Comparative Study of Krill Herd, Firefly and Cuckoo Search Algorithm

Cuckoo search via Levy flights, in: Proc. Of World Congress on Nature & Biologically Inspired Computing Optimizing the encounter rate in biological interactions
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I.J. Intelligent Systems and Applications, 2014, 03, 35-49 Published Online February 2014 in MECS (http://www.mecs-press.org/) DOI: 10.5815/ijisa.2014.03.04 Comparative Study of Krill Herd, Firefly and Cuckoo Search Algorithms for Unimodal and Multimodal Optimization Gobind Preet Singh, Abhay Singh Dept. of Computer Science, GTBIT, Guru Gobind Singh Indraprastha University, New Delhi, India E-mail: gobind75gmail.com;abhaysaysthisyahoo.co.in Abstract— Today, in computer science, a are intensification and diversification. The computational challenge exists in finding a globally intensification searches around the current best solution optimized solution from an enormously large search and selects the best candidate or solution. The space. Various meta-heuristic methods can be used for diversification ensures that the algorithm explores the finding the solution in a large search space. These search space more efficiently. Maintaining balance methods can be explained as iterative search processes between diversification and intensification is important that efficiently perform the exploration and exploitation because firstly we have to quickly find the regions in in the solution space. In this context, three such nature the large search space with high quality solutions and inspired meta-heuristic algorithms namely Krill Herd secondly not to waste too much time in regions of the Algorithm (KH), Firefly Algorithm (FA) and Cuckoo search space which are either already explored or which search Algorithm (CS) can be used to find optimal do not provide high quality solutions 3. solutions of various mathematical optimization In this paper, we have used three meta-heuristic problems. In this paper, the proposed algorithms were algorithms Krill Herd Algorithm (KH) 4, Firefly used to find the optimal solution of fifteen unimodal Algorithm (FA) 1 and Cuckoo Search Algorithm (CS) and multimodal benchmark test functions commonly 5.First is the Krill Herd Algorithm which was used in the field of optimization and then compare their developed by Amir Hossein Gandomi and Amir performances on the basis of efficiency, convergence, Hossein Alavi in 2011. The KH algorithm is based on time and conclude that for both unimodal and the simulation of the herding behaviour of Krill multimodal optimization Cuckoo Search Algorithm via individuals. Second is the Firefly Algorithm (FA) which Lé vy flight has outperformed others and for multimodal was developed by X.-S.Yang in 2007. It was inspired optimization Krill Herd algorithm is superior than by the flashing pattern of fireflies. Third algorithm is Firefly algorithm but for unimodal optimization Firefly the Cuckoo search Algorithm which was developed by is superior than Krill Herd algorithm. X.-S.Yang and S.Deb in 2009. It is based on the interesting breeding behaviour of certain species of cuckoos such as brood parasitism. Index Terms— Meta-heuristic Algorithm, Krill Herd Algorithm, Firefly Algorithm, Cuckoo Search This paper aims to provide the comparison study of Algorithm, Unimodal Optimization, Multimodal the Krill Herd Algorithm (KH) with Firefly Algorithm Optimization (FA) and Cuckoo Search (CS) Algorithm via L´ evy Flights against unimodal and multimodal test functions. Rest of the paper is organised as follows. First we will briefly explain the Krill Herd Algorithm, Firefly I. Introduction Algorithm, Cuckoo Search Algorithms and several Mathematical benchmark functions in section (II).Then In recent times, nature inspired meta-heuristic experimental settings and results will be shown in algorithms are being widely used for solving section (III) and then finally we will conclude the paper. optimization problems, including NP-hard problems which might need exponential computation time to solve in worst case scenario. In meta-heuristics methods 1, 9 we might compromise on finding an optimal II. Nature Inspired Algorithms and Optimization solutions just for the sake of getting good solutions in a specific period of time. The main aim of meta-heuristic 2.1 Krill Herd Algorithm algorithms are to quickly find solution to a problem, 2.1.1 Krill Swarm’s Herding Behavior this solution may not be the best of all possible solutions to the problem but still they stand valid as Many Research have been done in order to find the they do not require excessively long time to be solved. mechanism that lead to the development non- random Two crucial characteristics of meta-heuristic algorithms formation of groups by various marine animals Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 36 Comparative Study of Krill Herd, Firefly and Cuckoo Search Algorithms for Unimodal and Multimodal Optimization 11,12.The significant mechanisms identified are (2) feeding ability, protection from predators, enhanced reproduction and environmental condition 6. In (2) stands for maximum induced speed which is equal to 0.01 (m/s) 6 , is the direction of Krills from Antarctic region are one of the best motion induced which is estimated from target swarm researched marine animals. One of the most significant ability of krills is that they can form large swarms 13, density and local swarm density, is the inertia 14.Yet there are number of uncertainties about the weight of the motion induced, is the last motion mechanism that lead to distribution of krill herd induced. 