Model for evaluating the security of wireless sensor network with fuzzy number intuitionistic fuzzy information

Abstract

Many experts and scholars focus on the Maclaurin symmetric mean (MSM) equation, which can reflect the interrelationship among the multi-input arguments. It has been generalized to different fuzzy environments and put into use in various actual decision problems. The fuzzy data intuitionistic fuzzy numbers (FNIFNs) could well depict the uncertainties and fuzziness during the security evaluation of Wireless Sensor Network (WSN). And the WSN security evaluation is frequently viewed as the multiple attribute decision-making (MADM) issue. In this paper, we expand the generalized Maclaurin symmetric mean (GMSM) equation with FNIFNs to propose the fuzzy number intuitionistic fuzzy generalized MSM (FNIFGMSM) equation and fuzzy number intuitionistic fuzzy weighted generalized MSM (FNIFWGMSM) equation in this study. A few MADM tools are developed with FNIFWGMSM equation. Finally, taking WSN security evaluation as an example, this paper illustrates effectiveness of the depicted approach. Moreover, by comparing and analyzing the existing methods, the effectiveness and superiority of the FNIFWGMSM method are further certified.

Keywords

Multiple attribute decision making (MADM)fuzzy number intuitionistic fuzzy sets (FNIFSs)fuzzy number intuitionistic fuzzy GMSM (FNIFGMSM) equation fuzzy number intuitionistic fuzzy weighted GMSM (FNIFWGMSM) equation Wireless Sensor Network (WSN)

1 Introduction

In 1965, Zadeh [1] established a novel fuzzy set (FS) to deal with decision information in the fuzzy new domain [2 –5]. In recent years, more and more methods and models are proposed to solve the fuzzy decision-making issues [6 –12]. To extend novel FS, the intuitionistic fuzzy sets (IFSs) [13] was developed. Subsequently, FS and its related extension knowledges are exploited into the more and more decision domains [14 –21]. Li [22] built the GOWA operator to MADM using IFSs. Tan [23] constructed the Choquet integral-based TOPSIS method for IF-MADM. Wu and Zhang [24] built the IF-MADM based on weighted entropy. Zhao, Zheng and Wan [25] defined the Interactive intuitionistic fuzzy algorithms for multilevel programming problems. De and Sana [26] defined the The (p,q,r,l) method for random demand with Bonferroni mean under IFSs. Garg [3] proposed the improved cosine similarity measure for IFSs. Zhao, Wei, Chen and Wei [27] defined the intuitionistic fuzzy MABAC method based on cumulative prospect theory. Arya and Yadav [28] defined the intuitionistic fuzzy super-efficiency slack-based measure. Joshi, Kumar, Gupta and Kaur [29] defined the Jensen-alpha-Norm dissimilarity measure for IFSs. Xiao, Zhang, Wei, Wu, Wei, Guo and Wei [30] built the intuitionistic fuzzy Taxonomy method. Li, Chen, Yang and Li [31] defined the time-preference and VIKOR-based dynamic method for IF-MADM. Zhang, Gao, Wei and Chen [32] built the grey relational analysis method based on cumulative prospect theory for intuitionistic fuzzy MAGDM. [33]built the Taxonomy method for MAGDM based on interval-valued intuitionistic fuzzy with entropy. Furthermore, Liu and Yuan [34] built the fuzzy number IFSs (FNIFSs) to combine the IFSs with the triangular fuzzy sets (TFSs). Wang [35] built the geometric operators under FNIFSs. Verma [36] defined the GFNIFWBM operator under FNIFSs. Wang and Wang [37] defined the FNIFHCG operator under FNIFSs. Lu [38] built the IFNIFHCG operator for international competitiveness assessment. Fan [39] built the FNIFHPWG function to make evaluation about knowledge innovation ability. Wang and Yu [40] defined the FNIFHCA function to evaluate the rural landscape design projects.

Nevertheless, all the functions and tools proposed by the above scholars do not take into account the relationship between parameters [41 –45]. To conquer these shortcomings, the crucial purpose of the article is to connect the FNIFSs with GMSM operator [46, 47] to build several novel fused formulas under FNIFSs.

Consequently, the rest work would be depicted. Several basic concepts of FNIFSs and GMSM formulas would be depicted in the second chapter. The GMSM formulas with FNIFSs would be constructed in the third chapter. An instance about WSN security evaluation is given in the fourth chapter. The conclusions reached will be depicted the last chapter.

2 Preliminaries

In this section, we introduced the concept of fuzzy number intuitionistic fuzzy sets (FNIFSs) [32] and the GMSM operator[47].

2.1 Fuzzy number intuitionistic fuzzy set

Liu and Yuan [34] gave the definition of FNIFS, and the membership and non-membership are given in the form of TFNs.

Definition 1 [34]. Supposed E ={ e₁, e₂, ⋯ , e_n } is a fixed set, B is a FNIFS on E, and gave its’ expression form: $B = \in {〈 e, T_{B} (e), F_{B} (e) 〉 \in E}$ (1)

T_B (e), F_B (e) are two TFNs between 0 and 1, and T_B (e) = (X (e) , Y (e) , Z (e)) , e → [0, 1], F_B (e) = (A (e) , S (e) , D (e)) , E → [0, 1], 0 ⩽ Z (e) + D (e) ⩽ 1, ∀ e ∈ E.

