Adaptive Bacterial Foraging Optimization

and Applied Analysis 3 order to coordinate pattern emerges. In 18 , the proposed CBFO applied two cooperative approaches to the original BFO, namely, the serial heterogeneous cooperation on the implicit space decomposition level and the serial heterogeneous cooperation on the hybrid space decomposition level. In order to improve the BFO’s performance on complex optimization problems with high dimensionality, we apply two natural foraging strategies, namely, the producerscrounger foraging PSF and the area concentrated search ACS , to the original BFO, resulting in two new adaptive bacterial foraging optimization models ABFOs , namely, ABFO1 and ABFO2. Instead of the simple description of chemotactic behavior in BFO, the proposed algorithms can also adaptively strike a balance between the exploration and the exploitation of the search space during the bacteria evolution, by which the significant improvement can be gained. In order to evaluate the performance of the proposed algorithms, extensive studies based on a set of well-known benchmark functions have been carried out. For comparison purposes, the work also implemented a real-coded genetic algorithm GA , the standard particle swarm optimization PSO , and the original BFO on these functions respectively. The simulation results are encouraging. The ABFO algorithms have markedly superior search performance when compared to the original BFO, while maintaining the similar or even superior performance compared to PSO and GA in terms of accuracy, robustness, and convergence speed on all benchmark functions. The proposed ABFO1 and ABFO2 described in this paper enhance previous BFO works in the following aspects: i a new adaptive strategy, namely, the producer-scrounger foraging, to dynamically determine the chemotactic step sizes for the whole bacterial colony during a run, hence dividing the foraging procedure of artificial bacteria colony into multiple explore and exploit phases; ii a new self-adaptive foraging strategy, namely, the area concentrate search, to respectively tune the chemotactic step size for each single bacterium during its run, hence casting the bacterial foraging process into heterogeneous fashion; iii a comprehensive study comparing ABFO1 and ABFO2 with another two state-ofthe-art global optimization algorithms, namely, GA and PSO, on high dimensional functions; iv single and colonial bacterial behaviors in both ABFO1 and ABFO2 that were simulated respectively in order to analyze in depth the adaptive and self-adaptive foraging schemes in the proposed models; v new results on benchmark functions up to 300 dimensions. The rest of the paper is organized as follows. In Section 2, we will give brief reviews of the bacterial chemotaxis and the original BFO algorithm. The in-depth analysis of the influence of the run-length unit parameter on the bacterial behavior in BFO model is also presented here. In Section 3, two adaptive strategies employed by ABFOs are described and the state-of-the-art adaptation mechanism is summarized. Then, our adaptive bacterial optimization algorithms will be introduced, and its implementation details will be given in Section 4. In Section 5, the experiment studies of the proposed ABFO algorithms and other algorithms are presented with descriptions of the benchmark functions, experimental settings, and experimental results. Finally, Section 6 concludes the paper. 4 Abstract and Applied Analysis 2. The Classical BFO Algorithm Bacterial foraging algorithm is inspired by an activity called “chemotaxis” exhibited by bacterial foraging behaviors. Motile bacteria such as E. coli and salmonella propel themselves by rotation of the flagella. To move forward, the flagella rotates counterclockwise and the organism “swims” or “runs” while a clockwise rotation of the flagellum causes the bacterium to randomly “tumble” itself in a new direction and swim again 19 . Alternation between “swim” and “tumble” enables the bacterium to search for nutrients in random directions. Swimming is more frequent as the bacterium approaches a nutrient gradient. Tumbling, hence direction changes, is more frequent as the bacterium moves away from some food to search for more. Basically, bacterial chemotaxis is a complex combination of swimming and tumbling that keeps bacteria in places of higher concentrations of nutrients. 2.1. Bacterial Foraging Optimization The classical bacterial foraging optimization BFO system consists of three principal mechanisms, namely, chemotaxis, reproduction, and elimination-dispersal 1 . We briefly describe each of these processes as follows. 