DMQEA: Dual Multiobjective Quantum-inspired Evolutionary

Algorithm

Si-Jung Ryu

, Jong-Hwan Kim

and Ki-Baek Lee

Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea

Department of Electric Engineering, Kwangwoon University, Seoul 139-701, Korea

Keywords:

Multiobjective evolutionary Algorithm, Quantum-inspired evolutionary algorithm, Preference-based evolu-

tionary Algorithm.

Abstract:

This paper proposes dual multiobjective quantum-inspired evolutionary algorithm (DMQEA) with the dual-

stage of dominance check by introducing secondary objectives in addition to primary objectives. The sec-

ondary objectives are to maximize global evaluation values and crowding distances of the solutions in the

external global population obtained for the primary objectives and the previous archive obtained from the sec-

ondary objectives-based nondominated sorting. By employing the secondary objectives for sorting the solu-

tions in each generation, DMQEA can induce the balanced exploration of the solutions in terms of user’s pref-

erence and diversity to generate preferable and diverse nondominated solutions in the archive. To demonstrate

the effectiveness of the proposed DMQEA, empirical comparisons with MQEA, MQEA-PS, and NSGA-II are

carried out for benchmark functions.

1 INTRODUCTION

Multiobjective evolutionary algorithms (MOEAs) are

designed to solve multiobjective optimization prob-

lems to get Pareto-optimal solutions while maintain-

ing as diverse a distribution as possible. These are

well-known two goals, proximity to Pareto-optimal

front and diversity preservation, in ideal multiobjec-

tive optimization. Much research has been conducted

to enhance the solution quality and diversity (Lau-

manns et al., 2002; Cui et al., 2001; Bosman and

Thierens, 2003; Kim et al., 2009; Deb et al., 2002;

Lee and Kim, 2012).

The other issue is how to select a preferable so-

lution among the widely distributed solutions in the

Pareto-optimal front for the application of the real

world problem. To solve this issue, preference-based

solution selection algorithm (PSSA) was proposed

(Kim et al., 2012). It selects a solution considering

user’s preference for each objective, which is repre-

sented by the fuzzy measures. In PSSA, global eval-

uation value of a candidate solution is calculated by

the fuzzy integral of the partial evaluation values with

respect to the fuzzy measures. The solution with the

highest global evaluation value is selected out of the

candidate solutions.

Based on PSSA, multiobjective quantum-inspired

evolutionary algorithm with preference-based selec-

tion (MQEA-PS) was proposed (Kim et al., 2012).

In each archive generation process, MQEA-PS em-

ploys PSSA in MQEA for preference-based sorting

for the solutions in the external global population and

the previous archive. It means that the nondominated

solutions in the archive are obtained by preference-

based sorting instead of dominance-based sorting,

whereas the internal subpopulations are sorted by fast

nondominated sorting. In this way, the solutions that

reﬂect user’s preference for each objective can be ob-

tained in the archive. Furthermore, for considering

the diversity of the solutions as well as user’s prefer-

ence, crowding distance sorting after the preference-

based sorting in the archive generation process is de-

veloped (Ryu et al., 2012). However, the solutions are

lack of the proximity to the Pareto front.

In this paper, we propose dual multiobjective

quantum-inspired evolutionary algorithm (DMQEA)

by introducing secondary objectivesin addition to pri-

mary objectives that are given objectives in the prob-

lem. The proposed DMQEA has the dual-stage of

dominance check respectively for the primary and

secondary objectives. In the ﬁrst stage, the domi-

nated solutions with respect to primary objectives are

culled out by primary objectives-based nondominated

sorting (PONS). In the second stage, nondominated

207

Ryu S., Kim J. and Lee K..

DMQEA: Dual Multiobjective Quantum-inspired Evolutionary Algorithm.

DOI: 10.5220/0005071802070214

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2014), pages 207-214

ISBN: 978-989-758-052-9

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

sorting is applied for the secondary objectives for

the generation of archive, which is called secondary

objectives-based nondominated sorting (SONS). The

secondary objectives are to maximize global eval-

uation values and crowding distances of the solu-

tions in the previous archive and the external global

population obtained for the primary objectives. The

archive consists of ﬁrst-tier solutions obtained from

the SONS.

