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A Systematic and Meta-analysis Survey of Whale Optimization Algorithm

Hardi M. Mohammed, Shahla U. Umar, Tarik A. Rashid

arXiv:1903.08763v3cs.NE

TL;DR

The paper addresses the absence of a systematic survey of WOA despite its use across optimization problems. It synthesizes WOA’s mechanisms, limitations, modifications, hybridizations, applications, and comparative results, and proposes WOA-BAT. The survey reports strong WOA performance in exploration, exploitation, convergence, and local-optimum avoidance, while WOA-BAT improves on WOA across specified benchmark sets.

  • Problem

    No systematic survey search had examined WOA, motivating a synthesis of its algorithmic foundations, limitations, variants, hybridizations, applications, and comparative performance.

  • Method

    The paper conducts a systematic and meta-analysis survey of WOA studies, compares modifications and hybridizations, and proposes WOA-BAT using BAT for exploration and WOA for exploitation.

  • Results

    WOA performs well across unimodal, multimodal, and composite benchmark functions, while WOA-BAT outperforms WOA on 13 CEC2005 functions and 7 CEC2019 functions.

  • Takeaways & Limitations

    The survey provides a consolidated basis for using, modifying, and hybridizing WOA, including a competitive WOA-BAT approach for optimization benchmarks.

  • Takeaways & Limitations

    The authors state that WOA has poor convergence speed near the global optimum and that comparing all WOA variants is impossible because studies use different benchmark functions.

Abstract

from arXiv · show

Whale Optimization Algorithm (WOA) is a nature-inspired meta-heuristic optimization algorithm, which was proposed by Mirjalili and Lewis in 2016. This algorithm has shown its ability to solve many problems. Comprehensive surveys have been conducted about some other nature-inspired algorithms, such as ABC, PSO, etc.Nonetheless, no survey search work has been conducted on WOA. Therefore, in this paper, a systematic and meta analysis survey of WOA is conducted to help researchers to use it in different areas or hybridize it with other common algorithms. Thus, WOA is presented in depth in terms of algorithmic backgrounds, its characteristics, limitations, modifications, hybridizations, and applications. Next, WOA performances are presented to solve different problems. Then, the statistical results of WOA modifications and hybridizations are established and compared with the most common optimization algorithms and WOA. The survey's results indicate that WOA performs better than other common algorithms in terms of convergence speed and balancing between exploration and exploitation. WOA modifications and hybridizations also perform well compared to WOA. In addition, our investigation paves a way to present a new technique by hybridizing both WOA and BAT algorithms. The BAT algorithm is used for the exploration phase, whereas the WOA algorithm is used for the exploitation phase. Finally, statistical results obtained from WOA-BAT are very competitive and better than WOA in 16 benchmarks functions. WOA-BAT also outperforms well in 13 functions from CEC2005 and 7 functions from CEC2019.

1. Introduction

This paper surveys WOA’s biological foundations, search mechanisms, characteristics, limitations, modifications, hybridizations, and applications. It presents WOA as a metaheuristic that alternates exploration and exploitation to balance global search, convergence, and avoidance of local optima.

  • Survey scope: The survey reviews WOA studies, modifications, hybridizations, applications, and benchmark comparisons, then proposes WOA-BAT to improve exploration and convergence.WOA-BAT uses BAT for exploration and WOA for exploitation, with the stated goal of overcoming local optima and increasing convergence speed.
  • WOA foundations: WOA models humpback-whale hunting through exploitation mechanisms and random prey search during exploration.Exploitation uses encircling and spiral position updating, while exploration reorganizes locations around a randomly selected search agent.
  • WOA foundations: WOA selects shrinking encircling or spiral movement with 50% probability during exploitation.The spiral model imitates the helix-shaped movement of humpback whales, while shrinking encircling reduces the oscillating search range.
  • WOA foundations: The algorithm alternates between best-solution updating when |A| < 1 and random-agent updating when |A| > 1.The parameter a decreases from 2 to 0, supporting transitions between exploration and exploitation across iterations.
  • WOA characteristics: Independent exploration and exploitation stages help WOA avoid local optima while maintaining convergence speed.The paper contrasts this two-stage updating with PSO and GSA, which use one position-update format.
  • WOA limitations: WOA limitations include randomization-related computational cost, dependence on parameter a, weak escape from local optima, and restricted binary-space applicability.The paper also reports difficulties with complex environmental constraints and 0–1 knapsack problems without additional functions.

