Methods for evaluating the computer network security with fuzzy number intuitionistic fuzzy dual Hamy mean operators

Abstract

The concept of fuzzy number intuitionistic fuzzy sets (FNIFSs) is designed to effectively depict uncertain information in decision making problems which fundamental characteristic of the FNIFS is that the values of its membership function and non-membership function are depicted with triangular fuzzy numbers (TFNs). The dual Hamy mean (DHM) operator gets good performance in the process of information aggregation due to its ability to capturing the interrelationships among aggregated values. In this paper, we used the dual Hamy mean (DHM) operator and dual weighted Hamy mean (WDHM) operator with fuzzy number intuitionistic fuzzy numbers (FNIFNs) to propose the fuzzy number intuitionistic fuzzy dual Hamy mean (FNIFDHM) operator and fuzzy number intuitionistic fuzzy weighted dual Hamy mean (FNIFWDHM) operator. Then the MADM methods are proposed along with these operators. In the end, we utilize an applicable example for computer network security evaluation to prove the proposed methods.

Keywords

Multiple attribute decision making (MADM)dual weighted hamy mean (WDHM) operator fuzzy number intuitionistic fuzzy weighted dual hamy operators (FNIFWDHM)computer network security evaluation

1 Introduction

Zadeh [1] designed the fuzzy set (FS) and opened the door to a new field of fuzzy theory in 1965. As a very useful and effective tool, it is applicable to various fields [2]. It could be founded the footprint and research results from the engineering science and technology to the social and human sciences. To extend Zadeh’s classical fuzzy set, the intuitionistic fuzzy set (IFS), which takes into account the membership degree, the non-membership degree and the hesitancy degree [3] at the same time, was developed and studied by Atanassov [4, 5] in 1986 and 1989. Since its appearance, IFSs and their extension have been successfully applied by the more and more researchers into the various domains for solving the decision-making issues [6 –12]. Based on the normalized score matrix, Xu and Hu [13] proposed an entropy-based procedure to derive attribute weights. Liu, Liu and Chen [14] presented some novel intuitionistic fuzzy operators by extending the BM operator on the basis of the Dombi operations [15] and designed some MAGDM methods. Bao, Xie, Long and Wei [16] proposes the method based on prospect theory and the evidential reasoning approach, which aiming at analyzing MADM issues in which the attributes values are IFNs and the attributes weights is unknown. Li and Wang [17] investigated MADM problems through using IFSs, which the interval fractional programming model is used based on the TOPSIS methodology for solving such problems. Gan and Luo [18] defined the hybrid algorithms based on DEMATEL and IFSs to examine the cause-effect relationships among factors. Xu [19] investigated the MADM problems, in which attribute weights is incomplete and the attribute values are depicted in intuitionistic fuzzy numbers (IFNs). Chen [20] examined a comparative study of score functions in MCDM based on IFSs. Chen [21] proposed a useful model of relating optimism and pessimism to MCDM in the context of IFSs based on the unipolar bivariate model. Chen [22] defined a useful study for relating anchor dependency and accuracy functions to MADM problems in the context of IFSs. Chen [23] designed a prioritized aggregation operator-based approach to handling MCDA issues in which there exists a prioritization relationship over evaluative criteria. Wan and Li [24] built a new intuitionistic fuzzy programming model to solve heterogeneous multiattribute GDM problems with intuitionistic fuzzy truth degrees in which there are four types of attribute values such as IFSs, trapezoidal fuzzy numbers, intervals, and real numbers. Chen [25] proposed an extended TOPSIS method with an inclusion comparison method for addressing MAGDM problems in the framework of interval-valued IFSs. He, He and Huang [26] integrated the power averaging operators with IFSs and defined several intuitionistic fuzzy power interaction aggregation operators. Li and Ren [27] defined an effective model for solving MADM problems in which the attribute values were depicted with IF sets based on the TOPSIS, which took into the amount and the reliability of an IF set. Gupta, Arora and Tiwari [28] extended the fuzzy entropy [29] to IFSs with axiomatic justification and proposed importance of parameter alpha. Bao, Xie, Long and Wei [16] put forward a decision method depending on the prospect theory and the evidential reasoning method under IFSs. Zeng, Llopis-Albert and Zhang [30] introduced the weighted intuitionistic fuzzy IOWA weighted average (WIFIOWAWA) operator. Krishankumar, Ravichandran and Saeid [31] developed IFSP (intuitionistic fuzzy set based PROMETHEE) which was a novel ranking method. In order to make aggregate use of the advantages of both Schweizer-Sklar T-norm and T-conorm (SSTT) and Maclaurin symmetric mean (MSM), Wang and Liu [32] extended SSTT to IFNs and defined Schweizer-Sklar operational rules of IFNs. Gou, Xu and Lei [33] pointed out a novel exponential operational law about IFNs and offered a method which was utilized to aggregate intuitionistic fuzzy information. Lu and Wei [34] designed the TODIM method for performance appraisal on social-integration-based rural reconstruction under IVIFSs. Wu, Wei, Wu and Wei [35] proposed some interval-valued intuitionistic fuzzy Dombi Heronian mean operators for evaluating the ecological value of forest ecological tourism demonstration areas. Wan and Dong [36] designed a novel interval-valued intuitionistic fuzzy mathematical programming model for hybrid MCGDM considering alternative comparisons with hesitancy degrees, which the subjective preference relations between alternatives were expressed by each DM were formulated as the IVIFSs, IFSs, trapezoidal fuzzy numbers (TrFNs), linguistic variables, intervals and real numbers. Wu, Wang and Gao [37] designed the algorithms for competiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators. Zeng, Chen and Fan [38] defined the new MADM method based on the nonlinear programming (NLP) algorithms and the TOPSIS method and interval-valued intuitionistic fuzzy values (IVIFVs). Furthermore, to combine the intuitionistic fuzzy sets (IFSs) and the triangular fuzzy sets (TFSs), Liu and Yuan [39] proposed the fuzzy number intuitionistic fuzzy sets (FNIFSs) which fundamental feature of the FNIFSs is that the membership degree and non-membership degree are represented as triangular fuzzy numbers (TFNs) rather than exact values. Li, Niu and Zhang [40] defined some similarity measure and fuzzy entropy of FNIFSs. Wei, Lin, Zhao and Wang [41] proposed some MADM issues based on the induced choquet integral with FNIFSs and defined some aggregation operators. Wang [42] defined some geometric aggregation operators under FNIFSs. Verma [43] developed some generalized Bonferroni mean operator called generalized fuzzy number intuitionistic fuzzy weighted Bonferroni mean (GFNIFWBM) operator for aggregating the FNIFSs. Wang [44] proposed some operational laws of FNIFSs and developed some new arithmetic aggregation operators. Wang and Wang [45] designed the fuzzy number intuitionistic fuzzy Hamacher correlated geometric (FNIFHCG) operator for performance evaluation of communication network. Wang and Yu [46] defined the fuzzy number intuitionistic fuzzy Hamacher correlated average (FNIFHCA) operator for estimating the rural landscape design schemes. Chen and Wang [47] developed the induced fuzzy number intuitionistic fuzzy Hamacher OWA (IFNIFHOWA) operator for performance evaluation of projects loaned by international financial organizations. Lu [48] developed the induced fuzzy number intuitionistic fuzzy Hamacher correlated geometric (IFNIFHCG) operator for assessing the international competitiveness of financial system. Fan [49] defined the fuzzy number intuitionistic fuzzy Hamacher power weighted geometric (FNIFHPWG) operator for assessing the knowledge innovation ability of new ventures based on knowledge management. Zhao, Li and Zhang [50] proposed the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator for evaluating the software quality.

