Multi-objective Topology Optimization of 2D Trusses using Genetic

Multi-objective Topology Optimization of 2D Trusses using Genetic Algorithms Guido Rodriguez Zamalloa1, David Mauricio2 1

National University of Engineering (UNI), Department of Industrial and Systems Engineering. 210 Túpac Amaru Avenue, Lima 25, Lima, Peru 2 San Marcos Major National University (UNMSM), Department of Systems Engineering and Computer Science. Germán Amezaga Avenue, Lima 01, Lima, Peru 1 [email protected], [email protected] Abstract Topology optimization of truss structures is a strategic factor in several branches of the industry. The last developments with Genetic algorithms (GA) in order to achieve the minimum weight of the structures are encouraging. However, there have been problems: the high rate of unfeasible individuals, the computational high costs, and slow convergence. The most of these techniques follow two approaches: 1) Assuming a ground structure (a complete truss formed by all possible connections between nodes) and 2) Distributing material in the design domain through a binary representation (empty / full of material). In this paper we proposed a technique with a totally different approach from the representation; instead of nodes, bars or portions of matter in the space, we codify instructions of assemblage that are read by a "construction algorithm ", which based on pre-established rules, constructs the truss. Besides, this technique does not appeal to the formulation of maximum rigidity. We use the “Matrix Method of Calculation of Trusses with Articulated Nodes" instead of Finite Elements. The numeric experiments on test instances prove that the proposed method reduces computational costs and lessens weight of trusses. Keywords: genetic algorithms, topology design, truss optimization, topology optimization. 1.

Introduction

Topology optimization of structures has become a strategic factor in several branches of the industry. For the first time, in 1960 Schmidt [1] proposed the optimum design of structures as a problem of minimization with restrictions and its solution by means of non-linear programming. But Bendsøe and Kikuchi [2] in 1988 laid the foundations of, what today is called, Topology Optimization of Structures (TOS). Since then, TOS problems have been set by means of formulations of maximum rigidity trying to distribute a predetermined quantity of material in an enclosure, so that the rigidity is maximized or the energy of deformation is minimized for a certain state of loads, being the SIMP (Solid Isotropic Material with Penalty) the most used model which assigns to every element of the equivalent mesh of Finite Elements a relative density of the material, which varies from 0 to 1 (from emptiness to fullness) [3]. The densities are the variables of design and the volume is the restriction. The disadvantage of this method is that it needs to solve derivatives of higher order, which is unviable to certain problems. The most used structures are obtained tying up converged bars into nodes, so that they define little triangular and rigid cells, named generally, Trusses. Kalyanmoy [10] classify the optimization of these structures under three great categories: the measuring, the configuration and the topology. The problem we attempt to solve confronts these three categories. It consists in finding a set of nodes displaying connectivity and measurements according to needs of load, effort restrictions, deformation and geometry in order to answer to certain criteria of optimization. An alternative to overcome the difficulties of the classic methods for TOS is given by the evolutionary computation, which needs strictly neither of models of optimization nor of gradients. Its application in this area, also called Evolutionary Structural Optimization [9] (ESO) follows two principal currents. The first current named "Ground Structure", assumes that the truss is formed by all possible connections between nodes, from which elements are extracted by means of Boolean variables (1/0), and whose principal exponents are Ohsaki [8], Steven [9], Kalyanmoy et alt [10], Hajela [11]. The second way is that of "Distribution of mass” which inspired by the model SIMP, codifies the chromosome in a binary representation (0/1) distributing a quantity of matter in a limited space. It has been developed by Jakiela [5], Hansel [14], Capello [15], Aguilar [16] and Wang [7] among others. Works developed through Genetic Algorithms have showed encouraging results in the search of optimum structures. Nevertheless, when it is considered more than one of Kalyanmoy's categories [10], they present serious problems such as: high percentage of "unviable individuals” (bars with no connection, unstable structures, etc.), computational high costs and slow convergence. In this work we introduce an approach for trusses optimization by means of a new form of codifying of chromosome and the usage of the “Matrix Method of Calculation of Trusses with Articulated Nodes” for evaluation fitness instead of “Finite Elements Method” (FEM). The proposed method reduces dramatically the rate of unviable solutions when genetic operators are applied, and avoids the conditioning of the discretization on not having needed of equivalent mesh. This provides important advantages in the convergence and in the results. In addition it allows us consider multiple optimality criteria. This paper is organized in 7 sections. In section 2 we present a multi-objective model of trusses optimization. The strategy followed to solve the problem is showed in section 3. In section 4 are described the genetic algorithm, the codification, the genetic operators and the technique of evaluation by the method of calculation proposed. In section 5 we describe the OTEMOGA software, developed in MatLab 5.3 and in section 6, numerical experiments are realized. Finally, conclusions are presented in section 7.

