Test scheduling of network-on-chip using hybrid WOA-GWO algorithm

Abstract

The promising Network-on-Chip (NoC) model replaces the existing system-on-chip (SoC) model for complex VLSI circuits. Testing the embedded cores using NoC incurs additional costs in these SoC models. NoC models consist of network interface controllers, Internet Protocol (IP) data centers, routers, and network connections. Technological advancements enable the production of more complex chips, but longer testing times pose a potential problem. NoC packet switching networks provide high-performance interconnection, a significant benefit for IP cores. A multi-objective approach is created by integrating the benefits of the Whale Optimization Algorithm (WOA) and Grey Wolf Optimization (GWO). In order to minimize the duration of testing, the approach implements optimization algorithms that are predicated on the behavior of grey wolves and whales. The P22810 and D695 benchmark circuits are under consideration. We compare the test time with existing optimization techniques. We assess the effectiveness of the suggested hybrid WOA-GWO algorithm using fourteen established benchmark functions and an NP-hard problem. This proposed method minimizes the time needed to test the P22810 benchmark circuit by 69%, 46%, 60%, 19%, and 21% compared to the Modified Ant Colony Optimization, Modified Artificial Bee Colony, WOA, and GWO algorithms. In the same vein, the proposed method reduces the testing time for the d695 benchmark circuit by 72%, 49%, 63%, 21%, and 25% in comparison to the same algorithms. We experimented to determine the time savings achieved by adhering to the suggested procedure throughout the testing process.

Keywords

Test time whale optimization algorithm test access mechanism grey wolf optimization test scheduling network-on-chip

1. Introduction

Advances in semiconductor manufacturing and integrated circuit design have enabled the development of microelectronic devices that can perform multiple functions on a single chip. Developments in fabrication technology allow the production of ICs with billions of transistors. A System-on-Chip (SoC) is a semiconductor integrated circuit comprising one million transistors or one thousand package pins. Intellectual property (IP) cores are reusable design elements for a cell, logic, chip, or layout licensed or owned by a single party. The core may contain multiple cores, depending on its construction method. The NoC design paradigm faces a significant barrier because we can only evaluate cores after assembly. Another challenge is devising a method to test after fabrication. Because of its intricate nature, the Design for Testability (DFT) simplifies the testing of complex ICs. The NoC will return the completed test data to the sink for further processing. Core-based designs must simultaneously reduce testing time to evaluate as many cores as possible. The tests’ sequencing determines the order in which different CPU cores undergo testing. Scheduling NoC tests on hard or legacy cores is more difficult when they are hierarchical. The NoC, which is made up of many mega cores that each have their cores, is the highest level of testing for hierarchical cores. The equator’s objective function, the test time, must be the same [1].

\begin{aligned} T (W_{i}) = [1 + max (S_{i}, S_{o})] . t_{p i} + min (S_{i}, S_{o}) \end{aligned}

(1)

$S_{0}$ and $S_{i}$ are the outputs of the scan chain and inputs of length; _thiS is the pattern to be tested.

Table 1 represents the particulars of the ITC’02 IEEE Test Conference [2] NoC benchmarks.

Table 1

ITC’02 NoC benchmark details.

NoC	d695	p22810
No. of levels	2	3
No. of SFFs	6384	24723
Contributor	Duke University	Philips Semiconductors
I/O	1845	4283
Length of scan chain	137	196
No. of tests	10	30
No. of modules	11	29
Pattern count
Minimum	1	12
Maximum	12325	234
Average	830	88
Length of scan chain
Maximum	401	55
Average	126	46
Minimum	1	32

Each core test’s duration and start time impact a SOC’s total testing time. The TAM assignments determine the cores’ test times and the wrapper’s construction. We investigate the trade-offs between various design objectives by combining test scheduling, wrapper creation, and TAM assignment using a hybrid WOA-GWO technique. New methodologies and tools are required to address the challenges of chip test scheduling in NoC design, which is a complex optimization problem with numerous unknowns. We must adequately resolve these issues because NoCs are critical components of many modern electronic goods. To overcome these challenges, it is essential to consider the specific constraints of the NoC design process and devise effective workarounds. The motivation behind mixing the algorithms is to utilize the strengths of the WOA and GWO algorithms to increase the overall performance. The hybrid algorithm that combines WOA and GWO can outperform either approach alone.

This article proposes a new meta-heuristic algorithm to enhance the existing method by hybridizing WOA with GWO. The hybrid algorithm results in a rapid and high convergence rate, as well as overcoming the limitations of individual algorithms. The merging of WOA and GWO produces an efficient optimization algorithm that can reduce the testing time. The migration operator for Monarch Butterfly Optimization (MBO) favours local search [3]. The MBO method rapidly converges to local optima without conducting sufficient exploration, which is prone to suboptimal results. MBO parameters can have a significant impact on performance, necessitating substantial adjustments. In response to the changing hunting behaviors of slime mold organisms, researchers devised the Slime Mould Algorithm (SMA) [4]. The complexity of SMA algorithms may make them difficult to understand and implement when faced with significant challenges. Investigating numerous potential solutions in highly complex search regions may also be difficult. Photographing and analyzing moth wings led to the creation of the Moth Swarm Algorithm (MSA) [5]. Achieving a balance among exploitation and exploration in MSA can take time and effort, and in some instances, it may lead to a deadlock. Scalability becomes a concern when the complexity or proportion of problems in NoC systems increases, as it can harm performance. One suggestion was to conduct a Hunger Games Search (HGS) based on animal actions and behaviors [6]. HGS’s intricate ability to adapt to evolving circumstances during operation scheduling sets it apart. Perhaps hasty convergence is due to a need for more diverse solutions. Engineers frequently employ the high-precision, single-step Runge-Kutta (RUN) technique [7]. Initially developed for numerical analysis, the RUN algorithm may not be suitable for combinatorial optimization applications, including test scheduling. It may necessitate substantial processing resources, particularly for issues with many dimensions. The Colony Predation Algorithm (CPA) is based on how carnivorous plants deal with adversity by scavenging and fertilizing insects [8]. If calibration is inadequate, the CPA may fail to investigate specific potential opportunities in the exploration space. The complexities of the algorithm’s architecture may impede its practical implementation in real-world circumstances. The weighted mean of vectors (INFO) technique is a sophisticated optimization method that maximizes the process by utilizing multiple weighted average vector rules [9]. The Harris Hawks Optimization (HHO) program replicates predation actions in hawks [10]. In the HHO algorithm, inadequate exploration strategies can lead to the possibility of becoming ensnared in local optima. By modifying the algorithm settings, you can observe substantially divergent outcomes.

