les-flow around a sphere-.pdf

7/29/2019 LES-flow around a sphere-.pdf

1/19

This article was downloaded by: [41.201.96.229]On: 03 August 2012, At: 10:21Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Engineering Opt imizat ionPublication detai ls, including instructions for authors and

subscr ipt ion inf ormat ion:h t t p : / / w w w. ta n dfo nl i ne . co m/ l oi / ge no 20

An implementat ion of new selectionst rategies in a genetic algorithm populat ion recombinat ion and elit istrefinementNoh-Sung Kwak a & Jongsoo Lee a

aSchool of Mechanical Engineer ing, Yonsei Universit y, Seoul ,

120-749, Korea

Version of record f i rst publ i shed: 07 Jul 2011

To cite this article: Noh-Sung Kwak & Jongsoo Lee (2011): An im plem ent at ion of new select ion

st rat egies in a genet ic a lgor i t hm populat ion r ecombinat ion and el i t ist ref i nement , Engineer ing

Optimization, 43:12, 1367-1384

To link to this art icle: ht t p : / / dx .do i .o rg/ 10.1080/ 0305215X.2011.558577

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.
http://www.tandfonline.com/loi/geno20http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditionshttp://dx.doi.org/10.1080/0305215X.2011.558577http://www.tandfonline.com/loi/geno20


2/19

Engineering Optimization

Vol. 43, No. 12, December 2011, 13671384

An implementation of new selection strategies in a genetic

algorithm population recombination and elitist refinement

Noh-Sung Kwak and Jongsoo Lee*

School of Mechanical Engineering, Yonsei University, Seoul 120-749 Korea

(Received 8 August 2010; final version received 17 January 2011)

The present study aims to develop a new genetic algorithm utilizing population recombination and elitistrefinement. Population recombination determines how a population and its elitist sub-population evolve.A whole population consists of three major sub-populations: the first is the current generations elitist sub-population, the second is obtainedfrom the purecrossover of the elitist sub-population with another existingsub-population, and the third is the result of a crossover between the elitist and random sub-populations.Genetic operations such as reproduction and crossover are applied among sub-populations during theprocess of population recombination. The refinement of the elitist sub-population is then implementedin order to improve the converged solution that was obtained from the recombination. The refinement ofelitist sub-populations facilitates the locations of a more enhanced design by altering the binary valuesin chromosomes. The proposed method is verified through a number of nonlinear and/or multi-modalfunctions and constrained optimization problems.

Keywords: distributed genetic algorithm; population recombination; elitist sub-population; elitist refine-ment; tail evolution and fitting

1. Introduction

Genetic algorithms (GAs) and their advanced versions have been recognized as one of the most

efficient optimization tools and have been shown to be effective when the analysis model is

inherently nonlinear or when the design problem is represented by a mixture of continuous,

integer and/or discrete design variables (Hajela and Lee 1995, Lee et al. 2002). GA has the

promising capability of locating a global optimum without the need for derivative-basedsensitivity

information. GA operates through the evolution of multiple candidate solutions to design under

implicit parallelism. Biologically inspired operations of crossover and mutation facilitate the

production of competitive genes during genetic evolution. Such mechanisms in GAs improve

the current level of system performance and eventually enable a near-global optimum state for

the system under the given parameter environment (Deb 2003). The distinct features of GAs draw

upon diversity, discovery and adaptation. In GA, the system adapts for maximum performance

by discovering new competitive genes among the diverse individuals.

In general, GA develops their design candidate solutions based on random exploration. Such

randomness enables the accommodation of diversity and non-convexity in design candidate

*Corresponding author. Email: [email protected]

ISSN 0305-215X print/ISSN 1029-0273 online 2011 Taylor & Francishttp://dx.doi.org/10.1080/0305215X.2011.558577http://www.tandfonline.com


3/19

1368 N.-S. Kwak and J. Lee

solutions, whereas it works as disruptiveness in schema convergence when, for example, two

mating pairs for crossover are randomly selected. Such a phenomenon illustrates one of the main

reasons why GA is computationally expensive for use in locating the expected level of opti-

mized design solutions. There have been a number of developments in advanced GA focusing

on convergence speed increase through a new implementation of selection (Huber and Schell

2002, Dukkipati and Murty 2002, Chang et al. 2003, Yoshiaki et al. 2006, Li et al. 2006, Graff

et al. 2007) and crossover (Chakraborty and Chakraborty 2005, Liua and Chena 2009). Many

GA researchers have been interested in methods for recombining the population in which chro-

mosomes are included or excluded in the population (Thierens and Goldberg 1994, Angelov

and Wright 2000, Daoxiong and Xiaogang 2004, Christopher and Ian 2004, Mak et al. 2005,

Li et al. 2007). Some researchers are interested in new encoding methods or hybrid methods

combining GA with other algorithms or techniques (Wang et al. 2006, Hwang and He 2006).

