Multi-objective quantum hybrid evolutionary algorithms for enhancing quality-of-service in internet of things

Abstract

In the context of Internet of Things (IoT), optimizing quality of service (QoS) parameters is a critical challenge due to its heterogeneous and resource-constrained nature. This paper proposes a novel quantum-inspired multi-objective optimization algorithm for IoT service management. Traditional multi-objective optimization algorithms often face limitations such as slow convergence and susceptibility to local optima, reducing their effectiveness in complex IoT environments. To address these issues, we introduce a quantum-inspired hybrid algorithm that combines the strengths of Multi-Objective Grey Wolf Optimization Algorithm (MOGWOA) and Multi-Objective Whale Optimization Algorithm (MOWOA), enhanced with quantum principles. This novel integration overcomes the limitations of traditional algorithms by improving convergence speed and avoiding local optima. The hybrid algorithm enhances QoS in IoT applications by achieving superior optimization in terms of energy efficiency, latency reduction, convergence, and coverage cost. The incorporation of quantum-inspired mechanisms, such as quantum position and behavior, strengthens the exploration and exploitation capabilities of the algorithm, enabling faster and more accurate optimization. Extensive simulations and testing demonstrate the proposed method’s superior performance compared to existing algorithms, validating its effectiveness in addressing key IoT challenges.

Introduction

In recent years, the rapid growth of IoT networks has necessitated the development of sophisticated optimization algorithms to efficiently manage the multitude of devices and data traffic. As highlighted by Langley et al.¹, the Internet of Everything (IoE) paradigm emphasizes the importance of smart device management in modern business models. Fang et al.² introduced a dynamic multi-objective evolutionary algorithm tailored to the dynamic nature of IoT environments, showcasing its effectiveness in addressing IoT service management. These studies underscore the critical role hybrid algorithms play in managing the complexities of IoT networks.

Multi-objective optimization algorithms have proven effective in addressing challenges such as impractical regions, local fronts, and generating diverse optimal solutions. However, managing conflicting objectives and the complexity of metaheuristics within this framework remain significant challenges. Unlike traditional single-objective methods, many-objective optimization problems (MaOP) handle numerous conflicting objectives, leveraging Pareto-based techniques to provide decision-makers with optimal solutions that balance trade-offs^3,4,5,6,7. To address these gaps, this research proposes hybrid algorithms enhanced with quantum-inspired techniques to improve effectiveness in tackling complex optimization challenges.

The literature highlights various evolutionary approaches for multi-objective optimization, with selection methods such as mating and environmental selection playing a key role⁸. This work focuses on optimizing the radius configuration to extend IoT network lifetimes, employing an enhanced grey Wolf Optimization (GWO) algorithm combined with a novel energy-harvesting fitness function. Prior studies, such as Deshmukh et al.⁹, demonstrated the potential of quantum entanglement-inspired GWO in diverse applications, while Li et al.¹⁰ applied a chaotic quantum Whale Optimization Algorithm (WOA) to berth-crane allocation, and Alamir et al.¹¹ developed a quantum artificial rabbit optimizer for microgrid energy management. These adaptations, including the Multi-Objective Whale Optimization Algorithm (MOWOA), ensure Pareto-optimal solutions with enhanced diversity and convergence¹². Applications in energy systems, transportation, and design optimization further validate the efficiency of these quantum-inspired approaches in addressing complex IoT challenges^13,14.

Highlights of the problem description

The IoT ecosystem faces significant challenges in managing diverse QoS requirements efficiently, which is critical given the heterogeneous nature of IoT applications^6,7,15,16. These challenges are particularly acute in healthcare, where optimizing services involves balancing objectives such as cost, delay, and sensor energy. This research proposes an approach that surpasses existing methodologies in addressing these objectives¹⁷. Optimizing QoS parameters-including energy consumption, delay, convergence cost, and coverage cost-presents a complex, multi-dimensional challenge. Traditional evolutionary algorithms often struggle with issues like slow convergence and susceptibility to local optima, especially as the number of objectives and decision variables increases. This study addresses the following key problems:

• Slow convergence rates in multi-objective optimization problems for IoT networks^2,11.

• Local optima trapping, where MOWOA and MOGWOA algorithms fail to identify global optimal solutions effectively.

• Scaling challenges in managing the growing number of devices and data within IoT networks, while optimizing multiple conflicting QoS parameters to enhance overall performance.

Highlights of the author’s contributions

This paper introduces a hybrid optimization algorithm combining Multi-Objective Grey Wolf Optimization Algorithm (MOGWOA) and Multi-Objective Whale Optimization Algorithm (MOWOA), enhanced with quantum principles, to optimize QoS in IoT systems. MOGWOA leverages the social hierarchy-based leadership mechanism of grey wolf for superior exploration, while MOWOA employs its bubble-net hunting strategy to excel in exploitation. By integrating these complementary strengths, the hybrid algorithm effectively balances exploration and exploitation, making it well-suited for addressing the multi-dimensional challenges of IoT QoS optimization.

Traditional algorithms such as NSGA-II, MOPSO, and DE often encounter limitations such as slow convergence and susceptibility to local optima, especially in heterogeneous and resource-constrained IoT environments. The proposed hybrid algorithm overcomes these shortcomings by leveraging the advanced capabilities of MOGWOA and MOWOA, further enhanced by quantum-inspired mechanisms. These mechanisms significantly improve the algorithms’ ability to explore diverse solutions and avoid local optima, thereby increasing convergence speed and overall performance.

The proposed solution is designed to maximize QoS while ensuring stable network connectivity. Two validation tests have been conducted: (1) a smart IoT application-based test to evaluate the algorithm’s real-world effectiveness and (2) a detailed assessment of QoS characteristics, including energy efficiency, latency, and service cost^18,19,20. Experimental results confirm the feasibility and superior efficiency of the proposed approach in comparison to traditional methods.

The key contributions of this study are as follows:

• Development and implementation of quantum-enhanced MOGWOA and MOWOA algorithms to improve convergence speed and avoid local optima.

• Introduction of novel hybrid algorithms that merge the strengths of MOGWOA and MOWOA using quantum-inspired techniques, optimizing QoS parameters in IoT networks.

• Application of the hybrid algorithms to optimize multiple QoS parameters, such as energy consumption, delay, convergence cost, coverage cost, and fitness cost, showing significant performance improvements.

• Experimental evaluations using four-objective fitness functions and optimized service costs, benchmarking the proposed algorithms against standard algorithms to validate their effectiveness.

• Analysis of the Pareto front for 2-objective and 3-objective scenarios in IoT applications, providing insights into the performance and trade-offs of the proposed algorithms.

Article organization

The article is organized as follows: Section II provides a literature review of multi-objective evolutionary algorithms in the IoT context. Section III describes the IoT-based performance metric framework. Section IV introduces the proposed quantum-inspired hybrid method for IoT service optimization. Section V presents the experimental results and analysis, evaluating the performance of the proposed method. Finally, Section VI concludes the study and outlines future research directions.

Literature review

This literature review explores advancements in hybrid quantum-based multi-objective optimization algorithms and their applications in IoT networks, focusing on existing methods and their limitations.

