Solving Structural Optimization Problem Using Bare ...

1st International Conference on Sustainable Civil Engineering Structures and Construction Materials

Solving Structural Optimization Problem Using Bare-Bones Particle Swarm Optimization D. K. Wibowo Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan Department of Civil Engineering, Institut Teknologi Sepuluh November, Surabaya, Indonesia

M. - Y. Cheng Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan

D. Prayogo Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan Department of Civil Engineering, Institut Teknologi Sepuluh November, Surabaya, Indonesia

Abstract: In many practical applications of structural optimization, most problems are highly nonlinear due to a huge number of design variables and complex constraints on stresses, displacements, load carrying capability, and geometrical configuration. As a result, finding the optimal solution becomes an uneasy task and extremely challenging. The traditional local search algorithms, such as Nelder-Mead downhill methods, hill-climbing, and steepest descent, are often failed to search the optimum solution in nonlinear problem. In this study, a metaheuristic approach, namely bare-bones PSO (BBPSO), is introduced to solve a set of complex structural optimization problems. BBPSO is a variant of the particle swarm optimization in which the velocity and position-update rules are replaced with samples from Gaussian distribution. To validate the performance of BBPSO, several case studies of structural optimization problem are provided, including truss design, reinforce concrete beam design, and tubular column design. The results indicate the potential of BBPSO as an alternative approach to solve many structural optimization problems.

Keywords: Bare-bones PSO, Structural optimization, Metaheuristic.

1 INTRODUCTION

In the last three decades, structural design optimization has received significant attention from many researchers. The major obstacle factors to apply structural optimization are the large number of design variable, large size of search space and a great number of design constraints. Traditionally, various local search methods such as Nelder-Mead downhill methods, hill-climbing, and steepest descent have been developed in order to solve structural optimization problems. However, these methods are limited approach, often failed to search the optimum solution (Gandomi et al. 2011). Metaheuristic is used to solve computational (usually optimization) problem by black-box procedures in a hopefully efficient way. It is called black-box because it is hard to explain the reason why these methods can obtain global minimum. Two important characteristics of metaheuristic methods are intensification and diversification. Intensification applies to look for around the current best solutions and to choose the best candidate. Diversification enables the optimizer to explore the search space more efficiently. The

balance of these two characteristics is needed to attain global solution (Yang 2010). During this several past decades, many metaheuristic algorithms were developed such as Genetic Algorithm (GA) (Goldberg 1989), Particle Swarm Optimization (PSO) (Eberhart and Kennedy 1995), Ant Colony Optimization (ACO) (Dorigo et al. 1996), Differential Evolution (DE) (Storn and Price 1997), Firefly Algorithm (FA) (Yang 2009), and Cuckoo Search (CS) (Xin-She and Deb 2009). Many of these metaheuristic methods have been used to solve structural optimization problems such as: GA (Ali et al. 2003; Kahraman and Erbatur 2001), PSO (Jansen and Perez 2011; Yang and Hsieh 2011), CS (Gandomi et al. 2011), FA (Gandomi et al. 2011), etc. Among other algorithms, PSO has a number of advantages that make it as a promising tool to solve structural design optimization. This algorithm is robust, easy to implement, and well suited to handle non-linear problems. It can also deal with either continuous or discrete variable types. Compare to the others, PSO is more efficient, requires less number of evaluations, and produces a better or at least the same quality of the results (Hassan et al. 2005; Xiaohui et al. 2003). 1

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In the development of PSO, there are several types of PSO such as: Linear Decreasing Inertia Weight PSO (Shi and Eberhart 1998), Time Varying Acceleration Coefficient PSO (Ratnaweera et al. 2004), FullyInformed (Mendes et al. 2004), and Bare-bones PSO (Kennedy 2003). Among these algorithms, Barebones (BBPSO) is recognized as the simplest version of PSO (Feng et al. 2008). In BBPSO, velocity formula is eliminated and a Gaussian distribution is used to directly sample the next location. Thus, compared with original PSO, the BBPSO is more solid and there is no parameter tuning. This paper focuses on implementation and application of BBPSO, for structural optimization. The general explanation of PSO and BBPSO is introduced. Then, application of the algorithm to several structural optimization problems is shown. The development of the paper is as follow: Section 2 presents general information of PSO and improvements of basic algorithm. Section 3 presents the concept of BBPSO is introduced. Section 4 presents several structural optimization case studies. Finally, the paper is closed with some concluding remarks.



