Linear Subset Size Scheduling for Many-objective Optimization

using NSGA-II based on Pareto Partial Dominance

Makoto Ohki

Field of Technology, Tottori University,

4, 101 Koyama-Minami, Tottori, Tottori 680-8552, Japan

Keywords:

Many-Objective Evolutionary Algorithm, Pareto Partial Dominance, Subset Size Scheduling, NSGA-II,

0/1 Knapsack Problem.

Abstract:

This paper describes techniques for improving the solution search performance of a multi-objective evolution-

ary algorithm (MOEA) in many-objective optimization problems (MaOP). As an MOEA for MaOP, we focus

on NSGA-II based on Pareto partial dominance. NSGA-II based on Pareto partial dominance requires before-

hand a combination list of the number of objective functions to be used for Pareto partial dominance. More-

over, the contents of the combination list greatly inﬂuence the optimization result. We propose to schedule a

parameter r meaning the subset size of objective functions for Pareto partial dominance. This improvement not

only releases users from the schedule of the parameter r but also improves the convergence to Pareto optimal

solutions (P OS) and the diversity of the individual set obtained by the optimization. Moreover, we propose

to kill individuals of the archive set, where the individuals have the same contents as the individual created

by the mating. This improvement excludes individuals with the same contents which obtained relatively good

evaluations. The improved technique and other conventional techniques are applied to a many-objective 0/1

knapsack problem for veriﬁcation of the effectiveness.

1 INTRODUCTION

In the real world, there are many problems with

more than four objectives. Such the multi-objective

optimization problems (MOP) with objective num-

ber of four or more are called many-objective op-

timization problem (MaOP). MaOP is difﬁcult to

solve and is tackled by many researchers (Zitzler

and Thiele, 1998; Zitzler, 1999; Zitzler et al., 2001;

Deb et al., 2000; Deb, 2001; Coello et al., 2007).

Although SPEA2 (Zitzler and Thiele, 1998; Zitzler,

1999; Zitzler et al., 2001) and NSGA-II (Deb et al.,

2000; Deb, 2001) are well known as powerful al-

gorithm for MOPs, they do not work so effectively

for MaOPs (Purshouse and Fleming, 2003; Hughes,

2005; Aguirre and Tanaka, 2007). In this paper, we

handle the case of solving an MaOP by NSGA-II

based algorithm.

When applying NSGA-II or SPEA2 to MaOP, as

the objective number increases, most of the solutions

in the solution set, or population, become a relation

that is not superior or inferior to each other. This re-

lation is called non-dominated (ND) relationship. As

a result, the convergence of the obtained set of Pareto

Optimal Solutions (P OS) to the optimum Pareto front

remarkably decreases. Sato et al. have proposed a

Pareto partial dominance that makes it easier to deter-

mine the superiority/inferiority relationship between

solutions by using several objective functions instead

of all objective functions as an algorithm for such

MaOP (Sato et al., 2010). Since NSGA-II based on

Pareto partial dominance focuses on a relatively small

number of objectives, solutions are easy to decide su-

periority/inferiority even on MaOP, and an effective

selection pressure can be expected.

NSGA-II based on Pareto partial dominance has

the following three problems. The ﬁrst problem is

that a combination list of the number of objects to

be used for Pareto partial dominance must be speci-

ﬁed before the optimization. The second one is that

an appropriate number of selected objectives accord-

ing to the complexity of the problem in undecided.

Moreover, the contents of the combination list greatly

inﬂuence the optimization result. NSGA-II based on

Pareto partial dominance performs ND sorting using

all objective functions at a speciﬁc generation cycle,

and preserves parents as an archive set for the next

generation. This process generates child individuals

Ohki, M.

Linear Subset Size Scheduling for Many-objective Optimization using NSGA-II based on Pareto Partial Dominance.

