The Effects of Supermajority on Multi-Parent Crossover
Chuan-Kang Ting, Member, IEEE, and Chun-Cheng Chen
Abstract— Multi-parent crossover allows more than two parents participating in crossover. The increase of parents incrossover intensifies exploitation or exploration or both; however, the intensification is often unbalanced and consequently
causes deterioration in performance. In this paper we propose
the use of supermajority to address the issue of lopsided
intensification on exploitation and exploration in multi-parent
crossover. The proposed crossover, called biased occurrencebased scanning crossover (bOB), controls the tendency toward
exploitation or exploration by the threshold in supermajority.
Two adaptive strategies are developed to adjust the thresholdof bOB. Experimental results indicate that bOB can achievesignificant improvement on uniform crossover and occurrencebased scanning crossover in both solution quality and convergence speed. Precisely, the improvement in mean best fitness
ranges from 4–89% on our test problems. The preferable
results validate that bOB crossover can not only enhance theperformance but also provide an effective way to control the
exploitation and exploration in crossover.

I. I NTRODUCTION
Evolutionary algorithms (EAs) have shown their effectiveness in dealing with search and optimization problems.
The essence of EAs is simulating natural evolution, such as
selection, crossover, and mutation to find the near-optimal
solutions [7]. Crossover in EAs is ordinarily performed withtwo parents, by analogy with sexual reproduction in nature.
However, it is not necessary to limit the number of parents
in EAs. Multi-parent crossover allows more than two parents
participating in the crossover operation. In terms of thenumber of parents, multi-parent crossover can be viewed asa generalization of traditional 2-parent crossover.
Several multi-parent crossover operators have been proposed [2], [3], [10]. In general, the increase of parents brings
about a more comprehensive survey for determining theoffspring genes and leads to a stronger tendency towards exploitation or exploration or both [2], [12], [13]. An effective
EA, nevertheless, requires a balance of these two driving
forces: Strong exploitation leads to a rapid convergencebut may suffer from premature convergence, while strong
exploration helps to search more regions but pays the cost of
efficiency. Designing effective strategies to control the power
intensified by the increase of parents is therefore a key issue
in improving the performance of multi-parent crossover.
This paper presents a novel approach to address the issueof lopsided intensification on exploitation or exploration in
multi-parent crossover. Specifically, we introduce the concept of supermajority into the occurrence-based scanningcrossover (OB-Scan) [4] — a multi-parent generalization of
uniform crossover — to replace the simple majority system
The authors are with the Department of Computer Science and Information Engineering, National Chung Cheng University, Taiwan (e-mail:ckting@cs.ccu.edu.tw; ccch95m@cs.ccu.edu.tw).

for determining offspring genes among multiple parents.
The threshold in the supermajority affects the tendency of
OB-Scan towards exploitation or exploration. Two strategies
are further developed to adaptively adjust the threshold inresponse to the allele frequency. Experimental results show
the proposed approach can effectively balance exploitationand exploration and then achieves significant improvement
on OB-Scan in terms of convergence speed and solution
quality on our benchmark problems.
The remainder of this paper is organized as follows. Section 2 reviews related work about multi-parent crossover. In
Section 3, we detail the proposed multi-parent crossover andthe derivative strategies. Section 4 presents the experimental
results and performance evaluation. Finally, conclusions are
drawn in Section 5.
II. M ULTI -PARENT C ROSSOVER
Multi-parent crossover breaks through the convention ofusing two parents in crossover. The debut of multi-parent
crossover in evolutionary computation was as global recombination in evolutionary strategies (ES) [1], [10]. Global
recombination determines each offspring gene according to
the whole population rather than only two parents. Such a
global form allows an offspring to inherit from more than
two parents.
For genetic algorithms (GAs), several multi-parent
crossovers have been proposed. Eiben et al. proposed scanning crossover [4] and diagonal crossover [5], [6], which generalize uniform crossover [11] and one-point crossover [7],respectively. Depending on the heuristics applied to it,
scanning crossover has three variations: uniform scanningcrossover (U-Scan), occurrence based scanning crossover(OB-Scan), and fitness based scanning crossover (FB-Scan).
Experimental results show that both scanning crossoverand diagonal crossover can achieve better performance thantheir 2-parent versions in several test functions. Mühlenbein et al. [9], [18] introduced the concept of global recombination into GAs as gene pool recombination (GPR). Rather
than from two parents, GPR samples the genes for crossoverfrom the gene pool, which consists of several pre-selectedparents. The studies indicate that GPR and its variants areeasier to analyze and these methods can converge fasterthan 2-parent recombination. Tsutsui and Jain [16] proposed
multi-cut crossover (MX) and seed crossover (SX). Multicut crossover generalizes the classic two-point crossover andwas shown empirically to outperform diagonal crossover.
In addition to binary-coded GAs, Tsutsui and Ghosh [14],[15] presented three multi-parent crossover operators for realcoded GAs: center of mass crossover (CMX), multi-parent
feature-wise crossover (MFX), and seed crossover (SX).