range of 0 and 1, x pBest represents the best ever



particle position of particle i, xi (t ) represents the



position of particle at iteration t, xgBest corresponds to the global best position in the swarm up to iteration k. The remaining terms, c1 and c2 , represent “trust” parameters indicating how much confidence the current particle has in itself ( c1 or cognitive parameter) and how much confidence it has in the swarm ( c2 or social parameter). The value of c1 and

c2 are usually between 1 and 4. Meanwhile, inertia weight has important role because it directly impacts the current velocity. Large inertia weight allows for wide velocity to globally explore the search space, whereas small inertia weight value will make the velocity updates to neighboring regions of the search space. The update position of the swarm i at iteration k + 1 is shown in Equation (2) and illustrated in Figure 1.

   xi (t  1)  xi (t )  vi (k  1)

(2)

2 PARTICLE SWARM OPTIMIZATION

Particle swarm optimization (PSO) is an algorithm developed by Kennedy and Ebenhart in 1995 (Eberhart and Kennedy 1995) that mimic the social adaptation of biological creature in swarm. It is implementing the fundamental rules that enable large number of organism (birds, fishes, etc) to move synchronously, often changing direction rapidly, spreading out, and regrouping. Each particle in the swarm iteratively moves across the search space. To update the current position of each particle in the swarm, a velocity vector is used. The velocity vector is updated based on the “memory” gained by each particle not only from own experience but also information gained by the whole swarm. The particle is attract to the position (location) of the best fitness (evaluation of the objective function) historically achieved by the particle itself (local best; pBest) and by the best among the neighbors of the particle (global best; gBest). Numerically, the velocity vector of each particle i at iteration k + 1 is calculated as shown in Equation (1).       vi (k  1)  w  vi (k )  r1c1 ( x pBest  xi (t ))  r2c2 ( xgBest  xi (t )) (1) where w is an inertia weight to control influence of the previous velocity (usually between 0 and 1),  vi (k ) is the velocity vector at iteration k, r1 and r2 are two random numbers uniformly distributed in the 2

Figure 1. PSO update location and velocity 3 BARE-BONES PARTICLE SWARM OPTIMIZATION

Bare-bones PSO (BBPSO) was introduced by Kennedy in 2003 (Kennedy 2003). The BBPSO eliminates the velocity term and uses the Gaussian distribution (see Figure 2) to find the next location based on the global best (gBest) and the local best (pBest). To obtain value from Gaussian distribution, the mean and standard deviation are needed. The values of mean and standard deviation are obtained by using Equation (3) and (4).

1st International Conference on Sustainable Civil Engineering Structures and Construction Materials

i 

  xgBest  x pBest 2 



 i2  xgBest  x pBest

4.1 Cantilever Beam

(3) (4)

This case study related with cantilever beam (Chickermane and Gea 1996). The cantilever beam as shown in Figure 3 is made from five elements. Each element has a hollow cross section with fixed thickness. The beam is rigidly supported as shown, and there is vertical force acting at the free end of the cantilever. The problem is to minimize the weight of the beam. The design variables are the height (or width) xi of each beam element. The bound constraints are set as 0.01 ≤ xi ≤ 100. The problem is formulated using classical beam theory as follows: Minimize:

f ( X )  0.0624( x1  x2  x3  x4  x5 )

(6)

Subject to:

g( X )  Figure 2. Representation of Gaussian distribution in BBPSO

The deviation of gBest and pBest allows particles whose personal best position are far away from the global best position to make large step sizes towards the global best position. This may compose local best positions to move closer to the global best. While personal best position is close to the global best, step size is small to limit exploration in order to exploit area near global best. Equation (5) is based on the theoretical studies that each particle converges to a weighted average of its personal best and neighborhood best positions (van den Bergh and Engelbrecht 2006). Thus, Equation (1) and (2) are replaced by Equation (5):

 xi (t  1)  N (i ,  i2 )

(5)

The BBPSO is the simplest PSO, which has no parameters and require less computation cost. Moreover, it is easily to implement and combine with other algorithms (Feng et al. 2008).

