Pythagorean Fuzzy Sets-based VIKOR and TOPSIS-based multi-criteria decision-making model for mitigating resource deletion attacks in WSNs

Abstract

In Wireless Sensor Networks (WSNs), resource depletion attacks that focusses on the compromization of routing protocol layer is identified to facilitate a major influence over the network. These resource depletion attacks drain the batter power of the sensor nodes drastically with persistent network disruption. Several protocols were established for handling the impact of Denial of Service (DoS) attack, but majority of them was not able to handle it perfectly. In specific, thwarting resource depletion attack, a specific class of DoS attack was a herculean task. At this juncture, Multicriteria Decision Making Model (MCDM) is identified as the ideal candidate for evaluating the impact introduced by each energy depletion compromised sensor nodes towards the process of cooperation into the network. In this paper, A Pythagorean Fuzzy Sets-based VIKOR and TOPSIS-based multi-criteria decision-making model (PFSVT-MCDM) is proposed for counteracting with the impacts of resource depletion attacks to improve Quality of Service (QoS) in the network. This PFSVT-MCDM used the merits of Pythagorean Fuzzy Sets information for handling uncertainty and vagueness of information exchanged in the network during the process of data routing. It utilized VIKOR and TOPSIS for exploring the trust of each sensor nodes through the exploration of possible dimensions that aids in detecting resource depletion attacks. The experimental results of PFSVT-MCDM confirmed better throughput of 21.29%, enhanced packet delivery fraction of 22.38%, minimized energy consumptions 18.92%, and reduced end-to-end delay of 21.84%, compared to the comparative resource depletion attack thwarting strategies used for evaluation.

Keywords

Wireless sensor networks resource depletion attacks pythagorean fuzzy sets TOPSIS (Technique For Order Performance By Similarity To Ideal Solution)quality of service VIKOR (VlseKriterijumska Optimizacija Kompromisno Resenje)

1 Introduction

Mobile Wireless Sensor Networks (MWSNs) have their own features in contrast to wired networks for examining security related issues. A MWSNs includes a collection of sensor nodes that are self-organized and connect among themselves without a steady infrastructure or a centralised co-ordinator. The sensor nodes collaborate with adjoining devices, and nodes act as hosts and routers [1]. A node that is not in the communication range of the neighbouring node involves the intermediary nodes for conveying messages. These networks have an advantage over traditional networks in that they can be built efficiently and offer flexibility due to the node’s unbound characteristics. As the nodes are power constrained, it is not possible to constantly preserve the processing of malicious behavior detection on every node [2]. Furthermore, the existing malicious node detection systems do not consider the functioning environments, resulting in the observation of every node with unchanging probability irrespective of whether the node under observation has a malevolent description or not. Several anomalies are found in networks as they are dynamic and are without any infrastructure [3]. The causes for those irregularities include congestion in the network, faulty equipment, active attacks along with intrusion. Intrusion is a threatening abnormality that affects the service integrity as well as availability of networks. A renowned kind of network intrusion is the Denial of service (DoS) attack that worsens the service offered to authentic users [4]. DoS attacks can take many forms including Wormhole, Grayhole, Blackhole, and Flooding. They create numerous security breaches. Some factors that affect the performance of the wireless systems include connection disruption, flooding of traffic, blocking of access or system related issues. As MWSNSs offer open communication, they are more susceptible to attacks. The attacks may be active or passive [5].

For identifying active attacks, numerous techniques are proposed, though passive attacks are highly complex. Hence, many efforts are taken to guard the networks from assailants. The malicious node detection system is implemented directly to inspire detection on every node in the network [6]. The implementation may not be suitable for energy-constrained networks. To handle this issue, a general scheme which forms clusters in MWSNS and chooses a CH for every cluster is essential [7]. The CHs should have their malicious node detection system to defend the whole cluster. The theoretic models of malicious node detection system define network security with advanced decision making. Commonly, decision makers can be attackers or defenders which are incompatible [8]. An attacker focuses on interrupting the activities in the network. To handle the issues related to intrusion detection, Game theoretical models are commonly used. The models show the opposing goals of the attacker and defender in a seamless manner. The decision states are detailed and learnt using Game theory [9]. A dominant collection of tools is offered for intelligently learning and forecasting the outcome of complex associations amid rational entities [10].

Further, resource depletion attacks possess the capability of exploiting the control messages which are indispensable entities of the standard routing protocols contributed for WSNs since they do not necessitate authentication process [11]. In unattended scenarios, the energy of sensor nodes associated with WSN application cannot recharged resulting in network death rapid than the normal scenarios. This impossibility of sensor node energization results in network disruptions in the network. In particular, the network operators need to detect the uncertain impacts of energy depletion attacks and isolate them from the network [12]. This energy depletion attack can be categorized into two types that include stretch attack and route loop attack. In the stretch attack, the attackers attempt to plonging the normal routing length as much as possible to facilitate the data to pass through unnecessary sensor nodes several times without any potential objective [13]. This stretch attack increases the option of increasing the mean route length more than the expected level during the process of data transmission, resulting in unnecessary delay and packet drop in the network [14]. On the other hand, route loop attack also termed as carousel attack intentionally constructs the routing loops and induces the data packers to get propagated over the same loop for maximized number times without any necessity.

Furthermore, the energy depletion attacks are determined to slowly and systematic paralyze the network in small increments rather than generating huge amount of data. Detecting and preventing energy depletion attacks is a herculean task as it can be identified only during the data transmission phase before which significantly amount of network resources are already drained and exploited [15]. Moreover, the existing energy depletion attack detecting solutions that rely on cryptographic operations necessitates potential amount of energy and computational energy requirement from sensor nodes which are already energy-restricted resources [16]. In addition, only limited number of energy depletion attack detection mechanisms are existing in the literature to the best of our knowledge.

1.1 Motivation

In any decision-making process, Imprecision is considered as a vital factor. Several tools and strategies were introduced for resolving the imprecise environment under group decision making [17]. One of the potential and recent methodology used for handling imprecision is the Pythagorean fuzzy sets. This PFSs generalizes the sets of intuitionistic fuzzy sets using a broad scope of applications [18]. This wide of application of PFSs motivates the option of exploring into the cooperation of sensor nodes in the network towards reliable data dissemination under resource depletion attacks. In specific, multi-criteria decision-methods (MCDM) is determined to ideal for diversified field of applications in different areas that includes health management [19], logistic, engineering, etc. The core objective of MCDM concentrates on ranking the decision variables in the order of the best to the worst. It is more essential for identifying a reliable solution to the problem of decision [20]. But the rankings determined using different MCDM techniques are generally different, and there is a potential gap between them is assessing the reliability and accuracy of MCDM methods. The MCDM methods of TOPSIS and VIKOR are completely based on the function of aggregation that indicates the closeness with respect to the ideal solutions [21]. The method of TOPSIS identifies the coefficient of closeness with respect to the solution by considering the distance of each alternative from the positive and negative ideal solution [22]. On the other hand, VIKOR method determines the compromise solution by identifying a maximum group utilization for the majority that determines the minimum of an individual regret for the opponent [23].

