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OPTIMIZATION PROBLEM FORMULATION AND SOLUTION

TECHNIQUES

5.1 Introduction

There are many cases in practical applications where the variables of optimization are

not continuous. Some or all of the variables must be selected from a list of integer or

discrete values. For example, structural members may have to be designed using

sections available in standard sizes; member–cross–sectional dimensions may have to be

selected from the commercially available ones. Therefore, considerable interest was

shown for discrete variable engineering optimization problems since the late 1960s and

early 1970s. However, at that time even optimization methods for simpler continuous

nonlinear programming (NLP) problems were still in the process of development. In the

1970s and 80s, a major effort was put into development and evaluation of such

algorithms. Although research in this area continues to develop better methods,

especially for large scale problems, several reliable algorithms are now available for

NLP problems, including sequential quadratic programming (SQP) and

augmented Lagrangian methods. In recent years, the focus has shifted back to

V

Optimization Problem Formulation and Solution Techniques 129

applications to practical problems that naturally use discrete or mixed discrete

continuous variables in their formulation. Among the methods for discrete variable

nonlinear optimization problems, the following techniques have been most commonly

discussed: branch and bounds (BBM), zero–one variable techniques, rounding–off

techniques. Penalty function approach, sequential linear programming (SLP) and

random search techniques have also been applied to discrete optimization problems.

The purpose of this chapter is to define the basic formulation of the design

optimization problem, and survey the most relevant techniques to structural design

optimization. The basic simple genetic algorithm is also described in detail.

5.2 Minimum weight via optimization techniques

It is not just in civil engineering that the search for minimum weight is the main goal,

quantity of material is an important factor in most design fields. Everyone naturally tries

to achieve as much as possible using as little as possible. The ability of engineers to

produce better designs has been severely limited by the techniques available for design

optimization. Typically, much of the development effort has focused on simulation

programs to evaluate design parameters. Now the question arises why minimum weight

design for steel frameworks built for domestic and residential activities. This question

will be answered in the next section.

5.3 Why minimum weight design for steel structures?

5.3.1 Client br ief

Clients specify their requirements through a brief. It is essential for effective design to

understand the intentions of the client: the brief is the way in which the client expresses

and communicates these intentions. As far as the designer is concerned, the factors

Optimization Problem Formulation and Solution Techniques

130

which are most important are intended use, budget cost limits, time to completion and

quality. Once these are understood, a realistic basis for producing the design will be

established.

5.3.2 Cost considerations

The time taken to realise a steel building from concept to completion is generally less

than that for a reinforced concrete alternative (Owens et al.,1992). This reduces time–

related building costs, enables the building to be used earlier and produces an earlier

return on the capital invested. To gain full benefit from the manufacturer and

particularly the advantages of speed of construction, accuracy and lightness, the

cladding and finishes of the building must have similar attributes. In addition, because a

steel framework is made up of prefabricated components produced in a factory,

repetition of dimensions, shapes and details will streamline the manufacturing process

and are major factors in economic design.

The cost of steel frameworks is governed to a great extent by the degree of

simplicity and repetition embodied in the framework components and connections.

Typical cost breakdown is investigated by many authors, e.g. Owens et al. (1992). This

can be summarised in three major stages as follows:

1. Fabrication: this includes piling foundation, steel framework, brickwork, external

and internal cladding, sunscreens, etc. This stage may cost about 52% of the total

cost.

2. Finishes: this involves ceilings and floors, etc. This stage is calculated as 20% of the

total cost of the building.

3. Services: this entails electric facilities, lifts plumbing and sprinklers, etc. It is

estimated as 28 % of the total cost.


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The designer of a steel framework should aim to achieve minimum overall cost. This

is a balance between the capital cost of the framework and the improved revenue from

early occupation of the building through fast design, fabrication and erection. For a

domestic and residential building, the cost of welding and connections becomes the

same for different designs presupposing the method of design.

