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

Mohamed A. Tawhid; Ahmed F. Ali

doi:10.3934/naco.2017020

Article Contents

2017, Volume 7, Issue 3: 301-323. Doi: 10.3934/naco.2017020

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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
* 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: October 2017

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.

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Abstract

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:

Mathematics Subject Classification: Primary: 90C10, 90C47, 90C56; Secondary: 68U20, 68W20.

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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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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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Table 3. The properties of the Integer programming test functions

Function	Dimension (d)	Bound	Optimal
FI₁	5	[-100 100]	0
FI₂	5	[-100 100]	0
FI₃	5	[-100 100]	-737
FI₄	2	[-100 100]	0
FI₅	4	[-100 100]	0
FI₆	2	[-100 100]	-6
FI₇	2	[-100 100]	-3833.12

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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
FI₁	660.4	1988.35	227.3
FI₂	560.2	678.15	225.6
FI₃	4920.5	819.45	701.24
FI₄	2860.3	266.14	283.5
FI₅	1520.6	872.46	508.15
FI₆	3760.2	254.15	364.27
FI₇	1200.3	245.47	235.16

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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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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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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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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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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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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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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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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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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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