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A hybrid CBO–PSO algorithm for optimal design of truss structures with dynamic constraints
Accepted Manuscript Title: A hybrid CBO-PSO algorithm for optimal design of truss structures with dynamic constraints Author: A. Kaveh V.R. Mahdavi PII: DOI: Reference:
S1568-4946(15)00310-5 http://dx.doi.org/doi:10.1016/j.asoc.2015.05.010 ASOC 2963
To appear in:
Applied Soft Computing
Received date: Revised date: Accepted date:
10-12-2014 8-5-2015 10-5-2015
Please cite this article as: A. Kaveh, V.R. Mahdavi, A hybrid CBO-PSO algorithm for optimal design of truss structures with dynamic constraints, Applied Soft Computing Journal (2015), http://dx.doi.org/10.1016/j.asoc.2015.05.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Graphical Abstract
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A new hybrid CBO–PSO algorithm is presented by adding positive properties of the PSO algorithm to the CBO. The CBO is a recently developed meta-heuristic algorithm which does not use the
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internal parameter and memory in its formulation.
In the CBO-PSO the memory of the PSO is added to the CBO to improve the
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performance of the latter algorithm.
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The new algorithm is compared to other advanced meta-heuristic methods to illustrate the
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Revised for Applied Soft Computing on 7th of May 2015
A hybrid CBO-PSO algorithm
A. Kaveh*, V.R. Mahdavi
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ABSTRACT
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for optimal design of truss structures with dynamic constraints
The vibration domain of structures can be reduced by imposing some constraints on their natural
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frequencies. For this purpose optimal design of structures under frequency constraints is required which involves highly non-linear and non-convex problems. In this paper an efficient hybrid
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algorithm is developed for solving such optimization problems. This algorithm utilizes the recently developed colliding bodies optimization (CBO) algorithm as the main engine and uses
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the positive properties of the particle swarm optimization (PSO) algorithm to increase the efficiency of the CBO. The distinct feature of the present hybrid algorithm is that it requires no parameter tuning. The CBO is known for being parameter independent, and avoiding the use of
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the traditional penalty method to handle the constraints upholds this property. Two mathematical
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constrained functions taken from the literature are studied to verify the performance of the algorithm. The algorithm is then applied to optimize truss structures with frequency limitations.
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The numerical results demonstrate the efficiency of the presented algorithm for this class of problems.
Keywords: Colliding bodies optimization, Particle swarm optimization, Penalty approach, Structural design problems, Dynamic constrains.
A.Kaveh* (□), Professor of Structural Engineering Centre of Excellence for Fundamental Studies in Structural Engineering, Department of Civil Engineering, Iran University of Science and Technology, Tehran, Iran. E-mail: [email protected] (A. Kaveh) V.R. Mahdavi, PhD Student School of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran
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1. Introduction Optimization algorithms are utilized to maximize or minimize one/some objective functions under some specific limitations. In optimal design of structures, the main objective is to minimize the weight of the structures under some constraints, such as stresses and displacements.
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For structures subjected to dynamic excitations, the natural frequencies play important role in the dynamic response of structures and resonance phenomenon. In recent years, optimal design of
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structures under frequency limitations has been extensively investigated using different optimization algorithms by many researchers. This problem is high non-linear and non-convex
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with respect to the design variables [1].
Algorithms for optimization can be divided into two general categories of Gradient-based methods and metaheuristics. The formulation of metaheuristic algorithms is often inspired by
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either natural phenomena or physical laws. Every metaheuristic algorithm consists of two phases: exploration of the search space and exploitation of the best solutions found. One of the
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main problems in developing a good metaheuristic algorithm is to keep a reasonable balance between the exploration and exploitation abilities [2, 3]. Some of the well-known metaheuristic
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algorithms can be found in [4-9]. One of the difficulties with most of the metaheuristic algorithms is parameter tuning which is often time consuming. This problem becomes more
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severe when the parameters of penalty method for constraint handling are added. This is because the feasible solutions are extremely sensitive to setting the parameters. In practical structural
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design tasks, designers usually have limited knowledge about the appropriate values of these parameters. Therefore, the selection of proper values for the parameters and utilizing penalty approach becomes a tedious process.
One of the recently developed metaheuristics is the colliding bodies optimization (CBO) algorithm originated by the present authors [10]. Compared to conventional meta-heuristic optimization algorithms, simple formulation and no necessity for parameter tuning are the main advantages of this algorithm. On other hand, the exploitation phase of this algorithm is weak due having no memory for saving the best-so-far solution in its formulation (i.e. the best position of agents from the previous iterations). Particle swarm optimization (PSO) developed by Eberhart and Kennedy [4] is also a popular metaheuristic algorithm which is based on swarm intelligence. In this algorithm, the agents (particles) improve their position by using their data from the past generations. 4 Page 4 of 36
The main objective of the present study is to optimize the weight of truss structures with natural frequency constraints. Thus, an efficient and no parameter tuning algorithm is proposed based on hybridization of CBO and PSO algorithms for constrained optimization problem. In this algorithm, the used memories of the PSO, i.e. the local and global best, is added to the CBO. exploitation and the convergence rate for the best found solution.
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Thus in the new algorithm the bodies move toward the best saved bodies increasing the
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The present paper is organized as follows: In the next section, formulation of the optimal design of truss structures with frequency limitations is presented. In section 3, both standard
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algorithms are briefly introduced and the hybrid method is presented. This is followed by a section consisting of the study of optimization of two mathematical constrained functions and 5, respectively.
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2. Formulation of optimal design of structures
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three truss structures. A discussion and brief concluding remarks are presented in sections 4 and
Optimization problems can be classified as multi-objective and mono-objective. In structural
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optimization problems, the main objectives are minimizing the weight or cost of the structures
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while satisfying some constraints. In this class of optimization problems, the design variables can be considered as the cross-sectional areas of elements and/or the coordinates of nodes. The
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optimization problem for a truss structure can be stated as follows: Find
X = [x1,x2,x3,..,xN] ne
minimizing
W ( X ) i Ai li
(1)
i 1
satisfying
gj(X)0, j=1,2,…,m xlmin ≤ xl ≤ xlmax
where X is the vector of all design variables with N unknowns; W is the weight of truss structure; ρi, Ai and li are mass density, cross sectional area and length of the ith member, respectively; ne is number of the structural elements; gj is the jth constraint from m inequality constraints. Also, xlmin and xlmax are the lower and upper bounds of design variable vector, respectively.
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Since the solution of this multi-objective problem is rather complex, most researchers have considered this problem as mono-objective and the penalty function approach has been used for handling the constraints. Coello [11] compared different constraint-handling methods used in metaheuristic algorithms with comments on their advantages and disadvantages. These methods
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can generally be classified into five groups: penalty functions, special representations and operators, repair algorithms, separation of objectives and constraints, and hybrid methods. The
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most common and convenient method in metaheuristic algorithms to handle constraints is to use penalty functions. In this method, a constrained optimization problem is transformed into an
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unconstrained one by multiplying a certain value to objective function based on the amount of constraint violation appears in a problem. Therefore the merit (or pseudo objective) function
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which should be minimized is defined as: Mer(X) = W(X) ×fpenalty(X)
(2)
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where Mer(X) is the merit function; W(X) is the objective function (weight of truss structure); fpenalty(X) is the penalty function which results from the violations of the constraints
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corresponding to the response of the structure: m
f penalty ( X ) 1 p max(0, g j ( X )) 1 p G ( X )
(3)
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j 1
where γp is penalty parameter, and G(X) is the constraint function. In dynamic constraint
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problems the behavioral constraints are the restricted natural frequencies defined as follows:
n ( x) u n , g n ( x)
n ( x) 1 0 u n
for some natural frequencies n
( x) m ( x) lm , g m ( x) 1 m 0 for some natural frequencies m u m
(4)
where ωn and ωm are the nth and mth natural frequency, respectively; ωln and ωun are the lower bound and upper bound of the nth and mth natural frequency, respectively. As can be seen from Eq. (3), the penalty function method consists of a penalty parameter where the feasible solutions are dependent on right tuning of these parameters. By setting small values of the penalty parameter, the pseudo objective function becomes less sensitive to optimization constraints and the optimization process may converge to a violated solution. Conversely, setting the penalty parameter to a large value leads to amplify the effect of 6 Page 6 of 36
optimization constraints and the optimization process may get stuck in a local optimum. This dependency of the optimization algorithms performance on penalty parameters has led researchers to devise efficient method for constraint handling [12]. In this paper, we present a constraint handling method that does not dependent on any
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parameter tuning. The pair-wise comparison method which presented in Ref. [13] has been adopted in the presented algorithm. In this method, for ranking and comparison of the agents (or
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populations), three criteria are enforced: In comparison of (i) two feasible/inviolate agents, the one with better objective function value is preferred, (ii) one feasible/inviolate and one
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infeasible/violate agent, the feasible agent is preferred, and (iii) two infeasible/violate agents, the one with smaller constraint violation is preferred. features of the standard CBO and PSO algorithms
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In the next section the new CBO-PSO based hybrid algorithm is presented using the positive
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3. CBO-PSO based hybrid optimization algorithm
Inspired by the natural features of the CBO and PSO algorithms, we develop an efficient
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hybrid algorithm (CBO-PSO) in this section. In the following, both standard CBO and PSO
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algorithms are briefly introduced and then the proposed hybrid method is presented. 3.1. Colliding Bodies Optimization algorithm
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Colliding bodies optimization (CBO) is a recently efficient and robust metaheuristic optimization algorithm for simple implementation of complex computations. As CBO is simple to be implemented and can deal with complex problems without extensive mathematical computations and parameter tuning, it is widely used in various fields of optimization problems [14, 15]. The CBO mimics a 1-dimensional collision between two colliding bodies (CBs). In this case, each CB collides distinctly with its pair and updates its new velocity based on momentum and energy conservation law for 1-dimensional collision [16]. According to this law, the velocities of two bodies after a one-dimensional collision can be obtained as:
v1'
(m1 m2 )v1 (m2 m2 )v2 m1 m2
(5)
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v2'
(m2 m1 )v2 (m1 m1 )v1 m1 m2
(6)
where v1 is the velocity of the first object before impact, v2 is the velocity of the second
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object before impact, v1' is the velocity of the first object after impact, v2' is the velocity of the second object after impact, m1 is the mass of the first object, m 2 is the mass of the
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second object, is the coefficient of restitution (COR) of two colliding bodies, defined as the ratio of relative velocity of separation to the relative velocity of approach.
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The CBO algorithm consists of steps that are briefly described as follows:
Similar to other multi-agent methods, CBO has a number of individuals (or agents),
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named Colliding Body (CB), consisting of the variables of the problem. Each CB is treated as an object with specified mass and velocity and collides to others; after collision, each CB
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moves to a new position with new velocity (Fig. 1). CBO starts with an initial population consisting of 2n parent individuals created by means of a random initialization. Then, CBs are sorted in ascending order based on the value of cost function as shown in Fig. 2a. The sorted
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CBs are divided equally in to two groups. The first group is stationary and consists of good
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agents. This set of CBs is stationary and velocity of these bodies before collision is zero. The second group consists of moving agents which move toward the first group. Then (see Fig.
