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A multi-objective artificial sheep algorithm

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Abstract

In this paper, a novel multi-objective artificial sheep algorithm (MOASA) is proposed. The basic search idea of MOASA inherits from the BASA, which is inspired by the social behavior of sheep herd, while some modifications are made to extend the algorithm to multi-objective problems. The Pareto-based theory is adopted in the MOASA along with external archive and leader selection mechanism to bring about multi-objective optimization. Furthermore, a novel neighborhood search method is proposed and applied to the external archive to enhance the performance of the algorithm. The proposed MOASA is then tested on 17 multi-objective benchmark problems to verify its efficiency and effectiveness by comparing with six powerful multi-objective optimization algorithms (MOAs). Experimental results show that the MOASA is generally superior to its competitors in solving those benchmark problems in terms of convergence and Pareto front distribution.

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Abbreviations

SI:

Swarm intelligence

SOO:

Single-objective optimization

SOP:

Single-objective optimization problem

MOO:

Multi-objective optimization

MOP:

Multi-objective optimization problem

MOA:

Multi-objective optimization algorithm

GA:

Genetic algorithm

NSGA:

Non-dominated sorting genetic algorithm

SPEA:

Strength Pareto evolutionary algorithm

NSGA-II:

Fast and elitist multi-objective genetic algorithm

MOEA/D:

Multi-objective evolutionary algorithm based on decomposition

MOEA:

Multi-objective evolutionary algorithm

BASA:

Binary artificial sheep algorithm

MOPSO:

Multi-objective particle swarm optimization

MOGSA:

Multi-objective gravitational search algorithm

MOASA:

Multi-objective artificial sheep algorithm

MOABC:

Multi-objective artificial bee colony algorithm

MODA:

Multi-objective dragonfly algorithm

MOGWO:

Multi-objective gray wolf optimizer

MODA:

Multi-objective dragonfly algorithm

MOPSO:

Multi-objective particle swarm optimization

MOACO:

Multi-objective ant colony optimization

MOPSDE:

Multi-objective particle swarm-differential algorithm

GD:

Generational distance

IGD:

Inverted generational distance

MS:

Maximum spread

SF:

Superiority of feasible solution

SEEH:

Short-term economic environmental hydrothermal scheduling

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Acknowledgements

This paper is supported by the National Key Research and Development Program of China (2016YFC0401905) and the National Natural Science Foundation of China (Nos. 51679095, 51479076).

Author information

Authors and Affiliations

  1. School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Xinjie Lai, Chaoshun Li, Nan Zhang & Jianzhong Zhou

Authors
  1. Xinjie Lai
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  2. Chaoshun Li
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  3. Nan Zhang
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  4. Jianzhong Zhou
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Corresponding author

Correspondence to Chaoshun Li.

Appendices

Appendix 1

See Tables 17, 18 and 19.

Table 17 Mathematical representation of two-objective unconstrained test instances UF1–UF7
Full size table
Table 18 Mathematical representation of three-objective unconstrained test instances UF8–UF10
Full size table
Table 19 Mathematical representation of three-objective unconstrained test instances UF8–UF10
Full size table

Appendix 2

The initial parameters of MOASA are as follows:

MaxIter

Equal to 300,000 function evaluations

MaxTrial = 100

Max invalid trial

N = 100

Number of population

\(N_{\text{rep}} = \left\{ {\begin{array}{*{20}l} {100} \hfill \\ {150} \hfill \\ \end{array} } \right.\)

Maximum number of repository for bi-objective test problems

 

Maximum number of repository for tri-objective test problems

nGrid = 100

Number of grids per dimension

\(\alpha = 0.1\)

Grid inflation parameter

\(\beta = 2\)

Leader selection pressure

\(\gamma = 2\)

Deletion selection pressure

For MOGWO, the following initial parameters are chosen:

MaxIter

Equal to 300,000 function evaluations

N = 100

Number of population

\(N_{\text{rep}} = \left\{ {\begin{array}{*{20}l} {100} \hfill \\ {150} \hfill \\ \end{array} } \right.\)

Maximum number of repository for bi-objective test problems

 

Maximum number of repository for tri-objective test problems

nGrid = 10

Number of grids per dimension

\(\alpha = 0.1\)

Grid inflation parameter

\(\beta = 2\)

Leader selection pressure

\(\gamma = 2\)

Deletion selection pressure

For MOPSO, the initial parameters are as follows:

MaxIter

Equal to 300,000 function evaluations

N = 100

Number of population

\(N_{\text{rep}} = \left\{ {\begin{array}{*{20}l} {100} \hfill \\ {150} \hfill \\ \end{array} } \right.\)

Maximum number of repository for bi-objective test problems

 

Maximum number of repository for tri-objective test problems

nGrid = 10

Number of grids per dimension

\(\alpha = 0.1\)

Grid inflation parameter

\(\beta = 2\)

Leader selection pressure

\(\gamma = 2\)

Deletion selection pressure

\(\varphi_{1} = 2.05\)

 

\(\varphi_{2} = 2.05\)

 

\(\varphi = 4.1\)

 

