Comprehensive Technical Review of Recent Bio-Inspired Population-Based Optimization (BPO) Algorithms for Mobile Robot Path Planning

Over recent decades, the field of mobile robot path planning has evolved significantly, driven by the pursuit of enhanced navigation solutions. The need to determine optimal trajectories within complex environments has led to the exploration of diverse path planning methodologies. This paper focuses on a specific subset: Bio-inspired Population-based Optimization (BPO) methodologies. BPO methods play a pivotal role in generating efficient paths for path planning. Amidst the abundance of optimization approaches over the past decade, only a fraction of studies has effectively integrated these methods into path planning strategies. This paper’s focus is on the years 2014-2023, reviewing BPO techniques applied to mobile robot path planning challenges. Contributions include a comprehensive review of recent BPO methods in mobile robot path planning, along with an experimental methodology to compare them under consistent conditions. This encompasses the same environment, initial conditions, and replicates. A multi-objective function is incorporated to evaluate optimization methods. The paper delves into key concepts, mathematical models, and algorithm implementations of examined optimization techniques. The experimental setup, methodology, and benchmarking performance results are discussed. Based on the proposed experimental methodology, Improved Sparrow Search Algorithm (ISpSA) shows the best cost improvement percentage (7.87%), but suffers in terms of optimization time. On the other hand, Whale Optimization Algorithm (WOA) has lesser improvement percentage of 6.05% but better optimization time. In conclusion, the standardized approach for benchmarking BPO algorithms provides useful insights into their strengths and challenges in mobile robot path planning.


I. INTRODUCTION
In recent decades, substantial evolution has been witnessed in the field of mobile robot path planning, driven by the pursuit The associate editor coordinating the review of this manuscript and approving it for publication was Genoveffa Tortora .
of refined navigation solutions.The determination of optimal trajectories within intricate environments has prompted the exploration of a diverse range of path planning methodologies.This study centers its focus on a specific subset amidst this diversified landscape: Bio-inspired Populationbased Optimization (BPO) methodologies.With inspiration drawn from natural phenomena, these techniques endeavor to enhance the efficiency and adaptability of path planning strategies.Amidst the plethora of available methodologies, attention is channeled towards the application of BPO techniques in the domain of mobile robot path planning.
Building upon this foundation, this paper stands as an updated and extended contribution, which emphasized on experimental methodology to analyze the performance of recent BPO algorithms.

A. PAPER SCOPES AND OBJECTIVES
The scope of this technical review paper covers a comprehensive analysis of BPO methods applied to the domain of path planning over the past decade (2014)(2015)(2016)(2017)(2018)(2019)(2020)(2021)(2022)(2023).While the last 10 years have witnessed a multitude of optimization techniques, this review specifically narrows its focus to the category of recent BPO methods.Within this context, the emphasis is placed solely on strategies that have been employed for solving path planning challenges in robotics.Given that population-based metaheuristic methods rely on randomness, which leads to varying optimization outcomes, the paper aims to standardize experimental conditions when reviewing and analyzing the algorithms.

B. PAPER CONTRIBUTIONS
The contributions of this paper are as follows: 1) Firstly, the paper compiles a review of recent BPO methods applied to mobile robot path planning over the last ten years.
2) The second contribution of this paper is the design of an experimental methodology to compare various BPO methods under consistent circumstances, including similar environment, initial conditions, and replicates.
3) The third contribution pertains to the incorporation of a multi-objective function for the purpose of evaluating the efficacy of the selected BPO methods.