In the present study, a new process of population-handling is suggested such that a whole

population is decomposed into a number of sub-populations, andthe genetic evolution is conducted

between sub-populations. That is, the unique features in the proposed process is the selection (orreproduction) of the fittest sub-population out of several sub-populations, and the subsequent

crossover due to the mating of a chromosome in one sub-population with a chromosome in

another sub-population. Conventional GA (i.e. simple GA, SGA), on the other hand, works with

individual chromosomes within a single population. This proposed process is referred to as the

recombination of the population. Recombination aims to identify the superior sub-population,

even though one of the chromosomes in that sub-population might at times be disruptive. The

sub-population is a similar idea used in the distributed genetic algorithms, but, the evolutionary

processes are different from the existing distributed algorithms (Michalewicz 1995, Erick 2001).

The sub-populations in the distributed algorithms are used as isolated groups with respective

evolutions for the parallel searching strategies, and only a few individuals are migrated to eachother groups. But, the sub-populations collaborate on a single evolution in the proposed algorithm.

The detailed methods are described in the following chapter.

One of the significant features in GA is the ability to rapidly locate the near-optimal design

solutions, since GA analyses multiple individuals over the entire design space. However, GA

sometimes terminates due to a premature convergence, showing that the randomness in GA works

both positively and negatively. Hybrid GA methods combined with derivative-based methods have

been widely developed in order to enhance the near-optimal design solutions (Oh etal. 2004,Vieira

et al. 2005, Rao and Xiong 2005, Fisz et al. 2005, Yuan et al. 2008); however, such methods have

limitations due to the fact that they require the formulation of sensitivity information. With regard

to premature convergence, the present study suggests another search process such that once the

converged solution is obtained from the aforementioned recombination process, binary values in

some portions of the design candidate solutionssub-chromosomes are altered as either 0 to 1 or 1

to 0 in order to refine the design candidate solutions. This process is referred to as the refinement

of the elitist sub-population. Since this is a type of mutation-like precision search, an efficient

handling of the elitist refinement is described later in greater detail.

The proposed method was graphically examined via an unconstrained two-variable function

problem and was also verified through a number of nonlinear or multi-modal function problems

and constrained optimization problems.

2. Proposed method

The proposed method consists of two processes: one is population recombination and the other

is elitist refinement. A detailed description of the proposed method is given as follows:


4/19

Engineering Optimization 1369

2.1. Recombination of the population (Rp)

Step 1: Generate a randomly distributed initial population, P0, in which there are a total of 20 Nsub design candidate individuals (i.e. coded chromosomes), where Nsub is an integer

number (i.e. 5, 10 or 20) indicating the basic sub-population size. (20 Nsub may be anynumber just greater than 2 Nsub . The researchers recommend the number between 10and 20 for efficiency.)

Step 2: Evaluate and rank all of the design candidate individuals in terms of their unconstrained

and/or constrained objective function value.

Step 3: Choose a total of 2 Nsub top-ranked design candidate individuals from the population,and denote this population as P1.

Step 4: Divide P1 into the upper 50% and lower 50% design candidate individuals, and denote

them as sub-populations P1-Upper and P1-Lower, respectively. Subsequently, conduct

a full (100%) crossover operation by mating a chromosome in one sub-population with

a chromosome in the other sub-population; that is, an individual in P1-Upper should becrossed-over with one in P1-Lower. The present study uses a two-point crossover. The

resulting population from the crossover between P1-Upper and P1-Lower is referred to

as P2.

Step 5: Generate a random sub-population whose size is Nsub , and denote it as P1-Random.

Step 6: Conduct a full crossover operation between P1-Upper and P1-Random using the same

method described in Step 4. The resulted population from P1-Upper and P1-Random is

called P3.

Step 7: Construct a new population in which P-New = P1 + P2 + P3. The size of P-New is6 Nsub since P1, P2 and P3 commonly have a total of 2 Nsub individuals.

Step 8: Using P-New, repeat these steps beginning with Step 3, until the convergence condition

is met such that the elitist design candidate individual is unchanged over ten consecu-

tive generations. The final elitist sub-population is conducted using such elitist design

candidate individual. When the final elitist sub-population is converged, proceed to the

next step.

Step 9: Elitist refinement.

Step 9.1: Tail evolution.

Step 9.2: Tail fitting.

Step 10: Terminate the algorithm.

The procedure for the recombination process is shown in Figure 1. It is worthy of note

that the new population size at Step 7 turned out to be 6 Nsub , while the initial populationsize was 20 Nsub . A number of diminished sub-populations are indicated by dotted lines inFigure 1. The repeated processes between Step 3 and Step 8 thus worked with a population size

of 6 Nsub . During the aforementioned processes, Step 4 and Step 6 adopted full crossovers;all of the individuals participated in the crossover operation with a crossover probability of

1.0 (100%). A randomly generated sub-population was used at Step-5, instead of using muta-

tion. The new population in Step 7 consisted of P1, P2 and P3, products of genetic operations,

such as selection/reproduction, crossover and mutation. P1 was the result of reproduction, a

combination of the current generations elitist sub-population. P2 was obtained from a pure

crossover between P1-Upper and the other existing sub-population, and P3 was the result of

a crossover of P1-Upper with a mutated (i.e. randomly generated) sub-population. We mustemphasize that a new population, P-New, was introduced based on the current generations elitist

sub-population, P1-Upper. This extensive mutation-like operation was employed in Step 9, where

the final elitist sub-population was further enhanced through an altering of the binary values of its

sub-chromosomes.


5/19


Figure 1. Procedure for recombination process (GARp).