15.There are proposed conceptual models to explain observed formation of krill heard 16 and result (3) obtained from those models states that krill swarms form the basic unit of organization for this species. In (3) is the local effect due to the neighbors, and is the target direction effect due to the best Whenever predators (Penguins, Sea Birds) attack krill krill individual. swarms, they take individual krill which leads in reducing the krill density. After the attack by predators, Attractive or Repulsive effect of the neighbors on an formation of krill is a multi- objective process mainly individual krill movement can be formulated as: including two Goals: (1) Increasing Krill density and (2) Reaching food. Attraction of Krill to increase density (4) and finding food are used as objective function which finally lead the krills to herd around global minima. In (5) this mechanism, all individual krill moves towards the best possible solution while searching for highest (6) density and food. In (4) NN is the number of the neighbors. In (5) and 2.1.2 Krill Herd Algorithm (6) and are, the worst and the best fitness values of the krill individuals till now, represents the As Predator remove individual from Krill swarm, the fitness value of the ith krill individual, is the fitness average krill density and distance of krill swarm from of jth neighbor and X represents the related positions. the food location decreases. We assume this process as the initialization phase in the Krill Herd Algorithm 4. To choose the neighbor, using actual behavior of Value of objective function for each individual is Krill individual, a sensing distance (ds) is calculated supposed to be combination of distance from food and using highest density of krill swarm. Three essential actions (7) 6 considered by Krills to determine the time dependent position of an individual krill are: In (7) is the sensing distance for the ith krill (i) Movement induced by other krill individuals individual and N stands for number of krill individuals. (ii) Foraging activity Based on (7), two krill individuals are neighbor if the distance between them is less than . (iii) Random Diffusion The effect of the individual krill having the best We know that all the optimization algorithm should fitness on the ith individual krill is calculated as follow: have searching capability in space of arbitrary dimensionality. Hence we generalize Lanrangian model (8) of krill herding to n dimensional decision space. In (8) is the effective coefficient of the krill (1) with the best fitness and is defined as: Here is the motion induced by other krill (9) individuals, is the foraging motion and stands for physical diffusion for ith krill individuals. Where rand is a random value in the range 0, 1, I is the actual iteration number, and is the maximum number of iterations. 2.1.3 Motion Induced by other krill individual According to research krill individuals move due to the mutual effects by each other so as to maintain high 2.1.4 Foraging motion density 6.Movement for the krill individual is defined Foraging motion is developed in terms of two main as: effective parameters. One is the food location and the Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 Comparative Study of Krill Herd, Firefly and Cuckoo 37 Search Algorithms for Unimodal and Multimodal Optimization second one is the previous experience about the food (16) location. This motion can be explained for the krill individual as follow: In (16) is the maximum diffusion speed 8 and is the random directional vector and it includes (10) random values in range -1, 1. As krills position gets better, random motion is also reduced. Thus, another Where term is added to the physical diffusion formula to consider this effect. The effects of the motion induced (11) by other krill individuals and foraging motion gradually decrease increase in iterations. The physical diffusion is In (10) is the foraging speed, is the inertia a random vector hence it does not steadily reduces with weight of the foraging motion in the range 0, 1 and increase in number of iterations due to which another is the last foraging motion. In (11) and term is added to (16). This term, linearly decreases the are the food attractive effect and best fitness of the random speed with time and works on the basis of a krill so far respectively. Measured values of the geometrical