# A New Genetic Algorithm Methodology for Design Optimization of Truss Structures: Bipopulation-Based Genetic Algorithm with Enhanced Interval Search

Authors: Tugrul Talaslioglu
Journal: Modelling and Simulation in Engineering
 (2009)

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## Abstract

A new genetic algorithm (GA)methodology, Bipopulation-Based Genetic Algorithm with Enhanced Interval Search (BGAwEIS), is introduced and used to optimize the design of truss structures with various complexities. The results of BGAwEIS are compared with those obtained by the sequential genetic algorithm (SGA) utilizing a single population, a multipopulation-based genetic algorithm (MPGA) proposed for this study and other existing approaches presented in literature. This study has two goals: outlining BGAwEIS's fundamentals and evaluating the performances of BGAwEIS and MPGA. Consequently, it is demonstrated that MPGA shows a better performance than SGA taking advantage of multiple populations, but BGAwEIS explores promising solution regions more efficiently than MPGA by exploiting the feasible solutions. The performance of BGAwEIS is confirmed by better quality degree of its optimal designations compared to algorithms proposed here and described in literature.


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## Body

## 1. Introduction

The
steel structures consist of hot-rolled steel profiles with different
cross-sectional properties. The optimum design of steel structures is considered
as a constrained  optimization problem. Modern optimization methods used in the
design of steel structure as well as in a number of engineering design problems
are inspired by natural phenomenon, such as survival of the fittest, immune
system, swarm intelligence, simulating annealing, and ant colony (Saka [1]). These methods explore the problem space utilizing the global
or local search-based algorithms. Moreover, it is also possible both to
incorporate a local search algorithm into a global search, namely, hybridization
of algorithms (memetic algorithms) and to run them in parallel (Moscato [2],
Radcliffe and Surry [3],  Cantú-Paz [4]).The
powerful member of these algorithms is evolutionary algorithms (EAs). EAs mimic
the process of natural evolution. The evolutionary computation is achieved by  either
simultaneously examining and manipulating a set of possible candidate
individuals or using a special individual along with its neighbors in the
generation of new individuals.Genetic
algorithm (GA), a member of EAs, is a population-based global search technique
based on the  Darwinian evolutionary theory (Holland [5], Goldberg
[6]). The preliminary approach of GAs is SGA (see a pseudocode in  Algorithm 1). SGA guides the evolutionary search by a
single population Pi. The
size of Pi is denoted by SP. Individuals are encoded in a string
scheme associated with one of the codes of the binary, integer, and real. In
the evolutionary search, the promising individuals  Pi−sel and Pi+1−sel
are chosen from  the population by a selection operation (roulette wheel,
stochastic universal sampling, ranking, truncation, etc.). Then, the individuals
chosen are applied to recombination and mutation operation (one or multipoint
crossover and mutation, uniform crossover, etc.). These evolutionary operations
(mutation mut, crossover cr, and selection sel) are governed by their related evolutionary parameters Par (mutation and recombination
probability rates, selection pressure, etc.). The population
Pnew evolved by the
application of these evolutionary operators is decoded. Then, the fitness
values are computed by use of this population. The evolutionary search is
executed to transmit (migration) the individuals (emigrant and immigrants) to
the next populations until satisfying a predetermined stopping criteria (e.g.,
completion of a generation number NG).Algorithm 1:Pseudocode for SGA.
SGA(Pi,NG,SP,Fi,Parsel,Parmut,Parcr)If[Pi]=[], Initialize (Pi, SP, NDV)fori=1: NG[Pid]=PiIf required,[Pid]=Decoding(Pi)If[Fi]=[],[Fi]=Fitness_Calculation(Pid)[Pi−sel]=Selection(Pi,Fi,Parsel)[Pi+1−sel]=Selection(Pi,Fi,Parsel)[Pnew]=PnewU  Crossover(Pi−sel,Pi+1−sel,Parcr)U Mutation(Pi−sel,Pi+1−sel,Parmut)[Pi]=[Pnew]endSGA is more
flexible optimization tools. Therefore, it is possible to achieve a balance
between two main genetic features: exploration of promising locations in the
search space and exploitation of best solutions obtained. The accuracy of this
balance has a big effect in the determination of SGAs’ performance associating
with the quality of solution, speed of convergence and generation of feasible solutions,
and so forth. If this balance is not appropriately achieved throughout the
generations, a stagnation problem in the progression of evolutionary search is
occurred after an equilibrium state. This equilibrium state is called immature
convergence. Figure1 (depicted the local maxima by +'s) can be used for
highlighting the reason of immature convergence. As seen in Figure 1, exploration
of the trajectory to the global maxima is provided only by maintaining the
migration of the populations. That is, if keeping the variations among the
populations and the diversity within populations, then the quality of migration
will increase, and so, the exploration of the global maxima embedded in one of
the subregions will be more powerful in attributes of the geographically nearby
populations. In this regard, the task about how the computational cost required
for this balance to be minimized leads to the emergence of new GAs.Figure 1
General visualisation of the populations randomly scattered in search topography.In this study, a bipopulation-based genetic algorithm methodology named
BGAwEIS, whose crucial elements are developed by utilizing the fundamentals of
SGA, is applied for the design optimization of truss structures. BGAwEIS
utilizes feasible solutions to collect valuable genetic heredity from potential
ancestors and to transmit it to offspring. For this purpose, two populations
are employed for the transmission process. An intensive search of subregions of
entire solution region is provided by gradual exploration strategy developed
for BGAwEIS. Moreover, the dominance of similar feasible solutions in next
generations is prevented by recreation of the populations at certain generation
numbers. In order to asses the quality of optimal designations
generated by BGAwEIS, optimal design results obtained by both SGA and existing
approaches outlined in the literature are considered. Furthermore, a
multipopulation-based genetic algorithm (MPGA) approach is proposed to
investigate the effect of usage of multiple and single populations on the
quality degree of optimal designations. For this purpose, an optimization tool
called GEATbx coded in MATLAB is utilized to compute the evolutionary processes
of MPGA (Pohlheim [7]).This paper is organized as follows. The next section presents a
background concerned existing design optimization approaches; Sections3 and 4
contain the optimum design problem and main elements of BGAwEIS including the
basic principles of MPGA associated with GEATbx; design details and examples
are provided in Sections 5 and 6, sequentially; conclusion is presented in last
section following Section 7 that summarizes the discussion of results.


## 2. The Review of Major Design Optimization Approaches Used in the Design of
Steel Structures

A summary of major optimization approaches and their applications to the
design optimization of steel structures are presented by a brief introduction. In this regard, the first part reviews the preliminary studies. Second part
evaluates the evolutionary algorithms including their hybrid and parallel
models. In this summary, it is intended to present the most representative
works in a chronological order.


### 2.1. Preliminary Studies

The preliminary studies on
the design optimization of steel structures are based on gradient-based
mathematical programming techniques. Linear programming approach was widely
utilized for weight minimization of truss structures, considering structural
responses under both elastic and plastic behaviors (Cornell [8]), Bigelow and Gaylord [9]). Nonlinear programming was usedasan alternative method to linear
programming. Majid and Elliott [10] applied nonlinear programming to optimize
the weight of a two-bay four storey frame. Afterwards, sequential linear and
quadratic programming techniques (SLP and SQP) were widely used for the design
optimization of steel structures. Vanderplaats and Sugimoto [11] developed a design technique “automated design synthesis” utilizing the approaches of SLP and SQP. Automated design synthesis method was proposed for minimizing the weight of frames with various bays and stories under static and seismic loadings by Karihaloo and Kanagasundaram [12], and Gülay and Boduroğlu [13]. The optimization techniques based on nonlinear programming were used for generation of optimal designations for steel structures under different loading conditions
and design requirements (Lassen [14], Wang and Grandhi [15], Salajegheh [16],
Hernández[17]).Optimality criteria method (OCM) is
another challenging method. Hybridizing the nonlinear mathematical programming
with Lagrange multipliers for inclusion of constraints forms the basis of OCM
(Arora [18], Cameron et al. [19]). Saka and Hayalioglu [20] used OCM for optimization of geometrically nonlinear steel structures made of elastoplastic material. Hayalioglu and Saka [21] proposed OCM for design optimization of
frames with nonuniform cross-sections. Chan et al. [22, 23] carried out the
weigh minimization of three-dimensional steel structures with discrete
cross-sections by OCM. They devised a transformation process for continuous and discrete design variables. Saka [24] optimized a frame design with tapered members thereby firstly computing the member responses under external static loadings and combining them by Lagrange multipliers to generate depth variables. Saka and Kameshki [25] used OCM for design optimization of unbraced
rigid frames considering constraints imposed by sway deflections and member
stresses.


### 2.2. Evolutionary-Based Optimization Studies

Evolutionary computation based on
simulation of natural evolutionary is a new approach used in the design
optimization of steel structures. Due to being appropriate for both traditional
and novel computation applications in the field of structural engineering, evolutionary
approaches whose major members are GAs by Holland [5], evolutionary programming (GP) by Fogel et al. [26] and evolutionary strategies (ES) by Rochenberg [27] have been improved by new implementations, such as hybrid and parallel searches. Therefore, research developments on three major EAs are firstly reviewed. Then, subsection
provides an overview of recent developments concerned the issues of parallel
and hybrid implementations.GP is managed by programs defined by
point-labeled parse trees used to describe the node and elements in the steel
structure. The most important step in GP is the determination of the size and
shape of parse trees for a design problem (Keijzer and Bobovic[28, 29]). Cevik [30] used a GP methodology, namely, a gene-expression
programming, for determining rotation capacity of wide flange beams.ES uses a population of tentative design solutions and generates the populations using several genetic operators with self-adaptive parameters (Back and Schwefel [31, 32]). Cai and Thierauf [33] proposed an evolutionary strategy without self-adaptive parameters
for the design optimization of steel structures. Similar approaches were also
utilized for the design optimization, such as ES with self-adaptive parameters
for discrete and continuous design variables (Ebenau et al. [34], Rajasekaran
[35], and Baumann and Kost [36]).One evolutionary algorithm approach is the SGA. Due to its flexible
structure, its genetic components have been improved. Taking into account the
usage of genetic components, the studies are grouped into two general
categories.(i) Genetic Operators with Adjustable Parameters and Representation of Design Variables
Hajela [37] introduced a representation technique for discrete design
variables. Thus, the lower and upper bounds of continues design variables were
used to compute the values of discrete design variables with any determined
precision. He discussed the negative effect of higher precision values leading
to a larger-length binary representation. Adeli and Cheng [38] presented a decoding technique in order to use binary coded strings for continues design variables. Chen [39] discussed the lack of proportional selection operation leading to stagnation problem in evolutionary search. He showed that usage of both scaled fitness values which are computed considering certain statistical quantities of fitness values and crossover operator which was applied at different rates to the same individual improved the quality degree of optimal designations. Yang and Soh [40] pointed out that the tournament selection method was more efficient than the existing selection methods considering quality degree of optimal designations. The use of graph theory for GAs is one of the new developments in structural optimization problem. Wang and Tai [41]
devised a graph representation for the topology-related design variables in structural optimization problems. Kaveh and Kalatjari [42] utilized the graph theory for representation of the size-related design variables and force method for structural analysis. Thus, grouping of truss design variables according to magnitude and sign (compression and tension) of stress becomes a new challenging approach in the design optimization of steel structure (Saka [43],
Saka et al. [44], Toğan and Daloğlu [45]).(ii) Constraint Handling for Evaluation of Fitness Values
Camp et al. [46] devised a penalty function with several variables for
design optimization of two-dimensional steel structures. The negative effect of
this approach leading to an inaccurate penalization was shown by Rasheed [47]. In
order to cope with this task, Rasheed [47] utilized an adaptive approach for
handling the constraints. For this purpose, a penalty function was used to
compute the penalty values based on an adaptation of penalty coefficients with
respect to the penalty degree. Le Riche et al. [48] divided the population into
small groups and applied a penalty function with different variable
coefficients for each group. Coello [49] used two populations for the
generation of a new population thereby comparing the penalized values. Nanakorn and Meesomklin [50] improved the penalty function by an adaptive procedure.
One of the alternative approaches to
the penalty function is artificial immune-inspired model (Garrett [51]). Firstly, Yoo and Hajela [52] utilized this approach for solving design optimization
problem. They employed two populations: one population was used to compute
penalty values, while the other one measured the hamming distance between
penalized fitness values. Coello and Cruz Cortés [53] improved Yoo and Hajela’s
technique by devising an adaptive evolutionary mechanism against the necessity
for a penalty function.Hybrid and Parallel Search-Based Evolutionary Algorithm Approaches
In order to improve the flexibility and efficiency of evolutionary
algorithms, utilizing the hybrid or parallel models of evolutionary search
algorithms is one of the important attempts.
The hybridization concept is emerged by use of local search methods for evolutionary algorithms as a complementary tool. Local search methods are
proposed to propagate the genetic information obtained throughout evolutionary
process into the next generations. One powerful hybridization model is memetic
algorithm. This biological learning mechanism is associated with Dawkins’
notion of a meme defined as a unit of cultural evolution
(Dawkins [54]). Two important approaches, namely, Lamarckian and Baldwinian approaches
make use of this learning mechanism for their evolutionary processes. Whereas
the structure of chromosome and its fitness value are changed in the
Lamarckian’ approach, Baldwinian’ learning mechanism only affects the fitness
value of chromosome without any change in its structure. In the hybridization
of evolutionary algorithms with local search method, the converging speed to
the global optima may be lower than the case of using a pure evolutionary
search algorithm without any hybrid implementation (Ong and Keane [55]). In
order to increase the converging speed, a new memetic algorithm, namely
coevolving memetic algorithm, is developed. Its fundamentals as well as a
comprehensive review of the basic approaches based on this algorithm are
presented by Smith [56].
The concept of parallel search is introduced to evolutionary algorithms
thereby employing a number of computer processes with distributed or shared memories
for a global population or a divided global population into small populations
(subpopulations) (Cantú-Paz and Goldberg [57],Cantú-Paz [58, 59]). Parallel
systems not only preserve diversity within the current populations, but also
ensure a perpetual novelty for populations to be generated in a way of disseminating
the different characteristic features embedded in the chromosomes to next
populations. Among evolutionary algorithms the GAs are preferably chosen for the
parallel applications. The basic genetic models utilized in parallel
search are grouped into four main classes (Cantú-Paz [4]).
(a)Master-slave GA. It uses a single
population, while master processor is employed to collect valuable genetic information,
slave processors service are responsible to compute the fitness values for a
certain number of individuals (Grefenstette [60], Robbins [61] and Levine [62]).
(b)Fine grained GA or cellular GA. The larger
number of processors is assigned for fitness evaluation of subpopulations. Due
to the higher number of processors, one of difficulties is encountered during
the decision about how the computer processors to be designed and arranged
(Baluja [63]).
(c)Coarse grained GA or distributed GA. It has
several similar properties of fine grained GA (migration implementation and
multiple population usage) with an exception of using smaller number of
processors. The ease of designing and arranging the computer processors makes
this model more attractive compared to cellular GA (Herrera et al. [64]). The
distributed GA is performed depending on migration related parameters (policy,
topology, frequency etc.) (Cohoon et al. [65], Alba and Troya [66], Skolicki
and De Jong [67]). Moreover, the other important issues for migration policy are
the number and frequency of migration, replacement of immigrants, size of
populations and migration topology (Tanese [68, 69]). The migration related
processes are directly responsible for determining the excitation order of
computer processors. Moreover, if any computer processor waits to run the
migration process for exchanging the individuals, then this parallel search is
called as “synchronous,” otherwise “asynchronous” (Alba and Troya [70]).
One of the basic models utilized by the distributed GA is island model. Several distinct subpopulations are isolated with each other, but communicated
by a migration process. Evolutionary operators are applied to each
subpopulation. If the parameter values of evolutionary operators are adjustable
for each subpopulation being important for an independent exploration of
different region of search space, then island model of this type is named as
homogenous and nonhomogeneous distributed GA (Alba and Troya [70]).
(d)Hybrid GA. Distributed GA can be
straightforwardly implemented on the parallel systems consisted of a number of
computer processors providing a considerable profit to the evolutionary search
in a way of decreasing the computing time. Therefore, by itself, distributed GA
is hybridized with existing search methods at different hierarchical levels. The hierarchical level is determined according to the use of evolutionary tools
and operators for the structured population. In the determination of any
hierarchical scheme, one of the most important steps is how the population to
be structured according to a lattice-like topology.
The
cellular GA is also successfully used in the hybridization models (Martin et
al. [71]). However, the complexity degree of cellular GA is higher than distributed
GA due to the increased size of both its subpopulations and underlying grid
system consisted of computer processors. Alba and Troya [70] compared the
cellular, distributed GAs and their hybrid models. They showed that numeric
efficiency and resistance to scalability was increased by the distributed
versions of cellular GA. In order to improve the exploitation capability of the
cellular GA in a way of increasing the converging rate, it was hybridized with
a local search method, named as cellular memetic algorithm. (Folino et al. [72]). Afterwards, cellular memetic algorithm was enhanced by an implementation of an adaptive
mechanism. (Krasnogor and Smith [73]), Neri et al. [74], Caponio et al. [75],
and Quang et al. [76]).
Sakamoto and Oda [77] hybridized GAs with optimality criteria method for
topology and size optimization of truss structures. While the topology of truss
structure was evolved through GA, optimality criteria methods determine the
cross-sectional areas of truss bars. Soh and Yang [78] devised a GA approach that
is managed by the fuzzy-logic-based rules. This approach was applied for weight
minimization of structures and achieved to obtain more optimal designations
compared to SGA’s.
Adeli and Cheng [79] were developed a parallel GA called “concurrent GA.”
They utilized a number of computer processors in parallel for the design
optimization of truss and frame structures. Topping and Leite [80] utilized this
parallel GA for the design optimization of a bridge, considering a number of
constraints. Adeli and Kumar [81] used a network consisted of computer
processors for optimization of large-spaced steel structures. Sarma and Adeli [82]
hybridized the coarse grained GAs with the fuzzy logic search method, for
design optimization of three-dimensional frame structures.
Tanimura et al. [83] proposed an
island model for design optimization of truss structures taking into account
several constraints. They utilized a new penalty function and compared their
optimal designations with SGA. They showed that their island model was more
efficient than SGA. Kicinger et al. [84] utilized the island models for both
topology and size optimization of tall buildings made up with steel profiles. They used two migration topologies (ring and fully-connected topology) with
various migration strategies for the design optimization of two-dimensional
frame with the bracing elements of various types. They showed that the quality
degree of optimal designations is improved when island models were executed by using
higher number of subpopulations. Then, Kicinger and Arciszewski [85] made use
of MAs in the design optimization of same steel structure. Examining various
genetic operators and their related parameters, they showed that the MAs were
more successful than GAs.
Kaveh and Shahrouzi [86] proposed
implementing the graph theory for MAs. Lamarckian and Baldwinian approaches
were adapted to optimize the frame bracing layouts of steel frames. Moreover,
the application of these approaches is illustrated for a two-dimensional steel
frame. They compared their optimal results with SGAs and displayed that whereas
Lamarckian approach reduces the topological variance with a more converging
rate, the better results are obtained by an incorporation of a dynamic mutation
band control to the Baldwinian approach.
Karakasis et al. [87] devised a radial
basis function network for the distributed GA and applied it to an aerodynamic
shape optimization problem. They compared four variants of GA and concluded
that their distributed versions offer an additional advantage in the
exploration of the interconnected processor network. Then, in order to carry
out the shape optimization of same design problem, they devised a hierarchical
distributed evolutionary scheme thereby adapting both the aerodynamic design
formulation and a navier-stoke equation solver into a radial basis unction
network (Karakasis and et al. [88]). Liakopoulos et al. [89] utilized a grid
system consisted of a number of computer processors for performing the
hybridization of hierarchical and distributed algorithms.