2.2. Refinement of the elitist sub-population (Re)

The motivation behind this elitist refinement is the enablement of the final elitist sub-population

to move much closer to the global optimum. Moreover, this refinement process was created to

overcome the issues resulting from premature convergence (Hamzacebi 2008). Such a process is

performed by altering the binary values of the final elitist sub-population until no further improved

design candidate solution is detected. The refinement process facilitates the identification of

the improved design candidate solution in cases where the final elitist sub-population from the

recombination process turns out to be prematurely converged. Two schemes such as tail evolution

and tail fitting were also examined in following techniques.

Step 9.1: Tail evolution

For a large dimensionality design problem, it is computationally expensive to individually alter the

binary values of the final elitist sub-population. Such binary alteration cannot capture multimodal

behaviours toward the global optimum. Therefore, a group of elitist-variants was introduced so


6/19


as to efficiently search for a more improved optimized design solution. An individual consists

of several sub-chromosomes which is same to the number of design variables. The head part of

the sub-chromosomes takes most value of the design variable value, and the other part, the tail

part of them takes little of the design variable value. Since the final design candidate solution is

considered close to the optimal solution, controlling the head part of each sub-chromosome is

wasting time. The detailed sub-steps were as follows:

9.1.1: Construct a total of 6 Nsub elitist-variants by randomly altering the binary values located in the tail part ofeach sub-chromosome, as shown in Figure 2.

9.1.2: Conduct a single generation of conventional genetic evolution with the elitist-variants. As in genetic operations,an elitist selection and a full (100%) two-point crossover are used without mutation.

The resulting fittest chromosome is referred to as the fittest variant. The fittest-variant should

be as well as or better adapted than the final elitist.

Step 9.2: Tail fitting

The successive steps of tail fitting, a mutation-like precision search, are as follows:

9.2.1: Consider the first design variable value of the fittest-variant.

9.2.2: Both add and subtract the binary value of 1 to the above value.

9.2.3:If themodifieddesign candidatesolutionis betterthan the subtracted one in terms of a feasible-usable objectivefunction value, thencontinue to add the binary value until an improved design candidate solutionis no longer obtained.But if the subtracted design candidate solution is better, conduct the same process with subtraction of the binaryvalue.

Figure 2. Elitist-variants in tail evolution.


7/19


Figure 3. Overall procedure for GARp + TE + TF.

9.2.4: Perform Step 2 and Step 3 for the remaining design variables.

Tail fitting implicitly mimics a forced mutation or mutation-like operation. There are two

schemes, tail evolution

+tail fitting and tail fitting, in the refinement process. The first case,

the tail evolution + tail fitting scheme can be conducted with the sequential use of Steps 9.1 and9.2. In the second case, in which only tail fitting is considered, the tail fitting should begin with

the final elitist instead of the fittest-variant. The overall procedure for the proposed algorithm

is shown in Figure 3.

3. Graphical representation

The proposed method was graphically examined using Goldstein-Prices function with two design

variables (Pohlheim 2005) expressed as follows:

fGP(x) = (1 + (x1 + x2 + 1)2 (19 14x1 + 3x21 14x2 + 6x1x2 + 3x22 )) (30 + (2x1 3x2)2 (18 32x1 + 12x1 + 48x2 36x1x2 + 27x22 )) 4.095 xi 4.096, xi 0.001, i = 1, 2, (1)


8/19

Table 1. Condition parameters used in numerical examples.

Common setting GARp

Number of bits for

Number of

design total binary each binary

Results variables variables variable Nsub P-new size

Goldstein-Price Figures 4, 5 2 26 13 15 90 Rastrigin Figure 6 10 100 10 40 240

Table 2Easom Table 4 2 30 15 100 600

Rastrigin Table 5 10 100 10 100 600

Schwefel Table 6 10 170 17 30 180

Griewangk Table 7 10 170 17 70 420 Sum of different power Table 8 100 800 8 5 30

Ackleys path Table 9 10 130 13 70 420

Himmelblau Tables 10, 11 5 56 12 or 11 50 300

Ten-bar truss Tables 12, 13 10 100 10 70 420


9/19


where the global optimum is fGP = 3.0 at (x1 , x2 ) = (0.0,1.0). The condition parametersused in the numerical examples are shown in Table 1. As described in Step 1, an initial population

was randomly distributed (Figure 4(a)), wherein the exact solution of the global optimum is

indicated by an arrow. Next, three plots demonstrate the evolution of the design solutions P1,

P2 and P3 in P-New toward the final elitist sub-population. The first population of P-New is

shown in Figure 4(b), as are P1, P2 and P3, as well as the elitist sup-population. Design candidate

solutions for P1 at the third andeighth (final) generations converged more toward each generations

elitist sub-population, as shown in Figures 4(c) and Figures 4(d), respectively. Most of the design

candidate solutions for the eighth generation were as close as the final elitist sub-population

(Figures 4(d)). Moreover, the dotted region of P1, in which the elitist sub-population (i.e. P1-

Upper) and the elitist chromosome were located, decreased toward the optimum solution as the

generations increased. Successively, two schemes in the refinement process were performed,

starting with the final elitist as an initial chromosome. Their solution behaviours are presented in

Figure 5, wherein final elitist (eighth elitist) represents the final elitist from the recombination

process, and fittest variant is the result from Step 9.1.3. The solution in the tail fitting scheme

Figure 4. The change of the population in the generation progress.