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2b), the better and worse CBs, i.e. agents with upper fitness value, of each group will collide together to improve the positions of moving CBs and to push stationary CBs towards better positions. The change of the body position represents the velocity of CBs before collision as: i 1,..., n 0, Vi X i n X i , i n 1,...,2n
where
and
(7)
are the velocity and position vectors of the ith CB, respectively. 2n is the
number of population size. After the collision, the velocity of bodies in each group is evaluated using momentum and energy conservation law and the velocities before collision (Eq. (7)). The velocity of CBs after the collision is: 8 Page 8 of 36
(mi n mi n )Vi n , i 1,..., n m m i i n Vi ' (mi mi n )Vi , i n 1,...,2n mi mi n
and
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where
(8)
are the velocities of the ith CB before and after the collision, respectively; mi is
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the mass of the ith CB and defined as:
(9)
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1 fit (k ) mk n , k 1,2,...,2n 1 i 1 fit (i )
where fit(i) represents the objective function value of the ith agent. Obviously a CB with good
objective function, the term
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values exerts a larger mass and fewer moves than the bad ones. Also, for maximizing the 1 is replaced by fit (i ) . is the coefficient of restitution (COR) fit (i)
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and is defined as the ratio of the separation velocity of two agents after collision to approach
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velocity of two agents before collision. In this algorithm, this index is defined to control of the exploration and exploitation rates. For this purpose, COR decreases linearly from unit value to
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zero. Thus, is defined as:
1
iter itermax
(10)
where iter is the actual iteration number and itermax is the maximum number of iterations. COR having unit value and zero value represent the global and local search, respectively. In this way a good balance between the global and local search is achieved by increasing the number of iterations. The new positions of the CBs are evaluated using the generated velocities after the collision in position of stationary CBs:
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X i rand Vi ' , i 1,..., n X i n rand Vi ' , i n 1,...,2n
where
and
(11)
are the new position and the new velocity after the collision of the ith CB,
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X
new i
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respectively. 3.2. Particle Swarm Optimization
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The PSO is based on a metaphor of social interaction such as bird flocking and fish schooling, and is developed by Eberhart and Kennedy [4]. The PSO simulates a commonly
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observed social behavior, where members (particles) of a group (swarm) tend to follow the lead of the best of the group.
In this algorithm, position of the particles and their associated velocities are initialized with
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random positions. Then, the particles fly through the search space and their positions are updated k based on the best positions of individual particles denoted by p i and the best position among all
(12)
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X i ( k 1) X i ( k ) Vi (k 1)
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particles in the search space represented by p gk :
(13)
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Vi (k 1) Vi (k ) C1 r1 ( Pi (k ) X i (k )) C2 r2 ( Pg (k ) X i (k )
where X i (k ) and Vi (k ) are the position and velocity for the ith particle at iteration k; is an inertia weight to control the influence of the previous velocity; r1, and r2 are two random numbers uniformly distributed in the range of (0, 1); c1 and c2 are the cognitive and social scaling parameters, respectively; pi (k ) is the best position of the ith particle up to iteration k; pg (k ) is the best position among all particles in the swarm up to iteration k and the sign “ ” denotes element-by-element multiplication [17]. 3.3 A hybrid CBO-PSO based algorithm In this study, CBO and PSO are hybridized leading to a new optimization search strategy. The main algorithm is based on the CBO, where some features of the PSO are added. Here, three features are added to the formulation of the standard CBO: i) the stationary CBs move also 10 Page 10 of 36
toward better positions, ii) the CBs move toward local best, i.e. pi is added which is used in the PSO, iii) the old velocities are added to the new velocities of CBs before collision. In fact, the exploitation ability of CBO algorithm increases by adding the saved particles. The CBO-PSO algorithm can simply be described as follows:
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Step 1: Initialization: Initial position, velocities and best position of populations are created by means of a random initialization:
,
and
j 1,2,...2n
(14)
are the ith element of position, velocity and best position of the jth and
are the lower and upper bounds for the ith decisions variable,
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population, respectively;
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Where
&
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pii j xij [ xiL , xiU ] , i 1,2,...N j vi (arbitrary domain)
respectively; N and 2n are the number of decision variable and population size, respectively. Step 2: Arrangement of populations: As mentioned before, the arrangement of the CBs is
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performed such that three criteria are satisfied: In comparison of (i) two inviolate populations, the one with better objective function value is selected, (ii) one inviolate and one violate
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population, the inviolate population is selected, and (iii) two infeasible populations, the one with smaller constraint violation is preferred. For this purpose a matrix similar to Eq. (15) is
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constructed using the position, velocity and best local vectors, as well as with the pseudo
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objective function of each position vector F(X):
x1 12 x1 2n x1
x12 x22
x1N v11 xN2 v12
v12 v22
v1N pi11 vN2 pi12
pi21 pi22
x22 n
xN2 n v12 n
v22 n
vN2 n pi12 n
pi22 n
piN2 n F ( X 2 n ) 1 pi1N F ( X ) 2 piN2 F ( X )
(15)
This matrix is sorted in ascending order of pseudo objective function, that is . The pseudo objective function is defined to compare the feasible and/or infeasible population based on the three mentioned criteria as:
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f (X ) F(X ) f max G ( X )
if
G( X ) 0,
(16)
otherwise,
where f(X) and G(X) are the objective and constraint functions which are defined in Eqs. (1, 3); and fmax is the objective function of the worse inviolate CB in the current iteration. Thus, the
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fitness of an infeasible population not only depends on the amount of constraint violation, but also on the population of solutions at hand. However, the fitness of a feasible solution is always
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population, pi, is evaluated based the above mentioned criteria.
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fixed and is equal to its objective function value. It should be noted that the best position of
Step 3: Mating process: The CBs are divided to two parts as shown in Fig. 2. The first half
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and second half of this partition named as part 1 and part 2 agents, respectively. Step 4: Evaluation of velocities before collision: The part 1 and part 2 agents toward the best
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position of CBs of part 1 agents. The old velocities are also added to new velocities. Therefore, the velocity of CBs before collision is derived using:
and
are the velocity and position vectors of the jth CB in the kth iteration,
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where
(17)
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V j (k ) ( Pi j (k ) X j (k )), j 1,..., n V (k 1) j j n j j n 1,...,2n V (k ) ( Pi (k ) X (k )), j
respectively;
is the best position vector of the jth CBs up to iteration k.
Step 5: Evaluation of velocities after collision: Compute the velocity of CBs after the collision using Eqs. (5, 6) and Eq. (17):
(m j m j n ) * V j (k 1) (m j n m j n ) * V j (k 1) , j 1,..., n m j m jn m j m jn 'j V (k 1) j j n j j n j n j n (m m ) * V (k 1) (m m ) * V (k 1) , j n 1,...,2n m j m jn m j m j n
(18)
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where
and
respectively;
are the velocity vectors of the jth CB after and before the collision,
is the mass of the ith CB which is defined as Eq. (9). is the coefficient of
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restitution (COR) and is defined as Eq. (10).
following expression: Pi j (k ) rand V ' j (k 1) , j 1,..., n X (k 1) j n Pi (k ) rand V ' j (k 1) , j n 1,...,2n
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j
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Step 6: Evaluation of new position: The new positions of the CBs are evaluated using the
(19)
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where rand is a random number uniformly distributed in the range of (-1, 1);
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Step 7: Updating the velocities: The velocities of the CBs are updated as:
(20)
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V j (k 1) X j (k 1) X j (k )
Step 8: Termination criteria: The optimization is repeated from Step 2 until a termination
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criterion, as the maximum number of iteration, is satisfied.
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As it can be seen in equations, by sorting the populations based on feasible or infeasible solutions, the formulation of proposed algorithm needs no parameter setting or tuning. The following section provides examples and comparisons for validation of the presented hybrid algorithm.
4. Numerical examples
In this section the efficiency of the proposed algorithm, CBO-PSO, is studied through two mathematical function examples and three well-studied truss structures with frequency limitations taken from the optimization literature. Examples 1 and 2 show the applicability of the CBO-PSO for optimization of constraint problems. In example 3 a spatial truss structure is studied for finding the optimal cross sections. In example 4 the performance of the CBO-PSO is studied for finding the optimal size and shape of a spatial truss structures. In the last example, a planar truss structure with many variables is selected to show the importance of selection of 13 Page 13 of 36
optimization algorithm in the final optimal weight. These examples are independently optimized 20 times. The algorithm was coded in MATLAB, and structural analysis was performed with the direct stiffness method. A comparison study of the obtained results is performed for the
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considered examples and the next section consists of the discussions on the results. 4.1. Example 1: Constrained function I
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Optimization of the constrained function expressed in Eq. (22) is considered as the first example. This problem also has seven variables and four nonlinear inequality constraints. This
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problem is defined as:
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Find
To minimize
(22)
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f ( x) ( x1 10) 2 5( x2 12) 2 x34 3( x4 11) 2 10 x56 7 x62 x74 4 x6 x7 10 x6 8 x7
Subjected to:
(21)
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x1 , x2 , x3 , x4 , x5 , x6 , x7
g1 ( x) 127 2 x12 3 x24 x3 4 x42 5 x5 0, g 2 ( x) 282 7 x1 3 x2 10 x32 x4 x5 0,
(23)
g 3 ( x) 196 23 x1 x22 6 x62 8 x7 0, g 4 ( x) 4 x12 x22 3 x1 x2 2 x32 5 x6 11x7 0.
The bounds on the design variables are:
10 xi 10
(i 1 7 )
(24)
This problem has been solved in the literature using various global optimization techniques [13, 18]. Taking into account the probabilistic nature of metaheuristic algorithms, a number of independent runs were performed. In this example, the population size of the CBO-PSO and 14 Page 14 of 36
CBO are set to 40 and 60 individuals, respectively, and the maximum number of optimization iterations was limited to 600. The best solution vectors and statistical result of these runs obtained by CBO and CBO-PSO are compared with those obtained by other metaheuristic algorithms in Table 1. As it can be seen
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from this table, the best solution obtained by CBO-PSO is better than CBO and those quoted for the other algorithms. However, the CBO-PSO, CBO and HS needs 24,000, 36,000 and 160,000
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fittness function evalutions to find the optimum result, respectively. The objective function and constraint function of feseable population against generations are plotted in both sides of Fig. 3.
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As it can be seen, the constraint function is equal zero and hense the constrints are satisfed in the
4.2 Example 2: Constrained function II
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final iterations.
This is a 10-variable challenging optimization problem. The problem has also eight nonlinear
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inequality constraints, and can be defined as:
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To minimize
(25)
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x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10
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Find
f ( x) x12 x22 x1x2 14x1 16x2 ( x3 10)2 4( x4 5)2 ( x5 3)2 2( x6 1)2 5 x72 7( x8 11)2 2( x9 10)2 ( x10 7)2 45
(26)
Subjected to:
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g1 ( x) 105 4 x1 5 x2 3x7 9 x8 0, g 2 ( x) 10 x1 8 x2 17 x7 2 x8 0, g3 ( x) 8 x1 2 x2 5 x9 2 x10 12 0, g 4 ( x) 3( x1 2) 2 4( x2 3) 2 2 x32 7 x4 120 0,
(27)
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g5 ( x) 5 x12 8 x2 ( x3 6) 2 2 x4 40 0, g 6 ( x) x12 2( x2 2) 2 2 x1 x2 14 x5 6 x6 0, g 7 ( x) 0.5( x1 8) 2 2( x2 4) 2 3x52 x6 30 0,
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g8 ( x) 3x1 6 x2 12( x9 8) 2 7 x10 0. The bounds on the design variables are:
(i 1 10)
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10 xi 10
(28)
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This problem has been solved by Michalewicz and Schoenauer [19] using GA-based
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methods. Deb [13] utilzed also an efficient constraint handling method for the GA, and Lee and
Geem [18] employed the harmony search algorithm. In this case, CBO-PSO and CBO population sizes were set to 80 and 100 individuals, respectively. The maximum number of optimization iterations was set as 1,000.
Table 2 compares the optimized variables, the corresponding objective function and statistical results obtained by CBO and CBO-PSO with those obtained by other existing metaheuristic algorithms. It can be seen that the proposed algorithm is the best among the four standard algorithms. The CBO-PSO needs 80,000 objective function evaluations to find the optimum result while it is 100,000 and 230,000 for the CBO and HS as reported, respectively. Similar to the first example, the objective function and constraint function of feasible population versus generations are plotted in Fig. 4.
