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Applied Soft Computing 11 (2011) 881–890

Contents lists available at ScienceDirect

Applied Soft Computing

journa l homepage: www.e lsev ier .com/ locate /asoc

he effects of two new crossover operators on genetic algorithm performance

ustafa Kaya ∗

ksaray University, Faculty of Engineering, Adana Street, Aksaray, Turkey

r t i c l e i n f o

rticle history:eceived 13 November 2008

a b s t r a c t

In this study, two new crossover operators in genetic algorithm are proposed; sequential and random

eceived in revised form 14 January 2010ccepted 17 January 2010vailable online 25 January 2010

eywords:enetic algorithm

mixed crossover. In the first stage, existing and developed crossover operators were applied to twobenchmark problems, namely the reinforced concrete beam problem and the space truss problem. In thesecond stage, the developed crossover operators were applied to a deep beam problem and, a concretemix design problem.The fittest values obtained using developed crossover operators were higher thanthose were obtained with existing crossover operator after 30,000 generations. Moreover, in developedcrossover operators, the random mixed crossover yields higher fitness values than the existing crossoveroperators.

rossover operators

. Introduction

Genetic algorithms (GA) came into existence with thedaptation of developed biological processes to the computer envi-onment. They use units stored in the computer’s memory in theame way as those in natural populations. The initial population issed for the solution of optimization problems with GA.

The risk of capturing local optimum traps was less for GA inomparison with traditional optimization methods. The use of thenconstrained objective functions enabled the discovery of newombinations that have higher fitness values. GA use the bestbjective and fitness function values for problems in which designariables are complex and discontinuous furthermore, does notequire derivatives of the objective function.

The first study related to GA was the introduction of the basicomponents by Holland [1] in 1975 entitled “Machine Learning”.ater, a study on gas pipes, by Holland’s student, Goldberg, provedhat GA had practical uses. GA studies in engineering are generallyhe optimizations of topology, shape, and dimension [2].

Application studies of GA to optimization problems have beenndertaken Jenkins [3–5] and Rajeev and Krishnamoorty [6]

n which the effect of crossover operators on the behavior of

A was investigated. The crossover operator is as important asoding, selection, and mutation in GA. There are various exist-ng types of crossover including; one-point, two-point, uniform,ariable-to-variable, multi-point, mixed and direct design vari-ble exchange. Adeli and Cheng [7–9] applied the dimension

∗ Tel.: +90 382 2150953; fax: +90 382 2150592.E-mail address: kaya261174@hotmail.com.

568-4946/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.asoc.2010.01.008

© 2010 Elsevier B.V. All rights reserved.

optimization problems to three truss beams to numerically com-pare one-point, two-point, and uniform crossover operators. Thebest results were obtained from the two-point crossover oper-ator. Wu and Chow [10] compared the one-point, two-point,three-point, and four-point crossover operators and showed thattwo-point, three-point, and four-point crossover operators are bet-ter than the one-point crossover. Jenkins [11] argues in favor ofmulti-point crossover operator in term of fast progress becomesvery slow in case single-point crossover is used. Using one-pointcrossover, Dejong and Spears [12] introduced the relationshipbetween crossover operators and population size. They statethat two-point crossover is performs better in the problems inwhich the population is large, but uniform crossover is betterfor the small size populations. Syswerda [13] showed that theuniform crossover operator is more efficient when comparedwith two-point crossover. Erbatur and Hasancebi [14,15] sug-gested combining two crossover operators in their study aboutthe effects of crossover operators on the behavior of GA. Themixed crossover operator was also applied to the population inwhich crossover operators such as one-point, two-point, and three-point determined definite rates of generation numbers. Anothertechnique has been suggested that is a direct design variableexchange crossover operator. Each of the design variables waschanged to a probability function, which was formed empirically.The mixed crossover operator first suggested performed fairly wellin the study. The second technique direct design variable exchangecrossover gave the best result among existing crossover operators

[14,15].

In this study, the effect of sequential crossover and randommixed crossover operators on the behavior of GA was investigatedusing a reinforced concrete (RC) beam problem, space truss prob-lem, RC deep beam problem, and concrete mix design problem.

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82 M. Kaya / Applied Soft C

. Stages of the study

In the first stage of this study, existing and developed crossoverperators were applied to the RC beam and, space truss problems.nalysis of the benchmark problems was made for each of ninerossover operators. The programs in analyses were terminatedfter 30,000 generations.

In the second stage, only developed crossover operators appliedo a deep beam and the concrete mix design problem ana-yzed. Maximum fitness values obtained from application ofeveloped crossover operators were determined after 30,000 gen-rations.

The determination steps for the minimum cost on the RC beam,inimum weight of the space truss, the minimum reinforcement

iameters in RC deep beam and the minimum cement weight inoncrete mix design using GA were summarized schematically inig. 1. This figure consists of the following steps:

. Construction of the initial population randomly which comprisesnumbers.

. Decoding the permutation coding for the design variables of eachmember and finding their sequence numbers in the availablevariable list.