## 2.1. Preliminary Studies

The preliminary studies on
the design optimization of steel structures are based on gradient-based
mathematical programming techniques. Linear programming approach was widely
utilized for weight minimization of truss structures, considering structural
responses under both elastic and plastic behaviors (Cornell [8]), Bigelow and Gaylord [9]). Nonlinear programming was usedasan alternative method to linear
programming. Majid and Elliott [10] applied nonlinear programming to optimize
the weight of a two-bay four storey frame. Afterwards, sequential linear and
quadratic programming techniques (SLP and SQP) were widely used for the design
optimization of steel structures. Vanderplaats and Sugimoto [11] developed a design technique “automated design synthesis” utilizing the approaches of SLP and SQP. Automated design synthesis method was proposed for minimizing the weight of frames with various bays and stories under static and seismic loadings by Karihaloo and Kanagasundaram [12], and Gülay and Boduroğlu [13]. The optimization techniques based on nonlinear programming were used for generation of optimal designations for steel structures under different loading conditions
and design requirements (Lassen [14], Wang and Grandhi [15], Salajegheh [16],
Hernández[17]).Optimality criteria method (OCM) is
another challenging method. Hybridizing the nonlinear mathematical programming
with Lagrange multipliers for inclusion of constraints forms the basis of OCM
(Arora [18], Cameron et al. [19]). Saka and Hayalioglu [20] used OCM for optimization of geometrically nonlinear steel structures made of elastoplastic material. Hayalioglu and Saka [21] proposed OCM for design optimization of
frames with nonuniform cross-sections. Chan et al. [22, 23] carried out the
weigh minimization of three-dimensional steel structures with discrete
cross-sections by OCM. They devised a transformation process for continuous and discrete design variables. Saka [24] optimized a frame design with tapered members thereby firstly computing the member responses under external static loadings and combining them by Lagrange multipliers to generate depth variables. Saka and Kameshki [25] used OCM for design optimization of unbraced
rigid frames considering constraints imposed by sway deflections and member
stresses.


## 2.2. Evolutionary-Based Optimization Studies

Evolutionary computation based on
simulation of natural evolutionary is a new approach used in the design
optimization of steel structures. Due to being appropriate for both traditional
and novel computation applications in the field of structural engineering, evolutionary
approaches whose major members are GAs by Holland [5], evolutionary programming (GP) by Fogel et al. [26] and evolutionary strategies (ES) by Rochenberg [27] have been improved by new implementations, such as hybrid and parallel searches. Therefore, research developments on three major EAs are firstly reviewed. Then, subsection
provides an overview of recent developments concerned the issues of parallel
and hybrid implementations.GP is managed by programs defined by
point-labeled parse trees used to describe the node and elements in the steel
structure. The most important step in GP is the determination of the size and
shape of parse trees for a design problem (Keijzer and Bobovic[28, 29]). Cevik [30] used a GP methodology, namely, a gene-expression
programming, for determining rotation capacity of wide flange beams.ES uses a population of tentative design solutions and generates the populations using several genetic operators with self-adaptive parameters (Back and Schwefel [31, 32]). Cai and Thierauf [33] proposed an evolutionary strategy without self-adaptive parameters
for the design optimization of steel structures. Similar approaches were also
utilized for the design optimization, such as ES with self-adaptive parameters
for discrete and continuous design variables (Ebenau et al. [34], Rajasekaran
[35], and Baumann and Kost [36]).One evolutionary algorithm approach is the SGA. Due to its flexible
structure, its genetic components have been improved. Taking into account the
usage of genetic components, the studies are grouped into two general
categories.(i) Genetic Operators with Adjustable Parameters and Representation of Design Variables
Hajela [37] introduced a representation technique for discrete design
variables. Thus, the lower and upper bounds of continues design variables were
used to compute the values of discrete design variables with any determined
precision. He discussed the negative effect of higher precision values leading
to a larger-length binary representation. Adeli and Cheng [38] presented a decoding technique in order to use binary coded strings for continues design variables. Chen [39] discussed the lack of proportional selection operation leading to stagnation problem in evolutionary search. He showed that usage of both scaled fitness values which are computed considering certain statistical quantities of fitness values and crossover operator which was applied at different rates to the same individual improved the quality degree of optimal designations. Yang and Soh [40] pointed out that the tournament selection method was more efficient than the existing selection methods considering quality degree of optimal designations. The use of graph theory for GAs is one of the new developments in structural optimization problem. Wang and Tai [41]
devised a graph representation for the topology-related design variables in structural optimization problems. Kaveh and Kalatjari [42] utilized the graph theory for representation of the size-related design variables and force method for structural analysis. Thus, grouping of truss design variables according to magnitude and sign (compression and tension) of stress becomes a new challenging approach in the design optimization of steel structure (Saka [43],
Saka et al. [44], Toğan and Daloğlu [45]).(ii) Constraint Handling for Evaluation of Fitness Values
Camp et al. [46] devised a penalty function with several variables for
design optimization of two-dimensional steel structures. The negative effect of
this approach leading to an inaccurate penalization was shown by Rasheed [47]. In
order to cope with this task, Rasheed [47] utilized an adaptive approach for
handling the constraints. For this purpose, a penalty function was used to
compute the penalty values based on an adaptation of penalty coefficients with
respect to the penalty degree. Le Riche et al. [48] divided the population into
small groups and applied a penalty function with different variable
coefficients for each group. Coello [49] used two populations for the
generation of a new population thereby comparing the penalized values. Nanakorn and Meesomklin [50] improved the penalty function by an adaptive procedure.
One of the alternative approaches to
the penalty function is artificial immune-inspired model (Garrett [51]). Firstly, Yoo and Hajela [52] utilized this approach for solving design optimization
problem. They employed two populations: one population was used to compute
penalty values, while the other one measured the hamming distance between
penalized fitness values. Coello and Cruz Cortés [53] improved Yoo and Hajela’s
technique by devising an adaptive evolutionary mechanism against the necessity
for a penalty function.Hybrid and Parallel Search-Based Evolutionary Algorithm Approaches
In order to improve the flexibility and efficiency of evolutionary
algorithms, utilizing the hybrid or parallel models of evolutionary search
algorithms is one of the important attempts.
The hybridization concept is emerged by use of local search methods for evolutionary algorithms as a complementary tool. Local search methods are
proposed to propagate the genetic information obtained throughout evolutionary
process into the next generations. One powerful hybridization model is memetic
algorithm. This biological learning mechanism is associated with Dawkins’
notion of a meme defined as a unit of cultural evolution
(Dawkins [54]). Two important approaches, namely, Lamarckian and Baldwinian approaches
make use of this learning mechanism for their evolutionary processes. Whereas
the structure of chromosome and its fitness value are changed in the
Lamarckian’ approach, Baldwinian’ learning mechanism only affects the fitness
value of chromosome without any change in its structure. In the hybridization
of evolutionary algorithms with local search method, the converging speed to
the global optima may be lower than the case of using a pure evolutionary
search algorithm without any hybrid implementation (Ong and Keane [55]). In
order to increase the converging speed, a new memetic algorithm, namely
coevolving memetic algorithm, is developed. Its fundamentals as well as a
comprehensive review of the basic approaches based on this algorithm are
presented by Smith [56].
The concept of parallel search is introduced to evolutionary algorithms
thereby employing a number of computer processes with distributed or shared memories
for a global population or a divided global population into small populations
(subpopulations) (Cantú-Paz and Goldberg [57],Cantú-Paz [58, 59]). Parallel
systems not only preserve diversity within the current populations, but also
ensure a perpetual novelty for populations to be generated in a way of disseminating
the different characteristic features embedded in the chromosomes to next
populations. Among evolutionary algorithms the GAs are preferably chosen for the
parallel applications. The basic genetic models utilized in parallel
search are grouped into four main classes (Cantú-Paz [4]).
(a)Master-slave GA. It uses a single
population, while master processor is employed to collect valuable genetic information,
slave processors service are responsible to compute the fitness values for a
certain number of individuals (Grefenstette [60], Robbins [61] and Levine [62]).
(b)Fine grained GA or cellular GA. The larger
number of processors is assigned for fitness evaluation of subpopulations. Due
to the higher number of processors, one of difficulties is encountered during
the decision about how the computer processors to be designed and arranged
(Baluja [63]).
(c)Coarse grained GA or distributed GA. It has
several similar properties of fine grained GA (migration implementation and
multiple population usage) with an exception of using smaller number of
processors. The ease of designing and arranging the computer processors makes
this model more attractive compared to cellular GA (Herrera et al. [64]). The
distributed GA is performed depending on migration related parameters (policy,
topology, frequency etc.) (Cohoon et al. [65], Alba and Troya [66], Skolicki
and De Jong [67]). Moreover, the other important issues for migration policy are
the number and frequency of migration, replacement of immigrants, size of
populations and migration topology (Tanese [68, 69]). The migration related
processes are directly responsible for determining the excitation order of
computer processors. Moreover, if any computer processor waits to run the
migration process for exchanging the individuals, then this parallel search is
called as “synchronous,” otherwise “asynchronous” (Alba and Troya [70]).
One of the basic models utilized by the distributed GA is island model. Several distinct subpopulations are isolated with each other, but communicated
by a migration process. Evolutionary operators are applied to each
subpopulation. If the parameter values of evolutionary operators are adjustable
for each subpopulation being important for an independent exploration of
different region of search space, then island model of this type is named as
homogenous and nonhomogeneous distributed GA (Alba and Troya [70]).
(d)Hybrid GA. Distributed GA can be
straightforwardly implemented on the parallel systems consisted of a number of
computer processors providing a considerable profit to the evolutionary search
in a way of decreasing the computing time. Therefore, by itself, distributed GA
is hybridized with existing search methods at different hierarchical levels. The hierarchical level is determined according to the use of evolutionary tools
and operators for the structured population. In the determination of any
hierarchical scheme, one of the most important steps is how the population to
be structured according to a lattice-like topology.
The
cellular GA is also successfully used in the hybridization models (Martin et
al. [71]). However, the complexity degree of cellular GA is higher than distributed
GA due to the increased size of both its subpopulations and underlying grid
system consisted of computer processors. Alba and Troya [70] compared the
cellular, distributed GAs and their hybrid models. They showed that numeric
efficiency and resistance to scalability was increased by the distributed
versions of cellular GA. In order to improve the exploitation capability of the
cellular GA in a way of increasing the converging rate, it was hybridized with
a local search method, named as cellular memetic algorithm. (Folino et al. [72]). Afterwards, cellular memetic algorithm was enhanced by an implementation of an adaptive
mechanism. (Krasnogor and Smith [73]), Neri et al. [74], Caponio et al. [75],
and Quang et al. [76]).
Sakamoto and Oda [77] hybridized GAs with optimality criteria method for
topology and size optimization of truss structures. While the topology of truss
structure was evolved through GA, optimality criteria methods determine the
cross-sectional areas of truss bars. Soh and Yang [78] devised a GA approach that
is managed by the fuzzy-logic-based rules. This approach was applied for weight
minimization of structures and achieved to obtain more optimal designations
compared to SGA’s.
Adeli and Cheng [79] were developed a parallel GA called “concurrent GA.”
They utilized a number of computer processors in parallel for the design
optimization of truss and frame structures. Topping and Leite [80] utilized this
parallel GA for the design optimization of a bridge, considering a number of
constraints. Adeli and Kumar [81] used a network consisted of computer
processors for optimization of large-spaced steel structures. Sarma and Adeli [82]
hybridized the coarse grained GAs with the fuzzy logic search method, for
design optimization of three-dimensional frame structures.
Tanimura et al. [83] proposed an
island model for design optimization of truss structures taking into account
several constraints. They utilized a new penalty function and compared their
optimal designations with SGA. They showed that their island model was more
efficient than SGA. Kicinger et al. [84] utilized the island models for both
topology and size optimization of tall buildings made up with steel profiles. They used two migration topologies (ring and fully-connected topology) with
various migration strategies for the design optimization of two-dimensional
frame with the bracing elements of various types. They showed that the quality
degree of optimal designations is improved when island models were executed by using
higher number of subpopulations. Then, Kicinger and Arciszewski [85] made use
of MAs in the design optimization of same steel structure. Examining various
genetic operators and their related parameters, they showed that the MAs were
more successful than GAs.
Kaveh and Shahrouzi [86] proposed
implementing the graph theory for MAs. Lamarckian and Baldwinian approaches
were adapted to optimize the frame bracing layouts of steel frames. Moreover,
the application of these approaches is illustrated for a two-dimensional steel
frame. They compared their optimal results with SGAs and displayed that whereas
Lamarckian approach reduces the topological variance with a more converging
rate, the better results are obtained by an incorporation of a dynamic mutation
band control to the Baldwinian approach.
Karakasis et al. [87] devised a radial
basis function network for the distributed GA and applied it to an aerodynamic
shape optimization problem. They compared four variants of GA and concluded
that their distributed versions offer an additional advantage in the
exploration of the interconnected processor network. Then, in order to carry
out the shape optimization of same design problem, they devised a hierarchical
distributed evolutionary scheme thereby adapting both the aerodynamic design
formulation and a navier-stoke equation solver into a radial basis unction
network (Karakasis and et al. [88]). Liakopoulos et al. [89] utilized a grid
system consisted of a number of computer processors for performing the
hybridization of hierarchical and distributed algorithms.


## 3. Optimum Design Problem

In this study, the weight of steel structure is minimized by
taking the constraints of maximum allowable stresses and displacements into
account. The evolutionary operations are operated on a population of tentative
designations with binary, integer, and real codes which contains the design
variables of discrete and continues types. Genetic operators are carried out by
use of either phenotypic or genotypic representations of design variables. The
representations of design variables encoded in genotype level are either kept
in all levelsof
evolutionary computation or decoded for fitness evaluation in phenotypic level. The fitness values of tentative design solutions are adjusted according to the
violation of constraints. In case of constrain violation; the penalized value
is included into fitness value by a penalty function.The weight of truss system and constraints are formulated as(1)W=∑i=1nρ*Li*Ai, subject to (2)σi≤σmaxi=1,…,n,Uk≤Umaxk=1,…,m.Here,W represents weight of the truss system. ρ is the density of steel, Li and Ai are the length and cross-sectional
area of ith member,
respectively; n is the total number of members in the truss system. σi and σmax symbolize the stress and the maximum allowable
stress for ith member. Uk is the displacement at kth degree of freedom while m
is the total degree of freedom of nodes. Umax represents the maximum allowable displacement for kth degree of freedom. Constraints gs and gd controls the joint displacements and element stresses, considering the
allowable displacement and stress values. The number of constraints is
determined by smax and dmax which indicates the number of
joints and displacements to be constrained.The violation of constraint is penalized. The penalization process is
used to obtain a penalty value. Thus, the fitness valueF is obtained by the sum of weight of the truss system W and penalty value P. F is used in weight
minimization of truss system. The minimization process is formulated as (3)MinF=W+P, where
the
term “W” is given in (1) and P is (4)P=(ro*t)φ*(∑i=1ngi+∑k=1mgk)*f. In (4), the stress constraint is expressed as (5)gs={σiσmax−1:σi≤σmax0:σmax>σii=1,…,n,s=1,…,smax, and displacement constraint as (6)gd={UkUmax−1:σk≤max0:Umax>σkk=1,…,m,d=1,…,dmax.The values of the constants in the calculation of the penalty
valueP are taken as r0=0.50, φ=2,f=10, and t = current generation number as given in Hasançebi and
Erbatur [90].


## 4. An
Introduction of BGAwEIS and Multipopulation-Based Genetic Algorithm (MPGA)

The
main features of BGAwEIS are similar to the island models with respect to the
usage of multiple populations and static parameters in the evolutionary
operators. In order to asses the effect of multiple populations on the quality
of optimal designations, MPGA is proposed. It is able to perform the
evolutionary processes with one processor and also capable of performing the
evolutionary operations with static parameters on multiple populations. The
fundamentals of BGAwEIS and MPGA are summarized in the following sections.


