7th World Congress on Structural and Multidisciplinary Optimization COEX Seoul, 21 May – 25 May 2007, Korea
Application of Ant Colony Optimization Algorithms for Continuous and Discrete Structural Optimization Ali Rahmani Firoozjaee1, Ebrahim Jafari2, Mohammad Ghanbari Vandi3 1
Islamic Azad University, 2Islamic Azad University, 3Islamic Azad University
Faculty member, Civil engineering Department, Islamic Azad University of Ghaemshahr branch, Iran
[email protected] Civil engineering Department, Islamic Azad University, Ghaemshahr, Iran,
[email protected] Civil engineering Department, Islamic Azad University, Ghaemshahr, Iran,
[email protected] 1. Abstract: In this paper application of ant colony optimization (ACO) algorithms to continuous and discrete optimization problems is described and used for some structural problems such as plates and trusses. Since Ant colony optimization Algorithms are discrete optimization Algorithms, its application to continuous problems requires a process of search space discretization. A new method of discretization is developed in which its parameters are automatically refined using a stochastic process. Using this discretization method and Ant colony optimization algorithms, some structural benchmark problems are considered and results are presented and compared with the results of previous works. A Sensitivity analysis was also carried out to assess the effect of the Ant colony optimization and discretization method parameters on the final results. 2. Keywords: Ant Colony Optimization, Structural Optimization, Discretization Algorithms 3. Introduction Structural optimization is determination of design variables to minimize the cost or weight of the structure whereas stress and displacement constraints are satisfied. Design variables of structural problems can be continuous or discrete variables. In structural optimization problems there are several researches using mathematical programming methods[1-4]. These methods are developed for continuous variables and then in confront with discrete variables need some considerations. Hansen and Vanderplaats and Salajegheh and Vanderplaats proposed an approach to solve size and shape optimization problems using mathematical programming methods [5-6]. Generally these methods are gradient based and consequently their quality in multimodal optimization problem is not too high. In recent years metaheuristic optimization methods are developed. Generally these methods seek the solution by evolution of an initial population. Nowadays these methods are developing so rapidly. Genetic algorithm (GA), evolution strategy (ES), ant colony optimization (ACO), simulated annealing (SA) and particle swarm optimization (PSO) are in this category. Some of these methods such as evolution strategy and simulated annealing and particle swarm optimization methods are basically applied to continuous variables optimization problems and the others (genetic algorithm and ant colony optimization) are applied to discrete variables optimization problems [7]. It is obvious that each discrete optimization method for the solution of a continuous problem needs that the search space be discretized. The ant colony optimization (ACO) is a new metaheuristic to solve optimization problem by using principles of communicative behavior found in real ant colonies [8]. it has seen wide and successful application to many different optimization problems. In this study ant colony optimization method is applied to structural optimization problems and sensitivity analysis on some parameters is carried out. 4. Ant colony optimization Ant colony optimization (ACO) is an iterative probabilistic optimization inspired by the way real ants find short path between their nest and a food source. An important and interesting behavior of ant colonies their foraging behavior, and, in particular, how ants can find shortest path between food sources and their nest. While walking from food sources to the nest and vice versa, ants deposit on the ground a substance called pheromone. Ants can smell pheromone and, when choosing their way, they tend to choose, in probability, path marked by strong pheromone concentrations. over time, shorter path are reinforced with greater amount of pheromone they require lees time to be traversed, thus
becoming the dominant path for the colony to follow. In this study three algorithms of ACO are used that are described: 4.1. Ant system (AS) Ant system (AS) is the original and simplest ACO algorithm [9]. The decision policy used within AS is presented in Eq.(1) :
p ij (t ) =
[t
å [t lÎni
Where
][ ] a
ij
b
(t ) h ij
(1)
(t )] [h il ] a
il
b
t ij (t ) is the amount of pheromone trail on arc (i,j) at iteration t, h ij is the desirability factor, a and b are
two parameters that control the relative weight of pheromone trail and heuristic value,
ni is the set of neighbors of
node i. The pheromone updating equation in AS for each edge (i,j) is given by Eq.(2):
t ij (t + 1) = (1 - r )t ij (t ) + Dt ij (t )
Where
(2)
r is the pheromone trail decay coefficient r Î [0,1] and m
Dt ij = å Dt ijk (t )
(3)
k =1
in which m is the number of ants at each iteration 4.2. Ant colony system (ACS) In attempt to regulate the trade-off between exploitation of the current best solution and further exploration of the solution space, presented the ant colony system (ACS) [10]. In ACS decision policy can be expressed by Eq.(4):
a ij (t ) =
[t
å [t lÎni
If
ij
][ ]
b
(t ) h ij
(4)
(t )][h il ]
b
il
ì1 Þ j = arg max(a ij ) p ijk (t ) = í î0 Þ otherwise a ij (t ) p ijk (t ) = When q > q 0 å a il (t )
q £ q 0 then
q is random variable uniformly distributed over
[0.1]
(5)
(6)
lÎni
and
q 0 is the tunable exploration-exploitation factor
(0 £ q0 £ 1) . The operation to locally update pheromone edge (i,j) selected by an ant is: t ij ( t ) = ( 1 - r l )t ij ( t ) + ft 0
Where
(7)
r l is the local pheromone decay coefficient, and t 0 is the initial pheromone intensity laid on all edges. ACS
also involves global updating. The global updating rule is given by:
t ij ( t ) = ( 1 - r g )t ij ( t ) + r g Dt ij ( t ) Where
r g is the global pheromone decay coefficient (similar to r in AS) and Dt ij (t ) = 1
of the best tour since the beginning of the trail.
