Energy efficient clustering in industrial Iot using a quantum informed artificial hummingbird optimization algorithm

Abstract

In Industrial Internet of Things (IIoT), Clustering facilitates the proves of organizing similar types of devices or data points into different clusters for the objective of enhancing resource utilization, network management and data processing. This clustering in IIoT helps in addressing the challenges that are associated with the process of handling network complexity, satisfying requirements of real-time processing and dealing with massive data volumes. In specific, swarm intelligent optimization algorithms are used for selecting optimal CHs and determining reliable route through the network such that the parameters of data aggregation, delay and energy consumptions are handled with maximized performance. IIoT networks when blended with optimization algorithms-based clustering aids in improving scalability and energy efficiency which results in more cost-effective and reliable industrial applications. In this paper, Energy Efficient Quantum-Informed Artificial Hummingbird Optimization Algorithm (EEQIAHBOA) is proposed for maximizing the performance of IoT networks and addressing the energy preservation problem such that the information is gathered and sent to the base station for reactive decision making. This EEQIAHBOA approach is proposed as a reliable routing algorithm which is implementation with the determination of information heuristic factors and efficient encoding scheme. It is proposed as a significant clustering algorithm for the objective of achieving network lifetime such that the factors of residual energy, and distance between the cluster member IoT nodes and energy consumptions during the selection of Cluster Heads (CHs). The simulation experiments of EEQIAHBOA approach conducted with different network scenarios confirmed 32.12% improvement in energy efficiency and 35.62% enhancement in network lifetime under different live nodes compared to the baseline approaches used for investigation.

Introduction

In the industrial domain, wireless communication technology is gradually applied over the years for diversified number of operations due to its characteristics of easy deployment, high flexibility and low cost¹. This systematic incorporation of wireless communication technology into the industrial domain helped in realization and instantiation of its application with respect to Industrial Internet of Things (IIoT). The technology of Industrial Wireless Sensor Networks (IWSNs) is identified to constantly progressing with greatest maturity and widely attracted a broad market opportunity with respect to the growth in IIoT environment². This IWSNs compared to the classical wireless sensor network depends on required degree of Quality of Service (QoS) such as data packet loss rate and transmission delay with extended network lifetime since they still used battery power for their operation^3,4. In this context, it is specifically essential to design a significant approach which facilitates the option of reliable clustering and data collection in IWSNs for the need of prolonging the network lifetime with necessitated timeliness and guaranteed reliable process of data transmission⁵. The sensor nodes in IWSNs directly communicates with the base station, which is completely immovable in IWSNs, and hence utilizes more amount of energy utilization in the network during the data forwarding process^6,7. Hence, a hierarchical multi-hop routing method is utilized for data communication for preventing high degree of energy incurred by the sensor nodes which are situated far away from the base station since it can result in longer distance data signal transmission⁸. During the implementation of the hierarchical multi-hop routing protocol, the network nodes are partitioned into cluster member nodes and CHs in the network⁹. In specific, CHs possesses comparatively a greater amount of energy and merits of the location on par with the cluster member nodes^10,11. Hence the data sensed by the cluster member nodes are first forwarded to the CHs, and then it is sent to the immovable base station in a single or multi-hop manner¹². The adoption of the hierarchical multi-hop routing protocol-based clustering aids in making the energy consumptions in the network in a more uniform manner and extend the lifespan in IWSNs^13,14. But the selection of CHs and deriving the benefits of cluster construction strategy is highly challenging for guaranteeing the robustness of IWSNs since realistic IIoT environment comprises of sensor nodes distribution which is uneven and possesses complex network topology¹⁵.

Several researchers have contributed different clustering mechanisms over the recent years for identifying a good clustering framework which possesses the capability of enhancing the requirements of implementing a IWSN. Among the clustering frameworks available in the literature, Low-Energy Adaptive Clustering Hierarchy (LEACH) protocol is considered to be notable for randomly selecting the node as a CH in a probabilistic manner¹⁶. However, some of the potential nodes cannot become CHs due to the adoption of random selection strategy¹⁷. On the other hand, some of the clustering framework included the factors of energy consumptions or residual energy during the process of CHs selection for preventing the issue of random selection. Further heuristic and swarm intelligent algorithms-inspired clustering framework were introduced for obtaining the optimal set of CHs since the process of selecting potential nodes as CHs pertains to a combinatorial optimization NP problem¹⁸. Moreover, quality factors with respect to communication and network lifetime need to be considered during the selection of optimal CHs set in the network.

Motivation

The topology of IIoT is dynamic in nature since the input traffic and other data extracted from the implementation environment is determined to be uncertain. The data determined from the IIoT environment are highly prone to outages and unexpected overloads¹⁹. Fuzzy logic on one hand is a suitable candidate for handling the realistic problems of IIoT since it facilitates a strengthened mathematical foundation for addressing the degree of non-statistical uncertainty and imprecision in the real world. However, fuzzy logic-based clustering methods compared to the classical methods incurs an increased computational complexity²⁰. This increase in computational complexity is mainly due to the computation of membership degree with respect to the process of sensor nodes’clustering. On the other hand, swarm intelligent metaheuristic optimization algorithms-based clustering protocol is more ideal for IIoT since it facilitates the benefits of scalability, mobility, routing, self-organization and resource management²¹. These intelligent clustering protocols are potent in tackling the requirements of applications that are associated with the decisions of operating environment, location-based monitoring and real-time event detection²².

In addition, the working framework of the proposed EEQIAHBOA approach is presented as follows.

Fig. 1

Working framework of the proposed EEQIAHBOA approach.

Novelty of the proposed EEQIAHBOA approach

This proposed EEQIAHBOA approach used quantum hummingbird algorithm which enhanced the classical hummingbird algorithm by including the benefits of quantum computing which eventually improves the balance between the capabilities of exploration and exploitation. This approach used the principle of quantum mechanism to explore the complete solution space for determining optimized solutions with more effectiveness and potentiality. In the classical AHA, the population is initialized with random values while the adopted QAHA used chaotic mapping for the purpose of population initialization such that search space is well covered with better solution diversity. Compared to the existing quantum-based metaheuristic optimization algorithms²³, the utilized QAHA facilitates the option of rapid convergence such that better solutions with optimality can be achieved for high-dimensional complex problems like the clustering of IIoT nodes considered in this study.

Major contributions

The main contributions of this article can be listed as follows.

• i) Presented a multi-objective clustering model using the merits of QIAHBOA algorithm for selecting the potential CHs in the IIoT network a more accurate and reasonable way using the factors of packet loss, delay, energy consumption and residual energy into account.

• ii) Adopted a potential weight determination approach for each of the criteria used for evaluating the potential of IIoT nodes during the process of CHs selection and data forwarding to the base station for the objective of adapting flexibly according to the needs of the industry.

• iii) Proposed as a high-performance clustering framework for enhancing the clustering efficiency at one end and improve the clusters diversity with the capability of preventing the solution being struck into local point of optimality on the other end.

• iv) The performance of this multi-objective clustering framework is assessed using the metrics of network lifetime, alive operating nodes, packet delay rate, network delay and energy consumption rate with different number of nodes in the IIoT environment.

The remaining section of the article is structured as follows. Section 2 presents comprehensive survey on the different intelligent metaheuristic optimization algorithm-based clustering algorithm for achieving high performance colouring in IIoT with the pros and cons. Chapter 3 presents the detailed view of the network model and assumptions, energy consumption model, primitives of Artificial Humming Bird Optimization Algorithm (AHBOA) and quantum-informed QIAHBOA used for multi-objective clustering with the steps involved in the process of implementation. Section 4 presents the comparative analysis of the proposed EEQIAHBOA mechanism with the baseline clustering frameworks using the evaluation metrics of network lifetime, delay and optimized number of clusters. Section 5 presents the conclusions and possible scope of enhancement.

Related work

In this section, the comprehensive survey on the different intelligent metaheuristic optimization algorithm-based clustering algorithm for achieving high performance colouring in IIoT with the pros and cons.

Jannu et al.²⁴ proposed a Quantum-Informed Ant Colony Optimization-based clustering mechanism for maximizing energy and network stability in IIoT environment. This QIACOCM is proposed with the capability of maximizing the network lifetime for gathering the information and aggregation them for the objective of sending it to the base station of reactive decision making. It specifically used the merits of QACO algorithm-based clustering algorithm for organizing the IoT nodes such that potential CHs are identified through the evaluation of factors related to residual energy and distance between the IoT cluster members into account. It was proposed with a potential encoding strategy which helped in determination of factors related to heuristic information such that significant clustering and CHs is selected in the network. The results of this QIACOCM approach determined under different network scenarios conformed its potential in terms of live IoT nodes, network lifetime and network residual energy. Nagappan et al.²⁵ proposed a Bald Eagle Search (BESACRP) Algorithm-based clustering-based route planning approach which used the factors related to trust-aware multi-objective metaheuristic optimization for facilitating potential construction of clusters in the IIoT network. This BESACRP approach concentrated on the process of clustering and routing processes by computing a fitness function which included the factors of trust and energy efficiency into account. It used the sub-factors such as node degree, residual energy, communication cost and trust level during the process of CHs selection in the network. It included the parameters of link quality and queue length for the evaluation of fitness function that helped in the process of route selection. The results of BESACRP approach determined with different range of experiments confirmed better performance in terms of stability period, half node death period and network lifetime independent of number of IIoT nodes deployed in the network.

Avval et al.²⁶ proposed an Elephant Herd Optimization Algorithm-based IIoT nodes’clustering protocol (EHPACP) for preventing the process of long delays, energy consumption, communication and computation overhead in the network. This EHPACP approach concentrated on the process of minimizing delays incurred during the continuous cost of monitoring, and at the same time focused on reducing the cost of resources utilization in the IIoT network. It integrated the factors of clan updating and separating operator for handling the issues of industrial production scheduling. The experiments of EHPACP algorithm confirmed better performance in optimizing the process of planning with respect to the process of production facilitated in the IIoT environment. Chithaluru et al.²⁷ contributed a Fuzzy Logic and Neural Network-based Clustering Protocol (FLNNCP) for attaining superior resource management in IIoT network. This FLNNCP approach facilitated better resource management in different situations related to querying process, monitoring user decision and environmental conditions monitoring. It was proposed with the potential of fuzzy sets and neural network to determined suitable candidate solutions for reducing the degree of computation involved in the process of resource management in IIoT networks. The results of this FLNNCP approach confirmed better results in terms of reliability, energy consumption control and network lifetime independent to the number of alive IIoT nodes.