Parent 1 1 0 1 1 0 0 0 1
Parent 2 1 1 0 1 1 1 0 1
Offspring 1 1 0 1 0 1 0 1
majority
randomFig. 1.

Parent 1 1 0 1 1 0 0 0 1
Parent 2 1 0 0 1 1 1 0 1
Parent 3 0 0 1 0 1 1 0 0
Parent 4 0 1 1 0 1 0 1 1
Offspring 1 0 1 0 1 1 0 1

Examples of 2-parent OB-Scan (left) and 4-parent OB-Scan (right)

Experimental results demonstrate that these multi-parent
crossovers can lead to better performance although theperformance is problem-dependent. Another multi-parent
crossover, simplex crossover (SPX) [17], produces offspringon the basis of the simplex sampled from multiple parents.
Their results show that SPX performs well with three or fourparents for multimodal and epistatic problems. Moreover,
Kita et al. [8] introduced multiple parents into unimodal
normal distribution crossover (UNDX) to enhance the diversity of offspring. This multi-parent extension of UNDX
exhibits its improvement in search ability on highly epistatic
problems.
The theoretical work [12], [13] reveals that the increase ofparents in OB-Scan intensifies the exploitation of the search.
The one-sided intensification speeds up the convergence at
the risk of premature convergence, which is reflected in
the poor solution quality of OB-Scan using more than twoparents. The present paper is to address this issue throughsupermajority.
III. T HE P ROPOSED A PPROACH
In this paper we introduce the concept of supermajority
into OB-Scan to deal with the issue of unbalanced exploitation and exploration. A supermajority is a majority
required to exceed a specified percentage θ higher thansimple majority, e.g. θ = 60% of total votes. Through control
over the supermajority threshold θ, we are able to regulate
the tendency of OB-Scan towards exploitation or exploration.
Before describing the approaches, we give a definition of thechromosome and the population in binary-coded GAs.
Definition 1 (Chromosome and Population):def1) a chromosome c is encoded as a bit string, i.e. c =(c1 , . . . , cl ) ∈ {0, 1}l , where ci denotes a gene and lis the chromosome length;
def2) theof chromosomes: C = population C is a multisetlc1 , . . . , cm | ci ∈ {0, 1} , where m is called the population size.
OB-Scan is a multi-parent generalization of uniform
crossover. Rather than random, OB-Scan determines offspring genes depending on the occurrence of parental genes
at that locus. Specifically, it takes the simple majority ofparental genes as the offspring gene for each locus. Notethat in this paper OB-Scan is defined to break ties byrandomly1 choosing a parental gene at that locus. Figure 1
1 The original OB-Scan [4] breaks ties by directly inheriting the genefrom the first selected parent. However, random tie break conforms to thegeneralization of uniform crossover.

