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Showing papers on "Swarm intelligence published in 2004"


Book
01 Jan 2004
TL;DR: Ant colony optimization (ACO) is a relatively new approach to problem solving that takes inspiration from the social behaviors of insects and of other animals as discussed by the authors In particular, ants have inspired a number of methods and techniques among which the most studied and the most successful is the general purpose optimization technique known as ant colony optimization.
Abstract: Swarm intelligence is a relatively new approach to problem solving that takes inspiration from the social behaviors of insects and of other animals In particular, ants have inspired a number of methods and techniques among which the most studied and the most successful is the general purpose optimization technique known as ant colony optimization Ant colony optimization (ACO) takes inspiration from the foraging behavior of some ant species These ants deposit pheromone on the ground in order to mark some favorable path that should be followed by other members of the colony Ant colony optimization exploits a similar mechanism for solving optimization problems From the early nineties, when the first ant colony optimization algorithm was proposed, ACO attracted the attention of increasing numbers of researchers and many successful applications are now available Moreover, a substantial corpus of theoretical results is becoming available that provides useful guidelines to researchers and practitioners in further applications of ACO The goal of this article is to introduce ant colony optimization and to survey its most notable applications

6,861 citations


Journal ArticleDOI
TL;DR: An approach in which Pareto dominance is incorporated into particle swarm optimization (PSO) in order to allow this heuristic to handle problems with several objective functions and indicates that the approach is highly competitive and that can be considered a viable alternative to solve multiobjective optimization problems.
Abstract: This paper presents an approach in which Pareto dominance is incorporated into particle swarm optimization (PSO) in order to allow this heuristic to handle problems with several objective functions. Unlike other current proposals to extend PSO to solve multiobjective optimization problems, our algorithm uses a secondary (i.e., external) repository of particles that is later used by other particles to guide their own flight. We also incorporate a special mutation operator that enriches the exploratory capabilities of our algorithm. The proposed approach is validated using several test functions and metrics taken from the standard literature on evolutionary multiobjective optimization. Results indicate that the approach is highly competitive and that can be considered a viable alternative to solve multiobjective optimization problems.

3,474 citations


Journal ArticleDOI
TL;DR: A novel parameter automation strategy for the particle swarm algorithm and two further extensions to improve its performance after a predefined number of generations to overcome the difficulties of selecting an appropriate mutation step size for different problems.
Abstract: This paper introduces a novel parameter automation strategy for the particle swarm algorithm and two further extensions to improve its performance after a predefined number of generations. Initially, to efficiently control the local search and convergence to the global optimum solution, time-varying acceleration coefficients (TVAC) are introduced in addition to the time-varying inertia weight factor in particle swarm optimization (PSO). From the basis of TVAC, two new strategies are discussed to improve the performance of the PSO. First, the concept of "mutation" is introduced to the particle swarm optimization along with TVAC (MPSO-TVAC), by adding a small perturbation to a randomly selected modulus of the velocity vector of a random particle by predefined probability. Second, we introduce a novel particle swarm concept "self-organizing hierarchical particle swarm optimizer with TVAC (HPSO-TVAC)". Under this method, only the "social" part and the "cognitive" part of the particle swarm strategy are considered to estimate the new velocity of each particle and particles are reinitialized whenever they are stagnated in the search space. In addition, to overcome the difficulties of selecting an appropriate mutation step size for different problems, a time-varying mutation step size was introduced. Further, for most of the benchmarks, mutation probability is found to be insensitive to the performance of MPSO-TVAC method. On the other hand, the effect of reinitialization velocity on the performance of HPSO-TVAC method is also observed. Time-varying reinitialization step size is found to be an efficient parameter optimization strategy for HPSO-TVAC method. The HPSO-TVAC strategy outperformed all the methods considered in this investigation for most of the functions. Furthermore, it has also been observed that both the MPSO and HPSO strategies perform poorly when the acceleration coefficients are fixed at two.

2,753 citations


Journal ArticleDOI
TL;DR: A variation on the traditional PSO algorithm, called the cooperative particle swarm optimizer, or CPSO, employing cooperative behavior to significantly improve the performance of the original algorithm.
Abstract: The particle swarm optimizer (PSO) is a stochastic, population-based optimization technique that can be applied to a wide range of problems, including neural network training. This paper presents a variation on the traditional PSO algorithm, called the cooperative particle swarm optimizer, or CPSO, employing cooperative behavior to significantly improve the performance of the original algorithm. This is achieved by using multiple swarms to optimize different components of the solution vector cooperatively. Application of the new PSO algorithm on several benchmark optimization problems shows a marked improvement in performance over the traditional PSO.