The core objective targets on determining a reliable and useful method for identifying the most significant criterion and alternatives using the merits of Pythagorean Fuzzy Sets-based VIKOR and Pythagorean Fuzzy Sets-TOPSIS. The first important innovation introduced by this approach is the estimation of weights using Pythagorean fuzzy VIKOR, such that the value of the crips value determined by the decision makers are represented in the form of pair-wise comparison matrix. Secondly, the determined matrix is transformed into Pythagorean fuzzy number, such that finally Pythagorean Fuzzy Sets-TOPSIS can be employed for ranking the alternatives (sensor nodes) in the network to ensure better routing process through reliable nodes.

In this paper, A Pythagorean Fuzzy Sets-based VIKOR and TOPSIS-based multi-criteria decision-making model (PFSVT-MCDM) is proposed for counteracting with the impacts of resource depletion attacks to improve Quality of Service (QoS) in the network. This PFSVT-MCDM used the merits of Pythagorean Fuzzy Sets information for handling uncertainty and vagueness of information exchanged in the network during the process of data routing. It utilized VIKOR and TOPSIS for exploring the trust of each sensor nodes through the exploration of possible dimensions that aids in detecting resource depletion attacks. The experimental results of PFSVT-MCDM confirmed better throughput of 21.29%, enhanced packet delivery fraction of 22.38%, minimized energy consumptions 18.92%, and reduced end-to-end delay of 21.84%, compared to the comparative resource depletion attack thwarting strategies used for evaluation.

The rest of the paper is structured as follows. The comprehensive review of the existing works contributed to the literature for thwarting different types of energy depletion nodes is presented with merits and limitations in Section 2. The detailed view of the proposed FTFCCM scheme with its indispensable role in detecting and isolating energy depletion attacks is presented in Section 3. The simulation results of the proposed FTFCCM scheme and its benchmarked energy depletion mitigation approaches used for comparison is demonstrated in Section 4. Section 5 concludes the paper with major contributions and future scope of enhancement.

2 Related works

In this section, the different methods contributed to the literature of detecting resource depletion or exhaustion attack is presented with the merits and limitations.

2.1 Traditional methods

Initially, Channawar and Chavan [24] redefined resource exhaustion attack at routing layer. It always restricts the networks by rapidly draining the energy of nodes. The routing protocols are susceptible to vampire attacks that are risky, challenging to identify and are easy to perform involving only protocol-compliant messages. A single vampire increases the energy consumed in a network by O(n), where ‘n’ is the quantity of nodes. The proposed scheme determines the solution for carousal as well as stretch attacks to obtain improved security. Trust-based energy effective scheme is used to maintain the dynamic nature of the network. It also aids in identifying and circumventing malevolent nodes during routing. It keeps in track the details of the forwarded packets to avoid redundancy. It removes malevolent nodes depending on packet to energy ratio to evade additional amount of energy. It determines alternate paths when packets are dropped. Bhutada and Manisha [25] have explored resource depletion and packet forwarding at the network layer that draining the power of nodes’ power. In existing systems, the presence of vampire attack is carried out at the destination for every packet received. In the proposed system, a system that alleviates vampire attacks by conserving bandwidth, time as well as power is proposed. It validates the paths at every node to rapidly find the perseverance of vampire attacks. It finds a safe path for transferring data by finding the route topology and forwarding data packets securely using PLGP Protocol. The amount of energy used increases based on adversary per packet. Forwarding-phase attacks and traffic broadcasting are considered, such that the packet determines the path by circumventing traffic and conserving bandwidth using broadcasting messages. Cryptographic schemes on data are used to defend data from intruders. Indices are added to packets to overcome carousel attacks. In case of stretch attacks, route metric along with indices is verified. In case the index is more than the metric, it is concluded that stretch attacks have happened.

2.2 Stateful Protocols-based detection approaches

Kumari and Sharma [26] designed a mechanism to decrease the influence of vampire attack on AODV protocol. The directional antenna vampire attack on stateful protocol is identified and overcome. It reduces the amount of energy consumed by nodes that are not part of genuine path, thus dropping the impact of this attack to a particular extent. It uses an additional variable Unique identify of the node followed by some computation on the incoming packet and routing table to identify the attack. Every intermediate node verifies the status of ensuing node before sending the data packet. The performance of AODV and proposed Silver-Coated Scheme Algorithm (SCSA) protocol is assessed for few sensor nodes. The proposed scheme offers better Throughput and involves reduced energy in contrast to AODV routing protocol.

2.3 Fuzzy and MCDM-based detection methods

Balaji Srikaanth and Nagarajan [27] have formulated a scheme based on fuzzy trust relationship for overcoming vampire attack and applying limited energy drain rate of conspiring nodes. The proposed Fuzzy Trust Relationship Perspective-based Prevention Mechanism (FTRPPM) determines the relative trust as well as reputation of nodes. In addition, it enumerates the influence of factors which can persuade vampire attack. Lastly, it enables identification of vampire nodes depending on recognized ranges of threshold which are dynamically attuned depending on enumerated probability level. Empirical as well as simulation outcomes of FTRPPM is established to be excellent as it guarantees remarkable enhancement in terms of improved PDR and throughput for increasing amount of mobile nodes in contrast to present mitigation mechanisms. Then, an energy prediction scheme proposed by Srikaanth and Nagarajan [28] used grey theory which guarantees consistent network connectivity influenced by vampire nature of nodes in case of active communication. The proposed Semi-Markov Chain-based Grey Prediction-based Mitigation (SMCGPM) is an improved MC model which integrates characteristics of stochastic and grey theories for enhancing the efficiency in identifying stretch attack. The clarified data from every node are primarily modelled using Grey model. Residual error is computed amid predicted and detected amounts of energy of mobile nodes depending on rate of forwarding packets. It is capable of forecasting the feasible transition nature of nodes using assessed residual error obtained from MC matrices. From the outcomes, it is understood that it is better than standard prediction mechanisms by enabling an efficient detection rate, with improved correctness as well as accuracy for prediction through SMC stochastic features stimulated energy factor forecast.

Rajesh and Esther Rani [29] proposed a energy depletion attack scheme using probabilistic variable fuzzy rough set for prolonging network lifetime with the advantages of fuzzy membership ranks. It derives the benefits of fuzzy information system by including the method of approximation concept that decides upon the mitigation of energy depletion nodes through the formulation of lower and upper cooperation thresholds. Furthermore, it is proposed to handle the degree of uncertainty, missingness, and vagueness of data disseminated between the sensor nodes during the monitoring process in the network. The simulation results of the proposed MVA-PVFRS scheme is attained through ns-2 simulations with different number of sensor nodes, number of energy depletion nodes, pause time of simulation, number of packets, and packet sizes.

2.4 Deep Learning and Optimization Models-based detection approaches

Juneja et al. [3] proposed a energy depletion attack detection mechanism using two-fold strategy with the merits of policy gradient-based deep reinforcement learning (PGDRL-VADM). This deep learning strategy adopted a policy by handling the behavior of sensor nodes that drains energy during the transmission of data. It helped in reducing the impacts of energy depletion attack nodes that intentionally elongates the route between the sensing node and the base station which helps in reactive decision-making process. It was proposed as an integrated trust mechanism for selecting secure route that plays an indispensable role of data forwarding. It was identified to guarantee the selection of secure hop even under the existence of energy depletion behaviour of sensor nodes. The results of PGDRL-VADM confirmed better network lifetime and increased detection rate compared to the competitive energy depletion attack mitigation approaches.