The basic GWO requires further refinement to accurately estimate velocity and accuracy in varying degrees of partial shadow. Meta-heuristic algorithms are appealing to academics due to their applicability to a wide range of scenarios. The gray wolf algorithm is a meta-heuristic that leverages collective intelligence within a group. Gray wolves’ hunting strategies and behavioral characteristics inspired this algorithm. The classification of gray wolves into various groups depends on how much their daily survival relies on hunting [11,12]. The alpha wolves choose their prey. They also serve as the pack’s leaders. Betas in the second cohort collaborated with alphas to make decisions. The sentry wolf, often known as the omega, represents the third distinct class of wolves. The quick convergence of the algorithm, lack of population heterogeneity, and imbalance between exploration and exploitation all contributed to its failure [13].

The proposed hybridization of WOA and GWO is distinct from previous combinations of WOA and GWO. The proposed method differentiates itself from WOA and GWO by introducing a novel approach to updating the WOA exploration stage. During the exploitation phase, we also added the GWO hunting system to track the whale’s location. A recently proposed hybridization, WOA-GWO, improves WOA’s performance. The proposed WOA-GWO algorithm differs from GWO and WOA because it adds a new way to update the WOA exploration stage. Several optimization algorithms, such as the Modified ACO, Modified ABC, Modified Firefly, WOA, and GWO algorithms, reduce the time it takes to test the two benchmarks. Resolving the NoC test scheduling conundrum minimizes the provided cost function. This research aims to shorten the analysis of testing times for different TAM widths.

This paper proposes a novel way to test scheduling to minimize length and costs. The following individuals contributed to the paper: 1)

Hybrid WOA-GWO algorithms are much faster than other optimization techniques for scheduling tests.

Two NoC benchmark circuits determine how well the meta-heuristic algorithms work for different TAM widths.

We prepare the remainder of the paper as follows: Section 2 elaborates on the literature review. Section 3 discusses the proposed work. Section 4 elaborates and confirms our findings. Section 5 summarizes the conclusion.

2. Literature review

Numerous studies on test schedules assumed flat cores and a single TAM level. We can only achieve this if we can assemble the inner cores. We recommend a heuristic fixed-width design to reuse test data for cores in SoCs. Using a graph-based method, we solved the Tabu search problem. The more extensive wrapper design method approached and resolved the test scheduling challenge as a space allocation problem. We use a two-dimensional bin-packing model in the planning and execution of testing. We can use a rectangle packing approach to enhance the effectiveness of the grid for test scheduling. A novel test preparation strategy and a more efficient test execution method focus on stackable ICs. We compute test time analysis using both the Distribution and Multiplexing architecture. Multiplexing-type testing shares TAM bandwidth among all cores in a predefined manner. In this type of distribution, each core only gains a portion of the bandwidth because all the cores share the bandwidth. We use test scheduling algorithms to perform TAM and wrapper optimization. Three mechanisms can test core-based systems: ATE for external testing, BIST for internal testing, and a combination of BIST and ATE. We employ rectangular bins to evaluate cores, where the height aligns with the TAM width and the length aligns with the test period. We divide the width of TAM into multiple widths for test buses. We assign additional cores to various test buses during scheduling based on their TAM widths [14]. The Genetic Algorithm (GA) [15] performs test scheduling by evolution and mutation operations, thereby reducing the testing time.

The Simulated Annealing (SA) algorithm, as proposed in [16], solves a two-dimensional bin packing problem by optimizing the design of the wrapper and TAM. The SA algorithm selects every core-best layout among available arrangements. The foraging activity of a swarm of birds notifies Particle Swarm Optimization [17]. The term “particle” encompasses each member within the swarm. A given problem determines the particle’s dimension’s values, equivalent to several parameters. The Random Insertion (RAIN) algorithm uses a rectangle to represent a core [18], with the width and height of the rectangle representing the test time and TAM width, respectively. Rectangles plot the cores, representing a sequence pair. Ant colony optimization (ACO) [19] is established on ant activity, which ants follow to search for the shortest route from their place to the food. Pheromone is the hormone ants leave while seeking new directions for their food. ACO rapidly converges on suboptimal solutions due to a need for more diversity mechanisms. Because of the supplementary computational burden, the performance may decrease as the size of the problems increases due to the supplementary computational burden. In modified ACO, several ants decrease when the core count decreases [20]. So, the testing time is reduced. As a result, we can produce output more efficiently.

The ABC algorithm was inspired by honeybee foraging behaviour. The paradigm is based on three primary determinants: the unemployment rate, the job of foraging pollinators, and food availability [21]. Developing a balanced state of exploration and exploitation in ABC is often difficult, resulting in suboptimal outcomes. Performance may deteriorate as the problem or circumstance becomes more complex. In ABC, perturbation frequency is the parameter that influences the new solutions. By adjusting the perturbation frequency, we can improve the ABC algorithm, which we refer to as the modified ABC algorithm [22]. Fireflies use their flashing behavior to attract each other during mating, which inspired the Firefly Algorithm (FA) [23]. Flashing behavior serves the purpose of either attracting potential mates or luring prey. The modified Firefly algorithm overcomes FA constraints. Fireflies’ improved mobility and randomness facilitate increased exploitation and exploration [24]. If calibration is performed correctly, the Firefly Algorithm may be able to investigate alternative solutions. A system’s parameters significantly impact performance, varying considerably based on the specific challenges being addressed. The Modified Firefly algorithm scaled the randomization parameter ’a’ linearly down from $a_{0}$ to $a_{2}$ . Where $a_{0}$ and $a_{2}$ represent the starting and ending values of the iteration, respectively. We propose a bio-inspired algorithm [25] that uses sonar called an “echo” to detect prey and avoid obstacles. Each person listens to a sound pulse that bounces back from the object. The pulses vary depending on the species’ nature. We employ cooperative co-evolutionary algorithms to address multi-objective optimization problems, considering variable choice factors and yielding positive outcomes [26]. We are developing an MTO-DRA method to allocate computing resources better to demanding tasks. We designed a multi-swarm cooperative PSO [27] to address a multi-objective optimization problem, fine-tuning each sub-swarm to optimize performance for a single objective. Due to the limited diversity of particles in PSO, the system has a propensity to swiftly converge around local optima. The cognitive/social coefficients and inertia weight are two components that necessitate precise adjustments.