By employing SONS in each generation, DMQEA

can induce the balanced exploration of the solutions

in terms of user’s preference and diversity to pro-

duce preferable and diverse nondominated solutions

in the archive. The effectiveness of the proposed

DMQEA is demonstrated through statistical com-

parisons with MQEA, MQEA-PS, and NSGA-II for

benchmark functions. The experimental results con-

ﬁrm that the proposed DMQEA generates the so-

lutions with larger hypervolume while maintaining

user’s preference compared to the existing two algo-

rithms, MQEA and MQEA-PS.

The rest of this paper is organized as fol-

lows: quantum-inspired evolutionary algorithm

(QEA), preference-based solution selection algorithm

(PSSA), and crowding distance are brieﬂy described

in Section II. Section III proposes dual multiobjective

evolutionary algorithm (DMQEA). The experimental

results are presented in Section IV and concluding re-

marks follow in Section V.

2 PRELIMINARIES

2.1 QEA

Quantum-inspired evolutionary algorithm (QEA) is

an evolutionary algorithm, which employs the prob-

abilistic mechanism inspired by the concept and prin-

ciples of quantum computing, such as a quantum bit

and superposition of states (Han and Kim, 2002; Han

and Kim, 2004). Building block of classical digital

computer is represented by two binary states, ‘0’ or

‘1’, which is a ﬁnite set of discrete and stable state. In

contrast, QEA utilizes a novel representation, called a

Q-bit representation, for the probabilistic representa-

tion that is based on the concept of qubits in quantum

computing (Hey, 1999). Quantum system enables the

superposition of such state as follows:

α|0i+ β|1i (1)

where α and β are the complex numbers satisfying

|α|

+ |β|

= 1.

Qubit is shown in Fig. 1, which can be illustrated

as a unit vector on the two dimensional space as fol-

" #

Figure 1: Qubit described in two-dimensional space.

lows:





(2)

where |α|

+ |β|

= 1. Q-bit individual is deﬁned as a

string of Q-bits as follows:



j,m−1

j,m−2

··· α

j,0

j,m−1

j,m−2

··· β

j,0



(3)

where m is the string length of Q-bit individual, and

j = 1,2,...,n for the population size n. The population

of Q-bit individuals at generation t is represented as

follows:

Q(t) = {q

,··· ,q

}. (4)

Since Q-bit individual represents the linear super-

position of all possible states probabilistically, vari-

ous individuals are generated during the evolution-

ary process. The procedure of QEA and the over-

all structure for single-objective optimization prob-

lems are described in (Han and Kim, 2002). To solve

multiobjective optimization problems, multiobjective

quantum-inspired evolutionary algorithm (MQEA) is

also developed (Kim et al., 2006).

2.2 PSSA

Preference-based solution selection algorithm

(PSSA) selects a solution among the obtained non-

dominated solutions considering user’s preference

(Kim et al., 2012). The nondominated solutions

cannot be directly compared against each other, and

therefore a multicriteria decision making (MCDM)

algorithm is required to evaluate them. In PSSA,

the global evaluation value of a candidate solution is

calculated by the fuzzy integral, as an MCDM algo-

rithm, of the partial evaluation values with respect to

the fuzzy measures. The fuzzy measures represent

the degrees of consideration for objectives, and

the partial evaluation value indicates a normalized

objective function value. Overall procedure of global

evaluation is summarized in Algorithm 1. Each step

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208

in the algorithm is brieﬂy described in the following

and detail procedure of global evaluation is described

in (Kim et al., 2012). 1. Calculate λ-fuzzy Measures

Objectives are deﬁned as criteria in multi-

objective problem. λ-fuzzy measure represents the

degree of consideration for each criterion. To get the

values of λ-fuzzy measures, a pairwise comparison

matrix (P) is initially deﬁned by user for represent-

ing preference degrees between criteria. Secondly,

the normalized weights of P are calculated by adding

each value in the row of the pairwise comparison ma-

trix and dividing it by the total sum of the values in

the row. Lastly, λ-fuzzy measures are obtained using

the normalized weights (Bajwa et al., 2008).