Isaac and Jasper [30] proposed the security during data transmission is provided to the network by using the proposed Secure Atom Search Routing (SASR) algorithm, which is adopted from the behavior of molecular dynamics. For global optimization problems, this algorithm provides an effective solution based on the constraint and interaction force of atoms. Moreover, the performance of SASR is improved by providing a proper balance between exploitation and exploration. Since the knowledge base processes all the data, the computational complexity is reduced, and the lifetime of the network is increased. The simulation and performance are carried out for the proposed Knowledge and Intrusion Detection based Secure Atom Search Routing (KID-SASR) protocol and is compared with the existing methods based on the metrics trust, delay, throughput, energy, packet delivery ratio, network lifetime, trust detection rate, and communication cost. The results obtained show improvement in the overall performance of the system.

In addition, Table 1 presents the summary of the various existing resource depletion attack detection mechanisms with their merits and limitations.

Table 1
Summary of the state-of-the art resource depletion attack detection mechanisms

Schemes Proposed mechanism Merits Limitations

Channawar and Chavan [24] Trust factor-based carousal and stretch attack mitigation scheme It was proposed for maintaining the dynamic nature of the network. It keeps in track the details of the forwarded packets to avoid redundancy. It removes malevolent nodes depending on packet to energy ratio to evade additional amount of energy The comprehensive factors considered for mitigating the category of vampire attacks was not comprehensive

Bhutada and Manisha [25] UPLGP Protocol-based resource depletion node detection scheme It estimated the paths at every node to rapidly find the perseverance of vampire attacks. It finds a safe path for transferring data by finding the route topology and forwarding data packets securely using PLGP Protocol. The amount of energy used increases based on adversary per packet. Its focus was only on the factor of energy, and only focussed on the influential factors to the minimal level.

Kumari and Sharma [26] Silver-Coated Scheme Algorithm (SCSA) protocol It detected the impact of the directional antenna vampire attack based on the benefits of the stateful protocol. It minimized the amount of energy consumed by nodes that are not part of genuine path, thus dropping the impact of this attack to a particular extent. The stateful protocol was not able to accommodate the complete set of states that a sensor node can enter during its lifetime.

Balaji Srikaanth and Nagarajan [27] Fuzzy Trust Relationship -based vampire attack detection and isolation It assessed the relative trust as well as reputation of nodes for specifically determining the influence of factors which can persuade vampire attack. It achieved the identification of vampire nodes depending on recognized ranges of threshold which are dynamically attuned depending on enumerated probability level It still possesses a room of improvement with respect to its capability in handling uncertainty and vagueness

Srikaanth and Nagarajan [28] Grey theory-based energy prediction scheme for thwarting vampire nodes It adopted grey theory to ensure consistent network connectivity influenced by vampire nature of nodes in case of active communication. It is an improved MC model which integrates characteristics of stochastic and grey theories for enhancing the efficiency in identifying stretch attack. The performance of grey theory-based resource depletion attack detection strategy was comparatively lower that the state-of-the art approaches.

Juneja et al. [3] Policy gradient-based deep reinforcement learning (PGDRL-VADM)-based energy depletion attack detection mechanism It utilized a two-fold strategy with the merits of deep learning strategy by handling the behavior of sensor nodes that drains energy during the transmission of data. It helped in reducing the impacts of energy depletion attack nodes that intentionally elongates the route between the sensing node and the base station which helps in reactive decision-making process. The classification accuracy and time taken for detection of resource depleting nodes was comparatively less.

Isaac and Jasper [30] Secure Atom Search Routing (SASR) algorithm-based vampire attack detection scheme It was proposed based on the inspiration derived from the behavioral characteristics of molecular dynamics. It was proposed as an effective solution that aids in achieving global optimization depending on the constraints used for mitigation process. It was also proposed with the well-balanced trade-off between exploration and exploitation for improving the performance of this SASR protocol. This SASR still possesses a room of improvement in terms of throughput, communication overhead and energy consumptions.

Rajesh and Esther Rani [31] Energy depletion attack detection scheme using probabilistic variable fuzzy rough set It used the merits of probabilistic variable fuzzy rough set for targeting on the objective of extending network lifetime.It also adopted the potentialities of fuzzy information system through an approximation method that enforces improved probability in increasing the rate of detecting energy depleting nodes. It was not able to handle uncertainty and vagueness of data exchanged during the typical behavior of sensor nodes.

Schemes	Proposed mechanism	Merits	Limitations
Channawar and Chavan [24]	Trust factor-based carousal and stretch attack mitigation scheme	It was proposed for maintaining the dynamic nature of the network. It keeps in track the details of the forwarded packets to avoid redundancy. It removes malevolent nodes depending on packet to energy ratio to evade additional amount of energy	The comprehensive factors considered for mitigating the category of vampire attacks was not comprehensive
Bhutada and Manisha [25]	UPLGP Protocol-based resource depletion node detection scheme	It estimated the paths at every node to rapidly find the perseverance of vampire attacks. It finds a safe path for transferring data by finding the route topology and forwarding data packets securely using PLGP Protocol. The amount of energy used increases based on adversary per packet.	Its focus was only on the factor of energy, and only focussed on the influential factors to the minimal level.
Kumari and Sharma [26]	Silver-Coated Scheme Algorithm (SCSA) protocol	It detected the impact of the directional antenna vampire attack based on the benefits of the stateful protocol. It minimized the amount of energy consumed by nodes that are not part of genuine path, thus dropping the impact of this attack to a particular extent.	The stateful protocol was not able to accommodate the complete set of states that a sensor node can enter during its lifetime.
Balaji Srikaanth and Nagarajan [27]	Fuzzy Trust Relationship -based vampire attack detection and isolation	It assessed the relative trust as well as reputation of nodes for specifically determining the influence of factors which can persuade vampire attack. It achieved the identification of vampire nodes depending on recognized ranges of threshold which are dynamically attuned depending on enumerated probability level	It still possesses a room of improvement with respect to its capability in handling uncertainty and vagueness
Srikaanth and Nagarajan [28]	Grey theory-based energy prediction scheme for thwarting vampire nodes	It adopted grey theory to ensure consistent network connectivity influenced by vampire nature of nodes in case of active communication. It is an improved MC model which integrates characteristics of stochastic and grey theories for enhancing the efficiency in identifying stretch attack.	The performance of grey theory-based resource depletion attack detection strategy was comparatively lower that the state-of-the art approaches.
Juneja et al. [3]	Policy gradient-based deep reinforcement learning (PGDRL-VADM)-based energy depletion attack detection mechanism	It utilized a two-fold strategy with the merits of deep learning strategy by handling the behavior of sensor nodes that drains energy during the transmission of data. It helped in reducing the impacts of energy depletion attack nodes that intentionally elongates the route between the sensing node and the base station which helps in reactive decision-making process.	The classification accuracy and time taken for detection of resource depleting nodes was comparatively less.
Isaac and Jasper [30]	Secure Atom Search Routing (SASR) algorithm-based vampire attack detection scheme	It was proposed based on the inspiration derived from the behavioral characteristics of molecular dynamics. It was proposed as an effective solution that aids in achieving global optimization depending on the constraints used for mitigation process. It was also proposed with the well-balanced trade-off between exploration and exploitation for improving the performance of this SASR protocol.	This SASR still possesses a room of improvement in terms of throughput, communication overhead and energy consumptions.
Rajesh and Esther Rani [31]	Energy depletion attack detection scheme using probabilistic variable fuzzy rough set	It used the merits of probabilistic variable fuzzy rough set for targeting on the objective of extending network lifetime.It also adopted the potentialities of fuzzy information system through an approximation method that enforces improved probability in increasing the rate of detecting energy depleting nodes.	It was not able to handle uncertainty and vagueness of data exchanged during the typical behavior of sensor nodes.