A novel approach using the Rainbow DQN, an advanced DRL algorithm, to optimize task offloading in a device-to-device computing framework of Edge-Cloud [28,29,30]. This algorithm effectively improves energy efficiency. Compressing and exchanging tests within a single system reduces testing time. This strategy lengthens the optimization process as complexity increases. The study presents an SoC testing strategy that combines a test access design, a test wrapper, a testing schedule, and test data compression. This work employs a static test algorithm. We will adjust the algorithm to find an earlier start time for each exam and create a test schedule with a heat awareness test to shorten the test period. In SoC design, reconfigurable core wrappers reduce TAM design and test schedules [31]. Furthermore, it aims to reduce TAM through an expanded core design and hierarchically extract attributes. These studies use conventional and altered methods of reducing TAM, but the researchers note that there is still room for improvement.

3. Proposed methodologies

3.1. Whale optimization algorithm

Andrew Lewis and Seyedali Mirjalili proposed a technique that uses the Whale Optimization Algorithm (WOA) [32]. This technique provides a detailed description of the foraging techniques used by humpback whales. Changing the parameters from two to zero gives the algorithm a chance to exploit and explore, which is one of the most essential parts of WOA. When $| A | >$ 1, the best solution is reserved for the subsequent iteration. The current iteration’s value updates the immediate best solution if $| A |$ is 1. After completing the iterations, each iteration updates the best position based on various scenarios. The equations signify the updated technique:

\begin{aligned} D & = | C . X * (t) - X (t) | \end{aligned}

(2)

\begin{aligned} X (t + 1) & = X (t) - A . D \end{aligned}

(3)

Where ‘C’ and ‘A’ represent coefficients, t is the present iteration, and $X *$ represents the solution’s optimal position.

In every iteration, the $X *$ value would be updated if an improved solution was found than the current optimum solution. We measure the vectors A and C:

\begin{aligned} A & = 2 a r - a \end{aligned}

(4)

\begin{aligned} C & = 2 r \end{aligned}

(5)

The iteration reduces ‘a’ linearly ‘2 to 0’ and “r” becomes a vector with values ranging from ‘0 to 1’. We are developing two methods to represent the behavior of bubble-net all whales display quantitatively.

The Shrinking-Encircle feeding mechanism involves hunting prey by circling it in groups and then shrinking its circle by varying from two to zero. ‘A’ is a randomly chosen vector that takings values among ‘ $- a$ ’ and ‘a’. Adding these flexibilities results in multiple levels within the mathematical model, as shown in Fig. 1. It depicts all the areas moving from (X, Y) to ( $X *$ , $Y *$ ) in a two-dimensional space. We can also approach the problem in three-dimensional space. We treat the present location as (X, Y), and the target prey as ( $X *$ , $Y *$ ). Next step involves traversing the spiral path to reach the victim. The equations below depict the details of this process.

Figure 1.

Search agent’s position updating mechanism.

During the spiral updating state, it is feasible to observe the bubble net feeding scenarios. The algorithm is required to ascertain the distance between the prey’s present location and its current position. The current position is treated as (X, Y), and the target prey is at ( $X *$ , $Y *$ ). After this, it must be traversed through the spiral path to reach the victim. The details of the same are depicted in the equations given below.

\begin{aligned} X (t + 1) = D^{'} . e^{b l} . \cos (2 π l) + X * (t) \end{aligned}

(6)

The space among the whale and its sustenance is quantified by the optimal solution, given by $D = | X^{*} (t) + X (t) |$ . The constant b ensures the stability of the spiral trajectory. The vector ‘l’ is arbitrarily generated and spans the range of $-$ 1 to 1. A 50% probability is given for choosing between the above two methodologies of shirking circles and bubble-net for feeding. The model that consists of this treatment can be represented as follows:

\begin{aligned} X (t + 1) = {\begin{matrix} X^{*} (t) - A . D & i f p < 0.5 \\ D^{'} . e^{b l} . \cos (2 π l) & i f p ⩾ 0.5 \end{matrix} \end{aligned}

(7)

Where p is treated like a number chosen randomly between zero and 1, as in the shrinking circles and bubble-net feeding scenarios, the humpback whales may also be randomly prey searching, as shown in the below section.

\begin{aligned} D = | C * X_{r a n d} - X (t) | \end{aligned}

(8)

\begin{aligned} X (t + 1) = X_{r a n d} - A * D \end{aligned}

(9)

$X_{r a n d}$ is viewed as a randomly selected position vector from the current population. Figure 2 depicts a randomly designated point vector as of the present population.

Figure 2.

Bubble Net Mechanism for WOA.

Whale optimization is a flexible algorithm for achieving a global minimum due to the exploitation and exploration with multiple adjustment possibilities. Figure 3 shows the flowchart for WOA for test scheduling.

Figure 3.

WOA algorithm for test scheduling.

3.2. GREY wolf optimization

Mirjalili [33] introduces Grey Wolf Optimization (GWO). This algorithm takes its cues for success from the natural hunting hierarchies established by gray wolves. Figure 4 shows the gray wolf social hierarchy. The excellence of hunting is mainly due to the three best leaders: alpha ( $α$ ), delta ( $δ$ ), and beta $β$ ), and the supporting people in the group are omega $ω$ .

Figure 4.

Social hierarchy of grey wolf.

Figure 5.

Grey wolf hunting.

Figure 6.

Flowchart of the GWO algorithm for NoC test scheduling.

Mathematically, we refer to a group of grey wolves as a “population” and another group of grey wolves as an “iteration”. Figure 5 shows the grey wolf hunting. Consider a pack of gray wolves chasing their prey. It can be defined as finding a solution to a problem. Equations (10), (11), and (12), which track the excellent behavior of a group of wolves encircling the prey, mathematically approach us. Equations (10), (11), and (12) delineate the selection of the top wolves by calculating the updated distance between the top three wolves and the remaining wolves, using the following equation as a basis.

\begin{aligned} D_{α} & = | C_{1} * X_{α} - X | \end{aligned}

(10)

\begin{aligned} D_{β} & = | C_{1} * X_{β} - X | \end{aligned}

(11)

\begin{aligned} D_{δ} & = | C_{3} * X_{δ} - X | \end{aligned}

(12)

where

D_{α}

D_{β}

, and

D_{δ}

are the updated distance vectors among the best wolf’s position vectors alpha (

X_{α}

), beta (

X_{β}

), and delta (

X_{δ}

) and position to the other wolf’s vector (X), which are three coefficient vectors. We linearly adjust these vectors from 0 to 2 in each iteration to attain the optimal solution.

