EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5601 ISSN 1307-5543 – ejpam.com Published by New York Business Global Graphical Representation of Zadeh’s Max-Min Composition Operator for Two 3-Dimensional Quadratic Fuzzy Numbers Yong Sik Yun1, BongJu Lee2,∗ 1 Department of Mathematics. Jeju National University. Jeju 63243, Republic of Korea 2 Department of Mathematics Education. Kyungpook National University. Daegu 41566, Republic of Korea Abstract. We computed the extended operations for generalized quadratic fuzzy sets and ex- tended quadratic fuzzy numbers from R to R2. By defining parametric operations between two α-cuts, which are regions, we derived the parametric operations for two quadratic fuzzy numbers defined on R2. The outcomes of these parametric operations serve as a generalization of Zadeh’s extended algebraic operations. We demonstrated that the results obtained from the paramet- ric operations represent an extension of Zadeh’s extended algebraic operations. Additionally, we expanded quadratic fuzzy numbers initially defined in two dimensions to three dimensions and calculated Zadeh’s max-min composition operator for two extended three-dimensional quadratic fuzzy numbers. We presented an illustrative example of three-dimensional results along with cor- responding graphs. 2020 Mathematics Subject Classifications: 47S40, 03E72 Key Words and Phrases: Graphic representation, parametric operation, 3-dimensional quadraric fuzzy number 1. Introduction A quadratic fuzzy number expands upon the concept of traditional fuzzy numbers by incorporating quadratic functions to represent the degree of membership of an element in a set. Fuzzy numbers are utilized to model uncertainty and imprecision in various applications, and quadratic fuzzy numbers provide a more flexible representation through the use of quadratic functions. These numbers are commonly employed in decision-making processes, especially in scenarios where there is a need to model and analyze uncertain or imprecise information. They find applications in diverse fields such as optimization, control systems, and decision analysis [4, 7]. It’s essential to recognize that various researchers and practitioners may employ slightly different formulations and definitions for quadratic ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5601 Email addresses: yunys@jejunu.ac.kr (Y. S. Yun), leebj@knu.ac.kr (B. Lee) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 2 of 17 fuzzy numbers. Additionally, the specific context of their application can influence how they are defined and utilized. The membership function of a quadratic fuzzy number is composed of a quadratic function with the maximum value of 1. In contrast, a general quadratic fuzzy set is a quadratic fuzzy set that may not have a maximum value of 1. We calculated the extended operations for generalized quadratic fuzzy sets [9] and expanded the quadratic fuzzy numbers from R to R2 [2]. By defining parametric operations between two α-cuts, which are regions, we derived the parametric operations for two quadratic fuzzy numbers defined on R2. The outcomes of these parametric operations serve as a generalization of Zadeh’s extended algebraic operations. We have shown that the results of parametric operations represent a generalization of Zadeh’s max-min composition operations. Furthermore, we expanded the concept of general quadratic fuzzy sets from R to R2. We performed calculations for the parametric operations applied to two generalized 2-dimensional quadratic fuzzy sets [8]. Our evi- dence demonstrates that the parametric operations for two generalized quadratic fuzzy sets defined on R2 constitute a broader generalization of Zadeh’s max-min composition operations for two general quadratic fuzzy sets defined on R [6]. In this paper, we extend quadratic fuzzy numbers defined in two dimensions to three dimensions and calculate Zadeh’s max-min composition operator for two extended three- dimensional quadratic fuzzy numbers. We provide an illustrative example showcasing three-dimensional results and present graphs depicting the example. 2. max-min composition operations of Zadeh for generalized quadratic fuzzy sets defined on R We start by introducing the α-cut and α-set of the fuzzy set A on R with the member- ship function µA(x). An α-cut of the fuzzy number A is formally defined as Aα = {x ∈ R | µA(x) ≥ α} when α ∈ (0, 1] and A0 is determined as the closure of {x ∈ R | µA(x) > 0}. For α ∈ (0, 1), the set Aα = {x ∈ X | µA(x) = α} is referred to as the α-set of the fuzzy set A, where A0 represents the boundary of {x ∈ R | µA(x) > 0}, and A1 is equivalent to A1. Definition 1. [12] The extended addition A(+)B, extended subtraction A(−)B, extended multiplication A(·)B, and extended division A(/)B are fuzzy sets with membership func- tions as follows. For all x ∈ A and y ∈ B, µA(∗)B(z) = sup z=x∗y min{µA(x), µB(y)}, ∗ = +,−, ·, / Now, we extend the concept to encompass general quadratic fuzzy sets. A general quadratic fuzzy set is symmetric and may not necessarily attain the maximum value of 1. The membership function graph of a general quadratic fuzzy set exhibits symmetry with respect to a certain line defined by x = m. Definition 2. [9] A fuzzy set A with a membership function Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 3 of 17 µA(x) = { 0, x < x1, x2 ≤ x, −a(x− x1)(x− x2) = −a(x−m)2 + p, x1 ≤ x < x2, where m = x1+x2 2 , 0 < a, 0 < p ≤ 1, is called a generalized quadratic fuzzy set and denoted by [[x1, p, x2]] or [[a,m, p]]+. Theorem 1. [9] Let A = [[a,m, p]]+ = [[x1, p, x2]] and B = [[b, n, q]]+ = [[x3, q, x4]] be generalized quadratic fuzzy sets. Assume that p ≤ q and µB(x) ≥ p on [k1, k2]. We can then deduce the followings: (1) A(+)B is a fuzzy set with a membership function µA(+)B(x) =  0 (x < x1 + x3, x2 + x4 ≤ x) f1(x) (x1 + x3 ≤ x < m+ k1) p (m+ k1 ≤ x < m+ k2) f2(x) (m+ k2 ≤ x < x2 + x4) where f1(x) = 1 a2 − 2ab+ b2 ( −abm(a+ b+ an+ bn)− abn(am+ bm + an+ bn)− ab(p+ q) + a2q + b2p+ 2ab(am+ bm+ an + bn)x− ab(a+ b)x2 + 2ab(m+ n− x) · √ g1(x) ) , f2(x) = 1 a2 − 2ab+ b2 ( −abm(a+ b+ an+ bn)− abn(am+ bm + an+ bn)− ab(p+ q) + a2q + b2p+ 2ab(am+ bm+ an + bn)x− ab(a+ b)x2 − 2ab(m+ n− x) · √ g1(x) ) , and g1(x) = ab(m+ n)2 + (a− b)(p− q)− 2ab(m+ n)x+ abx2. (2) A(−)B is a fuzzy set with a membership function µA(−)B(x) =  0 (x < x1 − x4, x2 − x3 ≤ x) f3(x) (x1 − x4 ≤ x < m− k2) p (m− k2 ≤ x < m− k1) f4(x) (m− k1 ≤ x < x2 − x3) where f3(x) = 1 a2 − 2ab+ b2 ( −abm(am+ bm− an− bn)− abn(an+ bn − am− bm)− ab(p+ q) + a2q + b2p+ 2ab(am+ bm− an − bn)x− ab2x2 + 2ab(m− n− x) · √ g2(x) ) , Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 4 of 17 f4(x) = 1 a2 − 2ab+ b2 ( −abm(am+ bm− an− bn)− abn(an+ bn − am− bm)− ab(p+ q) + a2q + b2p+ 2ab(am+ bm− an − bn)x− ab2x2 − 2ab(m− n− x) · √ g2(x) ) , and g2(x) = ab(m− n)2 + (a− b)(p− q)− 2ab(m− n)x+ abx2. (3) If p = q, A(·)B is a fuzzy set with a membership function µA(·)B(x) = { 0 (x < x1x3, x2x4 ≤ x) f5(x) (x1x3 ≤ x < x2x4) where f5(x) = 1 2 (−am2 − bn2 + 2p)− √ abx+ 1 2 √ g3(x), g3(x) =− am2(am2 + 3bn2)− bn2(bn2 + 3am2) + 