Through five structural optimization problems, the ability of BBPSO was examined in this section. Optimization results were compared with data recently published in literature. For all cases, BBPSO used 30 particles and terminated after 200 iterations.

(7)

Figure 3. Cantilever beam

The best solutions obtained by BBPSO and various methods (Chickermane and Gea 1996; Gandomi et al. 2011) for solving this problem are listed in Table 1. As shown in the table, the solution found out by BBPSO is slightly better than other methods. Table 1. Best result of the cantilever beam design Methods 1

4 CASE STUDY

61 37 19 7 1     1 x13 x23 x33 x43 x53

CONLIN MMA1 GCA(I)1 GCA(II)1 CS2 BBPSO

x1

x2

x3

x4

x5

Fmin

6.0100 6.0100 6.0100 6.0100 6.0089 6.0261

5.3000 5.3000 5.3000 5.3000 5.3049 5.3034

4.4900 4.4900 4.4900 4.4900 4.5023 4.4881

3.4900 3.4900 3.4900 3.4900 3.5077 3.5074

2.1500 2.1500 2.1500 2.1500 2.1504 2.1487

NC 1.3400 1.3400 1.3400 1.33999 1.33996

1

Chickermane and Gea (1996), 2Gandomi et al. (2011)

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4.2 Tubular Column Design

A uniform column of tubular section should be designed, with hinge joints at both ends (see Figure 4) to carry a compressive load P=2500 kgf with minimum cost (Rao 2009). The column is made of a

y

material with a yield strength ( ) of 500 kgf/cm2, modulus of elasticity (E) of 0.85  106 kgf/cm2, and weight density (  ) of 0.0025 kgf/cm3. The length (L) of the column is 250 cm. The stress included in the column should be less than the buckling stress (constraint g1) and yield stress (constraint g2). The mean diameter of the column is restricted to lie between 2 and 14 cm, and columns with thicknesses outside the range 0.2 to 0.8 cm are not available in the market. The cost of the column includes material and construction costs and can be taken as 5W + 2d, where W is the weight in kilograms force and d is the mean diameter of the column in centimeters. It is taken as the objective function.

g 4  d  14  0

(12)

g 5  t  0.2  0

(13)

g 6  t  0.8  0

(14)

Table 2 compares the results obtained by BBPSO with previous methods reported in the literature (Gandomi et al. 2011; Hsu and Liu 2007; Rao 2009). It can be examined from table 2 that the best objective value achieved by other methods nevertheless it violate constraint ( g 2 ). Since BBPSO fulfills all of the constraint, thus, BBPSO offers the best result. Table 2. Best solution for the tubular column example

d t

g1 g2 g3 g4 g5 g6 Fmin a

Hsu and Liu (2007) 5.5407 0.292 499.9827

Rao (2009)

Gandomi et al. (2011) 5.44 5.45139 0.293 0.29196 499.2564 499.9879

PSO

BBPSO (Present study) 8.5756 5.4512 0.3365 0.292 275.7659 499.9369

0.0001a

0.0026a

-0.0001

-0.7768

-0.0001

-3.4507

-3.44

-3.45139 -6.5756

-3.4512

-8.5493

-8.56

-8.54861 -5.4244

-8.5488

-0.092

-0.093

-0.09196 -0.1365

-0.092

-0.508

-0.507

-0.50804 -0.4635

-0.508

26.4991

26.5323

26.53217 26.6012

26.4995

represents violated sets

4.3 Three-bar Truss Design

Figure 4. The tubular column

The optimization model of this problem is given as follows: Minimize:

f (d , t )  9.8dt  2d

This case considers 3-bar truss design problem (Ray and Saini 2001) as seen in Figure 5. The volume of a statistically loaded 3-bar truss is to be minimize subject to stress (  ) constraints. The objective is to evaluate the optimal cross sectional areas (A1, A2). The problem can be stated as follow: Minimize:

f ( A1 , A2 )  (2 2 A1  A2 )  l

(15)

(8) Subject to:

Subject to:

P g1  y dt

(9)

