This paper investigates the behaviors at different developmental stages in
Swarm intelligence is the emergent collective intelligent behaviors from a large number of autonomous individuals. It provides an alternative way to design novel intelligent algorithms to solve complex realworld problems. Different from conventional computing paradigms [
Nowadays, most of swarm intelligent optimization algorithms are inspired by the behavior of animals with higher complexity. Particle swarm optimization (PSO) [
As prokaryote, bacteria behave in a simple pattern which can be easily described. Inspired by the foraging behavior of
However, traditional bacterial behaviorbased algorithms (BFO or BC) only considered individual behaviors instead of social behaviors with swarm intelligence. Each individual in the colony independently searches for food by their own experience without any information exchange with others. What made the situation worse is the complicated characteristics of the original bacterial behaviorbased algorithms. Taking BFO for example, long period of time has been spent on random chemotaxis. Additionally, the chemotaxis, reproduction, elimination and dispersal processes are inner iterations that lead to high computation complexity. The frequency of chemotaxis, reproduction, elimination, and dispersal is predefined without considering the environmental dynamics.
To deal with the aforementioned problems, we propose a new bacterial behavior based optimization algorithm—bacterial colony optimization (BCO)—to formulate bacteria's behavior with swarm intelligence. The main contributions of this paper are described as follows.
A new description of artificial bacteria lifecycle is formulated, which include five basic behaviors and their corresponding models.
Newly created bacterial behavior model is proposed to simplify the bacteria optimization process.
A novel chemotaxis and communication mechanism is used as well to update the bacterium positions.
Two methods of communication: individual interaction and group exchange are introduced to improve the optimization efficiency.
The rest of the paper is organized as follows. Section
Bacteria swim by rotating whiplike flagella driven by a reversible motor embedded in the cell wall (Figure
Individual and social behavior of artificial bacteria.
The basic behavior of bacteria in the lifecycle can be simply divided into five parts: chemotaxis, elimination, reproduction, migration, and communication. The detail descriptions of those processes are given as follows.
A fascinating property of
Bacteria need to migrate up to the concentration gradient of nutrients. Hence, the alteration between two operations in chemotaxis must be well organized. A basic strategy used by microbes is to move in one direction for several steps. If the new environment cannot satisfy the bacteria, then they would tumble to pull themselves into a new direction and start a second run.
Based on the theory of nature selection, bacteria with poorer searching ability would have higher chance of being eliminated. In contrast, those who perform well in the chemotaxis process would obtain more energy for survival and thus have a high probability of reproduction. In our proposed model—bacteria colony optimization (BCO), artificial bacteria with high quality in searching for nutrition have the opportunity to be endowed a relevant level of energy grade. Whether a bacterium has the chance to reproduce or not would base on the level of its energy grade.
After a long time of chemotaxis, elimination, and reproduction in the same area or surroundings, the nutrition must be used up or cannot satisfy all the bacteria. At this time, some bacteria have to migrate into a new nutritious place, and this process is called “Migration.”
Communication is an essential behavior that exists in whole processes of bacterial life. Three basic communication mechanisms are employed in the bacteria colony optimization (see Figure
Communication mechanism.
Dynamic neighbor oriented
Random oriented
Group oriented
The behavior of artificial bacteria in this paper includes five parts, but those behaviors are continuous, mingle, and amalgamate. Chemotaxis behavior is always accompanied by communication along the whole lifecycle. Therefore, chemotaxis and communication are treated as one model in Bacterial Colony Optimization (BCO). Bacteria have two chooses after long times of chemotaxis and communication. They may die for the lack of food, or they may reproduce if they are capable of searching for food. Within the complicated environment, some individuals may run into dangerous place (go out of boundary). Specific situations like this also worth special treatment in lifecycle model (LCM). Migration conducts as an independent model, which involves energy depletion, group diversity, and chemotaxis efficiency. The overall model of bacterial lifecycle is shown in Figure
Lifecycle model.
The framework of this model is based on an agentenvironmentrule (AER) schema; that is, there are three fundamental elements: agent, environment, and rule. The detailed description is listed below:
A: artificial bacteria,
E: artificial environment,
R: the environment/organism interaction mechanisms.
LCM model is different from the original populationbased model in which all the individuals share the same state properties. LCM is a philosophy that embraces the uniqueness of the individuals in a system with multiple individuals that have its own set of state variables and parameters. Looking in state space, the population is akin to clouds of individuals with similar behaviors, and other clouds, amounting to separate individuals. Fundamentally, this allows for individuals to exist and speciation to occur and permits extinction. In general, the lifecycle model of artificial bacteria in BCO can be divided into four submodels: chemotaxis and communication model, reproduction model, elimination model, and migration model.
The detailed explanations of each submodel are formulated in the following.
Chemotaxis is accompanied with communication in the whole optimization process. Bacteria run and tumble in the competitive environment. However, they also have to offer their information to the colony in exchange of overall information which would guide them in direction and ways of movement. As is shown in Figure
Chemotaxis.
Bacterium runs or tumbles with communication process can be formulated as:
Actually, the above position updating equations only consider the relationship between the individuals and the group. The bacteria share information between individuals also merged into the communication model. Pseudocode for
Individual exchange
Each bacterium is marked with an energy degree based on its search capability. The higher level of energy indicates a better performance of bacterium. The level of energy decides the probability of elimination and reproduction. The distribution of bacterial energy degree was sorted and analyzed and then used as a criterion to judge the qualification of the bacteria. The details are summarized as
All behaviors of bacteria were restricted within a restrained area. As a general principle, individuals are not allowed to go out of the region, so boundary control is especially important. If bacteria move away from the feasible domain, at least two strategies will be performed based on experiences. One is to generate new individuals to replace the outer ones, and the other is to let the outer ones stay at boundary but change the forward direction to keep them effectiveness. In this paper, those outer individuals named “unhealthy” are put into a set which hold the candidate bacteria for elimination.
Naturally, bacteria could pursue more nutrition by migration. In optimization aspect, migration can avoid local optimum within some distance. Especially, the migration of artificial bacteria in BCO is not based on a certain given probability. It depends on a given condition. When condition is fulfilled, bacterium would migrate to a new random place, as described by
Migration
As illustrated above, bacteria chemotaxis all life time can be divided into two models: tumbling and swimming. In the process of tumbling, a stochastic direction participates into actually swimming process. Therefore, turbulent director and optimal searching director altogether influence the search orientation in tumbling, update the positions of each bacterium as (
Interactive exchange in BCO can be divided into individual exchange and group exchange as described above. Individual exchange also can specify dynamic neighbor oriented (Figure
Interactive exchange
Choose one in two neighbor bacteria, compare the fitness,
replace the poorer one
Randomly choose a bacterium from the group, compare the
fitness, replace the poorer one
Compute the group best, compare the fitness, replace the poor
one
In BCO algorithm, artificial bacterial behaviors are executed based on given conditions responding to the dynamic environment. The procedure of chemotaxis, communication, reproduction, elimination, and migration is not premeditated, but determined only when certain given conditions are reached.
As shown in Figure
The flowchart of BCO.
The overall procedure of bacterial colony optimization (BCO) is presented in Pseudocode
Bacterial Colony Optimization (BCO)
Compute the
Reproduction and elimination using (
Migration using (
To test the effectiveness of the new proposed BCO algorithm, twelve wellknown test functions with 15 dimensions and 40 dimensions are adopted. Test problems include two unimodal functions
To evaluate the performance of the proposed BCO, five other algorithms were used for comparisons: particle swarm optimization (PSO), genetic algorithm (GA), bacterial foraging optimization (BFO), bacterial foraging optimization with linear decreasing cemotaxis step (BFOLDC), and bacterial foraging optimization with nonlinear decreasing chemotaxis step (BFONDC). The parameters used for these five algorithms were recommended from [
Globe optimum, search ranges, and initialization ranges of test functions.
Function  Function name 