2(am2 + bn2 − 2p)2 + 8p(am2 + bn2 − p) + 8abmnx− 1 8 √ abx { −8(am2 + bn2 − 2p)3 + 8(am2 + bn2 − 2p)h1(x)− 16h2(x) } , h1(x) =am2(am2 + 2bn2) + bn2(bn2 + 2am2)− 6p(am2 + bn2 − p)− 4abmnx− 2abx2, h2(x) =abm2n2(am2 + bn2 − 4p)− am2p(am2 − 3p)− bn2p(bn2 − 3p) − 2p3 − 2abmn(am2 + bn2 − 2p)x+ ab(am2 + bn2 + 2p)x2. (4) A(/)B is a fuzzy set with a membership function µA(/)B(x) =  0 (x < x1/x4, x2/x3 ≤ x) f6(x) (x1/x4 ≤ x < m/k2) p (m/k2 ≤ x < m/k1) f7(x) (m/k1 ≤ x < x2/x3) where f6(x) = 1 b2 − 2abx2 + a2x4 ( −b2(am2 + p) + 2ab2mnx− ab(am2 + bn2 + p + q)x2 + 2a2bmnx3 − a2(bn2 − q)x4 + 2abx(m− nx) · √ g4(x) ) , f7(x) = 1 b2 − 2abx2 + a2x4 ( −b2(am2 + p) + 2ab2mnx− ab(am2 + bn2 + p + q)x2 + 2a2bmnx3 − a2(bn2 − q)x4 − 2abx(m− nx) · √ g4(x) ) , and g4(x) = b(am2 − p+ q)− 2abmnx+ a(bn2 + p− q)x2. Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 5 of 17 3. max-min composition operations of Zadeh for quadratic fuzzy numbers defined on R2 We extended the concept of quadratic fuzzy numbers from R to R2, introducing 2- dimensional quadratic fuzzy numbers. Additionally, we formulated parametric operations for two such 2-dimensional quadratic fuzzy numbers by employing region-valued α-cuts in R2. Definition 3. [2] A fuzzy set A with a membership function µA(x, y) = { 1− ( (x−x1)2 a2 + (y−y1)2 b2 ) , b2(x− x1) 2 + a2(y − y1) 2 ≤ a2b2, 0, otherwise, where a, b > 0 is reffered to as a 2-dimensional quadratic fuzzy number, denoted by [a, x1, b, y1] 2. The α-cut Aα of a 2-dimensional quadratic fuzzy number A = [a, x1, b, y1] 2 is the interior of an ellipse in an xy-plane, including the boundary Aα = { (x, y) ∈ R2 ∣∣∣ b2(x− x1) 2 + a2(y − y1) 2 ≤ a2b2(1− α) } = { (x, y) ∈ R2 ∣∣∣ (x− x1) 2 a2(1− α) + (y − y1) 2 b2(1− α) ≤ 1 } . Theorem 2. [3] Let A be a continuous convex fuzzy number defined on R2 and Aα = {(x, y) ∈ R2|µA(x, y) = α} be the α-set of A. Then for all α ∈ (0, 1), there exist continuous functions fα 1 (t) and fα 2 (t) defined on [0, 2π] such that Aα = {(fα 1 (t), f α 2 (t)) ∈ R2|0 ≤ t ≤ 2π}. Definition 4. [3] Let A and B be convex fuzzy numbers defined on R2 and Aα = {(fα 1 (t), f α 2 (t)) ∈ R2|0 ≤ t ≤ 2π}, Bα = {(gα1 (t), gα2 (t)) ∈ R2|0 ≤ t ≤ 2π} be the α-sets of A and B, respectively. For α ∈ (0, 1), the parametric addition, parametric subtraction, parametric multiplication, and parametric division are fuzzy numbers that have their α-sets as follows: (1) parametric addition A(+)pB: (A(+)pB)α = {(fα 1 (t) + gα1 (t), f α 2 (t) + gα2 (t)) ∈ R2|0 ≤ t ≤ 2π} (2) parametric subtraction A(−)pB: (A(−)pB)α = {(xα(t), yα(t)) ∈ R2|0 ≤ t ≤ 2π}, where Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 6 of 17 xα(t) = { fα 1 (t)− gα1 (t+ π), if 0 ≤ t ≤ π fα 1 (t)− gα1 (t− π), if π ≤ t ≤ 2π and yα(t) = { fα 2 (t)− gα2 (t+ π), if 0 ≤ t ≤ π fα 2 (t)− gα2 (t− π), if π ≤ t ≤ 2π (3) parametric multiplication A(·)pB: (A(·)pB)α = {(fα 1 (t) · gα1 (t), fα 2 (t) · gα2 (t)) ∈ R2|0 ≤ t ≤ 2π} (4) parametric division A(/)pB: (A(/)pB)α = {(xα(t), yα(t)) ∈ R2|0 ≤ t ≤ 2π}, where xα(t) = fα 1 (t) gα1 (t+ π) (0 ≤ t ≤ π), xα(t) = fα 1 (t) gα1 (t− π) (π ≤ t ≤ 2π) and yα(t) = fα 2 (t) gα2 (t+ π) (0 ≤ t ≤ π), yα(t) = fα 2 (t) gα2 (t− π) (π ≤ t ≤ 2π) For α = 0 and α = 1, (A(∗)pB)0 = limα→0+(A(∗)pB)α and (A(∗)pB)1 = limα→1−(A(∗)pB)α, where ∗ = +, −, ·, /. Theorem 3. [3] Let A = [a1, x1, b1, y1] 2 and B = [a2, x2, b2, y2] 2 be two 2-dimensional quadratic fuzzy numbers. Subsequently, the following results hold: (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 = {(xα(t), yα(t)) | 0 ≤ t ≤ 2π}, where xα(t) = x1x2 + (x1a2 + x2a1) √ 1− α cos t+ a1a2(1− α) cos2 t and yα(t) = y1y2 + (y1b2 + y2b1) √ 1− α sin t+ b1b2(1− α) sin2 t. (4) (A(/)pB)α = {(xα(t), yα(t)) | 0 ≤ t ≤ 2π}, where xα(t) = x1 + a1 √ 1− α cos t x2 − a2 √ 1− α cos t and yα(t) = y1 + b1 √ 1− α sin t y2 − b2 √ 1− α sin t . Therefore, A(+)pB and A(−)pB become 2-dimensional quadratic fuzzy numbers, whereas A(·)pB and A(/)pB do not qualify as 2-dimensional quadratic fuzzy numbers. Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 7 of 17 Example 1. [3] Consider A = [6, 3, 8, 5]2 and B = [4, 2, 5, 3]2. Subsequently, the following observations hold: (1) A(+)pB = [10, 5, 13, 8]2 (2) A(−)pB = [10, 1, 13, 2]2 (3) (A(·)pB)α = {(xα(t), yα(t)) | 0 ≤ t ≤ 2π}, where xα(t) = 6 + 24 √ 1− α cos t+ 24(1− α) cos2 t and yα(t) = 15 + 49 √ 1− α sin t+ 40(1− α) sin2 t. (4) (A(/)pB)α = {(xα(t), yα(t)) | 0 ≤ t ≤ 2π}, where xα(t) = 3 + 6 √ 1− α cos t 2− 4 √ 1− α cos t and yα(t) = 5 + 8 √ 1− α sin t 3− 5 √ 1− α sin t . Thus A(+)pB and A(−)pB become 2-dimensional quadratic fuzzy numbers, but A(·)pB and A(/)pB are not 2-dimensional quadratic fuzzy numbers. 4. max-min composition operations of Zadeh for quadratic fuzzy numbers defined on R3 We extended the concept of quadratic fuzzy numbers from R2 to R3, thereby intro- ducing 3-dimensional quadratic fuzzy numbers. Our objective is to formulate parametric operations between two such 3-dimensional quadratic fuzzy numbers. In R3, α-cuts take the form of cubics, which makes the traditional calculation methods between α-cuts in- feasible. Therefore, we adopted a novel approach to reinterpret the existing method and apply it to cubic-valued α-cuts in R3. Definition 5. A fuzzy set A with a membership function µA(x, y, z) =  1− ( (x−x1)2 a2 + (y−y1)2 b2 + (z−z1)2 c2 ) , if b2c2(x− x1) 2 + c2a2(y − y1) 2 +a2b2(z − z1) 2 ≤ a2b2c2, 0, otherwise, where a, b, c > 0 is called the 3-dimensional quadratic fuzzy number and denoted by [a, x1, b, y1, c, z1] 3. Note that µA(x, y) forms a cone in R2, but we can not determine the shape of µA(x, y, z) in R3. The α-cut Aα of a 3-dimensional quadratic fuzzy number A = [a, x1, b, y1, c, z1] 3 is defined as the following set Aα = { (x, y, z) ∈ R3 ∣∣∣ (x− x1) 2 a2 + (y − y1) 2 b2 + (z − z1) 2 c2 ≤ 1− α } = { (x, y, z) ∈ R3 ∣∣∣ (x− x1) 2 a2(1− α) + (y − y1) 2 b2(1− α) + (z − z1) 2 c2(1− α) ≤ 1 } . Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 8 of 17 Definition 6. A 3-dimensional fuzzy number, A defined on R3, is termed a convex fuzzy number if, for all α ∈ (0, 1), the α-cuts Aα = {(x, y, z) ∈ R3|µA(x, y, z) ≥ α} represent convex subsets in R3. Theorem 4. [10] Let A be a continuous convex fuzzy number defined on R3, and Aα = {(x, y, z) ∈ R3|µA(x, y, z) = α} be the α-set of A. Then, for all α ∈ (0, 1), there exist continuous functions fα 1 (s), f α 2 (s, t), and fα 3 (s, t)(0 ≤ s ≤ 2π, 0 ≤ t ≤ π 2 ) such that Aα = {(fα 1 (s), f α 2 (s, t), f α 3 (s, t)) ∈ R3|0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 }. Definition 7. Let A and B are two continuous convex fuzzy numbers defined on R3 and Aα = {(x, y, z) ∈ R3|µA(x, y, z) = α} = {(fα 1 (s), f α 2 (s, t), f α 3 (s, t)) ∈ R3|0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 }, Bα = {(x, y, z) ∈ R3|µB(x, y, z) = α} = {(gα1 (s), gα2 (s, t), gα3 (s, t)) ∈ R3|0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 } be the α-sets of A and B, respectively. For α ∈ (0, 1), we define the parametric addition, parametric subtraction, parametric multiplication, and parametric division of two fuzzy numbers A and B as fuzzy numbers with α-sets as follows: (1) parametric