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1 ........................................................................................................

2 .......................................................................................................

2 ......................................................................................................... . 1 2 .............................................................................................................................. . 1-1

6 ................................................................................................................. . 2-1

7 ............................................................................................... . 2 7 ........................................................................................................ . 1-2

7 ........................................... . 2-2

21 .................................................................................................................... . 3-2

21 .................................................................................................................... . 4-2

31 .............................................................................................................. . 5-2

31 .................................................................................................................. . 6-2

71 ........................................................................................ . 7-2

12 ................................................... . 8-2

22 ............................................................................................................. .1-8-2

22 ......................................................................................... . 3 42 ......................................................................................................................... . 1-3

42 ................................................................................................................. enilpS-B . 2-3

52 ....................................................................... )TSC( noitamrofsnarT-epahS-ssalC . 3-3

62 ................................................ )TRSC( noitamrofsnarT-tnemenifeR-epahS-ssalC . 4-3

22 ................................................................ 21032 . 2 82 ...................................................................................................... . 1-4

92 .......................................................................................................................... . 2-4

33 ........................................................................................................ . 3-4

13 .................................................................... .1-3-4

33 ................................................................... .2-3-4

43 .................................................................. .3-3-4

63 ................................................................. .4-3-4

73 ............................................................................... .5-3-4

02 ....................................................................................................... . 2

12 .......................................................................................................

11 .................................................................................................... OSPT 1-2 41 ...................................................................................................................... 1 2-2 41 ..................................................................................................................... 1 3-2 51 ..................................................................................................................... 2 4-2 51 ..................................................................................................................... 3 5-2 61 ...................................................................................................................... 4 6-2 61 ...................................................................................................................... 5 7-2 91 ........................................................... 35 8-2 32 ........................................................... 35 9-2 82 ...................... 51332 TNEULF LIOFX 1-4 13 ................................................................. 2-4 23 ....................................... 3-4 23 .............................................. 4-4 33 ................................................................. 5-4 33 ....................................... 6-4 43 .............................................. 7-4 43 ................................................................. 8-4 53 ....................................... 9-4 53 ............................................ 31-4 63 ............................................................... 11-4 63 ..................................... 21-4 73 ............................................ 31-4 83 ....................... 41-4 51-4

83 .................................................................................................................................................. 93 .... 61-4

31 ....................................................................................................... 1-2 81 ........................................................................ 2-2 81 ........................................................................... 3-2 32 ................................................... 4-2 33 ........................................................................................................ 1-4

13 ................................................................................ 2-4 13 ............................................................................................ 3-4

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1 10F :Rastrigin, Weierstrass, Griewank, Ackley and Sphere Function

2-7 2

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Function Dim TPSO PSO ARPSO FDRPSO CRIBS

Rastrigin

20 18.82 23.69 23.32 24.75 24.72

50 63.61 113.85 116.83 131.64 120.56

100 137.93 309.57 314.35 351.29 292.32

Michalewicz

20 -15.11 -13.33 -13.13 -12.75 -12.94

50 -33.49 -20.09 -18.98 -21.81 -20.29

100 -63.77 -31.16 -34.17 -34.61 -30.56

Schwefel

20 -1.44e+04 -5337.06 -5265.33 -5495.45 -5365.71

50 -2.68e+04 -1.31e+04 -1.31e+04 -1.37e+04 -1.30e+04

100 -5.12e+04 -2.58e+04 -2.65e+04 -2.77e+04 -2.61e+04

Griewangk

20 2.06e-01 1.31e-02 1.72e-02 1.11e-02 2.46e-02

50 3.55e-01 1.12e-03 4.68e-03 1.91e-02 1.09e-02

100 4.71e-01 3.56e-03 3.31e-03 7.66e-02 5.5e-03

Rosenbrock

20 18.17 38.27 34.12 27.56 37.27

50 46.02 130.09 135.49 253.42 158.82

100 99.60 345.12 346.24 1490.56 402.17

Ackley

20 8.88e-01 2.58e-05 2.82e-04 5.57e-02 1.22e-01

50 1.59 1.78e-01 2.44e-01 1.06 3.27e-01

100 2.03 1.01 1.21 1.98 5.75e-01

2-2

Function Dim TPSO PSO ARPSO FDRPSO CRIBS

TF 5 0.2 0.38 0.36 0.49 0.29

CF1 5 15.95 43.79 31.64 123.84 32.87

CF2 5 65.63 97.90 89.41 70.17 92.81

CF3 5 120.28 156.65 152.68 248.73 187.69

CF4 5 216.76 272.39 469.17 398.72 327.64

CF5 5 32.42 72.78 47.71 77.64 66.09

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MOTPSO Parameterization

Swarm Size: 100

Mutation polynomial

Archive Size: 100 individuals

: 0.5

Inertia weight: (0.9 to 0.4)