Now the question arises what the structural designer could do to provide the client

with an economic design presupposing the intend use of the building as a domestic and

residential activities. Assuming the location of the building is fixed, the cost of

foundation piling can be determined depending of the structural system and the bearing

capacity of the soil. In addition, the cost of finishes and services depends on the

intended use of the building and this can be easily determined. From this discussion, it

becomes clear that for a domestic and residential construction, the minimum weight of

construction becomes a major task for a structural designer. From this point of view,

many researches, among them Grierson and Pak (1993), Adeli and Kumar (1995),

Huang and Arora (1997), Jenkins (1997), Saka (1998) and Camp et al. (1998), have

investigated methods seeking minimum weight design and considering different

constraints.

The formulation of the optimization will be addressed and the concept of genetic

algorithms in structural optimization will be discussed.

5.4 Optimization problem formulation

The constrained optimization problem, which is more practical optimization problem,

has been formulated in terms of some parameters and restrictions. The parameters

chosen to describe the design of a structure are known as design variables while the


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restrictions are known as constraint conditions. Mathematicians formulated the

optimization problem in a standard mathematical function formula F(x) as described in

the following sections.

5.4.1 Design var iables

Implicitly, the notion of optimizing a structure presupposed some flexibility to change

the design elements. The potential in changing expressed in terms of ranges of

permissible changes of certain design variables denoted by a vector x = { x1 , x2 ,…, xn} .

The design variables in the structural optimization problem might be the cross–sectional

area, the node position, the second moment of inertia, etc. In other words, they are the

parameters that control the geometry of the optimized structure. The design variable can

take either continuous or discrete variables. A continuous variable is one that takes any

value in the range of the variation in its region. A discrete variable is one that takes only

isolated values, typically from a list of permissible values or a catalogue. Therefore,

these design variables can be expressed as

)21( TTTTJj ,,, xxxxx Λ= , Jj Λ,2,1=

jj,i Dx ∈ and (5.1)

)(21 λ,j

,...,,j

,,jj dddD = .

The vector of design variables x is divided into J sub–vectors Jx . The

components of these sub–vectors j,ix take values from a corresponding cataloguej

D , i

indicates the number of design variables in each sub–vector and λ is the number of

sections in each catalogue.


133

In structural steelwork design problem, the material design variables and sectional

properties from catalogue are often discrete. In the present study, the standard sections

of universal beams, universal columns and circular hollow sections suggested by the BS

4 and BS 4848 are used.

Although the discrete variable problem appears to be easier to solve than the

continuous one (since fewer possible solutions exist), in general, it is more difficult to

solve except in some trivial cases. This is due to the fact that the discrete design space is

disjoint and non–convex (Arora et al., 1994).

5.4.2 Objective function

The notation of optimization also implies that there are some merit function or functions

that can be improved and can also be used as a measure of effectiveness of the design.

The objective function, merit function, and cost function are names of the function F(x)

being optimized and this function measures the effectiveness of the design. This

function might be a formulation of a single objective f1(x) or multiple objectives as

follows:

F(x) = { f1(x), f2(x),…, fp(x)} . (5.2)

Optimization with more than one objective is generally referred to as multicriteria

optimization. For structural optimization problems, weight, displacements, stresses,

buckling loads, vibration frequency and cost or any combination of these can be used as

objective function. The multicriteria function has different ways commonly used for

reducing the number of functions to one. The first way is simply to generate a composite

objective function that replaces all the objectives. The second way, most common in the

formulation of design optimization problems, is to select the most important objective


134

function, for instance the total weight of the structure, and to consider this function to be

the goal of the optimization task. Then, imposed limits, like stresses in each member,

nodal displacements and critical buckling load, etc, are prescribed. The third way is

Pareto optimization in which a range of potential products designs is created to meet

conflecting objectives, thus allowing the requirements to be refined in the light of

further information (see Brandt, 1992).

5.4.3 Constraints

The limits, which take values for the design variables, are known as side constraints.

The side constraints are divided into two types. The first type, commonly used in the

design problem, is an inequality constraint:

1)(

~)(

≤x

x

s

s

G

G, s =1, 2, ss,Λ (5.3)

where )(xsG and )(~

xsG are the calculated and limited values of constraints and ss is

the number of inequality constraint functions.

In the design optimization problem, not all constraints are functions of one term

but they are functions of several terms. This can be expressed by

1)(

~)(

...)(

~)(

)(~

)(

,

,

2

21

1

≤+++x

x

x

x

x

x

sss

sss

s, s,

G

G

G

G

G

G

s,s,

(5.4)

where ss is the number of terms in the constraint function.