addition A(+)pB: (A(+)pB)α = {(fα 1 (s) + gα1 (s), f α 2 (s, t) + gα2 (s, t), f α 3 (s, t) + gα3 (s, t)) ∈ R3| 0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 } (2) parametric subtraction A(−)pB: (A(−)pB)α = {(fα 1 (s)− gα1 (s+ π), fα 2 (s, t)− gα2 (s+ π, t), fα 3 (s, t)− gα3 (s+ π, t)) ∈ R3|0 ≤ s ≤ π,−π 2 ≤ t ≤ π 2 }, (A(−)pB)α = {(fα 1 (s)− gα1 (s− π), fα 2 (s, t)− gα2 (s− π, t), fα 3 (s, t)− gα3 (s− π, t)) ∈ R3|π ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 } (3) parametric multiplication A(·)pB: (A(·)pB)α = {(fα 1 (s) · gα1 (s), fα 2 (s, t) · gα2 (s, t), fα 3 (s, t) · gα3 (s, t)) ∈ R3| 0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 } (4) parametric division A(/)pB: Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 9 of 17 (A(/)pB)α = {( fα 1 (s) gα1 (s+ π) , fα 2 (s, t) gα2 (s+ π, t) , fα 3 (s, t) gα3 (s+ π, t) ) ∈ R3| 0 ≤ s ≤ π,−π 2 ≤ t ≤ π 2 }, (A(/)pB)α = {( fα 1 (s) gα1 (s− π) , fα 2 (s, t) gα2 (s− π, t) , fα 3 (s, t) gα3 (s− π, t) ) ∈ R3| π ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 } For α = 0 and α = 1, (A(∗)pB)0 = limα→0+(A(∗)pB)α and (A(∗)pB)1 = limα→1−(A(∗)pB)α, where ∗ = +, −, ·, /. Theorem 5. Let A = [a1, x1, b1, y1, c1, z1] 3 and B = [a2, x2, b2, y2, c2, z2] 3 be two 3- dimensional quadratic fuzzy numbers. Subsequently, the following results hold: (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(·)pB)α = {(xα(s), yα(s, t), zα(s, t)) ∈ R3 | 0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 }, where xα(s) = x1x2 + (x1a2 + x2a1) √ 1− α cos s+ a1a2(1− α) cos2 s, yα(s, t) = y1y2 + (y1b2 + y2b1) √ 1− α sin s cos t+ b1b2(1− α) sin2 s cos2 t and zα(s, t) = z1z2 + (z1c2 + z2c1) √ 1− α sin s sin t+ c1c2(1− α) sin2 s sin2 t. (4) (A(/)pB)α = {(xα(s), yα(s, t), zα(s, t)) ∈ R3 | 0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 }, where xα(s) = x1 + a1 √ 1− α cos s x2 − a2 √ 1− α cos s yα(s, t) = y1 + b1 √ 1− α sin s cos t y2 − b2 √ 1− α sin s cos t and zα(s, t) = z1 + c1 √ 1− α sin s sin t z2 − c2 √ 1− α sin s sin t . Therefore, A(+)pB and A(−)pB become 3-dimensional quadratic fuzzy numbers, while A(·)pB and A(/)pB do not qualify as 3-dimensional quadratic fuzzy numbers. Proof. Since A and B are continuous convex fuzzy numbers defined on R3, by Theorem 4, there exists fα 1 (s), g α 1 (s), f α i (s, t), g α i (s, t) (i = 2, 3) such that Aα = {(fα 1 (s), f α 2 (s, t), f α 3 (s, t)) ∈ R3|0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 }, Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 10 of 17 and Bα = {(gα1 (s), gα2 (s, t), gα3 (s, t)) ∈ R3|0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 }. Since A = [a1, x1, b1, y1, c1, z1] 3 and B = [a2, x2, b2, y2, c2, z2] 3, we have fα 1 (s) = x1 + a1 √ 1− α cos s, fα 2 (s, t) = y1 + b1 √ 1− α sin s cos t fα 3 (s, t) = z1 + c1 √ 1− α sin s sin t and gα1 (s) = x2 + a2 √ 1− α cos s, gα2 (s, t) = y2 + b2 √ 1− α sin s cos t gα3 (s, t) = z2 + c2 √ 1− α sin s sin t. (1) Since fα 1 (s) + gα1 (s) = x1 + x2 + (a1 + a2) √ 1− α cos s fα 2 (s, t) + gα2 (s, t) = y1 + y2 + (b1 + b2) √ 1− α sin s cos t and fα 3 (s, t) + gα3 (s, t) = z1 + z2 + (c1 + c2) √ 1− α sin s sin t we have (A(+)pB)α = { (x, y, z) ∈ R3 ∣∣∣ (x− x1 − x2) 2 (a1 + a2)2(1− α) + (y − y1 − y2) 2 (b1 + b2)2(1− α) + (z − z1 − z2) 2 (c1 + c2)2(1− α) = 1 } . Thus A(+)pB = [ a1 + a2, x1 + x2, b1 + b2, y1 + y2, c1 + c2, z1 + z2 ]3 . (2) If 0 ≤ s ≤ π, fα 1 (s)− gα1 (s+ π) = x1 − x2 + (a1 + a2) √ 1− α cos s fα 2 (s, t)− gα2 (s+ π, t) = y1 − y2 + (b1 + b2) √ 1− α sin s cos t and fα 3 (s, t)− gα3 (s+ π, t) = z1 − z2 + (c1 + c2) √ 1− α sin s