Multi-attribute decision-making method based on normal T-spherical fuzzy aggregation operator

Abstract

T-spherical fuzzy numbers (FNs), which add an abstinence degree based on membership and non-membership degrees, can express neutral information conveniently and have a considerable large range of information expression. The normal FNs (NFNs) are very available to characterize normal distribution phenomenon widely existing in social life. In this paper, we first define the normal T-SFNs (NT-SFNs) which can combine the advantages of T-SFNs and NFNs. Then, we define their operational laws, score value, and accuracy value. By considering the interrelationship among multi-input parameters, we propose the Maclaurin symmetric mean operator with NT-SFNs (NT-SFMSM) and its weighted form (NT-SFWMSM). Furthermore, we study some characteristics and special cases of them. Based on the NT-SFWMSM operator, we put forward a novel multi-attribute decision-making (MADM) approach. Finally, some numerical examples are conducted to prove that the proposed approach is valid and superior to some other existing methods.

Keywords

MADM normal T-spherical fuzzy numbers normal distribution

1 Introduction

Multi-attribute decision-making (MADM) is a very important part of modern decision sciences, and it is commonly used in many fields, including engineering, technology, economy, and management. In a realistic decision environment, because of the uncertainty and fuzziness of information, fuzzy numbers (FNs) are more appropriate for expressing attribute values of MADM problems [1 –8]. In 1965, Zadeh [9] firstly gave the idea of the fuzzy set (FS) with membership degree (MD), which is a valid tool to address inaccurate and vague information. In 1986, Atanassov [1, 2] presented intuitionistic FS (IFS) by assigning the non-MD (NMD) to elements of the set. However, IFS cannot handle the situation where the decision-makers (DMs) might remain neutral. For example, in a voting process, the choices of voters are composed of four categories: affirmative vote, abstain vote, negative vote, refusal vote. Considering the inadequacy of IFS in this situation, Cuong [3] put forward the picture FS (PFS) which is an extension of IFS. The PFS uses a triplet (sd, id, dd) to present FNs, where sd, id, dd are the MD, neutral membership degree and NMD, respectively, under the condition that 0 ⩽ sd + id + dd ⩽ 1, and the refusal membership is denoted as r = 1 - (sd + id + dd). The importance of PFS is reflected in a situation where people’s opinions may be positive, negative, abstinence, or refusal. Obviously, this novel concept is more consistent with human cognition. Recently, many researches have been conducted on PFS [10 –14]. However, owing to the constraint sd + id + dd ⩽ 1, the domain of PFS is restricted. In some cases, the sum of three elements in PFSs may exceed one, such as the data (0.7, 0.15, 0.2) from the example in Section 5. To enlarge the range of information expression, Mahmood et al. [15] proposed a novel concept of T-spherical FS (T-SFS), in which the sum of the nth power of the MD, abstinence MD (AMD), and NMD is restricted to one. The T-SFSs are wider in expressing the uncertain information because the PFSs are special situations of T-SFSs. Recently, there have been more and more researches on the expansion of T-SFNs and their application to MADM [16 –20]. Liu et al. [21] proposed the T-spherical fuzzy power Muirhead mean (SPFPMM) operator and weighted SPFPMM operator based on some novel operational rules for spherical FNs (SFNs). Based on the characteristic of Maclaurin symmetric mean (MSM), Liu et al. [22] developed generalized MSM for T-SFSs (T-SFGMSM) and its weighted form (T- SFWGMSM).

Besides, in real life, a lot of phenomena, such as the useful lifespan of productions, random error, and so on, can conform to the normal distribution. So, it is of great significance to study the aggregation of normal distribution information (NDI). Yang and Ko [23] first put forward the normal FNs (NFNs). Li and Liu [24] pointed that the NFNs have two significant advantages compared with triangular and trapezoidal FNs: (1) Many random phenomena in real life obey or approximately obey normal distribution, and NFNs can represent these phenomena very well. (2) The NFNs have the membership function with continuous higher derivative, while other FNs do not have this property. They believe that if it is considered that fuzziness exists at both the macro and micro levels, and the concept is continuous on different scales, then the higher derivatives should also be continuous. Then, Wang et al. [25, 26] proposed the normal IFNs (NIFNs), which can be viewed as the combination of intuitionistic FNs (IFNs) and NFNs, and defined their operational laws and score value (SV). Wang et al. [27] introduced the NIFIGOWA operator for NIFNs and its application in MADM problems. Liu and Teng [28] defined normal interval-valued IFNs (NIVIFNs) in which the MD and NMD were expressed by interval FNs. Liu and Liu [29] proposed some Bonferroni mean (BM) for NIFNs and then to solve the MADM problems.

The information aggregation operator (AO) is an effective tool for solving the MADM problems. In recent years, many kinds of AOs emerged and attracted much attention from researchers. For instance, Xu [30] developed a new dependent OWA which is proposed by Yager [31], and can relieve the impact of biased data. Xu and Yager [32] developed some geometric operators for IFNs. Yager [33] introduced the power average (PA) which can reduce the influence of biased values. However, the above AOs only pay attention to each data or ordered position, ignored the interrelationship among the attributes. Yager [34] emphasized the importance of considering the interrelationship between the criteria. Furthermore, Xu and Yager [35] extended the BM to IFNs. Yu [36] investigated geometric Heronian means (HMs) for IFNs and discussed some properties of them. However, BM and HM operators can only consider the correlation between two input parameters. Compared with the above operators, the MSM operator, which was proposed by Maclaurin [37], and Detemple and Robertson [38], is more appropriate to aggregate information because it is more general than some existing operators considering the relationship among attributes. The most significant advantage of MSM is that it can take into account the relationship among multi-input parameters, and by the different parameter k, it can reduce to BM operator (p = q = 1). Furthermore, the MSM operator is decreasing monotonously as the parameter k increases, which can express the DMs’ risk preferences. Based on the above advantages, many scholars have done further research on MSM operator to deal with MADM problems [39 –48]. Liu and Liu [42] extended the MSM operator to process the IFNs and proposed the intuitionistic fuzzy interaction MSM (IFIMSM) operator and its weighted form. Wei and Lu [46] extended the MSM to a more general information environment and proposed some MSMs for Pythagorean FNs. Liu et al. [48] put forward the power MSM for q-ROFNs (q-ROFPMSM) and its weighted form (q-ROFPWMSM) and studied their application in MADM.

Based on the advantages of the T-SFNs and MSM, the motivations of this article are summarized as follows:

The T-SFNs have a larger scope of information expression, and they are considerably closer to human cognition with the addition of abstinence degree and refusal degree. Moreover, NDI extensively exists in social life and economic activities. So, it is meaningful to combine the NFNs and T-SFNs to propose a novel concept of normal T-SFNs (NT-SFNs) which can express people’s neutral attitudes while reflecting NDI.

The MSM operator has two main advantages, one is that it can consider the interrelationship among two or more input parameters; another is that it can reflect the risk attitude of DMs by adjusting the parameter k. The MSM operator under T-SFNs or NFNs has not been studied. In this paper, we proposed some MSM operators under NT-SFNs.

This study aims to develop a novel method to handle MADM problems. First, a novel information expression form named NT-SFNs is defined. Then, we choose the MSM operator to aggregate this information and further propose the MSM operator for NT-SFNs (NT-SFMSM) and its weighted form (NT-SFWMSM). Finally, we discuss their applications through a numerical example and perform a comparison to check the effectiveness and superiority of our method. In summary, the contribution of our work lies in the following three points:

This paper first proposes the NT-SFNs and their operational rules, SV, and accuracy value (AV). Compared with other information expression forms, the NT-SFNs are more effective to express NDI.

The paper proposes the NT-SFMSM and NT-SFWMSM operators, which consider the interrelationship of multiple attribute values depicted by NT-SFNs, and it also studies some properties and special examples of the NT-SFMSM and the NT-SFWMSM operators.

The paper proposes a novel MADM method based on NT-SFWMSM operator and provides detailed algorithms. A case study and comparative study demonstrate the validity and superiority of the proposed approach.

The remainder of this article is arranged as follows. Section 2 briefly reviews the NFNs, T-SFNs, and MSM operators. In Section 3, we define the NT-SFNs and some operational rules, SV, and AV of them. Section 4 proposes the NT-SFMSM operator and its weighted form NT-SFWMSM operator. In section 5, the NT-SFWMSM operator is utilized to develop a novel MADM method. In section 6, we verify the validity and advantages of our proposed approach by a numerical example and a comparative study. We also perform a sensitivity analysis of the parameters k and m. In section 7, we present a conclusion of our research.

2 Preliminaries

2.1 NFNs

Definition 1. [23] Let R be a real number set, x, η, σ ∈ R. A NFN can be expressed as A = (η, σ) when its MD meets: $A (x) = e^{- {(\frac{x - η}{σ})}^{2}} (σ > 0)$ (1) and N can be regarded as the set of all NFNs.

Definition 2. [49] Let Y, Z ∈ N, and Y = (η, σ) , Z = (b, β), then the operations of Y and Z are defined as follows: $λ Y = λ (η, σ) = (λ η, λ σ) (λ > 0)$ (2) $Y + Z = (η, σ) + (b, β) = (η + b, σ + β)$ (3)

2.2 T-SFS

Definition 3. [15] For any universal set X, a T-SFS is given as follow: $T = {x, sd (x), id (x), dd (x) for all x \in X}$ (4) where sd, id, dd : X → [0, 1] indicate the MD, AMD, and NMD respectively, and meet that 0 ⩽ sd (x) ⁿ + id (x) ⁿ + dd (x) ⁿ ⩽ 1 for some positive integer n, and $r (x) = \sqrt[n]{1 - sd {(x)}^{n} - id {(x)}^{n} - dd {(x)}^{n}}$ is the refusal degree of x in T. The triplet (sd, id, dd) is known as the T-SFN.

2.3 MSM operator

Definition 4. [37] Let a_i (i = 1, 2, …, n) be a group of non-negative real numbers, and k = 1, 2, …, n, then the aggregation function: ${MSM}^{(k)} (a_{1}, a_{2}, \dots, a_{n}) = {(\frac{\sum_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} \prod_{j = 1}^{k} a_{i_{j}}}{C_{n}^{k}})}^{\frac{1}{k}}$ (5) is called the MSM operator, where (i₁, i₂, …, i_k) traversals all the k-tuple combination of (1, 2, …, n), $C_{n}^{k}$ is the binomial coefficient.

