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Abstract

The computer network environment is very complex, and there are many factors that need to be considered in the process of network security evaluation. At the same time, various factors have complex nonlinear relationships. Neural networks are mathematical models that simulate the behavioral characteristics of animal neural networks. They process information by adjusting the connection relationships of internal nodes, and have a wide range of applications in solving complex nonlinear relationship problems. The computer network security evaluation is multiple attribute group decision making (MAGDM) problems. In this paper, based on projection measure and bidirectional projection measure, we shall introduce four forms projection models with q-rung orthopair fuzzy sets (q-ROFSs). Furthermore, combine projection measure and bidirectional projection measure with q-ROFSs, we develop four forms of projection models with q-ROFSs. Based on developed weighted projection measure models, the multiple attribute group decision making (MAGDM) model is established and all computing steps are simply depicted. Finally, a numerical example for computer network security evaluation is given to illustrate this new model and some comparisons are also conducted to verify advantages of the new built methods.

Keywords

Multiple attribute group decision making (MAGDM)q-rung orthopair fuzzy sets (q-ROFSs)projection model bidirectional projection model computer network security evaluation

1. Introduction

The high complexity of the network environment has led to an increasing number of network security threats faced by computers [1, 2]. Network attacks such as network intrusions, computer viruses, and illegal access are not only highly harmful in themselves, but also have a high correlation with each other [3, 4]. In computer security evaluation, it is often necessary to consider multiple complex factors and clarify the non-linear relationship between various factors, this undoubtedly brings higher requirements to the traditional computer security evaluation system [5, 6, 7]. Due to limitations in technology and other aspects, traditional security evaluation models cannot achieve sufficient accuracy. Computers have brought the information age to the people. On the contrary, in the information age, they have also promoted the development of computer networks, which have been widely used in various industries [8, 9, 10, 11]. They have brought great convenience to the people’s lives and high economic benefits to social production. At the same time, with the accelerated development of computer networks, network security issues have become increasingly prominent, causing adverse effects on the lives and social production of the people [12, 13]. Strengthening research on security evaluation is the primary task for computer professionals. Computer security evaluation can be understood as an assessment of the security of computer usage. Generally speaking, computer security evaluation includes the evaluation of multiple indicators such as data security, device stability, data integrity, and data confidentiality [14, 15, 16]. Physical security refers to the use of physical protection to ensure that a computer is in a stable and reliable state at all times during operation, preventing data loss caused by physical damage. Logical security refers to the availability, confidentiality, and integrity of information data, mainly achieved through software protection [17, 18]. From this, it can be seen that the network security evaluation of computers is multidimensional, including not only the security of hardware usage, but also the security of data transmission and software usage. The development of computer network technology has created infinite progress and wealth for society, but at the same time, it also hides huge security risks [19, 20]. According to statistics, a new computer may be discovered within 15 minutes of accessing the internet, and a network intrusion case is reported globally every 20 seconds, posing a threat to network security. In theory, network security includes two aspects: technical level and management level, with the former focusing on preventing external illegal attacks [21, 22]. The latter focuses on internal human management, and both complement each other and are indispensable. However, due to the openness, freedom of movement, and international nature of the network, users accessing information through the internet will inevitably increase their probability of being attacked, thus there are many security risks in computer networks [23, 24]. Computer network security evaluation is actually the analysis and evaluation of whether the use of computer networks is good, whether the security meets the requirements, and the classification of computer network security levels. With the continuous improvement of people’s living standards and the rapid popularization of computers, people are paying more and more attention to computer security and yearning for their information to be protected [25, 26]. At the same time, the network security of computers is influenced by multiple factors, and traditional linear network security evaluation cannot accurately capture vulnerabilities and timely reflect the operation status of computers. Therefore, it is necessary to apply neural networks to computer network security analysis to timely and accurately reflect the security status of computers and reduce user losses [27, 28].

In real society, no matter individuals or organizations, they are always faced with decision-making problems [29, 30, 31, 32, 33, 34]. The MADM problem based on different attributes to select several options is one of the most important problems in decision-making problems [35, 36, 37]. Similar to the MADM, the multicriteria Decision Analysis (MCDA) is the important research parts in practical management science [38, 39, 40, 41]. In many MCDA problems, due to the limitations which recognizing the practical decision-making problem, it is often impossible to depict the accurate information and could only be portrayed through the uncertainty which constitutes the uncertain MCDA issue [42, 43, 44, 45]. MAGDM has extensive practicability and applicability in many fields such as industry, economy and management [46, 47, 48, 49, 50]. Motivated by the theories of intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs) [51, 52, 53, 54, 55, 56], in order to portray more fuzzy and uncertain information, Yager [57] put forward the q-rung orthopair fuzzy sets (q-ROFSs), which meets the condition that the sum of q-th power of them is equal or less than 1. Evidently, the q-ROFSs is the generalization of IFS and PFSs, especially, when $q=1$ and $q=2$ , the q-ROFSs will reduce to IFSs and PFSs, this indicates q-ROFSs can be more suitable for dealing with MAGDM problems. The computer network security evaluation is classical MAGDM problems. The q-ROFSs [57] are used as a tool for characterizing uncertain information during the computer network security evaluation. The traditional projection measure [58] and bidirectional projection measure [59] has been used to cope with MAGDM issues. As far as authors know, there is no research conducted based on projection method to deal with MAGDM problems with q-ROFSs and so it is of great significance to study it. In this paper, based on traditional projection measure and bidirectional projection measure, we shall propose some projection measures for handling MAGDM problems under q-ROFSs. Finally, a numerical example for computer network security evaluation is given to verify this new model and some comparisons are also conducted to illustrate advantages of the new built method. The main research aim and motivation of this paper is constructed: (1) the projection measure and bidirectional projection measure is extended to the q-ROFSs; (2) some projection measures are constructed to handle MAGDM problems under q-ROFSs; (3) the information Entropy is used to obtain the weight values based on the score value and accuracy value under q-ROFSs; (4) Finally, a numerical case study for computer network security evaluation and some comparisons are conducted to validate the proposed method.

