Today we're going to revisit the genetic algorithm. If you haven't read Genetic Algorithms Part 1 yet, I strongly recommend reading that now. This article will skip over the fundamental concepts covered in part 1 -- so if you're new to genetic algorithms you'll definitely want to start there.

Just looking for the example?

The Problem

Given the weight and value of each element, which combination should you take to maximize the total value without exceeding the weight limit?

You're a scientist that has recently been framed for murder by an evil company. Before you flee the lab you have an opportunity to steal 1,000 pounds (or kilograms!) of pure elements from the chemical warehouse; your plan is to later sell them and survive off of the earnings.This is called the knapsack problem . The one above is a one-dimensional problem, meaning the only constraint is weight. We could complicate matters by also considering volume, but we need to start somewhere. Note that in our version of the problem only one piece of each element is available, and each piece has a fixed weight. There are some knapsack problems where you can take unlimited platinum or up to 3 pieces of gold or something like that, but here we only have one of each available to us.

Why is this problem tough to solve? We'll be using 118 elements. The brute-force approach would require that we test 2118 or 3.3 * 1035 different combinations of elements.

Greedy Algorithm

A quick benchmark we'll use for our solution is called the "greedy" solution. The greedy algorithm grabs the most valued items and puts them into the knapsack until it can't fit any more.

Sometimes this works great. Sometimes it doesn't. Imagine that there's a piece of gold in the warehouse that's valued at $1,000 but weighs 600 pounds. And there's also a piece of cadmium that has a value of $950 but only weighs 300 pounds, and there are a bunch of other elements that have a pretty high value but reasonably light weights. The greedy algorithm still tosses gold in there, and all that precious available weight is taken up by something that's not really worth it.

The "naive" greedy algorithm for our dataset will give us a total value of $3649 with a total weight of 998 pounds.

You may at this point be thinking "why don't we just figure out value per pound for each element and use that?" Sure, that works too! It will, in fact, work way better than the above.

Using that approach, the "weighted" greedy algorithm gives us a total value of $4901 and a total weight of 969.

So those are our numbers to beat: we should expect to handily beat $3,649, and we'll be happy if we also beat $4,901.

Why does the greedy algorithm work well for this type of problem? Because the greedy algorithm is solving for the "highest value per unit weight" and that's very close to what we want to fill our knapsack with. However, the greedy algorithm will not perform well in instances where there's a large range of weights and values. That is, the greedy algorithm will perform better if the range of values (and/or weights) of the elements is between $1 - $100, but will perform worse if the range is $1 - $500.

Respond in the comments: why would the greedy algorithm perform worse with a larger range of weights and values?

Our GA might lose to the greedy algorithm from time to time, but that's ok. The GA will continue to perform well as complexity increases, but the greedy algorithm will not.

So we're still tackling a pretty simplified and contrived problem here, but it's certainly more complex and useful than our "Hello, World!" from the last article. Let's get started.

Key Differences From "Hello World"

We can't use elements (genes) more than once, while "Hello, World!" can (must!) use letters more than once.

In the "Hello, World!" example, we knew that the string needed to be 13 characters long. Here we don't know how many elements to take.

We don't know the highest possible value ("fitness" or "score") for this problem. It could be $4,901 like the greedy algorithm guessed, or it could be $10,000, or $23,304

Representing the Chromosome

There are some major differences between our problem and the previous "Hello, World!":The "Hello, World!" algorithm represented the chromosome as a string. Mutating involved randomly changing a letter. Mating (or crossover, as it's really called) involved splicing two strings together at the halfway point. We need to do things a little differently here.

It turns out that representing our solution requires a little more finesse than "Hello, World!". Since we don't know how many elements to choose, we can't use a fixed length string. (Just the first of our problems!)

Instead, I propose we use a "bitmask" of sorts. We don't have to use an actual bitmask, but my proposal is to use a representation of all of the available elements, and set each "present" or not.

Our chromosome could look like this:

Helium: present Hydrogen: not present Lithium: not present

And et cetera for all 118 elements. Or if you want to go the bitmask route:

10000011000001000100000010000010010010000

Where each bit represents a single element and the value of the bit indicates whether that element is in the knapsack or not.