Let T_B (e) = (X (e) , Y (e) , Z (e)), F_B (e) = (A (e) , S (e) , D (e)), so Q (e) =〈 (X (e) , Y (e) , Z (e)) , (A (e) , S (e) , D (e)) 〉, Q (e) is view as a FNIFN.

Definition 2 [35, 48]. Q (e_i) =〈 (X (e_i) , Y (e_i) , Z (e_i)) , (A (e_i) , S (e_i) , D (e_i)) 〉 and Q (e_j) =〈 (X (e_j) , Y (e_j) , Z (e_j)) , (A (e_j) , S (e_j) , D (e_j)) 〉 are two FNIFNs,consequently, (1)

$Q (e_{i}) \oplus Q (e_{j}) = {(\begin{matrix} X (e_{i}) + X (e_{j}) - X (e_{i}) X (e_{j}), \\ Y (e_{i}) + Y (e_{j}) - Y (e_{i}) Y (e_{j}), \\ Z (e_{i}) + Z (e_{j}) - Z (e_{i}) Z (e_{j}) \end{matrix}), (\begin{matrix} A (e_{i}) A (e_{j}), \\ S (e_{i}) S (e_{j}), \\ D (e_{i}) D (e_{j}) \end{matrix})};$

$Q (e_{i}) \otimes Q (e_{j}) = {((\begin{matrix} X (e_{i}) X (e_{j}), \\ Y (e_{i}) Y (e_{j}), \\ Z (e_{i}) Z (e_{j}) \end{matrix})), ((\begin{matrix} A (e_{i}) + A (e_{j}) - A (e_{i}) A (e_{j}), \\ S (e_{i}) + S (e_{j}) - S (e_{i}) S (e_{j}), \\ D (e_{i}) + D (e_{j}) - D (e_{i}) D (e_{j}) \end{matrix}))};$

$λ Q (e_{i}) = {(\begin{matrix} 1 - {(1 - X (e_{i}))}^{λ}, \\ 1 - {(1 - Y (e_{i}))}^{λ}, \\ 1 - {(1 - Z (e_{i}))}^{λ} \end{matrix}), (\begin{matrix} {(A (e_{i}))}^{λ}, \\ {(S (e_{i}))}^{λ}, \\ {(D (e_{i}))}^{λ} \end{matrix})}, λ ⩾ 0;$

${(Q (e_{i}))}^{λ} = {(\begin{matrix} {(X (e_{i}))}^{λ}, \\ {(Y (e_{i}))}^{λ}, \\ {(Z (e_{i}))}^{λ} \end{matrix}), (\begin{matrix} 1 - {(1 - A (e_{i}))}^{λ}, \\ 1 - {(1 - S (e_{i}))}^{λ}, \\ 1 - {(1 - D (e_{i}))}^{λ} \end{matrix})}, λ ⩾ 0 .$

Definition 3 [35, 48]. Let Q (e) =〈 (X (e) , Y (e) , Z (e)) , (A (e) , S (e) , D (e)) 〉 be a given FNIFN, a score function of a FNIFN Q (e) can be depicted:

$\begin{matrix} SF (Q (e)) = \frac{X (e) + 2 Y (e) + Z (e)}{4} \\ - \frac{A (e) + 2 S (e) + D (e)}{4}, SF (Q (e)) \in [- 1, 1] . \end{matrix}$ (2)

Definition 4 [35, 48]. Let Q (e) =〈 (X (e) , Y (e) , Z (e)) , (A (e) , S (e) , D (e)) 〉 be a given FNIFN, an accuracy function of a FNIFN Q (e) can be defined:

$\begin{matrix} AH (Q (e)) = \frac{X (e) + 2 Y (e) + Z (e)}{4} \\ + \frac{A (e) + 2 S (e) + D (e)}{4}, AH (Q (e)) \in [- 1, 1] \end{matrix}$ (3)

Based on the SF (Q (e)) and AH (Q (e)), next, let’s look at the size comparison of the two FNIFN:

Definition 5 [35, 48]. Let Q (e₁) and Q (e₂) be two FNIFNs, then if SF (Q (e₁)) < SF (Q (e₂)), then Q (e₁) < Q (e₂); if SF (Q (e₁)) = SF (Q (e₂)), then (1)

if AH (Q (e₁)) = AH (Q (e₂)), then Q (e₁) = Q (e₂);

if AH (Q (e₁)) < AH (Q (e₂)), then Q (e₁) < Q (e₂).

2.2 GMSM operators

Maclaurin [47] proposed the generalized MSM (GMSM) considering the individual differences.

Definition 9 [40]. Let g_m (m = 1, 2, …, k) be is a real number greater than 0, and λ₁, λ₂, …, λ_k ⩾ 0. A GMSM operator of dimension n is a mapping GMSM^{(k,λ₁,λ₂,…,λ_k)} : (R⁺) ⁿ → R, and it can be defined as follows:

$\begin{matrix} {GMSM}^{(k, λ_{1}, λ_{2}, \dots, λ_{k})} (g_{1}, g_{2}, \dots, g_{k}) \\ = {(\frac{\sum_{1 ⩽ i_{1} < \dots < i_{k} ⩽ i_{n}} \prod_{j = 1}^{k} g_{i_{j}}^{λ_{j}}}{C_{n}^{k}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix}$ (4)

Then we called GMSM^{(k,λ₁,λ₂,…,λ_k)} the GMSM operator, where (i₁, i₂, …, i_k) traverses all the k-tuple combinations of (1, 2, …, n), $C_{n}^{k}$ is the binomial coefficient.

