A Fast Elitist Non-Dominated Sorting Genetic
Algorithm for Multi-Objective Optimization: NSGA-II
Kalyanmoy Deb, Samir Agrawal, Amrit Pratap, and T Meyarivan
Kanpur Genetic Algorithms Laboratory (KanGAL)
Indian Institute of Technology Kanpur
Kanpur, PIN 208 016, India
deb,samira,apratap,mary @iitk.ac.in
http://www.iitk.ac.in/kangal
KanGAL Report No. 200001
Abstract. Multi-objective evolutionary algorithms which use non-dominated sort-
ing and sharing have been mainly criticized for their (i) З´СЖїµ computational
complexity (where С is the number of objectives and Ж is the population size),
(ii) non-elitism approach, and (iii) the need for specifying a sharing parameter. In
this paper, we suggest a non-dominated sorting based multi-objective evolution-
ary algorithm (we called it the Non-dominated Sorting GA-II or NSGA-II) which
alleviates all the above three difficulties. Specifically, a fast non-dominated sort-
ing approach with З´СЖѕµ computational complexity is presented. Second, a
selection operator is presented which creates a mating pool by combining the
parent and child populations and selecting the best (with respect to fitness and
spread) Ж solutions. Simulation results on five difficult test problems show that
the proposed NSGA-II is able to find much better spread of solutions in all prob-
lems compared to PAES—another elitist multi-objective EA which pays special
attention towards creating a diverse Pareto-optimal front. Because of NSGA-II’s
low computational requirements, elitist approach, and parameter-less sharing ap-
proach, NSGA-II should find increasing applications in the years to come.
1 Introduction
Over the past decade, a number of multi-objective evolutionary algorithms (MOEAs)
have been suggested [9,3,5,13]. The primary reason for this is their ability to find
multiple Pareto-optimal solutions in one single run. Since the principal reason why a
problem has a multi-objective formulation is because it is not possible to have a single
solution which simultaneously optimizes all objectives, an algorithm that gives a large
number of alternative solutions lying on or near the Pareto-optimal front is of great
practical value.
The Non-dominated Sorting Genetic Algorithm (NSGA) proposed in Srinivas and
Deb [9] was one of the first such evolutionary algorithms. Over the years, the main
criticism of the NSGA approach have been as follows:
High computational complexity of non-dominated sorting: The non-dominated sort-
ing algorithm in use uptil now is З´СЖ¿µ which in case of large population sizes
2
Deb, Agrawal, Pratap, and Meyarivan
is very expensive, especially since the population needs to be sorted in every gen-
eration.
Lack of elitism: Recent results [12, 8] show clearly that elitism can speed up the per-
formance of the GA significantly, also it helps to prevent the loss of good solutions
once they have been found.
Need for specifying the sharing parameter × Ц
: Traditional mechanisms of insur-
ing diversity in a population so as to get a wide variety of equivalent solutions have
relied heavily on the concept of sharing. The main problem with sharing is that it
requires the specification of a sharing parameter (× Ц). Though there has been
some work on dynamic sizing of the sharing parameter [4], a parameterless diver-
sity preservation mechanism is desirable.
In this paper, we address all of these issues and propose a much improved version of
NSGA which we call NSGA-II. From the simulation results on a number of difficult test
problems, we find that NSGA-II has a better spread in its optimized solutions than PAES
[6]—another elitist multi-objective evolutionary algorithm. These results encourage the
application of NSGA-II to more complex and real-world multi-objective optimization
problems.
2 Elitist Multi-Objective Evolutionary Algorithms
In the study of Zitzler, Deb, and Theile [12], it was clearly shown that elitism helps in
achieving better convergence in MOEAs. Among the existing elitist MOEAs, Zitzler
and Thiele’s [13] strength Pareto EA (SPEA), Knowles and Corne’s Pareto-archived
evolution strategy (PAES) [6], and Rudolph’s [8] elitist GA are well known.
Zitzler and Thiele [13] suggested an elitist multi-criterion EA with the concept of
non-domination in their strength Pareto EA (SPEA). They suggested maintaining an
external population at every generation storing all non-dominated solutions discovered
so far beginning from the initial population. This external population participates in
genetic operations. At each generation, a combined population with the external and
the current population is first constructed. All non-dominated solutions in the com-
bined population are assigned a fitness based on the number of solutions they dominate
and dominated solutions are assigned fitness worse than the worst fitness of any non-
dominated solution. This assignment of fitness makes sure that the search is directed
towards the non-dominated solutions. A deterministic clustering technique is used to
ensure diversity among non-dominated solutions. Although the implementation sug-
gested in [13] is З´СЖ¿µ, with proper book-keeping the complexity of SPEA can be
reduced to З´СЖ¾µ. An important aspect of this study and subsequent studies [12, 11]
is that they clearly show the importance of introducing elitism in evolutionary multi-
criterion optimization.