5.4.4 Standard formulation

From the above sections, the final formulation of the optimization problem can be

mathematically represented by


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Minimize F(x)

subjected to: 1)(

~)(

≤x

x

s

s

G

G, s =1, 2, ss,Λ

),,2,1( TTTTJj xxxxx Λ= , Jj Λ,2,1= (5.5)

jji Dx ∈, and

)(21 λ,j

,...,j

,,jj dddD = .

The feasible solution of a nonlinear problem can be graphically represented. For

example, a nonlinear function F(x) of two design variables x1 and x2 with three

nonlinear constraints G1(x), G2(x),and G3(x) can depicted as shown in Figure 5.1.

5.5 Features of a design optimization problem

It is important to highlight some of the features of the discrete nonlinear problem (5.5).

First, any of the inequality constraints may not be active at the optimum point because

the constraint surface may not pass through any of the discrete points, i.e. in numerical

calculations only a point closest to the constraint boundary may be found. Second, there

is no simple criterion such as Kuhn-Tucker condition to terminate the iterative search

x1

F(x)

x2

G3(x)

G2(x)

G1(x)

Figure 5.1. Feasible region in nonlinear problem


136

process. Thus, local optimality of the solution point can not be assured unless an

exhaustive search is performed. Third, the size of discreteness and nature of the discrete

values may govern the behaviour of some of the numerical algorithms as well as the

final solution of the problem. Fourth, the design problem is highly non–linear problem

due to the nature of design variables and the relationships between the constraint

functions and design variables. Fifth, constraints have different formulation for different

members of the structure. For example, a structure has beams, columns, and a bracing

system. The constraints that control the design of beams are different from those of

bracing systems or columns. Moreover, the set of catalogue sections for beams are

different from those of bracing systems or columns. Sixth, the computational effort

needed to reach satisfactory results increases with the complexity of the treated design

problem. Therefore, it is important to review optimization techniques that deal with

discrete design variables. This is summarised in the following section.

5.6 Review of discrete optimization techniques

A review of the methods for discrete variable optimization was recently presented by

Bremicker et al. (1990), Vanderplaats and Thanedar (1991) and Arora et al. (1994).

Several algorithms for discrete optimization problems were developed, among them

branch and bound method, penalty function approach, rounding–off, cutting plane,

simulated annealing, genetic algorithms, neural networks, and Lagrangian relaxation

methods. It is observed that some of the methods for discrete variable optimization use

the structure of the problem to speed up the search for the discrete solution. This class of

methods is not suitable for implementation into a general purpose application (Arora et

al., 1994). The branch and bound method, simulated annealing, and genetic algorithm


137

are the most used methods. Herein, the literature review will be focused on these

methods in the following sections.

5.6.1 Branch and bound method

The branch and bound method BBM is perhaps the most widely known–method for

mixed–discrete optimization problems. The method was originally developed for LP,

however it is quite general and can be applied to nonlinear discrete and mixed variable

problems. It is basically an enumeration method where one first obtains a minimum

point for the problem assuming all variables to be continuous. Then, each variable is

assigned a discrete value in sequence and the problem is solved again in the remaining

variables. The process of assigning discrete values to variables need not start from a

continuous optimum point although this approach may reduce the number of times the

problem needs to be re–solved to obtain a feasible discrete point and subsequently the

optimum solution. It can be seen that the number of times the problem needs to be re–

solved increases exponentially with the number of variables. Several procedures have

been devised to reduce this number. The first use of the branch and bound method is

attributed to Land and Doig (1960) for linear problem. Other attempts, to use BBM to

solve integer LP problems related to the plastic design of frames, made by Reinschmidt

(1971).

BBM was combined with exterior penalty functions and SQP methods to treat the

mixed–discrete NLP problem. John et al. (1988) combine BBM with sequential

linearization for discrete optimal design of trusses. Hajela and Shih (1990) used BBM to

solve multi–objective optimization problems with discrete and integer variables.