. Calculating the value of unconstrained function �(x) for eachmember using Eqs. (11), (22), (30) and (37). Finding the max-imum and minimum values of this function in the population.

. Calculation of the fitness value for each member using Eqs. (12),(23), (31) and (38).

. Application of the sequential selection method. Copying themembers into the mating pool according to their fitness, andcouple them randomly.

. Generation of child using crossover operators and thus obtain-ing the new population. Seven existing and two new crossoveroperators were applied.

. Application of the mutation to each offspring in the new popu-lation with a specific probability.

. Replacing the initial population by the new population andrepeat steps 3–8 until the termination criteria is obtained.

.1. Coding

In this study, permutation coding was used because of the designariables consists of more than one variable group. The chromo-ome length is equal to the number of variable groups in this typef coding.

Fig. 1. Structure of genetic algorithm.

ing 11 (2011) 881–890

2.2. Formation of initial population

The GA searches inside the population consisting of pointsinstead of searching point-to-point [2]. While the initial popula-tion is being formed, its members must be given importance sothat the same members are not selected, since the members mustbe chosen randomly.

The suitable selection of population size significantly affectsthe performance of genetic algorithm. In various studies it wasobserved that the result was reached earlier in larger populations incomparison with small populations [2]. In this study, the populationsize (N) was selected as 100.

2.3. Evaluation

The GA basically finds the maximum of an unconstrainedobjective function. To solve a constrained objective minimizationfunction, two transformations need to be made. The first transformsthe original objective constrained function into an unconstrainedobjective function, using the concept of the penalty function.In the second transformation, the unconstrained objective func-tion is transformed to the fitness function [13]. In this study, forthe four problems, different constrained objective functions wereoccurred. These functions are given in Eqs. (10), (21), (29) and (34).After these equations were occurred, this functions transformedunconstrained objective functions. This unconstrained objectivefunctions are given in Eqs. (11), (22), (30) and (37). Finally, thisunconstrained objective functions is transformed into the fitnessfunctions, as given in Eqs. (12), (23), (31) and (38).

2.4. Selection

The members of the new population in each generation areselected by a process from members of the existing population. Theselection technique performs natural selection artificially and inthis study, a sequential selection method was used. In this method,members are set in order by a linearly decreasing function. Themembers with the low fitness value are removed from populationin a defined ratio and members with the high fitness values replacethose removed in the same ratio. In this study, members with thelowest fitness values constituting 25% of population were elimi-nated and replaced with the highest valued members in the sameratio.

2.5. Crossover operators

GA can rapidly identify discrete zones within a large searchspace area to concentrate the search for an optimum solution. Thistechnique changes mutually defined parts of two members selectedand obtains different members that give new points in the searchspace. The crossover operators used in the current study are sum-marized below.

2.5.1. Existing crossover operatorsThe crossover point in the one-point crossover operator is ran-

domly selected between 1 and L − 1 where L is the length ofthe chromosome. Two new members were obtained by relocat-ing parts, after this the cut-off point is matched in two members.The codes after 9th site of members given in Table 1 were changedusing the one-point crossover operator. In the two-point crossover

operators, two different cut-off points were randomly selectedbetween 1 and L − 1. New members were obtained by relocatingzones between the cut-off points of paired members. The codesbetween 7th and 13th sites of members given in Table 2 werechanged.

M. Kaya / Applied Soft Computing 11 (2011) 881–890 883

Table 1One-point crossover sample.

5 3 6 8 1 3 2 7 4 3 2 6 1 8 2 6 3 5Parent1

2 7 4 3 6 8 1 3 2 7 4 6 8 2 1 5 4 2Parent2

5 3 6 8 1 3 2 7 4 7 4 6 8 2 1 5 4 2Child1

2 7 4 3 6 8 1 3 2 3 2 6 1 8 2 6 3 5Child2

Table 2Two-point crossover sample.

5 3 6 8 1 3 2 7 4 3 2 6 1 8 2 6 3 5Parent1

2 7 4 3 6 8 1 3 2 7 4 6 8 2 1 5 4 2Parent2

5 3 6 8 1 3 1 3 2 7 4 6 1 8 2 6 3 5

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In multi-point crossover operators, many cut-off points areelected between 1 and L − 1. New members were obtained by relo-ating zones between these cut-off points of pairing members andhe codes of the members were changed in four sites as shown inable 3.

In variable-to-variable crossover operators, first the pairedembers (strings) are decomposed into their substrings. Next, a

ingle-point crossover is separately carried out on all the substringsTable 4.). In this way, each design variable amongst the memberss activated to separately accomplish the design exchange [14].

In uniform crossover operators, as a first stage, a randomember must be temporarily formed in the binary string. The chro-osome length of this member is equal to chromosome lengths of

he other members in the population. If the code of the nth site ofhe chromosome of the temporary member is 0, in this case, theode of 1st old member is used in the nth site of the 1st new mem-er and the code of 2nd old member is used in the nth site of thend new member. Equally, if the code of the nth site of the tem-orary member’s chromosome is 1, in this case, the code of thend old member is used in the nth site of the 1st new membernd the code of the 1st old member is used in the nth site of thend new member. The application of the uniform crossover oper-tor to the two members selected from the population is given inable 5 [13].

