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A Novel Hybrid GWO with WOA for Global Numerical Optimization and Solving Pressure Vessel Design

Hardi M. Mohammed, Tarik A. Rashid

arXiv:2003.11894v1cs.NE

TL;DR

WOA is described as weak in exploitation and prone to stagnation at local solutions. The paper proposes WOAGWO, combining GWO’s hunting mechanism in exploitation with a new exploration update; it reports favorable benchmark comparisons and a pressure-vessel solution better than WOA and FDO.

  • Problem

    WOA is reported to have low exploitation performance, difficulty escaping local optima, and insufficient solution improvement after iterations.

    Source: §1 · pp. 2–3

  • Method

    WOAGWO embeds GWO’s hunting mechanism in WOA’s exploitation phase and adds an exploration technique that compares new solutions with the best solution.

    Source: §1 · p. 3

  • Results

    WOAGWO was superior on 20 of 23 common functions versus WOA and GWO; Wilcoxon results were significant on 17 of 23 common, 13 CEC2005, and 6 of 10 CEC2019 functions.

    Source: §6 · pp. 23–24

  • Takeaways & Limitations

    The authors conclude that WOAGWO improves WOA on problems associated with poor performance and stagnation in exploration.

    Source: §6 · p. 24

Abstract

from arXiv · show

A recent metaheuristic algorithm, such as Whale Optimization Algorithm (WOA), was proposed. The idea of proposing this algorithm belongs to the hunting behavior of the humpback whale. However, WOA suffers from poor performance in the exploitation phase and stagnates in the local best solution. Grey Wolf Optimization (GWO) is a very competitive algorithm comparing to other common metaheuristic algorithms as it has a super performance in the exploitation phase while it is tested on unimodal benchmark functions. Therefore, the aim of this paper is to hybridize GWO with WOA to overcome the problems. GWO can perform well in exploiting optimal solutions. In this paper, a hybridized WOA with GWO which is called WOAGWO is presented. The proposed hybridized model consists of two steps. Firstly, the hunting mechanism of GWO is embedded into the WOA exploitation phase with a new condition which is related to GWO. Secondly, a new technique is added to the exploration phase to improve the solution after each iteration. Experimentations are tested on three different standard test functions which are called benchmark functions: 23 common functions, 25 CEC2005 functions and 10 CEC2019 functions. The proposed WOAGWO is also evaluated against original WOA, GWO and three other commonly used algorithms. Results show that WOAGWO outperforms other algorithms depending on the Wilcoxon rank-sum test. Finally, WOAGWO is likewise applied to solve an engineering problem such as pressure vessel design. Then the results prove that WOAGWO achieves optimum solution which is better than WOA and Fitness Dependent Optimizer (FDO).

1. Introduction

The introduction frames balancing exploration and exploitation as a key challenge for nature-inspired metaheuristics and motivates a WOA–GWO hybrid to address stated WOA weaknesses. WOAGWO combines GWO’s hunting mechanism in exploitation with an added exploration update.

1. Introduction

  • The introduction identifies exploration–exploitation balance as a central metaheuristic challenge and notes that algorithms should be evaluated on the target problem before being called best.It describes stochastic algorithms as randomized and distinguishes them from deterministic approaches,.Source: §1 · p. 2
  • Earlier WOA hybrids pair it with local search, BAT, CBO, brain-storm optimization, or PSO, using these additions for tasks including exploration, solution quality, or convergence –.The described approaches include a two-group explorer–imitator division in WOA-CBO.Source: §1 · pp. 2–3
  • GWO is reported as competitive in exploitation and to perform better on half of 29 functions, while its exploration–exploitation balance and updating mechanism remain concerns,,.The cited comparison includes PSO, DE, GSA, and FEP.Source: §1 · p. 3
  • WOA is described as weak in exploitation, prone to local optima, and insufficient at improving solutions after each iteration.The introduction identifies these issues as motivations for the hybridization, [28].Source: §1 · pp. 2–3
  • WOAGWO adds a condition and GWO’s hunting mechanism to WOA’s exploitation phase, while a new exploration update compares each solution with the best and retains improved positions.The paper presents the two changes as ways to improve WOA’s performance.Source: §1 · p. 3
  • The paper plans to evaluate WOAGWO on 23 common, 25 CEC2005, and 10 CEC2019 benchmark functions, compare it with other algorithms, and apply it to pressure-vessel design.The introduction also says statistical results will be presented.Source: §1 · p. 4

2. WOA

WOA models humpback-whale hunting through exploration and exploitation phases. Its position updates use encircling, spiral, or random-search mechanisms, while the paper notes prior modifications and hybridizations.