### 4.1. BGAwEIS

Parallel GAs are perfect
evolutionary tools due to its flexibility structure which is adaptable to
various environmental conditions. They utilize a number of processors and
populations simultaneously. Considering the elevated number of interacting characteristics,
it is said that parallel GAs have “complex mechanisms.” While using smaller
number of populations decreases this complexity, the quality of optimal
solutions drops due to insufficient exploitation of genetic heredity. On the other
hand, with increasing number of populations the adjustment of the
values of
related evolutionary parameters becomes increasingly difficult and cause a slow
down in the variation among populations. This effect prevents the exploration
of promising solution regions (Cantú-Paz [4]). Therefore, a balance between
exploitation of genetic heredity and exploration of promising solution regions
should be  achieved. For this purpose, an appropriate number of populations must
be used for transmitting of the genetic heredity extracted from high-quality
solutions. In this regard, a new GA, namely, bipopulation-based Genetic Algorithm with Enhanced Interval Search
(BGAwEIS), is developed. The basic features of BGAwEIS are itemized as follows.(i)
The design constraints may
increase the complexity of the search in the solution region (Eiben and Ruttkay
[91]). The largeness of the solution region affects the exploration efficiency
of the GA. If the feasible solutions are utilized in the exploration of the
solution region, then more promising individuals located in some other regions
may be obtained. Therefore, BGAwEIS utilizes feasible solutions in order to
compose the genetic heredity. The valuable genetic heredity obtained is adapted
to current populations by transmission processes called “extraction” and
“insertion-based transmission processes.”(ii)
Two populations called “outward”
and “inward” within a core population are used in transmission processes in
order to investigate the unknown subsolution regions and use the genetic
information obtained from previously visited candidates to explore better
candidates. Transmission process is achieved by regenerating a population
through migration among the feasible solutions taking into account of gradual
exploration strategy developed for utilizing the promising subsolution
regions of the entire solution region. Because, the exploration capacity is
increased by dividing the entire solution region into subsolution regions. As a
result, promising feasible solutions are used to explore more promising
solution regions.(iii)
The similar feasible solutions
which may be dominated in the search or feasible solutions obtained may be not
enough to explore the entire solution region. Therefore, the core population is
recreated at certain generation numbers.(iv)
The evolutionary processes are
governed by four parameters depending on the number of design variables, size
of solution region, and SGA-related mutation, crossover, selection parameters (Parmut,Parcr,Parsel):
size of populations, number of generations, number of subsolution before search
and number of generation for exploration strategy.The main elements of BGAwEIS are
described in the following sections. An example which clarifies how BGAwEIS
works is also included.Main Elements of BGAwEIS
BGAwEIS completes one generation after five interdependent procedures
with two populations within a core population (see the pseudocode in Algorithm2).
The number of subpopulations SubpopNum indicated by the number of populations which is obtained by dividing the global
population into small populations is taken as 1; the number of individuals
contained in each subpopulation SubpopNum is equal to the value of parameter SP.
The populations are called inward populationPinw, outward population Poutw  within core population Pcor. These populations have the same total number of
individuals and every individual in each population has the same number of
design variables assigned to it. The interdependent procedures are extraction-based
transmission, fitness calculation, insertion-based transmission, recreation of
the population and application of SGA operators. In addition, gradual
exploration strategy is applied for utilizing promising solution regions.
Several parameters are specified prior to the evolutionary process of
BGAwEIS considering size of solution region (SSR) and number of
design variables (NDV). These parameters are number of
generations (NG), size of
population (SP), number of
generations for gradual exploration strategy (NGGES), and number of subsolution regions before search (NSBS). The data outcome after the
completion of search is number of feasible solution (NFS) and number of subsolution regions after the search (NSAS).
The solution regions are composed of a design vectorX=(x1,x2,…,xn) which
consists of n design variables indicating the cross-sectional areas of the truss members. The design
variable has an upper bound Xmax and lower bound Xmin.  The
value of any discrete design variable in one-dimensional solution region will
be between Xmin=1 and Xmax. Xmax defines the total number of different cross-sectional
areas in the discrete design variables set.
BGAwEIS works on a multidimensional solution region which is divided into
one-dimensional subsolution regions and accomplishes the search within these
solution regions, simultaneously. In this regard, one-dimensional subsolution
region bounded byXmax and Xmin is divided into equal
segments as shown in  Figure 2. The number of segments is denoted by NSBS. The value of NSBS is proportional to SSR. The bounds of each segment are BVILi and BVIUi(i=1,…,NSBS). If desired, NSBS can be changed.
The boundaries of subsolution regions are gradually enlarged. This
approach is called “gradual exploration strategy” and activated by NGGES. The value of NGGES is specified by the ratio of NG to NSBS. NSBS is
proportional to parameter SSR. After the current generation number becomes equal to the value of NGGES, the value of NSBS is decreased. Thus, the bounds
of subsolution regions are enlarged.Algorithm 2:Pseudocode for BGAwEIS.
BGAwEIS (SubPopNum=1, SubPopIndNum = SP, SSR,NDV,NG, NGGES, NSBS, Parmut,Parcr, Parsel)Initialize (Pi, SupPopNum, SupPopIndNum)[Pcor]=Pi[Parcord]=PiIf required,[Parcord]=Decoding(Pcor),[Fcor,FeasPool]=Fitness_Calculation(Parcord)fori=1: NG[Pinw, Poutw] = Extraction_based_transmission(Fcor, Pcor, FeasPool)[Finw, Foutw, FeasPool] = Fitness_Calculation(Pinw, Poutw)[Pcor] = Insertion_based_transmission(Finw, Poutw, Pinw, Poutw, FeasPool)[Pcor, NSBS] = Re-creation_Population(Finw, Foutw, Pinw, Poutw, SSR, NSBS, FeasPool)[Pcor] = SGA(Pcor, NG=1, SP, Fcor, Parsel, Parmut, Parcr)[Parcord]=PcorIf required,[Parcord] = Decoding (Pcor),[Fcor, FeasPool] = Fitness_Calculation (Parcord)endAlgorithm 3:A Pseudocode for MPGA.
MPGA (SubPopNum, SubPopIndNum, SSR, NDV, NG=1, Epoch, Parrank, Parmig, Parcomp,Parmut, Parcr, Parsel)Initialize (Pi, SubPopNum, SubPopIndNum)[Pid] = Decoding (Pi)[Fi] = Fitness_Calculation (Pid)[Fi] = Ranking(Fi, SubPopNum, SubPopIndNum, Parrank)fori=1: Epoch[Pi] = Subpopulation_Order_Evaluation(Pi, Fi, SubPopNum, SubPopIndNum)for j=1: SubPopNum[Pi] = SGA (Pi, NG, SubPopIndNum, Fi, Parsel, Parmut, Parcr)End[Pi] = Control (Pi, SSR, NDV)[Pid] = Decoding (Pi)[Fi] = Fitness_Calculation (Pid)[Fi] = Ranking(Pi, SubPopNum, SubPopIndNum, Parrank)[Pi] = Competion_Process (Pi, Parcomp)[Pi] = Migration_Process (Pi, Parmig)endFigure 2
Visualization of the one-dimensional subsolution regions.


#### 4.1.1. Extraction Based Transmission

In this process,
the individuals that come from the core population are re-generated in order to
generate inwardPinwSP and outward PoutwSP populations. The number of individuals located in these populations is limited
by SP. In the construction of
the inward population, the individuals taken from core PcorSP population are regenerated by
converging them to the best solution of the feasible pool XBF.In the generation of the outward population, the individuals taken from
core population are regenerated by diverging them to the boundsXmin and Xmax of the design variable. Furthermore, in the
generation of the outward and inward populations, the one-dimensional solution
region is divided into equal segments. While these segments are used to
generate the outward population, the position of the best feasible solution
with respect to these segments is used to generate inward population. In order
to regenerate the individuals of   PinwSP and PoutwSP,
the corresponding segment BVILi and BVIUi used
for the individuals located in PcorSP is determined. This is followed by finding out the position of the best
feasible XBF solution in
the feasible solution pool to the corresponding segment BVILi and BVIUi. There are three possible locations for XBF relative to the corresponding segment (Figure 3): (i) below, (ii) above, and (iii)
within the corresponding segment.Figure 3
Graphical depiction
of the possibilities for the location of best feasible solution (XBF)
into the corresponding segment in extraction-based transmission.PoutwSP is regenerated by taking into account the segment  BVIiL and BVIiU used for the individuals of PcorSP. The individuals of PoutwSP are forced to simultaneously diverge to both Xmin and Xmax by taking into account XBF, BVIiL, BVIiU, and PcorSP. An
algorithm based on the possibilities shown in Figure 3 is developed for the
regeneration of PoutwSP from  PcorSP (see (7). In order to explore entire solution region, two individuals are
simultaneously produced for the regeneration of PoutwSP,
(7)SPPoutwNDV={Xmin+rand(BVILi-Xmin):(Indv.1)PcorNDVk+rand(Xmax−PcorNDVk):(Indv.2)if,BVILi<XBF<PcorNDVkXmin+rand(PcorNDVk-Xmin):(Indv.1)BVIUi+rand(Xmax-BVIUi):(Indv.2)else{if,BVILi<XBF<BVIUi(1.possibility)Xmin+rand(BVILi-Xmin):(Indv.1)XBF+rand(Xmax-XBF):(Indv.2){else if,XBF>BVIUi(k=1,…,SP)(2.possibility)Xmin+rand(XBF-Xmin):(Indv.1)BVIUi+rand(Xmax-BVIUi):(Indv.2){else(i=Number,…,of Current Intervals)(3.possibility)PinwSP is
regenerated by taking into account the segment BVIiL and BVIiU used for the individuals of the PcorSP. The individuals of PinwSP are forced to converge to the XBF by taking the BVIiL, BVIiU  and PcorSP into
consideration. An algorithm based on the possibilities shown in Figure 3 is
developed for the regeneration of  PinwSP from PcorSP:(8)SPPinwNDV={BVILi+rand(SPPcorNDV-BVILi)SPPcorNDV+rand(BiVIU−SPPcorNDV)ifBVILi<XBF<SPPcorNDVelseBVILi{if,BVILi<XBF<BVIUi(2.possibility)BVILi+rand(XBF-BVILi){if,XBF>BVIUi(3.possibility)XBF+rand(BVIUi-XBF){else(i=1,…,Number of Current Interval)(1.possibility)An
application of extraction-based transmission on two-dimensional solution region
represented by two design variables is graphically shown in Figure4. Only
design variable 1 that is bounded by (Xmin and Xmax) is visualized. The solution region has three optimum points. One of these three optimum points
is a global optimum symbolized by “•”. The remaining feasible
solutions are also indicated by “•”. The individuals from
the core population, which are indicated as “+” and enclosed in a thin dashed closed curve, are used to build
the inward and outward populations (Figure 4). In the regeneration of the
inward and outward populations, the bounds of segment corresponding to the
individual of the core population are in the range of (BVIiL and BVIiU). Outward population is symbolized by “*” and enclosed in a rectangle with dashed edges (Figure 4). The individuals of inward
population is symbolized as “x” and
enclosed in a rectangle with a thick continuous edge (Figure 4).Figure 4
Display of the extraction-based transmission.


#### 4.1.2. Fitness Calculation

The module of fitness calculation computes the
fitness values of individuals in each population by taking the constraint
violations into account. Thus, fitness values corresponding to populationsPinw, Poutw and Pcor are collected in the matrices Finw, Foutw, and Fcor. At the same time, feasible solutions obtained are chosen to
locate in feasible solution pool when its fitness values are lowest compared to
other feasible fitness values found in the feasible solution pool.


#### 4.1.3. Insertion-Based Transmission

This process involves construction of the core population with the
individuals coming from inward and outward populations. The core, inward and
outward populations all have the same number of individuals. Since one
population is generated from two populations, it is necessary to eliminate a
certain number of individuals. This is carried out by prioritizing certain
individuals according to their feasibility and fitness. All the feasible
solutions located in feasible solution pool are used in constructing the core
population. In order to adapt the feasible solution pool to the core population,
the inward or outward populations is divided into two equal parts. The
algorithm developed in this regard is managed by four cases based on the
position of the number of feasible solutions with respect to the inward or
outward populations with same number of individuals, as depicted in Figure5. These cases are as follows.Figure 5
Insertion-based transmission.Case 1
(C1) The number of feasible solutions in
inward populationNI1 is more than SP/2.Case 2
(C2) The number of feasible solutions in
inward populationNI2 is less than SP/2.Case 3
(C3) The number of feasible solutions in
outward populationNO1 is more than SP/2.Case 4
(C4) The number of feasible solutions in
outward populationNO2 is less than SP/2.The core population from the inward and outward populations is
constructed from the combination of these four cases. These combinations are
C1+C3, C1+C4, C2+C3, and C2+C4. They are explained as(i)
collect (NI1+NO1), (NI1+NO2),
(NI2+NO1), and (NI2+NO2) feasible solutions from inward or outward populations corresponding to the
combinations of “C1+C3,” “C1+C4,” “C2+C3,” and “C2+C4,”
respectively;(ii)
rank the collected feasible solutions in a descending order of
their fitness values, and then store it in a dummy column matrix;(iii)
if the number of
individuals in the combination of “C1+C3,” “C1+C4,” “C2+C3,” and “C2+C4”
is greater than SP,((NI1+NO1)−SP), ((NI1+NO2)−SP), ((NI2+NO1)−SP) and ((NI2+NO2)−SP) feasible solutions with
least fitness are discarded from the dummy column matrix. Otherwise, (SP−(NI1+NO1)), (SP−(NI1+NO2)), (SP−(NI2+NO1)) and (SP−(NI2+NO2)) feasible solutions with least fitness is discarded from the dummy matrix,
respectively. If the number of individuals in dummy matrix does not reach SP, some individuals are borrowed
from the inward and outward populations in the descending order of their fitness
values.


#### 4.1.4. Recreation of the Core Population

An algorithm is developed for the recreation of corePcorSP
population. This algorithm is managed
by three possibilities regarding the position of the boundaries, namely XFmin and XFmax, of the feasible
solution pool with respect to the center (cent) of the interval representing
the one-dimensional solution region. These possibilities are graphically
depicted in Figure 6. The recreation process is activated depending on NGGES and the enlargeable bounds of
subsolution regions.Figure 6
Graphical depiction of possibilities for the location of the interval (XFmin, XFmax) and
cent = (XFmin+XFmax)/2 in the recreation of the core population.The algorithm for the recreation of the core populationPcorSP based on
the possibilities given in Figure 6 is formulated by (9),
(9)SPPcorNDV={A.Case:1≤(SP*(30/100))B.Case:(SP*(30/100))<k≤(SP*(50/100))C.Case:(SP*(30/100))<k≤SPif,cent>XFmaxA.Case:1≤(SP*(40/100))B.Case:(SP*(40/100))<k≤(SP*(60/100))C.Case:(SP*(60/100))<k≤SPelse if,XFmin≤cent≤XFmax(k=1,…,SP)A.Case:1≤(SP*(50/100))B.Case:(SP*(50/100))<k≤(SP*(70/100))C.Case:(SP*(70/100))<k≤SPelse
where A  Case:Xmin+rand(XFmin−Xmin); B  Case:XFmin+rand(XFmax−XFmin); B  Case:XFmax+rand(Xmax−XFmax).


#### 4.1.5. Application of SGA Operators

SGA operators are
used to regenerate the core population in order to provide a variation for the
next generations. These are one-point crossover, mutation, and roulette wheel
selection operators. Also, some optional operators exist for the search
including what follows (Eiben and Ruttkay [91]).(i)
Multipoint mutation
and crossover operators, and the other selection operators (stochastic
universal sampling and stochastic remainder sampling),(ii)
the generation gap against genetic drift problem,(iii)
different fitness scaling
methods such as linear normalization, baseline windowing, sigma truncating,
linear scaling, and adaptive windowing are employed along with the elitist
selection scheme against the loosing of the valuable genetic heredity.The five interdependent procedures mentioned above are processed in one
of the real, integer, and binary coding schemes. For this reason, recoding of
the design variables is required before these processes are applied.In order to depict the gradual exploration strategy, let us consider a
planar truss with two bars as an example and construct a core population with
two individuals. The upper and lower bounds of the design variables are given
for continuous or discrete set of design variables (see Table1).Table 1
A preliminary demonstration of GAwEIS to a planar truss with two-bars.
Indiv.Section properitiesNSBSXminXmaxDVNVNDVSNBVILBVIUContinous design variablesIndiv. (1)[0.32 0.56]40.1110.3210.1000.32520.5630.5510.775Indiv. (2)[0.82 0.24]40.1110.8240.7761.00020.2410.1000.325Discrete design variablesIndiv. (1)[001011]4151110.1000.3252330.5510.775Indiv. (2)[100001]4151440.7761.0002110.1000.325NSBS: number of subsolution regions (the bumber of segment), VNDV: value of each design variable, DVN: design variable number, SN: subsolution region (segment) number.The solution region of each design variable is divided into 4 segments which
are used for subsolution regions obtained by dividing entire solution region
into small ones. Numerical values of design variables vary within the interval
(0.1, 1) for continuous type and discrete type of design variables. The
one-dimensional solution region is divided into one-dimensional subsolution
regions. One-dimensional subsolution regions are represented by intervals
(0.100, 0.325), (0.326, 0.550), (0.551, 0.775), and (0.776, 1.000). The
discrete design variables are coded by using three-binary-digits. Therefore,
the total number of digits is equal to 6. The continuous design variables are
used to find the corresponding intervals whereas the values of discrete design
variables represent the segment numbers.


### 4.2. MPGA

MPGA makes use of a migration process with several parameters in order to
provide a control for multiple populations and a communication between them. The evolutionary processes of MPGA are carried out by using GEATbx (Polheim [7]). GEATbx is the ability of running with multiple populations which is systematically
structured for adequately the execution of various evolutionary processes and
rich in options regarding different genetic operators and their related
parameters for real-valued variables. Considering the main elements of BGAwEIS, some elements of
GEATbx are appropriately activated in the implementation of MPGA procedure. The
crucial evolutionary operators of MPGA are presented via a pseudocode in Algorithm 3.The first step is the initialization
ofSubPopNum subpopulations. In the
beginning stage, a single population with (SubPopIndNum*SubPopNum) individuals to be settled to
the subpopulations is created. Following this step, the fitness values are
calculated by using the fitness functions. The fitness values are penalized if
they violate the constraints (see (3) and (4). The penalization values are
added to the fitness values.Ranking process governed by ranking
parameterParrank (see related parameters in Section 5) is based on
the redistribution of fitness values where artificial values are used instead
of the actual fitness values. In this way, dominance of the best solutions on
the other solutions is weakened. Ranking process is carried out in two separate
stages (Bäck and Hoffmeister [92]). In the first stage, the fitness values are
recreated by the linear or nonlinear scaling functions. In the second stage,
scaled fitness values are redistributed depending on the quality
of actual fitness values. Furthermore, the ranking share procedure is
applied where fitness values are rescaled according to their rank (Goldberg and
Richardson [93]).Following the ranking process,
evolutionary approaches are repeatedly executed in a loop until a predefined
loop numberepoch is reached. The
first inner loop is regarded with determining the order of population. The
subpopulations are ordered with respect to the fitness values of the
individuals. For this purpose, a simple competition process based on ranking
procedure is utilized (Polheim [7]). The rank of subpopulations has a big impact
on the migration process because a communication network comprised of
subpopulations is used for transmission of emigrants and immigrants.After ordering subpopulation by taking
into account the fitness values, SGA operators (selectionsel, mutation mut, and
crossover cr operators) and their
related parameters Parsel, Parmut, and Parcr (see
Section 5) are activated. These three operators are separately executed
for each subpopulation. The subsequent process is activated when the values of
design variable exceed the ranges of SSR. If this occurs, then related individuals are repaired.The competition process aims to move the robust
individuals to other subpopulations that exhibit relatively poor performance
(Schlierkamp-Voosen and Mühlenbein [94]). The competition of subpopulations is
governed by parameter Parcomp (see Section 5) and carried out in three steps: (i) determination of the capacity of each subpopulations for taking emigrant and
sending immigrant individuals, (ii) picking robust individuals according to
their fitness values, and (iii) the adjustment of the subpopulations size for
the settlement of the robust individuals (Polheim [7]).The transmission of immigrants to
the other subpopulations is accomplished by a migration process. The migration
process is regulated with parameterParmig which indicates
the several parameters, such as migration rate, interval, and topology (see Section 5).