To do so, the remainder of this paper is set out as follows. The concepts of q-ROFSs are introduced in Section 2. Section 3 combines traditional projection and bidirectional projection measures with q-ROFSs. In Section 4, two decision models based on our developed models is built to deal with MAGDM. In Section 5, we give a numerical example for computer network security evaluation with q-ROFNs, in addition, some comparisons are also conducted to verify advantages of the built methods. Section 6 concludes such paper.

2. Preliminaries

2.1 The q-ROFSs

The q-ROFSs [57] is introduced.

Definition 1 [57]. Let $\Theta$ be a fixed set. The q-ROFSs is constructed as:

$\displaystyle F=\{{\langle{\theta,({f\alpha(\theta),f\beta(\theta)})}\rangle|{% \theta\in\Theta}}\}$ (1)

where the function $f\alpha(\theta)$ indicates the membership and the function $f\beta(\theta)$ indicates the nonmembership, and, it holds that

$\displaystyle({f\alpha(\theta)})^{q}+({f\beta(\theta)})^{q}\leqslant 1,q% \geqslant 1.$ (2)

Definition 2 [57]. Assume that $f_{1}=({f\alpha_{1},f\beta_{1}})$ , $fp_{2}=({f\alpha_{2},f\beta_{2}})$ , and $f=({f\alpha,f\beta})$ be three q-rung orthopair fuzzy numbers (q-ROFNs), then the operation laws are expressed:

$\displaystyle(1)f_{1}\oplus f_{2}=({\sqrt[q]{({f\alpha_{1}})^{q}+({f\alpha_{2}% })^{q}-({f\alpha_{1}})^{q}\times({f\alpha_{2}})^{q}},f\beta_{1}\times f\beta_{% 2}});(2)f_{1}\otimes f_{2}=({f\alpha_{1}\times f\alpha_{2},\sqrt[q]{({f\beta_{% 1}})^{q}+({f\beta_{2}})^{q}-({f\beta_{1}})^{q}({f\beta_{2}})^{q}}});(3)\lambda f% =({\sqrt[q]{1-({1-f\alpha^{q}})^{\lambda}},f\beta^{\lambda}}),\lambda>0;(4)(f)% ^{\lambda}=({f\alpha^{\lambda},\sqrt[q]{1-({1-f\beta^{q}})^{\lambda}}}),% \lambda>0;(5)f^{c}=({f\beta,f\alpha}).$

Definition 3 [57]. Let $f_{1}=({f\alpha_{1},f\beta_{1}})$ , $fp_{2}=({f\alpha_{2},f\beta_{2}})$ , and $f=({f\alpha,f\beta})$ , according to the Definition 2, the operation laws have the following properties.

$\displaystyle(1)f_{1}\oplus f_{2}=f_{2}\oplus f_{1},f_{1}\otimes f_{2}=f_{2}% \otimes f_{1},({(f)^{\lambda_{1}}})^{\lambda_{2}}=(f)^{\lambda_{1}\lambda_{2}}% ;(2)\lambda({f_{1}\oplus f_{2}})=\lambda f_{1}\oplus\lambda f_{2},({f_{1}% \otimes f_{2}})^{\lambda}=({f_{1}})^{\lambda}\otimes({f_{2}})^{\lambda};(3)% \lambda_{1}f\oplus\lambda_{2}f=({\lambda_{1}+\lambda_{2}})f,(f)^{\lambda_{1}}% \otimes(f)^{\lambda_{2}}=(f)^{({\lambda_{1}+\lambda_{2}})}.$

Definition 4 [57, 60]. Let $f=({f\alpha,f\beta})$ be a q-ROFN, the score function and accuracy function are constructed:

$\displaystyle SF(f)=f\alpha-f\beta,SF(f)\in[{-1,1}].$ (3) $\displaystyle AF(f)=\frac{({1+f\alpha-f\beta})}{2},AF(f)\in[{0,1}].$ (4)

For two q-ROFNs $f_{1}=({f\alpha_{1},f\beta_{1}})$ and $fp_{2}=({f\alpha_{2},f\beta_{2}})$ , then (1) if $SF({f_{1}})<SF({f_{2}})$ , $f_{1}<f_{2}$ ; (2) if $SF({f_{1}})=SF({f_{2}})$ , if $AF({f_{1}})<AF({f_{2}})$ , $f_{1}<f_{2}$ ; if $AF({f_{1}})=AF({f_{2}})$ , $f_{1}=f_{2}$ .

2.2 Some q-rung orthopair fuzzy aggregation operators

In this section, the q-ROFWA and q-ROFWG operator [60] is introduced.