4.3 Example 3: A 72-bar space truss 16 Page 16 of 36
In this example a structural designer intends to evaluate the cross sections of a 72-bar space steel truss structure shown in Fig. 5 such that its weight becomes minimum, while the first two frequencies are limited to some specified values. The structural elements are also labeled in Fig. 5. The following values are used for material properties of the steel: the material density is 2770
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kg/m3 and the modulus of elasticity is 69,800 MPa. Four non-structural masses of 2270kg are attached to the nodes 1-4. This design case addresses a continuous treatment of the problem, in
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which the independent size variables are chosen from a real-valued design interval with lower bound equal to 0.645cm2. In this example, the first two natural frequency constraints are imposed
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as 1 4 HZ , 2 6 HZ . This example has been solved previously by Konzelman [20] using a dual method (DM) and by Sedaghati [21] using the force method (FM). Gomes [22] has
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investigated the problem using the PSO. Kaveh and Zolghadr [23, 24] have investigated the problem using the standard and an enhanced CSS. The population size (i.e. number of CBs) and number of iterations are defined as 2n=30 and iteration=200 for both CBO and CBO-PSO.
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Comparison of the optimal design results reported in the literature and those of the present work are detailed in Table 3. The PSO-CBO algorithm found the best weight as 324.377kg after
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6,000 analyses, with the standard deviation of 1.55kg. Though, the number of iteration are more than standard CSS and enhanced CSS, this solution is reported to be the optimum design of the
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problem reached in the present study, and it is the best solution amongst the existing literature
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results. A comparatively better design weight of 325.459kg was determined using CBO. Fig. 6 shows the feasible solution history for the obtained best results of 20 individual run. It can be seen that the best solution found by the proposed algorithm is 324.377kg which is attained at the 179th iterations.
4.4 Example 4: A 52-bar dome-like truss This example considers shape and size optimization of a dome-like space truss structure as shown in Fig. 7. The initial topology and the element numbering of this truss are shown in Fig. 7a. This design has also been investigated in Lingyun et al. [25] using the NHGA algorithm, and Gomes [22] utilizing PSO algorithm. The problem has also been optimized by Kaveh and Zolghadr [23, 26] using the CSS algorithm. The space truss has 52 bars and non-structural masses of m = 50kg are added to the free nodes. The material density is 7800kg/m3 and the modulus of elasticity is 210,000MPa. For this structure, the symmetry around x and y-axes is 17 Page 17 of 36
employed to group the 52 truss members into eight independent size variables, where all members in a group share the same material and cross-sectional properties. Table 4 shows each element group by member numbers. The range of the cross-sectional areas varies from 1 to 10cm2. The shape optimization is performed considering that the
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symmetry is preserved in the process of design. Each movable node is allowed to move ±2m. For the frequency constraints, 1 15.916HZ and 2 28.649HZ are considered. Hence, this truss
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optimization problem has 13 design variables (five shape variables plus eight size variables). The population size and number of iterations are defined as 2n=30, and iteration=200 for both CBO
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and CBO-PSO, respectively.
Table 5 compares the optimal cross sections, best weight, mean weight and standard
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deviation of the results obtained using CBO and CBO-PSO with the outcomes of other researches. As anticipated, CBO-PSO has led to a much better results than the others in term of the best result found by the CBO-PSO.
d
4.5 Example 5: A 200-bar planar truss
M
the best, the mean weight and also standard deviation. Fig. 8 provides the convergence rates of
In this problem, the objective is to achieve the minimum weight design of the 200-bar plane
te
truss shown in Fig. 9. This truss has been investigated using the standard CSS and CSS-BB-BC
Ac ce p
algorithms as a frequency constraint weight optimization problem by Kaveh and Zolghadr [23, 27]. The material density and modulus of elasticity of members are 7860kg/m3 and 210,000MPa, respectively. Non-structural masses of 100kg are attached to the upper nodes. A lower bound of 0.1 cm2 is assumed for the cross-sectional areas. In calculating the frequency constraints, the first three frequencies are restricted to 1 5HZ , 2 10HZ and 3 15 HZ . The elements are divided into 29 groups. For this example, the population size and number of iterations are also defined as 2n=30 and iteration=300 for both CBO and CBO-PSO, respectively. The elements grouping and results obtained using this research with those of the other researches are presented in Table 6. According to this table, the result obtained by the CBO-PSO is meaningfully lighter than that of the standard CSS, CSS-BB-BC and CBO algorithms. Fig. 10 shows the convergence curves for the proposed algorithm.
5. Discussion 18 Page 18 of 36
Further examples for the problem of optimal design of truss structures with frequency limitations show that CBO-PSO generally has a better performance than the standard CBO. The main reasons for the improvements of the proposed algorithm can be summarized as follows: Since parameter tuning in meta-heuristic algorithms affects the exploration and exploitation
ip t
process of algorithms, the efficiency of optimization algorithms is completely dependent on the right tuning of these algorithms. In the other hand, CBO is a mate-heuristic algorithm which in
cr
its formulation does not use the internal parameters and memories. Having no internal parameter tuning is a good feature of this algorithm. However, the implementation of penalty approach to
us
handle the constraints and therefore tuning of penalty parameter, makes the algorithm to be dependent on the penalty parameter. Consequently, by sorting the populations according to these objective and constraint functions values, one enforces the populations to move rapidly towards
an
the feasible solutions such that the constraints are satisfied, and then move toward the point with smaller objective functions. As demonstrated in the convergence curves of the examples, in the
M
course of implementing the CBO-PSO algorithm, the value of constraint function decreases with the increase in the number of iterations until it reaches zero. This means that CBO-PSO
d
algorithm has successfully optimized the objective function while satisfying all the specified constraints. It is evidence that the values of the objective and constraint functions do not
te
monotonically decrease with the increase of the number of iterations, which is completely different from observation in the common meta-heuristic algorithms using the penalty approach,
Ac ce p
where the value of the objective function (or pseudo objective function) monotonically decreases with the increase of number of iteration. Particularly, in early iterations of the optimization where the constraint function is still unsatisfied, the constraint function causes the optimization process not to solely rely on the objective function values. Therefore, in early iterations the global search is performed in the entire search space to find the feasible solutions and then the local search is achieved in the feasible search space to find the best objective function. Hence, the exploration and exploitation are balanced during the optimization process whiteout setting any parameter. As it can be seen from the details of these figures, in the later iterations, the final optimum value is achieved without violation of constraints. Moreover, a weakness of the CBO is that it does not use the memories in its formulation. In the CBO-PSO algorithm, old velocities and best local of populations are added to its formulation. From results of these examples, it can be 19 Page 19 of 36
concluded that the best, mean and standard division of 20 independent runs for the new hybrid algorithm are better than those of the other algorithms. 6. Concluding remarks
ip t
This study develops a hybridized search algorithm so called CBO-PSO for constrained optimization problems and in particular for finding optimal weight of truss structures with
cr
dynamic limitations .The proposed method is mainly based on collision event concept borrowed from colliding bodies optimization.
us
In comparison with the standard CBO, the hybrid CBO-PSO has the following advantages: 1. In the standard CBO, the better solutions are stationary and the worse solutions move
an
toward the better solutions in each iteration. But, in the CBO-PSO algorithm the bodies of the CBO move toward the saved best particles which are used in the PSO algorithm and collide to these for promoting the exploitation ability of the CBO.
M
2. In standard CBO, the solutions are sorted according to the defined pseudo objective function in the penalty method. But in the CBO-PSO, the solutions are sorted based on the
d
constraints and objective functions values. Consequently the penalty parameters are also
te
vanished.
Concerning the efficiency of the CBO-PSO in comparison to the standard CBO and based on
Ac ce p
the presented numerical examples, the idea of a new sorting method and adding the memory vectors help the algorithm to identify potentially suitable areas in the search space much faster while avoiding the stagnation and being trapped in local optimums. It should be noted that the application of the present method can be extended to other constrained mono- and multiobjective optimization problems.
Acknowledgement: The first author is grateful to Iran National Science Foundation for the support.
References
20 Page 20 of 36
1. S. Gholizadeh, E. Salajegheh, P. Torkzadeh, Structural optimization with frequency constraints by genetic algorithm using wavelet radial basis function neural network, J. Sound Vib. 312 (2008) 316-331. 2. J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press,
ip t
Ann Arbor, USA, 1975.
3. A. Kaveh, Advance in Metaheuristic Algorithms for Optimal Design of Structures, Springer,
cr
Switzerland, 2014.
4. R.C. Eberhart, J. Kennedy, A new optimizer using particle swarm theory, in: Proceedings of
us
the Sixth International Symposium on Micro Machine and Human 557 Science, Nagoya, Japan, 1995.
5. A.H. El-Maleh, A.T. Sheikh, S.M. Sait, Binary particle swarm optimization (BPSO) based
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state assignment for area minimization of sequential circuits, Appl. Soft Comput. 13 (12) (2013) 4832–4840.
M
6. W. Deng, R. Chen, B. He, Y.Q.L. Liu, F. Yin, J.H. Guo, A novel two-stage hybrid swarm intelligence optimization algorithm and application, Appl. Soft Comput. 16 (10) (2012)
d
1707–1722.
7. M. Dorigo, V. Maniezzo, A. Colorni, The ant system: optimization by a colony of
te
cooperating agents, IEEE Trans. Syst. Man Cybern. B. 26 (1) (1996) 29–41. 8. A. Kaveh, S. Talatahari, A novel heuristic optimization method: charged system search, Acta
Ac ce p
Mech. 213 (3-4) (2010) 267-289.
9. A. Kaveh and M. Khayatazad, A new meta-heuristic method: ray optimization, Comput. Struct. 112-113 (2012) 283–294.
10. A. Kaveh, V.R. Mahdavai, Colliding bodies optimization: A novel meta-heuristic method, Comput. Struct. 139 (2014)18–27. 11. C.A.C. Coelllo, Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art, Comput. Methods Appl. Mech. Eng. 191 (2002) 1245–87. 12. S. Gholizadeh, Layout optimization of truss structures by hybridizing cellular automata
and particle swarm optimization, Comput. Struct. 125 (2013) 86–99.
21 Page 21 of 36
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13. K. Deb, An efficient constraint handling method for genetic algorithms, Comput. Meth. Appl.
Mech. Eng. 186 (2-4) (2000) 311–338.
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14. A. Kaveh, V.R. Mahdavai, Colliding Bodies Optimization method for optimum design of truss structures with continuous variables, Adv. Eng. Softw. 70 (2014) 1–12.
us
15. A. Kaveh, V.R. Mahdavai, Colliding-Bodies Optimization for Truss Optimization with Multiple Frequency Constraints, J. Comput. Civ. Eng. (2014) 10.1061/(ASCE)CP.1943-5487. 0000402 . Lifshitz, Publisher: Butterworth: 224, 1976.
an
16. Landau LD. Mechanics: Volume 1 (Course of Theoretical Physics) Author: LD Landau, EM
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17. A. Kaveh, S. Talatahari, Hybrid charged system search and particle swarm optimization for engineering design problems, Eng. Comput. 28 (4) (2011) 423-440. 18. K.S. Lee, Z.W. Geem, A new meta-heuristic algorithm for continuous engineering
te
(36-38) (2005) 3902–3933.
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optimization: harmony search theory and practice, Comput. Methods Appl. Mech. Eng. 194 19. Z. Michalewicz, M. Schoenauer, Evolutionary algorithms for constrained parameter
Ac ce p
optimization problems, Evol. Comput, 4 (1) (1996) 1–13. 20. C.J. Konzelman, Dual methods and approximation concepts for structural optimization, M.Sc. thesis, Department of Mechanical Engineering, University of Toronto (1986). 21. R. Sedaghati, Benchmark case studies in structural design optimization using the force method, Int. J. Solids Struct. 42 (2006) 5848–5871. 22. M.H. Gomes, Truss optimization with dynamic constraints using a particle swarm algorithm, Expert Sys. Applics. 38 (2011) 957–968. 23. A. Kaveh, A. Zolghadr, Shape and size optimization of truss structures with frequency constraints using enhanced charged system search algorithm, Asian J. Civil Eng. 12 (2011) 487–509. 24. A. Kaveh and A. Zolghadr, Topology Optimization of Trusses Considering Static and Dynamic Constraints Using the CSS, Appl. Soft Comput. No. 5, 13(2013)2727-2734 22 Page 22 of 36
25. W. Lingyun, Z. Mei, W. Guangming, M. Guang, Truss optimization on shape and sizing with frequency constraints based on genetic algorithm, J. Comput. Mech. 25 (2005) 361–368. 26. A. Kaveh and A. Zolghadr, Democratic PSO for truss layout and size optimization with frequency constraints, Comput. Struct. 130(2014)10-21.