Figure 6 shows the flowchart for GWO-based test scheduling. The alpha wolf’s ( $α$ ) hunting position toward the prey is updated at each iteration by adjusting the coefficient vector frm ( $-$ 2a to 2a) in Eq. (13). The current distance ( $D_{α}$ ) is expressed as given below.

\begin{aligned} X_{1} = X_{α} - A_{1} * D_{α} \end{aligned}

(13)

Accordingly, beta wolf and delta wolf are updated using Eqs (14) and (15).

\begin{aligned} X_{2} & = X_{β} - A_{2} * D_{β} \end{aligned}

(14)

\begin{aligned} X_{3} & = X_{δ} - A_{3} * D_{δ} \end{aligned}

(15)

$A_{2}$ , $A_{3}$ is the coefficient vectors adjusted from ( $-$ 2) to (2) in each iteration to achieve the best search solution path. The best grey wolf replaces other grey wolves to better attack the prey. The decision of the three top wolves in a pack to encircle a wolf is expressed mathematically, as shown below.

\begin{aligned} X (t + 1) = \frac{X_{1} + X_{2} + X_{3}}{3} \end{aligned}

(16)

Where $X (t + 1)$ is the efficient grey wolf place for the subsequent iteration.

3.3. Hybrid WAO-GWO

Hybridizing the WOA and GWO algorithms involves integrating the beneficial aspects of both methods to enhance the efficacy of addressing optimization challenges, such as test scheduling in NoC systems. The hybrid WOA-GWO algorithm synergistically combines the exploitative power of GWO with the exploratory prowess of WOA. Initially, WOA employs global research to identify advantageous regions within the solution space. By concentrating on local inquiry in the vicinity of the most optimal locations, GWO can improve potential solutions. The hybrid method ensures the attainment of optimal solutions while simultaneously preserving variation within the population. In the hybrid WOA-GWO optimization, status refers to the current state of viable results within the search space. We determine the optimal position by assessing the adequacy of each potential solution. The algorithm preserves a record of the most effective solution discovered during the search. In the context of WOA, this procedure involves identifying the whale (solution) with the most favorable location, as determined by its fitness value. The best solution is the one with the most excellent fitness score. The algorithm enhances its most optimal solutions by considering the evaluations conducted by both methods during the iterations of WOA-GWO.

In NoC systems test scheduling, the fitness function assesses the efficacy of potential solutions by determining their capacity to satisfy the established objectives. The fitness function is essential for estimating the value of each solution and directing the optimization process. The primary goal is to reduce the total test duration. The assessment of the efficacy of processing and transmitting test signals through the NoC is necessary to reduce the overall duration of the testing process. The function can calculate the total time while ensuring that it does not exceed the limits of simultaneous processing. The resource utilization criterion ensures efficient bandwidth, memory, and processor allocation. Requirements for the fitness function may include evaluating all essential test cases, including test coverage, during the scheduling process. Utilizing a novel hybrid optimization technique that synergistically incorporates the GWO with the WOA improves the scheduling of tests in NoC systems. This innovative method enhances the efficacy of communication among interconnected components and simultaneously addresses two substantial obstacles: resource usage and test time. The proposed methodology surpasses the current leading method in complex NoC designs by integrating the search capabilities of WOA with GWO’s hierarchical search strategy. Hybridization substantially improves NoC testing methodologies, reduces testing overhead and improves scalability and adaptability to various systems. The humpback whale’s hunting strategy, which entails employing techniques such as encircling prey to navigate their hunting grounds effectively, is the basis of the WOA’s hunting strategy. The GWO emphasizes using leading and following mechanisms to improve solutions, drawing inspiration from grey wolves’ social hierarchy and hunting capabilities. Initializing a population of potential solutions is a component of the hybrid technique, which incorporates several techniques. Consequently, it implements GWO’s ranking methodology to construct a hierarchical framework among the available alternatives and implements WOA are encircling strategy to investigate potential solutions. The best test schedule is found through a process that repeats itself. GWO and WOA are used to find better solutions to balance the optimization approach.

Figure 7.

Flowchart of the hybrid WOA-GWO algorithm for NoC test scheduling.

To aid in test scheduling, researchers designed a hybrid WAO-GWO [34]. This method generates many random initial solutions before iteratively refining the best candidate using the current solution as a starting point. Figure 7 depicts the flowchart for the hybrid WOA-GWO technique. When it first starts, WOAGWO sets the quantity of agents in the population, which includes wolves and whales. If the agents leave the search space, the population undergoes a procedure to modify the agents.

Consequently, we determine the fitness function. If best score is greater than the fitness, health and alpha scores are the same. We then update the variables a, A, L, p, and C. It produces a random number as an outcome. The random number must have a result of less than 0.5 to meet the following conditions. If we meet this criterion, we determine the new position using Eq. (3). We will alter the old position to replicate the improvement in the latest one. However, if ( $/ A / >=$ 1), Eq. (2) determines the new position. They are considering the updated position’s fitness in comparison to the old. Put, if the new role is superior to the old one, the old one will gradually disappear. If the random number is 2, however, a different rule applies: if ( $A 1 >$ $-$ 1 $| | A <$ 1) is counted, it is added [35]. Therefore, we use Eq. (16) to ascertain the new coordinates. The stages of WAO-GWO are (i)

Initialization: Forming a randomly selected sleuthing crew (WGi, where I = 1, 2, 3, $\dots$ , k).

(ii)

Fitness evaluation: Estimate the fitness function and select the finest agent.

(iii)

If the prey is immobile, encircling it is the best option. We appraise the locations of other search agents to align with the most effective agent.

(iv)

Position updating, encirclement, spiralling, and size reduction are some exploitation tactics used. Reduced encircling considers both the unique position and the current optimal agent within the provided range [1, 1]. The second method uses the spiral approach to keep agents’ locations up-to-date.

(v)

Exploration: The location of another randomly selected agent determines a search agent’s path during an expedition.

(vi)

Termination: Upon a search agent’s departure from the search area, we adjust the ideal search agent’s value and initiate the next iteration. You can iterate this procedure to determine the best solution. We use the combined WOA-GWO to determine the best agent.

Table 2

Input parameter initialization for WOA, GWO, and hybrid WOA-GWO.

