COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Engineering Structural Optimization with an Improved Ant Colony
Algorithm
Yubing Gong *, Quanyong Li
Department of Electronic Machinery andTtraffic Engineering, Guilin University of Electronic Technology, Guilin,
541004 China
Email: [email protected], [email protected]
Abstract Ant colony optimization algorithm (ACO) is a newly developed random bionics algorithm in recent years.
The ACO algorithms have been applied, particularly starting from 1999 to several kinds of optimization problems as
the traveling salesman problem, sequential ordering, and management of communications networks and so on. In this
paper the ACO algorithms was applied to solve the engineering structural optimization problems. For gaining the
better optimal result, several parameters of the ACO algorithms had been improved. Three examples on structural
optimization were presented and solved by the improved ACO algorithms, Genetic Algorithm, Simulated Annealing
Algorithm and so on. The comparisons between ACO algorithms and other algorithm for the three examples have been
obtained in terms of efficiency and effectiveness. The comparisons show the improved ACO algorithm is a very
effective approach for solving structural optimization problems. And the improved ACO algorithm can greatly
reinforce the robustness of the optimal result.
Key words: ACO, Structural Optimization, Evolutionary Algorithm
INTRODUCTION
Engineering structural optimization has been and continues to be a large field of active research. Mathematical
programming (MP) based approach and optimality criterion methods have received great interest and application in
structural optimization successfully. Recently, some evolutionary algorithm, such as GA (Genetic Algorithm), ACO
(Ant Colony Optimization) algorithm has been adopted to effectively solve the structural optimization, which belongs
to soft computing techniques and can avoid the complexity of implementation of MP approach, such as the need of the
derivatives [1]. In our previous work [2], we have employed ACO to solve some specific structural optimization
problems. Even though the approaches could find the best solution in those simulated cases, the search efficiency
seemed not good enough. Especially, for some problems, the parameters in ACO algorithm must be elaborately tuned
in order to obtain the optimal result. Sometimes, the optimal result can not be found. Some reference had discussed
some principles on how to select those parameters. But generally, those parameters were pre-specified according to the
ones own experiments.
In this paper, the pheromone update rules of ACO algorithm were improved. Several engineering structural
optimizations were solved by the improved ACO algorithm. Simulation results were reported and the improved ACO
algorithm indeed has satisfying performance for tested problems.
THE FORMULATION OF ENGINEERING STRUCTURAL OPTIMIZATION
In this paper, the objective function of structural optimization model was weight. The model for space truss can be
stated as follows:
Find X to mininize
F(X )＝
∑ ρx l
(1)
i i
Subject to g j ( X ) ≤ 0
j = 1 , 2 L m . Li ≤ xi ≤ U i
⎯ 1019 ⎯
i = 1, 2Ln
Here X=｛x1, x2, …，xn｝, is the vector of design variable representing cross-sectional area, n is the number of the design
variables. And F(X) is the objective function, P is the density, li is the length of the i-th bar, gj(x) is the constraint
conditions and m is the number of constrain conditions. Ui is maximum area limit and Li is the lower area limit of the
i-th design variable.
THE BASICS OF ANT COLONY ALGORITHM
Ant colony algorithm is a class of heuristic search algorithms that have been successfully applied to solving NP hard
problem, such as the traveling salesman problem, sequential ordering, and management of communications networks
and so on. ACO algorithm is biologically inspired from the behavior of colonies of real ants, and in particularly how
they forage for food. It is an evolutionary approach where several generations of artificial ants in a cooperative way
search for good solutions. During the process of find feasible solutions, ants collect and store information in
pheromone trails. The pheromone will be evaporated over time and be released by ant in the search process. More
details can be found in Ref. [3]. The general procedure of ACO algorithm is stated as:
Procedure Set parameters, initialize pheromone, and so on
While (termination criterion was not met) do
Construct Solutions
Applying local search
/optional
Updating pheromone trail
End
End
In ACO [4], ants find solutions starting from a start node and moving to feasible neighbor nodes in the process of
construction solutions. During the process, information collected by ants is stored in the so-called pheromone trails.
