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Modified Krill Herd Optimization Algorithm using Chaotic Parameters Mahdi Bidar1, Edris Fattahi1 and Hamidreza Rashidy Kanan2 1
Department of Electrical, Computer and IT Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran 2 Department of Electrical Engineering, Bu-Ali Sina University, Hamedan, Iran {mhd_bidar; edris.fattahi}@qiau.ac.ir,
[email protected] and also cover other possible places. Various metaheuristic algorithms have being developed in these years. Some of them are briefly introduced in the following: Genetic algorithm (GA) is the most popular evolutionary algorithm which has obtained many successes in dealing with wide range of optimization problems [1]. Simulated annealing (SA) is a single agent optimization algorithm which was inspired by annealing process of metal [2]. Other meta-heuristic algorithms are, Ant Colony Optimization (ACO) algorithm [3], Particle Swarm Optimization (PSO) algorithm [4], Bee Colony Optimization (BCO) algorithm [5, 6], Imperialist Competitive Algorithm (ICA) [7] and Firefly Algorithm (FA) [8]. Recently, some other natural-inspired algorithms have been developed which most famous ones are Cuckoo search algorithm [9], Gravitational Search Algorithm [10], Bat-inspired algorithm [11], Krill herd optimization algorithm [12], Dolphin echolocation [13] and Bird mating optimizer [14]. The significant characteristic of meta-heuristic algorithms is their dependency to the values of their parameters in initial step and during searching process. The most important parameters are the controlling parameters which controls the exploration-exploitation balance. Allocating values to these parameters can be performed by different methods such as fuzzy systems and chaotic systems. Using chaotic systems and utilizing chaos theory in these algorithms bring dynamism and instability properties to them which by heightening the performance of random search help the algorithm to escape from local optimum traps. Chaotic system by producing and allocating chaotic sequence to the controlling parameters with aid of chaotic map increase the performance of the meta-heuristic algorithms. Krill Herd algorithm is famous newly developed optimization algorithm which has drawn many attentions in dealing with optimization problems. There is no doubt that each algorithm has some weaknesses and different researchers by different point of views trying to fix them. Recently some researchers by different point of views and by different ways try to heighten the performance of Krill Herd optimization algorithm [15,16,17]. In this paper, we have used chaos theory to tune inertia weights of the algorithm to bring dynamism property to the algorithm and improve the performance of Krill Herd algorithm. We examined the proposed method by some famous benchmark functions and the attained results indicate increased performance of Standard KH algorithm. The rest of the paper is organized as follows:
Abstract— Krill Herd optimization algorithm which is a new metaheuristic search algorithm mimics the herding behavior of krill individuals. The significant characteristic of metaheuristic algorithms is their ability in combination of local search and global search. This property can adjust the contribution of local search and global search in initial step and during searching process and plays crucial role in the algorithm performance. One hazard which threats the metaheuristic algorithms is getting stuck in local optimum traps. An appropriate solution to deal with this problem is using chaos theory which brings dynamism and instability properties to the algorithm so that by strengthening the performance of random search helps the algorithm to escape from local optimum traps. In this paper, we propose a new method called chaotic Krill Herd optimization algorithm which by adopting chaos theory in Krill Herd optimization algorithm heighten its performance in dealing with various optimization problems. The obtained results by the proposed method in comparison with those of the standard Krill Herd optimization algorithm indicates the higher performance of the proposed algorithm. Keywords: Metaheuristic Optimization Algorithms; Krill Herd Optimization Algorithm; Chaotic Krill Herd Optimization Algorithm; Exploration; Exploitation.
I.