Liang et al.⁸ introduced an evolutionary many-task optimization approach leveraging multisource knowledge transfer, significantly improving task optimization. Similarly, Mirjalili et al.²¹ proposed the foundational MOGWO, which has inspired further research in multi-objective optimization. Dev et al.³ combined Rider and Grey Wolf optimization to enhance IoT network lifetimes, demonstrating practical applications of hybrid optimization techniques.

Further contributions include Yue et al.⁴ and Tawhid and Ibrahim⁵, who explored hybrid algorithms, enhancing the robust framework of multi-objective optimization. Mirjalili and Lewis introduced the WOA²¹, inspired by the bubble-net hunting behavior of humpback whales. Later, the MOWOA adapted WOA to multi-objective problems by integrating non-dominated sorting and crowding distance techniques for handling multiple objectives²².

Quantum-inspired methods for optimization have gained prominence. Zheng and Chai²³ and Chai et al.¹⁵ advanced reference-point-based non-dominated sorting approaches for multitasking optimization, while Ran et al.²⁴ introduced a many-objective evolutionary algorithm leveraging heuristic search techniques. Jin et al.²² applied hybrid Wolf optimization to control strategies in electric motors, and Elsedimy et al.²⁵ developed a hybrid quantum support vector machine for intrusion detection systems, showcasing their versatility in cybersecurity.

Wang et al.²⁶ used a hybrid grey Wolf optimizer for hyperspectral image band selection. Ghorpade et al.⁶ applied quantum PSO in in Heterogeneous Industrial IoT to optimize configuration, while El-Shorbagy et al.²⁷ tackled dynamic wireless sensor network optimization using a novel PSO algorithm. Recent innovations include Xie et al.’s²⁸ dynamic transfer reference point-oriented MOEA/D, Huang et al.’s²⁹ whale optimization for mobile edge computing, and Gu et al.’s³⁰ improved NSGA-III algorithm.

Quantum-inspired approaches have also been applied in specific domains. Olvera et al.³¹ explored quantum evolutionary algorithms in continuous spaces, while Bilal et al.³² applied a quantum-enhanced grey Wolf optimizer for breast cancer diagnosis. Jain and Sharma¹⁷ introduced a hybrid SSA-GWO algorithm for cloud computing, and Elaziz et al.¹⁸ proposed a quantum artificial hummingbird algorithm for feature selection in social IoT.

Table 1 provides a summary of prior methods, their limitations, and the ways in which the proposed method addresses these gaps.

Table 1 Comparison of Existing Works

Despite these advancements, several limitations remain:

• Many existing algorithms struggle to maintain scalability and efficiency in high-dimensional IoT networks with conflicting objectives.

• Most algorithms lack a robust mechanism to balance exploration and exploitation, leading to convergence issues and local optima traps.

• Existing methods fail to comprehensively address the heterogeneity of IoT devices, including variations in energy, latency, and cost.

• While quantum-inspired algorithms have shown promise, their application in IoT optimization remains underexplored.

Background, system model and framework for QoS in IoT

Multiobjective optimization (MOP)

IoT service management can be modelled as a non-linear equation system because the various QoS parameters exhibit a non-linear relationship with each other. Optimization problems in IoT service management, such as resource allocation, routing, and load balancing often lead to non-linear constraints and objective functions. To locate optimal solution of Non-linear equations can be solved by Evolutionary Algorithms (EA) which consist of two steps. First, the transformation of non-linear into optimization problems and then in the second step optimization problem is solved using any optimization algorithms. These optimization algorithms can be categorised into single objective, constrained objectives, and multi-objective optimization. The MOP can be explained as follows³⁷:

$$min/max f(X) = \{f_1(X),f_2(X),.....f_M(x)\}$$

Where, $X=(x_1,x_2.....x_D)\in S$ represents the decision vector, D decision variables, $S\subseteq R^D$ decision space, and M is the number of Objective functions. In MOP, if there are two candidate solutions then they are compared by Pareto dominance. For example, suppose $X_1$ and $X_2$ are two candidate solutions, we can say $X_1$ is Pareto dominance to $X_2$ if:

• For every objective $i$ (where $i \in \{1, 2, \dots , M\}$), the objective function value of $X_1$, $f_i(X_1)$, is less than or equal to the $f_i(X_2)$:

  $$f_i(X_1) \le f_i(X_2) \quad \forall i \in \{1, 2, \dots , M\}$$

• There exists at least one objective $j$ (where $j \in \{1, 2, \dots , M\}$) for which $f_j(X_1)$ is strictly less than $f_j(X_2$:

  $$\exists j \in \{1, 2, \dots , M\} \text { such that } f_j(X_1) < f_j(X_2)$$

$X_1$ Pareto dominates $X_2$ if it is no worse in all objectives and strictly better in at least one objective. A candidate solution $X$ is termed a Pareto optimal solution if no other solution in the decision space Pareto dominates $X$.

The collection of all Pareto optimal solutions is called the Pareto set (PS), and it can be represented as:

$$\text {PS} = \{ X \mid X \text { is Pareto optimal} \}$$

The set containing the objective values corresponding to each solution in the Pareto set, denoted as $f(X)$ for $X \in \text {PS}$, is known as the Pareto front (PF):

$$\text {PF} = \{ f(X) \mid X \in \text {PS} \}$$

The MOGWOA and MOWOA are nature-inspired metaheuristic algorithms tailored to address multi-objective optimization problems. Both algorithms are known for their ability to handle trade-offs between objectives in multi-objective problems, making them popular choices for evolutionary multi-objective optimization.

Quantum computation

Quantum computing on the principles of quantum mechanics, offering unique advantages for complex optimization tasks. Unlike traditional computing, quantum computing leverages quantum bits (qubits)³⁸. Due to the phenomenon of superposition, qubits can exist in multiple states simultaneously, allowing quantum computers to perform certain operations more efficiently than classical computers, particularly in multi-objective optimization for IoT QoS.

A qubit’s state can be represented within a two-dimensional complex vector space called the Bloch sphere, which includes the basis states $|0\rangle$ and $|1\rangle$. Through superposition, a qubit can exist in a combination of these states, expressed as $|\Psi \rangle$:

$$|\Psi \rangle = \alpha |0\rangle + \beta |1\rangle$$

where $\alpha$ and $\beta$ are complex numbers representing the probability amplitudes of states $|0\rangle$ and $|1\rangle$, respectively. During measurement, the qubit collapses to either $|0\rangle$ or $|1\rangle$, with probabilities $|\alpha |^2$ and $|\beta |^2$. These amplitudes satisfy the normalization condition:

$$|\alpha |^2 + |\beta |^2 = 1$$

Hence, a qubit can be represented as a vector:

$$q = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}$$

In a similar manner, a multi-qubit system with $n$ qubits can represent $2^n$ states simultaneously. Such a system is represented as:

$$Q = \begin{bmatrix} q_1, q_2, \dots , q_n \end{bmatrix} = \begin{bmatrix} \alpha _1 & \alpha _2 & \dots & \alpha _n \\ \beta _1 & \beta _2 & \dots & \beta _n \end{bmatrix}$$

For example, a 2-qubit system can be represented by probability amplitudes as follows:

$$Q = \begin{bmatrix} \sqrt{\frac{1}{2}} & -\sqrt{\frac{3}{3}} \\ \sqrt{-\frac{1}{2}} & \sqrt{\frac{6}{3}} \end{bmatrix}$$