3 FNIFGMSM and FNIFWGMSM operators

3.1 The FNIFGMSM operator

Here we’re going to expand GMSM to coalesce all FNIFNs and establish the fuzzy number intuitionistic fuzzy GMSM (FNIFGMSM) operator.

Definition 7. Let Q (e_m) = 〈 (X (e_m) , Y (e_m) , Z (e_m)) , (A (e_m) , S (e_m) , D (e_m)) 〉 , m = 1, 2, ⋯ k be a set of given FNIFNs. The FNIFGMSM operator could be depicted as:

$\begin{matrix} {FNIFGMSM}^{(n)} (Q (e_{1}), Q (e_{2}), \dots, Q (e_{k})) \\ = {(\frac{\underset{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}}{\oplus} (\otimes_{l = 1}^{n} Z {(x_{m_{l}})}^{λ_{j}})}{C_{n}^{k}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix}$ (5)

Theorem 1. Q (e_m) = 〈 (X (e_m) , Y (e_m) , Z (e_m)) , (A (e_m) , S (e_m) , D (e_m)) 〉 , m = 1, 2, ⋯ k be a suite of given FNIFNs. The coalesced data abtained from FNIFGMSM equations is still an FNIFN.

$\begin{matrix} {FNIFGMSM}^{(n)} (Q (e_{1}), Q (e_{2}), \dots, Q (e_{k})) = {(\frac{\underset{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}}{\oplus} (\otimes_{l = 1}^{n} Z {(x_{m_{l}})}^{λ_{j}})}{C_{n}^{k}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \\ = {(\begin{matrix} {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(Y (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(Z (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - A (e_{m_{j}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - S (e_{m_{j}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - D (e_{m_{j}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix})} \end{matrix}$ (6)

Proof. According to definition 2, we can derive

${(Q (e_{m_{l}}))}^{λ_{j}} = {\begin{matrix} ({(X (e_{m_{l}}))}^{λ_{j}}, {(Y (e_{m_{l}}))}^{λ_{j}}, {(Z (e_{m_{l}}))}^{λ_{j}}), \\ (1 - {(1 - A (e_{m_{l}}))}^{λ_{j}}, 1 - {(1 - S (e_{m_{l}}))}^{λ_{j}}, 1 - {(1 - D (e_{m_{l}}))}^{λ_{j}}) \end{matrix}}$ (7)

Thus,

$\begin{matrix} \otimes_{l = 1}^{n} {(Q (e_{m_{l}}))}^{λ_{j}} \\ = {\begin{matrix} (\prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}}, \prod_{l = 1}^{n} {(Y (e_{m_{l}}))}^{λ_{j}}, \prod_{l = 1}^{n} {(Z (e_{m_{l}}))}^{λ_{j}}), \\ (1 - \prod_{l = 1}^{n} {(1 - A (e_{m_{l}}))}^{λ_{j}}, 1 - \prod_{l = 1}^{n} {(1 - S (e_{m_{l}}))}^{λ_{j}}, 1 - \prod_{l = 1}^{n} {(1 - D (e_{m_{l}}))}^{λ_{j}}) \end{matrix}} \end{matrix}$ (8)

Thereafter, $\begin{matrix} \underset{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}}{\oplus} (\otimes_{l = 1}^{n} {(Q (e_{m_{l}}))}^{λ_{j}}) \\ = {(\begin{matrix} 1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (\prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}}), \\ 1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (\prod_{l = 1}^{n} {(Y (e_{m_{l}}))}^{λ_{j}}), \\ 1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (\prod_{l = 1}^{n} {(Z (e_{m_{l}}))}^{λ_{j}}) \end{matrix}), (\begin{matrix} \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {(1 - A (e_{m_{l}}))}^{λ_{j}}), \\ \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {(1 - S (e_{m_{l}}))}^{λ_{j}}), \\ \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {(1 - D (e_{m_{l}}))}^{λ_{j}}) \end{matrix})} \end{matrix}$

Furthermore,

$\begin{matrix} \begin{matrix} \frac{\underset{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}}{\oplus} (\otimes_{l = 1}^{n} {(Q (e_{m_{l}}))}^{λ_{j}})}{C_{n}^{k}} \\ = {(\begin{matrix} 1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}}, \\ 1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(Y (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}}, \\ 1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(Z (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}} \end{matrix}), (\begin{matrix} \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - A (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}}, \\ \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - S (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}}, \\ \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {(1 - D (e_{m_{l}}))}^{λ_{j}}) \end{matrix})} \end{matrix} \end{matrix}$ (9)

Therefore, $\begin{matrix} \begin{matrix} {FNIFGMSM}^{(n)} (Q (e_{1}), Q (e_{2}), \dots, Q (e_{k})) = {(\frac{\underset{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}}{\oplus} (\otimes_{l = 1}^{n} Z {(x_{m_{l}})}^{λ_{j}})}{C_{n}^{k}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \\ = {(\begin{matrix} {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(Y (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(Z (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix}), \end{matrix} \end{matrix}$

$\begin{matrix} (\begin{matrix} 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - A (e_{m_{j}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - S (e_{m_{j}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - D (e_{m_{j}}))}^{λ_{j}})}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix})} \end{matrix}$ (10)

Hence, (6) is kept.

