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November  2018, 1(4): 349-368. doi: 10.3934/mfc.2018017

## An effective hybrid firefly algorithm with the cuckoo search for engineering optimization problems

 1 Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada V2C 0C8 2 Department of Computer Science, Faculty of Computers & Informatics, Suez Canal University, Ismailia, Egypt 3 Postdoctoral Fellow, Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada V2C 0C8

* Corresponding author: Ahmed F. Ali

The research of the 1st author is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Received  August 2018 Revised  October 2018 Published  December 2018

Firefly and cuckoo search algorithms are two of the most widely used nature-inspired algorithms due to their simplicity and inexpensive computational cost when they applied to solve a wide range of problems. In this article, a new hybrid algorithm is suggested by combining the firefly algorithm and the cuckoo search algorithm to solve constrained optimization problems (COPs) and real-world engineering optimization problems. The proposed algorithm is called Hybrid FireFly Algorithm and Cuckoo Search (HFFACS) algorithm. In the HFFACS algorithm, a balance between the exploration and the exploitation processes is considered. The main drawback of the firefly algorithm is it is easy to fall into stagnation when the new solution is not better than its previous best solution for several generations. In order to avoid this problem, the cuckoo search with Lèvy flight is invoked to force the firefly algorithm to escape from stagnation and to avoid premature convergence. The proposed algorithm is applied to six benchmark constrained optimization problems and five engineering optimization problems and compared against four algorithms to investigate its performance. The numerical experimental results show the proposed algorithm is a promising algorithm and can obtain the optimal or near optimal solution within a reasonable time.

Citation: Mohamed A. Tawhid, Ahmed F. Ali. An effective hybrid firefly algorithm with the cuckoo search for engineering optimization problems. Mathematical Foundations of Computing, 2018, 1 (4) : 349-368. doi: 10.3934/mfc.2018017
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##### References:
The general performance of the HFFACS algorithm with constrained optimization problems
Welded beam design problem
Pressure vessel design problem
Speed reducer design problem
Spring design problem
The general performance of the proposed HFFACS algorithm with engineering optimization problems
Constrained test problems
