Competitive genetic algorithms with application to reliability optimal design

Advances in Engineering Software 34 (2003) 773–785 www.elsevier.com/locate/advengsoft

Competitive genetic algorithms with application to reliability optimal design C.K. Dimou, V.K. Koumousis* Institute of Structural Analysis and Aseismic Research, National Technical University of Athens, Athens, Greece

Abstract Competition is introduced among the populations of a number of genetic algorithms (GAs) having different sets of parameters. The aim is to calibrate the population size of the GAs by altering the resources of the system, i.e. the allocated computing time. The co-evolution of the different populations is controlled at the level of the union of populations, i.e. the metapopulation, on the basis of statistics and trends of the evolution of every population. Evolution dynamics improve the capacity of the optimization algorithm to find optimum solutions and results in statistically better designs as compared to the standard GA with any of the fixed parameters considered. The method is applied to the reliability based optimal design of simple trusses. Numerical results are presented and the robustness of the proposed algorithm is discussed. q 2003 Elsevier Ltd. All rights reserved. Keywords: Structural optimization; Genetic algorithms; Competition; Population dynamics; Reliability analysis

1. Introduction Genetic algorithms (GAs) are search algorithms based on the concepts of natural selection and survival of the fittest. They guide the evolution of a set of randomly selected individuals towards good near optimal and/or optimal solutions. This is accomplished in a number of generations that are subjected to successive reproduction, crossover and mutation, based on the statistics of the generation. The efficiency of the whole process is problem dependent and relies heavily on the successful selection of a number of parameters, such as population size, probability of crossover and mutation, type of crossover, etc. In this work, a method is proposed that attempts to automate the evolution of population size through an adaptive process. This is based on the competition of populations, with different sets of GA parameters, struggling for the available resources of the system. Competition among different populations is common in natural systems. Populations evolve by adapting themselves to the environment where resources are limited. The process starts with the generation of number of populations with different sets of GA parameters. In * Corresponding author. Fax: þ 30-2107721651. E-mail address: [email protected] (V.K. Koumousis). 0965-9978/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0965-9978(03)00101-7

subsequent generations, as the populations evolve, a scheme of competing populations (CP) alters the population size in an adaptive manner based on the relative performance of the populations at the metapopulation level. By altering the available resources, competition is activated forcing the system to organize better its overall search strategy towards optimal solutions. The ability of every population to adapt to the artificial habitat is used to calculate the relative performance index at a particular generation, which represents a comparative measure between all the populations that comprise the metapopulation. Competition arises when the available resources are insufficient to sustain the entire metapopulation. This causes conflicts among the populations where the most-fit ones survive, whereas the less-fit populations shrink or become extinct. This coupled scheme manages to arrive at good near optimal solutions statistically faster, as compared to the same number of standard GAs with fixed parameters. Reliability optimal structural design problems are computationally intensive and represent a particular class of optimization problems. Therefore, application of the proposed algorithm to this class of problems is expected to reveal the particular features of the algorithm. The method is applied to the reliability based optimal design (RBOD) of simple trusses. Numerical results are presented illustrating the advantages of the proposed method, as

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C.K. Dimou, V.K. Koumousis / Advances in Engineering Software 34 (2003) 773–785

compared to the same number of standard GAs with fixed parameters.

2. Genetic algorithms A simple genetic algorithm (GA) based on binary coding is employed for every population [1]. No mixing among individuals of different populations is allowed, to preserve the characteristics of the populations. Reproduction is based on a ranking scheme [2], while elitism is adopted [3] allowing the pair of best individuals to reappear in the next generation. GAs are used in a wide range of structural optimization problems [4,5]. One-point and two-point crossover schemes are used. For the probability of mutation, decreasing functions with the number of generations are implemented. For every generation the entire system operates at two levels, i.e. the level of populations and the level of metapopulation [6], where all decisions about the characteristics of the next generation of populations are made. The optimal structural design problem is usually formulated in non-linear mathematical programming form as min Cij ðxÞ Subject where

gk ðxÞ # 0

k ¼ 1; …; Nc

ð1Þ

n

x[D

where Cij ðxÞ is the objective function for the ith individual of the jth population, gk ðxÞ is the kth inequality constraint, Nc is the number of constraints, x is the vector of design variables and Dn is the design space. An equivalent unconstrained maximization problem is determined following a penalty function formulation as 8 9 > > > > > > > > < = A # maxn fij ðxÞ ¼ " ð2Þ Nc > x[D > X > > > > > Cij ðxÞ þ ck TðgðxÞÞ > : ; k¼1

