The effects of a new selection operator on the performance of a genetic algorithm

Applied Mathematics and Computation 217 (2011) 7669–7678

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

The effects of a new selection operator on the performance of a genetic algorithm Mustafa Kaya Aksaray University, Faculty of Engineering, Aksaray, Turkey

a r t i c l e

i n f o

Keywords: Genetic operators Selection operators

a b s t r a c t In this study, the new back controlled (BCSO) selection operator for genetic algorithm (GA) is presented. In the first stage, six existing operators and the BCSO were applied to the traveling salesman and space truss benchmark problems. An analysis of the results of the benchmark problems was made for each of the seven selection operators. In the second stage, the BCSO, the Roulette Wheel, dominant selection the hybrid II and operators were used to determine the fittest beam section has been used. In this stage, the cost of beam for a 18.3 m span developed by Mid-Atlantic States Pre-stressed Concrete Committee for Economic Fabrication and using a genetic algorithm the cost of a beam using BCSO were compared. It was determined that the beam produced using the proposed BCSO selection operator would be lower than those obtained with the existing selection operators. Also the determined beam cost is compares favourably with the average cost of a beam sold by the precast industry in the USA. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Processes executed with a genetic algorithm (GA) came into existence with the adaptation of biological processes to the computer environment, using units stored in the computer’s memory in the same way as those in natural populations. A population known as the initial population is used for the solution of optimization problems with a GA. The risk of capturing local optimum traps was less for GAs in comparison with traditional optimization methods. The use of the unconstrained objective functions enabled the discovery of new combinations that have higher fitness values. A GA uses the best objective and fitness function values for problems in which design variables are complex and discontinuous. GA does not require derivatives of the objective function. The first study related to GA was in 1975 when Holland [1] introduced the basic components of GA called ‘‘Machine Learning’’. Later, an examination of gas pipes, by Goldberg, proved that GAs could have practical uses. GA studies in engineering are generally the optimizations of topology, shape, and dimension [2]. Studies on the application of GAs to optimization problems and the effect of operators on the behavior of GAs are presented in Refs. [3–6]. The selection operator is as important as encoding, crossover, and mutation in GAs. There are various existing types of selection operators including; Roulette Wheel, sequential, tournament, dominant, hybrid and, kin selection operators. There have been a number of studies on the effects of selection operators and these studies continue. Tang et al. used a parallel genetic algorithm (PGA) along with new genetic operator namely the kin selection operator [7]. Katayama et al. investigated two selection operators (Selection l and Selection 2) with relatively high selection pressures [8]. Jihua et al. proposed a dominant selection operator (DSO) which strengthens the action of dominant individuals and weakens the action of the inferior individuals in the process of evolution [9]. Stern et al. compared three popular selection E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.02.070

7670

M. Kaya / Applied Mathematics and Computation 217 (2011) 7669–7678

operators: proportional selection, ranking selection [10] and tournament selection [11,12]. Arumugam et al. proposed two hybrid selection operators that were a combination of both Roulette Wheel (RWS) and Tournament Selection operator (TS). They designed two types of hybrid selections: single level and two level hybrid selection operators. [13]. Ashutosh et al. proposed a kin selection operator. This selection operator is controlled by a kin probability variable [14]. In the first stage of this study, the six existing and the proposed selection operators were applied to the traveling salesman and space truss problems. An analysis of the benchmark problems was made for each of seven selection operators. In the second stage a back controlled selection operator (BCSO), RWS, DSO and the hybrid II selection operator were used to determine the fittest beam section has been used. In this stage, the cost of beam for a 18.3 m span developed by Mid-Atlantic States Pre-stressed Concrete Committee for Economic Fabrication (PCEF) using a GA and the cost of beam using BCSO were compared. 2. Stages of the study In this study, the program was terminated after 30,000 generations. The determination of the steps to achieve the shortest way, weight of steel space truss and the cost of pre-stressed precast beam using GA are given in Fig. 1. 2.1. Encoding The most significant feature that distinguishes the proposed GA from other operators is the use of codes to represent the design variables. As can be seen in Fig. 1, the first step in the application of the GA to a problem is determination of the most appropriate coding type. In this study, permutation coding was used which is the type preferred for problems in which the encoding design variables consists of more than one variable group. The chromosome length is equal to the number of variable groups in this type of encoding. 2.2. Formation of initial population While the initial population is being formed, its members must be given importance so that the same members are not selected, since the members must be selected randomly. 2.3. Evaluation The GA basically finds the maximum of an unconstrained objective function. To solve a constrained objective minimization function, it is necessary to make two transformations. The first transforms the original objective constrained function into an unconstrained objective function, using the concept of the penalty function. In the second transformation, the unconstrained objective function is transformed to the fitness function. 2.4. Selection Individuals of new population in each generation are selected from the individuals of the existing population with a selection operator after creating the initial population. This operator performs the natural selection artificially. The selection operators used are summarized below.