10/19


Figure 5. Two schemes in refinement process.

was modified by individually altering the final elitists binary value, while the tail evolution

+ tail fitting technique identified the fittest elitist beginning with the final elitist. A number offunction evaluations were required for tail evolution, since it works with a population size of

6 Nsub . In addition, tail evolution was able to accommodate the multimodal behaviour of theglobal optimization function.

The present study shows the advantages of the recombination process in terms of convergence

patterns. Convergence histories for a simple GA (SGA) and a GA with a recombination process

(GARp) were examined as well. In a simple GA, elitist and tournament selection strategies

are used, and the probabilities of crossover and mutation are 0.8 and 0.05, respectively. In this

numerical experiment, Rastrigins function with a total of ten design variables was considered as

follows:

fR (x) = 10 n +10

i=1(x2i 10 cos(2xi ))

5.11 xi 5.12, xi 0.01, i = 1, . . . , 10 (2)

The global optimum of this function is fR=

0.0 at (xi )

=(0.0). GARp and SGA were run

1000 times starting with different initial populations, with three randomly selected results from the1000 runs presented in Figure 6. In this figure, it is evident that GARp produces more consistently

reduced objective function values than does SGA. The optimization performances were further

compared and are summarized in Table 2, where + is the standard deviation. The randomness inSGA resulted from the initial population, and GARp randomly generated P1-Random, as well as


11/19


Figure 6. Convergence histories of SGA and GARp for Rastrigins function.

Table 2. Optimization of SGA and GARp for Rastrigins function.

Objective function Number of function evaluations

Mean Best value Mean

SGA 8.36 3.43 1.119 10413.52

GARp 2.35 2.66 0.059 10353.28

an initial population. As seen from the results in Table 2, the recombination process (GARp) was

superior to SGA in terms of objective function value and the number of function evaluations.

4. Numerical examples

The condition parameters used in the numerical examples are shown in Table 1. Most of the

parameters are same in all the algorithms. But, some parameters such as crossover rate and

population size cannot be equal since they use different types of populations and recombination

methods. Mutation rate is 0.05 in all algorithms. Crossover rate of all the examples is 0.8 in asimple GA (SGA), whereas it is 1.0 in the proposed methods as described in the section Proposed

method. The population size of each example in SGA is determined by considering the function

complexity or nonlinearity. The population size used in the proposed method is almost the same as

that used in SGA. That is, (P2 + P3) in GARp and Population size in SGA look alike as shown


12/19


in Table 1. Instead of using P-New = P1 + P2 + P3, only P2 + P3 participate into generatingnew function evaluations. P-New succeeds to P1 from the previous generation without function

evaluation.

4.1. Unconstrained function problems

A total of six nonlinear and/or multimodal unconstrained functions (Pohlheim 2005) were tested

in order to examine the optimization performance of the proposed method. Test functions such

as Easoms, Rastringins, Schwefels, Griewangks, sum of different powers and Ackleys path

functions are outlined in Table 3, wherein their side constraints and global optimum solutions are

presented as well.

4.2. Himmelblaus problem

The nonlinear constrained optimization problem described with five design variables is stated as

follows (Bhuvaneshwaran et al. 2006):

Minimize

fH(x) = 5.3578547x23 + 0.8356891x1x5 + 37.293239x1 40792.141 (3)subject to

0 1(x) 92

Table 3. Unconstrained functions.

Easoms function

fE (x) = cos(x1) cos(x2) e((x1)2+(x2 )2)100.00 xi 100.00, xi 0.01, i = 1, 2 fE = 1 at (x1 , x2 ) = (,)

Rastrigins function

fR (x) = 10 n +10

i=1(x2i 10 cos(2 xi ))

5.11 xi 5.12, xi 0.01, i = 1, . . . , 10 fR = 0.0 at (x1 , x2 ) = (0.0, 0.0)Schwefels function

fS(x) =10

i=1

xi sin(|xi |)500.0 xi 500.0, xi 0.01, i = 1, . . . , 10 fS = 10 418.9829 at (xi ) = (420.9687)

Griewangks function

fG(x) =10

i=1

x2i4000

10

i=1cos

xi

i

+ 1

600 xi 600, xi 0.01, i = 1, . . . , 10 fG = 0.0 at (xi ) = (0.0)Sum of different power function

fSD P(x) =100i=1

|xi |(i+1)

1.27 xi 1.28, xi 0.01, i = 1, . . . , 100 fSDP = 0.0 at (xi ) = (0.0)Ackleys path function

fAP(x) = 20 exp0.2

10i=1 x

2i

10

exp

10i=1 cos(2 xi )

10

+20.0 + exp(1.0) 32.768 xi 32.768, xi 0.01,i = 1, . . . , 10 fAP = 0.0 at (xi ) = (0.0)


13/19


90 2(x) 11020 3(x) 25

where

1(x) = 85.334407 + 0.0056858x2x5 + 0.00026x1x4 0.0022053x3x52(x) = 80.51249 + 0.0071317x2x5 + 0.0029955x1x2 + 0.0021813x233(x) = 9.300961 + 0.0047026x3x5 + 0.0012547x1x3 + 0.0019085x3x4

78 x1 102, 33 x2 4527 x3 45, 27 x4 45, 27 x5 45 xi 0.01.