The aforementioned-limitations of the existing state-of-the art resource depletion attack detection mechanisms formed the motivation behind the formulation and implementation of the proposed Pythagorean Fuzzy Sets-based VIKOR [32] and TOPSIS [33]-based multi-criteria decision-making model.

3 Proposed Pythagorean Fuzzy Sets-based VIKOR and TOPSIS-based multi-criteria decision-making model (PFSVT-MCDM)

In this section, the primitive definition of fuzzy linguistic information, Pythagorean Fuzzy Sets (PFSs), Pythagorean Fuzzy Sets-based VIKOR and TOPSIS-based multi-criteria decision-making model, and their role in detecting and isolated resource depletion attack node from WSNs.

3.1 Definition of fuzzy linguistic information

Several real-world problems can be accurately evaluated qualitatively and not quantitatively owing to inadequate information, imprecision and vagueness. Using fuzzy linguistic information offers effective outcomes in diverse areas. Use of linguistic information suggests methods for Computing with Words (CW). There are 2 traditional computing systems designed to perform CW, such as, Semantic model - Extension Principle-based Linguistic computing model: It performs processes on membership functions which support linguistic term semantics. This model uses extension principle that raises the ambiguity of results. Depending on the principle, Fuzzy numbers which do not match the preliminary linguistic terms are obtained as outcomes, and Symbolic model - Symbolic linguistic computing model: It performs computations on indices of linguistic terms using well-organized form of linguistic terms to enable essential symbolic computations. The intermediary outcomes are approximated numeric values obtained by using a function in every step which indicates related linguistic term’s index.

3.2 Introduction to Pythagorean Fuzzy Sets (PFSs)

Assessments on risks made by decision makers are unclear, specific and inaccurate causing fuzziness in the ordering of risks of Autonomous Vehicle (AV). Fuzzy Set (FS) theory along with linguistic variables which deals with unclear information while obtaining specific model outcomes [34]. For making decisions under uncertainties, Fuzzy tools are more suitable when compared to soft computing-based schemes. Linguistic values are employed for expressing experts’ opinions based on criteria and other options. Normal FSs are newly extended, like Intuitionistic FS (IFS) and Pythagorean FS (PFS) to show the imprecision of opinions made by decision makers [35]. The total membership as well as non-membership levels should be a maximum of 1, while the sum may be more than1, but for Pythagorean Fuzzy Sets (PFSs), the sum of squares may not be so. As IFSs are incapable of modelling complex vagueness in practical circumstances [23], In specific, PFSs can be determined as a generality of IFSs. PFSs differ from IFSs in their unique constraint conditions. The space consumed by PF value is more than that of IF. PFSs represent inexact information that Atanassov’s IFSs can gather and model ambiguous as well as vague information [36].

Traditional computing method like symbolic scheme is extensively applied to several decision-making problems due to the simplicity as well as increased interpretability. When PFSs and symbolic schemes are compared, it is found that more effort has to be taken to implement PFSs to signify verbal inclinations. Symbolic linguistic computing scheme causes information loss while performing approximation. The ensuing information loss is due to the illustration model of fuzzy linguistic scheme that is discrete in continuous domain. It leads to vague outcomes, and hence imprecision in real-life circumstances. To deal with the inadequacies of imprecision in re-translation, PFSs that are found to be commanding as well as flexible in handling the uncertainty problems are developed. Moreover, crisp and ordinary fuzzy numbers are not adequate for handling imprecision involved in the complex environment like AV safety. PF Numbers (PFNs) are employed for ordering risks in AVs as they define the assessments of experts more precisely and are effectively implemented in real-time.

Definition 3.1. Assume ’Y’ to be the space of discourse, ‘y’ an element belonging to ‘Y’. PFS ~P in ‘Y’ is single valued as defined in the following equations $\tilde{P} = {y, P (φ_{p}^{y}, v_{p}^{y}) 〉 | y \in Y}$ (1) where, $φ_{p}^{y} \in [0, 1]$ - Degree of Membership

$v_{p}^{y} \in [0, 1]$ - Degree of Non-membership ‘y’ to ‘~P’. $0 \leq (φ_{p}^{y})^{2} + (v_{p}^{y})^{2} \leq 1$ (2) For ‘~P’, the hesitancy level of ‘y’ to ~P is represented as shown in the ensuing Equation. $ρ_{p}^{x} = \sqrt{1 - (φ_{p}^{y})^{2} - (v_{p}^{y})^{2}}$ (3)

Definition 3.2. Assume ${\tilde{P}}_{1} = P (φ_{p}^{1}, v_{p}^{1})$ and ${\tilde{P}}_{2} = P (φ_{p}^{2}, v_{p}^{2})$ be 2 PF numbers, and ‘γ > 0’ ${\tilde{P}}_{1} \oplus {\tilde{P}}_{1} = P (\sqrt{(φ_{p}^{1})^{2} + (φ_{p}^{2})^{2} - (φ_{p}^{1})^{2} (φ_{p}^{2})^{2}}, v_{p}^{1} v_{p}^{2})$ (4) ${\tilde{P}}_{1} \oplus {\tilde{P}}_{1} = P (φ_{p}^{1} φ_{p}^{2}, \sqrt{(φ_{p}^{1})^{2} + (φ_{p}^{2})^{2} - (φ_{p}^{1})^{2} (φ_{p}^{2})^{2}})$ (5) $γ {\tilde{P}}_{1} = (\sqrt{1 - (1 - (φ_{p}^{1})^{2})^{γ}}), γ > 0$ (6) $({\tilde{P}}_{1})^{γ} = ((φ_{p}^{1})^{γ}, \sqrt{1 - (1 - (φ_{p}^{1})^{2})^{γ}}), γ > 0$ (7)

$\begin{matrix} {\tilde{P}}_{1} ⊖ {\tilde{P}}_{2} = (\sqrt{\frac{φ_{1}^{2} - φ_{2}^{2}}{1 - φ_{2}^{2}}, \frac{v_{1}}{v_{2}}}), \\ if φ_{p}^{1} \geq φ_{p}^{2}, v_{p}^{1} \leq \min {v_{p}^{2}, \frac{v_{p}^{2} ρ_{P}^{1}}{ρ_{P}^{2}}} \end{matrix}$ (8)

$\begin{matrix} \frac{{\tilde{P}}_{1}}{{\tilde{P}}_{2}} = (\frac{φ_{1}}{φ_{2}}, \sqrt{\frac{v_{1}^{2} - v_{2}^{2}}{1 - v_{2}^{2}}}), \\ if φ_{p}^{1} \leq \min {φ_{p}^{2}, \frac{φ_{P}^{2} ρ_{P}^{1}}{ρ_{P}^{2}}}, v_{P}^{1} \geq v_{P}^{2} \end{matrix}$ (9)

Definition 3.3. Assume ~P_i = P (φ_i, v_i) , i = (1, 2, …, m) be a group of PFNs. The aggregate using PF Weighted Averaging (PFWA) is shown below.

