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September  2017, 7(3): 301-323. doi: 10.3934/naco.2017020

A simplex grey wolf optimizer for solving integer programming and minimax problems

Mohamed A. Tawhid 1,2,, and  Ahmed F. Ali 3,4,

1. 

Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada V2C 0C8
2. 

Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Moharam Bey 21511, Alexandria, Egypt
3. 

Department of Computer Science, Faculty of Computers & Informatics, Suez Canal University, Ismailia, Egypt21511, Alexandria, Egypt
4. 

Postdoctoral fellow, Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada V2C 0C8

* Corresponding author: Mohamed A. Tawhid

The reviewing process of the paper was handled by Shengjie Li as Guest Editor

Received  April 2016 Revised  May 2017 Published  July 2017

Fund Project: We are thankful to the anonymous reviewers for constructive feedback and insightful suggestions which greatly improved this paper.The research of the 1st author is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). The postdoctoral fellowship of the 2nd author is supported by NSERC

In this paper, we propose a new hybrid grey wolf optimizer (GWO) algorithm with simplex Nelder-Mead method in order to solve integer programming and minimax problems. We call the proposed algorithm a Simplex Grey Wolf Optimizer (SGWO) algorithm. In the the proposed SGWO algorithm, we combine the GWO algorithm with the Nelder-Mead method in order to refine the best obtained solution from the standard GWO algorithm. We test it on 7 integer programming problems and 10 minimax problems in order to investigate the general performance of the proposed SGWO algorithm. Also, we compare SGWO with 10 algorithms for solving integer programming problems and 9 algorithms for solving minimax problems. The experiments results show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time.

Keywords: Grey wolf optimizer, Simplex Nelder-Mead method, integer programming problems, minimax problems.
Mathematics Subject Classification: Primary: 90C10, 90C47, 90C56; Secondary: 68U20, 68W20.
Citation: Mohamed A. Tawhid, Ahmed F. Ali. A simplex grey wolf optimizer for solving integer programming and minimax problems. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 301-323. doi: 10.3934/naco.2017020
References:
[1]

J. S. Arora, Introduction to Optimum Design, McGraw–Hill, New York, 1989.
[2]

N. Bacanin and M. Tuba, Artificial Bee Colony (ABC) algorithm for constrained optimization improved with genetic operators, Studies in Informatics and Control, 21 (2012), 137-146.

[3]

N. Bacanin, I. Brajevic and M. Tuba, Firefly Algorithm Applied to Integer Programming Problems, Recent Advances in Mathematics, 2013.
[4]

J. W. Bandler and C. Charalambous, Nonlinear programming using minimax techniques, Journal of Optimization Theory and Applications, 13 (1974), 607-619. doi: 10.1007/BF00933620.

[5]

B. Borchers and J. E. Mitchell, Using an interior point method in a branch and bound algorithm for integer programming, Technical Report, Rensselaer Polytechnic Institute, July 1992. doi: 10.1016/0305-0548(94)90024-8.

[6]

S. A. Chu, P. -W. Tsai and J. -S. Pan. Cat swarm optimization, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 4099 (2006), 854–858.
[7]

M. Dorigo, Optimization, Learning and Natural Algorithms, Ph. D. Thesis, Politecnico di Milano, Italy, 1992.
[8]

D. Z. Du and P. M. Pardalose, Minimax and Applications, Kluwer, 1995. doi: 10.1007/978-1-4613-3557-3.

[9]

R. Fletcher, Practical Method of Optimization, Vol. 1 & 2, John Wiley and Sons, 1980.

[10]

A. Glankwahmdee, J. S. Liebman and G. L. Hogg, Unconstrained discrete nonlinear programming, Engineering Optimization, 4 (1979), 95-107.

[11] P. E. Gill, W. Murray and M. H. Wright, Practical Optimization, Academic Press, London, 1981.
[12] J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975.
[13]

A. C. P. Isabel, E. Santo and E. Fernandes, Heuristics pattern search for bound constrained minimax problems, Computational Science and Its Applications, ICCSA, 6784 (2011), 174-184. doi: 10.1007/978-3-642-21931-3_15.

[14]

R. Jovanovic and M. Tuba, An ant colony optimization algorithm with improved pheromone correction strategy for the minimum weight vertex cover problem, Applied Soft Computing, 11 (2011), 5360-5366.

[15]

R. Jovanovic and M. Tuba, Ant colony optimization algorithm with pheromone correction strategy for minimum connected dominating set problem, Computer Science and Information Systems (ComSIS), 10 (2013), 133-149. doi: 10.2298/CSIS110927038J.

[16]

D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: artificial bee colony (abc) algorithm, Journal of Global Optimization, 39 (2007), 459-471. doi: 10.1007/s10898-007-9149-x.

[17]

J. Kennedy and R. C. Eberhart, Particle Swarm Optimization, Proceedings of the IEEE International Conference on Neural Networks, 4 (1995), 1942-1948.

[18]

E. C. Laskari, K. E. Parsopoulos and M. N. Vrahatis, Particle swarm optimization for integer programming, Proceedings of the IEEE 2002 Congress on Evolutionary Computation, Honolulu (HI), (2002), 1582–1587.
[19]

E. L. Lawler and D. W. Wood, Branch and bound methods: A survey, Operations Research, 14 (1966), 699-719.

[20]

X. L. Li, Z. J. Shao and J. X. Qian, An optimizing method based on autonomous animals: Fish-swarm algorithm, System Engineering Theory and Practice, 22 (2003), 32-38.

[21]

G. Liuzzi, S. Lucidi and M. Sciandrone, A derivative-free algorithm for linearly constrained finite minimax problems, SIAM Journal on Optimization, 16 (2006), 1054-1075. doi: 10.1137/040615821.

[22]

L. Lukan and J. Vlcek, Test problems for nonsmooth unconstrained and linearly constrained optimization, Technical report 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000.
[23]

V. M. Manquinho, J. P. Marques Silva, A. L. Oliveira and K. A. Sakallah, Branch and bound algorithms for highly constrained integer programs, Technical Report, Cadence European Laboratories, Portugal, 1997.
[24]

S. Mirjalili, S. M. Mirjalili and A. Lewis, Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61.

[25]

J. A. Nelder and R. Mead, A simplex method for function minimization, Computer Journal, 7 (1965), 308-313. doi: 10.1093/comjnl/7.4.308.

[26]

G. L. Nemhauser, A. H. G. Rinnooy Kan and M. J. Todd, Handbooks in OR & MS, volume 1. Elsevier, 1989.
[27]

K. E. Parsopoulos and M. N. Vrahatis, Unified particle swarm optimization for tackling operations research problems, in Proceeding of IEEE 2005 swarm Intelligence Symposium, Pasadena, USA, (2005), 53-59.
[28]

M. K. Passino, Biomimicry of bacterial foraging for distributed optimization and control, Control Systems, IEEE, 22 (2002), 52-67.

