A Comparative Study of Flower Pollination Algorithm and Bat Algorithm on Continuous Optimization Problems

Transcription

1 A Comparative Study Flower Pollination Algorithm Bat Algorithm on Continuous Optimization Problems Nazmus Sakib Department Dhaka-128, Md. Wasi Ul Kabir Department Dhaka-128, Md Subbir Rahman Department Dhaka-128, Mohammad Shafiul Alam Department Dhaka-128, ABSTRACT Nature is a rich source inspiration, which has inspired many researchers in many ways. Nowadays, new algorithms have been developed by the inspiration from nature. The flower pollination algorithm is based on the characteristics pollination process flowers plants. Pollination is a natural biological process mating in plants. In flowers, pollen is carried to stigma through some mechanisms that confirm a proper balance in the genetic creations the species. Another nature inspired algorithm the Bat algorithm is based on the echolocation behavior bats. In this paper, the Flower pollination algorithm is compared with the basic Bat algorithm. We have tested these two algorithms on both unimodal multimodal, low high dimensional continuous functions. Simulation results suggest that the Flower pollination algorithm can perform much better than the Bat algorithm on the continuous optimization problems. Keywords Flower pollination algorithm, Bat algorithm, Swarm intelligence, Meta-heuristic optimization. 1. INTRODUCTION Many real-world optimization problems are very complex challenging to solve, many applications have to deal with these problems. To solve such problems, approximate optimization algorithms have to be used, though there is no guarantee that the optimal solution can be obtained [1]. Nature has been solving many problems for billions years, many kinds biological systems have shown fascinating remarkable efficiency in problem solving [2] [3] [4]. Over the last few decades optimization algorithms have been applied in extensive numbers difficult problems. Several nature-inspired algorithms have been developed over the last few years by the scientific community [2] [4] [5]. Recently two new nature inspired algorithm, namely the Flower pollination algorithm (FPA) the Bat algorithm (), have been developed [1]. For more than hundred million years, flower plants have been evolving. There are over 2 hundreds types flower plants in the nature where flower species are three fourth all plant species. Surprisingly from the beginning the nature, flowering plants dominate the lscape [5] [6]. The reproduction flower is achieved via the pollination process. Flower pollination can be described as the distribution processes pollen through a wide range pollinators such as insects, birds, bats some other animals [7]. The Bat Algorithm is a swarm based algorithm which is established on the echolocation behavior bats. The bats use echolocation to find its prey [8]. The Bat algorithm is a very recent swarm intelligence algorithm, introduced in 2 [9]. After that a few improvements Bat algorithm have been proposed in the literature, e.g., [] [11] [12] [13]. In this paper, we compare the performance the flower pollination algorithm with the basic Bat algorithm [14] [9]. The rest this paper is organized as follows. Sections 2 3 describe the Flower Pollination Algorithm the Bat algorithm. Section 4 provides details the simulation analysis both these algorithms on the benchmarking problems, parameter settings the algorithms compares their results. Finally, section 5 draws the conclusion this paper with a few comments suggestions on future research. 2. FLOWER POLLINATION ALGORITHM 2.1 Characteristics Flower pollination Flower pollination algorithm is developed by the idea flower pollination process. Basically the pollination process carried the pollen from the male parts a flower to the female part called stigma a flower [6]. Pollinators can be very diverse. It is estimated that there are at least two hundred thous varieties pollinator exist in nature like insects, bats birds [7]. There are two types pollination Biotic pollination process Abiotic pollination process. 13