\(w = \frac{2}{{\varphi - 2 + \sqrt {\varphi^{2} - 4\varphi } }}\)

Inertia weight

\(c_{1} = \chi \cdot \varphi_{1}\)

Personal coefficient

\(c_{2} = \chi \cdot \varphi_{2}\)

Social coefficient

For MOEA/D, the following parameters are used:

MaxIter

Equal to 300,000 function evaluations

N = 100

Number of population

T = 0.1N

Number of neighbors

nr = 0.01N

Maximal copies of a new child in update

\(\delta = 0.9\)

Probability of selecting parents from the neighborhood

CR = 0.5

Crossover probability

\(\eta = 30\)

Distribution index

For NSGA-II, the following parameters are chosen:

MaxIter

Equal to 300,000 function evaluations

N = 100

Number of population

pool = 0.1N

Size of mating pool

\(\gamma = 0.01N\)

Crossover probability

\(\lambda = 0.9\)

Mutation probability

\(\mu_{\text{c}} = 0.5\)

Crossover distribution index

\(\mu_{\text{m}} = 30\)

Mutation distribution index

Appendix 3

The detailed results of hydrothermal system in scheme 9.

Period

Water discharge (104 m3/h)

Hydro output (MW)

Thermal output (MW)

Total output (MW)

Power loss (MW)

Load demand (MW)

Q h1

Q h2

Q h3

Q h4

P h1

P h2

P h3

P h4

P s1

P s1

P s1

1

14.073

6.000

22.231

6.000

97.768

49.000

34.173

131.880

175.000

132.374

140.127

760.32

10.32

750

2

5.000

6.000

23.887

6.000

51.575

50.164

19.183

129.027

175.000

227.987

139.113

792.05

12.05

780

3

8.276

6.000

21.012

6.000

76.800

51.296

30.012

125.744

170.142

126.998

128.458

709.45

9.45

700

4

5.000

6.000

18.199

6.000

52.426

52.934

41.047

121.625

175.000

127.801

87.812

658.64

8.64

650

5

5.032

6.000

18.108

6.000

53.114

54.500

39.425

115.822

174.992

106.277

135.484

679.61

9.61

670

6

5.000

6.000

20.965

6.415

53.021

55.504

26.578

136.565

174.648

229.842

135.760

811.92

11.92

800

7

12.546

6.000

18.243

6.000

96.914

55.994

36.271

145.619

174.999

299.944

154.678

964.42

14.42

950

8

10.277

6.000

18.685

10.098

87.269

55.994

32.821

210.380

175.000

299.999

163.339

1024.80

14.80

1010

9

12.199

9.133

19.518

18.531

93.567

76.432

26.735

296.256

174.910

296.424

140.053

1104.38

14.38

1090

10

15.000

6.000

17.535

20.000

96.716

55.929

34.681

305.176

171.815

288.536

141.127

1093.98

13.98

1080

11

7.859

9.603

18.625

18.403

71.809

79.897

30.501

295.909

175.000

300.000

162.039

1115.16

15.16

1100

12

12.161

6.000

19.518

18.685

92.550

57.074

26.891

297.679

175.000

300.000

218.604

1167.80

17.80

1150

13

7.402

10.543

19.988

19.518

69.561

85.232

27.637

303.076

174.608

299.732

165.474

1125.32

15.32

1110

14

5.000

6.000

19.518

18.428

52.103

56.819

28.891

295.926

174.084

297.296

139.027

1044.15

14.15

1030

15

8.658

12.425

19.048

18.155

80.610

93.139

33.176

293.151

174.924

217.749

129.469

1022.22

12.22

1010

16

6.884

10.783

19.518

19.988

69.357

84.741

30.021

305.499

161.799

277.997

143.675

1073.09

13.09

1060

17

5.000

11.362

19.098

19.096

54.258

85.636

31.011

299.504

172.058

287.195

133.983

1063.64

13.64

1050

18

11.862

10.543

14.201

19.518

98.296

79.564

43.821

303.076

174.989

295.885

138.708

1134.34

14.34

1120

19

12.770

6.000

10.000

19.048

99.805

50.512

46.386

300.085

163.584

281.699

141.076

1083.15

13.15

1070

20

5.000

7.880

10.000

19.518

53.445

63.467

48.269

303.076

175.000

213.709

208.193

1065.16

15.16

1050

21

5.000

15.000

10.000

19.098

53.609

91.248

51.741

300.407

174.543

110.439

138.645

920.63

10.63

910

22

5.000

6.000

10.000

17.991

53.911

47.640

54.754

292.856

171.548

119.004

130.407

870.12

10.12

860

23

5.000

14.305

10.000

18.331

54.302

87.791

55.256

291.519

174.894

145.548

50.000

859.31

9.31

850

24

5.000

12.425

18.671

17.879

54.705

78.281

49.446

279.941

171.379

124.894

50.000

808.65

8.65

800

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Lai, X., Li, C., Zhang, N. et al. A multi-objective artificial sheep algorithm. Neural Comput & Applic 31, 4049–4083 (2019). https://doi.org/10.1007/s00521-018-3348-x

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