The mixed crossover operator introduced by Hasancebi et al.14], uses a combination of single-point, two-point and three-point

rossovers for selected proportions of generations for a fixed gen-ration number. The idea behind the technique is to achieve anfficient search, which can be obtained by activating the positiveharacteristics of the existing techniques in the right appropriatelace in the search process [14].

able 3ulti-point crossover sample.

5 3 6 8 1 3 2 7 4Parent1

2 7 4 3 6 8 1 3 2Parent2

5 3 4 3 1 3 1 3 4Child1

2 7 6 8 6 8 2 7 2Child2

3 2 6 8 2 1 5 4 2

In the direct design variable exchange crossover operator, eachdesign variable (substring) is directly and separately exchangedbetween paired members according to a probability function. Thisfunction is empirically created, and is designed to achieve an effec-tive search during successive generations [14].

2.5.2. The new crossover operators2.5.2.1. Sequential crossover. In this new crossover operator, beforebeginning the process the existing crossover operators were num-bered 1–7 (Table 6). Then, the existing crossover operators wereapplied sequentially to the chromosome couples in the populationat in the same generation.

2.5.2.2. Random mixed crossover. In this operator, the existingseven crossover operators were numbered 1–7 as shown in Table 6were applied randomly to the chromosome couples in the popula-tion at the same generation.

2.6. Mutation operator

The mutation operator is critical to the success of geneticalgorithms since it determines the search directions and avoidsconvergence to local optima. In this operator, the gene selectedrandomly from the chromosome undergoes a change because ofthe intervening chromosomes that form the population from theoutside in a defined ratio [2]. In this study 0.1% and 0.5% and 1%

mutation ratios were applied to the chromosomes of the problems.The 0.1% and 0.5% mutation ratios did not adequately change thechromosomes, but the 1% mutation ratio gave a better result thanthe 0.1% and 0.5% mutation ratios. As a result the 1% mutation ratiowas applied on the chromosomes of all the problems.

3 2 6 1 8 2 6 3 5

7 4 6 8 2 1 5 4 2

3 4 6 1 8 1 5 3 5

7 2 6 8 2 2 6 4 2

884 M. Kaya / Applied Soft Computing 11 (2011) 881–890

Table 4Variable-to-variable crossover sample.

5 3 6 8 1 3 2 7 4 3 2 6 1 8 2 6 3 5Parent 1

2 7 4 3 6 8 1 3 2 7 4 6 8 2 1 5 4 2Parent2

5 7 6 3 1 8 2 3 4 7 2 6 1 2 2 5 3 5Child 1

2 3 4 8 6 3 1 7 2 3 4 6 8 8 1 6 4 5Child2

Table 5Uniform crossover sample.

0 1 1 0 0 1 1 0 1 1 1 0 1 0 1 0 0 1Temporary string

5 3 6 8 1 3 2 7 4 3 2 6 1 8 2 6 3 5Parent1

2 7 4 3 6 8 1 3 2 7 4 6 8 2 1 5 4 2Parent2

5 7 4 8 1 8 1 7 2 7 4 6 8 8 1 6 3 2

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instead of looking for the most economic ones [16]. The RC beamsection for this problem is given in Fig. 2.

For the purposes of this analysis, strength design procedureswere adopted because they have, among others, the followingadvantages [17].

Child1

2 3 6 3 6 3 2 3Child2

. Theoretical review of the crossover operators

The two main strategies that a solid crossover operator shouldse to locate the optimum are exploration and exploitation. Theacility for exploration in a crossover operator encourages theearch for a rapid and thorough discovery examination of the designpace. A robust exploration prevents the search from locating aocal peak. However, only using this type of search becomes ineffi-ient when making use of previously obtained points to reach moreppropriate points. On the other hand, a solid exploitation employshe previously found points to reach the optimum, however, in thisase a slow convergence is observed accompanied by an increasingisk of locating a local peak. In fact, these two strategies are contra-ictory, and a balanced use of both is vital for achieving an efficientearch through the crossover operator [14].

. Applications

In the first stage of this study, existing and developed crossoverperators were applied to the two benchmark problems, namely,he RC beam problem and the space truss problem. In the secondtage, the developed crossover operators were applied to a deepeam problem and a concrete mix design problem.

All the programs used to compare developed and existing

rossover operators were the same except for the crossover opera-ors. In all programs, population size (N = 100), permutation codingype, sequential selection method and the 1% mutation ratio weresed; therefore, the different results obtained from these analyses

able 6xistent crossover types and codes of these which used in sequential crossover andandom mixed crossover.

