A diverse versions and variations of Particle Swarm Optimization (PSO) algorithm for real world complex optimization problems

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 3, Issue 8, August 2013)

454

A diverse versions and variations of Particle Swarm

Optimization (PSO) algorithm for real world complex

optimization problems

T. Geetha

1

1 _{Assistant Professor, Department of Computer Science & Engineering, A.R. Engineering College, Villupuram, Tamilnadu,}

India.

Abstract— PSO is a robust and stochastic optimization technique based on the movement and the intelligence of the swarms. PSO is inspired with social behavior of bird flocking and fish schooling. PSO algorithm is applied to solve the real world complex optimization problems. So many real world optimization problems are solved by using different combinations and variations of PSO algorithms. In this paper, so many different versions and variations of the PSO algorithm was discussed in a clear and concise manner.

Keywords—Particle Swarm Optimization, Swarm Intelligence, Optimization, Robust, Stochastic.

I. INTRODUCTION

In the recent years, nature inspired computation plays a very major role and it got a great attention in solving complex problems. The most successful one is Swarm Intelligence. Swarm Intelligence (SI) comes under the computational intelligence. Swarm Intelligence focusing the animal collective behavior.SI is particularly applied to insects, but it can also be applied to any other animals that exhibits swarm behavior. SI is a population of boids interacting locally with one another and also with their environment.SI are Meta-Heuristic algorithm because they uses natural metaphor to solve the complex, combinatorial as well as numerical optimization problem to get optimal or near optimal solutions. It is either a local search or global search algorithm. SI includes two algorithms they are Ant Colony Optimization (ACO) and

Particle Swarm Optimization algorithm (PSO).SI

algorithms are suitable to solve ―NP-Hard‖ problems. The common NP-Hard problems are optimization problem, decision making problems and so on. In the few years , to solve the optimization problems many researchers are using SI algorithm especially PSO[1][2][3][4][5][6].In this paper we discussed the PSO algorithm and its different versions and the variations of the PSO algorithm.

The remainder of this paper organized as follows. Section II describes the related work. Section III, describes the different versions of PSO algorithm, in that a brief different versions and variations of PSO algorithm was neatly presented. In section IV, conclusion is discussed.

II. RELATED WORK

This section describes the particle swarm optimization algorithm history and its concept then its application along with their pseudocode.

A. Particle Swarm Optimization (PSO) Algorithm

PSO was developed by James Kennedy and Rusell Eberhart in the year 1995.PSO is inspired with social behavior of bird flocking and fish schooling.PSO is Robust and Stochastic optimization technique based on the movement and the intelligence of the swarms. PSO follows these basic principles. The first principle is that the population should respond to the quality factors pbest and gbest and the second principle describes the allocation of responses between pbest and gbest and it should ensures a diversity of response and the third principle is Stability describes that the population changes its state only when the gbest changes and the fourth principle Adaptive describes that the population is adaptive because it does change when gbest changes.PSO has the capability to solve continuous non-linear optimization problems and it is applied in various application areas like image processing,

pattern recognition, game design, reactive power

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International Journal of Emerging Technology and Advanced Engineering

455 In [2], Kennedy and Eberhart originally proposed PSO algorithm without any inertia weight factor. In order to improve the performance of the PSO algorithm, Eberhart and XiaohuiHu uses inertia weight [3, 4, 5, 7, 8, and 9].Paper [5] proposes by including inertia weight, particles have a capability to enlarge the search space. Inertia weight provides a balance between local search as well as global search. From [3] we can conclude that the inertia weight place a major role in the PSO algorithm. Inertia weight can be computed by the formula [3, 4, 5, 6, 7, 8 ] inertia weight,

ω = [0.5 + (rand ( ) / 2.0)]

If (ω < 0.8) is small, PSO finds global optimum solution very fast. If (ω > 1.2) is large, it makes use of new areas. If (0.8 < ω < 1.2) is medium, then the PSO algorithm failed to find the global optimum. In [7, 8] the authors said that if we linearly decrease the inertia weight value from 0.9 to 0.4 we can achieve the optimal performance value.

In [13] the authors compute inertia weight by the formula,

ω = ωmax – (k / k max) (ωmax – ωmin)

Where ωmax value is 0.9 and ωmin value is 0.9.