where ck is the penalty factor associated with the kth constraint of the problem which in this study was taken as 100 for all constraints, A is an arbitrary constant and operator T is given as ( pffiffiffiffiffiffiffi x21 x.1 TðxÞ ¼ ð3Þ 0 x#1 where 1 is the tolerance in violating the constraints. Throughout this study 1 ¼ 0:1: The maximization problem of Eq. (2) is appropriately normalized at the metapopulation level as follows 8 9 < = f ðxÞ ij maxn f^ij ðxÞ ¼ x[D : ½min fij ðxÞ ; i;j

ð4Þ

where mini;j {fij ðxÞ} is the objective of the less-fit individual in all populations. Moreover, the probability of mutation at a given generation t is given as 8 t # tinit > < Pinit 

 Ptmut ¼ t 2 tinit > t . tinit : Pfinal þ ðPinit 2 Pfinal Þexp 2 nhalf ð5Þ where Pinit and Pfinal is the initial and final mutation probability, respectively, tinit is the generation when the mutation probability starts to vary and nhalf is the parameter that controls the velocity of the variation of the mutation probability. Throughout this study these parameters have the fixed values: tinit ¼ 10 and nhalf ¼ 40:

3. Competition Competition is common in natural systems. Dimitrova and Vitanov [7] study the evolution of CPs through adaptation in a non-linear dynamical system with limited resources. In the book edited by Hanski and Gilpin [8], Nee et al. present the important parameters of interaction among different populations in a natural environment. Co-evolving populations of different species share the environment in a state of dynamic equilibrium. Competition among different species arises when they share the same resources, which are not sufficient to sustain all the populations. 3.1. Resources Assuming the computational time needed to process a single design as constant, the amount of resources required Rreq ; to process all the populations of the metapopulation consisting of N designs at a specific generation is given as Rreq ¼

Np X

Rj Nj

j¼1

N¼

Np X

Nj

ð6Þ

j¼1

where Rj and Nj are the resources per individual and the total number of individuals of the jth population, respectively, and Np is the number of populations in the system. The amount of available resources at generation t can be represented by a step like function with initial resources R Ravail ¼ R 2

m X

Hðt 2 tj ÞDRj

ð7Þ

j¼1

where m specifies the number of changes of the step-like function at particular instances, i.e. at generation ti with reduction of resources DRi ; and H is the Heaviside function. If the available resources are less than the required ones, conflict is introduced into the system. Abrupt changes on the available resources correspond to drastic events in

C.K. Dimou, V.K. Koumousis / Advances in Engineering Software 34 (2003) 773–785

775

the virtual environment that are expected to accelerate the adaptation of the CPs.

the average of the variation at all positions of the chromosome and is given as

3.2. Fitness

Dj ¼

The fitness of a population, in the metapopulation level, depends not only on the particular characteristics of its individuals but also on the profiles of all the other populations. Co-evolution defines the concurrent evolution of two or more populations where the fitness of a population depends also on the profiles of the remaining populations. Ficisi and Pollack [9] examine co-evolutionary algorithms from game theory viewpoint. Barbosa and Barreto [10] implement an interactive co-evolutionary GA in a graph layout problem. Riechmann [11] shows that in economics, where the fitness of a strategy is directly related to the adopted strategies of the remaining individuals, every GA is also a dynamic game. The fitness of a population in the metapopulation level can be expressed as the sum of the fitness of its individuals: Fj ¼

Nj X

f^ij ðxÞ

ð8Þ

i¼1

This expresses co-operation among individuals of the same population, which is common in natural systems and favors the expansion and survival of larger populations and of populations with many ‘good’ individuals. 3.3. Diversity The goal at this stage is to introduce a more stringent approach, as compared to the GA, that will process the emerging data and guide the next steps in an adaptive manner, while taking into account the uncertainties of the system. Diversity plays an important role for the GA to improve the existing best solutions [12,14]. Therefore, a diversity measure Dj of the chromosome of every population j is evaluated, based on descriptive statistics of the digits appearing at every position of the chromosome. Diversity is used as an estimate of the ‘age’ of a population. ‘Younger’ populations exhibit higher diversity as compared to ‘older’ ones and thus, they appear more promising to improve further the existing elite solution than ‘older’ ones. This factor is important in preserving the capacity of the metapopulation in finding the global optimum [13,14]. Considering that the digits at every position of the chromosome follow a binomial distribution, their variation at a specific generation is given as Var ½ fjm  ¼ Nj ·p·ð1 2 pÞ