2,038 citations


Proceedings ArticleDOI
19 Jun 2004
TL;DR: The results from this study show that DE generally outperforms the other algorithms, however, on two noisy functions, both DE and PSO were outperformed by the EA.
Abstract: Several extensions to evolutionary algorithms (EAs) and particle swarm optimization (PSO) have been suggested during the last decades offering improved performance on selected benchmark problems. Recently, another search heuristic termed differential evolution (DE) has shown superior performance in several real-world applications. In this paper, we evaluate the performance of DE, PSO, and EAs regarding their general applicability as numerical optimization techniques. The comparison is performed on a suite of 34 widely used benchmark problems. The results from our study show that DE generally outperforms the other algorithms. However, on two noisy functions, both DE and PSO were outperformed by the EA.

1,252 citations


Proceedings ArticleDOI
19 Jun 2004
TL;DR: The individual particle of a PSO system moving in a quantum multidimensional space is studied and a quantum delta potential well model for PSO is established and a trial method of parameter control and QDPSO is proposed.
Abstract: In this paper, inspired by the analysis of convergence of PSO, we study the individual particle of a PSO system moving in a quantum multidimensional space and establish a quantum delta potential well model for PSO. After that, a trial method of parameter control and QDPSO is proposed. The experiment result shows much advantage of QDPSO to the traditional PSO.

1,202 citations


Book ChapterDOI
17 Jul 2004
TL;DR: This paper proposes a definition to this newly emerging approach to swarm robotics, describing the desirable properties of swarm robotic systems, as observed in the system-level functioning of social insects, and proposing a definition for the term swarm robotics.
Abstract: Swarm robotics is a novel approach to the coordination of large numbers of relatively simple robots which takes its inspiration from social insects. This paper proposes a definition to this newly emerging approach by 1) describing the desirable properties of swarm robotic systems, as observed in the system-level functioning of social insects, 2) proposing a definition for the term swarm robotics, and putting forward a set of criteria that can be used to distinguish swarm robotics research from other multi-robot studies, 3) providing a review of some studies which can act as sources of inspiration, and a list of promising domains for the utilization of swarm robotic systems.

905 citations


Journal ArticleDOI
TL;DR: The approaches include transformations of the objective function through the recently proposed deflection and stretching techniques, as well as a repulsion source at each detected minimizer, resulting in an efficient algorithm which has the ability to avoid previously detected solutions and, thus, detect all global minimizers of a function.
Abstract: This paper presents approaches for effectively computing all global minimizers of an objective function. The approaches include transformations of the objective function through the recently proposed deflection and stretching techniques, as well as a repulsion source at each detected minimizer. The aforementioned techniques are incorporated in the context of the particle swarm optimization (PSO) method, resulting in an efficient algorithm which has the ability to avoid previously detected solutions and, thus, detect all global minimizers of a function. Experimental results on benchmark problems originating from the fields of global optimization, dynamical systems, and game theory, are reported, and conclusions are derived.

718 citations


Proceedings ArticleDOI
01 Dec 2004
TL;DR: A so-called mainstream thought of the population is introduced to evaluate the search scope of a particle and thus a novel parameter control method of QPSO is proposed.
Abstract: Based on the quantum-behaved particle swarm optimization (QPSO) algorithm, we formulate the philosophy of QPSO and introduce a so-called mainstream thought of the population to evaluate the search scope of a particle and thus propose a novel parameter control method of QPSO. After that, we test the revised QPSO algorithm on several benchmark functions and the experiment results show its superiority.

676 citations


Proceedings ArticleDOI
19 Jun 2004
TL;DR: This paper reviews the development of the particle swarm optimization method in recent years and modifications to adapt to different and complex environments are reviewed, and real world applications are listed.
Abstract: This paper reviews the development of the particle swarm optimization method in recent years. Included are brief discussions of various parameters. Modifications to adapt to different and complex environments are reviewed, and real world applications are listed.

