To simulate the freedom and uncertain individual behavior of krill herd, this paper introduces the opposition based learning (OBL) strategy and free search operator into krill herd optimization algorithm (KH) and proposes a novel oppositionbased free search krill herd optimization algorithm (FSKH). In FSKH, each krill individual can search according to its own perception and scope of activities. The free search strategy highly encourages the individuals to escape from being trapped in local optimal solution. So the diversity and exploration ability of krill population are improved. And FSKH can achieve a better balance between local search and global search. The experiment results of fourteen benchmark functions indicate that the proposed algorithm can be effective and feasible in both lowdimensional and highdimensional cases. And the convergence speed and precision of FSKH are higher. Compared to PSO, DE, KH, HS, FS, and BA algorithms, the proposed algorithm shows a better optimization performance and robustness.
As many optimization problems cannot be solved by the traditional mathematical programming methods, the metaheuristic algorithms have been widely used to obtain global optimum solutions. And the aim of developing modern metaheuristic algorithms is to increase the accessibility of the global optimum. Inspired by nature, many successful algorithms are proposed, for example, Genetic Algorithm (GA) [
Based on the simulation of the herding behavior of krill individuals, Gandomi and Alavi proposed the krill herd algorithm (KH) in 2012 [
In order to overcome the limited performance of standard KH on complex problems, a novel free search krill herd algorithm is proposed in this paper. The free search strategy has been introduced into the standard KH to avoid all krill individuals getting trapped into the local optima. The proposed algorithm can greatly enrich the diversity of krill population and improve the calculation accuracy, which leads to a good optimization performance. What is more, the new method can enhance the quality of solutions without losing the robustness.
The proposed FSKH algorithm is different from standard KH in two aspects. Firstly, in FSKH, the population of individuals is initialized using opposition based learning (OBL) strategy [
And secondly, the krill can do freedom and uncertain action using free search strategy. In standard KH, krill is influenced by its “neighbors” and the optimal krill, and the sensing distance of each krill is fixed. But in nature, even for the same krill, its sensitivity and range of activities will also change in different environment and different period. The proposed algorithm can simulate this freedom, uncertain individual behavior of the krill. The free search strategy allows nonzero probability for access to any location of the search space and highly encourages the individuals to escape from trapping in local optimal solution.
The remainder of this paper is organized as follows. In the Section
Krill herd (KH) is a novel metaheuristic swarm intelligence optimization method for solving optimization problems, which is based on the simulation of the herding behavior of krill individuals. The timedependent position of an individual krill in twodimensional surface is determined by the following three main actions:
movement induced by other krill individuals;
foraging activity;
physical diffusion.
KH algorithm used the Lagrangian model as follows:
For a krill individual, the motion induced by other krill individuals can be determined as follows:
The sensing distance for each krill individual is determined as follows:
The effect of the individual krill with the best fitness on the
The foraging motion is formulated in terms of two main effective parameters. The first is the food location and the second one is the previous experience about the food location. This motion can be expressed for the
In KH, the virtual center of food concentration is approximately calculated according to the fitness distribution of the krill individuals, which is inspired from “
The random diffusion of the krill individuals can be considered to be a random process in essence. This motion can be described in terms of a maximum diffusion speed and a random directional vector. It can be indicated as follows:
The crossover is controlled by a crossover probability
The mutation is controlled by a mutation probability
In general, the defined motions frequently change the position of a krill individual toward the best fitness. The foraging motion and the motion induced by other krill individuals contain two global and two local strategies. These are working in parallel which make KH a powerful algorithm. Using different effective parameters of the motion during the time, the position vector of a krill individual during the interval
It should be noted that
Sort the population krill from best to worst.
Sort the population krill from best to worst and find the current best.
Postprocessing the results and visualization.
Free search (FS) [
During the exploration, each krill achieves some favor (an objective function solution) and distributes a pheromone in amount proportional to the amount of the found favor (the quality of the solution). The pheromone is fully replaced with a new one after each walk.
Particularly, the animals in the algorithm are mobile. Each animal can operate either with small precise steps for local search or with large steps for global exploration. And each animal decides how to search (with small or with large steps) by itself. Explicit restrictions do not exist. The previous experience can be taken into account, but it is not compulsory.
The structure of the algorithm consists of three major events: initialization, exploration, and termination.
The modification strategy is
The individual behavior, during the walk, is modeled and described as
The pheromone generation is
The sensibility generation is
Selection and decision making for a start location
The steps are imitation of the ability for motion and action. The steps can be large or small and can vary. In the search process, the neighbor space is a tool for tuning rough and precise searches. So, search radius
There are two methods to set the value of search radius. The first one is that
The second method is that changing neighbor space
The detailed process of FS operator is described as in Algorithm
Initialize
Take initial walks
Generate the initial pheromone
Distribute the initial pheromone
Learn the initial achievements
Select start locations for a walk
Take exploration walks
Generate a pheromone
Distribute the pheromone
Learn the achievements
In KH algorithm, krill is influenced by its “neighbors” and the optimal krill, and the sensing distance of each krill is fixed. But in nature, the action of each krill is free and uncertain. In order to simulate this freedom, uncertain individual behavior of the krill, this paper introduces the free search strategy into the krill herd algorithm and proposes a novel free search krill herd algorithm (FSKH).
Population initialization has an important impact on the optimization results and global convergence; this paper introduces the initialization method of opposition based learning (OBL) strategy [
Select the
By utilizing OBL we can obtain fitter starting candidate solutions even when there is no knowledge about the solutions. This initialization method can make a more uniform distribution of the krill populations. Therefore, it is good for the method to get better optimization results. And by utilizing free search strategy, each krill individual in FSKH can decide how to search by itself (Algorithm
Generate uniformly distributed random population
Sort the population/krill from best to worst.
Motion induced by other individuals
Foraging motion
Free search operation
Take exploration walks
Modification strategy
Sort the population/krill from best to worst and find the current best.
Postprocessing the results and visualization.
In general, three main actions in standard KH algorithm can guide the krill individuals to search the promising solution space. But it is easy for the standard KH algorithm to be trapped into local optima, and the performance in highdimensional cases is unsatisfied. In the FSKH algorithm, the individual can search the promising area with small or large steps. So, the krill individuals can move step by step through multidimensional search space. In nature, the activity range of krill individuals is different.
Using the free search strategy, the krill individual can search any region of the search space. Each krill individual can search according to their perception and the scope of activities and can not only search around the global optimum, but also search around local optimum. When using larger step, it takes global search which can strengthen the weak global search ability of KH. Therefore, the proposed algorithm has better population diversity and convergence speed and can enhance the global searching ability of the algorithm. To achieve a better balance between local search and global search, FSKH algorithm includes both “exploration” process of FS and “exploitation” process of KH. When increasing the sensitivity, the krill individual will approach the whole population’s current best value (i.e., local search). While reducing the sensitivity, the krill individual can search around other neighborhood (i.e., global search) (Figure
Flowchart of the free search krill algorithm.
All the algorithms compared in this section are implemented in Matlab R2012a (7.14). And experiments are performed on a PC with a 3.01 GHz, AMD Athlon (tm) II X4 640 Processor, 3 GB of RAM, and Windows XP operating system. In the tests, population size is
In order to verify the effectiveness of the proposed algorithm, we select 14 standard benchmark functions [
Generally, the choice of parameters requires some experimenting. In this paper, after a lot of experimental comparison, the parameters of the algorithm are set as follows.
In KH and FSKH, the maximum induced speed
In BA, the parameters are generally set as follows: pulse frequency range is
The best, mean, worst, and Std. represent the optimal fitness value, mean fitness value, worst fitness value, and standard deviation, respectively. Bold and italicized results mean that FSKH is better, while the * results means that other algorithm is better.
For the lowdimensional case, the maximum number of iterations of each algorithm is
For benchmark functions in Table
Benchmark functions.
Category  Number  Name  Benchmark function  Scope 