Proposed algorithms	WOA		GWO		Hybrid WOA-GWO
Benchmarks	D695	P22810	D695	P22810	D695	P22810
Population count	20	60	20	60	20	60
No. of cores	10	30	10	30	10	30
TAM width	64 to 16
No. of iterations	100
A	$-$ 2 to 2		2 to 0		$-$ 2 to 2
$r_{1} / r_{2} / r$	0 or 1
C	$2 r$		$2 r_{2}$		$2 r$

4. Results and discussions

The proposed hybrid WOA-GWO algorithm to 14 well-established benchmark functions. Academics frequently use these functions to evaluate how well an algorithm works. Equations (17) to (30) (f1 to f14) test the hybrid WOA-GWO on these 14 benchmark functions. Each individual relies on this test code and gets information from other sources. Apply testing approaches to determine whether a local or global optimal solution exists. It is suitable for studying algorithm comparisons. Table 2 shows the input parameters for initializing WOA, GWO, and hybrid WOA-GWO for various TAM widths. The table represents the Benchmarks, No. of Cores, Population Count, No. of Iterations, TAM width, A, r1/r2/r, and C parameters.

\begin{aligned} f_{1} & = \sum_{i = 1}^{n} x_{i}^{2} \end{aligned}

(17)

\begin{aligned} f_{2} & = \sum_{i = 1}^{n} | x_{i} | + \prod_{i = 1}^{n} | x_{i} | \end{aligned}

(18)

\begin{aligned} f_{3} & = \sum_{i = 1}^{n} {(\sum_{j - 1}^{i} x_{j})}^{2} \end{aligned}

(19)

\begin{aligned} f_{4} & = max_{i} {| x_{i} |, 1 ⩽ i ⩽ n} \end{aligned}

(20)

\begin{aligned} f_{5} & = \sum_{i = 1}^{n - 1} [100 (x_{i + 1} - x_{i}^{2})^{2} - (x_{i} - 1)^{2}] \end{aligned}

(21)

\begin{aligned} f_{6} & = \sum_{i = 1}^{n} [(x_{i} + 0.5)^{2}] \end{aligned}

(22)

\begin{aligned} f_{7} & = \sum_{i = 1}^{n} i x_{i}^{4} + r a n d o m (0, 1) \end{aligned}

(23)

\begin{aligned} f_{8} & = \sum_{i = 1}^{n} - x_{i} \sin (\sqrt{| x_{i} |}) \end{aligned}

(24)

\begin{aligned} f_{9} & = \sum_{i = 1}^{n} [x_{i}^{2} - 10 \cos (2 π x_{i}) + 10] \end{aligned}

(25)

\begin{aligned} f_{10} & = - 20 \exp (- 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2}}) - \exp (\frac{1}{n} \sum_{i = 1}^{n} \cos (2 π x_{i})) + 20 + e \end{aligned}

(26)

\begin{aligned} f_{11} & = \frac{1}{4000} \sum_{i = 1}^{n} x_{i}^{2} - \prod_{i = 1}^{n} \cos (\frac{x_{i}}{\sqrt{i}}) + 1 \end{aligned}

(27)

\begin{aligned} f_{12} & = \frac{π}{n} {10 \sin (π y_{i}) + \sum_{i = 1}^{n - 1} (y_{i} - 1)^{2} [1 + 10 \sin^{2} (π y_{i + 1})] + (y_{n} - 1)^{2}} \\ + \sum_{i = 1}^{n} u (x_{i}, 10, 100, 4) \end{aligned}

(28)

\begin{aligned} y_{i} & = 1 + \frac{x_{i} + 1}{4} \\ u (x_{i}, a, k, m) & = {\begin{cases} k (x_{i} - a)^{m}; & x_{i} > a \\ 0 \\ k (- x_{i} - a)^{m}; & x_{i} < - a \end{cases} \end{aligned}

\begin{aligned} f_{13} & = 0.1 {\sin^{2} (3 π x_{1}) + \sum_{i = 1}^{n} (x_{i} - 1)^{2} [1 + \sin^{2} (3 π x_{i} + 1)] + (x_{n} - 1)^{2} [1 + \sin^{2} (2 π x_{n})]} \\ + \sum_{i = 1}^{n} u (x_{i}, 5, 100, 4) \end{aligned}

(29)

\begin{aligned} f_{14} & = - \sum_{i = 1}^{n} \sin (x_{i}) \cdot {(\sin (\frac{i x_{i}^{2}}{π}))}^{2 m}, m = 10 \end{aligned}

(30)

Table 3

Statistical measures of different techniques for the benchmark functions $f_{1}$ to $f_{14}$ .