Crossover type Code

One-point crossover 1Two-point crossover 2Uniform crossover 3Multi-point crossover 4Variable-to-variable crossover 5Mixed crossover 6Direct design variable exchange crossover (DDVECT) 7

4 3 2 6 1 2 2 5 4 5

were a consequence of the different crossover operators that wereused.

4.1. Benchmark problems

4.1.1. Reinforced concrete beam problemThe design of the RC beam is normally an iterative process, in

which the engineer assumes the self-weight of the beam before-hand, and a trial section is chosen. Then, the moment of resistanceof the section is determined to check its suitability against thegiven applied bending moment. The process is repeated until a suit-able trial section is found. This procedure often creates difficulty inexactly matching the moment due to the self-weight of the beam,which may be quite substantial in many cases. Therefore, the designprocess of a beam is not only slow, but also uneconomic, since theonly concern is to find any section suitable for the given conditions,

Fig. 2. Reinforced concrete beam section.

M. Kaya / Applied Soft Computing 11 (2011) 881–890 885

Table 7The result of runs obtained with different crossover techniques for reinforced concrete beam.

One-point Two-point Multi-point Variable-to-variable Uniform Mixed crossover DDVECT Sequential Random mixed

Cost of reinforced concrete beam ($)Run 1 53,59 52,84 51,13 50,64 45,81 47,92 44,95 45,21 42,76Run 2 54,10 51,89 49,89 49,43 48,43 46,72 47,16 44,93 43,95Run 3 51,64 52,64 50,89 50,38 48,17 45,71 46,92 43,76 43,76Run 4 54,10 53,35 48,92 51,13 48,92 48,17 47,43 44,46 42,00Run 5 52,64 50,89 51,38 49,68 47,16 46,92 46,22 42,18 43,00Run 6 52,64 51,89 50,13 48,67 47,16 47,67 46,22 43,49 42,25Run 7 51,64 52,64 50,89 50,38 48,17 45,71 46,92 43,76 43,76Run 8 54,10 53,35 48,92 51,13 48,92 48,17 47,43 44,46 42,00

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Strength design better predicts the strength of a section becauseof the recognition of the non-linearity of the stress strain curveat high stress level.Since the dead loads to which a structure is subjected are morecertainly determined than the live loads, it is unreasonable toapply the same factor of safety to both. Therefore, this approachallows the use of different safety factors for each type of loading.

The basic assumptions that are taken when using strengthesign are as follows [18].

Plane sections before bending remain plane after bending.The ultimate capacity, strain and stress are not proportional.The strain in the concrete is the proportional to the distance fromthe neutral axis.The tensile strength of the concrete is neglected in flexural com-putations.The ultimate concrete strain is 0.003. Please check all numbersfor, or.The average compressive stress in the concrete is 0,85. f ′

cThe average tensile stress in the reinforcement does not exceedfy.

The moment to be carried by the RC beam is given in Eq. (1). Inhis equation Mu: ultimate moment, Ø: strength reduction factor, �:teel ratio, fy: yield stress of steel, f ′

c: compressive stress of concrete;: weigh of the beam; d: height of the beam [17]:

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)(1)

Using extra reinforcement in the beam causes brittle behaviornder loading. The required reinforcement ratio of RC beam is given

n Eq. (2):

= (�(�fybd)(fc − 0.59fy))f ′c (2)

The most important parameter that affected the cost of theetermination of the cheapest beam is the weight of reinforcement

n the beam, the second parameter being the weight of con-rete. Taking this into account, the unit price of the reinforcementncreases the cost of the concrete 25 times. The fitness functionroposed for finding the minimum cost in the RC beam is given inq. (3). In this equation, F1, F2 are unit prices of steel and concrete;s and �c are unit weights of steel and concrete and L is the lengthf beam.

Since the number of variables representing the RC beam wasoo large, permutation coding was used where the chromosomeength is equal to the group number of design variables. In thisroblem the design variable groups were beam height, beam width,einforcement diameters and reinforcement numbers.

47,16 46,92 46,22 42,18 43,0047,16 47,67 46,22 43,49 42,25

45,81 45,71 44,95 42,18 42,00

For the RC beam, the objective function f(x) is expressed in Eq.(3), the penalty function is given in Eq. (9) and the constrainedobjective function is shown in Eq. (10):∑

min f (x) = bdL(F1��s + F2�c) (3)

ˇ1 = 0.85 − 0.05f ′c − 4000

1000(4)

�b = 0.85ˇ1fc87000fy(87000 + fy)

(5)

�max = 0.75�b (6)

If � ≥ 200fy

and � < �maxg(x) = bdL(F1��s + F2�c) (7)

Else If � <200fy

and � ≥ �maxg(x) = bdL(F1��s + F2�c)100 (8)

G =∑

g(x) (9)

�(s) = f (x)(1 + KG) (10)

where K is a coefficient selected for the problem taken to be 10 inthis study, g(x) is a penalty coefficient and calculated with Eq. (9).

