Abstract

In most real-world applications, single-valued neutrosophic fuzzy numbers are employed to make judgments in a fuzzy environment. As a result, single-valued neutrosophic fuzzy number ranking is crucial. This paper proposes a novel approach for ranking single-valued neutrosophic fuzzy numbers. The approach was created by combining the α, β, and γ cuts of neutrosophic fuzzy numbers in a convex way. In addition, the proposed approach has been used to prove a number of theorems. This paper also offers a number of numerical examples.

Keywords: Single valued neutrosophic fuzzy numbers, α-cut set, β-cut set, γ-cut set, ranking of neutrosophic fuzzy numbers, convex combination

24.1 Introduction

Real-life phenomena cannot be classified by real numbers due to the existence of linguistic concepts. Fuzzy numbers can be used to rate such issues because fuzzy numbers may deal with verbal words that are ambiguous or vague. As a result, numerous ways for ranking fuzzy numbers have been presented at various times. In addition, the ordering of fuzzy numbers has been thoroughly examined. In the literature, there are several research articles that describe various strategies for sorting fuzzy numbers. In a nutshell, no universally accepted method for sorting fuzzy numbers exists. Existing methods have been demonstrated to give outcomes that are counter-intuitive at times. Furthermore, some of these strategies are non-discrimi-natory and counter intuitive. In 1965, Zadeh [33] “was the first to introduce the notion of fuzzy set theory”. Many academics have worked hard to come up with new and better techniques to rank the unclear numbers. Jain [18, 19] introduced the first unorthodox efforts to the condition of “ordering fuzzy quantities using maximizing set” in 1976. After that, “some ranking methods of fuzzy numbers examined and compared” by Bortolan and Degani [4] and Burnelli and Mezei [5]. In addition, Wang and Kerre [30, 31], Dubois and Prade [15], Chen [6], Choobineh and Li [9], Yu and Dat [32], Chen and Sanguansat [7], Chen et al. [8], Shureshjani and Darehmiraki [25], Chutia [10], Chutia and Chutia [11], Chutia and Gogoi [12], and so forth proposed some ranking methods and some properties of the ranking methods. In 1998, Smarandache [26–28] did the first pioneering work on neutrosophic fuzzy numbers. Ulucay et al. [29], Karaaslan [23], Deli [13], Jana and Pal [21], Jana et al. [20, 22], and others utilized the neutrosophic number for a variety of decision-making situations. Several studies have been conducted on the ordering of single-valued neutrosophic numbers. Peng et al. [24] established a single-valued neutrosophic number ranking system that was subsequently used to “multi-criteria decision making issues”. To rank the single-valued neutrosophic numbers, Deli and Subas [14], Biswas et al. [3], and Aal et al. [1] “presented values and ambiguities of truth membership, indeterminacy-membership, and falsity-membership functions”. Garai et al. [34] “devised a single-valued neutrosophic number ranking system based on possible mean, variance, and standard deviation, which they subsequently applied to multi-attribute group decision-making difficulties”. Garai et al. [16] “proposed a weighted possibility means-based ranking method for single-valued neutrosophic numbers”. Bhaumik et al. [2] “proposed a new ranking method for single-valued neutrosophic numbers established on the (α, β, γ)-cut set, which was then used to the solution of bi-matrix games”.

Because of their linguistic terms, ranking fuzzy numbers plays a distinguishing part in the decision-making process. In some decision-making scenarios, rating single-valued neutrosophic numbers is crucial. Single-valued neutrosophic numbers can be ranked using a variety of approaches. Furthermore, existing single-valued neutrosophic fuzzy number ranking algorithms are incorrect in several instances. As a result, new ways are required to overcome present methods’ limits and downsides. This study makes an effort at doing so. The suggested technique is based on a convex combination of single-valued neutrosophic numbers from the α, β, and γ cut sets.

In Section 24.2, some fundamental definitions and arithmetic of single valued neutrosophic fuzzy numbers are covered. The convex combination of α-cut, β-cut, and γ-cut sets of a single valued neutrosophic fuzzy number are proposed in Section 24.3 as a novel ranking technique for single valued neutrosophic fuzzy numbers. With the aid of the suggested ranking mechanism, we establish certain theorems in Section 24.4. Some numerical examples of the suggested technique are presented in Section 24.5. The proposed method’s conclusion and future studies has been presented in Section 24.6.

24.2 Definition and Representations

This section looked at several definitions and representations for the idea of single-valued neutrosophic fuzzy numbers.

Definition 24.2.1 ([11]). “Let A represent a set and X represent the universal set, with A ∈ X. The set of ordered pairs A = {(x, µ_A(x)): x ∈ X, µ_A,: X → [0, 1]} is then referred to as a fuzzy set and µ_A is the membership function”.