Salajegheh and Vanderplaats (1993) used BBM for optimizing trusses with discrete

sizing and shape variables. Large storage space was needed and an exponential growth


138

in computational effort limits the applicability of BBM to solve higher dimensional

problems.

The usual BBM approach is to systematically search continuous solutions where

the discrete variables are forced to take discrete values from the specified set. The

logical structure for the set of solutions is that of a tree for each variable. Initially an

optimum point is obtained by treating all design variables as continuous. If this solution

is discrete, then the process is terminated. If one of the desired variables is not discrete,

then its value lies between two discrete values:

1,,, +<< λλ jjij dxd . (5.6)

Now two sub–problems are defined, one with the constraint λ,, jji dx ≤ and the

other with 1,, +≤ λjji dx . This process called branching. It basically eliminates some

portion of the continuous feasible region, which is not feasible for the discrete problem.

It does not, however, eliminate any of the discrete feasible solutions. The two sub–

problems are solved again, and the optimum solutions are stored as nodes of the tree

containing optimum values of the variables, the objective function and the appropriate

bounds on the variables. This process of branching and solving continuous problems is

continued until a feasible solution is obtained. The cost function corresponding to this

solution becomes an upper bound on the optimum solution. From this point, all the

nodes of the tree that have cost function values higher than the established upper bound

are eliminated from further consideration. Such nodes, known as fathomed nodes, when

their lowest point is reached and no further branching is necessary from them. This

process is known as bounding. The process of branching and bounding is repeated from

each of the unfathomed nodes. From each node, at most two new nodes may originate.

A better upper bound for the optimum objective function is established when a feasible


139

discrete solution is obtained with a value of objective function less than the current

upper bound. The nodes may be fathomed in any of the following three ways: (1) a

feasible discrete solution to the continuous solution with the cost function value higher

than the current upper bound, (2) an infeasible continuous problem and (3) the optimal

value of the objective function for the continuous problem higher than the upper bound.

The search is terminated when all the nodes have fathomed.

This method has been used successfully, however, for problems with a large

number of discrete design variables, the number of sub–problem nodes becomes large

making the method inefficient (Arora et al., 1994). This drawback is oppressive for

nonlinear optimization problems. Further detailed strategies and enhancements

developed can be found for the BB in Mesquita and Kamat (1987), Ringertz (1988),

Sandgren (1990) and Haftka and Gurdal (1993).

5.6.2 Simulated annealing

Simulated annealing (SA) is one of the techniques, which do not require derivatives of

the problem functions because it does not use any gradient or Hessian information. The

idea of SA originated in statistical mechanics by Metropolis et al. (1953). The approach

is suited to solving combinatorial optimization problems, such as discrete–integer

programming problems.

The use of simulated annealing for structural optimization is a quite recent

occurrence. Elperin (1988) applied the SA to design a ten–bar truss where member

cross–sectional area were to be selected from a set of discrete values. Kincaid and

Padula (1990) used SA for minimizing the distortion and internal forces in a truss

structure. Balling (1991) obtained the optimum design of three–dimensional steel

structures using SA. One year later, the same framework was studied using the filtered


140

simulated annealing algorithm by May and Balling (1992). Recently, Leite and Topping

(1996) proposed a parallel simulated annealing model for structural optimization.

5.6.3 Genetic algor ithms

5.6.3.1 Background

The famous naturalist Charles Darwin defined natural selection or survival of the fittest

in his book (Darwin, 1929) as the preservation of favourable individual differences and

variations, and the destruction of those that are injurious. In nature, individuals have to

adapt to their environment in order to survive in a process called evolution, in which

those features that make an individual more suitable to compete are preserved when it

reproduces, and those features that make it weaker are eliminated. Such features are

controlled by units called genes, which form sets known as chromosomes. Over

subsequent generations not only the fittest individuals survive, but also their genes

which are transmitted to their descendants during the sexual recombination process,

which is called crossover.

In the late 60s, John H. Holland became interested in the application of natural

selection to machine learning. He developed a technique known as reproductive plans

that allowed computer programs to mimic the process of evolution. This technique

became popular after the publication of his book (1975). He renamed this technique

using the term genetic algorithm (GA). The main goals of the research of Holland and

his students were

• to abstract and rigorously explain the adaptive processes of natural systems.