2. Compute Global Evaluation Value

First, the value of partial evaluation of each solu-

tion is calculated by normalizing the objective func-

tion value to 1. Global evaluation of each and every

solution is performed by the Choquet fuzzy integral

of the partial evaluation values with respect to the λ-

fuzzy measures, which are obtained from the previous

steps.

Algorithm 1: Procedure of global evaluation.

• l: No. of the solutions

• m: No. of the objectives

• C: A set of objectives C = {c

,...,c

}

• P: A power set of C

• f

): j-th Objective value of x

• h

): j-th partial evaluation value of x

• e(x

): Global evaluation value of x

—————————————————————

1. Calculate λ-fuzzy measures g’s of P(C)

1: Make a pairwise comparison matrix P

2: Calculate normalized weights of

3: Calculate λ-fuzzy measures of P(C).

2. Compute global evaluation value e

1: for k = 1 to l do

2: for j = 1 to m do

3: h

) = Normalize(f

))

4: end for

5: end for

6: for k = 1 to l do

7: e(x

) =

h◦g

8: end for

2.3 Crowding Distance

The crowding distance estimates the density of solu-

tions surrounding a particular solution in the popu-

lation (Deb et al., 2002). The crowding distance is

aimed to uniformly select the solutions in the front,

making the solutions in the most dense areas less

likely to be selected. The crowding distance is de-

ﬁned by the average distance of the closest points on

either side of the point for each objective. Therefore,

the crowding distance is inversely proportional to the

density of solutions. Boundary points for each objec-

tive have the maximum crowding distance, and they

are always selected. Calculation of crowding distance

is described in Algorithm 2.

Algorithm 2: Crowding distance assignment.

• l: No. of the solutions

• m: No. of the objectives

• f

): j-th objective value of x

• x

.CD: Crowding distance of the solution x

—————————————————————

1. Initialization

1: for k = 1 to l do

2: x

.CD = 0

3: end for

2. Calculate the crowding distances

1: for j = 1 to m do

2: for k = 1 to l do

3: Calculate the objective value f

)

4: end for

5: Sort the solutions using objective value

), x

= sort(x

)

6: x

.CD = x

.CD = ∞

7: for k = 2 to l −1 do

8: x

.CD = x

.CD+ |f

k+1

) − f

k−1

9: end for

10: end for

3 DMQEA

Dual multiobjective quantum-inspired evolutionary

algorithm (DMQEA) has the dual-stage of domi-

nance check for the primary and secondary objec-

tives. Primary objectives are the given objectives of

the problem. The secondary objectives are to maxi-

mize both the global evaluation values and crowding

distances of the solutions in the external global pop-

ulation obtained for the primary objectives and the

DMQEA:DualMultiobjectiveQuantum-inspiredEvolutionaryAlgorithm

209

Crowding distance

Solution in tier 1

Solution in tier 2

Solution in tier 3

Global evaluation value

Figure 2: Secondary objectives-based nondominated sort-

ing.

previous archive. In each archive generation process,

the secondary objectives are employed for sorting

the solutions, which is called secondary objectives-

based nondominated sorting (SONS). By the pro-

posed SONS, the archive stores ﬁrst-tier solutions.

3.1 SONS

SONS is to sort the solutions with the secondary ob-

jectives for maximizing the global evaluation value

(GEval) and crowding distance (CD). The SONS is

performed for the solutions in the external global pop-

ulation obtained for the primary objectives and the

previous archive. It means that in DMQEA, the solu-

tions are sorted by SONS that checks the dominance

relationship with respect to GEval and CD. By SONS,

the solutions that are not dominated by any other so-

lutions could be obtained as ﬁrst-tier solutions that are

stored in the archive.

The proposed SONS is depicted in Fig. 2. GEval

and CD of every solution in the external global pop-

ulation and the previous archive are calculated as ex-

plained in the previous section. Note that the global

evaluation value of a solution is calculated by the

fuzzy integral of the partial evaluation values with re-

spect to the fuzzy measures representing the user’s

preference for objectives. The solutions with higher

values of GEval and CD are better in terms of user’s

preference and diversity. For example, in the ﬁgure,

blue points are classiﬁed as ﬁrst-tier solutions to be

stored in the archive. The solutions in lower tiers are

discarded because they are dominated by the ﬁrst-tier

solutions.