DOI: 10.5220/0006905402770283

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, pages 277-283

ISBN: 978-989-758-321-6

277

having the same contents as the already existing indi-

vidual in the archive set in some cases. As a result, the

same individuals increases in the ﬁrst front set, which

disturbs effective ranking in the front selection. This

is the third problem. By consideration of these prob-

lems, this paper proposes a simple scheduling tech-

nique of partial objective set used for Pareto partial

dominance and a technique of killing individuals hav-

ing the same contents in preserving the archive set. In

order to verify the effectiveness of the proposed tech-

niques, we examine a many-objective 0/1 knapsack

problem(Zitzler and Thiele, 1998).

2 MANY-OBJECTIVE

OPTIMIZATION PROBLEM

MOP is a problem that optimizes, or maximizes in

this paper, multiple objective functions under several

constraints. Since the objective functions are in a

trade-off relationship with each other, it is not pos-

sible, in general, to obtain the only one solution that

completely satisﬁes all the objective functions. There-

fore, we require to obtain P OS of compromised solu-

tions without superiority or inferiority to each other.

For the objective function vector f consisting m objec-

tive functions, f

, the problem of ﬁnding the variable

vector x that maximizes the value of f

in the feasible

region S in the solution space is deﬁned as follows.



max. f(x) = [ f

(x), f

(x),··· , f

(x)]

s.t. x ∈ S

(1)

When the values of the objective function, f

, of two

solutions x and y satisfy the following relation, we say

that the solution x dominates the solution y.

f(x)  f(y) ,

∀i ∈ M : f

(x) = f

(y) ∧ ∃i ∈ M : f

(y) > f

(y) (2)

where M denotes a set of the indexes for the objec-

tive function, {1,2,..., m}. When there is no solu-

tion dominates a solution x, the solution x is called

non-inferior solution. A set of such the non-inferior

solutions is deﬁned as the following P OS.

P OS = {x ∈ S|¬∃y ∈ S.f(y)  f(x)} (3)

A Pareto front showing the the trade-off relation be-

tween the objective functions is deﬁned as follows.

F ront = {f(x)|x ∈ P OS} (4)

Several effective studies (Zitzler and Thiele, 1998;

Zitzler, 1999; Zitzler et al., 2001; Deb et al., 2000;

Deb, 2001; Coello et al., 2007) have been made on

MOP as deﬁned by Eq.(1). NSGA-II shown in Fig.1

is a powerful multi-objective optimization scheme as

a method proposed on one of these studies. NSGA-

II applies non-dominated sorting (ND sorting) to the

population Q, and the individuals are classiﬁed to sev-

eral ranked subsets, F

,· ··. While not exceed-

ing the size of the parent set P, the individuals of each

subset are moved to the parent set in order. Individ-

uals of the subset that exceeds the size of the parent

set is sorted using crowding distance (CD sorting) and

moved to the parent set. The individuals not selected

are culled. The mating operators generates the child

set C from the parent set P by using the crossover and

mutation operators.

Although NSGA-II effectively solves MOP with

less than four objective functions, as the objective

number m increases, an appropriate P OS could not be

obtained even by those methods containing the con-

ventional NSGA-II. When ND is performed based on

the conventional Pareto dominance using all m objec-

tive functions, as the number of objective function in-

creases, a subset of solutions satisfying Eq.(2) is dif-

ﬁcult to obtain (Tsuchida et al., 2009). Then most

solutions of the population become non-inferior solu-

tions. As a result, the superiority/inferiority relation-

ship between solutions is difﬁcult to determined, and

the selection pressure in the optimization is signiﬁ-

cantly reduced. This paper focuses to NSGA-II with

Pareto partial dominance shown in Fig.2 for solving

MaOP. Pareto partial dominance is based on a con-

cept of partially applying Pareto domination to r ob-

jective functions extracted from all m objective func-

tions. The Pareto partial dominance is deﬁned by the

following formula.

f(x) A f(y) ,

∀i ∈ R ⊂ M : f

(x) = f

(y)

∧∃i ∈ R ⊂ M : f

(y) > f

(y) (5)

where R denotes a set of r indexes selected from M.