[29]

Y. G. Petalas, K. E. Parsopoulos and M. N. Vrahatis, Memetic particle swarm optimization, Ann oper Res, 156 (2007), 99-127. doi: 10.1007/s10479-007-0224-y.

[30]

E. Polak, J. O. Royset and R. S. Womersley, Algorithms with adaptive smoothing for finite minimax problems, Journal of Optimization Theory and Applications, 119 (2003), 459-484. doi: 10.1023/B:JOTA.0000006685.60019.3e.

[31]

S. S. Rao, Engineering Optimization-Theory and Practice, Wiley: New Delhi, 1994.

[32]

G. Rudolph, An evolutionary algorithm for integer programming, in Parallel Problem Solving from Nature(eds. Y. Davidor, H-P. Schwefel, and R. Mnner), 3 (1994), 139-148.
[33]

E. Sandgen, Nonlinear integer and discrete programming in mechanical design optimization, Journal of Mechanical Design (ASME), 112 (1990), 223-229.

[34]

H. P. Schwefel, Evolution and Optimum Seeking, New York: Wiley, 1995.

[35]

R. Storn and K. Price, Differential evolution simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim., 11 (1997), 341-359. doi: 10.1023/A:1008202821328.

[36]

R. Tang, S. Fong, X. S. Yang and S. Deb, Wolf search algorithm with ephemeral memory, In Digital Information Management (ICDIM), 2012 Seventh International Conference on Digital Information Management, (2012), 165–172.
[37]

D. Teodorovic and M. DellOrco. Bee colony optimization cooperative learning approach to complex transportation problems, In Advanced OR and AI Methods in Transportation, Proceedings of 16th MiniEURO Conference and 10th Meeting of EWGT (13-16 September 2005). Poznan: Publishing House of the Polish Operational and System Research, (2005), 51–60.
[38]

M. Tuba, N. Bacanin and N. Stanarevic, Adjusted artificial bee colony (ABC) algorithm for engineering problems, WSEAS Transaction on Computers, 1 (2012), 111-120.

[39]

M. Tuba, M. Subotic and N. Stanarevic, Performance of a modified cuckoo search algorithm for unconstrained optimization problems, WSEAS Transactions on Systems, 11 (2012), 62-74.

[40]

B. Wilson, A Simplicial Algorithm for Concave Programming, PhD thesis, Harvard University, 1963.
[41]

S. Xu, Smoothing method for minimax problems, Computational Optimization and Applications, 20 (2001), 267-279. doi: 10.1023/A:1011211101714.

[42]

X. S. Yang, A new metaheuristic bat-inspired algorithm, Nature Inspired Cooperative Strategies for Optimization (NICSO 2010), (2010), 65-74.

[43]

X. S. Yang, Firefly algorithm, stochastic test functions and design optimisation, International Journal of Bio-Inspired Computation, 2 (2010), 78-84.

[44]

X. S. Yang and S. Deb, Cuckoo search via levy fights, in Nature & Biologically Inspired Computing, NaBIC 2009, World Congress on, IEEE, (2009), 210–214.
[45]

Z. Shen, A. Neumaier and M. C. Eiermann, Solving minimax problems by interval methods, BIT, 30 (1990), 742-751. doi: 10.1007/BF01933221.

show all references

References:
[1]

J. S. Arora, Introduction to Optimum Design, McGraw–Hill, New York, 1989.
[2]

N. Bacanin and M. Tuba, Artificial Bee Colony (ABC) algorithm for constrained optimization improved with genetic operators, Studies in Informatics and Control, 21 (2012), 137-146.

[3]

N. Bacanin, I. Brajevic and M. Tuba, Firefly Algorithm Applied to Integer Programming Problems, Recent Advances in Mathematics, 2013.
[4]

J. W. Bandler and C. Charalambous, Nonlinear programming using minimax techniques, Journal of Optimization Theory and Applications, 13 (1974), 607-619. doi: 10.1007/BF00933620.

[5]

B. Borchers and J. E. Mitchell, Using an interior point method in a branch and bound algorithm for integer programming, Technical Report, Rensselaer Polytechnic Institute, July 1992. doi: 10.1016/0305-0548(94)90024-8.

[6]

S. A. Chu, P. -W. Tsai and J. -S. Pan. Cat swarm optimization, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 4099 (2006), 854–858.
[7]

M. Dorigo, Optimization, Learning and Natural Algorithms, Ph. D. Thesis, Politecnico di Milano, Italy, 1992.
[8]

D. Z. Du and P. M. Pardalose, Minimax and Applications, Kluwer, 1995. doi: 10.1007/978-1-4613-3557-3.

[9]

R. Fletcher, Practical Method of Optimization, Vol. 1 & 2, John Wiley and Sons, 1980.

[10]

A. Glankwahmdee, J. S. Liebman and G. L. Hogg, Unconstrained discrete nonlinear programming, Engineering Optimization, 4 (1979), 95-107.

[11] P. E. Gill, W. Murray and M. H. Wright, Practical Optimization, Academic Press, London, 1981.
[12] J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975.
[13]

A. C. P. Isabel, E. Santo and E. Fernandes, Heuristics pattern search for bound constrained minimax problems, Computational Science and Its Applications, ICCSA, 6784 (2011), 174-184. doi: 10.1007/978-3-642-21931-3_15.

[14]

R. Jovanovic and M. Tuba, An ant colony optimization algorithm with improved pheromone correction strategy for the minimum weight vertex cover problem, Applied Soft Computing, 11 (2011), 5360-5366.

[15]

R. Jovanovic and M. Tuba, Ant colony optimization algorithm with pheromone correction strategy for minimum connected dominating set problem, Computer Science and Information Systems (ComSIS), 10 (2013), 133-149. doi: 10.2298/CSIS110927038J.

[16]

D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: artificial bee colony (abc) algorithm, Journal of Global Optimization, 39 (2007), 459-471. doi: 10.1007/s10898-007-9149-x.

[17]

J. Kennedy and R. C. Eberhart, Particle Swarm Optimization, Proceedings of the IEEE International Conference on Neural Networks, 4 (1995), 1942-1948.

[18]

E. C. Laskari, K. E. Parsopoulos and M. N. Vrahatis, Particle swarm optimization for integer programming, Proceedings of the IEEE 2002 Congress on Evolutionary Computation, Honolulu (HI), (2002), 1582–1587.
[19]

E. L. Lawler and D. W. Wood, Branch and bound methods: A survey, Operations Research, 14 (1966), 699-719.

[20]

X. L. Li, Z. J. Shao and J. X. Qian, An optimizing method based on autonomous animals: Fish-swarm algorithm, System Engineering Theory and Practice, 22 (2003), 32-38.