However, all above the aggregation operators and methods can’t pay attention to the relationship between arguments being aggregated. To overcome this shortcoming, the main aim of such paper is to combine the FNIFSs with dual Hamy mean (DHM) [51 –53] operator to propose some new aggregation operators with FNIFSs.

In order to do so, the rest of this paper is designed as follows. In the next section, we shall introduce the concept of FNIFSs and dual Hamy operators. In Section 3, we shall propose some DHM operators with FNIFSs: the fuzzy number intuitionistic fuzzy DHM (FNIFDHM) operator and the fuzzy number intuitionistic fuzzy weighted DHM (FNIFWDHM) operator. In Section 4, we shall present a numerical example for computer network security evaluation in order to illustrate the method proposed in this paper. Section 5 concludes the paper with some remarks.

2 Preliminaries

2.1 Fuzzy number intuitionistic fuzzy set

Liu and Yuan [39] designed the concept of FNIFS.

Definition 1 [39]. Given a fixed set X ={ x₁, x₂, ⋯ , x_n }, An FNIFS $\tilde{A}$ over X is an object having the following form: $\tilde{A} = {〈 x_{i}, {\tilde{t}}_{A} (x_{i}), {\tilde{f}}_{A} (x_{i}) 〉 | x_{i} \in X}$ (1) where ${\tilde{t}}_{A} (x_{i}) \subset [0, 1]$ and ${\tilde{f}}_{A} (x_{i}) \subset [0, 1]$ are TFNs, and ${\tilde{t}}_{A} (x_{i}) = (a (x_{i}), b (x_{i}), c (x_{i})), X \to [0, 1]$ , ${\tilde{f}}_{A} (x) = (l (x_{i}), m (x_{i}), p (x_{i})), X \to [0, 1]$ , 0 ⩽ c (x_i) + p (x_i) ⩽ 1, ∀ x ∈ X.

For convenience, let ${\tilde{t}}_{A} (x_{i}) = (a (x_{i}), b (x_{i}), c (x_{i}))$ , ${\tilde{f}}_{A} (x) = (l (x_{i}), m (x_{i}), p (x_{i}))$ , so $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ and we call $\tilde{a} (x_{i})$ a fuzzy number intuitionistic fuzzy number (FNIFN).

Definition 2 [42, 44]. Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ and $\tilde{a} (x_{j}) = 〈 (a (x_{j}), b (x_{j}), c (x_{j})), (l (x_{j}), m (x_{j}), p (x_{j})) 〉$ be two FNIFNs, then

$\tilde{a} (x_{i}) \oplus \tilde{a} (x_{j}) = {(\begin{matrix} a (x_{i}) + a (x_{j}) - a (x_{i}) a (x_{j}), \\ b (x_{i}) + b (x_{j}) - b (x_{i}) b (x_{j}), \\ c (x_{i}) + c (x_{j}) - c (x_{i}) c (x_{j}) \end{matrix}), (\begin{matrix} l (x_{i}) l (x_{j}), \\ m (x_{i}) m (x_{j}), \\ p (x_{i}) p (x_{j}) \end{matrix})};$

$\tilde{a} (x_{i}) \otimes \tilde{a} (x_{j}) = {((\begin{matrix} a (x_{i}) a (x_{j}), \\ b (x_{i}) b (x_{j}), \\ c (x_{i}) c (x_{j}) \end{matrix})), (\begin{matrix} l (x_{i}) + l (x_{j}) - l (x_{i}) l (x_{j}), \\ m (x_{i}) + m (x_{j}) - m (x_{i}) m (x_{j}), \\ p (x_{i}) + p (x_{j}) - p (x_{i}) p (x_{j}) \end{matrix})};$

$λ \tilde{a} (x_{i}) = {(\begin{matrix} 1 - {(1 - a (x_{i}))}^{λ}, \\ 1 - {(1 - b (x_{i}))}^{λ}, \\ 1 - {(1 - c (x_{i}))}^{λ} \end{matrix}), (\begin{matrix} {(l (x_{i}))}^{λ}, \\ {(m (x_{i}))}^{λ}, \\ {(p (x_{i}))}^{λ} \end{matrix})}, λ ⩾ 0;$

${(\tilde{a} (x_{i}))}^{λ} = {(\begin{matrix} {(a (x_{i}))}^{λ}, \\ {(b (x_{i}))}^{λ}, \\ {(c (x_{i}))}^{λ} \end{matrix}), (\begin{matrix} 1 - {(1 - l (x_{i}))}^{λ}, \\ 1 - {(1 - m (x_{i}))}^{λ}, \\ 1 - {(1 - p (x_{i}))}^{λ} \end{matrix})}, λ ⩾ 0 .$

Definition 3 [42, 44]. Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ be a FNIFN, a score function S of a FNIFN $\tilde{a} (x_{i})$ can be represented as follows:

$\begin{matrix} S (\tilde{a} (x_{i})) = & \frac{a (x_{i}) + 2 b (x_{i}) + c (x_{i})}{4} \\ - \frac{l (x_{i}) + 2 m (x_{i}) + p (x_{i})}{4}, \\ S (\tilde{a} (x_{i})) \in [- 1, 1] . \end{matrix}$ (2)Definition 4 [42, 44]. Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ be a FNIFN, an accuracy function H of a FNIFN $\tilde{a} (x_{i})$ can be represented as follows: $\begin{matrix} H (\tilde{a} (x_{i})) \\ = \frac{(a (x_{i}) + 2 b (x_{i}) + c (x_{i})) + (l (x_{i}) + 2 m (x_{i}) + p (x_{i}))}{4}, \\ H (\tilde{a} (x_{i})) \in [0, 1] . \end{matrix}$ (3) to evaluate the degree of accuracy of the FNIFN $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ , where $H (\tilde{a} (x_{i})) \in [0, 1]$ . The larger the value of $H (\tilde{a} (x_{i}))$ , the more the degree of accuracy of the FNIFN $\tilde{a} (x_{i})$ is.