NSGA-II SPEA2 AbYSS MOTPSO Prob

HV HV HV HV

6.61 e1 1.83 e4 1.03 e1 6.61e 1 7.72e3 1.05e 1 6.60e 1 8.69e 1 1.52e 1 6.59e 1 1.37e 2 3.70e 1 ZDT1

3.28 e1 6.10 e4 9.87 e2 3.28e 1 7.10e 3 1.07e 1 3.26e 1 8.73e 3 1.55e 1 3.26e 1 1.28e 2 3.81e 1 ZDT2

5.16e 1 9.13 e4 7.32 e1 5.16e 1 6.10e 3 7.09 e1 5.14e 1 9.72e 3 7.10e 1 5.15e 1 8.13e 1 7.47e 1 ZDT3

6.60e 1 8.86 e4 1.19 e1 6.55e 1 1.14e-2 1.27 e1 6.51e 1 3.42e 2 2.72e 1 6.56e 1 1.49e 2 4.02e 1 ZDT4

4.00e 1 9.48 e3 8.47e 1 4.01e 1 5.06e-3 8.99 e2 3.79e 1 2.42e 2 2.28e 1 3.88e 1 1.47e 2 3.56e 1 ZDT6

4.94e 1 2.49e 3 1.25e 1 4.86e 1 5.85e3 1.40e 1 4.89e 1 5.89e 3 1.81e 1 4.88e 1 7.13e 3 4.03e 1 DTLZ1

2.12e 1 7.85e 3 1.42e 1 2.12e 1 5.39e 3 1.09e 1 2.12e 1 7.34e 3 1.48e 1 2.11e 1 1.11e 2 3.84e 1 DTLZ2

2.12e 1 7.85e 3 2.81e 1 - 1.66e 0 7.55e 1 - 2.28e 0 1.07e 0 - 1.04e 0 9.53e 1 DTLZ3





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LIOFX LIOFX . TNEULF

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)100.0-,4.0-( 41

)100.0-,4.0-( 51

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liofriAL

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7845.1 1514.0 2000.0 29.131 2 .jbO tseB

3368.1 8299.0 8190.0 37.901 3 .jbO tseB

7622.2 7351.0 2950.0 02.68 4 .jbO tseB

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tnemevorpmI liofriA

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OSPTOM 3-4 2-4 . 51332

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23 51332 :

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33 51332 :

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6-2

43 51332 :

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41

[1 ] Buckley, P., H., Zhou, Y., B., Zingg, and W., D., 2010, "Airfoil Optimization Using Practical Aerodynamic Design Requirements," 47(5), pp. 13.

[2 ] Kuntawala, N. B., 2011, "Aerodynamic Shape Optimization of a Blended-Wing-Body Aircraft Configuration , " Ph.D. thesis, University of Toronto ,

[3 ] ,. ,1385 ," ," Ph.D. thesis , ,

[4 ] Ashrafizadeh, A., Raithby, G. D., and Stubley, G. D., 2002, "Direct Deign of Shape," Numerical Heat Transfer, 41(6), pp. 10.

[5 ] Ashrafizadeh, A., Raithby, G. D., and Stubley, G. D., 2003, "Direct Deign of Ducts," Fluids Engineering, ASME Transaction, 125(pp. 7.

[6 ] Ashrafizadeh, A., Raithby, G. D., and Stubley, G. D . ,2004 ," Direct Deign of Airfoil Shape with a Prescribed Surface Pressure," Numerical Heat Transfer, 46(6), pp. 22.

[7 ] ,. ,1386 ," ," Ph.D. thesis , ,

[8 ] Epstein, B., and Peigin, S., 2007, "Accurate Cfd Driven Optimization of Lifting Surfaces for Wing-Body Configuration," Journal of computers & fluids, 36(pp. 16.

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203(pp. 13. [10 ] Carpentieri, G., Koren, B., and Tooren, M. J. L. V., 2007, "Adjoint-Based Aerodynamic

Shape Optimization on Unstructured Meshes," Journal of Computational Physics ,224( pp. 21.

[11 ] Jameson, A., Martinelli, L., and Pierce, N. A., 1998, "Optimum Aerodynamic Design Using the NavierStokes Equations," Theoretical and Computational Fluid Dynamics, 10(1), pp. 213-237.

[12 ] Anderson, W. K., and Venkatakrishnan, V., 1999 ," Aerodynamic Design Optimization on Unstructured Grids with a Continuous Adjoint Formulation," Computers &

Fluids, 28(45), pp. 443-480. [13 ] Arias-Montao, A., Coello Coello, C., and Mezura-Montes, E., 2011, "Evolutionary

Algorithms Applied to Multi-Objective Aerodynamic Shape Optimization," 356(pp. 211-240.

