Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures

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Abstract

A heuristic particle swarm ant colony optimization (HPSACO) is presented for optimum design of trusses. The algorithm is based on the particle swarm optimizer with passive congregation (PSOPC), ant colony optimization and harmony search scheme. HPSACO applies PSOPC for global optimization and the ant colony approach is used to update positions of particles to attain the feasible solution space. HPSACO handles the problem-specific constraints using a fly-back mechanism, and harmony search scheme deals with variable constraints. Results demonstrate the efficiency and robustness of HPSACO, which performs better than the other PSO-based algorithms having higher converges rate than PSO and PSOPC.

Introduction

In the last decade, many new natural evolutionary algorithms have been developed for optimization of pin-connected structures, such as genetic algorithms (GAs) [1], [2], [3], [4], [5], particle swarm optimizer (PSO) [6], [7], ant colony optimization (ACO) [8], [9], [10] and harmony search (HS) [11], [12], [13]. These methods have attracted a great deal of attention, because of their high potential for modeling engineering problems in environments which have been resistant to solution by classic techniques. They do not require gradient information and possess better global search abilities than the conventional optimization algorithms [14]. Having in common processes of natural evolution, these algorithms share many similarities: each maintains a population of solutions which are evolved through random alterations and selection. The differences between these procedures lie in the representation technique utilized to encode the candidates, the type of alterations used to create new solutions, and the mechanism employed for selecting new patterns.

Compared to other evolutionary algorithms based on heuristics including evolutionary algorithms (EAs), evolutionary programming (EP) and evolution strategies (ES) [15], the advantages of PSO consist of easy implementation and smaller number of parameters to be adjusted. However, it is known that the original PSO (or SPSO) had difficulties in controlling the balance between exploration (global investigation of the search place) and exploitation (the fine search around a local optimum) [16]. In order to improve this character of PSO, it is hybridized with other approaches such as ACO or HS. PSACO (a hybrid particle swarm optimizer and ant colony approach) which was initially introduced by Shelokar et al. [17] for the solution of the continuous unconstrained problems and recently utilized for truss structures [18], is applied to PSO as a global search technique and the idea of ant colony approach is incorporated as a local search for updating the positions of the particles by applied pheromone-guided mechanism. HPSO (a hybrid particle swarm optimizer and harmony search scheme) was proposed by Li et al. [7] for truss design employed the harmony memory (HM) operator for controlling the variable constraints.

The present paper hybridizes PSO, ACO and HS, and it is based on the principles of those two methods with some differences. We have applied PSOPC (a hybrid PSO with passive congregation [19]) instead of PSO to improve the performance of the new method. The relation of standard deviation in ACO stage is different with that of Ref. [17], and the inertia weight is changed in PSOPC stage. New terminating criterion is employed to increase the probability of obtaining an optimum solution in a smaller number of iterations. In the proposed method, similar to HPSO, HS is utilized for controlling the variable constraint. The resulted method has a good control on the exploration and exploitation compared to PSO and PSOPC. It increases the exploitation, and guides the exploration, and as a result, the convergence rate of the proposed algorithm is higher than other heuristic approaches.

There are some problem-specific constraints in truss optimization problems that must be handled. The penalty function method has been the most popular constraint-handling technique due to its simple principle and ease of implementation. The main difficulty of the penalty function method lies in that the appropriate values of penalty factors are problem-dependent and a large amount of effort is needed for fine-tuning of the penalty factors. Therefore, several techniques have been incorporated to handle the constraints. Compared to other constraint-handling techniques, fly-back mechanism is relatively simple and easy to implement into the PSO [7]. Therefore, this paper handles the problem-specific constraints by using this mechanism.

The present paper is organized as follows: In Section 2, we describe the PSO, ACO and HS. Statement of the optimization design problem is formulated in Section 3. In Section 4, the fly-back mechanism is described. In Section 5, the new method is presented. Various examples are studied in Section 6. The efficiency of HPSACO is investigated in Section 7. Conclusions are derived in Section 8.

Section snippets

Introduction to PSO, ACO and HS

In order to make the paper self-explanatory, before proposing HPSACO for truss design optimization, the characteristics of PSO, ACO and HS, are briefly explained in the following three sections:

Statement of the optimization design problem

Size optimization of truss structures involves arriving at optimum values for member cross-sectional areas Ai that minimize the structural weight W. This minimum design also has to satisfy inequality constraints that limit design variable sizes and structural responses [11]. Thus, the optimal design problem may be expressed asminimizeW({x})=i=1nγi·Ai·Lisubject toδminδiδmaxi=1,2,,mσminσiσmaxi=1,2,,nσibσi0i=1,2,,ncAminAiAmaxi=1,2,,ngwhere W({x}) is the weight of the structure, n is

Fly-back mechanism

Fly-back mechanism has been introduced by He et al. [26]. For most of the structural optimization problems, the global minimum locates on or close to the boundary of a feasible design space. The particles are initialized in the feasible region. When the particles fly in the feasible space to search the solution, if any one of them flies into the infeasible region, it will be forced to fly-back to the previous position to guarantee a feasible solution. The particle which flies back to the

A heuristic particle swarm ant colony optimization for truss design

The framework of heuristic particle swarm ant colony optimization (HPSACO) algorithm is illustrated in Fig. 4. HPSACO algorithm applies PSOPC for global optimization, while ACO works as a local search, wherein, ants apply pheromone-guided mechanism to refine the positions found by particles in the PSOPC stage. In HPSACO, a simple pheromone-guided mechanism of ACO is proposed and employed for the local search. The proposed ACO algorithm handles P ants equal to the number of particles in PSOPC

Numerical examples

In this section, common truss optimization examples as benchmark problems are optimized with the proposed method. The final results are compared to the solutions of other methods to demonstrate the efficiency of the present approach.

For the proposed algorithm, a population of 50 individuals is used for both particles and ants; the value of constants c1 and c2 are set 0.8 and the passive congregation coefficient c3 is taken as 0.6. The value of inertia weight (ω(k)) decreases linearly from 0.9

Discussion on the efficiency of the HPSACO

Solution of a number of design examples shows that the performance of HPSACO is significantly better than other PSO-based algorithms. The major reasons for the improvements obtained by the present method can be summarized as follows:

  • (a)

    Increasing the exploitation: In truss optimization, usually there are some local optimums in the neighborhood of a desirable solution. Thus, the probability of finding a desirable optimum increases with additional searches around the local optimums. HPSACO does

Concluding remarks

In this paper HPSACO is developed for optimal design of trusses. HPSACO is based on PSOPC, ACO and HS. In this method, ACO helps PSO process not only to efficiently perform the global exploration for rapidly attaining the feasible solution space but also effectively helps to reach optimal or near optimal solution. In order to make the particles remain in the feasible space, fly-back mechanism and HS are used. Fly-back mechanism handles the problem-specific constraints, and the HS deals with the

Acknowledgement

The first author is grateful to the Iran National Science Foundation for the support.

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