An integrated firefly algorithm for the optimization of constrained engineering design problems

Abstract

As a nature-inspired metaheuristic method, the firefly algorithm (FA) arises more attentions in academic and engineering fields. However, too much attraction in FA’s global attraction model leads to low computational efficiency, and the stochastic model with fixed randomization parameter is hard to balance the exploitation and exploration of the algorithm. Thus, FA still needs improvement to deal with complex engineering problems. An integrated firefly algorithm (IFA) that combines two novel attractive models with a new stochastic model is proposed to improve the standard FA. Firstly, the attractive model and stochastic model of standard FA are investigated through theoretical analysis and numerical experiments. And the factors that affect the computational efficiency and accuracy of FA are revealed. Based on the analysis results, two new fitness-based update formulas for attractiveness parameter are constructed to avoid the invalidation. The proposed virtual attractive model and global best attractive model can reduce the computation complexity and enhance the exploitation ability. Moreover, an adaptive strategy is presented for the stochastic model to achieve a better balance between exploitation and exploration. The nonlinearly decreased model for the update of parameter α can adjust the population diversity through the iteration and ensure the convergence. Additionally, an adaptive penalty function method is developed to handle the constraints effectively. Then, the initial parameters are tested, and the best initial parameters corresponding to the optimal performance of IFA are obtained. The proposed algorithm is evaluated by CEC2015 hybrid composition and a set of classical functions. The numerical experimental results show that the proposed techniques can enhance the solution accuracy and accelerate the convergence speed. Finally, IFA and other metaheuristic algorithms are applied to solve five engineering design optimization problems with mixed variables and multiple constraint conditions. The results indicate that IFA with adaptive penalty function needs fewer fitness evaluations and costs less computational time to obtain the optimal solutions. Furthermore, it exhibits better accuracy and robustness than other algorithms.

Appendices

Appendix A: Spur speed reducer design

$$ \begin{aligned}& f({\mathbf{x}}) = 0.785x_{1} x_{2}^{2} (3.33x_{3}^{2} + 14.933x_{3} - 43.0934) - 1.508x_{1} (x_{6}^{2} + x_{7}^{2} )\\& \quad \qquad + 7.477(x_{6}^{3} + x_{7}^{3} ) + 0.7854(x_{4} x_{6}^{2} + x_{5} x_{7}^{2} ) \\& g_{1} ({\mathbf{x}}) = 27x_{1}^{ - 1} x_{2}^{ - 2} x_{3}^{ - 1} \le 1, \\& g_{2} ({\mathbf{x}}) = 397.5x_{1}^{ - 1} x_{2}^{ - 2} x_{3}^{ - 1} \le 1, \\& g_{3} ({\mathbf{x}}) = 1.93x_{2}^{ - 1} x_{3}^{ - 1} x_{4}^{3} x_{6}^{ - 4} \le 1, \\& g_{4} ({\mathbf{x}}) = 1.93x_{2}^{ - 1} x_{3}^{ - 1} x_{5}^{3} x_{7}^{ - 4} \le 1, \\& g_{5} ({\mathbf{x}}) = \frac{{A_{1} }}{{B_{1} }} \le 1, \, A_{1} = \left[ {\left( {\frac{{745x_{4} }}{{x_{2} x_{3} }}} \right)^{2} + (16.9 \times 10^{6} )} \right]^{1/2} , \, B_{1} = 110x_{6}^{3} \\ \end{aligned} $$

$$ \begin{aligned}& g_{6} ({\mathbf{x}}) = \frac{{A_{2} }}{{B_{2} }} \le 1, \, A_{2} = \left[ {\left( {\frac{{745x_{5} }}{{x_{2} x_{3} }}} \right)^{2} + (157.5 \times 10^{6} )} \right]^{1/2} ,\quad B_{2} = 85x_{7}^{3} \\& g_{7} ({\mathbf{x}}) = x_{2} x_{3} \le 40, \\& g_{8} ({\mathbf{x}}) = \frac{{x_{1} }}{{x_{2} }} \le 12, \\& g_{9} ({\mathbf{x}}) = \frac{{x_{1} }}{{x_{2} }} \ge 5, \\& g_{10} ({\mathbf{x}}) = (1.5x_{6} + 1.9)x_{4}^{ - 1} \le 1, \\& g_{11} ({\mathbf{x}}) = (1.1x_{7} + 1.9)x_{5}^{ - 1} \le 1, \\ \end{aligned} $$

where 2.6 ≤ x₁ ≤ 3.6, 0.7 ≤ x₂ ≤ 0.8, 17 ≤ x₃ ≤ 28, 7.3 ≤ x₄, x₅ ≤ 8.3, 2.9 ≤ x₆ ≤ 28, 5.0 ≤ x₇ ≤ 5.5.