2. WOA Modifications and Hybridizations

The survey reviews WOA modifications and hybridizations that adjust exploration, exploitation, convergence, and application-specific performance. It also describes WOA’s core update rules and reports improved results for several variants and hybrid algorithms.

  • Original WOA: The algorithm pseudocode updates parameters, positions, fitness values, and the best solution until the iteration limit is reached.Boundary violations are amended before fitness evaluation and best-solution updates.
  • Modifications: Adaptive WOA changes parameters according to fitness values, reducing parameter dependence and computational time while avoiding local optima through fast convergence.Self-adaptive WOA also improved performance over GSA, GA, ABC, PSO, and related methods for IoT cluster-head selection.
  • Modifications: Improved WOA variants replace the linear distance-control schedule with nonlinear functions, including sine, cosine, tangent, logarithmic, and square forms.The linear parameter is described as poorly adaptive to WOA’s nonlinear and complex search process.
  • Modifications: IWOA outperformed ABC, FOA, and WOA in mean and standard deviation across 31 benchmark functions using a population of 1000 and 30 iterations.Its mean values were lower than ABC and FOA on several listed functions, while convergence was faster than the compared algorithms.
  • Hybridizations: Hybridization extends WOA through chaotic local search, scheduling operators, heuristic initialization, or combinations with other optimization methods.Examples include MWOA for exploration, HWA-NEH for flow-shop scheduling, WGC for clustering, BS-WOA for cloud-data privacy, and ILWOA for cloud VM replacement.

3. Applications of WOA

WOA has been applied across engineering, scheduling, energy, cloud computing, and medical-image problems. These applications adapt WOA or combine it with local search and other techniques to address domain-specific optimization objectives.

  • Electrical and Electronics Engineering: WOA was used for capacitor placement in radial power-distribution systems to reduce line losses and improve system constancy and accuracy.The application targets voltage-ratio and power-factor issues caused by inductive loads.
  • Economic Scheduling: WOA was applied to economic dispatch on the IEEE 30-Bus test system to minimize power-plant generation costs.The results were compared with other metaheuristic approaches.
  • Economic Scheduling: A WOA-local-search hybrid addressed NP-hard flow-shop scheduling by reducing makespan and required time.Swap mutation increased schedule diversity, while insert-reverse block operations addressed local optima.
  • Applications: EWOA was tested on four truss and frame structural-optimization problems and showed better efficiency than standard WOA.The test cases included spatial 72-bar and 582-bar structures and 3-bay 15-story and 24-story frames.
  • Fuel and Energy; Medical Engineering: WOA-based methods were applied to photovoltaic modeling, solar-power extraction, and medical-image liver segmentation.Reported energy applications include chaotic and modified whale algorithms, while WOA was used for liver segmentation in MRI images.
  • Application scope: WOA applications include engineering, binary, multiobjective, and scheduling problems.A summary table is presented for several problem classes solved by WOA.

4. Benchmark Functions Experiment

The benchmark experiments assess WOA’s exploitation, exploration, local-optima avoidance, convergence, applications, and variants using standard functions and comparative tests. Results generally show competitive performance, while modifications such as AWOA and ILWOA improve specific outcomes.

  • Comparison and Discussion: 29 benchmark functions were used to validate WOA and its modifications across unimodal, multimodal, fixed-dimension multimodal, and composite categories.The experiments compare WOA with other metaheuristic algorithms and report average and standard-deviation results.
  • Capability Exploitation Assessment: WOA performs well on unimodal functions, ranking first for F1 and F2 and second on almost all remaining functions.These functions evaluate exploitation because they contain only one local optimum.
  • Capability Exploration Assessment: WOA ranks second on multimodal functions, demonstrating good exploration capability across landscapes with multiple local optima.The passage attributes this performance to WOA’s integrated exploration mechanism.
  • Escaping from Local Minima: WOA ranked first in three composite-function tests and was competitive with other optimization algorithms while balancing exploration and exploitation.The balance is presented as important for avoiding local optima in composite functions.
  • Convergence Behavior Analysis: WOA’s convergence rate was competitive on 29 benchmark functions, combining random exploration and spiral movement toward the best solution.The two phases occupy approximately half of the iterations each, supporting local-optima avoidance and rapid convergence.
  • WOA Variants: AWOA improved on WOA through faster convergence, randomness, and stochastic search, while ILWOA achieved the best host-machine-utilization results among the compared methods.ILWOA’s efficiency increased as the number of bin datasets increased.

5. Standard Bat Algorithm

The standard Bat Algorithm models microbat echolocation to search an optimization space. Its agents update positions using velocity, frequency, wavelength-related randomness, and loudness or pulse-emission behavior.