This system represents $2^2 = 4$ states, such as $|00\rangle , |01\rangle , |10\rangle ,$ and $|11\rangle$. The probability of observing $|00\rangle$ is:

$$\left( \sqrt{\frac{1}{2}} \right) ^2 \times \left( -\sqrt{\frac{3}{3}} \right) ^2 = \frac{1}{6}$$

Similarly, the probability of observing $|01\rangle$ is:

$$\left( \sqrt{\frac{1}{2}} \right) ^2 \times \left( \sqrt{\frac{6}{3}} \right) ^2 = \frac{1}{3}$$

Quantum gates and state manipulation for IoT optimization

Quantum gates, such as the NOT-gate, CNOT-gate, and Hadamard-gate, manipulate qubits by modifying their probability amplitudes, which is essential for quantum-inspired optimization algorithms. Quantum gates update the state of a qubit $q = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}$ to a new state $q' = \begin{bmatrix} \alpha ' \\ \beta ' \end{bmatrix}$ while maintaining normalization:

$$|\alpha '|^2 + |\beta '|^2 = 1$$

The quantum angle for a qubit $q_i$ can be determined as:

$$\theta _i = \arctan \left( \frac{\beta _i}{\alpha _i} \right)$$

Quantum population

The quantum population refers to the set of candidate solutions that are influenced by quantum-inspired updates during the optimization process. These solutions are updated based on quantum principles such as quantum superposition, allowing for a wider exploration of the solution space compared to traditional methods.

Quantum behavior coefficient

The quantum behavior coefficient is a parameter that governs the degree to which quantum-inspired behavior (such as quantum superposition and entanglement) influences the optimization process. It helps in balancing exploration and exploitation within the algorithm by probabilistically guiding the search toward promising regions in the solution space.

These quantum principles enable quantum-inspired algorithms to handle complex, multi-objective optimization problems, providing efficient exploration and convergence towards optimal solutions in IoT applications. This makes quantum-inspired methods highly effective in balancing diverse IoT QoS requirements, such as minimizing latency and energy consumption.

System model

The proposed framework shown in Figure 1 provides a comprehensive approach to evaluating and optimizing the performance of IoT applications using a quantum-based approach. The proposed IoT framework for multi-objective optimization integrates several key components to enhance system performance, incorporating quantum-inspired hybrid evolutionary algorithms. At the core is the Optimization Engine, which employs these advanced multi-objective optimization algorithms and objective functions to find optimal solutions. On the left, there are IoT devices, including sensors and actuators that collect data and execute actions. The Communication Infrastructure at the top handles data transmission and network connectivity. On the right, the Decision Support System provides decision-making tools and a user interface to interpret optimization results. The Monitoring and Feedback System collects and processes performance data, providing continuous feedback to the Optimization Engine. Data flows from IoT Devices to the Monitoring and Feedback System, which then processes the data and sends it to the Optimization Engine. Leveraging quantum principles, the engine processes this information, generates optimized configurations, and sends them back to the IoT Devices. The Decision Support System assists users in making decisions based on the optimization outcomes. This structure ensures a balanced approach to managing and optimizing multiple objectives in IoT systems, including energy efficiency, latency, and resource utilization, while avoiding local optima and ensuring faster convergence.

Key performance metrics considered include latency, reliability, throughput, energy consumption cost, delay cost, convergence cost, and coverage rate^19,20.

Fig. 1

IoT framework for multi-objective optimization.

Latency

Latency ($L(t)$) is the total time taken from data collection to decision-making. It can be decomposed into the following components.

$$\begin{aligned} L(t) = P(t) + T_d + D_m(t) \end{aligned}$$

(1)

where,

• $P(t)$ Processing time, potentially enhanced by quantum parallelism. It is define as time required for data processing at the IoT device or server. It is modeled as:

  $$P(t) = \frac{\text {Computational Load}}{\text {Processing Capacity}}$$

  where: - Computational Load is the number of operations required to process the data, - Processing Capacity is the computational power of the device or server in operations per second.

• $T_d$ Transmission delay: Transmission delay is the time taken for data to travel from the source to the destination. It is calculated as:

  $$T_d = \frac{\text {Data Size}}{\text {Bandwidth}} + \text {Propagation Delay}$$

  where: - Data Size is the size of the transmitted data in bits, - Bandwidth is the available network bandwidth in bits per second (bps), - Propagation Delay is the time taken for a signal to travel from the source to the destination.

• $D_m(t)$ is the decision-making delay, which can be reduced using quantum-inspired algorithms primarily through enhanced exploration and exploitation capabilities.

Reliability

Reliability ($R(t)$) refers to the probability that the system performs correctly over a specified period. It is modeled as:

$$\begin{aligned} R(t) = e^{-\lambda t} \end{aligned}$$

(2)

where $\lambda$ is the failure rate. Quantum error correction techniques can improve reliability by reducing the effective failure rate $\lambda$. These techniques protect quantum information from errors caused by decoherence and noise by detecting and correcting errors without collapsing the quantum state. By lowering the probability of errors, quantum error correction enhances the system’s fault tolerance and increases its reliability $R(t)$ over time, ensuring more dependable operation in IoT networks.

Throughput

Throughput ($\Theta$) is the rate at which data is successfully processed and transmitted. It is given by:

$$\begin{aligned} \Theta = \frac{N(t)}{L(t)} \end{aligned}$$

(3)

where $N(t)$ is the amount of data processed at time $t$. Quantum-inspired algorithms running on systems can increase $N(t)$ by enhancing the exploration and exploitation of the search space, using probabilistic mechanisms, and avoiding local optima. These improvements lead to faster data processing, resulting in higher throughput $\Theta$.

Energy consumption cost

Energy consumption cost ($E_c(t)$) is crucial due to the limited battery life of IoT devices. It is expressed as:

$$\begin{aligned} E_c(t) = P(t) \cdot P_e + T_d \cdot T_e \end{aligned}$$

(4)

where,

• $P_e$ is the power consumption rate during data processing.

• $T_e$ is the power consumption rate during data transmission.

Quantum-inspired algorithms can potentially reduce $P_e$ by optimizing computational processes, leading to more efficient use of energy during data processing.

Delay cost

Delay cost ($C_d(t)$) quantifies the penalty associated with time delays in data processing and transmission:

$$\begin{aligned} C_d(t) = \gamma L(t) \end{aligned}$$

(5)

where $\gamma$ is a weighting factor representing the importance of minimizing delay. Quantum-inspired algorithms can reduce $L(t)$ by accelerating data processing and optimizing transmission paths, thereby lowering the overall delay cost.

Convergence cost

Convergence cost ($C_c$) is related to the algorithm’s ability to reach an optimal solution efficiently. Quantum-inspired algorithms can significantly reduce this cost:

$$\begin{aligned} C_c = \delta \cdot N \end{aligned}$$

(6)

where:

• $\delta$ is a constant representing the computational cost per iteration.

• $N$ is the number of iterations, which can be reduced using quantum-inspired optimization techniques that enhance exploration and exploitation, utilize probabilistic jumps to escape local optima, and maintain population diversity. By converging faster to the global optimum, the overall computational cost per iteration is reduced, thereby lowering the total convergence cost $C_c$.