501 citations


Journal ArticleDOI
01 Apr 2004
TL;DR: This paper proposes a new framework for implementing ant colony optimization algorithms called the hyper-cube framework, which limits the pheromone values to the interval [0,1], and proves that in the ant system, the ancestor of all ant colony optimized algorithms, the average quality of the solutions produced increases in expectation over time when applied to unconstrained problems.
Abstract: Ant colony optimization is a metaheuristic approach belonging to the class of model-based search algorithms. In this paper, we propose a new framework for implementing ant colony optimization algorithms called the hyper-cube framework for ant colony optimization. In contrast to the usual way of implementing ant colony optimization algorithms, this framework limits the pheromone values to the interval [0,1]. This is obtained by introducing changes in the pheromone value update rule. These changes can in general be applied to any pheromone value update rule used in ant colony optimization. We discuss the benefits coming with this new framework. The benefits are twofold. On the theoretical side, the new framework allows us to prove that in the ant system, the ancestor of all ant colony optimization algorithms, the average quality of the solutions produced increases in expectation over time when applied to unconstrained problems. On the practical side, the new framework automatically handles the scaling of the objective function values. We experimentally show that this leads on average to a more robust behavior of ant colony optimization algorithms.

Journal ArticleDOI
TL;DR: A new evolutionary approach, particle swarm optimization, is adapted for single-slice 3D-to-3D biomedical image registration and a new hybrid particle swarm technique is proposed that incorporates initial user guidance.
Abstract: Biomedical image registration, or geometric alignment of two-dimensional and/or three-dimensional (3D) image data, is becoming increasingly important in diagnosis, treatment planning, functional studies, computer-guided therapies, and in biomedical research. Registration based on intensity values usually requires optimization of some similarity metric between the images. Local optimization techniques frequently fail because functions of these metrics with respect to transformation parameters are generally nonconvex and irregular and, therefore, global methods are often required. In this paper, a new evolutionary approach, particle swarm optimization, is adapted for single-slice 3D-to-3D biomedical image registration. A new hybrid particle swarm technique is proposed that incorporates initial user guidance. Multimodal registrations with initial orientations far from the ground truth were performed on three volumes from different modalities. Results of optimizing the normalized mutual information similarity metric were compared with various evolutionary strategies. The hybrid particle swarm technique produced more accurate registrations than the evolutionary strategies in many cases, with comparable convergence. These results demonstrate that particle swarm approaches, along with evolutionary techniques and local methods, are useful in image registration, and emphasize the need for hybrid approaches for difficult registration problems.

Book ChapterDOI
TL;DR: It is shown that the multi-swarm optimizer significantly outperforms single population PSO on this problem, and that multi-quantum swarms are superior to multi-charged swarms and SOS.
Abstract: Many real-world problems are dynamic, requiring an optimization algorithm which is able to continuously track a changing optimum over time. In this paper, we present new variants of Particle Swarm Optimization (PSO) specifically designed to work well in dynamic environments. The main idea is to extend the single population PSO and Charged Particle Swarm Optimization (CPSO) methods by constructing interacting multi-swarms. In addition, a new algorithmic variant, which broadens the implicit atomic analogy of CPSO to a quantum model, is introduced. The multi-swarm algorithms are tested on a multi-modal dynamic function – the moving peaks benchmark – and results are compared to the single population approach of PSO and CPSO, and to results obtained by a state-of-the-art evolutionary algorithm, namely self-organizing scouts (SOS). We show that our multi-swarm optimizer significantly outperforms single population PSO on this problem, and that multi-quantum swarms are superior to multi-charged swarms and SOS.

Book ChapterDOI
17 Jul 2004
TL;DR: This paper reflects on the “swarm” terminology to help clarify its association with various robotic concepts.
Abstract: The term “swarm” has been applied to many systems (in biology, engineering, computation, etc.) as they have some of the qualities that the English-language term “swarm” denotes. With the growth of the various area of “swarm” research, the “swarm” terminology has become somewhat confusing. In this paper, we reflect on this terminology to help clarify its association with various robotic concepts.