 problem f(x) Problem1 [13] $f_1=(x_1-2)^2+(x_2-1)^2$, Subject to $x_1=2x_2-1, \;\;\; \frac{x_1^2}{4}+x_2^2-1 \leq 0$ with $x_i \in [-100, 100], \;\;\; i=1, 2$ Problem2 [9] $f_2=(x_1-10)^3+(x_2-20)^3$, Subject to $100-(x_1-5)^2-(x_2-5)^2 \leq 0$, $(x_1-6)^2+(x_2-5)^2-82.81 \leq 0$, $13 \leq x_1 \leq 100, \;\;\; 0 \leq x_2 \leq 100$ Problem3 [14] $f_3=(x_1-10)^2+5(x_2-12)^2+x_3^4+ 3(x_4-11)^2+$ Subject to $10x_5^6+7x_6^2+x_7^4-4x_6 x_7-10x_6-8x_7$, $-127+2x_1^2+3x_2^4+x_3+4x_4^2+x_5 \leq 0$ $-282+7x_1+3x_2+10x_3^2+x_4-x_5 \leq 0$ $-196+23x_1+x_2^2+6x_6^2-8x_7 \leq 0$, $4x_1^2+x_2^2-3x_1 x_2+2x_3^2+5x_6-11x_7 \leq 0$, $-10 \leq x_i \leq 10$,$\; \; \; \ i=1, \ldots, 7$. Problem4 [14] $f_4=5.3578547x_3^2+0.8356891x_1 x_5 + 37.293239x_1-40792.141,$ Subject to $0\leq 85.334407+0.0056858T_1 +T_2 x_1 x_4-0.0022053x_3 x_5\leq 92$ $90 \leq 80.51249+0.0071317x_2 x_5+0.0021813x_3^2 \leq 110$ $20 \leq 9.300961+0.0047026x_3 x_5+0.0012547x_1 x_3+0.0019085x_3 x_4\leq 25$ $78 \leq x_1\leq 102, \;\;33\leq x_2 \leq 45, \;\; 27 \leq x_i \leq 45, \;\;\; i=3, 4, 5,$ Where $T_1=x_2 x_5$ and $T_5=0.0006262$ Problem5 [14] $f_5=5.3578547x_3^2+0.8356891x_1 x_5+ 37.293239x_1-40792.141$, Subject to $0\leq 85.334407+0.0056858T_1 +T_2 x_1 x_4-0.0022053x_3 x_5\leq 92$ $90 \leq 80.51249+0.0071317x_2 x_5+0.0021813x_3^2\leq 110$ $20\leq 9.300961+0.0047026x_3 x_5+0.0012547x_1 x_3+0.0019085x_3 x_4\leq 25$ $78 \leq x_1\leq 102, \;\; 33\leq x_2\leq 45, \;\; 27 \leq x_i\leq 45, \;\;i=3, 4, 5,$ Where $T_1=x_2 x_3$ and $T_5=0.00026$ Problem6 [25] $f_6=-10.5x_1-7.5x_2-3.5x_3-2.5x_4-1.5x_5-10x_6-0.5_(i=1)^5x_i^2$ Subject to $6x_1+3x_2+3x_3+2x_4+x_5-6.5 \leq 0$ $10x_1+10x_3+x_6\leq 20$ $0\leq x_i \leq 1$,$\; \; \; i=1, \ldots, 5,$ $70 \leq x_6 \leq 50$
 problem f(x) Problem1 [13] $f_1=(x_1-2)^2+(x_2-1)^2$, Subject to $x_1=2x_2-1, \;\;\; \frac{x_1^2}{4}+x_2^2-1 \leq 0$ with $x_i \in [-100, 100], \;\;\; i=1, 2$ Problem2 [9] $f_2=(x_1-10)^3+(x_2-20)^3$, Subject to $100-(x_1-5)^2-(x_2-5)^2 \leq 0$, $(x_1-6)^2+(x_2-5)^2-82.81 \leq 0$, $13 \leq x_1 \leq 100, \;\;\; 0 \leq x_2 \leq 100$ Problem3 [14] $f_3=(x_1-10)^2+5(x_2-12)^2+x_3^4+ 3(x_4-11)^2+$ Subject to $10x_5^6+7x_6^2+x_7^4-4x_6 x_7-10x_6-8x_7$, $-127+2x_1^2+3x_2^4+x_3+4x_4^2+x_5 \leq 0$ $-282+7x_1+3x_2+10x_3^2+x_4-x_5 \leq 0$ $-196+23x_1+x_2^2+6x_6^2-8x_7 \leq 0$, $4x_1^2+x_2^2-3x_1 x_2+2x_3^2+5x_6-11x_7 \leq 0$, $-10 \leq x_i \leq 10$,$\; \; \; \ i=1, \ldots, 7$. Problem4 [14] $f_4=5.3578547x_3^2+0.8356891x_1 x_5 + 37.293239x_1-40792.141,$ Subject to $0\leq 85.334407+0.0056858T_1 +T_2 x_1 x_4-0.0022053x_3 x_5\leq 92$ $90 \leq 80.51249+0.0071317x_2 x_5+0.0021813x_3^2 \leq 110$ $20 \leq 9.300961+0.0047026x_3 x_5+0.0012547x_1 x_3+0.0019085x_3 x_4\leq 25$ $78 \leq x_1\leq 102, \;\;33\leq x_2 \leq 45, \;\; 27 \leq x_i \leq 45, \;\;\; i=3, 4, 5,$ Where $T_1=x_2 x_5$ and $T_5=0.0006262$ Problem5 [14] $f_5=5.3578547x_3^2+0.8356891x_1 x_5+ 37.293239x_1-40792.141$, Subject to $0\leq 85.334407+0.0056858T_1 +T_2 x_1 x_4-0.0022053x_3 x_5\leq 92$ $90 \leq 