Further Gong et al.²⁸ proposed a Chaotic Multilevel Elite Clone Snake Optimization Algorithm-based Multi-objective High-Performance Clustering Framework (CMECSOACF) for facilitating the option of exchanging the information during the process of collection and transmission for robust operation in IIoT network. It was proposed as an imperative solution for addressing the problems of IIoT with respect to low QoS and shortened network lifetime. It was implemented as a multi-objective clustering model which adopted a fitness function using the factors of delay, packet loss rate, remaining energy and energy utilization into account for handling the problem of shortened network lifetime and low QoS in IIoT. It reasonably and accurately elected CHs for the objective of enhancing the overall performance such high-performance is guaranteed in IIoT network environments. It specifically used the merits of Chaotic Multilevel Elite Clone Snake Optimization Algorithm (CMECSOA) for enhancing the optimal clustering process. It included chaotic optimization for enhancing the degree of clustering efficiency to perform initial searching over the initial solutions that attribute towards diversity. It also prevented the possibility of the solutions from being struck into local point of optimality. It further used the merits of multilevel elite cloning strategy for enhancing the rate of convergence. The results of this CMECSOACF confirmed better results in terms of reliability, energy consumption control and network lifetime independent to the number of alive IIoT nodes. Wang et al.²⁹ proposed a Hybrid Cat Swarm and Monarch Butterfly Optimization Algorithm (HCSMBO)-based Cluster Routing Protocol (HCSMBOARP) for enhancing the degree of QoS with respect to application services in IIoT environment. This HCSMBOARP approach rather than adopting the blind search strategies focussed on the process of preventing low efficiency and wastage of computing resources in IIoT scenario. It was proposed with the capability of fast convergence, high stability and strengthened spatial search potential such that effective clustering of nodes is achieved in the IIoT environment for policing the process of production in the industry. It incorporated the benefits of Support Vector Machine (SVM) networks which focusses on the process of optimizing energy balance coefficient and coverage efficiency. It also identified global optimal solutions by adjusting the weight factors considered during the process of fitness function in clustering IIoT nodes. The results of this HCSMBOARP approach minimized energy consumptions, and at the same time improved the task scheduling stability and accuracy during the process of resource configuration and management.

Furthermore, Heidari et al.³⁰ proposed an artificial bee colony (ABC)-based Spanning Tree-based Cluster Routing Algorithm (ABCSTA) for effective reception and distribution of data between the IIoT nodes in the network. This ABCSTA approach specifically used the merits of density correlation degree and genetic operators for supporting the ABC algorithm such that reliable data aggregation can be achieved during the process of production in the industry environment. It was proposed with the capability of reconstructing the spanning tree whenever required such that mobility and failure of data routing is significantly addressed in the IIoT environment. It addressed the challenge of broken spanning tree which makes the process of data delivery to the base station very challenging. It computed the fitness value of the constructed spanning tree using the factors of mobility probability, devices’residual energy and hop count distance determined between the IIoT devices from the base station. The simulation results of this ABCSTA approach confirmed better performance in terms of distance, displacement probability, energy consumptions and reliability independent to the number of IIoT nodes in the network. Wang et al.³¹ proposed a Hybrid Chaotic Elite Artificial Bee Colony Algorithm-based clustering mechanism (HCEABCCM) which prevented the problem of single point of failure which is generally common during CHs selection in the IIoT scenario. This clustering protocol was proposed for attaining the problem of uneven data transmission and poor network dynamics. It focussed on the process of optimizing the challenges of network topology control, network lifespan and energy efficiency. It was proposed as a multi-objective clustering routing mechanisms that computed its fitness value during the process of CHs selection using the factors of QoS, system lifespan and network energy consumption. It was also proposed with the potential of comparatively reducing the time of convergence at one end, and at the same time, strengthening the search ability of the clustering process. It specifically incorporated the merits of chaotic strategy for preventing the challenges of premature convergence and preventing the solutions from being struck into local point of optimality. The results of this HCEABCCM approach confirmed better results in terms of enhancing the degree of QoS, system lifespan, and reducing the network energy utilization rate.

Antwi et al.³² Random-Neighbour-Selection-based intelligent data routing mechanism which concentrated on the process of enhancing the required level of throughput in the IIoT network. This RNSIDRM approach was proposed with the capability of blockchain for the process of enhancing the communication in the peer-to-peer environment of IIoT. It was proposed with the capability of randomly selecting the neighbours such that the process of Random-Neighbour-Selection increases the possibility of restricting the potential throughput. It was proposed with the capability of exploring and identifying the key features which influence the network performance to the maximized level. It was implemented by decoupling the network management process completely from the data of the blockchain. The results of this RNSIDRM approach improved the transactional throughput, and at the same time minimized the transaction time with reduced control messages exchanged during the process of data transmission. In addition, Kaur et al.³³ proposed an intelligent clustering protocol for preventing fault tolerance in IIoT since it possesses the challenge of hardware malfunctioning and failure that generally happens due to energy depletion. This intelligent clustering protocol was implemented for enhancing the network reliability by identifying the node faults in a more reactive manner. This clustering protocol confirmed enhanced recovery speed, communication delay, network lifetime, throughput, energy consumption and mean packet delivery.

Ying et al.³⁴ used a ACO and GA-based energy potential clustering protocol for maximizing the coverage rate and minimizing the degree of energy utilization during the process of industrial operation in IIoT. This GA-ACO hybrid approach computed the factors related to the number of neighbouring nodes, energy level, node coverage rate during the clustering and subsequent CHs selection process. It specifically used GA for selecting optimal CHs in the network and also used ACO for determining the paths of optimal data transmission in the network through the selected CHs. The results confirmed better satisfaction of CHs which can satisfy the requirements of data transmission such that network connectivity is maximized in the network. Then Yogarajan et al.³⁵ proposed an Integrated multi-criteria decision-making algorithm for the objective of balancing the energy consumptions among the IIoT nodes for extending the network lifetime. This decision-making approach explored different traffic patterns and heterogeneous energy levels such that energy potent clusters are constructed in the network. The results of this multi-criteria approach confirmed better delivery of data packets to the base station and improved network lifetime.

Banitalebi Dehkordi³⁶ proposed a hybrid IIoT architecture which handles the challenges that are associated with the energy balance and network lifetime under industrial operation. This hybrid IIoT architecture used the significance of Whale Optimization Algorithm (WOA) for selecting optimal CHs in the network. It used the benefits of blockchain technology which helped in guaranteeing secure data transmission using a distributed ledger that helps in determining trust and preventing tampering in IIoT networks. It considered the operation of smart car manufacturing factory as the case study for implementing this WOA-based IIoT architecture. The results confirmed minimized mean end-to-end delay, energy consumptions, CPU workload, throughput, bandwidth and data transmission latency. Reddy et al.³⁷ proposed a metaheuristic optimization algorithm-based dual CHs selection for efficient selection of operation in the industrial scenario. This optimization approach concentrated on the process of clustering the node and selecting optimal routes for the objective of transmitting the information in the network. It adopted a two-stage clustering approach for determining tentative and final CHs in the network. It also used cheetah optimization algorithm used the parameters of mean node distance and residual energy during the computation of fitness evaluation. It also used flower pollination algorithm for selecting the final CHs. It also used carnivorous plant algorithm which used the parameters of residual energy and distance to the sink for facilitating optimal route selection process.

In addition, Xeng et al.³⁸ proposed a data transmission approach which minimized computational complexity with reduced message overhead during industrial operations. This data transmission approach was implemented as a lightweight green security data aggregation model which facilitated better management of restricted energy available in the industrial operation implementation scenario. The results of this model satisfied the requirements of the energy consumption such that security, green and lightweight is guaranteed during its implementation. Antwi et al.³⁹ contributed a PSO-based swarm intelligent approach which optimized the process of implementing the peer-to-peer topology in a significant manner. It used minimized number of control messages such that transactional throughput is maximized in the network. The results of this PSO-based approach confirmed better results in terms of number of control messages used and the energy utilized in the network.

In addition, Table 1 presents the Literature review summary of the existing intelligent metaheuristic optimization algorithm-based clustering algorithm contributed for IIoT scenario with the merits and limitations.

Table 1 Literature review summary of the existing intelligent metaheuristic optimization algorithm-based clustering algorithm contributed for IIoT scenario.

Extract of the literature

The shortfalls identified from the comprehensive survey of the existing works of the literature is listed as follows.

• i) The heuristic fitness computing factors and the method computing the weights of those factors is not comprehensively defined and suitably adopted during the process of CHs selection and clustering process.

• ii) The optimal clustering mechanisms mainly focussed on energy efficiency and network lifetime improvement and ignored the factors of network connectivity and scheduling even though they are highly impact during the process of clustering.

• iii) The existing clustering approaches did not utilize the essential selection of potential nodes which contributes towards the achievement of each service with the required QoS to the user.

• iv) Many of the research works that used the concept of spanning tree-based cluster routing strategy did not include failure and mobility into account, and hence the use of key parameters such as distance, displacement probability, energy consumption and delay have the maximized the probability of data delivery for dynamic decision making at the base station.

Keeping the above pitfalls of the literature^40,41 in mind, EEQIAHBOA-based cluster routing framework is presented for attaining the objective of potential and CHs in the IIoT nodes with the required degree of QoS considered during the routing process.

Energy efficient quantum-informed artificial hummingbird optimization algorithm-based high-performance clustering framework

In this section, the network model, energy consumption radio model and the multi-objective fitness function-based CHs selection process utilized during the implementation of the proposed EEQIAHBOA-based cluster routing framework is presented.

Network model and assumptions

In IIoT, the collection and transmission of information is attained through the cooperation of different number of IoT nodes deployed for monitoring the industrial environment. In this model of clustering, the IoT nodes are categorized into cluster member nodes and CHs nodes. In specific, cluster member nodes are responsible for collecting information in the area of monitoring and forward them to the CHs of each of its cluster in which they exist. On the other hand, CH nodes are responsible for aggregating the information and forwarding it to the base station. Then the base station transmits the essential information directly to the server or satellite through the electromagnetic waves. These processes involved in the IIoT environment helps the client in retrieving and utilizing the data from the server on-demand whenever the tasks of decision-making need to be performed. In this EEQIAHBOA-based cluster routing framework, the network topology is mathematically modelled as a graph $\:\text{G}=(\text{V},\text{E})$, where $\:\text{V}$ and $\:\text{E}$ represents the network nodes (IIoT nodes) and communication links between the network nodes. In this implementation model, the base station is static and number of IIoT nodes ($\:n$) are deployed over the entire area of monitoring in a random manner.

