A hybrid meta-heuristic optimization algorithm based on SFLA

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1 2 nd International Conference on Engineering Optimization September 6-9, 2010, Lisbon, Portugal A hybrid meta-heuristic optimization algorithm based on SFLA Mohammadreza Farahani 1, Saber Bayat Movahhed 1, Seyyed Farid Ghaderi 1 1 Department of Industrial Engineering, Faculty of Engineering, University Of Tehran, Tehran, Iran, m_farahani@hotmail.com Abstract In the last decades, there has been widespread interaction between researchers seeking various evolutionary computation methods to conceive more accurate and expeditious solutions. We herein present an evolutionary algorithm based on the Shuffled Frog Leaping Algorithm (SFLA). Similar to the SFLA, our method partitions particles into different groups called memeplexes; however, the best particle in each memeplex thereafter determines its movement through the search space in each iteration of the algorithm toward the global best particle and the worst particle in each memeplex keeps track of its coordinates in the solution space by moving toward the local best particle (the best particle in the same memeplex). Not only does this method lessen computation costs and offer speedier solutions in comparison to the Particle Swarm Optimization, but it also has a distinct advantage over the SFLA in that it reduces the probability of the particle s being trapped in the local minima by directing the best local particle toward the global best particle. Our method was tested on two test functions in different circumstances and the results were subsequently compared with those of the other evolutionary methods such as the SFLA. Success rate and search time were the parameters considered to check our method efficiency. Keyword: Evolutionary algorithms (EA), SFLA, Optimization 1. Introduction Optimization algorithms have constituted some of the most significant subjects in mathematics and industry in the past few decades. For all the traditional algorithms available to seek best solutions to a given function, however, optimization continues to pose a challenge in most real world cases because of the huge and complex solution space [1]. Indeed, there are still large-scale optimization problems that necessitate speedy resolution in a time span between ten milliseconds and a few minutes, resulting in the exchange of optimality with speed gains. In the main, speed and precision are goals considered to be at variance, at least with respect to probabilistic algorithms: optimization accuracy can be enhanced only if there is more time available [2]. These obstacles, which are correlated with the utilization of mathematical optimization in large-scale problems, have gradually paved the way for the development of alternative solutions [3]. One of these alternative solutions is the Evolutionary Algorithm, first introduced by Holland [4]. The Evolutionary Algorithm was developed by mimicking or simulating processes found in nature and mainly includes Genetic Algorithms, Memetic Algorithms, Particle Swarm Optimization, Ant Colony Optimization, and Shuffled Frog Leaping Algorithm (SFLA).[1] The SFLA was recently devised as a novel meta-heuristic algorithm by Muzaffar Eusuff and Kevin Lansey through observing, imitating, and modeling the behavior of frogs searching for food placed on separate stones haphazardly positioned in a pond [5]. Since then, the SFLA has been tested on a large number of combinatorial problems and found to be efficient in finding global solutions [6]. Furthermore, the SFLA compares favorably with the Genetic Algorithm, the Ant Colony Optimization, and the Particle Swarm Optimization in terms of time processing [7]. The SFLA is a population-based cooperative search metaphor inspired by natural memetics [6] and consists of a frog leaping rule for local search and a memetic shuffling rule for global information exchange [8]. The SFLA comprises a set of interacting virtual population of frogs partitioned into different groups (memeplexes), referred to as memeplexes, searching for food. The different memeplexes are considered as different cultures of frogs, with each frog performing a local search. Within each memeplex, the individual frog with leading ideas can be influenced by the ideas of the other frogs and can evolve through a local search for a specific number of times [9]. The algorithm functions simultaneously as an independent local search in each memeplex. The local search is completed using a particle swarm optimization-like method adapted for discrete problems but emphasizing a local search [10]. The SFLA is also sometimes introduced as a combination of the determinacy method and randomicity method: while the determinacy strategy allows the algorithm to exchange messages effectively, the randomicity confers flexibility and robustness to the algorithm [11]. Recent years have witnessed the introduction of many hybrid algorithms based on the SFLA. A novel cognition component has been proposed to enhance the effectiveness of the SFLA, such that the frog adjusts its position