Ten-bar planar truss design

As an example of an engineering design problem, the ten-bar truss optimization shown in Figure 7

was explored. The design objective was to find cross sectional areas of truss members, Xi by

minimizing the total weight of a structure W (Xi ) subjected to stress constraints (Haftka and

Grdal 1993). The constraint is imposed such that each members tensile or compressive stress

should not exceed 25 kips under two static loadings of 1000 lb. For the 9th member, the stress

allowable is 75 ksi. As problem parameters, Youngs modulus of the truss is 1.0 104 psi, andthe material density is 0.1 lb/in3.

The optimization statement is written as follows:

Minimize

W (Xi ) (4)

Figure 7. Ten-bar planar truss.


14/19


subject to

Lowerj j Upperj , j= 1, . . . , 10

Xlower

i = 0.1 Xi Xupper

i = 10.0, i = 1, . . . , 10, (unit = in2

),

where j is a stress acting on the jth truss member, and Lower and Upper denote the lower and

upper bounds on the design variables, respectively.

5. Results and discussion

Easoms function problem with two design variables was first tested, and its optimization results

are shown in Table 4. Numerical experiments for all of the test functions were run 10,000 times,

using different initial populations. In terms of objective function value, the proposed methodGARp was better than the simple GA (SGA). When the tail evolution and tail fitting schemes were

employed to SGA, GARp+ TE + TF still resulted in improved design solutions, as expected (i.e.comparison between SGA + TE + TF and GARp + TE + TF). GARp and GARp + TE + TFalso provided more reliable solutions in terms of standard deviation. For the number of function

evaluations, GARp proved to be more efficientthan SGA. In Table 4, meanTFdenotes the average

number of function evaluations that performed the tail fitting scheme. That is, TF required a total of

12.8 function evaluations, while SGA+ TE + TF identified the solutions after 19395.02 functionevaluations. The ratio of TF to SGA+ TE + TF, referred to as the mean TF ratio, was 0.12% inthe Easoms function problem. This indicates that the precision search of TF occupies only 0.12%

of the total function evaluations. When such a precision search is used in the GARp

+TE

+TF

process, its average number of function evaluations (mean TF) and corresponding mean TF ratio

were more reduced compared to that of SGA + TE + TF, as was expected. The reason for thisconsiderable difference in the number of function evaluations between GA and GA + TE + TFis due the fact that a single genetic evolution of TE was conducted using a population size of

6 Nsub . For an analytical test function problem whose exact global optimum is already known,such a population size would be too large, and one can simply reduce the value of Nsub during the

tail evolution scheme. More generally, when unknown functions are to be explored, the adaptive

selection of Nsub according to the final elitist could be one of the most significant issues in the

context of computational cost savings.

The results of Rasrigins function with 10 design variables are shown in Table 5. The proposed

methods of GARp and GARpRe were superior to those of SGA and SGA + TE + TF in termsof both the mean and standard deviation in the objective function value. The proposed methods

proved advantageous in terms of the number of function evaluations as well. We detected that the

mean TF in SGA was better than that in GARp + TE + TF; however, their average differencewas less than 0.5, almost the same, since the order of function evaluations was O(104).

Table 4. Optimization performance for Easoms function.


Mean Best value Mean Mean TF Mean TF ratio

SGA 0.70 0.42 1 18494.28 SGA + TE + TF 0.80 0.40 1 19395.02 12.8 0.12%GARp 0.97 0.17 1 17803.28 GARp + TF 0.97 0.16 1 17874.22 1.22 0.01%GARp + TE + TF 0.97 0.17 1 18981.25 1.21 0.01%


15/19


Table 5. Optimization performance for Rastrigins function.



SGA 2.06 1.32 0 33089.92

SGA + TE + TF 1.92 1.27 0 33976.21 4.81 0.01%GARp 0.27 0.39 0 31895.12

GARp + TF 0.14 0.37 0 31869.07 5.07 0.02%GARp + TE + TF 0.13 0.36 0 33091.48 5.08 0.02%

Next, Schwefels function problem was tested and its optimization results are summarized in

Table 6. This problem also demonstrated the advantage of the proposed methods in terms of both

objective function value and the number of function evaluations. The results showed that GARp

and GARp

+TE

+TF generated design solutions with dramatically reduced standard deviations

compared to those of SGA and SGA + TE + TF. Accordingly, the dependence on the use of TFis attenuated in GARp + TF and GARp + TE + TF.

Similarly, the benefits of the proposed methods could be found in cases using Griewangks

function, the sum of different powers function and Ackleys path function, whose optimization

results are summarized in Tables 7, 8, and 9, respectively.

Table 6. Optimization performance for Schwefels function.



SGA 3997.62 122.64 4189.72119 22688.78 SGA + TE + TF 4057.74 117.26 4189.82887 27473.95 4586.96 16.38%GARp 4188.37 5.01 4189.82781 21581.51 GARp + TF 4189.79 2.05 4189.82887 22298.35 685.67 3.41%GARp + TE + TF 4189.78 2.37 4189.82887 22754.61 687.88 3.36%

Table 7. Optimization performance for Griewangks function.