2.1.1. Chemotaxis In the classical BFO, a unit walk with random direction represents a “tumble” and a unit walk with the same direction in the last step indicates a “run”. Suppose θ j, k, l represents the bacterium at jth chemotactic, kth reproductive, and lth elimination-dispersal step. C i , namely, the run-length unit parameter, is the chemotactic step size during each run or tumble. Then, in each computational chemotactic step, the movement of the ith bacterium can be represented as θ ( j 1, k, l ) θ ( j, k, l ) C i Δ i √ ΔT i Δ i , 2.1 whereΔ i is the direction vector of the jth chemotactic step. When the bacterial movement is run, Δ i is the same with the last chemotactic step; otherwise,Δ i is a random vector whose elements lie in −1, 1 . With the activity of run or tumble taken at each step of the chemotaxis process, a step fitness, denoted as J i, j, k, l , will be evaluated. 2.1.2. Reproduction The health status of each bacterium is calculated as the sum of the step fitness during its life, namely, ∑Nc j 1 J i, j, k, l , where Nc is the maximum step in a chemotaxis process. All bacteria are sorted in reverse order according to health status. In the reproduction step, only the first half of population survives, and a surviving bacterium splits into two identical ones, which are then placed in the same locations. Thus, the population of bacteria keeps constant. Abstract and Applied Analysis 5 2.1.3. Elimination and Dispersal The chemotaxis provides a basis for local search, and the reproduction process speeds up the convergence which has been simulated by the classical BFO, while, to a large extent, only chemotaxis and reproduction are not enough for global optima searching. Since bacteria may get stuck around the initial positions or local optima, it is possible for the diversity of BFO to change either gradually or suddenly to eliminate the accidents of being trapped into the local optima. In BFO, the dispersion event happens after a certain number of reproduction processes. Then, some bacteria are chosen, according to a preset probability Ped, to be killed and moved to another position within the environment.and Applied Analysis 5 2.1.3. Elimination and Dispersal The chemotaxis provides a basis for local search, and the reproduction process speeds up the convergence which has been simulated by the classical BFO, while, to a large extent, only chemotaxis and reproduction are not enough for global optima searching. Since bacteria may get stuck around the initial positions or local optima, it is possible for the diversity of BFO to change either gradually or suddenly to eliminate the accidents of being trapped into the local optima. In BFO, the dispersion event happens after a certain number of reproduction processes. Then, some bacteria are chosen, according to a preset probability Ped, to be killed and moved to another position within the environment. 2.2. Step-by-Step Algorithm In what follows, we briefly outline the original BFO algorithm step by step. Step 1. Initialize parameters n, S,Nc,Ns,Nre,Ned, Ped, C i i 1, 2, . . . , S , θ, where n: dimension of the search space, S: the number of bacterium, Nc: chemotactic steps, Ns: swim steps, Nre: reproductive steps, Ned: elimination and dispersal steps, Ped: probability of elimination, C i : the run-length unit i.e., the chemotactic step size during each run or tumble . Step 2. Elimination-dispersal loop: l l 1. Step 3. Reproduction loop: k k 1. Step 4. Chemotaxis loop: j j 1. Substep 4.1. For i 1 1, 2, . . . , S, take a chemotactic step for bacteria i as follows. Substep 4.2. Compute fitness function, J i, j, k, l . Substep 4.3. Let Jlast J i, j, k, l to save this value since we may find better value via a run. Substep 4.4. Tumble: generate a random vector Δ i ∈ R with each element Δm i , m 1, 2, . . . , S, a random number on −1, 1 . Substep 4.5. Move: let 2.1 . This results in a step of size C i in the direction of the tumble for bacteria i. Substep 4.6. Compute J i, j 1, k, l with θ j 1, k, l . 6 Abstract and Applied Analysis Substep 4.7. Swim: i let m 0 counter for swim length ii while m < Ns if not climbed down too long a let m m 1 b if J i, j 1, k, l < Jlast, let Jlast J i, j 1, k, l , Then, another step of size C i in this same directionwill be taken as 2.1 and use the new generated θ j 1, k, l to compute the new J i, j 1, k, l c else let m Ns. Substep 4.8. Go to next bacterium i 1 : if i / S go to b to process the next bacteria. Step 5. If j < Nc, go to Step 3. In this case, continue chemotaxis since the life of the bacteria is not over. Step 6. Reproduction. Substep 6.1. For the given k and l, and for each i 1, 2, . . . , S, let