Generally, the MSM operator has the following properties:

MSM^(k) (b, b, …, b) = b

MSM^(k) (x₁, x₂, …, x_n) ⩽ MSM^(k) (y₁, y₂, …, y_n) , if x_i ⩽ y_ifor all i

$min_{i} a_{i} ⩽ {MSM}^{(k)} (a_{1}, a_{2}, \dots, a_{n}) ⩽ max_{i} a_{i}$

3 NT-SFNs

Definition 5. Let X be an ordinary finite non-empty set and (η, σ) ∈ N,A =〈 (η, σ) , sd_A, id_A, dd_A 〉 is a NT-SFN, if its MD meets: ${sd}_{A} (x) = {sd}_{A} e^{- {(\frac{x - η}{σ})}^{2}}, x \in X$ (6) its AMD satisfies: ${id}_{A} (x) = 1 - (1 - {id}_{A}) e^{- {(\frac{x - η}{σ})}^{2}}, x \in X$ (7) and its NMD meets: ${dd}_{A} (x) = 1 - (1 - {dd}_{A}) e^{- {(\frac{x - η}{σ})}^{2}}, x \in X$ (8) where 0 ⩽ sd_A ⩽ 1, 0 ⩽ id_A ⩽ 1, 0 ⩽ dd_A ⩽ 1, $0 ⩽ {sd}_{A}^{n} + {id}_{A}^{n} + {dd}_{A}^{n} ⩽ 1$ . When id_A (x) = 0 and n = 1, the NT-SFN degrades into a NIFN. Compared with the NIFNs, the AMDs of NT-SFNs can express neutral opinions of DMs. Compared with T-SFNs and neutrosophic sets [50], the NT-SFNs have the advantage that can express data in the normal distribution. Besides, the NT-SFNs have a considerably large range of information expression and can avoid the loss of valid information. Thus, the NT-SFNs can be viewed as a more general and flexible information expression form.

Example 1. Suppose A =〈 (1.57, 0.78) , 0.7, 0.15, 0.2 〉 is the evaluation on a stock concerning the earnings per share, then (1.570.78) is the mean and standard deviation based on statistical data in a certain period, the degree of alternative A belonging to(1.57, 0.78) is 0.7, the degree of alternative A not belonging to (1.57, 0.78) is 0.2, and the degree of abstinence or neutral membership is 0.15.

Definition 6. Let A =〈 (η, σ_A) , sd_A, id_A, dd_A 〉 and B =〈 (b, σ_B) , sd_B, id_B, dd_B 〉 be any two NT-SFNs, then they have the following operational rules:

$A \oplus B = 〈 (η + b, σ_{A} + σ_{B}), {(1 - (1 - {sd}_{A}^{n}) (1 - {sd}_{B}^{n}))}^{\frac{1}{n}}, {id}_{A} {id}_{B}, {dd}_{A} {dd}_{B} 〉 (9)$

$A \otimes B = 〈 (η b, η b \sqrt{\frac{σ_{A}^{2}}{η^{2}} + \frac{σ_{B}^{2}}{b^{2}}}), {sd}_{A} {sd}_{B}, {(1 - (1 - {id}_{A}^{n}) (1 - {id}_{B}^{n}))}^{\frac{1}{n}}, {(1 - (1 - {dd}_{A}^{n}) (1 - {dd}_{B}^{n}))}^{\frac{1}{n}} 〉 (10)$

$λ A = 〈 (λ η, λ σ_{A}), {(1 - {(1 - {sd}_{A}^{n})}^{λ})}^{\frac{1}{n}}, {id}_{A}^{λ}, {dd}_{A}^{λ} 〉, λ ⩾ 0 (11)$

$A^{λ} = 〈 (η^{λ}, λ^{\frac{1}{2}} η^{λ - 1} σ_{A}), {sd}_{A}^{λ}, {(1 - {(1 - {id}_{A}^{n})}^{λ})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{A}^{n})}^{λ})}^{\frac{1}{n}} 〉, λ ⩾ 0 (12)$

Example 2. LetA =〈 (3, 0.5) , 0.6, 0.1, 0.3 〉,B =〈 (5, 0.4) , 0.7, 0.3, 0.2 〉, n = 3 and λ = 2, then: $A \oplus B = 〈 (3 + 5, 0.5 + 0.4), {(1 - (1 - 0 . 6^{3}) \times (1 - 0 . 4^{3}))}^{\frac{1}{3}}, 0.1 \times 0.3, 0.3 \times 0.2 〉 = 〈 (8, 0.9), 0.7856, 0.03, 0.06 〉$ $\begin{matrix} A \otimes B = 〈 (3 \times 5, 3 \times 5 \times \sqrt{\frac{0 . 5^{2}}{3^{2}} + \frac{0 . 4^{2}}{5^{2}}}), 0.6 \times 0.7, {(1 - (1 - 0 . 1^{3}) \times (1 - 0 . 3^{3}))}^{\frac{1}{3}}, {(1 - (1 - 0 . 3^{3}) \times (1 - 0 . 2^{3}))}^{\frac{1}{3}} 〉 \\ = 〈 (15, 2.7731), 0.42, 0.3036, 0.3264 〉 \end{matrix}$ $\begin{matrix} 2 A = 〈 (2 \times 3, 2 \times 0.5), {(1 - {(1 - 0 . 6^{3})}^{2})}^{\frac{1}{3}}, 0 . 1^{2}, 0 . 3^{2} 〉 = 〈 (6, 1), 0.7277, 0.01, 0.09 〉 \\ A^{2} = 〈 (3^{2}, 2^{\frac{1}{2}} \times 3^{2 - 1} \times 0.5), 0 . 6^{2}, {(1 - {(1 - 0 . 1^{3})}^{2})}^{\frac{1}{3}}, {(1 - {(1 - 0 . 3^{3})}^{2})}^{\frac{1}{3}} 〉 = 〈 (9, 2.1213), 0.36, 0.126, 0.3763 〉 \end{matrix}$

Suppose A =〈 (η, σ_A) , sd_A, id_A, dd_A 〉 and B =〈 (b, σ_B) , sd_B, id_B, dd_B 〉 are any two NT-SFNs, and λ, λ₁, λ₂ ⩾ 0, then we have the following operational properties:

A ⊕ B = B ⊕ A (13)

A ⊗ B = B ⊗ A (14)

λ (A ⊕ B) = λA ⊕ λB (15)

λ₁A ⊕ λ₂A = (λ₁ + λ₂) A (16)

(A ⊗ B) ^λ = A^λ ⊗ B^λ (17)

A^λ
₁ ⊗ A^λ
₂ = A^{(λ₁+λ₂)} (18)

For the proofs of the above properties, please refer to Appendix 1.

Definition 7. Let A =〈 (η, σ_A) , sd_A, id_A, dd_A 〉 be a NT-SFN, then its SVs are defined as follows: $S_{1} (A) = η ({sd}_{A}^{n} - {dd}_{A}^{n})$ (19) $S_{2} (A) = σ ({sd}_{A}^{n} - {dd}_{A}^{n})$ (20)

and its AVs are: $H_{1} = η ({sd}_{A}^{n} + {id}_{A}^{n} + {dd}_{A}^{n})$ (21) $H_{2} = σ ({sd}_{A}^{n} + {id}_{A}^{n} + {dd}_{A}^{n})$ (22)

Definition 8. Let A =〈 (η, σ_A) , sd_A, id_A, dd_A 〉 and B =〈 (b, σ_B) , sd_B, id_B, dd_B 〉 be two NT-SFNs, then:

If S₁ (A) > S₁ (B), thenA > B;

If S₁ (A) = S₁ (B) and H₁ (A) > H₁ (B), then A > B;

If S₁ (A) = S₁ (B) and H₁ (A) = H₁ (B), then

If S₂ (A) < S₂ (B), then A > B;

If S₂ (A) = S₂ (B) and H₂ (A) < H₂ (B), then A > B;

If S₂ (A) = S₂ (B) and H₂ (A) = H₂ (B), then A = B.

Example 3. Let A =〈 (3, 0.5) , 0.6, 0.1, 0.3 〉, B =〈 (5, 0.4) , 0.7, 0.3, 0.2 〉 and n = 2, then we can obtain S₁ (A) = 3 × (0 . 6² - 0 .3²) = 0.81, S₁ (B) = 5 × (0 . 7² - 0 .2²) = 2.25, so S₁ (B) > S₁ (A) and B > A.

4 The normal T-spherical fuzzy MSM operator and its weighted form

4.1 The normal T-spherical fuzzy MSM operator

In this subpart, we develop the NT-SFMSM operator and give some characteristics of it.

Definition 9. Let A_r = 〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 (r = 1, 2, …, m) be a group of NT-SFNs, and k = 1, 2, …, m. Then the NT-SFMSM operator can be defined as follows: $NT - {SFMSM}^{(k)} (A_{1}, A_{2}, \dots, A_{m}) = {(\frac{\underset{1 ⩽ r_{1} < \dots < r_{k} ⩽ m}{\oplus} (\otimes_{j = 1}^{k} A_{r_{j}})}{C_{m}^{k}})}^{\frac{1}{k}}$ (23)

Then we can drive Theorem 1 by the operational rules of NT-SFNs defined in section 3.

Theorem 1. Let A₁, A₂, …, A_m be NT-SFNs, where A_r = 〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 (r = 1, 2, …, m). Then the aggregated result by Equation (23) is also a NT-SFN, an $\begin{matrix} NT - {SFMSM}^{(k)} (A_{1}, A_{2}, \dots, A_{m}) \\ = 〈 ({(\frac{\sum_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}})}{C_{m}^{k}})}^{\frac{1}{k}}, \frac{1}{\sqrt{k}} {(\frac{\sum_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}})}{C_{m}^{k}})}^{\frac{1}{k} - 1} \frac{\sum_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} ((\prod_{j = 1}^{k} η_{r_{j}}) \sqrt{\sum_{j = 1}^{k} \frac{σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})}{C_{m}^{k}}) \\ {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {sd}_{r_{j}}^{n})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{nk}}, {(1 - {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} (1 - {id}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}}, \\ {(1 - {(1 - \prod_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} (1 - {dd}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}} 〉 \end{matrix}$ (24)

For the proof of Theorem 1, please refer to Appendix 2.