Knowles and Corne [6] suggested a simple MOEA using an evolution strategy (ES).
In their Pareto-archived ES (PAES) with one parent and one child, the child is compared
with respect to the parent. If the child dominates the parent, the child is accepted as the
next parent and the iteration continues. On the other hand, if the parent dominates the
child, the child is discarded and a new mutated solution (a new child) is found. However,
if the child and the parent do not dominate each other, the choice between the child and
Fast Elitist NSGA
3
the parent considers the second objective of keeping diversity among obtained solutions.
To maintain diversity, an archive of non-dominated solutions is maintained. The child
is compared with the archive to check if it dominates any member of the archive. If yes,
the child is accepted as the new parent and the dominated solution is eliminated from
the archive. If the child does not dominate any member of the archive, both parent and
child are checked for their
nearness with the solutions of the archive. If the child resides
in a least crowded region in the parameter space among the members of the archive, it
is accepted as a parent and a copy of added to the archive. Later, they suggested a multi-
parent PAES with similar principles as above. Authors have calculated the worst case
complexity of PAES for Ж evaluations as З´ СЖµ, where is the archive length. Since
the archive size is usually chosen proportional to the population size Ж, the overall
complexity of the algorithm is З´СЖ¾µ.
Rudolph [8] suggested, but did not simulate, a simple elitist multi-objective EA
based on a systematic comparison of individuals from parent and offspring popula-
tions. The non-dominated solutions of the offspring population are compared with that
of parent solutions to form an overall non-dominated set of solutions, which becomes
the parent population of the next iteration. If the size of this set is not greater than the
desired population size, other individuals from the offspring population are included.
With this strategy, he has been able to prove the convergence of this algorithm to the
Pareto-optimal front. Although this is an important achievement in its own right, the al-
gorithm lacks motivation for the second task of maintaining diversity of Pareto-optimal
solutions. An explicit diversity preserving mechanism must be added to make it more
usable in practice. Since the determinism of the first non-dominated front is З´СЖ¾
µ,
the overall complexity of Rudolph’s algorithm is also З´СЖ¾µ.
3 Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II)
The non-dominated sorting GA (NSGA) proposed by Srinivas and Deb in 1994 has
been applied to various problems [10, 7]. However as mentioned earlier there have been
a number of criticisms of the NSGA. In this section, we modify the NSGA approach in
order to alleviate all the above difficulties. We begin by presenting a number of different
modules that form part of NSGA-II.
3.1 A fast non-dominated sorting approach
In order to sort a population of size Ж according to the level of non-domination, each
solution must be compared with every other solution in the population to find if it is
dominated. This requires З´СЖµ comparisons for each solution, where С is the num-
ber of objectives. When this process is continued to find the members of the first non-
dominated class for all population members, the total complexity is З´СЖ¾µ. At this
stage, all individuals in the first non-dominated front are found. In order to find the
individuals in the next front, the solutions of the first front are temporarily discounted
and the above procedure is repeated. In the worst case, the task of finding of the second
front also requires З´СЖ¾µ computations. The procedure is repeated to find the sub-
sequent fronts. As can be seen the worst case (when there exists only one solution in
4
Deb, Agrawal, Pratap, and Meyarivan
each front) complexity of this algorithm is З´СЖ¿µ. In the following we describe a fast
non-dominated sorting approach which will require at most З´СЖ¾µ computations.
First, for each solution we calculate two entities: (i) Т , the number of solutions
which dominate the solution , and (ii) Л , a set of solutions which the solution domi-
nates. The calculation of these two entities requires З´СЖ¾µ comparisons. We identify
all those points which have Т
¼ and put them in a list ½. We call ½ the current
front. Now, for each solution in the current front we visit each member () in its set Л
and reduce its Т count by one. In doing so, if for any member the count becomes
zero, we put it in a separate list А. When all members of the current front have been
checked, we declare the members in the list ½ as members of the first front. We then
continue this process using the newly identified front А as our current front.