Since conditions satisfying Pareto partial dominance

are relaxed as compared with the conventional domi-

nance using all m objective functions, the population

is easier to rank ﬁnely in MaOP with large m.

Figure 1: The conventional NSGA-II.

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

278

In NSGA-II based on Pareto partial dominance,

ﬁrst of all, given r, the number of objective functions

to be considered in the partial ND sorting, a combi-

nation list of

selections is prepared beforehand.

For each I

generations, the combination of the ob-

jective functions to be considered for Pareto partial

dominance is changed, and R

g+1

is selected with per-

forming ND sorting on P

+A using all m objec-

tive functions and copied to the archive set A, where

+ denotes the direct sum.

Figure 2: NSGA-II with Pareto partial dominance.

3 IMPROVEMENT OF NSGA-II

BASED ON PARETO PARTIAL

DOMINANCE

NSGA-II based on Pareto partial dominance has the

following three problems. The ﬁrst problem is that

the subset size of the objective functions to be used

for Pareto partial dominance is required to beforehand

specify before the optimization in a form of a list, or

the combination list. The second one is that an appro-

priate value of the subset size according to the com-

plexity of the problem is unknown. The contents of

the combination list greatly inﬂuence the optimization

result. On the other hand, the creation of the combi-

nation list is a very troublesome and difﬁcult task for

the user. NSGA-II based on Pareto partial dominance

performs ND sorting using all objective functions at

a speciﬁc generation cycle, and preserves parents as

an archive set for the next generation. This process

generates child individuals having the same contents

as the already existing individual in the archive set

in some cases. As a result, the same individuals in-

creases in the ﬁrst front set, which disturbs effective

ranking in the front selection. This is the third prob-

lem. In order to avoid these problems, this paper pro-

poses two improvements. A block chart of the im-

proved NSGA-II based on Pareto partial dominance

is shown in Fig.3.

As the ﬁrst improvement, a subset size scheduling

is proposed for NSGA-II based on Pareto partial dom-

inance. NSGA-II based on Pareto partial dominance

treated in this paper does not use the combination list

for each I

generation cycle. The parameter r is given

by the following equations.

q =

g · m

+ rand int(2B + 1) − B, (6)

r =







B, q < B

q, B 5 q < m

m, q = m

(7)

where m denotes the number of the objective func-

tions, rand int(·) denotes a function returns a random

integer less than the argument, B denotes an integer

parameter larger than 1 and less than m/2, and G de-

notes the end generation. Fig.4 shows the possible

value of the selection number, r.

Figure 3: Improved NSGA-II with Pareto partial domi-

nance.

In NSGA-II based on Pareto partial dominance,

several individuals having the same contents as an in-

dividual already existing in the children, C

, or the

archive set, A, are generated and stored by the mating.

If the optimization proceeds while sustaining such the

individuals having relatively good evaluation, dupli-

cates of them increases within the population. If the

problem to be optimized is relatively simple, individ-

uals with the same content arefrequently generated

during the optimization. The second improvement is

killing such the individuals having the same contents

of an individual already existing in the children, C

and the archive set, A, after the mating. Since the

optimization problem treated in this paper is the max-

imizing problem, by setting the value of all objective

functions of such the individual to 0, the individual

are killed. The same content individual become the

worst individual. After killing the same content indi-

vidual, the mating does not reproduce the individual.

Linear Subset Size Scheduling for Many-objective Optimization using NSGA-II based on Pareto Partial Dominance

279

Figure 4: The selection number, r, probablistically takes a

value on the colored range according to the generation g,

where rand int(·) denotes a function returns a random inte-

ger less than the argument, B denotes an integer parameter

larger than 1 and less than m/2, and G denotes the ﬁnal

generation.