Based on the score function S and the accuracy function H, an order relation between two FNIFN is introduced as follows:

Definition 5 [42, 44]. Let $\tilde{a} (x_{i})$ and $\tilde{a} (x_{j})$ be two FNIFNs, $s (\tilde{a} (x_{i}))$ and $s (\tilde{a} (x_{j}))$ be the scores of $\tilde{a} (x_{i})$ and $\tilde{a} (x_{j})$ , respectively, and let $H (\tilde{a} (x_{i}))$ and $H (\tilde{a} (x_{j}))$ be the accuracy degrees of $\tilde{a} (x_{i})$ and $\tilde{a} (x_{j})$ , respectively, then if $s (\tilde{a} (x_{i})) < s (\tilde{a} (x_{j}))$ , then $\tilde{a} (x_{i})$ is smaller than $\tilde{a} (x_{j})$ , denoted by $\tilde{a} (x_{i}) < \tilde{a} (x_{j})$ ; if $s (\tilde{a} (x_{i})) = s (\tilde{a} (x_{j}))$ , then

(1) if $H (\tilde{a} (x_{i})) = H (\tilde{a} (x_{j}))$ , then $\tilde{a} (x_{i}) = \tilde{a} (x_{j})$ ; (2)if $H (\tilde{a} (x_{i})) < H (\tilde{a} (x_{j}))$ , then $\tilde{a} (x_{i}) < \tilde{a} (x_{j})$ .

2.2 DHM operator

Definition 6 [54]. The DHM operator is given as follows: $\begin{matrix} {DHM}^{(x)} (φ_{1}, φ_{2}, \dots, φ_{n}) \\ = {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (\frac{\sum_{j = 1}^{x} φ_{i_{j}}}{x}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}$ (4) where x is a parameter and x = 1, 2, …, n, i₁, i₂, …, i_x are x integer values taken from the set {1, 2, … , n } of k integer values, $C_{n}^{x}$ denotes the binomial coefficient and $C_{n}^{x} = \frac{n!}{x! (n - x)!}$ .

3 Some dual Hamy mean operators with FNIFNs

3.1 The FNIFDHM operator

In such section, based on DHM and FNIFNs, the fuzzy number intuitionistic fuzzy dual Hamy mean (FNIFDHM) operator is defined.

Definition 7. Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉 (i = 1, 2, \dots, n)$ be a set of FNIFNs. The FNIFDHM operator is:

$\begin{matrix} {FNIFDHM}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ = {(\underset{1 ⩽ i_{1} < \dots < i_{x} ⩽ n}{\otimes} (\frac{\oplus_{j = 1}^{k} \tilde{a} (x_{i_{j}})}{k}))}^{\frac{1}{C_{n}^{k}}} \end{matrix}$ (5)

Theorem 1. Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉 (i = 1, 2, \dots, n)$ be a set of FNIFNs. The aggregated value by using FNIFDHM operators is also a FNIFN where ${FNIFDHM}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = {(\underset{1 ⩽ i_{1} < \dots < i_{k} ⩽ n}{\otimes} (\frac{(\oplus_{j = 1}^{k} a (x_{i_{j}}))}{k}))}^{\frac{1}{C_{n}^{k}}}$

$\begin{matrix} = {(\begin{matrix} {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - a (x_{i_{j}})))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - b (x_{i_{j}})))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - c (x_{i_{j}})))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}} \end{matrix}), (\begin{matrix} 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} l (x_{i_{j}}))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} m (x_{i_{j}}))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} p (x_{i_{j}}))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}} \end{matrix})} \end{matrix}$ (6)

Proof:

$\oplus_{j = 1}^{k} a (x_{i_{j}}) = {\begin{matrix} (1 - \prod_{j = 1}^{k} (1 - a (x_{i_{j}})), 1 - \prod_{j = 1}^{k} (1 - b (x_{i_{j}})), 1 - \prod_{j = 1}^{k} (1 - c (x_{i_{j}}))), \\ (\prod_{j = 1}^{k} l (x_{i_{j}}), \prod_{j = 1}^{k} m (x_{i_{j}}), \prod_{j = 1}^{k} p (x_{i_{j}})) \end{matrix}}$ (7)

Thus,

$\frac{(\oplus_{j = 1}^{k} a (x_{i_{j}}))}{k} = {(\begin{matrix} 1 - {(\prod_{j = 1}^{k} (1 - a (x_{i_{j}})))}^{\frac{1}{k}}, \\ 1 - {(\prod_{j = 1}^{k} (1 - b (x_{i_{j}})))}^{\frac{1}{k}}, \\ 1 - {(\prod_{j = 1}^{k} (1 - c (x_{i_{j}})))}^{\frac{1}{k}} \end{matrix}), (\begin{matrix} {(\prod_{j = 1}^{k} l (x_{i_{j}}))}^{\frac{1}{k}}, \\ {(\prod_{j = 1}^{k} m (x_{i_{j}}))}^{\frac{1}{k}}, \\ {(\prod_{j = 1}^{k} p (x_{i_{j}}))}^{\frac{1}{k}} \end{matrix})}$ (8)

Thereafter,

$\begin{matrix} \underset{1 ⩽ i_{1} < \dots < i_{k} ⩽ n}{\otimes} (\frac{(\oplus_{j = 1}^{k} a (x_{i_{j}}))}{k}) \\ = {(\begin{matrix} \prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - a (x_{i_{j}})))}^{\frac{1}{k}}), \\ \prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - b (x_{i_{j}})))}^{\frac{1}{k}}), \\ \prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - c (x_{i_{j}})))}^{\frac{1}{k}}) \end{matrix}), (\begin{matrix} 1 - \prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} l (x_{i_{j}}))}^{\frac{1}{k}}), \\ 1 - \prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} m (x_{i_{j}}))}^{\frac{1}{k}}), \\ 1 - \prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} p (x_{i_{j}}))}^{\frac{1}{k}}) \end{matrix})} \end{matrix}$ (9)

Therefore,

$\begin{matrix} \begin{matrix} {FNIFDHM}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = {(\underset{1 ⩽ i_{1} < \dots < i_{k} ⩽ n}{\otimes} (\frac{(\oplus_{j = 1}^{k} a (x_{i_{j}}))}{k}))}^{\frac{1}{C_{n}^{k}}} \\ = {(\begin{matrix} {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - a (x_{i_{j}})))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - b (x_{i_{j}})))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - c (x_{i_{j}})))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}} \end{matrix}), (\begin{matrix} 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} l (x_{i_{j}}))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} m (x_{i_{j}}))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} p (x_{i_{j}}))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}} \end{matrix})} \end{matrix} \end{matrix}$ (10)

Hence, (6) is kept.

