Hybrid metaheuristic algorithm using butterfly and flower pollination base on mutualism mechanism for global optimization problems

Abstract

The butterfly optimization algorithm (BOA) is a new metaheuristic algorithm that is inspired from food foraging behavior of the butterflies. Because of its simplicity and effectiveness, the algorithm has been proved to be effective in solving global optimization problems and applied to practical problems. However, BOA is prone to local optimality and may lose its diversity, thus suffering losses of premature convergence. In this work, a hybrid metaheuristic algorithm using butterfly and flower pollination base on mutualism mechanism called MBFPA was proposed. Firstly, the flower pollination algorithm has good exploration ability and the hybrid butterfly optimization algorithm and the flower pollination algorithms greatly improve the exploration ability of the algorithm; secondly, the symbiosis organisms search has a strong exploitation capability in the mutualism phase. By introducing the mutualism phase, the algorithm's exploitation capability is effectively increased and the algorithm's convergence speed is accelerated. Finally, the adaptive switching probability is increased to increase the algorithm's balance in exploration and exploitation capabilities. In order to evaluate the effectiveness of the algorithm, in the 49 standard test functions, the proposed algorithm was compared with six basic metaheuristic algorithms and five hybrid metaheuristic algorithms. MBFPA has also been used to solve five classic engineering problems (three-bar truss design problem; multi-plate disc clutch brake design; welded beam design; pressure vessel design problem; and speed reducer design). The results show that the proposed method is feasible and has good application prospect and competitiveness.

Appendix A

1.1 A.1. Three-bar truss design

Minimize $f(x) = (2\sqrt 2 x_{1} + x_{2} ) \times l$.

Subject to $g_{1} (x) = \sqrt 2 x_{1} + x_{2} /\sqrt 2 x_{1}^{2} + 2x_{1} x_{2} P - \sigma \le 0;$

$$g_{2} (x) = x_{2} /\sqrt 2 x_{1}^{2} + 2x_{1} x_{2} P - \sigma \le 0;$$

$$g_{3} (x) = 1/\sqrt 2 x_{2} + x_{1} P - \sigma \le 0;$$

Variable range $0 \le x_{1} ,x_{2} \le 1$.

where $l = 100\,{\hbox{cm}};\;\,P = 2\,{\hbox{kN}}/{\hbox{cm}}^{2} ;r = 2\,{\hbox{kN}}/{\hbox{cm}}^{2}$.

1.2 A.2. Multi-plate disc clutch brake design problem

Minimize $f(x) = \pi (x_{2}^{2} - x_{1}^{2} )x_{3} (x_{5} + 1)\rho$.

Subject to:

$g_{1} (x) = x_{2} - x_{1} - \Delta R \ge 0$, $g_{2} (x) = L_{\max } - (x_{5} + 1)(x_{3} + \delta ) \ge 0$

$g_{3} (x) = P_{\max } - P_{rz} \ge 0$, $g_{4} (x) = P_{\max } v_{sr\max } - P_{rz} v_{sr} \ge 0$

$g_{5} (x) = v_{sr\max } - v_{sr} \ge 0$, $g_{6} (x) = T_{\max } - T \ge 0$

$g_{7} (x) = M_{h} - sM_{s} \ge 0$, $g_{8} (x) = T \ge 0$

where $M_{h} = \frac{2}{3}\mu x4x5\frac{{x_{2}^{3} - x_{1}^{3} }}{{x_{2}^{2} - x_{1}^{2} }},$ $w = \frac{\pi n}{{30}}rad/s$, $A = \pi (x_{2}^{2} - x_{1}^{2} )\,{\hbox{mm}}^{2} ,$

$P_{rz} = \frac{{x_{4} }}{A}\,{\hbox{ N}}/{\hbox{mm}}^{2} ,$ $V_{sr} = \frac{{\pi R_{sr} n}}{30}\,{\hbox{mm}}/{\hbox{s}}$, $R_{sr} = \frac{2}{3}\frac{{x_{2}^{3} - x_{1}^{3} }}{{x_{2}^{2} x_{1}^{2} }}\,{\hbox{mm}}$

$T = \frac{{I_{z} \pi n}}{{30(M_{h} + M_{f} )}}$ mm, $\Delta r = 20\,{\hbox{mm}},L\max = 30\,{\hbox{mm}},\mu = 0.6$

$T_{\max } = 15s,\mu = 0.5,s = 1.5,M_{s} = 40Nm,$

$p_{\max } = 1Mp_{a} ,\rho = 0.0000078\,{\hbox{kg}}/{\hbox{mm}}^{3}$, $v_{sr\max } = 10\,{\hbox{m}}/{\hbox{s}},\delta = 0.5\,{\hbox{mm}},s = 1.5$

$T_{\max } = 15s,n = 250\,{\hbox{rpm}},\;I_{z} = 55{\hbox{kg}}\,{\hbox{m}}^{2}$, $M_{s} = 40\,{\hbox{Nm}},\;M_{f} = 3\,{\hbox{Nm}}$

$60 \le x_{1} \le 80,90 \le x_{2} \le 110,1 \le x_{3} \le 3,$

$60 \le x_{4} \le 1000,2 \le x_{5} \le 9,$$i = 1,2,3,4,5$.

1.3 A.3. Pressure vessel design problem

Consider $x = [x_{1} ,x_{2} ,x_{3} ,x_{4} ] = [T_{s} ,T_{h} ,R,L]$.

