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Clever Algorithms: NatureInspired Programming RecipesBy Jason Brownlee PhD.Bacterial Foraging Optimization AlgorithmBacterial Foraging Optimization Algorithm, BFOA, Bacterial Foraging Optimization, BFO. TaxonomyThe Bacterial Foraging Optimization Algorithm belongs to the field of Bacteria Optimization Algorithms and Swarm Optimization, and more broadly to the fields of Computational Intelligence and Metaheuristics. It is related to other Bacteria Optimization Algorithms such as the Bacteria Chemotaxis Algorithm [Muller2002], and other Swarm Intelligence algorithms such as Ant Colony Optimization and Particle Swarm Optimization. There have been many extensions of the approach that attempt to hybridize the algorithm with other Computational Intelligence algorithms and Metaheuristics such as Particle Swarm Optimization, Genetic Algorithm, and Tabu Search. InspirationThe Bacterial Foraging Optimization Algorithm is inspired by the group foraging behavior of bacteria such as E.coli and M.xanthus. Specifically, the BFOA is inspired by the chemotaxis behavior of bacteria that will perceive chemical gradients in the environment (such as nutrients) and move toward or away from specific signals. MetaphorBacteria perceive the direction to food based on the gradients of chemicals in their environment. Similarly, bacteria secrete attracting and repelling chemicals into the environment and can perceive each other in a similar way. Using locomotion mechanisms (such as flagella) bacteria can move around in their environment, sometimes moving chaotically (tumbling and spinning), and other times moving in a directed manner that may be referred to as swimming. Bacterial cells are treated like agents in an environment, using their perception of food and other cells as motivation to move, and stochastic tumbling and swimming like movement to relocate. Depending on the cellcell interactions, cells may swarm a food source, and/or may aggressively repel or ignore each other. StrategyThe information processing strategy of the algorithm is to allow cells to stochastically and collectively swarm toward optima. This is achieved through a series of three processes on a population of simulated cells: 1) 'Chemotaxis' where the cost of cells is derated by the proximity to other cells and cells move along the manipulated cost surface one at a time (the majority of the work of the algorithm), 2) 'Reproduction' where only those cells that performed well over their lifetime may contribute to the next generation, and 3) 'Eliminationdispersal' where cells are discarded and new random samples are inserted with a low probability. ProcedureAlgorithm (below) provides a pseudocode listing of the Bacterial Foraging Optimization Algorithm for minimizing a cost function. Algorithm (below) provides the pseudocode listing for the chemotaxis and swing behaviour of the BFOA algorithm. A bacteria cost is derated by its interaction with other cells. This interaction function ($g()$) is calculated as follows: $g(cell_k) = \sum_{i=1}^S\bigg[d_{attr}\times exp\bigg(w_{attr}\times \sum_{m=1}^P (cell_m^k  other_m^i)^2 \bigg) \bigg] + \sum_{i=1}^S\bigg[h_{repel}\times exp\bigg(w_{repel}\times \sum_{m=1}^P cell_m^k  other_m^i)^2 \bigg) \bigg]$where $cell_k$ is a given cell, $d_{attr}$ and $w_{attr}$ are attraction coefficients, $h_{repel}$ and $w_{repel}$ are repulsion coefficients, $S$ is the number of cells in the population, $P$ is the number of dimensions on a given cells position vector. The remaining parameters of the algorithm are as follows $Cells_{num}$ is the number of cells maintained in the population, $N_{ed}$ is the number of eliminationdispersal steps, $N_{re}$ is the number of reproduction steps, $N_{c}$ is the number of chemotaxis steps, $N_{s}$ is the number of swim steps for a given cell, $Step_{size}$ is a random direction vector with the same number of dimensions as the problem space, and each value $\in [1,1]$, and $P_{ed}$ is the probability of a cell being subjected to elimination and dispersal. Input :