Then, some characteristics of the NT-SFMSM operator are given:

Property 1 (Idempotency). If A_r = A₀ =〈 (η₀, σ₀) , sd₀, id₀, dd₀ 〉 for all r (r = 1, 2, …, m), then $NT - {SFMSM}^{(k)} (A_{1}, A_{2}, \dots, A_{m}) = NT - {SFMSM}^{(k)} (A_{0}, A_{0}, \dots, A_{0}) = A_{0}$

Property 2 (Commutativity). Let A_r = 〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 (r = 1, 2, …, m) be a collection of NT-SFNs, and ${\tilde{A}}_{r} = 〈 ({\tilde{η}}_{r}, {\tilde{σ}}_{r}), {\tilde{sd}}_{r}, {\tilde{id}}_{r}, {\tilde{dd}}_{r} 〉$ be a permutation of A_r = 〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 (r = 1, 2, …, m), then $NT - {SFMSM}^{(k)} (A_{1}, A_{2}, \dots, A_{m}) = NT - {SFMSM}^{(k)} ({\tilde{A}}_{1}, {\tilde{A}}_{2}, \dots, {\tilde{A}}_{m}) .$

Property 3 (Monotonicity). Let A_r =〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 and ${\tilde{A}}_{r} = 〈 ({\tilde{η}}_{r}, {\tilde{σ}}_{r}), {\tilde{sd}}_{r}, {\tilde{id}}_{r}, {\tilde{dd}}_{r} 〉$ (r = 1, 2, …, m) be any two collections of NT-SFNs, if $η_{r} ⩽ {\tilde{η}}_{r}, σ_{r} ⩾ {\tilde{σ}}_{r}$ , ${sd}_{r} ⩽ {\tilde{sd}}_{r}, {id}_{r} ⩾ {\tilde{id}}_{r}, {dd}_{r} ⩾ {\tilde{dd}}_{r},$ then $NT - {SFMSM}^{(k)} (A_{1}, A_{2}, \dots, A_{m}) ⩽ NT - {SFMSM}^{(k)} ({\tilde{A}}_{1}, {\tilde{A}}_{2}, \dots, {\tilde{A}}_{m}) .$

Property 4 (Boundedness). Let A_r =〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 (r = 1, 2, …, m) be a collection of NT-SFNs, then $A^{-} ⩽ NT - {SFMSM}^{(k)} (A_{1}, A_{2}, \dots, A_{m}) ⩽ A^{+}$ where, $A^{+} = max_{r} A_{r}, A^{-} = min_{r} A_{r}$ .

All proofs are omitted.

Furthermore, some particular cases of the NT-SFMSM operator are given by setting different parameter k.

(1) When k = 1, the NT-SFMSM reduces to the normal T-spherical fuzzy average operator: $\begin{matrix} NT - {SFMSM}^{(1)} (A_{1}, A_{2}, . . ., A_{m}) \\ = 〈 ({(\frac{\sum_{1 ⩽ r_{1} ⩽ m} (\prod_{j = 1}^{1} η_{r_{j}})}{C_{m}^{1}})}^{\frac{1}{1}}, \frac{1}{\sqrt{1}} {(\frac{\sum_{1 ⩽ r_{1} ⩽ m} (\prod_{j = 1}^{1} η_{r_{j}})}{C_{m}^{1}})}^{\frac{1}{1} - 1} \frac{\sum_{1 ⩽ r_{1} ⩽ m} ((\prod_{j = 1}^{1} η_{r_{j}}) \sqrt{\sum_{j = 1}^{1} \frac{σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})}{C_{m}^{1}}), {(1 - \prod_{1 ⩽ r_{1} ⩽ m} {(1 - \prod_{j = 1}^{1} {sd}_{r_{j}}^{n})}^{\frac{1}{C_{m}^{1}}})}^{\frac{1}{n}}, \\ {(1 - {(1 - \prod_{1 ⩽ r_{1} ⩽ m} {(1 - \prod_{j = 1}^{1} (1 - {id}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{1}}})}^{\frac{1}{1}})}^{\frac{1}{n}}, {(1 - {(1 - \prod_{1 ⩽ r_{1} ⩽ m} {(1 - \prod_{j = 1}^{1} (1 - {dd}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{1}}})}^{\frac{1}{1}})}^{\frac{1}{n}} 〉 \\ = 〈 (\frac{\sum_{r = 1}^{m} η_{r}}{m}, \frac{\sum_{r = 1}^{m} σ_{r}}{m}), {(1 - \prod_{r = 1}^{m} {(1 - {sd}_{r}^{n})}^{\frac{1}{m}})}^{\frac{1}{n}}, {(\prod_{r = 1}^{m} {id}_{r})}^{\frac{1}{m}}, {(\prod_{r = 1}^{m} {dd}_{r})}^{\frac{1}{m}} 〉 \end{matrix}$

(2) When k = 2, the NT-SFMSM operator reduces to the normal T-spherical fuzzy BM operator: $\begin{matrix} NT - {SFMSM}^{(2)} (A_{1}, A_{2}, \dots, A_{m}) \\ = 〈 ({(\frac{\sum_{1 ⩽ r_{1} < r_{2} ⩽ m} (\prod_{j = 1}^{2} η_{r_{j}})}{C_{m}^{2}})}^{\frac{1}{2}}, \frac{1}{\sqrt{2}} {(\frac{\sum_{1 ⩽ r_{1} < r_{2} ⩽ m} (\prod_{j = 1}^{2} η_{r_{j}})}{C_{m}^{2}})}^{\frac{1}{2} - 1} \frac{\sum_{1 ⩽ r_{1} < r_{2} ⩽ m} ((\prod_{j = 1}^{2} η_{r_{j}}) \sqrt{\sum_{j = 1}^{2} \frac{σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})}{C_{m}^{2}}), \\ {(1 - \prod_{1 ⩽ r_{1} < r_{2} ⩽ m} {(1 - \prod_{j = 1}^{2} {sd}_{r_{j}}^{n})}^{\frac{1}{C_{m}^{2}}})}^{\frac{1}{2 n}}, {(1 - {(1 - \prod_{1 ⩽ r_{1} < r_{2} ⩽ m} {(1 - \prod_{j = 1}^{2} (1 - {id}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{2}}})}^{\frac{1}{2}})}^{\frac{1}{n}}, \\ {(1 - {(1 - \prod_{1 ⩽ r_{1} < r_{2} ⩽ m} {(1 - \prod_{j = 1}^{2} (1 - {dd}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{2}}})}^{\frac{1}{2}})}^{\frac{1}{n}} 〉 \\ = 〈 ({(\frac{\sum_{r_{1}, r_{2} = 1 r_{1} \neq r_{2}}^{m} η_{r_{1}} η_{r_{2}}}{m (m - 1)})}^{\frac{1}{2}}, \frac{1}{\sqrt{2}} {(\frac{\sum_{r_{1}, r_{2} = 1 r_{1} \neq r_{2}}^{m} η_{r_{1}} η_{r_{2}}}{m (m - 1)})}^{\frac{1}{2} - 1} \frac{\sum_{r_{1}, r_{2} = 1 r_{1} \neq r_{2}}^{m} (η_{r_{1}} η_{r_{2}} \sqrt{\frac{σ_{r_{1}}^{2}}{η_{r_{1}}^{2}} + \frac{σ_{r_{2}}^{2}}{η_{r_{2}}^{2}}})}{m (m - 1)}), {(1 - \prod_{r_{1}, r_{2} = 1 r_{1} \neq r_{2}}^{m} {(1 - {({sd}_{r_{1}} {sd}_{r_{2}})}^{n})}^{\frac{1}{m (m - 1)}})}^{\frac{1}{2 n}}, \\ {(1 - {(1 - \prod_{r_{1}, r_{2} = 1 r_{1} \neq r_{2}}^{m} {(1 - (1 - {id}_{r_{1}}^{n}) (1 - {id}_{r_{2}}^{n}))}^{\frac{1}{m (m - 1)}})}^{\frac{1}{2}})}^{\frac{1}{n}}, {(1 - {(1 - \prod_{r_{1}, r_{2} = 1 r_{1} \neq r_{2}}^{m} {(1 - (1 - {dd}_{r_{1}}^{n}) (1 - {dd}_{r_{2}}^{n}))}^{\frac{1}{m (m - 1)}})}^{\frac{1}{2}})}^{\frac{1}{n}} 〉 \\ = NT - {SFBM}^{1, 1} (A_{1}, A_{2}, \dots, A_{m}) \end{matrix}$

(3) When k = m, the NT-SFMSM reduces to the normal T-spherical fuzzy geometric mean operator. $\begin{matrix} NT - {SFMSM}^{(m)} (A_{1}, A_{2}, \dots, A_{m}) \\ = 〈 ({(\frac{\sum_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} (\prod_{j = 1}^{m} η_{r_{j}})}{C_{m}^{m}})}^{\frac{1}{m}}, \frac{1}{\sqrt{m}} {(\frac{\sum_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} (\prod_{j = 1}^{m} η_{r_{j}})}{C_{m}^{m}})}^{\frac{1}{m} - 1} \frac{\sum_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} ((\prod_{j = 1}^{m} η_{r_{j}}) \sqrt{\sum_{j = 1}^{m} \frac{σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})}{C_{m}^{m}}) \\ {(1 - \prod_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{m} {sd}_{r_{j}}^{n})}^{\frac{1}{C_{m}^{m}}})}^{\frac{1}{mn}}, {(1 - {(1 - \prod_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{m} (1 - {id}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{m}}})}^{\frac{1}{m}})}^{\frac{1}{n}}, \\ {(1 - {(1 - \prod_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{m} (1 - {dd}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{m}}})}^{\frac{1}{m}})}^{\frac{1}{n}} 〉 \\ = 〈 ({(\prod_{r = 1}^{m} η_{r})}^{\frac{1}{m}}, \frac{1}{\sqrt{m}} {(\prod_{r = 1}^{m} η_{r})}^{\frac{1}{m}} \sqrt{\sum_{r = 1}^{m} \frac{σ_{r}^{2}}{η_{r}^{2}}}), {(\prod_{r = 1}^{m} {sd}_{r})}^{\frac{1}{m}}, {(1 - {(\prod_{r = 1}^{m} (1 - {id}_{r}^{n}))}^{\frac{1}{m}})}^{\frac{1}{n}}, {(1 - {(\prod_{r = 1}^{m} (1 - {dd}_{r}^{n}))}^{\frac{1}{m}})}^{\frac{1}{n}} 〉 \end{matrix}$

4.2 The normal T-spherical fuzzy weighted MSM operator

Definition 10. Let A_r = 〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 (r = 1, 2, …, m) be a group of NT-SFNs, ω = (ω₁, ω₂, …, ω_m) ^T be the weight vector of A_r (r = 1, 2, …, m), and $ω_{r} \in [0, 1], \sum_{r = 1}^{m} ω_{r} = 1$ , then the normal T-spherical fuzzy weighted MSM (NT-SFWMSM) operator can be defined as follows: $NT - {SFWMSM}^{(k)} (A_{1}, A_{2}, \dots, A_{m}) = {(\frac{\underset{1 ⩽ r_{1} < \dots < r_{k} ⩽ m}{\otimes} (\otimes_{j = 1}^{k} A_{r_{j}}^{ω_{r_{j}}})}{C_{m}^{k}})}^{\frac{1}{k}}$ (25)

Theorem 2. Let A₁, A₂, …, A_m be NT-SFNs, where A_r = 〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 (r = 1, 2, …, m) and k = 1, 2, …, m. Then the result aggregated by Equation (25) is also a NT-SFN, and $\begin{matrix} NT - {SFWMSM}^{(k)} (A_{1}, A_{2}, \dots, A_{m}) = {(\frac{1}{C_{m}^{k}} \underset{1 ⩽ r_{1} < \dots < r_{k} ⩽ m}{\oplus} (\otimes_{j = 1}^{k} A_{r_{j}}^{ω_{r_{j}}}))}^{\frac{1}{k}} \\ = 〈 ({(\frac{\sum_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}})}{C_{m}^{k}})}^{\frac{1}{k}}, \frac{1}{\sqrt{k}} {(\frac{\sum_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}})}{C_{m}^{k}})}^{\frac{1}{k} - 1} \frac{\sum_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}}) \sqrt{\sum_{j = 1}^{k} \frac{ω_{r_{j}} σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}}}{C_{m}^{k}}), \\ {(1 - \prod_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {sd}_{r_{j}}^{n ω_{r_{j}}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{nk}}, {(1 - {(1 - \prod_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {(1 - {id}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}}, \end{matrix}$ $\begin{matrix} {(1 - {(1 - \prod_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {(1 - {dd}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}} 〉 \end{matrix}$ (26)

For the proof of Theorem 2, please refer to Appendix 3.