gives examples of 2-parent OB-Scan and 4-parent OB-Scan,
where 2-parent OB-Scan corresponds to uniform crossover.
Definition 2 (OB-Scan): Given n parents c1 , . . . , cn ∈ C,
OB-Scan reproduces an offspring c = (c1 , . . . , cl ) by, fork = 1, . . . , l,

n
1
1

0
n i=1 ci,k < 2 ,n11ck = 1i=1 ci,k > 2 ,n
cr,k otherwise,
where ci,k denotes the gene of parent ci at locus k, andr ∈ {1, . . . , n} is generated at random.
The way that OB-Scan decides offspring genes is a “simple
majority” voting in essence; that is, an allele gets elected if
more than half of the total parents hold it. The theoreticalanalysis [12], [13] indicates that increasing the number ofparents in OB-Scan leads to a higher probability for the major
alleles in the population to exist in every offspring, which
will cause the population to concentrate on a certain region
and thus lose the diversity rapidly. This situation is so-calledpremature convergence, a result of unbalanced exploitationand exploration. In other words, OB-Scan using more than
two parents suffers from premature convergence because it
excessively intensifies exploitation but lessens exploration.
However, OB-Scan provides the capability for other voting
system, namely, other criterion for determining offspring
genes. In this paper we use the “supermajority” in place ofthe simple majority to deal with the issues in OB-Scan. The
supermajority threshold θ represents the required level for the
candidates to be elected. In the supermajority voting system,
the winner requires a higher level (θ > 12 ) of support thanin simple majority voting system. The proposed approach iscalled biased occurrence-based scanning crossover (bOB) inthat the threshold for supermajority biases the requirement
for alleles 0 and 1 to be selected. The formal definition ofbOB is given below.
Definition 3 (bOB): Given n parents c1 , . . . , cn ∈ C andthe thresholds 0 ≤ θk ≤ 1, bOB reproduces an offspringc = (c1 , . . . , cl ) by, for k = 1, . . . , l,

n
1

0
n i=1 ci,k < θk ,n1ck = 1i=1 ci,k > θk ,n
cr,k otherwise,
where ci,k denotes the gene of parent ci at locus k, andr ∈ {1, . . . , n} is generated at random.

TABLE I
T EST FUNCTIONS (RAS, SCH, AND GRI)
Name

Functioni´2`
100 xi+1 − x2i + (xi − 1)2

N−1
P h

F2e

fF2e =

RAS

N ˆ
˜
P
x2i − 10 cos (2πxi )fRAS = 10N +i=1

fGRI = 1 +

GRI

N
P
i=1

i=1

x2
i

4000

−

N
Q
i=1

cos

“

x
√i
i

”

Remark 1: Uniform crossover and OB-Scan are special
cases of bOB: n-parent OB-Scan corresponds to n-parent
bOB with θk = 12 for all k ∈ {1, . . . , l}, while uniform
crossover corresponds to 2-parent OB-Scan.
By definition, bOB is a general form of uniform crossoverand OB-Scan. The additional parameter, θk , plays an important role on the decision criterion of bOB for offspring
genes. Restated, the value of θk biases bOB in favor of allele
0 or 1 at offspring locus k: The case θk > 12 gives rise toa supermajoritarian requirement for allele 1 to be elected,
which encourages bOB to choose 0 as the offspring gene.
Conversely, the case θk < 12 brings about a supermajority
for allele 0 so that bOB tends to chooses allele 1.
In view of the above effects, we develop two strategies tocontrol the parameter θk adaptively. The strategies are basedon the allele frequency (or gene frequency), which represents
the relative frequency of an allele in the population for alocus.
(α)
Definition 4 (Allele Frequency): Let Ck = {c ∈ C |ck = α} be the subset in which the chromosomes possessallele α at locus k. The allele frequency is defined by(α) def |Ck (α)|

pk

=

|C|

,

where | · | denotes the set cardinality.
(1)
For simplicity, we refer to the allele frequency pk as pk(0)
and refer to pk as (1 − pk ) since binary-coded GAs hold
(1)
(0)
(0)
only two types of allele frequency pk and pk with pk =(1)
1 − pk for any locus k.
The proposed strategies for determination of threshold θkare: for all k = 1, . . . , l,S TRATEGY 1: θk = pk
S TRATEGY 2: θk = 1 − pk
The bOB operators using the above strategies are denoted
as bOB(n, pk ) and bOB(n, 1 − pk ), respectively. The two
strategies result in contrary effects on the determination ofoffspring genes: bOB(n, pk ) makes the major alleles in the
population more difficult to be selected as offspring genes,
while bOB(n, 1 − pk ) makes them easier. That is to say,bOB(n, pk ) tends to pull the allele frequency back to theneural frequency 0.5. This effect is expected to allow more
exploration and relieve the premature convergence causedby the one-sided intensification on exploitation when using
more than two parents in OB-Scan. The bOB(n, 1 − pk ),Range of xi