2 not only according to the best individual within the memeplex or the global best of the population but also according to the thinking of the frog itself [12]. Another new memetic algorithm is the Shuffled Particle Swarm Optimization, which combines the learning strategy of the Particle Swarm Optimization and the shuffled strategy of the SFLA [13]. A modified form of the SFLA has also been introduced by incorporating a new search acceleration parameter into the formulation of the original algorithm [14]. Of significance among these recent developments is a new SFLA for continuous space optimization, which divides the population on the basis of the principle of uniform performance of memeplexes and thus enables all the frogs to participate in the evolution by keeping the inertia learning behaviors and learning from better ones selected randomly [15]. And finally, a hybrid evolutionary algorithm based on a combination of the Shuffled Particle Swarm Optimization and MSFLA has been proposed, called SAPSO SFLA, to optimally reconfigure radial distribution systems [16]. This paper proposes a new algorithm based on the SFLA, tests the robustness of the proposed algorithm on two test functions, and finally compares the results with those obtained by the original method. The remainder of this paper is organized as follows: the original SFLA is described in Section 2; the proposed memetic algorithm based on the SFLA is presented in Section 3; the two test functions, on the basis of which our method is tested, are introduced in Sections 4 and 5; the results and performance analyses of the several optimization functions are depicted in Section 6; and finally conclusions are offered in Section Original Shuffled Frog Leaping Algorithm In the SFLA, first an initial population of F frogs is created randomly. Next the population of F frogs is sorted in order of increasing performance level and separated into m memeplexes each holding n frogs (i.e. ) in such a way that the first frog goes to the first memeplex, the second frog goes to the second memeplex, the m th frog goes to the m th memeplex, and the (m+1) th frog goes back to the first[8]. The next step is the evaluation of each memeplex. In this step, each frog in the memeplex leaps toward the optimum location by learning from the best frog, so that the new position of the worst frog in the memeplex is calculated according to (1) (1) where is the position of the worst frog in the memeplex, is the position of the best frog in the memeplex, r is a random number between 0 and 1, and k is the iteration number of the memeplex [17]. If this evolution produces a better frog (solution), it replaces the older frog. Otherwise, is replaced by in (2), in which case the calculation of the new position can be expressed by (2): (2) If non-improvement occurs in this case, a random frog is generated to replace the old frog [16]. A pseudo code for the SFLA procedure is summarized in Appendix A. 3. Proposed Memetic Algorithm Based on SFLA Our new method is based on the identification of the weaknesses of the basic SFLA. To that end, the SFLA was initially applied to different functions and one of the fundamental weaknesses of this method was identified as the elimination of the effective frogs from the solving procedure in consequence of some memeplexes being wasted in the local minima. One way to reduce the probability of this occurrence in this method is to improve the guiding particle in each memeplex. This guiding particle in the SFLA is in each memeplex; therefore; in this method, the frog with the best position determines its movement through the search space toward the global best frog, as it expressed by (3) A pseudo code for the proposed method procedure is summarized in Appendix B. (3)

3 start Randomly produce the initial population Compute the performance value of the frogs g 2=1 Sort the frogs in order of the performance of each frog Partition the population into m memeplex p=1,g 1=1 Update the worst frog s position in memeplex by (1) Update the worst frog s update in the memeplex by (2) Update the best frog s position in the memeplex by (3) p>m p=p+1 Shuffle the memeplexes g 1=1+ g 1 g 1>G Stop Figure 1: Flow chart of the proposed method based on SFLA Where p counts the number of memeplexes and is compared with the total number of memeplexes (m), and g counts the number of iterations and is compared with the maximum number of steps (G). 4. Langermann s function The Langermann function is a multimodal test function. The local minima are unevenly distributed. The function has the following definition:

Figure 2: Langermann s function First, the minimum value of the drop wave function is obtained by value here is equal to -2.19334.

The given form of the function has only two variables and the following definition: Figure 3: Drop wave function [18].