80.51249+0.0071317x_2 x_5+0.0021813x_3^2\leq 110$ $20\leq 9.300961+0.0047026x_3 x_5+0.0012547x_1 x_3+0.0019085x_3 x_4\leq 25$ $78 \leq x_1\leq 102, \;\; 33\leq x_2\leq 45, \;\; 27 \leq x_i\leq 45, \;\;i=3, 4, 5,$ Where $T_1=x_2 x_3$ and $T_5=0.00026$ Problem6 [25] $f_6=-10.5x_1-7.5x_2-3.5x_3-2.5x_4-1.5x_5-10x_6-0.5_(i=1)^5x_i^2$ Subject to $6x_1+3x_2+3x_3+2x_4+x_5-6.5 \leq 0$ $10x_1+10x_3+x_6\leq 20$ $0\leq x_i \leq 1$,$\; \; \; i=1, \ldots, 5,$ $70 \leq x_6 \leq 50$
Parameter setting
 Parameters Definitions Values $P$ Population size 60 $\alpha$ Randomization parameter 0.5 $\beta_0$ Firefly attractiveness 0.2 $\gamma$ Light absorption coefficient 1 $P_a$ A fraction of worse solutions 0.25 MGN Maximum generation number 800
 Parameters Definitions Values $P$ Population size 60 $\alpha$ Randomization parameter 0.5 $\beta_0$ Firefly attractiveness 0.2 $\gamma$ Light absorption coefficient 1 $P_a$ A fraction of worse solutions 0.25 MGN Maximum generation number 800
Function properties
 Problem $f$ Optimal value Problem1 $f_1$ 1.3934651 Problem2 $f_2$ -6961.81381 Problem3 $f_3$ 680.630057 Problem4 $f_4$ -30665.538 Problem5 $f_5$ Unknown Problem6 $f_6$ -213
 Problem $f$ Optimal value Problem1 $f_1$ 1.3934651 Problem2 $f_2$ -6961.81381 Problem3 $f_3$ 680.630057 Problem4 $f_4$ -30665.538 Problem5 $f_5$ Unknown Problem6 $f_6$ -213
The function evaluation of the proposed HFFACS algorithm for functions $f_1-f_6$
 $f$ Best Worst Mean St.d Succ $f_1$ 9840 13, 200 11, 480 1681.428 30 $f_2$ 11, 520 16, 200 13, 800 2340.256 30 $f_3$ 40, 920 43, 800 42, 320 1441.66 30 $f_4$ 10, 200 16, 800 13, 400 3304.54 30 $f_5$ 8280 10, 800 9360 1297.99 30 $f_6$ 42, 000 53, 280 48, 360 5776.227 30
 $f$ Best Worst Mean St.d Succ $f_1$ 9840 13, 200 11, 480 1681.428 30 $f_2$ 11, 520 16, 200 13, 800 2340.256 30 $f_3$ 40, 920 43, 800 42, 320 1441.66 30 $f_4$ 10, 200 16, 800 13, 400 3304.54 30 $f_5$ 8280 10, 800 9360 1297.99 30 $f_6$ 42, 000 53, 280 48, 360 5776.227 30
The mean function evaluation of the cuckoo search, firefly and the proposed HFFACS algorithms
 $f$ CS FA HFFACS $f_1$ 12, 230 13, 420 11, 480 $f_2$ 14, 840 15, 460 13, 800 $f_3$ 42, 960 43, 450 42, 320 $f_4$ 14, 840 15, 640 13, 400 $f_5$ 10, 240 11, 840 9360 $f_6$ 48, 840 50, 450 48, 360
 $f$ CS FA HFFACS $f_1$ 12, 230 13, 420 11, 480 $f_2$ 14, 840 15, 460 13, 800 $f_3$ 42, 960 43, 450 42, 320 $f_4$ 14, 840 15, 640 13, 400 $f_5$ 10, 240 11, 840 9360 $f_6$ 48, 840 50, 450 48, 360
Comparison between RWMPSOg, RWMPSOl, PSOg, PSOl, and HFFACS for functions $f_1-f_6$
