^{1}

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.

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 [

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 [

SGA

If

for

If required,

If

end

SGA 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. Figure

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 [

This paper is organized as follows. The next section presents a
background concerned existing design optimization approaches; Sections

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.

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 [

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 [

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 [

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[

ES uses a population of tentative design solutions and generates the populations using several genetic operators with self-adaptive parameters (Back and Schwefel [

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.

Hajela [

Camp et al. [

One of the alternative approaches to
the penalty function is artificial immune-inspired model (Garrett [

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 [

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 [

(a)

(b)

(c)

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 [

(d)

The
cellular GA is also successfully used in the hybridization models (Martin et
al. [

Sakamoto and Oda [

Adeli and Cheng [

Tanimura et al. [

Kaveh and Shahrouzi [

Karakasis et al. [

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

Here,

The violation of constraint is penalized. The penalization process is
used to obtain a penalty value. Thus, the fitness value

The values of the constants in the calculation of the penalty
value

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.

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 [

The design constraints may
increase the complexity of the search in the solution region (Eiben and Ruttkay
[

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.

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.

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 (

The main elements of BGAwEIS are described in the following sections. An example which clarifies how BGAwEIS works is also included.

BGAwEIS completes one generation after five interdependent procedures
with two populations within a core population (see the pseudocode in Algorithm

The populations are called inward population

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 vector

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 by

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.

BGAwEIS (

Initialize (

If required,

for

If required,

end

MPGA (

Initialize (

for

end

Visualization of the one-dimensional subsolution regions.

In this process,
the individuals that come from the core population are re-generated in order to
generate inward

In the generation of the outward population, the individuals taken from
core population are regenerated by diverging them to the bounds

Graphical depiction
of the possibilities for the location of best feasible solution (

An
application of extraction-based transmission on two-dimensional solution region
represented by two design variables is graphically shown in Figure

Display of the extraction-based transmission.

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 populations

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 Figure

Insertion-based transmission.

(C1) The number of feasible solutions in
inward population

(C2) The number of feasible solutions in
inward population

(C3) The number of feasible solutions in
outward population

(C4) The number of feasible solutions in
outward population

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

collect (

rank the collected feasible solutions in a descending order of their fitness values, and then store it in a dummy column matrix;

if the number of
individuals in the combination of “C1+C3,” “C1+C4,” “C2+C3,” and “C2+C4”
is greater than SP,

An algorithm is developed for the recreation of core

Graphical depiction of possibilities for the location of the interval (

The algorithm for the recreation of the core population

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 [

Multipoint mutation and crossover operators, and the other selection operators (stochastic universal sampling and stochastic remainder sampling),

the generation gap against genetic drift problem,

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 Table

A preliminary demonstration of GAwEIS to a planar truss with two-bars.

Indiv. | Section properities | NSBS | DVN | VNDV | SN | |||||
---|---|---|---|---|---|---|---|---|---|---|

Continous design variables | Indiv. (1) | [0.32 0.56] | 4 | 0.1 | 1 | 1 | 0.32 | 1 | 0.100 | 0.325 |

2 | 0.56 | 3 | 0.551 | 0.775 | ||||||

Indiv. (2) | [0.82 0.24] | 4 | 0.1 | 1 | 1 | 0.82 | 4 | 0.776 | 1.000 | |

2 | 0.24 | 1 | 0.100 | 0.325 | ||||||

Discrete design variables | Indiv. (1) | [001011] | 4 | 1 | 5 | 1 | 1 | 1 | 0.100 | 0.325 |

2 | 3 | 3 | 0.551 | 0.775 | ||||||

Indiv. (2) | [100001] | 4 | 1 | 5 | 1 | 4 | 4 | 0.776 | 1.000 | |

2 | 1 | 1 | 0.100 | 0.325 |

NSBS: 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.

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 [

The first step is the initialization
of

Ranking process governed by ranking
parameter

Following the ranking process,
evolutionary approaches are repeatedly executed in a loop until a predefined
loop number

After ordering subpopulation by taking
into account the fitness values, SGA operators (selection

The competition process aims to move the robust
individuals to other subpopulations that exhibit relatively poor performance
(Schlierkamp-Voosen and Mühlenbein [

The transmission of immigrants to
the other subpopulations is accomplished by a migration process. The migration
process is regulated with parameter

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
parameters

Parameter set and related values proposed for SGA, BGAwEIS, and MPGA.