Functions	Statistics	Modified ACO	Modified ABC	Modified firefly	WOA	GWO	Hybrid WOA-GWO	p value- hybrid WOA-GWO
$f_{1}$	Std Dev	1.61E-004	7.15E+003	3.47E-001	3.52E-005	3.58E+003	1.11E-004	NA
	Mean	1.97E-004	4.52E+003	1.44E+000	5.25E-005	2.26E+003	1.10E-004
	Best	1.17E-004	2.24E+000	6.71E-001	9.57E-006	1.12E+000	1.05E-004
$f_{2}$	Std Dev	3.31E-002	2.94E+001	5.24E+001	1.66E-002	1.47E+001	1.01E-004	NA
	Mean	3.26E-002	4.42E+001	2.91E+001	1.63E-002	2.22E+001	1.01E-004
	Best	2.11E-003	6.60E-001	7.07E-001	1.01E-003	3.30E-001	1.01E-004
$f_{3}$	Std Dev	5.33E+001	1.64E+004	1.34E+002	2.67E+001	8.14E+003	1.49E-004	NA
	Mean	1.78E+002	3.31E+004	3.76E+002	8.88E+001	1.66E+004	1.66E-004
	Best	1.04E+002	9.83E+003	1.54E+002	5.10E+001	4.92E+003	1.15E-004
$f_{4}$	Std Dev	2.28E-001	6.66E+000	1.35E+000	1.15E-001	3.33E+000	5.70E-004	NA
	Mean	1.52E+000	6.86E+001	2.87E+000	7.51E-001	3.43E+001	1.61E-003
	Best	9.61E-001	5.17E+001	8.71E-001	4.82E-001	2.60E+002	8.21E-003
$f_{5}$	Std Dev	6.43E+002	1.28E+006	6.74E+001	2.72E+002	5.34E+005	4.08E-001	NA
	Mean	2.16E+003	3.04E+005	5.24E+003	6.71E+002	2.02E+005	3.90E+001
	Best	4.46E+002	4.51E+003	5.85E+002	4.67E+002	3.95E+003	3.89E+002
$f_{6}$	Std Dev	2.01E-003	6.15E+002	2.99E-002	6.34E-003	4.58E+002	2.48E-002	NA
	Mean	1.99E-003	3.98E+002	2.38E+001	3.33E+001	3.27E+004	8.66E+001
	Best	2.08E-005	4.42E+001	8.24E-002	5.16E+001	4.86E+001	8.30E+001
$f_{7}$	Std Dev	9.31E-001	9.47E+001	2.18E-003	5.16E-003	5.24E+001	3.03E-004	NA
	Mean	2.39E-001	4.48E+000	3.69E-002	1.20E-001	2.25E+000	2.14E-004
	Best	7.17E-002	1.02E-001	1.72E-002	3.60E-002	5.02E-002	1.02E-004
$f_{8}$	Std Dev	2.31E+004	2.17E+002	8.94E+001	4.41E+002	4.34E+002	5.51E+004	0.5491
	Mean	7.65E+004	2.13E+003	2.04E+004	2.20E+004	2.32E+004	2.49E+005
	Best	5.74E+004	2.37E+005	2.21E+005	2.13E+005	2.57E+005	2.77E+005
$f_{9}$	Std Dev	2.46E+002	5.25E+002	4.53E+002	6.27E+001	3.13E+002	2.01E-004	NA
	Mean	7.61E+001	2.18E+002	1.73E+002	3.81E+001	1.11E+002	1.01E-004
	Best	5.10E+001	1.19E+002	1.08E+002	2.55E+001	5.90E+001	1.01E-004
$f_{10}$	Std Dev	2.97E-001	5.91E+000	5.82E-001	1.48E-001	2.96E+000	1.01E-004	NA
	Mean	9.43E-002	1.71E+001	1.76E+000	4.72E-002	8.48E+000	1.01E-004
	Best	2.51E-003	1.11E+000	7.57E-001	1.21E-003	5.52E-001	1.01E-004
$f_{11}$	Std Dev	7.11E-004	5.76E+002	5.70E-003	5.81E-004	4.38E+002	5.71E-003	NA
	Mean	6.11E-003	4.41E+001	8.16E-001	1.17E-002	2.21E+001	1.74E-002
	Best	1.01E-004	1.02E+000	6.19E-001	2.36E-007	5.04E-001	1.01E-004
$f_{12}$	Std Dev	1.26E-002	6.84E+007	1.33E+000	7.96E-003	3.42E+007	3.71E-003	0.0579
	Mean	2.01E-003	1.93E+007	2.37E+000	1.36E-003	9.61E+006	9.01E-004
	Best	1.01E-004	3.78E+000	3.53E-002	8.93E-008	1.89E+000	1.01E-005
$f_{13}$	Std Dev	9.3E-003	5.49E+006	9.01E-001	3.51E-003	4.25E+006	5.19E-001	NA
	Mean	6.11E-003	2.05E+006	2.83E-002	3.87E-003	6.14E+005	8.23E-001
	Best	2.04E-004	2.74E+000	8.56E-001	2.93E-003	7.66E+001	3.86E-001
$f_{14}$	Std Dev	8.12E-002	2.30E+001	2.01E-003	3.56E-002	7.42E-002	1.01E-004	0.0103
	Mean	1.65E+000	1.51E+000	9.99E-001	1.33E+000	1.26E+000	9.98E-001
	Best	8.97E-002	8.86E-002	8.89E-002	9.16E-001	9.16E-001	9.98E-001

Figure 8.

Simulation results of test functions $f_{1}$ to $f_{14}$ .

We implemented a high-performance system equipped with a multi-core INTEL i5 and 16 GB RAM to assess the hybrid WOA-GWO approach and ensure the rapid and uninterrupted execution of all computations. We represented a variety of topologies, including mesh and torus configurations, using the simulated NoC architecture. We assessed the efficacy of the NoC under various test schedule scenarios using simulation techniques, which may have included our proprietary frameworks. We meticulously monitored performance metrics to determine the optimization technique’s effectiveness, including resource utilization and test duration. MATLAB version 2016b is currently in use. Table 3 compares the seven strategies used for improvement. We subject each benchmark function to 30 iterations of the WOA-GWO algorithm and other comparison methods. The standard deviation, mean, minimum, and maximum values are among the extensive scientific data that the experiment yields. We summarize the collected data in Table 3. By subjecting each approach to a predetermined maximum population size and a specific number of iterations, we assess its efficacy. We evaluate the hybrid WOA-GWO algorithm’s performance by comparing it to the WOA, GWO, modified firefly, modified ABC, and the original WOA method. Figure 8 of the program presents the results for test functions $f_{1}$ through $f_{14}$ .

Table 4

Evaluation of test time for d695 NoC.

TAM width (W)	W = 64	W = 56	W = 48	W = 40	W = 32	W = 24	W = 16
Modified ACO [17]	0.0359863	0.03665040	0.0367917	0.0382017	0.0360996	0.0376558	0.0377801
Modified ABC [19]	0.0211732	0.02120818	0.02131976	0.0213767	0.02141608	0.02147234	0.0215827
Modified Firefly [21]	0.0284903	0.02885020	0.0287198	0.0290803	0.0287055	0.0285490	0.0297982
WOA	0.0139984	0.01410120	0.0142384	0.0143523	0.0144712	0.0144892	0.0146049
GWO	0.0143216	0.01443140	0.0145471	0.0146513	0.0147665	0.0148714	0.0149125
Hybrid WOA-GWO	0.0114016	0.01249320	0.0126131	0.0127986	0.0128181	0.0129018	0.0131091

Table 5

Evaluation of test time for p22810 NoC.

TAM Width (W)	W = 64	W = 56	W = 48	W = 40	W = 32	W = 24	W = 16
Modified ACO [17]	0.02101820	0.021212	0.021340	0.021433	0.021563	0.021693	0.021735
Modified ABC [19]	0.01789550	0.018061	0.018169	0.018249	0.018360	0.018470	0.018506
Modified Firefly [21]	0.01957695	0.019758	0.019876	0.019964	0.020085	0.020205	0.020245
WOA	0.01453258	0.014667	0.014755	0.014820	0.014909	0.014999	0.015028
GWO	0.015013	0.015152	0.015243	0.015310	0.015402	0.015495	0.015525
Hybrid WOA-GWO	0.01201040	0.012121	0.012194	0.012248	0.012322	0.012396	0.012420

Figure 9.

Graphical illustration of test time for D695 NoC using various techniques.