annealing schedule: foraging speed 7 is 0.02( ). (17) The center of food is found at first and then food attraction is formulated. The virtual center of food concentration is estimated according to the fitness 2.1.6 Motion Process of the KH Algorithm distribution of the krill individuals, which is inspired from ‗‗center of mass‘‘. The center of food for each The defined motions frequently change the position iteration is formulated as: of a krill individual toward the best fitness. The foraging motion and the motion induced by other krill (12) individuals contain two global and two local strategies. KH a powerful algorithm as all these work in parallel. The formulations of these motions for the krill Hence, we can evaluate the food attraction for the krill individual by following equation: individual show that, if fitness value of each of the above mentioned effective factor like , , (13) , is better i.e. less than the fitness of the krill, it has an attractive effect else it is a repulsive effect. We can notice from the above formulations that In (13) is the food coefficient. As time passes the effect of food in the krill herding decrease and food better fitness has more effect on the movement of coefficient is evaluated as: krill individual. The position vector of a krill individual during the interval t to t + t is given by the following (14) equation: The attraction towards food is defined to attract the (18) krill swarm towards global optima. Based on this definition, the krill individuals normally herd around t should be carefully set according to the the global optima after some iteration. This can be optimization problem because this parameter works as a considered as an efficient global optimization strategy scale factor of the speed vector. t completely depends which helps improving the global optima of the KH on the search space and it can be obtained simply by the algorithm. following formula: The effect of the best fitness of the krill (19) individual is also handled using the following equation: In (19) NV is the total number of variables and (15) and are lower and upper bounds of the variables respectively. It is empirically found that is a In (15) is the best previously visited position of constant number between 0, 2. Lower the values of the krill individual. more carefully the krill individuals will search. 2.1.5 Physical diffusion 2.1.7 Crossover The physical diffusion of all the krills is basically a To improve the performance of the algorithm, genetic random process. We can express this motion in terms of reproduction mechanisms are incorporated into the maximum diffusion speed and a random directional algorithm. One such algorithm is crossover. Crossover vector. We can formulate it as follows: Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 38 Comparative Study of Krill Herd, Firefly and Cuckoo Search Algorithms for Unimodal and Multimodal Optimization is a genetic operator used to vary the programming of 2.2.2 Firefly Algorithm chromosomes from one generation to the next. In this The firefly (FA) algorithm 1, 9, 10 is a meta- Algorithm, an adaptive vectored crossover scheme is heuristic algorithm, inspired by the flashing behavior of employed. fireflies. The primary purpose for a firefly's flash is to We can control crossover by a crossover act as a signal system to attract other fireflies. probability , and actual crossover can be performed Xin-She Yang formulated this firefly algorithm by in two ways: binomial and exponential. The binomial taking three assumptions 1 scheme performs crossover on each of the d i. All fireflies are unisexual, so that one firefly will components or variables/parameters. By generating a be attracted to all other fireflies; uniformly distributed random number between 0 and 1, the component of , , is determined as: ii. Attractiveness is proportional to their brightness, and for any two fireflies, the less bright one will be attracted by (and thus move to) the brighter one; (20) however, the brightness can decrease as their distance increases; iii. If there are no fireflies brighter than a given firefly, (21) it will move randomly. In (20) r 1, 2,. . ., i _ 1, i + 1,. . .,N. With this new crossover probability, the crossover probability for 2.2.3 Light Intensity and Attractiveness the global best is equal to zero and it increases with Two core issues in firefly algorithm are (i) The decrease in fitness. variation of light intensity, (ii) The formulation of the attractiveness. 