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27. A. Kaveh and A. Zolghadr, Performance comparison of nine meta-heuristic algorithms for
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structural optimization with frequency constraints, Adv. Eng. Softw. 76(2014)9-30.
Lee and Geem
Present study
Present study
(x)
[18]
(CBO)
(CBO-PSO)
x1
2.323456
1.945693
1.951629
-0.44847
-0.49955
-0.454256
4.36192
4.385752
4.363343
-0.63008
-0.63714
-0.622787
1.03866
1.045751
1.051898
1.605384
1.590209
1.600022
680.6344
680.6413
680.6465
680.6331
680.6417
N/A
680.735
680.6436
Unavailable
x5
Best objective function
te
x7 Average objective function
Ac ce p
Std Dev
M
x3
x6
Fitness function evaluations
2.333539
1.951242
x2 x4
2.318452
an
Deb [13]
d
Optimal design variables
us
Table 1 Optimal variables obtained by different researchers for the constrained function I
N/A
N/A
0.1040
0.0101
350,070
160,000
36,000
24,000
Table 2 Optimal design variables obtained by different researchers for the constrained function II.
Optimal design variables (x)
Lee and Geem
Present study
Present study
[18]
(CBO)
(PSO-CBO)
2.155225
2.142755
2.163967
x2
2.407687
2.441786
2.387446
x3
8.778069
8.772559
8.761691
x4
5.102078
5.089189
5.070258
x5
0.967625
0.976804
0.987816
x6
1.357685
1.36545
1.413288
x1
Deb [13]
Unavailable
23 Page 23 of 36
1.287760
1.261765
1.303916
x8
9.800438
9.778372
9.813183
x9
8.187803
8.196755
8.221302
x10
8.256297
8.362651
8.284834
24.36679
24.38470
24.31977
Best objective function
24.37248
Average objective function
24.40940
N/A
24.86188
N/A
N/A
0.580431
350,070
230,000
100,000
Std Dev
24.34516 0.02794 80,000
cr
Fitness function evaluations
ip t
x7
Element group
Konzelman
Sedaghti
Gomes
[20]
[21]
[22]
Kaveh and Zolghadr [23] Standard
2.987
5-12
7.932
7.932
7.849
13-16
0.645
0.645
0.645
17-18
0.645
0.645
0.645
19-22
8.056
8.056
8.765
23-30
8.011
8.011
31-34
0.645
0.645
35-36
0.645
0.645
37-40
12.812
12.812
41-48
8.061
8.061
49-52
0.645
53-54
0.645
59-66 67-70 71-72
Best Weight (kg) Average (kg) Std dev
Weight
No. of analyses
CBO-PSO
2.528
2.522
3.294
3.490
8.704
9.109
8.575
7.821
0.645
0.648
0.645
0.645
0.645
0.645
0.645
0.645
8.283
7.946
9.044
8.060
8.153
7.888
7.703
7.579
7.994
0.645
0.645
0.647
0.645
0.645
0.645
0.645
0.6456
0.645
0.645
13.450
14.666
13.465
12.283
12.894
8.073
6.793
8.250
8.080
7.909
0.645
0.645
0.645
0.645
0.645
0.645
0.645
0.645
0.645
0.646
0.645
0.645
17.279
17.279
16.684
16.464
18.368
16.781
16.775
8.088
8.088
8.159
8.809
7.053
7.635
8.044
0.645
0.645
0.645
0.645
0.645
0.645
0.645
te
Ac ce p
55-58
CBO
CSS
M
3.499
Present Work
d
3.499
Enhanced
an
CSS
1-4
us
Table 3. Optimal cross-sectional areas (cm2) for the 72-bar space truss.
0.645
0.645
0.645
0.645
0.646
0.645
0.645
327.605
328.823
328.814
328.814
328.393
325.4593
324.3779
-
-
332.24
337.70
335.77
329.7996
325.7175
-
-
4.23
5.42
7.20
5.8408
1.553
-
-
42,840
4,000
6,000
6,000
6,000
Table 4 Element grouping for the 52-bar space truss.
Group number
Elements
1
1-4
2
5-8
3
9-16
24 Page 24 of 36
17-20
5
21-28
6
29-36
7
37-44
8
45-52
ip t
4
Table 5 Cross-sectional areas and nodal coordinates obtained by different researchers for the 52-bar space truss. Initial
Lingyun et al.
Gomes
Kaveh and
[25]
[22]
Zolghadr
5.8851
5.5344
5.2716
XB(m)
2.000
1.7623
2.0885
ZB(m)
5.700
4.4091
XF(m)
4.000
CBOPSO
6.0766
5.7274
1.5909
2.0000
2.1553
3.9283
3.7039
3.9106
3.6114
3.4406
4.0255
3.5595
4.000
3.8954
4.500
3.1874
A1(cm )
2.0
1.0000
A2(cm2)
2.0
2.1417
A3(cm )
2.0
1.4858
A4(cm2)
2.0
1.4018
A5(cm )
2.0
A6(cm2)
us
ZA(m)
6.000
CBO
an
[23]
Present Work
cr
Variable
2.5757
2.4240
1.0475
0.3696
1.0464
1.0580
1.0074
4.1912
1.7295
1.3974
1.2674
1.5123
1.6507
1.4299
1.3093
1.5620
1.5059
1.5275
1.4559
1.9110
1.9154
1.7210
1.5652
1.4228
2.0
1.0109
1.1315
1.0020
1.0006
1.0067
A7(cm )
2.0
1.4693
1.8233
1.7415
1.4244
1.6086
A8(cm2)
2.0
2.1411
1.0904
1.2555
1.3753
1.3839
236.046
228.381
205.237
199.066
195.721
-
-
234.3
213.101
202.138
202.501
-
-
5.22
7.391
6.235
4.290
-
-
11,270
4,000
6,000
6,000
2
2
2
Average Weight (kg)
Ac ce p
Std dev
te
338.69
Best Weight (kg)
d
2
M
2.4575
ZF(m)
No. of analyses
Table 6 Optimal cross-sectional areas (cm2) for the 200-bar planar truss.
Element number
Members in the group
Kaveh and Zolghadr
Present Work
[17] Standard
CSS-
CSS
BB-BC
1, 2, 3, 4
1.2439
0.2934
0.3268
0.2797
2
5, 8, 11, 14, 17
1.1438
0.5561
0.4502
0.6968
3
19, 20, 21, 22, 23, 24
0.3769
0.2952
0.1000
0.1000
4
18, 25, 56, 63, 94, 101, 132, 139, 170, 177
0.1494
0.1970
0.1000
0.1000
5
26, 29, 32, 35, 38
0.4835
0.8340
0.7125
0.5796
6
6, 7, 9, 10, 12, 13, 15, 16, 27, 28, 30, 31, 33, 34, 36, 37
0.8103
0.6455
0.8029
0.8213
1
CBO
CBO-PSO
25 Page 25 of 36
39, 40, 41, 42
0.4364
0.1770
0.1028
0.1279
8
43, 46, 49, 52, 55
1.4554
1.4796
1.4877
1.0152
9
57, 58, 59, 60, 61, 62
1.0103
0.4497
0.100
0.100
10
64, 67, 70, 73, 76
2.1382
1.4556
1.0998
1.5647
11
44, 45, 47, 48, 50, 51, 53, 54, 65, 66, 68, 69, 71, 72, 74,
0.8583
1.2238
0.8766
1.6465
0.1229
0.2296
2.9058
2.9007
0.100
0.100
3.5535
3.0133
1.3360
1.7175
1.6142
0.6289
an
0.100
0.2755
6.2338
5.0951
2.5793
0.6062
0.1102
0.100
3.0520
5.4393
5.8959
5.5172
1.8121
1.8435
2.1858
2.2032
1.2986
0.8955
0.5249
0.8659
77, 78, 79, 80
1.2718
0.2739
13
81, 84, 87, 90, 93
3.0807
1.9174
14
95,96, 97, 98, 99, 100
0.2677
0.1170
15
102, 105, 108, 111, 114
4.2403
3.9952
16
82, 83, 85, 86, 88, 89, 91, 92, 103, 104, 106, 107, 109,
2.0098
17
115, 116, 117, 118
18
119, 122, 125, 128, 131
4.8335
5.9423
19
133, 134, 135, 136, 137, 138
20
140, 143, 146, 149, 152
21
120, 121, 123, 124, 126, 127, 129, 130, 141, 142, 144,
110, 112, 113
M
d
145, 147, 148, 150, 151
1.5956
cr
12
us
75
ip t
7
153, 154, 155, 156
23
157, 160, 163, 166, 169
5.8810
8.1759
7.2676
7.6477
24
171, 172, 173, 174, 175, 176
0.2324
0.3209
0.1278
0.100
178, 181, 184, 187, 190
7.7536
10.98
7.8865
8.1273
158, 159, 161, 162, 164, 165, 167, 168, 179, 180, 182,
2.6871
2.9489
2.8407
2.9665
191, 192, 193, 194
12.5094
10.5243
11.7849
10.2386
195, 197, 198, 200
29.5704
20.4271
22.7014
20.6364
196, 199
8.2910
19.0983
7.884
11.6468
Weight (kg)
2559.86
2298.61
2203.212
No. of
10,000
10,000
9,000
26
Ac ce p
25
te
22
183, 185, 186, 188, 189
27 28 29
2195.469 9,000
analyses
26 Page 26 of 36
ip t cr
an
us
(a)
Ac ce p
te
d
M
(b)
(c)
Fig. 1 The collision between two objects, (a) before collision (b) same time collision (c) after collision.
Stationary
X1
X2
Moving
CBs
Xn-1
Xn
Xn+1
CBs
X2n
Fig. 2 The sorted CBs in an increasing order and the mating process for the collision.
27 Page 27 of 36
ip t cr us an
Ac ce p
te
d
M
(a)
(b) Fig. 3 The convergence curves for the constrained function I. (a) all iterations (b) 200-600 iterations.
28 Page 28 of 36
ip t cr us an
Ac ce p
te
d
M
(a)
(b) Fig. 4 The convergence curves for the constrained function II. (a) all iterations (b) 600-1000 iterations.
29 Page 29 of 36
ip t cr us an M
Ac ce p
te
d
Fig. 5 A 72-bar space truss
(a)
30 Page 30 of 36
ip t cr us an M (b)
Ac ce p
te
d
Fig. 6 The convergence curves for the 72-bar space truss example. (a) all iterations (b) 150-200 iterations.
31 Page 31 of 36
ip t cr us an M d te Ac ce p
(a) Top view
(b) Side view
32 Page 32 of 36
an
us
cr
ip t
Fig. 7 A 52-bar space truss.
Ac ce p
te
d
M
(a)
(b) Fig. 8 The convergence curves for the 52-bar space truss example. (a) all iterations (b) 200-200 iterations.
33 Page 33 of 36
ip t cr us an
Ac ce p
te
d
M
(a)
(b) Fig. 8 The convergence curves for the 52-bar space truss example. (a) all iterations (b) 200-300 iterations.
34 Page 34 of 36
ip t cr us an M d te Ac ce p
Fig. 9 A 200-bar planar truss
35 Page 35 of 36
ip t cr us an
Ac ce p
te
d
M
(a)
(b) Fig 10 The convergence curves for 200-bar planar truss. (a) all iterations (b) 200-300 iterations.