Example 1. Let $\tilde{a} (x_{1}) = {(0.2, 0.3, 0.4), (0.4, 0.5, 0.5)}, \tilde{a} (x_{2}) = {(0.1, 0.1, 0.3), (0.2, 0.4, 0.6)}, \tilde{a}$ $(x_{3}) = {(0.2, 0.2, 0.6), (0.4, 0.4, 0.6)} and \tilde{a} (x_{4}) = {(0.4, 0.5, 0.6), (0.5, 0.6, 0.7)}$ be four FNIFNs, and suppose k = 2, then according to (6), we have

$\begin{matrix} {FNIFDHM}^{(2)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = {(\underset{1 ⩽ i_{1} < \dots < i_{k} ⩽ n}{\otimes} (\frac{(\oplus_{j = 1}^{k} a (x_{i_{j}}))}{k}))}^{\frac{1}{C_{n}^{k}}} \\ = {(\begin{matrix} {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - a (x_{i_{j}})))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - b (x_{i_{j}})))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} (1 - c (x_{i_{j}})))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}} \end{matrix}), (\begin{matrix} 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} l (x_{i_{j}}))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} m (x_{i_{j}}))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{k} ⩽ n} (1 - {(\prod_{j = 1}^{k} p (x_{i_{j}}))}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}} \end{matrix})} \end{matrix}$

$\begin{matrix} = {\begin{matrix} (\begin{matrix} {(\begin{matrix} (1 - {((1 - 0.2) \times (1 - 0.1))}^{\frac{1}{2}}) \times (1 - {((1 - 0.2) \times (1 - 0.2))}^{\frac{1}{2}}) \times (1 - {((1 - 0.2) \times (1 - 0.4))}^{\frac{1}{2}}) \\ \times (1 - {((1 - 0.1) \times (1 - 0.2))}^{\frac{1}{2}}) \times (1 - {((1 - 0.1) \times (1 - 0.4))}^{\frac{1}{2}}) \times (1 - {((1 - 0.2) \times (1 - 0.4))}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}}, \\ {(\begin{matrix} (1 - {((1 - 0.3) \times (1 - 0.1))}^{\frac{1}{2}}) \times (1 - {((1 - 0.3) \times (1 - 0.2))}^{\frac{1}{2}}) \times (1 - {((1 - 0.3) \times (1 - 0.5))}^{\frac{1}{2}}) \\ \times (1 - {((1 - 0.1) \times (1 - 0.2))}^{\frac{1}{2}}) \times (1 - {((1 - 0.1) \times (1 - 0.5))}^{\frac{1}{2}}) \times (1 - {((1 - 0.2) \times (1 - 0.5))}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}}, \\ {(\begin{matrix} (1 - {((1 - 0.4) \times (1 - 0.3))}^{\frac{1}{2}}) \times (1 - {((1 - 0.4) \times (1 - 0.6))}^{\frac{1}{2}}) \times (1 - {((1 - 0.4) \times (1 - 0.6))}^{\frac{1}{2}}) \\ \times (1 - {((1 - 0.3) \times (1 - 0.6))}^{\frac{1}{2}}) \times (1 - {((1 - 0.3) \times (1 - 0.6))}^{\frac{1}{2}}) \times (1 - {((1 - 0.6) \times (1 - 0.6))}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}} \end{matrix}), \\ (\begin{matrix} 1 - {(\begin{matrix} (1 - {(0.4 \times 0.2)}^{\frac{1}{2}}) \times (1 - {(0.4 \times 0.4)}^{\frac{1}{2}}) \times (1 - {(0.4 \times 0.5)}^{\frac{1}{2}}) \\ \times (1 - {(0.2 \times 0.4)}^{\frac{1}{2}}) \times (1 - {(0.2 \times 0.5)}^{\frac{1}{2}}) \times (1 - {(0.4 \times 0.5)}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}}, \\ 1 - {(\begin{matrix} (1 - {(0.5 \times 0.4)}^{\frac{1}{2}}) \times (1 - {(0.5 \times 0.4)}^{\frac{1}{2}}) \times (1 - {(0.5 \times 0.6)}^{\frac{1}{2}}) \\ \times (1 - {(0.4 \times 0.4)}^{\frac{1}{2}}) \times (1 - {(0.4 \times 0.6)}^{\frac{1}{2}}) \times (1 - {(0.4 \times 0.6)}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}}, \\ 1 - {(\begin{matrix} (1 - {(0.5 \times 0.6)}^{\frac{1}{2}}) \times (1 - {(0.5 \times 0.6)}^{\frac{1}{2}}) \times (1 - {(0.6 \times 0.7)}^{\frac{1}{2}}) \\ \times (1 - {(0.6 \times 0.6)}^{\frac{1}{2}}) \times (1 - {(0.6 \times 0.7)}^{\frac{1}{2}}) \times (1 - {(0.6 \times 0.7)}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}} \end{matrix}) \end{matrix}} \\ = 〈 (0.2205, 0.2701, 0.4797), (0.3668, 0.4724, 0.5993) 〉 \end{matrix}$

Then we will give some properties of FNIFDHM operator.