The simulation experiments of the proposed EEQIAHBOA is conducted with the assumptions made over the network structure as described as follows.

• i) The initial energy for all the IoT nodes are initialized, and the base station is independent of the energy possessed by them.

• ii) The IIoT nodes and the base stations are considered to the static in the working environment.

• iii) All the IIoT nodes are identified by their distinct numbers and the position of these nodes are also determined at each point of time.

• iv) The transmission power of wireless signal is considered to increase with respective increase in the distance threshold.

• v) The neighbouring nodes of each IIoT nodes are identified with respect to the radius of coverage $\:R$, and a node is considered to a neighbour when they are within the network coverage radius as specified in Eq. (1)

$$\:{\text{V}}_{\text{J}\left(\text{N}\text{e}\text{i}\text{g}\text{h}\text{b}\text{o}\text{u}\text{r}\right)}=\left\{{\text{V}}_{\text{i}}|\left|{\text{V}}_{\text{i}}-{\text{V}}_{\text{i}}\right|\le\:\text{R},\:\text{j}\le\:\text{i},\:\text{j}\le\:\text{n}\right\}$$

(1)

The environment of IIoT is considered to be self-organized by different number of nodes for the objective of monitoring the industrial areas as depicted in Fig. 1. In this environment, each individual sensor nodes possesses an energy supply module, wireless transceiver module, processor memory and clock module. Further the CHs acts as the aggregator node and forwards the sensed packets to the network for testing purpose. Then the energy model is used by each of the node for computing the energy consumption, and it also calculates data related to packet loss rate and latency. Then the routing information is forwarded to the respective CH nodes termed as the aggregator node. In this context, the rate of packet loss is computed depending on the nodes past information. In other words, the packet loss is computed based on the proportion of data packets received by each node and the proportion of data forwarded by each node in the network. In addition, the CHs collects the primitive information associated with each individual node for generating a number of clusters in the network. Then the clustering process is optimized through the inclusion of the proposed EEQIAHBOA framework, such that the results determined by node can be shared in the network through the process of broadcasting. These methodologies are adopted for constructing the clustering architecture which helps in tracking or monitoring the operations that are performed in the industrial environment. In addition, the required degree of experimental indicators are simulated using the IIoT nodes functionality such that the assessment of the incorporated framework can be achieved as specified in Fig. 2.

Fig. 2

Model Architecture of the IIoT network considered for implementation.

Energy model

In IIoT environment, the energy consumption of nodes is mainly important during the process of data transmission and reception since it impacts the stability and lifetime of the network. When the nodes send and collects large data in clusters then it incurs a long time for achieving its objective of accomplishing the task. But each node collects the information from the industrial environment and forwards it to the CHs, and then these CHs forwards the aggregated data to the base data for decision making reduces the amount of energy utilization. Thus this proposed EEQIAHBOA-based cluster routing framework used the energy radio model contributed by Heinzelman et al.⁴² as depicted in Fig. 3a.

Fig. 3

(a) Radio energy dissipation model⁴². (b): Initial solution of an QAHA search agent used for routing. (c): Updated solution of an QAHA search agent used for routing.

This radio energy model is suitable for IIoT applications during the achievement of long-term operation with energy efficiency. This energy model aids in optimizing the process of network designs and application of communication protocols such that energy consumption is highly reduced with respect to resource-limited devices. It is also helpful for facilitating the option of deploying IIoT systems in different industrial settings which possesses less or no access to a power grid. According to this energy radio model, the energy incurred for sensing data packets is determined based on Eqs. (2), (3) and (4).

$$\:{{\text{E}}_{{\text{NG}}\left( {{\text{Tr}}} \right)}}({\text{q}},{\text{d}}) = \left\{ {\begin{array}{*{20}{c}} {{\text{q}}.{{\text{E}}_{{\text{Elec}}}} + \eta {\:_{{\text{amp}}\left( {\text{x}} \right)}}.{\text{q}}.{{\text{d}}^2}}&{\:{\text{d}} \leqslant \:{{\text{d}}_0}} \\ {\:{\text{q}}.{{\text{E}}_{{\text{Elec}}}} + \eta {\:_{{\text{amp}}\left( {\text{y}} \right)}}.{\text{q}}.{{\text{d}}^4}}&{\:{\text{d}}> {{\text{d}}_0}} \end{array}} \right.$$

(2)

$$\:{d}_{(x,y)}=\sqrt{{\left({x}_{1}-{x}_{2}\right)}^{2}+{\left({y}_{1}-{y}_{2}\right)}^{2}}$$

(3)

$$\:{\text{d}}_{0}=\sqrt{\frac{{{\upeta\:}}_{\text{a}\text{m}\text{p}\left(\text{x}\right)}}{{{\upeta\:}}_{\text{a}\text{m}\text{p}\left(\text{y}\right)}}}$$

(4)

Where $\:{{\upeta\:}}_{\text{a}\text{m}\text{p}\left(\text{x}\right)}$ and $\:{{\upeta\:}}_{\text{a}\text{m}\text{p}\left(\text{y}\right)}$ represents the characteristic parameters associated with the transmitter amplifier. Then $\:{\text{E}}_{\text{N}\text{G}\left(\text{T}\text{r}\right)}(\text{q},\text{d})$ and $\:{\text{E}}_{\text{E}\text{l}\text{e}\text{c}}\:$represents the amount of joules utilized during the process of signal transmission and amount of joules utilized by the electronic circuit. Further $\:d$ and $\:{\text{d}}_{0}$ represents the transmission distance and the threshold considered for transmission distance as specified in⁴³. Moreover, $\:q$ indicates the data packet size.

On the other hand, the energy used for data reception by each of the IIoT nodes is determined based on Eq. (5)

$$\:{\text{E}}_{\text{N}\text{G}\left(\text{R}\text{r}\right)}\left(\text{q}\right)=\text{q}.{\text{E}}_{\text{E}\text{l}\text{e}\text{c}}$$

(5)

Then the CHs aggregates the data received from the cluster member nodes, and thus minimizes the data forwarded to the base station for reducing the amount of energy used for data transmission. The amount of energy used by the CHs for data aggregation is represented using Eq. (6)

$$\:{\text{E}}_{\text{D}\text{A}}=\text{q}.{\text{E}\text{D}}_{\left(\text{p}\right)}$$

(6)

Where $\:{\text{E}}_{\text{D}\text{A}}\:$and $\:{\text{E}\text{D}}_{\left(\text{p}\right)}$ represents the amount of energy used for data aggregation and the energy utilized for aggregating 1-bit of data.

Information heuristic factor determination

In IIoT environment, the network nodes ate organized into clusters depending on the multiple factors of information heuristic factors (fitness factors used for evaluation). In specific, this proposed EEQIAHBOA approach used four important factors such as Mean Delay Function, Mean Packet Loss Rate Function, Energy Consumption Function and Residual Energy Function for computing the information Heuristic Factor (fitness) for solving the problem of Multi objective clustering.

i) Mean Delay Function: It is defined as the mean amount of time required by each individual cluster in the network to transmit information. This mean delay function is determined based on Eq. (7)

$$\:{\text{D}}_{\text{L}\text{Y}\left(\text{M}\text{e}\text{a}\text{n}\right)}=\frac{1}{\text{c}}\sum\:_{\text{i}=1}^{\text{c}}\left({\text{D}}_{\text{L}\text{Y}\left(\text{i}\right)}+{\text{D}}_{\text{B}\text{S}\:(\text{i}}\right)$$

(7)

Where $\:{\text{D}}_{\text{L}\text{Y}\left(\text{i}\right)}$ and $\:{\text{D}}_{\text{B}\text{S}\:(\text{i}}$ represents the delay incurred for each of the $\:{\text{i}}^{\text{t}\text{h}}\:$individual cluster, and the delay incurred from the $\:{\text{i}}^{\text{t}\text{h}}$ CH to the base station. In this case, delay (end-to-end delay) is the amount of time spent for collecting the information from the cluster member nodes, aggregation the collected information at the CHs and then forwarding the aggregated data to the base station. This time termed as delay includes processing delay ($\:{\text{D}}_{\text{L}\text{Y}\:\left(\text{p}\text{d}\right)}$), queuing delay, propagation delay ($\:{\text{D}}_{\text{L}\text{Y}\left(\text{q}\text{d}\right)}$), message transmission delay ($\:{\text{D}}_{\text{L}\text{Y}\left(\text{m}\text{r}\text{d}\right)})$, intermediate node forwarding delay ($\:{\text{D}}_{\text{L}\text{Y}\left(\text{i}\text{n}\text{f}\text{d}\right)}$), and so on. This delay is computed based on Eq. (8)

$$\:{\text{D}}_{\text{L}\text{Y}}={\text{D}}_{\text{L}\text{Y}\:\left(\text{p}\text{d}\right)}+{\text{D}}_{\text{L}\text{Y}\left(\text{q}\text{d}\right)}+{\text{D}}_{\text{L}\text{Y}\left(\text{m}\text{r}\text{d}\right)}+{\text{D}}_{\text{L}\text{Y}\left(\text{i}\text{n}\text{f}\text{d}\right)}$$

(8)

ii) Energy consumption function: It is defined as the amount of energy utilized for transmitting all the IP packets from each $\:{\text{i}}^{\text{t}\text{h}}\:$individual cluster to the base station (this energy utilization includes the energy spent by cluster member nodes for transmitting and receiving data packets, and the energy used by CHs for aggregating the data as well as transmitting and receiving the data) as specified in Eqs. (9) and (10)

$$\:{\text{E}}_{\text{C}\text{M}\text{N}}={\sum\:}_{\text{I}=1}^{\text{N}-\text{C}}{\text{E}}_{\text{N}\text{G}\left(\text{T}\text{r}-\text{i}\right)}+{\text{E}}_{\text{N}\text{G}\left(\text{R}\text{r}-\text{i}\right)}$$

(9)

$$\:{\text{E}}_{\text{C}\text{H}}={\sum\:}_{\text{I}=1}^{\text{C}}{{\text{E}}_{(\text{D}\text{A}-\text{i})}+\text{E}}_{\text{N}\text{G}\left(\text{T}\text{r}-\text{i}\right)}+{\text{E}}_{\text{N}\text{G}\left(\text{R}\text{r}-\text{i}\right)}$$

(10)

Where $\:{\text{E}}_{\text{N}\text{G}\left(\text{T}\text{r}-\text{i}\right)}$ and $\:{\text{E}}_{\text{N}\text{G}\left(\text{R}\text{r}-\text{i}\right)}$ represents the energy spent by cluster member nodes and CHs for transmitting and receiving data packets. In addition, $\:{\text{E}}_{(\text{D}\text{A}-\text{i})}$ indicates the energy used by CHs for aggregating the data.