4 MANY-OBJECTIVE 0/1

KNAPSACK PROBLEM

In order to verify the effectiveness of the improved

technique, a many-objective 0/1 knapsack Problem

(MaOKSP) is performed. MaOKSP composed of m

knapsacks and j items. The capacity of the i-th knap-

sack is c

. The weight and the price of the j-th item

are w

i j

and p

i j

respectively in the i-th knapsack. Let

an individual x ∈ 0, 1

be the n dimensional vector

that selects the items. MaOKSP is deﬁned by the fol-

lowing formula.











max. f(x) = [ f

(x), f

(x),· ·· , f

(x)]

s.t.

∑

j=1

i j

· x

5 c

(8)

(x) =

∑

j=1

i j

· x

for i = 1,2, ··· ,m (9)

P OS obtained by the optimization is evaluated by

using Maximum Spread (MS)(Zitzler, 1999) and

Norm(Sato et al., 2006).

MS expresses a measure showing the spread of

P OS distribution. On the other hand, Norm shows

a measure of the convergence to the optimal Pareto

front of P OS. These values are obtained by the fol-

lowing equations.

Norm(P OS) =

P OS

∑

j=1

∑

i=1

)

P OS

(10)

MS(P OS) = (11)

∑

i=1



max

P OS

j=1

) − min

P OS

j=1

)



The conventional NSGA-II, NSGA-II based on Pareto

partial dominance when r = 3, r = 6 and r = 8,

NSGA-II based on Pareto partial dominance in the

case of giving the combination list shown in Table1

and the improved technique are carried out for the

veriﬁcation. The optimization is performed by set-

ting the objective number to m = 4, 6,8, 10 and the

iterative generations to G = 1,000, 000.

Fig.5 shows transition of the number of individu-

als of the ﬁrst-front according to the generation in the

case that m = 10 and I

= 500. In the ﬁgure, “NSGA-

II” denotes the results by the conventional NSGA-II,

“PPD(r=*)” denotes the results by NSGA-II based on

Pareto partial dominance with the constant value of

r = ∗, “PPD(list)” denotes the results by NSGA-II

based on Pareto partial dominace with the combina-

tion list shown in Table1, and “Improved” denotes the

results by the algorithm proposed in this paper. The

conventional NSGA-II and NSGA-II based on Pareto

partial dominance in r = 8 has given large number

of the individuals of the ﬁrst-front set throughout the

optimization. NSGA-II based on Pareto partial domi-

nance with r = 6 has given the number next to them.

At the end of the optimization, the improved tech-

nique has caught up with these values. NSGA-II

based on Pareto partial dominance with the combi-

nation list is also similar.

Fig.6 shows Norm values values after the opti-

mization to the objective number m in the case that

= 500. In any technique, the convergence to

P OS increases as the number of objectives increases.

Although, regarding to the convergence, NSGA-II

based on Pareto partial dominance in the case that

r = 3, NSGA-II based on Pareto partial dominance

with the combination list and the improved technique

have given almost equivalent results, the conventional

NSGA-II has given relatively poor results.

Fig.7 shows MS values values after the optimiza-

tion to the objective number m in the case that I

500. The MS value, or the diversity of P OS, given

by NSGA-II based on Pareto partial dominance in

Table 1: The combination list for NSGA-II based on Pareto

partial dominance.

generation range

0 500k 900k

−500k −900k −1M

m r

4 2 3 4

6 3 5 6

generation rage

0 300k 600k 900k

−300k −600k −900k −1M

m r

8 3 5 7 8

10 3 6 8 10

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

280

Figure 5: Transition of the number of individuals of the

ﬁrst-front according to the generation.

the case that r = 3 decreases as the objective num-

ber increases, whereas it increases with the other three

techniques. In the improved technique, since r in-

creases as the generation progresses, the superior-

ity/inferiority relationship of solutions becomes dif-

ﬁcult to decide by Pareto partial dominance at the end

of the optimization, and many individuals belong to

the ﬁrst-front set. As a result, since most individu-

als of the parents are ranked by the CD sorting, and

it is considered that diversity has increased. NSGA-II

based on Pareto partial dominance with the combi-

nation list has shown diversity equal to or less than

that of the improved technique. The reason that suf-

ﬁcient diversity has not been obtained by NSGA-II

Figure 6: Comparison of Norm values to the object number

based on Pareto partial dominance in the case that

r = 3 is considered as because partial dominance by

using all objectives has not been performed only be-

tween 900,000-1 million generations. Regarding the

diversity of solutions, the conventional NSGA-II has

given the highest value.