Property 1. (Idempotency) If $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ (i = 1, 2, …, n) are equal, then ${FNIFDHM}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = \tilde{a} (x)$ (11)

Property 2. (Monotonicity) Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ (i = 1, 2, …, n) and ${\tilde{a}}^{'} (x_{i}) = 〈 (a^{'} (x_{i}), b^{'} (x_{i}), c^{'} (x_{i})), (l^{'} (x_{i}), m^{'} (x_{i}), p^{'} (x_{i})) 〉$ (i = 1, 2, …, n) be two sets of FNIFNs. If $\tilde{a} (x_{i}) ⩽ {\tilde{a}}^{'} (x_{i})$ hold for all i, then $\begin{matrix} {FNIFDHM}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ ⩽ {FNIFDHM}^{(k)} ({\tilde{a}}^{'} (x_{1}), {\tilde{a}}^{'} (x_{2}), \dots, {\tilde{a}}^{'} (x_{n})) \end{matrix}$ (12)

Property 3. (Boundedness) Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ (i = 1, 2, …, n) be a set of FNIFNs. If $\tilde{a} {(x_{i})}^{-} = min_{i} \tilde{a} (x_{i})$ , $\tilde{a} {(x_{i})}^{+} = max_{i} \tilde{a} (x_{i})$ , then

$\begin{matrix} \tilde{a} {(x)}^{-} & ⩽ {FNIFDHM}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ ⩽ \tilde{a} {(x)}^{+} \end{matrix}$ (13)

Property 4. (Commutativity) Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ (i = 1, 2, …, n) and ${\tilde{a}}^{'} (x_{i}) = 〈 (a^{'} (x_{i}), b^{'} (x_{i}), c^{'} (x_{i})), (l^{'} (x_{i}), m^{'} (x_{i}), p^{'} (x_{i})) 〉$ (i = 1, 2, …, n) be two sets of FNIFNs. Then $\begin{matrix} {FNIFDHM}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ = {FNIFDHM}^{(k)} ({\tilde{a}}^{'} (x_{1}), {\tilde{a}}^{'} (x_{2}), \dots, {\tilde{a}}^{'} (x_{n})) \end{matrix}$ (14) where ${\tilde{a}}^{'} (x_{j}) (j = 1, 2, \dots, n)$ is any permutation of $\tilde{a} (x_{j}) (j = 1, 2, \dots, n)$ .

3.2 The FNIFWDHM operator

In actual MADM, it’s important to consider attribute weights. This section design the fuzzy number intuitionistic fuzzy weighted dual Hamy mean (FNIFWDHM) operator.

Definition 8. Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉 (i = 1, 2, \dots, n)$ be a set of FNIFNs with their weight vector be w_i = (w₁, w₂, …, w_n) ^T, thereby satisfying w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then we can define the FNIFWDHM operator as follows:

$\begin{matrix} {FNIFWDHM}_{w}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ = {(\underset{1 ⩽ i_{1} < \dots < i_{k} ⩽ n}{\otimes} (\frac{\oplus_{j = 1}^{k} (w_{i_{j}} \tilde{a} (x_{i_{j}}))}{k}))}^{\frac{1}{C_{n}^{k}}} \end{matrix}$ (15)

Theorem 2. Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉 (i = 1, 2, \dots, n)$ be a set of FNIFNs. The aggregated value by using FNIFWDHM operator is also a FNIFN where

$\begin{matrix} \begin{matrix} {FNIFWDHM}_{w}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = {(\underset{1 ⩽ i_{1} < \dots < i_{k} ⩽ n}{\oplus} (\frac{\oplus_{j = 1}^{k} (w_{i_{j}} \tilde{a} (x_{i_{j}}))}{k}))}^{\frac{1}{C_{n}^{k}}} \\ = {(\begin{matrix} {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(1 - a (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{x}}}, \\ {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(1 - b (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{x}}}, \\ {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(1 - c (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}), (\begin{matrix} 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(l (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(m (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(p (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}} \end{matrix})} \end{matrix} \end{matrix}$ (16)

Proof. From definition 2, we can obtain,

$w_{i_{j}} \tilde{a} (x_{i_{j}}) = {(\begin{matrix} 1 - {(1 - a (x_{i_{j}}))}^{w_{i_{j}}}, \\ 1 - {(1 - b (x_{i_{j}}))}^{w_{i_{j}}}, \\ 1 - {(1 - c (x_{i_{j}}))}^{w_{i_{j}}} \end{matrix}), (\begin{matrix} {(l (x_{i_{j}}))}^{w_{i_{j}}}, \\ {(m (x_{i_{j}}))}^{w_{i_{j}}}, \\ {(p (x_{i_{j}}))}^{w_{i_{j}}} \end{matrix})}$ (17)

Thus,

$\oplus_{j = 1}^{k} (w_{i_{j}} \tilde{a} (x_{i_{j}})) = {(\begin{matrix} 1 - \prod_{j = 1}^{n} {(1 - a (x_{i_{j}}))}^{w_{i_{j}}}, \\ 1 - \prod_{j = 1}^{n} {(1 - b (x_{i_{j}}))}^{w_{i_{j}}}, \\ 1 - \prod_{j = 1}^{n} {(1 - c (x_{i_{j}}))}^{w_{i_{j}}} \end{matrix}) (\begin{matrix} \prod_{j = 1}^{n} {(l (x_{i_{j}}))}^{w_{i_{j}}}, \\ \prod_{j = 1}^{n} {(m (x_{i_{j}}))}^{w_{i_{j}}}, \\ \prod_{j = 1}^{n} {(p (x_{i_{j}}))}^{w_{i_{j}}} \end{matrix})}$ (18)

Therefore,

$\frac{\oplus_{j = 1}^{k} (w_{i_{j}} \tilde{a} (x_{i_{j}}))}{k} = {(\begin{matrix} 1 - {(\prod_{j = 1}^{n} {(1 - a (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}, \\ 1 - {(\prod_{j = 1}^{n} {(1 - b (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}, \\ 1 - {(\prod_{j = 1}^{n} {(1 - c (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}} \end{matrix}), (\begin{matrix} {(\prod_{j = 1}^{n} {(l (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}, \\ {(\prod_{j = 1}^{n} {(m (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}, \\ {(\prod_{j = 1}^{n} {(p (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}} \end{matrix})}$ (19)

Thereafter,

$\begin{matrix} \begin{matrix} \underset{1 ⩽ i_{1} < \dots < i_{k} ⩽ n}{\oplus} (\frac{\oplus_{j = 1}^{k} (w_{i_{j}} \tilde{a} (x_{i_{j}}))}{k}) \\ = {(\begin{matrix} \prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(1 - a (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}), \\ \prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(1 - b (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}), \\ \prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(1 - c (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}) \end{matrix}), (\begin{matrix} 1 - \prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(l (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}), \\ 1 - \prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(m (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}),, \\ 1 - \prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(p (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}), \end{matrix})} \end{matrix} \end{matrix}$ (20)

Furthermore,

Hence, (16) is kept.