3.2 Procedure of DMQEA

In an archive generation process, MQEA employs

dominance-basedsorting for primary objectivesof the

solutions in the external global population and the

previous archive. Most of them are nondominated by

the other solutions because primary objectives-based

Previous archive ( − 1)

SONS

Reference

subpopulation 



()

Previous higher-tier

subpopulation 



( − 1)

Binary subpopulation 



()

Q-bit subpopulation 



()

Q-gate

subpopulation

PONS

Update

Multiple

observations

Reference

subpopulation 



()

Previous higher-tier

subpopulation 



( − 1)

Binary subpopulation 



()

Higher-tier

subpopulation 



()

Q-bit subpopulation 



()

Q-gate

subpopulation

Update

Multiple

observations

Global population ()

Global

random migration

. . .

Archive ()

Higher-tier

subpopulation 



()

PONS

Figure 3: Overall procedure of DMQEA, where PONS: Pri-

mary objectives-based nondominated sorting, SONS: Sec-

ondary objectives-based nondominated sorting.

nondominated sorting (PONS) or fast nondominated

sorting is already performed in each subpopulation. It

means that the dominance-based sorting for the pri-

mary objectives might be an ineffective operation in

selecting solutions to be stored in the archive. In-

deed, in experiments, the external global population

almost consists of nondominated solutions. To solve

this problem, DMQEA employs SONS in the archive

generation process. By SONS, each solution is clas-

siﬁed into the corresponding tier and the solutions in

the ﬁrst tier are stored in the archive. These are used

for reference solutions through the global random mi-

gration process. The overall procedure of DMQEA is

summarized in Algorithm 3, and depicted in Fig. 3.

Each step is described in detail in the following.

1. Initialize Q

(t) and Generate Archive A(t)

(0) including q

, which consists of α

and β

, is

initialized with 1/

√

2, where i = 0,1,...,m − 1, j =

1,2,...,n, and k = 1, 2,...,s. Note that m is the string

length of Q-bit individual, n is the subpopulation size,

and s is the number of subpopulations. It means that

one Q-bit individual, q

, represents the linear super-

position of all possible states with same probabil-

ity. Binary solutions in P

(0) are produced by mul-

tiple observing the states of Q

(0), where P

(0) =

,...,x

} and x

= {x

j,m−1

j,m−2

,...,x

}, j =

1,2,...,n. A bit of one binary solution, x

, has a value

either ‘0’ or ‘1’ according to the probability either

|α

or |β

, where i = 0,1, ..., m−1, j = 1,2,...,n,

as follows:



0 if rand[0,1] ≥ |β

1 if rand[0,1] < |β

(5)

Multiple observation is performed on each and

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210

Algorithm 3: Procedure of DMQEA.

• P

(t) = {x

,...,x

}

• x

= {x

j,m−1

j,m−2

,...,x

}

• Q

(t) = {q

,··· ,q

}

• q



j,m−1

j,m−2

··· α

j,0

j,m−1

j,m−2

··· β

j,0



• R

(t) = {r

,...,r

}

• s = No. of subpopulations

• n = Size of subpopulation

• m = Q-bit string length

—————————————————————

1. Initialize Q

(t) and generate archive A(t)

1: t = 0

2: for k = 1 to s do

3: for j = 1 to n do

4: for i = 0 to m−1 do

5: α

= β

= 1/

√

6: end for

7: Make P

(t) by multiple observing the states

of Q

(t)

8: for each objective do

9: Evaluate the objective value from x

10: end for

11: Copy all solutions in P

(t) into P(t)

12: Store ﬁrst-tier solutions of P(t) by SONS in

the archive A(t)

13: end for

14: end for

2. Generate global population P(t)

1: t = t + 1

2: for k = 1 to s do

3: for j = 1 to n do

4: Make P

(t) by multiple observing the states

of Q

(t)

5: for each objective do

6: Evaluate the objective value from x

7: end for

8: end for

9: Run PONS for P

(t) ∪ B

(t −1)

10: Store n higher-tier solutions of P

(t) ∪B

(t −1)

into B

(t)

11: end for

12: Store all solutions in every B

(t) into P(t)