Minimize $f(x) = 0.6224x_{1} x_{3} x_{4} + 1.7781x_{2} x_{2}^{3} + 3.1661x_{1}^{2} x_{4} + 19.84x_{1}^{2} x_{3}$.

Subject to:

$g_{1} (x) = - x_{1} + 0.0193x_{3} \le 0$, $g_{2} (x) = - x_{3} + 0.00954z_{3} \le 0$

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

Variable range $0 \le x_{1} ,x_{2} \le 99$; $0 \le x_{3} ,x_{4} \le 200$.

1.4 A.4. Welded beam design problem

Consider ${\text{Z}} = [z_{1} ,z_{2} ,z_{3} ,z_{4} ] = [h,l,t,b]$.

Minimize $f(Z) = 1.10471z_{1}^{2} z_{2} + 0.04811z_{3} z_{4} (14.0 + z_{2} )$.

Subject to

$g_{1} (Z) = \tau ({\text{Z) - }}\tau_{{\max}} \le 0$,$g_{2} (Z) = \sigma (Z) - \sigma_{\max } \le 0$,$g_{3} (Z) = \delta (Z) - \delta_{\max } \le 0$

$g_{4} (Z) = z_{1} - z_{4} \le 0$,$g_{5} (Z) = P - P_{c} (Z) \le 0$,$g_{6} (Z) = 0.125 - z_{1} \le 0$

$g_{7} (Z) = 1.10471z_{1}^{2} + 0.04811z_{3} z_{4} (14.0 + z_{2} ) - 5.0 \le 0$

Variable range $0.05 \le z_{1} \le 2.00$,$0.25 \le z_{2} \le 1.30$, $2.00 \le z_{3} \le 15.0$

where $\tau (Z) = \sqrt {\tau ^{{\prime}{2}} + 2\tau ^{\prime}\tau ^{\prime\prime}\frac{{z_{2} }}{2R} + \tau ^{\prime\prime}2} ,\tau ^{\prime} = \frac{P}{{\sqrt 2 z_{1} z_{2} }},\tau ^{\prime\prime} = \frac{MR}{J},M = P(L + \frac{{z_{2} }}{2})$

$R = \sqrt {\frac{{z_{2}^{2} }}{4} + (\frac{{z_{1} + z_{3} }}{2})^{2} } ,J = 2\left\{ {\sqrt 2 z_{1} z_{2} \left[ {\frac{{z_{2}^{2} }}{12} + (\frac{{z_{1} + z_{3} }}{2})^{2} } \right]} \right\}$

$\sigma (Z) = \frac{6PL}{{z_{4} z_{3}^{2} }},\delta (Z) = \frac{{4PL^{3} }}{{Ez_{3}^{3} z_{4} }},P_{c} (Z) = \frac{{4.013E\sqrt {\frac{{z_{3}^{2} z_{4}^{6} }}{36}} }}{{L^{2} }}\left( {1 - \frac{{z_{3} }}{2L}\sqrt {\frac{E}{4G} }} \right)$

$P = 6000lb,L = 14in,E = 30 \times 10^{6} psi,G = 12 \times 10^{6} psi$.

1.5 A.5. Speed reducer

Minimize $f(x) = 0.785x_{1} x_{2}^{2} (3.333x_{3}^{2} + 14.9334x_{3} - 42.0934)$

$$- 1.508x_{1} (x_{6}^{2} + x_{7}^{2} ) + 7.4777x_{1} (x_{6}^{3} + x_{7}^{3} ) + 1.508x_{1} (x_{4} x_{6}^{2} + x_{5} x_{7}^{2} )$$

Subject to

$g_{1} (X) = \frac{27}{{x_{1} x_{2}^{2} x_{3} }} - 1 \le 0$, $g_{2} (X) = \frac{397.5}{{x_{1} x_{2} x_{3}^{2} }} - 1 \le 0$

$g_{3} (X) = \frac{{1.93x_{4}^{3} }}{{x_{1} x_{3} x_{6}^{4} }} - 1 \le 0$,$g_{4} (X) = \frac{{1.93x_{4}^{3} }}{{x_{1} x_{3} x_{7}^{4} }} - 1 \le 0$

$g_{5} (X) = \frac{1}{{110x_{6}^{3} }}\sqrt {(\frac{{745x_{4} }}{{x_{{{2} }} x_{{{3} }} }})^{2} + 16.9 \times 10^{6} } - 1 \le 0$ $g_{6} (X) = \frac{1}{{85x_{7}^{3} }}\sqrt {(\frac{{745x_{4} }}{{x_{2} x_{3} }})^{2} + 157.5 \times 10^{6} } - 1 \le 0$

$g_{7} (X) = \frac{x2x3}{{40}} - 1 \le 0$, $g_{8} (X) = \frac{5x2}{{x1}} - 1 \le 0$, $g_{9} (X) = \frac{x1}{{12x2}} - 1 \le 0$

$g_{10} (X) = \frac{1.5x6 + 1.9}{{x4}} - 1 \le 0$, $g_{11} (X) = \frac{1.1x7 + 1.9}{{x5}} - 1 \le 0$.

where $2.6 \le x1 \le 3.6,$$0.7 \le x2 \le 0.8,$$17 \le x3 \le 28,$$7.3 \le x4 \le 8.3$

$7.8 \le x5 \le 8.3,$$2.9 \le x6 \le 3.9,$$5.0 \le x7 \le 5.5$.