Then we need to prove that Eq. (6) is still an FNIFN. We need to prove two following conditions:

(X (e) , Y (e) , Z (e))⊆ [0, 1] , (A (e) , S (e) , D (e)) ⊆ [0, 1] ;

0 ⩽ Z (e) + D (e) ⩽ 1 .

Proof. ① Since 0 ⩽ X (e_{m
_l}) ⩽ 1, we get

$\begin{matrix} 0 & ⩽ \prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}} ⩽ 1 and 0 \\ ⩽ {(1 - \prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}} ⩽ 1 \end{matrix}$ (11)

Then, $0 ⩽ \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}} ⩽ 1$ (12)

Thus

$\begin{matrix} 0 ⩽ & (1 - \prod_{11 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} \\ {{(1 - \prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} ⩽ 1 \end{matrix}$ (13)

That is to say: $X (e) = (1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {{(1 - \prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \subseteq [0, 1]$ , similarly, we can get (Y (e) , Z (e)) ⊆ [0, 1] , and (A (e) , S (e) , D (e)) ⊆ [0, 1], so ① is maintained.

② For Z (e_{m
_l}) + D (e_{m
_l}) ⩽ 1, then we can derive Z (e_{m
_l}) ⩽ 1 - D (e_{m
_l}) , thus

$\begin{matrix} 0 ⩽ Z (e) + D (e) \\ = {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(X (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} + 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - D (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \\ ⩽ {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - D (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} + 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - D (e_{m_{l}}))}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \\ = 1 . \end{matrix}$

Next we explore some properties about the FNIFGMSM formula.

Property 1. (Idempotency) If Q (e_m) = 〈 (X (e_m) , Y (e_m) , Z (e_m)) , (A (e_m) , S (e_m) , D (e_m)) 〉 , m = 1, 2, ⋯ , k are equal, then

${FNIFGMSM}^{(n)} (Q (e_{1}), Q (e_{2}), \dots, Q (e_{k})) = Q (e)$ (14)

Property 2. (Monotonicity) Let Q (e_m) = 〈 (X (e_m) , Y (e_m) , Z (e_m)) , (A (e_m) , S (e_m) , D (e_m)) 〉 , m = 1, 2, ⋯ , k and Q (f_m) = 〈 (X (f_m) , Y (f_m) , Z (f_m)) , (A (f_m) , S (f_m) , D (f_m)) 〉 , m = 1, 2, ⋯ , k be two sets of given FNIFNs. If X (e_m) ⩽ X (f_m) , Y (e_m) ⩽ Y (f_m) , Z (e_m) ⩽ Z (f_m), A (e_m) ⩾ A (f_m) , S (e_m) ⩾ S (f_m) , D (e_m) ⩾ D (f_m) hold for all m, then $\begin{matrix} {FNIFGMSM}^{(n)} (Q (e_{1}), Q (e_{2}), \dots, Q (e_{k})) \\ ⩽ {FNIFGMSM}^{(n)} (Q (f_{1}), Q (f_{2}), \dots, Q (f_{k})) \end{matrix}$ (15)

Property 3. (Boundedness) Let Q (e_m) = 〈 (X (e_m) , Y (e_m) , Z (e_m)) , (A (e_m) , S (e_m) , D (e_m)) 〉 , m = 1, 2, ⋯ , k be a set of given FNIFNs. If $Q {(e)}^{+} = {\begin{matrix} (max (X (e_{m})), max (Y (e_{m})), max (Z (e_{m}))), \\ (min (A (e_{m})), min (S (e_{m})), min (D (e_{m}))) \end{matrix}} (m = 1, 2, \dots, k)$ and $Q {(e)}^{-} = {\begin{matrix} (min (X (e_{m})), min (Y (e_{m})), min (Z (e_{m}))), \\ (max (A (e_{m})), max (S (e_{m})), max (D (e_{m}))) \end{matrix}} (m = 1, 2, \dots, k)$ then

$\begin{matrix} Q {(e)}^{-} ⩽ {FNIFGMSM}^{(n)} \\ (Q (e_{q}), Q (e_{2}), \dots, Q (e_{n})) ⩽ Q {(e)}^{+} \end{matrix}$ (16)

Property 4. (Commutativity) Let Q (e_m) (m = 1, 2, …, k) be a set of given FNIFNs, and $Q (e_{m}^{'}) (m = 1, 2, \dots, k)$ is any permutation of Q (e_m) (m = 1, 2, …, k), then ${FNIFGMSM}^{(n)} (Q (e_{1}), Q (e_{2}), \dots, Q (e_{m})) = {FNIFGMSM}^{(n)} (Q (e_{1}^{'}), Q (e_{2}^{'}), \dots, (e_{m}^{'}))$ (17)

3.2 The FNIFWGMSM operator

In real-life MADM, it’s crucial to fully take attribute weights into account. we shall build fuzzy number intuitionistic fuzzy weighted GMSM (FNIFWGMSM) formula.

