Implementation of NSGA-II algorithm in form of a python library.

This implementation can be used to solve multivariate (more than one dimensions) multi-objective optimization problem. The number of objectives and dimensions are not limited. Some critical operators are chosen as: Binary Tournament Selection, Simulated Binary Crossover and Polynomial Mutation. Note that we didn't start everything from scratch but modified the source code from wreszelewski/nsga2. We are very thankful to Wojciech Reszelewski and Kamil Mielnik - authors of this original version. The following items are modified:

• Fix the crowding distance formula.
• Modify some parts of the code to apply to any number of objectives and dimensions.
• Modify the selection operator to Tournament Selection.
• Change the crossover operator to Simulated Binary Crossover.
• Change the mutation operator to Polynomial Mutation.

1. Class Problem

   Defined in problem.py.

   Using to define a multi-objective problem.

   Arguments:

   • objectives: A list of functions, representing the objective functions.
   • num_of_variables: An integer, representing the number of variables.
   • variables_range: A list of tuples of two elements, representing the lower and upper bound of each respective variables.
   • same_range: A boolean argument, default = False. If true, the range of all variables are the same (this case variables_range has only one element), otherwise each variable has its own range.
   • expand: A boolean argument, default = True. If true, input of functions are treated as list of respective variables (e.x. f(x,y,z)), otherwise they are treated as a vector (e.x. f([x1, x2, x3])).

2. Class Evolution

   Defined in evolution.py.

   Using to run NSGA-II.

   1. Arguments:
      • problem: An object of class Problem.
      • num_of_generations: An integer, default = 1000, representing the number of generations.
      • num_of_individuals: An integer, default = 100, representing the number of individuals, a.k.a the population size.
      • num_of_tour_particips: An integer, default = 2, representing the number of participants in tournament selection operator.
      • tournament_prob: A real number, default = 0.9, representing the probability used in tournament selection.
      • crossover_param: An integer, default = 2, representing the parameter used in simulated binary crossover.
      • mutation_param: An integer, default = 2, representing the paramenter used in polynomial mutation.
   2. Methods:
      • evolve:
        • Arguments: None.
        • Return: List of the best individuals in the last generation.

Examples about SCH problem and KUR problem are showed in sch.py and kur.py.

In addition, the results of some popular multi-objective problems is demonstrated as belows:

• SCH
• KUR
• ZDT1
• ZDT4
• VIENNET

• Pham Ngo Gia Bao, Ho Chi Minh University of Technology
• Tram Loi Quan, Ho Chi Minh University of Technology
• A/Prof. Quan Thanh Tho, Ho Chi Minh University of Technology (advisor)
• A/Prof. Akhil Garg, Shantou University (advisor)

We are very thankful to A/Prof. Tho and A/Prof. Akhil for helping and guiding us to finish this work.