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The effects of two new crossover operators on genetic algorithm performance
Applied Soft Computing 11 (2011) 881–890
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
The effects of two new crossover operators on genetic algorithm performance Mustafa Kaya ∗ Aksaray University, Faculty of Engineering, Adana Street, Aksaray, Turkey
a r t i c l e
i n f o
Article history: Received 13 November 2008 Received in revised form 14 January 2010 Accepted 17 January 2010 Available online 25 January 2010 Keywords: Genetic algorithm Crossover operators
a b s t r a c t In this study, two new crossover operators in genetic algorithm are proposed; sequential and random mixed crossover. In the first stage, existing and developed crossover operators were applied to two benchmark problems, namely the reinforced concrete beam problem and the space truss problem. In the second stage, the developed crossover operators were applied to a deep beam problem and, a concrete mix design problem.The fittest values obtained using developed crossover operators were higher than those were obtained with existing crossover operator after 30,000 generations. Moreover, in developed crossover operators, the random mixed crossover yields higher fitness values than the existing crossover operators. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Genetic algorithms (GA) came into existence with the adaptation of developed biological processes to the computer environment. They use units stored in the computer’s memory in the same way as those in natural populations. The initial population is used for the solution of optimization problems with GA. The risk of capturing local optimum traps was less for GA in comparison with traditional optimization methods. The use of the unconstrained objective functions enabled the discovery of new combinations that have higher fitness values. GA use the best objective and fitness function values for problems in which design variables are complex and discontinuous furthermore, does not require derivatives of the objective function. The first study related to GA was the introduction of the basic components by Holland [1] in 1975 entitled “Machine Learning”. Later, a study on gas pipes, by Holland’s student, Goldberg, proved that GA had practical uses. GA studies in engineering are generally the optimizations of topology, shape, and dimension [2]. Application studies of GA to optimization problems have been undertaken Jenkins [3–5] and Rajeev and Krishnamoorty [6] in which the effect of crossover operators on the behavior of GA was investigated. The crossover operator is as important as coding, selection, and mutation in GA. There are various existing types of crossover including; one-point, two-point, uniform, variable-to-variable, multi-point, mixed and direct design variable exchange. Adeli and Cheng [7–9] applied the dimension
∗ Tel.: +90 382 2150953; fax: +90 382 2150592. E-mail address: [email protected]. 1568-4946/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2010.01.008
optimization problems to three truss beams to numerically compare one-point, two-point, and uniform crossover operators. The best results were obtained from the two-point crossover operator. Wu and Chow [10] compared the one-point, two-point, three-point, and four-point crossover operators and showed that two-point, three-point, and four-point crossover operators are better than the one-point crossover. Jenkins [11] argues in favor of multi-point crossover operator in term of fast progress becomes very slow in case single-point crossover is used. Using one-point crossover, Dejong and Spears [12] introduced the relationship between crossover operators and population size. They state that two-point crossover is performs better in the problems in which the population is large, but uniform crossover is better for the small size populations. Syswerda [13] showed that the uniform crossover operator is more efficient when compared with two-point crossover. Erbatur and Hasanc¸ebi [14,15] suggested combining two crossover operators in their study about the effects of crossover operators on the behavior of GA. The mixed crossover operator was also applied to the population in which crossover operators such as one-point, two-point, and threepoint determined definite rates of generation numbers. Another technique has been suggested that is a direct design variable exchange crossover operator. Each of the design variables was changed to a probability function, which was formed empirically. The mixed crossover operator first suggested performed fairly well in the study. The second technique direct design variable exchange crossover gave the best result among existing crossover operators [14,15]. In this study, the effect of sequential crossover and random mixed crossover operators on the behavior of GA was investigated using a reinforced concrete (RC) beam problem, space truss problem, RC deep beam problem, and concrete mix design problem.
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2. Stages of the study
2.2. Formation of initial population
In the first stage of this study, existing and developed crossover operators were applied to the RC beam and, space truss problems. Analysis of the benchmark problems was made for each of nine crossover operators. The programs in analyses were terminated after 30,000 generations. In the second stage, only developed crossover operators applied to a deep beam and the concrete mix design problem analyzed. Maximum fitness values obtained from application of developed crossover operators were determined after 30,000 generations. The determination steps for the minimum cost on the RC beam, minimum weight of the space truss, the minimum reinforcement diameters in RC deep beam and the minimum cement weight in concrete mix design using GA were summarized schematically in Fig. 1. This figure consists of the following steps:
The GA searches inside the population consisting of points instead of searching point-to-point [2]. 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 chosen randomly. The suitable selection of population size significantly affects the performance of genetic algorithm. In various studies it was observed that the result was reached earlier in larger populations in comparison with small populations [2]. In this study, the population size (N) was selected as 100.
1. Construction of the initial population randomly which comprises numbers. 2. Decoding the permutation coding for the design variables of each member and finding their sequence numbers in the available variable list. 3. Calculating the value of unconstrained function (x) for each member using Eqs. (11), (22), (30) and (37). Finding the maximum and minimum values of this function in the population. 4. Calculation of the fitness value for each member using Eqs. (12), (23), (31) and (38). 5. Application of the sequential selection method. Copying the members into the mating pool according to their fitness, and couple them randomly. 6. Generation of child using crossover operators and thus obtaining the new population. Seven existing and two new crossover operators were applied. 7. Application of the mutation to each offspring in the new population with a specific probability. 8. Replacing the initial population by the new population and repeat steps 3–8 until the termination criteria is obtained. 2.1. Coding In this study, permutation coding was used because of the design variables consists of more than one variable group. The chromosome length is equal to the number of variable groups in this type of coding.
2.3. Evaluation The GA basically finds the maximum of an unconstrained objective function. To solve a constrained objective minimization function, two transformations need to be made. 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 [13]. In this study, for the four problems, different constrained objective functions were occurred. These functions are given in Eqs. (10), (21), (29) and (34). After these equations were occurred, this functions transformed unconstrained objective functions. This unconstrained objective functions are given in Eqs. (11), (22), (30) and (37). Finally, this unconstrained objective functions is transformed into the fitness functions, as given in Eqs. (12), (23), (31) and (38). 2.4. Selection The members of the new population in each generation are selected by a process from members of the existing population. The selection technique performs natural selection artificially and in this study, a sequential selection method was used. In this method, members are set in order by a linearly decreasing function. The members with the low fitness value are removed from population in a defined ratio and members with the high fitness values replace those removed in the same ratio. In this study, members with the lowest fitness values constituting 25% of population were eliminated and replaced with the highest valued members in the same ratio. 2.5. Crossover operators GA can rapidly identify discrete zones within a large search space area to concentrate the search for an optimum solution. This technique changes mutually defined parts of two members selected and obtains different members that give new points in the search space. The crossover operators used in the current study are summarized below.
Fig. 1. Structure of genetic algorithm.
2.5.1. Existing crossover operators 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 were 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 were changed using the one-point crossover operator. In the two-point crossover operators, two different cut-off points were randomly selected between 1 and L − 1. New members were obtained by relocating zones between the cut-off points of paired members. The codes between 7th and 13th sites of members given in Table 2 were changed.
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Table 1 One-point crossover sample. 5 Parent1
3
6
8
1
3
2
7
4
3
2
6
1
8
2
6
3
5
2 Parent2
7
4
3
6
8
1
3
2
7
4
6
8
2
1
5
4
2
5 Child1
3
6
8
1
3
2
7
4
7
4
6
8
2
1
5
4
2
2 Child2
7
4
3
6
8
1
3
2
3
2
6
1
8
2
6
3
5
Table 2 Two-point crossover sample. 5 Parent1
3
6
8
1
3
2
7
4
3
2
6
1
8
2
6
3
5
2 Parent2
7
4
3
6
8
1
3
2
7
4
6
8
2
1
5
4
2
5 Child1
3
6
8
1
3
1
3
2
7
4
6
1
8
2
6
3
5
2 Child2
7
4
3
6
8
2
7
4
3
2
6
8
2
1
5
4
2
In multi-point crossover operators, many cut-off points are selected between 1 and L − 1. New members were obtained by relocating zones between these cut-off points of pairing members and the codes of the members were changed in four sites as shown in Table 3. In variable-to-variable crossover operators, first the paired members (strings) are decomposed into their substrings. Next, a single-point crossover is separately carried out on all the substrings (Table 4.). In this way, each design variable amongst the members is activated to separately accomplish the design exchange [14]. In uniform crossover operators, as a first stage, a random member must be temporarily formed in the binary string. The chromosome length of this member is equal to chromosome lengths of the other members in the population. If the code of the nth site of the chromosome of the temporary member is 0, in this case, the code of 1st old member is used in the nth site of the 1st new member and the code of 2nd old member is used in the nth site of the 2nd new member. Equally, if the code of the nth site of the temporary member’s chromosome is 1, in this case, the code of the 2nd old member is used in the nth site of the 1st new member and the code of the 1st old member is used in the nth site of the 2nd new member. The application of the uniform crossover operator to the two members selected from the population is given in Table 5 [13]. The mixed crossover operator introduced by Hasancebi et al. [14], uses a combination of single-point, two-point and three-point crossovers for selected proportions of generations for a fixed generation number. The idea behind the technique is to achieve an efficient search, which can be obtained by activating the positive characteristics of the existing techniques in the right appropriate place in the search process [14].