Definition 8. Let Q (e_m) =〈 (X (e_m) , Y (e_m) , Z (e_m)) , (A (e_m) , S (e_m) , D (e_m)) 〉 (m = 1, 2, …, k) be a set of given FNIFNs with weight vector be ξ_m = (ξ₁, ξ₂, …, ξ_k) ^T, and ξ_m ∈ [0, 1], $\sum_{m = 1}^{k} ξ_{m} = 1$ . If

$\begin{matrix} {FNIFWGMSM}_{k ξ}^{(k, λ_{1}, λ_{2}, \dots, λ_{k})} \\ = {(\frac{\underset{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}}{\oplus} (\otimes_{l = 1}^{n} {(k ξ_{m_{l}} \otimes Q (e_{m_{l}}))}^{λ_{j}})}{C_{n}^{k}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix}$ (18)

Then we called ${FNIFWGMSM}_{k ξ}^{(k, λ_{1}, λ_{2}, \dots, λ_{k})}$ the fuzzy number intuitionistic fuzzy weighted GMSM (FNIFWGMSM) formula.

Theorem 2. Let Q (e_m) =〈 (X (e_m) , Y (e_m) , Z (e_m)) , (A (e_m) , S (e_m) , D (e_m)) 〉 (i = 1, 2, …, n) be a set of given FNIFNs. The coalesced data obtained from FNIFWGMSM formula is still a FNIFN. $\begin{matrix} {FNIFWGMSM}_{k ξ}^{(k, λ_{1}, λ_{2}, \dots, λ_{k})} = {(\frac{\underset{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}}{\oplus} (\otimes_{l = 1}^{n} {(k ξ_{m_{l}} \otimes Q (e_{m_{l}}))}^{λ_{j}})}{C_{n}^{k}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix}$

$\begin{matrix} {\begin{matrix} (\begin{matrix} {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(1 - Y (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(1 - Z (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(A (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(S (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(D (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix}) \end{matrix}} \end{matrix}$ (19)

Proof. According to definition 2, we could derive:

$k ξ_{m_{l}} \otimes Z (e_{m_{l}}) = {\begin{matrix} (1 - {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}}, 1 - {(1 - Y (e_{m_{l}}))}^{k ξ_{m_{l}}}, 1 - {(1 - Z (e_{m_{l}}))}^{k ξ_{m_{l}}}), \\ ({(A (e_{m_{l}}))}^{ξ_{m_{l}}}, {(S (e_{m_{l}}))}^{ξ_{m_{l}}}, {(D (e_{m_{l}}))}^{ξ_{m_{l}}}) \end{matrix}}$ (20)

Thus,

$\begin{matrix} {(k ξ_{m_{l}} \otimes Q (e_{m_{l}}))}^{λ_{j}} \\ = {\begin{matrix} ({(1 - {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}, {(1 - {(1 - Y (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}, {(1 - {(1 - Z (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}), \\ (1 - {(1 - {(A (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}, 1 - {(1 - {(S (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}, 1 - {(1 - {(D (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}) \end{matrix}} \end{matrix}$ (21)

Thereafter,

$\begin{matrix} \otimes_{l = 1}^{n} {(k ξ_{m_{l}} \otimes Q (e_{m_{l}}))}^{λ_{j}} \\ = {\begin{matrix} (\prod_{l = 1}^{n} {(1 - {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}, \prod_{l = 1}^{n} {(1 - {(1 - Y (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}, \prod_{l = 1}^{n} {(1 - {(1 - Z (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}), \\ (1 - \prod_{l = 1}^{n} {(1 - {(A (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}, 1 - \prod_{l = 1}^{n} {(1 - {(S (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}, 1 - \prod_{l = 1}^{n} {(1 - {(D (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}) \end{matrix}} \end{matrix}$ (22)

Furthermore,

$\begin{matrix} \underset{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}}{\oplus} (\otimes_{l = 1}^{n} {(k ξ_{m_{l}} \otimes Q (e_{m_{l}}))}^{λ_{j}}) = {\begin{matrix} (\begin{matrix} 1 - \prod_{1 ⩽ i_{1} < \dots 1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {(1 - {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}), \\ 1 - \prod_{1 ⩽ i_{1} < \dots 1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {(1 - {(1 - Y (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}), \\ 1 - \prod_{1 ⩽ i_{1} < \dots 1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {(1 - {(1 - Z (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}) \end{matrix}), \\ (\begin{matrix} \prod_{1 ⩽ i_{1} < \dots 1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {(1 - {(A (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}), \\ \prod_{1 ⩽ i_{1} < \dots 1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {(1 - {(S (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}), \\ \prod_{1 ⩽ i_{1} < \dots 1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {(1 - {(D (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}}) \end{matrix}) \end{matrix}} \end{matrix}$ (23)

Therefore,

$\begin{matrix} {FNIFWGMSM}_{k ξ}^{(k, λ_{1}, λ_{2}, \dots, λ_{k})} = {(\frac{\underset{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}}{\oplus} (\otimes_{l = 1}^{n} {(k ξ_{m_{l}} \otimes Q (e_{m_{l}}))}^{λ_{j}})}{C_{n}^{k}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \\ {\begin{matrix} (\begin{matrix} {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(1 - Y (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(1 - Z (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix}), \\ (\begin{matrix} 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(A (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(S (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}}, \\ 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} {(1 - {(D (e_{m_{l}}))}^{k ξ_{m_{l}}})}^{λ_{j}})}^{\frac{1}{C_{n}^{k}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \end{matrix}) \end{matrix}} \end{matrix}$ (24)

Hence, (19) is kept.

Then we could prove that Eq. (19) is an FNIFN. We need to prove two following conditions:

(X (e) , Y (e) , Z (e))⊆ [0, 1] , (A (e) , S (e) , D (e)) ⊆ [0, 1] ;

0 ⩽ Z (e) + D (e) ⩽ 1 .

