Single and multiobjective frame optimization by evolutionary algorithms and the auto-adaptive rebirth operator

Comput. Methods Appl. Mech. Engrg. 193 (2004) 3711–3743

www.elsevier.com/locate/cma

Single and multiobjective frame optimization byevolutionary algorithms and the auto-adaptive

rebirth operator

David Greiner, Jos�e Mar�ıa Emperador, Gabriel Winter *

Institute of Intelligent Systems and Numerical Applications in Engineering (IUSIANI), University of Las Palmas de Gran Canaria,

Las Palmas 35017, Spain

Received 27 February 2003; received in revised form 29 January 2004; accepted 3 February 2004

Abstract

The constrained minimum-mass problem of middle-size frames is taken into account, for both continuous and

discrete cases with ideal (without buckling effect and own gravitational load) and real (with both) models, comparing

three strategies of evolutionary algorithms. Some proposals to obtain appropriate results in middle-sized frames are

exposed: optimization considerations about coding and structure; and the introduction of the auto-adaptive rebirth

operator. Moreover, the introduction in the initial population of high quality single-optimization solutions obtained via

the auto-adaptive rebirth operator is proposed as a way to improve the final non-dominated fronts obtained in

structural frame multicriteria optimization (number of different cross-section types as second criteria). The results

through three test cases (55–35 bar-sized) show the advantageous use of the auto-adaptive rebirth operator in frame

optimization.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Evolutionary algorithms; Structural optimization; Multiobjective optimization; Frames; Rebirth

1. Introduction

Evolutionary algorithms (EAs) are global optimizers due to their populational search [1,2]. EAs may

equally handle single and multiobjective optimization and they have great advantages over traditional

methods for solving multiobjective optimization problems, since they can be applied simultaneously with

integer, discontinuous or discrete design variables. In structural optimization, where local optima and non-

connected domain zones are frequent it is very appropriate to solve the problems with EAs. Their suitability

in frame optimization because of their capability of global, discrete and multiobjective method, has beenexposed for small frames elsewhere, see e.g. [3]. This article handles with the case in which the frame size is

* Corresponding author.

E-mail addresses: [email protected] (D. Greiner), [email protected] (J.M. Emperador), [email protected]

(G. Winter).

0045-7825/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2004.02.001

mail to: [email protected]

3712 D. Greiner et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 3711–3743

greater, where refined and adequate considerations and tools are necessary for the evolutionary algorithmsin order to obtain proper results: The auto-adaptive rebirth operator (it allows to improve the convergence

results of the evolutionary algorithm) and some optimization considerations are presented.

The article is organized as follows: In the next section, the handled structural frame optimization

problem is exposed. Later, the rebirth operator is described. After that, the three tested strategies of

evolutionary algorithms (generational, steady-state and CHC) are exposed. Later, the 55 bar sized handled

test case is presented. Finally, the results of continuous and discrete minimization of constrained mass and

multicriteria optimization are shown. The article ends with the conclusions section and finally, with an

Appendix A with some computational and structural optimization considerations.

2. The structural problem

The minimization of the constrained mass is taken into account to minimize the raw material cost of the

designed structure. The constraints consider those conditions that allow the designed frame to carry out its

task without collapsing or deforming excessively. The constraints are:

(a) Stresses of the bars: where the limit stress depends on the frame material and the comparing stresstakes into account the axial and shearing stresses by means of the shear effort, and also the bending effort

(a common value for steel is of 260 MPa––S275JR steel), for each bar:

rco � rlim 6 0: ð1Þ(b) Compressive slenderness limit: where the klim value is 200 in the Spanish code EA-95 (to include the

buckling effect the evaluation of the beta factor, is based on Julian and Lawrence criteria applied to the

Spanish code). For each bar:

k� klim 6 0: ð2Þ(c) Displacements of joints (in each of the three possible degrees of freedom) or middle points of bars. In

the test cases, the maximum vertical displacement of each beam is limited according to the Spanish design

code guidelines (in the multiobjective test case the maximum vertical displacement of the beams is L=500, asis determined by the Spanish design code):

uco � ulim 6 0: ð3ÞThe first objective function constrained mass, results:

ObjectiveFunction1 ¼XNbars

i¼1

Ai � qi � li

" #1

"þ k �

XNviols

j¼1

ðviolj � 1Þ#; ð4Þ

where Ai ¼ bar i cross-section area; qi ¼ bar i specific mass; li ¼ length of bar i; k¼ constant that regulates

the equivalence between mass and constraints (its value is around one, value that recommends Goldberg in

[4]); violj ¼ for each of the violated constraints (stress, displacement or slenderness), is the quotient between

the value that violates the constraint limit: violated constraint value and its reference limit. The constraints

reference limits are chosen according to the Spanish design codes. So, constraints if violated, are integrated

into the mass of the whole structure as a penalty depending on the amount of the violation (for each

constraint violation the total mass is incremented). The limits are chosen according to the Spanish design

code guidelines.

violj ¼Violated Constraint Value

Constraint Limit:

D. Greiner et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 3711–3743 3713

Moreover, the minimization of a second function is considered as a multicriteria optimization problem:the number of different cross-section types, that supposes a condition of constructive order, and with

special relevancy in structures with high number of bars [5,6]. It helps to a better quality control during the

execution of the building site. It is a factor that has been also recently related with the life cost cycle

optimization of steel structures, as claimed in [7,8]. An advantageous application of the rebirth operator in

a multicriteria application example is presented at the end of Section 6.3.

3. Rebirth operator

The exploration/exploitation equilibrium is the main task to achieve in the resolution of search prob-

lems. An excessive exploitation of high quality obtained information implies a worse exploration of the

search space, being the population finite. Analogously, an excessive exploration could produce a perfor-

mance loss in the process, approaching it to a random search. The behaviour of the evolution of an evo-

lutionary algorithm can be provided with the point of view of this equilibrium [9].

A way to deal with the exploration/exploitation equilibrium in evolutionary algorithms is by means of

two main factors that control the evolution [10]: selection pressure and population diversity. Both are in-versely related: a high selection pressure implies a quick loss of population diversity, because of the

excessive focus of the evolutionary search on the best members of the population; on the contrary, the

maintenance of population diversity can neutralize the effects of an excessive selection pressure. Most

parameters that are used to adjust the strategies of an evolutionary search are indeed indirect terms of

tuning selection pressure and population diversity.

An operator that introduces diversity in the population is the reinitialization. This operator was sug-

gested in [1] and consists in the creation of a new starting population after the stagnation of the evolu-

tionary algorithm in which the best individual of the prior population is inserted. Thus, this individualprovides the knowledge obtained with the initial run and simultaneously the new randomly created indi-

viduals contribute to the population diversity; in this way, a further continuation in the evolution is

allowed. This reinitialization operator has been used in many evolutionary algorithms, such as the micro-

genetic algorithm [11] or the CHC algorithm [12,13].