Example 2. Let $\tilde{a} (x_{1}) = {(0.2, 0.3, 0.4), (0.4, 0.5, 0.5)}, \tilde{a} (x_{2}) = {(0.1, 0.1, 0.3), (0.2, 0.4, 0.6)}$ , $\tilde{a} (x_{3}) = {(0.2, 0.2, 0.6), (0.4, 0.4, 0.6)} and \tilde{a} (x_{4}) = {(0.4, 0.5, 0.6), (0.5, 0.6, 0.7)}$ be four FNIFNs, and suppose k = 2, w = (0.2, 0.1, 0.5, 0.2) ^T, then according to (16), we have

$\begin{matrix} {FNIFWDHM}_{w}^{(2)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = {(\underset{1 ⩽ i_{1} < \dots < i_{k} ⩽ n}{\oplus} (\frac{\oplus_{j = 1}^{k} (w_{i_{j}} \tilde{a} (x_{i_{j}}))}{k}))}^{\frac{1}{C_{n}^{k}}} \\ = {(\begin{matrix} {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(1 - a (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{x}}}, \\ {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(1 - b (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{x}}}, \\ {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(1 - c (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}), (\begin{matrix} 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(l (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(m (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}}, \\ 1 - {(\prod_{1 ⩽ i_{1} < \dots < i_{x} ⩽ n} (1 - {(\prod_{j = 1}^{n} {(p (x_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{k}}))}^{\frac{1}{C_{n}^{k}}} \end{matrix})} \\ = {(\begin{matrix} {(\begin{matrix} (1 - {({(1 - 0.2)}^{0.2} \times {(1 - 0.1)}^{0.1})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.2)}^{0.2} \times {(1 - 0.2)}^{0.5})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.2)}^{0.2} \times {(1 - 0.4)}^{0.2})}^{\frac{1}{2}}) \\ \times (1 - {({(1 - 0.1)}^{0.1} \times {(1 - 0.2)}^{0.5})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.1)}^{0.1} \times {(1 - 0.4)}^{0.2})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.2)}^{0.5} \times {(1 - 0.4)}^{0.2})}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}}, \\ {(\begin{matrix} (1 - {({(1 - 0.3)}^{0.2} \times {(1 - 0.1)}^{0.1})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.3)}^{0.2} \times {(1 - 0.2)}^{0.5})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.3)}^{0.2} \times {(1 - 0.5)}^{0.2})}^{\frac{1}{2}}) \\ \times (1 - {({(1 - 0.1)}^{0.1} \times {(1 - 0.2)}^{0.5})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.1)}^{0.1} \times {(1 - 0.5)}^{0.2})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.2)}^{0.5} \times {(1 - 0.5)}^{0.2})}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}}, \\ {(\begin{matrix} (1 - {({(1 - 0.4)}^{0.2} \times {(1 - 0.3)}^{0.1})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.4)}^{0.2} \times {(1 - 0.6)}^{0.5})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.4)}^{0.2} \times {(1 - 0.6)}^{0.2})}^{\frac{1}{2}}) \\ \times (1 - {({(1 - 0.3)}^{0.1} \times {(1 - 0.6)}^{0.5})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.3)}^{0.1} \times {(1 - 0.6)}^{0.2})}^{\frac{1}{2}}) \times (1 - {({(1 - 0.6)}^{0.5} \times {(1 - 0.6)}^{0.2})}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}} \end{matrix}), \\ (\begin{matrix} 1 - {(\begin{matrix} (1 - {(0 . 4^{0.2} \times 0 . 2^{0.1})}^{\frac{1}{2}}) \times (1 - {(0 . 4^{0.2} \times 0 . 4^{0.5})}^{\frac{1}{2}}) \times (1 - {(0 . 4^{0.2} \times 0 . 5^{0.2})}^{\frac{1}{2}}) \\ \times (1 - {(0 . 2^{0.1} \times 0 . 4^{0.5})}^{\frac{1}{2}}) \times (1 - {(0 . 2^{0.1} \times 0 . 5^{0.2})}^{\frac{1}{2}}) \times (1 - {(0 . 4^{0.5} \times 0 . 5^{0.2})}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}}, \\ 1 - {(\begin{matrix} (1 - {(0 . 5^{0.2} \times 0 . 4^{0.1})}^{\frac{1}{2}}) \times (1 - {(0 . 5^{0.2} \times 0 . 4^{0.5})}^{\frac{1}{2}}) \times (1 - {(0 . 5^{0.2} \times 0 . 6^{0.2})}^{\frac{1}{2}}) \\ \times (1 - {(0 . 4^{0.1} \times 0 . 4^{0.5})}^{\frac{1}{2}}) \times (1 - {(0 . 4^{0.1} \times 0 . 6^{0.2})}^{\frac{1}{2}}) \times (1 - {(0 . 4^{0.5} \times 0 . 6^{0.2})}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}}, \\ 1 - {(\begin{matrix} (1 - {(0 . 5^{0.2} \times 0 . 6^{0.1})}^{\frac{1}{2}}) \times (1 - {(0 . 5^{0.2} \times 0 . 6^{0.5})}^{\frac{1}{2}}) \times (1 - {(0 . 6^{0.2} \times 0 . 7^{0.2})}^{\frac{1}{2}}) \\ \times (1 - {(0 . 6^{0.1} \times 0 . 6^{0.5})}^{\frac{1}{2}}) \times (1 - {(0 . 6^{0.1} \times 0 . 7^{0.2})}^{\frac{1}{2}}) \times (1 - {(0 . 6^{0.5} \times 0 . 7^{0.2})}^{\frac{1}{2}}) \end{matrix})}^{\frac{1}{C_{4}^{2}}} \end{matrix})} \\ = 〈 (0 . 0602, 0 . 0748, 0 . 1542), (0 . 8013, 0 . 8392, 0 . 8873) 〉 \end{matrix}$

Then we will give some properties of FNIFWDHM operator.

Property 5. (Idempotency) If $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ (i = 1, 2, …, n) are equal, then

${FNIFWDHM}_{w}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = \tilde{a} (x)$ (22)

Property 6. (Monotonicity) Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ (i = 1, 2, …, n) and ${\tilde{a}}^{'} (x_{i}) = 〈 (a^{'} (x_{i}), b^{'} (x_{i}), c^{'} (x_{i})), (l^{'} (x_{i}), m^{'} (x_{i}), p^{'} (x_{i})) 〉$ (i = 1, 2, …, n) be two sets of FNIFNs. If $\tilde{a} (x_{i}) ⩽ {\tilde{a}}^{'} (x_{i})$ hold for all i, then $\begin{matrix} {FNIFWDHM}_{w}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ ⩽ {FNIFWDHM}_{w}^{(k)} ({\tilde{a}}^{'} (x_{1}), {\tilde{a}}^{'} (x_{2}), \dots, {\tilde{a}}^{'} (x_{n})) \end{matrix}$ (23)