(8)
l+
and
l + is the length
4.3. ASrank: ASrank algorithm is proposed by Bullnheimer et al [11]. The elitist rank ant system (ASrank) further develops the idea of elitist used in ASelitist to involve a rank-based updating scheme [12]: s -1
Dt ijr (t ) = å Dt ijk (t ) k =1
With
(9)
ìDt ijk ( t ) = 0 ï í Dt k ( t ) = ( s - k ) ïî ij lk(t )
if the ant with rank k has used arc (i, j) in its tour otherwise
(10)
in which l k (t ) is the length of the tour selected by the ant with rank k at iteration t. the updating rule is given by Eq.(11) :
t ij (t ) = (1 - r )t ij (t ) + sDt ij+ (t ) + Dt ij (t ) where
Dt (t ) = + ij
1
l + (t )
and
(11)
+
l being the length of the global best solution.
5. Stochastic adaptive discretization method Search space discretization is required when one is going to optimize a continuous problem using ACO. In this study a sequential adaptive discretization algorithm is proposed. This method is described in the following parts: 5.1. Initial Discretization The search space will be discretized to nc numbers of discrete spaces in which each discrete space includes ns intervals as Eq.(12): ì i = 1,2,...,n ï (12) Sik, j = xiL + xUi - xiL ´ Random(0,1 ) , í j = 1,2,...,ns ïk = 1,2,...,nc î
(
)
where S is the set of discretized values, xiL and xUi are the lower and upper bound of the ith design variable. n is the number of design variables, nc is the number of discrete spaces and ns is the number of intervals in each discretization space. Random(0,1) is a uniform random number between zero and one. It is clear that each discretized space will yield to an optimum solution using ACO.
(
X *i = x1*i , x2*i ,..., xn*i
)
T
*i
, i = 1,2 ,..., nc
(13)
th
where X is the vector of optimum design variables for the i discretized space. 5.2. Intermediate discretization The optimum solutions obtained by different discretized spaces are not necessarily identical. Therefore the mean and standard deviation of each optimum design variable may be obtained by Eqs.(14,15): xi* =
1 nc
nc
åx
å(x nc
s i*
*j i
=
,
i = 1,2 ,...,n
(14)
i = 1,2,..., n
(15)
j =1
*j i
- xi*
j =1
nc - 1
) ,
By these data the new nc adapted discretization spaces will be obtained by Eq.(16): Sik, j = xi* + N ( 0, s i* ) ,
ì i = 1,2,...,n ï í j = 1,2,...,ns ïk = 1,2,...,nc î
(16)
in which N (0, s i* ) is a normal random number with zero mean and s i* standard deviation. 5.3. Termination criteria After some discretization iterations, the optimum solutions obtained by discretized spaces will converge to a unique solution. In this situation s i* approaches zero. A suitable criteria to terminate the algorithm can be stated by Eq.(17): (17) s i* £ e , i = 1,2 ,...,n where e is a user defined small positive value. The flow chart of the discretization algorithm is illustrated in Fig. 1. 6. Numerical examples In this study some 2D and 3D structural optimization problems are solved using ant colony optimization
algorithms. Generating 1st discretized space
Generating 2nd discretized space
Optimization with ACO
Optimization with ACO
... ...