2.2 Firefly Algorithm For simplicity, it is assumed that the attractiveness of a firefly is determined by its brightness which in turn is 2.2.1 Behavior and nature of Fireflies associated with the encoded objective function of the Fireflies are the creatures that can generate light optimization problems. In the simplest case for inside of it. Light production in fireflies is due to a type maximum optimization problems, the brightness I of a of chemical reaction. The primary purpose for firefly‘s firefly for a particular location x could be chosen as I(x) flash is to act as a signal system to attract other ∝ f(x). Even so, the attractiveness β is relative, it fireflies. Although they have many mechanisms, the should be judged by the other fireflies. Thus, it will interesting issues are what they do for any differ with the distance between firefly i and firefly j. communication to find food and to protect themselves In addition, light intensity decreases with the distance from enemy hunters including their successful from its source, and light is also absorbed by the media, reproduction. There are around two thousand firefly so we should allow the attractiveness to vary with the species, and most of them produce short and rhythmic varying degree of absorption. flashes. The pattern observed for these flashes is unique specific species. The rhythm of the flashes, rate of Since a firefly‘s attractiveness is proportional to the flashing and the amount of time for which the flashes light intensity seen by adjacent fireflies, attractiveness β are observed together forms a pattern that attracts both of a firefly can be defined as the males and females to each other. Females of a species respond to individual pattern of the male of the , (m≥1), (22) same species. In (22), r or is the distance between the ith and jth The light intensity at a particular distance from the light source follows the inverse square law. That is as fireflies. Is the attractiveness at r = 0 and γ is a fixed the distance increases the light intensity decreases. light absorption coefficient. The distance between any Furthermore, the air absorbs light which becomes two fireflies ith and jth at and is the Cartesian weaker and weaker as there is an increase of the distance and can be calculated as: distance. There are two combined factors that make most fireflies visible only to a limited distance that is (23) usually good enough for fireflies to communicate each other. The flashing light can be formulated in such a way that it is associated with the objective function to In (23), is the kth component of the ith firefly ( ). be optimized. This makes it possible to formulate new The movement of ith firefly, to another more attractive meta-heuristic algorithms. (brighter) jth firefly, is determined by Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 Comparative Study of Krill Herd, Firefly and Cuckoo 39 Search Algorithms for Unimodal and Multimodal Optimization optimization and optimal search, and preliminary results show its promising capability 18, 20, 21, 22. (24) In (24) the second term is due to the attraction while 2.3.3 Cuckoo Search Algorithm the third term is the randomization with α being the Each egg in the nest represents solution, and Cuckoo randomization parameter. Rand is a random number egg represents new solution. The aim is to use the new generator uniformly distributed in the range of 0, 1. and potentially better solutions (Cuckoos) to replace For most cases in the implementation, = 1 and α ∈ not-so-good or inferior solution in the nests. In the 0, 1. Furthermore, the randomization term can easily simplest form, each nest has one egg. The algorithm 5 be extended to a normal distribution N (0, 1) or other can be extended to more complicated cases in which distributions. each nest has multiple eggs representing a set of solutions. 2.3 Cuckoo Search Algorithm via Lé vy Flight Cuckoo search is based on three idealized rules which states that 2.3.1 Cuckoo’s breeding behavior i. Each Cuckoo lays one egg, which represents a set Cuckoo is an interesting bird species, known not only of solution coordinates, at a time, and dumps it in a for the beautiful sound they make, but also for their random nest. aggressive reproductive strategy. Cuckoos are extremely diverse group of birds with regards to ii. A fraction of the nests containing the best eggs, or breeding system. Many Cuckoo species follow the solutions, will be carried over to the next strategy of brood parasitism by using host individuals generation. either of the same or different species to raise the young iii. The number of nests is fixed and there is a of the own. Cuckoo species such as Anis and Guira lay probability that a host can discover an alien egg. If their eggs in communal nest though they may remove this happens, the host can either discard the egg or others eggs to increase the survival probability of their the nest and this result in building a new nest in a own eggs. Some host birds can engage in direct conflict new location. with the intruding cuckoos. On recognition of parasitic eggs, the host may kick the parasites eggs out, or build a When generating new solutions for the ith new nest. Female parasitic cuckoos who are specialized Cuckoo, Lé vy Flight is performed. in mimicry lay eggs that closely which resemble the eggs of their host which reduces the probability of their (25) eggs being abandoned. In (25), α 0 is the step size which should be related Parasitic cuckoos often choose a nest where