INTRODUCTION
Recently, world has been facing with complicated problems with high-dimensional problem spaces which logical and classical methods unable to solve them efficiently. The problem space in these problems is like hilly landscape with lots of local optimums and just one global optimum. The Newton-Raphson algorithm is one of the popular classical gradient-based methods which has been proven that cannot deal with such mentioned problems and just works well in smooth and unimodal problems. Another classical method is hill-climbing method which follows the positive gradient to reach the peak, but it is crystal clear that it would be easily gets stuck in local optimums in hilly landscapes. Hill climbing method is a pure local search which locally searches the best solution. Another choice is random search which without considering the landscape’s signs and gained experiences in previous steps, blindly looking for the best solution. This method cannot succeed in finding the best solution too. Recently, the meta-heuristic algorithms which have gained reputation in dealing with NP-hard problems have drawn many attentions. If local search and random search be considered as two extremes, metaheuristic algorithms are between them which means these algorithms utilize both kind of searches simultaneously. So, they are not blind and while they are walking randomly, looking for some hints. Random search plays main role and take more contribution in searching tasks
,(((
In Section 2 the Standard Krill Herd algorithm is introduced. Section 3 contains the definition of the proposed method. In Section 4 obtained results are
evaluated and discussed. Finally Section 5 conclude the paper. II.
݂݀
is food attractive and ߚܾ݅݁ ݐݏis the best Where ߚ݅ fitness effect of i-th krill so far. Physical diffusion of krill is formulated as follow: ܫ ሻߜ (6) ݅ܦൌ ݔܽ݉ܦሺͳ െ
STANDARD KRILL HERD (SKH) OPTIMIZATION ALGORITHM
ݔܽ݉ܫ
Where ݔܽ݉ܦis the maximum diffusion speed, ߜ is the random directional vector and its arrays are values in (-1, 1), I and ݔܽ݉ܫare i-th and maximum iteration number. Location vector of krill during the interval t to t+1 is given by (7 ): ݀ܺ (7) ܺ݅ ሺ ݐ οݐሻ ൌ ܺ݅ ሺݐሻ ο݅ ݐ ݀ݐ Where ο ݐis an important constant that should be carefully set based on the optimization problem. After Motion calculation, to improve the performance of SKH, crossover and mutation operators added to algorithm [12].
The SKH algorithm is an optimization algorithm which is inspired by herding behavior of krill individuals developed by Gandomi [12] in 2012. The objective of krill individuals in their society is reaching food and keeping minimum distance to food source and the highest krill herd density. Kill individuals in their society face with predators such as penguins, seals and seabirds which form their behaviors and affect their lifestyle. Predators attack krill individuals and by removing individuals, reducing the krill herd density. This trend results in while they are trying to reach food, try to increase their herd density too which indicates they deal with multi-objective problem in their life. So their objective can be organized as follow: a) Reaching food b) Increasing krill herd