Additionally, if we were to allow more than one of each element, the representation could look like so:

Helium: 0 Hydrogen: 4 Lithium: 2

What would not work well is the following:

In Knapsack: Helium, Lithium, Lead, Tin

The above makes mating more difficult. You can make it work, but it feels like you'd be jumping through hoops to pull it off. Structure feels better for this problem.

We have a specific difficulty with mating: we need to make sure that even after mating and mutation we still only have at most one of each element in the list. Using the bitmask approach will help us to that end, but that's a common pitfall when trying the list approach above.

The difficulty of making sure something happens only once in a chromosome is a common one. If you're familiar with the traveling salesman problem, it's easy to imagine a scenario where you mate two solutions and end up visiting the same city twice. That city appeared in the first half of the first parent and the second half of the second parent -- therefore appearing nowhere in one child but twice in the other child.

Overweight Populations

For this problem we're going to keep track of three properties of the population: weight, value, and score.

Score is the same thing as value, with one difference: score accounts for the population being overweight.

You may be tempted to throw out overweight populations completely. It's a natural instinct because overweight solutions are not acceptable solutions! But there's a good, practical reason we don't want to throw out overweight chromosomes: there will be some slightly overweight (1,001 pounds) chromosomes that have very high values and just need to be "tweaked" a little to bring them within the weight range.

There could be a lot of potential in some of the overweight chromosomes. Rather than killing them, we'll penalize them just enough that they still get to reproduce, but are unlikely to be the #1 pick. That's what we'll use "score" for. If you're underweight, then your score is just your total value. If you're overweight, however, we'll penalize you 50 points for every pound over weight. Feel free to play with this number.

Evolutionarily, this "encourages" promising chromosomes to drop some weight. All they need is a little tweaking. There's no use in throwing out a potentially strong candidate!

We'll cover the specifics of mating and mutation and the important of death (really called "elitism") as we're looking at the code.

The Code

"Hydrogen":{ "weight":389, "value":400}, "Helium":{ "weight":309, "value":380}, "Lithium":{ "weight":339, "value":424}, "Beryllium":{ "weight":405, "value":387}, "Boron":{ "weight":12, "value":174},

First, let's take a look at the data set. I wrote a simple PHP script that generates random weights and values (ranged 1 - 500) for each element and outputs the set as JSON. It looks something like this:

And so on.

We then define three quick and easy helper functions:

function length(obj) { var length = 0; for (var i in obj) length++; return length; } function clone(obj) { obj = JSON.parse(JSON.stringify(obj)); return obj; } function pickRandomProperty(obj) { var result; var count = 0; for (var prop in obj) if (Math.random() < 1 / ++count) result = prop; return result; }

The 'length' property only exists for javascript arrays, so we create a length() function that works for objects.

We create a clone function that ensures our element objects aren't passed by reference.

Finally, we create a function that picks a random property of an object. This is an analog to PHP's 'array_rand' function, which returns a random array key.