$\begin{matrix} PFWA ({\tilde{P}}_{1}, {\tilde{P}}_{1} \dots, {\tilde{P}}_{m}) = \\ ({(1 - \prod_{i = 1}^{n} (1 - φ_{i}^{2})^{w_{i}})}^{\frac{1}{2}}, (\prod_{i = 1}^{n} (v_{i})^{w_{i}})) \end{matrix}$ (10) where w_i ∈ w₁, w₂, …, w_m - Weight vector of ‘~P, w_i ∈ [0, 1] and $\sum_{i = 1}^{n} = 1$

Definition 3.4. Assume ${\tilde{P}}_{1} = P (φ_{p}^{1}, v_{p}^{1})$ and ${\tilde{P}}_{2} = P (φ_{p}^{2}, v_{p}^{2})$ be 2 PFNs, quasi-ordering on PFNs is given by ~P₁ ≥ ~P₂, iff $φ_{p}^{1} \geq φ_{p}^{2}$ and $v_{p}^{1} \leq v_{p}^{2}$ .

Score Functions (SFs) are used for comparing the levels of 2 PFNs. The score or de-fuzzification functions are effective when PFNs are compared with MCDM problems. Score functions are used as de-fuzzification functions.

Definition 3.5. For PFN ${\tilde{P}}_{1} = P (φ_{p}^{1}, v_{p}^{1})$ , the SF of ‘~P₁’ is determined as follows $SF ({\tilde{P}}_{1}) = φ_{p}^{1} - v_{p}^{1}$ (11)

Definition 3.6. Assume ‘SF (~P₁)’ and ‘SF (~P₂)’ be SFs of ‘~P₁’ and ‘~P₂’ correspondingly. The ensuing rules are framed to compare 2 PFNs.

If SF (~P₁) < SF (~P₂), then ~P₁ < ~P₂

If SF (~P₁) > SF (~P₂), then ~P₁ > ~P₂

If SF (~P₁) = SF (~P₂), then ~P₁ ∼ ~P₂

3.3 Hybrid Multi-Criteria Decision Making (HMCDM)

Multi-Criteria Decision Making (MCDM) is used for systematically handling complex decision problems including several conflicting criteria. MCDM schemes find an ideal alternate or rank alternates, wherein highest rank is taken as the best substitute to decision makers. The central feature of AV is its vagueness in several aspects. As ordering of AV risks have incompatible criteria with a collection of favoured alternates, MCDM schemes can be efficiently applied to handle these problems. The proposed scheme includes 3 fundamental phases for ordering risks in AVs. The initial phase deals with defining the criteria along with alternates, and then weighing the criteria. Alternates and criteria are formed by going through the literature and based on expert knowledge, a Hierarchy Tree (HT) is formed during the primary 3 steps. Interval-valued PF Analytic Hierarchy Process (AHP) is implemented to quantify the comparative significance of leading and sub-criteria given from Steps 4 to 11. The next phase employs the proposed schemes involving the chosen criteria and alternates. PF-Technique for Order Preference by Similarity to an Ideal Solution (PF-TOPSIS) and PF-Vlse Kriterijumska Optimizacija I Kompromisno Resenje (PF-VIKOR) are executed by applying weights got from interval-valued PF AHP. The ensuing steps are related to the proposed HMCDM scheme. The next phase includes ordering of alternates, performing sensitivity analysis as well as comparing the outcomes. Subsequent hybrid MCDM schemes are applied to order risks in AVs:

•Hybrid IVPF-AHP - PF-TOPSIS method: Hybrid of Interval-valued PF AHP and PF TOPSIS

•Hybrid IVPF-AHP - PF-VIKOR method: Hybrid of Interval-valued PF AHP and PF VIKOR

3.3.1 Hybrid of Interval-valued PF AHP and PF TOPSIS

IVPF-AHP: The utilization of IVPF-AHP is more realistic in the evaluation of alternatives and criteria due to the uncertainty and impreciseness realized during the assessment of sensor nodes under the process of routing. Thus the adoption of IVPF-AHP introduces strong potential in preventing the uncertainty and vagueness in the assessment of sensor nodes under the impact of resource depletion attacks. Hence, IVPF-AHP possesses the capability of resolving the issues that are more common during the process of determining net values in VIKOR and TOPSIS. But, PF-AHP is not significant in handling the problem of uncertainty to the desired level, and still fails in determining the lower and upper threshold of detecting the cooperation degree of sensor nodes when it is used under resource depletion attacks.

AHP helps DMs to fix significances and take the finest decision. By gathering imprecise information using expert consultation. It is a standard scheme employed for prioritization and is implemented to a huge amount of problems in diverse fields. The main advantage of employing AHP is the simplicity of use. Pairwise comparisons permit operators to weigh the criteria as well as compare alternates with ease. Moreover, as it is scalable, AHP can certainly include DM problems of varying sizes in the hierarchical organisation. Finding the significance of criteria before problem solving aids in offering more consistent ranking of alternates that precisely reflects DMs’ inclinations. It is a common method for specific assessment of qualitative data. Instead, AHP method has some issues. The central criticism is with Pairwise Comparison Matrices (PCMs) and Principal Right Eigen Vector (PREV) capability to produce precise rankings. An added drawback of involving AHP in an integrated framework is that it demands dependence amid every criteria and alternates. This leads to discrepancies amid judgment as well as ranking standards. As per IVPF-AHP, experts use linguistic terms to assess the comparative significance of risk factors. The steps of PAHP are listed below:

Step 1: Find alternates (i = 1, 2, …, m)

Step 2: Find criteria (i = 1, 2, …, n)

Step 3: Build decision HT, the topmost level signifying the goal. Intermediary level shows main as well as sub-criteria for assessment. The lowest level comprises of alternates

Step 4: Construct negotiated PCM, R = (r_ik) _m×m based on experts’ linguistic assessments.