Another factor that influences the population diversity is the population size. The population size

plays an important role in the diversity of the population, and indeed in the exploratory task of the

algorithm: larger population sizes are associated with lower convergence speed of the algorithm, but also

with lesser premature stagnation of the population. Small population sizes can follow to prematurestagnation, and their lack of population diversity have to be corrected with other operators that increase

it, such as higher mutation rates or the reinitialization operator (micro-GA). A further study of the

influence of the population size and its interaction with rebirth applied to discrete optimization can be

seen in the results section.

An application for optimizing the estimation of optical parameters in liquid crystals uses the successive

reduction of the search intervals of the phenotypic variables as an improvement operator [14].

The Delta coding [15], is the most similar precursor of the rebirth: when convergence is achieved, the

algorithm takes the best value among those obtained at this point and makes a reinitialization. The newpopulation changes its codification and the phenotypic variables become incremental values respect of the

best individual, reducing the bits sizing (chromosome length) as well.

The rebirth operator consists in a reinitialization and a reduction of the phenotypic interval of the

variables of the chromosome (what implies a reduction of the search space) and the length of the chro-

mosome. It was introduced and used by Galante [5] in trusses optimization [4]. When the population

converges, it is applied and exterminates all the population members except the best individual. It creates a


new random population and recodes the interval of the variables, which is reduced and centred in thevariables of the best individual. A similar operator applied to structural trusses optimization denominated

‘Two Phase Method’ is used in [16], with variable length chromosomes.

Evolutionary algorithms are global optimization tools thanks to treat with a population of solutions

instead of a single individual, but due to their stochasticity, the finding of the global optimum is not

guaranteed. So, different executions to solve a problem will lead to different best solutions (variation of

results when several runs are launched). One alternative to reduce this variation, is to perform a further

search after the stagnation of the evolutionary algorithm. There are authors that perform a local search,

considering as starting point the best value found by the evolutionary algorithm. However, the rebirthoperator substitutes the local search by a evolutionary one, maintaining its global characteristic, focused in

a smaller space of the variables domain and centred in the variable values of the best individual.

Here the auto-adaptive rebirth [17] is introduced. The auto-adaptive rebirth operator is based on the

rebirth operator introduced by Galante in [5] an applied to truss optimization. It consists in a rebirth

applied consecutively to the same execution, as many times as it can improve the convergence. It is

applied after the saturation of the population repeatedly, as many times as necessary, to avoid the

saturation in a local minima and to improve the final results in terms of speed and accuracy. The

automatic criterion used to activate this proposed operator is the stagnation of the population (parameterRP that considers the number of fitness function evaluations without improvement in the best individual

fitness function value). An appropriate value of this parameter is essential to obtain the best convergence

behaviour. Considerations about how to select this parameter are discussed in the results section. An

obtained compromise value for the treated test cases is to choose 5000 evaluations. After this saturation

condition is satisfied, rebirth is applied. It starts a new population and a new search, after which rebirth

is applied again repeatedly, and so on. When after its n-times application the obtained solutions do not

improve, the stop criterion is reached. Comparative results showing the advantages of auto-adaptive

rebirth operator versus rebirth are presented in the results section, applied to discrete frame steelstructures optimization.

4. Proposed strategies of evolutionary algorithms

The way in which the operators are defined in evolutionary algorithms, the parameters that govern them

and the used genotypic representation contribute to the desired success of a specific algorithm. Some factors

that take part are the population size, the selection type, the crossover type, crossover and mutationprobabilities, etc.

The three adopted strategies, also considered in [3] are briefly described, for a common average popu-

lation of 50 individuals and a probability of mutation of 3%:

1. Generational [1] model of the evolutionary algorithm: with elitism (of two individuals), one point cross-

over, RWS selection and crossover probability of 80%.

2. Steady-state [10] model of evolutionary algorithm: without duplicates in the population (verified in the

initial population and also during the inclusion of new individuals each generation), where the insertionof two individuals is in substitution of the worst ones of the previous population: uniform crossover;

SUS selection and crossover probability of 100%.

3. CHC model [12,13] of evolutionary algorithm: dynamic incest prevention (25% and diminishing in 5%

intervals if there is no evolution); HUX crossover (it crosses exactly the half of the diverging bits between

individuals); truncation selection after evaluation; massive mutation in reinitialization (37% of random

bits) after the cyclic process of dynamic incest prevention.

Fig. 1. Frame test case.


5. Test case

The test case is represented in Fig. 1, based on a reference problem in [18]. The figure includes the

numbering of the bars, and the precise loads in tons. Moreover, in every beam there is a uniform load of

39,945 N/m. The lengths of the beams are 5.6 m and the heights of the columns are 2.8 m. The columns

belong to the HEB cross-section type series, and the beams belong to the IPE cross-section type series;

being the admissible stresses of 200 and 220 MPa, respectively. The maximum vertical displacement in the

middle point of each beam is established in l=300 ¼ 1:86� 10�2 m. The density and elasticity modulus are

the typical values of steel: 7850 kg/m3 and 2.1 · 105 MPa, respectively.

6. Experimental results

Experimental results of the proposed test case are exposed, for continuous and discrete optimization [19],

both for an ideal model (without buckling effect and without its own gravitational load) and for a real

model (considering a sidesway permitted buckling effect and the own gravitational load of the frame).

Finally, a real application case with multicriteria optimization and rebirth exploitation is presented.

6.1. Continuous optimization

The rebirth operator is designed for binary codification. It allows both the reduction of the search space,

and the reduction of the chromosome length. The main purpose of this paper is to show the advantages of

the auto-adaptive rebirth operator in discrete optimization, which is directly applicable to real designs. The

continuous optimization is performed for showing that successive rebirths can also improve this kind of

problems using binary coding.

The structural variables that characterize each cross-section type are approximated to the least-squaresapproximation: area (A), moment of inertia (I), modulus of section (W ), web area (Aw) and relation of beam

height (h1=h). The obtained relations are the following:


• IPE:

I ¼ 0:7918 � A2:32; W ¼ 0:727 � A1:667;

Aw ¼ 0:244542908759 � A1:0808017755; h1=h ¼ 0:687492831221� 0:0335326633632 � LnðAÞ:

• HEB:

I ¼ 0:148654970281 � A2:43355215167; W ¼ 0:31 � A1:73;

Aw ¼ 0:055 � A1:244; h1=h ¼ 0:397930760048 � A0:119:

Initially, the results obtained for the ideal model (without buckling effect and without the own gravi-

tational load) are shown, to be contrasted with the reference problem.Eight bits are codified for each variable and the following bars are joined as a unique variable for the

evolutionary algorithm: (a) Beams: 6–11–16, 7–12–17, 8–13–18, 9–14–19, 10–15–20; (b) Columns: 36–41,

37–42, 38–43, 39–44, 40–45; whose requirements are very similar. Therefore, 40 independent variables are

obtained. The search space is reduced to: 240�8 ¼ 2320 ffi 2� 1096. The codification interval of the variables is

reduced to 10�2 m2 and the Gray coding is introduced [20].