Encoding

Initial Population New Generation

gen=gen+1

Evaluation Mutation Selection

Crossover Fig. 1. Structure of genetic algorithm.

M. Kaya / Applied Mathematics and Computation 217 (2011) 7669–7678

7671

2.4.1. Existing selection operators 2.4.1.1. Roulette Wheel Selection operator (RWS). The simplest GA is the RWS. A circle is divided into N equal sectors by this operator. The ith sector in the circle presents the ith individual in the population. The wideness of this sector is equal to the selection probability of the same individual. In this case, the total widths of intervals in the circle are equal to one. The circle is turned N times at the selection step. The individual the Roulette Wheel indicates at each turning is copied to the new population [2]. 2.4.1.2. Sequential selection operator. In this selection operator individuals in the population are arranged in order from the best to the worst according to the fitness values. A copying number is assigned to the individuals with a decreasing function. The most common assignment function used is linear. The copying numbers assigned by this function are used to form a new population. 2.4.1.3. Tournament Selection operator (TS). This operator uses a Roulette selection N times to produce a tournament subset of chromosomes. Then the best chromosome in this subset is chosen as the selected chromosome. The individual having the best fitness value is copied to the new population. This process is repeated till the size reaches to the population size. In general, group size is taken as two, however, it is possible to increase this number [2]. 2.4.1.4. Dominant Selection operator (DSO). This selection operator includes the RWS, TS, top percent, and best selection. The use of a selection operator cannot produce a new schema, but can eliminate some less useful ones. This is the main reason that premature convergence occurs. Low fitness individuals tend to die before they are sufficiently mature to pass on their genes. The DSO strengthens the action of dominant individuals and weakens the effect of the inferior individuals in the process of evolution [6]. 2.4.1.5. Hybrid selection operator. The hybrid selection operator consists of a combination of the RWS and TS. In this study there were two types of hybrid selection operators; single level and two level hybrid selection operators. In the single level hybrid selection operator, for 50% of the population size the TS procedure is used and the RWS procedure is used in the remaining 50%. The two level hybrid selection operator uses TS for 25% of the population size followed the use of RWS for a further 25% and then repeats the process using the TS and for the final 25% using the RWS [11]. 2.4.1.6. Kin selection operator. The kin selection operator is controlled by a kin probability variable, which serves the same purpose as the crossover probability or mutation probability variables in the PGA. The chromosome to be sacrificed is selected randomly from the solutions pool. It is replaced by a copy of another chromosome having the least non-zero Hamiltonian distance from the original chromosome. In case of more than one such chromosome, the one with lower fitness values is selected. In case of more than one chromosome having the same lowest value, any one is selected randomly [12]. 2.4.2. Developed back controlled selection operator (BCSO) The selection process in existing selection operators is performed among individuals composing the population in the same generation. In these selection operators, the selection is carried out according to the fitness values of each individual. The fitness value of the individual in the existing generation is compared with fitness value in same generation. In the BSCO differs from existing selection operators because the fitness value of the individual is compared with the fitness value in previous generation which. If the fitness value of the individual is more than the one in the preceding generation, this individual would keep own position. Otherwise, if the fitness value of the same individual is less than or equal to the fitness value in the preceding generation, this individual would be discarded from the population. Then individual would be copied to the population in the preceding generation to replace this individual. 2.5. Crossover operators A GA can rapidly identify discrete zones within a large search space area to concentrate the search for an optimum solution. This operator changes the mutually defined parts of two members that are selected and obtains different members that give new points in the search space. The procedure for the one point crossover operator used in the current study is summarized here. The crossover point in the one-point crossover operator is randomly selected between 1 and L  1 where L is the length of the chromosome. Two new members are obtained by relocating parts, after this the cut-off point is matched in two members. The codes after 9th site of members given in Table 1 have been changed using the one-point crossover operator. 2.6. Mutation All genes form chromosomes that may be the same in the following generations. It is not possible to change a chromosome like this with crossover operator. In this case, the gene selected randomly from the chromosome undergoes a change because of the intervening chromosomes that form the population from the outside in a defined ratio [2].