36 Page 36 of 36
S1568-4946(15)00310-5 http://dx.doi.org/doi:10.1016/j.asoc.2015.05.010 ASOC 2963
To appear in:
Applied Soft Computing
Received date: Revised date: Accepted date:
10-12-2014 8-5-2015 10-5-2015
Please cite this article as: A. Kaveh, V.R. Mahdavi, A hybrid CBO-PSO algorithm for optimal design of truss structures with dynamic constraints, Applied Soft Computing Journal (2015), http://dx.doi.org/10.1016/j.asoc.2015.05.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Ac ce p
te
d
M
an
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cr
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Graphical Abstract
1 Page 1 of 36
Highlight of the paper -
A new hybrid CBO–PSO algorithm is presented by adding positive properties of the PSO algorithm to the CBO. The CBO is a recently developed meta-heuristic algorithm which does not use the
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-
internal parameter and memory in its formulation.
In the CBO-PSO the memory of the PSO is added to the CBO to improve the
cr
-
performance of the latter algorithm.
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The new algorithm is compared to other advanced meta-heuristic methods to illustrate the
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d
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effectiveness.
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2 Page 2 of 36
Revised for Applied Soft Computing on 7th of May 2015
A hybrid CBO-PSO algorithm
A. Kaveh*, V.R. Mahdavi
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ABSTRACT
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for optimal design of truss structures with dynamic constraints
The vibration domain of structures can be reduced by imposing some constraints on their natural
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frequencies. For this purpose optimal design of structures under frequency constraints is required which involves highly non-linear and non-convex problems. In this paper an efficient hybrid
an
algorithm is developed for solving such optimization problems. This algorithm utilizes the recently developed colliding bodies optimization (CBO) algorithm as the main engine and uses
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the positive properties of the particle swarm optimization (PSO) algorithm to increase the efficiency of the CBO. The distinct feature of the present hybrid algorithm is that it requires no parameter tuning. The CBO is known for being parameter independent, and avoiding the use of
d
the traditional penalty method to handle the constraints upholds this property. Two mathematical
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constrained functions taken from the literature are studied to verify the performance of the algorithm. The algorithm is then applied to optimize truss structures with frequency limitations.
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The numerical results demonstrate the efficiency of the presented algorithm for this class of problems.
Keywords: Colliding bodies optimization, Particle swarm optimization, Penalty approach, Structural design problems, Dynamic constrains.
A.Kaveh* (□), Professor of Structural Engineering Centre of Excellence for Fundamental Studies in Structural Engineering, Department of Civil Engineering, Iran University of Science and Technology, Tehran, Iran. E-mail: [email protected] (A. Kaveh) V.R. Mahdavi, PhD Student School of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran
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1. Introduction Optimization algorithms are utilized to maximize or minimize one/some objective functions under some specific limitations. In optimal design of structures, the main objective is to minimize the weight of the structures under some constraints, such as stresses and displacements.
ip t
For structures subjected to dynamic excitations, the natural frequencies play important role in the dynamic response of structures and resonance phenomenon. In recent years, optimal design of
cr
structures under frequency limitations has been extensively investigated using different optimization algorithms by many researchers. This problem is high non-linear and non-convex
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with respect to the design variables [1].
Algorithms for optimization can be divided into two general categories of Gradient-based methods and metaheuristics. The formulation of metaheuristic algorithms is often inspired by
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either natural phenomena or physical laws. Every metaheuristic algorithm consists of two phases: exploration of the search space and exploitation of the best solutions found. One of the
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main problems in developing a good metaheuristic algorithm is to keep a reasonable balance between the exploration and exploitation abilities [2, 3]. Some of the well-known metaheuristic
d
algorithms can be found in [4-9]. One of the difficulties with most of the metaheuristic algorithms is parameter tuning which is often time consuming. This problem becomes more
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severe when the parameters of penalty method for constraint handling are added. This is because the feasible solutions are extremely sensitive to setting the parameters. In practical structural
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design tasks, designers usually have limited knowledge about the appropriate values of these parameters. Therefore, the selection of proper values for the parameters and utilizing penalty approach becomes a tedious process.
One of the recently developed metaheuristics is the colliding bodies optimization (CBO) algorithm originated by the present authors [10]. Compared to conventional meta-heuristic optimization algorithms, simple formulation and no necessity for parameter tuning are the main advantages of this algorithm. On other hand, the exploitation phase of this algorithm is weak due having no memory for saving the best-so-far solution in its formulation (i.e. the best position of agents from the previous iterations). Particle swarm optimization (PSO) developed by Eberhart and Kennedy [4] is also a popular metaheuristic algorithm which is based on swarm intelligence. In this algorithm, the agents (particles) improve their position by using their data from the past generations. 4 Page 4 of 36
The main objective of the present study is to optimize the weight of truss structures with natural frequency constraints. Thus, an efficient and no parameter tuning algorithm is proposed based on hybridization of CBO and PSO algorithms for constrained optimization problem. In this algorithm, the used memories of the PSO, i.e. the local and global best, is added to the CBO. exploitation and the convergence rate for the best found solution.
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Thus in the new algorithm the bodies move toward the best saved bodies increasing the
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The present paper is organized as follows: In the next section, formulation of the optimal design of truss structures with frequency limitations is presented. In section 3, both standard
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algorithms are briefly introduced and the hybrid method is presented. This is followed by a section consisting of the study of optimization of two mathematical constrained functions and 5, respectively.
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2. Formulation of optimal design of structures
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three truss structures. A discussion and brief concluding remarks are presented in sections 4 and
Optimization problems can be classified as multi-objective and mono-objective. In structural
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optimization problems, the main objectives are minimizing the weight or cost of the structures
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while satisfying some constraints. In this class of optimization problems, the design variables can be considered as the cross-sectional areas of elements and/or the coordinates of nodes. The
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optimization problem for a truss structure can be stated as follows: Find
X = [x1,x2,x3,..,xN] ne
minimizing
W ( X ) i Ai li
(1)
i 1
satisfying
gj(X)0, j=1,2,…,m xlmin ≤ xl ≤ xlmax
where X is the vector of all design variables with N unknowns; W is the weight of truss structure; ρi, Ai and li are mass density, cross sectional area and length of the ith member, respectively; ne is number of the structural elements; gj is the jth constraint from m inequality constraints. Also, xlmin and xlmax are the lower and upper bounds of design variable vector, respectively.
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Since the solution of this multi-objective problem is rather complex, most researchers have considered this problem as mono-objective and the penalty function approach has been used for handling the constraints. Coello [11] compared different constraint-handling methods used in metaheuristic algorithms with comments on their advantages and disadvantages. These methods
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can generally be classified into five groups: penalty functions, special representations and operators, repair algorithms, separation of objectives and constraints, and hybrid methods. The
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most common and convenient method in metaheuristic algorithms to handle constraints is to use penalty functions. In this method, a constrained optimization problem is transformed into an
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unconstrained one by multiplying a certain value to objective function based on the amount of constraint violation appears in a problem. Therefore the merit (or pseudo objective) function
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which should be minimized is defined as: Mer(X) = W(X) ×fpenalty(X)
(2)
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where Mer(X) is the merit function; W(X) is the objective function (weight of truss structure); fpenalty(X) is the penalty function which results from the violations of the constraints
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corresponding to the response of the structure: m
f penalty ( X ) 1 p max(0, g j ( X )) 1 p G ( X )
(3)
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j 1
where γp is penalty parameter, and G(X) is the constraint function. In dynamic constraint
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problems the behavioral constraints are the restricted natural frequencies defined as follows:
n ( x) u n , g n ( x)
n ( x) 1 0 u n
for some natural frequencies n
( x) m ( x) lm , g m ( x) 1 m 0 for some natural frequencies m u m
(4)
where ωn and ωm are the nth and mth natural frequency, respectively; ωln and ωun are the lower bound and upper bound of the nth and mth natural frequency, respectively. As can be seen from Eq. (3), the penalty function method consists of a penalty parameter where the feasible solutions are dependent on right tuning of these parameters. By setting small values of the penalty parameter, the pseudo objective function becomes less sensitive to optimization constraints and the optimization process may converge to a violated solution. Conversely, setting the penalty parameter to a large value leads to amplify the effect of 6 Page 6 of 36
optimization constraints and the optimization process may get stuck in a local optimum. This dependency of the optimization algorithms performance on penalty parameters has led researchers to devise efficient method for constraint handling [12]. In this paper, we present a constraint handling method that does not dependent on any
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parameter tuning. The pair-wise comparison method which presented in Ref. [13] has been adopted in the presented algorithm. In this method, for ranking and comparison of the agents (or
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populations), three criteria are enforced: In comparison of (i) two feasible/inviolate agents, the one with better objective function value is preferred, (ii) one feasible/inviolate and one
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infeasible/violate agent, the feasible agent is preferred, and (iii) two infeasible/violate agents, the one with smaller constraint violation is preferred. features of the standard CBO and PSO algorithms
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In the next section the new CBO-PSO based hybrid algorithm is presented using the positive
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3. CBO-PSO based hybrid optimization algorithm
Inspired by the natural features of the CBO and PSO algorithms, we develop an efficient
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hybrid algorithm (CBO-PSO) in this section. In the following, both standard CBO and PSO
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algorithms are briefly introduced and then the proposed hybrid method is presented. 3.1. Colliding Bodies Optimization algorithm
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Colliding bodies optimization (CBO) is a recently efficient and robust metaheuristic optimization algorithm for simple implementation of complex computations. As CBO is simple to be implemented and can deal with complex problems without extensive mathematical computations and parameter tuning, it is widely used in various fields of optimization problems [14, 15]. The CBO mimics a 1-dimensional collision between two colliding bodies (CBs). In this case, each CB collides distinctly with its pair and updates its new velocity based on momentum and energy conservation law for 1-dimensional collision [16]. According to this law, the velocities of two bodies after a one-dimensional collision can be obtained as:
v1'
(m1 m2 )v1 (m2 m2 )v2 m1 m2
(5)
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v2'
(m2 m1 )v2 (m1 m1 )v1 m1 m2
(6)
where v1 is the velocity of the first object before impact, v2 is the velocity of the second
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object before impact, v1' is the velocity of the first object after impact, v2' is the velocity of the second object after impact, m1 is the mass of the first object, m 2 is the mass of the
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second object, is the coefficient of restitution (COR) of two colliding bodies, defined as the ratio of relative velocity of separation to the relative velocity of approach.
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The CBO algorithm consists of steps that are briefly described as follows:
Similar to other multi-agent methods, CBO has a number of individuals (or agents),
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named Colliding Body (CB), consisting of the variables of the problem. Each CB is treated as an object with specified mass and velocity and collides to others; after collision, each CB
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moves to a new position with new velocity (Fig. 1). CBO starts with an initial population consisting of 2n parent individuals created by means of a random initialization. Then, CBs are sorted in ascending order based on the value of cost function as shown in Fig. 2a. The sorted
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CBs are divided equally in to two groups. The first group is stationary and consists of good
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agents. This set of CBs is stationary and velocity of these bodies before collision is zero. The second group consists of moving agents which move toward the first group. Then (see Fig.