Example 4. Let A₁ =〈 (3, 0.5) , 0.6, 0.1, 0.3 〉, A₂ =〈 (4, 0.4) , 0.7, 0.3, 0.2 〉, A₃ =〈 (5, 0.4) , 0.6, 0.3, 0.4 〉, ω = (0.3, 0.4, 0.4),n = 3and k = 2, then: $\begin{matrix} NT - {SFWMSM}^{(2)} (A_{1}, A_{2}, A_{3}) \\ = 〈 ({(\frac{3^{0.3} \times 4^{0.4} + 3^{0.3} \times 5^{0.4} + 4^{0.4} \times 5^{0.4}}{C_{3}^{2}})}^{\frac{1}{2}}, \frac{1}{\sqrt{2}} \times {(\frac{3^{0.3} \times 4^{0.4} + 3^{0.3} \times 5^{0.4} + 4^{0.4} \times 5^{0.4}}{C_{3}^{2}})}^{\frac{1}{2} - 1} \\ \times \frac{3^{0.3} \times 4^{0.4} \sqrt{\frac{0.3 \times 0 . 5^{2}}{3^{2}} + \frac{0.4 \times 0 . 4^{2}}{4^{2}}} + 3^{0.3} \times 5^{0.4} \sqrt{\frac{0.3 \times 0 . 5^{2}}{3^{2}} + \frac{0.4 \times 0 . 4^{2}}{5^{2}}} + 4^{0.4} \times 5^{0.4} \sqrt{\frac{0.4 \times 0 . 4^{2}}{4^{2}} + \frac{0.4 \times 0 . 4^{2}}{5^{2}}}}{C_{3}^{2}}), \\ {(1 - {((1 - 0 . 6^{3 \times 0.3} \times 0 . 7^{3 \times 0.4}) \times (1 - 0 . 6^{3 \times 0.3} \times 0 . 6^{3 \times 0.4}) \times (1 - 0 . 7^{3 \times 0.4} \times 0 . 6^{3 \times 0.4}))}^{\frac{1}{C_{3}^{2}}})}^{\frac{1}{3 \times 2}}, \\ {(1 - {(1 - {((1 - {(1 - 0 . 1^{3})}^{0.3} \times {(1 - 0 . 3^{3})}^{0.4}) \times (1 - {(1 - 0 . 1^{3})}^{0.3} \times {(1 - 0 . 3^{3})}^{0.4}) \times (1 - {(1 - 0 . 3^{3})}^{0.4} \times {(1 - 0 . 3^{3})}^{0.4}))}^{\frac{1}{C_{3}^{2}}})}^{\frac{1}{2}})}^{\frac{1}{3}}, \\ {(1 - {(1 - {((1 - {(1 - 0 . 3^{3})}^{0.3} \times {(1 - 0 . 2^{3})}^{0.4}) \times (1 - {(1 - 0 . 3^{3})}^{0.3} \times {(1 - 0 . 4^{3})}^{0.4}) \times (1 - {(1 - 0 . 2^{3})}^{0.4} \times {(1 - 0 . 4^{3})}^{0.4}))}^{\frac{1}{C_{3}^{2}}})}^{\frac{1}{2}})}^{\frac{1}{3}} 〉 \\ = 〈 (1.6715, 0.1147), 0.8472, 0.1913, 0.2243 〉 \end{matrix}$

5 An approach to MADM based on NT-SFWMSM operator

In this part, we apply the NT-SFWMSM operator to address MADM problems with NT-SFNs, which can be described as follows: A = {A₁, A₂, . . . , A_m} is a collection of alternatives, C = {C₁, C₂, . . . , C_q} is the set of attributes, and the weight vector of attribute is ω = (ω₁, ω₂, …, ω_q), satisfying that ω_j ∈ [0, 1] (j = 1, 2, …, q) , $\sum_{j = 1}^{q} ω_{j} = 1$ . The evaluation on alternativeA_l (l = 1, 2, . . . , m) under the attribute C_j (j = 1, 2, . . . , q) is a NT-SFN represented by p_lj =〈 (η_lj, σ_lj) , sd_lj, id_lj, dd_lj 〉. Then we can obtain the initial decision matrix P = [p_lj] _m×q. The detailed decision steps are given as follows:

Step 1. Normalization. Generally, attributes in MADM are separated into two different types, cost and benefit attributes. To maintain their consistency, we need to convert the decision-making matrix P = [p_lj] _m×q into the normalization matrix $R = {[r_{lj}]}_{m \times q} = 〈 ({\bar{η}}_{lj}, {\bar{σ}}_{lj}), {\bar{sd}}_{lj}, {\bar{id}}_{lj}, {\bar{dd}}_{lj} 〉$ by the following formula:

For benefit attributes: ${\bar{η}}_{lj} = \frac{η_{lj}}{{max}_{l} (η_{lj})}, {\bar{σ}}_{lj} = \frac{σ_{lj}}{{max}_{l} (σ_{lj})} \frac{σ_{lj}}{a_{lj}}, {\bar{sd}}_{lj} = {sd}_{lj}, {\bar{id}}_{lj} = {id}_{lj}, {\bar{dd}}_{lj} = {dd}_{lj}$ (27)

For cost attributes: ${\bar{η}}_{lj} = \frac{{min}_{l} (η_{lj})}{η_{lj}}, {\bar{σ}}_{lj} = \frac{σ_{lj}}{{max}_{l} (σ_{lj})} \frac{σ_{lj}}{η_{lj}}, {\bar{sd}}_{lj} = {sd}_{lj}, {\bar{id}}_{lj} = {id}_{lj}, {\bar{dd}}_{lj} = {dd}_{lj}$ (28)

Step 2. Utilize the NT-SFWMSM operator to obtain the synthesized value by Equation (26), and have $R_{l} = 〈 (η_{l}, σ_{l}), {sd}_{l}, {id}_{l}, {dd}_{l} 〉 (l = 1, 2, \dots, m)$

Step 3. Calculate the SVs and AVs of R_l. By Equations (19)-(22), we can obtain the appropriate SVs S₁ (R_l) , S₂ (R_l) and the appropriate Avs H₁ (R_l) , H_l (R_l).

Step 4. Order all the alternatives and select the best one by the SVs and AVs.

6 The application of the proposed approach

6.1 A numerical example

Concerning the stock market, investors pay much attention to the analysis of stock investment value. It is of great significance to develop a scientific method to evaluate the values of stocks for improving the quality of listed companies and effectively guiding investors [51]. It is worth noting that many financial indexes can roughly obey normal distribution, so the NT-SFNs are very suitable to express stock valuation information. Then, the above MADM approach with NT-SFNs is applied to stock selection problems.

Suppose an investor in Shanghai wants to select a stock to buy. There are four listed companies A_i (i = 1, 2, 3, 4) to be selected from, and they are all from the electronic component manufacturing industry. In order to better evaluate the investment income of four stocks, the experts selected four extremely important financial indicators: (1) C₁ is earnings per share; (2) C₂ is net asset value per share; (3) C₃ is the undistributed profit per share; (4) C₄ is equity ratio (the ratio of total liabilities to total equity). The weight vector of the indexes is ω = (0.3, 0.25, 0.25, 0.2). The DMs use NT-SFNs to evaluate four indexes of the alternatives, and the evaluation results (i.e., decision matrix P) are listed in Table 1.

Table 1
The evaluation matrix P

C ₁ C ₂

A ₁ < (1.57,0.78),0.7,0.15,0.2> < (0.72,0.43),0.5,0.2,0.1>

A ₂ < (2.18,0.96),0.55,0.1,0.15> < (0.4,0.34),0.7,0.2,0.1>

A ₃ < (1.8,0.91),0.6,0.3,0.15> < (0.95,0.68),0.7,0.1,0.2>

A ₄ < (1.12,0.65),0.6,0.15,0.1> < (0.62,0.47),0.6,0.2,0.15>

C ₃ C ₄

A ₁ < (1.56,1.65),0.6,0.15,0.15> < (4.24,2.57),0.75,0.2,0.15>

A ₂ < (2.13,1.22),0.7,0.15,0.15> < (2.33,3.25),0.55,0.15,0.1>

A ₃ < (3.08,2.15),0.7,0.2,0.2> < (1.79,3.69),0.65,0.15,0.1>

A ₄ < (1.22,0.95),0.75,0.3,0.2> < (4.99,2.95),0.75,0.2,0.25>

	C ₁	C ₂
A ₁	< (1.57,0.78),0.7,0.15,0.2>	< (0.72,0.43),0.5,0.2,0.1>
A ₂	< (2.18,0.96),0.55,0.1,0.15>	< (0.4,0.34),0.7,0.2,0.1>
A ₃	< (1.8,0.91),0.6,0.3,0.15>	< (0.95,0.68),0.7,0.1,0.2>
A ₄	< (1.12,0.65),0.6,0.15,0.1>	< (0.62,0.47),0.6,0.2,0.15>
	C ₃	C ₄
A ₁	< (1.56,1.65),0.6,0.15,0.15>	< (4.24,2.57),0.75,0.2,0.15>
A ₂	< (2.13,1.22),0.7,0.15,0.15>	< (2.33,3.25),0.55,0.15,0.1>
A ₃	< (3.08,2.15),0.7,0.2,0.2>	< (1.79,3.69),0.65,0.15,0.1>
A ₄	< (1.22,0.95),0.75,0.3,0.2>	< (4.99,2.95),0.75,0.2,0.25>

Step 1. Obviously, all of these four attributes are benefit attributes. Then, based on Table 1 and Equation (27), we can get the normalized matrix $\tilde{R}$ showed in Table 2.