Definition 5 [60]. Let $f_{j}=({f\alpha_{j},f\beta_{j}})({j=1,2,\ldots,n})$ with weight $fw=(fw_{1},fw_{2},\ldots,fw_{n})^{T}$ which satisfies $fw_{j}>0$ , $\sum\nolimits_{j=1}^{n}{fw_{j}=1}$ , then q-ROFWA and q-ROFWG operator are given:

$\displaystyle\text{q-ROFWA}_{fw}({f_{1},f_{2},\ldots,f_{n}})=\mathop{\oplus}% \limits_{j=1}^{n}fw_{j}f_{j}$

(5) $\displaystyle{}=\left({\sqrt[q]{1-\prod\limits_{j=1}^{n}{({1-({f\alpha_{j}})^{% q}})^{fw_{j}}}},\prod\limits_{j=1}^{n}{({f\beta_{j}})^{fw_{j}}}}\right)$

and

$\displaystyle\text{q-ROFWG}_{fw}({f_{1},f_{2},\ldots,f_{n}})=\mathop{\otimes}% \limits_{j=1}^{n}({f_{j}})^{fw_{j}}$ (6) $\displaystyle{}=\left({\prod\limits_{j=1}^{n}{({f\alpha_{j}})^{fw_{j}},\sqrt[q% ]{1-\prod\limits_{j=1}^{n}{({1-({f\beta_{j}})^{q}})^{fw_{j}}}}}}\right)$
3. Projection measure of q-rung orthopair fuzzy numbers

In this section, combine the traditional projection, bidirectional projection and q-ROFNs, we shall give some projection measures and weighted projection measures under q-ROFSs.

3.1 Some projection measures with q-ROFNs

In this part, some projection measure and weighted projection measure under q-ROFSs are defined.

Definition 6. Let $f_{1}=({f\alpha_{1},f\beta_{1}})$ and $fp_{2}=({f\alpha_{2},f\beta_{2}})$ be two q-ROFNs, then the module of q-ROFNs $f_{1}=({f\alpha_{1},f\beta_{1}})$ and $fp_{2}=({f\alpha_{2},f\beta_{2}})$ can be constructed as:

$\displaystyle\|{f_{1}}\|=\sqrt{({({f\alpha_{1}})^{q}})^{2}+({({f\beta_{1}})^{q% }})^{2}}$ (7) $\displaystyle\|{f_{2}}\|=\sqrt{({({f\alpha_{2}})^{q}})^{2}+({({f\beta_{2}})^{q% }})^{2}}$ (8)

Definition 7. Let $f_{1}=({f\alpha_{1},f\beta_{1}})$ and $fp_{2}=({f\alpha_{2},f\beta_{2}})$ be two q-ROFNs, then the inner product $f_{1}\cdot f_{2}$ and the cosine of the included angle $\cos({f_{1},f_{2}})$ between $f_{1}=({f\alpha_{1},f\beta_{1}})$ and $fp_{2}=({f\alpha_{2},f\beta_{2}})$ are denoted:

$\displaystyle f_{1}\cdot f_{2}=f\alpha_{1}^{q}f\alpha_{2}^{q}+f\beta_{1}^{q}f% \beta_{2}^{q}$ (9) $\displaystyle\cos({fp_{1},fp_{2}})=\frac{f_{1}\cdot f_{2}}{\|{f_{1}}\|\|{f_{2}% }\|}$ (10)

Definition 8. Assume that $FX_{i}=({f_{i1},f_{i2},\ldots,f_{in}})({i=1,2,\ldots,m})$ be the i-th alternatives and $FX=({f_{1},f_{2},\ldots,f_{n}})$ be the ideal solution with q-ROFNs, where $f_{ij}=({f\alpha_{ij},f\beta_{ij}}),i=1,2,\ldots,m,j=1,2,\ldots,n$ and $f_{j}=({f\alpha_{j},f\beta_{j}}),j=1,2,\ldots,n$ . Then the projection of vector $FX_{i}$ on $F X$ can be denoted as:

$\displaystyle\textit{PRJ}_{FX}({FX_{i}})=\|{FX_{i}}\|\sum\limits_{j=1}^{n}{% \cos({FX_{i},FX})}$ (11) $\displaystyle{}=\|{FX_{i}}\|\cdot\frac{\sum\limits_{j=1}^{n}{({f\alpha_{ij}^{q% }f\alpha_{j}^{q}+f\beta_{ij}^{q}f\beta_{j}^{q}})}}{\|{FX_{i}}\|\|{FX}\|}=\frac% {1}{\|{FX}\|}\sum\limits_{j=1}^{n}{({f\alpha_{ij}^{q}f\alpha_{j}^{q}+f\beta_{% ij}^{q}f\beta_{j}^{q}})}$

where

$\displaystyle\|{fx}\|=\sqrt{\sum\limits_{j=1}^{n}{({({({f\alpha_{j}})^{q}})^{2% }+({({f\beta_{j}})^{q}})^{2}})}}$ (12)

Obviously, the greater the value $\textit{PRJ}_{FX}({FX_{i}})$ , the closer vector $FX_{i}$ to $F X$ , which indicates the better the alternatives $FX_{i}$ .

However, in practical decision making environment, the weighting values of evaluation information is an important factor which needs to be considered, thus, let $fw=({fw_{1},fw_{2},\ldots,fw_{n}})$ be the weights of q-ROFNs, the weighted projection of vector $FX_{i}$ on $F X$ can be developed as follows.