ð9Þ

where fjm is the population of digits of the jth population at the mth position and p is the probability of occurrence of 1 at the mth digit. When this probability is equal to 0.5 the variation takes its maximum value of Nj =4: The diversity measure over the length of the chromosome is calculated as

E

m¼1;…;k

{Var ½ fjm }

ð10Þ

where k is the length of the chromosome. 3.4. Relative performance The fitness of the fittest individual Bj is introduced as an additional parameter to measure the relative performance of the populations. The relative performance index of the jth population PIj ; is expressed as the product of the fitness of its best individual, the population fitness and the population diversity PIj ¼ ½Bj w ½Fj a ½Dj b

ð11Þ

where w; a and b are parameters, used to attenuate or intensify relative variations among the populations. This expression of relative performance index constitutes a comparative measure of the performance of a population. In this analysis a ¼ 1: Expressions like the one presented in Eq. (11) are frequently used in econometric models [15,16].

4. Engagement rules 4.1. Conflict The probability of conflict among two different populations i and j; when shortage of resources is observed, is given as 8 9 PIi 2 PIj >  > < = OF . OF DN i j PIi Pr½popi ; popj  ¼ T > N > : ; 0 OFi # OFj ðRreq 2 Ravail ÞE½Nj  ; E½Rj Nj  8 0 x,0 > > > < x  ¼ 0#x#d T½x d > > > : 1 x.d

DN ¼

ð12Þ

where d is a parameter controlling the transition from a state of no conflict to a state of conflict emerging between two populations, when lack of resources exists. In this analysis d ¼ 0:2: This relation assures that no conflict arises if the available resources are adequate. Moreover, the probability of conflict increases in proportion to the resource deficit, while stronger populations fight only weaker ones. The probability of conflict between two CPs increases linearly with the relative difference of their performance index. For every population only one conflict per generation is allowed. Further details that concern the scheme used for the selection of conflicting pairs are presented in

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the pseudocode description of the algorithm. Finally, conflicts cease when the available resources are adequate. The outcome of a conflict between populations i and j; determines the size of these populations in the next generation as follows

 PIi 2 PIJ tþ1 t  Ni ¼ Ni þ T eij PIi " ð13Þ # 1 PIi 2 PIJ tþ1 t  Nj ¼ Nj þ T 12 eij PIi where T is given as ( ) 8 > DN > > max g ;2 > > Np > > > > > >0 > < ( ) TðxÞ ¼ DN > 2max g ;2 > > Np > > > > ( ) > > > 2DN > > : 2max g N ; 4 p

0:5 þ f , x 0:5 # x , 0:5 þ f 0:5 2 f # x , 0:5

ð14Þ

x , 0:5 2 f

and eij is equal to eij ¼ 1 þ

eðrand 2 0:5Þ 0:5

ð15Þ

where e and f are parameters that handle the fuzziness of the outcome. For this analysis e ¼ 0:2 and f ¼ 0:2: Furthermore, the g factor is used to regulate the velocity of variation of the size of population. Typical values of g factor are around unity [17]. Furthermore, populations vanish if their population size drops to zero. 4.2. Termination criteria A convergence criterion is introduced that works as a trade-off between the variability of the population and the coefficient of Variation (COV) of the objective function. The variability factor measures the variability of schemata per design variable in the chromosome. This is calculated as the average of the variability factor of all design variables of the problem. Convergence of results and termination of