In the first transformation, the constrained objective function�(s) was transformed to unconstrained objective function �(x) asexpressed in Eq. (11):

�(x) =∑ �(s)

�(s)max(11)

In the second transformation in Eq. (12), the unconstrainedobjective function �(x) was converted to the fitness function F(s):

F(s) = �(x)max − �(x) (12)

In this problem, the RC beam was analyzed using existing anddeveloped crossover operators. The direct design variable exchangecrossover is the lowest cost with the single-point crossover beingthe most cost of the existing crossover operators. The cost found bysingle-point crossover is 21% higher than the cost found by directdesign variable exchange crossover. The random mixed crossovergives the lowest cost between existing and developed crossoveroperators. When the random mixed crossover and direct designvariable exchange crossover are compared, it can be seen that thecost found by the random mixed crossover operator is 7% less thanthe cost found by the direct design variable exchange crossoveroperator (Table 7).

Concerning the fitness values criteria of the RC beam problem

that terminated after 30,000 generations, the maximum fitnessvalue between existing crossover operators was obtained fromthe program that employs the direct design variable exchangecrossover operator. The maximum fitness value in the existing anddeveloped crossover operators was achieved using the program

886 M. Kaya / Applied Soft Computing 11 (2011) 881–890

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Fig. 3. The fitness values obtained after 30,

hat employs the random mixed crossover operator. The fitnessalue obtained by the developed random mixed crossover was 13%igher than the fitness value obtained by the existing direct designariable exchange crossover operator (Fig. 3).

.1.2. Space truss problemThe space truss problem is commonly chosen when solv-

ng problems about GA since the design space is medium sized6,7,9,14]. Also, in this study existing and developed crossoverperators were applied to the problem of determining the mostppropriate bar sections for the space truss (Fig. 4). In this prob-em chromosome length is 25 and this number is equal to designariable group number. By using existing and developed crossoverperators, the optimum design of the space truss was achieved inerms of weight.

In the space truss problem, consisting of 25 bar elements, thear sections were selected from profiles given in AISC [19]. In theested space truss, the yield strength was taken to be 2400 kg/cm2.

6 2

he elasticity module of steel was taken to be 2 × 10 kg/cm . A00-kN force was applied to the truss from two nodal points invertical direction. The truss was restricted to a maximum L/300orizontal drift. The codes used in the application vary from 1 to23 and the size of the designed space was 12325.

Fig. 4. 25-Bar space truss.

neration for the reinforced concrete beam.

For space truss; the objective function W(x) is expressed in Eq.(10), the penalty function is expressed Eq. (14) and the constrainedobjective function is expressed in Eq. (15):

min W(x) =Tk∑

k=1

AkLk�k (13)

where Ak is the cross-sectional area, �k the specific gravity of spacetruss elements, Lk is the space truss elements length, and Tk is thenumber of space truss elements.

Since steel structures have stability problems, the state of thestress caused by combined loading is considered to include bucklingand lateral buckling. The normalized forms of the constrains are asfollows.

Combined stress constrains:

gi(x) = �eb

�bem+ Cm�b(

1 − (�eb/�e))

�B

− 1 (14)

gi(x) = �eb

0.6�a+ �b

�B− 1 (15)

Eqs. (14) and (15) are required for members subjected to bendingmoment and, axial compression. For members subjected bendingmoment and axial stress are required to providing Eq. (9) is applied:

If�eb

�bem≤ 0.15 (16)

gi(x) = �eb

�bem+ �b

�B− 1 (17)

�eb: compression stress only subjected to axial force; �b: compres-sion stress only subjected to bending moment; �bem: allowablestress only subjected to axial force; �B: allowable compressionstress in the 1st member subjected only to bending moment; �e:Euler stress divided by the safety coefficient; �a: yield stress ofsteel; nm: number of members; Cm the adjustment coefficient andequal to 0.85 for the lateral displacement of members:

C =∑

ci (18)

If gi(x) > 0, ci = gi(x) (19)

If gi(x) ≤ 0, ci = 0 (20)

ci: negligence coefficient and calculated as follows:

�(s) = W(x)(1 + KC) (21)

K: a coefficient selected for the problem taken to be 10 in this study.In the first transformation, the constrained objective function

�(s) was transformed to an unconstrained objective function �(x)as expressed in Eq. (16):

�(x) =∑ �(s)

�(s)max(22)

M. Kaya / Applied Soft Computing 11 (2011) 881–890 887

Table 8The result of runs obtained with different crossover techniques for space truss.