C. PAPER OUTLINE
The paper's structure is organized as follows.Sections II through VI elaborate on key concepts and mathematical model of the recent BPO techniques that have been applied in mobile robot path planning.There are seven BPO methods discussed which are Grey Wolf Optimization (GWO), Whale Optimization Algorithm (WOA), Improved Whale Optimization Algorithm (IWOA), Grasshopper Optimization Algorithm (GOA), Salp Swarm Algorithm (SSA), Sparrow Search Algorithm (SpSA) and Improved Sparrow Search Algorithm (ISpSA).Since both Salp Swarm Algorithm and Sparrow Search Algorithm were being referred to SSA in multiple literature, to prevent confusion, the latter is replaced with SpSA.
The equations featured in these sections represent the finalized formulas commonly employed for each respective BPO method.Comprehensive derivations of these equations can be found in the original sources referenced for each method.For a thorough grasp of the derivations and the underlying mathematical concepts, interested readers are encouraged to refer to these sources.Given the potential variations in mathematical models and terminologies across diverse optimization methods, a table of nomenclature has been included at the beginning of each optimization section.This serves to facilitate convenient reference and comprehension of common terms used for each optimization method.Additionally, each section concludes with a detailed algorithm outlining the practical implementation of the respective optimization approach.
Section VII meticulously outlines the experimental setup.This encompasses the environment configuration (Section VII-A), population initialization (Section VII-B), postprocessing of generated paths (Section VII-C), the utilized objective functions (Section VII-D), the devised experimental design and its step-by-step execution (Section VII-E).Concluding this subsection, Section VII-F delves into the discussion of the achieved benchmarking performance results.
In Section VIII, an in-depth analysis of the simulation outcomes and the benchmarking results are presented.Finally, the paper is brought to a close in Section IX.

II. GREY WOLF OPTIMIZATION (GWO)
GWO was proposed by [8] based on Grey wolf (Canis lupus) that belongs to a Canidae family.Grey wolves are considered as apex predators, meaning that they are at the top of the food chain.Grey wolves mostly prefer to live in a pack.
In the GWO algorithm, the optimization process is guided by the alpha, beta, and delta wolves, mirroring the hunting behavior observed in actual wolf packs.These three wolves take the lead in exploring the search space and determining the direction of the optimization process.The omega wolves, representing subordinate members of the pack, follow the alpha, beta, and delta, aligning their movement and choices with those of the leaders.
By emulating the hierarchical structure of wolf packs in the GWO algorithm, the optimization process is influenced by the dominant wolves, while the subordinate wolves adapt and learn from their movements.This hierarchical framework allows the GWO algorithm to strike a balance between exploration and exploitation, leveraging the leadership of the alpha, beta, and delta wolves to guide the optimization towards better solutions.

A. MATHEMATICAL MODEL
To incorporate the social hierarchy of wolves into the GWO algorithm, a mathematical representation is used.The fittest solution in the population is designated as the alpha (α), representing the highest-ranking wolf.Similarly, the second and third best solutions are referred to as the (β) and delta (δ), respectively.The remaining candidate solutions are considered omega (ω), representing lower-ranking wolves within the pack.Table 1 shows the nomenclature used in GWO.

B. ENCIRCLING PREY
The mathematical model of the behaviour of grey wolves encircling prey during the hunt are represented as follows: where t indicates the current iteration, A and C are coefficient vectors, X p is the position vector of the prey, X indicates the position of a grey wolf.
This is accomplished by reducing the value of a in the Eq. ( 3).Note that the fluctuation range of A is likewise decreased by a.In other words A is a random number in the interval [−a, a] where a is decreased from 2 to 0 throughout iterations.

C. WOLF HUNTING
In order to mathematically simulate the hunting behavior of grey wolves, it is assumed that the alpha (best candidate solution), beta, and delta have better knowledge about the potential location of prey.Therefore, the first three best solutions obtained so far were saved and the other search agents (including the omegas) were prompted to update their positions according to the position of the best search agents as shown in Eq. ( 5) -Eq.( 7): The algorithm flow of GWO is shown in Algorithm 1. for each wolf X i do 9:

Algorithm 1 Grey Wolf Optimization Algorithm
Update the position of current wolf X i by Eq. ( Update the position X i if exceed boundaries Calculate the fitness of all search agents 14: Update X α , X β , and X δ 15: end while 16: return X α

III. WHALE OPTIMIZATION ALGORITHM (WOA)
WOA was initially proposed by [9].It is an optimization algorithm that is inpired by humpback whales.Once they have discovered their target, humpback whales possess the ability to encircle it.The WOA approach operates under the premise that the current best candidate solution is either the prey of interest or is very near to the optimal.This is because the placement of the optimal design inside the search space is a priori unknown.Once the best search agent has been selected, the remaining search agents will attempt to draw closer to it.Table 2 shows the nomenclature used in WOA.Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

A. MATHEMATICAL MODEL
The behaviour of the whale is demonstrated through Eq. ( 8)-Eq.( 9): where t represents the current iteration, A and C are coefficient vectors, X * represents the position vector of the best outcome so far, X represents the position vector, Two strategies are created to mathematically simulate humpback whale behaviour in bubble nets: The shrinking encircling mechanism is similar to GWO (refer to Eq. (1) -Eq.( 2)).

