Efficient network selection using multi fuzzy criteria for confidential data transmission in wireless body sensor networks

Abstract

In the last decade, due to wireless technology’s enhancement, the people’s interest is highly increased in Wireless Body Sensor Networks (WBSNs). WBSNs consist of many tiny biosensor nodes that are continuously monitoring diverse physiological signals such as BP (systolic and diastolic), ECG, EMG, SpO2, and activity recognition and transmit these sensed patients’ sensitive information to the central node, which is straight communicate with the controller. To disseminate this sensitive patient information from the controller to remote Medical Server (MS) needs to be prolonged high-speed wireless technology, i.e., LTE, UMTS, WiMAX, WiFi, and satellite communication. It is a challenging task for the controller to choose the optimal network to disseminate various patient vital signs, i.e., emergency data, normal data, and delay-sensitive data. According to the nature of various biosensor nodes in WBSNs, monitor patient vital signs and provide complete intelligent treatment when any abnormality occurs in the human body, i.e., accurate insulin injection when patient sugar level increased. In this paper, first, we select the optimal network from accessible networks using four different fuzzy attribute-based decision-making techniques (Triangular Cubic Hesistent Fuzzy Weighted Averaging Operator, Neutrosophic Linguistic TOPSIS method, Trangualar Cubic Hesistent Fuzzy Hamacher Weighted Averaging Operator and Cubic Grey Relational Analysis) depending upon the quality of service requirement for various application of WBSNs to prolong the human life, enhanced the society’s medical treatment and indorse living qualities of people. Similarly, leakage and misuse of patient data can be a security threat to human life. Thus, confidential data transmission is of great importance. For this purpose, in our proposed scheme, we used HECC for secure key exchange and an AES algorithm to secure patient vital signs to protect patient information from illegal usage. Furthermore, MAC protocol is used for mutual authentication among sensor nodes and Base Stations (BS). Mathematical results show that our scheme is efficient for optimal network selection in such circumstances where conflict arises among diverse QoS requirements for different applications of WBSNs.

Keywords

Wireless body sensor network quality of service security fuzzy logic decision making

1 Introduction

The WSN applications in various fields like natural disasters, habitat monitoring, battlefield, and other emergency services got the researchers’ attention [1], and WSN evolved to WBSNs for medical applications. In 1996 T.G. Zimmerman proposed the idea of WBSNs for the first time. These networks were initially called Wireless Personal Area Network (WPANs). A typical sensor node’s hardware consists of processor and memory, wireless communication stack, analog to digital converter, and sensing [2]. WBSNs network comprised of low power, low processing, small size, lightweight body sensors deployed on the patient body which regularly monitor Electroencephalogram (EEG), respiratory rate, heart rate, Blood Pressure (BP) then through controller forwarded the real-time sensed patient data to BS outside the body for onward transmitting to remote MS. After receiving patient data by MS, the ward physician gives feedback for the patient’s health care [3]. In the current era, it is a hit research area for the researcher to provide the best services to network users using selecting an optimal network to disseminate patient vital signs, i.e., emergency data, normal data, and delay-sensitive data toward the MS. WBSNs face two significant challenges due to their constrained natured environment. These are secure communication of sensitive patient information and high overhead in terms of computation and communication. To handle these issues, smart and secure cryptosystem along-with optimal network selection are required to minimize the latency and improve the network throughput for better network performance. In our proposed scheme, we used four different fuzzy attribute-based decision-making techniques (Triangular Cubic Hesistent Fuzzy Weighted Averaging Operator, Neutrosophic Linguistic TOPSIS method, Trangualar Cubic Hesistent Fuzzy Hamacher Weighted Averaging Operator, and Cubic Grey Relational Analysis) depending upon the QoS requirement for various applications of WBSNs to select the optimal network from accessible networks. Moreover, for secure communication between communicating parties, it is essential to share secret keys confidentially. In our proposed scheme Hyper Elliptic Curve Cryptosystem (HECC) is used for secure key exchange while Advanced Encryption Standard (AES) algorithm used for confidential data transmission and Message Authentication Code (MAC) protocol used for mutual authentication between sensor nodes and BS. The bandwidth for transmission in WBSNs is 10Kbps to 10Mbps [2]. The following Fig. 1, shown the general four-tier architecture of WBSNs.

Fig. 1

Our proposed three tier architecture for WBSNs.

