Solving the 0-1 Knapsack Problem with a Genetic Algorithm in Ruby
The Knapsack Problem is an NP combinatorial optimization problem in which items that have both value and weight are placed into a "knapsack" with a weight limit. The goal is to maximize the value of the items while keeping the total weight of the items below the weight limit threshold. A maximized solution can be approximated using a genetic algorithm.
Genetic algorithms are biologically inspired, using natural selection, reproduction, mutation, and other elements of evolution to obtain solutions. They are often used to solve optimization problems and model certain systems.
class Item attr_accessor :weight, :value def initialize(weight, value) @weight = weight @value = value end end
Item class has both a weight and a value, which will be set randomly within a range.
class Knapsack attr_accessor :chromosome, :total_weight, :total_value def initialize(chromosome) @chromosome = chromosome total_weight = 0.0 total_value = 0.0 end end
Knapsack class has a chromosome, or array of zeroes and ones representing whether a specific item is included in the knapsack, along with a total weight and a total value which will be calculated and stored based on its chromosome.
require_relative('item.rb') require_relative('knapsack.rb') if ARGV.length > 0 num_items = ARGV.chomp.to_i num_knapsacks = ARGV.chomp.to_i num_generations = ARGV.chomp.to_i verbose = ARGV if verbose == "true" verbose = true else verbose = false end end
Knapsack classes and accepts user input for the number of items, the number of knapsacks in the population, the maximum number of generations, and whether the script should run in verbose mode or silently. Note that there isn't really any validation here, so the script assumes correct user input.
items =  knapsacks =  generation = 1 # Generate random items num_items.times do ran_weight = (rand * 10).round(2) ran_value = (rand * 100).round(2) items << Item.new(ran_weight, ran_value) end # Generate initial knapsacks num_knapsacks.times do ran_items =  num_items.times do if rand < 0.1 ran_items << 1 else ran_items << 0 end end knapsacks << Knapsack.new(ran_items) end
The group of items is created with a pseudorandom weights between
10.0 and pseudorandom values between
100.0. then the initial generation of knapsacks is created each with a pseudorandom chromosome where each item has a 10% chance of being turned on.
# Main loop until generation > num_generations puts "==================================" if verbose puts "Begin generation: " + generation.to_s if verbose puts "==================================" if verbose sum_value = 0.0 best_value = 0.0 best_knapsack = 0 max_weight = 50.0 # Calculate value and weight knapsacks.each_with_index do |knapsack, index| total_weight = 0.0 total_value = 0.0 knapsack.chromosome.each_with_index do |gene, gene_index| if gene === 1 total_weight += items[gene_index].weight total_value += items[gene_index].value end end if total_weight <= max_weight if total_value > best_value best_value = total_value best_knapsack = index end else total_value = 0.0 end knapsack.total_weight = total_weight knapsack.total_value = total_value sum_value += total_value end
As the main loop begins, we calculate the total weight and total value of each knapsack in order to determine its fitness. If a knapsack is over the weight limit, its value becomes
0.0 which effectively will remove it from the population. Note that the best overall knapsack is stored, and the
sum_value of all knapsacks is calculated as well.
# Use Roulette wheel algorithm to proportionately create next generation new_generation =  elitist = Knapsack.new(knapsacks[best_knapsack].chromosome.clone) puts 'Elitist: ' + best_knapsack.to_s if verbose p elitist if verbose (num_knapsacks-1).times do rnd = rand(); rnd_sum = 0.0 rnd_selected = 0 knapsacks.each do |knapsack| rel_value = knapsack.total_value / sum_value rnd_sum += rel_value if rnd_sum > rnd break else rnd_selected += 1 end end new_generation << Knapsack.new(knapsacks[rnd_selected].chromosome.clone) end # Replace old generation with new knapsacks =  knapsacks = new_generation generation += 1
Here a Roulette Wheel style algorithm is used to create a new generation. Essentially a random number is chosen between
1.0, then each knapsack is looped through and their relative value is summed. The knapsack whose relative value is the one that puts the sum over the random value becomes a parent in the next generation. This has the effect of each knapsack occupying its own "slice" of a Roulette wheel, with its size proportionate to its share of value in the population.
# Randomly select two knapsacks rnd_knap_1 = (0...num_knapsacks-1).to_a.sample rnd_knap_2 = rnd_knap_1 until (rnd_knap_2 != rnd_knap_1) rnd_knap_2 = (0...num_knapsacks-1).to_a.sample end # Perform crossover split_point = (0...num_items-1).to_a.sample front_1 = knapsacks[rnd_knap_1].chromosome[0, split_point] front_2 = knapsacks[rnd_knap_2].chromosome[0, split_point] back_1 = knapsacks[rnd_knap_1].chromosome[split_point, num_items-1] back_2 = knapsacks[rnd_knap_2].chromosome[split_point, num_items-1] new_chr_1 = front_1 + back_2 new_chr_2 = front_2 + back_1 new_1 = Knapsack.new(new_chr_1) new_2 = Knapsack.new(new_chr_2) knapsacks[rnd_knap_1] = new_1 knapsacks[rnd_knap_2] = new_2
Now it is time to expand the search space, so we randomly choose two knapsacks from the new generation and perform crossover. A randomly determined point in the chromosome separates each chromosome into a head and a tail. The heads and tails are swapped between the two chromosomes, which represents a large jump in the search space.
# Perform mutation knapsacks.each do |knapsack| knapsack.chromosome.each_with_index do |gene, index| if rand < 0.01 puts 'Successful mutation at gene: ' + index.to_s if verbose gene == 0 ? gene = 1 : gene = 0 knapsack.chromosome[index] = gene end end end
Mutation is performed to make smaller "tweaks" in the search space. For each knapsack and each gene in its chromosome there is a 1% chance that it will flip, either going from
knapsacks << elitist puts 'Last knapsack:' if verbose p knapsacks[num_knapsacks-1] if verbose puts 'Best value:' p best_value end
Finally, the elitist knapsack is added back into the next population (after the crossover and mutations so that it is not effected) and the main loop ends with some useful output.
The solution provided by the genetic algorithm is not guaranteed to be an absolute maximum, but if it runs long enough to eventually cover the majority of the solution space then it will certainly provide a very good solution. There are lots of things that you can tweak in this algorithm as well, such as the percentage chance of mutation or the method used for crossover. You can remove the elitism if desired, or change it to include the best five. If you make any significant changes and want to share them, don't hesitate to contact me!
The full code is publicly available on GitHub, and may be more up to date than the code here although I will try to keep this post updated.