[21]

G. Liuzzi, S. Lucidi and M. Sciandrone, A derivative-free algorithm for linearly constrained finite minimax problems, SIAM Journal on Optimization, 16 (2006), 1054-1075. doi: 10.1137/040615821.

[22]

L. Lukan and J. Vlcek, Test problems for nonsmooth unconstrained and linearly constrained optimization, Technical report 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000.
[23]

V. M. Manquinho, J. P. Marques Silva, A. L. Oliveira and K. A. Sakallah, Branch and bound algorithms for highly constrained integer programs, Technical Report, Cadence European Laboratories, Portugal, 1997.
[24]

S. Mirjalili, S. M. Mirjalili and A. Lewis, Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61.

[25]

J. A. Nelder and R. Mead, A simplex method for function minimization, Computer Journal, 7 (1965), 308-313. doi: 10.1093/comjnl/7.4.308.

[26]

G. L. Nemhauser, A. H. G. Rinnooy Kan and M. J. Todd, Handbooks in OR & MS, volume 1. Elsevier, 1989.
[27]

K. E. Parsopoulos and M. N. Vrahatis, Unified particle swarm optimization for tackling operations research problems, in Proceeding of IEEE 2005 swarm Intelligence Symposium, Pasadena, USA, (2005), 53-59.
[28]

M. K. Passino, Biomimicry of bacterial foraging for distributed optimization and control, Control Systems, IEEE, 22 (2002), 52-67.

[29]

Y. G. Petalas, K. E. Parsopoulos and M. N. Vrahatis, Memetic particle swarm optimization, Ann oper Res, 156 (2007), 99-127. doi: 10.1007/s10479-007-0224-y.

[30]

E. Polak, J. O. Royset and R. S. Womersley, Algorithms with adaptive smoothing for finite minimax problems, Journal of Optimization Theory and Applications, 119 (2003), 459-484. doi: 10.1023/B:JOTA.0000006685.60019.3e.

[31]

S. S. Rao, Engineering Optimization-Theory and Practice, Wiley: New Delhi, 1994.

[32]

G. Rudolph, An evolutionary algorithm for integer programming, in Parallel Problem Solving from Nature(eds. Y. Davidor, H-P. Schwefel, and R. Mnner), 3 (1994), 139-148.
[33]

E. Sandgen, Nonlinear integer and discrete programming in mechanical design optimization, Journal of Mechanical Design (ASME), 112 (1990), 223-229.

[34]

H. P. Schwefel, Evolution and Optimum Seeking, New York: Wiley, 1995.

[35]

R. Storn and K. Price, Differential evolution simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim., 11 (1997), 341-359. doi: 10.1023/A:1008202821328.

[36]

R. Tang, S. Fong, X. S. Yang and S. Deb, Wolf search algorithm with ephemeral memory, In Digital Information Management (ICDIM), 2012 Seventh International Conference on Digital Information Management, (2012), 165–172.
[37]

D. Teodorovic and M. DellOrco. Bee colony optimization cooperative learning approach to complex transportation problems, In Advanced OR and AI Methods in Transportation, Proceedings of 16th MiniEURO Conference and 10th Meeting of EWGT (13-16 September 2005). Poznan: Publishing House of the Polish Operational and System Research, (2005), 51–60.
[38]

M. Tuba, N. Bacanin and N. Stanarevic, Adjusted artificial bee colony (ABC) algorithm for engineering problems, WSEAS Transaction on Computers, 1 (2012), 111-120.

[39]

M. Tuba, M. Subotic and N. Stanarevic, Performance of a modified cuckoo search algorithm for unconstrained optimization problems, WSEAS Transactions on Systems, 11 (2012), 62-74.

[40]

B. Wilson, A Simplicial Algorithm for Concave Programming, PhD thesis, Harvard University, 1963.
[41]

S. Xu, Smoothing method for minimax problems, Computational Optimization and Applications, 20 (2001), 267-279. doi: 10.1023/A:1011211101714.

[42]

X. S. Yang, A new metaheuristic bat-inspired algorithm, Nature Inspired Cooperative Strategies for Optimization (NICSO 2010), (2010), 65-74.

[43]

X. S. Yang, Firefly algorithm, stochastic test functions and design optimisation, International Journal of Bio-Inspired Computation, 2 (2010), 78-84.

[44]

X. S. Yang and S. Deb, Cuckoo search via levy fights, in Nature & Biologically Inspired Computing, NaBIC 2009, World Congress on, IEEE, (2009), 210–214.
[45]

Z. Shen, A. Neumaier and M. C. Eiermann, Solving minimax problems by interval methods, BIT, 30 (1990), 742-751. doi: 10.1007/BF01933221.