2 Related Work

In this section, we discussed the previous related techniques of WBSNs along with fuzzy logic. In scheme [4] proposed a new two-level Fuzzy Logic Control (FLC) architecture among biosensor nodes and coordinator nodes to improve quality of service in WBSNs and efficiently control wireless media for dissemination of patient information to enhance the network performance and reliability of patient sensitive medical information. Moreover, this paper used a co-operative co-evolutionary method for FLC’s efficient design in a resource-constrained environment of WBAN. In scheme [5] proposed an efficient dynamic protocol called (Fuzzy- TADMAC) for WBAN to overcome the energy consumption problem of biosensor node deployed on a patient body using BAN radio CC2420 radio chips. Furthermore, the proposed protocol is simulated using (OMNET ++) and then obtain results are compared with the existing protocol under two radio chips. Finally, results show that the proposed protocol is optimal in energy consumption than the TADMAC protocol. In scheme [6], the authors proposed a novel technique for network selection in heterogeneous public networks called utility-based non-linear (F-AHP). Moreover, fuzzy triangular numbers are applied to display the members’ comparison matrices for data (video, voice) and best-effort application. To improve the decision-making on data in this paper proposed a non-linear (F-AHP) method in which consistent weights are obtained than previous models. Various parameters of QoS are designed using parameterized utility functions. Using three MADM scores are calculates of SAW, TOPSIS, and MEW. Finally result shown that MEW, along with the utility function, gives the best scores than TOPSIS and SAW. In scheme [7], the authors proposed a heterogeneous wireless environment in which different radio-access technologies are used to disseminate data. In the recent era, the optimal network selection from available networks (WiMAX and LTE) for data transmission is the hit research area for WBSNs researchers. The authors proposed a novel algorithm for best network selection based on different parameters, i.e. (RSS, SNR, Throughput, bandwidth, and bit error rate). Swarm optimization technique used to optimize the relative-weight of decision-making attributes. Moreover, the Monte Carlo method is used for the calculation of satisfied users. The result has shown that the proposed network selection algorithm is efficient as compared to existing approaches. In scheme [8] proposed another heterogeneous wireless setting composed of (LTE and WiMAX). Which provide efficient QoS to the end-users according to their need and demand. In this paper, the authors proposed a new scheme called (C-P-F) for best network selection, network load balancing, and user satisfaction that used multimedia services. Using fuzzy algorithm and cost function unnecessary rate of network selection reduces and enhances QoS user satisfaction. In scheme [9], the author proposed HDS modular- design using fuzzy to handle the problem of the best selection of network among various networks in a heterogeneous setting to maximize the throughput and minimize the delay. The performance is calculated in terms of computation time and network selection. Using this method, we got much improvement in computation time. Various other schemes [10 –13] were designed for network selection and vertical-handoff to increase heterogeneous wireless networks’ efficiency when different users access the public network simultaneously for multimedia services. These approaches ensured seamless mobility in the combination of hotspots, WiFi, WiMax, and cellular networks. In scheme [14] proposed a cloud-based electronic health care system called (HC-AAS) for diagnostic patient sensitive medical information. This paper’s fuzzy model is used for controlling rules and system credentials in terms of inputs and outputs. The external users use the patient symptoms and diseases, i.e., doctor, nurse to maintain the patient health data and enhanced medical treatment quality in Egypt and KSA. Yun et al. [15] proposed the spherical fuzzy entropy to find the criteria’s unknown weights. Khan et al. [16] proposed the series of logarithmic aggregation operators: logarithmic picture fuzzy weighted averaging/geometric and logarithmic picture fuzzy ordered weighted averaging/geometric aggregation operators and characterized their desirable properties. Jamil et al. [17] introduced the bipolar neutrosophic Hamacher weighted averaging (BNHWA), bipolar neutrosophic Hamacher ordered weighted averaging (BNHOWA), and bipolar neutrosophic Hamacher hybrid averaging (BNHHA) along with their desirable properties. Khan et al. [18] proposed new Pythagorean trapezoidal uncertain linguistic fuzzy aggregation information. Namely, the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein weighted averaging (PTULFEWA) operator, the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein ordered weighted averaging (PTULFEOWA) operator, and the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein hybrid weighted averaging (PTULFEHWA) operator using the Einstein operational laws. Khan et al. [19] introduced the Pythagorean fuzzy set models using Pythagorean fuzzy Dombi aggregation operators. Ashraf et al. [20] proposed the single-valued neutrosophic hybrid aggregation operators as a tool for multi-criteria decision-making (MCDM) under the neutrosophic environment and discussed some properties. Son et al. [21] introduced some new hesitant fuzzy Hamacher power-aggregation operators for hesitant fuzzy information based on Hamacher t-norm and t-conorm. Baykasoğlu et al. [22] proposed the new interval type-2 fuzzy (IT2F) functions model to predict the current performance of alternatives based on the historical decision matrices. Zhu et al. [23] introduced experts to provide preference information concerning failure modes’ evaluation and risk factors’ weight. Han et al. [24] introduced the measure to calculate integrated weights by combining objective weights and subjective weights. Abdel-Basset et al. [25] proposed a combination of quality function deployment (QFD) with plithogenic aggregation operations. In mathematics, fuzzy sets (aka uncertain sets) are somewhat like sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh [26]. Fahmi et al. [27] developed the hamming distance for triangular cubic fuzzy numbers and weighted averaging operator. Fahmi et al. [28] proposed the cubic TOPSIS method and grey relational analysis set. Fahmi et al. [29] defined the triangular cubic fuzzy number and operational laws. The authors developed the triangular cubic fuzzy hybrid aggregation (TCFHA) administrator to all individual fuzzy choice structures provide by the decision-makers into the aggregate cubic fuzzy decision matrix. Fahmi et al. [30] defined aggregation operators for triangular cubic linguistic hesitant fuzzy sets, which include cubic linguistic fuzzy (geometric) operator, triangular cubic linguistic hesitant fuzzy weighted geometric (TCLHFWG) operator, triangular cubic linguistic hesitant fuzzy ordered weighted geometric (TCHFOWG) operator and triangular cubic linguistic hesitant fuzzy hybrid geometric (TCLHFHG) operator. Abbas et al. [31] introduced the score function, accuracy functions to compare CPFNs, Cubic Pythagorean fuzzy weighted averaging (CPFWA) operator, and Cubic Pythagorean fuzzy weighted geometric (CPFWG) operator. In this scheme, [32] hybrid set named cubic hesitant fuzzy set can express hesitancy fuzzy and uncertain material concurrently for decision-making problems of multiple attributes in manufacturing practice in uncertain decision environment. In scheme [33] author proposed a novel method to overcome the network congestion. Varying traffic, data dissemination from many to one node, and limited link capacity are the major causes of network congestion. Congestion negatively affects the network’s performance, i.e., losses of the data packet, increasing source to destination delay, and due to huge numbers of packet re-transmission wasting the energy consumption. In critical patient data transmission, delay causes the death of the patient. Moreover, to prolong human life and enhance society’s medical treatment, and indorse the living qualities of people, the author proposed a system that detects network congestion using local information, i.e., node rate and buffer size, while assigning different priorities sensitive medical information based upon their criticalness. The result has shown that the proposed scheme provides better QoS and efficient for disseminating patient critical vital signs in emergencies.

In scheme [34], the author proposed a novel and efficient idea to secure the patient-sensitive information in remote cloud using online/offline signcryption along with blockchain technologies. Furthermore, a heterogeneous infrastructure based on certificate less cryptosystem and public-key cryptosystem are applied to enhance the security and scalability of the BSNs. Additionally, the elementary security properties are proved using a formal security model.

In scheme [35] author’s proposed efficient and lightweight attribute-based scheme for multi-receiver to minimize the cost overhead using generalized signcryption along with HECC for resource-constrained environment of BSNs. Moreover, modified priority-based scheduling algorithm are applied to enhance the quality of service in BSN. The security of the scheme is proved using well known AVISPA tool. Besides, data privacy is achieved using attribute-based access policies. Only authorized users can access the patient’s sensitive information using their valid credentials.

In scheme [36], a novel protocol is designed to overcome the energy consumption in WBSNs using trust-based reliable communication along with remote patient monitoring in a desirable way. Additionally, the performance analysis show that the proposed scheme is efficient in terms of communication cost, computational cost, and energy consumption and suitable for resource-limited environment.

In scheme [37], a reliable co-operative along with fault-tolerant protocol are designed for efficient utilization of network resources to improve the network performance in terms of high throughput, less delay, and less bit error rate. Moreover, the obtained results show that the scheme provides better results as compared to other states of the arts schemes.

3 Proposed Scheme

In our proposed scheme, we used four novel fuzzy attribute-based decision-making methods that are Triangular Cubic Hesistent Fuzzy Weighted Averaging Operator, Neutrosophic Linguistic TOPSIS method, Trangualar Cubic Hesistent Fuzzy Hamacher Weighted Averaging Operator, and Cubic Grey Relational Analysis depending upon the QoS requirement for various applications of WBSNs to improve the decision level of selection optimal network from available networks for better QoS in WBSN.

From the security point of view, our proposed scheme consists of four phases which are: registration phase, authentication and key exchange phase, confidential data dissemination phase, and session key update phase. The notation guide of the proposed scheme is shown in the following Table 1.

Table 1
Scheme Notations

Notation Description

X_si, X_BS Random Numbers Select by Biosensor nodes and Base Station

PU_si, PU_BS Public Keys

t Time Stamp

D Divisor

S _ID Identification Number of Biosensor Node

BS _ID Identification Number of Base Station

E _k D _k Encryption/Decryption Keys

C C _i Encoded Information

X ₁ X ₂ Hash Value/ Compressed Hash Value

MAC Message authentication code

V _sgn Vital sign

M_ri - 1 Last round data

K_si - 1 Secret key data

AES Advanced Encryption Standard

Notation	Description
X_si, X_BS	Random Numbers Select by Biosensor nodes and Base Station
PU_si, PU_BS	Public Keys
t	Time Stamp
D	Divisor
S _ID	Identification Number of Biosensor Node
BS _ID	Identification Number of Base Station
E _k D _k	Encryption/Decryption Keys
C C _i	Encoded Information
X ₁ X ₂	Hash Value/ Compressed Hash Value
MAC	Message authentication code
V _sgn	Vital sign
M_ri - 1	Last round data
K_si - 1	Secret key data
AES	Advanced Encryption Standard

3.1 Registration Phase

In this phase, each biosensor node deployed on a patient body can pre-register with BS using their unique ID. Moreover, all external users doctors, nurses, government agencies, and researchers such as also pre-register with MS using their valid credentials.

3.2 Authentication and Key Exchange Phase

In this phase, mutual authentication performs between the biosensor nodes and BS, shown in the following Fig. 2.

Fig. 2

Process of authentication and key exchange.