7672

M. Kaya / Applied Mathematics and Computation 217 (2011) 7669–7678

Table 1 One point crossover sample. Parent 1 5 3 Parent 2 2 7 Child 1 5 3 Child 2 2 7

6

8

1

3

2

7

4

3

2

6

1

8

2

6

3

5

4

3

6

8

17

3

2

7

4

6

8

2

1

5

4

2

6

8

1

3

2

7

4

7

4

6

8

2

1

5

4

2

4

3

6

8

1

3

2

3

2

6

1

8

2

6

3

5

3. Applications 3.1. Benchmark problem In the first stage of this study, the six existing selection operators (RWS sequential, TS, DSO, hybrid-I and hybrid-II) and proposed BCSO were applied to the traveling salesman and space truss problems. In the second stage, the BCSO, RWS, DSO and the hybrid II selection operators were applied to a pre-stressed precast beam problem. In all programs, a permutation coding type, one point crossover operator (Table 1) and the 1% mutation ratio were applied (Table 2); therefore, the different results obtained from these analyses were a consequence of the different selection operators that had been used. 3.1.1. Travelling Salesman Problem (TSP) The Travelling Salesman Problem (TSP) is an integer program that gives the shortest route when the distance between every pair of points is known and each point is visited only once. It is one of the few integer algorithm based programs described in the literature [15–17]. The TSP is also a way of collecting and distributing the necessary parts of the objects in the modelling system, thus maximizing profit while minimizing cost over the shortest way. In this problem, a salesman wants to sell goods in n cities and he wants to visit all the cities by the shortest route. Hence, salesman has a choice for n cities of 1 ðn  1Þ! different routes. The salesman must travel to 32 cities, visiting each city only once, beginning by travelling from 2 city A to city B. This means there are 8.05  1032 different routes for the 32 cities (Fig. 2). For the TSP the objective function W(x) is expressed in Eq. 1

min WðxÞ ¼

nk X

Lk

ð1Þ

k¼1

where Lk is length of the distance between two cities, The constrained objective function was transformed to an unconstrained objective function /(s) as shown in Eq. 7, i: first city number j: second city number xi and yi: first city coordinates xj and yj: second city coordinates When the salesman goes the target city from the starting city then;

If xi < xj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ffi gðxÞ ¼ ðxi  xj Þ2 þ ðyi  yj Þ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ffi ðxi  xj Þ2 þ ðyi  yj Þ2

Else gðxÞ ¼

ð2Þ

x100

ð3Þ

Table 2 Bireyin mutasyona ug˘ratılması. Mutasyondan önce 1 7 2

8

1

5

3

6

4

8

6

3

4

7

2

1

5

4

2

Mutasyondan sonra 1 7 2

8

6

5

3

6

4

8

6

3

4

7

2

1

5

4

2

7673

M. Kaya / Applied Mathematics and Computation 217 (2011) 7669–7678

Fig. 2. Optimum solution for TSP by using GA.

0.8

0.74

Fitness value after 30.000 generations 0.68

0.7 0.6

0.57

0.6 Fitness value

0.51 0.5 0.4

0.41

0.38

0.3 0.2

0.1 0 Roulette Whell

Sequental Tournament Dominant

Hybrid-1

Hybrid-2

Back Controlled

Selection operators Fig. 3. The fitness values obtained after 30,000 generation for the traveling salesman problem.