the evolution of a particular population is considered when the variability factor is less than 30% and the COV less than 5%. Moreover a hard termination criterion of the GA optimization process is applied after 250 generations, which depends on the particular problem. For the problem addressed in this paper, most of the populations converge following the first criterion and very rarely the hard termination criterion was activated. Therefore, the proposed scheme at the level of metapopulation is based on the introduction of diversity measure given in Eq. (10), obtained from formal descriptive statistics on the different schemata of the population, the implementation of the conflict scheme and the fuzzy outcome of the conflict between pairs of populations. The proposed algorithm is briefly presented as follows. 4.3. Pseudocode of the proposed algorithm † Step 0: Start process. Read data for the competition scheme used in Eqs. (5) – (15). Read parameters for the GAs (reproduction scheme, type of crossover, probability of crossover and probability of mutation) for all populations i ¼ 1; …; Np : Specify the population size Nit of every population. Read parameters of termination criteria. † Step 1: Set t ¼ 0: Generate random initial populations for all populations ðNp Þ: Evaluate fitness for all designs, i.e. for i ¼ 1; …; Np and j ¼ 1; …; Nit call the RBOD code and calculate the fij ðxÞ: Perform the statistics of all populations ðNp Þ: † Step 2: Operate on the metapopulation level to decide for the new population size of all populations. Calculate the normalized fitness fij ðxÞ using Eq. (4). † Step 3: Decide about conflict. Calculate the required and available resources using Eqs. (6) and (7). If Np ¼ 1 or Rreq ¼ Ravail set Nitþ1 ¼ Nit for i ¼ 1; …; Np then go to Step 7 else; if Rreq , Ravail go to Step 6 else go to Step 4. † Step 4: Calculate the performance index of every population using Eqs. (8) – (11) and store it in a vector {A} in a sorted form. † Step 5: Assign the pairs of possible conflict. Select the strongest population {Popi } from vector {A}: For

Fig. 1. Statically determinate truss (loads and members).

C.K. Dimou, V.K. Koumousis / Advances in Engineering Software 34 (2003) 773–785

777

Fig. 2. Statically determinate truss (design variables).

†

†

†

†

†

†

the remaining populations assign selection probabilities based on a ranking. Step 5.1: Select randomly the ‘weak’ population {Popj }: Calculate the probability of conflict between {Popi } and {Popj } using Eq. (12). If Pr½Popi ; Popj  $ randð Þ then assign populations i and j as pair of conflicting populations and remove their indexes from vector {A} otherwise remove only {Popi }: If {A} contains more than one populations then go to Step 5 else, Step 5.2: For all pairs of conflicting populations calculate the Nitþ1 ; i ¼ 1; …; Np using Eqs. (13) – (15). For the remaining populations Nitþ1 ¼ Nit : Go to Step 7. Step 6: Distribute the resources in surplus evenly across the evolving populations and calculate the corresponding Nitþ1 ; i ¼ 1; …; Np : Step 7: Set t ¼ t þ 1: For i ¼ 1; …; Np and j ¼ 1; …; Nit call the standard GA routine using the new population size for all the populations and calculate the fitness fij ðxÞ of all the designs calling the RBOD code. Perform the statistics of all populations. Step 8: Check for convergence for all populations. For all populations i ¼ 1; …; Np check for the satisfaction of termination criteria. Freeze the available resources (resources cannot be re-allocated) for populations that satisfy the termination criteria. Update Np : If Np – 0 then go to Step 2 else go to Step 9. Step 9: End process.

5. Reliability optimal design The RBOD of a statically determinate 25-bar truss is considered (Fig. 1) [18]. The members of the structure form four groups, i.e. the lower and upper chords, the vertical struts and the diagonal members. The design variables are the cross-sectional areas of the most stressed members of the groups Ai ; i ¼ 1; …; 4; the seven heights hj ; j ¼ 1; …; 7 and two lengths lk ; k ¼ 1; 2 that control the shape of the truss (Fig. 2). Sixteen tubular cross-sections are considered for each of the four groups. These cross-sections are presented in Table 1. Heights h1 to h4 vary from 0.25 up to 3.00 m whereas h5 to h7 vary from 0.0 up to 0.90 m. Lengths l1 and l2 vary from 2.00 up to 3.75 m. For the design variables h1 to h7 ; 16 different values are considered, whereas for design variables l1 and l2 eight different values are taken into account. In total (4 £ 4 þ 7 £ 4 þ 2 £ 3) ¼ 50 bits are needed to fully describe a particular design. Therefore, the multiplicity of the design space enumerates 250 (1, 125, 899, 906, 842, 624) designs. The objective function of the problem is the average expected cost of the structure, i.e. construction cost plus the cost due to possible structural failure. The constraints of the problem are the failure probability of the entire structure Pf;s and the failure probabilities of its elements Pf;i :

Table 1 Cross-sections considered in the analysis A/A

Cross-section

d (mm)

t (mm)

A/A

Cross-section

d (mm)

t (mm)

1 2 3 4 5 6 7 8

TUBO-D21.3 £ 2.8 TUBO-D26.7 £ 2.9 TUBO-D33.7 £ 3.2 TUBO-D42.7 £ 3.2 TUBO-D48.4 £ 3.2 TUBO-D60.1 £ 3.2 TUBO-D76.1 £ 3.2 TUBO-D82.5 £ 3.2