Algorithm name | SGA | BGAwEIS | MPGA | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Population no. | 1 | 1 | 2 | 3 | 4 | 5 | ||||

Example 1 | 300 | 300 | 60 | 60 | 60 | 60 | 60 | |||

Population size | Example 2 | 500 | 500 | 100 | 100 | 100 | 100 | 100 | ||

Example 3 | 150 | 150 | 40 | 40 | 40 | 40 | 40 | |||

Operations | Variable type | Operator parameter name | ||||||||

Selection | All variables | Stochastic universal sampling | Insertion rate | 0.50 | 0.50 | 0.80 | 0.60 | 0.50 | 0.40 | 0.30 |

Pressure | — | — | 1.90 | 1.70 | 1.50 | 1.30 | 1.10 | |||

Ranking method | — | — | NL | NL | L | |||||

Crossover | Discrete variables | Single-point | Rate | 0.80 | 0.80 | 0.90 | 0.70 | 0.50 | 0.30 | 0.10 |

Continuous variables | Real type | Rate | 0.80 | 0.80 | 0.90 | 0.70 | 0.50 | 0.30 | 0.10 | |

Mutation | Discrete variables | Single-point | Rate | 0.70 | 0.70 | 1.00 | 0.80 | 0.60 | 0.40 | 0.20 |

Continuous variables | Real type | Rate | 0.7 | 0.70 | 0.100 | 0.80 | 0.60 | 0.40 | 0.20 | |

Range | 0.50 | 0.50 | 0.80 | 0.40 | 0.20 | 0.08 | 0.01 | |||

Competition | All variables | Competition of subpop. | Interval | — | — | 20 | 20 | 20 | 20 | 20 |

Rate | — | — | 0.05 | 0.06 | 0.07 | 0.08 | 0.10 | |||

Generation gap | All variables | — | — | 0.70 | 0.70 | 1.90 | 1.70 | 1.50 | 1.30 | 1.10 |

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 parameters

Introductions of the migration topologies for MPGA with five populations (Pop.) (a) ring-shaped topology, (b) neighborhood topology, and (c) unrestricted topology.

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 Table

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:

While

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.

This design problem is widely employed for the evaluation of various
optimization methods (Figure

Design and evolutionary data for BGAwEIS (spatial truss with 25-bars).

Design data | |||||
---|---|---|---|---|---|

Modulus of elasticity: | |||||

Density of material: 0.1 lb/ | |||||

Case number | Joint number | X (kips) | Y (kips) | Z (kips) | |

1 | 1 | 1 | −10 | −10 | |

2 | 0 | −10 | −10 | ||

3 | 0.5 | 0 | 0 | ||

6 | 0.6 | 0 | 0 | ||

Displacement constraints: | |||||

Stress constraints: | |||||

Elements of discrete sets
and their position number for | |||||

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 data | |||||

Number of design variables: 8 | |||||

Size of solution region: 30 | |||||

Number of generation: 400 | |||||

Size of inward population: 300 | |||||

Size of outward population: 300 | |||||

Size of core population: 300 | |||||

Cases | |||||

Case I | Case II | Case III | Case IV | ||

NGGES | 50 | 20 | 20 | 25 | |

NSBS | 20 | 50 | 20 | 15 | |

NSAS | 13 | 34 | 1 | 1 | |

NFS | 7 | 6 | 10 | 18 | |

Ratio 1 | 3.75 | 3.75 | 3.75 | 3.75 | |

Ratio 2 | 20 | 8 | 20 | 27 | |

Ratio 3 | 57 | 67 | 40 | 22 | |

Best feasible fitness value | 571.618 | 592.656 | 515.845 | 485.90 | |

Mean of feasible fitness values | 659.771 | 619.972 | 587.133 | 521.678 | |

Standard deviation of feasible fitness values | 59.591 | 18.856 | 67.096 | 45.880 |

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 Table

Variation of feasible fitness values according to design variables for spatial truss with 25-bars.