SGA 0.26 0.21 0.00002 47611.55

SGA + TE + TF 0.13 0.14 0 48266.45 135.10 0.36%GARp 0.10 0.10 0.00006 49065.44

GARp + TF 0.08 0.06 0 48983.78 55.37 0.15%GARp + TE + TF 0.08 0.06 0 49772.12 53.89 0.14%

Table 8. Optimization performance for sum of different power function.



SGA 25025.10 139527.68 0.04659 4353.32

SGA + TE + TF 0 0 0 5814.73 1374.45 28.28%GARp 132.39 939.39 0.02003 1632.00 GARp + TF 0 0 0 2942.76 1305.02 45.72%GARp + TE + TF 0 0 0 2991.95 1309.35 45.05%


16/19


Table 9. Optimization performance for Ackleys path function.



SGA 0.63 1.02 0 36213.63

SGA + TE + TF 0.54 0.98 0 36696.63 11.61 0.03%GARp 0.17 0.26 0.01318 34684.92

GARp + TF 0.04 0.24 0 34737.41 13.88 0.04%GARp + TE + TF 0.04 0.23 0 35510.85 13.65 0.04%

As a constrained minimization problem, Himmelblaus function was tested in the present study,

and its optimization results are shown in Tables 10 and 11. A constraint handling was formulated

via an exterior penalty method (Haftka and Grdal 1993). The proposed methods of GARp

and GARp

+TE

+TF were superior to those of the SGA-based methods in terms of the mean

and standard deviation of the objective function value and the number of function evaluations.Moreover, the TF precision search requires at most 10 function evaluations; the value of ten is

extremely small compared to the total number of function evaluations.

A 10-bar planar truss design problem working with 10 design variables and stress constraint

was also explored, and its solution results are summarized in Tables 12 and 13. Table 12 includes

five kinds of combinations using SGA, GARp, TE and TF to show the effects of each approach

in a 10-bar truss problem. As shown in Table 13, the best results of five kinds of combinations

are compared with an exact solution (Haftka and Grdal 1993) and results cited from references

(Andrew 2004, Rodrigo and Jun 2008). It is noted that solutions in a reference (Rodrigo and

Jun 2008) are obtained using the combination of sequential linear programming and GA and

a program, GA-Truss rebirth is used in a reference (Andrew 2004). Some of the proposedalgorithms outrun the existing exact solution, whereas the truss-specialized algorithms do not

reach the exact solutions.

The role of tail evolution (TE) was examined by comparing GARp + TF and GARp+ TE + TFfor each of the unconstrained and constrained minimization problems, as shown in Tables 4 to 11.

Tail evolution is derived from the concept of a traditional genetic evolution and can subsequently

provide solution refinement to the final elitist sub-population. But, the effect of the operation is

Table 10. Optimization performance for Himmelblaus function.



SGA 30693.6 144.10 31019.36055 12104.14 SGA + TE + TF 30714.53 143.43 31019.31501 12609.57 11.65 0.17%GARp 30879.66 74.55 31024.05873 11189.78 GARp + TF 30880.35 74.78 31022.13208 11229.52 1.34 0.01%GARp + TE + TF 30886.19 73.29 31022.96619 11822.18 4.40 0.04%

Table 11. Best optimized solutions for Himmelblaus function.

x1 x2 x3 x4 x5

SGA 78.0 33.041036 27.140694 45.00000 44.753786

SGA + TE + TF 78.0 33.076209 27.123107 44.94724 44.832926GARp 78.0 33.005862 27.079140 44.97362 44.956033GARp + TF 78.0 33.05276 27.105520 45.00000 44.868100GARp + TE + TF 78.0 33.00000 27.096727 45.00000 44.894480


17/19


Table 12. Optimization performance for ten-bar planar truss.


Mean Best value Worst value Mean Mean TF Mean TF ratio

SGA 1599.77 45.74 1520.30142 2109.46595 47381.31

SGA + TE + TF 1577.58 39.98 1506.20334 1953.48703 48493.75 67.94 0.17%GARp 1552.23 21.22 1499.83687 1736.02376 48694.16

GARp + TF 1548.62 20.58 1498.22222 1691.06556 48833.25 12.21 0.03%GARp + TE + TF 1548.58 20.89 1498.47368 1695.37798 49937.37 12.21 0.03%

Table 13. Best optimized solutions for 10-bar planar truss.