Minimum value  Range of 
Initialization range 


Sphere 

0 



Rosenbrock 

0 



Sum of different powers 

0 



Sin 

0 



Rastrigin 

0 



Griewank 

0 



Ackley 

0 



Weierstrass 

0 



RoRastrigin 

0 



RoGriewank 

0 



RoAckley 

0 



RoWeierstrass 

0 


Tables
Experimental results on benchmark functions
Methods  Function  

Unimodal  Unimodal  Multimodal  Multimodal  
Sphere  Rosenbrock  Sum of powers  Sin  
GA 




PSO 




BFO 




BFOLDC 




BFONDC 




BCO 




Experimental results on benchmark functions
Methods  Function  

Multimodal  Multimodal  Multimodal  Multimodal  
Rastrigin  Griewank  Ackley  Weierstrass  
GA 




PSO 




BFO 




BFOLDC 




BFONDC 




BCO 




Experimental results on benchmark functions
Methods  Function  

Unimodal  Unimodal  Multimodal  Multimodal  
Rotated Rastrigin  Rotated Griewank  Rotated Ackley  Rotated Weierstrass  
GA 




PSO 




BFO 




BFOLDC 




BFONDC 




BCO 




The median convergence characteristics of 15D test functions.
Sphere function
Rosenbrock function
Sum of powers function
Sin function
Rastrigin function
Griewank function
Ackley function
Weierstrass function
RORastrigin function
ROGriewank function
ROAckley function
ROWeierstrass function
The means and variances of the median run of each algorithm on
Pseudocode
Figure
The experiments conducted on 15D problems are repeated on the 40D problems. Similarly to the case in 15D, Tables
Experimental results on benchmark functions
Methods  Function  

Unimodal  Unimodal  Multimodal  Multimodal  
Sphere  Rosenbrock  Sum of powers  Sin  
GA 




PSO 




BFO 




BFOLDC 




BFONDC 




BCO 




Experimental results on multimodal benchmark functions
Methods  Function  

Multimodal  Multimodal  Multimodal  Multimodal  
Rastrigin  Griewank  Ackley  Weierstrass  
GA 




PSO 




BFO 




BFOLDC 




BFONDC 




BCO 




Experimental results on rotated benchmark functions
Methods  Function  

Unimodal  Unimodal  Multimodal  Multimodal  
Rotated Rastrigin  Rotated Griewank  Rotated Ackley  Rotated Weierstrass  
GA 




PSO 




BFO 




BFOLDC 




BFONDC 




BCO 




The median convergence characteristics of 40D test functions.
Sphere function
Rosenbrock function
Sum of powers function
Sin function
Rastrigin function
Griewank function
Ackley function
Weierstrass function
RORastrigin function
ROGriewank function
ROAckley function
ROWeierstrass function
According to the comparative experiments, the proposed BCO algorithm shows the superior searching abilities in most cases. In this section, simulation studies will conduct in a vary environment with nutrientnoxious distribution. The nutrient distribution of environment at
Nutrientnoxious environment.
From Figure
The average fitness values obtained with iterations.
The final optimal function values of each individual.
Figures
2D Position with the iteration process.
The process of finding the optimum.
The above figures only point out the group search ability without answering how microcommunities can adapt their behavior to nutrients. Figures
Four bacteria find the optimum when chemotaxis ranges between 1~100.
Optimal process of four bacteria when chemotaxis ranges between 1~100.
Figure
Single bacterium finds the optimum when chemotaxis ranges between 1~100.
Optimal process of one bacterium when chemotaxis ranges between 1~100.
In this paper, a lifecycle model concerned with modeling ecological and evolutionary processes of
Based on the results of the six algorithms on the twelve chosen test problems belonging to three classes, we can conclude that BCO gives the best performance on almost all the benchmarks problems irrespective of if they are unrotated or rotated when compared with five other algorithms.
However, BCO is still in its infant stage. Further work may focus on (i) incorporating a dynamic population size strategy to BCO, (ii) hybridizing BCO with other swarm intelligent algorithms, (iii) applying BCO to multiobjective problems.
Sphere function
Rosenbrock function
Sum of different powers function
Sin function
Rastrigin function
Griewank function
Ackley function
Weierstrass function
Rotated Rastrigin function
Rotated Griewank function
Rotated Ackley function
Rotated Weierstrass function
The first author would like to thank Dr Yujuan Chai for modifying the manuscript and giving many valuable comments. This work is supported by National Natural Science Foundation of China (Grant nos. 71001072, 71271140, and 71210107016), China Postdoctoral Science Foundation (Grant no. 20100480705), Science and Technology Project of Shenzhen (Grant no. JC201005280492A), and the Natural Science Foundation of Guangdong Province (Grant no. 9451806001002294 and S2012010008668).