Definition 24.2.2 ([11]). “Let a fuzzy number A = (x, µ_A(x)) = (a, b, c, d; w_A) and if the membership function µ_A(x) is represented as,

then the fuzzy number A is called a generalized trapezoidal fuzzy number (GTrFN)”.

Definition 24.2.3 ([11]). “An intuitionistic fuzzy set (IFS) A on ℝ is defined as A = ⟨x, µ_A(x), ν_A(x)⟩: x ∈ ℝ), where the function µ_A: ℝ → [0, 1] is known as the degree of membership and ν_A: ℝ → [0, 1] is known as the degree of non-membership of the element x ∈ ℝ, such that 0 ≤ µ_A(x) + ν_A(x) ≤ 1”.

Definition 24.2.4 ([11]). “Consider an intuitionistic fuzzy number (IFN) A = ⟨(a, b, c, d; w_A), (p, q, r, s; u_A)⟩, where c ≤ a, q ≤ b ≤ c ≤ r, d ≤ s are the parameters. The membership function and the non-membership function are defined as:

respectively, where w_A is the maximum degree of membership and u_A is the minimum degree of non-membership such that 0 ≤ w_A ≤ 1, 0 ≤ u_A ≤ 1 and 0 ≤ w_A + u_A ≤ 1”.

Definition 24.2.5 ([17]). “Assume that X is the universal set and that A is a set in X. The ordered pair set A = {⟨x, µ_A(x), ν_A(x), ρ_A(x)⟩: x ∈ X} is a single-valued neutrosophic set (SVNS), with µ_A(x) being the truth membership degree, defined as µ_A: X → [0, 1], ν_A(x) being the indeterminacy membership degree, defined as ν_A: X → [0, 1] and ρ_A(x) being the falsity membership degree is defined as ρ_A: X → [0, 1], of the element x to the set A, with the constraint 0 ≤ µ_A(x) + ν_A(x) + ρ_A(x) ≤ 3, ∀x ∈ X”.

Definition 24.2.6 ([17]). “A SVNS A = {⟨x, µ_A(x), ν_A(x), ρ_A(x)⟩∀x ∈ X} is said to be neutrosophic-normal if there exist at least three points a, b, c ∈ X where µ_A: a → {1}, ν_A: b → {1} and ρ_A: c → {1}”.

Definition 24.2.7 ([17]). “An (α, β, γ)-cut set of a SVNS A is a crisp subset over the set of real numbers ℝ which are defined as, A(α, β, γ) = {x,: µ_A(x) ≥ α, ν_A(x) ≤ β, ρ_A(x) ≤ γ}, with 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ α + β + γ ≤ 3”.

Definition 24.2.8 ([17]). “A SVNS, A = {⟨x, µ_A(x), ν_A(x), ρ_A(x)⟩∀x ∈ X} is called neutrosophic-convex if:

1. µ_A(λa + (1 − λ)b) ≥ min(µ_A(a), µ_A(b)),
2. ν_A(λa + (1 − λ)b) ≤ max(ν_A(a), ν_A(b)),
3. ρ_A(λa + (1 − λ)b) ≤ max(ρ_A(a), ρ_A(b)), for all a, b ∈ X and λ ∈ [0, 1]”.

Definition 24.2.9 ([17]). “A single-valued nutrosophic set {A = ⟨x, µ_A(x), ν_A(x), ρ_A(x)⟩∀x ∈ X} is said to be a single-valued neutrosophic number (SVNN) if,

1. A is neutrosophic-normal,
2. A is neutrosophic-convex,
3. µ_A is upper semi-continuous, ν_A, and ρ_A are lower semi-continuous,
4. Support (S(A)) of A is bounded, i.e., S(A) = {µ_A(x) > 0, ν_A < 1, ρ_A < 1, ∀x ∈ X}”.

Definition 24.2.10 ([17]). “A SVNS, A = {⟨x, µ_A(x), ν_A(x), ρ_A(x)⟩∀x ∈ X},= ⟨(a₁, b₁, c₁), (i₁, j₁, k₁), (p₁, q₁, r₁)⟩ in the set of real numbers is said to be single-valued triangular neutrosophic number (SVTNN), such that p₁ ≤ i₁ ≤ a₁ ≤ q₁ ≤ j₁ ≤ b₁ ≤ c₁ ≤ j₁ ≤ r₁ are the parameters. Then, the truth membership degree, the indeterminacy membership degree, and thee falsity membership degree of A can be defined as:”

Definition 24.2.11 ([2]). Cut Sets of SVTNN

“Let, a SVTNN A = {⟨µ_A(x), ν_A(x), ρ_A(x)⟩: x ∈ X} = ⟨(a₁, b₁, c₁), (i₁, j₁, k₁), (p₁, q₁, r₁)⟩, where a₁, b₁, c₁, i₁, j₁, k₁, p₁, q₁, r₁ ∈ R. Then, (α, β, γ)-cut sets of A are defined as:

1. α-cut set: A_α = {x: µ_A(x) ≥ α} = [L^α(A), R^α(A)] = [a₁ + α(i₁ − a₁), p₁ − α(p₁ − i₁)],
2. β-cut set: A_β = {x: ν_A(x) ≤ β} = [L^β(A), R^β(A)] = [b₁ + β(j₁ − b₁), b₃₁ − β(q₁ − j₁)],
3. γ-cut set: A_γ = {x: ρ_A(x) ≤ γ} = [L^γ(A), R^γ(A)] = [c₁ + γ(k₁ − c₁), r₁ − γ(r₁ − k₁)]”.

Definition 24.2.12 ([2]). Defined the Quantity Value

“Let A = {⟨µ_A(x), ν_A(x), ρ_A(x)⟩: x ∈ X} be a SVNN. Then,

1. The value of a SVTNN “A” for the truth membership function is denoted by V_µ(A) and is defined as
   

   where α ∈ [0, 1] and the weighted function f (α) = α satisfies the condition f (0) = 0 and f (1) = 1, such that .

2. The value of a SVTNN “A” for the indeterminacy membership function is denoted by V_ν(A) and is defined as
   

   where β ∈ [0, 1] and the weighted function g(β) = 1 − β satisfy the condition g(0) = 1 and g(1) = 0, such that .

3. The value of a SVTNN “A” for falsity membership function is denoted by V_ρ(A) and is defined as
   

   where γ ∈ [0, 1] and the weighted function h(γ) = 1 − γ satisfy the condition h(0) = 1 and h(1) = 0, such that .

Definition 24.2.13 ([2]). Arithmetic

Let two SVTNNs as A = {⟨µ_A(x), ν_A(x), ρ_A(x)⟩: x ∈ X} = ⟨(a₁, b₁, c₁), (i₁, j₁, k₁), (p₁, q₁, r₁)⟩ ∈ ℝ and B = {⟨µ_B(x), ν_B(x), ρ_B(x)⟩: x ∈ X} = ⟨(a₂, b₂, c₂), (i₂, j₂, k₂), (p₂, q₂, r₂)⟩ ∈ ℝ, where µ_A and µ_B are truth memberships, ν_A and ν_B are indeterminacy memberships, and ρ_A and ρ_B are falsity memberships of A and B, respectively. Then,

1. “A + B = ⟨(a₁ + a₂ − a₁a₂, b₁ + b₂ − b₁b₂, c₁ + c₂ − c₁c₂), (i₁i₂, j₁j₂, k₁k₂), (p₁p₂, p₁q₂, r₁r₂)⟩,
2. A × B = ⟨(a₁b₁, a₂b₂, c₁c₂), (i₁ + i₂ − i₁i₂, j₁ + j₂ − j₁j₂, k₁ + k₂ − k₁k₂), (p₁ + p₂ − p₁p₂, q₁ + q₂ − q₁q₂, r₁ + r₂ − r₁r₂)⟩,
3. κA = ⟨(1 − (1 − a₁)^κ, 1 − (1 − b₁)^κ, 1 − (1 − c₁)^κ), , where κ can be any real number,
4. A^κ = ⟨((a₁)^κ, (b₁)^κ, (c₁)^κ), (1−(1−i₁)^κ, 1−(1−j₁)^κ, 1−(1−k₁)^κ), ((p₁)^κ, 1−(1−q₁)^κ, 1−(1−r₁)^κ)⟩, where κ can be any real number”.

24.3 Proposed Method

Here, a new ranking method for single-valued neutrosophic fuzzy numbers (SVNFNs) has been discussed. Consider a fuzzy number A ∈ X to be a SVTNN, such as A = ⟨x, µ_A(x), ν_A(x), ρ_A(x)⟩ = ⟨[L^α(A), R^α(A)], [L^β(A), R^β(A)], [L^γ(A), R^γ(A)]⟩, 0 ≤ α, β, γ ≤ 1, “where [L^α(A), R^α(A)] is α- cut set, [L^β(A), R^β(A)] is β- cut set and [L^γ(A), R^γ(A)] is γ-cut set for truth membership function, indeterminacy membership function, and falsity membership function” of the SVTNN A, respectively.

Then, the proposed ranking index at α, β, γ cuts is defined as

where λ₁, λ₂, λ₃ ∈ [0, 1].

Thus, the ordering of the fuzzy numbers A, B ∈ X, based on the ranking index is defined as:

1. (A) > (B), then A ≻ B;
2. (A) < (B), then A ≺ B;
3. (A) = (B), then A ~ B;

where (0 ≤ λ₁, λ₂, λ₃ ≤ 1).

24.4 Theorems

Some crucial and fundamental theorems are discussed here.