Table 1 displays the results of best and worst of the benchmark function. According to Table 1, the hybrid WOA-GWO outperforms both of its competitors. The results show that the hybrid WOA-GWO is more successful at profit maximization. For the significance of statistical composite WOA-GWO data, we implement Wilcoxon’s Rank-Sum Test [35] with threshold of 5%. Table 3 shows the p values for hybrid WOA-GWO for all benchmarks ( $f_{1}$ through $f_{14}$ ). Table 3 indicates that this test could not evaluate a similar technique as statistically significant, with NA standing for “not applicable”. Figure 8 depicts the algorithm convergence curves for the functions $f_{1}$ through $f_{14}$ . The hybrid WOA-GWO approach, as shown in the diagram, has a faster convergence rate. In this scenario, it appears that the hybrid WOA-GWO technique is capable of efficiently locating a global solution when applied to benchmark functions. Results show that the hybrid WOA-GWO technique outperforms alternative optimization strategies. Most results indicate that the hybrid WOA-GWO algorithm outperforms other methods in terms of accuracy and convergence speed, which are contingent on evaluating and approving the convergence curves. All test results show that the hybrid WOA-GWO approach optimizes convergence accuracy and overall convergence speed. Ensuring optimization correctness and convergence rates increase the algorithm’s robustness.

Figure 10.

Graphical illustration of test time for P22810 NoC using various techniques.

Table 4 shows that the Modified ACO, Modified ABC, Modified Firefly, WOA, and GWO techniques reduce the testing time for the d695 NoC benchmark circuit by 69%, 46%, 60%, 19%, and 21%, respectively. Table 5 shows that the modified ACO algorithm reduces the p22810 NoC test time by 72%, the modified ABC algorithm by 49%, the modified Firefly algorithm by 63%, and the hybrid WOA-GWO technique by 25%. The findings indicate that the suggested hybrid WOA-GWO methodology significantly reduces testing time compared to current methods.

Figures 9 and 10 show the execution timings of several approaches for the p22810 and d695 NoC benchmarks, respectively. The graph clearly shows that the suggested WOA-GWO method significantly reduces testing time compared to other alternatives. Depending on the test outcomes, existing approaches may take longer to evaluate than the suggested method.

5. Conclusion

To improve meta-heuristic methodologies, we present a new hybrid WOA-GWO approach that combines the best features of WOA and GWO. We test the proposed optimization algorithms, hybrid WOA-GWO, WOA, and GWO, using the NoC benchmarks p22810 and d695. We estimate the testing times of the hybrid WOA-GWO, WOA, and GWO algorithms for various TAM widths. We demonstrate the superior performance of the recommended method using 14 commonly used benchmark functions and one NP-hard test scheduling problem for NoC. Experiments show that the proposed method for finding the best answer outperforms current methods and offers advantages for exploration. The hybrid WOA-GWO algorithm minimizes the test time to 72%, 49%, 63%, 21%, and 25% for p22810 and 69%, 46%, 60%, 19%, and 21% for p22810 SoC, respectively. The findings imply that the proposed methodology outperforms alternative tactics and increases research efficacy. The methods under examination have two advantages: they produce excellent results and speed up the convergence process. Further investigation will be required to identify the ideal list schedule, number of partitions, and most acceptable places. Unquestionably, the hybrid WOA-GWO approach supplied the best answers for the ITC’02 circuits. Algorithms like the artificial fish swarm algorithm, hybrid WOA-SA, and hybrid Harris hawks could potentially reduce test time in the future. Future research can focus on optimizing test processes for NoCs while keeping TAM width, 3D placement, and temperature constraints in mind.

Footnotes

Acknowledgments

I want to acknowledge this manuscript has not been published elsewhere.

ORCID iD

Sadesh S

Gokul Chandrasekaran

Rajasekaran Thangaraj

Neelam Sanjeev Kumar

Conflict of interest

The authors have no conflicts of interest.

Data availability statement

The article includes the data used to bolster the investigation’s conclusions.

Funding statement

This study was not funded.

References

Larsson

Fujiwara

, System-on-chip test scheduling with reconfigurable core wrappers, IEEE Transactions on Very Large Scale Integration (VLSI) Systems 14 (2006), 305–309. doi: https://doi.org/10.1109/tvlsi.2006.871757.

Marinissen

E.J.

Iyengar

V.S.

Chakrabarty

, A set of benchmarks for modular testing of SOCs, International Test Conference. (2002). doi: https://doi.org/10.1109/test.2002.1041802.

Wang

G.G.

Deb

Cui

, Monarch butterfly optimization, Neural Computing and Applications 31 (2015), 1995–2014. doi: https://doi.org/10.1007/s00521-015-1923-y.

Chen

Wang

Heidari

A.A.

Mirjalili

, Slime mould algorithm: A new method for stochastic optimization, Future Generation Computer Systems 111 (2020), 300–323. doi: https://doi.org/10.1016/j.future.2020.03.055.

Mohamed

A.A.A.

Mohamed

Y.S.

El-Gaafary

A.A.M.

Hemeida

A.M.

, Optimal power flow using moth swarm algorithm, Electric Power Systems Research 142 (2017), 190–206. doi: https://doi.org/10.1016/j.epsr.2016.09.025.

Yang

Chen

Heidari

A.A.

Gandomi

A.H.

, Hunger games search: Visions, conception, implementation, deep analysis, perspectives, and towards performance shifts, Expert Systems with Applications 177 (2021), 114864. doi: https://doi.org/10.1016/j.eswa.2021.114864.

Ahmadianfar

Heidari

A.A.

Gandomi

A.H.

Chu

Chen

, RUN beyond the metaphor: An efficient optimization algorithm based on Runge Kutta method, Expert Systems with Applications 181 (2021), 115079. doi: https://doi.org/10.1016/j.eswa.2021.115079.

Chen

Wang

Gandomi

A.H.

, The Colony Predation Algorithm, Journal of Bionic Engineering 18 (2021), 674–710. doi: https://doi.org/10.1007/s42235-021-0050-y.

Ahmadianfar

Heidari

A.A.

Noshadian

Chen

Gandomi

A.H.

, INFO: An efficient optimization algorithm based on weighted mean of vectors, Expert Systems with Applications 195 (2022), 116516. doi: https://doi.org/10.1016/j.eswa.2022.116516.

10.

Ramadan

Kamel

Korashy

Almalaq

Domínguez-García

J.L.

, An enhanced Harris Hawk optimization algorithm for parameter estimation of single, double and triple diode photovoltaic models, Soft Computing 26 (2022), 7233–7257. doi: https://doi.org/10.1007/s00500-022-07109-5.

11.

Makhadmeh

S.N.

Al-Betar

M.A.

Doush

I.A.

Awadallah

M.A.

Kassaymeh

Mirjalili

Zitar

R.A.

