{2} + (b_{1} + b_{2}) (1 - α) \\ sin s cos t \end{matrix}$

and $\begin{matrix} f_{3}^{α} (s, t) - g_{3}^{α} (s + π, t) = z_{1} - z_{2} + (c_{1} + c_{2}) (1 - α) \\ sin s sin t . \end{matrix}$

In the case of π ≤ s ≤ 2π, we have

$f_{1}^{α} (s) - g_{1}^{α} (s - π) = f_{1}^{α} (s) - g_{1}^{α} (s + π),$

$f_{2}^{α} (s, t) - g_{2}^{α} (s - π, t) = f_{2}^{α} (s, t) - g_{2}^{α} (s + π, t)$

and

$f_{3}^{α} (s, t) - g_{3}^{α} (s - π, t) = f_{3}^{α} (s, t) - g_{3}^{α} (s + π, t) .$

Thus

$(A (-)_{p} B)^{α} = {(x, y, z) \in ℝ^{3} | (\frac{x - x_{1} + x_{2}}{(a_{1} + a_{2}) (1 - α)})^{2} + (\frac{y - y_{1} + y_{2}}{(b_{1} + b_{2}) (1 - α)})^{2} + (\frac{z - z_{1} + z_{2}}{(c_{1} + c_{2}) (1 - α)})^{2} = 1},$

i.e.,

$\begin{matrix} A (-)_{p} B = (a_{1} + a_{2}, x_{1} - x_{2}, b_{1} + b_{2}, y_{1} - y_{2}, \\ c_{1} + c_{2}, z_{1} - z_{2})^{3} . \end{matrix}$

(3) Let $(A (\cdot)_{p} B)^{α} = {(x_{α} (s), y_{α} (s, t), z_{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}}$ . Then

$x_{α} (s) = f_{1}^{α} (s) \cdot g_{1}^{α} (s) = x_{1} x_{2} + (x_{1} a_{2} + x_{2} a_{1}) (1 - α) cos s + a_{1} a_{2} (1 - α)^{2} {cos}^{2} s, y_{α} (s, t) = f_{2}^{α} (s, t) \cdot g_{2}^{α} (s, t) = y_{1} y_{2} + (y_{1} b_{2} + y_{2} b_{1}) (1 - α) sin s cos t + b_{1} b_{2} (1 - α)^{2} {sin}^{2} s {cos}^{2} t$

and

$z_{α} (s, t) = f_{3}^{α} (s, t) \cdot g_{3}^{α} (s, t) = z_{1} z_{2} + (z_{1} c_{2} + z_{2} c_{1}) (1 - α) sin s sin t + c_{1} c_{2} (1 - α)^{2} {sin}^{2} s {sin}^{2} t .$

(4) Let $(A (/)_{p} B)^{α} = {(x_{α} (s), y_{α} (s, t), z_{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}}$ . Then $\begin{matrix} x_{α} (s) = \frac{x_{1} + a_{1} (1 - α) cos s}{x_{2} - a_{2} (1 - α) cos s}, \\ y_{α} (s, t) = \frac{y_{1} + b_{1} (1 - α) sin s cos t}{y_{2} - b_{2} (1 - α) sin s cos t} \end{matrix}$

and

$z_{α} (s, t) = \frac{z_{1} + c_{1} (1 - α) sin s sin t}{z_{2} - c_{2} (1 - α) sin s sin t} .$

The proof is now complete.

Example 3.7. Let A = (6, 3, 8, 5, 4, 7) ³ and B = (4, 2, 5, 3, 6, 4) ³. Then, by Theorem 3.5, we have the following:

(1) A (+) _pB = (10, 5, 13, 8, 10, 11) ³

(2) A (-) _pB = (10, 1, 13, 2, 10, 3) ³

(3) $(A (\cdot)_{p} B)^{α} = {(x_{α} (s), y_{α} (s, t), z_{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}},$ where

$x_{α} (s) = 6 + 24 (1 - α) cos s + 24 (1 - α)^{2} {cos}^{2} s,$ $\begin{matrix} y_{α} (s, t) = 15 + 49 (1 - α) sin s cos t + 40 (1 - α)^{2} \\ {sin}^{2} s {cos}^{2} t \end{matrix}$

and

$\begin{matrix} z_{α} (s, t) = 28 + 58 (1 - α) sin s sin t + 24 (1 - α)^{2} \\ {sin}^{2} s {sin}^{2} t . \end{matrix}$

(4) $(A (/)_{p} B)^{α} = {(x_{α} (s), y_{α} (s, t), z_{α} (s, t)) \in ℝ^{3} | 0 \leq s \leq 2 π, - \frac{π}{2} \leq t \leq \frac{π}{2}},$ where $\begin{matrix} x_{α} (s) = \frac{3 + 6 (1 - α) cos s}{2 - 4 (1 - α) cos s}, \\ y_{α} (s, t) = \frac{5 + 8 (1 - α) sin s cos t}{3 - 5 (1 - α) sin s cos t} \end{matrix}$

and

$z_{α} (s, t) = \frac{7 + 4 (1 - α) sin s sin t}{4 - 6 (1 - α) sin s sin t} .$

Therefore, A (+) _pB and A (-) _pB become 3-dimensional triangular fuzzy numbers, but A (·) _pB and A (/) _pB are not 3-dimensional triangular fuzzy numbers.
4 Conclusion

We generalize triangular fuzzy numbers from $ℝ^{2}$ to $ℝ^{3}$ . In addition, by defining parametric operations between two α-cuts, which are subsets of $ℝ^{3}$ , we obtain parametric operations for two triangular fuzzy numbers defined on $ℝ^{3}$ . It is meaningful to expand the dimension by unifying what Zadeh has defined as the max-min operation in two ways (see [12]). Moreover, since the calculations of the operations are the same, this will assist to expand the study of triangular fuzzy matrices in the future (see [1, 3]).

In [10], the results of 2-dimensional trapezoidal fuzzy sets have been illustrated. In the case of 3-dimension, we can not illustrate the results, but predict that A (+) B and A (-) B become 3-dimensional triangular fuzzy numbers. However, A (·) B and A (/) B do not. Since A (+) B and A (-) B are perfectly formed, they can be applied to many areas without any modification. On the other hand, if the forms of A (·) B and A (/) B are modified, they can be used in various applications. This result can be utilized in proving that the 3-dimensional case is the generalization of the 2-dimensional case. There have been several attempts to expand the dimension, but no studies have yet succeeded in expansion while maintaining Zadeh’s 1-dimensional results. In this sense, this paper will facilitate the development of applications of triangular fuzzy numbers to be an extension dimension of the application (see [7, 8]). Since there are few results for 3-dimensional case, this paper will be applied to 3-dimensional fuzzy operation. Research results have a possibility for real time application as well as a theoretical significance. This paper is expected to be applied to numerous fields from engineering such as robotics and software programming to public policy making and medicine.

Acknowledgments

The author would like to express sincere thanks to the anonymous referees for their carefully reading the paper and valuable comments and suggestions which have improved the original paper.

This research was supported by the 2020 scientific promotion program funded by Jeju National University, and the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2018R1D1A1B07040648).

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