Multi-trial Vector-based Whale Optimization Algorithm

Abstract

The Whale Optimization Algorithm (WOA) is a swarm intelligence metaheuristic inspired by the bubble-net hunting tactic of humpback whales. In spite of its popularity due to simplicity, ease of implementation, and a limited number of parameters, WOA’s search strategy can adversely affect the convergence and equilibrium between exploration and exploitation in complex problems. To address this limitation, we propose a new algorithm called Multi-trial Vector-based Whale Optimization Algorithm (MTV-WOA) that incorporates a Balancing Strategy-based Trial-vector Producer (BS_TVP), a Local Strategy-based Trial-vector Producer (LS_TVP), and a Global Strategy-based Trial-vector Producer (GS_TVP) to address real-world optimization problems of varied degrees of difficulty. MTV-WOA has the potential to enhance exploitation and exploration, reduce the probability of being stranded in local optima, and preserve the equilibrium between exploration and exploitation. For the purpose of evaluating the proposed algorithm's performance, it is compared to eight metaheuristic algorithms utilizing CEC 2018 test functions. Moreover, MTV-WOA is compared with well-stablished, recent, and WOA variant algorithms. The experimental results demonstrate that MTV-WOA surpasses comparative algorithms in terms of the accuracy of the solutions and convergence rate. Additionally, we conducted the Friedman test to assess the gained results statistically and observed that MTV-WOA significantly outperforms comparative algorithms. Finally, we solved five engineering design problems to demonstrate the practicality of MTV-WOA. The results indicate that the proposed MTV-WOA can efficiently address the complexities of engineering challenges and provide superior solutions that are superior to those of other algorithms.

Appendices

Appendix A

See Tables 8, 9, 10, 11, 12; Figs. 3, 4.

Table 8 The unimodal functions' average error

Table 9 The multimodal functions' average error

Table 10 The hybrid functions' average error

Table 11 The composition functions' average error

Table 12 The Friedman test results

Fig. 3

Convergence curves for some unimodal and multimodal functions

Fig. 4

Convergence curves for some hybrid and composition functions

Appendix B

See Figs. 5, 6; Tables 13, 14, 15.

Fig. 5

The impact analysis of using the proposed TVPs

Table 13 The comparison of MTV-WOA with well-stablished, recent, and WOA variant algorithms

Table 14 Results of Wilcoxon's test on D = 10

Fig. 6

Convergence comparison of MTV-WOA with flagship, recent, and WOA variant algorithms

Table 15 The results of using proposed TVPs for improving PSO and LSHADE-SPACMA algorithms

Appendix C

3.1 C.1. Pressure Vessel Design Problem

The major aim of this problem, represented in Fig. 7, is optimizing the cost of material, forming, and welding a vessel. The problem has four variables T_s, T_h, R, and L. The mathematical representation of this problem is provided in Eq. (24) (Figs. 7, 8, 9, 10, 11).

$${\text{Consider}}\;\vec{x} = \left[ {x_{1} x_{2} x_{3} x_{4} } \right] = \left[ {T_{s} { }T_{h} { }R{ }L} \right]$$

$${\text{Minimize}}\;f\left( {\vec{x}} \right) = 0.6224x_{1} x_{3} x_{4} + 1.7781x_{2} x_{3}^{2} + 3.1661x_{1}^{2} x_{4} + 19.84x_{1}^{2} x_{3}$$

${\text{Subject to}}\;g_{1} \left( {\vec{x}} \right) = - x_{1} + 0.0193x_{3} \le 0,\;g_{2} \left( {\vec{x}} \right) = - x_{2} + 0.00954x_{3} \le 0$,

$g_{3} \left( {\vec{x}} \right) = - \pi x_{3}^{2} x_{4} - \frac{4}{3}\pi x_{3}^{3} + 1,296,000 \le 0,\;g_{4} \left( {\vec{x}} \right) = x_{4} - 240 \le 0$,

$${\text{where}}\;0 \le x_{i} \le 100{\text{ for }}i = 1,{ }2\;{\text{and}}\;10 \le x_{i} \le 200{\text{ for }}i = 3,{ }4$$