III. CHAOTIC KRILL HERD (CKH) OPTIMIZATION ALGORITHM A. Chaotic system The general definition of chaos is bounded unstable dynamic behavior which indicates the sensitive dependence on initialization which includes infinite unstable periodic movements in nonlinear systems. In mathematic chaos means randomness which has generated by deterministic system. In fact randomness is result of sensitivity of chaotic systems to initial conditions which means small changes in a function or system breed big different future behaviors. A system can be easily changed to chaotic system by just make changes in one of their control parameter. Control parameters and their quantities play crucial role in gaining high performance and efficiency. Chaotic systems with aid of chaotic maps generate chaotic sequence. In follow some chaotic maps are briefly introduced:
Fitness of an individual is combination of distance from food and distance from highest krill density. Movement of krill individuals in 2-D space is affected by three main factors: 1) Movement induced by other krill individuals 2) Foraging motion 3) physical diffusion This movement is formulated as follow: ݀ܺ ݅ ݀ݐ
ൌ ܰ݅ ݅ܨ ݅ܦ
(1)
Where ܰ݅ is the movement induced by other krill individuals, ݅ܨis the foraging motion and ݅ܦis the physical diffusion of the i-th krill individuals. The motion induced by other krill individuals is defined as follow: ܰ݅݊݁ ݓൌ ܰ ݉ܽ ݅ߙ ݔ ߱݊ ݈ܰ݅݀ (2) Where ܰ ݉ܽ ݔis the maximum induced speed, ߱݊ is the inertia weight of the motion induced and ܰ݅ ݈݀is the last motion induced. Also ߙ݅ is the direction of movement which is estimated by local, target and repulsive effect and represented as follow: ݐ݁݃ݎܽݐ (3) ߙ݅ ൌ ߙ݈݅ ݈ܽܿ ߙ݅ Where ߙ݈݅ ݈ܽܿis the local effect provided by the neighbors ݐ݁݃ݎܽݐ the target direction effect provided by the best and ߙ݅ krill individual. Foraging motion is affected by two main factors, the first one is the current food location and the second one is the previous knowledge about food location. Foraging motion is formulated as follow: ݅ܨൌ ܸ݂ ߚ݅ ݂߱ ݈݀݅ܨ (4) Where ܸ݂ is the foraging speed, ݂߱ is the inertia weight of the foraging motion, ݈݀݅ܨis the last foraging motion. And also : ݂݀ ߚ݅ ൌ ߚ݅ ߚܾ݅݁ݐݏ (5)
1) Logistic map The logistic map is one of the simplest and thus more widely used chaotic maps. This map is defined as follow: ܺ݅ͳ ൌ ߙ ݅ܺ כሺͳ െ ܺ݅ ሻ (8) Where ܺ݅ is i-th chaotic number and i is iteration number. ܺ݅ אሺͲǡͳሻǡ ܺͲ בሼͲǡ ͲǤʹͷǡ ͲǤͷǡͲǤͷǡ ͳሽ and Ͳ ߙ Ͷ. 2) Tent map Tent map is defined as follow: ܺ݅ ܺ݅ ൏ ͲǤ ͲǤ ܺ݅ͳ ൌ ቐ ͳͲ ݁ݏ݅ݓݎ݄݁ݐ
(9)
͵ ݅ ܺכሺͳെܺ ݅ ሻ
3) Sinusoidal map This map is represented as following equation: ሺߨ ݅ܺ כሻ (10) ܺ݅ͳ ൌ כ ʹ ݅ܺ כן༌ ןൌ ʹǤ͵ܽ݊݀ܺͲ ൌ ͲǤ. The simplest form can be as follow: ܺ݅ͳ ൌ ༌ሺߨ ݅ܺ כሻ (11) It generates chaotic sequence in (0, 1). 4) Gauss map
This map is formulated as follow: Ͳܺ݅ ൌ Ͳ ܺ݅ͳ ൌ ቊ ͳ ݉݀ሺͳሻ ݁ݏ݅ݓݎ݄݁ݐ, ͳ
ܺ݅
݉݀ሺͳሻ ൌ ܺ݅
ͳ ܺ݅
ͳ
െቂ ቃ ܺ݅
(12) (13)
݅ܨൌ ܸ݂ ߚ݅ ݂߱ ݈݀݅ܨ
(19)
݈݅ܺ ܽ݉ܿ݅ݐݏ݅݃ͳ ൌ ݅ܺ כןሺͳ െ ܺ݅ ሻ ݄ܿܽ ݂ܹܿ݅ݐ ܹ݂ ሺ݅ ͳሻ ൌ ݂ܹ כןሺ݅ሻ כሺͳ െ ܹ݂ ሺ݅ሻሻ
(20)
՜ ݄ܿܽ ݅ܨܿ݅ݐൌ ܸ݂ ߚ݅
This map also produces chaotic sequence in (0, 1).
(22)
൬ ݂ܹ כןሺ݅ െ ͳሻ כቀͳ െ ܹ݂ ሺ݅ െ ͳሻቁ൰ ݈݀݅ܨ כ
5) Circle map
So, small changes in initial values of ܹ݊ and ܹ݂ breeds big changes in future behavior of the SKH algorithm. Equation (15) and (19) are deterministic and to gain dynamic characteristic initial values of ܹ݂ ሺͲሻܹܽ݊݀݊ ሺͲሻ must be as follow: ܹ݂ ሺͲሻܹܽ݊݀݊ ሺͲሻ בሼͲǡ ͲǤʹͷǡ ͲǤͷǡͲǤͷǡ ͳሽ and Ͳ ן Ͷ.