2 In Biotic pollination, pollen is carried to the stigma by insects animals [15] [16]. In Abiotic pollination, pollination occurs via wind or diffusion in water [15] [17]. Survey says that, % pollination takes Abiotic pollination process which does not require any pollinators. Pollination can be achieved by two ways. They are self-pollination cross-pollination. Cross-pollination process means pollination can be happened from pollen a flower a different plant. On contrary, self-pollination is the pollination one flower, from pollen the same flower or different flowers the same plant, which ten happens when there is no available pollinator [17] [15]. Biotic, cross-pollination may be happened in long distance. Bees, bats, birds flies are mostly used as pollinators which are able to fly a long distance. So, these pollinators are considered as the carrier the global pollination [5] [16]. Xin-She Yang describes this flower constancy pollinator behavior in the pollination process into the following four rules [7]: 1. Biotic cross-pollination is considered as global pollination process with pollen-carrying pollinators performing L evy flights. 2. Abiotic self-pollination are considered as local pollination. 3. Flower constancy can be considered as the reproduction probability is proportional to the similarity two flowers involved. 4. Local pollination global pollination is controlled by a switch probability p [, 1]. Besides the physical proximity other factors like wind water, local pollination can have a significant fraction p in the overall pollination process [18]. 2.2 Global pollination Local pollination Global pollination process Local pollination process are two key steps in Flower pollination algorithm [7]. In global pollination process, flower pollens are carried by pollinators. These pollens may travel over a long distance because pollinators can ten fly move in longer range. This global pollination can be represented mathematically as Here is the pollen i or solution vector at iteration t, is the current best solution found among all solutions at the current iteration. Here λ is a scaling factor to control the step size. Hence, parameter is also the step size which corresponds to the strength the pollination [5] [7] [18]. Since pollinators can be travelled over a long distance with different distance steps. Here, L evy flight can be used to imitate this travelling characteristic [18]. Assuming fom a L evy distribution Here, Γ(λ) is the stard gamma function L evy distribution is valid for long steps. Therefore, Rule 2 Rule 3 which are basically for the local pollination can be represented like Here, are pollen from different flowers the same plant species. The equation imitates flower constancy in limited neighborhoods [5]. Assuming in mathematically if (1) (2) (3) comes from the same species or selected from the same population, this equivalently becomes a local rom walk if a graph can be drawn from a uniform distribution in [, 1]. The pseudo code the flower pollination algorithm is given below. 2.3 Pseudo code Flower Pollination Algorithm 1. Objective min or max 2. Initialize a population n flowers/pollen gametes with rom solutions 3. Find the best solution in the initial population 4. Define a switch probability p [, 1] 5. while (t < MaxGeneration) 6. for i = 1 to n ( all n flowers in the population) 7. if r < p 8. Draw a (d-dimensional) step vector L which obeys a L evy distribution 9. Global pollination via. else 11. Draw ϵ from a uniform distribution in [,1] 12. Local pollination via 13. end if 14. Evaluate new solutions 15. if new solutions are better, update them in the population 16. end for 17. Find the current best solution 18. end while 3. T ALGORITHM Bat algorithm is developed on the basis bat s hunting behavior. Bats are fascinating animals whose are also mammals with wings. Bats are born with the appealing innovative capability echolocation. Microbats are insectivores. Echolocation is a special type sonar, used by the Microbats to avoid obstacles, detect prey, pinpoint their prey. Bats emit a high sound frequency to listen the echo that bounces back from the neighboring objects [13] [19] [2]. Bats use short, frequency-modulated signals to sweep through about an octave. Signal frequencies bats varies depends on the species [21] [22]. Echolocation characteristics Microbats emphasize on some approximate rules, which are given below [9]: i. Distance: Bats use echolocation to sense distance. They acknowledge the ranges/spaces between prey surrounded barriers in some miraculous ways. ii. Frequency: Bats fly romly with velocity at position with a fixed frequency, varying wavelength λ loudness to search for prey. They can automatically adjust the wavelength their emitted pulses adjust the rate pulse emission in the range [, 1], depending on the proximity their target [22]. iii. Loudness: Though loudness can vary in many ways. Here we like to assume that the loudness differs from a large A to a minimum constant value. 3.1 Initialization Bat Algorithm Initial population is generated romly by considering lower upper boundaries with the dimension the number bats from the real valued vectors. Where are upper lower boundaries for dimension. (4) 14