Fitness values | Design variable groups | |||||||
---|---|---|---|---|---|---|---|---|

1 | 2–5 | 6–9 | 10-11 | 12-13 | 14–17 | 18–21 | 22–25 | |

Feasible design variable values obtained from different generations | ||||||||

624.71 | 7 | 29 | 20 | 5 | 5 | 24 | 1 | 29 |

623.60 | 28 | 20 | 29 | 26 | 10 | 12 | 11 | 26 |

575.49 | 27 | 14 | 28 | 7 | 18 | 15 | 5 | 29 |

565.82 | 22 | 17 | 26 | 20 | 17 | 10 | 7 | 29 |

538.74 | 24 | 9 | 30 | 2 | 21 | 13 | 4 | 29 |

523.87 | 1 | 1 | 29 | 5 | 17 | 10 | 12 | 30 |

516.84 | 1 | 10 | 29 | 3 | 12 | 10 | 7 | 30 |

509.60 | 1 | 10 | 29 | 3 | 12 | 9 | 7 | 30 |

502.30 | 1 | 10 | 29 | 3 | 12 | 9 | 7 | 30 |

495.11 | 1 | 10 | 29 | 3 | 12 | 9 | 5 | 30 |

493.80 | 1 | 12 | 29 | 1 | 11 | 9 | 4 | 30 |

492.63 | 1 | 2 | 30 | 2 | 19 | 10 | 7 | 30 |

491.13 | 1 | 2 | 30 | 2 | 18 | 10 | 7 | 30 |

489.63 | 1 | 2 | 30 | 2 | 17 | 10 | 7 | 30 |

488.13 | 1 | 2 | 30 | 2 | 16 | 10 | 7 | 30 |

487.41 | 1 | 1 | 30 | 1 | 20 | 10 | 7 | 30 |

486.63 | 1 | 2 | 30 | 1 | 16 | 10 | 7 | 30 |

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 Table

Statistical analysis results of feasible fitness values obtained by use of parameter sets proposed for MPGA (spatial truss with 25-bars).

Parameter set | Best | Mean | Std | Parameter set | Best | Mean | Std |
---|---|---|---|---|---|---|---|