PSO is a population based cooperative search.PSO satisfies the local optimization but it is hard to express the robustness because of its iteration ability. In order to reduce the iteration, the operator such as crossover and mutation of GA are mix with PSO for a better solution.PSO is the only algorithm. The following example clearly describes the PSO algorithm. A group of birds are randomly searching some food in an area. There is only one piece of food in the area being searched. All the birds do not know where the food is, but they know how far the food is in each iteration. So the best way to find the food is follow the bird which is nearer to the food. In PSO, each single solution is a ‗bird‘ in the search space. Generally it is called as a ‗particle‘. All the particles are having some fitness value which is evaluated by the fitness function and have velocities which direct the flying of the particles. In general PSO is initialized with a group of random particles (Solutions) and then searches for optima by updating generations. Each particle is having two properties position and velocity. In every iteration, each particle is updated by two best values. The first one is the best solution (fitness) it has achieved so far.(the fitness value is stored).This value is called as pbest (personal best).Another best value is obtained so far any particle in the population, this best value is a gbest (global best).After finding the two best values, the particle updates its position and velocity.

Large inertia weight describes the greater global search ability and the small inertia weight describes the greater local search ability. Velocity, v is given by the formula

Where v is the velocity,

rand () is the random number is generated in between 0 & 1

C1 and c2 are learning factors The current particle is given by

Pseudocode for Particle Swarm Optimization (PSO) Algorithm:

1. Initialization: Initializing the velocity and position of particle.

2. Evaluate the fitness.

3. Evaluate the new position‘s fitness; for each particle, if the fitness of new particle is better than the original particle, swap it.

4. Compare with all pbest to find gbest. 5. Update velocity and position.

6. If the termination condition is satisfied then stop otherwise Goto step 2.

III. VERSIONS OF PARTICLE SWARM OPTIMIZATION (PSO)

ALGORITHM

This section classifies the PSO algorithm based on the variations and the combination of PSO algorithm with other evolutionary algorithms and swarm intelligence algorithms. The first one is the combination of PSO with other algorithms and the second one is the variation of PSO algorithm.

A. Combination Of Particle Swarm Optimization (PSO) With Other Algorithms

In [32] the author proposed hybrid GA and hybrid PSO algorithm to minimize the makespan for assembly job shop scheduling problem. In Hybrid PSO, the authors include the social learning factors c1, c2 values as 0.5 and Vmax as

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International Journal of Emerging Technology and Advanced Engineering

456 This increases the effectiveness of the algorithm. They newly include sigmoid function to transform the velocity from real number to probability space. The results are compared with ENS, MPNS- GRASP, HybGEN, GEN-PSO-ENS these algorithms, finally they concluded that HybGENPSO algorithm is best.

In [34] the authors proposed Hybrid GA and PSO for multimodal functions. In this paper, the authors proposed GA-PSO algorithm and they include both GA and PSO method. The authors include Selection, Crossover and Mutation operators of GA and the update mechanism of PSO algorithm. The results of GA-PSO algorithm is compared with CHA, ECTS, CGA, ESA, CRTS minimum, CRTS average, TS and Interopt algorithms and they finally concluded that GA-PSO is giving better results than other algorithms in finding the global optimum.

In [35] the authors proposed Hybrid GA and PSO

(HGAPSO) algorithm for solving bi-level linear

programming problem. The authors implemented crossover and mutation operators of GA to update the particles in PSO algorithm. The results of HGAPSO3 algorithm is compared with the algorithms like GA, PSO, HGAPSO 1, HGAPSO2 and they finally conclude that HGAPSO3 is getting smaller standard deviation value as well as it got better stability.

In [48] the authors proposed hybrid GA and PSO (HGAPSOA) algorithm for order clustering. The authors include GA operators as well as PSO mechanism and they calculate fitness by using Euclidean distance. The result of HGAPSOA outperforms GA, PSO, GKA, PSKO, GA-PSO and GA-PSKO algorithms.

In [53] the authors proposed GA-PSO algorithm (integrating GA and PSO algorithm) to solve non-linear optimization problem. The authors included both GA (operators like crossover, mutation, selection and elitist strategy) and PSO (constriction factor) methods. The results of GA-PSO algorithm are compared with GA and PSO algorithm and they concluded that GA-PSO algorithm is giving better global optimum solution than GA and PSO algorithms.

In [54] the authors proposed DCPG algorithm (integration of PSO and GA algorithm) for dynamic clustering. The authors included crossover and mutation to the PSO algorithm. The results of DCPG algorithm is compared with DCPSO, ACMPSO, and DCGA algorithms and they concluded that DCPG is converging fast and it gives better results in clustering.