host bird to the scales of the problem of interest. In most cases, have just laid their eggs .The cuckoo egg hatches earlier we can use α = O (L/10) where L is the characteristic as compared to the host's, and the cuckoo chick grows scale of the problem of interest. The above equation is faster than them;. In most cases the chick evicts the essentially the stochastic equation for a random walk. eggs laid by host species, which increases the cuckoo chick‘s share of food provided by its host bird. Some The product ⊕ means entry wise multiplications. cuckoo chick can even replicate the call of host chicks The Lé vy flight essentially provides a random walk to gain access to more feeding opportunity. whose random step length is drawn from a Lé vy distribution 2.3.2 Lé vy flight , (26) A Lé vy flight is a random walk in which the step- lengths have a probability distribution that is heavy- This has an infinite variance with an infinite mean. tailed. Research works have shown that flight behavior Here the steps essentially form a random walk process of many animals and insects demonstrated the typical with a power-law step-length distribution with a heavy characteristics of Lé vy flights 17, 18, 19, 20. Fruit tail. Some of the new solutions should be generated by flies or Drosophila melanogaster, explore their walk around the best solution obtained so far, this landscape using a series of straight flight paths will speed up the local search. To ensure that that the punctuated by a sudden 90 degree turn, leading to a system will not be trapped in a local optimum, a Lé vy -flight-style intermittent scale free search pattern substantial fraction of the new solutions should be was shown in a study conducted by Reynolds and Frye. generated by far field randomization whose locations Many researches shows that Lé vy flights interspersed should be far enough from the current best solution. with Brownian motion can describe the animals' hunting patterns 24, 25. Even light can be related to Lé vy flights 23. Latterly, such behavior has been applied to Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 40 Comparative Study of Krill Herd, Firefly and Cuckoo Search Algorithms for Unimodal and Multimodal Optimization 2.4 Testing Optimization Functions In Literature 26 there are many benchmark test functions which are designed to test the performance of optimization algorithms. In this paper we will compare and validate above mentioned algorithms against these benchmark functions. Seventeen functions 27,28,29 including many multimodal functions are used in this paper in order to compare and verify efficiency and convergence of all three above mentioned Nature Inspired Algorithms. Certain test functions used in our simulations are as follows: Ackley Function is multimodal function widely used for testing optimizat1ion algorithms. Fig. 2: The visualization of the Beale function With a global minimum at in the range of ∈ -32.768, 32.768, for all = 1, 2… d. Fig. 3: The visualization of the Branin function Colville is a unimodal test function. Fig.1: The visualization of the Ackley function Which has minimum in range ∈ -10, 10, for all i = 1, 2, 3, 4. Beale function is 2- dimensional multimodal function, with sharp peaks at the corners of the input domain. Dixon-Price’s unimodal test function Which has minimum in range ∈ -4.5, 4.5, for all i = 1, 2. With a global minimum at Branin function is multimodal with three global bin the range of ∈ -10, 10, for all i = 1, 2… d. minima. The recommended values of a, b, c, r, s and t are: a = 1, b = 5.1 ⁄ (4π2), c = 5 ⁄ π, r = 6, s = 10 and t = 1 ⁄ (8π). Easom function has several local minima. It is unimodal, and the global minimum has a small area relative to the search space. With a global minimum at in the range ∈ -5, 10 and x2 ∈ 0, 15. Which has minimum in range ∈ -100,100, for all i = 1, 2, Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 Comparative Study of Krill Herd, Firefly and Cuckoo 41 Search Algorithms for Unimodal and Multimodal Optimization Fig. 4: The visualization of the Dixon-Price‘s function Fig. 5: The visualization of the Easom function Shubert function a multimodal test function has Which has minimum in range several local minima and many global minima. Its ∈ -10, 10, for all i = 1, 2, equation is Fig. 6: The visualization of the Shubert function Levy Test function is multimodal function. Its equation is With a global minimum at which is evaluated in the range of ∈ -10, 10, for all = 1, 2… d. Fig. 7: The visualization of the Levy function Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 42 Comparative Study of Krill Herd, Firefly and Cuckoo Search Algorithms for Unimodal and Multimodal Optimization Rastrigin function has several local minima. It is highly multimodal, but locations of the minima are regularly distributed. With a global minimum at which is evaluated in the range of ∈ -5, 10, for all = 1, 2… d. With a global minimum at evaluated in the range of ∈ -5.12, Zakharov function has no