Figure 1.  Social hierarchy of grey wolf
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Figure 2.  The general performance of the proposed SGWO algorithm on integer programming problems
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Figure 3.  The general performance of the proposed SGWO algorithm on minimax problems
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Table 1.  Parameter setting
Parameters Definitions Values
n Search agents no (population size) 20
$\vec{a}$ Coefficient vector 2
$\vec{A}$ Coefficient vector $[-2a,2a]$
$\vec{r_1}, \vec{r_2}$ random vectors [0, 1]
$\vec{C}$ Coefficient vector $2\cdot \vec{r_2}$
$N_{elite}$ Number of best solution for final intensification 1
$Max_{itr}$ Maximum number of iterations 2d -3d
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Parameters Definitions Values
n Search agents no (population size) 20
$\vec{a}$ Coefficient vector 2
$\vec{A}$ Coefficient vector $[-2a,2a]$
$\vec{r_1}, \vec{r_2}$ random vectors [0, 1]
$\vec{C}$ Coefficient vector $2\cdot \vec{r_2}$
$N_{elite}$ Number of best solution for final intensification 1
$Max_{itr}$ Maximum number of iterations 2d -3d
Table 2.  Integer programming optimization test problems
Test problem Problem definition
Problem 1 [32] $FI_{1}(x)=\|x\|_1=|x_1|+\ldots+|x_n|$
Problem 2 [32] $FI_{2}(x)=x^Tx= \begin{bmatrix} x_1\cdots x_n \end{bmatrix} \begin{bmatrix} x_1\\ \vdots\\x_n\end{bmatrix}$
Problem 3 [10] $FI_{3}(x)= \begin{bmatrix} 15\;\;27\;\;36\;\;18\;\;12 \end{bmatrix}x+x^T \begin{bmatrix} 35&-20&-10&32&-10\\ -20&40&-6&-31&32\\ -10&-6&11&-6&-10\\ 32&-31&-6&38&-20\\ -10&32&-10&-20&31\\ \end{bmatrix}x$
Problem 4 [10] $FI_{4}(x)=(9x_1^2+2x_2^2-11)^2+(3x_1+4x_2^2-7)^2$
Problem 5 [10] $FI_{5}(x)=(x_1+10x_2)^2+5(x_3-x_4)^2+(x_2-2x_3)^4+10(x_1-x_4)^4$
Problem 6 [32] $FI_{6}(x)=2x_1^2+3x_2^2+4x_1x_2-6x_1-3x_2$
Problem 7 [10] $FI_{7}(x)=-3803.84-138.08x_1-232.92x_2+123.08x_1^2+203.64x_2^2+182.25x_1x_2$
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Test problem Problem definition
Problem 1 [32] $FI_{1}(x)=\|x\|_1=|x_1|+\ldots+|x_n|$
Problem 2 [32] $FI_{2}(x)=x^Tx= \begin{bmatrix} x_1\cdots x_n \end{bmatrix} \begin{bmatrix} x_1\\ \vdots\\x_n\end{bmatrix}$
Problem 3 [10] $FI_{3}(x)= \begin{bmatrix} 15\;\;27\;\;36\;\;18\;\;12 \end{bmatrix}x+x^T \begin{bmatrix} 35&-20&-10&32&-10\\ -20&40&-6&-31&32\\ -10&-6&11&-6&-10\\ 32&-31&-6&38&-20\\ -10&32&-10&-20&31\\ \end{bmatrix}x$
Problem 4 [10] $FI_{4}(x)=(9x_1^2+2x_2^2-11)^2+(3x_1+4x_2^2-7)^2$
Problem 5 [10] $FI_{5}(x)=(x_1+10x_2)^2+5(x_3-x_4)^2+(x_2-2x_3)^4+10(x_1-x_4)^4$
Problem 6 [32] $FI_{6}(x)=2x_1^2+3x_2^2+4x_1x_2-6x_1-3x_2$
Problem 7 [10] $FI_{7}(x)=-3803.84-138.08x_1-232.92x_2+123.08x_1^2+203.64x_2^2+182.25x_1x_2$
Table 3.  The properties of the Integer programming test functions
Function Dimension (d) Bound Optimal
FI15[-100 100]0
FI25[-100 100]0
FI35[-100 100]-737
FI42[-100 100]0
FI54[-100 100]0
FI62[-100 100]-6
FI72[-100 100]-3833.12
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Function Dimension (d) Bound Optimal
FI15[-100 100]0
FI25[-100 100]0
FI35[-100 100]-737
FI42[-100 100]0
FI54[-100 100]0
FI62[-100 100]-6
FI72[-100 100]-3833.12
Table 4.  The efficiency of invoking the Nelder-Mead method in the final stage of SGWO algorithm for FI1 −FI7 integer programming problems
Function Standard NM SGWO
GWO method
FI1660.41988.35227.3
FI2560.2678.15225.6
FI34920.5819.45701.24
FI42860.3266.14283.5
FI51520.6872.46508.15
FI63760.2254.15364.27
FI71200.3245.47235.16
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Function Standard NM SGWO
GWO method
FI1660.41988.35227.3
FI2560.2678.15225.6
FI34920.5819.45701.24
FI42860.3266.14283.5
FI51520.6872.46508.15
FI63760.2254.15364.27
FI71200.3245.47235.16
Table 5.  Experimental results (min, max, mean, standard deviation and rate of success) of function evaluation for FI1 − FI7 test problems
Function Algorithm Min Max Mean St.D Suc
FI1 RWMPSOg 17,160 74,699 27,176.3 8657 50
RWMPSOl 24,870 35,265 30,923.9 2405 50
PSOg 14,000 261,100 29,435.3 42,039 34
PSOl 27,400 35,800 31,252 1818 50
SGWO 225 275 227.3 24.95 50