• to design artificial systems software that retained the important mechanisms of

natural systems.


141

Basically, the central thrust of the research on genetic algorithms (GAs) has been

due to its robustness and the balance between efficiency and efficacy necessary for

survival in many different environments. GAs are search algorithms, which are based on

the mechanics of natural selection and survival of the fittest, and unlike many

mathematical programming algorithms they do not require the evaluation of gradients of

the objective function and constraints.

Koza (1992) provides the following definition of a GA:

The genetic algorithm is a highly parallel mathematical algorithm that

transforms a set (population of individual mathematical objectives typically

fixed–length character strings patterned after chromosome strings), each

with an associated fitness value, into a new population (i.e., the next

generation) using operations patterned after the Darwinian principle of

reproduction and survival of the fittest and after naturally occurring genetic

operations (notably sexual recombination).

The GA–based techniques accept discrete and/or continuous design variables and

therefore are very versatile. GAs are different from most optimization techniques in

many ways:

• GAs work on a coding of the design variables–binary bit string representation is one

of such coding, rather than the design variables themselves. This characteristic

allows the genetic algorithms to be extended to a design space consisting of a mix of

continuous, discrete and integer variables,

• GAs proceed from several points in the design space to another set of design points.

Consequently, GA techniques have a better chance of locating the global minima as

opposed to these schemes that proceed from one point to another,


142

• GAs work on the function evaluations alone and do not require any of the function

derivatives. Although derivative based techniques contribute to faster convergence

toward the optimum, the derivative also directs the search process towards a local

optimum, and

• GAs use probabilistic transition rules, which is an important advantage in guiding a

highly exploitative search. To this extent, GAs should not be considered as a variant

of the random walk approach.

The general features of the theory of the GA are widely accepted and applied,

which result in good solutions for different types of problems in different disciplines.

The following is a description of the features of natural evolution as observed by

Holland (1975).

1. Evolution is a process that operates on chromosomes rather than on the living beings

they encode.

2. Natural selection is the link between chromosomes and the performance of their

decoded structures. Processes of natural selection cause those chromosomes that

encode a successful structure to reproduce more often than those do not.

3. The process of reproduction is the point at which evolution takes place. Mutations

may cause the chromosomes of biological children to be different from those of their

biological parents and recombination processes may create quite different

chromosomes in the children by combining material from the chromosomes of two

parents.

4. Biological evolution has no memory. Whatever it knows about producing

individuals who will function well in their environment is contained in the gene

pool, the set of chromosomes carried by the current individuals, and in the structure

of the chromosome decoders.


143

5.6.3.2 Survival of the fittest

GAs are implicit enumeration procedures. A set of randomly created design alternatives

or individuals representing a population in a given generation are allowed to reproduce

and cross among themselves, with bias allocated to the most fit individuals. A

combination of the most desirable characteristics of mating members of the population

results in progenies that are more fit than their parents. Therefore, if a measure which

indicates the fitness of a generation is also the desired goal of a design process,

successive generations produce better values of the objective function.

5.6.3.3 Encoding the design var iables

The technique for encoding solutions may vary from problem to problem and from

genetic algorithm to genetic algorithm. In Holland’s work (Holland, 1975), encoding is

carried out using bit strings, 0 and 1. A major task is the encoding of different design

sets into chromosomes so that the GA can use them.

In structural design optimization, section properties of one member form a design

variable. The member section is then represented by a bit string. Each bit–string is then

merged to form chromosomes, which represent a design set. In the present study, the

possible cross–sections of each design variable are presented in binary strings. The bit–

string is associated with a position in the table and its corresponding sectional

properties. To make things clearer, assume a single–bay single–storey framework shown

in Figure 5.2. This frame has two design variables x1 and x2, which represent the

columns and the beam girder respectively. The variable x1 takes a position out of 32 UCs

from BS 4 while x2 takes a position out of 64 UBs.