Each such iteration requires З´Жµ computations. This process continues till all
fronts are identified. Since at most there can be Ж fronts, the worst case complexity of
this loop is З´Ж¾µ. The overall complexity of the algorithm now is З´СЖ¾µ ·З´Ж¾µ
or З´СЖ¾µ.
It is worth mentioning here that although the computational burden has reduced
from З´СЖ¿µ to З´СЖ¾µ by performing systematic book-keeping, the storage has
increased from З´Жµ to З´Ж¾µ in the worst case.
The fast non-dominated sorting procedure which when applied on a population И
returns a list of the non-dominated fronts .
fast-nondominated-sort(И)
for each Ф ¾ И
for each Х ¾ И
if ´Ф
Хµ then
if Ф dominates Х then
ЛФ
ЛФ
Х
include Х in ЛФ
else if ´Х
Фµ then
if Ф is dominated by Х then
ТФ
ТФ · ½
increment ТФ
if ТФ
¼ then
if no solution dominates Ф then
½
½
Ф
Ф is a member of the first front
½
while
А
for each Ф ¾
for each member Ф in
for each Х ¾ ЛФ
modify each member from the set ЛФ
ТХ
ТХ ½
decrement ТХ by one
if ТХ
¼ then А А Х
if ТХ is zero, Х is a member of a list А
· ½
А
current front is formed with all members of А
3.2 Density Estimation
To get an estimate of the density of solutions surrounding a particular point in the pop-
ulation we take the average distance of the two points on either side of this point along
each of the objectives. This quantity
ЧШ Т
serves as an estimate of the size of the
Fast Elitist NSGA
5
largest cuboid enclosing the point without including any other point in the population
(we call this the
crowding distance). In Figure 1, the crowding distance of the -th so-
lution in its front (marked with solid circles) is the average side-length of the cuboid
(shown with a dashed box). The following algorithm is used to calculate the crowding
Cuboid
f
f
1
2
i
i-1
i+1
0
l
Fig. 1. The crowding distance calculation is shown.
distance of each point in the set Б :
crowding-distance-assignment(Б)
Р
Б
number of solutions in Б
for each , set Б ℄ ЧШ Т
¼
initialize distance
for each objective С
Б = sort(Б Сµ
sort using each objective value
Б Ѕ℄ ЧШ Т
= Б Р℄ ЧШ Т
= ½
so that boundary points are always selected
for
¾ to ´Р Ѕµ
for all other points
Б ℄ ЧШ Т
= Б ℄ ЧШ Т
+ ´Б · Ѕ℄ С Б Ѕ℄ Сµ
Here Б ℄ С refers to the С-th objective function value of the -th individual in
the set Б. The complexity of this procedure is governed by the sorting algorithm. In
the worst case (when all solutions are in one front), the sorting requires З´СЖ РУ Жµ
computations.
3.3 Crowded Comparison Operator
The crowded comparison operator ( Т) guides the selection process at the various
stages of the algorithm towards a uniformly spread out Pareto-optimal front. Let us
assume that every individual in the population has two attributes.
1. Non-domination rank (Ц Т)
2. Local crowding distance ( ЧШ Т
)
We now define a partial order Т as :
Т
if (Ц Т
Ц Т) or ((Ц Т = Ц Т) and ( ЧШ Т
ЧШ Т
) )
6
Deb, Agrawal, Pratap, and Meyarivan
That is, between two solutions with differing non-domination ranks we prefer the
point with the lower rank. Otherwise, if both the points belong to the same front then
we prefer the point which is located in a region with lesser number of points (the size
of the cuboid inclosing it is larger).
3.4 The Main Loop
Initially, a random parent population И¼ is created. The population is sorted based on
the non-domination. Each solution is assigned a fitness equal to its non-domination
level (1 is the best level). Thus, minimization of fitness is assumed. Binary tournament
selection, recombination, and mutation operators are used to create a child population
Й¼ of size Ж. From the first generation onward, the procedure is different. The elitism
procedure for Ш
½ and for a particular generation is shown in the following:
КШ
ИШ
ЙШ
combine parent and children population
fast-nondominated-sort(КШ)
´ ½
¾
µ, all non-dominated
fronts of КШ
until ИШ·Ѕ
Ж
till the parent population is filled
crowding-distance-assignment( ) calculate crowding distance in
ИШ·Ѕ
ИШ·Ѕ
include -th non-dominated front in the parent pop
Sort(ИШ·Ѕ
Т)
sort in descending order using Т
ИШ·Ѕ
ИШ·Ѕ ¼ Ж℄
choose the first N elements of ИШ·Ѕ
ЙШ·Ѕ = make-new-pop(ИШ·Ѕ)
use selection,crossover and mutation to create
Ш
Ш · ½
a new population ЙШ·Ѕ
First, a combined population КШ
ИШ
ЙШ is formed. The population КШ will be
of size ¾Ж. Then, the population КШ is sorted according to non-domination. The new
parent population ИШ·Ѕ is formed by adding solutions from the first front till the size
exceeds Ж. Thereafter, the solutions of the last accepted front are sorted according to
Т and the first Ж points are picked. This is how we construct the population ИШ·Ѕ of
size Ж. This population of size Ж is now used for selection, crossover and mutation to
create a new population ЙШ·Ѕ of size Ж. It is important to note that we use a binary
tournament selection operator but the selection criterion is now based on the niched
comparison operator Т.