Theorem 24.4.1. Let be linear.

Proof. Let, A = ⟨(a₁, b₁, c₁), (i₁, j₁, k₁), (p₁, q₁, r₁)⟩ and B = ⟨(a₂, b₂, c₂), (i₂, j₂, k₂), (p₂, q₂, r₂)⟩ be two SVTNNs and κ be any real number. Then, it has to prove that . Then,

Theorem 24.4.2. If A, B, and C are three arbitrary SVTNNs and A ≲ B and B ≲ C are true, then A ≲ C is true.

Proof. Given that A ≲ B, then (A) ≤ (B) for 0 ≤ λ₁, λ₂, λ₃ ≤ 1. Also, given that B ≲ C, then (B) ≤ (C), for 0 ≤ λ₁, λ₂, λ₃ ≤ 1. This implies that (A) ≤ (C), for 0 ≤ λ₁, λ₂, λ₃ ≤ 1. Hence, the result follows that A ≲ C.

24.5 Numerical Examples

In this section, some numerical examples are discussed to assess how well the proposed method performs.

Example 24.5.1. Consider two SVTNNs A = ⟨(0.50, 0.65, 0.80), (0.10, 0.15, 0.30), (0.10, 0.20, 0.30)⟩, B = ⟨(0.10, 0.20, 0.30), (0.20, 0.30, 0.40), (0.40, 0.50, 0.70)⟩ and three weighted functions be P = (λ₁ = 0.3, λ₂ = 0.4, λ₃ = 0.5), Q = (λ₁ = 0.5, λ₂ = 0.5, λ₃ = 0.5) and S = (λ₁ = 1.0, λ₂ = 1.0, λ₃ = 0.9).

Then, the comparison is between two SVTNNs, A and B, as shown in Table 24.1.

Example 24.5.2. Consider two SVTNNs A = ⟨(0.30, 0.45, 0.50), (0.10, 0.20, 0.40), (0.10, 0.20, 0.30)⟩, B = ⟨(0.20, 0.30, 0.35), (0.10, 0.10, 0.10), (0.60, 0.70, 0.80)⟩ and three weighted functions P = (λ₁ = 1.0, λ₂ = 1.0, λ₃ = 1.0), Q = (λ₁ = 0.5, λ₂ = 0.5, λ₃ = 0.5) and S = (λ₁ = 0.1, λ₂ = 0.1, λ₃ = 0.1).

Then, the comparison is between the two SVTNNs, A and B, as shown in Table 24.2.

Example 24.5.3. Consider two SVTNNs A = ⟨(0.6, 0.7, 0.9), (0.1, 0.3, 0.4), (0.1, 0.4, 0.6)⟩, B = ⟨(0.1, 0.2, 0.3), (0.4, 0.5, 0.6), (0.7, 0.8, 0.9)⟩ and three weighted functions P = (λ₁ = 0.3, λ₂ = 0.4, λ₃ = 0.5), Q = (λ₁ = 0.3, λ₂ = 0.6, λ₃ = 0.5) and S = (λ₁ = 0.6, λ₂ = 0.4, λ₃ = 0.7). Then, the comparison is between the two SVTNNs, A and B, as shown in Table 24.3.

λ1,λ2,λ3,	(A)	(B)	Results
0.3,0.4,0.5	0.5217	0.5384	A ≺ B
0.5,0.5,0.5	0.5083	0.5083	A ~ B
1.0,1.0,0.9	0.4534	0.45502	A ≻ B

λ1,λ2,λ3,	(A)	(B)	Results
1.0,1.0,1.0	0.0669	0.5168	A ≺ B
0.5,0.5,0.5	0.4251	0.5460	A ≺ B
0.1,04,0.1	0.4719	0.5693	A ≺ B

λ1,λ2,λ3,	(A)	(B)	Results
0.3,0.4,0.5	0.6584	0.8101	A ≺ B
0.3,0.6,0.5	0.7550	0.7500	A ≻ B
0.6,0.4,0.7	0.7000	0.7401	A ≺ B

24.6 Conclusion

The majority of real-life decision-making is based on fuzzy numbers due to their language connotations. Ordering fuzzy numbers is much more important in order to solve such difficulties. There are several documented ranking systems for single-valued fuzzy numbers. This paper developed a novel ranking approach for single-valued fuzzy numbers based on convex combinations of the fuzzy numbers’ α, β, and γ cut sets. A number of theorems have also been proved using the proposed method. Furthermore, numerical examples have been provided to demonstrate the suggested strategy’s superior performance. This ranking method may be used to look at the rankings of intuitionistic fuzzy numbers, type-2 fuzzy numbers, hesitant fuzzy numbers, and other fuzzy numbers. Further, this proposed method can be applied to real-life problems like medicine, business, economics, computer sciences, etc.

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