Moreover, the rebirth operator is included into the best of the three previous strategies, applying it to the

fitness function evaluation number 25,000, with a phenotypic variable reduction to ±40, and a second one

in evaluation 60,000 with variable reduction to ±24. A considerable improvement is achieved with thisstrategy, as it can be seen in Figs. 2 (mean) and 3 (variance). The obtained optimum until 110,000 eval-

uations (from thirty independent executions) is 9444.54 kg and null constraints.

Having analysed the improved results obtained via the rebirth operator, the optimization is carried out

with a progressive and continuous rebirth during the whole evolution, representing the obtained results in

Fig. 4, for a reduced codification of 40 variables (rhombus without rebirth) and a complete of 55, both

coded with 8 bits per bar. The considered strategies are:

Forty variables (continuous line): Rebirth in the function evaluation numbers: (in thousands): 25, 60, 95,

130, 165, 195, 225, 255; with interval amplitudes, respectively of ±40, ±24, ±14.4, ±8.64, ±5.16, ±3.11,±1.87 and ±1.12 (10�4 m2) (progressive reductions of 60%).

Fig. 2. Average of 30 cases, ideal model. Number of function evaluations from 2000 to 110,000.

Fig. 3. Variance of 30 cases, ideal model. Number of function evaluations from 2000 to 110,000.

Fig. 4. Average of 30 cases, ideal model. Number of function evaluations from 2000 to 350,000.


Fifty-five variables (dots): Rebirth in the function evaluation numbers: (in thousands): 30, 65, 105, 145,185, 225, 265, 305; with interval amplitudes, respectively of ±40, ±20, ±10, ±5, ±2.5, ±1.25, ±0.625 and

±0.3125 (10�4 m2) (progressive reductions of 50%).

Obtaining a mass of 9327.16 kg as the best solution (constraint R ¼ 0:20 kg). This solution is considered

of a similar magnitude order as the reference solution [18], considering the slightly different least-square

approximations taken into account. It corresponds to continuous optimization, calculated only for the ideal


model, without own load and without buckling effect. In the reference, a value of 9234 kg is reported, butthe evaluation of the structure mass using the detailed areas of the reported solution, follows to a mass of

9289 kg.

It can be observed in Fig. 4, how the chromosome codification of 55 variables implies slower conver-

gence than the codification of 40 variables; however, the successive rebirth operator cancels this drawback,

permitting a faster and more accurate solution (in this figure the first stretch of the 40 variables without

rebirth convergence is also represented).

It is noteworthy how the evolutionary algorithms are capable to approximate precisely the stresses up to

the yield values imposed to the bars, as it can be observed in Table 1. The solution bar areas and theirstresses are detailed.

Next, results from the real model (including sidesway permitted buckling effect and gravitational own

load in all bars) are presented:

We consider first a codification of 8 bits per variable and joining the following bars as a unique variable

for the evolutionary algorithm: (a) Beams: 6–11–16, 7–12–17, 8–13–18, 9–14–19, 10–15–20; (b) Columns:

36–41, 37–42, 38–43, 39–44, 40–45, whose requirements are very similar.

The search space is reduced to: 240�8 ¼ 2320 ffi 2� 1096. The interval range of the variables is 100 cm2 and

a Gray code is used. The results are shown in Figs. 5 and 6.In the fourth applied evolutionary strategy, steady-state with rebirth, the evaluation functions where it is

applied, are: 25,000, with a phenotypic variable reduction to ±40 (10�4 m2) and a second rebirth in 60,000

with a phenotypic variable reduction to ±20 (10�4 m2). The obtained optimum until 110,000 evaluations

(from thirty independent executions) is 9674.73 kg.

Having analysed the improved results obtained via the rebirth operator also in the real model, analo-

gously to the ideal model cases, the optimization is carried out with a progressive and continuous rebirth

during the whole evolution, representing the obtained results in Fig. 7, for a reduced codification of 40

variables (rhombus without rebirth) and a total of 55, both coded with 8 bits per bar. The consideredstrategies are:

Forty variables (continuous line): Rebirth in the function evaluation numbers: (in thousands): 25, 60, 95,

130, 165, 195, 225, 255; with interval amplitudes, respectively of ±40, ±24, ±14.4, ±8.64, ±5.16, ±3.11,

±1.87 and ±1.12 (10�4 m2) (progressive reductions of 60%).

Fifty five variables(dots): Rebirth in the function evaluation numbers: (in thousands): 30, 65, 105, 145,

185, 225, 265, 305; with interval amplitudes, respectively of ±40, ±20, ±10, ±5, ±2.5, ±1.25, ±0.625 and

±0.3125 (10�4 m2) (progressive reductions of 50%).

Obtaining a mass of 9551.75 kg (constraint R ¼ 0:13 kg) as the best solution, which is 2.4% over the bestobtained solution belonging to the ideal model (9327.16 kg).

Similar behaviour of the tested strategies can be observed in the real model as in the ideal model,

coinciding in the advantages of the successive rebirths.

6.2. Discrete optimization

The used codification considers each bar (the total number of bars is 55) as an isolated variable for the

evolutionary algorithm, being every variable chosen from the 16 lower cross-section types of the series IPE(beams) and HEB (columns). Thus, the search space is equal to 255�4 ¼ 2220 ffi 1:7� 1066.

The rebirth operator is included, applying it to the fitness function evaluation number 120,000, with a

phenotypic variable reduction from 16 to 4. The reference value of the variable is the second of the four

possible values, so it is allowed a search between two lower cross-section types and a higher. When this

interval surpasses the inferior or superior limit of the initial 16 cross-section types, the four lower or higher

are selected. A second rebirth is here also applied to evaluation 145,000.

Table 1

Stresses and areas of beam and columns of the continuous ideal case best solution achieved

GA area (10�4 m2) Stress (10 MPa)

Beam number

1 56.84 2194.79

2 55.04 2198.28

3 54.94 2194.23

4 52.41 2198.52

5 53.28 2197.78

6 51.64 2199.53

7 51.21 2197.19

8 50.47 2199.52

9 50.50 2199.23

10 52.68 2199.45

11 52.15 2198.88

12 51.48 2199.83

13 51.23 2199.99

14 50.89 2198.36

15 51.17 2199.06

16 52.92 2199.78

17 51.29 2198.56

18 51.40 2199.52

19 50.42 2199.11

20 51.13 2199.64

21 49.74 2198.96

22 48.67 2196.58

23 49.13 2199.87

24 49.19 2198.31

25 51.11 2197.12

Column number

26 54.07 1999.78

27 61.60 1987.03

28 59.22 1999.77

29 43.56 1989.00

30 66.01 1999.71

31 82.76 2000.02

32 70.32 1998.03

33 57.03 1998.66

34 40.78 1998.62

35 26.18 1991.58

36 76.99 1999.57

37 63.11 1999.95

38 49.89 1999.85

39 37.22 1999.01

40 23.29 1997.47

41 78.40 1999.65

42 63.63 2000.04

43 50.95 1999.91

44 38.59 1999.04

45 23.64 1999.00

46 78.18 1999.54

47 62.69 1998.34

48 49.97 1998.64

49 39.26 1998.53

50 23.25 1996.03

(continued on next page)


Fig. 5. Average of 30 cases, real model. Number of function evaluations from 2000 to 110,000.