3.3 Confidential Patient Data Dissemination Phase

In this phase, each biosensor node deployed on a patient body sense patient vital signs if vital sign found in a critical condition then transmits that data immediately to the MS for further necessary actions. The confidentiality of the transmitted patient vital signs is maintain using the AES algorithm, while for data integrity first, we apply the hash function then the compressed hash. If data not found critical, discard the patient’s sensed data to overcome the transmission and energy cost of WBSNs. The following algorithm 1 is used for confidential data transmission.

Algorithm 1 Confidential patient data transmission

For each deployed bio-sensor (S_i) node ∈ patient (P)

If S_i sensed patient vital sign (V_sgn) = = Critical data

{

Calculate X₁ = Hash (V_sgn)

Calculate X₂ = Compressed Hash (X₁)

Calculate C_i = E_AES(V_sgn || X₂ )

Transmit C_i toward medical server

}

Else

Discard patient sensed data

End if

End for

3.4 Session Key Update Phase

Session key updating is a crucial job, especially when a new node joins or leaves the WBSNs. To ensure forward and backward secrecy, we used the session key updating phase to maintain the entire session’s secrecy by updating the session key after a specific time interval. Each session used a different secret key for the secure transmission of patient data. If new biosensor node want to join the WBSN so BS chooses a fresh time interval t_n+1 then transmit (BS_ID, PU_BS, tn + 1) back to the new node. Biosensor node selects secret key and computes: PU_si+1 = (X_si+1 · t_n+1) · D Then send (S_ID+1 · PU_si+1) to BS. The following algorithm 2 is used to apply for the session key updation purposes.

Algorithm 2 Session Key Update Phase

Sensor Node

Keep information of the previous session last round ( $M_{r_{ı}} - 1)$

Applied message digest SHA- 512 = h ( $M_{r_{ı}} - 1)$

Choose fresh time interval t_n₊₁

Transmitted ${B S}_{ID}$ , PU_BS, and t_n₊₁ to Base Station (BS)

Computes secret key $K_{si + 1}$ = (X_si₊₁. t_n₊₁). D

Communicated $(S_{ID + 1} \cdot K_{si + 1})$ to BS

Generated updated Session key ${K s}_{r ı} = {K_{sr ı}}_{- 1} oplus h$ ( $M_{r_{ı}} - 1)$

Base Station (BS)

Accept last data of last round ( $M_{r_{ı}} - 1)$

Computes h′ = h (M_{r
_i} - 1)

Compare message digest h′ = h

If matched h′ = h than accepted otherwise rejected

Received $(S_{ID + 1} \cdot K_{si + 1})$ from sensor node

Generated updated Session key ${K s}_{r ı} = {K_{sr ı}}_{- 1} oplus h$ ( $M_{r_{ı}} - 1)$

Applied secure session key ${K s}_{r ı}$ for secure patient data transmission

4 An approach to multiple attribute group decision making with triangular cubic hesitant fuzzy information

This section will consume the suggested Triangular Cubic Hesitant Fuzzy Aggregation (TCHFA) operators to develop an approach to multiple attribute group decision-making with triangular cubic hesitant fuzzy information. First, a multiple attribute group decision-making with triangular cubic linguistic hesitant fuzzy information can be described as follows. Let

Y = {Y₁, Y₂, ⋯ , Y_m}

be a set of m alternatives, G = {G₁, G₂, ⋯ , G_n}

a gathering of n attributes, whose weight vector is

$\ddot{Ψ} = ({\ddot{Ψ}}_{1}, {\ddot{Ψ}}_{2}, \dots, {\ddot{Ψ}}_{n})^{T}$ , with

${\ddot{Ψ}}_{i} \in [0, 1], i = 1, 2, \dots, n$ and $\sum_{i = 1}^{n} {\ddot{Ψ}}_{i} = 1$ . Let D = {D₁, D₂, ⋯ , D_l} be a set of l decision-makers, whose weight vector is τ = (τ₁, τ₂, ⋯ , τ_n) ^T, with τ_k ∈ [0, 1] , k = 1, 2, ⋯ , l and $\sum_{k = 1}^{l} τ_{k} = 1$ . Let $R^{k} = (r_{ij}^{k})_{m \times n}$

be the triangular cubic hesitant fuzzy decision matrix, where $r_{ij}^{k} = {γ_{ij}^{k} | γ_{ij}^{k} \in r_{ij}^{k}} = {\begin{matrix} 〈 [a^{-}, \\ b^{-}, c^{-}], \\ [a^{+}, b^{+}, \\ c^{+}], a, \\ b, c 〉 \end{matrix}}$

is a TCHFE given by the decision-maker. D_k ∈ D, where ${\begin{matrix} [a^{-}, \\ b^{-}, c^{-}], \\ [a^{+}, b^{+}, \\ c^{+}] \end{matrix}}$

indicates the possible interval value triangular hesitant fuzzy set range that the alternative Y_i ∈ Y satisfies the attribute G_j ∈ G, while [a, b, c] indicates the possible triangular hesitant fuzzy set range that the alternative Y_i ∈ Ydoes not satisfy the attribute G_j ∈ G. The Fig. 4 shows the overall process of optimal network selection, where we utilize the proposed operators to develop an approach to multiple attribute group decision making with triangular cubic hesitant fuzzy information, which includes the following steps:

Find the triangular cubic hesitant fuzzy information. $(r_{ij}^{k})^{r} = {γ_{ij}^{k} | γ_{ij}^{k} \in r_{ij}^{k}} = {\begin{matrix} 〈 s_{θ_{γ_{ij}^{k}}}, [a_{γ_{ij}^{k}}^{-}, \\ b_{γ_{ij}^{k}}^{-}, c_{γ_{ij}^{k}}^{-}], \\ [a_{γ_{ij}^{k}}^{+}, b_{γ_{ij}^{k}}^{+}, \\ c_{γ_{ij}^{k}}^{+}], a_{γ_{ij}^{k}}, \\ b_{γ_{ij}^{k}}, c_{γ_{ij}^{k}} 〉 \end{matrix}}$

Utilize the TCHFWA operator ${\begin{matrix} TCHFWA (p_{ij}^{1}, p_{ij}^{2}, \dots, p_{ij}^{l}) \oplus_{i = 1}^{n} (τ_{i} h_{i}) = 〈 \\ ⋃_{\begin{matrix} p \in a, q \in b, r \in c, \\ p^{-} \in a^{-}, q^{-} \in b^{-}, r^{-} \in c^{-}, \\ p^{+} \in a^{+}, q^{+} \in b^{+}, r^{+} \in c^{+} \end{matrix}} [1 - \prod_{i = 1}^{n} (1 - p_{i}^{-})^{τ_{i}}], \\ [1 - \prod_{i = 1}^{n} (1 - q_{i}^{-})^{τ_{i}}], \\ [1 - \prod_{i = 1}^{n} (1 - r_{i}^{-})^{τ_{i}}] [1 - \prod_{i = 1}^{n} (1 - p_{i}^{+})^{τ_{i}}], \\ [1 - \prod_{i = 1}^{n} (1 - q_{i}^{+})^{τ_{i}}], \\ [1 - \prod_{i = 1}^{n} (1 - r_{i}^{+})^{τ_{i}}], \prod_{i = 1}^{n} (p_{i})^{τ_{i}}, \prod_{i = 1}^{n} (q_{i})^{τ_{i}}, \\ \prod_{i = 1}^{n} (r_{i})^{τ_{i}} 〉 \end{matrix}}$

To aggregate the individual triangular cubic linguistic hesitant fuzzy decision matrix