3. Update archive A(t)

1: for each solution in A(t −1) ∪P(t) do

2: Evaluate GEval and CD

3: end for

4: Run SONS

5: Store the ﬁrst-tier solutions into the archive A(t)

every Q-bit individual in subpopulations, q

(0), k = 1,2,...,s. Each binary solution in P

(0)

4. Migrate and update Q

(t)

1: for k = 1 to s do

2: for j = 1 to n do

3: Select a solution in A(t) randomly

4: Store it into r

5: Update q

using Q-gates referring to the

solutions in r

6: end for

7: end for

5. Go back to Step 2 and repeat

is decoded to a real number if necessary, and its

objective value is calculated. All solutions in each

binary subpopulation P

(0) are copied to the external

global population P(0) and store ﬁrst tier solutions of

P(0) by SONS in the archive A(t).

2. Generate Global Population P(t)

Binary solutions are generated by multiple obser-

vations of Q-bit individuals in Q-bit subpopulation

(t). Each bit of binary solution x

, l = 1,2,...,o,

where o is the observation index is obtained. x

is assigned by the best among the observed binary

solutions x

, l = 1,2,...,o, from the multiple ob-

servations. And then, evaluation is performed to

(t), where k = 1,2,...,s. Therefore, objective

values of all solutions in each subpopulation are

obtained. The solutions in the previous higher-tier

subpopulation and the current binary subpopulation

(t) ∪ B

(t − 1) are sorted by PONS to select n

solutions in order from the ﬁrst tier to the lower

tiers. The n higher-tier solutions form B

(t), where

(t) = {b

,...,b

} that is to become the previous

higher-tier subpopulation in the next generation. To

update Q-bit individuals corresponding to higher-tier

subpopulation later, Q-bit subpopulation Q

(t) is

rearranged by replacing each q

in the subpopulation

by the Q-bit individual that has generated b

. All

higher-tier solutions in each subpopulation B

(t) are

copied to the external global population P(t).

3. Update Archive A(t)

Global evaluation values are calculated by the

fuzzy integral and crowding distance is also cal-

culated. The fuzzy integral reﬂects how much a

user prefers the solution, and crowding distance

denotes the density of the solutions. SONS with

GEval and CD for the solutions in the external global

population and the previous archive is performed.

The nondominated solutions in the ﬁrst tier are stored

into the archive A(t). The size of the archive might

be different each generation.

DMQEA:DualMultiobjectiveQuantum-inspiredEvolutionaryAlgorithm

211

4. Migrate and Update Q

(t)

The solutions in the archive A(t) are randomly

selected n times and they are globally migrated to

each reference subpopulation R

(t), where R

(t) =

,...,r

}. Note that the solutions in R

(t) are

employed as references to update Q-bit individuals,

each of which is corresponding to the solution in the

higher-tier subpopulation. Global random migration

procedure occurs at every generation. In the update

process of Q-bit individuals, the rotation gate is em-

ployed. r

and b

in each subpopulation are compared

bit-by-bit to decide the update directions of Q-bit in-

dividuals in the rotation gate U(∆θ), which is deﬁned

as follows:

= U(∆θ) ·q

t−1

(6)

with

U(∆θ) =



cos(∆θ) –sin(∆θ)

sin(∆θ) cos(∆θ)



where ∆θ is the rotation angle of each Q-bit as shown

in Fig. 1. Note that crossover and mutation operators

are not used in QEA.

5. Go back to Step 2 and Repeat

Go back to Step 2 and repeat until a termination

condition is satisﬁed.

4 EXPERIMENTAL RESULTS

4.1 Experimental Settings

The proposed DMQEA was compared with MQEA,

MQEA-PS, and NSGA-II. To evaluate the perfor-

mance of algorithms, we employed seven DTLZ func-

tions as benchmark functions. The number of vari-

ables for each DTLZ function was set to 9 for DTLZ1,

16 for DTLZ2 to DTLZ6, and 26 for DTLZ7. Param-

eters for DMQEA, MQEA-PS, MQEA, and NSGA-II

were equally set and given in Table 1. Belief mea-

sure (ξ = 0.25) for MQEA-PS and DMQEA was used.