Table 1 (continued)

GA area (10�4 m2) Stress (10 MPa)

51 66.79 1989.60

52 74.15 1999.56

53 72.13 1960.19

54 53.69 1999.49

55 74.13 1999.98


The optimum values from a set of 30 independent executions are:

(A) Ideal model: mass¼ 9852.32 kg; restriction¼ 0.0 kg.

(B) Real model: mass¼ 10128.7 kg; restriction¼ 1.38 kg.

Figs. 8 and 9 show the statistics of the evolution of the 30 executed cases, with comparison to the reduced

codification of 40 independent variables of the evolutionary algorithm (continuous line in the figures). The

slowdown of convergence can be observed in the case of 55 variables (dots in the figures), what certifies that

a greater search space (greater chromosome length) also implies slower convergence, although the func-

tional space of the problem is the same.Moreover, it is noteworthy, that in case of a reduced codification, the evolution is stopped in the fitness

function 70,000, and therefore, the figure has maintained this final value in order to compare it with a larger

and more accurate codification of 55 independent variables. It can be seen the clear solution gain in average

and also in variance reduction after the rebirth application. In the real model the improvement in the

obtained solution is also appreciable.

It has been verified through the gained performance of the rebirth operator, that combining simulta-

neously the injection of more diversity with the diminishing of the search space, it allows the stagnation of

the convergence to surpass and cause a slope increment in the evolution. The best average results and thebest results are achieved, and moreover a decrease in variance, as it can be seen in the figures (which is

applied either for continuous and discrete optimization) is reached.

Fig. 7. Average of 30 cases, real model. Number of function evaluations from 2000 to 350,000.

Fig. 6. Variance of 30 cases, real model. Number of function evaluations from 3000 to 110,000.


The utility of the rebirth operator can be increased if it is successively applied until reaching a definitiveconvergence without a further better evolution. The rebirth is applied automatically after satisfying a

stagnation condition in the population (parameter RP: certain number of fitness function evaluations

without improvement in the best value). Thus, here an auto-adaptive rebirth is implemented, which applies

successive rebirths as many times as population stagnates, allowing an improvement of the population

evolution to the optimum.

Fig. 8. Average of 30 cases. Number of function evaluations from 2000 to 180,000.

Fig. 9. Variance of 30 cases. Number of function evaluations from 4000 to 180,000.


When this method is evaluated in 30 independent executions in discrete optimization for the real and the

ideal models of the frame test case, we obtain average solutions that have a similar quality to the previous

tested case (two consecutive rebirths at certain points of the evolution), but needing less functional eval-uations. The encountered optima in a series of thirty executions are:

Ideal model: mass¼ 9852.32 kg; restriction¼ 0.0 kg.

Real model: mass¼ 10128.7 kg; restriction¼ 1.38 kg.


Figs. 10 and 11 show the compared evolutions of the auto-adaptive rebirth versus the case of Fig. 8, forboth real and ideal models.

It can be observed a better evolutionary performance with the auto-adaptive rebirth, characterized with

a smoother average convergence; although the variance is significantly incremented in the middle part of

evolution, because the application time of the rebirth operator is not simultaneous among the different

executions. Therefore, it causes a higher divergence in the reached solutions during evolution, what implies

Fig. 10. Average of 30 cases. Number of function evaluations from 5000 to 180,000.

Fig. 11. Variance of 30 cases. Number of function evaluations from 2000 to 180,000.


a higher variance. This effect is diminished when we bring near the final stages of the algorithm evolution,when each independent execution has converged to high quality and fast equal solutions.

If we compare the best discrete solution achieved directly by the evolutionary algorithm with the discrete

solution that is obtained by upper approximation of the continuous evolutionary optimum solution, we

appreciate the actual necessity of a discrete global frame optimization method, which the described evo-

lutionary algorithm can provide. Table 3 shows the difference in terms of mass of the solutions obtained by

each method and row D indicates the percentage the mass gains between both solutions. In Table 2 the

discrete solutions in the real model case are detailed, and highlighted in italics the differences.

6.2.1. Influence of the population size and the space reduction

First, we show some figures (Figs. 12–15) comparing a medium population size (50 individuals) with a

small one (eight individuals). We present some results varying the saturation parameter RP which defines

when auto-adaptive rebirth is applied. The values of RP¼ 3000, 6000, 9000 and 12,000 are chosen, defining

the number of function evaluations without change in the best individual of the population. Figures reflect

the average of 30 different executions and the real model of the discrete optimization test case is considered.

We observe that in each case the small population size allows an initial faster convergence, but a worse

final result, following to an increased amount of suboptima values (its convergence line is always upperrespect to the medium population size algorithm).

This can be explained because the quality of the initial best located solution previous to the first rebirth

application, depends highly on the population size, even more with greater search spaces: they need more

population diversity––it depends on the population size––to avoid premature stagnation (this test case has a

2220 search space).

Second, we show some figures (Figs. 16–18) comparing the algorithm with auto-adaptive rebirth and the

algorithm with only reinitialization in a population of eight individuals (auto-adaptive rebirth versus micro-

GA). Both methods use reinitialization as a way to increase the population diversity after the saturation ofthe population, but the micro-GA does not use a reduction of the search space, as the auto-adaptive rebirth

does. The beneficial effect of the space reduction introduced by the rebirth is clear shown in the graphs, both

in terms of speed convergence and in terms of quality of the final results––independently of the RP

parameter.

6.2.2. Influence of the population saturation parameter (RP)

The time to switch to a restart has to allow the algorithm to complete its convergence. If it is too short, it

might follow to a premature rebirth that will take a suboptima as the central point of the reduction of thesearch space, which could lead convergence to suboptima solutions. If it is too long, we are lengthening the

convergence.

The following figures reflect the average values (Fig. 19) and variances (Fig. 20) of 30 different executions

of the real model of the discrete optimization test case. Four different values of the saturation parameter RP

are chosen: 3000, 5000, 9000 and 12,000, defining the number of function evaluations without change in the

best individual of the population. The value of 3000 leads to more premature convergence and higher

variance than the others, but the value of 12,000 has the lowest convergence speed.

An appropriate selection of the saturation parameter is a value that has a fast convergence without lossof final accuracy.

The reported value of 5000 has a faster convergence speed than 9000 and 12,000 without significant loss

of accuracy, seeing Fig. 19 (however its variance is slightly worse compared with those two values).

To select an appropriate value of the saturation parameter, the chromosome length is the main factor to

be taken into account, because the search space size (in this case 220 bits) mainly determines the speed of

convergence. Shorter chromosome lengths allow shorter RP values.