Then the energy consumption function ($\:{\text{E}}_{\text{C}\text{F}}$) which represents the total energy utilized by the network is computed using $\:{\text{E}}_{\text{C}\text{M}\text{N}}$ and $\:{\text{E}}_{\text{C}\text{H}}$ as specified in Eq. (11)

$$\:{\text{E}}_{\text{C}\text{F}\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}={\text{E}}_{\text{C}\text{M}\text{N}}+{\text{E}}_{\text{C}\text{H}}$$

(11)

iii) Residual energy function: This function of residual energy ($\:{\text{R}}_{\text{E}\text{F}}$) is computed based on the cumulative sum of energy incurred by all the CHs in the current network as specified in Eq. (12)

$$\:{\text{R}}_{\text{E}\text{F}\:\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}={\sum\:}_{i=1}^{c}{\text{R}}_{\text{E}\text{n}\text{e}\text{r}\text{g}\text{y}\:\left(\text{i}\right)}$$

(12)

Where $\:\text{C}$ is the number of clusters and $\:{\text{R}}_{\text{E}\text{n}\text{e}\text{r}\text{g}\text{y}\:\left(\text{i}\right)}$ is the residual energy possessed by each of the CHs in the network.

iv) Packet Loss Rate Function: This function is defined as the number of packets lost in the network when the cluster member of each cluster forwards the information to the CHs. In specific, the communication quality of the complete set of network clusters can be reliably determined based on packet loss rate ($\:{\text{P}\text{L}}_{\text{R}\text{a}\text{t}\text{e}}$) as specified in Eq. (13)

$$\:{\text{P}\text{L}}_{\text{R}\text{a}\text{t}\text{e}}=1-\prod\:_{\:i\in\:CMN}\:(1-\text{P}\text{L}\text{R}(\text{i},\:\text{C}\text{H}\left)\right)$$

(13)

Where $\:\text{P}\text{L}\text{R}(\text{i},\:\text{C}\text{H}$) indicates the rate of packet loss with respect to each of the cluster member’$\:i$'to the selected CHs. In specific, indicates the set of sensor member nodes associated with each cluster. Moreover, PLR highlights the packet loss rate related to each cluster. This function of packet loss rate is mainly visualized when the cluster is lost or corrupted when it performs the process of sending and receiving IP packets as computed using Eq. (14)

$$\:{\text{P}\text{L}\text{R}}_{\text{M}\text{e}\text{a}\text{n}}=\frac{1}{\text{C}}\sum\:_{\text{i}=1}^{\text{C}}\left({\text{P}\text{L}\text{R}}_{\left(\text{i}\right)}\text{*}{\text{P}\text{L}\text{R}}_{\text{B}\text{S}\:\left(\text{i}\right)}\right)$$

(14)

The greater probability of packet loss highlights the maximized degree of data corruption. In specific, $\:{\text{P}\text{L}\text{R}}_{\text{B}\text{S}\:\left(\text{i}\right)}$ and $\:{\text{P}\text{L}\text{R}}_{\left(\text{i}\right)}\:$represents the packet loss rate identified when the packets are forwarded from the $\:{\text{i}}^{\text{t}\text{h}}\:$CHs to the base station, and the total number of packets lost in the $\:{\text{i}}^{\text{t}\text{h}}$ cluster.

However, these calculated four-fitness function evaluating factors differs in the numeric range, and hence these values may have an impact during the computation of weighted multi-objective function. Hence the method of normalization is used for data processing over the four fitness evaluation factors as mentioned in Eqs. (15), (16), (17) and (18).

$$\:{\text{D}}_{\text{L}\text{Y}\left(\text{M}\text{e}\text{a}\text{n}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)=\frac{{\text{D}}_{\text{L}\text{Y}\left(\text{M}\text{e}\text{a}\text{n}\right)}-{\text{D}}_{\text{L}\text{Y}\left(\text{M}\text{e}\text{a}\text{n}\right)}^{\text{M}\text{i}\text{n}})}{{\text{D}}_{\text{L}\text{Y}\left(\text{M}\text{e}\text{a}\text{n}\right)}^{\text{M}\text{a}\text{x}}-{\text{D}}_{\text{L}\text{Y}\left(\text{M}\text{e}\text{a}\text{n}\right)}^{\text{M}\text{i}\text{n}}}$$

(15)

$$\:{\text{E}}_{\text{C}\text{F}\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)=\frac{{\text{E}}_{\text{C}\text{F}\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}-{\text{E}}_{\text{C}\text{F}\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}^{\text{M}\text{i}\text{n}}}{{\text{E}}_{\text{C}\text{F}\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}^{\text{M}\text{a}\text{x}}-{\text{E}}_{\text{C}\text{F}\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}^{\text{M}\text{i}\text{n}}}$$

(16)

$$\:{\text{P}}_{LR\:\left(\text{M}\text{e}\text{a}\text{n}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)=\frac{{P}_{\text{L}\text{R}\left(\text{M}\text{e}\text{a}\text{n}\right)}-{P}_{\text{L}\text{R}\left(\text{M}\text{e}\text{a}\text{n}\right)}^{\text{M}\text{i}\text{n}})}{{\text{P}}_{\text{L}\text{R}\:\left(\text{M}\text{e}\text{a}\text{n}\right)}^{\text{M}\text{a}\text{x}}-{P}_{\text{L}\text{R}\:\left(\text{M}\text{e}\text{a}\text{n}\right)}^{\text{M}\text{i}\text{n}}}$$

(17)

$$\:{\text{R}}_{\text{E}\text{F}\:\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)=\frac{{\text{R}}_{\text{E}\text{F}\:\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}-{\text{R}}_{\text{E}\text{F}}^{\text{M}\text{i}\text{n}}}{{\text{R}}_{\text{E}\text{F}}^{\text{M}\text{a}\text{x}}-{\text{R}}_{\text{E}\text{F}}^{\text{M}\text{i}\text{n}}}$$

(18)

Where $\:{\text{D}}_{\text{L}\text{Y}\left(\text{M}\text{e}\text{a}\text{n}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)$, $\:{\text{E}}_{\text{C}\text{F}\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)$, $\:{\text{P}}_{LR\:\left(\text{M}\text{e}\text{a}\text{n}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)\:$ and $\:{\text{R}}_{\text{E}\text{F}\:\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)\:$ represented the normalized values with respect to the fitness evaluating factors of delay, energy consumptions, packet loss rate and residual energy, respectively.

Further, the information heuristic factor (fitness function) used for assessing the potential of the selected CHs (used multiple fitness evaluating factors and Pareto Optimization-based weight determination strategy) is formulated for solving the problem of clustering as specified in Eq. (19)

$$\:\underset{}{\text{min}}{\text{F}\text{i}\text{t}}_{\text{F}\text{n}}=\frac{{\text{W}}_{1}\text{*}{\text{D}}_{\text{L}\text{Y}\left(\text{M}\text{e}\text{a}\text{n}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)+{\text{W}}_{2}\text{*}{\text{E}}_{\text{C}\text{F}\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)+{\text{W}}_{3}\text{*}{\text{P}}_{LR\:\left(\text{M}\text{e}\text{a}\text{n}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)}{{{\text{W}}_{4}\text{*}\text{R}}_{\text{E}\text{F}\:\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)}$$

$$\:\left\{\begin{array}{c}1\ge\:{\text{D}}_{\text{L}\text{Y}\left(\text{M}\text{e}\text{a}\text{n}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)\ge\:0\\\:1\ge\:{\text{E}}_{\text{C}\text{F}\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)\ge\:0\\\:\begin{array}{c}1\ge\:{\text{P}}_{LR\:\left(\text{M}\text{e}\text{a}\text{n}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)\ge\:0\\\:\begin{array}{c}1\ge\:{\text{R}}_{\text{E}\text{F}\:\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)\ge\:0\\\:{\text{W}}_{1}+{\text{W}}_{2}+{\text{W}}_{3}+{\text{W}}_{4}=1\end{array}\end{array}\end{array}\right.$$

(19)

Such that.

Where $\:{\text{W}}_{1},\:{\text{W}}_{2},\:{\text{W}}_{3}\:and\:{\text{W}}_{4}$ indicates the weights related to the factors of Mean Delay Function, Mean Packet Loss Rate Function, Energy Consumption Function and Residual Energy Function used for computing the multi-objective function $\:{\text{F}\text{i}\text{t}}_{\text{F}\text{n}}$.

Furthermore, the weights associated with respect to each of the factors considered for fitness evaluation is determined based on Criteria Importance Through Intercriteria Correlation (CRITIC) method, This CRITIC method⁴⁴ aids in objectively determining the weight of the factors using the concept intensity which specific the degree of deviation that is identified between the criterion and how each of the criterion is correlated with other criteria used for evaluation. In specific, higher weights need to be provided for the factors which has greater deviation, and lower weight to the factors that facilitates less correlations with others. The weights associated with the factors of Mean Delay Function, Mean Packet Loss Rate Function, Energy Consumption Function and Residual Energy Function considered during the implementation of the proposed clustering scheme are 0.15, 0.15, 0.35 and 0.35, respectively.

In this CRITIC Method, the normalized values termed $\:{\text{D}}_{\text{L}\text{Y}\left(\text{M}\text{e}\text{a}\text{n}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)$, $\:{\text{E}}_{\text{C}\text{F}\left(\text{T}\text{o}\text{t}\text{a}\text{l}\right)}\left(\text{N}\text{o}\text{r}\text{m}\right)$, $\:{P}_{LR\:\left(Mean\right)}\left(Norm\right)\:$ and $\:{R}_{EF\:\left(Total\right)}\left(Norm\right)\:$ related to the normalized values with respect to the fitness evaluating factors of delay, energy consumptions, packet loss rate and residual energy presented in Eqs. (15)-(18) as mentioned in Eq. (20)

$$\:{\text{x}}_{\text{i}\text{j}}=\frac{{\text{x}}_{\text{i}\text{j}}-{\text{x}}_{\text{j}}^{\text{W}\text{o}\text{r}\text{s}\text{t}}}{{\text{x}}_{\text{j}}^{\text{B}\text{e}\text{s}\text{t}\:}-{\text{x}}_{\text{j}}^{\text{W}\text{o}\text{r}\text{s}\text{t}}}$$

(20)

Where $\:{\text{x}}_{\text{i}\text{j}}\:$is the normalized value of each IIoT sensor nodes ($\:i$) with respect to the specific criterion $\:\text{j}$. In addition, $\:{\text{x}}_{\text{j}}^{\text{B}\text{e}\text{s}\text{t}\:}$and $\:{\text{x}}_{\text{j}}^{\text{W}\text{o}\text{r}\text{s}\text{t}}\:$represents the best and worst score of each IIoT sensor nodes ($\:i$) with respect to the criterion $\:\text{j}$.