[14 ] Wickramasinghe, U. K., Carrese, R., and Xiaodong, L., 2010, "Designing Airfoils Using a Reference Point Based Evolutionary Many-Objective Particle Swarm

Optimization Algorithm," eds., pp. 1 -8. [15 ] Kennedy, J., and Eberhart, R., 1995, "Particle Swarm Optimization," eds., 4, pp. 1942-

1948 vol.4. [16 ] Eberhart, R., and Shi, Y., 1998, Evolutionary Programming Vii, Springer Berlin /

Heidelberg, Comparison between Genetic Algorithms and Particle Swarm Optimization.

[17 ] Poli, R., 2008, "Analysis of the Publications on the Applications of Particle Swarm Optimisation," J. Artif. Evol. App., 2008(pp. 1-10.

[18 ] Poli, R., Kennedy, J., and Blackwell, T., 2007, "Particle Swarm Optimization," Swarm Intelligence, 1(1), pp. 33-57.

42

[19 ] Mendes, R., 2004, "Population Topologies and Their Influence in Particle Swarm Performance," Ph.D. thesis, Departamento de Informatica, Escola de Engenharia,

Universidade do Minho, Portugal. [20 ] Riget, J., and Vesterstrm ,J. S., 2002, "A Diversity-Guided Particle Swarm Optimizer-

the Arpso," Technical Report No. Department of Computer Science, University of

Aarhus, Denmark. [21 ] Peram, T., Veeramachaneni, K., and Mohan, C. K., 2003, "Fitness-Distance-Ratio

Based Particle Swarm Optimization," eds., pp. 174-181. [22 ] Ben Ghalia, M., 2008, "Particle Swarm Optimization with an Improved Exploration-

Exploitation Balance," eds., pp. 759-762. [23 ] Monson, C. K., and Seppi, K. D., 2006, 1144006. [24 ] Liang, J. J., Suganthan, P. N . , and Deb, K., 2005, "Novel Composition Test Functions

for Numerical Global Optimization," eds., pp. 68-75. [25 ] Ahrari, A., Saadatmand, M. R., Shariat-Panahi, M., and Atai, A. A., 2010, "On the

Limitations of Classical Benchmark Functions for Evaluating Robustness of Evolutionary Algorithms," Applied Mathematics and Computation, 215(9), pp. 3222-

3229. [26 ] Gaviano, M., and Lera, D., 1998, "Test Functions with Variable Attraction Regions for

Global Optimization Problems," Journal of Global Optimization, 1 3(2) , pp. 207-223. [27 ] Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T., 2002, "A Fast and Elitist

Multiobjective Genetic Algorithm: Nsga-Ii," Evolutionary Computation, IEEE

Transactions on, 6(2), pp. 182-197. [28 ] Deb, K., 2001, Multi-Objective Optimization Using Evolutionary Algorithms John

Wiley & Sons , [29 ] Zitzler, E., Deb, K., and Thiele, L., 2000, "Comparison of Multiobjective Evolutionary

Algorithms: Empirical Results," Evolutionary Computation, 8(2), pp. 173-195. [30 ] Nebro, A. J., Luna, F . , Alba, E., Dorronsoro, B., Durillo, J. J., and Beham, A., 2008,

"Abyss: Adapting Scatter Search to Multiobjective Optimization," Evolutionary

Computation, IEEE Transactions on, 12(4), pp. 439-457. [31 ] Zitzler, E., Laumanns, M., and Thiele, L., 2001, "Spea2: Improving the Strength Pareto

Evolutionary Algorithm," Computer Engineering, TIK-Report(pp. 1--21. [32 ] Deb, K., Thiele, L., Laumanns, M., and Zitzler, E., 2005, Springer Berlin Heidelberg,

Scalable Test Problems for Evolutionary Multiobjective Optimization. [33 ] Knowles, J., Thiele, L., and Zitzler, E., 2006, "A Tutorial on the Performance

Assessment of Stochastic Multiobjective Optimizers," Technical Report No. 214,

Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of

Technology (ETH) Zurich , [34 ] Zitzler, E., and Thiele, L., 1999, "Multiobjective Evolutionary Algorithms: A

Comparative Case Study and the Strength Pareto Approach," Evolutionary

Computation, IEEE Transactions on, 3(4), pp. 257-271. [35 ] Durillo, J. J., and Nebro, A. J., 2011, "Jmetal: A Java Framework for Multi-Objective

Optimization," Advances in Engineering Software, 42(pp. 760-771. [36 ] Kulfan, B. M., 2008, "Universal Parametric Geometry Representation Method," 45(1),

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ab55-d74e980f724a [38 ] Drela, M., and Giles, M. B., 1987, "Viscous-Inviscid Analysis of Transonic and Low

Reynolds Number Airfoils," 25(10), pp .

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