2) SPIRAL UPDATING POSITION
The first step in this process is to determine the distance between the whale at position (X, Y) and the prey at position (X * , Y * ).A spiral equation (see Eq. (10)) is then built between the position of the whale and its prey in order to mimic the helix-shaped movement of humpback whales.
where D ′ is represented in Eq. (11) and indicates the distance between the i-th whale to the prey (best solution obtained so far), b is a constant for defining the shape of the logarithmic spiral, l is a random number in [−1, 1] and • is an elementby-element multiplication.
Note that humpback whales swim simultaneously in a diminishing circle and spiral pattern around their prey.To characterise this concurrent behaviour, we assume a 50 percent likelihood of picking either the shrinking encirclement mechanism or the spiral model to update the whales' position.In addition to the bubble-net method, humpback whales also search for prey at random.The mathematical model of the search is as follows.

C. SEARCH FOR PREY (EXPLORATION PHASE)
Prey search (exploration) can be done using the same approach based on an adjustment to the A vector.Humpback whales actually conduct random searches depending on their respective positions.Therefore, to force the search agent to move away from a reference whale, A with random values greater than 1 or lower than -1 were utilised.The position of a search agent is updated based on a randomly chosen search agent during the exploration phase as opposed to the best search agent so far discovered during the exploitation phase.With the help of this method and |A| > 1, which emphasise exploration, the WOA algorithm is able to conduct a global search.The mathematical model appears as follows: where X rand is a random position vector (a random whale) chosen from the current population.

D. MODIFICATIONS
Based on original WOA, the algorithm decides position update mechanism based on the value of p.The algorithm's weak global search capability and ease of falling into the local optimum are caused by the random nature of the variable p and the fact that the WOA only does a global search at p < 0.5.This problem was addressed by the changes that [42] suggested for WOA.From this point forward, the Improved Whale Optimization Algorithm (IWOA) will be used to refer to all of these adjustments.The first change is the addition of an adaptive dynamic adjustment mechanism to enhance the WOA's capabilities for both global and local optimization.
As seen in Eq. ( 14) parameter p is replaced with pp that changes dynamically depending on the ratio of maximum iteration t max and current iteration, t.
The position update formula for both the exploitation and exploration phases remained unchanged, with the sole modification being the adjustment of the conditions that trigger the activation of these methods (see Eq. ( 15) -Eq.( 16) and Algorithm 2).
The second modification is the shrinking behaviour of a.In the original WOA, A is a random number in the interval [−a, a] where a is decreased from 2 to 0 throughout iterations.Since behaviour of A is determined by value a, component k is added to introduce adjustibility of the algorithm inclination to either global optimization or local optimization.
where k is directly related to the iteration number t.
The final modification is to introduce Differential Evolution (DE) (refer Eq. ( 19)) to aid problems of low search efficiency and ease of falling into the local optimum.This strategy is employed after the whale updates its position.The value of ϱ (refer Eq. ( 20)) decreases as the number of iterations increases.After each iteration, the current individual position is compared with the position obtained after updating with the differential mutation evolution strategy, and take the optimal position before and after the change.
Algorithm 2 Whale Optimization Algorithm 1: Initialize the whales population Calculate the fitness of each search agent (whale) 3: X * = the best search agent 4: while t < t max do 5: for each search agent do 6: Update a, A, C, l, and p/pp Update the position of the current search agent by using Eq. ( 8) -Eq.( 9) 10: Select random search agent X rand 12: Update the position of the current search agent by Eq. ( 12) -Eq.( 13)