Chromosome Functions

```javascript var Chromosome = function(members) { this.members = members; for (var element in this.members) { if (typeof this.members[element]['active'] == 'undefined') { this.members[element]['active'] = Math.round( Math.random() ); } } this.mutate(); this.calcScore(); };

Chromosome.prototype.weight = 0;

Chromosome.prototype.value = 0;

Chromosome.prototype.members = [];

Chromosome.prototype.maxWeight = 1000;

Chromosome.prototype.mutationRate = 0.7;

Chromosome.prototype.score = 0;

The chromosome constructor takes an object of 'members'. In this case, we'll either be passing our original list of elements data when we're creating a brand new chromosome, <em>or</em> we'll be passing in the results of a mating operation. The constructor randomly activates elements if the 'active' property isn't yet defined. The end result is that this will create a random chromosome if we're creating one from scratch, and it'll leave a pre-configured chromosome alone. The prototype also specifies some defaults. The mutationRate property is the chance that a chromosome will mutate. ```javascript Chromosome.prototype.mutate = function() { if (Math.random() > this.mutationRate) return false; var element = pickRandomProperty(this.members); this.members[element]['active'] = Number(! this.members[element]['active']); };

The mutate method is most similar to the "Hello, World!" example. If the chromosome is to mutate then we simply pick an element at random and toggle its 'active' property. I cast to Number here. It would have been more semantic to cast the Math.random() in the constructor to Boolean. I'll ignore this, as I've already pasted all the code into this post.

Chromosome.prototype.calcScore = function() { if (this.score) return this.score; this.value = 0; this.weight = 0; this.score = 0; for (var element in this.members) { if (this.members[element]['active']) { this.value += this.members[element]['value']; this.weight += this.members[element]['weight']; } } this.score = this.value; if (this.weight > this.maxWeight) { this.score -= (this.weight - this.maxWeight) * 50; } return this.score; };

The calcScore method starts with a tiny performance optimization: if we've calculated the score already, just serve the cached score -- it's just a nice way to not have to worry about at which point in the chromosome life cycle to calculate the score.

We then look through the elements and add up the value and weights for the active ones. We then apply a penalty of 50 points per overweight pound.

Chromosome.prototype.mateWith = function(other) { var child1 = {}; var child2 = {}; var pivot = Math.round( Math.random() * (length(this.members) - 1) ); var i = 0; for (var element in elements) { if (i < pivot) { child1[element] = clone(this.members[element]); child2[element] = clone(other.members[element]); } else { child2[element] = clone(this.members[element]); child1[element] = clone(other.members[element]); } i++; } child1 = new Chromosome(child1); child2 = new Chromosome(child2); return [child1, child2]; };

In the "Hello, World!" example we picked the center point as the pivot point when mating two chromosomes; in this example we pick a random point instead.

This adds a little more randomness to the system and can help avoid local optima.

Once we've picked our pivot point we create two children by splicing the parents at the pivot and combining. We then use our chromosome constructor to generate chromosome objects and return them.

The Population

var Population = function(elements, size) { if ( ! size ) size = 20; this.elements = elements; this.size = size; this.fill(); }; Population.prototype.elitism = 0.2; Population.prototype.chromosomes = []; Population.prototype.size = 100; Population.prototype.elements = false;

The population constructor is straightforward: we give it the master list of elements and the desired population size. We also define the 'elitism' parameter; this is the percentage of chromosomes that will survive from one generation to the next.

Population.prototype.fill = function() { while (this.chromosomes.length < this.size) { if (this.chromosomes.length < this.size / 3) { this.chromosomes.push( new Chromosome( clone(this.elements) ) ); } else { this.mate(); } } };

We use the fill method to initialize the population; we'll also use it to fill the population after killing the weakest chromosomes. A little bit of logic determines whether we should create random chromosomes or fill the population through mating instead. If our population size is 20, the first 6 chromosomes will be random and the remaining will be generated by mating. If the population size ever dips below 30% (perhaps due to death/elitism), new random chromosomes will be created until the population is diverse enough to create babies through mating.

Yes, 'this.chromosomes.length' in a while loop is bad form. If you expect to use a large population size -- or want this to be highly optimized -- do this the right way and cache the length.

Population.prototype.sort = function() { this.chromosomes.sort(function(a, b) { return b.calcScore() - a.calcScore(); }); }; Population.prototype.kill = function() { var target = Math.floor( this.elitism * this.chromosomes.length ); while (this.chromosomes.length > target) { this.chromosomes.pop(); } };