In the direct design variable exchange crossover operator, each design variable (substring) is directly and separately exchanged between paired members according to a probability function. This function is empirically created, and is designed to achieve an effective search during successive generations [14]. 2.5.2. The new crossover operators 2.5.2.1. Sequential crossover. In this new crossover operator, before beginning the process the existing crossover operators were numbered 1–7 (Table 6). Then, the existing crossover operators were applied sequentially to the chromosome couples in the population at in the same generation. 2.5.2.2. Random mixed crossover. In this operator, the existing seven crossover operators were numbered 1–7 as shown in Table 6 were applied randomly to the chromosome couples in the population at the same generation. 2.6. Mutation operator The mutation operator is critical to the success of genetic algorithms since it determines the search directions and avoids convergence to local optima. In this operator, 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]. 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 the chromosomes, but the 1% mutation ratio gave a better result than the 0.1% and 0.5% mutation ratios. As a result the 1% mutation ratio was applied on the chromosomes of all the problems.
Table 3 Multi-point crossover sample. 5 Parent1
3
6
8
1
3
2
7
4
3
2
6
1
8
2
6
3
5
2 Parent2
7
4
3
6
8
1
3
2
7
4
6
8
2
1
5
4
2
5 Child1
3
4
3
1
3
1
3
4
3
4
6
1
8
1
5
3
5
2 Child2
7
6
8
6
8
2
7
2
7
2
6
8
2
2
6
4
2
884
M. Kaya / Applied Soft Computing 11 (2011) 881–890
Table 4 Variable-to-variable crossover sample. 5 Parent 1
3
6
8
1
3
2
7
4
3
2
6
1
8
2
6
3
5
2 Parent2
7
4
3
6
8
1
3
2
7
4
6
8
2
1
5
4
2
5 Child 1
7
6
3
1
8
2
3
4
7
2
6
1
2
2
5
3
5
2 Child2
3
4
8
6
3
1
7
2
3
4
6
8
8
1
6
4
5
Table 5 Uniform crossover sample. 0 Temporary string
1
1
0
0
1
1
0
1
1
1
0
1
0
1
0
0
1
5 Parent1
3
6
8
1
3
2
7
4
3
2
6
1
8
2
6
3
5
2 Parent2
7
4
3
6
8
1
3
2
7
4
6
8
2
1
5
4
2
5 Child1
7
4
8
1
8
1
7
2
7
4
6
8
8
1
6
3
2
2 Child2
3
6
3
6
3
2
3
4
3
2
6
1
2
2
5
4
5
3. Theoretical review of the crossover operators
were a consequence of the different crossover operators that were used.
The two main strategies that a solid crossover operator should use to locate the optimum are exploration and exploitation. The facility for exploration in a crossover operator encourages the search for a rapid and thorough discovery examination of the design space. A robust exploration prevents the search from locating a local peak. However, only using this type of search becomes inefficient when making use of previously obtained points to reach more appropriate points. On the other hand, a solid exploitation employs the previously found points to reach the optimum, however, in this case a slow convergence is observed accompanied by an increasing risk of locating a local peak. In fact, these two strategies are contradictory, and a balanced use of both is vital for achieving an efficient search through the crossover operator [14]. 4. Applications In the first stage of this study, existing and developed crossover operators were applied to the two benchmark problems, namely, the RC beam problem and the space truss problem. In the second stage, the developed crossover operators were applied to a deep beam problem and a concrete mix design problem. All the programs used to compare developed and existing crossover operators were the same except for the crossover operators. In all programs, population size (N = 100), permutation coding type, sequential selection method and the 1% mutation ratio were used; therefore, the different results obtained from these analyses
4.1. Benchmark problems 4.1.1. Reinforced concrete beam problem The design of the RC beam is normally an iterative process, in which the engineer assumes the self-weight of the beam beforehand, and a trial section is chosen. Then, the moment of resistance of the section is determined to check its suitability against the given applied bending moment. The process is repeated until a suitable trial section is found. This procedure often creates difficulty in exactly matching the moment due to the self-weight of the beam, which may be quite substantial in many cases. Therefore, the design process of a beam is not only slow, but also uneconomic, since the only concern is to find any section suitable for the given conditions, instead of looking for the most economic ones [16]. The RC beam section for this problem is given in Fig. 2. For the purposes of this analysis, strength design procedures were adopted because they have, among others, the following advantages [17].
Table 6 Existent crossover types and codes of these which used in sequential crossover and random mixed crossover. Crossover type
Code
One-point crossover Two-point crossover Uniform crossover Multi-point crossover Variable-to-variable crossover Mixed crossover Direct design variable exchange crossover (DDVECT)
1 2 3 4 5 6 7
Fig. 2. Reinforced concrete beam section.
M. Kaya / Applied Soft Computing 11 (2011) 881–890
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Table 7 The result of runs obtained with different crossover techniques for reinforced concrete beam. One-point
Two-point
Cost of reinforced concrete beam ($) Run 1 53,59 52,84 Run 2 54,10 51,89 Run 3 51,64 52,64 Run 4 54,10 53,35 Run 5 52,64 50,89 Run 6 52,64 51,89 Run 7 51,64 52,64 Run 8 54,10 53,35 Run 9 52,64 50,89 Run 10 52,64 51,89 Best
51,64
50,89
Multi-point
Variable-to-variable
Uniform
Mixed crossover
DDVECT
Sequential
Random mixed
51,13 49,89 50,89 48,92 51,38 50,13 50,89 48,92 51,38 50,13
50,64 49,43 50,38 51,13 49,68 48,67 50,38 51,13 49,68 48,67
45,81 48,43 48,17 48,92 47,16 47,16 48,17 48,92 47,16 47,16
47,92 46,72 45,71 48,17 46,92 47,67 45,71 48,17 46,92 47,67
44,95 47,16 46,92 47,43 46,22 46,22 46,92 47,43 46,22 46,22
45,21 44,93 43,76 44,46 42,18 43,49 43,76 44,46 42,18 43,49
42,76 43,95 43,76 42,00 43,00 42,25 43,76 42,00 43,00 42,25
48,92
48,67
45,81
45,71
44,95
42,18
42,00
• Strength design better predicts the strength of a section because of the recognition of the non-linearity of the stress strain curve at high stress level. • Since the dead loads to which a structure is subjected are more certainly determined than the live loads, it is unreasonable to apply the same factor of safety to both. Therefore, this approach allows the use of different safety factors for each type of loading.
For the RC beam, the objective function f(x) is expressed in Eq. (3), the penalty function is given in Eq. (9) and the constrained objective function is shown in Eq. (10):
The basic assumptions that are taken when using strength design are as follows [18].
b =
• 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 from the neutral axis. • The tensile strength of the concrete is neglected in flexural computations. • The ultimate concrete strain is 0.003. Please check all numbers for, or. • The average compressive stress in the concrete is 0,85. fc • The average tensile stress in the reinforcement does not exceed fy . The moment to be carried by the RC beam is given in Eq. (1). In this equation Mu : ultimate moment, Ø: strength reduction factor, : steel ratio, fy : yield stress of steel, fc : compressive stress of concrete; b: weigh of the beam; d: height of the beam [17]:
Mu = (fy bd2 )
1 − 0.59
fy fc
(1)
Using extra reinforcement in the beam causes brittle behavior under loading. The required reinforcement ratio of RC beam is given in Eq. (2): = ((fy bd)(fc − 0.59fy ))fc
(2)
The most important parameter that affected the cost of the determination of the cheapest beam is the weight of reinforcement in the beam, the second parameter being the weight of concrete. Taking this into account, the unit price of the reinforcement increases the cost of the concrete 25 times. The fitness function proposed for finding the minimum cost in the RC beam is given in Eq. (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 length of beam. Since the number of variables representing the RC beam was too large, permutation coding was used where the chromosome length is equal to the group number of design variables. In this problem the design variable groups were beam height, beam width, reinforcement diameters and reinforcement numbers.
min f (x) = bdL(F1 s + F2 c )
ˇ1 = 0.85 − 0.05
fc − 4000 1000
0.85ˇ1 fc 87000 fy (87000 + fy )
Else If G=
<
and 200 fy
(4) (5)
max = 0.75b 200 If ≥ fy
(3)
(6) < max g(x) = bdL(F1 s + F2 c ) and
g(x)
(s) = f (x)(1 + KG)
(7)
≥ max g(x) = bdL(F1 s + F2 c )100 (8) (9) (10)
where K is a coefficient selected for the problem taken to be 10 in this 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) as expressed in Eq. (11): (x) =
(s) (s)max
(11)
In the second transformation in Eq. (12), the unconstrained objective 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 and developed crossover operators. The direct design variable exchange crossover is the lowest cost with the single-point crossover being the most cost of the existing crossover operators. The cost found by single-point crossover is 21% higher than the cost found by direct design variable exchange crossover. The random mixed crossover gives the lowest cost between existing and developed crossover operators. When the random mixed crossover and direct design variable exchange crossover are compared, it can be seen that the cost found by the random mixed crossover operator is 7% less than the cost found by the direct design variable exchange crossover operator (Table 7). Concerning the fitness values criteria of the RC beam problem that terminated after 30,000 generations, the maximum fitness value between existing crossover operators was obtained from the program that employs the direct design variable exchange crossover operator. The maximum fitness value in the existing and developed crossover operators was achieved using the program
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M. Kaya / Applied Soft Computing 11 (2011) 881–890
Fig. 3. The fitness values obtained after 30,000 generation for the reinforced concrete beam.
that employs the random mixed crossover operator. The fitness value obtained by the developed random mixed crossover was 13% higher than the fitness value obtained by the existing direct design variable exchange crossover operator (Fig. 3). 4.1.2. Space truss problem The space truss problem is commonly chosen when solving problems about GA since the design space is medium sized [6,7,9,14]. Also, in this study existing and developed crossover operators were applied to the problem of determining the most appropriate bar sections for the space truss (Fig. 4). In this problem chromosome length is 25 and this number is equal to design variable group number. By using existing and developed crossover operators, the optimum design of the space truss was achieved in terms of weight. In the space truss problem, consisting of 25 bar elements, the bar sections were selected from profiles given in AISC [19]. In the tested space truss, the yield strength was taken to be 2400 kg/cm2 . The elasticity module of steel was taken to be 2 × 106 kg/cm2 . A 100-kN force was applied to the truss from two nodal points in a vertical direction. The truss was restricted to a maximum L/300 horizontal drift. The codes used in the application vary from 1 to 123 and the size of the designed space was 12325 .