sin t. In the case of π ≤ s ≤ 2π, we have fα 1 (s)− gα1 (s− π) = fα 1 (s)− gα1 (s+ π) fα 2 (s, t)− gα2 (s− π, t) = fα 2 (s, t)− gα2 (s+ π, t) Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 11 of 17 and fα 3 (s, t)− gα3 (s− π, t) = fα 3 (s, t)− gα3 (s+ π, t). Thus (A(−)pB)α = { (x, y, z) ∈ R3 ∣∣∣ (x− x1 + x2) 2 (a1 + a2)2(1− α) + (y − y1 + y2) 2 (b1 + b2)2(1− α) + (z − z1 + z2) 2 (c1 + c2)2(1− α) = 1 } , i.e., A(−)pB = [ a1 + a2, x1 − x2, b1 + b2, y1 − y2, c1 + c2, z1 − z2 ]3 . (3) Let (A(·)pB)α = {(xα(s), yα(s, t), zα(s, t)) | 0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 }. Since fα 1 (s) = x1 + a1 √ 1− α cos s, fα 2 (s, t) = y1 + b1 √ 1− α sin s cos t, fα 3 (s, t) = z1 + c1 √ 1− α sin s sin t and gα1 (s) = x2 + a2 √ 1− α cos s, gα2 (s, t) = y2 + b2 √ 1− α sin s cos t, gα3 (s, t) = z2 + c2 √ 1− α sin s sin t, we have xα(s) = fα 1 (s) · gα1 (s) = x1x2 + (x1a2 + x2a1) √ 1− α cos s + a1a2(1− α) cos2 s, yα(s, t) = fα 2 (s, t) · gα2 (s, t) = y1y2 + (y1b2 + y2b1) √ 1− α sin s cos t + b1b2(1− α) sin2 s cos2 t, zα(s, t) = fα 3 (s, t) · gα3 (s, t) = z1z2 + (z1c2 + z2c1) √ 1− α sin s sin t + c1c2(1− α) sin2 s sin2 t. (4) Let (A(/)pB)α = {(xα(s), yα(s, t), zα(s, t)) | 0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 }. Similarly, we have xα(s) = x1 + a1 √ 1− α cos s x2 − a2 √ 1− α cos s yα(s, t) = y1 + b1 √ 1− α sin s cos t y2 − b2 √ 1− α sin s cos t , zα(s, t) = z1 + c1 √ 1− α sin s sin t z2 − c2 √ 1− α sin s sin t . The proof is complete. Example 2. Consider A = [6, 3, 8, 5, 4, 7]3 and B = [4, 2, 5, 3, 6, 4]3. Subsequently, the following observations hold: (1) A(+)pB = [10, 5, 13, 8, 10, 11]3 Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 12 of 17 (2) A(−)pB = [10, 1, 13, 2, 10, 3]3 (3) (A(·)pB)α = {(xα(s), yα(s, t), zα(s, t)) | 0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 }, where xα(s) = 6 + 24 √ 1− α cos s+ 24(1− α) cos2 s, yα(s, t) = 15 + 49 √ 1− α sin s cos t+ 40(1− α) sin2 s cos2 t and zα(s, t) = 28 + 58 √ 1− α sin s sin t+ 24(1− α) sin2 s sin2 t. (4) (A(/)pB)α = {(xα(s), yα(s, t), zα(s, t)) | 0 ≤ s ≤ 2π,−π 2 ≤ t ≤ π 2 }, where xα(s) = 3 + 6 √ 1− α cos s 2− 4 √ 1− α cos s yα(s, t) = 5 + 8 √ 1− α sin s cos t 3− 5 √ 1− α sin s cos t and zα(s, t) = 7 + 4 √ 1− α sin s sin t 4− 6 √ 1− α sin s sin t . Thus A(+)pB and A(−)pB become 3-dimensional quadratic fuzzy numbers, but A(·)pB and A(/)pB are not 3-dimensional quadratic fuzzy numbers. The membership function of the 3-dimensional quadratic fuzzy number is a function defined on R3 with values in [0, 1]. In the case of the 3-dimensional quadratic fuzzy numbers A = [6, 3, 8, 5, 4, 7]3 and B = [4, 2, 5, 3, 6, 4]3, we depict the values of the membership function using colors, as illustrating in Figure 1 and Figure 2. Figure 1: A Figure 2: B Figure 3: A(+)pB Figure 4: A(−)pB Figure 5: A(·)pB Figure 6: A(/)pB The results of the example are depicted in Figures 3 to 6. The membership function values for points within A(+)pB and A(−)pB, when intersected by a plane, are demon- strated in Figures 7 to 12 and Figures 13 to 21, respectively. Although all graphs appear Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 13 of 17 similar in shape, a closer look at the bar graph showing the function values next to each figure uncovers notable differences. It becomes clear that the function values are not consistent, unlike what is typically seen in one or two dimensions. Fig. 7: A(+)pBz ≤ 8 Fig. 8: A(+)pBz ≤ 9 Fig. 9: A(+)pBz ≤ 10 Fig. 10: A(+)pBz ≤ 11 Fig. 11: A(+)pBz ≤ 12 Fig. 12: A(+)pBz ≤ 13 Fig. 13: A(−)pBz ≤ −0.15 Fig. 14: A(−)pBz ≤ 0.5 Fig. 15: A(−)pBz ≤ 1 5. Conclusion We are broadening the scope of quadratic fuzzy numbers from a two-dimensional space R2 to a three-dimensional space R3. By establishing parametric operations between two α-cuts, which are subsets of R3, we are able to formulate parametric operations for two quadratic fuzzy numbers within the R3 space. The significance of this dimensional ex- pansion lies in its incorporation of Zadeh’s defined max-min operation in two dimensions [10]. Moreover, as long as the computations of these operations remain consistent, this dimensional expansion is expected to further the research in fractional programming in the future [1]. Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 14 of 17 In [2], the results of quadratic fuzzy numbers in two dimensions have been detailed. When extended to three dimensions, the operations A(+)B and A(−)B evolve into 3- dimensional quadratic fuzzy numbers. However, this transformation does not apply to A(·)B and A(/)B. The inherent well-structured nature of A(+)B and A(−)B allows them to be utilized in a wide range of fields without needing any alterations. Conversely, by modifying the forms of A(·)B and A(/)B, they can be adapted for use in various applications. Fig. 16: A(−)pBz ≤ 2 Fig. 17: A(−)pBz ≤ 3 Fig. 18: A(−)pBz ≤ 4 Fig. 19: A(−)pBz ≤ 5 Fig. 20: A(−)pBz ≤ 5.5 Fig. 21: A(−)pBz ≤ 6 This result can be applied to demonstrate that the 3-dimensional case is a general- ization of the 2-dimensional case. While there have been various attempts to expand the dimension, no studies have successfully achieved expansion while preserving Zadeh’s re- sults for 1 and 2-dimensional quadratic fuzzy numbers. This paper aims to contribute to the advancement of applications of quadratic fuzzy numbers by extending their dimension. The application scope of this paper includes solving the 3-dimensional flow shop scheduling problem and quadratic fuzzy equations [5, 11], with expectations of further applications across numerous fields. Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this article. Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 15 of 17 References [1] S. Jain. Close interval approximation of piecewise quadratic fuzzy numbers for fuzzy fractional program. Iranian Journal of Operations Research, 2(1):77–88, 2010. [2] C. Kang and Y.S. Yun. A zadeh‘s max-min composition operator for two 2- dimensional quadratic fuzzy numbers. Far East Journal of Mathematical Sciences, 101(10):2185–2193, 2017. [3] C. Kim and Y.S. Yun. Zadeh’s extension principle for 2-dimensional triangular fuzzy numbers. Journal of Fuzzy Logic and Intelligent Systems, 25(2):197–202, 2015. [4] W. Kumam and A. Pongpullponsak. Optimization for estimation to medical service value of informal workers social security office in thailand. Journal of Information and Optimization Sciences, 37(1):125–154, 2016. [5] M. Landowski. Horizontal Fuzzy Numbers for Solving Quadratic Fuzzy Equation. Springer, Advances in Soft and Hard Computing, Advances in intelligent systems and computing 889, 2019. [6] J.W. Park and Y.S. Yun. The result of arithmetic operations applied on general quadratic fuzzy sets. Journal of Analysis and Applications, 20(1):2459–2471, 2022. [7] L. Platil and T. Tanaka. Optimization for estimation to medical office in thailand. Multi-criteria evaluation for intuitionistic fuzzy sets based on set-relations, 34:1–18, 2023. [8] Y.S. Yun. An algebraic operations for two generalized 2-dimensional quadratic fuzzy sets. Journal of the Chungcheong Mathematical