Table 2

The normalized decision matrix $\tilde{R}$

	C ₁	C ₂
A ₁	< (0.72,0.404),0.7,0.15,0.2>	< (0.753,0.378),0.5,0.2,0.1>
A ₂	< (1,0.4404),0.55,0.1,0.15>	< (0.421,0.425),0.7,0.2,0.1>
A ₃	< (0.826,0.479),0.6,0.3,0.15>	< (1,0.716),0.7,0.1,0.2>
A ₄	< (0.514,0.393),0.6,0.15,0.1>	< (0.653,0.524),0.6,0.2,0.15>
	C ₃	C ₄
A ₁	< (0.507,0.812),0.6,0.15,0.15>	< (0.85,0.422),0.75,0.2,0.15>
A ₂	< (0.697,0.325),0.7,0.15,0.15>	< (0.467,1.229),0.55,0.15,0.1>
A ₃	< (1,0.698),0.7,0.2,0.2>	< (0.359,2.062),0.65,0.15,0.1>
A ₄	< (0.396,0.344),0.75,0.3,0.2>	< (1,0.473),0.75,0.2,0.25>

Step 2. Use the NT-SFWMSM operator to aggregate the values ${\tilde{r}}_{lj}$ of jth line (j = 1, 2, 3, 4), then obtain the overall evaluation values ${\tilde{r}}_{l}$ of the company A_l (l = 1, 2, 3, 4) (suppose k = 2, n = 3). ${\tilde{r}}_{1} = 〈 (0.9120, 0.3830), 0.8933, 0.1103, 0.0976 〉,$ ${\tilde{r}}_{2} = 〈 (0.8935, 0.5309), 0.8899, 0.0959, 0.0811 〉,$ ${\tilde{r}}_{3} = 〈 (0.9387, 0.9558), 0.9024, 0.1263, 0.1070 〉,$ ${\tilde{r}}_{4} = 〈 (0.8766, 0.3248), 0.9057, 0.1362, 0.1125 〉$

Step 3. Obtain the SV $S_{1} ({\tilde{r}}_{l}) (l = 1, 2, 3, 4)$ of the overall values of the NT-SFNs ${\tilde{r}}_{l} (l = 1, 2, 3, 4)$ . $S_{1} ({\tilde{r}}_{1}) = 0.6492,$ $S_{1} ({\tilde{r}}_{2}) = 0.6292,$ $S_{1} ({\tilde{r}}_{3}) = 0.6886,$ $S_{1} ({\tilde{r}}_{4}) = 0.6499$

Step 4. According to the score $S_{1} ({\tilde{r}}_{l}) (l = 1, 2, 3, 4)$ , we can easily obtain A₃ ≻ A₄ ≻ A₁ ≻ A₂, where “ ≻ ” means “preferred to”. Thus, the best stock is A₃_.

6.2 The influence of the parameters k and n

For the NT-SFWMSM operator, different parameters k and n will produce different synthesized results, and then the ranking result may be affected. Therefore, based on the above example, we can observe the orders of the alternatives by setting different k and n.

First, we will observe the influence of the parameter k on the decision results. We set different values for the parameter k when nis fixed to 3. The final results are displayed in Table 3.

Table 3
Ranking results based on different k

k Ranking results

k = 1 A₃ ≻ A₁ ≻ A₄ ≻ A₂

k = 2 A₃ ≻ A₄ ≻ A₁ ≻ A₂

k = 3 A₃ ≻ A₄ ≻ A₁ ≻ A₂

k = 4 A₃ ≻ A₁ ≻ A₄ ≻ A₂

k	Ranking results
k = 1	A₃ ≻ A₁ ≻ A₄ ≻ A₂
k = 2	A₃ ≻ A₄ ≻ A₁ ≻ A₂
k = 3	A₃ ≻ A₄ ≻ A₁ ≻ A₂
k = 4	A₃ ≻ A₁ ≻ A₄ ≻ A₂

From Table 3, we can know that (1) with the increase of the parameter k, the SVs obtained by NT-SFWMSM operator decreases; (2) the best choice is always A₃ although the final results are slightly different. In the actual decision-making process, DMs could choose an appropriate value k by their risk preference. We usually take k = [q/2] for aggregation, where the symbol [·] means the round function, and q represents the number of attributes. Taking the intermediate value [q/2] is not only simple and intuitional, but also can reflect the neutral risk preferences of DMs, meanwhile, the interrelationship among the attributes can be fully considered. So, in general, if there are no preferences of DMs, we can take k = [q/2].

In the following part, the influence of the parameter n on the final results is explored by using different n to calculate the aggregated values. At this time, we take the intermediate value k = 2. The SVs and decision results are listed in Table 4 and Fig. 1.

Table 4

Ranking results based on different n

	Ranking results
n = 2	A₃ ≻ A₁ ≻ A₄ ≻ A₂
n = 3	A₃ ≻ A₄ ≻ A₁ ≻ A₂
n = 4	A₃ ≻ A₄ ≻ A₁ ≻ A₂
n = 5	A₃ ≻ A₄ ≻ A₁ ≻ A₂
n = 8	A₃ ≻ A₄ ≻ A₁ ≻ A₂
n = 10	A₃ ≻ A₄ ≻ A₁ ≻ A₂

Fig. 1

SVs based on different parameter n.

As indicated in Table 4, the decision results may be slightly different when we take different parametern, but the best alternative is always A₃. In addition, from Fig. 1, we can easily find that the SV is relatively large when the parameter nis relatively small (from 2 to 5); with the increase of n, the SVs S₁ (A) become smaller and smaller. So, when the parameter n takes 2 to 5, we believe that the attitude of DMs is relatively optimistic, while it is more pessimistic when the values of n is larger. Under the condition iⁿ + dⁿ + sⁿ ⩽ 1, the DMs can choose the proper value n according to their preferences.

6.3 Comparative analysis

In this subpart, we verify the effectiveness and superiority of our proposed approach by using five methods to deal with the above example, including the weighted averaging operator for NT-SFNs (NT-SFWA), the weighted geometric operator for NT-SFNs (NT-SFWG), the normal intuitionistic fuzzy weighted geometric BM (NIFWGBM) operator proposed by Liu and Liu [29], the T-spherical fuzzy weighted generalized MSM operator (T-SFWGMSM) proposed by Liu et al. [22] and the proposed method in this paper. Then we compare and analyze their ranking results.

The definitions of NT-SFWA and NT-SFWG are defined as follows:

Definition 11. Let A_r = 〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 (r = 1, 2, …, m) be a collection of NT-SFNs, ω_r = (ω₁, ω₂, …, ω_m) is the weight vector of A_r (r = 1, 2, …, m), and $ω_{r} > 0, \sum_{r = 1}^{m} ω_{r} = 1$ , then $\begin{matrix} NT - SFWA (A_{1}, A_{2}, \dots, A_{m}) = ω_{1} A_{1} \oplus ω_{2} A_{2} \oplus \dots \oplus ω_{m} A_{m} \\ = 〈 (\sum_{r = 1}^{m} ω_{r} η_{r}, \sum_{r = 1}^{m} ω_{r} σ_{r}), {(1 - \prod_{r = 1}^{m} {(1 - {sd}_{r}^{n})}^{ω_{r}})}^{\frac{1}{n}}, \prod_{r = 1}^{m} {id}_{r}^{ω_{r}}, \prod_{r = 1}^{m} {dd}_{r}^{ω_{r}} 〉 \end{matrix}$ (29) $\begin{matrix} NT - SFWG (A_{1}, A_{2}, \dots, A_{m}) = A_{1}^{ω_{1}} \otimes A_{2}^{ω_{2}} \otimes \dots \otimes A_{m}^{ω_{m}} \\ = 〈 (\prod_{r = 1}^{m} η^{ω_{r}}, \prod_{r = 1}^{m} η^{ω_{r}} \sqrt{\sum_{r = 1}^{m} \frac{ω_{r} σ_{r}^{2}}{μ_{r}^{2}}}), \prod_{r = 1}^{m} {sd}_{r}^{ω_{r}}, {(1 - \prod_{r = 1}^{m} {(1 - {id}_{r}^{n})}^{ω_{r}})}^{\frac{1}{n}}, {(1 - \prod_{r = 1}^{m} {(1 - {dd}_{r}^{n})}^{ω_{r}})}^{\frac{1}{n}} 〉 \end{matrix}$ (30)

In order to use Liu and Liu’s approach [29] based on the NIFWGBM operator and Liu et al.’s approach [22] based on the T-SFWGMSM operator, we need to adjust the data in Table 2 from two aspects. (1) Owing to the fact that the method in [29] cannot process these data, we do not consider the AMD in Table 2, that is, we take the AMD as zero, then it turns out that the sums of MD and NMD are all less than one. The adjusted data is shown in Table 5. (2) Because Liu et al.’s approach [22] based on the T-SFWGMSM operator cannot express normal distribution phenomenon, we put aside the mean and standard deviation to transform the NT-SFNs into T-SFNs. The adjusted matrix is presented in Table 6. Then the aggregated results and ranking results are presented in Table 7.

Table 5

Decision matrix expressed by NIFNs

	C ₁	C ₂
A ₁	< (0.72,0.404),0.7,0.2>	< (0.753,0.378),0.5,0.1>
A ₂	< (1,0.4404),0.55,0.15>	< (0.421,0.425),0.7,0.1>
A ₃	< (0.826,0.479),0.6,0.15>	< (1,0.716),0.7,0.2>
A ₄	< (0.514,0.393),0.6,0.1>	< (0.653,0.524),0.6,0.15>
	C ₃	C ₄
A ₁	< (0.507,0.812),0.6,0.15>	< (0.85,0.422),0.75,0.15>
A ₂	< (0.697,0.325),0.75,0.15>	< (0.467,1.229),0.55,0.1>
A ₃	< (1,0.698),0.7,0.2>	< (0.359,2.062),0.65,0.1>
A ₄	< (0.396,0.344),0.75,0.2>	< (1,0.473),0.75,0.25>

Table 6

Decision matrix expressed by T-SFNs

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.7,0.15,0.2)	(0.5,0.2,0.1)	(0.6,0.15,0.15)	(0.75,0.2,0.15)
A ₂	(0.55,0.1,0.15)	(0.7,0.2,0.1)	(0.75,0.15,0.15)	(0.55,0.15,0.1)
A ₃	(0.6,0.3,0.15)	(0.7,0.1,0.2)	(0.7,0.2,0.2)	(0.65,0.15,0.1)
A ₄	(0.6,0.15,0.1)	(0.6,0.2,0.15)	(0.75,0.3,0.2)	(0.75,0.2,0.25)