An ant-decision rule, made up of pheromone and heuristic information, governs ants’ search toward neighbor nodes
stochastically. The kth ant at time t positioned on node r move to next node s with the rule governed by
⎧arg{max[τ ru (t )α ηruβ ]}
s=⎨
⎩S
u = allownk (t ), when q ≤ q0
otherwise
(2)
Where τ ru (t ) is the pheromone trail at time t, η ru is the problem-specific heuristic information, α is a parameter
representing the importance of pheromone information, β is a parameter representing the importance of heuristic
information, q is a random number uniformly distributed in [0, 1], q0 is a pre-specified parameter, allownk (t ) is the
set of feasible nodes currently not assigned by ant k at time t, and S is an index of node selected from allownk (t )
according to probability distribution given by
⎧ τ rs (t )α η rsβ
⎪⎪
β
Prsk (t ) = ⎨ ∑τ ru (t )η ru
u∈allownk ( t )
⎪
⎪⎩0
if s ∈ allown k (t ),
(3)
otherwise
Pheromone updating is a process of decreasing the intensities of pheromone trails over time. This process is used to
avoid locally convergence and to explore more search space. The pheromone intensities of pheromone trails are
updated by applying the updating rules of the formula bellow:
τ ij (t + 1) = (1 − ρ )τ ij (t ) + ρΔτ ij
⎧1 / L
Δτ ij = ⎨ best
⎩0
(4)
if (i, j ) ∈ BestTour
else
Here Lbest is the length of the global best tour up to now, p is the pheromone trail evaporation rate.
THE IMPROVED ANT COLONY OPTIMIZATION ALGORITHM
How to make use of pheromone trails and heuristic information is the key of ACO algorithm. In order to promote the
efficiency of pheromone trails, the pheromone updating rules is improved as followed:
⎯ 1020 ⎯
τ ij (t + 1) = (1 − ρ )τ ij (t ) + ρΔτ ij
⎧λ *CC*(1− ρ)cycle*(cons/ L)γ
Δτij = ⎨
else
⎩0
if (i, j) ∈ feasibleTo
ur L
(5)
Herein, CC is the initial intensity of pheromone trails, cycle is the number of iteration and L is the value of objective
function. λ、γ
are two parameters, which distribute in [0－1]. It is noted that all the feasible tour were taken into account to release the
pheromone, including the best tour. And the relation among the number of iteration, the initial intensity of pheromone
trail and the increment of pheromone is built. In addition, the penalty of object is also applied.
In our practice, the improved pheromone updating rules can greatly reduce the computed effort and help reinforce the
robustness of the optimal result.
Accordingly, the above ant-decision rule was slightly modified as following.
⎧u
s=⎨
⎩rand
when q ≤ Psuk (t )
(6)
r ∈ allownk (t )
r
NUMERICAL EXAMPLES
1. Example 1: 10 bar truss [3] The 10 bar cantilever truss problem have been used as a benchmark problem to verify
the efficiency of diverse optimization methods. The initial geometry of the structure is shown in Fig1, and the loading
conditions, material properties and constraints are given in Table 1.
Figure 1: 10 bar truss, initial geometry
Table 1 Loading and design condition for the 10 bar problem shown in Fig. 1
Node
FX (lbs)
FY (lbs)
FZ (lbs)
2
0
−100,000
0
4
0
−100,000
0
Constants
Young’s modulus=1e7 psi
Density=0.1 lbs/in.3
Area range=0.1－35 in.2
Allowable stress=(+/−)25000psi
Allowable nodal displacement in x and y directions= (+/−) 2 in.
Length a=360 in.
The optimal design given by ACO, improved ACO and other algorithm are compared and listed in Table 2.
In our studies, the continuous value field of the design variables is equally divided to discrete segment.
⎯ 1021 ⎯
It can be also seen from the Table 2 that the optimal result (5070.49) given by ACO algorithm is just with a weight
about 0.06% higher than the best design obtained (5067.3) by Ringertz in Ref.8 by now. And the improved ACO
algorithm performs as same as ACO algorithm.
Table 2 Comparison of the design variables and objective values for 10 bar truss
Conf.
This paper
Ref. [3]
Ref. [5]
Ref. [5]
Ref. [6]
method
Improved ACO
ACO
zigzag
DDDU
PGA
1
30.812
30.812
30.9662
30.902
29.35533
2
0.100
0.100
0.1000
0.100
0.51720
3
23.483
23.483
23.7568
23.545
24.56669
4
14.758
14.758
14.9362
14.960
14.93669
5
0.100
0.100
0.1000
0.100
0.20820
6
0.449
0.449
0.3063
0.297
0.84602
7
7.778
7.778
7.4698
7.611
7.02227
8
21.040
21.040
21.2322
21.275
21.23720
9
21.389
21.389
21.1226
21.156
22.33451
10
0.100
0.100
0.1000
0.100
0.10030
weight
5070.49
5070.49
5067.71
5069.4
5116.416
2. Example 2. 25 bar truss [7] A 25 bar truss structure is shown in Fig 2. The listing of member grouping can be
found in Ref [5]. The material constants, loading and stress constraint values are also given in Table 3. ACO and
improved ACO algorithm was used to solve this problem, respectively. The optimal design given by ACO, improved
ACO and other algorithm are compared and listed in Table 4. It can be seen from Table 4 that the improved ACO
algorithm obtained the best design.