Society, 31(4):379–386, 2018. [9] Y.S. Yun and J.W. Park. The extended operations for generalized quadratic fuzzy sets. Journal of The Korean Institute of Intelligent Systems, 20(4):592–595, 2010. [10] L.A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning – i. Information Sciences, 8:199–249, 1975. [11] T. Zhou, H.E. Khalifa, S.E. Najafi, and S.A. Edalatpanah. Minimizing the machine processing time in a flow shop scheduling problem under piecewise quadratic fuzzy numbers. Discrete Dynamics in Nature and Society, page Article ID 3990534, 2022. [12] H.J. Zimmermann. Fuzzy Set Theory - and Its Applications. Kluwer-Nijhoff Publish- ing, Boston-Dordrecht-Lancaster, 1985. Appendix The Mathematica commands to obtain the above graphs are as follows. (Figure 1) DensityPlot3D[1 - ((x - 3)^2/6 + (y - 5)^2/8 + (z - 7)^2/4), {x, y, z} in Ellipsoid[{3, 5, 7}, {Sqrt[6], Sqrt[8], 2}], PlotPoints -> 100, ColorFunct ion -> "SunsetColors", OpacityFunction -> 0.05, BoxRatios -> {Sqrt[6], Sqr t[8], 2}, PlotLegends -> Automatic] Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 16 of 17 (Figure 3) DensityPlot3D[1 - ((x - 5)^2/10 + (y - 8)^2/13 + (z - 11)^2/10), {x, y, z} in Ellipsoid[{5, 8, 11}, {Sqrt[10], Sqrt[13], Sqrt[10]}], PlotPoints -> 10 0, ColorFunction -> "SunsetColors", OpacityFunction -> 0.05, BoxRatios -> {Sqrt[10], Sqrt[13], Sqrt[10]}, PlotLegends -> Automatic] (Figure 5) g[a_] := ParametricPlot3D[{6 + 24 Sqrt[1 - a] Cos[s] + 24 (1 - a) (Cos[s]) ^2, 15 + 49 Sqrt[1 - a] Sin[s] Cos[t] + 40 (1 - a) (Sin[s])^2 (Cos[t])^2, 28 + 58 Sqrt[1 - a] Sin[s] Sin[t] + 24(1 - a) (Sin[s])^2 (Sin[t])^2}, {s, 0, 2 Pi}, {t, -Pi/2, Pi/2}, PlotStyle -> Directive[RGBColor[0.2, 0.5 + a/2 , 0.5 + a/2], Opacity[0.3]], BoxRatios -> {1, 1, 1}]; tg = Table[g[i], {i, 0, 1.0, 0.01}]; Show[tg] (Figure 7) reg1 = ImplicitRegion[0 <= (x - 5)^2/10 + (y - 8)^2/13 + (z - 11)^2/10 <= 1 && z <= 8, {x, y, z}];DensityPlot3D[1 - ((x - 5)^2/10 + (y - 8)^2/13 + ( z - 11)^2/10), {x, y, z} in reg1, PlotPoints -> 100, ColorFunction -> "Sun setColors", OpacityFunction -> 1, BoxRatios -> {Sqrt[10], Sqrt[13], Sqrt[1 0]}, PlotLegends -> Automatic] (Figure 10) reg1 = ImplicitRegion[0 <= (x - 5)^2/10 + (y - 8)^2/13 + (z - 11)^2/10 <= 1 && z <= 11, {x, y, z}];DensityPlot3D[1 - ((x - 5)^2/10 + (y - 8)^2/13 + (z - 11)^2/10), {x, y, z} in reg1, PlotPoints -> 100, ColorFunction -> "Su nsetColors", OpacityFunction -> 1, BoxRatios -> {Sqrt[10], Sqrt[13], Sqrt[ 10]}, PlotLegends -> Automatic] (Figure 13) reg1 = ImplicitRegion[0 <= (x - 1)^2/10 + (y - 2)^2/13 + (z - 3)^2/10 <= 1 && z <= -0.15, {x, y, z}];DensityPlot3D[1 - ((x - 1)^2/10+ (y - 2)^2/13 + (z - 3)^2/10),{x, y, z} in reg1, PlotPoints -> 100, ColorFunction -> "Sun setColors", OpacityFunction -> 1, BoxRatios -> {Sqrt[10], Sqrt[13], Sqrt[1 0]}, PlotLegends -> Automatic] (Figure 16) Y. S. Yun, B. Lee / Eur. J. Pure Appl. Math, 18 (1) (2025), 5601 17 of 17 reg1 = ImplicitRegion[0 <= (x - 1)^2/10 + (y - 2)^2/13 + (z - 3)^2/10 <= 1 && z <= 2, {x, y, z}];DensityPlot3D[1 - ((x - 1)^2/10+ (y - 2)^2/13 + (z - 3)^2/10),{x, y, z} in reg1, PlotPoints -> 100, ColorFunction -> "SunsetCo lors", OpacityFunction -> 1, BoxRatios -> {Sqrt[10], Sqrt[13], Sqrt[10]}, PlotLegends -> Automatic] (Figure 19) reg1 = ImplicitRegion[0 <= (x - 1)^2/10 + (y - 2)^2/13 + (z - 3)^2/10 <= 1 && z <= 5, {x, y, z}];DensityPlot3D[1 - ((x - 1)^2/10+ (y - 2)^2/13 + (z - 3)^2/10),{x, y, z} in reg1, PlotPoints -> 100, ColorFunction -> "SunsetCo lors", OpacityFunction -> 1, BoxRatios -> {Sqrt[10], Sqrt[13], Sqrt[10]}, PlotLegends -> Automatic]