$(p_{ij}^{k}) = (α_{ij}^{k})_{m \times n}$ into the collective triangular cubic hesitant fuzzy decision matrix. $p_{ij}^{k} = p_{ij}^{k} = {α_{ij}^{k} | α_{ij}^{k} \in p_{ij}^{k}} = {\begin{matrix} 〈 [a_{γ_{ij}^{k}}^{-}, \\ b_{γ_{ij}^{k}}^{-}, c_{γ_{ij}^{k}}^{-}], \\ [a_{γ_{ij}^{k}}^{+}, b_{γ_{ij}^{k}}^{+}, \\ c_{γ_{ij}^{k}}^{+}], a_{γ_{ij}^{k}}, \\ b_{γ_{ij}^{k}}, c_{γ_{ij}^{k}} 〉 \end{matrix}} .$

Calculate the score values s (p_i) (i = 1, 2, ⋯ , m) of p (i = 1, 2, ⋯ , m): $\begin{matrix} s (p_{i}) = \\ {\frac{\begin{matrix} \frac{1}{9} ⋃_{\begin{matrix} p \in a, q \in b, r \in c, \\ p^{-} \in a^{-}, q^{-} \in b^{-}, r^{-} \in c^{-}, \\ p^{+} \in a^{+}, q^{+} \in b^{+}, r^{+} \in c^{+} \end{matrix}} \\ ([〈 s_{θ_{γ_{ij}^{k}}}, [p_{γ_{ij}^{k}}^{-} + q_{γ_{ij}^{k}}^{-} + r_{γ_{ij}^{k}}^{-}] + [p_{γ_{ij}^{k}}^{+} + q_{γ_{ij}^{k}}^{+} + r_{γ_{ij}^{k}}^{+}] \\ - [p_{γ_{ij}^{k}} + q_{γ_{ij}^{k}} + r_{γ_{ij}^{k}} 〉]) \end{matrix}}{# (p_{i})}} \end{matrix}$

Get the ranking of the alternatives Y_i (i = 1, 2, ⋯ , m) by ranking. s (p_i) (i = 1, 2, ⋯ , m).

Fig. 3

Proposed scheme for optimal network selection.

Fig. 4

First Proposed Method.

4.1 Numerical Application

Calculate the triangular cubic hesitant fuzzy decision matrix Table 2.

Utilize the TCHFWA operator

Calculate the score values s (p_i) (i = 1, 2, ⋯ , m) of p (i = 1, 2, ⋯ , m): s (p₁) =0.2323, s (p₂) =0.3123, s (p₃) =0.4563 .

Table 2
Triangular cubic hesitant fuzzy decision matrix

C ₁ C ₂ C ₃

A ₁ ${\begin{matrix} 〈 [0.2, 0.4, \\ 0.6], [0.4, \\ 0.6, 0.8], \\ 0.3, 0.5, \\ 0.7 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.103, 0.0105, \\ 0.0107], [0.0105, \\ 0.0107, 0.0109], \\ 0.0104, 0.0106, \\ 0.0108 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.103, 0.105, \\ 0.107], [0.105, \\ 0.107, 0.109], \\ 0.104, 0.106, \\ 0.108 〉 \end{matrix}}$

D1= A ₂ ${\begin{matrix} 〈 [0.1001, 0.1003, \\ 0.1005], [0.1003, \\ 0.1005, 0.1007], \\ 0.1002, 0.1004, \\ 0.1006 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.6, 0.8, \\ 0.10], [0.8, \\ 0.10, 0.12], \\ 0.7, 0.9, \\ 0.11 〉 \end{matrix}}$

A ₃ ${\begin{matrix} 〈 [0.6, 0.8, \\ 0.10], [0.8, \\ 0.10, 0.12], \\ 0.7, 0.9, \\ 0.11 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.11, 0.13, \\ 0.15], [0.13, \\ 0.15, 0.17], \\ 0.12, 0.14, \\ 0.16 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.404, 0.406, \\ 0.408], [0.406, \\ 0.408, 0.410], \\ 0.405, 0.407, \\ 0.409 〉 \end{matrix}}$

		C ₁	C ₂	C ₃
	A ₁	${\begin{matrix} 〈 [0.2, 0.4, \\ 0.6], [0.4, \\ 0.6, 0.8], \\ 0.3, 0.5, \\ 0.7 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.103, 0.0105, \\ 0.0107], [0.0105, \\ 0.0107, 0.0109], \\ 0.0104, 0.0106, \\ 0.0108 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.103, 0.105, \\ 0.107], [0.105, \\ 0.107, 0.109], \\ 0.104, 0.106, \\ 0.108 〉 \end{matrix}}$
D1=	A ₂	${\begin{matrix} 〈 [0.1001, 0.1003, \\ 0.1005], [0.1003, \\ 0.1005, 0.1007], \\ 0.1002, 0.1004, \\ 0.1006 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.6, 0.8, \\ 0.10], [0.8, \\ 0.10, 0.12], \\ 0.7, 0.9, \\ 0.11 〉 \end{matrix}}$
	A ₃	${\begin{matrix} 〈 [0.6, 0.8, \\ 0.10], [0.8, \\ 0.10, 0.12], \\ 0.7, 0.9, \\ 0.11 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.11, 0.13, \\ 0.15], [0.13, \\ 0.15, 0.17], \\ 0.12, 0.14, \\ 0.16 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.404, 0.406, \\ 0.408], [0.406, \\ 0.408, 0.410], \\ 0.405, 0.407, \\ 0.409 〉 \end{matrix}}$

Table 3

Triangular cubic hesitant fuzzy decision matrix

		C ₁	C ₂	C ₃
	A ₁	${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.2, 0.4, \\ 0.6], [0.4, \\ 0.6, 0.8], \\ 0.3, 0.5, \\ 0.7 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.202, 0.204, \\ 0.206], [0.204, \\ 0.206, 0.208], \\ 0.203, 0.205, \\ 0.207 〉 \end{matrix}}$
D2=	A ₂	${\begin{matrix} 〈 [0.202, 0.204, \\ 0.206], [0.204, \\ 0.206, 0.208], \\ 0.203, 0.205, \\ 0.207 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.5, 0.7, \\ 0.9], [0.7, \\ 0.9, 0.11], \\ 0.6, 0.8, \\ 0.10 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$
	A ₃	${\begin{matrix} 〈 [0.101, 0.103, \\ 0.105], [0.103, \\ 0.105, 0.107], \\ 0.102, 0.104, \\ 0.106 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.2032, 0.2034, \\ 0.2036], [0.2034, \\ 0.2036, 0.2038], \\ 0.2033, 0.2035, \\ 0.2037 〉 \end{matrix}}$

	C ₁	C ₂	C ₃
A ₁	${\begin{matrix} 〈 [0.0635, 0.1592, \\ 0.2752], [0.1592, \\ 0.2752, 0.4303], \\ 0.5696, 0.7243, \\ 0.8407 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.0456, 0.0996, \\ 0.1692], [0.0990, \\ 0.1692, 0.2768], \\ 0.3153, 0.3506, \\ 0.3519 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.0648, 0.0657, \\ 0.2628], [0.0657, \\ 0.2628, 0.0675], \\ 0.4622, 0.4649, \\ 0.4675 〉 \end{matrix}}$
A ₂	${\begin{matrix} 〈 [0.0635, 0.01592, \\ 0.2752], [0.1592, \\ 0.2752, 0.4303], \\ 0.5696, 0.7243, \\ 0.8407 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.0456, 0.0996, \\ 0.1692], [0.0990, \\ 0.1692, 0.2768], \\ 0.3153, 0.3506, \\ 0.3519 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.0648, 0.0657, \\ 0.2628], [0.0657, \\ 0.2628, 0.0675], \\ 0.4622, 0.4649, \\ 0.4675 〉 \end{matrix}}$
A ₃	${\begin{matrix} 〈 [0.1643, 0.0346, \\ 0.5235], [0.0346, \\ 0.5235, 0.0419], \\ 0.2897, 0.3059, \\ 0.1079 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.1050, 0.2196, \\ 0.3480], [0.2196, \\ 0.3480, 0.5010], \\ 0.1549, 0.2366, \\ 0.3098 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.3127, 0.3147, \\ 0.3236], [0.3147, \\ 0.3236, 0.3146], \\ 0.2869, 0.2877, \\ 0.2889 〉 \end{matrix}}$