3 Loudness (A) Pulse Rate (r) International Journal Applied Information Systems (IJAIS) ISSN : Solution, Frequency & Velocity Step size a solution in Bat algorithm is defined by the frequency. Initially, frequency Bat algorithm is set to rom value for each solution in range. Then velocity is updated using the frequency for each solution using following equations. Where ϵ indicates romly generated number, represents global best solution. In r (pulse rate) probability, a solution is selected among the best solution rom walk is applied in order to increase exploration. Thus a new cidate solution is generated. [9]. is average loudness all the bats, ϵ is uniform rom number [22]. 3.3 Loudness Pulse Emission Rate Loudness pulse emission rate are updated as iterations proceed. When the bat gets closer to its prey, the loudness usually decreases pulse emission rate increases [9] [22]. The loudness pulse emission rate are updated by using the following equations: (5) (6) (7) (8) (9) () Where α γ are constants which having the determined values for these equations. Here are rom values can be normally [1,2], while can be [, 1] normally [23] Iteration Number Iteration Number Fig.1: Loudness (A) Pulse Emission Rate (r) in Bat Algorithm 3.4 Pseudo code Bat Algorithm 1. Objective function: 2. Initialize bat population x i velocity 3. Define pulse frequency 4. Initialize pulse rate loudness 5. while ( <maximum number iterations) 6. Generate new solutions by adjusting frequency, updating velocities location. 7. if (r > ) 8. Select a solution among the best solutions 9. Generate a local solution around the selected best solution. end if 11. if (r < 12. Accept new solutions 13. increase, reduce 14. end if 15. Rank the bats find current best 16. end while 17. Display results. 4. SIMULATION AND ANALYSIS 4.1 Benchmark functions To compare the performance the Bat algorithm () [22] [24] the Flower pollination algorithm (FPA) [7], we have used benchmark functions, including five unimodal five multimodal functions. The analytical form each function, along with their names, bounds search space global minimum values the functions are shown in Table 1. Functions are tested with the dimension D =,. Both unimodal multimodal functions are used in this experiment. For both the algorithms, the population size is fixed to. Algorithms are tested with independent runs for each test function. All tests were carried out on a 64-bit operating system with a 2.1GHz CPU 2GB RAM. Table 1. Benchmark functions used in the experimental studies. Here, D: Dimensionality the function, S: search space, C: function characteristics with values U: Unimodal M: Multimodal. No. Name D C S Function Definition f min f 1 Griewank,, M [-6, 6] D. f 2 Schwefel,, M [-, ] D. 15

4 f 3 Ackley,, M [-32, 32] D e. f 4 Michalewicz,, M [, π] D - f 5 Rastrigin,, M [-15, 15] D. f 6 Sphere,, U [-5.12, 5.12] D. f 7 Step,, U [-, ] D. f 8 Zakharov,, U [-5, ] D. f 9 Rosenbrock,, U [-15, 15] D. f Quartic,, U [-1.28, 1.28] D. Fig. 2: 3D surface plots 2D Quartic function (left) Griewank function (right). 4.2 Parameter Settings Both the algorithms are tested with independent runs on each the test functions listed in Table 1. The swarm size (i.e., no. cidate solutions) is set to for both algorithms. The number generations is set to, 1 2 for D =, respectively for each run. Minimum frequency value f min is set to while the maximum value f max is set to 1 in the Bat algorithm. Both α λ in Bat algorithm are set to.9 in the simulation. We have used Matlab version R213a for the simulation. 4.3 Experimental Results For any swarm intelligence algorithm, there should be a balance between the degrees explorations exploitations with which the swarm members search across the search space. Too much exploitation at the expense too little exploration may cause the algorithm to be trapped around the locally optimal points, which makes it difficult or even impossible to find the global optimum. On the other h, too much exploration at the cost too little exploitation may make it difficult for the algorithm to converge thus slows down the convergence speed compromise the overall search performance. In the Bat algorithm, the loudness A pulse emission rate r are the control parameters that try to balance between explorations exploitations to find global optimum. But the pulse emission rate, r decreases while loudness A increases exponentially as the iterations proceed (Fig. 1). The condition in line 7 in Bat algorithm is satisfied only with a very low probability. So the algorithm gradually loses its exploration capability. Moreover, the algorithm generates new solutions using eq. (8) where the mutation step size very small. So the algorithm is unable to explore the whole search space is likely to fail to escape from the intermediate locally optimal points. 16