In [55] the authors‘ proposed Local search based hybrid PSO algorithm for multi objective optimization problems. The proposed method has three stages. The first one is PSO stage and the second one is GA stage and finally LS stage.

The use of LS is to improve the quality of the solution. They introduced Modified LS (MLS) scheme. From the results they concluded that proposed methodology is suitable to solve complex multi objective optimization problems and it has the capability to find a well distributed pareto optimal curve in the objective space. The proposed methodology track all the feasible solution and it does not having any restrictions on the number of pareto optimal solutions and then the local search speeds-up the search space and it assist in giving fine grained value for the objective functions.

In [38] the authors proposed Hybrid Particle Swarm Optimization – back-propagation algorithm for feed forward neural network training. The proposed Hybrid PSO-BP algorithm combines both the concept of PSO and Back-Propagation. Back-Propagation algorithm has the strong ability to find the local optimistic result but weak in finding global optima, for that reason Back-Propagation algorithm is applied with PSO algorithm. PSO-BP has less cpu time than PSO algorithm.

In [40] the authors proposed Hybrid self-adaptive particle Swarm Optimization and modified shuffled frog leaping algorithm for distribution feeder reconfiguration problem. The proposed SAPSO–MSFLA (Self Adaptive Particle Swarm Optimization – Modified Shuffled Frog Leaping algorithm) algorithm is the combination of Self Adaptive Particle Swarm Optimization – Modified Shuffled Frog Leaping algorithm. The self adaptive framework was introduced to improve the robustness of the PSO algorithm. The Modified Shuffled Frog Leaping was proposed to improve the local exploration of the SFLA algorithm. The result of SAPSO–MSFLA was compared with SAPSO and MSFLA algorithm and the author concluded that SAPSO-MSFLA algorithm produces better results than the two algorithms.

In [41] the authors proposed that Memetic PSO algorithm for Multi Modal optimization problem. The proposed Memetic PSO algorithm has three major operations. It includes new local PSO model, adaptive LS model and trigerred re-initialization scheme. From the result the authors concluded that Memetic PSO algorithm is suitable to solve Multi Modal Optimization Problems (MMOP).

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International Journal of Emerging Technology and Advanced Engineering

457 B. Variations In Particle Swarm Optimization (PSO)

Algorithm

In [31] the authors proposed Bare-Bones multi objective particle swarm optimization (BB-MOPSO) algorithm for environment/economic dispatch problems. This paper is having three features. The particle updating strategy does not require tuning up control parameters. The mutation operator is to expand the search capability and to avoid the premature convergence problem. The particle diversity is to update the global particle leaders. From the results, BB-MOPSO algorithm is generating excellent approximation of true pareto front and it is suitable to solve all types of multi objective optimization problems. BB-MOPSO algorithm is giving better results than other algorithms like SMOPSO, CMOPSO and TV-MOPSO in terms of convergence performance. BB-MOPSO algorithm is better than CMOPSO and TV-MOPSO algorithms in the distribution of solutions on the obtained Pareto front. BB-MOPSO is attaining smallest standard deviation value.

In [36] the authors proposed Hybrid PSO algorithm (CGPSO) for High Dimensional problems. The CGPSO algorithm is the combination of canonical PSO with chaotic and Gaussian local search procedure with ‗shrinking‖ strategy on the gbest particle. CGPSO is better than PSO in terms of scalability, search ability and convergence speed. From the results the authors said that the CGPSO algorithm is better than other algorithms like CPSO-H, DMS-L-PSO, DEa-hcSPX and MA-S2 algorithms.

In [37] the authors proposed Hybrid PSO algorithm for optimal task assignment in Distributed systems. Here they concentrate on velocity (the updated velocity is restricted by maximum threshold value Vmax). This algorithm

improved the solution quality of each particle by using the embedded hill-climbing heuristic.

In [39] the authors proposed Hybrid Swarm Intelligence Particle – Bee algorithm for construction site layout optimization. The operation used in PSO algorithm to find pbest is applied to find elite, best and random bees in bee algorithm.

In [42] the authors proposed a new modified PSO algorithm for adaptive equalization. In this they proposed new constriction factor (stabilizing effect) the time dependent linearly decreasing k, the k value is adjusted at each iteration by the recursion.

In [43] the authors proposed a novel and effective

particle swarm optimization like algorithm with

extrapolation technique. The proposed algorithm is extrapolated pso (ePSO), has two extrapolation coefficients e1 and e2 which is similar to inertia weight in pso algorithm.