Journal ArticleDOI
TL;DR: A new robotic concept called swarm-bot is introduced in which the collective interaction exploited by the swarm intelligence mechanism goes beyond the control layer and is extended to the physical level, which implies the addition of new mechanical functionalities on the single robot, together with new electronics and software to manage it.
Abstract: The swarm intelligence paradigm has proven to have very interesting properties such as robustness, flexibility and ability to solve complex problems exploiting parallelism and self-organization. Several robotics implementations of this paradigm confirm that these properties can be exploited for the control of a population of physically independent mobile robots. The work presented here introduces a new robotic concept called swarm-bot in which the collective interaction exploited by the swarm intelligence mechanism goes beyond the control layer and is extended to the physical level. This implies the addition of new mechanical functionalities on the single robot, together with new electronics and software to manage it. These new functionalities, even if not directly related to mobility and navigation, allow to address complex mobile robotics problems, such as extreme all-terrain exploration. The work shows also how this new concept is investigated using a simulation tool (swarmbot3d) specifically developed for quickly designing and evaluating new control algorithms. Experimental work shows how the simulated detailed representation of one s-bot has been calibrated to match the behaviour of the real robot.

Journal ArticleDOI
TL;DR: This paper introduces a self-assembling and self-organizing artifact, called a swarm-bot, composed of a swarm of s-bots, mobile robots with the ability to connect to and to disconnect from each other.
Abstract: In this paper, we introduce a self-assembling and self-organizing artifact, called a swarm-bot, composed of a swarm of s-bots, mobile robots with the ability to connect to and to disconnect from each other. We discuss the challenges involved in controlling a swarm-bot and address the problem of synthesizing controllers for the swarm-bot using artificial evolution. Specifically, we study aggregation and coordinated motion of the swarm-bot using a physics-based simulation of the system. Experiments, using a simplified simulation model of the s-bots, show that evolution can discover simple but effective controllers for both the aggregation and the coordinated motion of the swarm-bot. Analysis of the evolved controllers shows that they have properties of scalability, that is, they continue to be effective for larger group sizes, and of generality, that is, they produce similar behaviors for configurations different from those they were originally evolved for. The portability of the evolved controllers to real s-bots is tested using a detailed simulation model which has been validated against the real s-bots in a companion paper in this same special issue.

Journal ArticleDOI
01 May 2004
TL;DR: The results obtained seem to indicate that Particle Swarm Data Mining Algorithms are competitive, not only with other evolutionary techniques, but also with industry standard algorithms such as the J48 algorithm, and can be successfully applied to more demanding problem domains.
Abstract: Particle Swarm Optimisers are inherently distributed algorithms where the solution for a problem emerges from the interactions between many simple individual agents called particles. This article proposes the use of the Particle Swarm Optimiser as a new tool for Data Mining. In the first phase of our research, three different Particle Swarm Data Mining Algorithms were implemented and tested against a Genetic Algorithm and a Tree Induction Algorithm (J48). From the obtained results, Particle Swarm Optimisers proved to be a suitable candidate for classification tasks. The second phase was dedicated to improving one of the Particle Swarm optimiser variants in terms of attribute type support and temporal complexity. The data sources here used for experimental testing are commonly used and considered as a de facto standard for rule discovery algorithms reliability ranking. The results obtained in these domains seem to indicate that Particle Swarm Data Mining Algorithms are competitive, not only with other evolutionary techniques, but also with industry standard algorithms such as the J48 algorithm, and can be successfully applied to more demanding problem domains.

Book ChapterDOI
18 Sep 2004
TL;DR: In simulation tests, it is shown that AntHocNet can outperform AODV, one of the most important current state-of-the-art algorithms, both in terms of end-to-end delay and packet delivery ratio.
Abstract: In this paper we present AntHocNet, a new algorithm for routing in mobile ad hoc networks. Due to the ever changing topology and limited bandwidth it is very hard to establish and maintain good routes in such networks. Especially reliability and efficiency are important concerns. AntHocNet is based on ideas from Ant Colony Optimization. It consists of both reactive and proactive components. In a reactive path setup phase, multiple paths are set up between the source and destination of a data session, and during the course of the communication session, ants proactively test existing paths and explore new ones. In simulation tests we show that AntHocNet can outperform AODV, one of the most important current state-of-the-art algorithms, both in terms of end-to-end delay and packet delivery ratio.

Journal ArticleDOI
TL;DR: The PSO and its variants are applied to a synthetic test system of five types of candidate units with 6- and 14-year planning horizon and the results obtained are compared with dynamic programming in terms of speed and efficiency.