Property 7. (Boundedness) Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ (i = 1, 2, …, n) be a set of FNIFNs. If $\tilde{a} {(x_{i})}^{-} = min_{i} \tilde{a} (x_{i})$ , $\tilde{a} {(x_{i})}^{+} = max_{i} \tilde{a} (x_{i})$ , then

$\begin{matrix} \tilde{a} {(x)}^{-} & ⩽ {FNIFWDHM}_{w}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ ⩽ \tilde{a} {(x)}^{+} \end{matrix}$ (24)

Property 8. (Commutativity) Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ (i = 1, 2, …, n) and ${\tilde{a}}^{'} (x_{i}) = 〈 (a^{'} (x_{i}), b^{'} (x_{i}), c^{'} (x_{i})), (l^{'} (x_{i}), m^{'} (x_{i}), p^{'} (x_{i})) 〉$ (i = 1, 2, …, n) be two sets of FNIFNs. Then $\begin{matrix} {FNIFWDHM}_{w}^{(k)} (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ = {FNIFWDHM}_{w}^{(k)} ({\tilde{a}}^{'} (x_{1}), {\tilde{a}}^{'} (x_{2}), \dots, {\tilde{a}}^{'} (x_{n})) \end{matrix}$ (25) where ${\tilde{a}}^{'} (x_{j}) (j = 1, 2, \dots, n)$ is any permutation of $\tilde{a} (x_{j}) (j = 1, 2, \dots, n)$ .

4 Numerical example and comparative analysis

4.1 Numerical example

The network has become an essential information infrastructure of our country, and has confronted with frequent network attacks. In terms of response to the threat of net-work attacks, enterprise and security operators have deployed large number of network security devices, such as IDS, firewall. However, most of these network security devices are used separately in their own administrative domains, and suffer from a lack of information sharing. At the national and global levels, we come into the situation of lacking the global control of large-scale network security situation. Therefore, based on the current network security infrastructures and technologies, constructing a large-scale network security situation analysis system has now become an urgent necessity, especially on net-work security situation control, analysis and forecast. The evaluating problem of the computer network security is a classical MADM problem [55 –60]. Thus, in this section we shall present a numerical example to evaluate computer network systems with FNIFNs in order to illustrate the method proposed in this paper. There is a panel with five possible computer network systems A_i (i = 1, 2, 3, 4, 5) to select. The experts select four attributes to evaluate the five possible computer network systems: ➀G₁ is the tactics factor; ➁G₂ is the technology and economy factors; ➂G₃ is the logistics factor; ➃G₄ is the strategy factor. The five possible computer network systems A_i (i = 1, 2, 3, 4, 5) are to be evaluated using the FNIFNs by the decision maker under the above four attributes (whose weighting vector ω = (0.1, 0.2, 0.2, 0.5).)

In the following, we utilize the approach developed to evaluate the computer network systems.

Step 1. According to Table 1, we can aggregate all FNIFNs r_ij by using the FNIFWHM (FNIFWDHM) operator to get the overall FNIFNs A_i (i = 1, 2, 3, 4, 5) of the computer network systems A_i. Suppose that k = 2, then the aggregating results are shown in Table 2.

Table 1
FNIFN decision matrix (R)

G ₁ G ₂

A₁ 〈 (0.1, 0.1, 0.3) , (0.2, 0.4, 0.6)〉〈 (0.2, 0.3, 0.4) , (0.4, 0.5, 0.5)〉

A₂ 〈 (0.5, 0.7, 0.7) , (0.2, 0.5, 0.8)〉〈 (0.2, 0.5, 0.5) , (0.3, 0.4, 0.6)〉

A₃ 〈 (0.4, 0.7, 0.7) , (0.2, 0.3, 0.4)〉〈 (0.7, 0.7, 0.8) , (0.2, 0.2, 0.4)〉

A₄ 〈 (0.2, 0.2, 0.4) , (0.3, 0.3, 0.4)〉〈 (0.1, 0.4, 0.6) , (0.2, 0.3, 0.7)〉

A₅ 〈 (0.1, 0.1, 0.4) , (0.3, 0.3, 0.4)〉〈 (0.5, 0.5, 0.6) , (0.3, 0.3, 0.5)〉

G ₃ G ₄

A₁ 〈 (0.4, 0.5, 0.6) , (0.5, 0.6, 0.7)〉〈 (0.2, 0.3, 0.6) , (0.4, 0.4, 0.6)〉

A₂ 〈 (0.4, 0.4, 0.7) , (0.2, 0.5, 0.8)〉〈 (0.5, 0.6, 0.7) , (0.4, 0.6, 0.6)〉

A₃ 〈 (0.5, 0.6, 0.6) , (0.1, 0.2, 0.2)〉〈 (0.7, 0.7, 0.9) , (0.2, 0.4, 0.5)〉

A₄ 〈 (0.4, 0.6, 0.7) , (0.5, 0.6, 0.6)〉〈 (0.1, 0.4, 0.7) , (0.2, 0.5, 0.8)〉

A₅ 〈 (0.3, 0.4, 0.5) , (0.4, 0.5, 0.5)〉〈 (0.2, 0.2, 0.4) , (0.1, 0.2, 0.2)〉

	G ₁	G ₂
A₁	〈 (0.1, 0.1, 0.3) , (0.2, 0.4, 0.6)〉	〈 (0.2, 0.3, 0.4) , (0.4, 0.5, 0.5)〉
A₂	〈 (0.5, 0.7, 0.7) , (0.2, 0.5, 0.8)〉	〈 (0.2, 0.5, 0.5) , (0.3, 0.4, 0.6)〉
A₃	〈 (0.4, 0.7, 0.7) , (0.2, 0.3, 0.4)〉	〈 (0.7, 0.7, 0.8) , (0.2, 0.2, 0.4)〉
A₄	〈 (0.2, 0.2, 0.4) , (0.3, 0.3, 0.4)〉	〈 (0.1, 0.4, 0.6) , (0.2, 0.3, 0.7)〉
A₅	〈 (0.1, 0.1, 0.4) , (0.3, 0.3, 0.4)〉	〈 (0.5, 0.5, 0.6) , (0.3, 0.3, 0.5)〉
	G ₃	G ₄
A₁	〈 (0.4, 0.5, 0.6) , (0.5, 0.6, 0.7)〉	〈 (0.2, 0.3, 0.6) , (0.4, 0.4, 0.6)〉
A₂	〈 (0.4, 0.4, 0.7) , (0.2, 0.5, 0.8)〉	〈 (0.5, 0.6, 0.7) , (0.4, 0.6, 0.6)〉
A₃	〈 (0.5, 0.6, 0.6) , (0.1, 0.2, 0.2)〉	〈 (0.7, 0.7, 0.9) , (0.2, 0.4, 0.5)〉
A₄	〈 (0.4, 0.6, 0.7) , (0.5, 0.6, 0.6)〉	〈 (0.1, 0.4, 0.7) , (0.2, 0.5, 0.8)〉
A₅	〈 (0.3, 0.4, 0.5) , (0.4, 0.5, 0.5)〉	〈 (0.2, 0.2, 0.4) , (0.1, 0.2, 0.2)〉