N

Bits of xi

l[−2.048, 2.047]

10

12

120

[−5.12, 5.11]

10

10

100

[−512, 511]

10

10

100

TABLE IIPARAMETERS OF GA S USED IN THE EXPERIMENTS
Representation
GA modelPopulation size
Parent selection
CrossoverNumber of parents
Crossover rate pc
Mutation
Mutation rate pm
Survivor selection
Termination
Number of runs

Bit string
Generational GA
100
2-tournament
OB-Scan, bOB
2, . . . , 10
1.0
Bit-flip
1/lµ+λ
500 generations
100 runs per experimenton the other hand, encourages the preference for the major
alleles and pushes the allele frequency toward 0 or 1,
which will further increase the exploitation and thus cause
an faster but more likely premature convergence than OBScan. Through θk we can control the tendency of crossover
toward or against choosing major alleles as offspring genes;
consequently, the exploitation and exploration in bOB canbe regulated. The effects of the proposed bOB(n, pk ) andbOB(n, 1 − pk ) will be thoroughly examined in the next
section.
IV. P ERFORMANCE E VALUATIONThis work conducts a series of experiments to evaluate
the performance of the proposed bOB. To this end, we
compare the performance of GAs using uniform crossover,
OB-Scan, bOB(n, pk ), and bOB(n, 1−pk ), respectively. Notethat 2-parent OB-Scan corresponds to uniform crossover.
The effectiveness of bOB on enhancing OB-Scan will be
examined in terms of solution quality and convergence speed.
The test functions include the extended De Jong’s second function (F2e), the Rastrigin function (RAS), and the
Griewangk function (GRI). Table I describes of these test
functions and the related parameters. Note that the problem
is to find the global minimum of these test functions. The
setting of GAs for the experiments is summarized in Table II.
Each experiment includes 100 independent runs.
A. Solution Quality
Figure 2 compares the solution quality for OB-Scan,bOB(n, pk ), and bOB(n, 1 − pk ) using different numberof parents. The experimental results show that, in general,

70
OB(n)
bOB(n, pk)
bOB(n, 1-pk)

60

20
Fitness (RAS)

50
Fitness (F2e)OB(n)
bOB(n, pk)
bOB(n, 1-pk)

25

40
30

15

10

20
5
10
0

0
1

2

3

4

5
6
7
Number of parents (n)

8

9

10

11

1

2

3

4

5
6
7
Number of parents (n)(a) F2e

8

9

10

11

(b) RASOB(n)bOB(n, pk)
bOB(n, 1-pk)

1.8
1.6

Fitness (GRI)

1.4
1.2
1
0.8
0.6
0.4
0.2
0
1

2

3

4

5

6

7

8

9

10

11

Number of parents (n)(c) GRI
Fig. 2. Mean and standard error of the best fitness for GAs using OB-Scan, bOB(n, pk ), and bOB(n, 1 − pk ) with different number of parents on(a) the F2e, (b) the RAS, and (c) the GRI functionsbOB(n, pk ) outperforms OB-Scan while OB-Scan outperforms bOB(n, 1 − pk ) in terms of solution quality. The
preferable results of bOB(n, pk ) demonstrate its effectivenesson overcoming the issue of excessively intensified exploitation in OB-Scan. In addition, the performance of bOB(n, pk )becomes better as the number of parents increases. On theother hand, OB-Scan and bOB(n, 1 − pk ) using more than
two parents perform worse than their 2-parent versions. The
increase of parents worsens the resultant fitness for thesetwo crossovers. This poor outcome reflects the problem of
unbalance of exploitation and exploration in OB-Scan andespecially in bOB(n, 1 − pk ).
Table III accesses the improvement of bOB(n, pk ) on uniform crossover, which is the OB-Scan performing best in our
experiments. The table exhibits that GAs using bOB(n, pk )gain superiority in mean best fitness over GAs using uniform
crossover by 62–89% on the F2e, 16–38% on the RAS,
and 4–28% on the GRI functions. Moreover, we perform
a one-tailed t-test to check the statistical significance. With
confidence level α = 0.05, the p-values in Table III validatethat bOB(n, pk ) using 2 to 10 parents is capable of significant