For space truss; the objective function W(x) is expressed in Eq. (10), the penalty function is expressed Eq. (14) and the constrained objective function is expressed in Eq. (15): min W (x) =
Tk
Ak Lk k
(13)
k=1
where Ak is the cross-sectional area, k the specific gravity of space truss elements, Lk is the space truss elements length, and Tk is the number of space truss elements. 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 constrains are as follows. Combined stress constrains: gi (x) =
eb Cm b −1 + bem 1 − (eb /e ) B
(14)
gi (x) =
eb + b −1 0.6a B
(15)
Eqs. (14) and (15) are required for members subjected to bending moment and, axial compression. For members subjected bending moment and axial stress are required to providing Eq. (9) is applied: eb ≤ 0.15 bem
If
gi (x) =
(16)
eb + b −1 bem B
(17)
eb : compression stress only subjected to axial force; b : compression stress only subjected to bending moment; bem : allowable stress only subjected to axial force; B : allowable compression stress in the 1st member subjected only to bending moment; e : Euler stress divided by the safety coefficient; a : yield stress of steel; nm: number of members; Cm the adjustment coefficient and equal to 0.85 for the lateral displacement of members: C=
ci
(18)
If gi (x) > 0, If
gi (x) ≤ 0,
ci = gi (x)
(19)
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): Fig. 4. 25-Bar space truss.
(x) =
(s)
(s)max
(22)
M. Kaya / Applied Soft Computing 11 (2011) 881–890
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Table 8 The result of runs obtained with different crossover techniques for space truss. One-point Space truss weight (kN) Run 1 10,67 Run 2 10,28 Run 3 10,62 Run 4 10,77 Run 5 10,46 Run 6 10,48 Run 7 10,54 Run 8 10,41 Run 9 10,70 Run 10 10,32 Best
10,28
Two-point
Multi-point
Variable-to-variable
Uniform
Mixed crossover
DDVECT
Sequential
Random mixed
10,52 10,62 10,48 10,13 10,33 10,33 10,57 10,34 10,41 10,21
10,18 9,98 10,13 10,23 9,93 9,74 9,76 9,91 10,03 10,17
9,69 10,08 10,03 10,18 9,89 9,84 10,11 9,75 9,98 10,07
9,64 9,39 9,59 9,74 9,39 9,25 9,27 9,71 9,55 9,67
9,39 8,95 9,34 9,44 9,20 9,20 9,01 8,37 9,21 9,11
9,54 9,59 9,49 9,10 9,34 9,30 9,51 9,17 9,46 9,33
9,00 8,85 8,71 8,66 8,95 8,66 8,69 9,03 8,78 8,86
8,51 8,56 8,71 8,36 8,40 8,41 8,47 8,69 8,74 8,43
10,13
9,74
9,69
9,25
8,95
9,10
8,66
8,40
Fig. 5. The fitness values obtained after 30,000 generation for the space truss.
In the second transformation, the unconstrained objective function (x) was converted to an F(s) fitness function in Eq. (17): F(s) = (x)max − (x)
(23)
In this problem, the space truss beam was analyzed using existing and developed crossover operators. The mixed crossover operator gives the lowest truss beam weight, but the single-point crossover gives the most beam weight among the existing crossover operators. The random mixed crossover gives the lowest truss beam weight among all the crossover operators. The truss weight found by the mixed crossover is 15% heavier than the truss beam weight found by the random mixed crossover (Table 8). Concerning the fitness values criteria of the space truss problem that terminated after 30,000 generations, the maximum fitness value from the existing crossover operators was obtained from the direct design exchange crossover (DDVECT) operator. The maximum fitness value in the developed crossover operators was achieved using the random mixed crossover operator. The fitness value obtained by the developed random mixed crossover was 15% higher than the fitness value obtained from the existing direct design exchange crossover operator (Fig. 5).
beam in the bending calculation due to its L/h ratio being greater than 5/4, but less than 5 [22]. A deep beam of loaded from the top-edge and of L/h ratio of 4 is given as an example in Fig. 6. This beam is accepted as deep beam in calculations of shear force while it was accepted as normal beam for bending according to the ACI 318 [22]. In this study, the stresses on the deep beam caused by external forces were first determined through the finite elements method. Then, the steel bar diameters given in standards depending on the effects on these sections were determined using GA. The deep beam was tested under compressive and tensioning stress restrictions. In this study, the stresses on the bars caused by external forces were first determined through the finite elements method. Then, the bar sections given in standards depending on the effects on these sections were determined using GA. As a result of this behavior, problems occurred in the analysis of deep beams. The fittest horizontal and vertical reinforcement diameters were determined in a deep beam using the GA.
4.2. Proposed and analyzed problems 4.2.1. Deep beam problem Deep beams used in shear walls, folded roof plates, and silos show different behaviors when compared with normal beams owing to the fact that their depth–span ratio is higher than that of normal beams. Such studies confirm the usual hypothesis that plane sections before bending remain plane after bending, does not hold for deep beams. Significant warping of the cross-sections occurs because of the high shear stresses [20,21]. The deep beam, loaded from its top-edge, is accepted as a deep beam in the shearing calculation, while it is not accepted as deep
Fig. 6. The fittest reinforcement obtained for deep beam.
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For deep beam; the objective function W(x) is expressed in Eq. (18), the penalty function is expressed Eq. (14) and the constrained objective function is expressed in Eq. (21): min W (x) =
Cs
As Ls s
(24)
s=1
where As is the reinforcement area, s the specific gravity of reinforcement, Ls is the reinforcement length and, Cs number of zones in the deep beam. The constrained objective function was transformed to an unconstrained objective function (x) as expressed in Eq. (18):
gi (x) =
s Ls As ns fyd
−1
(25)
If gi (x) ≥ 0,
ci = gi (x)
(26)
If gi (x) < 0,
ci = gi (x) × 100
(27)
C=
Fei
ci
(28)
(s) = W (x)(1 + KC)
(29)
K: a coefficient selected for the problem taken to be 10 in this study and ci : negligence coefficient and calculated as follows: In the first transformation, the constrained objective function (s) was transformed to an unconstrained objective function (x) as expressed in Eq. (22): (x) =
(s) (s)max
(30)
where Fei is the required force met by reinforcements in the x or y direction in the ith zone, fyd is the yield strength of reinforcement, and 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 transformation was achieved using the maximum of the ith element of the unconstrained objective function. The fitness values of the members were calculated according to Eq. (23) as follows: F(s) = (x)max − (x)
(31)
Concerning the fitness values criteria of deep beam problem that terminated after 30,000 generations, the maximum fitness value from the existing crossover operators was obtained from the mixed crossover operator. The maximum fitness value in the developed crossover operators was achieved using the sequential crossover operator. The fitness value obtained by the sequential crossover was 12% higher than the fitness value obtained from the existing mixed crossover operator (Fig. 7). In this problem the deep beam was analyzed using existing and developed operators. The direct design variable exchange crossover
operator gives the smallest reinforcement diameter among the existing crossover operators. The random mixed crossover gives the smallest reinforcement diameter among all crossover operators. The reinforcement diameter found by direct design variable exchange crossover is 7% bigger than the reinforcement diameter found by the random mixed crossover (Table 9). In this example, diameters of horizontal and vertical reinforcement determined for 9 zones by using genetic algorithm are given in Table 10 and Fig. 6. In this figure, horizontal and vertical lines show that the horizontal and vertical shear reinforcement in those zones, respectively. The weights of reinforcement determined using the GA were compared with weight of reinforcement specified in accordance with ACI 318-99. As a result of comparisons, it can be seen that the weight 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 the resulting concrete has adequate strength, proper workability for placing and low cost. In this problem, the minimum amount of cement for 1 m3 high strength concrete (fc = 50 MPa) was determined by the GA and the Bolomey formula [23,24] given in Eq. (32). Then, amounts of cement which are determined using both methods were compared.The amount of cement used in concrete mixtures is directly proportional to the amount of water. Furthermore, the amount of water is inversely proportional to the fineness modulus of the aggregate when the amount of cement is minimum and the fineness modulus is at maximum:
fc =
fcc a
c 2
(32)
w
where c is the cement weight, w is the weight of the water, fc is the compressive strength of concrete, fcc is the compressive strength of cement and a is changes between 4 and 8. The compressive strength of the concrete (fc ), compressive strength (fcc) and a the coefficient of cement are fixed. The fineness modulus of the aggregate (k) will be determined as the minimum amount of cement. In this study, the number of design variable groups is equal to the number of sieves and varies between 1 and 8. The number of variables in all groups varied between 0 and 2100 kg. 0 kg represents that aggregate does not exist and 2100 kg represents that aggregate is fully present in 1 m3 concrete. The objective function W(cement) is expressed in Eq. (33) as: min W (cement) =
n max(k) =
1
fc a(10 − k)
%passed
fcc
or
100
Fig. 7. The fitness values obtained after 30,000 generation for the deep beam.