## 4.1. BGAwEIS

Parallel GAs are perfect
evolutionary tools due to its flexibility structure which is adaptable to
various environmental conditions. They utilize a number of processors and
populations simultaneously. Considering the elevated number of interacting characteristics,
it is said that parallel GAs have “complex mechanisms.” While using smaller
number of populations decreases this complexity, the quality of optimal
solutions drops due to insufficient exploitation of genetic heredity. On the other
hand, with increasing number of populations the adjustment of the
values of
related evolutionary parameters becomes increasingly difficult and cause a slow
down in the variation among populations. This effect prevents the exploration
of promising solution regions (Cantú-Paz [4]). Therefore, a balance between
exploitation of genetic heredity and exploration of promising solution regions
should be  achieved. For this purpose, an appropriate number of populations must
be used for transmitting of the genetic heredity extracted from high-quality
solutions. In this regard, a new GA, namely, bipopulation-based Genetic Algorithm with Enhanced Interval Search
(BGAwEIS), is developed. The basic features of BGAwEIS are itemized as follows.(i)
The design constraints may
increase the complexity of the search in the solution region (Eiben and Ruttkay
[91]). The largeness of the solution region affects the exploration efficiency
of the GA. If the feasible solutions are utilized in the exploration of the
solution region, then more promising individuals located in some other regions
may be obtained. Therefore, BGAwEIS utilizes feasible solutions in order to
compose the genetic heredity. The valuable genetic heredity obtained is adapted
to current populations by transmission processes called “extraction” and
“insertion-based transmission processes.”(ii)
Two populations called “outward”
and “inward” within a core population are used in transmission processes in
order to investigate the unknown subsolution regions and use the genetic
information obtained from previously visited candidates to explore better
candidates. Transmission process is achieved by regenerating a population
through migration among the feasible solutions taking into account of gradual
exploration strategy developed for utilizing the promising subsolution
regions of the entire solution region. Because, the exploration capacity is
increased by dividing the entire solution region into subsolution regions. As a
result, promising feasible solutions are used to explore more promising
solution regions.(iii)
The similar feasible solutions
which may be dominated in the search or feasible solutions obtained may be not
enough to explore the entire solution region. Therefore, the core population is
recreated at certain generation numbers.(iv)
The evolutionary processes are
governed by four parameters depending on the number of design variables, size
of solution region, and SGA-related mutation, crossover, selection parameters (Parmut,Parcr,Parsel):
size of populations, number of generations, number of subsolution before search
and number of generation for exploration strategy.The main elements of BGAwEIS are
described in the following sections. An example which clarifies how BGAwEIS
works is also included.Main Elements of BGAwEIS
BGAwEIS completes one generation after five interdependent procedures
with two populations within a core population (see the pseudocode in Algorithm2).
The number of subpopulations SubpopNum indicated by the number of populations which is obtained by dividing the global
population into small populations is taken as 1; the number of individuals
contained in each subpopulation SubpopNum is equal to the value of parameter SP.
The populations are called inward populationPinw, outward population Poutw  within core population Pcor. These populations have the same total number of
individuals and every individual in each population has the same number of
design variables assigned to it. The interdependent procedures are extraction-based
transmission, fitness calculation, insertion-based transmission, recreation of
the population and application of SGA operators. In addition, gradual
exploration strategy is applied for utilizing promising solution regions.
Several parameters are specified prior to the evolutionary process of
BGAwEIS considering size of solution region (SSR) and number of
design variables (NDV). These parameters are number of
generations (NG), size of
population (SP), number of
generations for gradual exploration strategy (NGGES), and number of subsolution regions before search (NSBS). The data outcome after the
completion of search is number of feasible solution (NFS) and number of subsolution regions after the search (NSAS).
The solution regions are composed of a design vectorX=(x1,x2,…,xn) which
consists of n design variables indicating the cross-sectional areas of the truss members. The design
variable has an upper bound Xmax and lower bound Xmin.  The
value of any discrete design variable in one-dimensional solution region will
be between Xmin=1 and Xmax. Xmax defines the total number of different cross-sectional
areas in the discrete design variables set.
BGAwEIS works on a multidimensional solution region which is divided into
one-dimensional subsolution regions and accomplishes the search within these
solution regions, simultaneously. In this regard, one-dimensional subsolution
region bounded byXmax and Xmin is divided into equal
segments as shown in  Figure 2. The number of segments is denoted by NSBS. The value of NSBS is proportional to SSR. The bounds of each segment are BVILi and BVIUi(i=1,…,NSBS). If desired, NSBS can be changed.
The boundaries of subsolution regions are gradually enlarged. This
approach is called “gradual exploration strategy” and activated by NGGES. The value of NGGES is specified by the ratio of NG to NSBS. NSBS is
proportional to parameter SSR. After the current generation number becomes equal to the value of NGGES, the value of NSBS is decreased. Thus, the bounds
of subsolution regions are enlarged.Algorithm 2:Pseudocode for BGAwEIS.
BGAwEIS (SubPopNum=1, SubPopIndNum = SP, SSR,NDV,NG, NGGES, NSBS, Parmut,Parcr, Parsel)Initialize (Pi, SupPopNum, SupPopIndNum)[Pcor]=Pi[Parcord]=PiIf required,[Parcord]=Decoding(Pcor),[Fcor,FeasPool]=Fitness_Calculation(Parcord)fori=1: NG[Pinw, Poutw] = Extraction_based_transmission(Fcor, Pcor, FeasPool)[Finw, Foutw, FeasPool] = Fitness_Calculation(Pinw, Poutw)[Pcor] = Insertion_based_transmission(Finw, Poutw, Pinw, Poutw, FeasPool)[Pcor, NSBS] = Re-creation_Population(Finw, Foutw, Pinw, Poutw, SSR, NSBS, FeasPool)[Pcor] = SGA(Pcor, NG=1, SP, Fcor, Parsel, Parmut, Parcr)[Parcord]=PcorIf required,[Parcord] = Decoding (Pcor),[Fcor, FeasPool] = Fitness_Calculation (Parcord)endAlgorithm 3:A Pseudocode for MPGA.
MPGA (SubPopNum, SubPopIndNum, SSR, NDV, NG=1, Epoch, Parrank, Parmig, Parcomp,Parmut, Parcr, Parsel)Initialize (Pi, SubPopNum, SubPopIndNum)[Pid] = Decoding (Pi)[Fi] = Fitness_Calculation (Pid)[Fi] = Ranking(Fi, SubPopNum, SubPopIndNum, Parrank)fori=1: Epoch[Pi] = Subpopulation_Order_Evaluation(Pi, Fi, SubPopNum, SubPopIndNum)for j=1: SubPopNum[Pi] = SGA (Pi, NG, SubPopIndNum, Fi, Parsel, Parmut, Parcr)End[Pi] = Control (Pi, SSR, NDV)[Pid] = Decoding (Pi)[Fi] = Fitness_Calculation (Pid)[Fi] = Ranking(Pi, SubPopNum, SubPopIndNum, Parrank)[Pi] = Competion_Process (Pi, Parcomp)[Pi] = Migration_Process (Pi, Parmig)endFigure 2
Visualization of the one-dimensional subsolution regions.


### 4.1.1. Extraction Based Transmission

In this process,
the individuals that come from the core population are re-generated in order to
generate inwardPinwSP and outward PoutwSP populations. The number of individuals located in these populations is limited
by SP. In the construction of
the inward population, the individuals taken from core PcorSP population are regenerated by
converging them to the best solution of the feasible pool XBF.In the generation of the outward population, the individuals taken from
core population are regenerated by diverging them to the boundsXmin and Xmax of the design variable. Furthermore, in the
generation of the outward and inward populations, the one-dimensional solution
region is divided into equal segments. While these segments are used to
generate the outward population, the position of the best feasible solution
with respect to these segments is used to generate inward population. In order
to regenerate the individuals of   PinwSP and PoutwSP,
the corresponding segment BVILi and BVIUi used
for the individuals located in PcorSP is determined. This is followed by finding out the position of the best
feasible XBF solution in
the feasible solution pool to the corresponding segment BVILi and BVIUi. There are three possible locations for XBF relative to the corresponding segment (Figure 3): (i) below, (ii) above, and (iii)
within the corresponding segment.Figure 3
Graphical depiction
of the possibilities for the location of best feasible solution (XBF)
into the corresponding segment in extraction-based transmission.PoutwSP is regenerated by taking into account the segment  BVIiL and BVIiU used for the individuals of PcorSP. The individuals of PoutwSP are forced to simultaneously diverge to both Xmin and Xmax by taking into account XBF, BVIiL, BVIiU, and PcorSP. An
algorithm based on the possibilities shown in Figure 3 is developed for the
regeneration of PoutwSP from  PcorSP (see (7). In order to explore entire solution region, two individuals are
simultaneously produced for the regeneration of PoutwSP,
(7)SPPoutwNDV={Xmin+rand(BVILi-Xmin):(Indv.1)PcorNDVk+rand(Xmax−PcorNDVk):(Indv.2)if,BVILi<XBF<PcorNDVkXmin+rand(PcorNDVk-Xmin):(Indv.1)BVIUi+rand(Xmax-BVIUi):(Indv.2)else{if,BVILi<XBF<BVIUi(1.possibility)Xmin+rand(BVILi-Xmin):(Indv.1)XBF+rand(Xmax-XBF):(Indv.2){else if,XBF>BVIUi(k=1,…,SP)(2.possibility)Xmin+rand(XBF-Xmin):(Indv.1)BVIUi+rand(Xmax-BVIUi):(Indv.2){else(i=Number,…,of Current Intervals)(3.possibility)PinwSP is
regenerated by taking into account the segment BVIiL and BVIiU used for the individuals of the PcorSP. The individuals of PinwSP are forced to converge to the XBF by taking the BVIiL, BVIiU  and PcorSP into
consideration. An algorithm based on the possibilities shown in Figure 3 is
developed for the regeneration of  PinwSP from PcorSP:(8)SPPinwNDV={BVILi+rand(SPPcorNDV-BVILi)SPPcorNDV+rand(BiVIU−SPPcorNDV)ifBVILi<XBF<SPPcorNDVelseBVILi{if,BVILi<XBF<BVIUi(2.possibility)BVILi+rand(XBF-BVILi){if,XBF>BVIUi(3.possibility)XBF+rand(BVIUi-XBF){else(i=1,…,Number of Current Interval)(1.possibility)An
application of extraction-based transmission on two-dimensional solution region
represented by two design variables is graphically shown in Figure4. Only
design variable 1 that is bounded by (Xmin and Xmax) is visualized. The solution region has three optimum points. One of these three optimum points
is a global optimum symbolized by “•”. The remaining feasible
solutions are also indicated by “•”. The individuals from
the core population, which are indicated as “+” and enclosed in a thin dashed closed curve, are used to build
the inward and outward populations (Figure 4). In the regeneration of the
inward and outward populations, the bounds of segment corresponding to the
individual of the core population are in the range of (BVIiL and BVIiU). Outward population is symbolized by “*” and enclosed in a rectangle with dashed edges (Figure 4). The individuals of inward
population is symbolized as “x” and
enclosed in a rectangle with a thick continuous edge (Figure 4).Figure 4
Display of the extraction-based transmission.


### 4.1.2. Fitness Calculation

The module of fitness calculation computes the
fitness values of individuals in each population by taking the constraint
violations into account. Thus, fitness values corresponding to populationsPinw, Poutw and Pcor are collected in the matrices Finw, Foutw, and Fcor. At the same time, feasible solutions obtained are chosen to
locate in feasible solution pool when its fitness values are lowest compared to
other feasible fitness values found in the feasible solution pool.


### 4.1.3. Insertion-Based Transmission

This process involves construction of the core population with the
individuals coming from inward and outward populations. The core, inward and
outward populations all have the same number of individuals. Since one
population is generated from two populations, it is necessary to eliminate a
certain number of individuals. This is carried out by prioritizing certain
individuals according to their feasibility and fitness. All the feasible
solutions located in feasible solution pool are used in constructing the core
population. In order to adapt the feasible solution pool to the core population,
the inward or outward populations is divided into two equal parts. The
algorithm developed in this regard is managed by four cases based on the
position of the number of feasible solutions with respect to the inward or
outward populations with same number of individuals, as depicted in Figure5. These cases are as follows.Figure 5
Insertion-based transmission.Case 1
(C1) The number of feasible solutions in
inward populationNI1 is more than SP/2.Case 2
(C2) The number of feasible solutions in
inward populationNI2 is less than SP/2.Case 3
(C3) The number of feasible solutions in
outward populationNO1 is more than SP/2.Case 4
(C4) The number of feasible solutions in
outward populationNO2 is less than SP/2.The core population from the inward and outward populations is
constructed from the combination of these four cases. These combinations are
C1+C3, C1+C4, C2+C3, and C2+C4. They are explained as(i)
collect (NI1+NO1), (NI1+NO2),
(NI2+NO1), and (NI2+NO2) feasible solutions from inward or outward populations corresponding to the
combinations of “C1+C3,” “C1+C4,” “C2+C3,” and “C2+C4,”
respectively;(ii)
rank the collected feasible solutions in a descending order of
their fitness values, and then store it in a dummy column matrix;(iii)
if the number of
individuals in the combination of “C1+C3,” “C1+C4,” “C2+C3,” and “C2+C4”
is greater than SP,((NI1+NO1)−SP), ((NI1+NO2)−SP), ((NI2+NO1)−SP) and ((NI2+NO2)−SP) feasible solutions with
least fitness are discarded from the dummy column matrix. Otherwise, (SP−(NI1+NO1)), (SP−(NI1+NO2)), (SP−(NI2+NO1)) and (SP−(NI2+NO2)) feasible solutions with least fitness is discarded from the dummy matrix,
respectively. If the number of individuals in dummy matrix does not reach SP, some individuals are borrowed
from the inward and outward populations in the descending order of their fitness
values.


### 4.1.4. Recreation of the Core Population

An algorithm is developed for the recreation of corePcorSP
population. This algorithm is managed
by three possibilities regarding the position of the boundaries, namely XFmin and XFmax, of the feasible
solution pool with respect to the center (cent) of the interval representing
the one-dimensional solution region. These possibilities are graphically
depicted in Figure 6. The recreation process is activated depending on NGGES and the enlargeable bounds of
subsolution regions.Figure 6
Graphical depiction of possibilities for the location of the interval (XFmin, XFmax) and
cent = (XFmin+XFmax)/2 in the recreation of the core population.The algorithm for the recreation of the core populationPcorSP based on
the possibilities given in Figure 6 is formulated by (9),
(9)SPPcorNDV={A.Case:1≤(SP*(30/100))B.Case:(SP*(30/100))<k≤(SP*(50/100))C.Case:(SP*(30/100))<k≤SPif,cent>XFmaxA.Case:1≤(SP*(40/100))B.Case:(SP*(40/100))<k≤(SP*(60/100))C.Case:(SP*(60/100))<k≤SPelse if,XFmin≤cent≤XFmax(k=1,…,SP)A.Case:1≤(SP*(50/100))B.Case:(SP*(50/100))<k≤(SP*(70/100))C.Case:(SP*(70/100))<k≤SPelse
where A  Case:Xmin+rand(XFmin−Xmin); B  Case:XFmin+rand(XFmax−XFmin); B  Case:XFmax+rand(Xmax−XFmax).


### 4.1.5. Application of SGA Operators

SGA operators are
used to regenerate the core population in order to provide a variation for the
next generations. These are one-point crossover, mutation, and roulette wheel
selection operators. Also, some optional operators exist for the search
including what follows (Eiben and Ruttkay [91]).(i)
Multipoint mutation
and crossover operators, and the other selection operators (stochastic
universal sampling and stochastic remainder sampling),(ii)
the generation gap against genetic drift problem,(iii)
different fitness scaling
methods such as linear normalization, baseline windowing, sigma truncating,
linear scaling, and adaptive windowing are employed along with the elitist
selection scheme against the loosing of the valuable genetic heredity.The five interdependent procedures mentioned above are processed in one
of the real, integer, and binary coding schemes. For this reason, recoding of
the design variables is required before these processes are applied.In order to depict the gradual exploration strategy, let us consider a
planar truss with two bars as an example and construct a core population with
two individuals. The upper and lower bounds of the design variables are given
for continuous or discrete set of design variables (see Table1).Table 1
A preliminary demonstration of GAwEIS to a planar truss with two-bars.
Indiv.Section properitiesNSBSXminXmaxDVNVNDVSNBVILBVIUContinous design variablesIndiv. (1)[0.32 0.56]40.1110.3210.1000.32520.5630.5510.775Indiv. (2)[0.82 0.24]40.1110.8240.7761.00020.2410.1000.325Discrete design variablesIndiv. (1)[001011]4151110.1000.3252330.5510.775Indiv. (2)[100001]4151440.7761.0002110.1000.325NSBS: number of subsolution regions (the bumber of segment), VNDV: value of each design variable, DVN: design variable number, SN: subsolution region (segment) number.The solution region of each design variable is divided into 4 segments which
are used for subsolution regions obtained by dividing entire solution region
into small ones. Numerical values of design variables vary within the interval
(0.1, 1) for continuous type and discrete type of design variables. The
one-dimensional solution region is divided into one-dimensional subsolution
regions. One-dimensional subsolution regions are represented by intervals
(0.100, 0.325), (0.326, 0.550), (0.551, 0.775), and (0.776, 1.000). The
discrete design variables are coded by using three-binary-digits. Therefore,
the total number of digits is equal to 6. The continuous design variables are
used to find the corresponding intervals whereas the values of discrete design
variables represent the segment numbers.


## 4.1.1. Extraction Based Transmission

In this process,
the individuals that come from the core population are re-generated in order to
generate inwardPinwSP and outward PoutwSP populations. The number of individuals located in these populations is limited
by SP. In the construction of
the inward population, the individuals taken from core PcorSP population are regenerated by
converging them to the best solution of the feasible pool XBF.In the generation of the outward population, the individuals taken from
core population are regenerated by diverging them to the boundsXmin and Xmax of the design variable. Furthermore, in the
generation of the outward and inward populations, the one-dimensional solution
region is divided into equal segments. While these segments are used to
generate the outward population, the position of the best feasible solution
with respect to these segments is used to generate inward population. In order
to regenerate the individuals of   PinwSP and PoutwSP,
the corresponding segment BVILi and BVIUi used
for the individuals located in PcorSP is determined. This is followed by finding out the position of the best
feasible XBF solution in
the feasible solution pool to the corresponding segment BVILi and BVIUi. There are three possible locations for XBF relative to the corresponding segment (Figure 3): (i) below, (ii) above, and (iii)
within the corresponding segment.Figure 3
Graphical depiction
of the possibilities for the location of best feasible solution (XBF)
into the corresponding segment in extraction-based transmission.PoutwSP is regenerated by taking into account the segment  BVIiL and BVIiU used for the individuals of PcorSP. The individuals of PoutwSP are forced to simultaneously diverge to both Xmin and Xmax by taking into account XBF, BVIiL, BVIiU, and PcorSP. An
algorithm based on the possibilities shown in Figure 3 is developed for the
regeneration of PoutwSP from  PcorSP (see (7). In order to explore entire solution region, two individuals are
simultaneously produced for the regeneration of PoutwSP,
(7)SPPoutwNDV={Xmin+rand(BVILi-Xmin):(Indv.1)PcorNDVk+rand(Xmax−PcorNDVk):(Indv.2)if,BVILi<XBF<PcorNDVkXmin+rand(PcorNDVk-Xmin):(Indv.1)BVIUi+rand(Xmax-BVIUi):(Indv.2)else{if,BVILi<XBF<BVIUi(1.possibility)Xmin+rand(BVILi-Xmin):(Indv.1)XBF+rand(Xmax-XBF):(Indv.2){else if,XBF>BVIUi(k=1,…,SP)(2.possibility)Xmin+rand(XBF-Xmin):(Indv.1)BVIUi+rand(Xmax-BVIUi):(Indv.2){else(i=Number,…,of Current Intervals)(3.possibility)PinwSP is
regenerated by taking into account the segment BVIiL and BVIiU used for the individuals of the PcorSP. The individuals of PinwSP are forced to converge to the XBF by taking the BVIiL, BVIiU  and PcorSP into
consideration. An algorithm based on the possibilities shown in Figure 3 is
developed for the regeneration of  PinwSP from PcorSP:(8)SPPinwNDV={BVILi+rand(SPPcorNDV-BVILi)SPPcorNDV+rand(BiVIU−SPPcorNDV)ifBVILi<XBF<SPPcorNDVelseBVILi{if,BVILi<XBF<BVIUi(2.possibility)BVILi+rand(XBF-BVILi){if,XBF>BVIUi(3.possibility)XBF+rand(BVIUi-XBF){else(i=1,…,Number of Current Interval)(1.possibility)An
application of extraction-based transmission on two-dimensional solution region
represented by two design variables is graphically shown in Figure4. Only
design variable 1 that is bounded by (Xmin and Xmax) is visualized. The solution region has three optimum points. One of these three optimum points
is a global optimum symbolized by “•”. The remaining feasible
solutions are also indicated by “•”. The individuals from
the core population, which are indicated as “+” and enclosed in a thin dashed closed curve, are used to build
the inward and outward populations (Figure 4). In the regeneration of the
inward and outward populations, the bounds of segment corresponding to the
individual of the core population are in the range of (BVIiL and BVIiU). Outward population is symbolized by “*” and enclosed in a rectangle with dashed edges (Figure 4). The individuals of inward
population is symbolized as “x” and
enclosed in a rectangle with a thick continuous edge (Figure 4).Figure 4
Display of the extraction-based transmission.