Coverage rate

Coverage rate ($C_r(t)$) is essential for ensuring comprehensive monitoring of the IoT system. It is defined as.

$$\begin{aligned} C_r(t) = \frac{\text {Area Covered at time } t}{\text {Total Area}} \end{aligned}$$

(7)

Multiobjective optimization-based fitness function

To optimize the QoS in IoT applications, we define a multiobjective fitness function based on the aforementioned metrics. Quantum-inspired optimization algorithms can enhance the efficiency of solving this multiobjective problem by leveraging quantum principles such as superposition and entanglement. These algorithms maintain a diverse set of potential solutions, improving the balance between exploration and exploitation. This approach allows for a more thorough search of the solution space, avoiding local optima and converging more quickly to optimal solutions. As a result, the overall efficiency in solving complex multiobjective optimization problems is significantly improved, leading to better QoS performance in IoT networks.

Objective 1: minimize latency and delay cost

$$\begin{aligned} f_1(t) = L(t) + C_d(t) \end{aligned}$$

(8)

Objective 2: minimize energy consumption cost

$$\begin{aligned} f_2(t) = E_c(t) \end{aligned}$$

(9)

Objective 3: minimize convergence cost

$$\begin{aligned} f_3 = C_c \end{aligned}$$

(10)

Objective 4: maximize coverage rate

$$\begin{aligned} f_4(t) = -C_r(t) \end{aligned}$$

(11)

Quantum-inspired multiobjective fitness function

The overall fitness function $F_q(t)$ incorporates quantum techniques for enhanced optimization. It combines the objectives using respective weights $w_1, w_2, w_3,$ and $w_4$:

$$\begin{aligned} F_q(t) = w_1 \cdot f_1(t) + w_2 \cdot f_2(t) + w_3 \cdot f_3 + w_4 \cdot f_4(t) \end{aligned}$$

(12)

where:

• $w_1, w_2, w_3,$ and $w_4$ are weights reflecting the relative importance of each objective.

• Quantum parallelism and entanglement are used to explore multiple solutions simultaneously, reducing convergence time.

• Quantum error correction enhances reliability.

• Quantum optimization algorithms, such as MOPSO, MOWOA, etc., improve the efficiency of finding optimal solutions.

The weighted sum method in Eq. (12) serves to construct a comprehensive fitness function, $F_q(t)$, that combines multiple objectives into a single scalar value for enhanced optimization. This step simplifies the comparative evaluation of solutions by reflecting the relative importance of different objectives through their respective weights $w_1, w_2, w_3, w_4$. This approach allows us to integrate quantum-inspired mechanisms, such as quantum parallelism and entanglement, into the optimization process to enhance convergence and exploration capabilities.

While the weighted sum method consolidates objectives, the optimization problem inherently remains multi-objective due to the conflicting nature of the individual objectives (e.g., energy efficiency, latency, and coverage cost). MOGWOA and MOWOA are specifically designed to handle such conflicting objectives by balancing exploration and exploitation during the search process. Their use ensures that the algorithm effectively explores the Pareto front of potential solutions before finalizing the single-objective representation via the weighted sum method.

The adoption of quantum techniques within MOGWOA and MOWOA enhances their multi-objective optimization capabilities by enabling simultaneous exploration of multiple solutions (quantum parallelism) and ensuring diversity in the search space (quantum entanglement). These mechanisms significantly reduce convergence time and improve the algorithm’s ability to avoid local optima, making it suitable for solving complex IoT-related optimization problems.

The integration of quantum principles increases computational overhead due to quantum position calculations and state updates. Experiments showed a 15-20% increase in computational cost for the quantum-inspired MOGWOA and MOWOA compared to their conventional versions. Despite this, the improved optimization performance justifies the cost. Scalability tests on IoT networks of varying sizes (100-500 devices) demonstrated a linear increase in computational time with network size, while energy efficiency and QoS improvements scaled proportionally, proving the method’s applicability to large-scale IoT environments.

Table 2 reports the terminologies and notations of the proposed algorithm.

Table 2 Terminologies and Notations

Proposed methodology

In this section, we designed the hybrid methodology, which incorporates IoT-based applications. The proposed methodology has three algorithms: the first algorithm is the MOWOA, the second is the MOGWOA, the third is the hybrid algorithm with IoT-based QoS. Here is a more detailed description:

Multi-objective whale optimization algorithm (MOWOA) using quantum approach

The MOWOA using an Artificial Based Quantum approach is designed to optimize multiple objectives simultaneously. The detailed description is shown in Algorithm 1.

Initially³⁹, the population of whales $W$ is initialized with random positions $X_i$ for $i = 1, 2, \ldots , n$. A quantum population $Q$ is also initialized. The algorithm defines maximum iterations $MaxIter$ and fitness functions $f_1, f_2, \ldots , f_k$. The iteration counter $t$ is set to 0. In each iteration (while $t < MaxIter$), the fitness values $f_j(X_i)$ for each whale $X_i$ and for each objective $j = 1, 2, \ldots , k$ are calculated. The position of each whale $X_i$ is then updated based on a random number $r$ generated from a uniform distribution between 0 and 1. If $r < 0.7$, the position is updated using (13):

$$\begin{aligned} X_i(t+1) = X_i(t) + \alpha \cdot (Q_i(t) - X_i(t)) \end{aligned}$$

(13)

as shown (13), where $\alpha$ is a control parameter. The quantum position $Q_i(t)$ is updated using a quantum delta potential well:

$$\begin{aligned} Q_i(t+1) = Q_i(t) + \beta \cdot (X_i(t) - Q_i(t)) \end{aligned}$$

(14)

as shown in (14), where $\beta$ is a scaling factor. If $r \ge 0.7$, the position is updated using conventional WOA steps. If $r < 0.7$ within this branch, the position is updated using the encircling prey mechanism:

$$\begin{aligned} X_i(t+1) = X^* - A \cdot L \end{aligned}$$

(15)

where

$$\begin{aligned} L = |C \cdot X^* - X_i| \end{aligned}$$

(16)

where $L = |C \cdot X^* - X_i|$ as shown in (15) and (16). Here, $X^*$ is the best solution found so far. The coefficient vectors $A$ and $C$ are calculated as:

$$\begin{aligned} A= & 2a \cdot r_1 - a \end{aligned}$$

(17)

$$\begin{aligned} C= & 2 \cdot r_2 \end{aligned}$$

(18)

as shown in equations (17) and (18), where $a$ decreases linearly from 2 to 0 over the course of iterations, and $r_1$ and $r_2$ are random vectors in $[0, 1]$. Otherwise, the position is updated using the bubble-net attacking mechanism:

$$\begin{aligned} X_i(t+1) = L' \cdot e^{bl} \cdot \cos (2\pi l) + X^* \end{aligned}$$

(19)

where

$$\begin{aligned} L' = |X^* - X_i| \end{aligned}$$

(20)

as shown in (19) and (20). $b$ is a constant defining the shape of the logarithmic spiral, and $l$ is a random number in $[-1, 1]$. The iteration counter $t$ is then incremented by 1. This process repeats until the maximum number of iterations $MaxIter$ is reached. The algorithm then returns the best solutions found.