FI2 RWMPSOg 252 912 578.5 136.5 50
RWMPSOl 369 1931 773.9 285.5 50
PSOg 400 1000 606.4 119 50
PSOl 450 1470 830.2 206 50
SGWO 215 250 225.6 18.52 50
FI3 RWMPSOg 361 41,593 6490.6 6913 50
RWMPSOl 5003 15,833 9292.6 2444 50
PSOg 2150 187,000 12,681 35,067 50
PSOl 4650 22,650 11,320 3803 50
SGWO 715 750 701.24 37.52 50
FI4 RWMPSOg 76 468 215 97.9 50
RWMPSOl 73 620 218.7 115.3 50
PSOg 100 620 369.6 113.2 50
PSOl 120 920 390 134.6 50
SGWO 275 290 283.5 7.63 50
FI5 RWMPSOg 687 2439 1521.8 360.7 50
RWMPSOl 675 3863 2102.9 689.5 50
PSOg 680 3440 1499 513.1 43
PSOl 800 3880 2472.4 637.5 50
SGWO 510 540 508.15 16.07 50
FI6 RWMPSOg 40 238 110.9 48.6 50
RWMPSOl 40 235 112 48.7 50
PSOg 80 350 204.8 62 50
PSOl 70 520 256 107.5 50
SGWO 355 370 361.6 7.63 50
FI7 RWMPSOg 72 620 242.7 132.2 50
RWMPSOl 70 573 248.9 134.4 50
PSOg 100 660 421.2 130.4 50
PSOl 100 820 466 165 50
SGWO 215 250 235.16 18.02 50
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Function Algorithm Min Max Mean St.D Suc
FI1 RWMPSOg 17,160 74,699 27,176.3 8657 50
RWMPSOl 24,870 35,265 30,923.9 2405 50
PSOg 14,000 261,100 29,435.3 42,039 34
PSOl 27,400 35,800 31,252 1818 50
SGWO 225 275 227.3 24.95 50
FI2 RWMPSOg 252 912 578.5 136.5 50
RWMPSOl 369 1931 773.9 285.5 50
PSOg 400 1000 606.4 119 50
PSOl 450 1470 830.2 206 50
SGWO 215 250 225.6 18.52 50
FI3 RWMPSOg 361 41,593 6490.6 6913 50
RWMPSOl 5003 15,833 9292.6 2444 50
PSOg 2150 187,000 12,681 35,067 50
PSOl 4650 22,650 11,320 3803 50
SGWO 715 750 701.24 37.52 50
FI4 RWMPSOg 76 468 215 97.9 50
RWMPSOl 73 620 218.7 115.3 50
PSOg 100 620 369.6 113.2 50
PSOl 120 920 390 134.6 50
SGWO 275 290 283.5 7.63 50
FI5 RWMPSOg 687 2439 1521.8 360.7 50
RWMPSOl 675 3863 2102.9 689.5 50
PSOg 680 3440 1499 513.1 43
PSOl 800 3880 2472.4 637.5 50
SGWO 510 540 508.15 16.07 50
FI6 RWMPSOg 40 238 110.9 48.6 50
RWMPSOl 40 235 112 48.7 50
PSOg 80 350 204.8 62 50
PSOl 70 520 256 107.5 50
SGWO 355 370 361.6 7.63 50
FI7 RWMPSOg 72 620 242.7 132.2 50
RWMPSOl 70 573 248.9 134.4 50
PSOg 100 660 421.2 130.4 50
PSOl 100 820 466 165 50
SGWO 215 250 235.16 18.02 50
Table 6.  SGWO and other meta-heuristics and swarm intelligence algorithms embedded with NM algorithm for FI1 − FI7 integer programming problems
Function GA+NM DE+NM PSO+NM FF+NM CS+NM SGWO
FI1 Avg 1106.56 935.48 1160.12 975.23 845.68 227.3
SD 457.96 256.12 423.56 235.69 115.48 24.95
FI2 Avg 947.42 914.58 1125.56 911.25 984.69 225.6
SD 110.07 246.24 285.46 115.48 254.89 18.52
FI3 Avg 2053.43 1846.23 1915.35 1825.23 1115.12 701.24
SD 41.64 115.47 235.69 245.56 158.42 37.52
FI4 Avg 866.73 745.78 875.69 796.23 414.26 283.5
SD 284.48 125.26 145.36 175.28 118.54 7.63
FI5 Avg 954.54 923.18 1115.24 960.43 1142.58 508.15
SD 74.93 126.21 114.56 124.56 215.48 16.07
FI6 Avg 554.35 515.24 535.46 498.75 458.49 361.6
SD 115.14 125.36 223.52 113.58 118.35 7.63
FI7 Avg 485.14 454.36 514.12 435.47 390.78 235.16
SD 117.12 148.12 90.14 142.58 118.62 18.02
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Function GA+NM DE+NM PSO+NM FF+NM CS+NM SGWO
FI1 Avg 1106.56 935.48 1160.12 975.23 845.68 227.3
SD 457.96 256.12 423.56 235.69 115.48 24.95
FI2 Avg 947.42 914.58 1125.56 911.25 984.69 225.6
SD 110.07 246.24 285.46 115.48 254.89 18.52
FI3 Avg 2053.43 1846.23 1915.35 1825.23 1115.12 701.24
SD 41.64 115.47 235.69 245.56 158.42 37.52
FI4 Avg 866.73 745.78 875.69 796.23 414.26 283.5
SD 284.48 125.26 145.36 175.28 118.54 7.63
FI5 Avg 954.54 923.18 1115.24 960.43 1142.58 508.15
SD 74.93 126.21 114.56 124.56 215.48 16.07
FI6 Avg 554.35 515.24 535.46 498.75 458.49 361.6
SD 115.14 125.36 223.52 113.58 118.35 7.63
FI7 Avg 485.14 454.36 514.12 435.47 390.78 235.16
SD 117.12 148.12 90.14 142.58 118.62 18.02
Table 7.  Experimental results (mean, standard deviation and rate of success) of function evaluation between BB and SGWO for FI1 − FI7 test problems
Function Algorithm Mean St.D Suc
FI1 BB 1167.83 659.8 30
SGWO 223.15 22.35 30
FI2 BB 139.7 102.6 30
SGWO 220.26 15.26 30
FI3 BB 4185.5 32.8 30
SGWO 690.55 34.56 30
FI4 BB 316.9 125.4 30
SGWO 279.85 8.56 30
FI5 BB 2754 1030.1 30
SGWO 498.25 15.48 30
FI6 BB 211 15 30
SGWO 361.75 9.45 30