144

The bit–string of each design variable implies a position in the corresponding

table and thus the properties of this section can be selected. The cross section, selected

from the catalogue, can be represented in the binary code according to the number of the

available cross sections. The string length of each design variable vnλ should be

evaluated by

vnλ = n2 . (5.7)

For instance, there are 64 types of UBs, so the number of bits required to distinguish the

range is 6. A part of the encoded variables for UBs are listed in Table 5.1. Similar

encodings for the utilised UCs and CHS can be drawn. Hence, for the given framework

shown Figure 5.2, the chromosomes given in Figure 5.3 represent an individual in the

population. This can be read as a design in which the design variable x1 takes the

position of number 8 in the table of UCs while x2 takes the position of number 6 in the

table of UBs.

0 0 1 1 1 0 0 0 1 0 1

x1 x2

Figure 5.3. Chromosomes of a design set using binary representation

x1 x1

x2

Figure 5.2. Single–bay single–storey framework


145

Table 5.1. Part of the encoded variables for UBs

Second moment of area (cm4) Catalogue

position Encode var iable

Cross section Area (cm2) about

major axis about

minor axis

1 000000 914 × 419 × 388 UB 494 719000 45400

2 000001 914 × 419 × 343 UB 437 625000 39200

3 000010 914 × 305 × 289 UB 369 505000 15600

4 000011 914 × 305 × 253 UB 323 437000 13300

5 000100 914 × 305 × 224 UB 285 376000 11200

6 000101 914 × 305 × 201 UB 256 326000 9430

7 000110 838 × 292 × 226 UB 289 340000 11400

8 000111 838 × 292 × 194 UB 247 279000 9070

9 001000 838 × 292 × 176 UB 224 246000 7790

10 001001 762 × 267 × 197 UB 251 240000 8170

11 001010 762 × 267 × 173 UB 220 205000 6850

12 001011 762 × 267 × 147 UB 188 169000 5470

13 001100 686 × 254 × 170 UB 217 170000 6620

14 001101 686 × 254 × 152 UB 194 150000 5780

Different ways of encoding the variables were implemented. For example,

Goldberg (1990) presented a theory of convergence for real–coded (floating–point)

GAs, and also real numbers and other alphabets have been proposed by Wright (1991).

The term “ floating” may seem misleading since the position of the implied decimal

point is at a fixed position, and the term fixed point representation seems to be more

appropriate. However, the reason is that the variable, representing a parameter to be

optimized, may have a point at any position along the string. This means that even when

the point is fixed for each gene, it is not necessarily fixed along the chromosomes.

Therefore, some variables could have a precision of 32 decimal places, while others are

integers. As Eshelman and Schaffer (1993) point out, many researchers in the GA

The

res

t of

cros

s se

ctio

nal p

rope

rtie

s fr

om th

e ca

talo

gue


146

community agreed to use real coded GAs for numerical optimization despite the fact

that there are theoretical arguments that seem to show that small alphabets should be

more effective than large alphabets. Muhlenbein and Schilierkamp–Voosen (1993) also

used real numbers directly for continuous function optimization.

5.6.3.4 Why bit str ing encoding?

Over other encodings, bit strings have several advantages that can be summarised as

follows:

1. They are simple to create and manipulate.

2. They are theoretically tractable, in that their simplicity makes it easy to prove

theorems.

3. Performance theorems have been proved for bit string chromosomes that

demonstrate the power of natural selection on bit string encodings.

4. Just about anything can be encoded in bit strings, so one–point crossover and

mutation operators can be applied to a wide range of problems.

5.6.3.5 The anatomy of a simple GA

In a simple GA, one starts with a randomly created set of designs. From this set, new

and better designs are reproduced using the fittest members of the set. The entire process

is similar to a natural population of biological creatures, where successive generations

are conceived, born and raised until they are ready to reproduce. A simple GA is

composed of three operations. These are reproduction, crossover and mutation.

Reproduction is an operation where an old string is copied into the new population

according to the string fitness. Here, fitness is defined according to the objective

function value. More fit strings, i.e. those with smaller objective function values, receive


147

higher numbers of offsprings. The reproduction operator may be implemented in

algorithmic form in a number of ways. Perhaps the easiest way is to create a biased

roulette wheel (see DeJong, 1975) where each current string in the population has a

roulette wheel slot sized in proportion to its fitness. Consequently, each time we require

offspring, a simple spin of the weighted wheel yields the reproduction candidate. In this

way, more highly fit strings have a higher number of offspring based on the probability

of selection seliP in the succeeding generation as given by

�=

=p

selN

1jj

ii

F

FP (5.8)

where Fi is the value of the objective function of the individual i–th and pN is the

number of individuals in the population and is known as population size.