local minima except the 5.12, for all = 1, 2… d. global one. It‘s a unimodal function and its equation is With a global minimum at which is evaluated in the range of ∈ -5, 10, for all = 1, 2… d. Fig. 8: The visualization of the Rastrigin function Fig. 10: The visualization of the Zakharov function Griewank function has many widespread local minima, which are regularly distributed. It is a multimodal test function. Fig. 9: The visualization of the Rosenbrock function With a global minimum at Rosenbrock function is unimodal, and the global which is evaluated in the range of ∈ -600, 600, for minimum lies in a narrow, parabolic valley. all i = 1, 2… d. Fig. 11: The visualization of the Griewank function Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 Comparative Study of Krill Herd, Firefly and Cuckoo 43 Search Algorithms for Unimodal and Multimodal Optimization Trid function has no local minimum except the III. Implementation and Numerical Experiments global one. It is a unimodal function. In this section we will compare the performance of Krill Herd algorithm Firefly algorithm, Cuckoo search algorithm for various benchmark test functions. The benchmarks function include both unimodal and multimodal with both low and high dimensional With a global minimum for d=6 and problems are described in Section 2.4 and for evaluation at d=10 which is evaluated in the range all computational procedures described above has been of ∈ - , , for all = 1,2, …, d. implemented in MATLAB™ computer program. In order to compare these algorithms we have carried out extensive simulations and each algorithm has been run 50 times so as to carry out meaningful analysis. The maximum number of function evaluations is set as 10,000 for high dimensional functions and 1000 for low dimensional functions. Here for Krill Herd Algorithm is set to 0.5 and the inertia weights ( ) are equal to 0.9 at beginning of search and it linearly decreases to 0.1 at the end. For Firefly algorithm certain constants are fixed as , and for simulation. For cuckoo search algorithm probability for host bird is fixed as . For simulation we have used various population sizes Fig. 12: The visualization of the Trid function from n = 10 to 150, and found that for most problems, it is sufficient to use n = 15 to 50. Therefore, we have used a fixed population size of n = 50 in all our Powell Function is a unimodal function simulations for comparison. Now we will divide this section in two parts comprising comparison of algorithms for unimodal test function in first section and comparison of algorithms This function is usually evaluated on the region x ∈ i for multimodal test functions in another. For both the -4, 5, for all i = 1, 2… d. having minima at section we will be comparing KH algorithm, FA Algorithm and CS Algorithm via Lé vy Flights on the basis of three criteria i.e. Optimization fitness (efficiency), Convergence and processing time. Eggholder function is a difficult function to optimize, because of the large number of local minima. It is multimodal test function. 3.1 Optimization for Unimodal Test Function In this section we have compared KH algorithm, FA Algorithm and CS Algorithm via Lé vy Flights on Eight Unimodal benchmark functions popular for Which has minimum optimization. Unimodal functions are those function in range which have only single local minima and these function easier to optimize. ∈ -512, 512, for all i = 1, 2. 3.1.1 Optimization fitness Here we have calculated the mean fitness value using the above mentioned algorithms for all unimodal test functions mentioned above. Optimized fitness result where global optima is reached are summarized in Table 1. Fig. 13: The visualization of the Eggholder function Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 44 Comparative Study of Krill Herd, Firefly and Cuckoo Search Algorithms for Unimodal and Multimodal Optimization Table 1: Comparison of Optimization Fitness for Unimodal Test Functions Function/Algorithms KH Algorithm FFA Algorithm CS Algorithm Dixon-Price(d=20) 0.6975 0.6668 0.6667 31.89 14.49 2.63e-14 Rosenbrock (d=20) Zakharov(d=20) 5.973 3.19e-06 2.77e-28 Powell(d=20) 0.0299 0.002269 5.27e-09 Trid(d=20) 9.34e+2 1.52e+03 1.52e+03 Beale 2.92e-11 5.49e-10 1.34e-30 -0.96 -0.86 -1 Easom Colville -1.40e+09 -1.56e+07 -7.57e+17 Here we can see that Cuckoo Search algorithm has Herd algorithm for high dimensional functions but for outperformed both Krill Herd and Firefly algorithm. For low dimensional function result using Krill Herd all Unimodal test function, fitness value of CS algorithm are better than Firefly algorithm. Also in Krill algorithm is much closer to global optima as compared Herd algorithm we have varied dimensions from d= 5 to to other two algorithms. But if we just compare the 20 and observed that as we increases the dimensions, other two algorithms i.e. Firefly and Krill herd fitness value for function decreases. algorithm, their result are very close to each other, but on average, results obtained using