2. WOA

  • WOA models whale hunting in exploration and exploitation phases, with random prey search and encircling or spiral attacks.The whale algorithm description presents these as its two phases.Source: §2 · p. 4

2.2 Searching for Prey

  • During exploration, whales move relative to a randomly selected whale rather than the best-known position.The account says this random-search strategy is intended to support global optimization and overcoming local optima.Source: §2.2 · p. 5

2.3 WOA Modifications and Hybridizations

  • The paper surveys WOA modifications and hybridizations, with Table 1 titled as a summary of WOA modifications.The supplied passage says Table 2 presents several WOA hybridizations.Source: §2.3 · p. 6, Table 1 · p. 6

WOA in neural networks

The supplied passage identifies a table on WOA hybridizations.

WOA in neural networks

  • Table 2 is titled “WOA Hybridizations.”The supplied passage does not describe the table's contents.Source: Table 2 · p. 6

3. Grey Wolf Optimization

GWO represents candidate solutions through a wolf hierarchy and updates their positions using the three leading solutions. Its account distinguishes exploitation from exploration and reviews prior modifications and hybridizations.

3. Grey Wolf Optimization

  • GWO ranks candidate solutions as alpha (best), beta (second-best), delta (third-best), and omega (the remaining candidates).The account says the alpha, beta, and delta positions represent the prey location used in position updates.Source: §3 · p. 8, §3.2 · p. 8

3.1 Encircling prey

  • In encircling, wolves estimate prey distance using coefficient vectors A and C, whose values control the nearby search area.The parameter a decreases from 2 to 0 during the iterations.Source: §3.1 · p. 8

3.2 Hunting

  • During hunting, alpha, beta, and delta retain the best solutions and guide updates to the wolves’ positions.The paper describes these three ranked wolves as locating the prey.Source: §3.2 · p. 8

3.3 Attacking Prey (Exploitation)

  • For exploitation, wolves attack when A lies between -1 and 1; the paper identifies stagnation in local optima as a GWO problem.When A falls outside that range, search agents diverge from the prey.Source: §3.3 · p. 8, §3.4 · p. 9

3.4 Search for Prey (Exploration)

  • For exploration, agents diverge from the prey when A > 1, while the algorithm iteratively updates positions and the alpha, beta, and delta solutions.The algorithm description returns the alpha solution after the iterations.Source: §3.4 · p. 9

3.5 GWO Modifications and Hybridizations

  • The review summarizes prior GWO modifications and hybridizations and states that GWO had not been hybridized with WOA.The section points to Tables 3 and 4 for examples of modifications and hybridizations.Source: §3.5 · p. 10

4. Our Approach: WOAGWO

WOAGWO combines WOA and GWO to improve exploitation and adds an exploration-phase condition intended to improve solutions after each iteration. Its pseudocode initializes and updates search agents, checks their fitness, and returns the best solution.

4. Our Approach: WOAGWO

  • WOAGWO adds GWO equations and a new condition to WOA’s exploitation phase, and a separate exploration condition intended to improve solutions after each iteration.The exploitation condition is described as avoiding local optima; the exploration condition moves the current solution toward the best and avoids accepting a worse position.Source: §4 · p. 12

Algorithm 3 WOAGWO

  • WOAGWO initializes a population of whales and wolves, evaluates fitness, repairs out-of-range positions, updates the best solution, and returns it.The pseudocode repeats fitness evaluation and best-solution updates during the iteration loop.Source: §4 · p. 12, Algorithm 3 WOAGWO · p. 13
  • For p<0.5, WOAGWO calculates a search agent’s new location by Eq. (1) when |A|<1, otherwise by Eq. (2), and updates position only when current fitness is better than previous fitness.Source: Algorithm 3 WOAGWO · p. 13

4. Our Approach: WOAGWO

  • When the exploitation condition holds, WOAGWO uses the Alpha, Beta, and Delta positions with Eq. 16 to calculate a new position.The algorithm describes this condition using A1, A2, and A3.Source: §4 · p. 12, Algorithm 3 WOAGWO · p. 13

5. Experimental Result and Discussion

The section evaluates WOAGWO on classical, CEC2005, and CEC2019 benchmark functions against WOA, GWO, and other optimizers. It also applies the algorithm to pressure-vessel design.