Fig.8 shows Norm values to the generation g in

the case that m = 10 and I

= 500. In NSGA-II based

on Pareto partial dominance, the convergence to P OS

tends to decrease as the value of the parameter r in-

creases. In this technique, when r approaches m, the

solutions are hard to dominated by the partial domi-

nance, so a large number of individuals are selected

as the ﬁrst-front set. As a result, sufﬁcient ranking

is not made in the non-dominated sorting, and the

convergence has deteriorated. On the other hand, al-

though the improved technique has shown the highest

convergence at the beginning of the optimization, the

convergence has declined at the ﬁnal stage. In the

improved technique, since the value of r increases as

the generation progresses, the solutions become hard

to dominated by the partial dominance. As a result,

sufﬁcient ranking is not made in the non-dominated

sorting, and the convergence has deteriorated in the

ﬁnal stage.

Fig.9 shows MS values to the generation g in the

case that m = 10 and I

= 500. Although the diversity

in the cases of the conventional NSGA-II and NSGA-

II based on Pareto partial dominance in r = 8, main-

tains a high value throughout, the convergence is low

as shown in Fig.8, so it is not necessary to pay atten-

tion to them. On the other hand, the diversity is ris-

ing as the optimization progress in the case of the im-

proved technique. Moreover, the improved technique

Figure 7: Comparison of MS values to the object number

Linear Subset Size Scheduling for Many-objective Optimization using NSGA-II based on Pareto Partial Dominance

281

Figure 8: Comparison of the Norm values to the generation

Figure 9: Comparison of the MS values to the generation g.

brings relatively high convergence as shown in Fig.8,

so that the superiority of the improved technique is

shown overall.

5 CONCLUSION

In this paper, the improvement of NSGA-II based

on Pareto partial dominance has been proposed with

the aim of improving the solution search performance

of MOEA for MaOP. In the improvement, we have

proposed the simple scheduling of the number r of

the objective functions for Pareto partial dominance

and killing the individuals of the archive set, where

the individual has the same contents as the individ-

ual created by the mating. The improved technique

and other conventional techniques are applied to the

many-objective 0/1 knapsack problem for veriﬁcation

of the effectiveness. The improved technique has

given the higher diversity than other techniques as the

number of the objective functions of the problem in-

creases. On the other hand, the improved technique

has given the convergence equal to or higher than the

other techniques even when the number of the objec-

tive functions becomes large. By means of the pro-

posed simple scheduling of the parameter r, sufﬁcient

convergence has been obtained in the early genera-

tions with the smaller r, and the diversity has been

supplemented in the generations with the larger r at

the end of the optimization.

Since the improved technique still has given in-

sufﬁcient results in terms of the diversity, we need to

improve this point while maintaining the current con-

vergence. Although each technique has been applied

to the relatively simple many-objective 0/1 knapsack

problem in this paper, we need to apply to more com-

plicated problems and verify the effectiveness. And

we also need to pursue an optimal combination list

for NSGA-II based on Pareto partial dominance with

the selection list.

ACKNOWLEDGEMENTS

This research work has been supported by JSPS

KAKENHI Grant Number JP17K00339.

The author would like to thank to her families, the

late Miss Blackin’, Miss Blanc, Miss Caramel, Mr.

Civita, Miss Marron, Miss Markin’, Mr. Yukichi and

Mr. Ojarumaru, for bringing her daily healing and

good research environment.

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