## 4.1.2. Fitness Calculation

The module of fitness calculation computes the
fitness values of individuals in each population by taking the constraint
violations into account. Thus, fitness values corresponding to populationsPinw, Poutw and Pcor are collected in the matrices Finw, Foutw, and Fcor. At the same time, feasible solutions obtained are chosen to
locate in feasible solution pool when its fitness values are lowest compared to
other feasible fitness values found in the feasible solution pool.


## 4.1.3. Insertion-Based Transmission

This process involves construction of the core population with the
individuals coming from inward and outward populations. The core, inward and
outward populations all have the same number of individuals. Since one
population is generated from two populations, it is necessary to eliminate a
certain number of individuals. This is carried out by prioritizing certain
individuals according to their feasibility and fitness. All the feasible
solutions located in feasible solution pool are used in constructing the core
population. In order to adapt the feasible solution pool to the core population,
the inward or outward populations is divided into two equal parts. The
algorithm developed in this regard is managed by four cases based on the
position of the number of feasible solutions with respect to the inward or
outward populations with same number of individuals, as depicted in Figure5. These cases are as follows.Figure 5
Insertion-based transmission.Case 1
(C1) The number of feasible solutions in
inward populationNI1 is more than SP/2.Case 2
(C2) The number of feasible solutions in
inward populationNI2 is less than SP/2.Case 3
(C3) The number of feasible solutions in
outward populationNO1 is more than SP/2.Case 4
(C4) The number of feasible solutions in
outward populationNO2 is less than SP/2.The core population from the inward and outward populations is
constructed from the combination of these four cases. These combinations are
C1+C3, C1+C4, C2+C3, and C2+C4. They are explained as(i)
collect (NI1+NO1), (NI1+NO2),
(NI2+NO1), and (NI2+NO2) feasible solutions from inward or outward populations corresponding to the
combinations of “C1+C3,” “C1+C4,” “C2+C3,” and “C2+C4,”
respectively;(ii)
rank the collected feasible solutions in a descending order of
their fitness values, and then store it in a dummy column matrix;(iii)
if the number of
individuals in the combination of “C1+C3,” “C1+C4,” “C2+C3,” and “C2+C4”
is greater than SP,((NI1+NO1)−SP), ((NI1+NO2)−SP), ((NI2+NO1)−SP) and ((NI2+NO2)−SP) feasible solutions with
least fitness are discarded from the dummy column matrix. Otherwise, (SP−(NI1+NO1)), (SP−(NI1+NO2)), (SP−(NI2+NO1)) and (SP−(NI2+NO2)) feasible solutions with least fitness is discarded from the dummy matrix,
respectively. If the number of individuals in dummy matrix does not reach SP, some individuals are borrowed
from the inward and outward populations in the descending order of their fitness
values.


## 4.1.4. Recreation of the Core Population

An algorithm is developed for the recreation of corePcorSP
population. This algorithm is managed
by three possibilities regarding the position of the boundaries, namely XFmin and XFmax, of the feasible
solution pool with respect to the center (cent) of the interval representing
the one-dimensional solution region. These possibilities are graphically
depicted in Figure 6. The recreation process is activated depending on NGGES and the enlargeable bounds of
subsolution regions.Figure 6
Graphical depiction of possibilities for the location of the interval (XFmin, XFmax) and
cent = (XFmin+XFmax)/2 in the recreation of the core population.The algorithm for the recreation of the core populationPcorSP based on
the possibilities given in Figure 6 is formulated by (9),
(9)SPPcorNDV={A.Case:1≤(SP*(30/100))B.Case:(SP*(30/100))<k≤(SP*(50/100))C.Case:(SP*(30/100))<k≤SPif,cent>XFmaxA.Case:1≤(SP*(40/100))B.Case:(SP*(40/100))<k≤(SP*(60/100))C.Case:(SP*(60/100))<k≤SPelse if,XFmin≤cent≤XFmax(k=1,…,SP)A.Case:1≤(SP*(50/100))B.Case:(SP*(50/100))<k≤(SP*(70/100))C.Case:(SP*(70/100))<k≤SPelse
where A  Case:Xmin+rand(XFmin−Xmin); B  Case:XFmin+rand(XFmax−XFmin); B  Case:XFmax+rand(Xmax−XFmax).


## 4.1.5. Application of SGA Operators

SGA operators are
used to regenerate the core population in order to provide a variation for the
next generations. These are one-point crossover, mutation, and roulette wheel
selection operators. Also, some optional operators exist for the search
including what follows (Eiben and Ruttkay [91]).(i)
Multipoint mutation
and crossover operators, and the other selection operators (stochastic
universal sampling and stochastic remainder sampling),(ii)
the generation gap against genetic drift problem,(iii)
different fitness scaling
methods such as linear normalization, baseline windowing, sigma truncating,
linear scaling, and adaptive windowing are employed along with the elitist
selection scheme against the loosing of the valuable genetic heredity.The five interdependent procedures mentioned above are processed in one
of the real, integer, and binary coding schemes. For this reason, recoding of
the design variables is required before these processes are applied.In order to depict the gradual exploration strategy, let us consider a
planar truss with two bars as an example and construct a core population with
two individuals. The upper and lower bounds of the design variables are given
for continuous or discrete set of design variables (see Table1).Table 1
A preliminary demonstration of GAwEIS to a planar truss with two-bars.
Indiv.Section properitiesNSBSXminXmaxDVNVNDVSNBVILBVIUContinous design variablesIndiv. (1)[0.32 0.56]40.1110.3210.1000.32520.5630.5510.775Indiv. (2)[0.82 0.24]40.1110.8240.7761.00020.2410.1000.325Discrete design variablesIndiv. (1)[001011]4151110.1000.3252330.5510.775Indiv. (2)[100001]4151440.7761.0002110.1000.325NSBS: number of subsolution regions (the bumber of segment), VNDV: value of each design variable, DVN: design variable number, SN: subsolution region (segment) number.The solution region of each design variable is divided into 4 segments which
are used for subsolution regions obtained by dividing entire solution region
into small ones. Numerical values of design variables vary within the interval
(0.1, 1) for continuous type and discrete type of design variables. The
one-dimensional solution region is divided into one-dimensional subsolution
regions. One-dimensional subsolution regions are represented by intervals
(0.100, 0.325), (0.326, 0.550), (0.551, 0.775), and (0.776, 1.000). The
discrete design variables are coded by using three-binary-digits. Therefore,
the total number of digits is equal to 6. The continuous design variables are
used to find the corresponding intervals whereas the values of discrete design
variables represent the segment numbers.


## 4.2. MPGA

MPGA makes use of a migration process with several parameters in order to
provide a control for multiple populations and a communication between them. The evolutionary processes of MPGA are carried out by using GEATbx (Polheim [7]). GEATbx is the ability of running with multiple populations which is systematically
structured for adequately the execution of various evolutionary processes and
rich in options regarding different genetic operators and their related
parameters for real-valued variables. Considering the main elements of BGAwEIS, some elements of
GEATbx are appropriately activated in the implementation of MPGA procedure. The
crucial evolutionary operators of MPGA are presented via a pseudocode in Algorithm 3.The first step is the initialization
ofSubPopNum subpopulations. In the
beginning stage, a single population with (SubPopIndNum*SubPopNum) individuals to be settled to
the subpopulations is created. Following this step, the fitness values are
calculated by using the fitness functions. The fitness values are penalized if
they violate the constraints (see (3) and (4). The penalization values are
added to the fitness values.Ranking process governed by ranking
parameterParrank (see related parameters in Section 5) is based on
the redistribution of fitness values where artificial values are used instead
of the actual fitness values. In this way, dominance of the best solutions on
the other solutions is weakened. Ranking process is carried out in two separate
stages (Bäck and Hoffmeister [92]). In the first stage, the fitness values are
recreated by the linear or nonlinear scaling functions. In the second stage,
scaled fitness values are redistributed depending on the quality
of actual fitness values. Furthermore, the ranking share procedure is
applied where fitness values are rescaled according to their rank (Goldberg and
Richardson [93]).Following the ranking process,
evolutionary approaches are repeatedly executed in a loop until a predefined
loop numberepoch is reached. The
first inner loop is regarded with determining the order of population. The
subpopulations are ordered with respect to the fitness values of the
individuals. For this purpose, a simple competition process based on ranking
procedure is utilized (Polheim [7]). The rank of subpopulations has a big impact
on the migration process because a communication network comprised of
subpopulations is used for transmission of emigrants and immigrants.After ordering subpopulation by taking
into account the fitness values, SGA operators (selectionsel, mutation mut, and
crossover cr operators) and their
related parameters Parsel, Parmut, and Parcr (see
Section 5) are activated. These three operators are separately executed
for each subpopulation. The subsequent process is activated when the values of
design variable exceed the ranges of SSR. If this occurs, then related individuals are repaired.The competition process aims to move the robust
individuals to other subpopulations that exhibit relatively poor performance
(Schlierkamp-Voosen and Mühlenbein [94]). The competition of subpopulations is
governed by parameter Parcomp (see Section 5) and carried out in three steps: (i) determination of the capacity of each subpopulations for taking emigrant and
sending immigrant individuals, (ii) picking robust individuals according to
their fitness values, and (iii) the adjustment of the subpopulations size for
the settlement of the robust individuals (Polheim [7]).The transmission of immigrants to
the other subpopulations is accomplished by a migration process. The migration
process is regulated with parameterParmig which indicates
the several parameters, such as migration rate, interval, and topology (see Section 5).


## 5. Design Details

Due to the differentiation in
parameters of the proposed algorithms, a number of parameter sets have to be
tested to determine those with higher performance. The best way to accomplish
this is to focus on their basic operators. In order to make an unbiased
comparison among these proposed algorithms, the values of common operator
parametersParsel, Parmut, Parcr
and some evolutionary parameters SP
and GN are kept constant for
all algorithms. Operators of these algorithms and their related parameter
values for each example problem are tabulated on Table 2. According to Table 2,
while crossover rates indicate the number of individuals to be combined, mutation
rates and ranges are used to define the number of variables per individuals to
be mutated and the range of mutation steps for each variable, respectively. In
addition, the selection operator, namely stochastic universal sampling, is able
to run with any ranking method, namely, linear and nonlinear ranking using a
ranking-related parameters Parrank. In the
approach of MPGA, the fitness assignment provided by the linear or nonlinear
ranking method is assumed according to a certain value of its parameter, namely,
selection pressure. Furthermore, the competition of subpopulations is governed
by the parameters, competition interval and rate denoted by Parcomp. While competition interval determines the
frequency of competition process, the number of migrated individuals with lower
performance to be removed from the subpopulations is determined by the
competition rate.Table 2
Parameter set and related values proposed for SGA, BGAwEIS, and MPGA.
Algorithm nameSGABGAwEISMPGAPopulation no.11(b)12345Example 13003006060606060Population sizeExample 2500500100100100100100Example 31501504040404040OperationsVariable typeOperator(a)
nameOperator parameter
nameSelectionAll variablesStochastic universal samplingInsertion rate0.500.500.800.600.500.400.30Pressure——1.901.701.501.301.10Ranking method——NL(c)NLNLL(c)LCrossoverDiscrete variablesSingle-pointRate0.800.800.900.700.500.300.10Continuous variablesReal typeRate0.800.800.900.700.500.300.10MutationDiscrete variablesSingle-pointRate0.700.701.000.800.600.400.20Continuous variablesReal typeRate0.70.700.1000.800.600.400.20Range0.500.500.800.400.200.080.01CompetitionAll variablesCompetition of subpop.Interval——2020202020Rate——0.050.060.070.080.10Generation gapAll variables——0.700.701.901.701.501.301.10(a)See details in Polheim
[7].(b)Altough existing two populations
within a core population, genetic operators are applied to the core
population.(c)NL: Nonlinear L: Linear.In the arrangement of operators,
various parameter sets are proposed for each algorithm. BGAwEIS uses two basic
parameters, namely, NGGES and NSBS. In order to investigate the relation
between two parameters, the first parameter values are specified as “50, 20, 20,
and 25,” while the values of second parameter are fixed by “20, 50, 20, and 15.”
Thus, four parameter sets, namely (50, 20), (20, 50), (20, 20), and (25, 15),
are devised for design tests.MPGA is governed basically by
migration related parametersParmig such as
migration topology (MT), interval (MI), and rate (MR). The individuals with
higher quality are migrated into five populations. In order to determine the highest
qualified parameter set through examining the parameter values, MI and MR are
taken as “2, 10, 15, 5” and “0.10, 0.01, 0.10, 0.40”,
while migration topologies are chosen either as unrestricted (denoted by 0) or
of neighborhood type (denoted by 1) or ring shaped (denoted by 2) (see the
depiction of proposed migration topologies for five populations (Pop) in Figure 7). Thus, a total of 48 parameter sets are tackled to assess the performance
of MPGA.Introductions of the migration topologies for MPGA with five populations (Pop.)  (a) ring-shaped topology, (b) neighborhood topology, and (c) unrestricted topology.
(a)(b)(c)


## 6. Design Examples

The design examples are presented in
the increasing order of complexity degree indicated by the number of truss bars
and nodes. Three design examples with 24, 72, and 200 bars with one or two
loading cases are employed for application of SGA, BGAwEIS, and MPGA. BGAwEIS
and MPGA are compared considering their optimal designations obtained by using
different parameter sets. The performance of SGA is evaluated with respect to
its optimal designation generated by using one parameters set (see Table2).The dominant evaluation criteria will not only be the feasible solutions
that form the basis of BGAwEIS’ control mechanism but some statistical measures
are also included into the performance assessment. Besides, two interacting
features of genetic search, exploration and exploitation, are utilized for the
evaluation of proposed algorithms. Exploration causes a random moving on the
solution space, but exploitation involves an intensive search of promising
solution region explored previously. In this regard, while exploration leads to
a lower increase in fitness values, exploitation is responsible for a higher
increase.BGAwEIS is initially applied to observe the interdependence of its
parameters with the output associated with three ratios:Ratio 1
R1: (Size of Solution Region SSR)/(Number of Design Variables NDV),Ratio 2
R2:
(number of generation NG)/(Number of Subsolution Regions before Search NSBS),Ratio 3
R3:
(Number of Generation NG)/(Number of Feasible Solution NFS).WhileR1 is indicative about the
quantities of feasible solutions, R2 and R3 are used to measure the performance
of gradual exploration strategy. Moreover, R1 is important for specifying NG and SP. Optimal designations
generated by use of four parameters sets are both tabulated for their output including
corresponding statistical data and displayed for their convergence histories. Statistical data are computed by use of feasible solutions deserved to collect
in feasible solution pool.The optimal designations generated
by MPGA are reported considering 48 parameter sets. The output data is both
listed and displayed associated with feasible solutions obtained. For this
purpose, the standard deviation and mean values of feasible solutions are
computed. In order to comparatively present the designations, the parameter
sets which achieve to obtain lower and higher quality of optimal designations are
chosen among 48 parameter sets. Besides, activated frequencies of these
parameter sets are also presented.