This map is defined as follow: ߙ (14) ܺ݅ͳ ൌ ܺ݅ ߚ െ ቀ ቁ כሺʹߨܺ݅ ሻ ݉݀ሺͳሻ ʹߨ Whereߙ ൌ ͲǤͷ and ߚ ൌ ͲǤʹ. It produces chaotic sequence in (0, 1) [18, 19, 20, 21, 22]. B. The Proposed Algorithm Meta-heuristic algorithms are natural-inspired algorithms which can be mentioned as parameter-based algorithms which their performance and efficiency are highly dependent on their controlling parameters and starting values. As it’s mentioned in previous sections, meta-heuristic algorithms utilize the combination of exploration or random search and exploitation or local search. This combination creates a special property for meta-heuristic algorithms so that classical, logical and heuristic methods cannot compete with them. But the point is adjusting controlling parameters to determine the contribution amount of exploration and exploitation to gain exploration-exploitation balance. SKH algorithm is a newly developed meta-heuristic algorithm so there is no exception about it. The most important application of chaotic systems is their parameter-tuner role in metaheuristic algorithms which is performed by generating and allocating chaotic sequences to their controlling parameters. These process dedicates the dynamic and unstable behavior to meta-heuristic algorithms which able them to easily escape from local optimums. The controlling parameters in SKH algorithm are ܹ݂ and ܹ݊ which control the exploration-exploration balance. So by allocating chaotic sequence to these parameters, SKH algorithm catch dynamic behavior and gain explorationexploitation balance which help the algorithm so as not to being trapped in local optimums. As it mentioned previously there are various maps which can be used for producing chaotic sequence so that one of the most popular and simplest of them is logistic map. In this paper logistic map is considered to produce the chaotic sequence. The controlling parameters in (2) and (4) have been changed to chaotic form with use of logistic map as follow:
IV. SIMULATION AND EXPERIMENL RESULTS A. Parameter Settings Matlab 2010 software was used for implementation of our proposed method. In order to examine the performance of our proposed method, we have applied it to deal with eight high dimensional famous benchmark functions which are presented in Table I, attained from the literature [23], and obtained results have been compared with the results obtained from the SKH and also another state-of-art method, PSO algorithm. Stochastic treatment of meta-heuristic algorithms are inevitable and it causes different results at different runs. So to eliminate stochastic difference, we run the program 50 times involving 50 different initial solution. We have considered the 200 iterations, 25 krill individuals, upper bound to 10 and lower bound to -10 and we have assigned value of ܸ݂ , ݔܽ݉ܦand ܰ݉ܽ ݔto 0.02, 0.005 and 0.01 respectively. Also as it mentioned in previous section, inertia weights (ܹ݊ ǡ ܹ݂ ሻ get their values from chaotic sequence generating by chaotic map. TABLE II. BENCHMARK FUNCTIONS High dimensional functions Name
Functions
݂ሺݔሻ ݊
Ackley
ൌ െܽ ቌെͲǤͲʹඨ݊െͳ ݅ൌͳ
െ ൬݊െͳ Griewank Quartic
ܰ݅݊݁ ݓൌ ܰ ݉ܽ ݅ߙ ݔ ܹ݊ ݈ܰ݅݀
(15)
Rastrigin
݈݅ܺ ܽ݉ܿ݅ݐݏ݅݃ͳ ൌ ߙ ݅ܺ כሺͳ െ ܺ݅ ሻ ݄ܿܽ ݊߱ܿ݅ݐ: ܹ݊ ሺ݅ ͳሻ ൌ ܹ݊ כןሺ݅ሻ כሺͳ െ ܹ݊ ሺ݅ሻሻ
(16) (17)
Rosenbrock
՜ ݄ܿܽ ݓ݁݊݅ܰܿ݅ݐൌ ܰ ݉ܽ ݅ߙ ݔ ሺ ܹ݊ כןሺ݅ െ ͳሻ כሺͳ െ ܹ݊ ሺ݅ െ ͳሻሻሻ ݈݀݅ܰ כ
(21)
݊
ሺʹߨ ݅ݔሻ൰ ܽ ݁
݅ൌͳ ݊
݂ሺݔሻ ൌ ͳ
݊ ͳ ݅ݔ ʹ݅ݔെ ෑ