Table 7

The SVs and final results for different approaches

Methods	Score values	Ranking results
the NT-SFWA operator(when n = 3)	$\begin{matrix} S_{1} (A_{1}) = 0.1947, S_{1} (A_{2}) = 0.1736 \\ S_{1} (A_{3}) = 0.2371, S_{1} (A_{4}) = 0.1926 \end{matrix}$	A₃ ≻ A₁ ≻ A₄ ≻ A₂
the NT-SFWG operator(when n = 3)	$\begin{matrix} S_{1} (A_{1}) = 0.1677, S_{1} (A_{2}) = 0.1495 \\ S_{1} (A_{3}) = 0.2157, S_{1} (A_{4}) = 0.1668 \end{matrix}$	A₃ ≻ A₁ ≻ A₄ ≻ A₂
the NIFWGBM operator proposed by Liu and Liu’s [29] (whenp = 1, q = 1)	$\begin{matrix} S_{1} (A_{1}) = 0.7782, S_{1} (A_{2}) = 0.7654 \\ S_{1} (A_{3}) = 0.8061, S_{1} (A_{4}) = 0.7542 \end{matrix}$	A₃ ≻ A₁ ≻ A₂ ≻ A₄
the T-SFWGMSM operator proposed by Liu et al.’s [22] (when k = 4, r = 2, n = 3, λ₁ = 1, λ₂ = 1)	$\begin{matrix} S_{1} (A_{1}) = 0.2624, S_{1} (A_{2}) = 0.2454 \\ S_{1} (A_{3}) = 0.2887, S_{1} (A_{4}) = 0.3093 \end{matrix}$	A₄ ≻ A₃ ≻ A₁ ≻ A₂
the NT-SFWMSM operator proposed in this paper (when n = 3, k = 1)	$\begin{matrix} S_{1} (A_{1}) = 0.6619, S_{1} (A_{2}) = 0.6362 \\ S_{1} (A_{3}) = 0.6938, S_{1} (A_{4}) = 0.6617 \end{matrix}$	A₃ ≻ A₁ ≻ A₄ ≻ A₂
the NT-SFWMSM operator proposed in this paper (whenn = 3, k = 2)	$\begin{matrix} S_{1} (A_{1}) = 0.6492, S_{1} (A_{2}) = 0.6292 \\ S_{1} (A_{3}) = 0.6886, S_{1} (A_{4}) = 0.6499 \end{matrix}$	A₃ ≻ A₄ ≻ A₁ ≻ A₂
the NT-SFWMSM operator proposed in this paper (when n = 1, k = 2)	$\begin{matrix} S_{1} (A_{1}) = 0.7778, S_{1} (A_{2}) = 0.7653 \\ S_{1} (A_{3}) = 0.8059, S_{1} (A_{4}) = 0.7540 \end{matrix}$	A₃ ≻ A₁ ≻ A₂ ≻ A₄

Table 7 shows that the methods based on the NT-SFWA and the NT-SFWG operators have the same ranking results with our method (when n = 3, k = 1), they are all A₃ ≻ A₁ ≻ A₄ ≻ A₂. However, when we take k = 2, the ranking result turns to A₃ ≻ A₄ ≻ A₁ ≻ A₂. This is because when k = 1, our proposed approach can process the situation in which the attributes are independent of each other, just like the NT-SFWA and NT-SFWG operator. So, the same decision result when k = 1 shows the effectiveness

of our approach, and when k = 2, the different decision result can explain the superiority of our proposed approach because the proposed method can process the interrelated attribute values by adjusting the parameter k. We can also see that the final result obtained by the NIFWGBM operator from [29] (when p = 1, q = 1) is the same with the proposed method in this paper (when n = 1, k = 2), which is A₃ ≻ A₁ ≻ A₂ ≻ A₄, because they can both consider the relationship between two attributes. Thus, the effectiveness of our method is verified.

However, the decision result obtained by the T-SFWGMSM operator (whenk = 4, r = 2, n = 3, λ₁ = 1, λ₂ = 1) is obviously different from the proposed method (whenn = 3, k = 2). The main reason is that the approach based on the T-SFWGMSM operator does not take the mean and the standard deviation into account. From Equations (19)-(22), we can know that the values of the SVs and AVs are directly affected by the mean values and the standard deviation. Generally, the smaller the standard deviation of a set of data, the more stable the set of data. In addition, for benefit attributes, the greater the mean of attribute values, the more desirable the alternatives. Obviously, these impacts cannot be considered using the T-SFWGMSM operator. So, our method is more practical when analyzing the normal distribution phenomenon. Then, we will show the main superiority of our method by comparing it with different ones.

(1) Compare with the approaches based on the NT-SFWA and NT-SFWG operator.

Although the WA and WG operators are elementary and their calculations are very simple, they have the main disadvantage that cannot consider the interrelationship among multiple attributes. However, we can find out that when we take k = 1, our approach can also process the decision problems where all attributes are independent. Obviously, our method is more general and flexible than the approaches based on the NT-SFWA and NT-SFWG operators.

(2) Compared with Liu and Liu’s approach [29] based on the NIFWGBM operator.

The main advantage of Liu and Liu’s approach [29] is that it can process the NIFNs and can capture the interrelationship between two attributes. But the shortcomings of this approach are also obvious. On the one hand, it cannot consider the AMD in the decision-making process. Owing to the condition that the sum of MD and NMD is less than one, its scope of information expression is much narrower than our method. On the other hand, the NIFWGBM operator cannot handle the interrelationship only for two attributes. In the above analysis, it turns out that our approach based on NT-SFWMSM can deal with the interrelationship among multi-attributes by the parameter k, and when k = 2, our proposed approach can reduce to Liu and Liu’s approach [29]. This verifies that our approach is more reliable and general than Liu and Liu’s approach [29] related to the NIFWGBM operator.

(3) Compared with Liu et al.’s method [22] based on the T-SFWGMSM operator.

Liu et al.’s method [22] has advantages that it can process T-spherical fuzzy information and the GMSM operator can be converted into the other operators flexibly by a parameter. However, it cannot deal with the normal distribution phenomenon which widely exists in the reality. The central limit theorem tells us, when the number of random variables is large enough, the mean always obeys the normal distribution. Compared with the T-SFNs, NT-SFNs is of much more practical significance. Moreover, our method has lower computational complexity, because the novel NT-SFWMSM operator has only two parameters while the T-SFWGMSM operator has five. So, the proposed method is more reliable and convenient in the evaluation of the normal distribution phenomenon.

In Table 8, we summarized and compared the main characteristics of the above methods. We can intuitively see the superiority of the proposed method from the table.

Table 8

Comparison of different methods

	capture interrelationship between two attributes	capture interrelationship among multiple attributes	express distribution information	express a wider range of information
Method based on NT-SFWA	No	No	Yes	Yes
Method based on NT-SFWG	No	No	Yes	Yes
Liu and Liu’s approach [29]	Yes	No	Yes	No
Liu et al.’s approach [22]	Yes	Yes	No	Yes
The proposed approach	Yes	Yes	Yes	Yes

7 Conclusions

In this paper, we first defined the NT-SFNs and their operational laws, SV, and AV. Then we proposed the NT-SFMSM and the NT-SFWMSM operators. Furthermore, we discussed some characteristics and particular examples of developed aggregation operators. Based on the NT-SFWMSM operator, we proposed a new MADM approach. To be clear, the main advantages of this study are summarized as follows: (1) this study proposes the novel information form named NT-SFNs, which can express both NDI and the abstinence of DMs while has a considerable wide range of expression; (2) this study extends the MSM operators to NT-SFNs, which can consider the interrelationship among more than two attributes; (3) the proposed method in this paper is applied to solve the selection of stocks, which provides a general solution for many MADM problems. However, the proposed method also has a limitation that the attributes weight vector is given rather than obtained by a certain method. In the future, we will try to combine the NT-SFNs with weighting methods such as entropy method, CRITIC method [52], and maximizing deviation method [53]; we can also study the combination of linguistic information [54, 55] or complex information [56, 57] with the NT-SFNs, and apply them to some other decision-making situation, such as group decision-making, consensus model [58, 59] and so on.

Footnotes

Appendix 1

Suppose A =〈 (η, σ_A) , sd_A, id_A, dd_A 〉 and B =〈 (b, σ_B) , sd_B, id_B, dd_B 〉 are any two NT-SFNs, and λ, λ₁, λ₂ ⩾ 0, then we have the following operational properties:

A ⊕ B = B ⊕ A (13)

A ⊗ B = B ⊗ A (14)

λ (A ⊕ B) = λA ⊕ λB (15)

λ₁A ⊕ λ₂A = (λ₁ + λ₂) A (16)

(A ⊗ B) ^λ = A^λ ⊗ B^λ (17)

A^{λ
₁} ⊗ A^{λ
₂} = A^{(λ₁+λ₂)} (18)

Proof.

(1) Equation (13) is obviously true according to the operation law (1) defined in Equation (9).

(2) Equation (14) is obviously true according to the operation law (2) defined in Equation (10).

(3) For the left of the Equation (15), we have $\begin{matrix} λ (A \oplus B) \\ = λ 〈 (η + b, σ_{A} + σ_{B}), {(1 - (1 - {sd}_{A}^{n}) (1 - {sd}_{B}^{n}))}^{\frac{1}{n}}, {id}_{A} {id}_{B}, {dd}_{A} {dd}_{B} 〉 \\ = 〈 (λ (η + b), λ (σ_{A} + σ_{B})), {(1 - {(1 - {(1 - (1 - {sd}_{A}^{n}) (1 - {sd}_{B}^{n}))}^{\frac{1}{n}})}^{n})}^{λ}, {id}_{A}^{λ} {id}_{B}^{λ}, {dd}_{A}^{λ} {dd}_{B}^{λ} 〉 \\ = 〈 (λ η + λ b, λ σ_{A} + λ σ_{B}), {(1 - {(1 - {sd}_{A}^{n})}^{λ} {(1 - {sd}_{B}^{n})}^{λ})}^{\frac{1}{n}}, {id}_{A}^{λ} {id}_{B}^{λ}, {dd}_{A}^{λ} {dd}_{B}^{λ} 〉 \end{matrix}$

For the right of the Equation (15), we have $λ A = 〈 (λ η, λ σ_{A}), {(1 - {(1 - {sd}_{A}^{n})}^{λ})}^{\frac{1}{n}}, {id}_{A}^{λ}, {dd}_{A}^{λ} 〉,$ $λ B = 〈 (λ b, λ σ_{B}), {(1 - {(1 - {sd}_{B}^{n})}^{λ})}^{\frac{1}{n}}, {id}_{B}^{λ}, {dd}_{B}^{λ} 〉$

then

$\begin{matrix} λ A \oplus λ B \\ = 〈 (λ η + λ b, λ σ_{A} + λ σ_{B}), {(1 - (1 - {({(1 - {(1 - {sd}_{A}^{n})}^{λ})}^{\frac{1}{n}})}^{n}) (1 - {({(1 - {(1 - {sd}_{B}^{n})}^{λ})}^{\frac{1}{n}})}^{n}))}^{\frac{1}{n}}, {id}_{A}^{λ} {id}_{B}^{λ}, {dd}_{A}^{λ} {dd}_{B}^{λ} 〉 \end{matrix}$ $= 〈 (λ η + λ b, λ σ_{A} + λ σ_{B}), {(1 - {(1 - {sd}_{A}^{n})}^{λ} {(1 - {sd}_{B}^{n})}^{λ})}^{\frac{1}{n}}, {id}_{A}^{λ} {id}_{B}^{λ}, {dd}_{A}^{λ} {dd}_{B}^{λ} 〉$

So, we can get λ (A ⊕ B) = λA ⊕ λB.