I 

Sphere 




Step 


 

Rosenbrock 


 

Quartic 


 

Rastrigin 


 

Ackley 


 

Quadric 


 

Griewank 


 

Alpine 


 

Zakharov 


 


II 

Schaffer’s F6 




Drop wave 





Sixhump camel back 





Easom 



Simulation results for
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Note: In this paper, both KH and KH II represent the KH with crossover operator.
Simulation results for
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Convergence curves for
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Convergence curves for
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ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
In order to validate the optimization ability of algorithms in highdimensional case, this paper set
The comparison results for highdimensional case are shown in Table
Simulation results for
Benchmark function  Result  Method  

PSO  DE  KH II  FSKH  HS  FS  BA  

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Figures
Convergence curves for
Convergence curves for
Convergence curves for
Convergence curves for
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Convergence curves for
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Convergence curves for
Convergence curves for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
ANOVA tests of the global minimum for
In order to overcome the shortcomings of krill herd algorithm (e.g., poor population diversity, low precision of optimization, and poor optimization performance in high dimensional case). This paper introduces the free search strategy into krill herd algorithm and proposes a novel free search krill herd algorithm (FSKH). The main improvement is that the krill individual can search freely and the diversity of krill population is enriched. The proposed algorithm (FSKH) achieves a better balance between the global search and local search. Experiment simulation and comparison results with other algorithms show that the optimization precision, convergence speed, and robustness of FSKH are all better than other algorithms for most benchmark functions. So FSKH is a more feasible and effective way for optimization problems.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Science Foundation of China under Grant no. 61165015, the Key Project of Guangxi Science Foundation under Grant no. 2012GXNSFDA053028, the Key Project of Guangxi High School Science Foundation under Grant no. 20121ZD008, and the Funded by Open Research Fund Program of Key Lab of Intelligent Perception and Image Understanding of Ministry of Education of China under Grant no. IPIU01201100.