A
75in
1
100in
2
75in
3
6
4
5
100in
75in
7
10
8
20
0i
n
9
20
n
0i
Figure 2: 25 bar truss, initial geometry
Table 3 Loading and design condition for the 25 bar space truss problem shown in Fig. 3
Case
1
Node
FX(lbs)
FY(lbs)
FZ(lbs)
1
1000
−10000
−10000
2
0
−10000
−10000
3
500
0
0
6
600
0
0
Constants
Young’s modulus = 1×107 psi, Density = 0.1 lbs/in3
Arearange={0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4,
2.5, 2.6, 2.8, 3.0, 3.2, 3.4} in2
Allowable stress=(+/−)40000psi, Allowable displacement = 0.35 in
⎯ 1022 ⎯
Table 4 Comparison of the design variables and objective values for 25 bar truss
Conf.
This paper
ACO
Ref7
method
Improve ACO
1
0.1
0.1
0.1
2
0.5
0.2
1.8
3
3.4
3.2
2.3
4
0.1
0.1
0.2
5
1.5
2.0
0.1
6
0.9
0.8
0.8
7
0.6
1.2
1.8
8
3.4
3.4
3.0
weight
486.29
505.818
546.01
GA
3. Example 3: 72 bar space truss A 72 bar truss structure is shown in Fig 3. The listing of member grouping can be
found in Ref. [5]. The material constants, load case and constraint values are also given in Table 5.
Figure 3: 72 bar space truss, initial geometry
Table 5 Loading and design condition for the 72 bar space truss problem shown in Fig3.
Case
Node
FX(lbs)
FY(lbs)
FZ(lbs)
1
1
5000
5000
−5000
1
0
0
−5000
2
0
0
−5000
3
0
0
−5000
4
0
0
−5000
2
Constants
Young’s modulus = 1×107 psi
Density = 0.1 lbs/in.3
Area range = 0.1-35 in2
Allowable stress = (+/−)25000 psi
Allowable nodal displacement in x, y and z directions= (+/−) 0.25 in (relative tolerance 5%)
Length a = 120in, b = 60in.
The optimal design given by ACO, the improved ACO and other algorithm are compared and listed in Table 6. It is can
be seen from the Table 6 that both ACO algorithms behaved as well as other algorithms. And the improved ACO
algorithm obtained the better design than that by ACO.
⎯ 1023 ⎯
Table 6 Comparison of the design variables and objective values for 72 bar truss
Section number Improved ACO
ACO
Ref. [5]
Ref. [8]
1
0.16
0.154
0.1564
0.1585
2
0.71
0.602
0.5457
0.5936
3
0.33
0.514
0.4106
0.3414
4
0.51
0.517
0.5692
0.6076
5
0.46
0.406
0.5237
0.2643
6
0.505
0.517
0.5171
0.5408
7
0.1
0.100
0.1000
0.1000
8
0.1075
0.154
0.1001
0.1509
9
0.96
1.252
1.2683
1.1067
10
0.5125
0.514
0.5116
0.5792
11
0.1
0.100
0.1000
0.1000
12
0.1
0.154
0.1000
0.1000
13
1.7375
1.810
1.8862
2.0784
14
0.47
0.514
0.5123
0.5034
15
0.1
0.100
0.1000
0.1000
16
0.1
0.100
0.1000
0.1000
Weight
373.467
387.84
379.62
388.63
Acknowledgements
This research was financially supported by the National Natural Science Foundation of China (Grant No. 60166001).
REFERENCES
1. Jose H et al. in Proceedings of 6th World Congress on Structural and Multidisciplinary Optimization, Rio de
Janeiro, 2005.
2. Colorni A, Dorigo M, Mariezzo V. Distributed optimization by ant colonies. Proc. of the 1st Euro. Conf. on
Artificial Life, Elsevier Pub., France, 1991, pp.134-142.
3. Gong YB, Li QY et al. The application of ant algorithm in structural optimization. Proceedings of WCSM06, No.
4101, 2005.
4. Lee ZJ, Lee CY. A hybrid search algorithm with heuristic for resource allocation problem. Information Science,
2005; 173: 155-167.
5. Qian LX. Optimization design for engineering structures. Hydro-Electric Press, Beijing, China, 1983 (in
Chinese).
6. Wu JY, Wang XC. A parallel genetic design method with coarse grain. Chinese Journal of Computational
Mechanics, 2002; 19(2): 148-153 (in Chinese).
7. Rajeev S, Krishnamoorthy CS. Discrete optimization of structures using genetic algorithm. J. Struct. Engng
ASCE, 1992; 118(5): 1233-1250.
8. Schmit LA, Farshi B. Some approximation concept for structural synthesis. AIAA J, 1974; 12(2): 231-233.
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