(4) For the left of the Equation (16), we have $λ_{1} A = 〈 (λ_{1} η, λ_{1} σ_{A}), {(1 - {(1 - {sd}_{A}^{n})}^{λ_{1}})}^{\frac{1}{n}}, {id}_{A}^{λ_{1}}, {dd}_{A}^{λ_{1}} 〉,$ $λ_{2} A = 〈 (λ_{2} η, λ_{2} σ_{A}), {(1 - {(1 - {sd}_{A}^{n})}^{λ_{2}})}^{\frac{1}{n}}, {id}_{A}^{λ_{2}}, {dd}_{A}^{λ_{2}} 〉$

then $\begin{matrix} λ_{1} A \oplus λ_{2} A \\ = 〈 (λ_{1} a + λ_{2} a, λ_{1} σ_{A} + λ_{2} σ_{A}), {(1 - (1 - {({(1 - {(1 - s_{A}^{n})}^{λ_{1}})}^{\frac{1}{n}})}^{n}) (1 - {({(1 - {(1 - s_{A}^{n})}^{λ_{2}})}^{\frac{1}{n}})}^{n}))}^{\frac{1}{n}}, i_{A}^{λ_{1}} i_{A}^{λ_{2}}, d_{A}^{λ_{1}} d_{A}^{λ_{2}} 〉 \\ = 〈 ((λ_{1} + λ_{2}) η, (λ_{1} + λ_{2}) σ_{A}), {(1 - {(1 - {sd}_{A}^{n})}^{λ_{1}} {(1 - {sd}_{A}^{n})}^{λ_{2}})}^{\frac{1}{n}}, {id}_{A}^{λ_{1} + λ_{2}}, {dd}_{A}^{λ_{1} + λ_{2}} 〉 \\ = 〈 ((λ_{1} + λ_{2}) η, (λ_{1} + λ_{2}) σ_{A}), {(1 - {(1 - {sd}_{A}^{n})}^{λ_{1} + λ_{2}})}^{\frac{1}{n}}, {id}_{A}^{λ_{1} + λ_{2}}, {dd}_{A}^{λ_{1} + λ_{2}} 〉 \end{matrix}$ and for the right of the Equation (16), we have $(λ_{1} + λ_{2}) A = 〈 ((λ_{1} + λ_{2}) η, (λ_{1} + λ_{2}) σ_{A}), {(1 - {(1 - {sd}_{A}^{n})}^{λ_{1} + λ_{2}})}^{\frac{1}{n}}, {id}_{A}^{λ_{1} + λ_{2}}, {dd}_{A}^{λ_{1} + λ_{2}} 〉 = λ_{1} A \oplus λ_{2} A$

Thus, the proof is complete.

(5) For the left of the Equation (17), we have $\begin{matrix} {(A \otimes B)}^{λ} = {〈 (η b, ab \sqrt{\frac{σ_{A}^{2}}{η^{2}} + \frac{σ_{B}^{2}}{b^{2}}}), {sd}_{A} {sd}_{B}, {(1 - (1 - {id}_{A}^{n}) (1 - {id}_{B}^{n}))}^{\frac{1}{n}}, {(1 - (1 - {dd}_{A}^{n}) (1 - {dd}_{B}^{n}))}^{\frac{1}{n}} 〉}^{λ} \\ = 〈 ({(η b)}^{λ}, λ^{\frac{1}{2}} {(η b)}^{λ - 1} η b \sqrt{\frac{σ_{A}^{2}}{η^{2}} + \frac{σ_{B}^{2}}{b^{2}}}), {({sd}_{A} {sd}_{B})}^{λ}, (1 - {(1 - {({(1 - (1 - {id}_{A}^{n}) (1 - {id}_{B}^{n}))}^{\frac{1}{n}})}^{n})}^{λ}), \\ (1 - {(1 - {({(1 - (1 - {dd}_{A}^{n}) (1 - {dd}_{B}^{n}))}^{\frac{1}{n}})}^{n})}^{λ}) 〉 \\ = 〈 (η^{λ} b^{λ}, λ^{\frac{1}{2}} η^{λ} b^{λ} \sqrt{\frac{σ_{A}^{2}}{η^{2}} + \frac{σ_{B}^{2}}{b^{2}}}), {sd}_{A}^{λ} {sd}_{B}^{λ}, {(1 - {(1 - {id}_{A}^{n})}^{λ} {(1 - {id}_{B}^{n})}^{λ})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{A}^{n})}^{λ} {(1 - {dd}_{B}^{n})}^{λ})}^{\frac{1}{n}} 〉 \end{matrix}$

for the right of the Equation (17), we have $A^{λ} = 〈 (η^{λ}, λ^{\frac{1}{2}} η^{λ - 1} σ_{A}), {sd}_{A}^{λ}, {(1 - {(1 - {id}_{A}^{n})}^{λ})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{A}^{n})}^{λ})}^{\frac{1}{n}} 〉,$ $B^{λ} = 〈 (b^{λ}, λ^{\frac{1}{2}} b^{λ - 1} σ_{B}), {sd}_{B}^{λ}, {(1 - {(1 - {id}_{B}^{n})}^{λ})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{B}^{n})}^{λ})}^{\frac{1}{n}} 〉$

then $\begin{matrix} A^{λ} \otimes B^{λ} = 〈 (η^{λ} b^{λ}, η^{λ} b^{λ} \sqrt{\frac{{(λ^{\frac{1}{2}} η^{λ - 1} σ_{A})}^{2}}{η^{2 λ}} + \frac{{(λ^{\frac{1}{2}} b^{λ - 1} σ_{B})}^{2}}{b^{2 λ}}}), {sd}_{A}^{λ} {sd}_{B}^{λ}, \\ {(1 - (1 - {({(1 - {(1 - {id}_{A}^{n})}^{λ})}^{\frac{1}{n}})}^{n}) (1 - {({(1 - {(1 - {id}_{B}^{n})}^{λ})}^{\frac{1}{n}})}^{n}))}^{\frac{1}{n}}, \\ {(1 - (1 - {({(1 - {(1 - {dd}_{A}^{n})}^{λ})}^{\frac{1}{n}})}^{n}) (1 - {({(1 - {(1 - {dd}_{B}^{n})}^{λ})}^{\frac{1}{n}})}^{n}))}^{\frac{1}{n}} 〉 \\ = 〈 (η^{λ} b^{λ}, η^{λ} b^{λ} \sqrt{\frac{λ η^{2 λ - 2} σ_{A}^{2}}{η^{2 λ}} + \frac{λ b^{2 λ - 2} σ_{b}^{2}}{b^{2 λ}}}), {sd}_{A}^{λ} {sd}_{B}^{λ}, {(1 - {(1 - {id}_{A}^{n})}^{λ} {(1 - {id}_{B}^{n})}^{λ})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{A}^{n})}^{λ} {(1 - {dd}_{B}^{n})}^{λ})}^{\frac{1}{n}} 〉 \\ = 〈 (η^{λ} b^{λ}, λ^{\frac{1}{2}} η^{λ} b^{λ} \sqrt{\frac{σ_{A}^{2}}{η^{2}} + \frac{σ_{b}^{2}}{b^{2}}}), {sd}_{A}^{λ} {sd}_{B}^{λ}, {(1 - {(1 - {id}_{A}^{n})}^{λ} {(1 - {id}_{B}^{n})}^{λ})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{A}^{n})}^{λ} {(1 - {dd}_{B}^{n})}^{λ})}^{\frac{1}{n}} 〉 \\ = {(A \otimes B)}^{λ} \end{matrix}$

Thus, the proof of Equation (17) is complete.

(6) For the left of the Equation (18), we have $\begin{matrix} A^{λ_{1}} = 〈 (η^{λ_{1}}, λ_{1}^{\frac{1}{2}} η^{λ_{1} - 1} σ_{A}), {sd}_{A}^{λ_{1}}, {(1 - {(1 - {id}_{A}^{n})}^{λ_{1}})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{A}^{n})}^{λ_{1}})}^{\frac{1}{n}} 〉, \\ A^{λ_{2}} = 〈 (η^{λ_{2}}, λ_{2}^{\frac{1}{2}} η^{λ_{2} - 1} σ_{A}), {sd}_{A}^{λ_{2}}, {(1 - {(1 - {id}_{A}^{n})}^{λ_{2}})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{A}^{n})}^{λ_{2}})}^{\frac{1}{n}} 〉 \end{matrix}$

then $\begin{matrix} A^{λ_{1}} \otimes A^{λ_{2}} \\ = 〈 (η^{λ_{1}} η^{λ_{2}}, η^{λ_{1}} η^{λ_{2}} \sqrt{\frac{{(λ_{1}^{\frac{1}{2}} η^{λ_{1} - 1} σ_{A})}^{2}}{η^{2 λ_{1}}} + \frac{{(λ_{2}^{\frac{1}{2}} η^{λ_{2} - 1} σ_{A})}^{2}}{η^{2 λ_{2}}}}), {sd}_{A}^{λ_{1}} {sd}_{A}^{λ_{2}}, \\ {(1 - (1 - {({(1 - {(1 - {id}_{A}^{n})}^{λ_{1}})}^{\frac{1}{n}})}^{n}) (1 - {({(1 - {(1 - {id}_{A}^{n})}^{λ_{2}})}^{\frac{1}{n}})}^{n}))}^{\frac{1}{n}}, \\ {(1 - (1 - {({(1 - {(1 - {dd}_{A}^{n})}^{λ_{1}})}^{\frac{1}{n}})}^{n}) (1 - {({(1 - {(1 - {dd}_{A}^{n})}^{λ_{2}})}^{\frac{1}{n}})}^{n}))}^{\frac{1}{n}} 〉 \\ = 〈 (η^{λ_{1} + λ_{2}}, η^{λ_{1} + λ_{2}} {(λ_{1} + λ_{2})}^{\frac{1}{2}} \sqrt{\frac{σ_{A}^{2}}{η^{2}}}), {sd}_{A}^{λ_{1}} {sd}_{A}^{λ_{2}}, {(1 - {(1 - {id}_{A}^{n})}^{λ_{1}} {(1 - {id}_{A}^{n})}^{λ_{2}})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{A}^{n})}^{λ_{1}} {(1 - {dd}_{A}^{n})}^{λ_{2}})}^{\frac{1}{n}} 〉 \\ = 〈 (η^{λ_{1} + λ_{2}}, η^{λ_{1} + λ_{2} - 1} {(λ_{1} + λ_{2})}^{\frac{1}{2}} σ_{A}), {sd}_{A}^{λ_{1}} {sd}_{A}^{λ_{2}}, {(1 - {(1 - {id}_{A}^{n})}^{λ_{1} + λ_{2}})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{A}^{n})}^{λ_{1} + λ_{2}})}^{\frac{1}{n}} 〉 \end{matrix}$

for the right of the Equation (18), we have $\begin{matrix} A^{λ_{1} + λ_{2}} = 〈 (η^{λ_{1} + λ_{2}}, {(λ_{1} + λ_{2})}^{\frac{1}{2}} η^{λ_{1} + λ_{2} - 1} σ_{A}), {sd}_{A}^{λ_{1} + λ_{2}}, {(1 - {(1 - {id}_{A}^{n})}^{λ_{1} + λ_{2}})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{A}^{n})}^{λ_{1} + λ_{2}})}^{\frac{1}{n}} 〉 \\ = A^{λ_{1}} \otimes A^{λ_{2}} \end{matrix}$

which completes the proof of Equation (18).