Algorithm 1

MOWOA using artificial based quantum approach.

The multi-objective grey wolf optimization algorithm (MOGWOA)

The MOGWOA using an Artificial Based Quantum Approach is designed to optimize multiple objectives $f_1, f_2, \ldots , f_k$ simultaneously. Initially, a population of grey wolf $W$ is initialized with random positions $X_i$ for $i = 1, 2, \ldots , n$, and a quantum population $Q$ is also initialized. The algorithm¹⁷ operates for a maximum number of iterations $\textit{MaxIter}$, where it iteratively improves the solutions found. Each grey wolf $X_i$ computes its fitness values $f_j(X_i)$ for each objective $j$. The top three solutions are identified as $\alpha , \beta ,$ and $\delta$, representing the alpha, beta, and delta wolf, respectively, which are updated dynamically throughout the iterations. During each iteration, grey wolf update their positions based on a random selection between quantum behavior and conventional grey Wolf Optimization (GWO) steps. For quantum behavior, if a random number $r < 0.7$, the position update is expressed by:

$$\begin{aligned} X_i(t+1) = X_i(t) + \alpha _q \cdot (Q_i(t) - X_i(t)), \end{aligned}$$

(21)

where $\alpha _q$ is a control parameter and $Q_i(t)$ is the quantum position of grey wolf $i$ at time $t$. The quantum population $Q$ is updated using:

$$\begin{aligned} Q_i(t+1) = Q_i(t) + \beta _q \cdot (X_i(t) - Q_i(t)), \end{aligned}$$

(22)

where $\beta _q$ is a scaling factor.

Alternatively, if $r \ge 0.7$, grey wolf update their positions using conventional GWO steps. Each grey wolf $X_i$ calculates distances $L_{i,\alpha }, L_{i,\beta },$ and $L_{i,\delta }$ to the alpha, beta, and delta wolf, as shown in Algorithm 2, where $C_1, C_2,$ and $C_3$ are coefficient vectors. The positions are updated as shown in Algorithm 2, where $A_1, A_2,$ and $A_3$ are coefficient vectors. Finally, the grey wolf updates its position as the average of these three positions:

$$\begin{aligned} X_i(t+1) = \frac{X_{i,1} + X_{i,2} + X_{i,3}}{3}. \end{aligned}$$

(23)

The algorithm iterates until $\textit{MaxIter}$ is reached, returning the best solutions found. This approach effectively combines quantum-inspired mechanisms with traditional grey Wolf Optimization, enhancing the algorithm’s ability to find optimal solutions across multiple objectives.

Algorithm 2

Multi-objective grey wolf optimization algorithm (MOGWOA) using artificial based quantum approach.

The proposed hybrid methodology

The hybrid approach combines the strengths of MOWOA and MOGWOA by leveraging the quantum-inspired mechanisms in both algorithms. During each iteration, the hybrid algorithm dynamically chooses between quantum-enhanced updates and conventional updates based on a probabilistic selection criterion. This approach aims to enhance the exploration-exploitation trade-off and improve the convergence speed towards optimal solutions across multiple objectives. The proposed Algorithm 3, integrates the MOWOA Algorithm and the MOGWOA Algorithm using an Artificial Based Quantum Approach. Initially, populations of whales $W$ and grey wolf $G$ are initialized with random positions $X_i$ and $Y_i$ for $i = 1, 2, \ldots , n$, respectively. Quantum populations $Q_W$ and $Q_G$ are also initialized for whales and grey wolf, enhancing their exploration-exploitation capabilities. The algorithm operates for a specified number of iterations $\textit{MaxIter}$, during which fitness values $f_j(X_i)$ and $f_j(Y_i)$ for each objective $j$ are evaluated. The top solutions $\alpha _W, \beta _W, \delta _W$ for whales and $\alpha _G, \beta _G, \delta _G$ for grey wolf are identified and updated dynamically throughout the iterations. Each iteration involves a probabilistic selection mechanism where whales and grey wolf update their positions based on quantum-inspired behavior optimization steps. For quantum behavior, a random number $r_W$ or $r_G$ determines whether the quantum-enhanced update or conventional update is applied. Quantum updates for whales and grey wolf are governed by equations similar to (14) and (22), respectively. Conventional updates follow the WOA and GWO methodologies, adjusting positions based on distance calculations to top solutions.

The hybrid approach aims to exploit the complementary strengths of MOWOA and MOGWOA, leveraging quantum-inspired mechanisms to enhance exploration and convergence speed across multiple objectives. Algorithm 3 encapsulates these principles, integrating MOWOA’s quantum-enhanced exploration with MOGWOA’s robust optimization capabilities, thereby improving the overall efficiency and effectiveness of multi-objective optimization tasks.

Algorithm 3

Hybrid algorithm: MOWOA and MOGWOA with IoT-based QoS.

Criteria for switching between quantum and conventional updates:

The decision to use quantum or conventional updates is guided by a threshold probability, $r_W$ for whales and $r_G$ for gray wolves, where $r_W, r_G \in [0,1]$.

If $r_W < 0.7$ for whales or $r_G < 0.7$ for gray wolves, quantum behavior is applied. This threshold ensures a balanced exploration and exploitation process by probabilistically alternating between quantum-inspired updates and conventional algorithmic steps.

These probabilities were chosen based on empirical studies, ensuring an optimal trade-off between enhanced exploration (via quantum behavior) and refinement of solutions (via conventional steps).

The proposed hybrid algorithm with IoT-based QoS

The hybrid algorithm, as shown in Algorithm 3, integrates the MOWOA and the MOGWOA using an Artificial Based Quantum Approach. Initially, populations of whales $W$ and grey wolf $G$ are initialized with random positions $X_i$ and $Y_i$ for $i = 1, 2, \ldots , n$, respectively. Quantum populations $Q_W$ and $Q_G$ are also initialized for whales and grey wolf, enhancing their exploration-exploitation capabilities. The algorithm operates for a specified number of iterations $\textit{MaxIter}$, during which fitness values $f_j(X_i)$ and $f_j(Y_i)$ for each objective $j$ are evaluated. These objectives include IoT-specific QoS metrics such as latency $L(t)$, reliability $R(t)$, throughput $\Theta$, energy consumption $E_c(t)$, and delay cost $C_d(t)$. For each whale and grey wolf, a random number determines whether the position will be updated using quantum behavior optimization steps from WOA and GWO, respectively. Quantum updates involve adjusting the position based on a quantum factor, which helps in exploring the search space more effectively. After updating positions, the algorithm evaluates the Pareto dominance of the new solutions. Non-dominated solutions (i.e., those that are not outperformed by any other solution in all objectives) are added to the Pareto front set $P$, while dominated solutions are removed. This ensures that $P$ always contains the best trade-off solutions found so far. Finally, the algorithm returns the Pareto front set $P$, representing the optimal trade-offs between the different objectives, thereby providing a set of solutions that balance energy consumption and delay cost-effectively. The proposed algorithm  3 improves further the efficiency and effectiveness of multi-objective optimization tasks in IoT environments.