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2.

The Optimization Model

Ω GS

Consider a set

formed by all points of a 2D discrete and limited space and consider a set

ΨGS

of all possible combinations

that can be made by connecting those points; it constitutes the “space of solutions " - Ground Structure (GS). Let groups selected points from the discrete space ( Ν

Ν

be the set which

⊂ Ω GS ); Ψ , the set which groups all the possible connections that can be established with these nodes, and Β , the set of bars subset of Ψ which groups the selected connections called “the bars or elements of the structure Β ⊂ Ψ ⊂ ΨGS ”.Then the problem consists in finding the set of nodes Ν and the set of bars Β such that satisfies the restrictions of effort, deformation and geometry, in order to answer to certain criteria of optimization.

Ω GS

The nodes are selected from using the decision variable

Z ij

using the decision variable

to form the set

Sj

to conform the set

Ν

. Bars are selected from the subset

Ψ

Β . Restrictions of effort and deformation are given by limit values. Geometric

constraints are given by the location of the "basic nodes" such that correspond to the supporting nodes, loaded or frozen nodes. The mathematical model is following described: Minimize:

(

d1 = ∑∑ f ij × 2 (x j − xi ) + ( y j − yi ) × Z ij n

n

j =1 i =1 n

2

2

)

(1)

n

d2 = ∑

∑Z

j =1 i =1 : f ij < 0

ij

∀ i, j ∈ [1, n ]

d k = φ k ( f ij Z ij , lij Z ij , ∂ i N j ) ..... M

(2) (3)

∀ i, j ∈ [1, n ]

(4)

∀f ij ≥ 0 y i, j = {1, 2, ..., n}

(5)

d nco = φ nco ( f ij Bij , lij Bij , ∂ i N j )

subject to t f ij × Z ij ≤ f adm

∂ j × S j ≤ δ adm

j = {1, 2, ..., n}

(x − x ) + ( y − y ) × Z (x − x ) + (y − y ) × Z ≤ L − f × ((x − x ) + ( y − y ) )× Z ≤ F Lmin ≤

2

2

j

2

i

j

2

2

j

i

j

i

ij

2

ij

j

2

i

j

i

x j ≤ xj ≤ x j

j = {1, 2, ..., n}

y j ≤ yj ≤ y j

j = {1, 2, ..., n}

n

j =1 i =1

∃C

-1

i, j = {1, 2, ..., n}

max

i, j = {1, 2, ..., n}

ij

c cr

∀f ij < 0 y i, j = {1, 2, ..., n}

(7) (8) (9) (10) (11)

n

n

∑∑Z

ij

2

i

(6)

ij

= 2× ∑S j − 3 j =1

∀i,j ∈ [1, n], such that: c kl = ϕ ( xi , y i , Z ij ) / k , l = {1, 2, ..., m}, i, j = {1, 2, ..., n}

Z ij = {1,0},

i, j = {1, 2, ..., n}

S j = {1,0}

j = {1, 2, ..., n}

(12)

(13) (14) (15)

where:

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xj

Coordenate X of node

yj

Coordenate Y of node

xj xj yj yj

j j

Maximum length of bar

Minimum value of node Maximum value of node

j in X

∂ adm

Minimum value of node

j in Y

t f adm

Maximum value of node

j in Y

Displacement of node

f ij

Axial tension of bar

lij

Length of bar

C c kl

i

m

ij

ij , such that

lij = 2 (x j − xi ) + ( y j − y i ) 2

δ ij nco

Minimum length of bar

Lmax

j in X

∂i

Deformation of bar

ij

Number of optimality criteria

2

Ω GS = number of points

n Lmin

Cardinality of

Admissible deformation of structure Admissible axial tension Matrix of coefficients Elements of matrix