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Fig. 7

Pressure vessel design

3.2 C.2. Three-Bar Truss Problem

This issue's purpose is to manufacture a truss with the least amount of weight while still adhering to three limitations. Regarding Fig. C.2, two design variables, x₁ and x₂, should be chosen while taking into account limits on stress, deflection, and buckling. Equation (25) is the mathematical representation of this problem.

$${\text{Consider}}\;\vec{x} = \left[ {x_{1} x_{2} } \right] = \left[ {A_{1} { }A_{2} } \right]$$

$${\text{Minimize}}\;f\left( {\vec{x}} \right) = \left( {2\sqrt 2 x_{1} + x_{2} } \right) \times l$$

$${\text{Subject to}}\;g_{1} \left( {\vec{x}} \right) = \frac{{\sqrt 2 x_{1} + x_{2} }}{{\sqrt 2 x_{1}^{2} + 2x_{1} x_{2} }}{\text{P}} - {\upsigma } \le 0,g_{2} \left( {\vec{x}} \right) = \frac{{x_{2} }}{{\sqrt 2 x_{1}^{2} + 2x_{1} x_{2} }}{\text{P}} - {\upsigma } \le 0,$$

$$g_{3} \left( {\vec{x}} \right) = \frac{1}{{\sqrt 2 x_{2} + x_{1} }}{\text{P}} - {\upsigma } \le 0$$

$${\text{where}}\;0 \le x_{1} ,{ }x_{2} \le 1,\;l = 100\;{\text{cm}},{ }P = 2\;{\text{kN}}/{\text{cm}}^{2} ,\;\sigma = 2\;{\text{kN}}/{\text{cm}}^{2}$$

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Fig. 8

Three-bar truss design

3.3 C.3. Welded Beam Problem

Determining the minimum cost to fabricate a welded beam is the subject of this design problem. It has four design factors that need to be optimized as shown in Fig. C.10 and four restrictions that should be considered. Equation (26) is the mathematical representation of this problem.

$${\text{Consider}}\;\vec{x} = \left[ {x_{1} x_{2} x_{3} x_{4} } \right] = \left[ {h{ }l{ }t{ }b} \right]$$

$${\text{Minimize}}\;f\left( {\vec{x}} \right) = 1.10471x_{1}^{2} x_{2} + 0.04811x_{3} x_{4} \times \left( {14.0 + x_{2} } \right)$$

${\text{Subject to}}\;g_{1} \left( {\vec{x}} \right) = {\uptau }\left( {\vec{x}} \right) - \tau_{max} \le 0,g_{2} \left( {\vec{x}} \right) = {\upsigma }\left( {\vec{x}} \right) - \sigma_{max} \le 0,g_{3} \left( {\vec{x}} \right) = {\updelta }\left( {\vec{x}} \right) - \delta_{max} \le 0$,

$$g_{4} \left( {\vec{x}} \right) = x_{1} - x_{4} \le 0,g_{5} \left( {\vec{x}} \right) = {\text{P}} - P_{{\text{c}}} \left( {\vec{x}} \right) \le 0,g_{6} \left( {\vec{x}} \right) = 0.125 - x_{1} \le 0$$

$g_{7} \left( {\vec{x}} \right) = 1.10471x_{1}^{2} + 0.04811x_{3} x_{4} \times \left( {14.0 + x_{2} } \right) - 0.5 \le 0$

$${\text{where}}\;0.1 \le x_{i} \le 2{\text{ for }}i = 1,{ }2\;{\text{and}}\;0.1 \le x_{i} \le 10{\text{ for }}i = 3,{ }4$$

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Fig. 9

Welded beam design

3.4 C.4. Tension/compression Spring Design Problem

The major goal of this design problem is to reduce the weight of the tension/compression spring. This problem has three design factors, as shown in Fig. C.11. Equation (27) is the mathematical representation of this problem.

$${\text{Consider}}\;\vec{x} = \left[ {x_{1} x_{2} x_{3} } \right] = \left[ {d{ }D{ }N} \right]$$

$${\text{Minimize }}f\left( {\vec{x}} \right) = \left( {x_{3} + 2} \right)x_{2} x_{1}^{2}$$

$${\text{Subject to }}g_{1} \left( {\vec{x}} \right) = 1 - \frac{{x_{2}^{3} x_{3} }}{{71785x_{1}^{2} }} \le 0,g_{2} \left( {\vec{x}} \right) = \frac{{4x_{2}^{2} - x_{1} x_{2} }}{{12566\left( {x_{2} x_{1}^{3} - x_{1}^{4} } \right)}} + \frac{1}{{5108x_{1}^{2} }} \le 0$$