Best design variables

Objective X1 X2 X3 X4 X5 X6 X7 X8 X9 X10

(Haftka and Grdal 1993) 1499.0 7.900 0.100 8.100 3.900 0.100 0.100 5.800 5.510 3.68 0.140

SGA 1520.3 7.116 0.874 8.839 3.197 0.100 0.874 6.913 4.445 2.916 1.232

SGA + TE + TF 1506.2 7.765 0.235 8.2 3.758 0.100 0.235 5.955 5.374 3.506 0.332GARp 1499.8 7.842 0.139 8.123 3.855 0.100 0.139 5.829 5.452 3.661 0.197

GARp + TF 1498.2 7.871 0.119 8.094 3.884 0.100 0.119 5.800 5.490 3.642 0.168GARp + TE + TF 1498.5 7.890 0.100 8.074 3.903 0.100 0.100 5.771 5.519 3.700 0.139SLP + GA (Rodrigo and Jun

2008)1509.0 7.900 0.100 8.090 3.900 0.100 0.100 5.780 5.520 3.860 0.130

GA rebirth (Andrew 2004) 1527.0 7.437 0.574 3.437 8.576 0.101 0.576 6.469 4.853 3.230 0.813

slight, since it is imposed to the tails of design variables. Therefore, a designer can get the benefits

of TE slightly in terms of the mean and standard deviation of the objective function values. But,the number of function evaluations grows up for TE. According to the design problem, a designer

can determine whether TE should be included during the refinement process. If the precision of

the result is more important than the expense of the function evaluation, TE helps get more precise

designs.

6. Conclusion

One of the significant features in GA is the ability to rapidly locate the near-optimal design

solutions, since GA analyzes multiple individuals over the entire design space. However, GA

sometimes terminates due to a premature convergence, showing that the randomness in GA

works both positively and negatively. With regard to premature convergence, the present study

suggests a new search process of GARp + TE + TF that efficiently recombines individuals inthe population and then refines the final elitist sub-population based on a forced mutation-like

precision search. Population recombination focuses on how the population and its elitist sub-

population are constructed during evolution. In this process, a whole population consists of three

major sub-populations: the first is obtained via the simple reproduction of the current generations

elitist sub-population, the second comes from the pure crossover of an elitist sub-population with

another existing sub-population, and third sub-population is a result of a crossover between the

elitist and random sub-populations. The recombination process results from genetic operationsamong sub-populations. The objective of elitist refinement is to successively improve the final

elitist sub-population obtained through the recombination process. The refinement process has

two schemes: tail evolution and tail fittings. Through a number of unconstrained and constrained

test-bed function problems, the proposed methods of recombination and refinement proved that


18/19


their optimization performances are efficient in terms of both the mean and standard deviation of

the objective function value and the number of function evaluations compared to those obtained

from existing GA-based methods. The tail fitting scheme is quite exhaustive due to the individual

alteration of binary values, but its corresponding number of function evaluations is very small

compared to the total number of function evaluations.

References

Andrew, B., 2004. Truss optimization using genetic algorithms. Proceedings of genetic and evolutionary computationconference (GECCO 2004), 2630 June 2004, Seattle, Washington, USA. Berlin: Springer.

Angelov, P.P. and Wright, J.A., 2000. A center-of-gravity-based recombination operator for genetic algorithms. Proceed-ings of 26th IEEE annual conference of the IEEE, Industrial Electronics Society (IECON2000), 2228 October,Nagoya, Japan. Piscataway, NJ: IEEE, 259264.

Bhuvaneshwaran,V., Langari, R. and Temponi, C., 2006. A genetic algorithm and fuzzy approach (GAFA) for constrained

nonlinear optimization in design. International Journal for Computational Methods in Engineering Science andMechanics, 7 (3), 173189.Chakraborty, G. and Chakraborty, B., 2005. Rank and proximity based crossover (RPC) to improve convergence in genetic

search. Proceedings of IEEEcongress on evolutionary computation (CEC2005), 1922 July, Edinburgh, UK.Munich,Germany: IEEE, 13111316.

Chang, M. et al., 2003. Group selection and its application to constrained evolutionary optimization. Proceedings ofIEEE congress on evolutionary computation (CEC2003), 2427 June, Newport Beach, CA. Piscataway, NJ: IEEE,684691.

Christopher, B. and Ian, C.P., 2004. Developments of the cluster oriented genetic algorithm (COGA). EngineeringOptimization, 36 (2), 249279.

Daoxiong, G. and Xiaogang, R., 2004. A new multi-parent recombination genetic algorithm. Proceedings of the 6th worldcongress on intelligent control and automation, 2123 June, Dalian, China. Piscataway, NJ: IEEE, 20992103.

Deb, K., 2003. Multi-objective optimization using evolutionary algorithm. Chichester: John Wiley & Sons, Ltd.Dukkipati, A. and Murty, M.N., 2002. Selection by parts: Selection in two episodes in evolutionary algorithms. Pro-

ceedings of IEEE congress on evolutionary computation (CEC2002), 1217 May, Honolulu, Hawali. Piscataway,NJ: IEEE, 657662.

Erick, C., 2001. Migration policies, selection pressure, and parallel evolutionary algorithms. Journal of Heuristics, 7 (4),311334.

Fisz, J.J., et al., 2005. Genetic algorithms optimization approach supported by the first-order derivative and Newton-Raphson methods: Application to fluorescence spectroscopy. Chemical Physics Letters, 407 (13), 812.

Graff, M., Poli, R. and Moraglio, A., 2007. Linear selection. Proceedings of IEEE congress on evolutionary computation(CEC2007), 2528 September, Singapore, 25982605.

Haftka, R.T. and Grdal, Z., 1993. Elements of structural optimization. The Netherlands: Kluwer Academic Publishers.Hajela, P. and Lee, E., 1995. Genetic algorithms in truss topological optimization. International Journal of Solids and

Structures, 32 (22), 33413357.Hamzacebi, C., 2008. Improving genetic algorithms performance by local search for continuous function optimization.

