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World Journal of Engineering
Review of algorithms of structural optimization with discrete variables
Yancang Li, , Beibei Heng, , Lingren Kong, , Weijuan Yang,
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Yancang Li, , Beibei Heng, , Lingren Kong, , Weijuan Yang, (2011) "Review of algorithms of structural optimization with
discrete variables", World Journal of Engineering, Vol. 8 Issue: 3, pp.231-236,
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World Journal of
World Journal of Engineering 8(3) (2011) 231-236
Review of algorithms of structural
optimization with discrete variables
Downloaded by California State University Fresno At 09:57 25 October 2017 (PT)
Yancang Li, Beibei Heng, Lingren Kong and Weijuan Yang
College of Civil Engineering, Hebei University of Engineering, Handan 056038, China
(Received 22 July 2010; accepted 17 December 2010)
Much work has been done on the optimization of the discrete variable structure design. In order
to handle the optimization problem effectively, the related theories, methods, and other beneficial
results were summarized. On the basis of analyzing the predecessors’ research, the development
direction was introduced. Then, some practical methods, including their improvements, for
discrete structural optimization were analyzed. Finally, the Ant Colony Optimization algorithms
were shown as promising methods. This work has significance in theory and practice for the
development of structural optimization.
Key words: Discrete variable structure, Optimization, Review, Ant colony algorithms
1. Introduction
In the actual structural design, not all of the
design variables can be handled as limited discrete
values (Guo, 2000). The usual optimization design
methods must deal with the results to meet the
engineering design specifications and various
technical standards, but the discrete results treated
by continuous variable optimization method is not
necessarily feasible or optimal (Zhang et al., 2003).
This situation has signified the study of related
optimization algorithms.
This paper presented the new results of the
structural discrete variable optimal design theory
including its advantages and disadvantages, and
pointed out its direction of development. This work
will give rise to the study of structural design
The paper is organized as follows. In section 2,
the characteristics of discrete structural optimization
were introduced. In the following part, attention
was paid to the methods of discrete structural
optimization. Then a prediction of discrete variable
structural design optimization was proposed. Finally,
the advantages of ant colony algorithm were
2. Characteristics of discrete structural
In the structural optimization, the objective
function and constraints of the mathematical model
of optimum design are discontinuous. As a result,
the ordinary methods, such as various gradient
algorithms of sensitivity analysis, K-T conditions,
and so on, are not valid.
The difficulties in structural optimization design
of discrete variables are as follows.
(1) Traditional mathematical analysis is not
valid. In traditional mathematics, the mixeddiscrete variable optimization problem
belongs to combinational optimization
Downloaded by California State University Fresno At 09:57 25 October 2017 (PT)
Yancang Li et al./World Journal of Engineering 8(3) (2011) 231-236
problem. With the increase of design
variables, the number of combination
increases exponentially, namely, called as
the “combination explosion” with the NPhard problems. The difficulty of NP-hard
problem is the number of combination
increasing rapidly with the increasing of
design variables (Li et al., 2010).
Shape optimization mainly includes section
variable optimization and shape (coordinates)
variable optimization. It is difficult to
process the two kinds of variable optimization
Topological optimization is similar to the
shape optimization. They both have a
topology between nodes and no-bar
connections. In dealing with different
types of optimization problems, optimal
topological solutions can’t be obtained
without the rods recovery strategy.
It is difficult to handle the layout
optimization, which deals with three nature
variables simultaneously: cross-section,
shape and topology optimization.
Little work has been done on the
optimization of structure type, which is a
high level of structural optimization.
3. Discrete structural optimization
Algorithms for discrete variable structural
optimization design can be divided into three types:
accurate algorithms, approximate algorithms and
heuristic algorithms. Sun introduced the traditional
methods of optimization design such as the
enumeration method, implicit enumeration method,
rounded method, branch and bound method, cut
plane method, the penalty function method, Bala’s
method, integer gradient method, etc. (Sun et al.,
2002). Here we mainly deal with the newly
developed heuristic algorithms.
3.1. Genetic algorithm (GA)
As a randomized, parallel and adaptive
optimization algorithm, GA is based on the idea of
“the survival of the fittest”. It includes selection,
crossover and mutation processes. The basic idea is
that the problem will be expressed by encoded
chromosomes, chromosome groups through
reproduction, crossover and mutation and other
operations, continuously evolved and converged to
solve the optimal problem. Chromosome length is
positively correlated with coding accuracy. When
selecting parameters the number of aspects should
be considered.
GA is a recent rise heuristic algorithm, especially
for the complex and nonlinear problems. Therefore,
it is more suitable for discrete optimization.
Burczynski optimized the continuous structures on
the topology optimization (Burczynski and Koko,
1998). Zhang and Wang employed it to optimize the
shape of the structure (Zhang and Wang, 2001).
The application results show the unique
advantage of GA in the structural optimization. But,
GA has the defects of too many times re-analysis of
structure, lower search efficiency and prematurity.
Many experts and scholars have extensively studied
and proposed various improved genetic algorithms,
such as layered GA, a restructuring big foreign
across generation mutant algorithm, self-adaptive
genetic algorithm, parallel genetic algorithm based
on niche genetic algorithm, and so on. These
improvements have been used successfully in the
3.2. Simulated annealing method
Simulated annealing is based on the principle of
annealing process of solids, with Monte Carlo
iterative solution strategy and an optimal algorithm.