TCHFWA Operator
A ₁	${\begin{matrix} 〈 [0.726, 0.1603, 0.086], \\ [0.046, 0.083, 0.121], \\ 0.0098, 0.0290, 0.0582 〉 \end{matrix}}$
A ₂	${\begin{matrix} 〈 [0.4431, 0.288, 0.0393], \\ [0.099, 0.0.171, 0.1137], \\ 0.0210, 0.073, 0.1919 〉 \end{matrix}}$
A ₃	${\begin{matrix} 〈 [0.0596, 0.2315, 0.6596], \\ [0.05963, 0.223, 0.56941], \\ 0.04012, 0.01178, 0.0241 〉 \end{matrix}}$

5 Neutrosophic Linguistic TOPSIS Method

In this section, we apply the linguistic TOPSIS method to the neutrosophic set. We define a new extension of the neutrosophic linguistic TOPSIS method by using the linguistic TOPSIS method.

Suppose a neutrosophic linguistic TOPSIS method decision-making problem under multiple attributes has m students and n decision attributes. The framework of neutrosophic linguistic TOPSIS decision matrix can be exhibit as follows:

The following Fig. 4 shows the process of the first proposed method.

$β = {\begin{matrix} s_{θ_{11}} [r_{11}, s_{11}, t_{11}] s_{θ_{12}} [r_{12}, s_{12}, t_{12}] \dots \dots s_{θ_{n 1}} [r_{n 1}, s_{n 1}, t_{n 1}] \\ s_{θ_{21}} [r_{21}, s_{21}, t_{21}] s_{θ_{22}} [r_{22}, s_{22}, t_{22}] \dots \dots s_{θ_{n 2}} [r_{n 2}, s_{n 2}, t_{n 2}] \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ s_{θ_{m 1}} [r_{m 1}, s_{m 1}, t_{m 1}] s_{θ_{m 1}} [r_{m 2}, s_{m 2}, t_{m 2}] \dots \dots s_{θ_{mn}} [r_{mn}, s_{mn}, t_{mn}] \end{matrix}}$

Construct the normalized neutrosophic linguistic TOPSIS method decision matrix R = [β_ij] . The normalized value r_ij is calculated as: $β = \frac{s_{θ}}{\sqrt{\sum_{i = 1}^{n} (s_{θ})^{2}}} [\frac{r}{\sqrt{\sum_{i = 1}^{n} (r_{ij})^{2}}}, \frac{s}{\sqrt{\sum_{i = 1}^{n} (s_{ij})^{2}}}, \frac{t}{\sqrt{\sum_{i = 1}^{n} (t_{ij})^{2}}}]$

Make the weighted normalized neutrosophic linguistic TOPSIS decision matrix by multiplying the normalized neutrosophic linguistic TOPSIS matrix by its associated weights. The weight vector W = (w₁, w₂, ⋯ , w_n) collected of the isolated weights w_j (j = 1, 2, 3, ⋯ , n) for each attribute C_j satisfying $\sum_{j = 1}^{n} W_{j} = 1 .$ The weighted normalized value is deliberate by B_j = v_ijw_j where 0 ≤ B_j ≤ 1 i = 1, 2, 3 ⋯ , m and j = 1, 2, 3, ⋯ , n .

The neutrosophic linguistic TOPSIS positive ideal solution and the neutrosophic linguistic TOPSIS negative-ideal solution is shown as, $\begin{matrix} α_{i}^{+} = [s_{θ_{1}} (r_{1}, s_{1}, t_{1}), s_{θ_{2}} (r_{2}, s_{2}, t_{2}) \dots s_{θ n} (r_{n}, s_{n}, t_{n})] \\ = max_{i} s_{θ ij}, max_{i} (r_{ij}, s_{ij}, t_{ij}), \\ α_{i}^{-} = [s_{θ_{1}} (r_{1}, s_{1}, t_{1}), s_{θ_{2}} (r_{2}, s_{2}, t_{2}) \dots s_{θ n} (r_{n}, s_{n}, t_{n})] \\ = min_{i} s_{θ ij}, min_{i} (r_{ij}, s_{ij}, t_{ij}) . \end{matrix}$

Calculate the separation measures using the n-dimensional Euclidean distance. The separation of each candidate from the NLPIS $q_{i}^{*} 〈 [B^{-}, B^{+}], η 〉$ is given as; $\begin{matrix} q_{i}^{+} 〈 [B^{-}, B^{+}], η 〉 = 〈 \frac{1}{3} (| s_{ij} - s_{j} | + | r_{ij} - r_{j} | + | s_{ij} - s_{j} | + | t_{ij} - t_{j} |) 〉 \end{matrix}$

The separation of each candidate from the NLNIS $q_{i}^{-} 〈 [B^{-}, B^{+}], η 〉$ is given as;

$q_{i}^{-} 〈 [B^{-}, B^{+}], η 〉 = 〈 \frac{1}{9 n} (\begin{matrix} | r_{ij} - r_{j} | + | s_{ij} - s_{j} | + | t_{ij} - t_{j} |, \\ max | B_{ij} - B_{j}^{-} |, | B_{ij} - B_{j}^{+} |, | η_{ij} - η_{j} | \end{matrix}), 〉 . ∥$

Calculate the closeness coefficients to an ideal solution. This progression comprehends the similitudes to an ideal solution of equations. $Z_{i} = \frac{q_{i}^{-} 〈 [B^{-}, B^{+}], η 〉}{q_{i}^{-} 〈 [B^{-}, B^{+}], η 〉 + q_{i}^{+} 〈 [B^{-}, B^{+}], η 〉} .$

The following Fig. 5 shows the process of the first proposed method.

Fig. 5

Second proposed method.

5.1 Numerical Application

Construct the normalized neutrosophic linguistic TOPSIS method decision matrix R = [β_ij] .the normalized value r_ij is calculated as:

Make the weighted normalized neutrosophic linguistic TOPSIS decision matrix by multiplying the normalized neutrosophic linguistic TOPSIS matrix by its associated weights.

The neutrosophic linguistic TOPSIS positive ideal solution and the neutrosophic linguistic TOPSIS negative ideal solution are shown as, $\begin{matrix} α_{i}^{+} = [s_{0.337}, (0.006, 0.035, 0.018)], [s_{0.449}, (0.018, 0.045, 0.027)], [s_{0.5}, (0.027, 0.004, 0.036)], \\ α_{i}^{-} = [s_{0.112}, (0.02, 0.009, 0.05)], [s_{0.224}, (0.03, 0.002, 0.06)], [s_{0.224}, (0.002, 0.02, 0.006)] \end{matrix}$

Calculate the separation measures using the n-dimensional Euclidean distance. The separation of each candidate from the NLPIS $q_{i}^{+}$ is given as $q_{1}^{+} = s_{0.562} (0.107), q_{2}^{+} = s_{0.388} (0.151), q_{1}^{+} = s_{0.49} (0.18),$

The separation of each candidate from the NLNIS $q_{i}^{-}$ is given as $q_{1}^{-} = s_{0.451} (0.173), q_{2}^{-} = s_{0.511} (0.201), q_{3}^{-} = s_{0.228} (0.214),$

Calculate similarity to the ideal solution. This progression comprehends the similitudes to an ideal solution of equations. Z₁ = s_0.561, Z₂ = s_0.638, Z₃ = s_0.356 .