4 Figure 2: Langermann s function First, the minimum value of the drop wave function is obtained by value here is equal to Second, the experiment results that meet a precision of as successful., and the minimum are considered 5. Drop wave function This is a multimodal test function. The given form of the function has only two variables and the following definition: Figure 3: Drop wave function [18].First, the minimum value of the drop wave function is obtained by value here is equal to Second, the experiment results that meet a precision of as successful. Also in both functions the number of variables is considered to be 20., and the minimum are considered 6. Results The runtime of the proposed method is more than the original SFLA owing to the fact that the proposed method has an extra step compared with the original method. In order to overcome this weakness, the following parameter settings are designed for algorithms: Original SFLA:

5 Proposed method: Consequently, the processing time is approximately equal in both algorithms. Tables 1 and 2 show the results of the two test functions carried out by the SFLA and our proposed method. Considering the results of the experiment, one can draw the following conclusion. The proposed method will have better or at least equal performance of global search. Interestingly, Langermann s function, compared with Drop wave function, reflects the difference between the SFLA and our proposed method more clearly since there is a higher probability that frogs are trapped in the local minima in this function. In the same fashion, Langermann s function can highlight the weaknesses of the original SFLA more efficiently than Drop wave. Langermann Success rate Mean value domain Proposed method Original SFLA Proposed method Original SFLA Table 1: experiments results of Langermann s function Drop Wave Success rate Mean value domain Proposed method Original SFLA Proposed method Original SFLA Table 2: experiments results of Drop wave function In order to demonstrate that the proposed method has the advantage of being far away from the local minima, 100 experiment results were gained by both methods with the following setting: domain=400 for Langermann s function and they were showed as an histogram in Fig 4,5. As shown in Fig 4, 5 fifteen experiment results are between (-1,-1.2). This range is one of the local minima of Langermann s function owing to this fact it was concluded that 15 percent of solutions are degenerate while in the proposed method less than 10 percent of solutions are degenerate in the same range.

6 Figure 4: The original SFLA histogram Figure 5: The proposed method histogram 7. Conclusion In this study initially several continuous functions were used to identify the weaknesses of the original SFLA in order to modify the original method by dealing with those weaknesses. After several simulations, it was figured out that the best way to modify the original SFLA was that the best particle in each memeplex should determine its movement through the search space toward the global best particle. To justify this methodical modification, two optimization test problems were solved using both algorithms. The comparative results also were presented. According to simulation results on two test functions in the proposed algorithm, the frogs were not trapped any longer. Compared with the standard SFLA with two Functions, the proposed method is verified to be superior in terms of robustness and stability.

7 Appendix A. A pseudo code for Shuffled Frog Leaping Algorithm Begin; Generate random population of P solutions (frogs) Feasible zone; For each individual i P: calculate fitness (i); Sort the population P in descending order of their fitness; Divide P into m memeplexes; For i=1 to number of generations For each memeplex; Determine the best and worst frogs; Improve the worst frog position using Eqs. (1), (2) Combine the evolved memeplexes; Sort the population P in descending order of their fitness; Check if termination=true; Appendix B. A pseudo code for improved Shuffled Frog Leaping Algorithm Begin; Generate random population of P solutions (frogs) Feasible zone; For each individual i P: calculate fitness (i); Sort the population P in descending order of their fitness; Divide P into m memeplexes; For i=1 to number of generations For each memeplex; Determine the best and worst frogs; Improve the worst frog position using Eqs. (1), (2) Improve the best frog in each memeplex toward the global best using Eqs. (3) Combine the evolved memeplexes; Sort the population P in descending order of their fitness; Check if termination=true; REFRENCES [1] Antariksha Bhaduri, A Clonal Selection Based Shuffled Frog Leaping Algorithm, 4th Annual IEEE Conference. International Advance Computing Conference, IACC, Patiala, India, March 2009 (ISBN: ). [2] Thomas Weise, Global Optimization Algorithms - Theory and Application, 2nd ed., self-published, 2009, Online available at [3] M. M. Eusuff, K. E. Lansey, F. Pasha, Shuffled frog-leaping algorithm: a memetic meta heuristic for discrete optimization, Engineering Optimization, 2006, vol. 38, no. 2, pp [4] Kezong TANG, Jingyu YANG, Shang GAO, Tingkai SUN, A self-adaptive linear evolutionary algorithm for solving constrained optimization problems, Journal of Control Theory and Applications, March 28, 2010 [5] Xue-hui, YANG Ye, LI Xia,Solving TSP with Shuffled Frog-Leaping Algorithm, Eighth International Conference on Intelligent Systems Design and Applications, ISDA, Kaohsiung, Taiwan, November 26-28,2008 ( ISBN: ). [6] M. M. Eusuff, K. E. Lansey, Optimization of water distribution network design using the shuffled frog leaping algorithm, Journal of Water Resource Planning and Management, vol. 129, issue 3, ,2003 [7] Emad Elbeltagi, Tarek Hegazy, and Donald Grierson, Comparison among five evolutionary-based optimization algorithms, Advanced Engineering Informatics, volume 19, issue 1, 43-53, January 2005