༌ ሺ ሻ ͶͲͲͲ ݅ൌͳ ݅ൌͳ ξ݅ ݊
݂ሺݔሻ ൌ ݊
݅ൌͳ
݂ሺݔሻ ൌ ͳͲ݊ ݊െͳ
݂ሺݔሻ ൌ ݅ൌͳ
Sphere
ʹ݅ݔቍ
݅ൌͳ
݅݅ݔͶ ܴܽ݊݀
ሺ ʹ݅ݔെ ͳͲ
༌ ሺʹߨ ݅ݔሻሻ
ሺͳͲͲሺ݅ݔͳ െ ʹ݅ݔሻʹ ሺ ݅ݔെ ͳሻʹ ሻ ݊
݂ሺݔሻ ൌ หȁݔȁหǡ หȁݔȁห ൌ ඨ ݅ൌͳ
(18)
Schwefel2.26 Schwefel2.22
And similarly ݄ܿܽ ݂ܹܿ݅ݐwould be:
݊
݂ሺݔሻ ൌ ݅ൌͳ ݊
݂ሺݔሻ ൌ ݅ൌͳ
ʹ݅ݔ
െ ݅ݔ༌ ሺඥȁ ݅ݔȁሻ ݊
ȁ ݅ݔȁ ෑ
݅ൌͳ
ȁ ݅ݔȁ
B. Results Table III shows the results of best and average cost for eight benchmark functions mentioned at previous sections. It is must be mentioned that the results have been
obtained from 50 times runs to guarantee that obtained results shows the real performance of the algorithm. There are also the results obtained by SKH and PSO algorithms to make the comparison of their performance and efficiency. TABLE IV. COSTS RESULTS OF BEST AND AVERAGE OF LAST ITERATION OF SKH, CKH AND PSO SKH
Function Ackley Griewank Quartic Rastrigin Rosenbrock Sphere Schwefel 2.26 Schwefel 2.22
CKH
PSO
Best Cost
Average Cost
Best Cost
Average Cost
Best Cost
Average Cost
0.00033910 3.31E-07 0.00325584 0.036824 0.000252 0.024716 0.00044 0.06782
0.3804307 0.0241097 0.0258426 0.38952 0.002248 0.35674 0.0546812 0.256200
0.0001268 1.32E-07 0.0017681 0.03072 3.983E-05 0.0195 0.0001874 0.05035
0.1859435 1.44E-02 0.0130789 0.265235 0.000165 0.24187 0.022151 0.10816
0.1275691 0.15687475 0.00510535 10.9708 18.3249 0.17211 0.10259 0.12416
1.15595160 0.348540502 0.011397253 14.9309 35.9886 0.36019 0.33802 0.80417
As it can be seen in Table V, results show superiority of CKH to SKH and PSO. The full convergence behavior of first three functions for best and average of 50 runs values are shown in Fig. 1, Fig. 2 and Fig. 3. In all figures, it is clear that performance of CKH is better than SKH and PSO.
Figure 3. Convergence comparison of best and average 50 runs values of SKH, CKH and PSO for Quartic function.
Allocating chaotic sequence to ܹ݂ and ܹ݊ , heightens the performance of random search and it results in gaining the exploration-exploration balance and also gaining appropriate instability and dynamism that helps the algorithm easily escape from local optimum traps. V. Figure 1. Convergence comparison of best and average 50 runs values of SKH, CKH and PSO for Ackley function.
CONCLUSION AND FUTURE WORKS
Exploitation-exploration balance is an important factor in meta-heuristic algorithms to obtain optimum points in search space. So, in this paper we proposed a new method called Chaotic Krill Herd optimization algorithm which uses chaotic sequence to tune the controlling parameters of Krill Herd optimization algorithm. It works by generating and allocating chaotic sequence to controlling parameters of standard krill herd which bring dynamic behavior to the Krill Herd algorithm which improve the performance of the algorithm in dealing with problems. Obtained results on eight benchmark problems show superiority of our proposed method to its standard one and PSO algorithm. REFERENCES [1] Campanini, R., Di Caro, G., Villani, M., D’Antone, I., & Giusti, G. (1994). Parallel architectures and intrinsically parallel algorithms: Genetic algorithms. International Journal of Modern Physics C, 5(1), 95–112. [2] Kirkpatrick, S.; Gelatt, C. D.; and Vecchi, M. P. "Optimization by Simulated Annealing." Science 220, 671-680, 1983.
Figure 2. Convergence comparison of best and average 50 runs values of SKH, CKH and PSO for Griewank function.
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