Proof. ① Since 0 ⩽ X (e_{m
_l}) ⩽ 1, we get

$\begin{matrix} 0 & ⩽ {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}} ⩽ 1 and 0 \\ ⩽ \prod_{l = 1}^{n} (1 - {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}}) ⩽ 1 \end{matrix}$ (25)

Then,

$0 ⩽ \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} (1 - {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}}))}^{\frac{1}{C_{k}^{n}}} ⩽ 1$ (26)

Thus,

$\begin{matrix} 0 ⩽ & (1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} \\ {{(1 - {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}}))}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} ⩽ 1 \end{matrix}$ (27)

That means $X (e) = (1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} (1 - \prod_{l = 1}^{n} {{∥ (1 - {(1 - X (e_{m_{l}}))}^{k ξ_{m_{l}}}))}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \subseteq [0, 1]$ , similarly, we can get (Y (e) , Z (e)) ⊆ [0, 1] , and (A (e) , S (e) , D (e)) ⊆ [0, 1], so ① is maintained.

② For Z (e_{m
_l}) + D (e_{m
_l}) ⩽ 1, then we can derive D (e_{m
_l}) ⩽ 1 - Z (e_{m
_l}) , thus $\begin{matrix} 0 ⩽ Z (e) + D (e) = {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} (1 - {(1 - Z (e_{m_{l}}))}^{k ξ_{m_{l}}}))}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \\ + 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} (1 - {(D (e_{m_{l}}))}^{k ξ_{m_{l}}}))}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \\ ⩽ {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} (1 - {(1 - Z (e_{m_{l}}))}^{k ξ_{m_{l}}}))}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} \\ + 1 - {(1 - \prod_{1 ⩽ m_{1} < \dots < m_{n} ⩽ m_{k}} {(1 - \prod_{l = 1}^{n} (1 - {(1 - Z (e_{m_{l}}))}^{k ξ_{m_{l}}}))}^{\frac{1}{C_{k}^{n}}})}^{\frac{1}{\sum_{j = 1}^{k} λ_{j}}} = 1 . \end{matrix}$

Then we will discuss some properties of FNIFWGMSM operator.

Property 5. (Idempotency) If Q (e_m) = 〈 (X (e_m) , Y (e_m) , Z (e_m)) , (A (e_m) , S (e_m) , D (e_m)) 〉 (m = 1, 2, …, k) are equal, then

${FNIFWGMSM}_{k ξ}^{(n)} (Q (e_{1}), Q (e_{2}), \dots, Q (e_{k})) = Q (e)$ (28)

Property 6. (Monotonicity) Let Q (e_m) = 〈 (X (e_m) , Y (e_m) , Z (e_m)) , (A (e_m) , S (e_m) , D (e_m)) 〉 , m = 1, 2, ⋯ , k and Q (f_m) = 〈 (X (f_m) , Y (f_m) , Z (f_m)) , (A (f_m) , S (f_m) , D (f_m)) 〉 , m = 1, 2, ⋯ , k be two sets of given FNIFNs. If X (e_m) ⩽ X (f_m) , Y (e_m) ⩽ Y (f_m) , Z (e_m) ⩽ Z (f_m), A (e_m) ⩾ A (f_m) , S (e_m) ⩾ S (f_m) , D (e_m) ⩾ D (f_m), hold for all m, then $\begin{matrix} {FNIFWGMSM}_{k ξ}^{(n)} (Q (e_{1}), Q (e_{2}), \dots, Q (e_{k})) \\ ⩽ {FNIFWGMSM}_{k ξ}^{(n)} (Q (f_{1}), Q (f_{2}), \dots, Q (f_{k})) \end{matrix}$ (29)

Property 7. (Boundedness) Let Q (e_m) = 〈 (X (e_m) , Y (e_m) , Z (e_m)) , (A (e_m) , S (e_m) , D (e_m)) 〉 , m = 1, 2, ⋯ , k be a set of given FNIFNs. If $Q {(e)}^{+} = {\begin{matrix} (max (X (e_{m})), max (Y (e_{m})), max (Z (e_{m}))), \\ (min (A (e_{m})), min (S (e_{m})), min (D (e_{m}))) \end{matrix}} (m = 1, 2, \dots, k)$ and $Q {(e)}^{-} = {\begin{matrix} (min (X (e_{m})), min (Y (e_{m})), min (Z (e_{m}))), \\ (max (A (e_{m})), max (S (e_{m})), max (D (e_{m}))) \end{matrix}} (m = 1, 2, \dots, k)$ then

$\begin{matrix} Q {(e)}^{-} ⩽ {FNIFWGMSM}_{k ξ}^{n} (Q (e_{q}), Q (e_{2}), \dots, \\ Q (e_{k})) ⩽ Q {(e)}^{+} \end{matrix}$ (30)

Property 8. (Commutativity) Let Q (e_m) (m = 1, 2, …, k) be a set of FNIFNs, and $Q (e_{m}^{'}) (m = 1, 2, \dots, k)$ is any permutation of Q (e_m) (m = 1, 2, …, k), then ${FNIFWGMSM}_{k ξ}^{n} (Q (e_{1}), Q (e_{2}), \dots, Q (e_{m})) = {FNIFWGMSM}_{k ξ}^{n} (Q (e_{1}^{'}), Q (e_{2}^{'}), \dots, Q (e_{m}^{'}))$ (31)