Then compute the standard deviation with respect to each criterion considered for fitness computation using Eq. (21)

$$\:{\text{S}\text{D}}_{\left(\text{j}\right)}=\sqrt{\frac{{\left(\sum\:_{\text{i}=1}^{\text{m}}{\text{x}}_{\text{i}\text{j}}-\stackrel{-}{{\text{x}}_{\text{j}}}\right)}^{2}}{\text{m}-1}}$$

(21)

Where $\:m$ and $\overrightarrow x$ is the number of IIoT sensor nodes in the network, and average value of the criterion $\:j$, respectively.

Further the conflicting association or relationship between the criteria considered for evaluation is determined using the method of Pearson correlation⁴³. But the method of Pearson correlation has the problem of capturing the actual relation between the different criterions in an accurate manner. Thus, the method of distance method is used for preventing the problem that generally occurs during the inclusion of Pearson correlation. This distance method which focusses on the process of reducing errors during the process of determining the final weights is computed based on Eq. (22)

$$\:{Cr}_{d}({C}_{i},\:{C}_{j})=\frac{{Cov}_{d}({\text{C}}_{\text{i}},\:{\text{C}}_{\text{j}})}{\sqrt{V{ar}_{d}\left({\text{C}}_{\text{i}}\right)\bullet\:V{ar}_{d}\left({\text{C}}_{\text{j}}\right)}}$$

(cdo22)

Where $\:V{ar}_{d}\left({\text{C}}_{\text{i}}\right)$and $\:V{ar}_{d}\left({\text{C}}_{\text{j}}\right)$ represents the variance identified with respect to the criterion $\:i$ and $\:j$, respectively with $\:{Cov}_{d}({\text{C}}_{\text{i}},\:{\text{C}}_{\text{j}}$) as the covariance identified between the criterion $\:i$ and $\:j$, respectively.

In this context, the value of $\:{Cov}_{d}({\text{C}}_{\text{i}},\:{\text{C}}_{\text{j}})$ is computed based on Rquation (23) for the purpose of identifying the information content at the next level.

$$\:{Cov}_{d}\left({\text{C}}_{\text{i}},\:{\text{C}}_{\text{j}}\right)=\sum\:_{i=1}^{n}\frac{\left({x}_{j}-\overbrace{x}\right)\left({x}_{k}-\overbrace{x}\right)}{n}$$

(23)

Further compute the content of information of each of the IIoT nodes using Eq. (24) based on the criterion $\:j$,

$$\:{\text{I}\text{C}}_{\left(\text{j}\right)}={\text{S}\text{D}}_{\left(\text{j}\right)}\sum\:_{k=1}^{n}\left(1-{Cr}_{d}({C}_{i},\:{C}_{j})\right)$$

(24)

Finally, the weights associated with the factors used for computing the fitness function is determined based on Eq. (25)

$$\:{W}_{j}=\frac{{\text{I}\text{C}}_{\left(\text{j}\right)}}{\sum\:_{j=1}^{m}{\text{I}\text{C}}_{\left(\text{j}\right)}}\:\text{W}\text{h}\text{e}\text{r}\text{e}\:1\le\:j\le\:n$$

(25)

Where $\:{\text{I}\text{C}}_{\left(\text{j}\right)}\:$is the content of information associated with each criterion considered for fitness function evaluation during the clustering process.

Further the initial solution encoding process used during the process of routing is explained as follows. For illustrating the encoding scheme, let us consider the IIoT network scenario with 5 CHs such as $\:{\text{C}\text{H}}_{\text{S}\left(\text{S}\text{e}\text{t}\right)}=\{{\text{C}\text{H}}_{\text{S}\left(1\right)},\:{\text{C}\text{H}}_{\text{S}\left(2\right),},\:{\text{C}\text{H}}_{\text{S}\left(3\right)},\:{\text{C}\text{H}}_{\text{S}\left(4\right)}$ and $\:{\text{C}\text{H}}_{\text{S}\left(5\right)}$, respectively. In this context, the solution encoding string length is 5. In this encoding process, the edge $\:{\text{C}\text{H}}_{\left(i\right)}\to\:{\text{C}\text{H}}_{j}$ represents that $\:{\text{C}\text{H}}_{\left(i\right)}$ can utilize $\:{\text{C}\text{H}}_{j}\:$as the next hop CH through which the data can reach the sink for reactive decision making. At the same time, when $\:{\text{C}\text{H}}_{\left(i\right)}$ is in close proximity to the sink than $\:{\text{C}\text{H}}_{\left(j\right)}$ and $\:{\text{C}\text{H}}_{\left(j\right)}$ is determined within the communication range of $\:{\text{C}\text{H}}_{\left(i\right)}$. The solution encoding presented in Fig. 3 (b) indicates that $\:{\text{C}\text{H}}_{\left(1\right)}$ is very close to the sink node. Further $\:{\text{C}\text{H}}_{\left(2\right)}$ can utilize $\:{\text{C}\text{H}}_{\left(3\right)}$ and $\:{\text{C}\text{H}}_{\left(5\right)}$ as one of the CHs depending on the distance determined between the selected CHs in the network.

Then Fig. 3 (c) presents the solution encoding for updating the solution of an QAHA search agent used for routing.

The updated solution encoding pattern exhibited that the sink is in close proximity to the $\:{\text{C}\text{H}}_{\left(1\right)}\:$and $\:{\text{C}\text{H}}_{\left(4\right)}$, and thus the initial solution is updated depending on the distance determined between the selected CHs in the network.

In the subsequent section, the primitives of AHBOA and Quantum-informed AHBOA used for optimizing the process of clustering and CHs selection in IIoT is presented as follows.

Artificial hummingbird algorithm (AHBOA)

This AHBOA is a renowned Swarm Intelligence (SI)-based algorithm proposed by Zhao et al.⁴⁵. It simulates the smart behaviour of Hummingbird which is considered to be highly intelligent with a small body in contrast to its intelligence level. Its intelligence is modelled as an optimization process. The steps of AHBOA are detailed below. The algorithm commences with initialization of population along with visit table, determination of flight behaviour and selection of foraging approach. Finally, the iterative algorithm terminates after every iteration.

Step 1: Construction of population and visit table.

Like other population-based algorithms, AHBOA initially develops its population which includes’$\:\text{n}$'hummingbirds positioned at’$\:n$'food sources. The positions ($\:P$) of hummingbirds are initialised using Eq. ('26)

$$\:{\text{P}}_{\text{i}}=\text{l}\text{b}+\dot{r}.(\text{u}\text{b}-\text{l}\text{b})$$

(26)

Where’$\:{\text{P}}_{\text{i}}$'refers to location of’$\:{\text{i}}^{\text{t}\text{h}}$'source allocated to’$\:{\text{i}}^{\text{t}\text{h}}$'hummingbird, $\:\text{i}\:=\:1,\:2,\:.\:.\:.\:,\text{n}$ and $\:\mathop r\limits^. \in \:[0,\:1]\:$is a random vector involving numbers. Then’$\:\text{l}\text{b}$‘and’$\:\text{u}\text{b}$'are search space limits, and’$\:n$'represents population size. In addition,'$\:\text{l}\text{b}$','$\:\text{u}\text{b}$','$\mathop r\limits^.$‘and’$\:{\text{P}}_{\text{i}}$'are vectors based on the problem and are of’$\:d$'dimension.

Besides initialization of population, the other features of hummingbirds include their intelligence as well as strong memory. Every hummingbird is accountable for a Food Source (FS) from which details including Nectar-Refilling Rate (NFR) can be gained. Moreover, it is capable of moving or migrating to get food from other sources of food. It memorizes 2 pieces of information; one is based on the FS it is assigned to, while the other focuses on visiting the sources last time. To simulate this, the birds’memories are presented as a table called Visiting Table (VT) that signifies the visiting level of FSs. With increase in the level, the time of the present visit since the former visit increases. In the’$\:n\times\:n$'visit table, the number of rows and columns signify the quantity of birds and FSs respectively. Table is filled based on Eq. (27) which includes two important cases that includes.

• i) $\:\mathbf{i}=\mathbf{j}$: It refers to hummingbird with the allocated FS. Visit level is not assigned, and hence value is null.

• ii) $\:\mathbf{i}\:\ne\:\mathbf{j}$: The FSs are assigned a value of’0'representing the latest visit as this is an initial step. For every hummingbird, the level of food sources is’0', except the source for which it is accountable which is’Null’.

$$\:{\text{V}\text{T}}_{\text{i},\text{j}}=\left\{\begin{array}{c}Null,\:if\:i=j\:\:\\\:0,\:if\:i\:\ne\:j\:\end{array}\right.$$

(27)

Where $\:\text{i}=\text{j}=1,\:2,\:.\:.\:.\:,\:\text{n}$.

Step 2: Find flight behaviour.

Normally, the birds’flying behaviour is omnidirectional. Nevertheless, for hummingbirds, the following flying behaviours are noticed: Axial, Diagonal and Omnidirectional flights. In case of Axial flight, they fly along the axes, whereas in Diagonal flight, they fly from a corner to opposite corner. Finally, in Omnidirectional flight, the bird’s movement is driven by the axes drive. However, a random number’$\:\text{r}\in\:[0,\:1]$'is used for flight selection based on Eq. (28).

$$\:\text{F}\text{T}=\left\{\begin{array}{c}Diagonal,\:if\:r<\frac{1}{3}\:\\\:Omnidirectional,\:if\:r>\frac{2}{3}\\\:Axial,\:Else\:\end{array}\right.$$

(28)

Based on the chosen type, an Equation from Eqs. (29)'(31) will be applied. The main goal is to define the Direction switch vector ($\:D$) that includes’$\:0s$‘and’1s’, wherein’$\:1$'represents a move related to’$\:{\text{i}}^{\text{t}\text{h}}$'dimension.