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2b), the better and worse CBs, i.e. agents with upper fitness value, of each group will collide together to improve the positions of moving CBs and to push stationary CBs towards better positions. The change of the body position represents the velocity of CBs before collision as: i 1,..., n 0, Vi X i n X i , i n 1,...,2n
where
and
(7)
are the velocity and position vectors of the ith CB, respectively. 2n is the
number of population size. After the collision, the velocity of bodies in each group is evaluated using momentum and energy conservation law and the velocities before collision (Eq. (7)). The velocity of CBs after the collision is: 8 Page 8 of 36
(mi n mi n )Vi n , i 1,..., n m m i i n Vi ' (mi mi n )Vi , i n 1,...,2n mi mi n
and
ip t
where
(8)
are the velocities of the ith CB before and after the collision, respectively; mi is
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the mass of the ith CB and defined as:
(9)
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1 fit (k ) mk n , k 1,2,...,2n 1 i 1 fit (i )
where fit(i) represents the objective function value of the ith agent. Obviously a CB with good
objective function, the term
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values exerts a larger mass and fewer moves than the bad ones. Also, for maximizing the 1 is replaced by fit (i ) . is the coefficient of restitution (COR) fit (i)
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and is defined as the ratio of the separation velocity of two agents after collision to approach
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velocity of two agents before collision. In this algorithm, this index is defined to control of the exploration and exploitation rates. For this purpose, COR decreases linearly from unit value to
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zero. Thus, is defined as:
1
iter itermax
(10)
where iter is the actual iteration number and itermax is the maximum number of iterations. COR having unit value and zero value represent the global and local search, respectively. In this way a good balance between the global and local search is achieved by increasing the number of iterations. The new positions of the CBs are evaluated using the generated velocities after the collision in position of stationary CBs:
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X i rand Vi ' , i 1,..., n X i n rand Vi ' , i n 1,...,2n
where
and
(11)
are the new position and the new velocity after the collision of the ith CB,
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X
new i
cr
respectively. 3.2. Particle Swarm Optimization
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The PSO is based on a metaphor of social interaction such as bird flocking and fish schooling, and is developed by Eberhart and Kennedy [4]. The PSO simulates a commonly
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observed social behavior, where members (particles) of a group (swarm) tend to follow the lead of the best of the group.
In this algorithm, position of the particles and their associated velocities are initialized with
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random positions. Then, the particles fly through the search space and their positions are updated k based on the best positions of individual particles denoted by p i and the best position among all
(12)
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X i ( k 1) X i ( k ) Vi (k 1)
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particles in the search space represented by p gk :
(13)
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Vi (k 1) Vi (k ) C1 r1 ( Pi (k ) X i (k )) C2 r2 ( Pg (k ) X i (k )
where X i (k ) and Vi (k ) are the position and velocity for the ith particle at iteration k; is an inertia weight to control the influence of the previous velocity; r1, and r2 are two random numbers uniformly distributed in the range of (0, 1); c1 and c2 are the cognitive and social scaling parameters, respectively; pi (k ) is the best position of the ith particle up to iteration k; pg (k ) is the best position among all particles in the swarm up to iteration k and the sign “ ” denotes element-by-element multiplication [17]. 3.3 A hybrid CBO-PSO based algorithm In this study, CBO and PSO are hybridized leading to a new optimization search strategy. The main algorithm is based on the CBO, where some features of the PSO are added. Here, three features are added to the formulation of the standard CBO: i) the stationary CBs move also 10 Page 10 of 36
toward better positions, ii) the CBs move toward local best, i.e. pi is added which is used in the PSO, iii) the old velocities are added to the new velocities of CBs before collision. In fact, the exploitation ability of CBO algorithm increases by adding the saved particles. The CBO-PSO algorithm can simply be described as follows:
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Step 1: Initialization: Initial position, velocities and best position of populations are created by means of a random initialization:
,
and
j 1,2,...2n
(14)
are the ith element of position, velocity and best position of the jth and
are the lower and upper bounds for the ith decisions variable,
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population, respectively;
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Where
&
cr
pii j xij [ xiL , xiU ] , i 1,2,...N j vi (arbitrary domain)
respectively; N and 2n are the number of decision variable and population size, respectively. Step 2: Arrangement of populations: As mentioned before, the arrangement of the CBs is
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performed such that three criteria are satisfied: In comparison of (i) two inviolate populations, the one with better objective function value is selected, (ii) one inviolate and one violate
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population, the inviolate population is selected, and (iii) two infeasible populations, the one with smaller constraint violation is preferred. For this purpose a matrix similar to Eq. (15) is
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constructed using the position, velocity and best local vectors, as well as with the pseudo
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objective function of each position vector F(X):
x1 12 x1 2n x1
x12 x22
x1N v11 xN2 v12
v12 v22
v1N pi11 vN2 pi12
pi21 pi22
x22 n
xN2 n v12 n
v22 n
vN2 n pi12 n
pi22 n
piN2 n F ( X 2 n ) 1 pi1N F ( X ) 2 piN2 F ( X )
(15)
This matrix is sorted in ascending order of pseudo objective function, that is . The pseudo objective function is defined to compare the feasible and/or infeasible population based on the three mentioned criteria as:
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f (X ) F(X ) f max G ( X )
if
G( X ) 0,
(16)
otherwise,
where f(X) and G(X) are the objective and constraint functions which are defined in Eqs. (1, 3); and fmax is the objective function of the worse inviolate CB in the current iteration. Thus, the
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fitness of an infeasible population not only depends on the amount of constraint violation, but also on the population of solutions at hand. However, the fitness of a feasible solution is always
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population, pi, is evaluated based the above mentioned criteria.
cr
fixed and is equal to its objective function value. It should be noted that the best position of
Step 3: Mating process: The CBs are divided to two parts as shown in Fig. 2. The first half
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and second half of this partition named as part 1 and part 2 agents, respectively. Step 4: Evaluation of velocities before collision: The part 1 and part 2 agents toward the best
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position of CBs of part 1 agents. The old velocities are also added to new velocities. Therefore, the velocity of CBs before collision is derived using:
and
are the velocity and position vectors of the jth CB in the kth iteration,
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where
(17)
te
d
V j (k ) ( Pi j (k ) X j (k )), j 1,..., n V (k 1) j j n j j n 1,...,2n V (k ) ( Pi (k ) X (k )), j
respectively;
is the best position vector of the jth CBs up to iteration k.
Step 5: Evaluation of velocities after collision: Compute the velocity of CBs after the collision using Eqs. (5, 6) and Eq. (17):
(m j m j n ) * V j (k 1) (m j n m j n ) * V j (k 1) , j 1,..., n m j m jn m j m jn 'j V (k 1) j j n j j n j n j n (m m ) * V (k 1) (m m ) * V (k 1) , j n 1,...,2n m j m jn m j m j n
(18)
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where
and
respectively;
are the velocity vectors of the jth CB after and before the collision,
is the mass of the ith CB which is defined as Eq. (9). is the coefficient of
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restitution (COR) and is defined as Eq. (10).
following expression: Pi j (k ) rand V ' j (k 1) , j 1,..., n X (k 1) j n Pi (k ) rand V ' j (k 1) , j n 1,...,2n
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j
cr
Step 6: Evaluation of new position: The new positions of the CBs are evaluated using the
(19)
an
where rand is a random number uniformly distributed in the range of (-1, 1);
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Step 7: Updating the velocities: The velocities of the CBs are updated as:
(20)
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V j (k 1) X j (k 1) X j (k )
Step 8: Termination criteria: The optimization is repeated from Step 2 until a termination
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criterion, as the maximum number of iteration, is satisfied.
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As it can be seen in equations, by sorting the populations based on feasible or infeasible solutions, the formulation of proposed algorithm needs no parameter setting or tuning. The following section provides examples and comparisons for validation of the presented hybrid algorithm.
4. Numerical examples
In this section the efficiency of the proposed algorithm, CBO-PSO, is studied through two mathematical function examples and three well-studied truss structures with frequency limitations taken from the optimization literature. Examples 1 and 2 show the applicability of the CBO-PSO for optimization of constraint problems. In example 3 a spatial truss structure is studied for finding the optimal cross sections. In example 4 the performance of the CBO-PSO is studied for finding the optimal size and shape of a spatial truss structures. In the last example, a planar truss structure with many variables is selected to show the importance of selection of 13 Page 13 of 36
optimization algorithm in the final optimal weight. These examples are independently optimized 20 times. The algorithm was coded in MATLAB, and structural analysis was performed with the direct stiffness method. A comparison study of the obtained results is performed for the
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considered examples and the next section consists of the discussions on the results. 4.1. Example 1: Constrained function I
cr
Optimization of the constrained function expressed in Eq. (22) is considered as the first example. This problem also has seven variables and four nonlinear inequality constraints. This
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problem is defined as:
an
Find
To minimize
(22)
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d
f ( x) ( x1 10) 2 5( x2 12) 2 x34 3( x4 11) 2 10 x56 7 x62 x74 4 x6 x7 10 x6 8 x7
Subjected to:
(21)
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x1 , x2 , x3 , x4 , x5 , x6 , x7
g1 ( x) 127 2 x12 3 x24 x3 4 x42 5 x5 0, g 2 ( x) 282 7 x1 3 x2 10 x32 x4 x5 0,
(23)
g 3 ( x) 196 23 x1 x22 6 x62 8 x7 0, g 4 ( x) 4 x12 x22 3 x1 x2 2 x32 5 x6 11x7 0.
The bounds on the design variables are:
10 xi 10
(i 1 7 )
(24)
This problem has been solved in the literature using various global optimization techniques [13, 18]. Taking into account the probabilistic nature of metaheuristic algorithms, a number of independent runs were performed. In this example, the population size of the CBO-PSO and 14 Page 14 of 36
CBO are set to 40 and 60 individuals, respectively, and the maximum number of optimization iterations was limited to 600. The best solution vectors and statistical result of these runs obtained by CBO and CBO-PSO are compared with those obtained by other metaheuristic algorithms in Table 1. As it can be seen
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from this table, the best solution obtained by CBO-PSO is better than CBO and those quoted for the other algorithms. However, the CBO-PSO, CBO and HS needs 24,000, 36,000 and 160,000
cr
fittness function evalutions to find the optimum result, respectively. The objective function and constraint function of feseable population against generations are plotted in both sides of Fig. 3.
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As it can be seen, the constraint function is equal zero and hense the constrints are satisfed in the
4.2 Example 2: Constrained function II
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final iterations.
This is a 10-variable challenging optimization problem. The problem has also eight nonlinear
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inequality constraints, and can be defined as:
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To minimize
(25)
te
x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10
d
Find
f ( x) x12 x22 x1x2 14x1 16x2 ( x3 10)2 4( x4 5)2 ( x5 3)2 2( x6 1)2 5 x72 7( x8 11)2 2( x9 10)2 ( x10 7)2 45
(26)
Subjected to:
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g1 ( x) 105 4 x1 5 x2 3x7 9 x8 0, g 2 ( x) 10 x1 8 x2 17 x7 2 x8 0, g3 ( x) 8 x1 2 x2 5 x9 2 x10 12 0, g 4 ( x) 3( x1 2) 2 4( x2 3) 2 2 x32 7 x4 120 0,
(27)
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g5 ( x) 5 x12 8 x2 ( x3 6) 2 2 x4 40 0, g 6 ( x) x12 2( x2 2) 2 2 x1 x2 14 x5 6 x6 0, g 7 ( x) 0.5( x1 8) 2 2( x2 4) 2 3x52 x6 30 0,
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cr
g8 ( x) 3x1 6 x2 12( x9 8) 2 7 x10 0. The bounds on the design variables are:
(i 1 10)
an
10 xi 10
(28)
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This problem has been solved by Michalewicz and Schoenauer [19] using GA-based
Ac ce p
te
d
methods. Deb [13] utilzed also an efficient constraint handling method for the GA, and Lee and
Geem [18] employed the harmony search algorithm. In this case, CBO-PSO and CBO population sizes were set to 80 and 100 individuals, respectively. The maximum number of optimization iterations was set as 1,000.
Table 2 compares the optimized variables, the corresponding objective function and statistical results obtained by CBO and CBO-PSO with those obtained by other existing metaheuristic algorithms. It can be seen that the proposed algorithm is the best among the four standard algorithms. The CBO-PSO needs 80,000 objective function evaluations to find the optimum result while it is 100,000 and 230,000 for the CBO and HS as reported, respectively. Similar to the first example, the objective function and constraint function of feasible population versus generations are plotted in Fig. 4.