When the salesman come back from the target city to the starting city then;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðxÞ ¼ ððxi  xj Þ2 þ ðyi  yj Þ2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Else gðxÞ ¼ ððxi  xj Þ2 þ ðyi  yj Þ2 Þ x100 X G¼ gðxÞ

If xi > xj

/ðsÞ ¼ WðxÞð1 þ KGÞ

ð4Þ ð5Þ ð6Þ ð7Þ

K: a coefficient selected for the problem taken to be 10 in this study. g(x): a negligence coefficient and calculated as follows; In the first transformation, the constrained objective function was transformed to an unconstrained objective function / (x) as expressed in Eq. 8:

/ðxÞ ¼

X

/ðsÞ=/max

ð8Þ

In the second transformation in Eq. 9 the unconstrained objective function /(x) was converted to the fitness function F(s).

FðsÞ ¼ /max  /ðxÞ

ð9Þ

The existing selection operators and the proposed BSCO were used in analyses to determine the shortest route. The fitness values obtained from 30,000 generations performed in the analyses were compared. The lowest and the highest fitness values were obtained from the analysis using the RWS and BCSO, respectively. The fitness value obtained from the analysis using the BCSO selection operator is 95% higher than the fitness value obtained from the RWS (Fig. 3).

7674

M. Kaya / Applied Mathematics and Computation 217 (2011) 7669–7678

Table 3 The result of runs obtained with different selection operators for traveling salesman problem (TSP). Roulette Whell Total way (km) Run 1 2944 Run 2 2907 Run 3 2818 Run 4 2865 Run 5 2870 Run 6 2886 Run 7 2827 Run 8 2923 Run 9 2899 Run 10 2918 Statistics Best 2818

Sequential

Tournament

Dominant

Hybrid-1

Hybrid-2

Controlled

2774 2863 2902 2821 2826 2842 2785 2801 2780 2897

2844 2881 2920 2839 2792 2860 2915 2837 2883 2805

2847 2759 2886 2805 2811 2826 2763 2878 2765 2781

2863 2899 2938 2857 2811 2878 2911 2832 2923 2927

2813 2855 2894 2766 2818 2834 2813 2889 2769 2778

2787 2808 2847 2766 2772 2720 2739 2817 2861 2780

2774

2792

2759

2811

2766

2720

z

1

m

0m

300

2

3 6

4

m

0m

300

5

y 7

10

8

400

0m

m

0m

m 9

400

x Fig. 4. Optimum solution for space truss by using GA.

The longest and shortest routes were obtained from the RWS and the BCSO, respectively. The route determined by using RWS 4% longer than the one using BCSO (Table 3). 3.1.2. Space truss problem In this study the six existing selection operators and the proposed BCSO were applied to the problem of determining the minimum weight of the space truss (Fig. 4). In this problem the chromosome length is 25 and this number is equal to a design variable group number. The selection operators were used to achieve the optimum design of the space truss in terms of the weight. In the space truss problem, the bar sections were selected from profiles given in the American Institute of Steel Construction [18]. In the tested space truss, the yield strength was taken to be 240 N/mm2. The elasticity module of steel was taken to be 2.105 N/mm2. A 100 kN force was applied to the truss from two nodal points (point 1 and point 2) in a vertical direction. The truss was restricted to a maximum L/300 horizontal drift. The codes used in the application vary from 1 to123 and the size of the designed space was 12325. For space truss; the objective function W(x) and the constrained objective function is expressed in Eq. 10, and the penalty function is given in (14).

min WðxÞ ¼

Tk X

Ak :Lk :qk

ð10Þ

k¼1

where Ak is the cross section area, qk the specific gravity of space truss elements, Lk is the length of the space truss elements, and Tk is the number of space truss elements.