C

Number of elements of matrix

1 Sj =  0 1 Z ij =  0

C

if the point j is active other case if bar ij is used other case

Objective - functions: Relations (1) and (2) request to minimize the weight of the structure (expressed as the product of axial force and length of bar) and the number of bars subject to compression, respectively. The relations (3) and (4) express possibility of increasing optimality criteria. Restrictions: Relations (5) and (6) are respectively effort restrictions and admissible deformation. Relations (7) to (9) are length and slenderness restrictions of bars that assure efficiency of Euler's formulae for bulge calculation. Expressions (10) and (11) express the restrictions from nodes´ geometric condition. Relation (12) is a requirement of truss´ internal isostasity, expressed as a relation between number of bars and number of nodes (Grubber's relation). Relation (13) indicates that the matrix of coefficients or invariant must has an inverse, so that the structure could be solved by the Matrix Method of articulated Trusses. Finally, relations (14) and (15) express the integrity constraint of decision variables 3. Scheme of problem solution After entering the geometric conditions, material, section bars and system of loads, we set the values of effort and deformation, parameters of optimality and their weighting. Limit values (special techniques)

Optimamity parameters

Material and section Loads systems

OTEMO

Geometry conditions Transform

OTEMO M

Evolutionary Process (GA)

Solution S

End

S = Ok

Fig 1 – Scheme of problem solution

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To solve the problem we transform the Multi-objective problem into mono-objective and by means of an evolutionary process rested on Genetic Algorithms we obtain efficient solutions. If solution is acceptable for Decisor, the program concludes, otherwise the parameters are readjusted and the program runs again until results become satisfactory (see Fig.1). 4. Description of Genetic Algorithm 4.1. Representation We raise representation from a different approach, instead of representing bars or nodes in space; we codify orders of assemblage which are interpreted by a constructer algorithm that assembles this structure. Chromosome is represented by a fixed long list of integer numbers, which consists of a representative and an intronic (non- representative) zone, and a connection of three bits of information that do not take part in genetic operations. The intronic zone brings resistance to individuals after genetic operations, allowing to modify freely the representative zone (see the Fig.2). Alleles take integer values that correspond to nodes labels or the zero value (0) of intronic zone. Location of genes in chromosomal chain contributes information to the assemblage.

Fig 2 Representation scheme 4.2. The Assembler Algorithm Any triangle formed with straight bars and articulated nodes, submitted to a system of exterior loads mutually neutralizants, behaves as a stable structure. Internally, forces are offset, which proves the first Newton's law of balance. Extending this condition to a network of triangles submitted to a system of loads mutually neutralizants, loads will be distributed inside the truss, neutralizing itself triangle to triangle. Then we can assure that a triangular structure with articulated nodes and correctly supported can remain in balance for any loads system inside an acceptable range of efforts and deformation. Under this principle the algorithm reads the chromosome and proceeds to assemble the bars forming triangular and isostatic structures. 4.3. Evaluation Trusses can be solved without need to appeal to the FEM thanks to “Matrix Method of Calculation of Trusses with Articulated Nodes”, which supposes 4 hypotheses (nodes are joints with no reinforcing, loads act exclusively on nodes, bars are straight and the nodal displacements are small) which that allows it to reduce number of incognitos to only one for element (the axial effort). These 4 hypotheses are respected by the algorithm assembler. Structural solution is a game of values of efforts and lengths of the elements and nodal displacements, which are submitted to a fitness function. Competition and assignment of survival probabilities of each individual is done through technique of Binary Contests, proposed by Wetzel [18] in its probabilistic version. 4.4. Genetic Operators Crossing: We raise Crossing 2PMP function that is a variation from Order Crossover (OX) operator for permutations, proposed by Davis [17], modified to resist combinations instead of permutations, considering that the parents chromosomes can contain different alleles, not only in order but also in number and value. Mutation: We establish 5 operators of mutation: Combinatorial Insertion, Combinatorial Exchange, Elimination, Insertion of Permutation and Exchange of Permutation. The first three ones produce combinatorial effects and the last three produce permutation effects on chromosomal chain.