Table 2

Detailed cross-section types of real model discrete solutions

Bar number

(areas in 10�4 m2)

Area continuous EA

(M ¼ 9551:75 kg)

Area upper discrete

approximation (M ¼ 10343:78 kg)

Area discrete EA

(M ¼ 10128:7 kg)

1 57.00 IPE330 IPE330

2 55.30 IPE330 IPE330

3 55.07 IPE330 IPE330

4 52.79 IPE300 IPE300

5 53.78 IPE300 IPE330

6 51.99 IPE300 IPE300

7 51.60 IPE300 IPE300

8 51.00 IPE300 IPE300

9 50.48 IPE300 IPE300

10 52.82 IPE300 IPE300

11 52.41 IPE300 IPE300

12 51.81 IPE300 IPE300

13 51.53 IPE300 IPE300

14 51.06 IPE300 IPE300

15 51.26 IPE300 IPE300

16 53.15 IPE300 IPE300

17 51.81 IPE300 IPE300

18 51.96 IPE300 IPE300

19 50.98 IPE300 IPE300

20 51.93 IPE300 IPE300

21 49.98 IPE300 IPE300

22 48.65 IPE300 IPE300

23 49.67 IPE300 IPE300

24 49.79 IPE300 IPE300

25 51.95 IPE300 IPE300

26 55.21 HEB180 HEB160

27 62.10 HEB180 HEB180

28 59.68 HEB180 HEB160

29 44.58 HEB160 HEB140

30 65.69 HEB200 HEB180

31 83.19 HEB220 HEB220

32 72.41 HEB200 HEB200

33 58.40 HEB180 HEB180

34 44.57 HEB160 HEB160

35 31.87 HEB120 HEB120

36 78.93 HEB220 HEB200

37 67.09 HEB200 HEB200

38 54.37 HEB180 HEB160

39 42.49 HEB140 HEB140

40 30.38 HEB120 HEB120

41 79.52 HEB220 HEB220

42 67.57 HEB200 HEB200

43 54.73 HEB180 HEB160

44 42.95 HEB140 HEB140

45 30.43 HEB120 HEB120

46 80.69 HEB220 HEB220

47 68.20 HEB200 HEB200

48 55.47 HEB180 HEB160

49 43.40 HEB160 HEB140

50 30.48 HEB120 HEB120

51 69.55 HEB200 HEB180

(continued on next page)


Table 3

Best mass of different optimization methods

Optimization method Ideal model (kg) Real model (kg) Mass real model/mass ideal

model (%)

A: Continuous by EA 9327.16 9551.75 2.4

B: Discrete upper approximation

of continuous solution

10031.00 10343.80 3.1

C: Discrete by EA 9852.32 10128.70 2.8

D: B/C 1.8% 2.1%

Fig. 12. Influence of population size. Average of 30 independent runs. Discrete frame optimization problem, RP¼ 3000.

Table 2 (continued)

Bar number

(areas in 10�4 m2)

Area continuous EA

(M ¼ 9551:75 kg)

Area upper discrete

approximation

(M ¼ 10343:78 kg)

Area discrete EA

(M ¼ 10128:7 kg)

52 73.82 HEB200 HEB200

53 71.35 HEB200 HEB180

54 54.12 HEB160 HEB180

55 72.81 HEB200 HEB200


Two additional frame structure test cases are analysed here. The second one has 45 bars and 30 nodes,

and it is represented in Fig. 21, considering that dimensions, loads and characteristics are the same as the

test case represented in Fig. 1 (including the uniform load in the beams). The best value found in 30

independent executions is 8290.6 kg with a restriction of 5.9 kg. The average results of median and variance

are represented in Figs. 22 and 23.A third frame structure test case is also analysed. It has 35 bars and 24 nodes, and it is represented in

Fig. 24, considering that dimensions, loads and characteristics are the same as the test case represented in




Fig. 1 (including the uniform load in the beams). The best value found in 30 independent executions is 6405.9

kg with a restriction of 9.7 kg. The average results of median and variance are represented in Figs. 25 and 26.

The two test cases reinforce the fact, how the auto-adaptive rebirth can obtain not only faster and lower

massed optimized solutions (in terms of the average figures), but also more robust solutions (in terms of the

variance figures), as can be seen in the graphs 22 and 23 (corresponding to the second test case), 25 and 26

(corresponding to the third test case).

Fig. 16. Influence of space reduction. Average of 30 independent runs. Discrete frame optimization problem, RP¼ 1000.

Fig. 15. Influence of population size. Average of 30 independent runs. Discrete frame optimization problem, RP¼ 12,000.


6.2.3. Comparing rebirth versus auto-adaptive rebirth

Compared with rebirth, an auto-adaptive rebirth operator, that applies rebirth consecutively improves

the convergence behaviour, as can be seen in the following Figs. 27–32. Figures reflect the average of 30

different executions and the real model of the first discrete optimization test case is considered. Different

values of the saturation parameter RP are considered (it defines the fitness function evaluations without

improvements in the best individual fitness––saturation criterion, after which rebirth is applied).

Fig. 17. Influence of space reduction. Average of 30 independent runs. Discrete frame optimization problem, RP¼ 6000.

Fig. 18. Influence of space reduction. Average of 30 independent runs. Discrete frame optimization problem, RP¼ 12,000.


Figs. 28–32 show how the auto-adaptive rebirth cause a faster and more accurate convergence than thesimple application of one rebirth and also than the standard algorithm.

The procedure of space reduction is applied after the saturation of the algorithm. The rebirth operator is

designed as a tool that allows a further search after the saturation of the algorithm, precisely, for avoiding

the premature stagnation and the loss of the global optimum. It applies a new search process after the

evolutionary algorithm saturation, where the best found individual is maintained, and the rest of the

Fig. 19. Influence of RP parameter. Average of 30 independent runs. Discrete frame optimization problem.

Fig. 20. Influence of RP parameter. Variance of 30 independent runs. Discrete frame optimization problem.


population is reinitialized (increasing the diversity in the population: exploration emphasis), but reducing

the search space around the values of the variables of this best individual (exploitation emphasis). It is

possible, however, after its application and after achieving the saturation of the population, that the global

optimum is still not reached. The auto-adaptive rebirth implements successive applications of rebirth to

reduce this effect.

Fig. 21. Second frame test case.

Fig. 22. Influence of RP parameter. Average of 30 independent runs. Second discrete frame optimization problem.


So, the falling in a local minima region is consequence of a bad convergence of the evolutionaryalgorithm. It could be induced by a premature application of the rebirth if the saturation parameter RP is

too short and does not allow a complete convergence. The auto-adaptive rebirth operator, by successive

applications of rebirth contributes to reduce this possible effect (as can be seen, for example, in Figs. 27 and

28), where rebirth leads to suboptima values (however, a correct choose of this parameter obviates this

effect).

Fig. 23. Influence of RP parameter. Variance of 30 independent runs. Second discrete frame optimization problem.

Fig. 24. Third frame test case.


6.3. Multiobjective optimization

As a real application case of structural optimization, we suppose a symmetric configuration of the test

case. The calculation of the supported loads and the applied combination coefficients are taken into ac-

count as explained in the Spanish regulation. With these considerations, we obtain seven load cases (one

static load, two live loads, one snow load, one wind load and two earthquake loads) and 13 load hypothesis

to solve. The distance between porticos is of 5.6 m. Having been supposed to have an office use, and being

free of buildings in its perimeter, it will suffer wind loadings from both sides and have sidesway buckling

Fig. 25. Influence of RP parameter. Average of 30 independent runs. Third discrete frame optimization problem.