Axial flight

Commencing with direction generation, a random number’$\:\text{r}\text{a}\text{n}\left(\right[1,\:\text{d}\left]\right)$'is generated which represents the dimension to which the bird can fly. Hence, selected dimension is set to’1', while the rest to’0'.

$$\:{\text{D}}^{\text{i}}=\left\{\begin{array}{c}1,\:i=ran\left(\right[1,\:d\left]\right)\\\:0,\:Else\:\end{array}\right.$$

(29)

Where, $\:\text{i}\:=\:1,\:.\:.\:.\:,\:\text{d}$

Diagonal flight

It is based on more than one axes ($\:2\:\text{t}\text{o}\:\text{d}-1$) as shown in Eq. (20). Hence, permutation random vector generator function ($\:{\text{V}}_{\text{P}}$) produces a different dimension.'$\:{\text{V}}_{\text{P}}$'is provided by’$\:\text{j}$'which is the number of permutation axes,'$\:j\in\:[1,\:k]$', $\:{\text{k}} \in \:[2,\:\left\lceil {\mathop {{{\mathop r\limits^. }_1}.\left( {{\text{d}} - 2} \right) + 1]}\limits^. } \right\rceil$ and $\:0 < {\mathop r\limits^. _1} \leqslant \:1$.

$$\:{\text{D}}^{\text{i}}=\left\{\begin{array}{c}1,\:if\:i={V}_{P}^{j},\:j\in\:\left[1,\:\text{k}\right],\:{\text{V}}_{\text{P}}=ranper\left(k\right)\:\:\\\:0,\:Otherwise\:\end{array}\right.$$

(30)

Omnidirectional flight

It presents movement for every dimension.'$\:D$'is assigned a value of 1 for all dimensions.

$$\:{\text{D}}^{\text{i}}=1$$

(31)

Step 3: Find foraging behaviour.

Based on availability as well as NF level, hummingbirds use any of the formerly mentioned flights to reach good sources of food. There are 3 types of foraging namely, guided, territorial and migration foraging.

Guided foraging

The birds are guided as its name states. The birds use pre-defined knowledge for guiding them to food source. Knowledge is gathered in the memory or VT as AHBOA representation. The birds return to VT and pick the source with maximum level (the one that was not visited for longer period of time). In case more than a source has equivalent visit level, decision is based on NFR, where the birds visit source with maximum level as well as NFR. This method of foraging is expressed as shown in Eq. (32).

$$\:{\rm{FS}}_{\rm{i}}^{{\rm{New}}}({\rm{t}} + 1) = {\rm{FS}}_{\rm{i}}^{{\rm{Tar}}}\left( {\rm{t}} \right) + \overline g .{\rm{D}}.\left( {{{\rm{P}}_{\rm{i}}}\left( {\rm{t}} \right) - {\rm{FS}}_{\rm{i}}^{{\rm{Tar}}}\left( {\rm{t}} \right)} \right)$$

(32)

Where’$\:{\text{F}\text{S}}_{\text{i}}^{\text{N}\text{e}\text{w}}(\text{t}+1)$'is the location of new FS for ensuing iteration $\:(\text{t}+1)$.'$\:{\text{F}\text{S}}_{\text{i}}^{\text{T}\text{a}\text{r}}\left(\text{t}\right)$‘and’$\:{\text{P}}_{\text{i}}\left(\text{t}\right)$'represent target FS and present location of bird at iteration’$\:\text{t}$'.'$\:\text{D}$'is established by flight type seen in Eqs. (29)-(31). The parameter’$\:g$'is a guiding factor which follows normal distribution ($\:\text{n}(0,\:1)$).

Territorial foraging

After visiting identified FSs and feeding on nectar, the birds commence searching for improved FSs in neighbouring areas (Territory). In specific, Eq. (33) will be used for allocating’$\:{\text{F}\text{S}}_{\text{i}}^{\text{N}\text{e}\text{w}}(\text{t}+1)$‘based on’$\:{\text{P}}_{\text{i}}\left(\text{t}\right)$‘at iteration’$\:t$‘and’$\:D$'. Further, foraging is based on territorial factor ($\mathop t\limits^.$) as presented in Eq. (34).

$$\:{\rm{FS}}_{\rm{i}}^{{\rm{New}}}({\rm{t}} + 1) = {{\rm{P}}_{\rm{i}}}\left( {\rm{t}} \right) + \mathop t\limits^. .{\rm{D}}.{{\rm{P}}_{\rm{i}}}\left( {\rm{t}} \right)$$

(33)

$$\:\mathop t\limits^.{\sim} {\rm{n}}\left( {{\rm{0,1}}} \right)$$

(34)

In case of both above-mentioned foraging, the birds fly to’$\:{FS}_{i}^{New}(t+1)$'only if NFR is better in contrast to that of present location $\:\left({P}_{i}\left(t\right)\right)$. The NFR is considered as the objective function of optimization problem $\:\left( {{\text{f}}(.)} \right)$. Hence, locations of birds at iteration’$\:{\text{P}}_{\text{i}}\left(\text{t}+1\right)$'is determined using Eq. (35).'$\:{\text{f}(\text{P}}_{\text{i}}\left(\text{t}\right))$'is NFR at present locations of birds, and’$\:\text{f}\left({\text{F}\text{S}}_{\text{i}}^{\text{N}\text{e}\text{w}}(\text{t}+1)\right)$'is NFR for new FS. Further, foraging behavior is chosen based on’$\mathop r\limits^. \:(\mathop r\limits^. \in \:[0,\:1])$'. If’$\mathop r\limits^. < 0.5$', guided foraging takes place; else, territorial foraging occurs. Furthermore, on determining FS, VT is updated. The level of selected source in VT is set to 1, and other levels of FSs are incremented by 1.

$$\:{\text{P}}_{\text{i}}\left(\text{t}+1\right)=\left\{\begin{array}{c}{\text{P}}_{\text{i}}\left(\text{t}\right),\:{\text{f}(\text{P}}_{\text{i}}\left(\text{t}\right))\le\:f\left({\text{F}\text{S}}_{\text{i}}^{\text{N}\text{e}\text{w}}(\text{t}+1)\right)\:\\\:{\text{F}\text{S}}_{\text{i}}^{\text{N}\text{e}\text{w}}(t+1),\:{\text{f}(\text{P}}_{\text{i}}\left(\text{t}\right))>f\left({\text{F}\text{S}}_{\text{i}}^{\text{N}\text{e}\text{w}}(\text{t}+1)\right)\end{array}\right.$$

(35)

Migration foraging

On examining the possible known and adjacent FSs, final alternate is to migrate. Decision to migrate is taken after identifying the deficiency of food. If the number of iterations reaches the Migration co-efficient$\:\left({M}_{Co-eff}\right)$, migration occurs.'$\:{M}_{Co-eff}$'is taken as $\:2\text{n}$). The locations of hummingbirds are updated to travel to arbitrary region that is same as developing the locations in initial population. Thus, new location of the bird $\:\left({P}_{Wor}(t+1)\right)$ is determined using Eq. (36).'$\:Wor$'represents the worst situation which makes the birds to migrate from their current positions. Once the source of food is reached, the table is updated.

$$\:{{\rm{P}}_{{\rm{Wor}}}}\left( {{\rm{t}} + 1} \right) = {\rm{lb}} + \mathop r\limits^. ({\rm{ub}} - {\rm{lb}})$$

(36)

Step 4: Termination.

Diverse stopping criteria are used for terminating the iterative optimization algorithm like defining maximum number of iterations ($\:{\text{I}\text{t}\text{e}\text{r}}_{\text{M}\text{a}\text{x}}$) and value of target objective. In case of AHBOA,'$\:{\text{I}\text{t}\text{e}\text{r}}_{\text{M}\text{a}\text{x}}$'is used as stopping criterion. The 2 previous steps are iteratively repeated until’$\:{\text{I}\text{t}\text{e}\text{r}}_{\text{M}\text{a}\text{x}}$'is reached.

Quantum-based optimization

The fundamental aspects of Quantum Based Optimization (QBO) are provided. Binary number signifies features which are to be chosen (1) or removed (0). Every feature is signified by a quantum bit ($\:\text{Q}-\text{b}\text{i}\text{t}$) denoted by’$\:\text{q}$'. It represents superposition of binary values denoted by a complex number as specified in Eq. (37)

(37)

Where, the Q-bit’s value being either’0‘or’1'is represented as’$\:\vartheta\:$‘and’'correspondingly. Parameter’$\:{\uptheta\:}$'represents angle of’$\:\text{q}$'which is modified using’'.Then determining the updated value of’$\:\text{q}$'plays a major role and it is found by computing’$\:\varDelta\:{\uptheta\:}$'as Eq. (38)

(38)

Where’$\:\text{R}\left(\varDelta\:{\uptheta\:}\right)$'represents Rotational Matrix ($\:RM$) related to a modification in’$\:\varDelta\:{\uptheta\:}$'defined based on Eq. (39)

$$\:\text{R}\text{M}\left(\varDelta\:{\uptheta\:}\right)\:-\:\left[\begin{array}{cc}\text{C}\text{o}\text{s}(\varDelta\:{\uptheta\:})&\:-\text{S}\text{i}\text{n}(\varDelta\:{\uptheta\:})\\\:\text{S}\text{i}\text{n}(\varDelta\:{\uptheta\:})&\:\text{C}\text{o}\text{s}(\varDelta\:{\uptheta\:})\end{array}\right]$$

(39)

Where the value of’$\:\varDelta\:{\uptheta\:}$'is defined based on best solution ($\:{\text{F}\text{S}}_{\text{B}\text{e}\text{s}\text{t}}$).

Quantum-improved QIAHBOA algorithm-based CHs selection

In this section, Quantum-improved QIAHBOA algorithm is used for improving the.

the capability to obtain a balance between phases of exploration and exploitation when searching for a possible solution. The proposed method of feature selection called QIAHBOA divides data into training (70%) as well as testing (30%) sets. The fitness of every individual is determined based on random values. Individual with least fitness is chosen as best agent. The solution is updated through AHBOA exploitation. The individual is updated repeatedly until stop criteria is reached. The dimension of testing set is decreased based on best solution. Further QIAHBOA is evaluated based on several measures. This QIAHBOA approach is implemented in three stages as possible.

First stage

In this stage, agents signifying the population are produced. Every solution includes’$\:{\text{n}}_{\text{f}}$'Q-bits, where’$\:{\text{n}}_{\text{f}}$'is the number of features. Solution’$\:{\text{F}\text{S}}_{\text{i}}$'is expressed using Eq. (40)

$$\:{\text{F}\text{S}}_{\text{i}}=\left[{\text{q}}_{\text{i}}^{1}\left|{\text{q}}_{\text{i}}^{2}\right.\left|\dots\:\right.\left|{\text{q}}_{\text{i}}^{{\text{n}}_{\text{f}}}\right.\right]=\left[{{\uptheta\:}}_{\text{i}}^{1}\left|{{\uptheta\:}}_{\text{i}}^{2}\right.\left|\dots\:\right.\left|{{\uptheta\:}}_{\text{i}}^{{\text{n}}_{\text{f}}}\right.\right]$$

(40)

Where’$\:{\text{F}\text{S}}_{\text{i}}$'refers to superposition of feature probabilities which may be selected or ignored with $\:\text{i}\:=\:1,\:2,\:.\:.\:.,\text{n}$.