4.3 Example 3: A 72-bar space truss 16 Page 16 of 36
In this example a structural designer intends to evaluate the cross sections of a 72-bar space steel truss structure shown in Fig. 5 such that its weight becomes minimum, while the first two frequencies are limited to some specified values. The structural elements are also labeled in Fig. 5. The following values are used for material properties of the steel: the material density is 2770
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kg/m3 and the modulus of elasticity is 69,800 MPa. Four non-structural masses of 2270kg are attached to the nodes 1-4. This design case addresses a continuous treatment of the problem, in
cr
which the independent size variables are chosen from a real-valued design interval with lower bound equal to 0.645cm2. In this example, the first two natural frequency constraints are imposed
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as 1 4 HZ , 2 6 HZ . This example has been solved previously by Konzelman [20] using a dual method (DM) and by Sedaghati [21] using the force method (FM). Gomes [22] has
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investigated the problem using the PSO. Kaveh and Zolghadr [23, 24] have investigated the problem using the standard and an enhanced CSS. The population size (i.e. number of CBs) and number of iterations are defined as 2n=30 and iteration=200 for both CBO and CBO-PSO.
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Comparison of the optimal design results reported in the literature and those of the present work are detailed in Table 3. The PSO-CBO algorithm found the best weight as 324.377kg after
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6,000 analyses, with the standard deviation of 1.55kg. Though, the number of iteration are more than standard CSS and enhanced CSS, this solution is reported to be the optimum design of the
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problem reached in the present study, and it is the best solution amongst the existing literature
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results. A comparatively better design weight of 325.459kg was determined using CBO. Fig. 6 shows the feasible solution history for the obtained best results of 20 individual run. It can be seen that the best solution found by the proposed algorithm is 324.377kg which is attained at the 179th iterations.
4.4 Example 4: A 52-bar dome-like truss This example considers shape and size optimization of a dome-like space truss structure as shown in Fig. 7. The initial topology and the element numbering of this truss are shown in Fig. 7a. This design has also been investigated in Lingyun et al. [25] using the NHGA algorithm, and Gomes [22] utilizing PSO algorithm. The problem has also been optimized by Kaveh and Zolghadr [23, 26] using the CSS algorithm. The space truss has 52 bars and non-structural masses of m = 50kg are added to the free nodes. The material density is 7800kg/m3 and the modulus of elasticity is 210,000MPa. For this structure, the symmetry around x and y-axes is 17 Page 17 of 36
employed to group the 52 truss members into eight independent size variables, where all members in a group share the same material and cross-sectional properties. Table 4 shows each element group by member numbers. The range of the cross-sectional areas varies from 1 to 10cm2. The shape optimization is performed considering that the
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symmetry is preserved in the process of design. Each movable node is allowed to move ±2m. For the frequency constraints, 1 15.916HZ and 2 28.649HZ are considered. Hence, this truss
cr
optimization problem has 13 design variables (five shape variables plus eight size variables). The population size and number of iterations are defined as 2n=30, and iteration=200 for both CBO
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and CBO-PSO, respectively.
Table 5 compares the optimal cross sections, best weight, mean weight and standard
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deviation of the results obtained using CBO and CBO-PSO with the outcomes of other researches. As anticipated, CBO-PSO has led to a much better results than the others in term of the best result found by the CBO-PSO.
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4.5 Example 5: A 200-bar planar truss
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the best, the mean weight and also standard deviation. Fig. 8 provides the convergence rates of
In this problem, the objective is to achieve the minimum weight design of the 200-bar plane
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truss shown in Fig. 9. This truss has been investigated using the standard CSS and CSS-BB-BC
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algorithms as a frequency constraint weight optimization problem by Kaveh and Zolghadr [23, 27]. The material density and modulus of elasticity of members are 7860kg/m3 and 210,000MPa, respectively. Non-structural masses of 100kg are attached to the upper nodes. A lower bound of 0.1 cm2 is assumed for the cross-sectional areas. In calculating the frequency constraints, the first three frequencies are restricted to 1 5HZ , 2 10HZ and 3 15 HZ . The elements are divided into 29 groups. For this example, the population size and number of iterations are also defined as 2n=30 and iteration=300 for both CBO and CBO-PSO, respectively. The elements grouping and results obtained using this research with those of the other researches are presented in Table 6. According to this table, the result obtained by the CBO-PSO is meaningfully lighter than that of the standard CSS, CSS-BB-BC and CBO algorithms. Fig. 10 shows the convergence curves for the proposed algorithm.
5. Discussion 18 Page 18 of 36
Further examples for the problem of optimal design of truss structures with frequency limitations show that CBO-PSO generally has a better performance than the standard CBO. The main reasons for the improvements of the proposed algorithm can be summarized as follows: Since parameter tuning in meta-heuristic algorithms affects the exploration and exploitation
ip t
process of algorithms, the efficiency of optimization algorithms is completely dependent on the right tuning of these algorithms. In the other hand, CBO is a mate-heuristic algorithm which in
cr
its formulation does not use the internal parameters and memories. Having no internal parameter tuning is a good feature of this algorithm. However, the implementation of penalty approach to
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handle the constraints and therefore tuning of penalty parameter, makes the algorithm to be dependent on the penalty parameter. Consequently, by sorting the populations according to these objective and constraint functions values, one enforces the populations to move rapidly towards
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the feasible solutions such that the constraints are satisfied, and then move toward the point with smaller objective functions. As demonstrated in the convergence curves of the examples, in the
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course of implementing the CBO-PSO algorithm, the value of constraint function decreases with the increase in the number of iterations until it reaches zero. This means that CBO-PSO
d
algorithm has successfully optimized the objective function while satisfying all the specified constraints. It is evidence that the values of the objective and constraint functions do not
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monotonically decrease with the increase of the number of iterations, which is completely different from observation in the common meta-heuristic algorithms using the penalty approach,
Ac ce p
where the value of the objective function (or pseudo objective function) monotonically decreases with the increase of number of iteration. Particularly, in early iterations of the optimization where the constraint function is still unsatisfied, the constraint function causes the optimization process not to solely rely on the objective function values. Therefore, in early iterations the global search is performed in the entire search space to find the feasible solutions and then the local search is achieved in the feasible search space to find the best objective function. Hence, the exploration and exploitation are balanced during the optimization process whiteout setting any parameter. As it can be seen from the details of these figures, in the later iterations, the final optimum value is achieved without violation of constraints. Moreover, a weakness of the CBO is that it does not use the memories in its formulation. In the CBO-PSO algorithm, old velocities and best local of populations are added to its formulation. From results of these examples, it can be 19 Page 19 of 36
concluded that the best, mean and standard division of 20 independent runs for the new hybrid algorithm are better than those of the other algorithms. 6. Concluding remarks
ip t
This study develops a hybridized search algorithm so called CBO-PSO for constrained optimization problems and in particular for finding optimal weight of truss structures with
cr
dynamic limitations .The proposed method is mainly based on collision event concept borrowed from colliding bodies optimization.
us
In comparison with the standard CBO, the hybrid CBO-PSO has the following advantages: 1. In the standard CBO, the better solutions are stationary and the worse solutions move
an
toward the better solutions in each iteration. But, in the CBO-PSO algorithm the bodies of the CBO move toward the saved best particles which are used in the PSO algorithm and collide to these for promoting the exploitation ability of the CBO.
M
2. In standard CBO, the solutions are sorted according to the defined pseudo objective function in the penalty method. But in the CBO-PSO, the solutions are sorted based on the
d
constraints and objective functions values. Consequently the penalty parameters are also
te
vanished.
Concerning the efficiency of the CBO-PSO in comparison to the standard CBO and based on
Ac ce p
the presented numerical examples, the idea of a new sorting method and adding the memory vectors help the algorithm to identify potentially suitable areas in the search space much faster while avoiding the stagnation and being trapped in local optimums. It should be noted that the application of the present method can be extended to other constrained mono- and multiobjective optimization problems.
Acknowledgement: The first author is grateful to Iran National Science Foundation for the support.
References
20 Page 20 of 36
1. S. Gholizadeh, E. Salajegheh, P. Torkzadeh, Structural optimization with frequency constraints by genetic algorithm using wavelet radial basis function neural network, J. Sound Vib. 312 (2008) 316-331. 2. J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press,
ip t
Ann Arbor, USA, 1975.
3. A. Kaveh, Advance in Metaheuristic Algorithms for Optimal Design of Structures, Springer,
cr
Switzerland, 2014.
4. R.C. Eberhart, J. Kennedy, A new optimizer using particle swarm theory, in: Proceedings of
us
the Sixth International Symposium on Micro Machine and Human 557 Science, Nagoya, Japan, 1995.
5. A.H. El-Maleh, A.T. Sheikh, S.M. Sait, Binary particle swarm optimization (BPSO) based
an
state assignment for area minimization of sequential circuits, Appl. Soft Comput. 13 (12) (2013) 4832–4840.
M
6. W. Deng, R. Chen, B. He, Y.Q.L. Liu, F. Yin, J.H. Guo, A novel two-stage hybrid swarm intelligence optimization algorithm and application, Appl. Soft Comput. 16 (10) (2012)
d
1707–1722.
7. M. Dorigo, V. Maniezzo, A. Colorni, The ant system: optimization by a colony of
te
cooperating agents, IEEE Trans. Syst. Man Cybern. B. 26 (1) (1996) 29–41. 8. A. Kaveh, S. Talatahari, A novel heuristic optimization method: charged system search, Acta
Ac ce p
Mech. 213 (3-4) (2010) 267-289.
9. A. Kaveh and M. Khayatazad, A new meta-heuristic method: ray optimization, Comput. Struct. 112-113 (2012) 283–294.
10. A. Kaveh, V.R. Mahdavai, Colliding bodies optimization: A novel meta-heuristic method, Comput. Struct. 139 (2014)18–27. 11. C.A.C. Coelllo, Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art, Comput. Methods Appl. Mech. Eng. 191 (2002) 1245–87. 12. S. Gholizadeh, Layout optimization of truss structures by hybridizing cellular automata
and particle swarm optimization, Comput. Struct. 125 (2013) 86–99.
21 Page 21 of 36
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13. K. Deb, An efficient constraint handling method for genetic algorithms, Comput. Meth. Appl.
Mech. Eng. 186 (2-4) (2000) 311–338.
cr
14. A. Kaveh, V.R. Mahdavai, Colliding Bodies Optimization method for optimum design of truss structures with continuous variables, Adv. Eng. Softw. 70 (2014) 1–12.
us
15. A. Kaveh, V.R. Mahdavai, Colliding-Bodies Optimization for Truss Optimization with Multiple Frequency Constraints, J. Comput. Civ. Eng. (2014) 10.1061/(ASCE)CP.1943-5487. 0000402 . Lifshitz, Publisher: Butterworth: 224, 1976.
an
16. Landau LD. Mechanics: Volume 1 (Course of Theoretical Physics) Author: LD Landau, EM
M
17. A. Kaveh, S. Talatahari, Hybrid charged system search and particle swarm optimization for engineering design problems, Eng. Comput. 28 (4) (2011) 423-440. 18. K.S. Lee, Z.W. Geem, A new meta-heuristic algorithm for continuous engineering
te
(36-38) (2005) 3902–3933.
d
optimization: harmony search theory and practice, Comput. Methods Appl. Mech. Eng. 194 19. Z. Michalewicz, M. Schoenauer, Evolutionary algorithms for constrained parameter
Ac ce p
optimization problems, Evol. Comput, 4 (1) (1996) 1–13. 20. C.J. Konzelman, Dual methods and approximation concepts for structural optimization, M.Sc. thesis, Department of Mechanical Engineering, University of Toronto (1986). 21. R. Sedaghati, Benchmark case studies in structural design optimization using the force method, Int. J. Solids Struct. 42 (2006) 5848–5871. 22. M.H. Gomes, Truss optimization with dynamic constraints using a particle swarm algorithm, Expert Sys. Applics. 38 (2011) 957–968. 23. A. Kaveh, A. Zolghadr, Shape and size optimization of truss structures with frequency constraints using enhanced charged system search algorithm, Asian J. Civil Eng. 12 (2011) 487–509. 24. A. Kaveh and A. Zolghadr, Topology Optimization of Trusses Considering Static and Dynamic Constraints Using the CSS, Appl. Soft Comput. No. 5, 13(2013)2727-2734 22 Page 22 of 36
25. W. Lingyun, Z. Mei, W. Guangming, M. Guang, Truss optimization on shape and sizing with frequency constraints based on genetic algorithm, J. Comput. Mech. 25 (2005) 361–368. 26. A. Kaveh and A. Zolghadr, Democratic PSO for truss layout and size optimization with frequency constraints, Comput. Struct. 130(2014)10-21.
ip t
27. A. Kaveh and A. Zolghadr, Performance comparison of nine meta-heuristic algorithms for
cr
structural optimization with frequency constraints, Adv. Eng. Softw. 76(2014)9-30.