FI7 BB 358.6 14.7 30
SGWO 233.45 21.45 30
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Function Algorithm Mean St.D Suc
FI1 BB 1167.83 659.8 30
SGWO 223.15 22.35 30
FI2 BB 139.7 102.6 30
SGWO 220.26 15.26 30
FI3 BB 4185.5 32.8 30
SGWO 690.55 34.56 30
FI4 BB 316.9 125.4 30
SGWO 279.85 8.56 30
FI5 BB 2754 1030.1 30
SGWO 498.25 15.48 30
FI6 BB 211 15 30
SGWO 361.75 9.45 30
FI7 BB 358.6 14.7 30
SGWO 233.45 21.45 30
Table 8.  Minimax optimization test problems
Test problem Problem definition
Problem 1 [41] $FM_{1}(x)=\text{max}\;\; {f_i(x)}, \;\; i=1,2,3,$
$f_1(x)=x_1^2+x_2^4,$
$f_2(x)=(2-x1)^2+(2-x_2)^2,$
$f_3(x)=2exp(-x_1+x_2)$
Problem 2 [41] $FM_{2}(x)=\text{max}\;\; {f_i(x)}, \;\; i=1,2,3,$
$f_1(x)=x_1^4+x_2^2$
$f_2(x)=(2-x1)^2+(2-x_2)^2,$
$f_3(x)=2exp(-x_1+x_2)$
Problem 3 [41] $FM_{3}(x)=x_1^2+x_2^2+2x_3^2+x_4^2-5x_1-5x_2-21x_3+7x_4,$
$g_2(x)=-x_1^2-x_2^2-x_3^3-x_4^2-x_1+x_2-x_3+x_4+8,$
$g_3(x)=-x_1^2-2x_2^2-x_3^2-2x_4+x_1+x_4+10,$
$g_4(x)=-x_1^2-x_2^2-x_3^2-2x_1+x_2+x_4+5$
Problem 4 [41] $FM_{4}(x)=\text{max}{f_i(x)}\;\; i=1,\ldots,5 $
$f_{1}(x)=(x_1-10)^2+5(x_2-12)^2+x_3^4+3(x_4-11)^2$
$+10x_5^6+7x_6^2+x_7^4-4x_6x_7-10x_6-8x_7,$
$f_2(x)=f_1(x)+10(2x_1^2+3x_2^4+x_3+4x_4^2+5x_5-127),$
$f_3(x)=f_1(x)+10(7x_1+3x_2+10x_3^2+x_4-x_5-282),$
$f_4(x)=f_1(x)+10(23x_1+x_2^2+6x_6^2-8x_7-196),$
$f_5(x)=f_1(x)+10(4x_1^2+x_2^2-3x_1x_2+2x_3^2+5x_6-11x_7$
Problem 5 [34] $FM_{5}(x)=\text{max}\;\; {f_i(x)},\;\;i=1,2,$
$f_1(x)=|x_1+2x_2-7|$,
$f_2(x)=|2x_1+x_2-5|$
Problem 6 [34] $FM_{6}(x)=\text{max} \;\; {f_i(x)}, $
$f_i(x)=|x_i|,\;\;i= 1,\ldots,10$
Problem 7 [22] $FM_{7}(x)= \text{max}\;\; {f_i(x)},\;\;i=1,2,$
$f_1(x)=(x_1-\sqrt{(x_1^2+x_2^2)}cos\sqrt{x_1^2+x_2^2})^2+0.005(x_1^2+x_2^2)^2,$
$f_2(x)=(x_2-\sqrt{(x_1^2+x_2^2)}sin\sqrt{x_1^2+x_2^2})^2+0.005(x_1^2+x_2^2)^2$
Problem 8 [22] $FM_{8}(x)= \text{max}\;\; {f_i(x)},\;\;i=1,\ldots,4,$
$f_1(x)=\big(x_1-(x_4+1)^4\big)^2+\Big(x_2-\big(x_1-(x_4+1)^4\big)^4\Big)^2+2x_3^2+x_4^2$
$\;\;\;\;\;\;\;\;\;\;-5\big(x_1-(x_4+1)^4\big)-5\Big(x_2-\big(x1-(x_4+1)^4\big)^4\Big)-21x_3+7x_4,$
$f_2(x)=f_1(x)+10\Big[\big(x_1-(x_4+1)^4\big)^2+\Big(x_2-\big(x_1-(x_4+1)^4\big)^4\Big)^2+x_3^2+x_4^2$
$\;\;\;\;\;\;\;\;\;\;+\big(x_1-(x_4+1)^4\big)-\Big(x_2-\big(x_1-(x_4+1)^4\big)^4\Big)+x_3-x_4-8\Big],$
$f_3(x)=f_1(x)+10\Big[\big(x_1-(x_4+1)^4\big)^2+2\Big(x_2-\big(x_1$
$\;\;\;\;\;\;\;\;\;\;-(x_4+1)^4\big)^4\Big)^2+x_3^2+2x_4^2-\big(x_1-(x_4+1)^4\big)-x_4-10\Big]$,
$f_4(x)=f_1(x)+10\Big[\big(x_1-(x_4+1)^4\big)^2+\Big(x_2-\big(x_1-(x_4+1)^4\big)^4\Big)^2 $
$\;\;\;\;\;\;\;\;\;\;+x_3^2+2\big(x_1-(x_4+1)^4\big)-\Big(x_2-\big(x_1-(x_4+1)^4\big)^4\Big)-x_4-5\Big]$
Problem 9 [22] $FM_{9}(x)= \text{max}\;\; {f_i(x)},\;\;i=1,\ldots,5, $
$f_1(x)=(x_1-10)^2+5(x_2-12)^2+x_3^4+3(x_4-11)^2+10x_5^6+7x_6^2+x_7^4 $
$\; \; \; \; \; \; \; \; \; \ -4x_6x_7-10x_6-8x_7$,
$f_2(x)=-2x_1^2-2x_3^4-x_3-4x_4^2-5x_5+127$,
$f_3(x)=-7x_1-3x_2-10x_3^2-x_4+x_5+282$,
$f_4(x)=-23x_1-x_2^2-6x_6^2+8x_7+196$,
$f_5(x)=-4x_1^2-x_2^2+3x_1x_2-2x_3^2-5x_6+11x_7$
Problem 10 [22] $FM_{10}(x)=\text{max} {|f_i(x)|}, \;\;\; i=1,\ldots,21,$
$f_i(x)=x_1exp(x_3t_i)+x_2exp(x_4t_i)-\frac{1}{1+t_i}, \ t_i=-0.5+\frac{i-1}{20}$
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Test problem Problem definition
Problem 1 [41] $FM_{1}(x)=\text{max}\;\; {f_i(x)}, \;\; i=1,2,3,$
$f_1(x)=x_1^2+x_2^4,$
$f_2(x)=(2-x1)^2+(2-x_2)^2,$
$f_3(x)=2exp(-x_1+x_2)$
Problem 2 [41] $FM_{2}(x)=\text{max}\;\; {f_i(x)}, \;\; i=1,2,3,$
$f_1(x)=x_1^4+x_2^2$
$f_2(x)=(2-x1)^2+(2-x_2)^2,$
$f_3(x)=2exp(-x_1+x_2)$
Problem 3 [41] $FM_{3}(x)=x_1^2+x_2^2+2x_3^2+x_4^2-5x_1-5x_2-21x_3+7x_4,$
$g_2(x)=-x_1^2-x_2^2-x_3^3-x_4^2-x_1+x_2-x_3+x_4+8,$
$g_3(x)=-x_1^2-2x_2^2-x_3^2-2x_4+x_1+x_4+10,$
$g_4(x)=-x_1^2-x_2^2-x_3^2-2x_1+x_2+x_4+5$
Problem 4 [41] $FM_{4}(x)=\text{max}{f_i(x)}\;\; i=1,\ldots,5 $
$f_{1}(x)=(x_1-10)^2+5(x_2-12)^2+x_3^4+3(x_4-11)^2$
$+10x_5^6+7x_6^2+x_7^4-4x_6x_7-10x_6-8x_7,$
$f_2(x)=f_1(x)+10(2x_1^2+3x_2^4+x_3+4x_4^2+5x_5-127),$
$f_3(x)=f_1(x)+10(7x_1+3x_2+10x_3^2+x_4-x_5-282),$
$f_4(x)=f_1(x)+10(23x_1+x_2^2+6x_6^2-8x_7-196),$
$f_5(x)=f_1(x)+10(4x_1^2+x_2^2-3x_1x_2+2x_3^2+5x_6-11x_7$