Let us now look at the complexity of one iteration of the entire algorithm. The
basic operations being performed and the worst case complexities associated with are
as follows:
1. Non-dominated sort is З´СЖ¾µ,
2. Crowding distance assignment is З´СЖ РУ Жµ, and
3. Sort on Т is ЗґѕЖ РУ ґѕЖµµ.
As can be seen, the overall complexity of the above algorithm is З´СЖ¾µ.
The diversity among non-dominated solutions is introduced by using the crowding
comparison procedure which is used in the tournament selection and during the popula-
tion reduction phase. Since solutions compete with their crowding distance (a measure
Fast Elitist NSGA
7
of density of solutions in the neighborhood), no extra niching parameter (such as × Ц
needed in the NSGA) is required here. Although the crowding distance is calculated in
the objective function space, it can also be implemented in the parameter space, if so
desired [1].
It is interesting to note here the connection of this algorithm with the algorithm
proposed by Rudolph [8]. Since the non-dominated front finding algorithm used in
Rudolph’s algorithm is З´СЖ¾µ for each front, Rudolph control’s the complexity of
his algorithm by working with just the first few fronts in the parent and the child pop-
ulations and treating the rest of the individuals in the child population at par. With the
availability of a fast non-domination sorting algorithm we can now afford to combine
the parent and child populations and do a complete sort to identify all the fronts and
allocate fitness accordingly.
4 Results
We compare NSGA-II with PAES on five test problems (minimization of both objec-
tives):
MOP2:
½ґЬµ ½ ЬФ ИТ
½ Ь ½ФТ
¾
Ь½ Ь¾ Ь¿
¾ґЬµ ½ ЬФ ИТ
½ Ь ·
½ФТ
¾
(1)
MOP3:
½ґЬµ
¢
Ѕ·ґ ½ ½µ
¾
· ´ ¾ ¾µ
¾£
¾ґЬµ
¢
´Ь · їµ¾ · ´Э · Ѕµ¾£
(2)
where
½
¼ × Т ½ ¾
УЧ ½ · × Т ¾ ½
УЧ ¾
¾
½ × Т ½
УЧ ½ · ¾ × Т ¾ ¼
УЧ ¾
½
¼ × Т Ь ¾
УЧЬ · × Т Э ½
УЧЭ
¾
½ × Т Ь
УЧ Ь · ¾ × Т Э ¼
УЧЭ
MOP4:
´
½ґЬµ
ИТ ½
½
Ѕј ЬФ ¼ ¾
Х
Ь¾ · Ь¾
·Ѕ
Ь½ Ь¾ Ь¿
¾ґЬµ
ИТ
½
Ь ¼ · × ТґЬ µ¿¡
(3)
EC4:
´
½ґЬµ Ь½
¼
Ь½
½
¾ґЬµ
½
Х
ЬЅ
Ь¾
ЬЅј
(4)
where
ґЬµ Ѕ·
Ѕј
¾
Ь¾ Ѕј
УЧґ Ь µ
¡
EC6:
½ґЬµ ½ ЬФґ Ь½µ × Т ´ Ь½µ ¼
Ь
½
½
Ѕј
¾ґЬµ
½ ´ ½
µ¾¡
(5)
where
ґЬµ Ѕ·
Ѕј
¾
Ь
јѕ
Since the diversity among optimized solutions is an important matter in multi-
objective optimization, we devise a measure based on the consecutive distances among
8
Deb, Agrawal, Pratap, and Meyarivan
the solutions of the best non-dominated front in the final population. The obtained set
of the first non-dominated solutions are compared with a uniform distribution and the
deviation is computed as follows:
¡
½
½
½
(6)
In order to ensure that this calculation takes into account the spread of solutions in the
entire region of the true front, we include the boundary solutions in the non-dominated
front ½. For discrete Pareto-optimal fronts, we calculate a weighted average of the
above metric for each of the discrete regions. In the above equation,
is the Euclidean
distance between two consecutive solutions in the first non-dominated front of the final
population in the objective function space. The parameter
is the average of these
distances.