One-point Two-point Multi-point Variable-to-variable Uniform Mixed crossover DDVECT Sequential Random mixed

Space truss weight (kN)Run 1 10,67 10,52 10,18 9,69 9,64 9,39 9,54 9,00 8,51Run 2 10,28 10,62 9,98 10,08 9,39 8,95 9,59 8,85 8,56Run 3 10,62 10,48 10,13 10,03 9,59 9,34 9,49 8,71 8,71Run 4 10,77 10,13 10,23 10,18 9,74 9,44 9,10 8,66 8,36Run 5 10,46 10,33 9,93 9,89 9,39 9,20 9,34 8,95 8,40Run 6 10,48 10,33 9,74 9,84 9,25 9,20 9,30 8,66 8,41Run 7 10,54 10,57 9,76 10,11 9,27 9,01 9,51 8,69 8,47Run 8 10,41 10,34 9,91 9,75 9,71 8,37 9,17 9,03 8,69Run 9 10,70 10,41 10,03 9,98 9,55 9,21 9,46 8,78 8,74Run 10 10,32 10,21 10,17 10,07 9,67 9,11 9,33 8,86 8,43

Best 10,28 10,13 9,74 9,69 9,25 8,95 9,10 8,66 8,40

fter 30

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As a result of this behavior, problems occurred in the analysisof deep beams. The fittest horizontal and vertical reinforcementdiameters were determined in a deep beam using the GA.

Fig. 5. The fitness values obtained a

In the second transformation, the unconstrained objective func-ion �(x) was converted to an F(s) fitness function in Eq. (17):

(s) = �(x)max − �(x) (23)

n this problem, the space truss beam was analyzed using existingnd developed crossover operators. The mixed crossover operatorives the lowest truss beam weight, but the single-point crossoverives the most beam weight among the existing crossover oper-tors. The random mixed crossover gives the lowest truss beameight among all the crossover operators. The truss weight found

y the mixed crossover is 15% heavier than the truss beam weightound by the random mixed crossover (Table 8).

Concerning the fitness values criteria of the space truss prob-em that terminated after 30,000 generations, the maximum fitnessalue from the existing crossover operators was obtained from theirect design exchange crossover (DDVECT) operator. The max-

mum fitness value in the developed crossover operators waschieved using the random mixed crossover operator. The fitnessalue obtained by the developed random mixed crossover was5% higher than the fitness value obtained from the existing directesign exchange crossover operator (Fig. 5).

.2. Proposed and analyzed problems

.2.1. Deep beam problemDeep beams used in shear walls, folded roof plates, and

ilos show different behaviors when compared with normaleams owing to the fact that their depth–span ratio is higherhan that of normal beams. Such studies confirm the usualypothesis that plane sections before bending remain plane after

ending, does not hold for deep beams. Significant warping ofhe cross-sections occurs because of the high shear stresses20,21].

The deep beam, loaded from its top-edge, is accepted as a deepeam in the shearing calculation, while it is not accepted as deep

,000 generation for the space truss.

beam in the bending calculation due to its L/h ratio being greaterthan 5/4, but less than 5 [22].

A deep beam of loaded from the top-edge and of L/h ratio of 4 isgiven as an example in Fig. 6. This beam is accepted as deep beamin calculations of shear force while it was accepted as normal beamfor bending according to the ACI 318 [22].

In this study, the stresses on the deep beam caused by externalforces were first determined through the finite elements method.Then, the steel bar diameters given in standards depending on theeffects on these sections were determined using GA. The deep beamwas tested under compressive and tensioning stress restrictions.In this study, the stresses on the bars caused by external forceswere first determined through the finite elements method. Then,the bar sections given in standards depending on the effects onthese sections were determined using GA.

Fig. 6. The fittest reinforcement obtained for deep beam.

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88 M. Kaya / Applied Soft C

For deep beam; the objective function W(x) is expressed in Eq.18), the penalty function is expressed Eq. (14) and the constrainedbjective function is expressed in Eq. (21):

in W(x) =Cs∑

s=1

AsLs�s (24)

here As is the reinforcement area, �s the specific gravity of rein-orcement, Ls is the reinforcement length and, Cs number of zonesn the deep beam.

The constrained objective function was transformed to annconstrained objective function �(x) as expressed in Eq. (18):

i(x) =(

�sLsAsnsfyd

Fei

)− 1 (25)

f gi(x) ≥ 0, ci = gi(x) (26)

f gi(x) < 0, ci = gi(x) × 100 (27)

=∑

ci (28)

(s) = W(x)(1 + KC) (29)

: a coefficient selected for the problem taken to be 10 in this studynd ci: negligence coefficient and calculated as follows:

In the first transformation, the constrained objective function(s) was transformed to an unconstrained objective function �(x)s expressed in Eq. (22):

(x) =∑ �(s)

�(s)max(30)

here Fei is the required force met by reinforcements in the x or yirection in the ith zone, fyd is the yield strength of reinforcement,nd ns is the number of reinforcements in the ith zone.

In the 2nd transformation, the unconstrained objective function(x) was converted to a fitness function F(s). This transformationas achieved using the maximum of the ith element of the uncon-

trained objective function. The fitness values of the members werealculated according to Eq. (23) as follows:

(s) = �(x)max − �(x) (31)

Concerning the fitness values criteria of deep beam problem thaterminated after 30,000 generations, the maximum fitness valuerom the existing crossover operators was obtained from the mixedrossover operator. The maximum fitness value in the developedrossover operators was achieved using the sequential crossover

perator. The fitness value obtained by the sequential crossoveras 12% higher than the fitness value obtained from the existingixed crossover operator (Fig. 7).In this problem the deep beam was analyzed using existing and

eveloped operators. The direct design variable exchange crossover

Fig. 7. The fitness values obtained after 30

ing 11 (2011) 881–890

operator gives the smallest reinforcement diameter among theexisting crossover operators. The random mixed crossover givesthe smallest reinforcement diameter among all crossover opera-tors. The reinforcement diameter found by direct design variableexchange crossover is 7% bigger than the reinforcement diameterfound by the random mixed crossover (Table 9).