The law comes from statistical mechanics. First, a
high temperature system is determined for faster
search, to define low-energy area. Declining with
temperature, accuracy of search for the optimal
solutions is also rising. Using the probability jump
to avoid the local optimal solution, the chance of
getting the global optimal solution is increased. The
temperature attenuation parameters and step of
optimization are both applied to the optimal speed
and precision (Shung et al., 2011). Because it does
not need any gradient information and Hessian
matrix, the algorithm is more suitable for
optimization design of discrete variables.
Much effort has been made to improve the
performance of the classic simulated annealing
algorithm. Du proposed mixed simulated annealing
algorithm, which can improve the speed of the
algorithm’s convergence (Du et al., 2001). Gao put
forward an improved and simulated annealing
algorithm (Gao et al., 2002), Gao combined the
simplex method with the simulated annealing
algorithm and proposed improvement measures,
giving the algorithm a better solution efficiency and
global optimization ability(Gao, 2007); Wu
Yancang Li et al./World Journal of Engineering 8(3) (2011) 231-236
Downloaded by California State University Fresno At 09:57 25 October 2017 (PT)
proposed a hybrid discrete variable optimization
method of simulated annealing (Wu, 1997).
3.3. Artificial neural network
As a heuristic search method, the Hopfield
Artificial Neural Network was successfully used in
structural optimization design (Hopfield, 1982).
Dhingra used Hopfield to solve the three rod nonstatically structural optimization and welding
structural optimization problem; Lu employed
Hopfield neural network model to solve the discrete
structural optimization problem (Lu, 1997). Now,
the chaotic cellular neural networks model, fuzzy
neural network model, and other improvements
have had been employed to the structural
optimization design.
3.4. Tabu search
Local search in the optimization process easily
falls into local optimal solution, and can’t get global
optimal solutions. The Tabu search algorithm was
put forward on the problem. Its main idea is that in
the search process, the searched local optimal
solutions are marked and will not be searched again
in the next search process, so as to ensure the
effective search path is different (Marcos et al.,
2008). Bland applied it to the optimization of truss
design (Bland, 1995). The method of Tabu search,
neural network and genetic algorithm were
commonly used in combination in order to
overcome the weakness of the basic algorithm (Luo
and Huo, 2005).
3.5. Ant colony algorithm
Biologists found that the ants in the foraging
process will release some special secretionspheromones. The ants within the scope can feel
the pheromones and tend to move toward the intensity
route. This behavior of ants is a kind of information
feedback phenomenon. Inspired by such biological
phenomena, Marco Dorigo, and other Italian scholars,
put forward the basic ACO (Ant Colony
Optimization) (Blum, 2005; Colorm et al., 1991). The
algorithm was successfully used to solve the famous
TSP (Traveling Salesman Problem) (Lao, 2009).
After ten years of development, some
improvements were proposed (Yang et al., 2009;
Hong et al., 2010; Dorigo et al., 2002; Zhou et al.,
2004). Numerical simulation results demonstrated
that the algorithm has many advantages, and it can
be used as a new heuristic method in solving the
combinatorial optimization problems (Balseiro et al.,
2011; Kaji, 2001; Karaboga et al., 2004).
In recent years, as stochastic optimization methods,
the ACO algorithms have been widely used in
single-objective optimization problems and multiobjective optimization problems. In the combinational
optimization it has gradually show its unique
advantages (Li and Li, 2007). ACO algorithms have
become very promising heuristic algorithms.
3.6. Other methods
At present, many scholars are engaged in the
structural optimization design of discrete variables
research. Guo proposed a full stress design method
based on GA (Guo, 2004). Zhang proposed unilateral
search algorithms (Zhang et al., 2005); Li proposed
a two-stage algorithm for structural optimization (Li
and Zhang, 2006). In 2004, the self-organization
behavior of search algorithm was applied to discrete
structural optimization (Li and Gong, 2004). In 2006,
the particle swarm algorithm was employed in the
discrete structural optimization (Wang and Liu,
2006). In 2007, the artificial fish colony optimization
algorithm was introduced into the application of
discrete structural optimization (Zhang et al., 2007).
Many scholars are trying to apply these intelligent
algorithms to the structural optimization design, but
we still have a long way to go.
4. Prediction of discrete variables in
structural optimization design
The direction of the discrete variables structure
optimization design can be shown as follows.
(1) Specific problem. How to reduce the
difference between the mathematic model
and the practical engineering structure is a
big problem for the structure optimization.
(2) Multi-objective problem. In reality, many
optimization problems have multiple targets
and contradictions. How to deal with the
interaction of the constraint is another
direction we should pay attention to.
(3) Structural optimization based on reliability.
Structural reliability has gradually become a
criterion of modern structure optimization
design. The structural optimization based on
the reliability will be an important research
(4) The study on the high-efficiency
optimization methods of handling the shape
(geometric), topology, layout optimization
simultaneity will be a promising direction
too (Jiang et al., 2007)
Yancang Li et al./World Journal of Engineering 8(3) (2011) 231-236
5. Conclusions
Structural optimization with discrete variables
has become a promising direction in civil
engineering. At present, much work has been done,
but further study is still needed. In order to deal with
this NP-hard problem more efficiently, we
summarized the common methods of the structural
optimization, especially the heuristic algorithms. As
a novel heuristic algorithm, the ant colony algorithm
will gradually show its advantage in this field.
Downloaded by California State University Fresno At 09:57 25 October 2017 (PT)
The work was supported by the Fund of Building
Technology of Hebei Province (2009–128),
Foundation of Hebei Educational Committee and
Hebei University of Engineering Funds for
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