$Problem_{size}$, $Cells_{num}$, $N_{ed}$, $N_{re}$, $N_{c}$, $N_{s}$, $Step_{size}$, $d_{attract}$, $w_{attract}$, $h_{repellant}$, $w_{repellant}$, $P_{ed}$
Output :
$Cell_{best}$
Population $\leftarrow$ InitializePopulation ($Cells_{num}$, $Problem_{size}$)For ($l=0$ To $N_{ed}$)For ($k=0$ To $N_{re}$)For ($j=0$ To $N_{c}$)ChemotaxisAndSwim (Population , $Problem_{size}$, $Cells_{num}$, $N_{s}$, $Step_{size}$, $d_{attract}$, $w_{attract}$, $h_{repellant}$, $w_{repellant}$)For (Cell $\in$ Population )If (Cost (Cell ) $\leq$ Cost ($Cell_{best}$))$Cell_{best}$ $\leftarrow$ Cell End End End SortByCellHealth (Population )Selected $\leftarrow$ SelectByCellHealth (Population , $\frac{Cells_{num}}{2}$)Population $\leftarrow$ Selected Population $\leftarrow$ Selected End For (Cell $\in$ Population )If (Rand () $\leq$ $P_{ed}$)Cell $\leftarrow$ CreateCellAtRandomLocation ()End End End Return ($Cell_{best}$)Input :
Population , $Problem_{size}$, $Cells_{num}$, $N_{s}$, $Step_{size}$, $d_{attract}$, $w_{attract}$, $h_{repellant}$, $w_{repellant}$
For (Cell $\in$ Population )$Cell_{fitness}$ $\leftarrow$ Cost (Cell ) + Interaction (Cell , Population , $d_{attract}$, $w_{attract}$, $h_{repellant}$, $w_{repellant}$)$Cell_{health}$ $\leftarrow$ $Cell_{fitness}$ $Cell'$ $\leftarrow$ $\emptyset$ For ($i=0$ To $N_{s}$)RandomStepDirection $\leftarrow$ CreateStep ($Problem_{size}$)$Cell'$ $\leftarrow$ TakeStep (RandomStepDirection , $Step_{size}$)${Cell'}_{fitness}$ $\leftarrow$ Cost ($Cell'$) + Interaction ($Cell'$, Population , $d_{attract}$, $w_{attract}$, $h_{repellant}$, $w_{repellant}$)If (${Cell'}_{fitness}$ > $Cell_{fitness}$)$i \leftarrow$ $N_{s}$ Else Cell $\leftarrow$ $Cell'$$Cell_{health}$ $\leftarrow$ $Cell_{health}$ + ${Cell'}_{fitness}$ End End End Heuristics
Code ListingListing (below) provides an example of the Bacterial Foraging Optimization Algorithm implemented in the Ruby Programming Language. The demonstration problem is an instance of a continuous function optimization that seeks $\min f(x)$ where $f=\sum_{i=1}^n x_{i}^2$, $5.0\leq x_i \leq 5.0$ and $n=2$. The optimal solution for this basin function is $(v_0,\ldots,v_{n1})=0.0$. The algorithm is an implementation based on the description on the seminal work [Passino2002]. The parameters for cellcell interactions (attraction and repulsion) were taken from the paper, and the various loop parameters were taken from the 'Swarming Effects' example. def objective_function(vector) return vector.inject(0.0) {sum, x sum + (x ** 2.0)} end def random_vector(minmax) return Array.new(minmax.size) do i minmax[i][0] + ((minmax[i][1]  minmax[i][0]) * rand()) end end def generate_random_direction(problem_size) bounds = Array.new(problem_size){[1.0,1.0]} return random_vector(bounds) end def compute_cell_interaction(cell, cells, d, w) sum = 0.0 cells.each do other diff = 0.0 cell[:vector].each_index do i diff += (cell[:vector][i]  other[:vector][i])**2.0 end sum += d * Math.exp(w * diff) end return sum end def attract_repel(cell, cells, d_attr, w_attr, h_rep, w_rep) attract = compute_cell_interaction(cell, cells, d_attr, w_attr) repel = compute_cell_interaction(cell, cells, h_rep, w_rep) return attract + repel end def