6 Triangular Cubic Hesitant Fuzzy Information with Hamacher Operator

In this section, we consume triangular cubic hesitant fuzzy aggregation operators to develop an approach to multiple attribute group decision-making with triangular hesitant fuzzy information. First, a multiple attribute group decision-making with triangular cubic hesitant fuzzy information can be described as follows.

Let Y = {Y₁, Y₂, ⋯ , Y_m} be a set of m alternatives, G = {G₁, G₂, ⋯ , G_n} a gathering of n attributes, whose weight vector is $\ddot{Ψ} = ({\ddot{Ψ}}_{1}, {\ddot{Ψ}}_{2}, \dots, {\ddot{Ψ}}_{n})^{T}$ , with ${\ddot{Ψ}}_{i} \in [0, 1]$ , i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} {\ddot{Ψ}}_{i} = 1 .$

Let D = {D₁, D₂, ⋯ , D_l} be a set of l decision-makers, whose weight vector is τ = (τ₁, τ₂, ⋯ . , τ_n) ^T, with $τ_{k} \in [0, 1], k = 1, 2, \dots, l, \sum_{k = 1}^{l} τ_{k} = 1$ and

Let $R^{k} = (r_{ij}^{k})_{m \times n}$ be the triangular cubic hesitant fuzzy decision matrix,where, $r_{ij}^{k} = {γ_{ij}^{k} | γ_{ij}^{k} \in r_{ij}^{k}} = {\begin{matrix} 〈 [a^{-}, b^{-}, c^{-}], \\ [a^{+}, b^{+}, c^{+}], \\ a, b, c 〉 \end{matrix}}$ a TCHFE given by the decision $D_{k} \in D, where {\begin{matrix} [a^{-}, b^{-}, c^{-}], \\ [a^{+}, b^{+}, c^{+}] \end{matrix}}$ -maker, indicating the possible interval value triangular hesitant fuzzy set range that the alternative Y_i ∈ Y satisfies G_j ∈ Gwhile[a, b, c] indicates the possible triangular hesitant fuzzy set range that the alternative Y_i ∈ Ydoes not satisfy the attributeG_j ∈ G.

		C ₁	C ₂	C ₃
	A ₁	s₂, [0.2, 0.4, 0.6]	s₁, [0.1, 0.2, 0.3]	s₂, [0.3, 0.5, 0.7]
D1=	A ₂	s _1,	s₄, [0.4, 0.5, 0.6]	s₅, [0.4, 0.6, 0.8]
	A ₃	s₃, [0.11, 0.13, 0.15]	s₂, [0.6, 0.7, 0.8]	s₄, [0.1, 0.3, 0.5]

	C ₁	C ₂	C ₃
A ₁	s_0.224, (0.030, 0.06, 0.09)	s_0.112, [0.01, 0.030, 0.04]	s_0.224, [0.01, 0.019, 0.02]
A ₂	s _0.112,	s_0.449, [0.06, 0.07, 0.09]	s_0.56, [0.09, 0.10, 0.12]
A ₃	s_0.337, [0.04, 0.07, 0.10]	s_0.224, [0.06, 0.09, 0.12]	s_0.449, [0.01, 0.04, 0.07]

	C ₁	C ₂	C ₃
A ₁	s_{_0.224}, [0.006, 0.012, 0.018]	s_0.112, [0.002, 0.008, 0.014]	s_{_0.224}, [0.002, 0.004, 0.006]
A ₂	s _0.112,	s_0.449, [0.018, 0.021, 0.027]	s_0.56, [0.027, 0.03, 0.036]
A ₃	s_0.337, [0.02, 0.035, 0.05]	s_0.224, [0.03, 0.045, 0.06]	s_0.449, [0.005, 0.02, 0.035]

Calculate the decision matrix $A_{ij}^{k} = a_{ij}^{k} = {α_{ij}^{k} | α_{ij}^{k} \in A_{ij}^{k}} = {\begin{matrix} 〈 [a_{γ_{ij}^{k}}^{-}, b_{γ_{ij}^{k}}^{-}, c_{γ_{ij}^{k}}^{-}], \\ [a_{γ_{ij}^{k}}^{+}, b_{γ_{ij}^{k}}^{+}, c_{γ_{ij}^{k}}^{+}], \\ a_{γ_{ij}^{k}}, b_{γ_{ij}^{k}}, c_{γ_{ij}^{k}} 〉 \end{matrix}}$

Utilize the TCHFHWA operator TCHFHWA

$({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = ⋃_{\begin{matrix} p \in a, q \in b, r \in c, \\ p^{-} \in a^{-}, q^{-} \in b^{-}, r^{-} \in c^{-}, \\ p^{+} \in a^{+}, q^{+} \in b^{+}, r^{+} \in c^{+} \end{matrix}} [\begin{matrix} \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) p_{j}^{-})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - p_{j}^{-})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) p_{j}^{-})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - p_{j}^{-})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) q_{j}^{-})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - q_{j}^{-})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) q_{j}^{-})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - q_{j}^{-})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{j}^{-})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - r_{j}^{-})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{j}^{-})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - r_{j}^{-})}^{w_{j}}} \end{matrix}]$

To aggregate all individual triangular cubic hesitant fuzzy decision matrices. $(A_{ij}^{k}) = (α_{ij}^{k})_{m \times n}$

For the collective triangular cubic hesitant fuzzy decision matrix.

$A_{ij}^{k} = a_{ij}^{k} = {α_{ij}^{k} | α_{ij}^{k} \in A_{ij}^{k}} = {\begin{matrix} 〈 [a_{γ_{ij}^{k}}^{-}, b_{γ_{ij}^{k}}^{-}, c_{γ_{ij}^{k}}^{-}], \\ [a_{γ_{ij}^{k}}^{+}, b_{γ_{ij}^{k}}^{+}, c_{γ_{ij}^{k}}^{+}], \\ a_{γ_{ij}^{k}}, b_{γ_{ij}^{k}}, c_{γ_{ij}^{k}} 〉 \end{matrix}} .$

Utilize the TCHFHWA operator

TCHFHWA $({\ddot{h}}_{1}, {\ddot{h}}_{2}, \dots, {\ddot{h}}_{n}) = ⋃_{\begin{matrix} p \in a, q \in b, r \in c, \\ p^{-} \in a^{-}, q^{-} \in b^{-}, r^{-} \in c^{-}, \\ p^{+} \in a^{+}, q^{+} \in b^{+}, r^{+} \in c^{+} \end{matrix}}$ $[\begin{matrix} \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) p_{j}^{-})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - p_{j}^{-})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) p_{j}^{-})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - p_{j}^{-})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) q_{j}^{-})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - q_{j}^{-})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) q_{j}^{-})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - q_{j}^{-})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{j}^{-})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - r_{j}^{-})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{j}^{-})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - r_{j}^{-})}^{w_{j}}} \end{matrix}], [\begin{matrix} \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) p_{j}^{+})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - p_{j}^{+})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) p_{j}^{+})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - p_{j}^{+})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) q_{j}^{+})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - q_{j}^{+})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) q_{j}^{+})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - q_{j}^{+})}^{w_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{j}^{+})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - r_{j}^{+})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{j}^{+})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - r_{j}^{+})}^{w_{j}}} \end{matrix}],$ $[\begin{matrix} \frac{γ \prod_{j = 1}^{n} p_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - p_{j}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} p_{j}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} q_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - q_{j}))}^{w_{j}} (γ - 1) \prod_{j = 1}^{n} q_{j}^{w_{j}}}, \frac{γ \prod_{j = 1}^{n} r_{j}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - r_{j}))}^{w_{j}} (γ - 1) \prod_{j = 1}^{n} r_{j}^{w_{j}}} \end{matrix}]$