Step 5: Compute the Difference Matrix D = (d_ik) _m×m a mid lower as well as upper memberships along with non-membership function values $d_{ik}^{L} = (φ_{ik}^{L})^{2} - (v_{ik}^{U})^{2}$ (12) $d_{ik}^{U} = (φ_{ik}^{U})^{2} - (v_{ik}^{L})^{2}$ (13) Step 6: Calculate interval multiplicative matrix I = (SF_ik) _m×m ${SF}_{ik}^{L} = \sqrt{1000^{d} L}$ (14) ${SF}_{ik}^{U} = \sqrt{1000^{d} U}$ (15) Step 7: Calculate determinacy value ϑ = (ϑ_ik) _m×m. $ϑ_{ik} = 1 - ((φ_{ik}^{U})^{2} - (φ_{ik}^{U})^{2}) - ((v_{ik}^{U})^{2} - (v_{ik}^{L})^{2})$ (16)Step 8: Determine the weight matrix W = (w_ik) _m×m by multiplying determinacy value with S = (SF_ik) _mxm. $t_{ik} = (\frac{{SF}_{ik}^{L} + {SF}_{{ik}_{{ik}_{U}}}^{U}}{2}) ϑ_{ik}$ (17)Step 9: Calculate the standardised priority weight (ω_i) $ω_{i} = \frac{\sum_{j = 1}^{m} ω_{ij}}{\sum_{j = 1}^{m} \sum_{j = 1}^{m} ω_{ij}}$ (18)

Consistency Consistency Ratio (CR) is utilised for assessing the steadiness of experts and complete hierarchy. Saaty’s standard CR process is applied as shown in the ensuing equation $CI = \frac{γ_{\max} - M}{M - I}, CR = \frac{CI}{RI}$ (19) where, n - Amount of matrix elements

γ_max - Maximum Eigen value

CI - Consistency Index

RI - Random Index

If CR ≤ threshold (0.1), then it is adequate. Else, the decision matrix is considered to be inconsistent and experts are recommended to reassess the matrix.

PF-TOPSIS TOPSIS aids in determining the best alternate and ranking order of alternates using a sequence of mathematical calculations. As PFS are widely used for focussing on the uncertainty as well as imprecision in MCDM problems, TOPSIS is enhanced under PF setting (PF-TOPSIS). In PF-TOPSIS, the chosen alternate involves the shortest and farthest distances from optimal and negative optimal solutions respectively that are based on PFSs. The steps of the algorithm are listed below:

Step 10: Build PF based decision matrix $D = (C_{j}^{(y_{i})})_{m \times n}$ where, C_j - Criteria

y_i - Alternates

Decision matrix with PFNs is shown below: $D = (C_{j}^{(y_{i})})_{m \times n}$ (20)Step 11: Combine the PF decision matrix using PFWA as shown in Equation (10).

Step 12: Compute PF Positive Ideal Solution (PIS) as well as Negative Ideal Solution (NIS) $\begin{matrix} y^{+} = {max_{i} 〈 SF (y_{i}) 〉} = {〈 P (u_{1}^{+}, v_{1}^{+}) 〉, \\ 〈 P (u_{2}^{+}, v_{2}^{+}) 〉, \dots, 〈 P (u_{n}^{+}, v_{n}^{+}) 〉} \end{matrix}$ (21) $\begin{matrix} y^{-} = {max_{i} 〈 SF (y_{i}) 〉} = {〈 P (u_{1}^{-}, v_{1}^{-}) 〉, \\ 〈 P (u_{2}^{-}, v_{2}^{-}) 〉, \dots, 〈 P (u_{n}^{-}, v_{n}^{-}) 〉} \end{matrix}$ (22) Step 13: Calculate distances from PF PIS as well as NIS $\begin{matrix} D (y_{i}, y^{+}) = \sum_{j = 1}^{n} w_{j} d (C_{j}^{(y_{i})}, C_{j}^{(y^{+})}) \\ = \frac{1}{2} \sum_{j = 1}^{n} w_{j} (| (φ_{ij})^{2} - φ_{j}^{+})^{2} | + | (v_{ij})^{2} - (v_{j}^{+})^{2} | \\ + | (ρ_{ij})^{2} - (ρ_{j})^{2} |) \end{matrix}$ (23) $\begin{matrix} D (y_{i}, y^{-}) = \sum_{j = 1}^{n} w_{j} d (C_{j}^{(y_{i})}, C_{j}^{(y^{-})}) \\ = \frac{1}{2} \sum_{j = 1}^{n} w_{j} (| (φ_{ij})^{2} - φ_{j}^{-})^{2} | + | (v_{ij})^{2} - (v_{j}^{-})^{2} | \\ + | (ρ_{ij})^{2} - (ρ_{j})^{2} |) \end{matrix}$ (24) Smaller the value of ‘D (y_i, y⁺)’, better the alternate ‘x_i’ and larger the value of ‘D (y_i, y^-)’, better the alternate ‘x_i’.

Let, $\begin{matrix} D_{\min} (y_{i}, y^{+}) = & min_{1 \leq i \leq m} D (y_{i}, y^{+}) \\ D_{\max} (y_{i}, y^{-}) = & max_{1 \leq i \leq m} D (y_{i}, y^{-}) \end{matrix}$ Step 14: Determine the modified closeness ‘μ (y_i)’ of the alternate ‘y_i’ $μ (y_{i}) = \frac{D (y_{i}, y^{-})}{D_{\max} (y_{i}, y^{-})} - \frac{D (y_{i}, y^{+})}{D_{\min} (y_{i}, y^{+})}$ (25)Step 15: Find the best order of ranking of alternates, wherein the best alternate is the one with maximum modified closeness ‘μ (y_i)’.

PF-VIKOR VIKOR is a MCDM scheme proposed by Opricovic [26] for MC optimization of complex systems. The scheme offers a negotiated solution that incorporates the weighted relation amid maximum group efficacy along with minimum specific regret. PF-VIKOR is used in the study.

Step 16: Find PFPIS, $a_{j}^{+} = (φ_{j}^{+}, v_{j}^{+})$ and PFNIS, $a_{j}^{-} = (φ_{j}^{-}, v_{j}^{-})$ of alternates for every criterion $a_{j}^{+} = {\begin{matrix} \max_{i} . a_{ij}, & if Benefit criterion \\ \min_{i} . a_{ij}, & if Cost criterion \end{matrix}$ (26) $a_{j}^{-} = {\begin{matrix} \min_{i} . a_{ij}, & if Benefit criterion \\ \max_{i} . a_{ij}, & if Cost criterion \end{matrix}$ (27)Step 17: Calculate the utility measure (S_i) and Regret measure (R_i) for alternate using Generalized PF Ordered Weighted Standardized Distance (GPFOWSD) operator [27]:

$\begin{matrix} S_{i} = GPFOWSD (〈 u_{j}^{+}, u_{j}^{-}, u_{i 1} 〉, \dots 〈 u_{j}^{+}, u_{j}^{-}, u_{im} 〉) \\ = (\sum_{k = 1}^{n} w_{k} d_{k}^{- γ}) \end{matrix}$ (28) $R_{i} = (max_{k} w_{k} d_{k}^{- γ})^{\frac{1}{γ}}$ (29) Where,

$d_{k}^{- γ}$ - ‘k^th’ largest of standardized PF distance

w_k - Weights of criteria

γ - Operator

Step 18: Compute PF-VIKOR-specific index (P_i) $P_{i} = ω \frac{S_{i} - S^{+}}{S^{-} - S^{+}} + (1 - ω) \frac{R_{i} - R^{+}}{R^{-} - R^{+}}$ (30) Where,

$S^{+} = min_{i} (S_{i})$

$S^{-} = max_{i} (S_{i})$

$R^{+} = min_{i} (R_{i})$

$R^{-} = max_{i} (R_{i})$

ω - Weight of maximum group utility

1 - ω - Weight of specific regret

Step 19: Order the alternates based on ‘S_i’, ‘R_i’ and ‘P_i’ in ascending order.