(33)
M. Kaya / Applied Soft Computing 11 (2011) 881–890
889
Table 9 The result of runs obtained with different crossover techniques for deep beam. One-point Deep beam weight (kN) Run 1 80,85 Run 2 81,59 Run 3 80,47 Run 4 77,87 Run 5 79,36 Run 6 79,36 Run 7 79,41 Run 8 81,40 Run 9 80,136 Run 10 78,16 Best
77,87
Two-point
Multi-point
Variable-to-variable
Uniform
Mixed crossover
DDVECT
Sequential
Random mixed
79,73 78,24 79,36 80,47 78,24 76,75 78,37 7,23 79,34 76,80
77,12 75,63 76,75 77,49 75,26 73,77 76,79 74,47 73,86 77,35
73,40 76,38 76,00 77,12 73,40 74,51 73,30 77,10 76,43 75,41
73,02 71,16 72,65 73,77 71,18 70,04 72,15 71,22 73,65 70,78
72,28 70,79 71,91 72,65 68,92 70,42 72,55 69,97 70,17 72,50
71,16 67,81 70,79 71,53 69,67 69,67 68,71 70,11 68,87 70,70
68,18 69,30 65,94 67,06 67,81 65,57 65,83 66,80 69,13 66,80
64,45 63,34 65,57 63,34 64,83 63,71 66,30 65,34 66,22 64,51
76,75
73,77
73,40
70,04
68,92
67,81
65,57
63,34
Table 10 Codes and reinforcement diameters obtained for deep beam. Zone No Code of horizontal Diameter of horizontal Code of vertical Diameter of vertical
1 4 ∅14 5 ∅16
2 5 ∅18 3 ∅14
3 4 ∅14 5 ∅14
4 4 ∅16 4 ∅14
The constrained objective function was transformed to an unconstrained objective function (s) as given in Eq. (34) and (x) was given in Eq. (37). Where n is the number of sieves:
n
(s) =
1
10 −
%passed
(34)
100 then (s) = 0
(35)
If (s) > 7.17 then (s) = 0
(36)
If (s) < 4.60
(x) =
(s)max (s)
5 6 ∅14 3 ∅14
6 4 ∅16 4 ∅16
7 3 ∅16 3 ∅16
8 3 ∅16 3 ∅14
9 3 ∅16 3 ∅14
Table 12 Concrete mix design. Component name
Weight (N)
Cement (KPC 42.5) Crushed aggregate (5–12 mm) Crushed aggregate (12–15 mm) Sand (0–5 mm) Marble flour Water Additive (Sikament 300)
3900 4800 4800 5300 3700 1500 81
(37)
In the 2nd transformation, the unconstrained objective function (x) was converted to a fitness function F(s). This transformation was achieved using the maximum of the ith element of the unconstrained objective function. The fitness values of the members were calculated according to Eq. (38) as follows: F(s) = (x)max − (x)
(38)
Concerning the fitness values criteria of the concrete mix design problem that terminated after 30,000 generations, the maximum fitness value from the existing crossover operators was obtained from the direct design variable exchange crossover (DDVECT) operator. The maximum fitness value in the developed crossover operators was achieved using the random mixed crossover operator. The fitness value obtained by the developed the random mixed
crossover operator was 8% higher than the fitness value obtained from the existing the direct design variable exchange crossover operator (Fig. 8). The concrete mix design prepared using existing and developed crossover operators. The uniform crossover gives the lightest amount of cement among the existing crossover operators. The sequential crossover operator gives the lightest amount of cement all the crossover operators. The amount of cement found by the sequential crossover operator is 8% smaller than the amount of cement 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 genetic algorithm. This concrete mix design (Table 12) was used to construct experiment members for the study on “post tensioned column-to-beam connections in precast structures” [25].
Table 11 The result of runs obtained with different crossover techniques for concrete mix design. One-point Cement weight (kN) Run 1 4,979 Run 2 5,025 Run 3 4,956 Run 4 4,795 Run 5 4,887 Run 6 4,887 Run 7 4,891 Run 8 5,012 Run 9 4,935 Run 10 4,813 Best
4,795
Two-point
Multi-point
Variable-to-variable
Uniform
Mixed crossover
DDVECT
Sequential
Random mixed
4,910 4,818 4,887 4,956 4,818 4,726 4,825 4,810 4,885 4,730
4,749 4,657 4,726 4,772 4,635 4,543 4,728 4,585 4,548 4,763
4,520 4,704 4,680 4,749 4,520 4,588 4,514 4,747 4,707 4,644
4,382 4,175 4,359 4,405 4,290 4,290 4,231 4,317 4,241 4,354
4,451 4,359 4,428 4,474 4,244 4,336 4,468 4,309 4,321 4,465
4,497 4,382 4,474 4,543 4,383 4,313 4,443 4,386 4,535 4,359
3,968 3,900 4,037 3,900 3,992 3,923 4,083 4,023 4,078 3,972
4,198 4,267 4,060 4,129 4,175 4,037 4,054 4,114 4,257 4,114
4,726
4,543
4,520
4,175
4,244
4,313
3,900
4,037
890
M. Kaya / Applied Soft Computing 11 (2011) 881–890
Fig. 8. The fitness values obtained after 30,000 generation for the concrete mix design.
5. Conclusion
References
In this study, the effect of seven existing crossover operators and two developed crossover operators on the performance of the GA were compared. In the first stage a RC beam and space truss problems were used in the comparison. In the analyses made for these problems, the maximum fitness value was obtained from the analysis of the direct design crossover operator selected from the existing crossover operators for the RC beam problem. The maximum fitness value was obtained from the analysis of the mixed crossover operator from the existing crossover operators used in the space truss beam problem. However, obtained fitness values are approximately equal to each other. When the results obtained from the developed and existing crossover types were compared, the two highest fitness values were obtained from the analysis of the random mixed crossover and sequential crossover operator. In the second stage of the study, using deep beam and concrete mix design problems, the effect of sequential crossover and random mixed crossover operators, developed using deep beam and concrete mix design problems, on the performance of the GA was investigated. The maximum fitness value was obtained by the analysis of the random mixed crossover operator used in deep beam and concrete mix design problems. While the weights of reinforcement determined from the use of the random mixed crossover and sequential crossover operators, were approximately equal to each other in the deep beam problem, the weights of cement determined using same operators were different in the concrete mix design problem. The use of the GA produced successful results in the determination of the lowest cost of the RC beam problem, of the minimum weight of steel truss beam, of the weight of reinforcement which is independent from dimensions of deep beams, and of the weight of the minimum amount cement in the concrete The operators developed for this study (random mixed crossover and sequential), obtaining higher fitness values than the existing crossover operators, shows that developed crossover operators thus GA give better results in solving problems in which design variables are discontinuous. The application of GA to the four different problems which are encountered in civil engineering and the successful results achieved from the analyses of these problems shows that GA could be applied to other/further problems in civil engineering
[1] J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975. [2] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA, 1989. [3] W.M. Jenkins, Towards structural optimization via the genetic algorithm, Computers and Structures 40 (5) (1991) 1321–1327. [4] W.M. Jenkins, Structural optimization with the genetic algorithm, Computers and Structures 69 (24) (1991) 418–422. [5] W.M. Jenkins, Plane frame optimum design environment based on genetic algorithm, Journal of Structural Engineering ASCE 118 (11) (1992) 3103–3112. [6] S. Rajeev, C.S. Krishnamoorty, Discrete optimization of structures using genetic algorithms, Journal of Structural Engineering ASCE 118 (5) (1992) 1233–1250. [7] H. Adeli, N.T. Cheng, Integrated genetic algorithm for optimization of space structures, Journal of Aerospace Engineering ASCE 6 (4) (1993) 315–328. [8] H. Adeli, N.T. Cheng, Concurrent genetic algorithms for optimization of large structures, Journal of Aerospace Engineering ASCE 7 (3) (1994) 276–296. [9] H. Adeli, N.T. Cheng, Augmented Lagrangian genetic algorithm for structural optimization, Journal of Aerospace Engineering ASCE 7 (1) (1994) 104–118. [10] S.J. Wu, P.T. Chow, Steady-state genetic algorithm for discrete optimization of trusses, Computers and Structures 56 (6) (1995) 979–991. [11] W.M. Jenkins, On the application of natural algorithms to structural design optimization, Engineering Structure 19 (4) (1997) 302–308. [12] K. Dejong, W.M. Spears, An analysis of the interacting roles of population sizes and crossover in genetic function optimization, in: H.P. Schwefel, R. Manner (Eds.), Proceedings of Parallel Problem Solving from Nature, Springer, Berlin, 1990, pp. 38–47. [13] G. Syswerda, Uniform crossover in genetic algorithms, in: J.D. Schaffer, M. Kaufman (Eds.), Proceedings of the Third International Conference on Genetic Algorithms, 1989, pp. 2–9. [14] O. Hasanc¸ebi, F. Erbatur, Evaluation of crossover operators in genetic algorithms based optimum structural design, Computers and Structures 1999 78 (2000) 435–448. [15] D. Zaharie, Influence of crossover on the behavior of differential evolution algorithms, Applied Soft Computing 9 (June (3)) (2009) 1126–1138. [16] C.C. Coello, S.F. Hernandez, F.A. Farrera, Optimal design of reinforced concrete beams using genetic algorithms, Expert Systems with Applications Algorithms 12 (1) (1997) 101–108. [17] N.J. Everard, J.L. Tanner, Theory and Problems of Reinforced Concrete Design, 2nd ed., McGraw-Hill, New York, 1993. [18] ACI 318-99, American Concrete Institute, 1999. [19] Manual of Steel Construction Allowable Stress Design, American Institute of Steel Construction (AISC), 1989. [20] L. Chow, H.D. Conway, G. Winter, Stresses in deep beams, Journal of Transportation Engineering ASCE 118 (1953) 686. [21] H.A.P. Depaiva, C.P. Siess, Strength and behavior of deep beams in shear, Journal of Structural Engineering ASCE 91 (ST5) (1965) 1–19. [22] M. Kaya, Design of reinforced concrete deep beams using genetic algorithms, M.Sc. Thesis, Gazi University, Ankara, 2001. [23] S. Akman, Construction Materials, Istanbul Technical University, Istanbul, 1990 (in Turkish). [24] M.S. Güner, Concrete Technology, Nobel Press, Istanbul, 1999 (in Turkish). [25] M. Kaya, Performance analysis of post-tensioning beam to column connections under cycling loading in prefabric structures, Ph.D. Thesis, Gazi University, Ankara, 2007.