Book ChapterDOI
05 Sep 2004
TL;DR: How the Ant Colony Optimization (ACO) metaheuristic can be extended to continuous search domains and applied to both continuous and mixed discrete-continuous optimization problems is presented.
Abstract: This paper presents how the Ant Colony Optimization (ACO) metaheuristic can be extended to continuous search domains and applied to both continuous and mixed discrete-continuous optimization problems. The paper describes the general underlying idea, enumerates some possible design choices, presents a first implementation, and provides some preliminary results obtained on well-known benchmark problems. The proposed method is compared to other ant, as well as non-ant methods for continuous optimization.


Journal ArticleDOI
TL;DR: A new competitive approach is developed for learning agents to play two-agent games that uses particle swarm optimizers (PSO) to train neural networks to predict the desirability of states in the leaf nodes of a game tree.
Abstract: A new competitive approach is developed for learning agents to play two-agent games. This approach uses particle swarm optimizers (PSO) to train neural networks to predict the desirability of states in the leaf nodes of a game tree. The new approach is applied to the TicTacToe game, and compared with the performance of an evolutionary approach. A performance criterion is defined to quantify performance against that of players making random moves. The results show that the new PSO-based approach performs well as compared with the evolutionary approach.

Journal Article
TL;DR: This paper describes a procedure that uses particle swarm optimization (PSO) combined with the Lagrangian Relaxation (LR) framework to solve a power-generator scheduling problem known as the unit commitment problem (UCP).
Abstract: This paper describes a procedure that uses particle swarm optimization (PSO) combined with the Lagrangian Relaxation (LR) framework to solve a power-generator scheduling problem known as the unit commitment problem (UCP). The UCP consists of determining the schedule and production amount of generating units within a power system subject to operating constraints. The LR framework is applied to relax coupling constraints of the optimization problem. Thus, the UCP is separated into independent optimization functions for each generating unit. Each of these sub-problems is solved using Dynamic Programming (DP). PSO is used to evolve the Lagrangian multipliers. PSO is a population based search technique, which belongs to the swarm intelligence paradigm that is motivated by the simulation of social behavior to manipulate individuals towards better solution areas. The performance of the PSO-LR procedure is compared with results of other algorithms in the literature used to solve the UCP. The comparison shows that the PSO-LR approach is efficient in terms of computational time while providing good solutions.

Journal ArticleDOI
25 Jul 2004
TL;DR: An architecture which automates the design of recurrent neural networks using a new evolutionary learning algorithm based on a hybrid of particle swarm optimization (PSO) and evolutionary algorithm (EA) is applied.
Abstract: To predict the 100 missing values from the time series consisting of 5000 data given for the IJCNN 2004 time series prediction competition, we applied an architecture which automates the design of recurrent neural networks using a new evolutionary learning algorithm. This new evolutionary learning algorithm is based on a hybrid of particle swarm optimization (PSO) and evolutionary algorithm (EA). By combining the searching abilities of these two global optimization methods, the evolution of individuals is no longer restricted to be in the same generation, and better performed individuals may produce offspring to replace those with poor performance. The novel algorithm is then applied to the recurrent neural network for the time series prediction. The experimental results show that our approach gives good performance in predicting the missing values from the time series.

Posted Content
TL;DR: An extended model is presented in order to deal with digital image habitats, in which artificial ants could be able to react to the environment and perceive it, and inspired by the work of Chialvo and Millonas.
Abstract: Some recent studies have pointed that, the self-organization of neurons into brain-like structures, and the self-organization of ants into a swarm are similar in many respects. If possible to implement, these features could lead to important developments in pattern recognition systems, where perceptive capabilities can emerge and evolve from the interaction of many simple local rules. The principle of the method is inspired by the work of Chialvo and Millonas who developed the first numerical simulation in which swarm cognitive map formation could be explained. From this point, an extended model is presented in order to deal with digital image habitats, in which artificial ants could be able to react to the environment and perceive it. Evolution of pheromone fields point that artificial ant colonies could react and adapt appropriately to any type of digital habitat. KEYWORDS: Swarm Intelligence, Self-Organization, Stigmergy, Artificial Ant Systems, Pattern Recognition and Perception, Image Segmentation, Gestalt Perception Theory, Distributed Computation.