Step 20: Recommend a negotiated solution wherein the alternate ‘A¹’ has minimum value of ‘P_i’ as solution.

To ensure the individuality of ideal solution, ensuing conditions should be fulfilled:

Acceptance Advantage: $P (A_{2}) - P (A_{1}) \geq \frac{1}{m - 1}$ Where,

A₁ and A₂ - Alternates with first 2 positions in the list of ‘P’

m - Quantity of alternates

Adequate stability: Alternates are ordered by ‘S_i’ and ‘R_i’. The negotiated solution ‘P_i’ is stable under MCDM and may be assessed using Voting using Majority rule (ω>0.5), Consensus (ω≈ 0.5) and Veto rule (ω<0.5)

It is assumed that the negotiated solution is distinctive after computations. In case of several negotiated solutions following computations, the quantity of neighbouring sensor nodes who assess the criteria as well as alternates may be improved owing to obtaining diverse views. In this manner, the resource depleting attack nodes are identified, and isolated from the network. In addition, Fig. 1 represents the comprehensive view of the proposed mechanism in the mitigation of resource depleting node attack in the network.

Fig. 1

Systematic steps included in the proposed PFSVT-MCDM scheme.

4 Simulation results and discussion

The simulation experiments of the proposed PFSVT-MCDM scheme and the baseline BFIPROMETHEE-RDAM [31], MVA-PVFRS [29] and KID-SASR [30] approaches are conducted using ns-2.34. In the simulation process, about 70% - 90% of packets are dropped by the sensor nodes when they behave in the malicious and selfish manner. In specific, about 10% - 40% of nodes among the complete number of sensor nodes in the network behave as non-cooperative nodes that include malevolent and selfish nodes. Initially, the degree cooperation rendered by each sensor node during the start of the simulation is 1. When the degree of communication is less than 0.4, the sensor nodes are found to be non-cooperative and isolated from the path, (complete node ranking is dependent on the degree of cooperation. The constant bit rate (CBR) utilized during the process of continuous flow rate in the process of simulation is 5 packets/sec for the objective of sending packets among source and destination. Table 2 shows the list of parameters used for simulation of the existing and proposed PFSVT-MCDM scheme schemes.

Table 2
Simulation Parameters for implementing PFSVT-MCDM scheme

Parameters Values

Simulator ns 2.34

Nodes 200

Amount of source- destination pairs 40

Routing protocol AODV

Simulation Area 1000 x1000 sqms

Transmission range 250 m

Transmission capacity 2 kb/s

Simulation time 11.04 min

Nodes’ mobility rate 10 m/s

Data packet size 512 Bytes

Trust for malevolent node identification 0.4

Mobility model Random Way Point

Pause time 10 ms

Data traffic type CBR

Parameters	Values
Simulator	ns 2.34
Nodes	200
Amount of source- destination pairs	40
Routing protocol	AODV
Simulation Area	1000 x1000 sqms
Transmission range	250 m
Transmission capacity	2 kb/s
Simulation time	11.04 min
Nodes’ mobility rate	10 m/s
Data packet size	512 Bytes
Trust for malevolent node identification	0.4
Mobility model	Random Way Point
Pause time	10 ms
Data traffic type	CBR

The performance of the standard and proposed PFSVT-MCDM schemes is analysed based on throughput, detection rate and time, energy consumption, Packet Delivery Rate (PDR) and mean end-to-end delay for varying number of nodes, energy depleting nodes [37], and simulation time.

4.1 Performance Evaluation of the proposed PFSVT-MCDM scheme using number of sensor nodes

In the first method of examination, throughput, energy depletion node detection rate, mean amount of energy consumed and mean end-to-end delay of the standard and proposed mechanisms are compared. Figures 3 shows the mean throughput along with rate of energy depletion node detection of the propounded and standard mechanisms for varying amounts of nodes respectively. The mean throughput of the propounded mechanism is found to be declining with a conforming upsurge in the number of nodes which introduce a greater number of packets in the network. Nevertheless, the proposed mechanism is efficient in sustaining the network’s average PDR as it comprises of the advantages of TOPSIS AND VIKOR for complete assessment of alternates to determine the genuine nodes during routing. The rate of detection of malevolent nodes by the proposed scheme is found to be sustained at an increased level, even with upsurge in the number of nodes. It includes diverse PF values which show the probabilities and irrelevances amid trust estimators of nodes. Improvement in mean throughput as well as rate of energy depletion node identification is possible by enclosing TOPSIS AND VIKOR which helps in partial as well as comprehensive node ranking along the routing path. The proposed mechanism for varying number of nodes is found to offer 11.42%, 13.57%, and 15.67% better mean throughput in contrast to the standard TBSRPVA, CTMC-VAMS and FTRPPVA schemes. Likewise, the rate of energy depletion node identification of proposed mechanism for varying amounts of nodes is found to be enhanced by 12.45%, 13.72% and 15.67% in contrast to the baseline schemes.

Fig. 2

PFSVT-MCDM scheme-Mean throughput with different set of sensor nodes in the network.

Fig. 3

PFSVT-MCDM scheme-Detection rate with different set of sensor nodes in the network.

Figures 5 highlights the mean energy consumption as well as average end-to-end delay involved by the propounded as well as standard schemes for varying amounts of nodes. Both the parameters show an increase with consistent upsurge in the number of nodes as the number of packets that should be sent by every node steadily increases. Nevertheless, the proposed mechanism detects the co-operative node along the path with increased accuracy as well as reduced detection time facilitating trusted transmission of data in the network. Further, the proposed mechanism uses outranking model of PROMETHEE-I for determining nodes along the path with varying dimensions, thus circumventing needless packet transmissions and drops. The average energy consumption as well as end-to-end delay reduce with inclusion of MADMs in TOPSIS and VIKOR in the propounded mechanism. The mean energy consumption of the propounded mechanism with regular upsurge in the count of mobile nodes is significantly reduced by 11.72%, 13.84% and 15.67% when compared to the standard schemes. Likewise, the mean end-to-end delay of proposed mechanism with a monotonic rise in number of nodes is reduced by 12.72%, 13.68% and 15.72% when compared to the standard schemes.

Fig. 4

PFSVT-MCDM scheme-Mean energy consumptions with different set of sensor nodes in the network.

Fig. 5

PFSVT-MCDM scheme-Packet delay with different set of sensor nodes in the network.

4.2 Performance evaluation of the proposed PFSVT-MCDM scheme using number of resource depleting nodes

In the second fold of investigation, the performances of standard and proposed mechanisms are assessed in terms of mean throughput, detection rate of malicious nodes, mean energy consumption and packet latency for varying amount of resource depleting sensor nodes. Figures 7 presents the mean throughput as well as the detection time experienced by the propounded as well as benchmarked schemes in detecting the non-cooperative nodes with a regular upsurge in amount of energy depletion nodes. The mean throughput of the proposed as well as benchmarked mechanisms for varying amount of energy depletion nodes is found to decrease as the number of packets dropped proportionately increases with a conforming rise in the amount of energy depletion nodes. Nevertheless, the proposed mechanism prevents dropping of packets by the malevolent nodes to a predictable level by precisely assessing CD using PROMETHEE-II which derives comprehensive node ranking during the process of routing. Instead, the detection time with a regular upsurge in non-cooperating nodes is found to upsurge with the time consumed for determining the features of nodes. The proposed mechanism reduces the detection time by imposing Analytical Hierarchy Process (AHP)-based CWs which decreases the time involved in ranking the communicating nodes along the path which are under observation. The throughput of the proposed mechanism for increasing amounts of energy depletion nodes is found to be improved by 9.14%, 11.67% and 12.89% in contrast to the standard schemes. The detection time of the proposed mechanism for increasing amount of non-cooperative nodes is reduced by 11.39%, 13.67% and 14.97% when compared to the standard schemes.