IV. GRASSHOPPER OPTIMIZATION ALGORITHM (GOA)
GOA was proposed by [10].Grasshoppers exhibit distinctive swarming behavior throughout their life cycle, encompassing both the nymph and adult stages.During the nymph phase, millions of grasshoppers move in a coordinated manner, resembling rolling cylinders, as they consume vegetation along their path.In adulthood, they form aerial swarms that facilitate long-distance migration.
The swarming characteristics vary between the nymph and adult stages.Nymph swarms exhibit slow movement and take small steps, while adult swarms display abrupt and wideranging movements.These swarms actively search for food sources, aligning with the underlying principles of natureinspired algorithms, which involve a balance between exploration and exploitation.Grasshoppers naturally exhibit these tendencies, with abrupt movements during exploration and more localized movements during exploitation.Furthermore, grasshoppers demonstrate target-seeking behavior.Table 3 shows the nomenclature used in GOA.

A. MATHEMATICAL MODEL
Eq. ( 21) represents the equation utilized to update the position of the grasshopper: where ub d is the upper bound in the d-th dimension, lb d is the lower bound in the d-th dimension, T d is best solution found so far, and c is a decreasing coefficient to shrink the comfort zone, repulsion zone, and attraction zone (refer Eq. ( 22)).
The social component s(r) is defined as Eq.(23).
where r = |x d j − x d i |, l is the attraction force and f is the repulsion force.Algorithm 3 shows the optimization flow of GOA.

V. SALP-SWARM ALGORITHM (SSA)
SSA was proposed by [11].In order to establish a mathematical representation of the salp chains, the population was initially divided into two distinct groups: the leader and the followers.The leader corresponds to the salp positioned at the forefront of the chain, while the remaining salps are considered followers.
As their names suggest, the leader directs the swarm, while the followers adhere to each other, either directly or indirectly.

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Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Update c by Eq. ( 22) 6: for each grasshopper X i do 7: Normalize the distance between the grasshoppers 8: Update the position of current grasshopper X i by Eq. ( 21) 9: Update the position X i if exceed boundaries 10: end for 11: Update T d if there is a better solution 12: end while 13: return T d It is assumed that there exists a food source referred to as F within the search space, which serves as the population's target.Table 4 shows the nomenclature used in SSA.

A. MATHEMATICAL MODEL
The coefficient c 1 is the most important parameter in SSA because as it balances exploration and exploitation.Coefficient c 1 is defined as Eq. ( 24): Here, l represents the current iteration, while L denotes the maximum number of iterations.The parameters c 2 and c 3 are random numbers generated uniformly within the interval [0, 1].These parameters play a crucial role in determining whether the next position in the j-th dimension should be directed towards positive infinity or negative infinity, as well as influencing the step size.
To update the leader's position x 1 j , the following Eq.(25) were proposed: where x 1 j shows the position of the first salp (leader) in the j-th dimension, F j is the position of the food source in the j-th dimension, ub j indicates the upper bound of j-th dimension, lb j indicates the lower bound of j-th dimension, c 1 , c 2 , and c 3 are random numbers.Due to conflicting literature regarding the value of c 3 in the range of c 3 ≥ 0.5 versus c 3 ≥ 0, the equation presented in Eq. ( 25) adheres to the boundaries specified in the author's MATLAB code [43].
To update the position of the followers, the following Eq.( 26) is utilized: where i ≥ 2 and x j i refer to the position of the i-th follower salp in the j-th dimension.Algorithm 4 shows the optimization flow of SSA.Update c 1 by Eq. ( 24) 6: for each salp x i do 7: Update the position of the leading salp by Eq. ( 25) 9: Update the position of the follower salp by Eq. ( 26) 11: end for 13: Update the position x i if exceed boundary 14: Update F if there is a better solution 15: end while 16: return F