### 6.1. Design Example 1 (25-Bars Space Truss)

This design problem is widely employed for the evaluation of various
optimization methods (Figure8). The members of space truss linked in 8 groups
are selected from a discrete set of 30 available sections (Table 3).Table 3
Design and evolutionary data for BGAwEIS (spatial truss with 25-bars).
Design dataMaterial propertiesModulus of elasticity:104 ksiDensity of material: 0.1 lb/in.3Loading dataCase numberJoint numberX (kips)Y (kips)Z (kips)111−10−1020−10−1030.50060.600Constraint dataDisplacement constraints:Uk≤0.35 inc (k=1,2) for X and Y directionsStress constraints:−40≤σi≤40 ksi (I=1,…,25)Elements of discrete sets
and their position number forAi (I=1,…,25)0.1(1),0.2(2),0.3(3),0.4(4),0.5(5),0.6(6),0.7(7),0.8(8),0.9(9),1.0(10),1.1(11),1.2(12),1.3(13),1.4(14),1.5(15),1.6(16),1.7(17),1.8(18),1.9(19),2.0(20),2.1(21),2.2(22),2.3(23),2.4(24),2.5(25),2.6(26),2.8(27),
3.0(28),3.2(29),3.4(30)Evolutionary dataInputNumber of design variables: 8Size of solution region: 30Number of generation: 400Size of inward population: 300Size of outward population: 300Size of core population: 300CasesCase ICase IICase IIICase IVNGGES50202025NSBS20502015OutputNSAS133411NFS761018Ratio 1R13.753.753.753.75Ratio 2R22082027Ratio 3R357674022Best feasible fitness value571.618592.656515.845485.90Mean of feasible fitness values659.771619.972587.133521.678Standard deviation of feasible fitness values59.59118.85667.09645.880Figure 8
Geometry of spatial truss with 25-bars.The design and evolutionary data for BGAwEIS (as an input and output
obtained by use of four parameter sets) are listed on Table3. The variation in
feasible solutions corresponding to design variables is presented on Table 4. The convergence history of feasible fitness values are displayed in Figure 9.Table 4
Variation of feasible fitness values
according to design variables for spatial truss with 25-bars.
Fitness valuesDesign variable
groups12–56–910-1112-1314–1718–2122–25Feasible design
variable values obtained from different generations624.71729205524129623.602820292610121126575.4927142871815529565.82221726201710729538.742493022113429523.871129517101230516.841102931210730509.60110293129730502.30110293129730495.11110293129530493.80112291119430492.63123021910730491.13123021810730489.63123021710730488.13123021610730487.41113012010730486.63123011610730485.9113011910730Figure 9
Convergence history of
feasible solutions obtained by use of parameters sets proposed for BGAwEIS
(spatial truss with 25-bars).From the tests of MPGA, the optimal designations obtained by use of 48
parameter sets are listed including their statistical analysis results (mean
and standard deviations of feasible fitness values) in Table5. The high and
low values of these quantities are indicated by shaded boxes. Considering the
parameter sets chosen, convergence history of feasible fitness values and
activated frequencies of their populations are presented in Figures 10 and 11.Table 5
Statistical analysis results of feasible fitness values obtained by use of parameter sets proposed for MPGA (spatial truss
with 25-bars).
Parameter setBestMeanStdParameter setBestMeanStdMT=0, MR=0.01, MI=2502,104577,03867,707MT=1, MR=0.10, MI=2497,679553,20452,961MT=0, MR=0.01, MI=10491,290572,76573,076MT=1, MR=0.10, MI=10486,295535,33949,407MT=0, MR=0.01, MI=15510,594594,44468,822MT=1, MR=0.10, MI=15492,001564,70761,902MT=0, MR=0.01, MI=5501,384589,60668,165MT=1, MR=0.10, MI=5490,972582,35673,748MT=0, MR=0.05, MI=2493,103553,96957,707MT=1, MR=0.40, MI=2493,464559,50454,353MT=0, MR=0.05, MI=10495,631564,71659,424MT=1, MR=0.40, MI=10494,279568,88263,133MT=0, MR=0.05, MI=15493,015558,27848,711MT=1, MR=0.40, MI=15492,607564,30671,255MT=0, MR=0.05, MI=5492,631559,88358,218MT=1, MR=0.40, MI=5488,096561,87254,620MT=0, MR=0.10, MI=2495,242579,44467,091MT=2, MR=0.01, MI=2510,554602,50877,442MT=0, MR=0.10, MI=10487,050571,73573,567MT=2, MR=0.01, MI=10488,320574,90772,809MT=0, MR=0.10, MI=15504,839585,48363,716MT=2, MR=0.01, MI=15505,944589,97768,911MT=0, MR=0.10, MI=5494,356581,33064,624MT=2, MR=0.01, MI=5512,587578,08759,429MT=0, MR=0.40, MI=2491,887546,26154,684MT=2, MR=0.05, MI=2493,299574,08544,792MT=0, MR=0.40, MI=10496,481570,75054,856MT=2, MR=0.05, MI=10489,609551,74863,178MT=0, MR=0.40, MI=15511,631583,17765,891MT=2, MR=0.05, MI=15493,907558,57653,586MT=0, MR=0.40, MI=5488,049549,87461,681MT=2, MR=0.05, MI=5493,523563,00760,136MT=1, MR=0.01, MI=2520,044583,86170,871MT=2, MR=0.10, MI=2491,125572,50660,291MT=1, MR=0.01, MI=10522,371607,75567,601MT=2, MR=0.10, MI=10495,555581,44279,229MT=1, MR=0.01, MI=15506,457582,05257,692MT=2, MR=0.10, MI=15492,714570,81868,788MT=1, MR=0.01, MI=5506,226602,78960,236MT=2, MR=0.10, MI=5511,718584,62260,603MT=1, MR=0.05, MI=2489,218580,43475,958MT=2, MR=0.40, MI=2498,955561,76050,194MT=1, MR=0.05, MI=10507,234556,62453,246MT=2, MR=0.40, MI=10492,850574,76663,040MT=1, MR=0.05, MI=15500,155562,18753,653MT=2, MR=0.40, MI=15491,468566,39669,870MT=1, MR=0.05, MI=5491,397561,55656,276MT=2, MR=0.40, MI=5501,608577,68851,228Figure 10
Convergence history of feasible solutions obtained by use of parameters sets proposed for MPGA (spatial truss with 25-bars).Figure 11
Activated numbers of each population obtained by MPGA (spatial truss with 25-bars).The optimal designations obtained by proposed algorithms and existing
ones outlined in literature are presented in Table6 including the critical
values of stress and displacement corresponding to the optimal designations.Table 6
Comparison of optimum designs, critical deflection, and stress
values for BGAwEIS (spatial
truss with 25-bars).
Ref.Best weightDesign variable groups12–56–910-1112-1314–1718–2122–25Rajeev and Krishnam. [95]546.010.101.802.300.200.100.801.803.00Zhu [96]562.930.101.902.600.100.100.802.102.60Erbatur et al. [97]493.800.101.203.200.101.100.900.403.40Coello** et al. [98]493.94————————Wu** and Chow [99]491.72————————SGA814.640.103.002.802.402.201.902.802.40MPGA486.290.100.503.400.101.500.900.603.40BGAwEIS485.900.100.103.400.101.901.000.703.40Max. displacement. inx, y and z directions: 0.1206, 0.3498, 0.0462 at node1Max. element stress: 20.2311 at element 25**Design variable groups are not presented in the references.


### 6.2. Design Example 2 (72-Bars Spatial Truss)

The transmission tower with 72 members is also used by many researchers
as a benchmark problem. This steel structure has 16 independent design
variables and subjected to two different loading conditions (Figure12).Figure 12
Geometry of the spatial truss
with 72-bars.The design and evolutionary data for BGAwEIS (as an input and output
obtained by use of four parameter sets) are listed on Table7. The variation of
feasible solutions values through generation number are displayed in Figure 13.Table 7
Design and evolutionary data for BGAwEIS (spatial truss with 72-bars).
Design dataMaterial propertiesModulus of elasticity:104 ksiDensity of material: 0.1 lb/in.3Loading dataCase numberJoint numberX (kips)Y (kips)Z (kips)1155−52100−5200−5300−5400−5Constraint dataDisplacement constraints:Uk≤0.25  inc (k=1,…,20) for X and Y directionsStress constraints:−25≤σi≤25 ksi (i=1,…,72)Elements of continuous sets0.1≤Ai≤2 (i=1,…,72)Evolutionary dataInputNumber of design variables: 16Size of solution region:∞Number of generation: 600Size of inward population: 500Size of outward population: 500Size of core population: 500CasesCase ICase IICase IIICase IVNGGES50204015NSBS20501550OutputNSAS835223NFS871112Ratio 1R1∞∞∞∞Ratio 2R230124012Ratio 3R375855550Best feasible fitness value427.753452.286380.784381.08Mean of feasible fitness values694.574842.722891.234882.993Standard deviation of feasible fitness values290.294329.609409.814351.210Figure 13
Convergence history of feasible solutions obtained by use of parameters sets
proposed for BGAwEIS (spatial truss with 72-Bars).Optimal designations generated by use of 48 parameter set for MPGA are
tabulated including their statistical analysis results (mean and standard
deviations of feasible fitness values) (Table8). High and low values of these
quantities are indicated by shaded boxes. The convergence history of feasible
fitness values corresponding to these parameter sets and activated frequencies
of their populations are shown in Figures 14 and 15. The optimal designations
obtained by proposed algorithms and existing methods outlined in the literature
are reported in Table 9 including the critical values of stress and
displacement corresponding to the optimal designations.Table 8
Statistical analysis results of feasible fitness values obtained by use of parameter sets proposed for MPGA (spatial truss
with 72-bars).
Parameter setBestMeanStdParameter setBestMeanStdMT=0, MR=0.01, MI=2640,229907,646147,346MT=1, MR=0.10, MI=2683,537912,827109,804MT=0, MR=0.01, MI=10734,629913,598113,454MT=1, MR=0.10, MI=10594,811951,129119,593MT=0, MR=0.01, MI=15711,954913,401126,805MT=1, MR=0.10, MI=15598,955952,984124,680MT=0, MR=0.01, MI=5695,537922,956113,759MT=1, MR=0.10 MI=5753,755943,440110,412MT=0, MR=0.05, MI=2767,134951,538123,862MT=1, MR=0.40, MI=2618,419890,768140,968MT=0, MR=0.05, MI=10760,033917,419107,376MT=1, MR=0.40, MI=10693,387935,066116,045MT=0, MR=0.05, MI=15633,658935,527136,514MT=1, MR=0.40, MI=15603,390871,091137,827MT=0, MR=0.05, MI=5723,858898,141114,732MT=1, MR=0.40, MI=5717,659948,091118,160MT=0, MR=0.10, MI=2615,721926,908105,704MT=2, MR=0.01, MI=2620,216893,702146,986MT=0, MR=0.10, MI=10689,895891,536148,994MT=2, MR=0.01, MI=10622,736902,485148,753MT=0, MR=0.10, MI=15732,047934,492113,616MT=2, MR=0.01, MI=15644,256929,181129,546MT=0, MR=0.10, MI=5696,988904,547115,401MT=2, MR=0.01, MI=5656,687921,091119,474MT=0, MR=0.40, MI=2704,639915,372111,372MT=2, MR=0.05, MI=2731,287922,980121,985MT=0, MR=0.40, MI=10673,424874,583125,405MT=2, MR=0.05, MI=10729,340941,674116,160MT=0, MR=0.40, MI=15678,748890,891127,018MT=2, MR=0.05, MI=15618,836877,379138,408MT=0, MR=0.40, MI=5682,176916,081127,354MT=2, MR=0.05, MI=5625,641885,707151,493MT=1, MR=0.01, MI=2669,352943,534122,819MT=2, MR=0.10, MI=2690,889867,153149,447MT=1, MR=0.01, MI=10604,188917,024136,592MT=2, MR=0.10, MI=10695,986920,233100,479MT=1, MR=0.01, MI=15621,338882,202134,963MT=2, MR=0.10, MI=15697,875909,407108,374MT=1, MR=0.01, MI=5667,580895,065133,153MT=2, MR=0.10, MI=5638,491870,591133,380MT=1, MR=0.05, MI=2735,599945,279113,778MT=2, MR=0.40, MI=2706,990943,478133,583MT=1, MR=0.05, MI=10625,002902,000121,989MT=2, MR=0.40, MI=10718,084939,940130,951MT=1, MR=0.05, MI=15673,220923,570114,286MT=2, MR=0.40, MI=15698,415906,278118,242MT=1, MR=0.05, MI=5679,263932,444122,463MT=2, MR=0.40, MI=5695,139944,974125,174Table 9
Comparison of optimum designs, critical deflection, and stress values for BGAwEIS (spatial truss with 72-bars).
Design variablesReferencesVenkaya [100]Gellatly and Berke [101]Renwei and Peng [102]Schmit and Farshi [103]Erbatur  et al. [97]SGAMPGABGAwEIS1–40.1610.14920.16410.15850.1610.8730.6750,1565–120.5570.77330.55520.59360.5441.6810.2530,55513–160.3770.45340.41870.34140.3790.1000.6010,37017-180.5060.34170.57580.60760.5211.4180.4370,51019–220.6110.55210.53270.26430.5350.9860.8410,62023–300.5320.60840.52560.54800.5351.5300.8610,53031–340.1000.1000.1000.1000.1031.9820.4600,10035-360.1000.1000.1000.15090.1111.1211.5130,10037–401.2461.02351.28931.10671.3101.5891.9101,25041–480.5240.54210.52010.57930.4981.9870.7890,52349–520.1000.1000.1000.1000.1101.0830.1320,10153-540.1000.1000.1000.1000.1031.8560.9360,10555–581.8181.4641.91732.07841.9100.2681.8401,86059–660.5240.52070.52070.50340.5251.4730.8990,51367–700.1000.1000.1000.1000.1220.8490.2440,10017–720.1000.1000.1000.1000.1031.4690.1830,100Best Weight381.28395.97379.66388.65383.1201196.89594.811380.7830.0091, 0.0091, 0.2391 at node1 for Case 1;Max. displacement. inx, y and z directions:0.2499, 0.2499, 0.0718 at node1 for Case 2Max. element stress:16.2519 at element 1 for Case 1;24.9371 at element 1 for Case 2Figure 14
Convergence history of feasible solutions obtained by use of parameters sets proposed for MPGA (spatial truss with 72-bars).Figure 15
Activated numbers of each population obtained by MPGA (spatial truss with 72-bars).


### 6.3. Design Example 3 (200-Bars Planar Truss)

The plane truss shown in Figure8
involves both continuous as well as discrete design variables (Ponterosso and
Fox [104]). It has 200 independent design variables (Figure 16).Figure 16
Geometry of planar truss with 200-bars.The design and evolutionary data for BGAwEIS (as an input and
output obtained by four parameter sets) are listed on Table10. The variation
of feasible fitness values through generation number are shown in Figure 17.Table 10
Design and evolutionary data for BGAwEIS (planar truss with 200-bars).
The design dataMaterial propertiesModulus of elasticity:30×103 ksiDensity of material: 0.283 lb/in.3Loading dataCase numberJoint numberX (kips)Y (kips)Z (kips)11,6,15,20,29,34,43,48,57,10062,711,2,3,4,5,6,8,10,12,14,15,0−10016,17,18,19,71,72,73,74,75Constraint dataDisplacement constraints:Uk≤0.50 inc (k=1,…,77) for X and Y directionsStress constraints:−30≤σi≤30 ksi (i=1,…,200)Elements of discrete sets
and their position number forAi (I=1,…,200)0.100(1),0.347(2),0.440(3),0.539(4),0.954(5),1.081(6),1.174(7),1.333(8),1.488(9),1.764(10),2.142(11),2.697(12),2.800(13),3.131(14),3.565(15),3.813(16),4.805(17),5.952(18),6.572(19),7.192(20),8.525(21),9.300(22),10.850(23),13.330(24),14.290(25),17.170
(26),19.180(27),23.680(28),28.080(29),33.700(30)Evolutionary dataInputNumber of design variables: 150Size of solution region: 30Number of generation: 200Size of inward population: 150Size of outward population: 150Size of core population: 150CasesCase ICase IICase IIICase IVNGGES50204015NSBS2050515OutputNSAS174211NFS561916Ratio 1 R10.150.150.150.15Ratio 2 R21044013Ratio 3 R340331113Best feasible fitness value37659.68339480.88133405.94935128.000Mean of feasible fitness values146652.109130204.259182974.671153277.013Standard deviation of feasible fitness values98588.65288555.56987725.05659388.735Figure 17
Convergence history of feasible solutions obtained by use of parameters sets
proposed for BGAwEIS (planar truss with 200-bars).Optimal designations obtained by MPGA, considering 48 parameter
sets are summarized including statistical analysis results (mean and standard
deviations of feasible fitness values) (Table11). The convergence history of
feasible fitness values obtained by use of these parameter sets chosen and
activated frequencies of their populations are displayed in Figures 18 and 19. The optimal designations with higher performance are presented for proposed
algorithms and existing approaches outlined in literature in Table 12 including
the critical values of stress and displacement corresponding to the optimal
designations. Design variables that belong to optimal designation obtained by
BGAwEIS are presented in the appendix.Table 11
Statistical analysis results of feasible fitness values obtained by use of parameter sets proposed for MPGA (planar truss with 200-bars).
Parameter setBestMeanStdParameter setBestMeanStdMT=0, MR=0.01, MI=272633,88080630,5698571,450MT=1, MR=0.10, MI=243405,94946677,3386914,943MT=0, MR=0.01, MI=1048093,57456656,4194216,594MT=1, MR=0.10, MI=1046110,34152166,1914860,834MT=0, MR=0.01, MI=1546658,76353303,6814797,318MT=1, MR=0.10, MI=1546797,20553567,5546498,843MT=0, MR=0.01, MI=543563,41749063,0163737,226MT=1, MR=0.10 MI=545121,76754525,7026203,645MT=0, MR=0.05, MI=247659,68353501,9634633,370MT=1, MR=0.40, MI=244316,22048637,4253729,784MT=0, MR=0.05, MI=1044562,84452044,2666328,921MT=1, MR=0.40, MI=1043141,91448152,4885033,876MT=0, MR=0.05, MI=1548956,63055827,1415594,529MT=1, MR=0.40, MI=1546041,41654705,6724620,251MT=0, MR=0.05, MI=540079,50746459,5475301,405MT=1, MR=0.40, MI=548661,99754604,3993207,818MT=0, MR=0.10, MI=240355,65446811,3174671,882MT=2, MR=0.01, MI=244005,45952233,3116203,747MT=0, MR=0.10, MI=1046765,12755515,5874599,817MT=2, MR=0.01, MI=1050810,97557383,1924752,537MT=0, MR=0.10, MI=1547207,68452014,0784626,456MT=2, MR=0.01, MI=1544926,93452888,6023982,791MT=0, MR=0.10, MI=556928,67360909,0243266,486MT=2, MR=0.01, MI=543645,02850901,1065238,565MT=0, MR=0.40, MI=241540,47046590,1984894,845MT=2, MR=0.05, MI=249959,85755518,2845472,687MT=0, MR=0.40 MI=1045410,65853645,7164960,128MT=2, MR=0.05, MI=1051736,32962136,4576177,860MT=0, MR=0.40, MI=1549511,15953964,2203839,889MT=2, MR=0.05, MI=1548048,71957350,7485234,649MT=0, MR=0.40, MI=549983,42753905,6033265,335MT=2, MR=0.05, MI=543096,14750628,6877155,731MT=1, MR=0.01, MI=263494,76364043,462350,856MT=2, MR=0.10, MI=270763,44573702,8542774,557MT=1, MR=0.01, MI=1058124,13862066,3604194,507MT=2, MR=0.10, MI=1046612,62551832,5074416,433MT=1, MR=0.01, MI=1553912,54458640,5744411,781MT=2, MR=0.10, MI=1545523,32350852,6764438,879MT=1, MR=0.01, MI=543427,51148630,3065504,416MT=2, MR=0.10, MI=552091,56457823,1823014,238MT=1, MR=0.05, MI=249480,88149870,6245751,019MT=2, MR=0.40, MI=248312,17354232,3836997,833MT=1, MR=0.05, MI=1041201,48348559,5515957,630MT=2, MR=0.40, MI=1046408,77152875,3033739,149MT=1, MR=0.05, MI=1544122,01449935,0793517,734MT=2, MR=0.40, MI=1546437,70256239,8936986,506MT=1, MR=0.05, MI=543731,88651337,6506528,667MT=2, MR=0.40, MI=543413,18753022,9545995,418Table 12
Comparison of optimum designs, critical deflection, and stress values for BGAwEIS (planar truss with 200-bars).
ReferencesPonterosso and Fox [104]SGAMPGABGAwEISMinimum weight35394.00122047.1440079.50733405.949Max. displacement. inx, y and z directions:0.4959, 0.2745, 0.4998 at node1 for Case 1;0.0948, 0.4738, 0.0428 at node1 for Case 2Max. element Stress:20.9676 at element 90 for Case 1;6.4872 at element 185 for Case 2Figure 18
Convergence history of feasible solutions obtained by use of parameters sets proposed for MPGA (planar truss with 200-bars).Figure 19
Activated numbers of each population obtained by MPGA (planar truss with 200-bars).