Once a string has been selected for reproduction, an exact replica of the string is

made. This string is then entered into a mating pool, a tentative new population, for

further genetic operator action.

Other selection schemes can be used, among them stochastic remainder selection

suggested by Brindle (1981), stochastic universal selection proposed by Baker (1987).

Ranking selection is presented by Baker (1985) in which the population is sorted from

best to worst, and each individual is copied as many times as possible. According to a

non–increasing assignment function, and then proportionate selection is performed

according to that assignment. Goldberg and Deb (1991) implemented tournament

selection in which the population is shuffled and then is divided into groups of gn

elements from which the best individual, i.e. the fittest, will be chosen. The number of

parents N, which are selected can be evaluated by


148

g

P

n

NN = . (5.9)

After being selected, crossover takes place. Crossover corresponds to allowing

selected members of the population to exchange characteristics of the design among

themselves (Arora et al., 1994). Simple crossover may proceed in two steps. First,

members of the newly reproduced strings in the mating pool are mated at random.

Second, each pair of strings undergoes crossover. Several ways of performing crossover

are used in the literature, the most simple one termed one–point crossover Goldberg

(1989). This can be illustrated as follows: an integer position b along the string as

indicated in Figure 5.4a is selected uniformly at random between 1 and the string length

less one [1, 1v

−nλ ]. Two new strings termed children are created by swapping all

characters between positions b+1 and vnλ of the parents inclusively. Figure 5.4b shows

a similar way of presenting the crossover. This is termed two–point crossover.

In order to find an effective search, Eshelman et al. (1989) and Syswerda (1989)

use different types of crossover such as segment crossover, uniform crossover, shuffle

crossover. Multi–point traditional crossover is also presented by DeJong (1975) as

natural extension of two–point crossover. He treats the chromosome in the multi–point

crossover as a ring, which the crossover points cut into segments.

Mutation is the third step in simple GA, and this step safeguards the process from

a complete premature loss of valuable genetic material during reproduction and

crossover. In terms of binary string, this step corresponds to selecting a few members

of the population, determining at random a location on the strings, and switching the 0

or 1 at that location. To illustrate mutation in an example, assume two crossed–over

children as given in Figure 5.5. The 2nd bit was randomly selected over the child 1


149

while it is the 6th bit over the child 2. Then, the procedure is to change the 1 to a 0 and

vice versa as shown in Figure 5.5.

Parent 1 0 0 0 1 0 1 1 1 1 0 1 0 1 1 1 0 Parent 2

Child 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 Child 2

Parent 1 0 0 0 1 0 1 1 1 1 0 1 0 1 1 1 0 Parent 2

Child 1 0 0 1 0 1 1 1 1 1 0 0 1 0 1 1 0 Child 2

1 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0

1 1 1 1 0 1 1 1 0 0 0 0 1 0 1 0

Figure 5.5. Assembling the mutation stage

Cross–point Cross–point

(a) Single–point crossover

Cross–points Cross–points

Figure 5.4. Most used crossover operators in the binary strings

(b) Two–point crossover

Crossed–over child 1 Crossed–over child 2

Mutated child 1 Mutated child 2


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It is observed that the mutation is a random walk through the string space. When

used sparingly with reproduction and crossover, it is an insurance policy against

premature loss of important notions (Golgberg, 1989). Consequently, it plays a

secondary role in GAs.

The foregoing three steps are repeated for successive generations of the

population until no further improvement in the fitness is attainable. The member in this

generation with the highest level of fitness is the optimum design.

Figure 5.6 illustrates the flowchart for a simple GA linked to a structural design

problem. At the beginning, all the necessary data – GA parameters and structural

geometry – will be read and the process of the GA will start for the first generation.

The initial population will be generated randomly. Then, the objective function

regarded as the weight of the structure as well as the constraint functions, which are

reflected on the design criteria requested by BS 5950, are computed. At this stage, the

average, maximum and the fittest design are obtained. Convergence criteria described

later are also checked. The GA process is terminated if the convergence is achieved.

