The Learning Curve

learning_curve.rb

=begin
  One of my favorite equations from business school: the learning curve.  
  Human experience has shown that every time our experience with a single 
  task doubles, our efficiency increases by 20%.  Our learning curve is
  actually around 20%, with a 95% confidence rate around 10% and 30%.

  This was the equation that founded BCG as they measured everything 
  related to the then-new aviation industry.  It's been used to measure
  all sorts of human tasks and has proven pretty resilient for people of
  all kinds of backgrounds and aptitudes.  

  One interesting extension to this little class would be to calculate
  the actual learning curve, given two measurements, but I did this little
  exercise to deal with insomnia, and I'm finally feeling sleepy again.
=end
class LearningCurve
  
  RATE = 0.2
  
  attr_accessor :initial_cost
  
  def initialize(initial_cost=1)
    @initial_cost = initial_cost
  end
  
  attr_writer :n
  def n
    @n ||= 1
  end
  
  def times_doubled
    Math.log2(n)
  end
  
  def unit_cost
    ((1 - RATE) ** times_doubled) * initial_cost
  end
  
  def assert_initial_cost_assuming_unit_cost_of(unit_cost)
    @initial_cost = unit_cost / ((1 - RATE) ** times_doubled)
  end
  
  # Had to break out the algebra for this one.
  # unit_cost = ((1 -rate)^log base 2(n)) * initial_cost
  # unit_cost / initial_cost = (1-rate)^log base 2(n)
  # log base(1-rate)(unit_cost / initial_cost) = log base 2(n)
  # 2^(log base(1-rate)(unit_cost / initial_cost)) = n
  # since log base(1-rate)(unit_cost / initial_cost) = log(unit_cost / initial_cost) / log(1-rate)
  # I get this final form, rounded up (since the experience must be finished)
  def experience_needed_for_unit_cost_under(unit_cost)
    (2**(Math.log(unit_cost.to_f / initial_cost) / Math.log(1 - RATE))).ceil
  end
end

learning_curve_spec.rb

require File.expand_path('../spec_helper', __FILE__)

describe LearningCurve do

  before do
    @curve = LearningCurve.new
  end
  
  it "should return 1.0 unit_cost and 0.0 times_doubled at n == 1" do
    @curve.n = 1
    @curve.unit_cost.should be_within(1.0e-05).of(1.0)
    @curve.times_doubled.should eql(0.0)
  end
  
  it "should return 0.8 unit_cost and 1.0 times_doubled at n == 2" do
    @curve.n = 2
    @curve.unit_cost.should be_within(1.0e-05).of(0.8)
    @curve.times_doubled.should eql(1.0)
  end

  it "should return 0.64 unit_cost and 2.0 times_doubled at n == 4" do
    @curve.n = 4
    @curve.unit_cost.should be_within(1.0e-05).of(0.64)
    @curve.times_doubled.should eql(2.0)
  end

  it "should return 0.512 unit_cost and 3.0 times_doubled at n == 8" do
    @curve.n = 8
    @curve.unit_cost.should be_within(1.0e-05).of(0.512)
    @curve.times_doubled.should eql(3.0)
  end
  
  it "should take into account the initial cost" do
    initial_cost = 15
    unit_rate = 0.64
    @curve = LearningCurve.new(initial_cost)
    @curve.n = 4
    @curve.unit_cost.should be_within(1.0e-05).of(initial_cost * unit_rate)
    @curve.times_doubled.should eql(2.0)
  end
  
  it "should be able to recompute the initial_cost" do
    @curve.n = 10
    @curve.assert_initial_cost_assuming_unit_cost_of(2)
    # A little algebra to solve for the initial cost
    expected = 2 / ((1 - LearningCurve::RATE) ** Math.log2(10))
    @curve.initial_cost.should be_within(1.0e-05).of(expected)
  end
  
  it "should be able to calculate the number of experiences needed to have a specific unit cost" do
    @curve.n = 10
    initial_cost = 2 / ((1 - LearningCurve::RATE) ** Math.log2(10))
    @curve.initial_cost = initial_cost
    # Had to break out the algebra for this one.
    expected = (2**(Math.log(1.0 / initial_cost) / Math.log(1 - LearningCurve::RATE))).ceil
    @curve.experience_needed_for_unit_cost_under(1).should be_within(1.0e-05).of(expected)
  end

end

usage.rb

@curve = LearningCurve.new
@curve.n = 10 # Assuming 10 bowties have been made
@curve.assert_initial_cost_assuming_unit_cost_of(2) # Gives us 4.197184791733325
@curve.experience_needed_for_unit_cost_under(1) # Gives us 87, or 77 more bowties