evaluate(cell, cells, d_attr, w_attr, h_rep, w_rep) cell[:cost] = objective_function(cell[:vector]) cell[:inter] = attract_repel(cell, cells, d_attr, w_attr, h_rep, w_rep) cell[:fitness] = cell[:cost] + cell[:inter] end def tumble_cell(search_space, cell, step_size) step = generate_random_direction(search_space.size) vector = Array.new(search_space.size) vector.each_index do i vector[i] = cell[:vector][i] + step_size * step[i] vector[i] = search_space[i][0] if vector[i] < search_space[i][0] vector[i] = search_space[i][1] if vector[i] > search_space[i][1] end return {:vector=>vector} end def chemotaxis(cells, search_space, chem_steps, swim_length, step_size, d_attr, w_attr, h_rep, w_rep) best = nil chem_steps.times do j moved_cells = [] cells.each_with_index do cell, i sum_nutrients = 0.0 evaluate(cell, cells, d_attr, w_attr, h_rep, w_rep) best = cell if best.nil? or cell[:cost] < best[:cost] sum_nutrients += cell[:fitness] swim_length.times do m new_cell = tumble_cell(search_space, cell, step_size) evaluate(new_cell, cells, d_attr, w_attr, h_rep, w_rep) best = cell if cell[:cost] < best[:cost] break if new_cell[:fitness] > cell[:fitness] cell = new_cell sum_nutrients += cell[:fitness] end cell[:sum_nutrients] = sum_nutrients moved_cells << cell end puts " >> chemo=#{j}, f=#{best[:fitness]}, cost=#{best[:cost]}" cells = moved_cells end return [best, cells] end def search(search_space, pop_size, elim_disp_steps, repro_steps, chem_steps, swim_length, step_size, d_attr, w_attr, h_rep, w_rep, p_eliminate) cells = Array.new(pop_size) { {:vector=>random_vector(search_space)} } best = nil elim_disp_steps.times do l repro_steps.times do k c_best, cells = chemotaxis(cells, search_space, chem_steps, swim_length, step_size, d_attr, w_attr, h_rep, w_rep) best = c_best if best.nil? or c_best[:cost] < best[:cost] puts " > best fitness=#{best[:fitness]}, cost=#{best[:cost]}" cells.sort{x,y x[:sum_nutrients]<=>y[:sum_nutrients]} cells = cells.first(pop_size/2) + cells.first(pop_size/2) end cells.each do cell if rand() <= p_eliminate cell[:vector] = random_vector(search_space) end end end return best end if __FILE__ == $0 # problem configuration problem_size = 2 search_space = Array.new(problem_size) {i [5, 5]} # algorithm configuration pop_size = 50 step_size = 0.1 # Ci elim_disp_steps = 1 # Ned repro_steps = 4 # Nre chem_steps = 70 # Nc swim_length = 4 # Ns p_eliminate = 0.25 # Ped d_attr = 0.1 w_attr = 0.2 h_rep = d_attr w_rep = 10 # execute the algorithm best = search(search_space, pop_size, elim_disp_steps, repro_steps, chem_steps, swim_length, step_size, d_attr, w_attr, h_rep, w_rep, p_eliminate) puts "done! Solution: c=#{best[:cost]}, v=#{best[:vector].inspect}" end Download: bfoa.rb. Unit test available in the github project
ReferencesPrimary SourcesEarly work by Liu and Passino considered models of chemotaxis as optimization for both E.coli and M.xanthus which were applied to continuous function optimization [Liu2002]. This work was consolidated by Passino who presented the Bacterial Foraging Optimization Algorithm that included a detailed presentation of the algorithm, heuristics for configuration, and demonstration applications and behavior dynamics [Passino2002]. Learn MoreA detailed summary of social foraging and the BFOA is provided in the book by Passino [Passino2005]. Passino provides a followup review of the background models of chemotaxis as optimization and describes the equations of the Bacterial Foraging Optimization Algorithm in detail in a Journal article [Passino2010]. Das et al. present the algorithm and its inspiration, and go on to provide an in depth analysis the dynamics of chemotaxis using simplified mathematical models [Das2009]. Bibliography

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