Otherwise, the GA process resumes. By creating the mating pool and applying the GA

operators, the next population is created. The GA process will proceed until either the

convergence is achieved or the maximum number of generations is reached.


151

Yes

No

Yes

No

Generation 1: Randomly generate the initial population

Design set i

Decode binary values to integer values

Analyse the framework, compute the weight of the structure, and investigate the constraint violation

(see Figure 2.15)

Design set =Np?

Convergence occurred?

Store the best individuals, and impose them into the next generation and carry out the crossover and mutation

New generation New design

Select the cross sectional properties from the proper catalogue for each design variable

Evaluate the objective and penalised functions for each design set

Start

Stop

Figure 5.6. Flowchart for genetic algorithm linked to structural design problem

Input data files: GA parameters, structural

geometry, etc


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5.6.3.6 Constraints management

GAs have traditionally been applied to unconstrained problems as they have no built–in

method to handle constraints. Constraints can be classified as two types: explicit and

implicit. Explicit constraints are those that can be checked without a system simulation.

Cost is often one example of an explicit constraint. Implicit constraints require a system

simulation i.e. analysis and design checks. For example, cross sections have design

criteria as requested by the code of practice, therefore, a system simulation must be run

before this information can be ascertained. Several approaches have been used to handle

constraints including:

1. using specialised operators that maintain feasibility,

2. allowing only feasible solutions in the population and

3. applying a penalty to those solutions that violate one or more constraints.

Specialised operators work only for explicit constraints, and are useful for those

problems such as the travelling salesman problem. The second and third approach can

be used with explicit or implicit constraints, or a combination of both. The second

approach, eliminating those designs from the population that violate one or more

constraints, can be very ineffective for large problems that have few viable solutions

compared with the number of infeasible ones. The most prevalent technique for coping

with constraint violations is to penalise a population member for one or more violations.

The main difficulty in applying penalty functions is that they are generally problem

dependent. Different techniques of employing penalty functions are used in the literature

among them Moe (1973), Fletcher (1975), Haftka and Starnes (1976), Shin et al. (1990),

Hajela and Yoo (1995), Huang and Arora (1997), and Camp et al. (1998). Generally, the

problems attempted using GAs are all of the constrained optimization type, and


153

consequently the optimization problem must be converted into unconstrained problems

This can be dealt with using a penalty–based transformation method (Hajela and Yoo,

1995), resulting in the following problem:

Minimize ))()(()()( xxxx H,G,rPFr,F−

+= (5.10)

where −F is the modified objective function that also contains the penalty term P ,

which brings the constraint functions into the problem and r is called a penalty

multiplier. The way in which the penalty parameters and the constraint functions are

combined and the rules for updating the penalty parameters specify the particular

method.

In the present work, the design optimization problem has been attacked differently

because careful consideration must be given to the selection of the penalty function, and

in the present context, the "exact" penalty function is used. This results in the following

definition of the fitness function combined with the simple "exact" penalty function:

Maximize ��

=violatedsconstraintofany0

satisfiedsconstraintall)(C)(

,

,F-F

xx (5.11)

where C is a constant evaluated at each generation. The technique used for penalty

function is described in Chapter 6.

5.6.3.7 Convergence cr iter ia and termination conditions

Convergence criteria have to be evolved to decide when to terminate the process of

optimization. In the present study, three criteria are used and if any of them are satisfied,

then the process will terminate. These criteria are:


154

• If the fittest design has not changed for 30 successive generations, or if the

difference between the fittest design cuF of the current generation and that of 30

generations before is very small value .cuC This could be expressed in the form

cucu

30cucu

CF

FF≤

− −

. (5.12)

• As we proceed with more generation the population gets filled by more fit

individuals, with perhaps a very small deviation from the fitness of the best

individuals. Consequently, the average fitness comes very close to the fitness of the

best design. This could result in another convergence criterion such that the

percentage difference between the average fitness avF of the current population and

the current fitness of the best design cuF reaches a very small value avC . This can

be expressed by

avcu

avcu

CF

FF≤

−. (5.13)

• The simplest one is when a total allocated number of generations ( 200max =gen )

are reached.

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