Contents lists available at ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
The effects of two new crossover operators on genetic algorithm performance Mustafa Kaya ∗ Aksaray University, Faculty of Engineering, Adana Street, Aksaray, Turkey
a r t i c l e
i n f o
Article history: Received 13 November 2008 Received in revised form 14 January 2010 Accepted 17 January 2010 Available online 25 January 2010 Keywords: Genetic algorithm Crossover operators
a b s t r a c t In this study, two new crossover operators in genetic algorithm are proposed; sequential and random mixed crossover. In the first stage, existing and developed crossover operators were applied to two benchmark problems, namely the reinforced concrete beam problem and the space truss problem. In the second stage, the developed crossover operators were applied to a deep beam problem and, a concrete mix design problem.The fittest values obtained using developed crossover operators were higher than those were obtained with existing crossover operator after 30,000 generations. Moreover, in developed crossover operators, the random mixed crossover yields higher fitness values than the existing crossover operators. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Genetic algorithms (GA) came into existence with the adaptation of developed biological processes to the computer environment. They use units stored in the computer’s memory in the same way as those in natural populations. The initial population is used for the solution of optimization problems with GA. The risk of capturing local optimum traps was less for GA in comparison with traditional optimization methods. The use of the unconstrained objective functions enabled the discovery of new combinations that have higher fitness values. GA use the best objective and fitness function values for problems in which design variables are complex and discontinuous furthermore, does not require derivatives of the objective function. The first study related to GA was the introduction of the basic components by Holland [1] in 1975 entitled “Machine Learning”. Later, a study on gas pipes, by Holland’s student, Goldberg, proved that GA had practical uses. GA studies in engineering are generally the optimizations of topology, shape, and dimension [2]. Application studies of GA to optimization problems have been undertaken Jenkins [3–5] and Rajeev and Krishnamoorty [6] in which the effect of crossover operators on the behavior of GA was investigated. The crossover operator is as important as coding, selection, and mutation in GA. There are various existing types of crossover including; one-point, two-point, uniform, variable-to-variable, multi-point, mixed and direct design variable exchange. Adeli and Cheng [7–9] applied the dimension
∗ Tel.: +90 382 2150953; fax: +90 382 2150592. E-mail address: [email protected]. 1568-4946/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2010.01.008
optimization problems to three truss beams to numerically compare one-point, two-point, and uniform crossover operators. The best results were obtained from the two-point crossover operator. Wu and Chow [10] compared the one-point, two-point, three-point, and four-point crossover operators and showed that two-point, three-point, and four-point crossover operators are better than the one-point crossover. Jenkins [11] argues in favor of multi-point crossover operator in term of fast progress becomes very slow in case single-point crossover is used. Using one-point crossover, Dejong and Spears [12] introduced the relationship between crossover operators and population size. They state that two-point crossover is performs better in the problems in which the population is large, but uniform crossover is better for the small size populations. Syswerda [13] showed that the uniform crossover operator is more efficient when compared with two-point crossover. Erbatur and Hasanc¸ebi [14,15] suggested combining two crossover operators in their study about the effects of crossover operators on the behavior of GA. The mixed crossover operator was also applied to the population in which crossover operators such as one-point, two-point, and threepoint determined definite rates of generation numbers. Another technique has been suggested that is a direct design variable exchange crossover operator. Each of the design variables was changed to a probability function, which was formed empirically. The mixed crossover operator first suggested performed fairly well in the study. The second technique direct design variable exchange crossover gave the best result among existing crossover operators [14,15]. In this study, the effect of sequential crossover and random mixed crossover operators on the behavior of GA was investigated using a reinforced concrete (RC) beam problem, space truss problem, RC deep beam problem, and concrete mix design problem.
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M. Kaya / Applied Soft Computing 11 (2011) 881–890
2. Stages of the study
2.2. Formation of initial population
In the first stage of this study, existing and developed crossover operators were applied to the RC beam and, space truss problems. Analysis of the benchmark problems was made for each of nine crossover operators. The programs in analyses were terminated after 30,000 generations. In the second stage, only developed crossover operators applied to a deep beam and the concrete mix design problem analyzed. Maximum fitness values obtained from application of developed crossover operators were determined after 30,000 generations. The determination steps for the minimum cost on the RC beam, minimum weight of the space truss, the minimum reinforcement diameters in RC deep beam and the minimum cement weight in concrete mix design using GA were summarized schematically in Fig. 1. This figure consists of the following steps:
The GA searches inside the population consisting of points instead of searching point-to-point [2]. 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 chosen randomly. The suitable selection of population size significantly affects the performance of genetic algorithm. In various studies it was observed that the result was reached earlier in larger populations in comparison with small populations [2]. In this study, the population size (N) was selected as 100.
1. Construction of the initial population randomly which comprises numbers. 2. Decoding the permutation coding for the design variables of each member and finding their sequence numbers in the available variable list. 3. Calculating the value of unconstrained function (x) for each member using Eqs. (11), (22), (30) and (37). Finding the maximum and minimum values of this function in the population. 4. Calculation of the fitness value for each member using Eqs. (12), (23), (31) and (38). 5. Application of the sequential selection method. Copying the members into the mating pool according to their fitness, and couple them randomly. 6. Generation of child using crossover operators and thus obtaining the new population. Seven existing and two new crossover operators were applied. 7. Application of the mutation to each offspring in the new population with a specific probability. 8. Replacing the initial population by the new population and repeat steps 3–8 until the termination criteria is obtained. 2.1. Coding In this study, permutation coding was used because of the design variables consists of more than one variable group. The chromosome length is equal to the number of variable groups in this type of coding.
2.3. Evaluation The GA basically finds the maximum of an unconstrained objective function. To solve a constrained objective minimization function, two transformations need to be made. 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 [13]. In this study, for the four problems, different constrained objective functions were occurred. These functions are given in Eqs. (10), (21), (29) and (34). After these equations were occurred, this functions transformed unconstrained objective functions. This unconstrained objective functions are given in Eqs. (11), (22), (30) and (37). Finally, this unconstrained objective functions is transformed into the fitness functions, as given in Eqs. (12), (23), (31) and (38). 2.4. Selection The members of the new population in each generation are selected by a process from members of the existing population. The selection technique performs natural selection artificially and in this study, a sequential selection method was used. In this method, members are set in order by a linearly decreasing function. The members with the low fitness value are removed from population in a defined ratio and members with the high fitness values replace those removed in the same ratio. In this study, members with the lowest fitness values constituting 25% of population were eliminated and replaced with the highest valued members in the same ratio. 2.5. Crossover operators GA can rapidly identify discrete zones within a large search space area to concentrate the search for an optimum solution. This technique changes mutually defined parts of two members selected and obtains different members that give new points in the search space. The crossover operators used in the current study are summarized below.
Fig. 1. Structure of genetic algorithm.
2.5.1. Existing crossover operators 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 were 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 were changed using the one-point crossover operator. In the two-point crossover operators, two different cut-off points were randomly selected between 1 and L − 1. New members were obtained by relocating zones between the cut-off points of paired members. The codes between 7th and 13th sites of members given in Table 2 were changed.