7675

M. Kaya / Applied Mathematics and Computation 217 (2011) 7669–7678

Since steel structures have stability problems, the state of the stress caused by combined loading is considered to include buckling and lateral buckling. The normalized forms of the constraints are as follows: Combined stress constraints;

g i ðxÞ ¼

reb  C m rb þ 1 rbem 1  reb rB

ð11Þ

re

g i ðxÞ ¼

reb 0:6ra

þ

rb 1 rB

ð12Þ

Eqs. (11)–(14) are required for members subjected to bending moment and axial compression.

reb 6 0:15 rbem reb rb þ 1 g i ðxÞ ¼ rbem rB

If

ð13Þ ð14Þ

reb: compression stress only subjected to axial force, rb: compression stress only subjected to bending moment, rbem: allowable stress only subjected to axial force, rB: allowable compression stress in the 1st member subjected only to bending moment, re: Euler stress divided by the safety coefficient, ra: yield stress of steel nm: number of members. Cm the adjustment coefficient and equal to 0.85 for the lateral displacement of members.

C¼

X

ci

ð15Þ

If g i ðxÞ > 0 ci ¼ g i ðxÞ

ð16Þ

If g i ðxÞ 6 0 ci ¼ 0

ð17Þ

ci: negligence coefficient and calculated as follows; K: a coefficient selected for the problem taken to be 10 in this study. In the first transformation, the constrained objective function / (s) as expressed in Eq. 18 was transformed to an unconstrained objective function /(x) as given in (19)

/ðsÞ ¼ WðxÞð1 þ KCÞ X /ðxÞ ¼ /ðsÞ=/ðsÞmax

ð18Þ ð19Þ

In the second transformation, the unconstrained objective function /(x) was converted to an F(s) fitness function in (20)

FðsÞ ¼ /ðxÞmax  /ðxÞ

ð20Þ

0.8 Fitness value after 30,000 generations

0.72

0.7

Fitness value

0.5

0.64

0.62

0.61 0.6

0.54

0.47 0.41

0.4 0.3 0.2

0.1 0 Roulette Whell

Sequental Tournament Dominant

Hybrid-1

Hybrid-2

Back Controlled

Selection operators Fig. 5. The fitness values obtained after 30,000 generation for the space truss problem.

7676

M. Kaya / Applied Mathematics and Computation 217 (2011) 7669–7678

Table 4 The result of runs obtained with different selection operators for space truss. Roulette Whell Total weight (kN) Run 1 10.63 Run 2 10.31 Run 3 10.59 Run 4 10.74 Run 5 10.51 Run 6 10.54 Run 7 10.47 Run 8 10.81 Run 9 10.63 Run 10 11.89 Statistics Best 10.31