P  2 E (d 2  t 2 ) g2   0 dt 8L2

(10)

g 3  d  2  0

(11)

4

g1 

2 A1  A2 P  2 A12  2 A1 A2

(16)

g2 

A2 P  2 A  2 A1 A2

(17)

2 1

1st International Conference on Sustainable Civil Engineering Structures and Construction Materials

g3 

1 A1  2 A2

P 

(18)

where 0  A1 , A2  1; l = 100 cm, P = 2 KN/cm2, and  = 2 KN/cm2.

assumed to be simply supported with a span of 30 ft and subjected to a live load of 2.0 klbf and a dead load of 1.0 klbf, including the weight of the beam. The concrete compressive strength (  c ) is 5 ksi and the yield stress of the reinforcing steel (  y ) is 50 ksi. The cost of concrete is 0.02/in2/linear ft and the cost of steel is $1.0/in2/linear ft. The area of the reinforcement ( As ), the width of the beam ( b ), and the depth of the beam ( h ) have to be determined such that the total cost of structure is minimized. In this paper, the cross-sectional area of the reinforcing bar ( As ) is taken as a discrete type variable that must be chosen from the standard bar dimensions listed in ref. the width of the concrete beam ( b ) is assumed to be an integer variable. The depth of the beam is a continuous variable. The effective depth is assumed to be 0.8 b .

Figure 5. Three-bar truss

Table 3. Best solutions for the three-bar truss design

A1 A2 g1 g2 g3

Ray and Saini (2001) 0.795

Tsai (2005)

Gandomi PSO et al. (2011)

BBPSO (Present study)

0.788

0.78867

0.6237

0.7885

0.395

0.408

0.40902

0.1342

0.4088

1.9966 2.0016a 1.9994

1.7059

0.5934

0.5191 0.5364

0.5365

0.5772

0.5365

1.4775 1.4652

1.4629

1.1287

1.4635

Fmin 264.36 263.68 a

Figure 6. Illustration of reinforced concrete beam

The structure should be proportioned to have a required strength based upon the ACI building code 318-77 as follows: M u  0.9 As  y (0.8)(1.0  0.59

As  y 0.8bh c

263.9716 264.1858 263.8959

represents violated sets

This design is a nonlinear fractional programming problem. Table 3 represents the solutions obtained by several researchers (Gandomi et al. 2011; Ray and Saini 2001; Tsai 2005). The solutions obtained by BBPSO are ( A1 , A2 ) = (0.7885, 0.4088) with the objective value equal to 263.8959. Actually, the best objective value was reported by Tsai. However the result is not feasible because the first constraint is violated. Hence, it can be concluded that the results obtained by BBPSO are better than those of the previous studies. 4.4 Reinforced Concrete Beam Structure

A simplified optimization of the total cost of reinforced concrete beam, as shown in Figure 6, was presented by Amir and Hasegawa. The beam is

)  1.4M d  1.7 M l

(19) in which M u , M d , and M l , respectively, are the flexural strength, dead load, and live load moments of the beam. In this case, M d = 1350 in kip and M l = 2700 in kip. The depth ratio of the beam is restricted to be less than or equal to 4. The optimization problem can be stated as: Minimize:

f ( As , b, h)  29.4 As  0.6bh

(20)

Subject to:

g1 

b 4 h

As2 g 2  180  7.375  As b h

(21)

(22) 5

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Proceedings of the 1st SCESCM

Table 4. Result of the reinforced concrete beam example Reference

Method

Fmin As b h g1 g2

Amir and Hasegawa (1989) SD-RC 374.2

Shih and Yang (2002)

Yun (2005)

Gandomi et al. (2011)

Present study

GHN-ALM 362.2455

GHN-EP 362.0065

GA 366.1459

FLC-AHGA 364.8541

CS 359.2080

BBPSO 359.2080

7.8

6.6

6.32

7.2

6.16

6.32

6.32

31 7.79 -0.0205

33 8.4952 -0.1155

34 8.6372 -0.0635

32 8.0451 -0.0224

35 8.75 0

34 8.5 0

34 8.5 0

-4.2012

0.0159

-0.7745

-2.8779

-3.6173

-0.2241

-0.2241

The variables bound are As : {6.0, 6.16, 6.32, 6.6, 7.0,

g1  Ts  0.0193R  0

(24)

g 2  Th  0.0095 R  0

(25)