Problem 5 [34] $FM_{5}(x)=\text{max}\;\; {f_i(x)},\;\;i=1,2,$
$f_1(x)=|x_1+2x_2-7|$,
$f_2(x)=|2x_1+x_2-5|$
Problem 6 [34] $FM_{6}(x)=\text{max} \;\; {f_i(x)}, $
$f_i(x)=|x_i|,\;\;i= 1,\ldots,10$
Problem 7 [22] $FM_{7}(x)= \text{max}\;\; {f_i(x)},\;\;i=1,2,$
$f_1(x)=(x_1-\sqrt{(x_1^2+x_2^2)}cos\sqrt{x_1^2+x_2^2})^2+0.005(x_1^2+x_2^2)^2,$
$f_2(x)=(x_2-\sqrt{(x_1^2+x_2^2)}sin\sqrt{x_1^2+x_2^2})^2+0.005(x_1^2+x_2^2)^2$
Problem 8 [22] $FM_{8}(x)= \text{max}\;\; {f_i(x)},\;\;i=1,\ldots,4,$
$f_1(x)=\big(x_1-(x_4+1)^4\big)^2+\Big(x_2-\big(x_1-(x_4+1)^4\big)^4\Big)^2+2x_3^2+x_4^2$
$\;\;\;\;\;\;\;\;\;\;-5\big(x_1-(x_4+1)^4\big)-5\Big(x_2-\big(x1-(x_4+1)^4\big)^4\Big)-21x_3+7x_4,$
$f_2(x)=f_1(x)+10\Big[\big(x_1-(x_4+1)^4\big)^2+\Big(x_2-\big(x_1-(x_4+1)^4\big)^4\Big)^2+x_3^2+x_4^2$
$\;\;\;\;\;\;\;\;\;\;+\big(x_1-(x_4+1)^4\big)-\Big(x_2-\big(x_1-(x_4+1)^4\big)^4\Big)+x_3-x_4-8\Big],$
$f_3(x)=f_1(x)+10\Big[\big(x_1-(x_4+1)^4\big)^2+2\Big(x_2-\big(x_1$
$\;\;\;\;\;\;\;\;\;\;-(x_4+1)^4\big)^4\Big)^2+x_3^2+2x_4^2-\big(x_1-(x_4+1)^4\big)-x_4-10\Big]$,
$f_4(x)=f_1(x)+10\Big[\big(x_1-(x_4+1)^4\big)^2+\Big(x_2-\big(x_1-(x_4+1)^4\big)^4\Big)^2 $
$\;\;\;\;\;\;\;\;\;\;+x_3^2+2\big(x_1-(x_4+1)^4\big)-\Big(x_2-\big(x_1-(x_4+1)^4\big)^4\Big)-x_4-5\Big]$
Problem 9 [22] $FM_{9}(x)= \text{max}\;\; {f_i(x)},\;\;i=1,\ldots,5, $
$f_1(x)=(x_1-10)^2+5(x_2-12)^2+x_3^4+3(x_4-11)^2+10x_5^6+7x_6^2+x_7^4 $
$\; \; \; \; \; \; \; \; \; \ -4x_6x_7-10x_6-8x_7$,
$f_2(x)=-2x_1^2-2x_3^4-x_3-4x_4^2-5x_5+127$,
$f_3(x)=-7x_1-3x_2-10x_3^2-x_4+x_5+282$,
$f_4(x)=-23x_1-x_2^2-6x_6^2+8x_7+196$,
$f_5(x)=-4x_1^2-x_2^2+3x_1x_2-2x_3^2-5x_6+11x_7$
Problem 10 [22] $FM_{10}(x)=\text{max} {|f_i(x)|}, \;\;\; i=1,\ldots,21,$
$f_i(x)=x_1exp(x_3t_i)+x_2exp(x_4t_i)-\frac{1}{1+t_i}, \ t_i=-0.5+\frac{i-1}{20}$
Table 9.  Minimax test functions properties.
Function Dimension(d) Desired error goal
FM1 2 1.95222245
FM2 2 2
FM3 4 -40.1
FM4 7 247
FM5 2 10−4
FM6 10 10−4
FM7 2 10−4
FM8 4 -40.1
FM9 7 680
FM10 4 0.1
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Function Dimension(d) Desired error goal
FM1 2 1.95222245
FM2 2 2
FM3 4 -40.1
FM4 7 247
FM5 2 10−4
FM6 10 10−4
FM7 2 10−4
FM8 4 -40.1
FM9 7 680
FM10 4 0.1
Table 10.  The efficiency of invoking the Nelder-Mead method in the final stage of SGWO for FM1 − FM10 minimax problems
Function Standard NM SGWO
GWO method
FM1 2940.2 290.35 210.23
FM2 3740.1 286.47 195.15
FM3 1120.2 537.46 381.75
FM4 4940.3 19,147.15 806.45
FM5 3520.4 273.36 215.36
FM6 2080.3 18,245.48 1602.18
FM7 1020.4 736.14 138.62
FM8 1620.4 1652.17 373.25
FM9 3760.5 19,857.69 942.45
FM10 1630.4 867.26 349.46
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Function Standard NM SGWO
GWO method
FM1 2940.2 290.35 210.23
FM2 3740.1 286.47 195.15
FM3 1120.2 537.46 381.75
FM4 4940.3 19,147.15 806.45
FM5 3520.4 273.36 215.36
FM6 2080.3 18,245.48 1602.18
FM7 1020.4 736.14 138.62
FM8 1620.4 1652.17 373.25
FM9 3760.5 19,857.69 942.45
FM10 1630.4 867.26 349.46
Table 11.  Evaluation function for the minimax problems FM1 − FM10
Algorithm Problem Avg SD %Suc
HPS2 FM1 1848.7 2619.4 99
FM2 635.8 114.3 94
FM3 141.2 28.4 37
FM4 8948.4 5365.4 7
FM5 772.0 60.8 100
FM6 1809.1 2750.3 94
FM7 4114.7 1150.2 100
FM8 - - -
FM9 283.0 123.9 64
FM10 324.1 173.1 100
UPSOm FM1 1993.8 853.7 100
FM2 1775.6 241.9 100
FM3 1670.4 530.6 100
FM4 12,801.5 5072.1 100
FM5 1701.6 184.9 100
FM6 18,294.5 2389.4 100
FM7 3435.5 1487.6 100
FM8 6618.50 2597.54 100
FM9 2128.5 597.4 100
FM10 3332.5 1775.4 100
RWMPSOg FM1 2415.3 1244.2 100
FM2 - - -
FM3 3991.3 2545.2 100
FM4 7021.3 1241.4 100
FM5 2947.8 257.0 100
FM6 18,520.1 776.9 100
FM7 1308.8 505.5 100
FM8 - - -
FM9 - - -
FM10 4404.0 3308.9 100
SGWO FM1 210.23 25.54 100
FM2 195.15 36.69 100
FM3 381.75 15.39 100
FM4 806.45 249.55 100
FM5 215.36 75.68 100
FM6 1602.18 425.18 100
FM7 932.6 12.6 100
FM8 138.62 15.23 100
FM9 942.45 55.68 100
FM10 349.46 25.45 100
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Algorithm Problem Avg SD %Suc
HPS2 FM1 1848.7 2619.4 99
FM2 635.8 114.3 94
FM3 141.2 28.4 37
FM4 8948.4 5365.4 7
FM5 772.0 60.8 100
FM6 1809.1 2750.3 94
FM7 4114.7 1150.2 100
FM8 - - -