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5. Software OTEMOGA: (“Topology Optimization of Multi-Objective Structures using Genetic Algorithms” in Spanish) developed with Matlab 5.3, uses a file (*.m) for principal program and a library of functions (*.m) to process and to prepare data. Genetic operators are also functions in files (*.m) that are defined in the “entry data” and summoned from the principal program. In MatLab 5.3 the program has been ran in platform Windows XP, in an Intel (R) Pentium (R) M equipment with a processor of 1600 Mz, 512 MB of RAM. The file containing data type (*.mat) is generated using Excel. Structuration and syntax follow the same rules of program “GA Demo” and its library of functions, which keeps expectation on OTEMOGA joining at some time. The system must fulfill the minimal requirements requested by MatLab 5.3 and Windows 2000 or Superior, RAM 512K, Processor Pentium IV and must keep at least 3MB free. 6. Numerical Experiments Elements submitted to compression must be measured considering geometry of transverse section and length of element and not only their area, what in effect is done in elements with traction. Unlike elements with traction, those with compression fail because of bulge and not because of exceeding the limit of flowing of the material. Since there is no available information in any of the studied instances, we have loaded our system of measuring with the topologic solution and compared these results with the obtained by our topology, using the same geometry of transverse section (round rod) with the characteristics of material described in every instance. When instance has a hyperstatic anchorage (more than three reactions for flat system of efforts) we must insert a mobile support instead of a fixed support, and add an extra bar that connects both supports, so that the liberated reaction is assumed by this one (Fig 3). There have been selected two instances that correspond to respective ways of representation, resolved each one for several groups of researchers. We have compared the technique proposed with each of the approaches. 360 in

360 in

360 in

360

360 in

360 in

360 in 360 in Klb

360 in 100Klb

100Klb

tica

100Klb

Solicitación Isostática 100Klb

Solicitación Isostá

100Klb

100Klb

360 in

Solicitación Hiperestática

Solicitación Isostática

Fig. 3 Modification of supports 6.1. Instance 1 Problem consists in finding a minimum weight structure inside 2D spatial segment of 720 x 360 inches with two intermediate vertical loads of 100,000 pounds each one. Structure is supported by two fixed supports as it is showed in figure 3. This instance has been solved by Kalyanmoy et al [10], Hajela et al [11] and others using "Ground Structure" technique.

360 in

360 in

360 in

360 in

360 in

100Klb

Solicitación Hiperestática

360 in

100Klb

100Klb

Solicitación Isostática

100Klb

Fig 4 – Test instance 1 Technical specifications: E= 104 KSI (Young’s module *) Density= 0.1e-3 Klbs/in3 Admissible effort=25.0 KSI FSC=1 (F. de Security to Compression)

FST=1 (F. Security to Traction) Permissible Displacement= 2 in Minimum area= 0.09 in2 (boundary of instance) * KSI = Klbs / in2

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6.2. Results for first instance In figure 6 we show results obtained by Hajela et al [11] and Kalyanmoy et al [10], respectively. Second group presents improvements regarding the first one: they join two phases of solution in only one reducing time of calculation, but solution they present is lightly heavier (11.80 Klbs against 8.02 Klb obtained by P. Hajela). These techniques have worked by populations between 300 and 450 individuals and about 190 generations.

P.Hajela y E.Lee [11]

Kalyanmoy Deb and Surendra Gulati [10]

Fig. 5 Solutions for First Instance by other authors Results with OTEMOGA: results obtained by the proposed technique contribute a topology different from others, with two bars less and a scarce weight of 5.61 Klb (Fig 6), and in addition, with a population of 20 individuals and 10 generations.