MT=0, MR=0.01, MI=2 | 502,104 | 577,038 | 67,707 | MT=1, MR=0.10, MI=2 | 497,679 | 553,204 | 52,961 |

MT=0, MR=0.01, MI=10 | 491,290 | 572,765 | 73,076 | ||||

MT=0, MR=0.01, MI=15 | 510,594 | 594,444 | 68,822 | MT=1, MR=0.10, MI=15 | 492,001 | 564,707 | 61,902 |

MT=0, MR=0.01, MI=5 | 501,384 | 589,606 | 68,165 | MT=1, MR=0.10, MI=5 | 490,972 | 582,356 | 73,748 |

MT=0, MR=0.05, MI=2 | 493,103 | 553,969 | 57,707 | MT=1, MR=0.40, MI=2 | 493,464 | 559,504 | 54,353 |

MT=0, MR=0.05, MI=10 | 495,631 | 564,716 | 59,424 | MT=1, MR=0.40, MI=10 | 494,279 | 568,882 | 63,133 |

MT=0, MR=0.05, MI=15 | 493,015 | 558,278 | 48,711 | MT=1, MR=0.40, MI=15 | 492,607 | 564,306 | 71,255 |

MT=0, MR=0.05, MI=5 | 492,631 | 559,883 | 58,218 | MT=1, MR=0.40, MI=5 | 488,096 | 561,872 | 54,620 |

MT=0, MR=0.10, MI=2 | 495,242 | 579,444 | 67,091 | MT=2, MR=0.01, MI=2 | 510,554 | 602,508 | 77,442 |

MT=2, MR=0.01, MI=10 | 488,320 | 574,907 | 72,809 | ||||

MT=0, MR=0.10, MI=15 | 504,839 | 585,483 | 63,716 | MT=2, MR=0.01, MI=15 | 505,944 | 589,977 | 68,911 |

MT=0, MR=0.10, MI=5 | 494,356 | 581,330 | 64,624 | MT=2, MR=0.01, MI=5 | 512,587 | 578,087 | 59,429 |

MT=0, MR=0.40, MI=2 | 491,887 | 546,261 | 54,684 | MT=2, MR=0.05, MI=2 | 493,299 | 574,085 | 44,792 |

MT=0, MR=0.40, MI=10 | 496,481 | 570,750 | 54,856 | MT=2, MR=0.05, MI=10 | 489,609 | 551,748 | 63,178 |

MT=0, MR=0.40, MI=15 | 511,631 | 583,177 | 65,891 | MT=2, MR=0.05, MI=15 | 493,907 | 558,576 | 53,586 |

MT=0, MR=0.40, MI=5 | 488,049 | 549,874 | 61,681 | MT=2, MR=0.05, MI=5 | 493,523 | 563,007 | 60,136 |

MT=1, MR=0.01, MI=2 | 520,044 | 583,861 | 70,871 | MT=2, MR=0.10, MI=2 | 491,125 | 572,506 | 60,291 |

MT=1, MR=0.01, MI=15 | 506,457 | 582,052 | 57,692 | MT=2, MR=0.10, MI=15 | 492,714 | 570,818 | 68,788 |

MT=1, MR=0.01, MI=5 | 506,226 | 602,789 | 60,236 | MT=2, MR=0.10, MI=5 | 511,718 | 584,622 | 60,603 |

MT=1, MR=0.05, MI=2 | 489,218 | 580,434 | 75,958 | MT=2, MR=0.40, MI=2 | 498,955 | 561,760 | 50,194 |

MT=1, MR=0.05, MI=10 | 507,234 | 556,624 | 53,246 | MT=2, MR=0.40, MI=10 | 492,850 | 574,766 | 63,040 |

MT=1, MR=0.05, MI=15 | 500,155 | 562,187 | 53,653 | MT=2, MR=0.40, MI=15 | 491,468 | 566,396 | 69,870 |

MT=1, MR=0.05, MI=5 | 491,397 | 561,556 | 56,276 | MT=2, MR=0.40, MI=5 | 501,608 | 577,688 | 51,228 |

Convergence history of feasible solutions obtained by use of parameters sets proposed for MPGA (spatial truss with 25-bars).

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 Table

Comparison of optimum designs, critical deflection, and stress values for BGAwEIS (spatial truss with 25-bars).

Ref. | Best weight | Design variable groups | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2–5 | 6–9 | 10-11 | 12-13 | 14–17 | 18–21 | 22–25 | ||

Rajeev and Krishnam. [ | 546.01 | 0.10 | 1.80 | 2.30 | 0.20 | 0.10 | 0.80 | 1.80 | 3.00 |

Zhu [ | 562.93 | 0.10 | 1.90 | 2.60 | 0.10 | 0.10 | 0.80 | 2.10 | 2.60 |

Erbatur et al. [ | 493.80 | 0.10 | 1.20 | 3.20 | 0.10 | 1.10 | 0.90 | 0.40 | 3.40 |

493.94 | — | — | — | — | — | — | — | — | |

491.72 | — | — | — | — | — | — | — | — | |

SGA | 814.64 | 0.10 | 3.00 | 2.80 | 2.40 | 2.20 | 1.90 | 2.80 | 2.40 |

MPGA | 486.29 | 0.10 | 0.50 | 3.40 | 0.10 | 1.50 | 0.90 | 0.60 | 3.40 |

^{**}Design variable groups are not presented in the references.

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 (Figure

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 Table

Design and evolutionary data for BGAwEIS (spatial truss with 72-bars).

Design data | |||||
---|---|---|---|---|---|

Modulus of elasticity: | |||||

Density of material: 0.1 lb/ | |||||

Case number | Joint number | X (kips) | Y (kips) | Z (kips) | |

1 | 1 | 5 | 5 | ||

2 | 1 | 0 | 0 | ||

2 | 0 | 0 | |||

3 | 0 | 0 | |||

4 | 0 | 0 | |||

Displacement constraints: | |||||

Stress constraints: | |||||

Evolutionary data | |||||

Number of design variables: 16 | |||||

Size of solution region: | |||||

Number of generation: 600 | |||||

Size of inward population: 500 | |||||

Size of outward population: 500 | |||||

Size of core population: 500 | |||||

Cases | |||||

Case I | Case II | Case III | Case IV | ||

NGGES | 50 | 20 | 40 | 15 | |

NSBS | 20 | 50 | 15 | 50 | |

NSAS | 8 | 35 | 2 | 23 | |

NFS | 8 | 7 | 11 | 12 | |

Ratio 1 | |||||

Ratio 2 | 30 | 12 | 40 | 12 | |

Ratio 3 | 75 | 85 | 55 | 50 | |

Best feasible fitness value | 427.753 | 452.286 | 380.784 | 381.08 | |

Mean of feasible fitness values | 694.574 | 842.722 | 891.234 | 882.993 | |

Standard deviation of feasible fitness values | 290.294 | 329.609 | 409.814 | 351.210 |

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) (Table

Statistical analysis results of feasible fitness values obtained by use of parameter sets proposed for MPGA (spatial truss with 72-bars).