Result and analysis

This section evaluates the proposed approach within IoT framework scenarios focusing on QoS. The methodology is applied to IoT service scenarios, and its performance is compared with established algorithms such as MOEA-D²⁸, NSGA-III⁴⁰, MOPSO⁴¹, and MOWOA⁴². The evaluation considers key metrics like energy consumption, delay, coverage rate, and service cost. The MOWOA and NSGA-III are selected as baseline algorithms due to their strong relevance in multi-objective optimization for IoT applications. NSGA-III is a widely recognized Pareto-based evolutionary algorithm suited for high-dimensional optimization, while MOWOA has demonstrated effectiveness in handling constrained and unconstrained multi-objective problems. Other hybrid or quantum-enhanced approaches were not considered due to their primary focus on different domains or the lack of standardized benchmarks for IoT optimization.

Comparative analysis is conducted using different generation sizes and population testing to ensure a comprehensive performance evaluation. The following outlines the step-by-step result analysis for the newly developed proposed hybrid algorithm:

• 1. Result analysis of overall fitness cost of QoS optimization from IoT Networks.

• 2. Test evaluations generate solutions randomly within the search space for each individual in the population $P$, then evaluate all the objectives to minimize both the energy consumption cost and the delay cost, while maximizing the coverage rate.

• 3. Result analysis of Multi-objective optimization algorithms for Pareto front performance analysis from IoT Applications.

Experimental setup

The simulations were implemented in Python 3.11, utilizing libraries such as NumPy (v1.24), SciPy (v1.10), and Matplotlib (v3.7) for numerical computations, optimization, and visualization. The experiments were conducted on a machine with an Intel Core i7-12700H CPU, 16GB RAM, and running Ubuntu 22.04. This setup ensured efficient computation and reproducibility of the results.

We establish an IoT framework of size $100 \times 100$, distributing 125 sensors equitably as service requests within this experimental setup as shown in figure 2. The grid size of 100$\times$100 was selected to simulate a mid-sized urban or smart-city environment, which is common in IoT applications such as smart traffic management, disaster monitoring, and energy distribution. This grid size offers a balance between computational feasibility and sufficient complexity to evaluate the performance of the proposed hybrid algorithm. Similarly, the deployment of 125 sensors reflects a realistic IoT network density, commonly seen in smart-city applications where devices are distributed to monitor environmental parameters or support IoT-enabled infrastructure. This number was carefully chosen to maintain diversity in sensor placements while ensuring manageable computational loads during the simulation. The control parameters for the hybrid algorithm were also rigorously defined and tuned. The population size was set to 50, ensuring sufficient diversity while maintaining computational efficiency. The maximum iterations (750) were chosen based on experimental evaluations to allow the algorithm to converge to high-quality solutions without excessive runtime. Quantum parameters, including $\alpha$q = 0.5 and $\beta$q = 0.3, were selected based on prior studies in quantum-inspired optimization and fine-tuned to enhance the balance between exploration and exploitation. Other parameters, such as the crossover rate (0.8) and mutation rate (0.2), were set to commonly used values in multi-objective optimization literature, ensuring adequate variation across generations. In terms of IoT-based QoS metrics, the simulation parameters were grounded in realistic ranges observed in IoT systems. Transmission delay was modeled within 10-100 milliseconds, reflecting real-world network latencies, while processing time ranged between 50-200 milliseconds to capture computational delays in resource-constrained IoT devices. Energy consumption cost was set between 0.5-5.0 joules per operation, representing typical energy usage for battery-powered IoT devices. The delay cost was designed as a function of both transmission delay and processing time, aligning with real-world service-level agreements in IoT. We set the control parameter of the proposed algorithm with other standard evolutionary algorithms as shown in Table 3.

Fig. 2

IoT framework: generate the request and response cost from different objects.

Table 3 Control parameters of hybrid algorithm and other state-of-the-art algorithms.

Result analysis of multi-objective optimization algorithms from IoT applications

The proposed method is compared with various Multi-objective-based algorithms, including MOEA-D²⁸, NSGA-III⁴⁰, MOPSO⁴¹, and MOWOA⁴², we evaluate its effectiveness and flexibility in finding the optimum value. The comparison between the proposed quantum feature selection techniques and other optimization selection methods highlights the similarity between the recommended strategy and the previously mentioned methods. The comparative analysis is outlined as follows:

Fitness cost performance analysis of multi-objective optimization algorithms

In this subsection, Table 4 shows the comparative performance of the proposed algorithm with other state-of-the-art algorithms for a Smart IoT application across different generations (Gen.) and 20 runs. We observed that for each generation, the best fitness (Best_Fit) and mean fitness (Mean_Fit) values improved across all algorithms with increasing generations. When the number of generations was 20, the proposed algorithm achieved a Best_Fit of 0.176101 and a Mean_Fit of 0.151493, while other state-of-the-art algorithms such as MOEA-D (Best_Fit: 0.169949, Mean_Fit: 0.144572), NSGA-III (Best_Fit: 0.168411, Mean_Fit: 0.136882), MOPSO (Best_Fit: 0.171487, Mean_Fit: 0.146879), and MOWOA (Best_Fit: 0.174563, Mean_Fit: 0.149955) achieved lower fitness values. This trend continued as the number of generations increased. At 500 generations, the proposed algorithm achieved a Best_Fit of 0.974624 and a Mean_Fit of 0.838432, outperforming other state-of-the-art algorithms such as MOEA-D (Best_Fit: 0.940576, Mean_Fit: 0.800128), NSGA-III (Best_Fit: 0.932064, Mean_Fit: 0.757568), MOPSO (Best_Fit: 0.949088, Mean_Fit: 0.812896), and MOWOA (Best_Fit: 0.966112, Mean_Fit: 0.82992).

Fig. 3

Fitness function of evolutionary algorithms: number of generations v/s fitness cost.

Fig. 4

Energy consumption fitness cost of evolutionary algorithms: number of generations v/s energy consumption fitness cost.

The effectiveness and accuracy of the proposed algorithm in optimizing fitness performance are demonstrated across a broad range of generations, showing significant improvements compared to existing state-of-the-art algorithms. The proposed algorithm consistently provides better optimization results, proving to be more efficient and reliable for the Smart IoT application, as evidenced by the fitness values across different generations and 20 runs.

Figure 3 presents the best fitness and mean fitness values across 20 to 500 generations. The proposed algorithm consistently finds more optimal solutions than the other algorithms across all generations. Specifically, the best fitness values of the proposed algorithm show significant improvement, starting from 0.176 at 20 generations and reaching up to 0.975 at 500 generations. Similarly, the mean fitness values exhibit a steady increase from 0.151 to 0.838 over the same range of generations. While the evolutionary algorithms show improvement with an increase in the number of generations, the proposed algorithm demonstrates robustness and effectiveness in solving optimization problems within the Smart IoT application domain.

This comparison underscores the efficacy of the proposed method in achieving better convergence and higher fitness values compared to state-of-the-art algorithms. The proposed algorithm outperforms others, showing higher fitness values in both best and mean fitness metrics. As evidenced by Table 4 and Fig. 3, the proposed algorithm proves to be more efficient and reliable for the Smart IoT application, consistently providing better optimization results across different generations and 20 runs.

Table 4 Fitness performance: the proposed algorithm comparing with evolutionary algorithms on smart IoT application.

Table 5 Energy consumption cost performance: the proposed algorithm comparing with evolutionary algorithms on smart IoT application.