## 6.1. Design Example 1 (25-Bars Space Truss)

This design problem is widely employed for the evaluation of various
optimization methods (Figure8). The members of space truss linked in 8 groups
are selected from a discrete set of 30 available sections (Table 3).Table 3
Design and evolutionary data for BGAwEIS (spatial truss with 25-bars).
Design dataMaterial propertiesModulus of elasticity:104 ksiDensity of material: 0.1 lb/in.3Loading dataCase numberJoint numberX (kips)Y (kips)Z (kips)111−10−1020−10−1030.50060.600Constraint dataDisplacement constraints:Uk≤0.35 inc (k=1,2) for X and Y directionsStress constraints:−40≤σi≤40 ksi (I=1,…,25)Elements of discrete sets
and their position number forAi (I=1,…,25)0.1(1),0.2(2),0.3(3),0.4(4),0.5(5),0.6(6),0.7(7),0.8(8),0.9(9),1.0(10),1.1(11),1.2(12),1.3(13),1.4(14),1.5(15),1.6(16),1.7(17),1.8(18),1.9(19),2.0(20),2.1(21),2.2(22),2.3(23),2.4(24),2.5(25),2.6(26),2.8(27),
3.0(28),3.2(29),3.4(30)Evolutionary dataInputNumber of design variables: 8Size of solution region: 30Number of generation: 400Size of inward population: 300Size of outward population: 300Size of core population: 300CasesCase ICase IICase IIICase IVNGGES50202025NSBS20502015OutputNSAS133411NFS761018Ratio 1R13.753.753.753.75Ratio 2R22082027Ratio 3R357674022Best feasible fitness value571.618592.656515.845485.90Mean of feasible fitness values659.771619.972587.133521.678Standard deviation of feasible fitness values59.59118.85667.09645.880Figure 8
Geometry of spatial truss with 25-bars.The design and evolutionary data for BGAwEIS (as an input and output
obtained by use of four parameter sets) are listed on Table3. The variation in
feasible solutions corresponding to design variables is presented on Table 4. The convergence history of feasible fitness values are displayed in Figure 9.Table 4
Variation of feasible fitness values
according to design variables for spatial truss with 25-bars.
Fitness valuesDesign variable
groups12–56–910-1112-1314–1718–2122–25Feasible design
variable values obtained from different generations624.71729205524129623.602820292610121126575.4927142871815529565.82221726201710729538.742493022113429523.871129517101230516.841102931210730509.60110293129730502.30110293129730495.11110293129530493.80112291119430492.63123021910730491.13123021810730489.63123021710730488.13123021610730487.41113012010730486.63123011610730485.9113011910730Figure 9
Convergence history of
feasible solutions obtained by use of parameters sets proposed for BGAwEIS
(spatial truss with 25-bars).From the tests of MPGA, the optimal designations obtained by use of 48
parameter sets are listed including their statistical analysis results (mean
and standard deviations of feasible fitness values) in Table5. The high and
low values of these quantities are indicated by shaded boxes. Considering the
parameter sets chosen, convergence history of feasible fitness values and
activated frequencies of their populations are presented in Figures 10 and 11.Table 5
Statistical analysis results of feasible fitness values obtained by use of parameter sets proposed for MPGA (spatial truss
with 25-bars).
Parameter setBestMeanStdParameter setBestMeanStdMT=0, MR=0.01, MI=2502,104577,03867,707MT=1, MR=0.10, MI=2497,679553,20452,961MT=0, MR=0.01, MI=10491,290572,76573,076MT=1, MR=0.10, MI=10486,295535,33949,407MT=0, MR=0.01, MI=15510,594594,44468,822MT=1, MR=0.10, MI=15492,001564,70761,902MT=0, MR=0.01, MI=5501,384589,60668,165MT=1, MR=0.10, MI=5490,972582,35673,748MT=0, MR=0.05, MI=2493,103553,96957,707MT=1, MR=0.40, MI=2493,464559,50454,353MT=0, MR=0.05, MI=10495,631564,71659,424MT=1, MR=0.40, MI=10494,279568,88263,133MT=0, MR=0.05, MI=15493,015558,27848,711MT=1, MR=0.40, MI=15492,607564,30671,255MT=0, MR=0.05, MI=5492,631559,88358,218MT=1, MR=0.40, MI=5488,096561,87254,620MT=0, MR=0.10, MI=2495,242579,44467,091MT=2, MR=0.01, MI=2510,554602,50877,442MT=0, MR=0.10, MI=10487,050571,73573,567MT=2, MR=0.01, MI=10488,320574,90772,809MT=0, MR=0.10, MI=15504,839585,48363,716MT=2, MR=0.01, MI=15505,944589,97768,911MT=0, MR=0.10, MI=5494,356581,33064,624MT=2, MR=0.01, MI=5512,587578,08759,429MT=0, MR=0.40, MI=2491,887546,26154,684MT=2, MR=0.05, MI=2493,299574,08544,792MT=0, MR=0.40, MI=10496,481570,75054,856MT=2, MR=0.05, MI=10489,609551,74863,178MT=0, MR=0.40, MI=15511,631583,17765,891MT=2, MR=0.05, MI=15493,907558,57653,586MT=0, MR=0.40, MI=5488,049549,87461,681MT=2, MR=0.05, MI=5493,523563,00760,136MT=1, MR=0.01, MI=2520,044583,86170,871MT=2, MR=0.10, MI=2491,125572,50660,291MT=1, MR=0.01, MI=10522,371607,75567,601MT=2, MR=0.10, MI=10495,555581,44279,229MT=1, MR=0.01, MI=15506,457582,05257,692MT=2, MR=0.10, MI=15492,714570,81868,788MT=1, MR=0.01, MI=5506,226602,78960,236MT=2, MR=0.10, MI=5511,718584,62260,603MT=1, MR=0.05, MI=2489,218580,43475,958MT=2, MR=0.40, MI=2498,955561,76050,194MT=1, MR=0.05, MI=10507,234556,62453,246MT=2, MR=0.40, MI=10492,850574,76663,040MT=1, MR=0.05, MI=15500,155562,18753,653MT=2, MR=0.40, MI=15491,468566,39669,870MT=1, MR=0.05, MI=5491,397561,55656,276MT=2, MR=0.40, MI=5501,608577,68851,228Figure 10
Convergence history of feasible solutions obtained by use of parameters sets proposed for MPGA (spatial truss with 25-bars).Figure 11
Activated numbers of each population obtained by MPGA (spatial truss with 25-bars).The optimal designations obtained by proposed algorithms and existing
ones outlined in literature are presented in Table6 including the critical
values of stress and displacement corresponding to the optimal designations.Table 6
Comparison of optimum designs, critical deflection, and stress
values for BGAwEIS (spatial
truss with 25-bars).
Ref.Best weightDesign variable groups12–56–910-1112-1314–1718–2122–25Rajeev and Krishnam. [95]546.010.101.802.300.200.100.801.803.00Zhu [96]562.930.101.902.600.100.100.802.102.60Erbatur et al. [97]493.800.101.203.200.101.100.900.403.40Coello** et al. [98]493.94————————Wu** and Chow [99]491.72————————SGA814.640.103.002.802.402.201.902.802.40MPGA486.290.100.503.400.101.500.900.603.40BGAwEIS485.900.100.103.400.101.901.000.703.40Max. displacement. inx, y and z directions: 0.1206, 0.3498, 0.0462 at node1Max. element stress: 20.2311 at element 25**Design variable groups are not presented in the references.


## 6.2. Design Example 2 (72-Bars Spatial Truss)

The transmission tower with 72 members is also used by many researchers
as a benchmark problem. This steel structure has 16 independent design
variables and subjected to two different loading conditions (Figure12).Figure 12
Geometry of the spatial truss
with 72-bars.The design and evolutionary data for BGAwEIS (as an input and output
obtained by use of four parameter sets) are listed on Table7. The variation of
feasible solutions values through generation number are displayed in Figure 13.Table 7
Design and evolutionary data for BGAwEIS (spatial truss with 72-bars).
Design dataMaterial propertiesModulus of elasticity:104 ksiDensity of material: 0.1 lb/in.3Loading dataCase numberJoint numberX (kips)Y (kips)Z (kips)1155−52100−5200−5300−5400−5Constraint dataDisplacement constraints:Uk≤0.25  inc (k=1,…,20) for X and Y directionsStress constraints:−25≤σi≤25 ksi (i=1,…,72)Elements of continuous sets0.1≤Ai≤2 (i=1,…,72)Evolutionary dataInputNumber of design variables: 16Size of solution region:∞Number of generation: 600Size of inward population: 500Size of outward population: 500Size of core population: 500CasesCase ICase IICase IIICase IVNGGES50204015NSBS20501550OutputNSAS835223NFS871112Ratio 1R1∞∞∞∞Ratio 2R230124012Ratio 3R375855550Best feasible fitness value427.753452.286380.784381.08Mean of feasible fitness values694.574842.722891.234882.993Standard deviation of feasible fitness values290.294329.609409.814351.210Figure 13
Convergence history of feasible solutions obtained by use of parameters sets
proposed for BGAwEIS (spatial truss with 72-Bars).Optimal designations generated by use of 48 parameter set for MPGA are
tabulated including their statistical analysis results (mean and standard
deviations of feasible fitness values) (Table8). High and low values of these
quantities are indicated by shaded boxes. The convergence history of feasible
fitness values corresponding to these parameter sets and activated frequencies
of their populations are shown in Figures 14 and 15. The optimal designations
obtained by proposed algorithms and existing methods outlined in the literature
are reported in Table 9 including the critical values of stress and
displacement corresponding to the optimal designations.Table 8
Statistical analysis results of feasible fitness values obtained by use of parameter sets proposed for MPGA (spatial truss
with 72-bars).
Parameter setBestMeanStdParameter setBestMeanStdMT=0, MR=0.01, MI=2640,229907,646147,346MT=1, MR=0.10, MI=2683,537912,827109,804MT=0, MR=0.01, MI=10734,629913,598113,454MT=1, MR=0.10, MI=10594,811951,129119,593MT=0, MR=0.01, MI=15711,954913,401126,805MT=1, MR=0.10, MI=15598,955952,984124,680MT=0, MR=0.01, MI=5695,537922,956113,759MT=1, MR=0.10 MI=5753,755943,440110,412MT=0, MR=0.05, MI=2767,134951,538123,862MT=1, MR=0.40, MI=2618,419890,768140,968MT=0, MR=0.05, MI=10760,033917,419107,376MT=1, MR=0.40, MI=10693,387935,066116,045MT=0, MR=0.05, MI=15633,658935,527136,514MT=1, MR=0.40, MI=15603,390871,091137,827MT=0, MR=0.05, MI=5723,858898,141114,732MT=1, MR=0.40, MI=5717,659948,091118,160MT=0, MR=0.10, MI=2615,721926,908105,704MT=2, MR=0.01, MI=2620,216893,702146,986MT=0, MR=0.10, MI=10689,895891,536148,994MT=2, MR=0.01, MI=10622,736902,485148,753MT=0, MR=0.10, MI=15732,047934,492113,616MT=2, MR=0.01, MI=15644,256929,181129,546MT=0, MR=0.10, MI=5696,988904,547115,401MT=2, MR=0.01, MI=5656,687921,091119,474MT=0, MR=0.40, MI=2704,639915,372111,372MT=2, MR=0.05, MI=2731,287922,980121,985MT=0, MR=0.40, MI=10673,424874,583125,405MT=2, MR=0.05, MI=10729,340941,674116,160MT=0, MR=0.40, MI=15678,748890,891127,018MT=2, MR=0.05, MI=15618,836877,379138,408MT=0, MR=0.40, MI=5682,176916,081127,354MT=2, MR=0.05, MI=5625,641885,707151,493MT=1, MR=0.01, MI=2669,352943,534122,819MT=2, MR=0.10, MI=2690,889867,153149,447MT=1, MR=0.01, MI=10604,188917,024136,592MT=2, MR=0.10, MI=10695,986920,233100,479MT=1, MR=0.01, MI=15621,338882,202134,963MT=2, MR=0.10, MI=15697,875909,407108,374MT=1, MR=0.01, MI=5667,580895,065133,153MT=2, MR=0.10, MI=5638,491870,591133,380MT=1, MR=0.05, MI=2735,599945,279113,778MT=2, MR=0.40, MI=2706,990943,478133,583MT=1, MR=0.05, MI=10625,002902,000121,989MT=2, MR=0.40, MI=10718,084939,940130,951MT=1, MR=0.05, MI=15673,220923,570114,286MT=2, MR=0.40, MI=15698,415906,278118,242MT=1, MR=0.05, MI=5679,263932,444122,463MT=2, MR=0.40, MI=5695,139944,974125,174Table 9
Comparison of optimum designs, critical deflection, and stress values for BGAwEIS (spatial truss with 72-bars).
Design variablesReferencesVenkaya [100]Gellatly and Berke [101]Renwei and Peng [102]Schmit and Farshi [103]Erbatur  et al. [97]SGAMPGABGAwEIS1–40.1610.14920.16410.15850.1610.8730.6750,1565–120.5570.77330.55520.59360.5441.6810.2530,55513–160.3770.45340.41870.34140.3790.1000.6010,37017-180.5060.34170.57580.60760.5211.4180.4370,51019–220.6110.55210.53270.26430.5350.9860.8410,62023–300.5320.60840.52560.54800.5351.5300.8610,53031–340.1000.1000.1000.1000.1031.9820.4600,10035-360.1000.1000.1000.15090.1111.1211.5130,10037–401.2461.02351.28931.10671.3101.5891.9101,25041–480.5240.54210.52010.57930.4981.9870.7890,52349–520.1000.1000.1000.1000.1101.0830.1320,10153-540.1000.1000.1000.1000.1031.8560.9360,10555–581.8181.4641.91732.07841.9100.2681.8401,86059–660.5240.52070.52070.50340.5251.4730.8990,51367–700.1000.1000.1000.1000.1220.8490.2440,10017–720.1000.1000.1000.1000.1031.4690.1830,100Best Weight381.28395.97379.66388.65383.1201196.89594.811380.7830.0091, 0.0091, 0.2391 at node1 for Case 1;Max. displacement. inx, y and z directions:0.2499, 0.2499, 0.0718 at node1 for Case 2Max. element stress:16.2519 at element 1 for Case 1;24.9371 at element 1 for Case 2Figure 14
Convergence history of feasible solutions obtained by use of parameters sets proposed for MPGA (spatial truss with 72-bars).Figure 15
Activated numbers of each population obtained by MPGA (spatial truss with 72-bars).


## 6.3. Design Example 3 (200-Bars Planar Truss)

The plane truss shown in Figure8
involves both continuous as well as discrete design variables (Ponterosso and
Fox [104]). It has 200 independent design variables (Figure 16).Figure 16
Geometry of planar truss with 200-bars.The design and evolutionary data for BGAwEIS (as an input and
output obtained by four parameter sets) are listed on Table10. The variation
of feasible fitness values through generation number are shown in Figure 17.Table 10
Design and evolutionary data for BGAwEIS (planar truss with 200-bars).
The design dataMaterial propertiesModulus of elasticity:30×103 ksiDensity of material: 0.283 lb/in.3Loading dataCase numberJoint numberX (kips)Y (kips)Z (kips)11,6,15,20,29,34,43,48,57,10062,711,2,3,4,5,6,8,10,12,14,15,0−10016,17,18,19,71,72,73,74,75Constraint dataDisplacement constraints:Uk≤0.50 inc (k=1,…,77) for X and Y directionsStress constraints:−30≤σi≤30 ksi (i=1,…,200)Elements of discrete sets
and their position number forAi (I=1,…,200)0.100(1),0.347(2),0.440(3),0.539(4),0.954(5),1.081(6),1.174(7),1.333(8),1.488(9),1.764(10),2.142(11),2.697(12),2.800(13),3.131(14),3.565(15),3.813(16),4.805(17),5.952(18),6.572(19),7.192(20),8.525(21),9.300(22),10.850(23),13.330(24),14.290(25),17.170
(26),19.180(27),23.680(28),28.080(29),33.700(30)Evolutionary dataInputNumber of design variables: 150Size of solution region: 30Number of generation: 200Size of inward population: 150Size of outward population: 150Size of core population: 150CasesCase ICase IICase IIICase IVNGGES50204015NSBS2050515OutputNSAS174211NFS561916Ratio 1 R10.150.150.150.15Ratio 2 R21044013Ratio 3 R340331113Best feasible fitness value37659.68339480.88133405.94935128.000Mean of feasible fitness values146652.109130204.259182974.671153277.013Standard deviation of feasible fitness values98588.65288555.56987725.05659388.735Figure 17
Convergence history of feasible solutions obtained by use of parameters sets
proposed for BGAwEIS (planar truss with 200-bars).Optimal designations obtained by MPGA, considering 48 parameter
sets are summarized including statistical analysis results (mean and standard
deviations of feasible fitness values) (Table11). The convergence history of
feasible fitness values obtained by use of these parameter sets chosen and
activated frequencies of their populations are displayed in Figures 18 and 19. The optimal designations with higher performance are presented for proposed
algorithms and existing approaches outlined in literature in Table 12 including
the critical values of stress and displacement corresponding to the optimal
designations. Design variables that belong to optimal designation obtained by
BGAwEIS are presented in the appendix.Table 11
Statistical analysis results of feasible fitness values obtained by use of parameter sets proposed for MPGA (planar truss with 200-bars).
Parameter setBestMeanStdParameter setBestMeanStdMT=0, MR=0.01, MI=272633,88080630,5698571,450MT=1, MR=0.10, MI=243405,94946677,3386914,943MT=0, MR=0.01, MI=1048093,57456656,4194216,594MT=1, MR=0.10, MI=1046110,34152166,1914860,834MT=0, MR=0.01, MI=1546658,76353303,6814797,318MT=1, MR=0.10, MI=1546797,20553567,5546498,843MT=0, MR=0.01, MI=543563,41749063,0163737,226MT=1, MR=0.10 MI=545121,76754525,7026203,645MT=0, MR=0.05, MI=247659,68353501,9634633,370MT=1, MR=0.40, MI=244316,22048637,4253729,784MT=0, MR=0.05, MI=1044562,84452044,2666328,921MT=1, MR=0.40, MI=1043141,91448152,4885033,876MT=0, MR=0.05, MI=1548956,63055827,1415594,529MT=1, MR=0.40, MI=1546041,41654705,6724620,251MT=0, MR=0.05, MI=540079,50746459,5475301,405MT=1, MR=0.40, MI=548661,99754604,3993207,818MT=0, MR=0.10, MI=240355,65446811,3174671,882MT=2, MR=0.01, MI=244005,45952233,3116203,747MT=0, MR=0.10, MI=1046765,12755515,5874599,817MT=2, MR=0.01, MI=1050810,97557383,1924752,537MT=0, MR=0.10, MI=1547207,68452014,0784626,456MT=2, MR=0.01, MI=1544926,93452888,6023982,791MT=0, MR=0.10, MI=556928,67360909,0243266,486MT=2, MR=0.01, MI=543645,02850901,1065238,565MT=0, MR=0.40, MI=241540,47046590,1984894,845MT=2, MR=0.05, MI=249959,85755518,2845472,687MT=0, MR=0.40 MI=1045410,65853645,7164960,128MT=2, MR=0.05, MI=1051736,32962136,4576177,860MT=0, MR=0.40, MI=1549511,15953964,2203839,889MT=2, MR=0.05, MI=1548048,71957350,7485234,649MT=0, MR=0.40, MI=549983,42753905,6033265,335MT=2, MR=0.05, MI=543096,14750628,6877155,731MT=1, MR=0.01, MI=263494,76364043,462350,856MT=2, MR=0.10, MI=270763,44573702,8542774,557MT=1, MR=0.01, MI=1058124,13862066,3604194,507MT=2, MR=0.10, MI=1046612,62551832,5074416,433MT=1, MR=0.01, MI=1553912,54458640,5744411,781MT=2, MR=0.10, MI=1545523,32350852,6764438,879MT=1, MR=0.01, MI=543427,51148630,3065504,416MT=2, MR=0.10, MI=552091,56457823,1823014,238MT=1, MR=0.05, MI=249480,88149870,6245751,019MT=2, MR=0.40, MI=248312,17354232,3836997,833MT=1, MR=0.05, MI=1041201,48348559,5515957,630MT=2, MR=0.40, MI=1046408,77152875,3033739,149MT=1, MR=0.05, MI=1544122,01449935,0793517,734MT=2, MR=0.40, MI=1546437,70256239,8936986,506MT=1, MR=0.05, MI=543731,88651337,6506528,667MT=2, MR=0.40, MI=543413,18753022,9545995,418Table 12
Comparison of optimum designs, critical deflection, and stress values for BGAwEIS (planar truss with 200-bars).
ReferencesPonterosso and Fox [104]SGAMPGABGAwEISMinimum weight35394.00122047.1440079.50733405.949Max. displacement. inx, y and z directions:0.4959, 0.2745, 0.4998 at node1 for Case 1;0.0948, 0.4738, 0.0428 at node1 for Case 2Max. element Stress:20.9676 at element 90 for Case 1;6.4872 at element 185 for Case 2Figure 18
Convergence history of feasible solutions obtained by use of parameters sets proposed for MPGA (planar truss with 200-bars).Figure 19
Activated numbers of each population obtained by MPGA (planar truss with 200-bars).