Fig. 26. Influence of RP parameter. Variance of 30 independent runs. Third discrete frame optimization problem.


effect. The considered yield stress is of 260 MPa and the limit displacement in the middle point of each beamis of l=500––what is equivalent to 1.12 · 10�2 m, because it has to support partition walls. The real model is

taken into account (considering own load and buckling effect).

Only 30 different variables are necessary, because of the imposed symmetry of the frame. Therefore, the

following bars are grouped into the same evolutionary variable (they have the same cross-section type in the

final solution designed frame): 1–21, 2–22, 3–23, 4–24, 5–25, 6–16, 7–17, 8–18, 9–19, 10–20, 11, 12, 13, 14,

Fig. 27. Rebirth versus auto-adaptive rebirth. Average of 30 independent runs. Discrete frame optimization problem, RP¼ 1000.



15, 26–51, 27–52, 28–53, 29–54, 30–55, 31–46, 32–47, 33–48, 34–49, 35–50, 36–41, 37–42, 38–43, 39–44,

40–45. The search space is: 230�4 ¼ 2120 ffi 1:3� 1036.

Moreover, a multicriteria optimization [21–28] is computationally implemented and programmed,

minimizing simultaneously the constrained mass (cost requirement) and the minimum number of cross-

section types (constructive requirement). The non-dominated sorting genetic algorithm (NSGA) of Srinivas




and Deb [29] and Deb [30], have been implemented and associated by a elitist operator and certain

duplicates elimination in the offspring population creation [3,31]. This strategy is defined as follows:

Elitism maintaining the half of the parent population and inserting the best half of the offspring popu-

lation; the insertion of duplicated individuals is avoided; population size of 100 individuals; Gray code

Fig. 31. Rebirth versus auto-adaptive rebirth. Average of 30 independent runs. Discrete frame optimization problem, RP¼ 12,000.

Fig. 32. Rebirth versus auto-adaptive rebirth. Average of 30 independent runs. Discrete frame optimization problem, RP¼ 15,000.


codification of the chromosome [20]; uniform crossover; Roulette Wheel Selection (RWS) [32] with linealprobabilities allocation based in the NSGA ranking; crossover rate of 80% and mutation rate of 0.8% [31].

Well known evolutionary multiobjective optimization algorithms have been applied to the multiobjective

frame optimization problem handled in this article with success in a 4 bars frame structure in [33]. The

introduction of the above described method, with certain duplicates elimination is satisfactorily showed in

[34], compared to methods without this operator. Here is presented a further improvement, which consists

in introducing in the initial population a high quality solution in terms of the constrained mass optimi-

Fig. 33. Optimum front reached in the case of initial random population.


zation problem (first fitness function). This high quality solution can be provided by the auto-adaptive

rebirth operator, leading to better final fronts.

Two independent executions are tried, with a population size of 100 individuals, 800 generations and a

mutation rate of 0.8%. The optimum front obtained from this executions is shown in Fig. 33.

Fig. 34. Optimum front reached in the case of initial random population with included solution with auto-adaptive rebirth operator.

Table 4

Final front with initial random population. Multiobjective real application case

Weight 9743.29 9778.46 9814.1 9902.87 10034.7 10017.1

Relative weight 1.000 1.004 1.007 1.016 1.030 1.028

Constraints mass 0.0 0.0 10.7 0.0 0.0 485.9

Objective function: constrained mass 9743.29 9778.46 9824.8 9902.87 10034.7 10503

Different cross-section types number N ¼ 9 N ¼ 8 N ¼ 7 N ¼ 6 N ¼ 5 N ¼ 4

Bar number 1–21 IPE330 IPE330 IPE330 IPE330 IPE330 IPE330










Bar number 11 IPE300 IPE300 IPE300 IPE300 IPE300 IPE330





Bar number 26–51 HEB160 HEB160 HEB160 HEB180 HEB180 HEB180
















It is noticeable how the optimum solution in terms of constrained mass, that for nine different cross-section types, is far away from the best single criteria solution, obtained with auto-adaptive rebirth.

In order to benefit from the collected information obtained via the auto-adaptive rebirth, and present in

this best solution, it will be inserted in the initial random population of the multicriteria elitist NSGA

algorithm, and two independent runs executed (same parameter values as the previous case). The results

show that the final front has improved significantly, and in certain circumstances, a high quality solution

can produce a better convergence of a multicriteria algorithm.

Fig. 34 shows a comparative of both obtained optimum fronts, in rhombus the elitist NSGA with initial

random population, and in crosses the elitist NSGA in which a solution obtained with the steady-statestrategy and auto-adaptive rebirth has been inserted in the initial population. It can be observed how this

obtained rebirth solution leads to an improved and more usable front for the engineer designer decision (to

choose the more suitable considered frame; here from 5 to 9 different cross-section types, varying from a

mass of 9605 and 9915 kg). In Tables 4 and 5 the complete detailed solutions of the fronts represented in

Figs. 27 and 28 are shown.

Table 5

Final front with initial population including auto-adaptive rebirth solution. Multiobjective real application case

Weight 9605.7 9640.9 9689.6 9753.4 9915.6 10017.1

Relative weight 1.000 1.004 1.009 1.015 1.032 1.043

Constraints mass 0.0 0.0 9.0 39.9 0.6 485.9

Objective function: constrained mass 9605.7 9640.9 9698.6 9793.3 9916.2 10503.0

Different cross-section types number N ¼ 9 N ¼ 8 N ¼ 7 N ¼ 6 N ¼ 5 N ¼ 4
































7. Conclusions

This article has handled with medium-sized frame structures (test cases of 55, 45 and 35 bars are con-

sidered), where the evolutionary optimization requires more fitness function evaluations than in frames of

smaller sizes (because of the increment in search space and chromosome length). The rebirth operator, and

more concretely the auto-adaptive rebirth has been introduced here as a necessary tool to improve the

optimization and obtain good solutions. Also general recommendations about code optimization have been

described, in terms of algorithm implementation and coupling between the evolutionary algorithm and theframe calculation code, whose recommended structure is detailed, saving CPU execution time (they follow

in Appendix A).

Considering the above-mentioned, the results have evidenced the necessity of a global and discrete

optimization procedure which can be provided by the evolutionary algorithms: the penalty of using an

upper discrete approximation obtained from a continuous frame optimization in comparison with the

direct discrete optimization is evaluated here in about 2% of mass.


Analysing the obtained results exposed in this article and having compared three different strategies ofevolutionary algorithms, the best strategy is the steady-state evolutionary algorithm, what coincides with

other results in smaller frames [3,17]. It is noteworthy how the auto-adaptive rebirth operator applied to

this best tested strategy permits to accelerate the convergence of the algorithm, compared to the standard

rebirth operator, and also allows to obtain solutions of improved precision. This improvement occurs

independently of the considered model (both in the ideal and real models) and independently of the cod-

ification used to solve the problem (continuous and discrete optimization with real cross-section types).