improvement on uniform crossover in mean best fitness,
except bOB(2, pk ) on the GRI function. The results also
reveal that using more than two parents in bOB(n, pk )achieves more significant improvement on uniform crossover.
The significance, nevertheless, does not get greater absolutely
with the increase of parents: In all the three test functions,even though the p-values decreases as the number of parents
increases from 2 to 6, the tendency of p-values becomesinconclusive for bOB(n, pk ) using more than 6 parents.
B. Convergence Speed
In terms of convergence speed, we first look into the
influence of the number of parents on the convergence of
GAs using OB-Scan, bOB(n, pk ), and bOB(n, 1 − pk ),respectively. Due to the similarity of results on different
problems, we present here the results on the F2e function
only. Figure 3 demonstrates that the increase of parents has
different effects on the convergence speed of GAs using
these crossovers. In OB-Scan, increasing parents gains fasterconvergence in the very beginning, retards the convergencesoon, and leads to a worse solution quality in the end. SuchTABLE III
C OMPARISON OF THE SOLUTION QUALITY BETWEEN UNIFORM CROSSOVER AND bOB(n, pk ). T HE p- VALUES REPRESENT THE t- TEST RESULTS( SIGNIFICANT IMPROVEMENT OF bOB(n, pk ) ON UNIFORM CROSSOVER IS MARKED WITH BOLDFACE )Crossover
UniformbOB(n, pk )

n
2
2
3
4
5
6
7
8
9
10

Avg.
20.89
8.01
4.75
4.66
3.69
2.76
3.46
2.89
2.33
2.73

F2e (opt: 0)
Std.
P -value
16.70
8.26
2.34E-10
4.96
2.18E-15
6.85
8.70E-15
5.00
1.07E-16
3.28
2.06E-18
5.56
8.86E-17
3.78
4.16E-18
3.28
5.78E-19
3.48
2.14E-18

200

RAS (opt: 0)
Std.
P -value
2.81
2.12
2.55E-02
1.69
2.52E-05
1.82
9.21E-04
1.58
3.80E-06
1.63
9.56E-07
1.74
1.04E-04
1.81
4.56E-05
1.88
5.69E-04
1.73
6.11E-06

Avg.
4.30
3.61
2.91
3.23
2.78
2.66
3.03
2.94
3.17
2.78

Avg.
0.23860.22860.21050.18390.18040.17760.18110.17170.18360.2095

GRI (opt: 0)
Std.
P -value
0.09620.0822
2.15E-01
0.0746
1.14E-02
0.06594.36E-06
0.06299.48E-07
0.0623
3.14E-07
0.0745
3.73E-06
0.0742
1.45E-07
0.0731
7.58E-06
0.1043
2.13E-02

500

OB(2)
OB(3)
OB(4)
OB(5)
OB(10)bOB(2, 1-pk)
bOB(3, 1-pk)
bOB(4, 1-pk)
bOB(5, 1-pk)
bOB(10, 1-pk)

400

Fitness (F2e)Fitness (F2e)

150

100

300

200

50
100

0

0
0

50

100

150

200
250
300
Generations

350

400

450

500

0

50

100

150

200
250
300
Generations

350

400

450

500

(b) bOB(n, 1−pk )(a) OB-Scan(n)30bOB(2, pk)
bOB(3, pk)
bOB(4, pk)
bOB(5, pk)
bOB(10, pk)

25

Fitness (F2e)

20

15

10

5

0
0

50

100

150

200
250
300
Generations

350

400

450

500

(c) bOB(n, pk )Fig. 3. Convergence of GAs using (a) OB-Scan, (b) bOB(n, 1 − pk ), and (c) bOB(n, pk ) with different number of parents on the F2e function (Notethe different y-scales)

situation is a common symptom of premature convergence,caused by over-intensified exploitation. The bOB(n, 1 −pk ) further aggravates this problem and consequently haseven worse solution quality, compared to OB-Scan andbOB(n, pk ). However, bOB(n, pk ) can benefit from theincrease of parents: Figure 3(c) shows that both convergence
speed and solution quality improve with the number ofparents, ranging from 2 to 10. The preferable results confirm
the merit of using more parents in the proposed bOB(n, pk ).