FM9 283.0 123.9 64
FM10 324.1 173.1 100
UPSOm FM1 1993.8 853.7 100
FM2 1775.6 241.9 100
FM3 1670.4 530.6 100
FM4 12,801.5 5072.1 100
FM5 1701.6 184.9 100
FM6 18,294.5 2389.4 100
FM7 3435.5 1487.6 100
FM8 6618.50 2597.54 100
FM9 2128.5 597.4 100
FM10 3332.5 1775.4 100
RWMPSOg FM1 2415.3 1244.2 100
FM2 - - -
FM3 3991.3 2545.2 100
FM4 7021.3 1241.4 100
FM5 2947.8 257.0 100
FM6 18,520.1 776.9 100
FM7 1308.8 505.5 100
FM8 - - -
FM9 - - -
FM10 4404.0 3308.9 100
SGWO FM1 210.23 25.54 100
FM2 195.15 36.69 100
FM3 381.75 15.39 100
FM4 806.45 249.55 100
FM5 215.36 75.68 100
FM6 1602.18 425.18 100
FM7 932.6 12.6 100
FM8 138.62 15.23 100
FM9 942.45 55.68 100
FM10 349.46 25.45 100
Table 12.  SGWO and other meta-heuristics and swarm intelligence algorithms for FM1 − FM10 minimax problems
Function GA+NM DE+NM PSO+NM FF+NM CS+NM SGWO
FM1 Avg 486.25 458.47 490.78 445.42 391.16 210.23
  SD 153.69 114.58 128.87 98.47 95.48 25.54
FM2 Avg 469.58 459.28 485.46 483.47 346.58 195.15
  SD 115.45 112.86 135.486 115.78 125.48 36.69
FM3 Avg 635.48 590.46 610.76 598.48 359.42 381.75
  SD 186.92 211.48 184.35 115.46 112.58 15.39
FM4 Avg 2158.69 2214.78 1985.46 1965.48 1846.35 806.45
  SD 354.76 387.45 453.84 536.44 458.75 249.55
FM5  Avg 476.58 436.48 469.85 456.48 315.36 215.36
SD 114.79 113.58 135.48 112.47 114.56 75.68
FM6 Avg 5383.49 4952.36 5148.46 4856.24 2952.14 1602.18
  SD 486.58 425.85 415.68 364.58 358.45 425.18
FM7  Avg 487.48 495.48 496.58 468.12 295.48 138.62
SD 127.85 142.36 185.26 169.35 85.34 12.6
FM8 Avg 2180.35 2049.15 2185.46 1954.15 1665.28 373.25
  SD 487.54 475.69 519.48 413.68 98.62 15.23
FM9  Avg 5982.48 5846.48 5948.47 5634.65 3158.46 942.45
SD 487.14 356.84 458.36 368.47 256.48 55.68
FM10  Avg 845.71 795.26 876.29 863.45 563.58 349.46
SD 248.27 195.47 112.84 158.58 158.16 25.45
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Function GA+NM DE+NM PSO+NM FF+NM CS+NM SGWO
FM1 Avg 486.25 458.47 490.78 445.42 391.16 210.23
  SD 153.69 114.58 128.87 98.47 95.48 25.54
FM2 Avg 469.58 459.28 485.46 483.47 346.58 195.15
  SD 115.45 112.86 135.486 115.78 125.48 36.69
FM3 Avg 635.48 590.46 610.76 598.48 359.42 381.75
  SD 186.92 211.48 184.35 115.46 112.58 15.39
FM4 Avg 2158.69 2214.78 1985.46 1965.48 1846.35 806.45
  SD 354.76 387.45 453.84 536.44 458.75 249.55
FM5  Avg 476.58 436.48 469.85 456.48 315.36 215.36
SD 114.79 113.58 135.48 112.47 114.56 75.68
FM6 Avg 5383.49 4952.36 5148.46 4856.24 2952.14 1602.18
  SD 486.58 425.85 415.68 364.58 358.45 425.18
FM7  Avg 487.48 495.48 496.58 468.12 295.48 138.62
SD 127.85 142.36 185.26 169.35 85.34 12.6
FM8 Avg 2180.35 2049.15 2185.46 1954.15 1665.28 373.25
  SD 487.54 475.69 519.48 413.68 98.62 15.23
FM9  Avg 5982.48 5846.48 5948.47 5634.65 3158.46 942.45
SD 487.14 356.84 458.36 368.47 256.48 55.68
FM10  Avg 845.71 795.26 876.29 863.45 563.58 349.46
SD 248.27 195.47 112.84 158.58 158.16 25.45
Table 13.  Experimental results (mean, standard deviation and rate of success) of function evaluation between SQP and SGWO for FM1 − FM10 test problems
Function Algorithm Mean St.D Suc
FM1 SQP 4044.5 8116.6 24
SGWO 211.45 31.56 30
FM2 SQP 8035.7 9939.9 18
SGWO 191.58 45.36 30
FM3 SQP 135.5 21.1 30
SGWO 385.75 27.42 30
FM4 SQP 20,000 0.0 0.0
SGWO 825.36 250.36 30
FM5 SQP 140.6 38.5 30
SGWO 210.45 85.65 30
FM6 SQP 611.6 200.6 30
SGWO 1648.23 512.34 30
FM7 SQP 15,684.0 7302.0 10
SGWO 152.34 15.48 30
FM8 SQP 20,000 0.0 0.0
SGWO 380.26 36.89 30
FM9 SQP 20,000 0.0 0.0
SGWO 936.48 62.35 30
FM10 SQP 4886.5 8488.4 22
SGWO 356.89 39.85 30
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Function Algorithm Mean St.D Suc
FM1 SQP 4044.5 8116.6 24
SGWO 211.45 31.56 30
FM2 SQP 8035.7 9939.9 18
SGWO 191.58 45.36 30
FM3 SQP 135.5 21.1 30
SGWO 385.75 27.42 30
FM4 SQP 20,000 0.0 0.0
SGWO 825.36 250.36 30
FM5 SQP 140.6 38.5 30
SGWO 210.45 85.65 30
FM6 SQP 611.6 200.6 30
SGWO 1648.23 512.34 30
FM7 SQP 15,684.0 7302.0 10
SGWO 152.34 15.48 30
FM8 SQP 20,000 0.0 0.0
SGWO 380.26 36.89 30
FM9 SQP 20,000 0.0 0.0
SGWO 936.48 62.35 30
FM10 SQP 4886.5 8488.4 22
SGWO 356.89 39.85 30
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