Parameter set | Best | Mean | Std | Parameter set | Best | Mean | Std |
---|---|---|---|---|---|---|---|

MT=0, MR=0.01, MI=2 | 640,229 | 907,646 | 147,346 | MT=1, MR=0.10, MI=2 | 683,537 | 912,827 | 109,804 |

MT=0, MR=0.01, MI=10 | 734,629 | 913,598 | 113,454 | ||||

MT=0, MR=0.01, MI=15 | 711,954 | 913,401 | 126,805 | MT=1, MR=0.10, MI=15 | 598,955 | 952,984 | 124,680 |

MT=0, MR=0.01, MI=5 | 695,537 | 922,956 | 113,759 | MT=1, MR=0.10 MI=5 | 753,755 | 943,440 | 110,412 |

1 | MT=1, MR=0.40, MI=2 | 618,419 | 890,768 | 140,968 | |||

MT=0, MR=0.05, MI=10 | 760,033 | 917,419 | 107,376 | MT=1, MR=0.40, MI=10 | 693,387 | 935,066 | 116,045 |

MT=0, MR=0.05, MI=15 | 633,658 | 935,527 | 136,514 | MT=1, MR=0.40, MI=15 | 603,390 | 871,091 | 137,827 |

MT=0, MR=0.05, MI=5 | 723,858 | 898,141 | 114,732 | MT=1, MR=0.40, MI=5 | 717,659 | 948,091 | 118,160 |

MT=0, MR=0.10, MI=2 | 615,721 | 926,908 | 105,704 | MT=2, MR=0.01, MI=2 | 620,216 | 893,702 | 146,986 |

MT=0, MR=0.10, MI=10 | 689,895 | 891,536 | 148,994 | MT=2, MR=0.01, MI=10 | 622,736 | 902,485 | 148,753 |

MT=0, MR=0.10, MI=15 | 732,047 | 934,492 | 113,616 | MT=2, MR=0.01, MI=15 | 644,256 | 929,181 | 129,546 |

MT=0, MR=0.10, MI=5 | 696,988 | 904,547 | 115,401 | MT=2, MR=0.01, MI=5 | 656,687 | 921,091 | 119,474 |

MT=0, MR=0.40, MI=2 | 704,639 | 915,372 | 111,372 | MT=2, MR=0.05, MI=2 | 731,287 | 922,980 | 121,985 |

MT=0, MR=0.40, MI=10 | 673,424 | 874,583 | 125,405 | MT=2, MR=0.05, MI=10 | 729,340 | 941,674 | 116,160 |

MT=0, MR=0.40, MI=15 | 678,748 | 890,891 | 127,018 | MT=2, MR=0.05, MI=15 | 618,836 | 877,379 | 138,408 |

MT=0, MR=0.40, MI=5 | 682,176 | 916,081 | 127,354 | ||||

MT=1, MR=0.01, MI=2 | 669,352 | 943,534 | 122,819 | MT=2, MR=0.10, MI=2 | 690,889 | 867,153 | 149,447 |

MT=1, MR=0.01, MI=10 | 604,188 | 917,024 | 136,592 | ||||

MT=1, MR=0.01, MI=15 | 621,338 | 882,202 | 134,963 | MT=2, MR=0.10, MI=15 | 697,875 | 909,407 | 108,374 |

MT=1, MR=0.01, MI=5 | 667,580 | 895,065 | 133,153 | MT=2, MR=0.10, MI=5 | 638,491 | 870,591 | 133,380 |

MT=1, MR=0.05, MI=2 | 735,599 | 945,279 | 113,778 | MT=2, MR=0.40, MI=2 | 706,990 | 943,478 | 133,583 |

MT=1, MR=0.05, MI=10 | 625,002 | 902,000 | 121,989 | MT=2, MR=0.40, MI=10 | 718,084 | 939,940 | 130,951 |

MT=1, MR=0.05, MI=15 | 673,220 | 923,570 | 114,286 | MT=2, MR=0.40, MI=15 | 698,415 | 906,278 | 118,242 |

MT=1, MR=0.05, MI=5 | 679,263 | 932,444 | 122,463 | MT=2, MR=0.40, MI=5 | 695,139 | 944,974 | 125,174 |

Comparison of optimum designs, critical deflection, and stress values for BGAwEIS (spatial truss with 72-bars).