As the preferred objectives, two objectives among the

ﬁve objectives in DTLZ functions were selected. The

preference degrees or the degrees of consideration for

ﬁve objectives was set as f

: f

= 1 : 10 :

1 : 10 : 1. The normalized weights from the pairwise

comparison matrix were calculated as (0.0435, 0.435,

0.0435, 0.435, 0.0435).

4.2 Performance Metrics

Two performance metrics, the size of dominated

space and the diversity, were employed to evaluate

Table 1: Parameter setting of MQEA, MQEA-PS, and

DMQEA for DTLZ functions

Algorithms Parameters Values

MQEA,

MQEA-PS,

DMQEA

The population size (N = n·s) 100

No. of generations 3000

Subpopulation size(n) 25

No. of subpopulations (s) 4

No. of multiple observations 10

The rotation angle (∆θ) 0.23π

NSGA-II

The population size (N) 100

No. of generations 3000

Mutation probability 0.1

the performances of MQEA, MQEA-PS, DMQEA,

and NSGA-II (Zitzler, 1999). The size of dominated

space,

S , is deﬁned by the hypervolume of the ﬁnally

obtained global population. The quality of the ob-

tained global population is high if this space is large.

The diversity,

D, is to evaluate the spread of nondom-

inated solutions, which is deﬁned as follows (Li et al.,

2004):

D =

∑

k=1

( f

(max)

− f

(min)

)

∑

i=1

−

(7)

where N

is the set of nondominated solutions, d

the minimal distance between the i-th solution and

the nearest neighbor, and

d is the mean value of all

. f

(max)

and f

(min)

represent the maximum and min-

imum objective function values of the k-th objective,

respectively. A larger value means a better diversity

of the nondominated solutions.

4.3 Results

The proposed DMQEA generated the optimized so-

lutions concentrated on the selected preferred objec-

tives, f

and f

. The hypervolume and diversity of

MQEA, MQEA-PS, DMQEA, and NSGA-II are sum-

marized in Tables 2. The results in Table 2 are aver-

aged ones by repeating the simulation 50 times.

For statistical analysis, t-test was employed to sta-

tistically compare the performance metrics of algo-

rithms. The t-test is a statistical hypothesis test in

which the test statistic follows a t distribution if the

null hypothesis H

is supported. If the null hypothesis

is rejected, the alternative hypothesis is supported.

t-test was used to determine whether two comparison

groups were signiﬁcantly different from each other.

The t-test was carried out with the two-tailed test. Ta-

bles 3 showt-value(and p-value) for the hypervolume

with MQEA, MQEA-PS, and NSGA-II.

As shown in Table 3, the proposed DMQEA had

larger hypervolume than MQEA for DTLZ1, DTLZ3,

DTLZ6, and DTLZ7. For the fair comparison of hy-

pervolume, the size of the obtained global populations

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Table 2: Comparisons of hypervolume and diversity among

MQEA, MQEA-PS, and DMQEA for seven DTLZ func-

tions.

(a) Average hypervolume of nondominated solutions

Problem NSGA-II MQEA MQEA-PS DMQEA

DTLZ1 99999 99255 99740 99748

DTLZ2 99998 99796 97139 99202

DTLZ3 66773 N/A 71911 79015

DTLZ4 99999 95119 92309 94898

DTLZ5 99158 98578 95877 98388

DTLZ6 96169 67967 89411 72915

DTLZ7 63202 10907 38574 40500

(b) Average diversity of nondominated solutions

Problem NSGA-II MQEA MQEA-PS DMQEA

DTLZ1 103.95 144.99 92.13 75.99

DTLZ2 132.95 71.05 93.96 79.19

DTLZ3 104.36 55.05 60.93 52.73

DTLZ4 132.53 80.62 105.24 137.61

DTLZ5 179.29 129.39 278.32 139.29

DTLZ6 143.12 73.35 65.26 63.46

DTLZ7 164.78 115.96 136.27 135.55

for three algorithms are set to the same value. In com-

parison with MQEA-PS, DMQEA had larger hyper-

volume for all DTLZ functions except for DTLZ1 and

DTLZ6. It means DMQEA found more optimized so-

lutions close to the Pareto-optimal front. However,

in comparison with NSGA-II, DMQEA had better

performance only for DTLZ3. This is because the

proposed DMQEA generated the optimized solutions

concentrated on the selected preferred objectives, f

and f

. Due to the property of the hypervolume,

DMQEA that generates the dense solutions in a small

region has a lower value of hypervolume compared to

NSGA-II for DTLZ1, DTLZ2, DTLZ5.