To aggregate all preference values a_ij (j = 1, 2, ⋯ , n) in the i line A, and then derive the collective overall preference value $A_{ij}^{k} = a_{ij}^{k} = {α_{ij}^{k} | α_{ij}^{k} \in A_{ij}^{k}} = 〈 [a_{γ_{ij}^{k}}^{-}, b_{γ_{ij}^{k}}^{-}, c_{γ_{ij}^{k}}^{-}], [a_{γ_{ij}^{k}}^{+}, b_{γ_{ij}^{k}}^{+}, c_{γ_{ij}^{k}}^{+}], a_{γ_{ij}^{k}}, b_{γ_{ij}^{k}}, c_{γ_{ij}^{k}} 〉 (i = 1, 2, \dots, m)$ of the alternative Y_i (i = 1, 2, ⋯ , m), where $\ddot{ξ} = ({\ddot{ξ}}_{1}, {\ddot{ξ}}_{2}, \dots, {\ddot{ξ}}_{n})^{T}$ the weight vector of the TCHFHWA operator, with ${\ddot{ξ}}_{j} \in [0, 1], j = 1, 2, \dots, n, \sum_{j = 1}^{n} ξ_{j} = 1$ and.

Calculate the score values s (a_i) (i = 1, 2, ⋯ , m) of a (i = 1, 2, ⋯ , m): $s (a_{i}) = {\frac{\begin{matrix} \frac{1}{9} ⋃_{\begin{matrix} p \in a, q \in b, r \in c, \\ p^{-} \in a^{-}, q^{-} \in b^{-}, r^{-} \in c^{-}, \\ p^{+} \in a^{+}, q^{+} \in b^{+}, r^{+} \in c^{+} \end{matrix}} \\ ([[p_{γ_{ij}^{k}}^{-} + q_{γ_{ij}^{k}}^{-} + r_{γ_{ij}^{k}}^{-}] + [p_{γ_{ij}^{k}}^{+} + q_{γ_{ij}^{k}}^{+} + r_{γ_{ij}^{k}}^{+}] \\ - [p_{γ_{ij}^{k}} + q_{γ_{ij}^{k}} + r_{γ_{ij}^{k}} 〉]) \end{matrix}}{# (p_{i})}}$

Get the priority of the alternatives Y_i (i = 1, 2, ⋯ , m)by ranking. s (a_i) (i = 1, 2, ⋯ , m) .

6.1 Numerical Application

Calculate the decision matrix.

Utilize the TCHFHWA operator

Utilize TCHFHWA operator again

7 Cubic GRA Set

In this part, we apply a cubic set to improve decision performance using grey analysis. We apply a novel version of the Cubic GRA Set by applying cubic sets. Let us explain the Cubic GRA Set. Let $\begin{matrix} Y = {[(y_{0}^{-}, y_{0}^{+}), y_{0}] [(y_{1}^{-}, y_{1}^{+}), y_{1}] \dots, \\ [(y_{i}^{-}, y_{i}^{+}), y_{i}] \dots, [y_{m}^{-}, y_{m}^{+}), y_{m}]}, \end{matrix}$

and be a sequence (candidate) of two sets, $y_{0}^{-} y_{0}^{+}, y_{0}$ denotes the referential sequences, $y_{i}^{-}, y_{i}^{+}$ y_i are comparative sequences. Let $y_{0 p}^{-}, y_{op}^{+}, x_{0 p}$ , and x_ip represent the respective value at point / factor p, p = 1, 2, 3 ⋯⋯ , n, for $y_{0}^{-}, y_{0}^{+} y_{0} y_{i}^{-}, y_{i}^{+}, y_{i},$ and. The cubic grey relation coefficients $α (y_{0 p}^{-}, y_{ij}) β (y_{op}^{+}, y_{ip}) γ (y_{0 p}, y_{ip})$ of these candidates at a point p can be computed by $\begin{matrix} α (y_{0 j}^{-}, y_{ij}) β (y_{op}^{+}, y_{ip}) γ (y_{0 p}, y_{ip}) \\ = 〈 \frac{\min_{i} \min_{p} {[| y_{0 p}^{-} - y_{ip} |, | y_{0 p}^{+} - y_{ip} |], (| y_{0 p} - y_{ip} |)} + ρ \max_{i} \max_{p} {[| y_{0 p}^{-} - y_{ip} |, | y_{0 p}^{+} - y_{ip} |], (| y_{0 p} - y_{ip} |)}}{{[| y_{0 p}^{-} - y_{ip} |, | y_{0 p}^{+} - y_{ip} |], (| y_{0 p} - y_{ip} |)} + ρ \max_{i} \max_{i} {[| y_{0 p}^{-} - y_{ip} |, | y_{0 p}^{+} - y_{ij} |], (| y_{0 p} - y_{ip} |)}} 〉 ∥ \end{matrix}$

Table 4
Triangular Cubic Hesitant Fuzzy Decision-1

C ₁ C ₂ C ₃

A ₁ ${\begin{matrix} 〈 [0.4, 0.6, \\ 0.8], [0.6, \\ 0.8, 0.10], \\ 0.5, 0.7, \\ 0.9 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.3, 0.5, \\ 0.7], [0.5, \\ 0.7, 0.9], \\ 0.4, 0.6, \\ 0.8 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$

D1= A ₂ ${\begin{matrix} 〈 [0.4, 0.6, \\ 0.8], [0.6, \\ 0.8, 0.10], \\ 0.5, 0.7, \\ 0.9 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.8, 0.10, \\ 0.12], [0.10, \\ 0.12, 0.14], \\ 0.9, 0.11, \\ 0.13 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.201, 0.203, \\ 0.205], [0.203, \\ 0.205, 0.207], \\ 0.202, 0.204, \\ 0.206 〉 \end{matrix}}$

A ₃ ${\begin{matrix} 〈 [0.02, 0.04, \\ 0.06], [0.04, \\ 0.06, 0.08], \\ 0.03, 0.05, \\ 0.07 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$ ${\begin{matrix} 〈 [0.8, 0.10, \\ 0.12], [0.10, \\ 0.12, 0.14], \\ 0.9, 0.11, \\ 0.13 〉 \end{matrix}}$

		C ₁	C ₂	C ₃
	A ₁	${\begin{matrix} 〈 [0.4, 0.6, \\ 0.8], [0.6, \\ 0.8, 0.10], \\ 0.5, 0.7, \\ 0.9 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.3, 0.5, \\ 0.7], [0.5, \\ 0.7, 0.9], \\ 0.4, 0.6, \\ 0.8 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$
D1=	A ₂	${\begin{matrix} 〈 [0.4, 0.6, \\ 0.8], [0.6, \\ 0.8, 0.10], \\ 0.5, 0.7, \\ 0.9 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.8, 0.10, \\ 0.12], [0.10, \\ 0.12, 0.14], \\ 0.9, 0.11, \\ 0.13 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.201, 0.203, \\ 0.205], [0.203, \\ 0.205, 0.207], \\ 0.202, 0.204, \\ 0.206 〉 \end{matrix}}$
	A ₃	${\begin{matrix} 〈 [0.02, 0.04, \\ 0.06], [0.04, \\ 0.06, 0.08], \\ 0.03, 0.05, \\ 0.07 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.8, 0.10, \\ 0.12], [0.10, \\ 0.12, 0.14], \\ 0.9, 0.11, \\ 0.13 〉 \end{matrix}}$