In this example, diameters of horizontal and vertical reinforce-ment determined for 9 zones by using genetic algorithm are givenin Table 10 and Fig. 6. In this figure, horizontal and vertical linesshow that the horizontal and vertical shear reinforcement in thosezones, respectively.

The weights of reinforcement determined using the GA werecompared with weight of reinforcement specified in accordancewith ACI 318-99. As a result of comparisons, it can be seen that theweight of reinforcement determined using the GA is 5% less [22].

4.3. High strength concrete mix design problem

The various components of a mix are proportioned so that theresulting concrete has adequate strength, proper workability forplacing and low cost. In this problem, the minimum amount ofcement for 1 m3 high strength concrete (fc′ = 50 MPa) was deter-mined by the GA and the Bolomey formula [23,24] given in Eq.(32). Then, amounts of cement which are determined using bothmethods were compared.The amount of cement used in concretemixtures is directly proportional to the amount of water. Further-more, the amount of water is inversely proportional to the finenessmodulus of the aggregate when the amount of cement is minimumand the fineness modulus is at maximum:

fc =(

fcc

a

(c

w

)2)

(32)

where c is the cement weight, w is the weight of the water, fc′ is thecompressive strength of concrete, fcc is the compressive strength ofcement and a is changes between 4 and 8. The compressive strengthof the concrete (fc′), compressive strength (fcc) and a the coefficientof cement are fixed. The fineness modulus of the aggregate (k) willbe determined as the minimum amount of cement.

In this study, the number of design variable groups is equal to thenumber of sieves and varies between 1 and 8. The number of vari-ables in all groups varied between 0 and 2100 kg. 0 kg representsthat aggregate does not exist and 2100 kg represents that aggregateis fully present in 1 m3 concrete.

The objective function W(cement) is expressed in Eq. (33) as:(fc′a(10 − k)

)

min W(cement) =

fccor

max(k) =(∑n

1%passed

100

)(33)

,000 generation for the deep beam.

M. Kaya / Applied Soft Computing 11 (2011) 881–890 889

Table 9The result of runs obtained with different crossover techniques for deep beam.

One-point Two-point Multi-point Variable-to-variable Uniform Mixed crossover DDVECT Sequential Random mixed

Deep beam weight (kN)Run 1 80,85 79,73 77,12 73,40 73,02 72,28 71,16 68,18 64,45Run 2 81,59 78,24 75,63 76,38 71,16 70,79 67,81 69,30 63,34Run 3 80,47 79,36 76,75 76,00 72,65 71,91 70,79 65,94 65,57Run 4 77,87 80,47 77,49 77,12 73,77 72,65 71,53 67,06 63,34Run 5 79,36 78,24 75,26 73,40 71,18 68,92 69,67 67,81 64,83Run 6 79,36 76,75 73,77 74,51 70,04 70,42 69,67 65,57 63,71Run 7 79,41 78,37 76,79 73,30 72,15 72,55 68,71 65,83 66,30Run 8 81,40 7,23 74,47 77,10 71,22 69,97 70,11 66,80 65,34Run 9 80,136 79,34 73,86 76,43 73,65 70,17 68,87 69,13 66,22Run 10 78,16 76,80 77,35 75,41 70,78 72,50 70,70 66,80 64,51

Best 77,87 76,75 73,77 73,40 70,04 68,92 67,81 65,57 63,34

Table 10Codes and reinforcement diameters obtained for deep beam.

Zone No 1 2 3 4 5 6 7 8 94 6 4 3 3 3

∅16 ∅14 ∅16 ∅16 ∅16 ∅164 3 4 3 3 3

∅14 ∅14 ∅16 ∅16 ∅14 ∅14

uw

�

I

I

�

�ws

E

F

pfifoot

Table 12Concrete mix design.