Definition 9. Assume that $FX_{i}=({f_{i1},f_{i2},\ldots,f_{in}})({i=1,2,\ldots,m})$ be the i-thalternatives and $FX=({f_{1},f_{2},\ldots,f_{n}})$ be the ideal solution with weigh information $fw=({fw_{1},fw_{2},\ldots,fw_{n}})$ , where $f_{ij}=({f\alpha_{ij},f\beta_{ij}}),i=1,2,\ldots,m,j=1,2,\ldots,n$ and $f_{j}=({f\alpha_{j},f\beta_{j}}),j=1,2,\ldots,n$ . Then the weighted projection of vector $FX_{i}$ on $F X$ can be denoted as:

$\displaystyle\textit{WPRJ}_{FX}({FX_{i}})=\|{FX_{i}}\|_{fw}\sum\limits_{j=1}^{% n}{\cos({FX_{i},FX})_{fw}}$ $\displaystyle{}=\|{FX_{i}}\|_{fw}\cdot\frac{\sum\limits_{j=1}^{n}{fw_{j}^{2q}(% {f\alpha_{ij}^{q}f\alpha_{j}^{q}+f\beta_{ij}^{q}f\beta_{j}^{q}})}}{\|{FX_{i}}% \|_{fw}\|{FX}\|_{fw}}$ (13) $\displaystyle{}=\frac{1}{\|{FX}\|_{fw}}\sum\limits_{j=1}^{n}{fw_{j}^{2q}({f% \alpha_{ij}^{q}f\alpha_{j}^{q}+f\beta_{ij}^{q}f\beta_{j}^{q}})}$

where

$\displaystyle\|{FX}\|_{fw}=\sqrt{\sum\limits_{j=1}^{n}{({({({fw_{j}f\alpha_{j}% })^{q}})^{2}+({({fw_{j}f\beta_{j}})^{q}})^{2}})}}$ (14)

where $fw_{j}({j=1,2,\ldots,n})$ satisfies $0\leqslant fw_{j}\leqslant 1$ , $\sum\nolimits_{j=1}^{n}{fw_{j}}=1$ . Obviously, the greater the value $\textit{WPRJ}_{FX}({FX_{i}})$ , the closer vector $FX_{i}$ to $F X$ , which indicates the better the alternatives $FX_{i}$ .

3.2 Some bidirectional projection measures with q-ROFNs

In this section, we shall introduce the q-rung orthopair fuzzy bidirectional projection measure and q-rung orthopair fuzzy weighted bidirectional projection measure to overcome the limitation of the general projection measure of q-ROFNs.

Definition 10. Assume that $FX_{i}=({f_{i1},f_{i2},\ldots,f_{in}})({i=1,2,\ldots,m})$ be the i-th alternatives and $FX=({f_{1},f_{2},\ldots,f_{n}})$ be the ideal solution with q-ROFNs, where $f_{ij}=({f\alpha_{ij},f\beta_{ij}}),i=1,2,\ldots,m,j=1,2,\ldots,n$ and $f_{j}=({f\alpha_{j},f\beta_{j}}),j=1,2,\ldots,n$ . Then the bidirectional projection of vector $FX_{i}$ on $F X$ can be denoted as:

$\displaystyle\textit{BPROJ}({FX_{i},FX})=\frac{1}{1+\left|{\frac{FX_{i}\cdot FX% }{\|{FX_{i}}\|}-\frac{FX_{i}\cdot FX}{\|{FX}\|}}\right|}$

(15) $\displaystyle{}=\frac{\|{FX_{i}}\|\|{FX}\|}{\|{FX_{i}}\|\|{FX}\|+|{\|{FX_{i}}% \|-\|{FX}\|}|({FX_{i}\cdot FX})}$

where

$\displaystyle\|{FX}\|=\sqrt{\sum\limits_{j=1}^{n}{({({({f\alpha_{j}})^{q}})^{2% }+({({f\beta_{j}})^{q}})^{2}})}}$ (16) $\displaystyle\|{FX_{i}}\|=\sqrt{\sum\limits_{j=1}^{n}{({({({f\alpha_{ij}})^{q}% })^{2}+({({f\beta_{ij}})^{q}})^{2}})}}$ (17) $\displaystyle(FX_{i}\cdot FX)=\sum\limits_{j=1}^{n}(f\alpha_{ij}^{q}f\alpha_{j% }^{q}+f\beta_{ij}^{q}f\beta_{j}^{q})$ (18)

Obviously, the greater the value $\textit{BPROJ}({FX_{i},FX})$ , the closer vector $FX_{i}$ to $F X$ , which indicates the better the alternatives $FX_{i}$ .

Consider the weighting vector of q-ROFNs, the weighted bidirectional projection of vector $FX_{i}$ on $F X$ can be developed.

Definition 11. Assume that $FX_{i}=({f_{i1},f_{i2},\ldots,f_{in}})({i=1,2,\ldots,m})$ be the i-th alternatives and $FX=({f_{1},f_{2},\ldots,f_{n}})$ be the ideal solution with weigh information $fw=({fw_{1},fw_{2},\ldots,fw_{n}})$ , where $f_{ij}=({f\alpha_{ij},f\beta_{ij}}),i=1,2,\ldots,m,j=1,2,\ldots,n$ and $f_{j}=({f\alpha_{j},f\beta_{j}}),j=1,2,\ldots,n$ . Then the weighted bidirectional projection of vector $FX_{i}$ on $F X$ can be denoted as:

$\displaystyle\textit{WBPROJ}({FX_{i},FX})_{fw}=\frac{1}{1+\left|{\frac{({FX_{i% }\cdot FX})_{fw}}{\|{FX_{i}}\|_{fw}}-\frac{({FX_{i}\cdot FX})_{fw}}{\|{FX}\|_{% fw}}}\right|}$ (19) $\displaystyle{}=\frac{\|{FX_{i}}\|_{fw}\|{FX}\|_{fw}}{\|{FX_{i}}\|_{fw}\|{FX}% \|_{fw}+|{\|{FX_{i}}\|_{fw}-\|{FX}\|_{fw}}|({FX_{i}\cdot FX})_{fw}}$

where

$\displaystyle\|{FX}\|_{fw}=\sqrt{\sum\limits_{j=1}^{n}{({({({fw_{j}f\alpha_{j}% })^{q}})^{2}+({({fw_{j}f\beta_{j}})^{q}})^{2}})}}$ (20) $\displaystyle\|{FX_{i}}\|_{fw}=\sqrt{\sum\limits_{j=1}^{n}{({({({fw_{j}f\alpha% _{ij}})^{q}})^{2}+({({fw_{j}f\beta_{ij}})^{q}})^{2}})}}$ (21) $\displaystyle({FX_{i}\cdot FX})_{w}=\sum\limits_{j=1}^{n}{fw_{j}^{2q}({f\alpha% _{ij}^{q}f\alpha_{j}^{q}+f\beta_{ij}^{q}f\beta_{j}^{q}})}$ (22)

where $fw_{j}({j=1,2,\ldots,n})$ satisfies $0\leqslant fw_{j}\leqslant 1$ , $\sum\nolimits_{j=1}^{n}{fw_{j}}=1$ . Obviously, the greater the value $\textit{WBPRJ}_{FX}({FX_{i}})$ , the closer vector $FX_{i}$ to $F X$ , which indicates the better the alternatives $FX_{i}$ .
4. The projection decision method with q-ROFNs information

Suppose there are $m$ alternatives $\{{FA_{1},FA_{2},\ldots,FA_{m}}\}$ , $n$ attributes $\{{FG_{1},FG_{2},\ldots,FG_{n}}\}$ with weight $fw=\{fw_{1},fw_{2},\ldots,fw_{n}\}$ and $\lambda$ experts $\{{FD_{1},FD_{2},\ldots,FD_{t}}\}$ with weight $f\omega=\{f\omega_{1},f\omega_{2},\linebreak\ldots f\omega_{\lambda}\}$ which satisfy $fw_{j}\in[{0,1}]$ , $f\omega_{k}\in[{0,1}]$ and $\sum\nolimits_{j=1}^{n}{fw_{j}}=1$ , $\sum\nolimits_{k=1}^{\lambda}{f\omega_{k}}=1$ . construct the evaluation matrix with q-ROFNs $FR=[{FR_{ij}^{\lambda}}]_{m\times n}=({f\alpha_{ij}^{\lambda},f\beta_{ij}^{% \lambda}})f\alpha_{ij}^{\lambda}\in[{0,1}]$ indicates the membership of $FA_{i}$ satisfies $FG_{j}$ given by DM, $f\beta_{ij}^{\lambda}\in[{0,1}]$ indicates the membership of $FA_{i}$ don’t satisfies $FG_{j}$ given by the decision maker, then the decision-making steps of q-rung orthopair fuzzy projection method can be expressed.

Step 1. Construct the matrix $FR=[{FR_{ij}^{\lambda}}]_{m\times n}$ with q-ROFNs.

Step 2. According to q-ROFWA operator, we can use overall $FR_{ij}^{\lambda}$ to $FR_{ij}$ to obtain the fused q-ROFNs matrix $FR_{i}=({FR_{ij}})_{m\times n}=({f\alpha_{ij},f\beta_{ij}})_{m\times n}$ ;

Step 3. Compute the attributes weight by using information entropy.

The weight is important for MAGDM. Many scholars focused on to obtain the weight information under different environment [61, 62, 63, 64, 65]. Entropy [66] is a conventional theory to derive weight. Firstly, the normalized q-ROFN-matrix $\textit{q-ROFN}_{ij}$ is obtained:

$\displaystyle\textit{q-ROFN}_{ij}=\frac{SF({f\alpha_{ij},f\beta_{ij}})+AF({f% \alpha_{ij},f\beta_{ij}})}{\sum\limits_{i=1}^{m}{({SF({f\alpha_{ij},f\beta_{ij% }})+AF({f\alpha_{ij},f\beta_{ij}})})}},$ (23)

Then, the q-ROFN Shannon entropy $\textit{q-ROFNSE}=({\textit{q-ROFNSE}_{1},\textit{q-ROFNSE}_{2},\ldots,\textit% {q-ROFNSE}_{n}})$ is defined by Eq. (17):

$\displaystyle\textit{q-ROFNSE}_{j}=-\frac{1}{\ln m}\sum\limits_{i=1}^{m}{% \textit{q-ROFN}_{ij}\ln\textit{q-ROFN}_{ij}}$ (24)

and $\textit{q-ROFN}_{ij}\ln\textit{q-ROFN}_{ij}=0$ if $\textit{q-ROFN}_{ij}=0$ .

Then, the weights $fw=({fw_{1},fw_{2},\ldots,fw_{n}})$ is defined:

$\displaystyle fw_{j}=\frac{1-\textit{q-ROFNSE}_{j}}{\sum\limits_{j=1}^{n}{({1-% \textit{q-ROFNSE}_{j}})}},j=1,2,\ldots,n.$ (25)

Step 4. Obtain the ideal alternative $FR^{+}=({FR_{j}^{+}})({j=1,2,\ldots,n})$ according to the Eq. (4).