5 Table 2. Comparison between FPA on stard benchmark functions. All algorithms are run different times on each the functions. The best result for each function with each dimensionality is marked with boldface font Fun Name Algorithm Dim Best Worst Mean Median SD f 1 f 2 f 3 f 4 f 5 f 6 f 7 Griewank Schwefel Ackley Michalewicz Rastrigin Sphere Step 5.32E+ 2.79E E E E+ FPA 1.92E E E E E E E+2 9.9E E E+1 FPA 2.64E+ 1.89E E+ 6.59E+ 3.37E+ 9.63E+1 2.2E E E+2 3.5E+1 FPA 1.38E E E E E+ 2.25E+3 3.6E E E E+2 FPA 8.12E E+3 1.9E+3 1.9E E E+3 1.5E E E E+2 FPA 4.37E E+3 5.3E+3 5.4E E E E E E E+2 FPA 8.73E+3 1.5E E+3 9.E E E+ 1.55E E E E+ FPA 1.1E E+ 9.86E-1 8.5E E E E E E E+ FPA 2.49E+ 7.96E+ 5.6E+ 5.8E+ 1.57E+ 1.36E+1 1.7E E E+1 7.2E-1 FPA 3.E+ 9.15E+ 5.95E+ 5.85E+ 1.17E E E E E+ 8.41E-1 FPA -7.67E+ -6.2E+ -7.E E+ 3.81E E+1-1.1E E E E+ FPA -1.59E E E E E E E E E E+ FPA -2.41E E+1-2.4E+1-2.3E E+ 6.68E E E+2 1.8E+2 3.8E+1 FPA 1.29E E E E+1 5.6E+ 3.E+2 6.8E E E E+1 FPA 1.4E E E E E E+2 1.7E E E+2 1.6E+2 FPA 1.95E E E+2 2.8E E E+ 1.49E+1 4.8E+ 4.73E+ 2.69E+ FPA 6.66E-6 2.2E E E E E E E E E+ FPA 6.19E E+ 2.12E+ 1.94E+ 1.14E+ 3.5E E E E+1 8.9E+ FPA 3.E+ 9.72E+ 5.78E+ 5.59E+ 1.82E+ 6.1E+2 4.E+3 2.E E E+2 FPA.E+ 1.E E+.E+ 2.53E+ 4.93E E+4 1.8E+4 1.6E E+3 FPA 1.89E+2 2.E E E E+2 1.2E E E E+4 4.9E+3 17

6 Fmin Fmin International Journal Applied Information Systems (IJAIS) ISSN : f 8 f 9 f Zakharov Rosenbrock Quartic FPA 8.7E E E E+3 1.1E E E E E E+3 FPA 1.2E E E-3 5.9E E-3 6.4E E E+7 1.E E+8 FPA 2.66E E E E E E E E+ 2.7E E+ FPA 1.5E E E E E E E E E E+4 FPA 3.76E+ 3.88E+1 1.6E E+ 7.47E+ 1.25E E E E E+5 FPA 1.3E E E E E E E E E E+5 FPA 3.37E E+4 3.E E E+4 1.2E+ 2.12E+ 1.82E+ 1.85E+ 2.25E-1 FPA 9.59E E+ 2.13E+ 2.23E+ 2.96E-1 9.7E+ 1.75E E E E+ FPA 9.51E+ 1.14E+1 1.4E+1 1.3E E-1 2.8E+1 4.6E E E+1 5.2E+ FPA 1.8E E E E E FPA 4 4 FPA Generations Generations Fig. 3: Convergence characteristics FPA on the Step function f 7 Rastrigin function f 5 On the other h, the Flower pollination algorithm controls the degrees explorations exploitations with a switch probability p. In FPA the global local pollination technique is used to balance between explorations exploitations. In global pollination, the Lévy distribution is applied to generate new solutions; while in local pollination new solutions are generated using romly selected local solutions. The Lévy distribution has the capability to generate new solutions with bigger mutation step size. Thus the algorithm is more likely to escape from the locally optimal points. From the experimental results, we can see that the Flower pollination algorithm show much improved results than the Bat algorithm on the continuous optimization problems. For simulation, ten benchmark functions have been used in this experiment. Among them five functions are unimodal five are multimodal. A unimodal function is one that has a single local minimum while a multimodal function is a function with many local minima. In Table 2, we can see that the solution quality (mean value) is much better for FPA than on both multimodal unimodal functions. Also the value the stard deviation indicates that the result FPA is more consistent. Bat algorithm gets trapped (Fig. 3) in some poor local minima, far from the global minimum. Thus the convergence characteristic FPA is also much better than. 5. CONCLUSION This paper presents a comparative study between two swarm intelligence based algorithms the Flower Pollination algorithm [7] [25] the Bat algorithm [9]. Both these algorithms use swarm intelligence to find the global optimum value the continuous optimization problems. Numerical results on the stard benchmark problems for Bat algorithm Flower pollination algorithms demonstrate the effectiveness competitiveness the algorithms. The flower pollination algorithm outperforms the Bat algorithm in terms the final solution quality in most the functions. The main reason the 18