The extrapolation coefficients values are decreasing from 0.999 to 0.36. The extrapolated pso algorithm does not need a velocity equation. The results are compared with five other methods including cPSO, GLBest PSO, RCGA, HRCGA and FA. From the results it is concluded that ePSO algorithm has the capability in solving optimization problems and it is competitive to other PSO methods as well as other EA‘s.

In [44] the authors proposed a practical algorithm based on PSO for haplotype reconstruction. In this paper, the authors newly introduced four operators Sub, Add, Mul and XOR and they newly introduced velocity and position formulas. PSO-Q algorithm uses binary string (0, 1) as to represent particle. From the results the authors concluded that the running time of P-MEC algorithm is shorter than the PSO-Q algorithm.

In [45] the authors proposed a similar Particle Swarm

Optimization algorithm for permutation flowshop

scheduling problem to minimize makespan. The proposed SPSOA algorithm introduced one crossover operator. In that each particle of the swarm shares their mutual information globally and they get the benefits from their discoveries and the previous experiences of all their colleagues during the search process.

In [47] the authors proposed an effective immune based symbiotic particle swarm optimization learning algorithm for TSK – type neuro fuzzy networks design. The proposed ISPSO algorithm embeds the symbiotic evolution scheme in IA and it utilizes the concept of PSO in order to improve the mutation mechanisms.

In [49] the authors proposed Automatic kernel clustering with a multi-elitist particle swarm optimization algorithm. The proposed MEPSO algorithm defined β growth rate for each particle and it introduced elitist concept. The elitist is introduced to prevent the swarm from reaching the global best too early in the searching process.

In [50] the authors proposed communication latency tolerant parallel algorithm for particle swarm optimization. The authors proposed parallel PSO for delayed exchange parallelization (DEP) problem. In globally synchronized parallel PSO the gbest fitness is determined by two factors. The first one is to determine the partial best fitness and the second is to exchange their partial best fitness to one another and then it calculates the global best fitness.

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International Journal of Emerging Technology and Advanced Engineering

458 VePSOgc has the global convergence characteristics and it is based on the value exchange. From the results the authors conclude that OePSOgc outperforms the other versions of PSO algorithm.

In [52] the authors proposed fault diagnosis of sensor by chaos particle swarm optimization algorithm and SVM. The authors include chaos behavior, to enrich the searching behavior and it avoids being trapped into local optimum. The chaos behavior improves the PSO algorithm and it helps to search the optimum value very quickly.

In [56] the authors proposed the modification of the fish swarm algorithm with particle swarm optimization formulation and communication behavior. This paper focusing the several improvements of the fish swarm algorithm and it includes 3 main things. The first one is reformulating the fish swarm algorithm using particle swarm optimization algorithm. The second one is integrating the communication behavior into the fish swarm algorithm and the third one is creating formulas for major fish swarm algorithm parameters.

In [57] the authors proposed optimal parameters estimation and input subset for grey model based on chaotic particle swarm optimization algorithm. This paper proposed chaotic particle swarm optimization method. The algorithm includes the basic steps like chaotic initialization, adaptive inertia weight factor, chaotic disturbance and finally chaotic local search.

In [58] the authors proposed PSO based algorithm for TSP and generalized TSP problem. In the Discrete PSO algorithm for TSP problem, the authors redefined the PSO operators by introducing the concepts of ―Swap Operator ―and ― Swap Sequence‖ and then it introduced the permutation concept. The authors redefined the random numbers as real vectors. In the Discrete PSO algorithm for Generalized TSP problem, the authors changed both the velocity and the position formulas.

In [60] the authors‘ proposed Short term scheduling of cascade reservoir by using an immune algorithm based PSO. This paper combines the immune information processing mechanism with an original PSO algorithm. The special concentration selection mechanism and immune vaccination improves the ability to find the global optimal result and it increases the convergence speed.

In [61] the authors proposed Simulation optimization using PSO algorithm with applications to assembly line design. This paper proposed PSOMS algorithm it combines the inertia weight method with the mutation operator.

IV. CONCLUSION

In this paper, Particle Swarm Optimization (PSO) algorithm was briefly discussed along their different versions and variations. PSO algorithm was applied in various real world optimization problem and its results proved that PSO algorithm gives better optimal results when compared to other algorithms. Therefore, we can apply PSO algorithm to solve complex optimization problems in future also.

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