Fig. 6

PFSVT-MCDM scheme-Mean throughput with different resource depleting nodes in the network.

Fig. 7

PFSVT-MCDM scheme-Detection rate with different resource depleting nodes in the network.

Figures 9 exhibits the plots of the mean energy consumption as well as average end-to-end delay of standard and proposed mechanisms with upsurge in the number of energy depleting sensor nodes. Both the parameters show an increase with the amount of energy depletion nodes as a greater number of packets are dropped by these nodes. Further, the proposed mechanism includes TOPIS and VIKOR for attaining precise assessment of cooperation degree which assures routing only through unaffected nodes. The phenomenal feature of the propounded mechanism reduces the number of retransmissions along with the energy consumed for needless data transmissions, thereby dropping the delay involved in data transmission amid source as well as destination. The average energy consumption of the proposed mechanism with orderly upsurge in the amount of non-cooperative nodes is considerably dropped by 10.18%, 12.96% and 14.82% when compared to the standard schemes. The average end-to-end delay of propounded mechanism for increasing amount of energy depletion nodes is reduced by 9.61%, 11.86% and 13.69% in contrast to the standard schemes.

Fig. 8

PFSVT-MCDM scheme-Mean energy consumptions with different resource depleting nodes in the network.

Fig. 9

PFSVT-MCDM scheme-Packet delay with different resource depleting nodes in the network.

4.3 Performance assessment of the proposed mechanism for varying amounts of simulation time

In the last variant of investigation, the proposed mechanism is assessed in terms of packet drop rate, PDR, total communication overhead as well as control overhead for varying simulation times. Figures 11 depicts the plots of PDR and total overhead of standard and proposed schemes for varying simulation times. The PDR of the standard and proposed schemes decrease with raise in simulation time, as the quantity of packets added to network shows an increase proportionately. Instead, the proposed mechanism is effectual in maintaining the PDR independent of raise in simulation time. Moreover, it incorporates multiple criteria and variations of PROMETHEE for efficient classification of nodes into malevolent and genuine nodes, so that they can be included or excluded during the process of routing. Instead, the total overhead of standard and proposed mechanism increases with the regular increase in the number of packets produced, communicated and retransmitted in the network. Nevertheless, the proposed mechanism reduces the number of retransmissions by including AHP which determines the weight of diverse criteria leading to reduction in the total overhead. The PDR of the proposed mechanism is enhanced by 7.19%, 8.78% and 9.72% in contrast to the standard schemes. Furthermore, the total overhead of the proposed mechanism is reduced by 8.74%, 9.38% and 10.12% in contrast to the standard mechanisms considered for investigation.

Fig. 10

PFSVT-MCDM scheme-Mean throughput with increasing simulation time.

Fig. 11

PFSVT-MCDM scheme-Total communication overhead with increasing simulation time.

With respect to the integrated Fuzzy TOPSIS and Fuzzy VIKOR method, it is identified to handle the degree of rank reversal and bias problem common in MADM approaches, and thereby realized to be better than the standalone Fuzzy TOPSIS and Fuzzy VIKOR. This capability of the proposed integrated Fuzzy TOPSIS and Fuzzy VIKOR (PFSVT-MCDM) method facilitated maximized throughput and at the same time minimized the total communication overhead, which resulted in maximized delivery of data packets to the destination. In specific, the mean throughput confirmed by the proposed integrated Fuzzy TOPSIS and Fuzzy VIKOR (PFSVT-MCDM) is improved on an average by 21.45%, and 24.56%, better than the standalone Fuzzy TOPSIS and Fuzzy VIKOR approaches. Moreover, the total communication cost during the implementation of the integrated Fuzzy TOPSIS and Fuzzy VIKOR (PFSVT-MCDM) method is reduced by 20.98%, and 23.49%, better than the baseline approaches.

Figures 13 portrays the plots of packet drop rate and total overhead incurred by the proposed scheme and the baseline schemes under varying simulation times. The packet drops rate of the proposed scheme and the baseline approaches reduced with upsurge in simulation time, as the number of packets in network shows a proportional increase. Nevertheless, the proposed scheme is effective in sustaining the PDR independent of upsurge in simulation times. Moreover, it incorporates multi-criteria as well as variations of TOPSIS for efficient classification of nodes into genuine as well as malevolent nodes. Likewise, the total overhead of standard and proposed schemes increases with the amount of generated, communicated and retransmitted packets in the network. Nevertheless, the proposed mechanism can reduce the number of retransmissions by including a BPF mechanism which finds the weights of varying criteria leading to the reduction in the total overhead. The PDR of the proposed is increased by 7.89%, 8.59% and 9.72% in contrast to the standard schemes. Furthermore, the total overhead involved by the proposed mechanism is reduced by 8.74%, 9.45% and 10.79% in contrast to standard schemes considered for investigation.

Fig. 12

PFSVT-MCDM scheme-Packet drop rate with increasing simulation time.

Fig. 13

PFSVT-MCDM scheme-Control overhead with increasing simulation time.

From the dimension of the proposed PFSVT-MCDM technique which is a combination of Fuzzy TOPSIS and Fuzzy VIKOR, it is considered to be better than the standalone Fuzzy TOPSIS and Fuzzy VIKOR since the degree of uncertainty handled by the integration of Fuzzy TOPSIS and Fuzzy VIKOR handled the imprecision and uncertainty to a maximized level independent to the factors considered for improvement. Thus the packet drop rate and control overhead realized by the integrated Fuzzy TOPSIS and Fuzzy VIKOR is significantly minimized by 18.92%, and 21.65%, better than the standalone Fuzzy TOPSIS and Fuzzy VIKOR approaches.

5 Conclusion

The proposed PFSVT-MCDM technique counteracted the influence of resource depletion attacks with the merits of VIKOR and TOPSIS to prevent information uncertainty that improves QoS in the network. This PFSVT-MCDM used the merits of Pythagorean Fuzzy Sets information for handling uncertainty and vagueness of information exchanged in the network during the process of data routing. It utilized VIKOR and TOPSIS for exploring the trust of each sensor nodes through the exploration of possible dimensions that aids in detecting resource depletion attacks. The experimental results of PFSVT-MCDM confirmed better throughput of 21.29 enhanced packet delivery fraction of 22.38%, minimized energy consumptions 18.92%, and reduced end-to-end delay of 21.84%, compared to the comparative resource depletion attack thwarting strategies used for evaluation. As the part of future scope of research, it is decided to formulate and implement a AHP and ELECTRE-based model, and compare it with the proposed PFSVT-MCDM technique with homogeneous and heterogeneous conditions of network.

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