8 [8] Thai-Hoang Huynh, Duc-Hoang Nguyen, Fuzzy Controller Design Using A New Shuffled Frog Leaping Algorithm, Proceedings of the IEEE International Conference on Industrial Technology, ICIT,Australia, Gippsland, VIC,2009 (ISBN: ). [9] Yinghai Li, Jianzhong Zhou, Junjie Yang, Li Liu, Hui Qin and Li Yang, The Chaos-based Shuffled Frog Leaping Algorithm and Its Application, Fourth International Conference on Natural Computation, ICNC, China, Jinan 2008 (ISBN: ). [10] Gonggui Chen, Combined Economic Emission Dispatch Using SFLA, International Conference on Information Engineering and Computer Science,ICNC, China, Wuhan, 2009 (ISBN: ). [11] Mei YUE, Tao HU, Baoping GUO, The Research Base on Memetic Meta-heuristic Shuffled Frog-leaping Algorithm, 2nd International Conference on Power Electronics and Intelligent Transportation System, PEITS, China, Shenzhen, 2009 (ISBN: ). [12] X. C. Zhang, X. M. Hu, G. Z. Cui, et al, An improved shuffled frog leaping algorithm with cognitive behavior, In Proceedings of the 7 th World Congress on Intelligent Control and Automation, WCICA, Chongqing, China, 2008 (ISBN: ). [13] Z. Y. Zhen, Z. S. Wang, Z. Gu, Y. Y. Liu, A novel memetic algorithm for global optimization based on PSO and SFLA, Lecture Notes in Computer Science, Springer Berlin / Heidelberg, , 2007 [14] E. Elbeltagi, T. Hegazy, D. Grierson, A modified shuffled frog-leaping optimization algorithm: applications to project management, Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycl, vol. 3, issue 1, 53-60, 2007 [15] Ziyang Zhen, Daobo Wang, Yuanyuan Liu, Improved Shuffled Frog Leaping Algorithm for Continuous Optimization Problem, IEEE Congress on Evolutionary Computation, CEC, Norway, Trondheim 2009 (ISBN: ). [16] Taher Niknam, Ehsan Azad Farsani, A hybridself-adaptive particle swarm optimization and modified shuffled frog leaping algorithm for distribution feeder reconfiguration, Engineering Applications of Artificial Intelligence,2010 [17] Jianping Luo Min-Rong Chen Xia Li,A Novel Hybrid Algorithm for Global Optimization Based on EO and SFLA, 4th Conference on Industrial Electronics and Applications, ICIEA China, Xi'an,2009 (ISBN: ). [18] Marcin Molga, Czesław Smutnicki, Test functions for optimization needs, 3 kwietnia 2005

International Journal of Current Research and Modern Education (IJCRME) ISSN (Online): & Impact Factor: Special Issue, NCFTCCPS -

International Journal of Current Research and Modern Education (IJCRME) ISSN (Online): & Impact Factor: Special Issue, NCFTCCPS - TO SOLVE ECONOMIC DISPATCH PROBLEM USING SFLA P. Sowmya* & Dr. S. P. Umayal** * PG Scholar, Department Electrical and Electronics Engineering, Muthayammal Engineering College, Rasipuram, Tamilnadu ** Dean