Firefly algorithm are slightly better than results obtained using Krill Herd 3.1.2 Processing time algorithm. As per results from Table 1 it is visible that Here we will compare above mentioned algorithms performance of Krill Herd algorithm is better for low on basis on processing time. Processing time is dimensional functions and as we move from low basically time consumed by algorithm to process single dimensional function to high dimensional functions simulation. It includes time consumed by fixed number fitness value for krill herd decreases i.e. distance from of iteration to solve the problem. global minima increases. According to results in Table 1 performance of Firefly algorithm is better than Krill Table 2: Comparison of processing time in seconds for Unimodal Test Functions Function/Algorithms KH Algorithm FFA Algorithm CS Algorithm Dixon-Price(d=20) 141.73 125.56 19.69 Rosenbrock (d=20) 137.18 123.44 21.58 137.66 125.66 22.92 Zakharov (d=20) Powell(d=20) 172.44 138.95 63.35 Trid(d=20) 155.12 146.04 30.72 13.034 11.18 1.64 Beale Easom 12.69 12.13 2.46 Colville 13.33 12.92 2.59 From Table 2 it is quite easily visible that time taken Firefly, Cuckoo Search are compared for fixed number or consumed by Cuckoo search algorithm is much less of iteration i.e. 10,000 iterations for high dimensional than the other two algorithms We can also compute function and 1000 iteration for low dimensional from the Table 2 that time consumed by Firefly function. Here we will estimate which algorithm gives algorithm is less than Krill Herd Algorithm although potentially better and quicker convergence towards difference is not much, In term of processing time optimality. Below In Figure 14-21 Convergence Graph Cuckoo search algorithm again outperform other two is plotted for all above mentioned Unimodal benchmark algorithms. functions. 3.1.3 Convergence In this section convergence plots of the benchmark functions for three different algorithm i.e. Krill Herd, Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 Comparative Study of Krill Herd, Firefly and Cuckoo 45 Search Algorithms for Unimodal and Multimodal Optimization Fig. 18: Convergence Plot for Easom Function Fig. 14: Convergence Plot for Beale Function Fig. 19: Convergence Plot for Colville Function Fig. 15: Convergence Plot for Rosenbrock Function Fig. 16: Convergence Plot for Zakharov Function Fig. 20: Convergence Plot for Powel Function Fig. 21: Convergence Plot for Trid Function Fig. 17: Convergence Plot for Dixon-Price Function Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 46 Comparative Study of Krill Herd, Firefly and Cuckoo Search Algorithms for Unimodal and Multimodal Optimization In Fig.18, Fig.19 and Fig.21 we have plotted the algorithm converges faster than cuckoo search absolute value of the fitness function. For Trid, Easom, algorithm. Colville function, value of global minima is negative and so to plot them on convergence graph we took their absolute vale. 3.2 Optimization for Multimodal Test Function From Fig 14-21 we can interpret that the algorithm In this section we have compared KH algorithm, FA that quickly converges to its optimal solution is Krill Algorithm and CS Algorithm via Lé vy Flights on Seven Herd Algorithm. When we compare them on the Multimodal benchmark functions popular for number of iteration, Krill Herd Algorithm takes least optimization. Multimodal functions are those function number of iteration to converge whereas solution of which have many number local minima and these other two algorithm are better in terms of fitness value. function comparatively more difficult to optimize. It can also be seen from Fig. 14-21 that for most of the test functions, the other two algorithms i.e. Firefly Algorithm and Cuckoo search algorithm do not 3.2.1 Optimization fitness converge till 10,000 iterations and for functions for Here we have calculated the mean fitness value using which these two algorithm converges before 10,000 the above mentioned algorithms for all multimodal test iterations, it is the cuckoo search algorithm which functions mentioned in Section 2.4. Optimized fitness converges quickly than firefly for more of function as result where global optima is reached are summarized compare to number of function for which firefly in Table 3. Table 3: Comparison of Optimization Fitness for Multimodal Test Functions Function/Algorithms KH Algorithm FFA Algorithm CS Algorithm Ackley(d=20) 1.19e-05 7.36e-03 4.44e-15 Levy(d=20) 0.066 2.28e-06 1.14e-06 Rastrigin(d=20) 8.69 20.36 2.84 Griewank(d=20) 1.37e-09 5.89e-04 0 Branin 0.3979 0.3981 0.3981 Shubert -186.7309 -186.7309 -186.7309 Eggholder -893.0205 -930.2513 -959.6407 From Table 3 we can see that Cuckoo Search the algorithms are almost equivalent, for most of the algorithm has outperformed both Krill Herd and Firefly time both are able to find the global optima but for high algorithm in multimodal optimization as well. For all dimensional multimodal functions fitness value Multimodal test function, fitness