Second stage

This stage focuses on updating agents until the stop criteria is reached by following a sequence of steps. Initially, binary of every’$\:{\text{F}\text{S}}_{\text{i}}$'is obtained using the ensuing Eq. (41)

(41)

Where,'$\:\text{r}\text{a}\text{n}\in\:[0,\:1]$'is a random value.

In specific, KNN classifier is trained with five neighbours as model hyper-parameters by using training features which represents’1s’in’$\:\text{B}{\text{F}\text{S}}_{\text{i}}^{\text{j}}$'. The fitness value is determined based on Eq. (42)

$$\:\text{F}\text{i}{\text{t}}_{\text{i}}={\uprho\:}\times\:{\upxi\:}+(1-{\uprho\:})\times\:\frac{\text{B}{\text{F}\text{S}}_{\text{i}}^{\text{j}}}{{\text{n}}_{\text{f}}}$$

(42)

In Eq. (42),'$\:\text{B}{\text{F}\text{S}}_{\text{i}}^{\text{j}}$'represents the number of chosen features,'$\:{\upxi\:}$'shows the error in classification observed when KNN classifier is employed.'$\:{\uprho\:}\in\:[0,\:1]\:$'balances fitness of both sections. KNN is efficient, simple and involves only one parameter. Moreover, it offers improved performance in contrast to other classifiers as it saves data from training set. The best agent ($\:{\text{F}\text{S}}_{\text{B}\text{e}\text{s}\text{t}}$) with greatest’$\:{\text{F}\text{i}\text{t}}_{\text{B}\text{e}\text{s}\text{t}}$'is determined.

Third stage

The size of testing set is dropped by choosing only features which correspond to’1s’in binary form of’$\:{\text{F}\text{S}}_{\text{B}\text{e}\text{s}\text{t}}$'. Trained KNN classifier is applied to the reduced testing set and output is determined. The quality of the output is assessed based on diverse indicators. Steps of algorithm 1 are shown below.

In the forthcoming algorithm, the steps involved in the process of QIAHBOA-based multi-objective clustering and CHs selection is presented.

Algorithm 1

QIAHBOA-based multi-objective clustering and CHs selection.

In addition, the flowchart depicting the implementation details of the proposed QIAHBOA-based multi-objective clustering and CHs selection is presented in Fig. 4.

Fig. 4

Flowchart QIAHBOA-based multi-objective clustering and CHs selection.

In addition, the detailed implementation of the proposed EEQIAHBOA mechanism with its included steps is presented in the forthcoming section.

Proposed EEQIAHBOA mechanism implementation

Proposed EEQIAHBOA mechanism functions in 2 phases which include setup and steady stages. Heterogeneous nodes with normal, intermediate and advanced energy levels are arbitrarily deployed. The locations of nodes are modified by employing proposed EEQIAHBOA algorithm from second iteration based on AHBOA combined with quantum, principles, and enhanced using cognitive portion and natural selection approach. Decision maker is accountable for choosing the configuration to be modified in subsequent steps. The sink or BS is positioned at centre of the network. In addition, CHs are chosen after setup phase to establish clusters such that CHs are assigned to appropriate member nodes based on fitness function assessed during algorithm implementation. In particular, SGOA and EWOA are integrated to support CH selection. In the steady-state phase, in every iteration, the algorithm runs for particular number of rounds. In this phase, distance threshold is compared with distance between nodes and sink for enabling communication. This phase is vital for making decision which aids in determining whether communication takes place in a single or multiple hops. The CHs play dominant role in processing and forwarding data to sink when data is received from members of clusters. In addition, nodes are examined for their intrinsic energy. In case the energy of a node is not more than’0', the nodes are found to be dead. The nodes are considered to be dead as they do not take part in communication.

In the process of reactive decision making, the base station and the IIoT nodes use the real time data from the sensors and the devices which are connected with one another such that data analytics can be achieved for triggering the specified action in the industry. This real-time triggering operation is achieved based on thresholds or predefined rules which helps in attaining the specified objective of industrial operation. This reactive decision making at the base station helps in facilitating rapid responses under uncertain conditions and situations in which the use of industrial resources need to be optimized for reducing the downtime during operation.

Results and discussion

The simulation experiments of the proposed EEQIAHBOA approach and the baseline approaches Such as CMECSOACF, HCEABCCM, HCSMBOARP and ABCSTA are conducted based the communication standard of IEEE 802.15.4. These simulation experiments are conducted over the system with Windows 10 operating system and the software platform that incorporated MATLAB R2021a.The performance assessment of the proposed EEQIAHBOA approach over the benchmarked approaches are conducted using four different scenarios termed Scenario-1, Scenario-2, Scenario-3 and Scenario-4 with the associated details specified in Table 2. These simulation experimental scenarios are considered to be different in terms of CHs ratio, network scale (number of IIoT nodes) and initial energy. In specific, the network dimension considered for simulation experiments is 100’×'100 square meters with the base station position set in the network as (50,150). The size of the population considered with respect to QAHS is 50 with the maximum number of iterations is also set to 50. In addition, the weights associated with the fitness evaluating factors used during the process of CHs selection in the proposed EEQIAHBOA approach with respect to the weight computation method adopted is 0.15, 0.35, 0.15 and 0.35, respectively.

Table 2 Experimental settings of the proposed EEQIAHBOA approach.

Performance assessment of the proposed EEQIAHBOA approach using number of surviving nodes with different rounds

Initially, the performance of the proposed EEQIAHBOA approach and the CMECSOACF, HCEABCCM, HCSMBOARP and ABCSTA methods is assessed using the number of surviving nodes with different rounds considered for implementation. This investigation is mainly to realize the impact of the proposed EEQIAHBOA approach over the benchmarked approaches with respect to network lifetime and energy stability. This is because the number of surviving nodes in the network exemplars how potent is the clustering and CHs selection mechanism is, in extending the energy stability and network lifetime. Figures 5 and 6 depicts the plots of the number of surviving nodes identified in the network during the implementation of the proposed EEQIAHBOA approach and the CMECSOACF, HCEABCCM, HCSMBOARP and ABCSTA methods in the presence of scenario-1 and scenario-2 with different rounds. The surviving nodes realized in the IIoT network during the application of the proposed EEQIAHBOA approach is determined to be comparatively higher even through the energy possessed by the nodes are completely different, The proposed EEQIAHBOA approach used the merits of CRITIC method for determining the actual weight of the factors considered for fitness evaluating during clustering process. This helped the optimal clustering process in maintaining a greater number of surviving nodes as higher weight is provided to the energy consumptions rate and residual energy. At the same time, the baseline CMECSOACF scheme used pareto optimal set weight determination approach, while HCEABCCM used trust factors-based weight estimation approach This weight approaches used by the benchmarked adopted by these CMECSOACF and HCEABCCM approach even through achieved the objective but fall short in sustaining a greater number of surviving nodes in the network. On the other hand, HCSMBOARP and ABCSTA methods maintained a number of surviving nodes which are still lower than the benchmarked CMECSOACF and HCEABCCM approaches since the method of CHs selection and phenomenon of clustering could not maintain a greater number of surviving nodes in the network.

Fig. 5

Number of surviving nodes with different rounds (Scenario-1).

Fig. 6

Number of surviving nodes with different rounds (Scenario-2).

Fig. 7

Number of surviving nodes with different rounds (Scenario-3).

Fig. 8

Number of surviving nodes with different rounds (Scenario-4).

Thus, the proposed EEQIAHBOA approach under scenario-1 confirmed maximized number of Surviving nodes at the maximized level of 16.21%, 18.64%, 20.86% and 21.38%, better than the benchmarked CMECSOACF, HCEABCCM, HCSMBOARP and ABCSTA methods. Moreover, the proposed EEQIAHBOA approach under scenario-1 confirmed maximized number of Surviving nodes at the maximized level of 17.46%, 19.52%, 21.94% and 23.18%, better than the benchmarked schemes.

Further Figs. 7 and 8 demonstrates the results of the proposed EEQIAHBOA approach and the CMECSOACF, HCEABCCM, HCSMBOARP and ABCSTA methods using number of surviving nodes with different rounds under the presence of scenario-3 and scenario-4. The pattern of surviving nodes realized in the network with different rounds during the application of the proposed clustering protocol was always superior since it used the combination of Pearson correlation and correlation density factors into the CRITIC method which helped in identifying the better weight to the fitness evaluation factors. On contrary, the benchmarked approaches such as CMECSOACF and HCEABCCM were traditional in the weight computation process. While HCSMBOARP and ABCSTA used random weights during CHs selection in IIoT environment. Hence the proposed EEQIAHBOA approach under scenario-3 confirmed maximized number of Surviving nodes at the Superior level of 17.18%, 19.44%, 21.18% and 23.76%, better than the benchmarked CMECSOACF, HCEABCCM, HCSMBOARP and ABCSTA methods. Moreover, the proposed EEQIAHBOA approach under scenario-4 confirmed maximized number of Surviving nodes at the Superior degree of 15.64%, 17.86%, `19.54% and 21.68%, better than the benchmarked schemes.