Appendix 2

Theorem 1. Let A₁, A₂, . . . , A_m be NT-SFNs, where A_r = 〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 (r = 1, 2, . . . , m). Then the aggregated result from (23) is also a NT-SFN, and (24) $\begin{matrix} NT - {SFMSM}^{(k)} (A_{1}, A_{2}, . . ., A_{m}) \\ = 〈 ({(\frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}})}{C_{m}^{k}})}^{\frac{1}{k}}, \frac{1}{\sqrt{k}} {(\frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}})}{C_{m}^{k}})}^{\frac{1}{k} - 1} \frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} ((\prod_{j = 1}^{k} η_{r_{j}}) \sqrt{\sum_{j = 1}^{k} \frac{σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})}{C_{m}^{k}}) \\ {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {sd}_{r_{j}}^{n})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{nk}}, {(1 - {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} (1 - {id}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}}, \\ {(1 - {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} (1 - {dd}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}} 〉 \end{matrix}$

Proof. By the operations of NT-SFNs, we have $\otimes_{j = 1}^{k} A_{r_{j}} = 〈 (\prod_{j = 1}^{k} η_{r_{j}}, (\prod_{j = 1}^{k} η_{r_{j}}) \sqrt{\sum_{j = 1}^{k} \frac{σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}}), \prod_{j = 1}^{k} {sd}_{r_{j}}, {(1 - \prod_{j = 1}^{n} (1 - {id}_{r_{j}}^{n}))}^{\frac{1}{n}}, {(1 - \prod_{j = 1}^{n} (1 - {dd}_{r_{j}}^{n}))}^{\frac{1}{n}} 〉$

and $\begin{matrix} \underset{1 ⩽ r_{1} < . . . < r_{k} ⩽ m}{\oplus} (\otimes_{j = 1}^{k} A_{r_{j}}) = 〈 (\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}), \sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} ((\prod_{j = 1}^{k} η_{r_{j}}) \sqrt{\sum_{j = 1}^{k} \frac{σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})), {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (1 - \prod_{j = 1}^{k} {sd}_{r_{j}}^{n}))}^{\frac{1}{n}}, \\ \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{n} (1 - {id}_{r_{j}}^{n}))}^{\frac{1}{n}}, \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{n} (1 - {dd}_{r_{j}}^{n}))}^{\frac{1}{n}} 〉 \end{matrix}$

then we can obtain $\begin{matrix} \frac{1}{C_{m}^{k}} \underset{1 ⩽ r_{1} < . . . < r_{k} ⩽ m}{\oplus} (\otimes_{j = 1}^{k} A_{r_{j}}) = 〈 (\frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}})}{C_{m}^{k}}, \frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} ((\prod_{j = 1}^{k} η_{r_{j}}) \sqrt{\sum_{j = 1}^{k} \frac{σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})}{C_{m}^{k}}), {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {sd}_{r_{j}}^{n})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{n}}, \\ \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{n} (1 - {id}_{r_{j}}^{n}))}^{\frac{1}{{nC}_{m}^{k}}}, \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{n} (1 - {dd}_{r_{j}}^{n}))}^{\frac{1}{{nC}_{m}^{k}}} 〉 \end{matrix}$

therefore, $\begin{matrix} NT - {SFMSM}^{(k)} (A_{1}, A_{2}, . . ., A_{m}) = {(\frac{1}{C_{m}^{k}} \underset{1 ⩽ r_{1} < . . . < r_{k} ⩽ m}{\oplus} (\otimes_{j = 1}^{k} A_{r_{j}}))}^{\frac{1}{k}} \\ = 〈 ({(\frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}})}{C_{m}^{k}})}^{\frac{1}{k}}, \frac{1}{\sqrt{k}} {(\frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}})}{C_{m}^{k}})}^{\frac{1}{k} - 1} \frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} ((\prod_{j = 1}^{k} η_{r_{j}}) \sqrt{\sum_{j = 1}^{k} \frac{σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})}{C_{m}^{k}}) \\ {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {sd}_{r_{j}}^{n})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{nk}}, {(1 - {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} (1 - {id}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}}, \\ {(1 - {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} (1 - {dd}_{r_{j}}^{n}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}} 〉 \end{matrix}$

Appendix 3

Theorem 2 Let A₁, A₂, …, A_m be NT-SFNs, where A_r = 〈 (η_r, σ_r) , sd_r, id_r, dd_r 〉 (r = 1, 2, . . . , m) and k = 1, 2, ⋯ , m. Then the result aggregated by Equation (25) is also a NT-SFN, an $\begin{matrix} NT - {SFWMSM}^{(k)} (A_{1}, A_{2}, . . ., A_{m}) = {(\frac{1}{C_{m}^{k}} \underset{1 ⩽ r_{1} < . . . < r_{k} ⩽ m}{\oplus} (\otimes_{j = 1}^{k} A_{r_{j}}^{ω_{r_{j}}}))}^{\frac{1}{k}} \\ = 〈 ({(\frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}})}{C_{m}^{k}})}^{\frac{1}{k}}, \frac{1}{\sqrt{k}} {(\frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}})}{C_{m}^{k}})}^{\frac{1}{k} - 1} \frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}}) \sqrt{\sum_{j = 1}^{k} \frac{ω_{r_{j}} σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}}}{C_{m}^{k}}), \end{matrix}$ (26) $\begin{matrix} {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {sd}_{r_{j}}^{n ω_{r_{j}}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{nk}}, {(1 - {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {(1 - {id}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}}, \\ {(1 - {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {(1 - {dd}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}} 〉 \end{matrix}$

Proof. By the operations of NT-SFNs, we have $A_{r_{j}}^{ω_{r_{j}}} = 〈 (η_{r_{j}}^{ω_{r_{j}}}, ω_{r_{j}}^{\frac{1}{2}} η_{r_{j}}^{ω_{r_{j}} - 1} σ_{r_{j}}), {sd}_{r_{j}}^{ω_{r_{j}}}, {(1 - {(1 - {id}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{n}}, {(1 - {(1 - {dd}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{n}} 〉$

and $\otimes_{j = 1}^{k} A_{r_{j}}^{ω_{r_{j}}} = 〈 (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}}, (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}}) \sqrt{\sum_{j = 1}^{k} \frac{ω_{r_{j}} σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}}), \prod_{j = 1}^{k} {sd}_{r_{j}}^{ω_{r_{j}}}, {(1 - \prod_{j = 1}^{k} {(1 - {id}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{n}}, {(1 - \prod_{j = 1}^{k} {(1 - {dd}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{n}} 〉$

and $\begin{matrix} \underset{1 ⩽ r_{1} < . . . < r_{k} ⩽ m}{\oplus} (\otimes_{j = 1}^{k} A_{r_{j}}^{ω_{r_{j}}}) = 〈 (\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}}), \sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} ((\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}}) \sqrt{\sum_{j = 1}^{k} \frac{ω_{r_{j}} σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})), \\ {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (1 - \prod_{j = 1}^{k} {sd}_{r_{j}}^{n ω_{r_{j}}}))}^{\frac{1}{n}}, \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {(1 - {id}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{n}}, \\ \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {(1 - {dd}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{n}} 〉 \end{matrix}$

then we can obtain $\begin{matrix} \frac{1}{C_{m}^{k}} \underset{1 ⩽ r_{1} < . . . < r_{k} ⩽ m}{\oplus} (\otimes_{j = 1}^{k} A_{r_{j}}^{ω_{r_{j}}}) = 〈 (\frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}})}{C_{m}^{k}}, \frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} ((\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}}) \sqrt{\sum_{j = 1}^{k} \frac{ω_{r_{j}} σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})}{C_{m}^{k}}), \\ {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {sd}_{r_{j}}^{n ω_{r_{j}}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{n}}, \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {(1 - {id}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{{nC}_{m}^{k}}}, \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {(1 - {dd}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{{nC}_{m}^{k}}}, 〉 \end{matrix}$

therefore, $\begin{matrix} NT - {SFWMSM}^{(k)} (A_{1}, A_{2}, . . ., A_{m}) = {(\frac{1}{C_{m}^{k}} \underset{1 ⩽ r_{1} < . . . < r_{k} ⩽ m}{\oplus} (\otimes_{j = 1}^{k} A_{r_{j}}^{ω_{r_{j}}}))}^{\frac{1}{k}} \\ = 〈 ({(\frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}})}{C_{m}^{k}})}^{\frac{1}{k}}, \frac{1}{\sqrt{k}} {(\frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} (\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}})}{C_{m}^{k}})}^{\frac{1}{k} - 1} \frac{\sum_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} ((\prod_{j = 1}^{k} η_{r_{j}}^{ω_{r_{j}}}) \sqrt{\sum_{j = 1}^{k} \frac{ω_{r_{j}} σ_{r_{j}}^{2}}{η_{r_{j}}^{2}}})}{C_{m}^{k}}), \\ {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {sd}_{r_{j}}^{n ω_{r_{j}}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{nk}}, {(1 - {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {(1 - {id}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}}, \\ {(1 - {(1 - \prod_{1 ⩽ r_{1} < . . . < r_{k} ⩽ m} {(1 - \prod_{j = 1}^{k} {(1 - {dd}_{r_{j}}^{n})}^{ω_{r_{j}}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}^{\frac{1}{n}} 〉 \end{matrix}$

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (No. 71771140), Project of cultural masters and “the four kinds of a batch” talents, the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), Major bidding projects of National Social Science Fund of China (19ZDA080).

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Multi-attribute decision-making method based on normal T-spherical fuzzy aggregation operator

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 NFNs

4.1 The normal T-spherical fuzzy MSM operator

6.1 A numerical example

Table 3 Ranking results based on different k k Ranking results k = 1 A3 ≻ A1 ≻ A4 ≻ A2 k = 2 A3 ≻ A4 ≻ A1 ≻ A2 k = 3 A3 ≻ A4 ≻ A1 ≻ A2 k = 4 A3 ≻ A1 ≻ A4 ≻ A2

Footnotes

Appendix 1

Appendix 2

Appendix 3

Acknowledgment

References

Table 3
Ranking results based on different k

k Ranking results

k = 1 A₃ ≻ A₁ ≻ A₄ ≻ A₂

k = 2 A₃ ≻ A₄ ≻ A₁ ≻ A₂

k = 3 A₃ ≻ A₄ ≻ A₁ ≻ A₂

k = 4 A₃ ≻ A₁ ≻ A₄ ≻ A₂