value of CS algorithm obtained using Krill Herd is better than fitness value is much closer to global optima as compared to other obtained using Firefly Algorithm. Results in Table 3 are two algorithms. But if we together compare the other in contradiction with results in Table 1 as in unimodal two algorithms i.e. Firefly and Krill herd algorithm for test function optimization fitness for high dimensional multimodal test functions, result for them are in function using firefly was better than krill herd but in contradiction with results in previous section for multimodal function optimization fitness using krill unimodal test functions. In multimodal optimization for herd algorithm is better than firefly algorithm for high most of the low dimensional function ,result for both dimensional function. Table 4: Comparison of processing time in seconds for Multimodal Test Functions Function/Algorithms KH Algorithm FFA Algorithm CS Algorithm 138.06 123.49 24.78 Ackley(d=20) Levy(d=20) 169.37 129.03 38.86 Rastrigin(d=20) 139.55 126.31 24.58 Griewank(d=20) 137.18 123.44 21.58 Branin 13.36 11.28 1.91 Shubert 13.83 11.33 2.56 Eggholder 13.04 11.23 2.83 Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 Comparative Study of Krill Herd, Firefly and Cuckoo 47 Search Algorithms for Unimodal and Multimodal Optimization 3.2.2 Processing time In this section we will compare Krill Herd, Firefly, Cuckoo Search algorithms on basis on processing time for multimodal optimization functions. Processing time is basically time consumed by algorithm to process single simulation. It includes time consumed by fixed number of iteration to solve the problem. Results in Table 4 are quite similar with the results in Table 2. In multimodal optimization function as well, time taken or consumed by Cuckoo search algorithm is much less than the other two algorithms It can also be interpreted from the Table 4 that time consumed by Fig. 24: Convergence Plot for Levy function Firefly algorithm is less than Krill Herd Algorithm although difference is not much, In term of processing time Cuckoo search algorithm again outperform other two algorithms. 3.2.3 Convergence Here convergence plots of the benchmark functions for three different algorithm i.e. Krill Herd, Firefly, Cuckoo Search are compared for fixed number of iteration i.e. 10,000 iterations for high dimensional function and 1000 iteration for low dimensional functions. Here we will estimate which algorithm gives potentially better and quicker convergence towards optimality. Convergence Graph for all above mentioned Multimodal benchmark functions are plotted in Figure Fig. 25: Convergence Plot for Rastrigin Function 22-28. Fig. 22: Convergence Plot for Branin Function Fig. 26: Convergence Plot for Griewank Function Fig. 23: Convergence Plot for Ackley Function Fig. 27: Convergence Plot for Eggholder Function Copyright © 2014 MECS I.J. Intelligent Systems and Applications, 2014, 03, 35-49 48 Comparative Study of Krill Herd, Firefly and Cuckoo Search Algorithms for Unimodal and Multimodal Optimization low dimensional functions whereas for unimodal optimization FFA algorithm is superior than KH Algorithm for High dimensional function but KH algorithm is superior for low dimensional functions but in terms of time processing FFA Algorithm is surpasses KH Algorithm for both unimodal and multi modal optimization. When we compare these algorithms on basis of convergence Krill Herd is the fastest of all to converge to its optimal solution after which comes the Cuckoo search algorithm and at last comes the Firefly algorithm which do not converge for most of the function to it optimal solution. During simulation we also noticed that as we increase the dimension, fitness value or optimization fitness for Fig. 28: Convergence Plot for Shubert Function KH algorithm decreases although it outperformed FFA algorithm for high dimensional multimodal function but on average as dimension increases optimization fitness In Fig.27, and Fig.28 we have plotted the absolute for KH algorithm decreases. value of the fitness function. In multimodal optimization Shubert and Eggholder function value of As all these powerful optimization strategy are able global minima is negative and so to plot them on to optimize both unimodal and multimodal test function convergence graph we took their absolute vale. effectively hence we can easily extend them to study multi objective optimization applications with various From Fig 22-28 we can interpret that although for constraints and even to NP-hard problems. many function all the algorithms are not able to converge before 10,000th iteration but for test functions like Rastrigin, Branin, Griewank, Levy and Eggholder, Krill herd algorithm is the fastest to converge to its References optimal solution. When we compare them on the 1 X. S. Yang, ―Nature-Inspired Metaheuristic number of iteration, Krill Herd Algorithm takes least number of iteration to converge. 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