VI. SPARROW SEARCH ALGORITHM (SPSA)
SpSA was proposed by [12] inspired by sparrow.In the avian context, producers are sparrows known for their high energy reserves, which they use to offer navigation cues and foraging guidelines to all scroungers.This role involves identifying zones of ample food resources.The energy level of producers are determined by the fitness values.
When a potential predator is detected, the sparrows respond by emitting a chorus of alarm chirps.Once this alarm value surpasses a predetermined safety threshold, it triggers the responsibility of producers to evacuate all scroungers to secure areas.Notably, each sparrow can become a producer upon diligently exploring better food sources, while maintaining a constant ratio of producers to scroungers across the population.Also, some famished scroungers are more inclined to seek alternate feeding spots to restore their energy.Scroungers follow producers who excel at locating optimal food sources for foraging.
Simultaneously, scroungers are tasked with the vigilant surveillance of producers and participate in competitive interactions to improve their predation rate.When faced with threats, sparrows in close proximity to a group swiftly reposition themselves to safer locations, thereby optimizing their positions.In contrast, individuals situated at the group's center exhibit stochastic movement patterns, with random movement strategies aimed at maintaining proximity to fellow group members.For reference, Table 5 provides an overview of the nomenclature utilized in SpSA.

A. MATHEMATICAL MODEL
The update formula for the producer's location is presented as Eq. ( 27): where t represents the current iteration number, X i,j represents the position information of the i-th sparrow with jth dimension, α is a random number in the range of [0,1].R(R ∈ [0, 1]) and S(S ∈ [0.5, 1]) represent warning and safety parameters, respectively, where R is a random number and S is a given constant.When R < S, no danger is found in the population, the search environment is safe; and the producers can conduct a wide range of searches.When R > S, the scouts find danger, adjust the search strategy, and quickly move closer to the safe area.Q is a random number that follows the normal distribution.L represents an all-one matrix of dimensions 1 × d.The location update formula of the scoungers is given by Here, X p denotes the current best position of the producer, and X t worst represents the current worst position.The symbol A corresponds to a matrix with dimensions 1 × d, where each element takes a value of either 1 or -1.Additionally, A + = A T (AA T ) −1 .When i > n/2, it signifies that the ith scrounger, characterized by having the worst fitness value, is more likely to be in a starving state.Drawing inspiration from the behavior of sparrows, it is assumed that the scout, which accounts for approximately 10% to 20% of the sparrow population, possesses the ability to detect danger.The formula for updating the location of the scout is represented as Eq. ( 29): In this context, X best denotes the current global optimal position.The symbol β represents a step length control parameter, which is obtained by drawing a random number from a normal distribution with a mean of 0 and a variance of 1. K is another random number chosen from the interval [−1, 1].Furthermore, f i stands for the current fitness value of an individual sparrow, while f g and f w denote the current global optimal and worst fitness values, respectively.To avoid encountering undefined results, a very small constant, denoted as ε, has been introduced.

B. MODIFICATIONS
In the original SpSA, the individual sparrow position update is influenced by its past position.Yet, a challenge arises if the current fitness level is not optimal.This leads to a degraded fitness outcome in the updated position.To address this, [38] proposed an upgraded version known as Improved Sparrow Search Algorithm (ISpSA), where in each sparrow position update considers its present personal-best fitness value.The improved position update formula for the producer is given by Eq. ( 30): where X t e represents the fittest position at X t−1 i,j .Consequently, the position of scrounger (Eq.( 31)) and scouts (Eq.( 32)) are modified by replacing the current position value X t i,j with X t e .
To further enhance the solution, ISpSA additionally incorporates a neighborhood search approach (refer Algorithm 5).

VII. EXPERIMENTAL SETUP
Based on the existing literature, seven optimization methodologies for path planning have been selected for analysis.These studies share typical attributes during the construction of the path planner.Therefore, particular studies such as [34] and [39] are excluded from the analysis due to their different path planning initialization approaches.The combined experimental configuration from the relevant literature using the most recent BPO algorithms for path planning is shown in Table 6.These literature sources provide a concise summary of the key advantages and drawbacks of each BPO methods.The experimental simulation setup, along with the implementation of all tested optimization methods, was developed and coded using MATLAB version 2023a.

A. ENVIRONMENT SETUP
To ensure experiment standardization, a 20 by 20 grid environment map [38] was chosen for testing all optimization methods.Environment configurations are depicted in Figure 1

B. POPULATION INITIALIZATION
Similarly, to ensure standardized results, the population initialization for each optimization approach is kept uniform.The MATLAB RNG ''twister'' function, which utilizes the Marsenne Twister generator, is employed for this purpose.This strategy ensures replicable results, as all randomized numbers adhere to the identical sequence dictated by the specified seeds.