21.3 26.7 33.7 42.7 48.4 60.1 76.1 82.5

2.8 2.9 3.2 3.2 3.2 3.2 3.2 3.2

9 10 11 12 13 14 15 16

TUBO-D88.9 £ 3.2 TUBO-D101.6 £ 3.6 TUBO-D108.0 £ 3.6 TUBO-D114.3 £ 3.6 TUBO-D127.0 £ 4.0 TUBO-D133.0 £ 4.0 TUBO-D139.7 £ 4.0 TUBO-D152.4 £ 4.0

88.9 101.6 108.0 114.3 127.0 133.0 139.7 152.4

3.2 3.6 3.6 3.6 4.0 4.0 4.0 4.0

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C.K. Dimou, V.K. Koumousis / Advances in Engineering Software 34 (2003) 773–785

The optimization problem is formulated as follows: Nt X min FðAi ; hj ; lk Þ ¼ Vm ðAi ; hj ; lk ÞCmat þ Pf;s Cfail m¼1

Table 2 Probabilistic data for the statistically determinate truss

ð16Þ

i ¼ 1; …; 4; j ¼ 1; …; 7; k ¼ 1; 2 Subject to gj ðAi ; hj ; lk Þ ¼ gs ðAi ; hj ; lk Þ ¼

Pf;j 2 1:0 # 0; Pj;lim

ð17Þ

Pf;s 2 1:0 # 0 Ps;lim

H gs;1 ðH4 Þ ¼ 4 2 0:15 # 0 2L

ð18Þ

H7 2 0:05 # 0 2L

The probability of failure of an element is given as Pf;i ¼ Pr½M # 1

ð19Þ

where M is the safety margin which is expressed as follows RðAi Þ M¼ Fðhi Þ

ð20Þ

where RðAi Þ and Fðhi Þ are the ultimate resistance and the applied load, respectively, given as RðAi Þ ¼ su Ai

fi ðhi Þ $ 0

RðAi Þ ¼ xsu Ai

fi ðhi Þ , 0

Distribution type

Average

COV (%)

Load P (kN) Ultimate strength (MPa) Cross-section (cm2) x Parameter

Lognormal Lognormal Lognormal Lognormal

20, 30, 40 275.0 Variable Calculated

12.5 7.00 10.0 0a , 5 , 10

a

where Nt is the number of elements of the truss, Vm is the volume of the mth element, Cmat and Cfail is the cost per unit volume of the structure and the cost of a potential structural failure, respectively, Pf;s is the overall failure probability, Pf;j and Pj;lim are the failure probability of the jth element and the maximum failure probability, respectively, and Pf;s and Ps;lim are the failure probability and its limit value for the entire structure. Two shape constraints that control the height to length ratio are also considered. These constraints are given as:

gs;2 ðH7 Þ ¼

Variable

Fðhi Þ ¼ fi ðhi ÞP

ð21Þ

and fi ðhi Þ are the influence coefficients that result from the analysis of the truss for a unit load and depend on the geometry of the structure. The parameter x reduces the compressive strength of the members due to buckling considerations and is based on the provisions of Eurocode 3 [19]. The load P, the ultimate yield stress su ; the crosssectional areas Ai ; and the parameter x are considered as random variables for the problem. When these variables are log normally distributed, an analytical solution of the failure probability can be obtained [20]. Characteristics of these random variables are presented in Table 2. For a statically determinate structure, failure of one of its elements results in failure of the entire structure. The structure is modeled as a series system of weakly correlated elements. Ditlevsen bounds are used to obtain estimates of the system reliability [20,21].

Deterministic variable.

From Eqs. (20) and (21), the safety margin is given as: M¼

xsu A ) lnðMÞ ¼ lnðsu ÞþlnðAÞþlnðxÞ2lnðf Þ2lnðPÞ fP ð22Þ

The mean and variance of the natural logarithm of the safety margin with respect to its components are given as: E½lnðMÞ ¼ E½lnðsu Þ þ E½lnðAÞ þ E½lnðxÞ 2 E½lnðf Þ 2 E½lnðPÞ V½lnðMÞ ¼ V½lnðsu Þ þ V½lnðAÞ þ V½lnðxÞ þ V½lnðPÞ ð23Þ The mean and variance of the terms of Eq. (23) are given as:

pffiffiffiffi  VX V½lnðXÞ ¼ ln þ1 E½X ð24Þ 1 E½lnðXÞ ¼ lnðE½XÞ 2 V½lnðXÞ 2 Substituting Eqs. (22) – (24) into Eq. (19), the failure probability of an element, considering that the natural logarithm of the safety margin follows a normal distribution, is given as