Lee and Geem
Present study
Present study
(x)
[18]
(CBO)
(CBO-PSO)
x1
2.323456
1.945693
1.951629
-0.44847
-0.49955
-0.454256
4.36192
4.385752
4.363343
-0.63008
-0.63714
-0.622787
1.03866
1.045751
1.051898
1.605384
1.590209
1.600022
680.6344
680.6413
680.6465
680.6331
680.6417
N/A
680.735
680.6436
Unavailable
x5
Best objective function
te
x7 Average objective function
Ac ce p
Std Dev
M
x3
x6
Fitness function evaluations
2.333539
1.951242
x2 x4
2.318452
an
Deb [13]
d
Optimal design variables
us
Table 1 Optimal variables obtained by different researchers for the constrained function I
N/A
N/A
0.1040
0.0101
350,070
160,000
36,000
24,000
Table 2 Optimal design variables obtained by different researchers for the constrained function II.
Optimal design variables (x)
Lee and Geem
Present study
Present study
[18]
(CBO)
(PSO-CBO)
2.155225
2.142755
2.163967
x2
2.407687
2.441786
2.387446
x3
8.778069
8.772559
8.761691
x4
5.102078
5.089189
5.070258
x5
0.967625
0.976804
0.987816
x6
1.357685
1.36545
1.413288
x1
Deb [13]
Unavailable
23 Page 23 of 36
1.287760
1.261765
1.303916
x8
9.800438
9.778372
9.813183
x9
8.187803
8.196755
8.221302
x10
8.256297
8.362651
8.284834
24.36679
24.38470
24.31977
Best objective function
24.37248
Average objective function
24.40940
N/A
24.86188
N/A
N/A
0.580431
350,070
230,000
100,000
Std Dev
24.34516 0.02794 80,000
cr
Fitness function evaluations
ip t
x7
Element group
Konzelman
Sedaghti
Gomes
[20]
[21]
[22]
Kaveh and Zolghadr [23] Standard
2.987
5-12
7.932
7.932
7.849
13-16
0.645
0.645
0.645
17-18
0.645
0.645
0.645
19-22
8.056
8.056
8.765
23-30
8.011
8.011
31-34
0.645
0.645
35-36
0.645
0.645
37-40
12.812
12.812
41-48
8.061
8.061
49-52
0.645
53-54
0.645
59-66 67-70 71-72
Best Weight (kg) Average (kg) Std dev
Weight
No. of analyses
CBO-PSO
2.528
2.522
3.294
3.490
8.704
9.109
8.575
7.821
0.645
0.648
0.645
0.645
0.645
0.645
0.645
0.645
8.283
7.946
9.044
8.060
8.153
7.888
7.703
7.579
7.994
0.645
0.645
0.647
0.645
0.645
0.645
0.645
0.6456
0.645
0.645
13.450
14.666
13.465
12.283
12.894
8.073
6.793
8.250
8.080
7.909
0.645
0.645
0.645
0.645
0.645
0.645
0.645
0.645
0.645
0.646
0.645
0.645
17.279
17.279
16.684
16.464
18.368
16.781
16.775
8.088
8.088
8.159
8.809
7.053
7.635
8.044
0.645
0.645
0.645
0.645
0.645
0.645
0.645
te
Ac ce p
55-58
CBO
CSS
M
3.499
Present Work
d
3.499
Enhanced
an
CSS
1-4
us
Table 3. Optimal cross-sectional areas (cm2) for the 72-bar space truss.
0.645
0.645
0.645
0.645
0.646
0.645
0.645
327.605
328.823
328.814
328.814
328.393
325.4593
324.3779
-
-
332.24
337.70
335.77
329.7996
325.7175
-
-
4.23
5.42
7.20
5.8408
1.553
-
-
42,840
4,000
6,000
6,000
6,000
Table 4 Element grouping for the 52-bar space truss.
Group number
Elements
1
1-4
2
5-8
3
9-16
24 Page 24 of 36
17-20
5
21-28
6
29-36
7
37-44
8
45-52
ip t
4
Table 5 Cross-sectional areas and nodal coordinates obtained by different researchers for the 52-bar space truss. Initial
Lingyun et al.
Gomes
Kaveh and
[25]
[22]
Zolghadr
5.8851
5.5344
5.2716
XB(m)
2.000
1.7623
2.0885
ZB(m)
5.700
4.4091
XF(m)
4.000
CBOPSO
6.0766
5.7274
1.5909
2.0000
2.1553
3.9283
3.7039
3.9106
3.6114
3.4406
4.0255
3.5595
4.000
3.8954
4.500
3.1874
A1(cm )
2.0
1.0000
A2(cm2)
2.0
2.1417
A3(cm )
2.0
1.4858
A4(cm2)
2.0
1.4018
A5(cm )
2.0
A6(cm2)
us
ZA(m)
6.000
CBO
an
[23]
Present Work
cr
Variable
2.5757
2.4240
1.0475
0.3696
1.0464
1.0580
1.0074
4.1912
1.7295
1.3974
1.2674
1.5123
1.6507
1.4299
1.3093
1.5620
1.5059
1.5275
1.4559
1.9110
1.9154
1.7210
1.5652
1.4228
2.0
1.0109
1.1315
1.0020
1.0006
1.0067
A7(cm )
2.0
1.4693
1.8233
1.7415
1.4244
1.6086
A8(cm2)
2.0
2.1411
1.0904
1.2555
1.3753
1.3839
236.046
228.381
205.237
199.066
195.721
-
-
234.3
213.101
202.138
202.501
-
-
5.22
7.391
6.235
4.290
-
-
11,270
4,000
6,000
6,000
2
2
2
Average Weight (kg)
Ac ce p
Std dev
te
338.69
Best Weight (kg)
d
2
M
2.4575
ZF(m)
No. of analyses
Table 6 Optimal cross-sectional areas (cm2) for the 200-bar planar truss.
Element number
Members in the group
Kaveh and Zolghadr
Present Work
[17] Standard
CSS-
CSS
BB-BC
1, 2, 3, 4
1.2439
0.2934
0.3268
0.2797
2
5, 8, 11, 14, 17
1.1438
0.5561
0.4502
0.6968
3
19, 20, 21, 22, 23, 24
0.3769
0.2952
0.1000
0.1000
4
18, 25, 56, 63, 94, 101, 132, 139, 170, 177
0.1494
0.1970
0.1000
0.1000
5
26, 29, 32, 35, 38
0.4835
0.8340
0.7125
0.5796
6
6, 7, 9, 10, 12, 13, 15, 16, 27, 28, 30, 31, 33, 34, 36, 37
0.8103
0.6455
0.8029
0.8213
1
CBO
CBO-PSO
25 Page 25 of 36
39, 40, 41, 42
0.4364
0.1770
0.1028
0.1279
8
43, 46, 49, 52, 55
1.4554
1.4796
1.4877
1.0152
9
57, 58, 59, 60, 61, 62
1.0103
0.4497
0.100
0.100
10
64, 67, 70, 73, 76
2.1382
1.4556
1.0998
1.5647
11
44, 45, 47, 48, 50, 51, 53, 54, 65, 66, 68, 69, 71, 72, 74,
0.8583
1.2238
0.8766
1.6465
0.1229
0.2296
2.9058
2.9007
0.100
0.100
3.5535
3.0133
1.3360
1.7175
1.6142
0.6289
an
0.100
0.2755
6.2338
5.0951
2.5793
0.6062
0.1102
0.100
3.0520
5.4393
5.8959
5.5172
1.8121
1.8435
2.1858
2.2032
1.2986
0.8955
0.5249
0.8659
77, 78, 79, 80
1.2718
0.2739
13
81, 84, 87, 90, 93
3.0807
1.9174
14
95,96, 97, 98, 99, 100
0.2677
0.1170
15
102, 105, 108, 111, 114
4.2403
3.9952
16
82, 83, 85, 86, 88, 89, 91, 92, 103, 104, 106, 107, 109,
2.0098
17
115, 116, 117, 118
18
119, 122, 125, 128, 131
4.8335
5.9423
19
133, 134, 135, 136, 137, 138
20
140, 143, 146, 149, 152
21
120, 121, 123, 124, 126, 127, 129, 130, 141, 142, 144,
110, 112, 113
M
d
145, 147, 148, 150, 151
1.5956
cr
12
us
75
ip t
7
153, 154, 155, 156
23
157, 160, 163, 166, 169
5.8810
8.1759
7.2676
7.6477
24
171, 172, 173, 174, 175, 176
0.2324
0.3209
0.1278
0.100
178, 181, 184, 187, 190
7.7536
10.98
7.8865
8.1273
158, 159, 161, 162, 164, 165, 167, 168, 179, 180, 182,
2.6871
2.9489
2.8407
2.9665
191, 192, 193, 194
12.5094
10.5243
11.7849
10.2386
195, 197, 198, 200
29.5704
20.4271
22.7014
20.6364
196, 199
8.2910
19.0983
7.884
11.6468
Weight (kg)
2559.86
2298.61
2203.212
No. of
10,000
10,000
9,000
26
Ac ce p
25
te
22
183, 185, 186, 188, 189
27 28 29
2195.469 9,000
analyses
26 Page 26 of 36
ip t cr
an
us
(a)
Ac ce p
te
d
M
(b)
(c)
Fig. 1 The collision between two objects, (a) before collision (b) same time collision (c) after collision.
Stationary
X1
X2
Moving
CBs
Xn-1
Xn
Xn+1
CBs
X2n
Fig. 2 The sorted CBs in an increasing order and the mating process for the collision.
27 Page 27 of 36
ip t cr us an
Ac ce p
te
d
M
(a)
(b) Fig. 3 The convergence curves for the constrained function I. (a) all iterations (b) 200-600 iterations.
28 Page 28 of 36
ip t cr us an
Ac ce p
te
d
M
(a)
(b) Fig. 4 The convergence curves for the constrained function II. (a) all iterations (b) 600-1000 iterations.
29 Page 29 of 36
ip t cr us an M
Ac ce p
te
d
Fig. 5 A 72-bar space truss
(a)
30 Page 30 of 36
ip t cr us an M (b)
Ac ce p
te
d
Fig. 6 The convergence curves for the 72-bar space truss example. (a) all iterations (b) 150-200 iterations.
31 Page 31 of 36
ip t cr us an M d te Ac ce p
(a) Top view
(b) Side view
32 Page 32 of 36
an
us
cr
ip t
Fig. 7 A 52-bar space truss.
Ac ce p
te
d
M
(a)
(b) Fig. 8 The convergence curves for the 52-bar space truss example. (a) all iterations (b) 200-200 iterations.
33 Page 33 of 36
ip t cr us an
Ac ce p
te
d
M
(a)
(b) Fig. 8 The convergence curves for the 52-bar space truss example. (a) all iterations (b) 200-300 iterations.
34 Page 34 of 36
ip t cr us an M d te Ac ce p
Fig. 9 A 200-bar planar truss
35 Page 35 of 36
ip t cr us an
Ac ce p
te
d
M
(a)
(b) Fig 10 The convergence curves for 200-bar planar truss. (a) all iterations (b) 200-300 iterations.
36 Page 36 of 36