## 7. Discussion

In
this section, BGAwEIS, MPGA, and SGA are evaluated, considering the effect of
different parameter sets on the quality degree of optimal designations and then
their performance is investigated taking into account the exploration and
exploitation features of genetic search. However, due to fact that evolutionary
parameters of SGA are fixed for design examples, the evaluation of SGA is
skipped here.For ease of presentation, the values of the parameters discussed below are presented using a vector-like notation like (•,•,•) where the 1st, 2nd,
and 3rd value within the parentheses correspond to examples 1, 2, and
3, respectively.


### 7.1. Consideration of Variation in the Values of Evolutionary Parameters of GAwEIS and MPGA

Regarding with BGAwEIS
(i) Generation number and population size
are proportional toR1. In this work, R1 has the values (3.75, ∞,
0.15) (Tables 3, 7, and 10). Corresponding to R1
value set, generation number is specified as (400, 600, 200), and
population size as (300, 500, 150).
(ii) There is a direct proportionality between R1 and output NFS computed using the feasible
solution pool (Tables3,  7, and 10). For example, considering the parameter sets
with higher performance, the values of R1 corresponding to the value of the
output NFS set of (18, 11, 19)
are equal to (3.75, ∞,
0.15).
(iii) The
best optimal designations are obtained when using parameters NGGES and NSBS set (25, 15),
(40, 15), and (40, 5) for each design example.The
gradual exploration strategy is activated by parameter NGGES. The parameters NSBS and NSAS are indicative of the activated frequency of gradual
exploration strategy (Tables3, 7, and 10). Considering parameter value sets
with better performance, the value set of output NSAS corresponding to the value set of parameter NGGES (25, 40, 40) are (1, 2, 1). The values of
parameter NSBS
are gradually decreased and reach the value of parameter NSAS eventually. This shows that evolutionary
search is successfully completed after gradually enlarging of the bounds of
subsolutions regions. Moreover, R2 also indicates about the activated frequency
of the gradual exploration strategy. The output NFS is proportionally increased by the activation of gradual
exploration strategy. In this regard, if R2 is close to or higher than R3, then
the gradual exploration strategy is successfully applied, that is, a feasible
solution is obtained once the bounds of subsolution regions are enlarged. For
example, an R2 value set (27, 40, 40) for Cases IV, III, and III of design
examples 1, 2, and 3 corresponds to an R3 value set (22, 55, 11). In design
example 2, although the value of R2 is lower than R3, the value of output NSAS is obtained as 2. This indicates that the
bounds of subsolution regions can be further enlarged.Considering
the convergence history of feasible fitness values corresponding to the four
cases of each design example, the success of parameter sets is also confirmed
by consistently decreased trend lines in Figures9, 13, and 17.Regarding with MPGA
(i) In the
sensitivity analysis of basic parameters of MPGA, a total of 48 parameter
sets composed of various values of parameters MT, MI, and MR are
considered. The parameter values with higher performance for each design
examples are indicated by a dark-shaded box and obtained as (1, 0.10, 10),
(1, 0.10, 10), and (0, 0.05, 5), each of which is denoted by MT, MI, and MR,
respectively (see Tables5, 8, and 11). These results show that migration
interval and rates varies proportionally with the generation numbers and
population size. For example, the migration interval (10, 10, 5) increases
with the generation number (400, 600, 200). This indicates that the rate
of migration interval to generation numbers varies within a range of
(0.015 (or 10/600)–0.025
(or 5/200) or % (1.5–2.5).  Migration rate varies within a range % (0.05–0.10)
of SP. It appears that the
number of migrated individuals is increased with the population size. For
example, the number of migrated individuals is (30, 50, 10) where the
migration rates are (0.10, 0.10, 0.05) and population sizes (300, 500,
200). The migration topologies referred to as unrestricted and dominantly
neighborhood perform well.
(ii) Considering
the lower and upper values of statistical data of results obtained by
parameter sets, several parameter sets are chosen and indicated by shaded
boxes (see Tables5, 8, and 11). The convergence history of feasible
fitness values corresponding to parameter sets chosen are shown in Figures
10, 14, and 18. The trend lines are consistently decreased for design
example 1, but inconsistently for design example 2 and 3. Particularly, it
is observed that the feasible solutions are generated in the beginning of
evolutionary search. Then, generation of feasible solutions stagnates at
the remaining generation numbers in design examples 2 and 3.
(iii) The activated numbers of populations obtained by use of parameter sets chosen
are shown by bars in Figures11, 15, and 19. From these figures, it is
obvious that the distribution of activated numbers of populations
corresponding to these parameter sets with higher performance is more
regular.


### 7.2. Performance Investigation of BGAwEIS and MPGA in Design Optimization

Considering the various  parameter
sets proposed for BGAwEIS and MPGA, a parameter set with high performance is
determined for each algorithm. The results obtained with these parameter sets
are to be examined according to exploration and exploitation features of
genetic search discussed previously and the quality of existing optimal
solutions outlined in literature.(i) The exploration and exploitation features of genetic
search cause a lower and higher increase in the fitness values,
respectively. This is easily confirmed for BGAwEIS by observing the change
in fitness values obtained for design example 1 (see Table4). While the
difference between the first and second feasible fitness values is equal
to 1.11 (or 624.71–623.60),
it increases to 27.08 (or 565.82–538.74)
for the fourth and fifth fitness values. This issue is also observed in
Figure 9, considering the trend lines corresponding to the best parameter
set denoted by Case IV. The exploration is dominant both in the beginning
and towards the end of search (after generation number 200). The
exploitation becomes dominant within a certain interval between generation
numbers 40 and 200. A balance between exploration and exploitation is relatively established for
Case III in design example 2 (Figure 13). In Case III of design example 3,
exploration is dominant in the beginning of evolutionary search, but then
exploitation begins to control the evolutionary search (Figure 17).  Starting with a lower fitness values for the first generation causes a
decrease in the mean and standard deviation of feasible fitness values
(see Figure 9, 13, and 17 along with Tables 3, 7, and 10).MPGA is managed by a migration dominated evolutionary
process. Considering the parameter sets with high performance, the dominancy of
exploration and exploitation is consistently observed at different generation
numbers in design example 1 that has a small number of bars and nodes, but
variably for design examples 2 and 3 where an increased number of bars and
nodes are present. Especially, evolutionary search ends up with stagnation
while searching the feasible solutions in design example 2 and 3 (see Figures10, 14, and 18). As in BGAwEIS, the initialization of evolutionary search with
higher fitness values causes an increase in the mean and standard deviations of
feasible fitness values (see Figures
10, 14, and 18 along with Tables 5, 8, and
11).(ii) Investigating the optimal designations obtained by
BGAwEIS, MPGA, SGA, and existing solution methods outlined in literature,
it can be said that BGAwEIS is more efficient in improving the quality of
optimal designations (see Tables6, 9, and 12). Particularly, in order to
cope with the complexity arising from the increase in the truss elements
and nodes, the values of parameter NSBS
associated with the value of parameters NGGES and NG are elevated. MPGA improves the quality of optimal
designations using unrestricted and dominantly neighborhood migration
topologies along with a migration interval about % (1.5–2.5) of NG and a migration rate about % (0.05–0.10) of SP.


## 7.1. Consideration of Variation in the Values of Evolutionary Parameters of GAwEIS and MPGA

Regarding with BGAwEIS
(i) Generation number and population size
are proportional toR1. In this work, R1 has the values (3.75, ∞,
0.15) (Tables 3, 7, and 10). Corresponding to R1
value set, generation number is specified as (400, 600, 200), and
population size as (300, 500, 150).
(ii) There is a direct proportionality between R1 and output NFS computed using the feasible
solution pool (Tables3,  7, and 10). For example, considering the parameter sets
with higher performance, the values of R1 corresponding to the value of the
output NFS set of (18, 11, 19)
are equal to (3.75, ∞,
0.15).
(iii) The
best optimal designations are obtained when using parameters NGGES and NSBS set (25, 15),
(40, 15), and (40, 5) for each design example.The
gradual exploration strategy is activated by parameter NGGES. The parameters NSBS and NSAS are indicative of the activated frequency of gradual
exploration strategy (Tables3, 7, and 10). Considering parameter value sets
with better performance, the value set of output NSAS corresponding to the value set of parameter NGGES (25, 40, 40) are (1, 2, 1). The values of
parameter NSBS
are gradually decreased and reach the value of parameter NSAS eventually. This shows that evolutionary
search is successfully completed after gradually enlarging of the bounds of
subsolutions regions. Moreover, R2 also indicates about the activated frequency
of the gradual exploration strategy. The output NFS is proportionally increased by the activation of gradual
exploration strategy. In this regard, if R2 is close to or higher than R3, then
the gradual exploration strategy is successfully applied, that is, a feasible
solution is obtained once the bounds of subsolution regions are enlarged. For
example, an R2 value set (27, 40, 40) for Cases IV, III, and III of design
examples 1, 2, and 3 corresponds to an R3 value set (22, 55, 11). In design
example 2, although the value of R2 is lower than R3, the value of output NSAS is obtained as 2. This indicates that the
bounds of subsolution regions can be further enlarged.Considering
the convergence history of feasible fitness values corresponding to the four
cases of each design example, the success of parameter sets is also confirmed
by consistently decreased trend lines in Figures9, 13, and 17.Regarding with MPGA
(i) In the
sensitivity analysis of basic parameters of MPGA, a total of 48 parameter
sets composed of various values of parameters MT, MI, and MR are
considered. The parameter values with higher performance for each design
examples are indicated by a dark-shaded box and obtained as (1, 0.10, 10),
(1, 0.10, 10), and (0, 0.05, 5), each of which is denoted by MT, MI, and MR,
respectively (see Tables5, 8, and 11). These results show that migration
interval and rates varies proportionally with the generation numbers and
population size. For example, the migration interval (10, 10, 5) increases
with the generation number (400, 600, 200). This indicates that the rate
of migration interval to generation numbers varies within a range of
(0.015 (or 10/600)–0.025
(or 5/200) or % (1.5–2.5).  Migration rate varies within a range % (0.05–0.10)
of SP. It appears that the
number of migrated individuals is increased with the population size. For
example, the number of migrated individuals is (30, 50, 10) where the
migration rates are (0.10, 0.10, 0.05) and population sizes (300, 500,
200). The migration topologies referred to as unrestricted and dominantly
neighborhood perform well.
(ii) Considering
the lower and upper values of statistical data of results obtained by
parameter sets, several parameter sets are chosen and indicated by shaded
boxes (see Tables5, 8, and 11). The convergence history of feasible
fitness values corresponding to parameter sets chosen are shown in Figures
10, 14, and 18. The trend lines are consistently decreased for design
example 1, but inconsistently for design example 2 and 3. Particularly, it
is observed that the feasible solutions are generated in the beginning of
evolutionary search. Then, generation of feasible solutions stagnates at
the remaining generation numbers in design examples 2 and 3.
(iii) The activated numbers of populations obtained by use of parameter sets chosen
are shown by bars in Figures11, 15, and 19. From these figures, it is
obvious that the distribution of activated numbers of populations
corresponding to these parameter sets with higher performance is more
regular.


## 7.2. Performance Investigation of BGAwEIS and MPGA in Design Optimization

Considering the various  parameter
sets proposed for BGAwEIS and MPGA, a parameter set with high performance is
determined for each algorithm. The results obtained with these parameter sets
are to be examined according to exploration and exploitation features of
genetic search discussed previously and the quality of existing optimal
solutions outlined in literature.(i) The exploration and exploitation features of genetic
search cause a lower and higher increase in the fitness values,
respectively. This is easily confirmed for BGAwEIS by observing the change
in fitness values obtained for design example 1 (see Table4). While the
difference between the first and second feasible fitness values is equal
to 1.11 (or 624.71–623.60),
it increases to 27.08 (or 565.82–538.74)
for the fourth and fifth fitness values. This issue is also observed in
Figure 9, considering the trend lines corresponding to the best parameter
set denoted by Case IV. The exploration is dominant both in the beginning
and towards the end of search (after generation number 200). The
exploitation becomes dominant within a certain interval between generation
numbers 40 and 200. A balance between exploration and exploitation is relatively established for
Case III in design example 2 (Figure 13). In Case III of design example 3,
exploration is dominant in the beginning of evolutionary search, but then
exploitation begins to control the evolutionary search (Figure 17).  Starting with a lower fitness values for the first generation causes a
decrease in the mean and standard deviation of feasible fitness values
(see Figure 9, 13, and 17 along with Tables 3, 7, and 10).MPGA is managed by a migration dominated evolutionary
process. Considering the parameter sets with high performance, the dominancy of
exploration and exploitation is consistently observed at different generation
numbers in design example 1 that has a small number of bars and nodes, but
variably for design examples 2 and 3 where an increased number of bars and
nodes are present. Especially, evolutionary search ends up with stagnation
while searching the feasible solutions in design example 2 and 3 (see Figures10, 14, and 18). As in BGAwEIS, the initialization of evolutionary search with
higher fitness values causes an increase in the mean and standard deviations of
feasible fitness values (see Figures
10, 14, and 18 along with Tables 5, 8, and
11).(ii) Investigating the optimal designations obtained by
BGAwEIS, MPGA, SGA, and existing solution methods outlined in literature,
it can be said that BGAwEIS is more efficient in improving the quality of
optimal designations (see Tables6, 9, and 12). Particularly, in order to
cope with the complexity arising from the increase in the truss elements
and nodes, the values of parameter NSBS
associated with the value of parameters NGGES and NG are elevated. MPGA improves the quality of optimal
designations using unrestricted and dominantly neighborhood migration
topologies along with a migration interval about % (1.5–2.5) of NG and a migration rate about % (0.05–0.10) of SP.


## 8. Conculusion

In this work, a new genetic
algorithm method, namely, (BGAwEIS) is presented to be used withthe design optimization of pin-jointed
structures. In order to evaluate the capability and efficiency of BGAwEIS, the
optimal designations obtained by SGA and solution methods outlined in
literature are not only examined but also an MPGA is proposed to assess the
influence of multiple populations on the quality of optimal designations. The
tests are performed on three design examples having 25, 72, and 200 bars. The
following conclusions are drawn from the results of design examples considered.(i) It is shown that bipopulation approach
proposed by BGAwEIS achieves effective usage of exploration and exploitation
features of genetic search simultaneously compared to MPGA with multiple
populations. Particularly, it is shown that the gradual exploration strategy
has a significant impact on BGAwEIS’ performance causing an increase in the
values of NSBS
with respect to NGGES
and NG. It is displayed that MPGA is able to improve quality of its optimal
designations by use of migration topologies called unrestricted and dominantly
neighborhood along with a migration interval about % (1.5–2.5) of generation
numbers and a migration rate about % (0.05–0.10) of population size. Furthermore, the activated numbers of populations obtained by use of these
parameter sets are shown to be more homogeneous compared to other ones.(ii) Although it is shown that MPGA is
successful in providing an equal distribution of activated frequencies for each
population, it has difficulties in directing the evolutionary search for
exploration of new solution regions because purely using the migration process
causes the certain individuals to be dominant during evolutionary search. This
negativity leads to stagnation on the generation of promising individuals. However, considering MPGA ability of using multiple populations with different
parameters, it is possible to improve its performance by the implementation of
genetic operations proposed by BGAwEIS.(iii)
It is demonstrated that BGAwEIS is able to
obtain more convergent results compared to existing methods outlined in
literature and optimal results obtained by MPGA and SGA.(iv) The search in BGAwEIS is initiated with
either a randomized or a user-defined population. Although a randomized
population is used in this work, it is noted that the utilization of the
user-defined population provides an advantage in the search for the offspring
on the promising subsolution regions.(v) The comparison of BGAwEIS, MPGA, and SGA is
carried out by keeping several evolutionary parameters within certain limits. If population size and generation number is increased thereby assigning
different values for the evolutionary parameters of these proposed algorithms,
it is possible to improve quality of optimal designations.In the future, the efficiency of BGAwEIS will be investigated
thereby carrying out the several applications as follows.(i) Statistical tests,
such as parametric or nonparametric tests of hypotheses and variance analysis,
will be performed for evaluation of the results generated by BGAwEIS. Thus, the
best combination of parameter values will be determined considering the optimal
designations with more convergent thereby including the decisions about the
population distributions.(ii) The possibilities
used in extraction or insertion-based transmission and recreation of core
population will be arranged for a self-adaptive usage.(iii) MPGA will be
modified to implement the main components of BGAwEIS. Moreover, the parallel
and hybrid models of this improved algorithm will be also proposed to observe
how the quality degree of optimal designations varies.



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*Source: 615162-2009-07-12.xml*