Generating ncth discretized space Optimization with ACO
Evaluation of mean and standard deviation of above optimum results
Termination criteria
No
Yes End
Figure 1- the flow chart of discretization algorithm 6.1. Ten -bar planar truss The cantilever truss, shown in Fig. 2, was previously analyzed using various mathematical methods by Schmit and Farshi [13], Schmit and Miura [14], Stander et al. [15], Xu and Grandhi [16], and Lamberti and 3
Pappalettere [17,18]. The material density was 0.1 lb / in and the modulus of elasticity was 10,000 ksi. The members were subjected to stress limitations of ±25 ksi, and displacement limitations of ±2.0 in were imposed on all nodes in both directions (x and y). No design-variable linking was used; thus there are 10 independent design variables. In this example, loading condition of p1 = 100 kips and p 2 = 0 is considered. The prescribed discrete value set is {0.88, 1.118, 1.37, 1.606, 1.85, 2.35, 2.748, 3.49, 4.867, 5.93, 6.4, 6.89, 7.53, 8.504, 10.945, 12.2, 16.693, 20, 23.62, 27.56, 31.49}
Figure 2- Ten-bar truss problem and its loading This problem is solved using ant system, ant colony system and ASrank ant algorithms using 50 ants, r = 0.1, a = b = 1.0 , q 0 = 0.1 and 4000 maximum number of iterations, optimum design variables and the optimum weight of the truss are presented in table 1. To assess the effects of the pheromone decay coefficient ( r ) on the solution a sensitivity analysis is carried out using 25 ants with 4000 maximum number of iterations. Effects of r on the optimum Weight of the 10-bar truss is presented in table 2 for each algorithm. The above mentioned problem is solved with several numbers of ants using ant colony system (ACS) algorithm and results are presented in table 3.
Table 1- optimum cross section area and the weight 10-bar truss AS ACS ASrank A(1) 27.56 31.49 31.49 A(2) 0.88 0.88 0.88 A(3) 23.62 23.62 23.62 A(4) 16.693 16.693 16.693 A(5) 1.37 0.88 0.88 A(6) 1.37 0.88 0.88 A(7) 6.89 8.504 8.504 A(8) 27.56 23.62 23.62 A(9) 23.62 20.0 20.0 A(10) 0.88 0.88 0.88 Weight(lb) 5574.99 5378.471 5378.471 Table 2- sensitivity analysis for AS ACS ρ = 0.0 6914.91 6706.43 ρ = 0.1 5851.54 5413.19 ρ = 0.2 5642.72 5408.58 ρ = 0.3 5716.98 5414.11 ρ = 0.4 6356.74 5469.17
r ASrank 5455.13 5446.06 5455.13 5467.25 5467.25
Table 3- weight of the 10-bar truss and the number of ants using ACS Number of ants 5 10 15 20 25 Weight(lb) 5643.4 5503.06 5446.15 5415.35 5413.19
35 5378.47
6.2. Twenty-Five-bar space truss The 25-bar transmission tower space truss, shown in Fig. 3, has been size optimized by many researchers These include Schmit and Farshi [13], Schmit and Miura [14], Stander et al. [15], Xu and Grandhi [16], and Lamberti and Pappalettere [17,18]. In these studies, the material density, modulus of elasticity, Stress limits and displacement limit was 3
0.1 lb / in , 10,000 ksi,
± 40000 psi and 0.35 in respectively. The loading is presented in table 4.
Figure 3- The 25-bar transmission tower space truss The loading is presented in table 4 and the prescribed discrete value set is {0.1, 0.2, 0.3, 0.4, …, 3.5, 3.6, 3.7, 3.8}. The
problem is solved with 50 ants and 2000 maximum iteration numbers. Table 4- loading data of 25-bar truss Joint F x (kips) F y (kips) F z (kips) number 1 1.0 -10.0 -10.0 2 0.0 -10.0 -10.0 3 0.5 0.0 0.0 6 0.6 0.0 0.0 This problem is solved using three above mentioned ant colony optimization algorithms and the optimum solution and optimum weight of the structure are presented in table 5. Table 5- optimum cross section area and the weight 25-bar truss AS ACS ASrank A1 0.1 0.1 0.1 A2-A5 0.7 0.7 0.6 A6-A9 3.1 3.3 3.3 A10-A11 0.1 0.2 0.1 A12-A13 1.0 1.5 1.4 A14-A17 1.1 0.9 0.9 A18-A21 0.5 0.1 0.2 A22-A25 3.6 3.8 3.8 Weight(lb) 494.34 479.09 478.11 6.3. Plate with a hole optimization Optimum design of a plate with 2 design variables is desired to achieve the minimum weight of the structure. The material density, Poisson ratio, modulus of elasticity, compressive and tensile stress limits are 2400 kg 2
2
/ m3 , 0.15,
2
2×106 kg / m , 7×105 kg / m and 7×106 kg / m respectively. Upper and lower limits of design variables are considered zero and 1.0 respectively. Other information is illustrated in Fig. 4.