Design variables | References | |||||||
---|---|---|---|---|---|---|---|---|

Venkaya [ | Gellatly and Berke [ | Renwei and Peng [ | Schmit and Farshi [ | Erbatur et al. [ | SGA | MPGA | BGAwEIS | |

1–4 | 0.161 | 0.1492 | 0.1641 | 0.1585 | 0.161 | 0.873 | 0.675 | |

5–12 | 0.557 | 0.7733 | 0.5552 | 0.5936 | 0.544 | 1.681 | 0.253 | |

13–16 | 0.377 | 0.4534 | 0.4187 | 0.3414 | 0.379 | 0.100 | 0.601 | |

17-18 | 0.506 | 0.3417 | 0.5758 | 0.6076 | 0.521 | 1.418 | 0.437 | |

19–22 | 0.611 | 0.5521 | 0.5327 | 0.2643 | 0.535 | 0.986 | 0.841 | |

23–30 | 0.532 | 0.6084 | 0.5256 | 0.5480 | 0.535 | 1.530 | 0.861 | |

31–34 | 0.100 | 0.100 | 0.100 | 0.100 | 0.103 | 1.982 | 0.460 | |

35-36 | 0.100 | 0.100 | 0.100 | 0.1509 | 0.111 | 1.121 | 1.513 | |

37–40 | 1.246 | 1.0235 | 1.2893 | 1.1067 | 1.310 | 1.589 | 1.910 | |

41–48 | 0.524 | 0.5421 | 0.5201 | 0.5793 | 0.498 | 1.987 | 0.789 | |

49–52 | 0.100 | 0.100 | 0.100 | 0.100 | 0.110 | 1.083 | 0.132 | |

53-54 | 0.100 | 0.100 | 0.100 | 0.100 | 0.103 | 1.856 | 0.936 | |

55–58 | 1.818 | 1.464 | 1.9173 | 2.0784 | 1.910 | 0.268 | 1.840 | |

59–66 | 0.524 | 0.5207 | 0.5207 | 0.5034 | 0.525 | 1.473 | 0.899 | |

67–70 | 0.100 | 0.100 | 0.100 | 0.100 | 0.122 | 0.849 | 0.244 | |

17–72 | 0.100 | 0.100 | 0.100 | 0.100 | 0.103 | 1.469 | 0.183 | |

Best Weight | 381.28 | 395.97 | 379.66 | 388.65 | 383.120 | 1196.89 | 594.811 | |

Convergence history of feasible solutions obtained by use of parameters sets proposed for MPGA (spatial truss with 72-bars).

Activated numbers of each population obtained by MPGA (spatial truss with 72-bars).

The plane truss shown in Figure

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 Table

Design and evolutionary data for BGAwEIS (planar truss with 200-bars).

The design data | |||||
---|---|---|---|---|---|

Modulus of elasticity: | |||||

Density of material: 0.283 lb/ | |||||

Case number | Joint number | X (kips) | Y (kips) | Z (kips) | |

1 | 1,6,15,20,29,34,43,48,57, | 1 | 0 | 0 | |

62,71 | |||||

1,2,3,4,5,6,8,10,12,14,15, | 0 | 0 | |||

16,17,18,19,71,72,73,74,75 | |||||

Displacement constraints: | |||||

Stress constraints: | |||||

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 data | |||||

Number of design variables: 150 | |||||

Size of solution region: 30 | |||||

Number of generation: 200 | |||||

Size of inward population: 150 | |||||

Size of outward population: 150 | |||||

Size of core population: 150 | |||||

Cases | |||||

Case I | Case II | Case III | Case IV | ||

NGGES | 50 | 20 | 40 | 15 | |

NSBS | 20 | 50 | 5 | 15 | |

NSAS | 17 | 42 | 1 | 1 | |

NFS | 5 | 6 | 19 | 16 | |

Ratio 1 R1 | 0.15 | 0.15 | 0.15 | 0.15 | |

Ratio 2 R2 | 10 | 4 | 40 | 13 | |

Ratio 3 R3 | 40 | 33 | 11 | 13 | |

Best feasible fitness value | 37659.683 | 39480.881 | 33405.949 | 35128.000 | |

Mean of feasible fitness values | 146652.109 | 130204.259 | 182974.671 | 153277.013 | |

Standard deviation of feasible fitness values | 98588.652 | 88555.569 | 87725.056 | 59388.735 |

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) (Table

Statistical analysis results of feasible fitness values obtained by use of parameter sets proposed for MPGA (planar truss with 200-bars).