 E½lnðMÞ Pf ¼ P½M , 1 ¼ P½lnðMÞ , 0 ¼ F 2 pffiffiffiffiffiffiffiffiffiffiffiffi V½lnðMÞ ¼ Fð2bÞ

ð25Þ

where b is the reliability index. The members of every design group are considered fully correlated. In this case, the failure probability is given as: Pf;G ¼ max {Pf;i }

ð26Þ

i[G

Substituting Eq. (25) into Eq. (26), the failure probability of every group is given as: Pf;G ¼ max {P½Mi , 1} ¼ P½max {Mi } , 1 i[G

i[G



 E½lnðMi Þ ¼ F max 2 pffiffiffiffiffiffiffiffiffiffiffiffi i[G V½lnðMi Þ

ð27Þ

The failure probability of the structure can be obtained from the failure probabilities of its groups. The Ditlevsen bounds

C.K. Dimou, V.K. Koumousis / Advances in Engineering Software 34 (2003) 773–785

for the system’s failure probability are given as: Pf;s #

n X

P½Mi , 1 2

i¼1

n X i¼2

Pf;s $ P½M1 , 1 þ

n X

having the same values of b; w exponents in Eq. (11). These parameters can be set equal to unity indicating no preference either for the fitness of the population, or the diversity or the fitness of the elite individual. Although the values of the parameters presented in Table 4 are not the optimal, they are selected on the basis of qualitative criteria for improved performance. For example for the decreasing RVS 1 and 2, the average population size is expected to decrease in time and thus, parameter b is set to a value of 5/6 bigger than parameter w which is set to a value of 0.5 to favor the evolution of populations that exhibit higher diversity. The minimum cost solutions for the load case E½P ¼ 30 kN and the x parameter considered as deterministic, random with COV 5 and 10%, respectively, are presented in Fig. 4. The optimal solutions for x parameter with COV 5% for three different loads are presented in Fig. 5. In Table 5 the optimal parameters that control the shape of the trusses are presented. It is observed that the variation of the x parameter is not affecting the optimal cost considerably. The shape of the truss changes and the height to length ratio decreases as the COV of the x parameter increases. The different average load affects considerably the shape, the cross-section areas and the expected cost of the optimal truss, following an almost linear relation between the average load and the optimal cost. With regard to the shape of the truss, an increase in the height to length ratio is observed as the average load increases. In addition, the l1 design variable and the sum of l1 þ l2 decrease as the average load increases. The evolution of the objective function of the best individual for E½P ¼ 40 kN; x parameter treated as random variable with a COV 10% for an initial population size equal to 80, that corresponds to RVS 1, is presented in Fig. 6. Population 12 finds the optimal solution at generation 58 and its evolution is terminated at generation 71. Population 3 follows a path similar to that of population 12. Its evolution is terminated at generation 58 and a near optimal solution within 8.9% of the computed optimum is obtained at generation 55. Population 9 converges at a non-optimal solution, within 13.5% of the computed optimum, at generation 88. The evolution of the population size of the 12 different GAs is presented in Fig. 7. Competition starts at generation 10 when the first resource reduction is imposed. For population 12 the population size varies slightly (less than 15% of the initial population size). Similar behavior is observed for population 9 until generation 50, where two

max P½Mi , 1 > Mj , 1 j,i



max

ð28Þ

P½Mi , 1

i¼2

2

iX 21

  P½Mi , 1 > Mj , 1 ; 0

j¼1

For the joint probability of events the following expression is used: P½Mi , 1 > Mj , 1 ¼ F2 ð2bi ; 2bj ; rÞ