Table 2

The aggregating results of the computer network systems by the FNIFWDHM operator

	FNIFWDHM
A₁	< (0.0602,0.0748,0.1542), (0.8013,0.8392,0.8873)>
A₂	< (0.1137,0.1686,0.2195), (0.7502,0.8553,0.9136)>
A₃	< (0.1995,0.2306,0.3011), (0.6642,0.7436,0.7972)>
A₄	< (0.0444,0.1219,0.2114), (0.7417,0.8245,0.9105)>
A₅	< (0.0748,0.0825,0.1397), (0.7173,0.7589,0.7932)>

Step 3. According to the aggregating results shown in Table 2 and the score functions of the computer network systems are shown in Table 3.

Step 4. According to the score functions shown in Table 3 and the comparison formula of score functions, the ordering of the computer network systems are shown in Table 4. Note that “> ” means “preferred to”. As we can see, depending on the aggregation operators used, the best computer network system is A₃.

Table 3

The score functions of the computer network systems

	FNIFWDHM
A₁	–0.7508
A₂	–0.6760
A₃	–0.4967
A₄	–0.7004
A₅	–0.6622

Table 4

Ordering of the computer network systems

	Ordering
FNIFWDHM	A₃ > A₅ > A₂ > A₄ > A₁

4.2 Influence of the parameter on the final result

In order to show the effects on the ranking results by changing parameters of k in the FNIFWDHM operators, all the results are shown in Table 5.

Table 5
Ranking results for different operational parameters of the FNIFWDHM operator

s(A₁) s(A₂) s(A₃) s(A₄) s(A₅) Ordering

k = 1 –0.7880 –0.7082 –0.5554 –0.7431 –0.7226 A₃ > A₂ > A₅ > A₄ > A₁

k = 2 –0.7508 –0.6760 –0.4967 –0.7004 –0.6622 A₃ > A₅ > A₂ > A₄ > A₁

k = 3 –0.7359 –0.6602 –0.4713 –0.6844 –0.6320 A₃ > A₅ > A₂ > A₄ > A₁

k = 4 –0.7273 –0.6506 –0.4562 –0.6759 –0.6113 A₃ > A₅ > A₂ > A₄ > A₁

	s(A₁)	s(A₂)	s(A₃)	s(A₄)	s(A₅)	Ordering
k = 1	–0.7880	–0.7082	–0.5554	–0.7431	–0.7226	A₃ > A₂ > A₅ > A₄ > A₁
k = 2	–0.7508	–0.6760	–0.4967	–0.7004	–0.6622	A₃ > A₅ > A₂ > A₄ > A₁
k = 3	–0.7359	–0.6602	–0.4713	–0.6844	–0.6320	A₃ > A₅ > A₂ > A₄ > A₁
k = 4	–0.7273	–0.6506	–0.4562	–0.6759	–0.6113	A₃ > A₅ > A₂ > A₄ > A₁

4.3 Comparative analysis

Then, we compare our proposed method with other existing methods including FNIFWA [33] operator and FNIFWG [35] operator. The comparative results are shown in Table 6.

Table 6
Ordering of the computer network systems

Ordering

FNIFWA [33] A₃ > A₅ > A₂ > A₄ > A₁

FNIFWG [35] A₃ > A₂ > A₅ > A₄ > A₁

FNIFWDHM A₃ > A₅ > A₂ > A₄ > A₁

	Ordering
FNIFWA [33]	A₃ > A₅ > A₂ > A₄ > A₁
FNIFWG [35]	A₃ > A₂ > A₅ > A₄ > A₁
FNIFWDHM	A₃ > A₅ > A₂ > A₄ > A₁

From above, Compare the values of our proposed FNIFWDHM operators with FNIFWA and FNIFWG operators, the results in ranking of alternatives are slightly different and the best alternatives are same. However, the existing aggregation operators, such as FNIFWA and FNIFWG operators, do not consider the information about the relationship between arguments being aggregated, and thus cannot eliminate the influence of unfair arguments on decision result. Our proposed FNIFWDHM operator consider the information about the relationship between arguments being aggregated.

5 Conclusion

In this paper, we investigate the MADM problems with FNIFNs. Then, we utilize the Hamy mean (HM) operator, weighted Hamy mean (WHM) operator, dual Hamy mean (DHM) operator and dual weighted Hamy mean (WDHM) operator to develop some Hamy mean aggregation operators with FNIFNs: fuzzy number intuitionistic fuzzy Hamy mean (FNIFHM) operator, fuzzy number intuitionistic fuzzy weighted Hamy mean (FNIFWHM) operator, fuzzy number intuitionistic fuzzy dual Hamy mean (FNIFDHM) operator, fuzzy number intuitionistic fuzzy weighted dual Hamy mean (FNIFWDHM) operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the MADM problems with FNIFNs. Finally, a practical example for evaluate the computer network systems is given to verify the developed approach and to demonstrate its practicality and effectiveness. In the future, the application of the proposed aggregating operators of FNIFNs needs to be explored in the uncertain and incomplete decision making [61 –70], risk uncertain analysis [71 –73] and many other research fields under uncertain settings [74 –83].

Footnotes

Acknowledgments

This work is supported by the Collaborative innovation project of “Integration of Moss, digital intelligence” funded by the innovation fund of University production, learning and research of the science and technology development center of the Ministry of Education.

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Methods for evaluating the computer network security with fuzzy number intuitionistic fuzzy dual Hamy mean operators

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Fuzzy number intuitionistic fuzzy set

3.1 The FNIFDHM operator

4.1 Numerical example

Table 6 Ordering of the computer network systems Ordering FNIFWA [33] A3 > A5 > A2 > A4 > A1 FNIFWG [35] A3 > A2 > A5 > A4 > A1 FNIFWDHM A3 > A5 > A2 > A4 > A1

Footnotes

Acknowledgments

References

Table 6
Ordering of the computer network systems

Ordering

FNIFWA [33] A₃ > A₅ > A₂ > A₄ > A₁

FNIFWG [35] A₃ > A₂ > A₅ > A₄ > A₁

FNIFWDHM A₃ > A₅ > A₂ > A₄ > A₁