7 performance difference is the relatively smaller amount search space explorations by the Bat algorithm, which may lead it towards premature convergence around the locally optimal points, failing to locate the globally optimum point. There has been a few research works (e.g., [8] [11] [26]) that try to improve the performance the original Bat algorithm. In future we plan to compare these algorithms with the Flower Pollination algorithm as well as some more evolutionary swarm intelligence based algorithms, such as the Artificial Bee Colony algorithm [27], Differential Evolution [28] [29]. 6. REFERENCES [1] Y. Xin-She, Optimization: An Introduction with Metaheuristic Application, Wiley, 2. [2] H. A. Abbass R. Sarker, "The Pareto diffential evolution algorithm," Int. J. Artificial Intelligence Tools, vol. 11(4), pp , 22. [3] Y. Xin-She, "Nature-inspired Metaheuristic Algorithms," Luniver Press, 28. [4] K. Deb, Multi-Objective optimization using evolutionary algorithms, New York: John Wiley & Sons, 21. [5] Y. Xin-She, K. Mehmet H. Xingshi, "Multi-objective Flower Algorithm for Optimization," in International Conference on Computational Science, ICCS 213, 213. [6] M. Walker, "How flowers conquered the world," BBC Earth News, July 29. [7] Y. Xin-She, "Flower pollination algorithm for global optimization," Unconventional Computation Natural Computation, Lecture Notes in, vol. 7445, p , 212. [8] G. Wang L. Guo, "A Novel Hybrid Bat Algorithm with Harmony Search for Global Numerical Optimization," Journal Applied Mathematics, vol. 213, p. 21, 213. [9] Y. Xin-She, "A New Metaheuristic Bat-Inspired Algorithm, Nature Inspired Cooperative Strategies for Optimization (NISCO 2)," Springer, vol. 284, no. Springer Berlin,, pp , 2. [] K. Khan A. Sahai, "A Comparison, GA, PSO, BP LM for Training Feed forward Neural Networks in e-learning Context," I.J. Intelligent Systems Applications, pp , June 212. [11] N. S. S. M. M. S. A. Md. Wasi Ul Kabir, "A Novel Adaptive Bat Algorithm to Control Explorations nd Exploitations for Continuous Optimization Problems," International Journal Computer Applications, vol. 94, no. 13, 214. [12] A Rekaby, "Directed Artificial Bat Algorithm (DA) A New Bio-Inspired Algorithm," in International Conference on Advances in Computing, Communications Informatics (ICACCI), Cairo, 213. [13] Y. Selim U. K. Ecir, "Improved Bat Algorithm (I) on Continuous Optimization Problems," Lecture Notes on Stware, vol. 1, no. 3, pp , 213. [14] G. Q. Huang, W. J. Zhao Q. Q. Lu, "Bat algorithm with global convergence for solving large scale optimization problem," Application Research Computers, vol., no. 3, pp. 1-, 213. [15] K. Gaganpreet D. S. Dr., "Pollination Based Optimization or Color Image Segmentation," International Journal Computer (IJCET), vol. 3, no. 2, pp , July-September 212. [16] K. S, "Pollination based optimization," in 6th International Multi Conference on Intelligent Systems, Sustainable, New Renewable Energy Nanotechnology (IISN212), March 16-18,212. [17] N. Waser, "Flower constancy: definition, cause measurement," The American Naturalist, [18] A.-R. Osama, A.-B. Mohamed E.-h. Ibrahim, "A Novel Hybrid Flower Pollination Algorithm with Chaotic Harmony Search for Solving Sudoku Puzzles," International Journal Trends (IJETT), vol. 7, no. 3, pp , January 214. [19] J. Altringham, Bats: Biology Behaviour, Oxford University Press, [2] T. Colin, The Varienty Life, Oxford University Press, 2. [21] A. Faritha C. Chrasekar, "An optimized approach modified bat algorithm to record deduplication," International Journal Computer Applications, vol. 62, no. 1, pp. -15, 212. [22] Y. Xin-She, "Bat Algorithm for Multiobjective Optimization," International Journal Bio-Inspired Computation, vol. 3, no. 5, pp , 211. [23] X. S. Yang, "Harmony Search as a Metaheuristic Algorithm,Music-Inspired Harmony Search Algorithm," Theory Applications,Studies in Computational Intelligence, vol. 191, pp. 1-14, 29. [24] X. S. Yang J. R. Gonzalez, ""A New Metaheuristic Bat-Inspired Algorithm" in Nature Inspired Cooperative Strategies for Optimization (NISCO 2)," Springer Press, vol. 284, pp , 2. [25] S. Yang X, "Nature-inspired Metaheuristic Algorithms," Luniver Press, 28. [26] Y. Selim K. E. U., "Improved Bat Algorithm (I) on Continuous Optimization Problems," Lecture Notes on Stware, vol. 1, no. 3, pp , August 213. [27] K. Dervis O. Celal, "A novel clustering approach: Artificial Bee Colony (ABC) algorithm," Applied St Computing, Elsevier, vol. 11, pp , 211. [28] S. Rainer P. Kenneth, "Differential Evolution A Simple Efficient Heuristic for Global Optimization over Continuous Spaces," Journal Global Optimization, Kluwer Academic Publishers, pp , [29] R. Poli, J. Kennedy T. Blackwell, "Particle Swarm Optimization," Springer Science, vol. 1, pp , 1 August