Fig 6 OTEMOGA solution for the First Instance 6.3. Instance 2. Problem consists in solving a girder in horizontal cantilever whose length is the double of its height submitted to a punctual load in the mid of the end. The aim is to reach a minimum weight structure. Anchorage in cantilever is changed to an isostatic system (a fixed support and mobile one). This instance, solved by Mancuso [15], Kalyanmoy [10] and others, has been approached with representations based on conception of “distribution or assignment of mass “in a bi-dimensional constant domain. Solution needs of an interpretation of the groups of points and relations between them. 24 in

12 in 10Klb

Solicitación Isostática

Fig 7 – Test instance 2 Technical specifications E= 10000.00 KSI(Klbs/in2) Admissible effort = 25.0 KSI FSC=1 (F. security to Compression)

FST=1 (F. Security to Traction) Permissible displacement= 2 in

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6.4. Results of second instance Solutions from both groups of researchers show different topologies. In effect, structure by Cappello and Mancuso [15] with 7 bars is 0.77 Klb weight, whereas that of Mark J. Jakiela’s group [5] has scarcely 3 bars but a total of 0.92 Klb. In figure 8 we can observe the topologies from groups and the respective trusses.

F. Cappello, A. Mancuso [15]

Mark J. Jakiela, y Otros [5]

F. Cappello, A. Mancuso [15]

Mark J. Jakiela, y Otros [5]

Fig 8 Preceding solutions for Second Instance OTEMOGA solution: With a population of scarcely 20 individuals and 10 generations we obtain a radically different topology and a minor weight (0.70 Klb.), see Fig.9.

Fig. 9 –OTEMOGA solution for Second Instance 6.5. Comparative analysis We show comparative tables with obtained results (values in brackets indicate that the bar is working with compression).

Table 1. Comparison of results for First Instance Autor: Barra

1 2 3 4 5 6 7 Total

Kalyanmoy Deb Y Surendra Gulati [10] Área T. (in2)

4.00 40.62 62.38 81.24 5.66 8.00 5.66

Longitud (in)

360.00 360.00 509.12 720.00 509.12 360.00 509.12

Peso (Klbs)

0.14 1.46 3.48 5.85 0.29 0.29 0.29 11.80

Tensión1 (Klbs)

100.00 (100.00) (141.42) (100.00) 141.42 200.00 141.42

Técnica Propuesta

P.Hajela y E.Lee [11] Área T. (in2)

4.00 57.45 68.32 5.66 8.00 40.62 5.66

Longitud (in)

360.00 360.00 509.12 509.12 360.00 360.00 509.12

Peso (Klbs)

0.14 2.07 3.48 0.29 0.29 1.46 0.29 8.02

Tensión1 (Klbs)

100.00 (200.00) (141.42) 141.42 200.00 (100.00) 141.42

Área T. (in2)

0.09 70.36 5.66 8.94 57.45

Longitud (in)

360.00 360.00 509.12 804.98 360.00

Peso (Klbs)

2.53 0.29 0.72 2.07 5.61

Table 2. Comparison of results for Second Instance

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Tensión1 (Klbs)

(300.00) 141.42 223.61 (200.00)

Autor: Barra

1 2 3 4 5 6 7 Total

F. Cappello, A. Mancuso [15] Área T. (in2)

2.00 1.66 1.61 2.83 6.00 1.60 4.47

Longitud (in)

12.00 12.00 16.97 16.97 12.00 13.42 13.42

Peso (Klbs)

0.07 0.06 0.08 0.13 0.20 0.06 0.17 0.77

Tensión1 (Klbs)

50.00 (150.00) (70.71) 70.71 150.00 (111.80) 111.80

Mark J. Jakiela, y Otros [5] Área T. (in2)

2.00 4.01 8.25

Longitud (in)

12.00 24.74 24.74

Peso (Klbs)

0.07 0.28 0.57

0.92

Técnica Propuesta

Tensión1 (Klbs)

50.00 (206.16) 206.16

Área T. (in2)

0.09 1.91 3.77 5.50 1.85

Longitud (in)

12.00 12.00 16.97 24.74 13.42

Peso (Klbs)

0.01 0.06 0.18 0.38 0.07

Tensión1 (Klbs)