4 g 3  R 2 L  R 3  1296000  0 3

(26)

g4  L  240  0

(27)

2

7.11, 7.2, 7.8, 7.9, 8.0, 8.4} in , b : {28, 29, 30, 31, . . . . . , 38, 39, 40} in, and 5 ≤ h ≤ 10 in. The constrained functions of g1 ( x) and g 2 ( x) as derived by Liebman et al. (1981) then, was used in this paper also. (Liebman et al. 1981) Table 4 presents the optimum designs of this problem and the parameters used. Several comparisons with previous research SD-RC (Amir and Hasegawa 1989), GHN-ALM and GHN-EP (Shih and Yang 2002), GA and FLC-AHGA (Yun 2005), CS (Gandomi et al. 2011). In this case study, both CS (Gandomi et al. 2011) and present study obtain the same results.

where 1 0.0625  Ts , Th  99  0.0625 , 10  R  200, and 10  L  240. Unlike the usual limit of 200 in considered in literature, the upper bound of design variable L was increased to 240 in to expand the search space.

4.5 Pressure Vessel Design

A cylindrical pressure vessel capped at both ends by hemispherical heads as shown in Figure 7. This compressed air tank has a working pressure of 3000 psi and a minimum volume of 750 ft3, and is designed according to the ASME boiler and pressure vessel code. The total cost, including a combination of single 60o welding cost, material, and forming cost, is to be minimized. The involved variables are the thickness ( Ts ), thickness of the head ( Th ), the inner radius ( R ), and the length of the cylindrical section of the vessel ( L ). The thickness of the variables are discrete values which are interger multiplies of 0.0625 in. Then, the optimization problem can be expressed as follows: Minimize: f (Ts , Th , R, L)  0.6224Ts RL  1.7781Th R 2  3.1661Ts L  19.84Th L 2

(23)

Subject to:

6

2

Figure 7. The pressure vessel

Optimization results are presented in Table 5. BBPSO found the global optimum in 5418.205. Comparison with other methods such as PSO-GA (Dimopoulos 2007), HS (Mahdavi et al. 2007), SA-DS (Hedar and Fukushima 2006), and FA (Gandomi et al. 2011). It can be seen that designs optimized by PSO-GA and HS methods are not feasible because the third constraints ( g 3 ) is violated. SA-DS and FA satisfy all constraints but correspond to high cost overall. Therefore, BBPSO find the best result.

1st International Conference on Sustainable Civil Engineering Structures and Construction Materials Table 5 Pressure vessel problem, comparison BBPSO with literature Reference Method

Fmin Ts Th R L g1 g2 g3 g4 a

Dimopoulos (2007) PSO-GA 5850.383

Mahdavi et al. (2007) HS 5849.7617

Hedar and Fukushima (2006) SA-DS 5868.76484

Gandomi et al. (2011) FA 5850.38306

Present study

0.75

0.75

0.7683

0.75

1.25

0.375

0.375

0.3797

0.375

0.625

38.8601 221.36549 -0.0000

38.8601 221.36553 -0.0000

39.80962 207.22555 -0.0000

38.8601 221.36547 -0.0000

64.7651 11.9964 -0.00003

-0.0043

-0.0043

-0.0000

-0.0043

-0.0097

a

a

-10.7065

-0.0134

-2.1710

-32.7744

-18.6345

-228.0036

0.0446

-18.6345

0.2713

-18.6345

BBPSO 5418.205

represents violated sets

5 CONCLUSION

This paper introduces the application of BBPSO to solve structural optimization problem. The advantages of BBPSO are very simple to code and require no tuning parameters. In this study, the BBPSO was tested in five structural optimization problems taken from literatures including cantilever beam, tubular column design, three-bar truss design, reinforced concrete beam structure, and pressure vessel design. Moreover, the result obtained by BBPSO was compare with the previous studies. The results indicate that BBPSO produces a better result than the other metaheuristic algorithms such as GA, PSO, HS, CS, and FA in every case study. Although BBPSO is very effective to solve structural optimization, some improvement should be done to make BBPSO more powerful. Furthermore, BBPSO can be modified solve multi-objective optimization problem. Hybridization of BBPSO with other metaheuristic algorithms also may be an interesting direction for further research. Finally, more comprehensive studies should be carried out in order to identify the strength points and weakness of BBPSO. REFERENCES

