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Full Paper Proc. of Int. Conf. on Advances in Computer Engineering 2012

Chaotic Shuffled Frog Leaping Optimization Algorithm Vahid Rashtchi, Meisam Hatami, Mahdi Sabouri Electrical Engineering Department, Faculty of Engineering, University of Zanjan, Zanjan, Iran Rashtchi@znu.ac.ir, Ma_hatami@yahoo.com ,M.Sabouri@znu.ac.ir Shuffled frog leaping algorithm(SFLA), developed by Eusuff and Lansey in 2000, is a population based heuristic for combinatorial optimization [10]. In this algorithm evolution of memes is driven by exchange of information among the interactive individuals. It combines the advantages of the genetic-based memetic algorithm (MA) and the social behavior-based Particle Swarm Optimization (PSO) algorithm[10]. SFLA has been tested on several combinatorial problems and found to be effective in searching the global solutions [11],[12].

Abstract– As a novel optimization technique, chaos has gained much attention and some applications during the past decade .For a given energy or cost function, by following chaotic ergodic orbits, a chaotic dynamic system may eventually reach the global optimum or its good approximation with high probability. To enhance the performance of Shuffled Frog Leaping Algorithm (SFLA) which meta-heuristic search method inspired from the memetic evolution of a group of frogs when seeking for food, hybrid SFLA is proposed by incorporating chaos. SFLA and chaos are hybridized to form a chaotic SFLA (CSFLA), which reasonably combines the population-based evolutionary searching ability of SFLA and chaotic searching behavior. Simulation results show that the CSFLA can effectively enhance the searching efficiency and greatly improve the searching quality.

II. OVERVIEW SHUFFLED FROG LEAPING ALGORITHM

The SFL algorithm is a memetic meta-heuristic that is designed to seek a global optimal solution by performing an informed heuristic search using a heuristic function. It is based on evolution of memes carried by interactive individuals and a global exchange of information among the population. The SFL algorithm progresses by transforming ‘‘frogs’’ in a memetic evolution. In this algorithm, frogs are seen as hosts for memes and described as a memetic vector. Each meme consists of a number of memotypes. The memotypes represent an idea in a manner similar to a gene representing a trait in a chromosome in a genetic algorithm. The SFL does not change the physical characteristics of an individual rather it progressively improves the ideas held by each frog in a socalled virtual population .The frogs can communicate with each other, and can improve their memes by infecting (passing information) each other. Improvement of memes results in changing an individual frog’s position by adjusting its leaping step size. Based on this abstract model of virtual frogs, the SFL algorithm draws on PSO as a local search tool and the idea of competitiveness and mixing information from parallel local searches to move toward a global solution from the shuffled complex evolution (SCE) algorithm [15]. The sample of virtual frogs constitutes a population. The population is partitioned into subsets described as memeplexes. The memeplexes can be perceived as a set of parallel frog cultures attempting to reach some goal. Each frog culture proceeds towards their goal exchanging ideas independently in parallel. Frog leaping improves an individual’s meme and enhances its performance towards the goal. Within each memeplex, the individual frogs hold information can be infected by other’s ideas, and hence they experience a memetic evolution. After a defined number of memetic evolution steps, information is passed between memeplexes in a shuffling process.

Index Terms– Chaotic Shuffled Frog Leaping Algorithm (CSFLA), SFLA, Chaos, Optimization I. INTRODUCTION

Chaos is a kind of characteristic of non-linear systems, which is a bounded unstable dynamic behavior that exhibits sensitive dependence on initial conditions and includes infinite unstable periodic motions. Although it appears to be stochastic, it occurs in a deterministic non-linear system under deterministic conditions. In recently years, growing interests from physics, chemistry, biology and engineering have stimulated the studies of chaos for control [1,2,3], synchronization [4] and optimization [5]–[7]. Due to the easy implementation and special ability to avoid being trapped in local optima, chaos has been a novel optimization technique and chaos-based searching algorithms have aroused intense interests [8]. Currently, there are two ways of application of chaos for optimization. The first one is chaotic neural network (CNN) [9] by incorporating chaotic dynamics into neural network. Through the rich non-equilibrium dynamics with various concomitant attractors, chaotic neuron-dynamics can be used to continually search for the global optimum by following chaotic ergodic orbits. The other one is chaotic optimization algorithm (COA) [6],[7] based on chaotic evolution of variables. The simple philosophy of the COA includes two main steps: firstly mapping from the chaotic space to the solution space, and then searching optimal regions using chaotic dynamics instead of random search. However, simple CNN and COA often need a large number of iterations to reach the global optimum and are sensitive to the initial conditions. © 2012 ACEEE DOI: 02.ACE.2012.03.9