Component name Weight (N)

Cement (KPC 42.5) 3900Crushed aggregate (5–12 mm) 4800Crushed aggregate (12–15 mm) 4800Sand (0–5 mm) 5300Marble flour 3700

TT

Code of horizontal 4 5 4Diameter of horizontal ∅14 ∅18 ∅14Code of vertical 5 3 5Diameter of vertical ∅16 ∅14 ∅14

The constrained objective function was transformed to annconstrained objective function �(s) as given in Eq. (34) and �(x)as given in Eq. (37). Where n is the number of sieves:

(s) =(

10 −(∑n

1%passed

100

))(34)

f �(s) < 4.60 then �(s) = 0 (35)

f �(s) > 7.17 then �(s) = 0 (36)

(x) = �(s)max

�(s)(37)

In the 2nd transformation, the unconstrained objective function(x) was converted to a fitness function F(s). This transformationas achieved using the maximum of the ith element of the uncon-

trained objective function.The fitness values of the members were calculated according to

q. (38) as follows:

(s) = �(x)max − �(x) (38)

Concerning the fitness values criteria of the concrete mix designroblem that terminated after 30,000 generations, the maximum

tness value from the existing crossover operators was obtained

rom the direct design variable exchange crossover (DDVECT)perator. The maximum fitness value in the developed crossoverperators was achieved using the random mixed crossover opera-or. The fitness value obtained by the developed the random mixed

able 11he result of runs obtained with different crossover techniques for concrete mix design.

One-point Two-point Multi-point Variable-to-variable

Cement weight (kN)Run 1 4,979 4,910 4,749 4,520Run 2 5,025 4,818 4,657 4,704Run 3 4,956 4,887 4,726 4,680Run 4 4,795 4,956 4,772 4,749Run 5 4,887 4,818 4,635 4,520Run 6 4,887 4,726 4,543 4,588Run 7 4,891 4,825 4,728 4,514Run 8 5,012 4,810 4,585 4,747Run 9 4,935 4,885 4,548 4,707Run 10 4,813 4,730 4,763 4,644

Best 4,795 4,726 4,543 4,520

Water 1500Additive (Sikament 300) 81

crossover operator was 8% higher than the fitness value obtainedfrom the existing the direct design variable exchange crossoveroperator (Fig. 8).

The concrete mix design prepared using existing and devel-oped crossover operators. The uniform crossover gives the lightestamount of cement among the existing crossover operators. Thesequential crossover operator gives the lightest amount of cementall the crossover operators. The amount of cement found by thesequential crossover operator is 8% smaller than the amount ofcement found by the uniform crossover operator (Table 11).

In this example, the concrete with fc′ = 50 MPa was produced

according to the minimum w/c ratio determined by using geneticalgorithm. This concrete mix design (Table 12) was used to con-struct experiment members for the study on “post tensionedcolumn-to-beam connections in precast structures” [25].

Uniform Mixed crossover DDVECT Sequential Random mixed

4,382 4,451 4,497 3,968 4,1984,175 4,359 4,382 3,900 4,2674,359 4,428 4,474 4,037 4,0604,405 4,474 4,543 3,900 4,1294,290 4,244 4,383 3,992 4,1754,290 4,336 4,313 3,923 4,0374,231 4,468 4,443 4,083 4,0544,317 4,309 4,386 4,023 4,1144,241 4,321 4,535 4,078 4,2574,354 4,465 4,359 3,972 4,114

4,175 4,244 4,313 3,900 4,037

890 M. Kaya / Applied Soft Computing 11 (2011) 881–890

0,000

5

tosyotp

tuv

cwa

mdci

rmmcdo

twit

cead

aab

[

[

[

[

[

[

[

[

[[

[

[

[

Fig. 8. The fitness values obtained after 3

. Conclusion

In this study, the effect of seven existing crossover opera-ors and two developed crossover operators on the performancef the GA were compared. In the first stage a RC beam andpace truss problems were used in the comparison. In the anal-ses made for these problems, the maximum fitness value wasbtained from the analysis of the direct design crossover opera-or selected from the existing crossover operators for the RC beamroblem.

The maximum fitness value was obtained from the analysis ofhe mixed crossover operator from the existing crossover operatorssed in the space truss beam problem. However, obtained fitnessalues are approximately equal to each other.

When the results obtained from the developed and existingrossover types were compared, the two highest fitness valuesere obtained from the analysis of the random mixed crossover

nd sequential crossover operator.In the second stage of the study, using deep beam and concrete

ix design problems, the effect of sequential crossover and ran-om mixed crossover operators, developed using deep beam andoncrete mix design problems, on the performance of the GA wasnvestigated.

The maximum fitness value was obtained by the analysis of theandom mixed crossover operator used in deep beam and concreteix design problems. While the weights of reinforcement deter-ined from the use of the random mixed crossover and sequential

rossover operators, were approximately equal to each other in theeep beam problem, the weights of cement determined using sameperators were different in the concrete mix design problem.

The use of the GA produced successful results in the determina-ion of the lowest cost of the RC beam problem, of the minimumeight of steel truss beam, of the weight of reinforcement which is

ndependent from dimensions of deep beams, and of the weight ofhe minimum amount cement in the concrete

The operators developed for this study (random mixedrossover and sequential), obtaining higher fitness values than thexisting crossover operators, shows that developed crossover oper-tors thus GA give better results in solving problems in which

esign variables are discontinuous.

The application of GA to the four different problems whichre encountered in civil engineering and the successful resultschieved from the analyses of these problems shows that GA coulde applied to other/further problems in civil engineering

[

[[

generation for the concrete mix design.

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