# Solving the 0-1 Knapsack Problem with a Genetic Algorithm in Ruby

## The Knapsack Problem

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 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.

## Solution

### item.rb

1class Item 2 3 attr_accessor :weight, :value 4 5 def initialize(weight, value) 6 @weight = weight 7 @value = value 8 end 910end

The `Item`

class as both a weight and a value, which will be set randomly within a range.

### knapsack.rb

1class Knapsack 2 3 attr_accessor :chromosome, :total_weight, :total_value 4 5 def initialize(chromosome) 6 @chromosome = chromosome 7 total_weight = 0.0 8 total_value = 0.0 9 end1011end

The `Knapsack`

class has a chromosome, or array of `0`

s and `1`

s 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.

### genetic.rb

1require_relative('item.rb') 2require_relative('knapsack.rb') 3 4if ARGV.length > 0 5 num_items = ARGV[0].chomp.to_i 6 num_knapsacks = ARGV[1].chomp.to_i 7 num_generations = ARGV[2].chomp.to_i 8 verbose = ARGV[3] 9 if verbose == "true"10verbose = true 11 else 12 verbose = false 13 end 14end

Loads the `Item`

and `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.

1items = [] 2knapsacks = [] 3generation = 1 4 5# Generate random items 6num_items.times do 7 ran_weight = (rand * 10).round(2) 8 ran_value = (rand * 100).round(2) 9 items << Item.new(ran_weight, ran_value)10end 11 12# Generate initial knapsacks 13num_knapsacks.times do 14 ran_items = [] 15 num_items.times do 16 if rand < 0.1 17 ran_items << 1 18 else 19 ran_items << 020end 21 end 22 knapsacks << Knapsack.new(ran_items) 23end

The group of items is created with a pseudorandom weights between `0.0`

and `10.0`

and pseudorandom values between `0.0`

and `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.

1# Main loop 2until generation > num_generations 3 4 puts "==================================" if verbose 5 puts "Begin generation: " + generation.to_s if verbose 6 puts "==================================" if verbose 7 8 sum_value = 0.0 9 best_value = 0.010best_knapsack = 0 11 max_weight = 50.0 12 13 # Calculate value and weight 14 knapsacks.each_with_index do |knapsack, index| 15 total_weight = 0.0 16 total_value = 0.0 17 knapsack.chromosome.each_with_index do |gene, gene_index| 18 if gene === 1 19 total_weight += items[gene_index].weight20total_value += items[gene_index].value 21 end 22 end 23 if total_weight <= max_weight 24 if total_value > best_value 25 best_value = total_value 26 best_knapsack = index 27 end 28 else 29 total_value = 0.030end 31 knapsack.total_weight = total_weight 32 knapsack.total_value = total_value 33 sum_value += total_value 34 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.

1 # Use Roulette wheel algorithm to proportionately create next generation 2 new_generation = [] 3 elitist = Knapsack.new(knapsacks[best_knapsack].chromosome.clone) 4 puts 'Elitist: ' + best_knapsack.to_s if verbose 5 p elitist if verbose 6 (num_knapsacks-1).times do 7 rnd = rand(); 8 rnd_sum = 0.0 9 rnd_selected = 0;1011 knapsacks.each do |knapsack| 12 rel_value = knapsack.total_value / sum_value 13 rnd_sum += rel_value 14 if rnd_sum > rnd 15 break 16 else 17 rnd_selected += 1 18 end 19 end20new_generation << Knapsack.new(knapsacks[rnd_selected].chromosome.clone) 21 end 22 23 # Replace old generation with new 24 knapsacks = [] 25 knapsacks = new_generation 26 generation += 1

Here a Roulette Wheel style algorithm is used to create a new generation. Essentially a random number is chosen between `0.0`

and `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.

1 # Randomly select two knapsacks 2 rnd_knap_1 = (0...num_knapsacks-1).to_a.sample 3 rnd_knap_2 = rnd_knap_1 4 until (rnd_knap_2 != rnd_knap_1) 5 rnd_knap_2 = (0...num_knapsacks-1).to_a.sample 6 end 7 8 # Perform crossover 9 split_point = (0...num_items-1).to_a.sample10front_1 = knapsacks[rnd_knap_1].chromosome[0, split_point]; 11 front_2 = knapsacks[rnd_knap_2].chromosome[0, split_point]; 12 back_1 = knapsacks[rnd_knap_1].chromosome[split_point, num_items-1]; 13 back_2 = knapsacks[rnd_knap_2].chromosome[split_point, num_items-1]; 14 new_chr_1 = front_1 + back_2 15 new_chr_2 = front_2 + back_1 16 new_1 = Knapsack.new(new_chr_1) 17 new_2 = Knapsack.new(new_chr_2) 18 19 knapsacks[rnd_knap_1] = new_120knapsacks[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.

1 # Perform mutation 2 knapsacks.each do |knapsack| 3 knapsack.chromosome.each_with_index do |gene, index| 4 if rand < 0.01 5 puts 'Successful mutation at gene: ' + index.to_s if verbose 6 gene == 0 ? gene = 1 : gene = 0 7 knapsack.chromosome[index] = gene 8 end 9 end10end

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 `0`

to `1`

or `1`

to `0`

.

1 knapsacks << elitist 2 3 puts 'Last knapsack:' if verbose 4 p knapsacks[num_knapsacks-1] if verbose 5 puts 'Best value:' 6 p best_value 7 8end

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.

## Conclusion

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.