Ali, N., Behdinan, K., and Fawaz, Z. (2003). "Applicability and viability of a GA based finite element analysis architecture for structural design optimization." Computers & Structures, 81(22– 23), 2259-2271. Amir, H., and Hasegawa, T. (1989). "Nonlinear Mixed‐Discrete Structural Optimization." Journal of Structural Engineering, 115(3), 626-646.

Chickermane, H., and Gea, H. C. (1996). "Structural Optimization Using A New Local Approximation Method." International Journal for Numerical Methods in Engineering, 39(5), 829-846. Dimopoulos, G. G. (2007). "Mixed-variable engineering optimization based on evolutionary and social metaphors." Computer Methods in Applied Mechanics and Engineering, 196(4–6), 803-817. Dorigo, M., Maniezzo, V., and Colorni, A. (1996). "Ant system: optimization by a colony of cooperating agents." Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on, 26(1), 29-41. Eberhart, R., and Kennedy, J. "A new optimizer using particle swarm theory." Proc., Micro Machine and Human Science, 1995. MHS '95., Proceedings of the Sixth International Symposium on, 39-43. Feng, P., Xiaohui, H., Eberhart, R., and Yaobin, C. "An analysis of Bare Bones Particle Swarm." Proc., Swarm Intelligence Symposium, 2008. SIS 2008. IEEE, 1-5. Gandomi, A., Yang, X.-S., and Alavi, A. (2011). "Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems." Engineering with Computers, 1-19. Gandomi, A. H., Yang, X.-S., and Alavi, A. H. (2011). "Mixed variable structural optimization using Firefly Algorithm." Computers & Structures, 89(23–24), 2325-2336. Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley Longman Publishing Co., Inc.

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Hassan, R., Cohanim, B., De Weck, O., and Venter, G. "A Comparison Of Particle Swarm Optimization And The Genetic Algorithm." Proc., 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 1-13. Hedar, A.-R., and Fukushima, M. (2006). "DerivativeFree Filter Simulated Annealing Method for Constrained Continuous Global Optimization." J. of Global Optimization, 35(4), 521-549. Hsu, Y.-L., and Liu, T.-C. (2007). "Developing a fuzzy proportional–derivative controller optimization engine for engineering design optimization problems." Engineering Optimization, 39(6), 679-700. Jansen, P. W., and Perez, R. E. (2011). "Constrained structural design optimization via a parallel augmented Lagrangian particle swarm optimization approach." Computers & Structures, 89(13–14), 1352-1366. Kahraman, M., and Erbatur, F. (2001). "- A GA Approach for Simultaneous Structural Optimization." Structural Engineering, Mechanics and Computation, A. Zingoni, ed., Elsevier Science, Oxford, 1147-1154. Kennedy, J. "Bare bones particle swarms." Proc., Swarm Intelligence Symposium, 2003. SIS '03. Proceedings of the 2003 IEEE, 80-87. Liebman, J. S., Chanaratna, V., and Khachaturian, N. (1981). "Discrete Structural Optimization." Journal of the Structural Division, 107( 11), 2177-2197. Mahdavi, M., Fesanghary, M., and Damangir, E. (2007). "An improved harmony search algorithm for solving optimization problems." Applied Mathematics and Computation, 188(2), 1567-1579. Mendes, R., Kennedy, J., and Neves, J. (2004). "The fully informed particle swarm: simpler, maybe better." Evolutionary Computation, IEEE Transactions on, 8(3), 204-210. Rao, S. S. (2009). "Introduction to Optimization." Engineering Optimization, John Wiley & Sons, Inc., 1-62. Ratnaweera, A., Halgamuge, S. K., and Watson, H. C. (2004). "Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients." Evolutionary Computation, IEEE Transactions on, 8(3), 240-255. Ray, T., and Saini, P. (2001). "Engineering Design Optimization Using A Swarm with An Intelligent

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