ð29Þ

For the evaluation of the joint probability of Eq. (29) the following bounds are used [20] maxð pi ; pj Þ # P½Mi , 1 > Mj , 1 , pi þ pj r . 0 0 # P½Mi , 1 > Mj , 1 , minðpi ;pj Þ

r,0

ð30Þ

where the parameters pi and pj are given as pi ¼ Fð2bi Þ·Fð2gj Þ

pj ¼ Fð2bji Þ·Fð2gi Þ

ð31Þ

and the gi and gj factors are given by the following relation:

bi 2 rij ·bj gi ¼ qffiffiffiffiffiffiffiffiffi 1 2 r2ij

bj 2 rij ·bi gj ¼ qffiffiffiffiffiffiffiffiffi 1 2 r2ij

779

ð32Þ

5.1. Numerical results A number of problems are solved for L ¼ 10 m and a ratio of costs Cfail =Cmat equal to 20,000, that corresponds to an estimation of real costs, starting with an initial population size of 40, 60, 80 and 100 designs. The limit probabilities considered during the analysis are Pj;lim ¼ 1026 and Ps;lim ¼ 5 £ 1026 : Ten runs with different random seeds are performed to produce data for a statistical evaluation of the proposed scheme. Twelve different populations are considered. Their characteristics are presented in Table 3. The available resources vary according to the resource variation schemes (RVS) 1, to 7 as shown in Fig. 3 associated with the parameters of Table 4. These RVSs are classified in four different groups namely, the decreasing ones (RVS 1 and 2), the alternating (RVS 3 and 4), the initially increasing and subsequently decreasing (RVS 5 and 7) and one scheme (RVS 6) with alternating – decreasing resources, each Table 3 GA parameters Population

1

2

3

4

5

6

7

8

9

10

11

12

Crossover prob. Crossover type Mutation scheme Scheme A: Pinit ¼ 1%; Pfinal ¼ 1‰

0.7 SP A Scheme B: Pinit ¼ 2%; Pfinal ¼ 2‰

0.7 SP B

0.8 SP A

0.8 SP B

0.7 DP A

0.7 DP B

0.8 DP A

0.8 DP B

0.9 SP B

0.9 SP A

0.9 DP B

0.9 DP A

780

C.K. Dimou, V.K. Koumousis / Advances in Engineering Software 34 (2003) 773–785

Fig. 3. Evolution of the resource variation schemes.

significant increases of the population size are observed, followed by small variations near the end of the optimization process. Finally, the sharp drop observed at generation 72 is attributed to a reduction of the available resources. Due to intense competition, populations 7 and 10 are forced to converge prematurely at generations 37 and 53, respectively. In Table 6, the schemes that produced the optimal results are presented. For the case where E½P ¼ 30 kN and RVS 6, the initial population size of 100 individuals produced the best design, whereas for E½P ¼ 40 kN and RVS 1 the initial population size of 80 or 100 individuals suggested the best design. For E½P ¼ 20 kN; mixed results are observed and in the case where x is treated as a random variable with a COV equal to 10% the best design was obtained from the standard GA scheme. With regard to the probability of crossover, probability of mutation and one- or two-point crossover schemes and for the specific range used in this analysis, no trends favoring a particular set of parameters are observed. Single point crossover is used in 6 out of 9 cases as it can be easily checked from Table 3 from the corresponding column ‘PopID’. Moreover, for the crossover probability the three different crossover probabilities appear 4, 3 and 2 times each for values of 0.7, 0.8 and 0.9, respectively. Mutation scheme

B is applied in 6 out of 9 cases. The number of generations required to derive the optimum solution shows considerable variability from a minimum of 24 generations to a maximum of 161 generations, which are, respectively, extended to a minimum of 29 generations and a maximum of 218 generations until termination. Also, from the different initial random seeds that gave the optimal results, no trends favoring a particular seed are observed as expected, which makes the statistics of the results valid. A more general statistical evidence of the overall performance of the proposed algorithm is depicted in Figs. 8– 11. The minimum, maximum and average ratio of computing time for seven different RVSs, for all problems, with respect to the standard GA is presented in Fig. 8. Moreover, Table 4 Problem parameters Parameter

b

w

RVS 1 RVS 2 RVS 3 RVS 4 RVS 5 RVS 6 RVS 7

5/6 5/6 0.5 0.5 5/6 5/6 5/6

0.5 0.5 1.0 1.0 0.5 0.75 0.5

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Fig. 4. Optimal solutions ðE½P ¼ 30 kNÞ:

Fig. 5. Optimal solutions (x random COV equal to 5%).

the number of problems where the proposed algorithm outperformed the classical GA with regard to the quality of the best design is also presented. The most computationally expensive schemes are RVS 3 followed by RVS 4, whereas the least computationally intensive scheme is RVS 2 followed by RVS 1. With regard to the quality of the best

design it can be observed that for all RVSs expect RVS 4 the proposed algorithm manages to produce a better design than the best optimal design of the std. GA in the majority of the problems examined. The comparison of groups shows that RVS 1 is more robust than RVS 2. Equivalently RVS 3 is more robust than RVS 4 and RVS 7 is more robust than

