ncxiao / nsga-ii

Multiobjective optimization using nondominated sorting-based genetic algorithm II (NSGA-II)

Multiobjective optimization using nondominated sorting-based genetic algorithm (NSGA-II)

This repository includes a notebook that shows a Python implementation of NSGA-II developed by Deb et al in 2002. The complete implementation is available in a Python module called nsga2.py. We show the use of this module below. For more detail about genetic algorithms (GAs) in general, please see another repo called genetic-algorithms.

Files

• README.md: this file.
• NSGA-II.ipynb is the Jupyter notebook detailing how the algorithms are implemented.
• NSGA-II.html is the HTML export of the notebook.
• nsga2.py: the Python file that contains the implementation of both NSGA-II and a binary coded genetic algorithm.
• Figure_1.png is a screenshot of running the application (see below).

Usage

We use an example problem called SCH in the original paper by Deb et al. to demonstrate how to use the nsga2 module. To use it, it is necessary to develop a new evaluator function since it is problem-specific. Here, we have one decision variable, $x$, that ranges from -4 to 4, and two objective functions defined as

$f_1(x) = x^2$
$f_2(x) = (x-2)^2$

And below is how we can apply nsga to this problem.

>>> from nsga2 import *
>>> def evaluator(vars): # this should reflect the equations of the problem
...     # need 1 variable
...     x = vars[0]
...     f1 = x**2
...     f2 = (x-2)**2
...     return [f1, f2]
... 
>>> N = 100             # the number of solutions in a population
>>> T = 10              # the number of generations
>>> numvar = 1          # the number of decision variables in a binary string
>>> varlen = 10         # the length of each variable in bits
>>> bounds = [[-4, 4]]  # the bounds of each variable
>>>
>>> crossover_prob = 0.9
>>> mutation_prob = 1/numvar/varlen
>>> P = nsga2_x(N, T, crossover_prob, mutation_prob, numvar, varlen, bounds, decoder, evaluator)

The following shows how to plot the results.

>>> import matplotlib.pyplot as plt
>>> 
>>> P['10K random'] = [solution(numvar, varlen, bounds=bounds, decoder=decoder, evaluator=evaluator) for _ in range(10000)]
>>> for i in P['10K random']:
...     i.evaluate()
... 
>>> toplot = ['10K random', 0, T//3, 2*T//3, T]
>>> for i in toplot:
...     x1 = [(p.obj[0], p.obj[1]) for p in P[i]]
...     x1.sort()
...     marker = 'o'
...     if i == toplot[-1]:
...         marker = 'o-'
...     plt.plot([i[0] for i in x1], [i[1] for i in x1], marker, label=f'generation {i}')
>>> plt.xlabel('f1')
>>> plt.ylabel('f2')
>>> plt.legend(loc='upper left')
>>> plt.show()

Below is an example of the plot.