A discrete multicriteria frame optimization (minimizing constrained mass and number of different cross-

section types) is performed successfully via evolutionary algorithms, which provide a complete front in onesingle run, even for this frame structure size. Moreover, the introduction in the initial population of high

quality single-optimization solutions obtained via the auto-adaptive rebirth operator is proposed as a way

to improve the final non-dominated fronts obtained in structural frame multicriteria optimization.

The auto-adaptive rebirth is a general purpose operator in evolutionary algorithms, so it would be worth

extending its application to other kind of problems and optimization fields––where evolutionary algorithms

are suitable––which can benefit from the advantages of its use in terms of faster and improved average

results, and also enhanced robustness.

Acknowledgements

Authors gratefully thank the funding of this research by the AP2001-3797 grant of the Education and

Universities Secretary of Spanish Government.

Appendix A. Optimization considerations

The fitness function of the evolutionary algorithm is needed to be evaluated multiple times, therefore any

effort made to reduce the execution time of the problem resolution will be worthwhile benefited from the

evolutionary algorithm CPU time necessities. The time performance improvement will be carried out by

two ways:

1. Optimizing the algorithm programming, using properly the possibilities that the programming language

ANSI C offers: computational considerations.2. Optimizing the objective function algorithms and its coupling with the evolutionary algorithm: struc-

tural considerations.

A.1. Computational considerations

The programming optimization consists often in a balance between execution time and occupied

memory or code size. Whenever the first factor is improved, the second is penalized and vice versa.

The programming language used to implement and develop the evolutionary and structural codes isANSI C. The great majority of evolutionary algorithms are developed in the C/C++ language environment.

Moreover, its additional characteristics of low-level language allows to perform operations at the bit level,

which are also the most computationally economical ones. When possible, during the evolutionary algo-

rithms programming, it is advisable to use this characteristic, so for example, using the internal repre-

sentation of the computer (binary) to store the chromosomes can accelerate the execution time

tremendously (an unsigned long int variable in a conventional Pentium architecture is interpreted as a

binary string of 32 bits).


We show a computational implementation of uniform binary mutation. The mutation operator can beimplemented through a mutation mask where a 0 implies no mutation and a 1 implies mutation. When

generating the 1s with pm rate for each bit, a XOR between the mutation mask and the chromosome

performs the mutation of the chromosome. A proposed way with low cost is the successive combination of

randomly generated masks with the AND operator, so reducing progressively the 1s appearance until the

desired probability. The C code of the subroutine is showed, being:

mutationmask: the mutation mask.

nument: number of long int contained in the chromosome.numindivfil: number of individuals in the offspring population.

nprob: number that depends on the mutation rate.

random32(): 32 bits pseudo-random generator.

ðnuevmemberþ jÞ ! fraccromos þ k: pointer to the k fraction of the chromosome belonging to the jindividual of the offspring population (nuevmember).

mutationmask ¼ ðunsigned long int�Þmallocðnument � sizeofðunsigned long intÞÞ;forðj ¼ 0; j < numindivfil; jþþÞ{

forðk ¼ 0; k < nument; k þþÞ{

�ðmutationmaskþ kÞ ¼� 0;forði ¼ 0; i < nprob; iþþÞ{

�ðmutationmaskþ kÞ& ¼ random32ðÞ;}

�ððnuevmemberþ jÞ ! fraccromos þ kÞ^ ¼ �ðmutationmaskþ kÞ;}

}

free(mutationmask);

Among the different strategies of computational optimization the following can be cited:

Use of pointers instead of arrays, use of inline functions in spite of the growth size of the code size (of

great utility when functional calls exist inside loops with a great number of iterations), use of the storage

variable type register; Loop operations, such as: sorting of nested loops from external to internal from low

to high iterations, so reducing the initialization and termination condition numbers; Elimination of

invariant code inside loops, which avoids the repetition of a sentence inside a loop unnecessarily when its

execution is possible externally; Loops unfolding can be sometimes beneficial, like in case of a constantexternal loop, which can be mixed with the internal, or a combination of various loops if they have the same

execution limits; Elimination of redundant test inside loops.

Moreover, it is noteworthy to use those compiler options oriented to optimize the code speed. In the

particular case of using a Linux based operation system and the gnu compiler, they are grouped in the –O

options.

A.2. Structural considerations

In order to achieve as many reduced execution times as possible, the evaluation process governed by the

fitness function should be speeded up. This applies in our handled problem to the direct stiffness method,

which gives the information to evaluate each candidate solution of the evolutionary algorithm.


Therefore, those operations that are required only once by the direct stiffness calculation and that arevalid for every frame independently of the cross-section type of their bars should be calculated only once; so

these operations will be saved as many times as the number of fitness evaluations.

The direct stiffness calculation will be split in two blocks:

First block: It includes initial operations that are calculated only ONCE in each evolutionary algorithm

execution and that are valid for each frame. They are the following:

• Reading of the files containing the data which define the frame to optimize. This reading is analogous to

what does a conventional structural calculation program.• Calculation of bar length and global angles among bars.

• Calculation of the optimized bar numbering via the inverse Cuthill–McKee algorithm.

• Calculation of the string that stores the active or effective equations of the frame.

• Calculation of the bandwidth of the frame matrix.

• Calculation of the global vector of forces for each load case, including its assembly (excluding the own

load case and the settlement load case which depend on the particular frame and are calculated later).

Second block: It includes those operations that are evaluated as MANY TIMES as function evaluationsare required by the evolutionary algorithm, saving all the operations that are calculated in the first block.

The remaining operations to complete the direct stiffness method are the following:

• Calculation of the effective stiffness matrix of the total frame.

• Consideration of the own load and settlement load as special load cases.

• Resolution of the system K � U ¼ F and the achievement of the node displacements.

• Calculation of the efforts of the structure.

• Calculation of each bar slenderness, and also of displacements and stresses for each load hypothesis.

Depending on the available information given by the matrix calculation program via the direct stiffness

method, the fitness function evaluation for each candidate solution can be reached, measuring its adap-

tation to the environment, what will be the tool to achieve the optimization of the handled frame.

The calculation of the second fitness function: the number of different cross-section types is done by

successive comparisons of the existing cross-section types in a certain structure. The developed algorithm

requires in its most unfavourable case (all the cross-section types different), a total of: ðn� 1Þ þ ðn� 2Þþðn� 3Þ þ � � � þ 2þ 1 ¼ n � ðn� 1Þ � 0:5 comparisons, being n the number of different codified bars in theevolutionary algorithm. In the most favourable case, when all the cross-section types of the bars are equal, a

total of ðn� 1Þ comparisons is required. A pseudo-code of the algorithm can be obtained in [33]. So, the

evaluation of the second fitness function varies between OðnÞ and Oðn2=2Þ.

References

[1] D.E. Goldberg, Genetic Algorithms for Search, Optimization, and Machine Learning, Addison Wesley, Reading, MA, 1989.