Next, we examine the impacts of crossover on the convergence speed of GAs. Figure 4 compares the convergenceof GAs using OB-Scan, bOB(n, pk ), and bOB(n, 1 − pk )on the three test functions. Here the adopted number ofparents for each crossover is the number that leads to thebest solution quality respectively. The results in Fig. 4 showthat bOB(n, pk ) outperforms both OB-Scan and bOB(n, pk )in convergence speed as well as solution quality. Nonetheless,
GA using bOB(n, 1−pk ) is inferior to that using bOB(n, pk )

200

80

OB(2)
bOB(9, pk)
bOB(3, 1-pk)

60
Fitness (RAS)

150
Fitness (F2e)OB(2)
bOB(5, pk)
bOB(2, 1-pk)

70

100

50

50
40
30
20
10

0

0
0

20

40

60

80
100
120
Generations

140

160

180

200

0

20

40

60

(a) F2e80100120Generations

140

160

180

200

(b) RAS40

OB(2)bOB(8, pk)
bOB(2, 1-pk)

Fitness (GRI)

30

20

10

0
0

20

40

60

80

100

120

140

160

180

200

Generations(c) GRI
Fig. 4.

Comparison of convergence for GAs using OB-Scan, bOB(n, pk ), and bOB(n, 1 − pk ) on (a) the F2e, (b) the RAS, and (c) the GRI functions

or OB-Scan. These outcomes reconfirm the effectiveness ofsupermajority strategy in control of exploitation and exploration in the multi-parent crossover: bOB(n, pk ) is highly
capable of balancing the two forces but bOB(n, 1−pk ) exacerbates the problems of OB-Scan in lopsided intensification
on exploitation.
V. C ONCLUSION
The paper proposes the use of supermajority to deal with
the issue of exploitation and exploration in multi-parent
crossover. The proposed approach, called biased occurrencebased scanning crossover (bOB), determines offspring genes
using the θ-majority in place of the simple majority in OBScan. Through control over the threshold θ, we are able
to regulate the level of preference for major alleles in the
crossover. Two variants of bOB crossover, viz bOB(n, pk )and bOB(n, 1 − pk ), are developed on the basis of allele
frequency pk . These two crossovers determine offspring
genes according to the adaptive majority thresholds pk and1−pk , instead of fixed 12 in the OB-Scan. Hence, bOB(n, pk )makes major alleles more difficult while bOB(n, 1 − pk )makes them easier to be chosen as offspring genes.