Parameter set | Best | Mean | Std | Parameter set | Best | Mean | Std |
---|---|---|---|---|---|---|---|

MT=1, MR=0.10, MI=2 | 43405,949 | 46677,338 | 6914,943 | ||||

MT=0, MR=0.01, MI=10 | 48093,574 | 56656,419 | 4216,594 | MT=1, MR=0.10, MI=10 | 46110,341 | 52166,191 | 4860,834 |

MT=0, MR=0.01, MI=15 | 46658,763 | 53303,681 | 4797,318 | MT=1, MR=0.10, MI=15 | 46797,205 | 53567,554 | 6498,843 |

MT=0, MR=0.01, MI=5 | 43563,417 | 49063,016 | 3737,226 | MT=1, MR=0.10 MI=5 | 45121,767 | 54525,702 | 6203,645 |

MT=0, MR=0.05, MI=2 | 47659,683 | 53501,963 | 4633,370 | MT=1, MR=0.40, MI=2 | 44316,220 | 48637,425 | 3729,784 |

MT=0, MR=0.05, MI=10 | 44562,844 | 52044,266 | 6328,921 | MT=1, MR=0.40, MI=10 | 43141,914 | 48152,488 | 5033,876 |

MT=0, MR=0.05, MI=15 | 48956,630 | 55827,141 | 5594,529 | MT=1, MR=0.40, MI=15 | 46041,416 | 54705,672 | 4620,251 |

MT=1, MR=0.40, MI=5 | 48661,997 | 54604,399 | 3207,818 | ||||

MT=0, MR=0.10, MI=2 | 40355,654 | 46811,317 | 4671,882 | MT=2, MR=0.01, MI=2 | 44005,459 | 52233,311 | 6203,747 |

MT=0, MR=0.10, MI=10 | 46765,127 | 55515,587 | 4599,817 | MT=2, MR=0.01, MI=10 | 50810,975 | 57383,192 | 4752,537 |

MT=0, MR=0.10, MI=15 | 47207,684 | 52014,078 | 4626,456 | MT=2, MR=0.01, MI=15 | 44926,934 | 52888,602 | 3982,791 |

MT=0, MR=0.10, MI=5 | 56928,673 | 60909,024 | 3266,486 | MT=2, MR=0.01, MI=5 | 43645,028 | 50901,106 | 5238,565 |

MT=0, MR=0.40, MI=2 | 41540,470 | 46590,198 | 4894,845 | MT=2, MR=0.05, MI=2 | 49959,857 | 55518,284 | 5472,687 |

MT=0, MR=0.40 MI=10 | 45410,658 | 53645,716 | 4960,128 | MT=2, MR=0.05, MI=10 | 51736,329 | 62136,457 | 6177,860 |

MT=0, MR=0.40, MI=15 | 49511,159 | 53964,220 | 3839,889 | MT=2, MR=0.05, MI=15 | 48048,719 | 57350,748 | 5234,649 |

MT=0, MR=0.40, MI=5 | 49983,427 | 53905,603 | 3265,335 | MT=2, MR=0.05, MI=5 | 43096,147 | 50628,687 | 7155,731 |

MT=2, MR=0.10, MI=2 | 70763,445 | 73702,854 | 2774,557 | ||||

MT=1, MR=0.01, MI=10 | 58124,138 | 62066,360 | 4194,507 | MT=2, MR=0.10, MI=10 | 46612,625 | 51832,507 | 4416,433 |

MT=1, MR=0.01, MI=15 | 53912,544 | 58640,574 | 4411,781 | MT=2, MR=0.10, MI=15 | 45523,323 | 50852,676 | 4438,879 |

MT=1, MR=0.01, MI=5 | 43427,511 | 48630,306 | 5504,416 | MT=2, MR=0.10, MI=5 | 52091,564 | 57823,182 | 3014,238 |

MT=1, MR=0.05, MI=2 | 49480,881 | 49870,624 | 5751,019 | MT=2, MR=0.40, MI=2 | 48312,173 | 54232,383 | 6997,833 |

MT=1, MR=0.05, MI=10 | 41201,483 | 48559,551 | 5957,630 | MT=2, MR=0.40, MI=10 | 46408,771 | 52875,303 | 3739,149 |

MT=1, MR=0.05, MI=15 | 44122,014 | 49935,079 | 3517,734 | MT=2, MR=0.40, MI=15 | 46437,702 | 56239,893 | 6986,506 |

MT=1, MR=0.05, MI=5 | 43731,886 | 51337,650 | 6528,667 | MT=2, MR=0.40, MI=5 | 43413,187 | 53022,954 | 5995,418 |

Comparison of optimum designs, critical deflection, and stress values for BGAwEIS (planar truss with 200-bars).

References | ||||
---|---|---|---|---|

Ponterosso and Fox [ | SGA | MPGA | ||

Minimum weight | 35394.00 | 122047.14 | 40079.507 | |