4 Data instance and comparative analysis

4.1 Data instance

A wireless sensor network (WSN) is a network constructed through the self-organization of a-large number of sensor nodes. Considering ongoing network security problems, considerable research is focusing on the security of WSNs from different perspectives, with the goal of ensuring that these networks are operating effectively. In terms of security issues, the problems encountered by WSNs and traditional wireless networks can widely differ based on the characteristics of the individual network. Since the resources for sensor nodes are limited, processing power, storage space, energy, and other factors can prevent the direct application of effective security protocols and algorithms to wireless sensor networks, resulting in greater security risks. Given the popular application of wireless sensor networks, the security problems experienced by WSNs are becoming increasingly concerning. Therefore, determining the most effective measures to ensure the safety of wireless sensor networks has become a common concern. The WSN security evaluation could de regarded as the MADM or MAGDM problem [49 –54]. A point in case about the WSN security evaluation with FNIFNs would be utilized to illustrate the above MADM methods. We shall give 5 possible computer network systems H_i (i = 1, 2, 3, 4, 5) to choose. The experts select four attributes to evaluate the WSN security of these computer network systems: ① J₁ represents tactics; ② J₂ means technology; ③ J₃ represents economy; ④ J₄ means the logistics and strategy. Several computer network systems shall be depicted with FNIFNs by the DMs on the strength of 4 criterions (whose weighting vector ξ = (0.25, 0.20, 0.15, 0.40)), the FNIFN decision matrix as depicted in Table 1.

Table 1
The FNIFN DM

J ₁ J ₂

H1 〈 (0.5, 0.6, 0.6) , (0.1, 0.2, 0.2)〉〈 (0.6, 0.6, 0.7) , (0.1, 0.1, 0.2)〉

H2 〈 (0.4, 0.6, 0.6) , (0.1, 0.2, 0.3)〉〈 (0.1, 0.4, 0.5) , (0.2, 0.3, 0.4)〉

H3 〈 (0.3, 0.4, 0.4) , (0.4, 0.5, 0.5)〉〈 (0.2, 0.2, 0.4) , (0.1, 0.2, 0.2)〉

H4 〈 (0.4, 0.5, 0.5) , (0.3, 0.4, 0.4)〉〈 (0.2, 0.3, 0.3) , (0.4, 0.4, 0.6)〉

H5 〈 (0.4, 0.4, 0.6) , (0.2, 0.2, 0.3)〉〈 (0.5, 0.6, 0.6) , (0.2, 0.3, 0.3)〉

J ₃ J ₄

H1 〈 (0.4, 0.5, 0.6) , (0.2, 0.3, 0.3)〉〈 (0.6, 0.6, 0.7) , (0.1, 0.1, 0.2)〉

H2 〈 (0.2, 0.2, 0.4) , (0.3, 0.3, 0.4)〉〈 (0.1, 0.3, 0.4) , (0.2, 0.3, 0.5)〉

H3 〈 (0.1, 0.1, 0.4) , (0.3, 0.3, 0.4)〉〈 (0.4, 0.5, 0.5) , (0.3, 0.3, 0.4)〉

H4 〈 (0.1, 0.2, 0.3) , (0.2, 0.5, 0.6)〉〈 (0.2, 0.3, 0.3) , (0.2, 0.5, 0.5)〉

H5 〈 (0.5, 0.7, 0.7) , (0.1, 0.2, 0.2)〉〈 (0.3, 0.4, 0.5) , (0.1, 0.2, 0.4)〉

	J ₁	J ₂
H1	〈 (0.5, 0.6, 0.6) , (0.1, 0.2, 0.2)〉	〈 (0.6, 0.6, 0.7) , (0.1, 0.1, 0.2)〉
H2	〈 (0.4, 0.6, 0.6) , (0.1, 0.2, 0.3)〉	〈 (0.1, 0.4, 0.5) , (0.2, 0.3, 0.4)〉
H3	〈 (0.3, 0.4, 0.4) , (0.4, 0.5, 0.5)〉	〈 (0.2, 0.2, 0.4) , (0.1, 0.2, 0.2)〉
H4	〈 (0.4, 0.5, 0.5) , (0.3, 0.4, 0.4)〉	〈 (0.2, 0.3, 0.3) , (0.4, 0.4, 0.6)〉
H5	〈 (0.4, 0.4, 0.6) , (0.2, 0.2, 0.3)〉	〈 (0.5, 0.6, 0.6) , (0.2, 0.3, 0.3)〉
	J ₃	J ₄
H1	〈 (0.4, 0.5, 0.6) , (0.2, 0.3, 0.3)〉	〈 (0.6, 0.6, 0.7) , (0.1, 0.1, 0.2)〉
H2	〈 (0.2, 0.2, 0.4) , (0.3, 0.3, 0.4)〉	〈 (0.1, 0.3, 0.4) , (0.2, 0.3, 0.5)〉
H3	〈 (0.1, 0.1, 0.4) , (0.3, 0.3, 0.4)〉	〈 (0.4, 0.5, 0.5) , (0.3, 0.3, 0.4)〉
H4	〈 (0.1, 0.2, 0.3) , (0.2, 0.5, 0.6)〉	〈 (0.2, 0.3, 0.3) , (0.2, 0.5, 0.5)〉
H5	〈 (0.5, 0.7, 0.7) , (0.1, 0.2, 0.2)〉	〈 (0.3, 0.4, 0.5) , (0.1, 0.2, 0.4)〉

Then, the FNIFWDMSM operator is used to deal with WSN security evaluation with FNIFNs.