Performance assessment of the proposed EEQIAHBOA approach using rounds (network lifetime) with different energy consumptions percentage

In this section, the performance assessment results of the proposed EEQIAHBOA approach realized in IIoT environment using the evaluation metrics of rounds (network lifetime0 with different percentage of energy consumptions is presented. and the inferences are justified. This performance assessment process is conducted with 200 number of IIoT nodes with the initial energy of 2 joules and distance threshold of 87 m in the network. These performance assessment results presented in Figs. 9 and 10 portrayed that the proposed EEQIAHBOA approach compared to the baseline approaches performed well independent to the percentage of energy consumptions in the IIoT network. This proposed EEQIAHBOA approach ensured a comparatively better number of rounds (network lifetime) since the assignment of weights during the process of CHs selection helped in determining potential IoT nodes as data aggregator in the IIoT network. This inclusion of weight determination strategy helped in identifying the impact of four fitness factors considered for selecting CHs in the network. The use of QIAHBOA played an indispensable role in utilizing the characteristics of quantum computing and enhancing the capabilities of exploration that determined significant optimal solutions in the search space. This inclusion of QIAHBOA helped in enhancing the network lifetime independent to the scenario-1 and 2 considered for investigation. On the other hand, the baseline CMECSOACF performed relatively equivalent to the proposed EEQIAHBOA approach but the imbalance between the exploration and exploitation still needs a scope of improvement. But the proposed EEQIAHBOA approach possessed an extra edge through the inclusion of quantum computing which enhanced the possibility of determining the solutions with maximized diversity. At the same time, HCEABCCM and HCSMBOARP schemes even though handled the problems of optimal clustering, but its performance was identified to be marginal with scalable increase in the number of IIoT nodes. Moreover, the benchmarked ABCSTA approach during the construction of spanning tree incurred more computational complexity which increases maximized degree of overhead during the process of clustering. With respect to scenario-1 and scenario-2, the proposed EEQIAHBOA approach with different percentage of energy consumptions guaranteed increased rounds (network lifetime) to the maximized level of 9.32%, 10.54%, 11.62% and 12.88%, better than the benchmarked approaches used for comparison.

Fig. 9

Rounds (network lifetime achieved with different energy consumption (scenario-1).

Fig. 10

Rounds (network lifetime achieved with different energy consumption (scenario-2).

Fig. 11

Rounds (network lifetime achieved with different energy consumption (scenario-3).

Fig. 12

Rounds (network lifetime achieved with different energy consumption (scenario-4).

Then Figs. 11 and 12 also exhibited the same pattern during the implementation of the proposed EEQIAHBOA approach with different percentage of energy consumptions under scenario-3 and scenario-4. When the IIoT nodes possesses 0.1 joules individually under scenario-3, the proposed EEQIAHBOA approach performed well by establishing a well-balance in the process of optimal clustering and optimal CHs selection. Moreover, similar kind of behaviour was introduced by the proposed EEQIAHBOA approach with scenario 4 in which each of the nodes possessed 0.1 Joules in the network. With respect to scenario-3 and scenario-4, the proposed EEQIAHBOA approach with different percentage of energy consumptions guaranteed increased rounds (network lifetime) to the maximized level of 8.54%, 9.42%, 10.58 and 11,94%, better than the benchmarked approaches used for comparison.

Performance assessment of the proposed EEQIAHBOA approach using mean data transmission delay with different four different scenarios considered for experimentation

In this section, the performance assessment of the proposed EEQIAHBOA approach and the baseline approaches such as CMECSOACF, HCEABCCM, HCSMBOARP and ABCSTA conducted using mean data transmission delay, packet loss rate, residual energy and throughout with different four scenarios of implementation is presented, and the interpretation with reasons are also depicted. Figures 13 and 14 highlighted the plots of mean data transmission delay and packet loss rate incurred by the proposed EEQIAHBOA approach and the baseline approaches such as CMECSOACF, HCEABCCM, HCSMBOARP and ABCSTA techniques with different scenarios of implementation.

Fig. 13

Mean data transmission delay with different scenarios of implementation.

Fig. 14

Packet loss rate (in %) with different scenarios of implementation.

Fig. 15

Residual energy (joules) with different scenarios of implementation.

Fig. 16

Throughput (kbps) with different scenarios of implementation.

In addition, Figs. 15 and 16 exemplars the plots of residual energy and throughput confirmed by the proposed EEQIAHBOA approach and the baseline approaches such as CMECSOACF, HCEABCCM, HCSMBOARP and ABCSTA techniques with different scenarios of implementation. Independent to the scenarios considered for implementation, the proposed EEQIAHBOA approach handled the limitations that could be possibly visualized during the process of nodes’clustering in a more reactive and realistic manner. This potentiality of the proposed multi-objective clustering method is mainly realized since the IIoT network need to take decision in real-time in order to counteract with the problems that could evolve in the implementation environment. On the other hand, CMECSOACF and HCEABCCM were equally capable in attaining optimal number of clusters, faced challenge in handling the issues that are related to the uneven distribution of IIoT nodes in the network. Thus, these approaches could cope up similar to the potentiality of the proposed EEQIAHBOA approach. Further the baseline HCSMBOARP and ABCSTA schemes were determined to focus on the same objective as concentrated by the proposed EEQIAHBOA approach, but the identification of candidate solutions during the process of CHs selection needs phenomenal change with respect to the rate of exploration and exploitation. Thus, the proposed EEQIAHBOA approach with different scenarios of implementation guaranteed higher Sustenance of residual energy to the maximized level of 7.86%, 8.76%, 10.64% and 13.24%, better than the benchmarked approaches used for comparison. In addition, the proposed EEQIAHBOA approach independent of the different scenarios of implementation guaranteed higher throughput of about 8.54%, 9.59%, 11.34% and 12.64%, better than the benchmarked approaches used for comparison.

Performance assessment of the proposed EEQIAHBOA approach with respect to scalability of nodes

In this section, the proposed EEQIAHBOA approach and the baseline approaches are evaluated and compared using packet delivery rate, cluster count, clustering efficiency and communication overhead with different number of nodes in the network, and the results are presented. The result presented in Fig. 17 proved that the proposed EEQIAHBOA approach is potent enough in addressing the issue of clustering Such that optimized clusters are constructed with improved network lifetime. This capability of the proposed methodology synchronized the CHs selection optimally Such that better route between the IIoT nodes are established during the industrial operations. This establishment of routes helped in attaining maximized degree of packet delivery such that the objective of monitoring in the specified targeted region of industry is achieved. Thus packet delivery rate confirmed by the proposed EEQIAHBOA approach is improved by 13.98%, 15.25%, 17.56% and 18.74%, better than the baseline approaches used for comparison. According to Fig. 18, the number of clusters count realized in the network during the application of the proposed EEQIAHBOA approach is comparatively lower than the baseline approaches since it used the factors of energy and distance for clustering such that data reaches the base station with the minimized number of hops in the network. The proposed clustering strategy confirmed better intra-clustering process using a suitable encoding scheme which played a vital role in sustaining maximized degree of energy in the network. However, the baseline CMECSOACF approach used energy efficient clustering, but the number of clusters constructed needs improvement since it comparatively generates a bit a greater number of clusters compared to the proposed method.

Fig. 17

Packet delivery rate with different scalable number of nodes.

Fig. 18

Cluster Count with different scalable number of nodes.

Fig. 19

Clustering Efficiency with different scalable number of nodes.

Fig. 20

Communication Overhead with different scalable number of nodes.

At the same time, HCEABCCM and HCSMBOARP approaches faced struggle in maintaining energy in the network since they were not able to optimize the objective factors in a parallel manner which resulted in more overhead. These limitations of the baseline approaches are addressed by the proposed scheme, and hence the clusters count generated by the proposed EEQIAHBOA approach is improved by 13.98%, 15.25%, 17.56% and 18.74%, better than the baseline approaches used for comparison.

Furthermore Figs. 19 and 20 presents the clustering efficiency and communication overhead achieved by the proposed EEQIAHBOA approach and the baseline approaches with different scalable number of nodes in the network. These results confirmed that the proposed EEQIAHBOA approach used heuristic information factor Such that clustering optimization is attained with significance. On the other hand, number of packets used for data transmission systematically gets increased with the number of nodes in the network. But the proposed method handled the situation in a proactive manner Such that communication overhead is comparatively minimized with better clustering efficiency. Hence the clustering efficiency confirmed by the proposed EEQIAHBOA approach is improved by 12.64%, 14.98%, 16.54% and 18.38%, better than the baseline approaches used for comparison. Moreover, communication overhead is minimized by 14.28%, 16.84%, 18.62% and 20.86%, better than the benchmarked approaches used for comparison.

In addition, Tables 3 and 4 presents the packet delay and normalized energy consumptions incurred by the proposed EEQIAHBOA approach and the baseline approaches with different scalable number of nodes in the network. These results confirmed that the proposed EEQIAHBOA approach used a better encoding strategy which helped in mapping the resources in the IIoT environment such that energy is optimally utilized in the network.

Table 3 Packet delay under different number of nodes in the network.

Table 4 Normalized energy consumptions under different number of nodes.

Time complexity analysis

The time complexity of the proposed EEQIAHBOA approach specifically depends in four important parameters which includes the phase of initialization, number of search agents, maximized number of iterations, and number of factors for heuristic information factor updating. The total number of dimensions related to all the individuals in the population is initialized before the application of the iteration process. In this context, the time incurred for initialization is determined based on $\:O(n\times\:d)$. The adopted QAHA algorithm is applied in three stages, thus the computational complexity for position updating in each phase is determined using $\:O(0.20\times\:{Iter}_{Max}\times\:n\times\:d)$ for stage 1, $\:O(0.35\times\:{Iter}_{Max}\times\:n\times\:d)$ for stage 2,and$\:O(0.45\times\:{Iter}_{Max}\times\:n\times\:d)$ for stage 3, respectively. Then the update strategy achieved in specific dimension is achieved with the complexity of $\:O({Iter}_{Max}\times\:n)$. Then the final time complexity of the proposed EEQIAHBOA approach is determined as $\:O\left(\text{E}\text{E}\text{Q}\text{I}\text{A}\text{H}\text{B}\text{O}\text{A}\right)=O\left(n\times\:d\right)+\left(0.20\times\:{Iter}_{Max}\times\:n\times\:d\right)+O\left(0.35\times\:{Iter}_{Max}\times\:n\times\:d\right)+O\left(0.45\times\:{Iter}_{Max}\times\:n\times\:d\right)+O({Iter}_{Max}\times\:n)$..

Conclusions

The proposed EEQIAHBOA approach achieved better clustering process and determined optimized number of clusters for sustaining maximized degree of energy in the network. It used the heuristic information factors which integrated the objective function using weights such that it is optimized during the clustering and CHs selection process. It used a superior encoding method which selected the optimal route from the CHs to the sink depending on its distance estimated between a specific CHs and the sink node and the communication range existing between two CHs in the network. It optimized the clustering process through the determination of factors related to residual energy, and distance between the cluster member IoT nodes and energy consumptions. It also used the characteristics of CRITIC method for determining the weights of the factors considered for heuristic information updating process. The simulation results of the proposed EEQIAHBOA approach with different percentage of energy consumptions guaranteed increased rounds (network lifetime) to the maximized level of 14.78%, better than the benchmarked approaches used for comparison. In addition, the proposed EEQIAHBOA approach independent of the different scenarios of implementation guaranteed higher residual energy and throughput of about 10.64% and 11.28%, better than the benchmarked approaches used for comparison.