M. Kaya / Applied Soft Computing 11 (2011) 881–890
883
Table 1 One-point crossover sample. 5 Parent1
3
6
8
1
3
2
7
4
3
2
6
1
8
2
6
3
5
2 Parent2
7
4
3
6
8
1
3
2
7
4
6
8
2
1
5
4
2
5 Child1
3
6
8
1
3
2
7
4
7
4
6
8
2
1
5
4
2
2 Child2
7
4
3
6
8
1
3
2
3
2
6
1
8
2
6
3
5
Table 2 Two-point crossover sample. 5 Parent1
3
6
8
1
3
2
7
4
3
2
6
1
8
2
6
3
5
2 Parent2
7
4
3
6
8
1
3
2
7
4
6
8
2
1
5
4
2
5 Child1
3
6
8
1
3
1
3
2
7
4
6
1
8
2
6
3
5
2 Child2
7
4
3
6
8
2
7
4
3
2
6
8
2
1
5
4
2
In multi-point crossover operators, many cut-off points are selected between 1 and L − 1. New members were obtained by relocating zones between these cut-off points of pairing members and the codes of the members were changed in four sites as shown in Table 3. In variable-to-variable crossover operators, first the paired members (strings) are decomposed into their substrings. Next, a single-point crossover is separately carried out on all the substrings (Table 4.). In this way, each design variable amongst the members is activated to separately accomplish the design exchange [14]. In uniform crossover operators, as a first stage, a random member must be temporarily formed in the binary string. The chromosome length of this member is equal to chromosome lengths of the other members in the population. If the code of the nth site of the chromosome of the temporary member is 0, in this case, the code of 1st old member is used in the nth site of the 1st new member and the code of 2nd old member is used in the nth site of the 2nd new member. Equally, if the code of the nth site of the temporary member’s chromosome is 1, in this case, the code of the 2nd old member is used in the nth site of the 1st new member and the code of the 1st old member is used in the nth site of the 2nd new member. The application of the uniform crossover operator to the two members selected from the population is given in Table 5 [13]. The mixed crossover operator introduced by Hasancebi et al. [14], uses a combination of single-point, two-point and three-point crossovers for selected proportions of generations for a fixed generation number. The idea behind the technique is to achieve an efficient search, which can be obtained by activating the positive characteristics of the existing techniques in the right appropriate place in the search process [14].
In the direct design variable exchange crossover operator, each design variable (substring) is directly and separately exchanged between paired members according to a probability function. This function is empirically created, and is designed to achieve an effective search during successive generations [14]. 2.5.2. The new crossover operators 2.5.2.1. Sequential crossover. In this new crossover operator, before beginning the process the existing crossover operators were numbered 1–7 (Table 6). Then, the existing crossover operators were applied sequentially to the chromosome couples in the population at in the same generation. 2.5.2.2. Random mixed crossover. In this operator, the existing seven crossover operators were numbered 1–7 as shown in Table 6 were applied randomly to the chromosome couples in the population at the same generation. 2.6. Mutation operator The mutation operator is critical to the success of genetic algorithms since it determines the search directions and avoids convergence to local optima. In this operator, 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]. 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 the chromosomes, but the 1% mutation ratio gave a better result than the 0.1% and 0.5% mutation ratios. As a result the 1% mutation ratio was applied on the chromosomes of all the problems.
Table 3 Multi-point crossover sample. 5 Parent1
3
6
8
1
3
2
7
4
3
2
6
1
8
2
6
3
5
2 Parent2
7
4
3
6
8
1
3
2
7
4
6
8
2
1
5
4
2
5 Child1
3
4
3
1
3
1
3
4
3
4
6
1
8
1
5
3
5
2 Child2
7
6
8
6
8
2
7
2
7
2
6
8
2
2
6
4
2
884
M. Kaya / Applied Soft Computing 11 (2011) 881–890
Table 4 Variable-to-variable crossover sample. 5 Parent 1
3
6
8
1
3
2
7
4
3
2
6
1
8
2
6
3
5
2 Parent2
7
4
3
6
8
1
3
2
7
4
6
8
2
1
5
4
2
5 Child 1
7
6
3
1
8
2
3
4
7
2
6
1
2
2
5
3
5
2 Child2
3
4
8
6
3
1
7
2
3
4
6
8
8
1
6
4
5
Table 5 Uniform crossover sample. 0 Temporary string
1
1
0
0
1
1
0
1
1
1
0
1
0
1
0
0
1
5 Parent1
3
6
8
1
3
2
7
4
3
2
6
1
8
2
6
3
5
2 Parent2
7
4
3
6
8
1
3
2
7
4
6
8
2
1
5
4
2
5 Child1
7
4
8
1
8
1
7
2
7
4
6
8
8
1
6
3
2
2 Child2
3
6
3
6
3
2
3
4
3
2
6
1
2
2
5
4
5
3. Theoretical review of the crossover operators
were a consequence of the different crossover operators that were used.
The two main strategies that a solid crossover operator should use to locate the optimum are exploration and exploitation. The facility for exploration in a crossover operator encourages the search for a rapid and thorough discovery examination of the design space. A robust exploration prevents the search from locating a local peak. However, only using this type of search becomes inefficient when making use of previously obtained points to reach more appropriate points. On the other hand, a solid exploitation employs the previously found points to reach the optimum, however, in this case a slow convergence is observed accompanied by an increasing risk of locating a local peak. In fact, these two strategies are contradictory, and a balanced use of both is vital for achieving an efficient search through the crossover operator [14]. 4. Applications In the first stage of this study, existing and developed crossover operators were applied to the two benchmark problems, namely, the RC beam problem and the space truss problem. In the second stage, the developed crossover operators were applied to a deep beam problem and a concrete mix design problem. All the programs used to compare developed and existing crossover operators were the same except for the crossover operators. In all programs, population size (N = 100), permutation coding type, sequential selection method and the 1% mutation ratio were used; therefore, the different results obtained from these analyses
4.1. Benchmark problems 4.1.1. Reinforced concrete beam problem The design of the RC beam is normally an iterative process, in which the engineer assumes the self-weight of the beam beforehand, and a trial section is chosen. Then, the moment of resistance of the section is determined to check its suitability against the given applied bending moment. The process is repeated until a suitable trial section is found. This procedure often creates difficulty in exactly matching the moment due to the self-weight of the beam, which may be quite substantial in many cases. Therefore, the design process of a beam is not only slow, but also uneconomic, since the only concern is to find any section suitable for the given conditions, instead of looking for the most economic ones [16]. The RC beam section for this problem is given in Fig. 2. For the purposes of this analysis, strength design procedures were adopted because they have, among others, the following advantages [17].
Table 6 Existent crossover types and codes of these which used in sequential crossover and random mixed crossover. Crossover type
Code
One-point crossover Two-point crossover Uniform crossover Multi-point crossover Variable-to-variable crossover Mixed crossover Direct design variable exchange crossover (DDVECT)
1 2 3 4 5 6 7
Fig. 2. Reinforced concrete beam section.
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Table 7 The result of runs obtained with different crossover techniques for reinforced concrete beam. One-point
Two-point
Cost of reinforced concrete beam ($) Run 1 53,59 52,84 Run 2 54,10 51,89 Run 3 51,64 52,64 Run 4 54,10 53,35 Run 5 52,64 50,89 Run 6 52,64 51,89 Run 7 51,64 52,64 Run 8 54,10 53,35 Run 9 52,64 50,89 Run 10 52,64 51,89 Best
51,64
50,89
Multi-point
Variable-to-variable
Uniform
Mixed crossover
DDVECT
Sequential
Random mixed
51,13 49,89 50,89 48,92 51,38 50,13 50,89 48,92 51,38 50,13
50,64 49,43 50,38 51,13 49,68 48,67 50,38 51,13 49,68 48,67
45,81 48,43 48,17 48,92 47,16 47,16 48,17 48,92 47,16 47,16
47,92 46,72 45,71 48,17 46,92 47,67 45,71 48,17 46,92 47,67
44,95 47,16 46,92 47,43 46,22 46,22 46,92 47,43 46,22 46,22
45,21 44,93 43,76 44,46 42,18 43,49 43,76 44,46 42,18 43,49
42,76 43,95 43,76 42,00 43,00 42,25 43,76 42,00 43,00 42,25
48,92
48,67
45,81
45,71
44,95
42,18
42,00
• Strength design better predicts the strength of a section because of the recognition of the non-linearity of the stress strain curve at high stress level. • Since the dead loads to which a structure is subjected are more certainly determined than the live loads, it is unreasonable to apply the same factor of safety to both. Therefore, this approach allows the use of different safety factors for each type of loading.