Table 5 Optimal values of the shape variables (m) Design

h1

h2

h3

h4

h5

h6

h7

l1

l2

a b c d e f

1.00 0.90 0.80 1.00 0.90 1.30

1.50 1.30 1.20 1.40 1.30 1.80

1.85 1.425 1.30 1.80 1.425 2.30

2.00 1.475 1.375 2.025 1.475 2.65

0.0225 0.0 0.275 0.025 0.0 0.0125

0.25 0.275 0.575 0.20 0.275 0.10

0.425 0.575 0.80 0.45 0.575 0.325

3.5 3.0 3.25 3.5 3.0 2.75

3.25 3.75 3.75 3.5 3.75 3.50

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Fig. 6. Evolution of the objective function (E½P ¼ 40 kN; (x RV 10%)).

Fig. 7. Evolution of the size of the population (E½P ¼ 40 kN; ( x RV 10%)).

Table 6 Schemes that produced the optimal results Case

Scheme

20 kN

x; Deta x; RVb 5% x; RV 10%

RVS 3 RVS 4 GA

80 80 60

30 kN

x; Det x; RV 5% x; RV 10%

RVS 4 RVS 6 RVS 6

40 kN

x; Det x; RV 5% x; RV 10%

RVS 1 RVS 1 RVS 1

a b

Det, deterministic variable. RV, random variable.

Pop size

Pop ID

Gen-opt

Gen-term

Seed

4 5 3

161 46 69

218 51 94

2 9 2

100 100 100

2 9 8

49 63 121

67 63 122

4 4 8

100 80 80

2 6 12

95 24 58

100 33 71

4 1 7

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Fig. 8. Average, minimum and maximum values of CE and frequency of surpassing the best design of the std. GA.

RVS 5. Moreover, RVS 3 managed to outperform the classical GA in all problems, but one, whereas very good results are also observed for RVS 6 and RVSs 1 and 7. Scattering of the cost for the various RVSs is presented in Figs. 9 – 11. In addition, the computational efficiency (CE) as compared to the standard GA and the probability of obtaining better solutions than the GA are presented, for different values of the average load and x parameter. In Fig. 9, for resource variation schemes RVS 3 to RVS 6, the probability to obtain better solutions than the standard

GA, is greater than 50% with a maximum of 72.5% for RVS 4. The probability for RVS 1 is equal to 50% and thus, the quality of results is equivalent to those of the GA but these results are obtained in considerably less time (62% of the computational time of the standard GA). From these results it is observed that moderate reductions of the resources are expected to maximize the performance of the algorithm. Moreover, schemes exhibiting an increase of available resources at the early stages of the optimization process, such as RVSs 5 and 7 are expected to produce statistically better results.

Fig. 9. Objective value scattering, CE and probability for better solutions than GA (E½P ¼ 20 kN; ( x RV 10%)).

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Fig. 10. Objective value scattering, CE and probability for better solutions than GA (E½P ¼ 30 kN; x deterministic).

Fig. 11. Objective value scattering, CE and probability for better solutions than the GA (E½P ¼ 40 kN; (x RV 5%)).

6. Conclusions From the above analysis it becomes evident that the proposed competitive algorithm controls satisfactorily the evolution process favoring the expansion of ‘promising’ populations and the contraction of ‘weak’ ones in a statistical sense. The descriptive statistics at the metapopulation level together with the rules of conflict guide the utilization of resources towards the most competent GAs. With regard to the algorithm’s capacity to produce good solutions, the proposed algorithm was able to trace better

designs as compared to the standard GA. RVSs that combine resource reduction with an alternating scheme, or schemes that are characterized by an increase of resources at the early stages of the optimization process are found to produce the best results. The proposed scheme succeeds in finding good ‘near’ optimal and optimal solutions in a robust way and in most cases faster than a standard GA, with all the sets of parameters considered for the specific problem. Based on the results presented in Fig. 8 the following suggestions can be made. In the case of problems with high computational

C.K. Dimou, V.K. Koumousis / Advances in Engineering Software 34 (2003) 773–785

cost RVS 1 can be used, since it combines low computational cost with increased robustness. For problems where computational cost is not of major importance RVS 3 can be used since at a small additional cost the robustness is increased by a factor of 8. Moreover, RVSs 6 and 7 present also good alternatives since they combine smaller computational cost with increased performance.

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