Sequential

Tournament

Dominant

Hybrid-1

Hybrid-2

Back controlled

10.52 10.62 10.48 10.13 10.33 10.33 11.17 10.79 10.89 11.05

10.18 9.98 10.13 10.23 9.93 9.74 10.37 10.21 9.81 11.20

9.69 10.08 10.03 10.18 9.89 9.84 9.93 10.97 11.45 11.21

9.64 10.88 9.59 9.74 9.39 9.25 10.73 9.99 10.64 10.67

9.54 10.59 9.49 9.10 9.34 9.30 9.31 9.89 9.67 10.12

9.39 8.95 9.34 9.44 9.20 9.20 10.47 9.61 10.27 9.31

10.13

9.74

9.10

9.25

9.69

8.95

The seven selection operators were used determine the minimum truss beam weight using a GA. The fitness values obtained from 30,000 generations performed in the analyses were compared. The lowest and the highest fitness values were obtained from the use of the Hybrid-1 selection operator and BCSO, respectively. The fitness value from the use of the Hybrid-1 selection operator is 76% higher than the value obtained from the use of the BCSO (Fig. 5). The maximum and minimum truss beam weights were obtained from the analyses using dominant selection operator and BCSO, respectively. The weight determined by BCSO is 2% less than the weight determined by DSO (Table 4). 3.2. Pre-stressed precast beam problem In this stage the section, which has an 18.3 m span and was selected from different 162 cross-sections that are up 54.86 m length, 2.41 m high at 9 different high, 3 different flange width and 3 different top and bottom flanges, was determined by the PCEF [19]. H30S24 heavy vehicle load was applied to the beam [20]. Concrete with a compressive strength of fck = 50 N/ mm2 was used in the design and normal structure steel (fyk = 420 N/mm2) was used in the beam. A 270 K pre-stressing strand with a diameter of 12.70 mm and having low relaxation was used. Four selection operators; BCSO, RWS, DSO and the hybrid II selection operators were applied to the lowest cost of precast pre-stressed beam problem. The most important parameter that affects the cost of the precast pre-stressing beam is the weight of the pre-stressing strand in the beam, the second parameter being the weight of the concrete. The fitness function proposed to find the minimum cost in the beam is given in Eq. 21. In this equation, F1, F2 are the unit prices of strand and concrete; cs and cc are the unit weights of the strand and concrete and L is the length of beam [15]. In this problem the design variable groups were depth, beam flange thickness, top and bottom flange and number of strands. For the precast pre-stressing beam, the objective function f(x) is given in Eq. 21, the constrained objective function /(s) is shown in (27)

X

min f ðxÞ ¼ F1  cs  L  Ap þ F 2  cc L  A a 0 z ¼ h   d 2 1; 15  M u fp ¼ Apstrand  n If fp 6 0; 7  fpu gðxÞ ¼ F1  cs  L  Ap þ F 2  cc L  A

ð21Þ ð22Þ ð23Þ ð24Þ

Else fp > 0; 7  fpu gðxÞ ¼ F1  cs  L  Ap þ F 2  cc L  A  100 X G¼ gðxÞ

ð25Þ

/ðsÞ ¼ f ðxÞð1 þ K:GÞ

ð27Þ

ð26Þ

K: a coefficient selected for the problem taken to be 10 in this study. g(x): a penalty coefficient and calculated with Eq. 9; In the first transformation, the constrained objective function /(s) was transformed to an unconstrained objective function /(x) as given in Eq. 28,

/ðxÞ ¼

X

/ðsÞ=/ðsÞmax

ð28Þ

In the second transformation in Eq. 29, the unconstrained objective function /(x) was converted to the fitness functionF(s). (Fig. 6)

FðsÞ ¼ /ðxÞmax  /ðxÞ

ð29Þ

7677

M. Kaya / Applied Mathematics and Computation 217 (2011) 7669–7678

Fig. 6. PCEF 45 deep bulb tee standard section [19].

0.9

Fitness value after 30.000 generations

0.79

0.8

0.69

0.7

Fitness value

0.6 0.5 0.4

0.34

0.31

0.3 0.2 0.1 0

Roulette Whell

Dominant

Hybrid-2

Back Controlled

Selection operators Fig. 7. The fitness values obtained after 30,000 generation for the prestressed precast beam problem. Table 5 The result of runs obtained with different selection operators for prestressed precast concrete beam.

Run 1 Run 2 Run 3 Run 4 Run 5 Run 6 Run 7 Run 8 Run 9 Run 10 Statistics Best

Roulette Whell

Dominant

Hybrid-2

Back controlled

1042 1010 1038 1053 1030 1033 1026 1059 1042 1165

987 1103 1024 1063 967 1091 1061 1017 976 1051

935 1038 930 911 915 911 912 969 948 992

920 877 915 925 892 897 1026 942 1006 912

1026

1103

911

892

Concerning the fitness value criterion of the pre-stressed precast beam problem that terminated after 30,000 generation, the maximum fitness value was obtained from the BCSO (Fig. 7). The RWS gave the lowest fitness value among four selection operators. The fitness value obtained by the RWS was 57% lower than the fitness value obtained from the BCSO (Fig. 7).