For the RC beam, the objective function f(x) is expressed in Eq. (3), the penalty function is given in Eq. (9) and the constrained objective function is shown in Eq. (10):
The basic assumptions that are taken when using strength design are as follows [18].
b =
• 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 from the neutral axis. • The tensile strength of the concrete is neglected in flexural computations. • The ultimate concrete strain is 0.003. Please check all numbers for, or. • The average compressive stress in the concrete is 0,85. fc • The average tensile stress in the reinforcement does not exceed fy . The moment to be carried by the RC beam is given in Eq. (1). In this equation Mu : ultimate moment, Ø: strength reduction factor, : steel ratio, fy : yield stress of steel, fc : compressive stress of concrete; b: weigh of the beam; d: height of the beam [17]:
Mu = (fy bd2 )
1 − 0.59
fy fc
(1)
Using extra reinforcement in the beam causes brittle behavior under loading. The required reinforcement ratio of RC beam is given in Eq. (2): = ((fy bd)(fc − 0.59fy ))fc
(2)
The most important parameter that affected the cost of the determination of the cheapest beam is the weight of reinforcement in the beam, the second parameter being the weight of concrete. Taking this into account, the unit price of the reinforcement increases the cost of the concrete 25 times. The fitness function proposed for finding the minimum cost in the RC beam is given in Eq. (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 length of beam. Since the number of variables representing the RC beam was too large, permutation coding was used where the chromosome length is equal to the group number of design variables. In this problem the design variable groups were beam height, beam width, reinforcement diameters and reinforcement numbers.
min f (x) = bdL(F1 s + F2 c )
ˇ1 = 0.85 − 0.05
fc − 4000 1000
0.85ˇ1 fc 87000 fy (87000 + fy )
Else If G=
<
and 200 fy
(4) (5)
max = 0.75b 200 If ≥ fy
(3)
(6) < max g(x) = bdL(F1 s + F2 c ) and
g(x)
(s) = f (x)(1 + KG)
(7)
≥ max g(x) = bdL(F1 s + F2 c )100 (8) (9) (10)
where K is a coefficient selected for the problem taken to be 10 in this 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) as expressed in Eq. (11): (x) =
(s) (s)max
(11)
In the second transformation in Eq. (12), the unconstrained objective 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 and developed crossover operators. The direct design variable exchange crossover is the lowest cost with the single-point crossover being the most cost of the existing crossover operators. The cost found by single-point crossover is 21% higher than the cost found by direct design variable exchange crossover. The random mixed crossover gives the lowest cost between existing and developed crossover operators. When the random mixed crossover and direct design variable exchange crossover are compared, it can be seen that the cost found by the random mixed crossover operator is 7% less than the cost found by the direct design variable exchange crossover operator (Table 7). Concerning the fitness values criteria of the RC beam problem that terminated after 30,000 generations, the maximum fitness value between existing crossover operators was obtained from the program that employs the direct design variable exchange crossover operator. The maximum fitness value in the existing and developed crossover operators was achieved using the program
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Fig. 3. The fitness values obtained after 30,000 generation for the reinforced concrete beam.
that employs the random mixed crossover operator. The fitness value obtained by the developed random mixed crossover was 13% higher than the fitness value obtained by the existing direct design variable exchange crossover operator (Fig. 3). 4.1.2. Space truss problem The space truss problem is commonly chosen when solving problems about GA since the design space is medium sized [6,7,9,14]. Also, in this study existing and developed crossover operators were applied to the problem of determining the most appropriate bar sections for the space truss (Fig. 4). In this problem chromosome length is 25 and this number is equal to design variable group number. By using existing and developed crossover operators, the optimum design of the space truss was achieved in terms of weight. In the space truss problem, consisting of 25 bar elements, the bar sections were selected from profiles given in AISC [19]. In the tested space truss, the yield strength was taken to be 2400 kg/cm2 . The elasticity module of steel was taken to be 2 × 106 kg/cm2 . A 100-kN force was applied to the truss from two nodal points in a vertical direction. The truss was restricted to a maximum L/300 horizontal drift. The codes used in the application vary from 1 to 123 and the size of the designed space was 12325 .
For space truss; the objective function W(x) is expressed in Eq. (10), the penalty function is expressed Eq. (14) and the constrained objective function is expressed in Eq. (15): min W (x) =
Tk
Ak Lk k
(13)
k=1
where Ak is the cross-sectional area, k the specific gravity of space truss elements, Lk is the space truss elements length, and Tk is the number of space truss elements. 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 constrains are as follows. Combined stress constrains: gi (x) =
eb Cm b −1 + bem 1 − (eb /e ) B
(14)
gi (x) =
eb + b −1 0.6a B
(15)
Eqs. (14) and (15) are required for members subjected to bending moment and, axial compression. For members subjected bending moment and axial stress are required to providing Eq. (9) is applied: eb ≤ 0.15 bem
If
gi (x) =
(16)
eb + b −1 bem B
(17)
eb : compression stress only subjected to axial force; b : compression stress only subjected to bending moment; bem : allowable stress only subjected to axial force; B : allowable compression stress in the 1st member subjected only to bending moment; e : Euler stress divided by the safety coefficient; a : yield stress of steel; nm: number of members; Cm the adjustment coefficient and equal to 0.85 for the lateral displacement of members: C=
ci
(18)
If gi (x) > 0, If
gi (x) ≤ 0,
ci = gi (x)
(19)
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): Fig. 4. 25-Bar space truss.
(x) =
(s)
(s)max
(22)
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Table 8 The result of runs obtained with different crossover techniques for space truss. One-point Space truss weight (kN) Run 1 10,67 Run 2 10,28 Run 3 10,62 Run 4 10,77 Run 5 10,46 Run 6 10,48 Run 7 10,54 Run 8 10,41 Run 9 10,70 Run 10 10,32 Best
10,28
Two-point
Multi-point
Variable-to-variable
Uniform
Mixed crossover
DDVECT
Sequential
Random mixed
10,52 10,62 10,48 10,13 10,33 10,33 10,57 10,34 10,41 10,21
10,18 9,98 10,13 10,23 9,93 9,74 9,76 9,91 10,03 10,17
9,69 10,08 10,03 10,18 9,89 9,84 10,11 9,75 9,98 10,07
9,64 9,39 9,59 9,74 9,39 9,25 9,27 9,71 9,55 9,67
9,39 8,95 9,34 9,44 9,20 9,20 9,01 8,37 9,21 9,11
9,54 9,59 9,49 9,10 9,34 9,30 9,51 9,17 9,46 9,33
9,00 8,85 8,71 8,66 8,95 8,66 8,69 9,03 8,78 8,86
8,51 8,56 8,71 8,36 8,40 8,41 8,47 8,69 8,74 8,43
10,13
9,74
9,69
9,25
8,95
9,10
8,66
8,40
Fig. 5. The fitness values obtained after 30,000 generation for the space truss.
In the second transformation, the unconstrained objective function (x) was converted to an F(s) fitness function in Eq. (17): F(s) = (x)max − (x)
(23)
In this problem, the space truss beam was analyzed using existing and developed crossover operators. The mixed crossover operator gives the lowest truss beam weight, but the single-point crossover gives the most beam weight among the existing crossover operators. The random mixed crossover gives the lowest truss beam weight among all the crossover operators. The truss weight found by the mixed crossover is 15% heavier than the truss beam weight found by the random mixed crossover (Table 8). Concerning the fitness values criteria of the space truss problem that terminated after 30,000 generations, the maximum fitness value from the existing crossover operators was obtained from the direct design exchange crossover (DDVECT) operator. The maximum fitness value in the developed crossover operators was achieved using the random mixed crossover operator. The fitness value obtained by the developed random mixed crossover was 15% higher than the fitness value obtained from the existing direct design exchange crossover operator (Fig. 5).
beam in the bending calculation due to its L/h ratio being greater than 5/4, but less than 5 [22]. A deep beam of loaded from the top-edge and of L/h ratio of 4 is given as an example in Fig. 6. This beam is accepted as deep beam in calculations of shear force while it was accepted as normal beam for bending according to the ACI 318 [22]. In this study, the stresses on the deep beam caused by external forces were first determined through the finite elements method. Then, the steel bar diameters given in standards depending on the effects on these sections were determined using GA. The deep beam was tested under compressive and tensioning stress restrictions. In this study, the stresses on the bars caused by external forces were first determined through the finite elements method. Then, the bar sections given in standards depending on the effects on these sections were determined using GA. As a result of this behavior, problems occurred in the analysis of deep beams. The fittest horizontal and vertical reinforcement diameters were determined in a deep beam using the GA.
4.2. Proposed and analyzed problems 4.2.1. Deep beam problem Deep beams used in shear walls, folded roof plates, and silos show different behaviors when compared with normal beams owing to the fact that their depth–span ratio is higher than that of normal beams. Such studies confirm the usual hypothesis that plane sections before bending remain plane after bending, does not hold for deep beams. Significant warping of the cross-sections occurs because of the high shear stresses [20,21]. The deep beam, loaded from its top-edge, is accepted as a deep beam in the shearing calculation, while it is not accepted as deep
Fig. 6. The fittest reinforcement obtained for deep beam.