$\displaystyle SF({FR_{j}^{+}})=SF({f\alpha_{j}^{+},f\beta_{j}^{+}})$ (26) $\displaystyle{}=\mathop{\max}\limits_{i}SF({FR_{ij}})=\mathop{\max}\limits_{i}% SF({f\alpha_{ij},f\beta_{ij}})$

Step 5. Compute the weighted projection measure $\textit{WPRJ}({FR_{i},FR^{+}})$ and weighted bidirectional projection measure $\textit{WBPRJ}({FR_{i},FR^{+}})$ between each alternative $FR_{i}$ and the ideal solution alternative $FR^{+}$ according to the Eqs (3.1) and (3.2) to rank all alternatives.

Step 6. Determine the ordering of alternatives based on the results of $\textit{WPRJ}({FR_{i},FR^{+}})$ and $\textit{WBPRJ}({FR_{i},FR^{+}})$ in a descending order. The greater value of these measures, the better alternative $FA_{i}$ .

5. Numerical example and comparative analysis

5.1 Numerical example

The rapid development of computer network technology has led to a rapid increase in various network service applications, and the diversity of network services has increased the difficulty of network security maintenance, posing a huge threat to computer network security. The frequent occurrence of viruses and network hacker attacks has had a very adverse impact on the stability of the computer network environment, greatly increasing the difficulty of network security defense. There are many factors that affect computer network security, and a network security evaluation system must objectively and accurately evaluate security risks. The so-called accuracy refers to the high reliability and accuracy of computer network security evaluation indicators, which can accurately present the level of network security in different states. Computer network security evaluation should follow four principles Accuracy is the foundation of computer network security evaluation. Only by ensuring the accuracy of computer network security evaluation can network security evaluation be valuable, otherwise computer network security evaluation will lose its due value. The so-called integrity refers to the fact that when evaluating computer network security, the security status of the computer network presented by various indicators should be complete. In the process of selecting network security evaluation indicators in practice, it is necessary to have a deep understanding of the connotation of each evaluation indicator, ensure the rationality and effectiveness of the evaluation indicators, avoid useless indicators and duplicate selection of evaluation indicators, and provide strong data support for accurate evaluation of network security. The so-called independence refers to the need to avoid duplicate indicators when constructing computer network security evaluation indicators, so that there is no overlap between the selected evaluation indicators and the correlation between the indicators is eliminated. Ensuring the independence of the selected computer network security evaluation indicators can greatly reduce the amount of data for security evaluation and improve the accuracy of network security evaluation. The so-called simplicity refers to the fact that the selected network security evaluation indicators should be concise and concise, with obvious characteristics in the process of selecting evaluation indicators, which is helpful for practical evaluation operations. The frequent occurrence of computer network security accidents has led to the deterioration of the computer network environment. During the development of computer networks, viruses and hacker attacks have always existed, and often have high concealment. Meanwhile, with the development of computer network viruses, their destructive power has greatly increased. For computer network users, their own network security awareness is insufficient, lacking network security awareness, which also makes it difficult to maintain computer network security, creating conditions for criminals to attack computer networks. The computer network security evaluation is classical MADM problems. In this section, we shall provide a numerical example for computer network security evaluation by using our developed projection model with q-ROFNs. Assume that five possible computer networks $FA_{i}({i=1,2,3,4,5})$ to be selected and four evaluation criteria to assess these computer networks: ⟀ $FG_{1}$ is the environment security of computer network systems; ⟁ $FG_{2}$ is the organization system of computer network systems; ⟂ $FG_{3}$ is the safe practice measures of computer network systems; ⟃ $FG_{4}$ is the communication security of computer network systems; ⟄ $FG_{5}$ is the management system of computer network systems; ⟅ $FG_{6}$ is entity security of computer network systems. The five possible computer network systems $FA_{i}({i=1,2,3,4,5})$ are evaluated with linguistic scale (See Table 1 [67]) with the six criteria by three invited experts $FD^{k}$ (expert’s weight $f\omega=$ (0.35, 0.30, 0.35), attributes weight $fw=$ (0.4, 0.2, 0.1, 0.3)).

Table 1
Linguistic scale and q-ROFNs

Linguistic terms	q-ROFNs
Exceedingly Terrible-FET	(0.10, 0.80)
Very Terrible-FVT	(0.20, 0.70)
Terrible-FT	(0.30, 0.60)
Medium-FM	(0.50, 0.50)
Well-FW	(0.65, 0.30)
Very Well-FVW	(0.75, 0.20)
Exceedingly Well-FEW	(1.0, 0.0)

The weighted projection measure is used to solve the computer network security evaluation.

Step 1. Construct the q-ROFNs matrix $FR=[{FR_{ij}^{t}}]_{5\times 6}({t=1,2,3})$ (See Tables 2–4).