5.1 Benchmark Functions; 5.2 Experimental Setup; 5.3 Evaluation Criteria

  • Tests cover 23 classical, 25 CEC2005, and 10 CEC2019 functions; evaluation included averages, standard deviations, algorithm comparisons, and box plots, with random initialization in Matlab R2017b on Windows 10.The benchmark suite includes multimodal, unimodal, expanded multimodal, and hybrid composition functions.Source: §5.1 · p. 14, §5.2 · p. 14, §5.3 · p. 15

5.4 WOAGWO Vs. WOA

  • On 23 functions, WOAGWO beats WOA on all seven unimodal functions and 13 of 16 multimodal functions; WOA does well on f16 and f17.The paper identifies unimodal functions as tests of exploitation and multimodal functions as tests of exploration.Source: Table 6 · p. 15, §5.4 · p. 15
  • On CEC2005, WOAGWO improves on WOA in 19 functions, WOA is better in 4, and they tie in 1.Source: Table 7 · p. 16, §5.4 · p. 16
  • On CEC2005, WOAGWO is efficient in 16 functions against GWO; GWO is better in 8, and they tie in f7.Source: Table 7 · p. 16, §5.4 · p. 16
  • On CEC2019, WOAGWO beats WOA in 7 functions and ties in 1, while it beats GWO in 5, ties in 2, and trails in 3.Source: Table 8 · p. 17, §5.4 · p. 17

5.5 Statistical Test

  • Wilcoxon tests show significant results against WOA in 13 of 25 CEC2005 functions and 6 of 10 CEC2019 functions; on the 23-function set, functions 9 and 11 are exceptions.Source: Table 9 · p. 19, §5.5 · p. 19

5.6 Comparing WOAGWO with Hybrid and Metaheuristic Algorithms

  • On CEC2005, WOAGWO and BSO each achieve best results in 9 of 25 functions, while WOAGWO raises WOA's exploration-exploitation count from 4 to 9.The comparison also describes WOAGWO as better at balancing exploration and exploitation than BSO, WOA, ABC, and DE.Source: §5.6 · p. 21, Table 11 · p. 22

5.7 WOAGWO for Solving Pressure Vessel Design Problem

  • WOAGWO outperforms WOA and FDO on pressure-vessel design, with Ts=1.63, Th=1.43, R=67.07, and L=10.The paper reports these values for shell thickness, head thickness, inner radius, and cylindrical length, respectively.Source: Table 12 · p. 23, §5.7 · p. 23

6. Conclusion

WOAGWO combines GWO hunting in WOA’s exploitation phase with a new exploration update. It outperformed WOA and GWO across many benchmark functions and achieved a pressure-vessel solution better than WOA and FDO.

6. Conclusion

  • WOAGWO outperformed WOA and GWO on 20 of 23 benchmark functions, tied WOA on 1, and was slightly worse than WOA on 2.On CEC2005, it performed well on 14 functions, tied WOA on 1, and WOA was better on 3; on CEC2019, WOAGWO was better than WOA on 7, tied on 1, and worse on 2.Source: §6 · pp. 23–24
  • Wilcoxon rank-sum testing found significant results for WOAGWO on 17 of 23 benchmark functions, 13 CEC2005 functions, and 6 of 10 CEC2019 functions.Source: §6 · p. 24
  • Against DE, ABC, BSO, and WOA, BSO was also better on 9 benchmark functions, while WOA was competitive on 4.The authors conclude that WOAGWO improved exploration, solution quality after each iteration, and avoidance of local optima.Source: §6 · p. 24
  • For pressure-vessel design, WOAGWO attained an optimum solution better than WOA and FDO.Source: §6 · p. 24
  • WOAGWO embeds GWO exploitation equations and beta- and delta-wolf solutions in WOA, while adding an exploration condition to reject updates that do not improve the previous position.The authors attribute its improved performance over WOA to these exploitation and exploration changes.Source: §6 · p. 24
  • The authors identify medical and engineering applications, other hybridizations, and chaotic maps as potential future research directions.Source: §6 · p. 24
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