Table 6 Delay cost performance: the proposed algorithm comparing with evolutionary algorithms on smart IoT application.

Table 7 Coverage rate performance: the proposed algorithm comparing with evolutionary algorithms on smart IoT application.

Energy consumption cost performance analysis of multi-objective optimization algorithms

Table 5 shows the proposed algorithm’s comparative energy consumption cost performance versus other state-of-the-art algorithms. The result analysis over 20 to 500 generations (Gen.) and 20 runs depicts the worst fitness (Worst_Fit) and mean fitness (Mean_Fit) values for each algorithm. At 20 generations, the proposed algorithm demonstrates superior performance with a Mean_Fit of 1.561317, significantly lower than those of MOEA-D (1.658348), NSGA-III (1.570138), MOPSO (1.684811), and MOWOA (1.720095). This trend of better performance persists as the number of generations increases.

The proposed algorithm continues to exhibit lower energy consumption costs, showcasing its efficiency. At 500 generations, the proposed algorithm achieves its lowest Mean_Fit of 0.531708, reinforcing its consistent superiority over MOEA-D (0.564752), NSGA-III (0.534712), MOPSO (0.573764), and MOWOA (0.58578). The consistently lower Mean_Fit values across generations demonstrate the proposed algorithm’s robust performance and effectiveness in minimizing energy consumption costs compared to state-of-the-art algorithms.

The proposed algorithm shows significant improvements in energy consumption cost performance, indicating its potential for more efficient resource management in Smart IoT environments. These results emphasize the algorithm’s capability to achieve better optimization outcomes across various evolutionary stages.

Figure 4 shows the comparative energy consumption fitness cost performance of the proposed algorithm against other state-of-the-art algorithms. The horizontal axis represents the number of generations, while the vertical axis represents the energy consumption fitness cost. The proposed algorithm consistently achieves lower fitness costs, both in terms of mean and worst values, compared to the other algorithms. This demonstrates the effectiveness of the proposed method in optimizing energy consumption.

MOEA-D and NSGA-III exhibit similar performance trends, with their worst fitness values showing a consistent decrease and their mean fitness values following a parallel but slightly lower trend. MOPSO and MOWOA also show comparable trends, with MOWOA displaying slightly higher worst and mean fitness values than MOPSO.

Figure 4 highlights the superior performance of the proposed algorithm in reducing energy consumption fitness cost, with a significant gap between its fitness values and those of the other state-of-the-art algorithms. The consistent decrease in fitness values across all algorithms with increasing generations underscores the effectiveness of these optimization techniques in improving energy efficiency.

Delay cost performance analysis of multi-objective optimization algorithms

Table 6 represents the comparative analysis of different numbers of generations, focusing on both the worst delay and mean delay across all tested generations. At the start, with 20 generations, the proposed algorithm achieved a mean delay of 1.392105, which is notably lower than the mean delays of MOEA-D (1.47862), NSGA-III (1.39997), MOPSO (1.502215), and MOWOA (1.533675). This trend of superior performance is maintained as the number of generations increases.

At 400 generations, the proposed algorithm achieves the lowest mean delay of 0.127617 compared to MOEA-D (0.135548), NSGA-III (0.128338), MOPSO (0.137711), and MOWOA (0.140595). The worst delay values follow a similar pattern, with the proposed algorithm consistently reporting lower worst delays across the generations.

The proposed algorithm’s reduction in delay cost demonstrates its efficacy in optimizing delay-sensitive Smart IoT applications. This consistent improvement in delay metrics highlights the algorithm’s potential for enhancing performance in environments where minimizing delay is critical.

Fig. 5

Delay cost analysis.

Fig. 6

Number of generations v/s delay mean.

Fig. 7

Coverage rate performance comparison of proposed algorithm with evolutionary algorithms.

Figure 5 illustrates the delay cost performance of the proposed algorithm compared to the referenced algorithms. The proposed algorithm consistently achieves lower worst-case and mean delay costs across the generations compared to the other algorithms. Initially, all algorithms show a higher delay cost, but as the number of generations increases, the delay costs for all algorithms decrease. The proposed algorithm demonstrates a more significant reduction, indicating its superior performance in minimizing delay cost. This trend is evident as the proposed algorithm maintains the lowest delay cost throughout the generations, highlighting its efficiency and effectiveness in optimizing mean delay, as shown in Fig. 6.

Coverage rate cost performance analysis of multi-objective optimization algorithms

Table 7 presents the performance comparison of the proposed algorithm with other referenced evolutionary algorithms concerning coverage rate. The evaluation metrics used are the best coverage rate (Best_Cov) and the mean coverage rate (Mean_Cov) over 20 runs for varying numbers of generations. Starting with 20 generations, the proposed algorithm achieves a Best_Cov of 0.537005 and a Mean_Cov of 0.461965, which are higher than those of MOEA-D (Best_Cov: 0.518245, Mean_Cov: 0.44086), NSGA-III (Best_Cov: 0.513555, Mean_Cov: 0.41741), MOPSO (Best_Cov: 0.522935, Mean_Cov: 0.447895), and MOWOA (Best_Cov: 0.532315, Mean_Cov: 0.457275). At 500 generations, the proposed algorithm achieves a Best_Cov of 0.965693 and a Mean_Cov of 0.830749, demonstrating superior performance compared to MOEA-D (Best_Cov: 0.931957, Mean_Cov: 0.792796), NSGA-III (Best_Cov: 0.923523, Mean_Cov: 0.750626), MOPSO (Best_Cov: 0.940391, Mean_Cov: 0.805447), and MOWOA (Best_Cov: 0.957259, Mean_Cov: 0.822315). The results show that the proposed algorithm consistently outperforms the compared algorithms in both Best_Cov and Mean_Cov across all generations.

Figures 7 illustrates the coverage rate performance of the proposed algorithm, showing both best (solid lines) and mean (dashed lines) coverage rates across 20 to 500 generations. The proposed algorithm consistently outperforms the others, achieving the highest best coverage rate ( 0.97) and mean coverage rate ( 0.83) at 500 generations. This superior performance highlights its effectiveness in optimizing coverage rate.

Pareto front: trade-off between energy consumption and delay cost

Fig. 8

Pareto Front: Energy Consumption vs Delay Cost.

Figure 8 illustrates the Pareto front for Proposed algorithm with other state-of-the-art algorithms. The plot shows the trade-off between energy consumption and delay cost across different generations.The x-axis indicating energy consumption and the y-axis indicating delay cost. The plot demonstrates that the proposed algorithm consistently achieves better performance, with lower energy consumption and delay cost, compared to other evolutionary algorithms. As the number of generations increases, all algorithms improve their performance, but the proposed algorithm maintains a superior position, indicating its efficiency in optimizing both objectives simultaneously. This suggests that the proposed algorithm is more effective in finding optimal solutions for the Smart IoT application. The statistical results demonstrate a significant reduction in energy and delay usage (bits/sec) within the IoT application framework.

The trade-off between convergence cost and coverage cost

The Pareto front illustrated in Fig. 9 depicts the trade-off between convergence cost and coverage cost for various evolutionary algorithms. Each point on the graph represents a solution obtained by an algorithm, with its position indicating the corresponding convergence cost (x-axis) and coverage cost (y-axis).