Full Paper Proc. of Int. Conf. on Advances in Computer Engineering 2012 It can be seen that SFLA is used for exploration by updating particle swarm, and chaos dynamic is applied for exploitation by locally modified the best particle resulted by SFLA. Besides, to maintain population diversity,several new particles are randomly generated and incorporated in the new population. Especially, the region for generating new particles is dynamically decreased so as to speed up convergence.

Shuffling enhances the meme quality after being infected by the frogs from different memeplexes, ensures that the cultural evolution towards any particular interest is free from bias. After this, this local search and shuffling process continues until defined convergence criteria are satisfied. The SFL algorithm is a combination of eterministic and random approaches. The deterministic strategy allows the algorithm to use response surface information effectively to flexibility and robustness of the search pattern. The SFL algorithm starts with an initial population of “P” frogs created randomly within the feasible space &!. For D-dimensional problems, the position of the ‘‘ith’’ frog is represented as Xi =(xi1, xi2, . . . , xiD). Afterwards the performance of each frog is computed based on its position. The frogs are sorted in a descending order according to their fitness. Then, the entire population is divided into m memeplexes, each containing n frogs (i.e., P=m×n). In this process, the first frog goes to the first memeplex, the second frog goes to the second memeplex,frog m goes to the mth memeplex, and frog m+1 goes to the first memeplex, and so on. Within each local memeplex, the frogs with the best and the worst fitness are identified as Xb and Xw, respectively.Also, the frog with the global best fitness is identified as Xg. Then, an evolutionary process is applied to improve only the frog with the worst fitness (not all frogs) in each cycle. Accordingly, each frog updates its position to catch up with the best frog as follows: Step size (Si) = Rand()× (Xb-Xw ) (1) New position (Xw)=Current position(Xw)+Si (2) -Smind Sd Smax (3) Where Rand() is a random number in the range[0,1], and Smax is the maximum step size allowed to be adopted by a frog after being infected. If this process produces a better solution, it replaces the worst frog. Otherwise, the calculations in Equations (1) and (2) are repeated with respect to the global best frog (i.e., Xg replaces Xb). If no improvement becomes possible in this case, then a new solution is randomly generated to replace the worst frog. The calculations then continue for a specific number of iterations [13,14]. The flowchart of the SFLA is illustrated in Fig. 1.

Figure 1. SFLA flow chart. IV. SIMULATION RESULTS

III. CHOATIC SFLA

Results of applying the proposed algorithm on benchmark functions are presented in this section to show its effectiveness in function optimization. The results are compared to the ones of [9] and are shown in Table I, Table II for the same number of fitness evaluation.

Based on proposed SLFA and the chaotic local search ,a two iterative strategy named Choatit Shuffed Frog Leaping algorithm (CSFLA) is proposed ,in which SFLA is applied to perform global exploration and CLS is employed to perform locally oriented search (exploitation) for the solutions resulted by SFLA. The procedure of CSFLA is described in Fig. 2.

Full Paper Proc. of Int. Conf. on Advances in Computer Engineering 2012

Figure 2. Steps of CSFLA algorithm Table I. AVERAGE OF THE BEST FITNESS FUNCTION

Table I (continue). AVERAGE

OF THE BEST FITNESS FUNCTION

Full Paper Proc. of Int. Conf. on Advances in Computer Engineering 2012 V. CONCLUSION

APENDIX BENCH MARK FUNCTIONS

The Chaotic Shuffled Frog Leaping algorithm (CSFLA) has been proposed to solve high dimension function optimization problems. However CSFLA performance in high dimension function optimization shows power of the algorithm and its flexibility in solving different kind of problems.

1) Sphere function :

REFERENCES

2) Schwefel‘s problem 2.22 :

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3) Schwefel‘s problem 1.2 :

4) Schwefel‘s problem 2.21:

5) Generalized Rosenbrock function :

6) Quadric function :

Note: This is a noisy fitness function. There is a random measurement noise in each fitness evaluation. 7) Generalized Rastrigin function :

8) Generalized Griewank function :