Experimental results demonstrate the effects of the proposed crossovers on the performance of GAs. The 2-parentbOB(n, pk ) achieves significant improvement on n-parent
OB-Scan with n = 2, . . . , 10 in terms of solution quality
and convergence speed. Increasing the number of parents inbOB(n, pk ) further enhances the performance. In terms ofmean best fitness, GAs using bOB(n, pk ) outperform GAs
using uniform crossover, i.e. the OB-Scan performing bestin out experiments, by 62–89% on the F2e, 16–38% on the
RAS, and 4–28% on the GRI functions. These preferable
results of bOB(n, pk ) confirm the merits of supermajority
in overcoming the issue of over-intensified exploitation in
OB-Scan. The results also validate that the intensificationof exploitation and exploration by the increase of parents
in crossover can be advantageous — if they can be wellcontrolled, as bOB(n, pk ) does. On the other hand, theexperimental results exhibit that bOB(n, 1 − pk ) suffers frompremature convergence seriously, which reveals its excessive
intensification on exploitation over OB-Scan.
Several directions remain for future work. First, the present
work evaluates the performance of bOB crossover withmerely three test functions. To validate the generality ofbOB’s advantages, experiments on more test functions should
be carried out systematically. Second, bOB provides aneffective way by adjusting the threshold θ to control exploitation and exploration in crossover. Therefore, it establishes
a promising direction for future study on design of (multiparent) crossover for balance of the two driving forces andfor enhancement of the performance of GAs. Work toward
self-adaptive strategies and analysis on the influences of θ isunderway.
ACKNOWLEDGMENT
This work was partially supported by the National Science
Council of Taiwan.
R EFERENCES[1] T. Bäck and H.-P. Schwefel. An overview of evolutionary algorithms
for parameter optimization. Evolutionary Computation, 1(1):1–23,
1993.
[2] A.E. Eiben. Multiparent recombination. In Evolutionary Computation
1: Basic Algorithms and Operators, pages 289–307. Institute ofPhysics Publishing, 2000.
[3] A.E. Eiben. Multiparent recombination in evolutionary computing. In
Advances in Evolutionary Computing, pages 175–192. Springer, 2002.
[4] A.E. Eiben, P-E. Raué, and Zs. Ruttkay. Genetic algorithms withmulti-parent recombination. In Parallel Problem Solving from Nature– PPSN III, volume 866 of LNCS, pages 78–87. Springer, 1994.
[5] A.E. Eiben and C.H.M. van Kemenade. Diagonal crossover ingenetic algorithms for numerical optimization. Journal of Control
and Cybernetics, 26(3):447–465, 1997.
[6] A.E. Eiben, C.H.M. van Kemenade, and J.N. Kok. Orgy in the computer: Multi-parent reproduction in genetic algorithms. In Proceedingsof the Third European Conference on Artificial Life, volume 929, pages
934–945, 1995.
[7] J. Holland. Adaptation in Natural and Artificial Systems. Universityof Michigan Press, 1975.
[8] H. Kita, I. Ono, and S. Kobayashi. Multi-parental extension of theunimodal normal distribution crossover for real-coded genetic algorithms. In Proceedings of the Congress on Evolutionary Computation,
volume 2, pages 1581–1588. IEEE Press, 1999.
[9] H. Mühlenbein, M. Schomisch, and J. Born. The parallel genetic
algorithm as function optimizer. In Proceedings of the 4th International Conference on Genetic Algorithms, pages 271–278. Morgan
Kaufmann, 1991.
[10] H.-P. Schwefel. Evolution and Optimum Seeking. John Wiley & Sons,
Inc., 1995.
[11] G. Syswerda. Uniform crossover in genetic algorithms. In Proceedingsof the 3rd International Conference on Genetic Algorithms, pages 2–9.
Morgan Kaufmann, 1989.
[12] C.K. Ting. On the convergence of multi-parent genetic algorithms.
In Proceedings of the 2005 Congress on Evolutionary Computation,
volume 1, pages 396–403. IEEE Press, 2005.
[13] C.K. Ting. On the mean convergence time of multi-parent genetic
algorithms without selection. In Proceedings of the Eighth European
Conference on Artificial Life, volume 3630 of LNAI, pages 403–412.
Springer-Verlag, 2005.
[14] S. Tsutsui. Multi-parent recombination in genetic algorithms withsearch space boundary extension by mirroring. In Parallel Problem
Solving from Nature – PPSN V, volume 1498 of Lecture Notes in
Computer Science, pages 428–437. Springer, 1998.
[15] S. Tsutsui and A. Ghosh. A study on the effect of multi-parent
recombination in real coded genetic algorithms. In Proceedings ofInternational Conference on Evolutionary Computation, pages 828–
833, 1998.
[16] S. Tsutsui and L.C. Jain. On the effect on multi-parent recombinationin real coded genetic algorithms. In Proceedings of the 2nd International Conference on Knowledge-based Intelligent electronic systems,
pages 155–160, 1998.
[17] S. Tsutsui, M. Yamamura, and T. Higuchi. Multi-parent recombination
with simplex crossover in real coded genetic algorithms. In Proceedings of the Genetic and Evolutionary Computation Conference,
volume 1, pages 657–664. Morgan Kaufmann, 1999.

[18] H.-M. Voigt and H. Mühlenbein. Gene pool recombination and theutilization of covariances for the breeder genetic algorithm. In Proceedings of the Second IEEE International Conference on Evolutionary
Computation, pages 172–177. IEEE Press, 1995.