C. SEQUENTIAL LINEAR PATHS (SLP)
In this experiment, lightweight SLP method proposed by [38] was used to improve the smoothness of the path and reduce the path generation time.Figure 2 shows the resultant plot that utilizes SLP.Interested readers are encouraged to read [38] and [45] for in depth explanation of the working mechanism of SLP.

D. OBJECTIVE FUNCTIONS
For this experiment, there will be two objective functions to be tested.The first is to find the path cost.The path cost is determined by using Eq.(33).
where (r k , c k ), (k = 1, 2, . . ., m) is the sequence of grid points inside path with total number of sequence m.
The second objective function is to get the smoothness of the path.The smoothness of each path can be calculated using Eq.(38).The path is viewed as a sequence of segments and the angle of the triangle it forms is calculated.There are a total segments of m − 1.
The value cos ρ for each segment i can be determined as follows: Rank the fitness values and find the current best individual X best and the current worst individual X worst . 6: X best = the best search agent Update the position based on R value using Eq. ( 27) for SpSA or Eq. ( 30) for ISpSA.Update the position using Eq. ( 28) for SpSA or Eq. ( 31) for ISpSA. 13: end for 14: for j = 1:SD do 15: Update the position using Eq. ( 29) for SpSA or Eq. ( 32) for ISpSA.The smoothness of the whole path can be calculated by taking the sum of the smoothness of all the segments.The path smoothness function is then determined by taking the average value of all the ρ(X i ).
It is important to note that a lower value of the smoothness function indicates a smoother path.The smoothness function does not have a specific unit of measurement.
In the experiment, there are two types of resultant objective function f (X) to be tested (refer Eq. ( 39)).The first objective v1 considers only the path cost whilst the second objective v2 takes consideration the smoothness component of the path on top of the path cost.The resultant objective function v2 combines both path cost and smoothness cost using weighted average function: where w 1 and w 2 is the weightage of each objective.Based on preliminary tuning result, the selected w 1 and w 2 are 0.7 and 0.3 respectively.It should be noted that for SSA 20950 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
method, in order to standardize the objective functions across all tests, the weighted average method was chosen instead of Multi-objective SSA (MSSA) method proposed in [11].

E. EXPERIMENTAL DESIGN
An experimental configuration using a factorial design was executed to investigate the performance of the optimization technique under varying parameter combinations.Each trial underwent three replications.The total number of experimental runs amounted to 84, with each optimization method being subjected to 12 runs.The presented algorithm, labeled as Algorithm 6, outlines the procedural steps involved in conducting a benchmarking experiment to evaluate different optimization methods for path planning.The purpose of this algorithm is to systematically assess the performance of various optimization techniques in the context of path planning tasks.
The experiment was organized in a structured manner, involving multiple nested loops.The outermost loop was iterated through three replicates, which represent different random number generator (RNG) seeds to ensure reliable and consistent results across trials.Within each replicate, a sequence of actions was undertaken to evaluate the optimization methods comprehensively.For each replicate, the Marsenne Twister RNG was reset and reinitialized to establish a consistent starting point for randomization.This step is crucial to eliminate any potential biases originating from random number generation.Subsequently, an inner loop was iterated through two different environments to simulate various path planning scenarios.Within each environment, another loop was iterated through two different types of objective functions, reflecting different optimization goals.For every combination of environment and objective type, the algorithm initialized the chosen objective function and the population of candidate solutions.It then calculated the fitness of these solutions and recorded the initial best cost.Within the objective function and environment context, the algorithm entered the most intricate loop.This loop was iterated through seven different optimization methods, each representing a distinct path planning approach.For each method, the optimization process was executed, resulting in a path planning solution.The algorithm recorded the benchmarking performance metrics to evaluate the efficiency and effectiveness of the optimization method.Reset and reinitialize Marsenne Twister   Four performance outputs were recorded which include optimization best cost, optimization time, optimization cost improvement percentage and optimization convergence rate.The equation to calculate the cost improvement percentage is shown in Eq. (40).Since it is a minimization problem, it is assumed that the initial cost is equal to or greater than the final optimized cost.