Table 2

Evaluation information by $FD_{1}$

	FG₁	FG₂	FG₃	FG₄	FG₅	FG₆
FA₁	ZW	ZM	ZVT	ZVW	ZVW	ZT
FA₂	ZM	ZT	ZVW	ZW	ZM	ZVW
FA₃	ZW	ZVW	ZVT	ZM	ZM	ZT
FA₄	ZVT	ZVW	ZM	ZVT	ZVT	ZM
FA₅	ZVW	ZVT	ZM	ZT	ZVW	ZW

Table 3

Evaluation information by $FD_{2}$

	FG₁	FG₂	FG₃	FG₄	FG₅	FG₆
FA₁	ZM	ZT	ZVW	ZW	ZVW	ZM
FA₂	ZVT	ZM	ZVW	ZT	ZVT	ZM
FA₃	ZT	ZW	ZM	ZVW	ZM	ZVT
FA₄	ZVW	ZW	ZVT	ZM	ZVT	ZVW
FA₅	ZM	ZW	ZVT	ZVW	ZVW	ZVT

Table 4

Evaluation information by $FD_{3}$

	FG₁	FG₂	FG₃	FG₄	FG₅	FG₆
FA₁	ZM	ZVT	ZT	ZM	ZW	ZT
FA₂	ZM	ZT	ZM	ZW	ZVW	ZT
FA₃	ZW	ZVT	ZM	ZVT	ZVT	ZM
FA₄	ZW	ZM	ZW	ZM	ZT	ZVW
FA₅	ZT	ZVW	ZT	ZW	ZVW	ZVT

Then according to q-ROFWA, the $FR=[{FR_{ij}}]_{5\times 6}$ is obtained (See Table 5).

Table 5

The $FR=[{FR_{ij}}]_{5\times 6}$

	ZG1	ZG2	ZG3
FA₁	(0.29, 0.81)	(0.46, 0.69)	(0.38, 0.72)
FA₂	(0.78, 0.32)	(0.47, 0.54)	(0.36, 0.68)
FA₃	(0.53, 0.48)	(0.49, 0.56)	(0.39, 0.73)
FA₄	(0.49, 0.51)	(0.43, 0.67)	(0.34, 0.79)
FA₅	(0.67, 0.42)	(0.28, 0.76)	(0.49, 0.52)
	ZG4	ZG1	ZG2
FA₁	(0.33, 0.79)	(0.54, 0.53)	(0.46, 0.54)
FA₂	(0.83, 0.28)	(0.74, 0.35)	(0.65, 0.36)
FA₃	(0.54, 0.52)	(0.32, 0.76)	(0.34, 0.78)
FA₄	(0.56, 0.49)	(0.49, 0.53)	(0.54, 0.59)
FA₅	(0.69, 0.43)	(0.68, 0.43)	(0.72, 0.45)

Step 3. Compute the weight values (See Table 6).

Table 6

The attribute weight information

	FG₁	FG₂	FG₃	FG₄	FG₅	FG₆
$f w$	0.1654	0.2273	0.1641	0.1014	0.1605	0.1813

Step 4. Obtain the ideal alternative $FR^{+}=(FR_{j}^{+})({j=1,2,\ldots,6})$ (See Table 7).

Table 7

The ideal alternative $FR^{+}=({FR_{j}^{+}})({j=1,2,\ldots,6})$

	ZG1	ZG2	ZG3
$FR^{+}$	(0.78, 0.32)	(0.49, 0.56)	(0.49, 0.52)
	ZG4	ZG1	ZG2
$FR^{+}$	(0.83, 0.28)	(0.68, 0.43)	(0.72, 0.45)

Step 5. Compute the weighted projection measure $\textit{WPRJ}({FR_{i},FR^{+}})$ and weighted bidirectional projection measure $\textit{WBPRJ}({FR_{i},FR^{+}})$ between each computer network systems $FR_{i}$ and the ideal computer network system $FR^{+}$ according to the Eqs (3.1) and (3.2) (See Table 8).

Table 8

The weighted projection measure $\textit{WPRJ}({FR_{i},FR^{+}})$ and weighted bidirectional projection measure $\textit{WBPRJ}({FR_{i},FR^{+}})$

	$\textit{WPRJ}({FR_{i},FR^{+}})$	$\textit{WBPRJ}({FR_{i},FR^{+}})$
FA₁	0.5465	0.5605
FA₂	0.9436	0.9953
FA₃	0.7306	0.7770
FA₄	0.6039	0.6524
FA₅	0.8565	0.8941

Step 6. Determine the ordering of computer network systems based on the results of $\textit{WPRJ}({FR_{i},FR^{+}})$ and $\textit{WBPRJ}({FR_{i},FR^{+}})$ in a descending order (See Table 9).

Table 9

The order of different alternatives

	$\textit{WPRJ}({FR_{i},FR^{+}})$	$\textit{WBPRJ}({FR_{i},FR^{+}})$
FA₁	5	5
FA₂	1	1
FA₃	3	3
FA₄	4	4
FA₅	2	2

Finally, the decision order could be obtained: $FA_{2}\succ FA_{5}\succ FA_{3}\succ FA_{4}\succ FA_{1}$ , and thus the optimal computer network systems is $FA_{2}$ .

5.2 Comparative analysis

Then, the weighted projection measure under q-ROFNs is compared with q-ROFWA operator [60], q-ROFWG operator [60], q-rung orthopair fuzzy weighted exponential aggregation (q-ROFWEA) operator [68], q-rung orthopair fuzzy Dombi weighted average (q-ROFDWA) operator [69], q-rung orthopair fuzzy Dombi weighted geometric (q-ROFDWG) operator [69], q-rung orthopair fuzzy EDAS (q-ROF-EDAS) method [70], q-rung orthopair fuzzy VIKOR (q-ROF-VIKOR) method [71] and q-rung orthopair fuzzy TODIM (q-ROF-TODIM) method [72]. The comparative decision results are shown in Table 10.