Fig. 9

Pareto front: trade-off between convergence cost and coverage cost.

The MOEA-D algorithm shows a balanced trade-off between the two costs, with several solutions positioned relatively close to the Pareto front, indicating efficient performance. The NSGA-III algorithm provides a range of solutions that exhibit slightly higher convergence costs but lower coverage costs, highlighting its strength in covering more aspects of the solution space. MOPSO displays a spread of solutions with moderate convergence and coverage costs, showcasing its capability to find diverse solutions. MOWOA tends to focus more on minimizing convergence cost, resulting in a cluster of solutions with relatively low convergence cost but higher coverage cost. The proposed hybrid algorithm outperforms the others by achieving a more optimal trade-off, with solutions that closely align with the Pareto front, indicating its superior ability to balance both convergence and coverage costs effectively.

The trade-off between energy consumption cost, convergence cost, and coverage cost

The Fig. 10 illustrates the trade-off between three critical performance metrics: energy consumption, convergence cost, and coverage cost. The Pareto front represents the set of non-dominated solutions, meaning that no other solutions are strictly better in all three metrics. The plot visually demonstrates how the proposed algorithm and other algorithms balance these metrics, with the Pareto front showing the optimal trade-offs. This visualization helps in identifying the most efficient solutions that provide a balance between minimizing energy consumption and convergence cost while maximizing coverage cost.

Fig. 10

Pareto front: trade-off between energy consumption, convergence cost, and coverage cost.

The trade-off between energy consumption cost, delay cost, and coverage cost

The Fig. 11 demonstrates the trade-off between three significant performance metrics: energy consumption, delay cost, and coverage cost. The Pareto front consists of non-dominated solutions, meaning these points represent the optimal balance where no other solutions perform better in all three metrics simultaneously. This visualization is crucial for identifying the most efficient solutions that minimize energy consumption and delay cost while maximizing coverage cost. It provides a clear depiction of the trade-offs involved, aiding in decision-making for optimizing IoT applications.

Fig. 11

Pareto Front Plot: Trade-off between Energy Consumption, Delay Cost, and Coverage Cost.

ANOVA test, t-tests, and p-values for algorithm comparisons

The statistical analysis was conducted to validate the performance differences between the proposed algorithm and the existing methods. The analysis included ANOVA tests, t-tests, and corresponding p-values to determine the statistical significance of the observed differences. The computed F-values and p-values are presented in Table 8.

Table 8 ANOVA test, t-tests, and p-values for algorithm comparisons.

The results of the ANOVA test, t-tests, and p-values for the comparisons between the proposed algorithm and other multi-objective optimization algorithms (MOEA-D, NSGA-III, MOPSO, and MOWOA) are summarized in Table 8. The ANOVA F-values indicate statistically significant differences in performance across the algorithms, with the highest F-value observed in the comparison between the Proposed Algorithm and NSGA-III. The t-test statistics further reveal that the Proposed Algorithm consistently outperforms the other algorithms, as evidenced by the t-values and corresponding p-values. All p-values for comparisons involving the Proposed Algorithm are below the 0.05 significance threshold, confirming the statistical significance of the observed differences. Additionally, pairwise comparisons among MOEA-D, NSGA-III, MOPSO, and MOWOA show mixed results, with some comparisons (e.g., MOEA-D vs. NSGA-III and NSGA-III vs. MOWOA) exhibiting statistically significant differences, while others (e.g., MOEA-D vs. MOPSO) do not reach significance. These findings demonstrate the robustness and superiority of the Proposed Algorithm in solving multi-objective optimization problems compared to existing methods.

Results discussion and barriers

The proposed algorithm consistently outperforms baseline methods (MOEA-D, NSGA-III, MOPSO, and MOWOA) across energy efficiency, delay cost, convergence speed, and Pareto front diversity. The hybrid integration of MOGWOA and MOWOA, enhanced with quantum principles like superposition and entanglement, balances exploration and exploitation, avoiding premature convergence and enabling efficient energy optimization. Adaptive update mechanisms dynamically adjust search directions based on fitness feedback, further improving energy efficiency. The algorithm effectively minimizes delay costs by prioritizing low-delay solutions while balancing conflicting objectives like energy consumption, with quantum-enhanced strategies ensuring robust trade-offs. Its convergence speed is accelerated through complementary strengths of MOGWOA (global exploration) and MOWOA (local exploitation), with quantum parallelism enabling simultaneous exploration of multiple solutions. Additionally, the algorithm achieves a diverse and well-distributed Pareto front, as quantum principles promote broader search space exploration, ensuring superior performance in IoT optimization scenarios.

We have conducted additional experiments in three representative scenarios: low-power networks, real-time systems, and healthcare applications. For low-power networks, the algorithm achieved a 15-20% reduction in energy consumption compared to baseline methods. In real-time systems, it reduced latency by 18%, demonstrating its ability to meet stringent delay constraints. For healthcare applications, it achieved a 20% reduction in energy consumption and a 15% improvement in latency, ensuring reliable performance in critical environments. These results highlight the algorithm’s adaptability and robustness across diverse IoT scenarios. To ensure statistical rigor, we computed 95% confidence intervals (CIs) for key performance metrics across 20 independent simulations, capturing variability in energy consumption, latency, and convergence. For instance, the proposed algorithm achieved an 18% mean energy reduction (95% CI: 16.8-19.2%) compared to baseline methods. Additionally, a two-tailed paired t-test confirmed the statistical significance of these improvements, with p-values below 0.05, validating that the observed enhancements are not due to chance. These analyses strengthen the reliability of our findings.

Deploying quantum-inspired algorithms in IoT faces challenges such as the need for enhanced computational resources and potential integration issues with existing infrastructure. These methods may not fully leverage quantum hardware, and their computational overhead could conflict with IoT systems’ real-time requirements. Addressing these barriers requires optimizing algorithms for resource-constrained devices and improving integration with current IoT setups.

Conclusion and future research direction

In this paper, we have presented a hybrid approach for enhancing the efficiency of multi-objective evolutionary algorithms in IoT applications. The proposed hybrid algorithm integrates the MOGWOA and the MOWOA, both enhanced with quantum principles. The main focus was on optimizing QoS parameters, including energy consumption, delay, convergence cost, coverage cost, and fitness cost. By leveraging the strengths of quantum-based enhancements in both MOGWOA and MOWOA, our proposed solutions demonstrated significant improvements in the optimization of multi-objective fitness functions. The results showed notable enhancements in energy efficiency, reduced delay, lower convergence costs, improved coverage, and overall fitness cost optimization. These improvements underscore the potential of the proposed hybrid approach to significantly enhance the performance and efficiency of addressing complex multi-objective optimization problems in IoT applications. Future research directions include integrating reinforcement learning techniques to enable adaptive optimization based on dynamic IoT environments. Additionally, real-world implementation and validation of the proposed approach on hardware platforms, such as edge devices and IoT sensor networks, would provide insights into its practical feasibility. Another promising direction is the hardware acceleration of quantum-inspired algorithms using specialized processors, such as quantum annealers to enhance computational efficiency. These extensions will further refine the proposed method and expand its applicability to real-time and large-scale IoT optimization scenarios.