B. ANALYSIS: OPTIMIZATION BEST COST
The boxplot featured in Figure 7 illustrates the assessment of performance among the various optimization algorithms examined.In Trial 1, ISpSA achieved the most favorable cost, followed by IWOA, GOA, WOA, GWO, and SpSA, while SSA exhibited the highest median best cost.The optimization consistency is notably observed with ISpSA and GWO, which present the least spread and a reliable performance trend across distinct RNG replicates.In contrast, IWOA displays the highest spread, indicative of a wider range of performance outcomes and potential sensitivity to input variations.Moving to Trial 2, ISpSA consistently maintains the lowest median, indicating its superior performance.However, it's worth noting that GOA's performance demonstrates more variance compared to the scenario involving the 1st objective function across diverse RNG replicates.Transitioning to Trial 3, which poses increased complexity compared to Environment 1, all optimization algorithms yield slightly higher optimal costs.Notably, ISpSA, WOA, and IWOA exhibit comparable median values.Among these, ISpSA stands out with its lesser variability across RNG trials.Upon testing Trial 4, ISpSA continues to lead with the lowest median value, while SSA lags in performance.Similar trends are observed in terms of optimization variability, with ISpSA  consistently displaying a dependable performance pattern, and IWOA presenting the most variation.

C. ANALYSIS: OPTIMIZATION TIME
Figure 8 shows the boxplot analysis of optimization time for various optimization algorithms.Understanding these computational characteristics is essential for selecting appropriate optimization strategies based on the specific requirements of an application.
The analysis of optimization time across the four trials has provided valuable insights into the computational performance of various optimization methods.In Trial 1, both WOA 20954 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.and SSA demonstrated the lowest median optimization times, highlighting their efficiency.IWOA and SpSA followed suit with slightly higher median times, while GWO, GOA, and ISpSA exhibited progressively higher median times.Trial 2 reaffirmed the prowess of SSA, which displayed the lowest median optimization time.Similarly, WOA and IWOA sustained their efficiency.In contrast, SpSA and GWO exhibited higher median times, and ISpSA and GOA showed the highest median optimization times.Trial 3 witnessed consistent trends, with both WOA and SSA maintaining their low median optimization times, while GWO, GOA, and ISpSA showed progressively higher median times.Trial  4 echoed the earlier trials, showcasing SSA's efficiency, followed by WOA and IWOA.Subsequently, SpSA and GWO exhibited higher median times, while ISpSA and GOA recorded the highest median optimization times.In all trials, spread analyses were aligned with these patterns, underscoring the methods' consistency and variability in computational performance.