Table 4 shows the result for the diversity and Ta-

ble 5 presents average objective values of preferred

solutions ﬁnally selected by PSSA among the solu-

tions obtained from MQEA-PS and DMQEA, respec-

tively. As Table 4 shows, DMQEA has a lower value

of diversity compared with MQEA, MQEA-PS, and

NSGA-II, and Table 5 shows DMQEA generated the

solutions that effectively reﬂect preferred objectives.

It means that DMQEA could generate the solutions

emphasized on preference objectives, f

, f

in dense

area. In other word, the proposed DMQEA could ﬁnd

more optimized solutions for the preferred objectives

compared with the other algorithms.

5 CONCLUSION

In this paper, dual multiobjective quantum-inspired

evolutionary algorithm (DMQEA) was proposed by

introducing secondary objectives in addition to pri-

mary objectives. The proposed DMQEA had the

dual-stage of dominance check for the primary and

Table 3: The hypothesis test on S of the three algorithms

: S

DMQEA

−S

MQEA

= 0

t-value (p-value) Reject H

DTLZ1 6.731 (0.000) NO S

DMQEA

−S

MQEA

> 0

DTLZ2 -14.524 (0.000) YES S

DMQEA

−S

MQEA

< 0

DTLZ3 67.767 (0.000) YES S

DMQEA

−S

MQEA

> 0

DTLZ4 -0.557 (0.580) NO N/A

DTLZ5 -7.233 (0.000) YES S

DMQEA

−S

MQEA

< 0

DTLZ6 8.871 (0.000) YES S

DMQEA

−S

MQEA

> 0

DTLZ7 57.875 (0.000) YES S

DMQEA

−S

MQEA

> 0

: S

DMQEA

−S

MQEA-PS

= 0

t-value (p-value) Reject H

DTLZ1 0.252 (0.802) NO N/A

DTLZ2 15.134 (0.000) YES S

DMQEA

−S

MQEA-PS

> 0

DTLZ3 3.367 (0.001) YES S

DMQEA

−S

MQEA-PS

> 0

DTLZ4 6.287 (0.000) YES S

DMQEA

−S

MQEA-PS

> 0

DTLZ5 12.742 (0.000) YES S

DMQEA

−S

MQEA-PS

> 0

DTLZ6 -37.855 (0.000) YES S

DMQEA

−S

MQEA-PS

< 0

DTLZ7 6.709 (0.000) YES S

DMQEA

−S

MQEA-PS

> 0

: S

DMQEA

−S

NSGA-II

= 0

t-value (p-value) Reject H

DTLZ1 -10.460 (0.800) YES S

DMQEA

−S

NSGA-II

< 0

DTLZ2 -17.689 (0.000) YES S

DMQEA

−S

NSGA-II

< 0

DTLZ3 4.771 (0.000) YES S

DMQEA

−S

NSGA-II

> 0

DTLZ4 -25.450 (0.000) YES S

DMQEA

−S

NSGA-II

< 0

DTLZ5 -31.727 (0.000) YES S

DMQEA

−S

NSGA-II

< 0

DTLZ6 -160.96 (0.000) YES S

DMQEA

−S

NSGA-II

< 0

DTLZ7 -136.51 (0.000) YES S

DMQEA

−S

NSGA-II

< 0

Table 4: The hypothesis test on D of the three algorithms.