Table 10
Order of the different methods

Methods	Order
q-ROFWA operator [60]	$FA_{2}\succ FA_{5}\succ FA_{3}\succ FA_{4}\succ FA_{1}$
q-ROFWG operator [60]	$FA_{2}\succ FA_{5}\succ FA_{4}\succ FA_{3}\succ FA_{1}$
q-ROFWEA operator [68]	$FA_{2}\succ FA_{5}\succ FA_{3}\succ FA_{4}\succ FA_{1}$
q-ROFDWA operator [69]	$FA_{2}\succ FA_{5}\succ FA_{3}\succ FA_{4}\succ FA_{1}$
q-ROFDWG operator [69]	$FA_{2}\succ FA_{5}\succ FA_{4}\succ FA_{3}\succ FA_{1}$
q-ROF-EDAS method [70]	$FA_{2}\succ FA_{5}\succ FA_{3}\succ FA_{4}\succ FA_{1}$
q-ROF-VIKOR method [71]	$FA_{2}\succ FA_{5}\succ FA_{3}\succ FA_{4}\succ FA_{1}$
q-ROF-TODIM method [72]	$FA_{2}\succ FA_{5}\succ FA_{3}\succ FA_{4}\succ FA_{1}$

In accordance with RW coefficients [73], the RW coefficient calculation between q-ROFWA operator [60], q-ROFWG operator [60], q-rung orthopair fuzzy weighted exponential aggregation (q-ROFWEA) operator [68], q-rung orthopair fuzzy Dombi weighted average (q-ROFDWA) operator [69], q-rung orthopair fuzzy Dombi weighted geometric (q-ROFDWG) operator [69], q-rung orthopair fuzzy EDAS (q-ROF-EDAS) method [70], q-rung orthopair fuzzy VIKOR (q-ROF-VIKOR) method [71], q-rung orthopair fuzzy TODIM (q-ROF-TODIM) method [72] and the proposed weighted projection measure under q-ROFNs is 1.0000, 0.9167, 1.0000, 1.0000, 0.9167, 1.0000, 1.0000, 1.0000, respectively. The RW coefficient calculation shows that the order result of q-ROFWA operator [60], q-rung orthopair fuzzy weighted exponential aggregation (q-ROFWEA) operator [68], q-rung orthopair fuzzy Dombi weighted average (q-ROFDWA) operator [69], q-rung orthopair fuzzy EDAS (q-ROF-EDAS) method [70], q-rung orthopair fuzzy VIKOR (q-ROF-VIKOR) method [71] and q-rung orthopair fuzzy TODIM (q-ROF-TODIM) method [72] is same with the order result of the proposed weighted projection measure under q-ROFNs, however, The RW coefficient calculation shows that the order result of q-ROFWG operator [60] and q-rung orthopair fuzzy Dombi weighted geometric (q-ROFDWG) operator [69] is slightly different form the order result of the proposed weighted projection measure under q-ROFNs. From the above detailed analysis, it could be seen that the order of these models is slightly different, however, these models have the same optimal computer network system and worst computer network system. This shows the proposed weighted projection measure under q-ROFNs is reasonable and effective.

6. Conclusion

The rapid development of computer network technology has made it widely applied in all aspects of national economic life. While citizens enjoy the convenience brought by computer networks, network security issues have also received attention. Criminals exploit security vulnerabilities in computer networks to steal confidential information from computer networks through methods such as Trojan viruses and network intrusions, resulting in huge economic losses for enterprises or individuals. Improving the security of computer networks and ensuring the stability of learning and work for network users is a challenge faced by the development of computer networks. The computer network security evaluation is classical MADM problems. Combine the advantages of the traditional projection measure and q-ROFSs, in this paper, based on projection measure and bidirectional projection measure, we propose four forms of projection measure with q-ROFNs, such as: q-rung orthopair fuzzy projection measure, q-rung orthopair fuzzy weighted projection measure, q-rung orthopair fuzzy bidirectional projection measure and q-rung orthopair fuzzy weighted bidirectional projection measure. According to the two weighed form projection models, we build the decision-making model to deal with MAGDM. Finally, a numerical example for computer network security evaluation is given to testify the scientific and effective of our developed models, moreover, some comparative analysis is conducted to demonstrate the advantages of our proposed methods. The main research contribution of this paper is constructed: (1) the projection measure and bidirectional projection measure is extended to the q-ROFSs; (2) some projection measures are constructed to handle MAGDM problems under q-ROFSs; (3) the information Entropy is used to obtain the weight values based on the score value and accuracy value under q-ROFSs; (4) Finally, a numerical case study for computer network security evaluation and some comparisons are conducted to validate the proposed method. In the future, we shall continue to investigate the MAGDM problems under q-ROFNs and extend our developed projection measure and bidirectional projection measure to other uncertain environments.

Footnotes

Acknowledgments

This work was supported by Vocational Education Working Committee of China Communications Society, Intelligent recommendation based on big data Application in “Precision Employment” Using research and the research on the digital transformation path of small and medium-sized enterprises empowered by artificial Intelligence. (RKX033).

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Projection measure-driven optimization of q-rung orthopair fuzzy MAGDM for computer network security evaluation

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 The q-ROFSs

3.1 Some projection measures with q-ROFNs

5.1 Numerical example

Table 1 Linguistic scale and q-ROFNs

Table 10 Order of the different methods

Footnotes

Acknowledgments

References

Table 1
Linguistic scale and q-ROFNs

Table 10
Order of the different methods