[2] G. Winter, M. Gal�an, P. Cuesta, D. Greiner, Genetic algorithms: a stochastic improvement technique. Tools, skills, pitfalls and

examples, in: G. Winter, J. Periaux, P. Cuesta, M. Gal�an (Eds.), Genetic Algorithms in Engineering and Computer Science, John

Wiley & Sons, 1995, pp. 217–249.

[3] D. Greiner, G. Winter, J.M. Emperador, Optimizing frame structures by different strategies of genetic algorithms, Finite Elem.

Anal. Des. 37-5 (2001) 381–402.

[4] D.E. Goldberg, M.P. Santani, Engineering optimization via genetic algorithm, in: Proceedings of the Ninth Conference on

Electronic Computation, ASCE, New York, 1986, pp. 471–482.

[5] M. Galante, Genetic algorithms as an approach to optimize real-world trusses, Int. J. Numer. Meth. Engrg. 39 (1996) 361–382.


[6] P. Hajela, C. Lin, Genetic search strategies in multi-criterion optimal design, Struct. Optimizat. 4 (1992) 99–107.

[7] K. Sarma, H. Adeli, Life-cycle cost optimization of steel structures, Int. J. Numer. Meth. Engrg. 55 (2002) 1451–1462.

[8] M. Liu, S.A. Burns, Y.K. Wen, Optimal seismic design of steel frame buildings based on life cycle cost considerations, Earthquake

Engrg. Struct. 32-9 (2003) 1313–1332.

[9] J.H. Holland, Genetic Algorithms, Scientific American, 1992, pp. 38–45.

[10] D. Whitley, The GENITOR algorithm and selection pressure: why rank-based allocation of reproductive trials is best, in: D.

Schaffer (Ed.), Proceedings of the Third International Conference on Genetic Algorithms, Morgan Kaufmann Publishers, 1989,

pp. 116–121.

[11] K. Krishnakumar, Micro-genetic algorithms for stationary and non-stationary function optimization, SPIE: Intell. Control

Adapt. Syst. 1196 (1989).

[12] L. Eshelman, The CHC adaptive search algorithm: how to have safe search when engaging in nontraditional genetic

recombination, in: G. Rawlings (Ed.), I Foundations of Genetic Algorithms, Morgan Kaufmann Publishers, 1990, pp. 265–284.

[13] L. Eshelman, J.D. Schaffer, Preventing premature convergence in genetic algorithms by preventing incest, in: R.K. Belew, L.B.

Booker (Eds.), Proceedings of the Fourth International Conference on Genetic Algorithms, Morgan Kaufmann Publishers, 1991,

pp. 115–121.

[14] D.J. Mikulin, Using genetic algorithms to fit HLGM data, Ph.D. Thesis, University of Exeter, 1997.

[15] D. Whitley, K. Mathias, P. Fitzhorn, Delta coding: an iterative search strategy for genetic algorithms, in: R.K. Belew, L.B. Booker

(Eds.), Proceedings of the Fourth International Conference on Genetic Algorithms, Morgan Kaufmann Publishers, 1991, pp. 77–

84.

[16] S. Rajeev, C.S. Krishnamoorthy, Genetic algorithms-based methodologies for design optimization trusses, J. Struct. Engrg.––

ASCE 123-3 (1997) 350–358.

[17] D. Greiner, G. Winter, J.M. Emperador, Genetic algorithm application in plane frame optimization problems, Ms. Thesis,

University of Las Palmas de Gran Canaria, Spain, 2000.

[18] S. Hern�andez Ib�aez, Methods for optimum design of structures, Coleccin Seinor, Colegio de Ingenieros de Caminos, Canales y

Puertos, Madrid, 1990 (in Spanish).

[19] S. Rajeev, C.S. Krishnamoorthy, Discrete optimization of structures using genetic algorithms, J. Struct. Engrg.––ASCE 118-5

(1992) 1233–1250.

[20] D. Whitley, S. Rana, R. Heckendorn, Representation issues in neighborhood search and evolutionary algorithms, in: D.

Quagliarella, J. Periaux, C. Poloni, G. Winter (Eds.), Genetic Algorithms and Evolution Strategies in Engineering and Computer

Science, John Wiley & Sons, 1997, pp. 39–57.

[21] K. Deb, Multiobjective Optimization using Evolutionary Algorithms, in: Series in Systems and Optimization, John Wiley & Sons,

2001.

[22] C. Coello, D. Van Veldhuizen, G. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, in: GENA Series,

Kluwer Academic Publishers, 2002.

[23] D. Greiner, B. Galv�an, G. Winter, Safety systems optimum design by multicriteria evolutionary algorithms, Lect. Notes Comput.

Sci. 2632 (2003) 722–736.

[24] C.A.C. Coello, A short tutorial on evolutionary multiobjective optimization, Lect. Notes Comput. Sci. 1993 (2001) 21–40.

[25] C.M. Fonseca, P.J. Fleming, E. Zitzler, K. Deb, L. Thiele (Eds.), Evolutionary Multi-Criterion Optimization, Springer Verlag,

2003.

[26] E. Zitzler, K. Deb, L. Thiele, C.C. Coello, D. Corne (Eds.), Evolutionary Multi-Criterion Optimization, Springer Verlag, 2003.

[27] R.F. Coelho, H. Bersini, P. Bouillard, Parametrical mechanical design with constraints and preferences: application to a purge

valve, Comput. Methods Appl. Mech. Engrg. 192 (39–40) (2003) 4355–4378.

[28] L. Costa, P. Oliveira, An adaptive sharing elitist evolution strategy for multiobjective optimization, Evolut. Comput. 11-4 (2003)

417–438.

[29] N. Srinivas, K. Deb, Multiobjective optimization using non-dominated sorting in genetic algorithms, Evolut. Comput. 2–3 (1995)

221–248.

[30] K. Deb, Evolutionary algorithms for multi-criterion optimization in engineering design, in: K. Miettinen, M. M€akel€a, P.

Neittaanm€aki, J. Periaux (Eds.), Evolutionary Algorithms in Engineering and Computer Science, John Wiley & Sons, 1999, pp.

135–161.

[31] D. Greiner, J.M. Emperador, G. Winter, Multiobjective optimization of bar structures by Pareto GA, in: Proceedings of the

European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, CIMNE, 2000.

[32] W. Mahfoud, Niching methods for genetic algorithms, Illigal Rep. no. 95001, 1995.

[33] D. Greiner, G. Winter, J.M. Emperador, B. Galv�an, A comparative analysis of controlled elitism in the NSGA-II applied to frame

optimization, in: T. Burczynski, A. Osyczka (Eds.), Proceedings of IUTAM Symposium on Evolutionary Methods in Mechanics,

Kluwer Academic Publishers, 2002.

[34] D. Greiner, G. Winter, J.M. Emperador, Searching for an efficient method in multiobjective frame optimization using

evolutionary algorithms, in: K. Bathe (Ed.), Computational Solid and Fluid Mechanics, Elsevier, 2003, pp. 2285–2290.

Documents

Single and multiobjective frame optimization by evolutionary algorithms and the auto-adaptive rebirth operator