(200.00) 94.28 137.44 149.07

0.70

7. Conclusions When it is intended to optimize in more of one of Kalyanmoy's categories [10], TOS problem gets more complex regarding conventional representations, because these allow codification of “non-structure” individuals. Amount of requirements that a set of nodes and bars or accumulations of matter must fulfill to be considered as a structure, reduces probabbilities of survival of viable individuals and increases those of the unviable ones, degenerating rapidly a population. On having raised a method of calculation that does not need of discretizing the structure to avoid the conditioning described by Navarrina [4], we have complicated the problem by increasing structuration requirements, which is owed to solicitation of 4 hypotheses of calculation, conditions of isostasity that demands an exact relation between number of nodes and bars and the multiple optimality criteria. Nevertheless, the approach of the representation and the contribution of intronic zones inside the chromosome, have reduced dramatically the rate of inviability in early and late populations and it has allowed us to come to more efficient results with small populations and in a minor number of generations. Numerical results showed in last section are encouraging, it invites us to deal with much more complex instances, to investigate in new genetic specializing operators for this approach, and to contemplate other aspects besides the minimum weight criteria, such as: final cost, facility of assembly, seismic response, and specializing assembler algorithms.

References 1.

L.A. Schmidt, Structural design by systematic synthesis, Proceedings of the Second SCE Conference on Electronic Computation, 105–122, Pittsburgh, USA (1960).

2.

Bendsøe M.P., Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, (1988); 71: 197-224.

3.

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4.

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5.

Mark J. Jakiela, Colin Chapman, James Duda, Adenike Adewuya Kazuhiro Saitou. “Continuum structural topology design with genetic algorithms.” Department of Mechanical Engineering, Washington University. ELSEVIER, Comput. Methods Appl. Mech. Engrg. 186 p. 339 a 356. 2000

6.

S.Y. Wang, K. Tai. “Graph representation for structural topology optimization using genetic algorithms”. Elsevier, Computers and Structures 82 (2004) 1609–1622.

7.

S.Y. Wang, K. Tai. “Structural topology design optimization using Genetic Algorithms with a bit-array representation” Elsevier, Comput. Methods Appl. Mech. Engrg. 194 (2005) 3749–3770

8.

M. Ohsaki. “genetic algorithm for topology optimization of trusses”. Pergamon, Compulers & Swucrurr Vol. 57. No. 2. pp. 219-225. 1995

9.

Grant Steven A, Osvaldo Querin, Mike Xie. “Evolutionary structural optimisation (ESO) for combined topology and size optimisation of discrete structures”. Elsevier, Comput. Methods Appl. Mech. Engrg. 188 (2000) 743-754

10. Kalyanmoy Deb*, Surendra Gulati. “Design of truss-structures for minimum weight using genetic algorithms”. Elsevier, Finite Elements in Analysis and Design (2001) 447-465 11. P. Hajela and e. Lee. “Genetic Algorithms in Truss Topological Optimization”, Pergamon, 0020-7683, (1994) 00306 – 8 12. Holland, John H. “Adaptation in Natural and Artificial Systems”. Ann Harbor: The University of Michigan Press. 1975 13. D.E. Goldberg, M.P. Samtani, Engineering optimization via genetic algorithms, Proceedings of the Ninth Conference on Electronic Computations, ASCE, Birmingham, Alabama, 1986, pp. 471}482. 14. Wilfried Hansel A, Andre Treptow B, Wilfried Becker A, Bernd Freisleben. “A heuristic and a genetic topology optimization algorithm for weight-minimal laminate structures. Elsevier , Composite Structures 58 (2002) 287–294.

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15. F.Cappello y A. Mancuso “A genetic algorithim for combined topology an shape optimisations”. Departamento de Mecánica y Aeronaútica de la Universidad de Palermo – Italia Dic’2002. ELSEVIER, CAD 35 (2003) 761-769 16. J.F. Aguilar Madeira, H. Rodrigues , Heitor Pina. “Multi-objective optimization of structures topology by genetic algorithms”. Advances in Engineering Software 36. P. 21–28. 2005 17. LAWRENCE DAVIS. “Job Shop Scheduling with Genetic Algorithms”. In John J. Grefenstette, editor, Proceedings of the First International Conference on Genetic Algorithms, pages 136–140. Lawrence Erlbaum Associates, Hillsdale, New Jersey, July 1985. 18. A. WETZEL. “Evaluation of the effectiveness of genetic algorithms in combinational optimization”. University of Pittsburgh, Pittsburgh (unpublished), 1983.

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