  • Echolocation of Microbats: The Bat Algorithm was developed by Xin-She Yang in 2010 from the echolocation capabilities of microbats.Bats use emitted sound pulses and returning echoes to detect prey or avoid obstacles.
  • Standard Bat Algorithm: Bat agents fly randomly while updating positions and velocities through wave frequency, random wavelength variation, and loudness parameters.The algorithm uses position x_i, velocity v_i, frequency f_i, random vector β, and the best solution x*.
  • Standard Bat Algorithm: The random vector β is sampled from a regular distribution within [0, 1], while each bat initially receives a frequency between f_min and f_max.The frequency limits depend on the optimization problem’s domain size.
  • Standard Bat Algorithm: As a bat finds prey or food, its loudness decreases and its pulse-emission rate increases.This parameter behavior is part of the algorithm’s echolocation-inspired search process.

6. Hybrid WOA-BAT Algorithm

The WOA-BAT hybrid addresses WOA’s slow convergence by embedding BAT-based exploration into WOA and retaining a replacement condition for improved candidate positions. It is evaluated against WOA on multiple benchmark suites.

  • Hybrid Design: WOA-BAT uses the BAT algorithm to improve WOA’s exploration capability and obtain better results in fewer iterations.BAT is partially embedded in WOA’s search phase, and a new position replaces the old one when it is better.

7. Implementation and Results

The study implements WOA-BAT and evaluates it against WOA and other metaheuristics across mathematical, CEC2005, and CEC2019 benchmark functions. WOA-BAT generally improves performance, especially across 13 CEC2005 functions and 7 CEC2019 functions.

  • Implementation and evaluation: WOA-BAT combines WOA and BAT procedures and evaluates them on 23 mathematical functions, CEC2005, and CEC2019 benchmarks.The evaluation uses average and standard deviation, box-and-whisker comparisons with WOA, and comparisons with other metaheuristics.
  • Comparison with WOA: WOA-BAT performs better than WOA on f3–f7, while WOA remains better on f1 and f2 in the exploitation tests.For multimodal tests, WOA-BAT has the optimum average in 10 functions, whereas WOA does so in 4 functions.
  • CEC2005 comparison: 13 of 25 CEC2005 functions favor WOA-BAT over WOA, while two functions show the same performance.WOA-BAT is better on f1, f2, f3, f4, f6, f9, f10, f12, f13, f18, f19, f22, and f25.
  • Comparison with metaheuristics: WOA-BAT ranks first among GA, DE, ABC, and BSO on CEC2005, outperforming well in 13 functions.Its overall ranking is 1.6, compared with 2.6 for BSO; ABC has the best standard deviation in 8 functions.

8. Conclusion

The conclusion finds that WOA performs well across unimodal, multimodal, and composite benchmarks, while WOA-BAT improves results and ranks strongly against competing algorithms. It also identifies limitations and several directions for future research.

  • Conclusion: WOA performs well on exploitation in unimodal functions and exploration in multimodal functions, while composite functions assess balance between both capabilities.The study also reports that WOA uses independent exploration and exploitation stages in each iteration.
  • Limitations: WOA cannot solve every optimization problem and has poor convergence speed while searching around the global optimum.The conclusion presents these as limitations of the original WOA.
  • Limitations: Comparing all WOA modifications and hybridizations is difficult because they use different benchmark functions and cannot all be evaluated against one another.The paper proposes a shared platform for uploading programs to facilitate systematic comparisons.
  • Conclusion: WOA-BAT outperforms WOA on 13 CEC2005 functions and has lower averages on 7 of 10 CEC2019 functions.WOA-BAT also ranks first among GA, DE, ABC, and BSO in the CEC2005 comparison.
  • Future research: Future research includes adaptive exploration–exploitation control, constrained and discrete optimization, machine-learning training, IoT cluster-head selection, and text or feature mining.The paper also suggests hybridizing WOA-BAT with other optimization algorithms and applying it to business and healthcare problems.

Composition Comments

The composition comments request verification of algorithm citation placement, author name display, and ORCID information for all listed authors.

  • Citation placement: Algorithm citations should be checked and relocated if necessary to match the required citation style.The comment specifically asks for confirmation of citation placement within the text.
  • Author information: The authors’ given names and last names should be confirmed for correct display on websites and indexing databases.The listed authors are Hardi M. Mohammed, Shahla U. Umar, and Tarik A. Rashid.
  • ORCID information: Each author should provide an ORCID to address name ambiguity in scholarly communications.The comments direct authors to register through the manuscript tracking system.
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