Table 5

Triangular cubic hesitant fuzzy decision

		C ₁	C ₂	C ₃
	A ₁	${\begin{matrix} 〈 [0.8, 0.10, \\ 0.12], [0.10, \\ 0.12, 0.14], \\ 0.9, 0.11, \\ 0.13 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.10, 0.12, \\ 0.14], [0.12, \\ 0.14, 0.16], \\ 0.11, 0.13, \\ 0.15 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.02, 0.04, \\ 0.06], [0.04, \\ 0.06, 0.08], \\ 0.03, 0.05, \\ 0.07 〉 \end{matrix}}$
D2=	A ₂	${\begin{matrix} 〈 [0.101, 0.103, \\ 0.105], [0.103, \\ 0.105, 0.107], \\ 0.102, 0.104, \\ 0.106 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$
	A ₃	${\begin{matrix} 〈 [0.01, 0.03, \\ 0.05], [0.03, \\ 0.05, 0.07], \\ 0.02, 0.04, \\ 0.06 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.3, 0.5, \\ 0.7], [0.5, \\ 0.7, 0.9], \\ 0.4, 0.6, \\ 0.8 〉 \end{matrix}}$	${\begin{matrix} 〈 [0.1, 0.3, \\ 0.5], [0.3, \\ 0.5, 0.7], \\ 0.2, 0.4, \\ 0.6 〉 \end{matrix}}$

	C ₁	C ₂	C ₃
A ₁	${〈 \begin{matrix} [0.818, 0.636, \\ 0.694], [0.636, \\ 0.694, 0.496], \\ 1.93, 0.383, \\ 0.359 \end{matrix} 〉}$	${〈 \begin{matrix} [0.563, 0.689, \\ 0.689], [0.689, \\ 0.689, 0.680], \\ 0.107, 0.163, \\ 0.245 \end{matrix} 〉}$	${〈 \begin{matrix} [0.361, 0.490, \\ 0.571], [0.490, \\ 0.571, 0.637], \\ 0.046, 0.094, \\ 0.148 \end{matrix} 〉}$
A ₃	${〈 \begin{matrix} [0.509, 4.88, \\ 0.281], [4.88, \\ 0.281, 0.297], \\ 0.192, 0.246, \\ 0.441 \end{matrix} 〉}$	${〈 \begin{matrix} [0.535, 0.393, \\ 0.478], [0.393, \\ 0.478, 0.553], \\ 0.464, 0.159, \\ 0.424 \end{matrix} 〉}$	${〈 \begin{matrix} [0.352, 0.470, \\ 0.546], [0.470, \\ 0.546, 0.605], \\ 0.121, 0.293, \\ 0.576 \end{matrix} 〉}$
A ₃	${〈 \begin{matrix} [0.1393, 0.061, \\ 0.0156], [0.061, \\ 0.0156, 0.1017], \\ 0.1221, 0.3116, \\ 0.6405 \end{matrix} 〉}$	${〈 \begin{matrix} [0.239, 0.449, \\ 0.773], [0.449, \\ 0.773, 0.877], \\ 0.612, 0.665, \\ 0.973 \end{matrix} 〉}$	${〈 \begin{matrix} [0.348, 0.1848, \\ 0.2549], [0.1848, \\ 0.2549, 0.3351], \\ 0.6846, 0.2609, \\ 0.3334 \end{matrix} 〉}$

TCHFHWA operator
A ₁	${\begin{matrix} 〈 [0.726, 0.1603, 0.086], \\ [0.046, 0.083, 0.121], \\ 0.0098, 0.0290, 0.0582 〉 \end{matrix}}$
A ₃	${\begin{matrix} 〈 [0.4431, 0.288, 0.0393], \\ [0.099, 0.0171, 0.1137], \\ 0.0210, 0.073, 0.1919 〉 \end{matrix}}$
A ₃	${\begin{matrix} 〈 [0.0596, 0.2315, 0.6596], \\ [0.0596, 0.223, 0.5694], \\ 0.0401, 0.0117, 0.0241 〉 \end{matrix}}$

Where ρ is the identification coefficient in a different case: $\begin{matrix} ρ \in [0, 1], i \in l = {1, 2, 3 \dots . m}, \\ p \in P = {1, 2, 3 \dots n} . \end{matrix}$

Later attaining cubic grey relation coefficients, the grade of cubic grey relations $α (y_{0 p}^{-}, y_{ip}) β (y_{op}^{+}, y_{ip}) γ (y_{0 p}, y_{ij})$ between y₀ altogether, $y_{0}^{-}, y_{0}^{+}$ and $y_{i}^{-}, y_{i}^{+}, y_{i}$ can be deliberated: $\begin{matrix} α (y_{0}^{-}, y_{i}) β (y_{0}^{+}, y_{i}) γ (y_{0}, y_{i}) = \\ 〈 \sum_{p = 1}^{n} wp α (y_{0 p}^{-}, y_{ip}) \sum_{p = 1}^{n} w_{p} β (y_{op}^{+}, y_{ip}) \\ \sum_{p = 1}^{n} w_{p} γ (y_{0 p}, y_{ij}) . \sum_{p = 1}^{n} w_{p} = 1 〉 . \end{matrix}$

8 Performance Analysis

In this section, we analyze the performance of different techniques used for network selection to improve the QoS for various applications of WBSN.

Moreover, the transmitted data are separated into three different traffic, i.e., EM, DS, and GM. Various MADM techniques are used, such as TCHFWA, NLTM, TCHFHWA, and Cubic GRA, to find optimal network selection for all types of transmitted data accordingly QoS and preference of different users. Moreover, two other MADM techniques, fuzzy AHP and AHP, are also used here to compute the attribute weights, which are further used in TCHFWA, NLTM, TCHFHWA, and Cubic GRA. The following graphs showed the obtain results in three different cases where Fig. 6 shows the process of the third proposed method and Fig. 7 shows the comparison of different data set. Furthermore Figs. 8, 9, and 10 shows AHP attribute weight for EM,DS, and GM. In Figs. 11, 12, and 13 we can show weight for EM,DS, and GM, While Figs. 14, 15, and 16 show fuzzy AHP attribute weight for EM, DS, and GM.

Fig. 6

Third proposed method.

Fig. 7

Comparison of different data sets.

Fig. 8

Network Selection with AHP attribute weights for EM.

Fig. 9

Network Selection with AHP attribute weights for DS.

Fig. 10

Network Selection with AHP attribute weights for GM.

Fig. 11

Network Selection with AHP Entropy attribute weights for EM.

Fig. 12

Network Selection with AHP Entropy attribute weights for DS.

Fig. 13

Network Selection with AHP Entropy attribute weights for GM.

Fig. 14

Network Selection with fuzzy AHP attribute weights for EM.

Fig. 15

Network Selection with fuzzy AHP attribute weights for DS.

Fig. 16

Network Selection with fuzzy AHP attribute weights for GM.

9 Conclusion

In this paper, we define four fuzzy attribute-based novel decision-making methods which are Triangular Cubic Hesistent Fuzzy Weighted Averaging Operator, Neutrosophic Linguistic TOPSIS method, Trangualar Cubic Hesistent Fuzzy Hamacher Weighted Averaging Operator and Cubic Grey Relational Analysis for optimal network selection from the available networks to improve the QoS of WBSNs for efficient transmission of patient vital signs. This leads to efficient resource utilization in the constrained environment of WBSNs. Keeping in view the importance of secure communication, we have used HECC for secure key exchange, the AES algorithm for confidential data transmission to protect patient medical data from adversaries’ attacks, and MAC protocol for mutual authentication. We have analyzed the network selection optimality through numerical examples from about four methods and have proved that our scheme is efficient in fuzzy-based decision-making for optimal network selection.

In the future, we will extend this approach for S-box image encryptions and proposed decision support models under the spherical fuzzy numbers. Also, we will work on network selection problems wireless body area network under spherical fuzzy numbers.

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