D. ANALYSIS: OPTIMIZATION PERCENTAGE IMPROVEMEMENT
The examination of optimization percentage improvement in Trial 1 reveals distinct trends.ISpSA and the upper quartile of WOA exhibit the highest median percentage improvement, indicating their superior performance.WOA's lower quartile, along with SpSA, achieve the fourth-highest median percentage improvement, emphasizing noteworthy advancements.IWOA secures the second-highest median, underscoring its effective enhancement.Notably, GOA and GWO share a median that ties for the third-highest improvement, signifying substantial solution quality improvement.SSA, GOA, and ISpSA exhibit median spreads, suggesting moderate variability.Moreover, GWO and WOA demonstrate larger spreads, suggesting more diverse enhancement outcomes.The widest spread is observed in IWOA, implying a broad range of improvement percentages.
Meanwhile, in Trial 2, GWO exhibits no improvement, while ISpSA stands out with the highest improvement percentage, signifying remarkable progress.Following  The analysis of optimization percentage improvement in Trial 4 yields compelling insights into the efficacy of the evaluated optimization methods.The assessment of median improvement percentages underscores ISpSA's leadership with the highest median improvement, closely followed by IWOA.Intriguingly, GWO and GOA share a similar median improvement, highlighting their comparable performance.Subsequently, the sequence of median improvements continues with SpSA and concludes with SSA, which exhibits the lowest median improvement.Notably, IWOA displays the most extensive spread in improvement percentages, suggesting significant variability in its performance outcomes.These BPO methods showcase progressively diminishing spread values starting from IWOA, followed by WOA, GWO, GOA, ISpSA, SSA and finally SpSA.remarkable performance, showcasing the fastest convergence rate and consistently achieving the lowest attained cost among all evaluated methods.This robust and reliable behavior positions ISpSA as a standout optimization approach for path planning challenges.
Moreover, the trials unveiled intriguing trends in convergence behavior for other optimization methods.The GOA method exhibited rapid convergence in most trials, yet often plateaued at certain stages, limiting its ability to achieve further improvements.The IWOA method showcased consistent convergence but at a slower pace, ultimately converging to costs similar to ISpSA.Meanwhile, the GWO method displayed distinctive behavior by plateauing in early stages before converging later, indicating its sensitivity to initial conditions.Similarly, the SpSA method exhibited rapid convergence followed by plateau phases, resulting in costs comparable to other methods.
Throughout the trials, the WOA method consistently exhibited steady and consistent convergence, securing costs comparable to ISpSA.The SSA and GWO methods, however, consistently displayed the absence of convergence, suggesting limitations in their application for the considered path planning scenarios.While it was not within the scope of the experiment, it is worth noting that the convergence rate of the SSA may yield varying results for the tests conducted using Objective v2 (Trial 2 and Trial 4), should the method be replaced with the repository function of the MSSA.In our future work, we will consider the integration of MSSA in the experimental setup.

IX. CONCLUSION
This paper compiles recent BPO algorithms applied in path planning methodology and introduces a standardized approach for reviewing and evaluating BPO algorithms from a benchmarking standpoint.Table 9 shows the BPO ranking for improvement percentage across all trials and replicates.The experimental findings lead to the conclusion that ISpSA demonstrates superior performance in optimizing the best cost and achieving the highest improvement percentage at 7.87%.Nonetheless, it is important to acknowledge that this optimization method requires more processing time compared to other alternatives.An alternative optimization technique WOA, offers faster optimization time but has lower improvement percentage of 6.05% compared to ISpSA.However, it's worth noting that WOA exhibits greater variability in benchmark outcomes compared to ISpSA.By standardizing the methodology used to analyze and review BPO algorithms, this paper aims to provide fellow researchers with clear insights for choosing an appropriate method tailored to their specific needs in the field of path planning.

Algorithm 6
Experiment Steps in Benchmarking BPO Method for Path Planning 1: for replicate=1:3 do 2: RESULT AND ANALYSIS A. SIMULATION RESULT: PLANNED PATH The resulting planned paths for individual replicate are depicted in the respective figures: Figure 3 for Trial 1, Figure 4 for Trial 2, Figure 5 for Trial 3, and Figure 6 for Trial 4.

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FIGURE 7 .
FIGURE 7. Boxplot analysis of optimization best cost for various optimization algorithms (lower values indicate better performance).

FIGURE 8 .
FIGURE 8. Boxplot analysis of optimization time for various optimization algorithms (lower values indicate better performance).

FIGURE 9 .
FIGURE 9. Boxplot analysis of optimization improvement percentage for various optimization algorithms (higher values indicate better performance).

E
Figure10shows the convergence plot of optimization best cost sampled from one of the replicates.The conducted trials encompassed a comprehensive evaluation of various optimization methods in the context of path planning in different environment with different objective functions.The convergence plot analysis yielded valuable insights into the behaviors and performances of these methods across different trials.Notably, the ISpSA method consistently demonstrated

TABLE 1 .
Common terms used in GWO.

TABLE 2 .
Common terms used in WOA.

TABLE 3 .
Common terms and parameters used in GOA.

TABLE 4 .
Common terms used in SSA.

TABLE 5 .
Key terms and parameters used in SpSA.

TABLE 6 .
Experimental setup of BPO methods in path planning ( * refers to the reviewed setup).

Table 8
illustrates the benchmark test parameters employed across all assessed algorithms, excluding GWO and SSA, both of which did not require preset parameters.

TABLE 9 .
BPO ranking for improvement percentage across all trials and replicates.