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non_domination_sort_mod.m
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non_domination_sort_mod.m
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function f = non_domination_sort_mod(x, M, V)
%% function f = non_domination_sort_mod(x, M, V)
% This function sort the current popultion based on non-domination. All the
% individuals in the first front are given a rank of 1, the second front
% individuals are assigned rank 2 and so on. After assigning the rank the
% crowding in each front is calculated.
% Copyright (c) 2009, Aravind Seshadri
% All rights reserved.
%
% Redistribution and use in source and binary forms, with or without
% modification, are permitted provided that the following conditions are
% met:
%
% * Redistributions of source code must retain the above copyright
% notice, this list of conditions and the following disclaimer.
% * Redistributions in binary form must reproduce the above copyright
% notice, this list of conditions and the following disclaimer in
% the documentation and/or other materials provided with the distribution
%
% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
% AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
% IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
% ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
% LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
% CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
% SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
% INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
% CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
% ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
% POSSIBILITY OF SUCH DAMAGE.
[N, m] = size(x);
clear m
% Initialize the front number to 1.
front = 1;
% There is nothing to this assignment, used only to manipulate easily in
% MATLAB.
F(front).f = [];
individual = [];
%% Non-Dominated sort.
% The initialized population is sorted based on non-domination. The fast
% sort algorithm [1] is described as below for each
% • for each individual p in main population P do the following
% – Initialize Sp = []. This set would contain all the individuals that is
% being dominated by p.
% – Initialize np = 0. This would be the number of individuals that domi-
% nate p.
% – for each individual q in P
% * if p dominated q then
% · add q to the set Sp i.e. Sp = Sp ? {q}
% * else if q dominates p then
% · increment the domination counter for p i.e. np = np + 1
% – if np = 0 i.e. no individuals dominate p then p belongs to the first
% front; Set rank of individual p to one i.e prank = 1. Update the first
% front set by adding p to front one i.e F1 = F1 ? {p}
% • This is carried out for all the individuals in main population P.
% • Initialize the front counter to one. i = 1
% • following is carried out while the ith front is nonempty i.e. Fi != []
% – Q = []. The set for storing the individuals for (i + 1)th front.
% – for each individual p in front Fi
% * for each individual q in Sp (Sp is the set of individuals
% dominated by p)
% · nq = nq?1, decrement the domination count for individual q.
% · if nq = 0 then none of the individuals in the subsequent
% fronts would dominate q. Hence set qrank = i + 1. Update
% the set Q with individual q i.e. Q = Q ? q.
% – Increment the front counter by one.
% – Now the set Q is the next front and hence Fi = Q.
%
% This algorithm is better than the original NSGA ([2]) since it utilize
% the informatoion about the set that an individual dominate (Sp) and
% number of individuals that dominate the individual (np).
%
for i = 1 : N
% Number of individuals that dominate this individual
individual(i).n = 0;
% Individuals which this individual dominate
individual(i).p = [];
for j = 1 : N
dom_less = 0;
dom_equal = 0;
dom_more = 0;
for k = 1 : M
if (x(i,V + k) < x(j,V + k))
dom_less = dom_less + 1;
elseif (x(i,V + k) == x(j,V + k))
dom_equal = dom_equal + 1;
else
dom_more = dom_more + 1;
end
end
if dom_less == 0 && dom_equal ~= M
individual(i).n = individual(i).n + 1;
elseif dom_more == 0 && dom_equal ~= M
individual(i).p = [individual(i).p j];
end
end
if individual(i).n == 0
x(i,M + V + 1) = 1;
F(front).f = [F(front).f i];
end
end
% Find the subsequent fronts
while ~isempty(F(front).f)
Q = [];
for i = 1 : length(F(front).f)
if ~isempty(individual(F(front).f(i)).p)
for j = 1 : length(individual(F(front).f(i)).p)
individual(individual(F(front).f(i)).p(j)).n = ...
individual(individual(F(front).f(i)).p(j)).n - 1;
if individual(individual(F(front).f(i)).p(j)).n == 0
x(individual(F(front).f(i)).p(j),M + V + 1) = ...
front + 1;
Q = [Q individual(F(front).f(i)).p(j)];
end
end
end
end
front = front + 1;
F(front).f = Q;
end
[temp,index_of_fronts] = sort(x(:,M + V + 1));
for i = 1 : length(index_of_fronts)
sorted_based_on_front(i,:) = x(index_of_fronts(i),:);
end
current_index = 0;
%% Crowding distance
%The crowing distance is calculated as below
% • For each front Fi, n is the number of individuals.
% – initialize the distance to be zero for all the individuals i.e. Fi(dj ) = 0,
% where j corresponds to the jth individual in front Fi.
% – for each objective function m
% * Sort the individuals in front Fi based on objective m i.e. I =
% sort(Fi,m).
% * Assign infinite distance to boundary values for each individual
% in Fi i.e. I(d1) = ? and I(dn) = ?
% * for k = 2 to (n ? 1)
% · I(dk) = I(dk) + (I(k + 1).m ? I(k ? 1).m)/fmax(m) - fmin(m)
% · I(k).m is the value of the mth objective function of the kth
% individual in I
% Find the crowding distance for each individual in each front
for front = 1 : (length(F) - 1)
% objective = [];
distance = 0;
y = [];
previous_index = current_index + 1;
for i = 1 : length(F(front).f)
y(i,:) = sorted_based_on_front(current_index + i,:);
end
current_index = current_index + i;
% Sort each individual based on the objective
sorted_based_on_objective = [];
for i = 1 : M
[sorted_based_on_objective, index_of_objectives] = ...
sort(y(:,V + i));
sorted_based_on_objective = [];
for j = 1 : length(index_of_objectives)
sorted_based_on_objective(j,:) = y(index_of_objectives(j),:);
end
f_max = ...
sorted_based_on_objective(length(index_of_objectives), V + i);
f_min = sorted_based_on_objective(1, V + i);
y(index_of_objectives(length(index_of_objectives)),M + V + 1 + i)...
= Inf;
y(index_of_objectives(1),M + V + 1 + i) = Inf;
for j = 2 : length(index_of_objectives) - 1
next_obj = sorted_based_on_objective(j + 1,V + i);
previous_obj = sorted_based_on_objective(j - 1,V + i);
if (f_max - f_min == 0)
y(index_of_objectives(j),M + V + 1 + i) = Inf;
else
y(index_of_objectives(j),M + V + 1 + i) = ...
(next_obj - previous_obj)/(f_max - f_min);
end
end
end
distance = [];
distance(:,1) = zeros(length(F(front).f),1);
for i = 1 : M
distance(:,1) = distance(:,1) + y(:,M + V + 1 + i);
end
y(:,M + V + 2) = distance;
y = y(:,1 : M + V + 2);
z(previous_index:current_index,:) = y;
end
f = z();
%% References
% [1] *Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, and T. Meyarivan*, |A Fast
% Elitist Multiobjective Genetic Algorithm: NSGA-II|, IEEE Transactions on
% Evolutionary Computation 6 (2002), no. 2, 182 ~ 197.
%
% [2] *N. Srinivas and Kalyanmoy Deb*, |Multiobjective Optimization Using
% Nondominated Sorting in Genetic Algorithms|, Evolutionary Computation 2
% (1994), no. 3, 221 ~ 248.