, Recent advances in Grey Wolf Optimizer, its versions and applications: Review, IEEE Access. (2023), 1–1. doi: https://doi.org/10.1109/access.2023.3304889.

12.

Gharehchopogh

F.S.

Gholizadeh

, A comprehensive survey: Whale Optimization Algorithm and its applications, Swarm and Evolutionary Computation 48 (2019), 1–24. doi: https://doi.org/10.1016/j.swevo.2019.03.004.

13.

Liu

Asarry

Hassan

M.K.

Hairuddin

A.A

Mohamad

, Review of the grey wolf optimization algorithm: variants and applications, Neural Computing and Applications 36 (2023). doi: https://doi.org/10.1007/s00521-023-09202-8.

14.

Mohammed

Rashid

, A novel hybrid GWO with WOA for global numerical optimization and solving pressure vessel design, Neural Computing and Applications 32 (2020), 14701–14718. doi: https://doi.org/10.1007/s00521-020-04823-9.

15.

Zhan

Xunsheng

Bing

Guangze

Nan

, A Test Scheduling Scheme for Core-Based SoCs Using Genetic Algorithm, IEEE International Conference on Embedded Software and Systems Symposia. (2008). doi: https://doi.org/10.1109/icess.symposia.2008.65.

16.

Zou

Reddy

S.M.

Pomeranz

Huang

, SOC test scheduling using simulated annealing, IEEE VLSI Test Symposium. (2003). doi: https://doi.org/10.1109/vtest.2003.1197670.

17.

Mahi

Baykan

Ö.K.

Kodaz

, A new hybrid method based on Particle Swarm Optimization, Ant Colony Optimization and 3-Opt algorithms for Traveling Salesman Problem, Applied Soft Computing 30 (2015), 484–490. doi: https://doi.org/10.1016/j.asoc.2015.01.068.

18.

J.B.

Chun

Kim

J.H.

Kang

, RAIN (RAndom Insertion) Scheduling Algorithm for SoC Test, IEEE Test Symposium. (2005). doi: https://doi.org/10.1109/ats.2004.71.

19.

Dorigo

Birattari

Stutzle

, Ant colony optimization, IEEE Computational Intelligence Magazine. 1 (2006), 28–39. doi: https://doi.org/10.1109/MCI.2006.329691.

20.

Wang

, Ant Colony Optimization-Based Thermal-Aware Adaptive Routing Mechanism for Optical NoCs, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 40 (2021), 1836–1849. doi: https://doi.org/10.1109/tcad.2020.3029132.

21.

Karaboga

Akay

, A comparative study of Artificial Bee Colony algorithm, Applied Mathematics and Computation 214 (2009), 108–132. doi: https://doi.org/10.1016/j.amc.2009.03.090.

22.

Tiunov

I.V.

Vasilyev

N.O.

, The Modification of the ABC Synthesis Algorithm for FPGAs Considering Architecture Features, 2021 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (ElConRus). (2021). doi: https://doi.org/10.1109/elconrus51938.2021.9396324.

23.

Gandomi

A.H.

Yang

X.S.

Talatahari

Alavi

A.H.

, Firefly algorithm with chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013), 89–98. doi: https://doi.org/10.1016/j.cnsns.2012.06.009.

24.

Amin

Hussain

Anjum

Khan

Baloch

N.K.

Nain

Kim

S.W

, Performance Evaluation of Application Mapping Approaches for Network-on-Chip Designs, IEEE Access 8 (2020), 63607–63631. doi: https://doi.org/10.1109/access.2020.2982675.

25.

Yang

X.S.

, Bat algorithm: literature review and applications, International Journal of Bio-Inspired Computation 5 (2013), 141. doi: https://doi.org/10.1504/ijbic.2013.055093.

26.

Gong

Zhang

Yang

Wang

Fan

Zhang

, Cooperative co-evolutionary algorithm for multi-objective optimization problems with changing decision variables, Information Sciences 607 (2022), 278–296. doi: https://doi.org/10.1016/j.ins.2022.05.123.

27.

Zhang

Gong

Ding

, Handling multi-objective optimization problems with a multi-swarm cooperative particle swarm optimizer, Expert Systems with Applications. 38 (2011). doi: https://doi.org/10.1016/j.eswa.2011.04.200.

28.

Moghaddasi

Rajabi

Gharehchopogh

F.S.

Hosseinzadeh

, An Energy-Efficient Data Offloading Strategy for 5G-Enabled Vehicular Edge Computing Networks Using Double Deep Q-Network, Wireless Personal Communications 133 (2023), 2019–2064. doi: https://doi.org/10.1007/s11277-024-10862-5.

29.

Moghaddasi

Rajabi

Gharehchopogh

F.S.

Ghaffari

, An Advanced Deep Reinforcement Learning Algorithm for Three-layer D2D-Edge-Cloud Computing Architecture for Efficient Task Offloading in Internet of Things, Sustainable Computing Informatics and Systems 43 (2024), 100992–100992. doi: https://doi.org/10.1016/j.suscom.2024.100992.

30.

Moghaddasi

Rajabi

Gharehchopogh

F.S.

, Multi-Objective Secure Task Offloading Strategy for Blockchain-Enabled IoV-MEC Systems: A Double Deep Q-Network Approach, IEEE Access 12 (2024), 3437–3463. doi: https://doi.org/10.1109/access.2023.3348513.

31.

Goel

Marinissen

E.J.

Sehgal

Chakrabarty

, Testing of SoCs with Hierarchical Cores: Common Fallacies, Test Access Optimization, and Test Scheduling, IEEE Transactions on Computers 58 (2009), 409–423. doi: https://doi.org/10.1109/tc.2008.169.

32.

Mirjalili

Lewis

, The Whale Optimization Algorithm, Advances in Engineering Software 95 (2016), 51–67. doi: https://doi.org/10.1016/j.advengsoft.2016.01.008.

33.

Mirjalili

S.M.

Lewis

, Grey Wolf Optimizer, Advances in Engineering Software. 69 (2014), 46–61. doi: https://doi.org/10.1016/j.advengsoft.2013.12.007.

34.

Singh

Hachimi

, A New Hybrid Whale Optimizer Algorithm with Mean Strategy of Grey Wolf Optimizer for Global Optimization, Mathematical and Computational Applications 23 (2018), 14. doi: https://doi.org/10.3390/mca23010014.

35.

Perolat

Couso

Loquin

Strauss

, Generalizing the Wilcoxon rank-sum test for interval data, International Journal of Approximate Reasoning 56 (2015), 108–121. doi: https://doi.org/10.1016/j.ijar.2014.08.001.