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For deep beam; the objective function W(x) is expressed in Eq. (18), the penalty function is expressed Eq. (14) and the constrained objective function is expressed in Eq. (21): min W (x) =
Cs
As Ls s
(24)
s=1
where As is the reinforcement area, s the specific gravity of reinforcement, Ls is the reinforcement length and, Cs number of zones in the deep beam. The constrained objective function was transformed to an unconstrained objective function (x) as expressed in Eq. (18):
gi (x) =
s Ls As ns fyd
−1
(25)
If gi (x) ≥ 0,
ci = gi (x)
(26)
If gi (x) < 0,
ci = gi (x) × 100
(27)
C=
Fei
ci
(28)
(s) = W (x)(1 + KC)
(29)
K: a coefficient selected for the problem taken to be 10 in this study and ci : negligence coefficient and calculated as follows: In the first transformation, the constrained objective function (s) was transformed to an unconstrained objective function (x) as expressed in Eq. (22): (x) =
(s) (s)max
(30)
where Fei is the required force met by reinforcements in the x or y direction in the ith zone, fyd is the yield strength of reinforcement, and 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 transformation was achieved using the maximum of the ith element of the unconstrained objective function. The fitness values of the members were calculated according to Eq. (23) as follows: F(s) = (x)max − (x)
(31)
Concerning the fitness values criteria of deep beam problem that terminated after 30,000 generations, the maximum fitness value from the existing crossover operators was obtained from the mixed crossover operator. The maximum fitness value in the developed crossover operators was achieved using the sequential crossover operator. The fitness value obtained by the sequential crossover was 12% higher than the fitness value obtained from the existing mixed crossover operator (Fig. 7). In this problem the deep beam was analyzed using existing and developed operators. The direct design variable exchange crossover
operator gives the smallest reinforcement diameter among the existing crossover operators. The random mixed crossover gives the smallest reinforcement diameter among all crossover operators. The reinforcement diameter found by direct design variable exchange crossover is 7% bigger than the reinforcement diameter found by the random mixed crossover (Table 9). In this example, diameters of horizontal and vertical reinforcement determined for 9 zones by using genetic algorithm are given in Table 10 and Fig. 6. In this figure, horizontal and vertical lines show that the horizontal and vertical shear reinforcement in those zones, respectively. The weights of reinforcement determined using the GA were compared with weight of reinforcement specified in accordance with ACI 318-99. As a result of comparisons, it can be seen that the weight 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 the resulting concrete has adequate strength, proper workability for placing and low cost. In this problem, the minimum amount of cement for 1 m3 high strength concrete (fc = 50 MPa) was determined by the GA and the Bolomey formula [23,24] given in Eq. (32). Then, amounts of cement which are determined using both methods were compared.The amount of cement used in concrete mixtures is directly proportional to the amount of water. Furthermore, the amount of water is inversely proportional to the fineness modulus of the aggregate when the amount of cement is minimum and the fineness modulus is at maximum:
fc =
fcc a
c 2
(32)
w
where c is the cement weight, w is the weight of the water, fc is the compressive strength of concrete, fcc is the compressive strength of cement and a is changes between 4 and 8. The compressive strength of the concrete (fc ), compressive strength (fcc) and a the coefficient of cement are fixed. The fineness modulus of the aggregate (k) will be determined as the minimum amount of cement. In this study, the number of design variable groups is equal to the number of sieves and varies between 1 and 8. The number of variables in all groups varied between 0 and 2100 kg. 0 kg represents that aggregate does not exist and 2100 kg represents that aggregate is fully present in 1 m3 concrete. The objective function W(cement) is expressed in Eq. (33) as: min W (cement) =
n max(k) =
1
fc a(10 − k)
%passed
fcc
or
100
Fig. 7. The fitness values obtained after 30,000 generation for the deep beam.
(33)
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Table 9 The result of runs obtained with different crossover techniques for deep beam. One-point Deep beam weight (kN) Run 1 80,85 Run 2 81,59 Run 3 80,47 Run 4 77,87 Run 5 79,36 Run 6 79,36 Run 7 79,41 Run 8 81,40 Run 9 80,136 Run 10 78,16 Best
77,87
Two-point
Multi-point
Variable-to-variable
Uniform
Mixed crossover
DDVECT
Sequential
Random mixed
79,73 78,24 79,36 80,47 78,24 76,75 78,37 7,23 79,34 76,80
77,12 75,63 76,75 77,49 75,26 73,77 76,79 74,47 73,86 77,35
73,40 76,38 76,00 77,12 73,40 74,51 73,30 77,10 76,43 75,41
73,02 71,16 72,65 73,77 71,18 70,04 72,15 71,22 73,65 70,78
72,28 70,79 71,91 72,65 68,92 70,42 72,55 69,97 70,17 72,50
71,16 67,81 70,79 71,53 69,67 69,67 68,71 70,11 68,87 70,70
68,18 69,30 65,94 67,06 67,81 65,57 65,83 66,80 69,13 66,80
64,45 63,34 65,57 63,34 64,83 63,71 66,30 65,34 66,22 64,51
76,75
73,77
73,40
70,04
68,92
67,81
65,57
63,34
Table 10 Codes and reinforcement diameters obtained for deep beam. Zone No Code of horizontal Diameter of horizontal Code of vertical Diameter of vertical
1 4 ∅14 5 ∅16
2 5 ∅18 3 ∅14
3 4 ∅14 5 ∅14
4 4 ∅16 4 ∅14
The constrained objective function was transformed to an unconstrained objective function (s) as given in Eq. (34) and (x) was given in Eq. (37). Where n is the number of sieves:
n
(s) =
1
10 −
%passed
(34)
100 then (s) = 0
(35)
If (s) > 7.17 then (s) = 0
(36)
If (s) < 4.60
(x) =
(s)max (s)
5 6 ∅14 3 ∅14
6 4 ∅16 4 ∅16
7 3 ∅16 3 ∅16
8 3 ∅16 3 ∅14
9 3 ∅16 3 ∅14
Table 12 Concrete mix design. Component name
Weight (N)
Cement (KPC 42.5) Crushed aggregate (5–12 mm) Crushed aggregate (12–15 mm) Sand (0–5 mm) Marble flour Water Additive (Sikament 300)
3900 4800 4800 5300 3700 1500 81
(37)
In the 2nd transformation, the unconstrained objective function (x) was converted to a fitness function F(s). This transformation was achieved using the maximum of the ith element of the unconstrained objective function. The fitness values of the members were calculated according to Eq. (38) as follows: F(s) = (x)max − (x)
(38)
Concerning the fitness values criteria of the concrete mix design problem that terminated after 30,000 generations, the maximum fitness value from the existing crossover operators was obtained from the direct design variable exchange crossover (DDVECT) operator. The maximum fitness value in the developed crossover operators was achieved using the random mixed crossover operator. The fitness value obtained by the developed the random mixed
crossover operator was 8% higher than the fitness value obtained from the existing the direct design variable exchange crossover operator (Fig. 8). The concrete mix design prepared using existing and developed crossover operators. The uniform crossover gives the lightest amount of cement among the existing crossover operators. The sequential crossover operator gives the lightest amount of cement all the crossover operators. The amount of cement found by the sequential crossover operator is 8% smaller than the amount of cement 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 genetic algorithm. This concrete mix design (Table 12) was used to construct experiment members for the study on “post tensioned column-to-beam connections in precast structures” [25].
Table 11 The result of runs obtained with different crossover techniques for concrete mix design. One-point Cement weight (kN) Run 1 4,979 Run 2 5,025 Run 3 4,956 Run 4 4,795 Run 5 4,887 Run 6 4,887 Run 7 4,891 Run 8 5,012 Run 9 4,935 Run 10 4,813 Best
4,795
Two-point
Multi-point
Variable-to-variable
Uniform
Mixed crossover
DDVECT
Sequential
Random mixed
4,910 4,818 4,887 4,956 4,818 4,726 4,825 4,810 4,885 4,730
4,749 4,657 4,726 4,772 4,635 4,543 4,728 4,585 4,548 4,763
4,520 4,704 4,680 4,749 4,520 4,588 4,514 4,747 4,707 4,644
4,382 4,175 4,359 4,405 4,290 4,290 4,231 4,317 4,241 4,354
4,451 4,359 4,428 4,474 4,244 4,336 4,468 4,309 4,321 4,465
4,497 4,382 4,474 4,543 4,383 4,313 4,443 4,386 4,535 4,359
3,968 3,900 4,037 3,900 3,992 3,923 4,083 4,023 4,078 3,972
4,198 4,267 4,060 4,129 4,175 4,037 4,054 4,114 4,257 4,114
4,726
4,543
4,520
4,175
4,244
4,313
3,900
4,037
890
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Fig. 8. The fitness values obtained after 30,000 generation for the concrete mix design.
5. Conclusion
References
In this study, the effect of seven existing crossover operators and two developed crossover operators on the performance of the GA were compared. In the first stage a RC beam and space truss problems were used in the comparison. In the analyses made for these problems, the maximum fitness value was obtained from the analysis of the direct design crossover operator selected from the existing crossover operators for the RC beam problem. The maximum fitness value was obtained from the analysis of the mixed crossover operator from the existing crossover operators used in the space truss beam problem. However, obtained fitness values are approximately equal to each other. When the results obtained from the developed and existing crossover types were compared, the two highest fitness values were obtained from the analysis of the random mixed crossover and sequential crossover operator. In the second stage of the study, using deep beam and concrete mix design problems, the effect of sequential crossover and random mixed crossover operators, developed using deep beam and concrete mix design problems, on the performance of the GA was investigated. The maximum fitness value was obtained by the analysis of the random mixed crossover operator used in deep beam and concrete mix design problems. While the weights of reinforcement determined from the use of the random mixed crossover and sequential crossover operators, were approximately equal to each other in the deep beam problem, the weights of cement determined using same operators were different in the concrete mix design problem. The use of the GA produced successful results in the determination of the lowest cost of the RC beam problem, of the minimum weight of steel truss beam, of the weight of reinforcement which is independent from dimensions of deep beams, and of the weight of the minimum amount cement in the concrete The operators developed for this study (random mixed crossover and sequential), obtaining higher fitness values than the existing crossover operators, shows that developed crossover operators thus GA give better results in solving problems in which design variables are discontinuous. The application of GA to the four different problems which are encountered in civil engineering and the successful results achieved from the analyses of these problems shows that GA could be applied to other/further problems in civil engineering
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