. Figure 4- plate and its loading Considering nc=3 and ns= 25 , the continuous optimization problem is solved using ACS algorithm. The results are presented and compared with the results of [19] in table 6. Table 6- comparison of ACO with previous mathematical methods Linear Adaptive linear Ant colony Approximation Approximation optimization
x1 ( m) x2 ( m)
0.34
0.34
0.35
0.26
0.26
0.21
Weight (kg)
196.44
196.44
193.25
7. Conclusion In this study ant colony optimization algorithms (ant system, ant colony system and ASrank) are applied to structural optimization problems. In discrete optimization problems ant colony algorithms are used to find optimum solution directly. Results confirm that ACS and ASrank have higher quality than AS. Comparison of ant algorithms and sensitivity analysis on parameters is carried out for the 10-bar truss optimization problem. In case that the problem has a continuous search space, discretization of the search space is required to find an optimum solution by ACO algorithms. A discretization method named, stochastic adaptive discretization method, is presented. This method can tune its parameters in optimization process. A plane stress plate with a continuous search space is optimized with combination of stochastic adaptive discretization method and ant colony optimization algorithms. The results show better solution than previous works. 8. References [1] G. M. Fadel, M. F. Riley and J. F. M. Barthelemy, Two-Point Exponential Approximation Method for Structural Optimization , Structural Optimization , Vol. 2 ,117-124 ,1990. [2] L. P. Wang and R. V. Grandhi, Efficient Safety Index Calculation for Structural Reliability Analysis, Computers and Structures, Vol. 52, 103- 111 ,1994. [3] L. P. Wang and R.. V. Grandhi, Improved Two - Point Function Approximations for Design Optimization, AIAA Journal, Vol. 33, 1720- 1727, 1995. [4] E. Salajegheh, Optimum design of plate structures using three-point approximation, Structural Optimization, VOL. 13, 142-147, 1997. [5] S. Hansen, and G. Vanderplaats, Approximation method for configuration optimization of trusses, AIAA Journal, vol. 28, pp. 161–168,1990. [6] E. Salajegheh, and G. Vanderplaats, Optimum design of trusses with discrete sizing and shape variables, Structural Optimization, vol. 6, pp. 79–85, 1993. [7] T. Back, Evolutionary Algorithms in Theory and Practice, Oxford University Press, New York, 1996. [8] Dorigo, M., Di Caro, G. & Gambardella, L.M. (1999). Ant algorithms for discrete optimisation. Artificial Life, 5(2), 137–172. [9] Dorigo, M., Maniezzo, V. & Colorni, A. (1996). The ant system: Optimisation by a colony of cooperating agents.IEEE Transactions on Systems, Man, and Cybernetics—Part B, 26(1), 29–41. [10] Dorigo, M. & Gambardella, L.M. (1997). Ant colony system: A cooperative learning approach to TSP. IEEE Transactions on Evolutionary Computation, 1(1), 53–66. [11] Bullnheimer, B., Hartl, R. F. & Strauss, C. (1999). A new rank based version of the Ant System: A computational study. Central European Journal for Operations Research and Economics, 7(1), 25–38. [12] Aaron C. Zecchina, , Holger R. Maier, Angus R. Simpson, Michael Leonard, and John B. Nixon, "Ant Colony Optimisation Applied to Water Distribution System Design: A Comparative Study of Five Algorithms" ,Journal of Water Resources Planning and Management, ASCE [13] Schmit Jr LA, Farshi B. Some approximation concepts for structural synthesis. AIAA J 1974;12(5):692–9. [14] Schmit Jr LA, Miura H. Approximation concepts for efficient structural synthesis. NASA CR-2552, Washington, DC: NASA; 1976. [15] Stander N, Snyman JA, Coster JE. On the robustness and efficiency of the S.A.M. algorithm for structural optimization. Int J Numer Methods Eng 1995;38:119–35. [16] Xu S, Grandhi RV. Effective two-point function approximation for design optimization. AIAA J 1998;36(12):2269–75. [17] Lamberti L, Pappalettere C. Comparison of the numerical efficiency of different sequential linear programming based on algorithms for structural optimization problems. Comput & Structures 2000;76:713–28. [18] Lamberti L, Pappalettere C. Move limits definition in structural optimization with sequential linear programming– Part II: Numerical examples. Comput & Structures 2003;81:215–38. [19] Salajegheh E. and Rahmani A., optimum design of plates with stress and displacement, fifth international conference of mechanical engineering (1998), IUST, Iran. (in Persian)