 $f$ Mean StD Succ $f_1$ RWMPSOg 1.832 0.474 25 RWMPSOl 1.427 0.061 30 PSOg 2.042 0.865 24 PSOl 1.454 0.078 30 HFFACS 1.393 1.234e-04 30 $f_2$ RWMPSOg -6961.283 0.380 30 RWMPSOl -6960.717 1.798 30 PSOg -6960.668 1.043 24 PSOl -6939.627 58.789 22 HFFACS -6961.813 3.818e-05 30 $f_3$ RWMPSOg 680.915 0.178 30 RWMPSOl 680.784 0.062 30 PSOg 681.254 0.245 30 PSOl 680.825 0.077 30 HFFACS 680.6301 3.464e-08 30 $f_4$ RWMPSOg -30665.550 0.000 30 RWMPSOl -30665.550 0.000 30 PSOg -60665.550 0.000 30 PSOl -30665.550 0.000 30 HFFACS -30665.518 2.545e-07 30 $f_5$ RWMPSOg -31021.173 11.506 30 RWMPSOl -31026.435 0.000 30 PSOg -31021.140 12.617 30 PSOl -31026.440 0.000 30 HFFACS -31026.427 1.767e-06 30 $f_6$ RWMPSOg -212.616 1.043 30 RWMPSOl -212.047 0.002 30 PSOg -211.833 1.840 30 PSOl -212.933 0.365 30 HFFACS -212.962 0.002 30
 $f$ Mean StD Succ $f_1$ RWMPSOg 1.832 0.474 25 RWMPSOl 1.427 0.061 30 PSOg 2.042 0.865 24 PSOl 1.454 0.078 30 HFFACS 1.393 1.234e-04 30 $f_2$ RWMPSOg -6961.283 0.380 30 RWMPSOl -6960.717 1.798 30 PSOg -6960.668 1.043 24 PSOl -6939.627 58.789 22 HFFACS -6961.813 3.818e-05 30 $f_3$ RWMPSOg 680.915 0.178 30 RWMPSOl 680.784 0.062 30 PSOg 681.254 0.245 30 PSOl 680.825 0.077 30 HFFACS 680.6301 3.464e-08 30 $f_4$ RWMPSOg -30665.550 0.000 30 RWMPSOl -30665.550 0.000 30 PSOg -60665.550 0.000 30 PSOl -30665.550 0.000 30 HFFACS -30665.518 2.545e-07 30 $f_5$ RWMPSOg -31021.173 11.506 30 RWMPSOl -31026.435 0.000 30 PSOg -31021.140 12.617 30 PSOl -31026.440 0.000 30 HFFACS -31026.427 1.767e-06 30 $f_6$ RWMPSOg -212.616 1.043 30 RWMPSOl -212.047 0.002 30 PSOg -211.833 1.840 30 PSOl -212.933 0.365 30 HFFACS -212.962 0.002 30
The general performance (function evaluations) of HFFACS with engineering optimization problems
 Design problem Best Mean Worst Std Welded beam design 21, 000 22, 253.4 23, 760 1397.31 Pressure vessel design 39, 360 40, 120 40, 800 723.32 Speed reducer design 87, 840 90, 200 92, 640 2401 Three-bar truss design 6120 7440 8400 1181.86 spring design 36, 120 38, 560 40, 080 2134.291
 Design problem Best Mean Worst Std Welded beam design 21, 000 22, 253.4 23, 760 1397.31 Pressure vessel design 39, 360 40, 120 40, 800 723.32 Speed reducer design 87, 840 90, 200 92, 640 2401 Three-bar truss design 6120 7440 8400 1181.86 spring design 36, 120 38, 560 40, 080 2134.291
The general performance (function values) of HFFACS with engineering optimization problems
 Design problem Best Mean Worst Std Welded beam design 2.380957 2.380957 2.380957 1.82e-8 Pressure vessel design 6059.71433 6059.71433 6059.71433 4.54e-12 Speed reducer design 2.994471 2.994471 2.994471 0.00 Three-bar truss design 263.895843 263.895843 263.895843 3.54e-11 spring design 0.012665 0.012665 0.012665 5.77e-12
 Design problem Best Mean Worst Std Welded beam design 2.380957 2.380957 2.380957 1.82e-8 Pressure vessel design 6059.71433 6059.71433 6059.71433 4.54e-12 Speed reducer design 2.994471 2.994471 2.994471 0.00 Three-bar truss design 263.895843 263.895843 263.895843 3.54e-11 spring design 0.012665 0.012665 0.012665 5.77e-12
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