Appendix B: Vehicle side impact design

Minimize:$f({\mathbf{x}}) = Weight.$

Subject to:

$$ g_{1} (x) = F_{a} \left( {\text{load in abdomen}} \right) \le 1kN, $$

$g_{2} ({\mathbf{x}}) = VC_{u} \left( {\text{dummy upper chest}} \right) \le 0.32{\text{m/s,}}$

$$ g_{3} ({\mathbf{x}}) = VC_{m} \left( {\text{dummy middle chest}} \right) \le 0.32{\text{m/s,}} $$

$g_{4} ({\mathbf{x}}) = VC_{l} \left( {\text{dummy lower chest}} \right) \le 0.32{\text{m/s,}}$

$$ g_{5} ({\mathbf{x}}) = \Delta_{ur} \left( {\text{upper rib deflection}} \right) \le 32{\text{mm,}} $$

$g_{6} ({\mathbf{x}}) = \Delta_{mr} \left( {\text{middle rib deflection}} \right) \le 32{\text{mm,}}$

$$ g_{7} ({\mathbf{x}}) = \Delta_{lr} \left( {\text{lower rib deflection}} \right) \le 32{\text{mm,}} $$

$g_{8} ({\mathbf{x}}) = F_{p} \left( {\text{Pubic force}} \right) \le 4{\text{kN,}}$

$$ g_{9} ({\mathbf{x}}) = V_{{{\text{MBP}}}} \left( {\text{Velocity of V - pillar at middle point}} \right) \le 9.9{\text{mm/ms,}} $$

$g_{10} ({\mathbf{x}}) = V_{{{\text{FD}}}} \left( {\text{Velocity of front door at V - Pilar}} \right) \le 15.7{\text{mm/ms}}{.}$

To simplify the analytical formulation of the optimization problem and speed up computations, the structural weight and response to impact can be approximated using global response surface methodology. The simplified models are defined as follows:

$$ Weight = 1.98 + 4.90x_{1} + 6.67x_{2} + 6.98x_{3} + 4.01x_{4} + 1.78x_{5} + 2.73x_{7} , $$

$$ F_{a} = 1.16 - 0.3717x_{2} x_{4} - 0.009931x_{2} x_{10} - 0.484x_{3} x_{9} + 0.01343x_{6} x_{10} , $$

$$ VC_{l} = 0.74 - 0.61x_{2} - 0.163x_{3} x_{8} + 0.001232x_{3} x_{10} - 0.166x_{7} x_{9} + 0.227x_{2}^{2} , $$

$$ \begin{aligned} VC_{m} & = 0.214 + 0.00817x_{5} - 0.131x_{1} x_{8} - 0.0704x_{1} x_{9} + 0.03099x_{2} x_{6} + 0.018x_{2} x_{7} + 0.0208x_{3} x_{8} + 0.121x_{3} x_{9} \\ & - 0.00364x_{5} x_{6} + 0.0007715x_{5} x_{10} - 0.0005354x_{6} x_{10} + 0.00121x_{8} x_{11} + 0.00184x_{9} x_{10} - 0.02x_{{_{2} }}^{2} , \\ \end{aligned} $$

$$ \begin{gathered} VC_{u} = 0.261 - 0.0159x_{1} x_{2} - 0.188x_{1} x_{8} - 0.019x_{2} x_{7} + 0.0144x_{3} x_{5} + 0.0008757x_{5} x_{10} + 0.08045x_{6} x_{9} + 0.00139x_{8} x_{11} \hfill \\ \, + 0.00001575x_{10} x_{11} , \hfill \\ \end{gathered} $$

$$ \Delta_{ur} = 28.98 + 3.818x_{3} - 4.2x_{1} x_{2} + 0.0207x_{5} x_{10} + 6.63x_{6} x_{9} - 7.7x_{7} x_{8} + 0.32x_{9} x_{10} , $$

$$ \Delta_{mr} = 33.86 + 2.95x_{3} + 0.1792x_{10} - 5.057x_{1} x_{2} - 11.0x_{2} x_{8} - 0.0215x_{5} x_{10} - 9.98x_{7} x_{8} + 22.0x_{8} x_{9} , $$

$$ \Delta_{lr} = 46.36 - 9.9x_{2} - 12.9x_{1} x_{8} + 0.1107x_{3} x_{10} , $$

$$ F_{p} = 4.72 - 0.5x_{4} - 0.19x_{2} x_{3} - 0.0122x_{4} x_{10} + 0.009325x_{6} x_{10} + 0.000191x_{{_{11} }}^{2} , $$

$$ V_{{{\text{MBP}}}} = 10.58 - 0.674x_{1} x_{2} - 1.95x_{2} x_{8} + 0.02054x_{3} x_{10} - 0.0198x_{4} x_{10} + 0.028x_{6} x_{10} , $$

$$ V_{{{\text{FD}}}} = 16.45 - 0.489x_{3} x_{7} - 0.843x_{5} x_{6} + 0.0432x_{9} x_{10} - 0.0556x_{9} x_{11} - 0.000786x_{{_{11} }}^{2} . $$

where 0.5 ≤ x₁, x₃, x₄ ≤ 1.5, 0.45 ≤ x₂ ≤ 1.35, 0.875 ≤ x₅ ≤ 2.625, 0.4 ≤ x₆, x₇ ≤ 1.2, x₈, x₉ ∈ {0.192, 0.345}, -30 ≤ x_10, x₁₁ ≤ 30.