The sort function above is just a helper; note that we use the calcScore method instead of accessing the 'score' property directly. If the score hasn't been calculated by this point, it will be now; if the score was already calculated we just use calcScore as an accessor.

After sorting, the kill method removes the weakest chromosomes from the bottom of the list by popping them until we reach our elitism value.

Population.prototype.mate = function() { var key1 = pickRandomProperty(this.chromosomes); var key2 = key1; while (key2 == key1) { key2 = pickRandomProperty(this.chromosomes); } var children = this.chromosomes[key1].mateWith(this.chromosomes[key2]); this.chromosomes = this.chromosomes.concat(children); };

The mate method is always called after the kill method, so the only chromosomes allowed to reproduce are the elite ones in the population (in our example, the best 20%). Rather than mating only the best two chromosomes (like we did in "Hello, World!"), we pick any two random chromosomes to mate -- with the exception that we won't mate a chromosome with itself.

Again, this serves to add a little more randomness to the system, and will avoid stasis if the top two chromosomes remain the same for many generations -- we see that happen sometimes in the "Hello, World!" example and fix it here.

Population.prototype.generation = function(log) { this.sort(); this.kill(); this.mate(); this.fill(); this.sort(); };

Then we define a "generation". The generation starts by sorting the chromosomes in terms of score. We then kill the weakest members.

Then we do something a little intriguing: we call mate() once and then call fill(), which we know will also call mate(). The reason we call mate() explicitly is as a bit of insurance: if the elitism parameter is less than 0.3 we want to mate at least once before potentially "polluting" the population with random members. This really depends on the elitism value; if you keep it over 0.3 you don't have to call mate() explicitly, because fill() will do that for you. But if you have elitism = 0.2 like we do, then we want to run at least one mating routine that involves only the elite, and not the new random chromosomes we introduce with fill().

Finally, we sort once more at the end of the generation. We could just as easily leave this part out, but it's nice to see the chromosomes in order after every generation if you're debugging.

Running and Stopping

I briefly introduced this idea in the "Hello, World!" example: we don't know the best possible score in this problem, therefore we won't know when to stop.

The technique (one of many) that we'll use is to stop when you've had 100 (or 1,000 or 1,000,000) generations with no improvement. We'll just call this the "threshold" or the "stop threshold".

Population.prototype.run = function(threshold, noImprovement, lastScore, i) { if ( ! threshold ) threshold = 1000; if ( ! noImprovement ) noImprovement = 0; if ( ! lastScore ) lastScore = false; if ( ! i ) i = 0; if (noImprovement < threshold) { lastScore = this.chromosomes[0].calcScore(); this.generation(); if (lastScore >= this.chromosomes[0].calcScore()) { noImprovement++; } else { noImprovement = 0; } i++; if (i % 10 == 0) this.display(i, noImprovement); var scope = this; setTimeout(function() { scope.run(threshold, noImprovement, lastScore, i) }, 1); return false; } this.display(i, noImprovement); };

The run method is an iterative function. It doesn't need to be. The only reason it is in this example is because writing to the DOM in the middle of a fast-moving loop doesn't work -- the DOM doesn't update until execution is done. It's just a javascript thing.

To get around the DOM limitation, we use a short setTimeout and have the run method call itself iteratively until we're done. In general, however, this function could just use a while loop instead of calling itself -- but in that case you'd either need to use console.log or just wait until the loop is done to watch the results.

Other than that DOM messiness, the run method is straightforward. We compare last generation's best score to this generation's best. If there was improvement, then we reset the 'noImprovement' counter. If we have noImprovement equal to our stop threshold, we stop.

Not shown is a simple display method used to print the results to a table on the page. We only call it every 10 generations, and then once again when we're done.

In Action

Our JSFiddle is below. You're given a slider that lets you set the stop tolerance and a big "Go" button. You then get to watch the population's evolution.

Even with a 10-generation stop threshold, the GA consistently beats the naive greedy algorithm, though this is to be expected.

At a 50-generation stop threshold, the GA also consistently beats the weighted greedy algorithm. This is a happy result. There are some datasets where the GA will never beat the greedy algorithm, and other datasets where the greedy algorithm doesn't perform well at all. This seems like an in-between. We don't crush the greedy algorithm, but we do outperform it by a significant margin.

The best score I've observed so far is 5944, with a weight of 992. Please let me know in the comments below if you find a better score, and I'll post the updates here.

Update: Sylvain Zimmer below found a solution with a score of 5968 and weight 998.

As always, feel free to fork and play with this example. I would encourage you to experiment with different values of population size, elitism, and mutation rate and observe the results.