Posted Content
TL;DR: The present ant clustering system (ACLUSTER) avoids not only short-term memory based strategies, as well as the use of several artificial ant types, present in some recent approaches, and is also the first application of ant systems into textual document clustering.
Abstract: Social insect societies and more specifically ant colonies, are distributed systems that, in spite of the simplicity of their individuals, present a highly structured social organization As a result of this organization, ant colonies can accomplish complex tasks that in some cases exceed the individual capabilities of a single ant The study of ant colonies behavior and of their self-organizing capabilities is of interest to knowledge retrieval/management and decision support systems sciences, because it provides models of distributed adaptive organization which are useful to solve difficult optimization, classification, and distributed control problems, among others In the present work we overview some models derived from the observation of real ants, emphasizing the role played by stigmergy as distributed communication paradigm, and we present a novel strategy to tackle unsupervised clustering as well as data retrieval problems The present ant clustering system (ACLUSTER) avoids not only short-term memory based strategies, as well as the use of several artificial ant types (using different speeds), present in some recent approaches Moreover and according to our knowledge, this is also the first application of ant systems into textual document clustering KEYWORDS: Swarm Intelligence, Ant Systems, Unsupervised Clustering, Data Retrieval, Data Mining, Distributed Computing, Document Maps, Textual Document Clustering

Proceedings ArticleDOI
26 Aug 2004
TL;DR: The fundamental and standard algorithm is introduced, the work on the algorithm improvement during the past years is surveyed, as well as the applications on the multi-objective optimization, neural networks and electronics, etc.
Abstract: Particle swarm optimization (PSO) explores global optimal solution through exploiting the particle's memory and the swarm's memory. Its properties of low constraint on the continuity of objective function and joint of search space, and ability of adapting to dynamic environment make PSO become one of the most important swarm intelligence methods and evolutionary computation algorithms. The fundamental and standard algorithm is introduced firstly. Then the work on the algorithm improvement during the past years is surveyed, as well as the applications on the multi-objective optimization, neural networks and electronics, etc. Finally, the problems remaining unresolved and some directions of PSO research are discussed.

Book ChapterDOI
05 Sep 2004
TL;DR: The best performing ACO algorithms implemented, when combined with a fine-tuned local search procedure, reach excellent performance on a set of well known benchmark instances.
Abstract: In this paper we present a study of several Ant Colony Optimization (ACO) algorithms for the Set Covering Problem. In our computational study we emphasize the influence of different ways of defining the heuristic information on the performance of the ACO algorithms. Finally, we show that the best performing ACO algorithms we implemented, when combined with a fine-tuned local search procedure, reach excellent performance on a set of well known benchmark instances.

Book ChapterDOI
05 Sep 2004
TL;DR: A particle swarm optimization algorithm (PSO) to solve the permutation flowshop sequencing problem (PFSP) with makespan criterion is presented and a total of 195 out of 800 best-known solutions in the literature is improved by the VNS version of the PSO algorithm.
Abstract: This paper presents a particle swarm optimization algorithm (PSO) to solve the permutation flowshop sequencing problem (PFSP) with makespan criterion. Simple but very efficient local search based on the variable neighborhood search (VNS) is embedded in the PSO algorithm to solve the benchmark suites in the literature. The results are presented and compared to the best known approaches in the literature. Ultimately, a total of 195 out of 800 best-known solutions in the literature is improved by the VNS version of the PSO algorithm.

Proceedings ArticleDOI
19 Jun 2004
TL;DR: A concurrent PSO (CONPSO) algorithm is proposed to alleviate the premature convergence problem of PSO algorithm, a type of parallel algorithm in which modified PSO and FDR-PS algorithms are simulated concurrently with frequent message passing between them.
Abstract: In this paper, a concurrent PSO (CONPSO) algorithm is proposed to alleviate the premature convergence problem of PSO algorithm. It is a type of parallel algorithm in which modified PSO and FDR-PS algorithms are simulated concurrently with frequent message passing between them. This algorithm avoids the possible crosstalk effect of pbest and gbest terms with nbest term in FDR-PSO. Thereby, search efficiency of proposed algorithm is improved. In order to demonstrate the effectiveness of the proposed algorithm, experiments were conducted on six benchmarks continuous optimization problems. Results clearly demonstrate the superior performance of the proposed algorithm in terms of solution quality, average computation time and consistency. This algorithm is very much suitable for the implementation in parallel computer.