: D

DMQEA

−D

MQEA

= 0

t-value (p-value) Reject H

DTLZ1 -5.966 (0.000) YES D

DMQEA

−D

MQEA

< 0

DTLZ2 9.492 (0.000) YES D

DMQEA

−D

MQEA

> 0

DTLZ3 -0.445 (0.658) NO N/A

DTLZ4 3.794 (0.000) YES D

DMQEA

−D

MQEA

> 0

DTLZ5 3.328 (0.002) YES D

DMQEA

−D

MQEA

> 0

DTLZ6 -10.177 (0.000) YES D

DMQEA

−D

MQEA

< 0

DTLZ7 3.439 (0.001) YES D

DMQEA

−D

MQEA

> 0

: D

DMQEA

−D

MQEA-PS

= 0

t-value (p-value) Reject H

DTLZ1 -2.959 (0.004) YES D

DMQEA

−D

MQEA-PS

< 0

DTLZ2 -11.512 (0.000) YES D

DMQEA

−D

MQEA-PS

< 0

DTLZ3 -2.002 (0.051) NO N/A

DTLZ4 1.997 (0.051) NO N/A

DTLZ5 -9.052 (0.000) YES D

DMQEA

−D

MQEA-PS

< 0

DTLZ6 -0.644 (0.522) NO N/A

DTLZ7 -0.119 (0.906) NO N/A

: D

DMQEA

−D

NSGA-II

= 0

t-value (p-value) Reject H

DTLZ1 -3.871 (0.000) YES D

DMQEA

−D

NSGA-II

< 0

DTLZ2 -63.8581 (0.000) YES D

DMQEA

−D

NSGA-II

< 0

DTLZ3 1.773 (0.082) NO N/A

DTLZ4 4.527 (0.000) YES D

DMQEA

−D

NSGA-II

< 0

DTLZ5 -4.076 (0.000) YES D

DMQEA

−D

NSGA-II

< 0

DTLZ6 -65.703 (0.000) YES D

DMQEA

−D

NSGA-II

< 0

DTLZ7 -6.241 (0.000) YES D

DMQEA

−D

NSGA-II

< 0

secondary objectives. The secondary objectives are

to maximize global evaluation values and crowding

distances of the solutions. The global evaluation of

a solution was carried out by the fuzzy integral of

the partial evaluation values with respect to the fuzzy

DMQEA:DualMultiobjectiveQuantum-inspiredEvolutionaryAlgorithm

213

Table 5: The objective function values of preferred solu-

tions each selected by PSSA among the solutions obtained

from MQEA-PS and DMQEA, respectively.

(a) Average objective values of a preferred so-

lution ﬁnally selected by PSSA among the solu-

tions obtained from MQEA-PS

DTLZ1 0.0599 0.0002 0.1402 0.0002 0.4596

DTLZ2 0.0000 0.0000 0.0000 0.0000 1.0000

DTLZ3 0.0021 0.0002 0.0027 0.1232 5.5938

DTLZ4 1.0000 0.0000 0.0000 0.0000 1.0000

DTLZ5 0.0000 0.0000 0.0000 0.0000 1.0000

DTLZ6 0.0002 0.0002 0.0002 0.0003 1.8851

DTLZ7 0.6302 0.0022 0.6022 0.0005 9.2351

(b) Average objective values of a preferred so-

lution ﬁnally selected by PSSA among the solu-

tions obtained from DMQEA

DTLZ1 0.0685 0.0005 0.1401 0.0005 0.5994

DTLZ2 0.0000 0.0000 0.0000 0.0000 1.0003

DTLZ3 0.0000 0.0000 0.0000 0.0000 4.4475

DTLZ4 1.0251 0.0000 0.0000 0.0000 1.0000

DTLZ5 0.0000 0.0000 0.0000 0.0000 1.0001

DTLZ6 0.0004 0.0002 0.0004 0.0006 4.1782

DTLZ7 0.1235 0.0017 0.3517 0.0026 10.7801

measures representing user’s preference for objec-

tives. By employing the secondary objectives-based

nondominated sorting in each archive generation pro-

cess, DMQEA could generate the preferable and di-

verse solutions. For the performance comparisons

among MQEA, MQEA-PS, DMQEA, and NSGA-

II, seven DTLZ functions were used as benchmark

functions, and hypervolume and diversity were em-

ployed as performance metrics. The experimental re-

sults conﬁrmed that the proposed DMQEA was able

to generate more optimized solutions for the preferred

objectives compared with the other algorithms.

ACKNOWLEDGEMENTS

This work was supported by the Technology Innova-

tion Program, 10045252, Development of robot task

intelligence technology, funded by the Ministry of

Trade, Industry & Energy (MOTIE, Korea).

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