7678

M. Kaya / Applied Mathematics and Computation 217 (2011) 7669–7678

The BCSO gives the lowest cost of the beam, the RWS gives the highest cost among four selection operators. The cost of the beam obtained using RWS was 15% higher than the cost of beam found by the BCSO (Table 5). The cost of beam determined BCSO was 9% higher than cost of beam determined by the PCEF. 4. Conclusions On completion of the analysis of travelling salesman, space truss and pre-stressed precast beam problems it was determined that the proposed BCSO yields the highest fitness value compared with the other six selection operators. The BCSO produced successful results in the determination of the shortest travel distance in the traveling salesman problem and gave the minimum weight of steel truss beam, the minimum cost in the pre-stressed precast concrete beam problems. However, the cost of the pre-stressed precast beam specified for this span was a little higher than the cost stated by the PCEF. It should be noted that the analyses using the BCSO were completed over a longer period than the analyses using the six existing selection operators. It can be said that, the BCSO facilitated a more extensive search of the design space than existing selection operators. Furthermore, the BCSO give better results in solving problems than existing selection operators that were used in this research. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

J.H. Holland, Adaptation in natural and artificial systems, University of Michigan Press, Ann Arbor, 1975. D.E. Goldberg, Genetic algorithms in search optimization and machine learning, Addison-Wesley, Reading, MA, 1989. WM. Jenkins, Towards structural optimisation via the genetic algorithm, Computers and Structures 40 (5) (1991) 1321–1327. WM. Jenkins, Structural optimisation with the genetic algorithm, Computers and Structures 69 (24) (1991) 418–422. WM. Jenkins, Plane frame optimum design environment based on genetic algorithm, Journal of Structural Engineering ASCE 118 (11) (1992) 3103– 3112. S. Rajeev, C.S. Krishnamoorthy, Discrete optimization of structures using genetic algorithms, Journal of Structural Engineering ASCE 118 (5) (1992) 1233–1250. L. Tang, J. Liu, A. Rong, Z. Yang, Modelling and a genetic algorithm solution for the slab stack shuffling problem when implementing steel rolling schedules, International Journal of Production Research 40 (7) (2002) 1583–1595. K. Katayama, H. Hlrabayashı, H. Narıhısa, Analysis of crossovers and selections in a coarse-grained parallel genetic algorithm, Mathematical and Computer Modeling 38 (2003) 1275–1282. Z. Jihua, Z. Jian, D. Haifeng, W. Su’an, Self-organizing genetic algorithm based tuning of PID controllers, Information Sciences 179 (2009) 1007–1018. H. Stern, Y. Chassidim, M. Zofi, Multi-agent visual area coverage using a new genetic algorithm selection scheme, European Journal of Operation Research 175 (2006) 1890–1907. E.B. James, Adaptive selection operators for genetic algorithms, in: J.J. Grefenstette (Ed.), Proceedings of the First International Conference on Genetic Algorithms and their Applications, Lawrence Erlbaum Associates, Hillsdale, NJ, 1985, pp. 101–111. B.L. Miller, D.E. Goldberg, Genetic algorithms, tournament selection and the effects of noise, Complex Systems (1995) 193–212. S. Arumugam, MVC. Rao, R. Palaniappan, New hybrid genetic operators for real coded genetic algorithm to compute optimal control of a class of hybrid systems, Applied Soft Computing (2005) 38–52. K. Ashutosh, S. Srinivas, MK. Tiwari, Steel rolling schedules and its solution using improved parallel genetic algorithms, International Journal of Production Economics 91 (2004) 135–147. L. Bianchi, J. Knowles, N. Bowler, Local search for the probabilistic traveling salesman problem Correction to the 2-p-opt and 1-shift algorithms, European Journal of Operational Research 162 (1) (2005) 206–219. O. Kwon, B. Golden, E. Wasil, Estimating the length of the optimal TSP tour an empirical study using regression and neural networks, Computers & Operations Research 22 (10) (1995) 1039–1046. H. Gzyl, R. Jimenez, The physicist’s approach to the travelling salesman problem, Mathematical and Computer Modelling 12 (6) (1989) 667–670. American Institute of Steel Construction Manual of steel construction allowable stress design, (AISC), 1989. last accessed 14/02/2011. Mid- Atlantic states pre-stressed concrete committee for economic fabrication (PCEF meeting, Buckeystown, MD, 2006. last accessed 14/02/2011.