Applied Mathematics and Computation, 196 (1), 309317.Huber, R. and Schell, T., 2002. Mixed size tournament selection. Soft Computing, 6, 449455.

Hwang, S.F. and He, R.S., 2006. Engineering optimization using a real-parameter genetic-algorithm-based hybrid method.Engineering Optimization, 38 (7), 833852.

Lee, J., Kim, S. and Kang, S., 2002. Evolutionary fuzzy modeling in global approximate optimization. InternationalJournal of Vehicle Design, 28 (4), 339355.

Li, J.P., Balazs, M.E. and Parks, G.T., 2007. Engineering design optimization using species-conserving genetic algorithms.

Engineering Optimization, 39 (2), 147161.Li, N., Gu, J. and Liu, B., 2006. A new genetic algorithm based on negative selection. Proceedings of international

conference on machine learning and cybernetics, 1316 August, Dalian, China. Piscataway, NJ: IEEE, 42974299.Liua, J.L. and Chena, C.M., 2009. Improved intelligent genetic algorithm applied to long-endurance airfoil optimization

design. Engineering Optimization, 41 (2), 137154.Mak, K.L., Lau, J.S.K. andWei, C., 2005.A Markov chain analysis of genetic algorithms with individuals having different

birth and survival rates. Engineering Optimization, 37 (6), 571589.Michalewicz, Z., 1995. Genetic algorithms + Data structures= Evolution programs. London: Springer-Verlag.Oh, I., Lee, J. and Moon, B., 2004. Hybrid genetic algorithms for feature selection.IEEE Transactions on Pattern Analysis

and Machine Intelligence, 26 (11), 14241437.Pohlheim, H., 2005. GEATbx examples: Examples of objective functions [online]. GEATbx version 3.7 (Genetic and

evolutionary algorithm toolbox for use with Matlab). Available from: http://www.geatbx.com [Accessed 30 October2006].

Rao, S.S. and Xiong, Y., 2005. A hybrid genetic algorithm for mixed-discrete design optimization. Transactions of theASME, Journal of Mechanical Design, 127 (6), 11001112.


19/19


Rodrigo, P.D.S. and Jun, S.O.F., 2008. Optimum truss design under failure constraints combining continuous and integerprogramming. Proceedings of international conference on engineering optimization (EngOpt 2008), 15 June 2008,Rio de Janeiro, Brazil. Berlin: Springer.

Thierens, D. and Goldberg, D., 1994. Elitist recombination:An integrated selection recombination GA. Proceedings of the1st IEEE conference on evolutionary computation, 2729 June, Orlando, FL, USA. Piscataway, NJ: IEEE, 508512.

Vieira, D.G., et al., 2005. A hybrid approach combining genetic algorithm and sensitivity information extracted from aparallel layer perceptron. IEEE Transactions on Magnetics, 41 (5), 17401743.

Wang, Y., et al., 2006. A new encoding based genetic algorithm for the Traveling Salesman Problem. EngineeringOptimization, 38 (1), 113.

Yoshiaki, S., Noriyuki, T. and Yukinobu, H., 2006. A fuzzy clustering based selection method to maintain diversity ingenetic algorithms. IEEE congress on evolutionary computation, 1621 July, Vancouver, Canada. Piscataway, NJ:IEEE, 30073012.

Yuan, Q., He, Z. and Leng, H., 2008. A hybrid genetic algorithm for a class of global optimization problems with boxconstraints. Applied Mathematics and Computation, 197 (2), 924929.

Glossary

Elitist sub-population The most highly ranked sub-population from P-New. The size is

2 Nsub .Elitist-variants A randomly generated population seeded from the final elitist. It was

used as an initial population in the tail evolution scheme. The size is

6 Nsub .Final elitist The fittest design obtained from the recombination process, GARp.

Fittest-variant The fittest design from the tail evolution process.

GARp Proposed genetic algorithm due to population recombination.

GARp + TE + TF Proposed genetic algorithm due to population recombination and elitistrefinement.P1 The fittest sub-population from P-New consisting of P1-Lower and

P1-Upper. The size is 2 Nsub .P1-Lower The lower-ranked half of the sub-population in P1. The size is Nsub .

P1-Random Randomly generated sub-population whose size was Nsub . This was

matched to P1-Upper in order to conduct a crossover.

P1-Upper The upper-ranked half of the sub-population in P2. The size is Nsub .

P2 Resulting sub-population from crossover between P1-Lower and

P1-Upper. The size is 2 Nsub .P3 Resulting sub-population from crossover between P1-Random and

P1-Upper. The size is 2

Nsub .

P-New The sum of P1, P2 and P3. This was used as an updated population afterinitialization. The size is 6 Nsub .

Nsub A user-specified integer indicating the basic sub-population size.

Recombination A process that consists of evolved sub-populations.

Refinement A process that refines the final elitist.

Sub-chromosome(s) A part of a string of chromosomes.

Tail evolution A single generation of evolution that experiences reproduction and

crossover using elitist-variants.

Tail fitting A mutation-like precision search starting from the final elitist or

fittest-variant.

les-flow around a sphere-.pdf

Documents