Step 1. According to Table 1, we can fuse all FNIFNs r_ij by FNIFWMSM, let U = (1, 2, 3, 4)) and FNIFWDMSM operator to calculate global FNIFNs H_i (i = 1, 2, 3, 4, 5) of the computer network systems H_i, the coalesced values are shown in Table 2 (n = 2)

Table 2

The coalesced values by the FNIFWMSM operators

Aggregation operator	computer network systems	Operation results
FNIFWGMSM	H₁	< (0.5190,0.5639,0.6309),(0.1771,0.1916,0.2442)>
	H2	< (0.2063,0.3927,0.4752),(0.2084,0.2913,0.4111)>
	H3	< (0.2557,0.3101,0.4128),(0.2687,0.3376,0.3745)>
	H4	< (0.2415,0.3384,0.3539),(0.3096,0.4573,0.5250)>
	H5	< (0.4060,0.4962,0.5756),(0.1868,0.2538,0.3237)>

Step 2. The SF of computer network systems are calculated in Table 3.

Step 3. According to Table 3, the order of the computer network systems are depicted in Table 4. Note that “> ” means “preferred to”. The best computer network systems is H₁.

6.2 Comparative analysis

In this section, we will compare the technology depicted with other technologies, and the conclusions are shown as Table 6.

Table 3
The SF of the computer network systems

FNIFWMSM

H₁ 0.3683

H2 0.0662

H3 –0.0074

H4 –0.1192

H5 0.2390

	FNIFWMSM
H₁	0.3683
H2	0.0662
H3	–0.0074
H4	–0.1192
H5	0.2390

Table 4

Order of the computer network systems

	Order
FNIFWGMSM	H₁ > H₅ > H₂ > H₃ > H₄

Table 5

Ranking results for different parameters of the FNIFWMSM operator

	SF(H₁)	SF(H₂)	SF(H₃)	SF(H₄)	SF(H₅)	Ordering
n = 1	0.4416	0.0866	0.0572	–0.0943	0.2745	H₁ > H₅ > H₂ > H₃ > H₄
n = 2	0.3683	0.0662	–0.0074	–0.1192	0.2390	H₁ > H₅ > H₂ > H₃ > H₄
n = 3	0.3657	0.0583	–0.0249	–0.1293	0.2291	H₁ > H₅ > H₂ > H₃ > H₄
n = 4	0.3658	0.0405	–0.0551	–0.1541	0.2168	H₁ > H₅ > H₂ > H₃ > H₄

Table 6

Comparative analysis

	Order
FNIFWG operator [35]	H₁ > H₅ > H₂ > H₃ > H₄
FNIFWA operator [48]	H₁ > H₅ > H₂ > H₃ > H₄
FNIFWGMSM	H₁ > H₅ > H₂ > H₃ > H₄

From above analysis, Compared the result of the proposed FNIFWGMSM operator with FNIFWA and FNIFWG formulas, these schemes rank a little differently and the optimal alternative is not different. The FNIFWA and FNIFWG formulas, don’t consider the relationship between given arguments, which can’t correctly estimate the effect of different values of n arguments on the end results. The FNIFWGMSM formula could perfectly consider the relationship between different values of n being fused.

7 Conclusion

With the development of science and technology, performance of sensors has been greatly improved. Wireless sensor networks have a wide range of applications in the fields of smart grid, intelligent transportation, aerospace and so on. However, wireless sensor networks are more vulnerable to malicious cyber-attacks and more frangible in security due to the intrinsic interconnections among the sensors. Therefore, ensuring the security of wireless sensor networks is more important and urgent than ever. At present, three categories of cyber-attacks are commonly investigated: denial-of-service attacks, false data injection attacks and replay attacks. In this paper, we study the MADM issues with FNIFNs. and utilize the GMSM formula to build several GMSM fused formulas with FNIFNs: FNIFGMSM operator and FNIFWGMSM operator. The characteristic of these two operators is also deliberated. The FNIFWGMSM formula is utilized to cope with the MADM issues with FNIFNs. Finally, taking WSN security evaluation as an example, this paper illustrates effectiveness of the depicted approach. Moreover, by comparing and analyzing the existing methods, the effectiveness and superiority of the FNIFWGMSM method are further certified. In our future research, the proposed methods and algorithms will be needful and meaningful to apply to work out other real decision making problems [54 –59] and the developed approaches can also be extended to other unpredictable and uncertain information [60 –68].

Footnotes

Acknowledgments

This work was supported by Characteristic innovation projects in Guangdong Province (2019GKTSCX037).

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Model for evaluating the security of wireless sensor network with fuzzy number intuitionistic fuzzy information

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Fuzzy number intuitionistic fuzzy set

3.1 The FNIFGMSM operator

4.1 Data instance

Table 3 The SF of the computer network systems FNIFWMSM H1 0.3683 H2 0.0662 H3 –0.0074 H4 –0.1192 H5 0.2390

Footnotes

Acknowledgments

References

Table 3
The SF of the computer network systems

FNIFWMSM

H₁ 0.3683

H2 0.0662

H3 –0.0074

H4 –0.1192

H5 0.2390