$g_{3} \left( {\vec{x}} \right) = 1 - \frac{{140.45x_{1} }}{{x_{2}^{2} x_{3} }} \le 0,g_{4} \left( {\vec{x}} \right) = \frac{{x_{1 + } x_{2} }}{1.5} - 1 \le 0$

$${\text{where }}0.05 \le x_{1} \le 2.00,0.25 \le x_{2} \le 1.30,2.00 \le x_{3} \le 15.0$$

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Fig. 10

Tension/compression spring design

3.5 C.5. Speed Reducer Design Problem

Taking into consideration the bending stress of the gear teeth, the surface stress, the transverse deflections, and the stresses in the shafts, the goal of this restricted optimization issue is to minimize the weight of the speed reducer. This problem has seven variables, as shown in Fig. C.3. The mathematical representation of this problem shown in Eq. (28).

Consider

$${\text{Minimize}}\;\;f\left( {\vec{x}} \right) = 0.7854x_{1} x_{2}^{2} \left( {3.3333x_{3}^{2} + 14.9334x_{3} - 43.0934} \right) - 1.508x_{1} \left( {x_{6}^{2} + x_{7}^{2} } \right) + 7.4777\left( {x_{6}^{3} + x_{7}^{3} } \right) + 0.7854\left( {x_{4} x_{6}^{2} + x_{5} x_{7}^{2} } \right)$$

$\begin{aligned} {\text{Subject to}}\; & \;g_{1} \left( {\vec{x}} \right) = \frac{27}{{x_{1} x_{2}^{2} x_{3} }} - 1 \le 0,\;\;g_{2} \left( {\vec{x}} \right) = \frac{397.5}{{x_{1} x_{2}^{2} x_{3}^{2} }} - 1 \le 0. \\ & g_{3} \left( {\vec{x}} \right) = \frac{{1.93x_{4}^{3} }}{{x_{2} x_{6}^{4} x_{3} }} - 1 \le 0 \\ \end{aligned}$,

$g_{4} \left( {\vec{x}} \right) = \frac{{1.93x_{5}^{3} }}{{x_{2} x_{7}^{4} x_{3} }} - 1 \le 0,g_{5} \left( {\vec{x}} \right) = \frac{{\left[ {\left( {745\left( {x_{4} /x_{2} x_{3} } \right)} \right)^{2} + 16.9 \times 10^{6} } \right]^{1/2} }}{{110x_{6}^{3} }} - 1 \le 0$,

$$g_{6} \left( {\vec{x}} \right) = \frac{{\left[ {\left( {745\left( {x_{5} /x_{2} x_{3} } \right)} \right)^{2} + 157.5 \times 10^{6} } \right]^{1/2} }}{{85x_{7}^{3} }} - 1 \le 0$$

$g_{7} \left( {\vec{x}} \right) = \frac{{x_{2} x_{3} }}{40} - 1 \le 0,g_{8} \left( {\vec{x}} \right) = \frac{{5x_{2} }}{{x_{1} }} - 1 \le 0,g_{9} \left( {\vec{x}} \right) = \frac{{x_{1} }}{{12x_{2} }} - 1 \le 0$

$g_{10} \left( {\vec{x}} \right) = \frac{{1.5x_{6} + 1.9}}{{x_{4} }} - 1 \le 0,g_{11} \left( {\vec{x}} \right) = \frac{{1.1x_{7} + 1.9}}{{x_{5} }} - 1 \le 0$,

$${\text{where}}\;\;2.6 \le x_{1} \le 3.6,{ }0.7 \le x_{2} \le 0.8,{ }17 \le x_{3} \le 28,{ }7.3 \le x_{4} \le 8.3,$$

$$7.3 \le x_{5} \le 8.3,{ }2.9 \le x_{6} \le 3.9,{ }5.0 \le x_{7} \le 5.5$$

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Fig. 11

Speed reducer design