Convergence history of feasible solutions obtained by use of parameters sets proposed for MPGA (planar truss with 200-bars).

Activated numbers of each population obtained by MPGA (planar truss with 200-bars).

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 (

(i) Generation number and population size
are proportional to

(ii) There is a direct proportionality between R1 and output NFS computed using the feasible
solution pool (Tables

(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 (Tables

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 Figures

(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 Tables

(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 Tables

(iii) The activated numbers of populations obtained by use of parameter sets chosen
are shown by bars in Figures

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 Table

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 Figures

(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 Tables

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.

The position numbers corresponding to optimal design for example 3 are [10, 14, 13, 14, 6, 2, 2, 7, 7, 13, 5, 23, 9, 23, 1, 20, 1, 11, 13, 9, 2, 2, 17, 19, 18, 9, 15, 13, 12, 5, 4, 10, 15, 14, 7, 22, 17, 16, 21, 5, 3, 11, 7, 23, 6, 4, 8, 27, 15, 15, 13, 17, 21, 9, 26, 8, 8, 8, 14, 8, 6, 4, 19, 8, 15, 14, 4, 17, 17, 15, 21, 2, 17, 13, 8, 7, 17, 9, 7, 19, 9, 10, 4, 9, 6, 8, 16, 1, 13, 5, 22, 12, 7, 7, 5, 11, 3, 2, 1, 16, 17, 24, 10, 5, 20, 17, 2, 18, 7, 7, 14, 9, 15, 8, 1, 4, 8, 5, 5, 2, 8, 27, 1, 8, 17, 8, 19, 23, 23, 4, 7, 20, 9, 8, 4, 9, 7, 7, 12, 16, 15, 6, 16, 14, 1, 14, 6, 3, 16, 12, 20, 18, 15, 7, 3, 2, 6, 11, 3, 15, 10, 22, 8, 17, 14, 19, 17, 3, 18, 11, 15, 5, 17, 8, 20, 8, 18, 8, 4, 8, 20, 21, 6, 12, 3, 19, 16, 7, 17, 15, 11, 13, 13, 11, 11, 23, 22, 10, 18, 22;]

Density of steel

Length of member

Cross-sectional area

Member stress

Maximum allowable stress

Joint displacement

Maximum allowable displacement

Fitness value

Penalty value

Weight of truss system

Design vector

Penalty constants

Current generation number

Values of interval bounds

Upper bound of interval

Lower bound of interval

Fitness values of inward, outward and core populations

Parameters regarded with ranking mutation, crossover and selection operations

Inward population

Outward population

Core population

Lower bound of feasible solution pool

Upper bound of feasible solution pool

Upper bound of design variable

Lower bound of design variable

Best feasible design variable

Feasible solution pool used to collect feasible solutions

Design variable number

Number of design variables

Number of feasible solution collected in feasible solution pool

Number of generations for gradual exploration strategy

Number of generations

Number of subsolution regions after search

Number of subsolution regions (number of segment) before search

Size of population

Size of solution region

Number of subpopulations

Number of individuals contained each subpopulation

Value of each design variable

Subsolution region (segment) number.