The deviation measure ¡ of these consecutive distances is then calculated for each
run. An average of these deviations over 10 runs is calculated as the measure (¡) for
comparing different algorithms. Thus, it is clear that an algorithm having a smaller ¡
is better, in terms of its ability to widely spread solutions in the obtained front.
For all test problems and with NSGA-II, we use a population of size 100, a crossover
probability of 0.8, a mutation probability of ½ Т (where Т is the number of variables).
We run NSGA-II for 250 generations. The variables are treated as real numbers and
the simulated binary crossover (SBX) [2] and the real-parameter mutation operator are
used. For the (1+1)-PAES, we have used an archive size of 100 and depth of 4 [6].
A mutation probability of ¼ јЅ is used. In order to make the comparisons fair, we have
used 25,000 iterations in PAES, so that total number of function evaluations in NSGA-II
and in PAES are the same.
Table 1 shows the deviation from an ideal (uniform) spread (¡) and its variance
in 10 independent runs obtained using NSGA-II and PAES. We show two columns
for each test problem. The first column presents the ¡ value of 10 runs and the second
column shows its variance. It is clear from the table that in all five test problems NSGA-
II has found much smaller ¡, meaning that NSGA-II is able to find a distribution of
solutions closer to a uniform distribution along the non-dominated front. The variance
columns suggest that the obtained ¡ values are consistent in all 10 runs.
Table 1. Comparison of mean and variance of deviation measure ¡obtained using NSGA-II and
PAES
Algorithm
MOP2
MOP3
MOP4
EC4
EC6
NSGA-II 0.361 0.00068 0.445 0.00043 0.387 0.00164 0.383 0.00099 0.365 0.01613
PAES 1.609 0.00671 1.341 0.00495 1.087 0.00687 1.563 0.05723 1.195 0.05151
In order to have a better understanding of how these algorithms are able to spread so-
lutions over the non-dominated front, we present the entire non-dominated front found
Fast Elitist NSGA
9
by NSGA-II and PAES in two of the above five test problems. Figures 2 and 3 show
that NSGA-II is able to find a much better distribution than PAES on MOP4.
In EC4, converging to the global Pareto-optimal front is a difficult task. As reported
elsewhere [11], SPEA converged to a front with
¼ in at least one out of five
different runs. With NSGA-II, we find a front with
¿ in one out of five different
-12
-10
-8
-6
-4
-2
0
-20
-19
-18
-17
-16
-15
-14
f_2
f_1
NSGA-II
Fig. 2. Non-dominated solutions obtained us-
ing NSGA-II on MOP4.
-12
-10
-8
-6
-4
-2
0
-20
-19
-18
-17
-16
-15
-14
f_2
f_1
PAES
Fig. 3. Non-dominated solutions obtained us-
ing PAES on MOP4.
runs.
Figure 4 shows the non-dominated solutions obtained using NSGA-II and PAES for
EC6. Once again, it is clear that the NSGA-II is able to better distribute its population
along the obtained front than PAES. It is worth mentioning here that with similar num-
ber of function evaluations, SPEA, as reported in [11], had found only five different
solutions in the non-dominated front.
5 Conclusions
In this paper, we have proposed a computationally fast elitist multi-objective evolution-
ary algorithm based on non-dominated sorting approach. On five difficult test problems
borrowed from the literature, it has been found that the proposed NSGA-II outperforms
PAES—another multi-objective EA with the explicit goal of preserving spread on the
non-dominated front. With the properties of a fast non-dominated sorting procedure, an
elitist strategy, and a parameterless approach, NSGA-II should find increasing attention
and applications in the near future.
Acknowledgements
Authors acknowledge the support provided by All India Council for Technical Educa-
tion, India during the course of this study.
10
Deb, Agrawal, Pratap, and Meyarivan
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f_2
f_1
Pareto-Optimal Front
NSGA-II
PAES
Fig. 4. Obtained non-dominated solutions with NSGA-II and PAES on EC6.
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