A Mating Strategy for Multi-parent Genetic Algorithms
by Integrating Tabu Search
Chuan-Kang Ting†

International Graduate School of Dynamic Intelligent Systems
Department of Computer Science, Electrical Engineering, and Mathematics
University Paderborn, 33098 Paderborn, Germany{ckting, kbcsl}@upb.de

Abstract- Multi-parent crossovers have been validated
their outperformance on several optimization problems.
However, there are two issues to be considered – thenumber of parents and the disruptiveness caused bymultiple parents. This paper presents a tabu
multi-parent genetic algorithm (TMPGA) to addressthese two issues by integrating tabu search into themating of multi-parent genetic algorithms. TMPGA
utilizes the tabu restriction and the aspiration criterion
to sift selected parents in consideration of population
diversity and selection pressure. Furthermore, theresulting mating validity further adjusts the number ofparents participating in a mating. Experiments are
conducted with four common test functions. The results
indicate that TMPGA can achieve better performance
than both two-parent GA and multi-parent GA with the
diagonal crossover.

1.

Hans Kleine Büning†

Introduction

Genetic algorithms (GAs) have been validated theiroutstanding performance on a variety of optimization
problems. The basic idea of GA is to simulate themechanisms in biological evolution such as selection,
crossover, and mutation [15]. Crossover is one of the mostsalient features in GA. It reproduces offspring by
exchanging and recombining genetic information from two
selected parents. This operation is believed capable of
exploring the problem space effectively.
The number of parents selected to crossover istraditionally set to two. This is reasonable because allsexual organisms on earth adopt two parents to reproduce
their offspring. However, it is possible for GAs to breakthrough such a limitation. Beyond two parents, Eiben et al.
proposed two multi-parent crossovers: scanning crossover[7] and diagonal crossover [8, 9]. These two methods
generalize uniform crossover and 1-point crossover
respectively. With several common test functions, theexperimental results indicate both scanning crossover anddiagonal crossover can achieve better performance thantheir two-parent versions, namely uniform and 1-pointcrossover. The results also reveal that diagonal crossoverholds a positive correlation between the accuracy and thenumber of parents from 11 to 15 parents. Such a correlation,
however, does not hold on the scanning crossover.

Moreover, Schippers [21] gave an advanced analysis on
the influence of scanning crossover upon genetic drift.
Besides, Mühlenbein and Voigt [19, 30] introduced thegene pool, which consists of several pre-selected parents.
Gene pool recombination (GPR) samples the genes used
for recombination from the whole gene pool instead of twoparents. The studies show that GPR and its variants areeasier to analyze and can converge faster than two-parent
recombination. Tsutsui and Jain [26] proposed multi-cut
crossover (MX) and seed crossover (SX). Multi-cut
crossover generalizes 2-point crossover and performs
better than diagonal crossover on De Jong’s test functions.
In addition to binary-coded GAs, Tsutsui and Ghosh [26,27] presented several multi-parent crossovers forreal-coded GAs: the center of mass crossover (CMX),
multi-parent feature-wise crossover (MFX), and seedcrossover (SX). Their experimental results reveal thatmulti-parent crossovers can lead to better performance inspite of its dependence upon problem types. Simplex
crossover (SPX) [29] was proposed to reproduce bysampling a simplex formed from multiple parents. The
results show its well performance with three or four parents
for multimodal or epistatic problems. Kita et al. [16]introduced multiple parents into the unimodal normal
distribution crossover (UNDX) to enhance the diversity ofoffspring. This multi-parent extension of UNDX
demonstrates its improvement in search ability on highly
epistatic problems.
For multi-objective optimization problems, Lis andEiben [17] introduced the concept of sex into multi-parent
crossover. The sex or gender, which is appended in eachchromosome, indicates a specific criterion to optimize; thatis, the fitness is evaluated by the corresponding objectivefunction of sex. Esquivel et al. [11, 12] extended thismethod to MSPC-GA by allowing multiple parents per sex
and multiple crossovers per mating. This approach obtains
a satisfactory result in the number of non-dominatedsolutions on the Pareto front. The study [18] further
enhances this method by incorporating local search andachieves better results.
Two excellent overviews of multi-parent crossovers canbe found in Eiben’s studies [4, 5]. These researches havedemonstrated the effectiveness of multi-parent crossovers
in binary-coded or real-coded GAs on functional
optimization problems as well as multi-objective problems.
However, there are two advanced issues for multi-parent

crossovers. The first issue is the increasing disruptivenesscaused by multiple parents. This disruptiveness on the one
hand leads to a more diverse exploration which can prevent
premature convergence, but on the other hand, it slows
convergence speed at the same time. Second, the number ofparents adopted is more than two but nevertheless, istraditionally set fixed in multi-parent crossovers. Adaptive
methods for two-parent crossover or mutation haveobtained many successful results in GAs [6, 14, 23].
Accordingly, the idea of adaptively tuning the number ofparents emerges. In this paper, we integrate the strategy oftabu search (TS) into multi-parent crossover to addressthese two issues. The proposed method, called tabu
multi-parents genetic algorithm (TMPGA), uses the tabu
restriction and the aspiration criterion to sift the mating ofmultiple parents. The advantage of such a strategy has beenverified for two-parent crossover to harmonize selection
pressure and diversity maintenance [24, 25]. Furthermore,
the tabu strategy adjusts the number of parents according to
the condition of mating pool. Consequently, thedisruptiveness caused by multi-parent crossover iscontrolled in consideration of exploitation and exploration.
The effectiveness of TMPGA is examined by several
experiments and comparisons with traditional two-parent
crossover and multi-parent diagonal crossover on fourcommon test functions.
The rest of this paper is organized as follows. In the next
section, we give a detailed description of the proposed
TMPGA. Section 3 presents performance comparisons andresult analysis of TMPGA. Finally, conclusions are drawn
in Section 4.

2.

Tabu Multi-Parent GA (TMPGA)

TMPGA integrates the strategy of tabu search into themating strategy of multi-parent genetic algorithms. First,
the tabu list restricts the mating in an incest-prevention
manner to maintain diversity. Second, the aspiration
criterion releases the restriction on mating to supply
selection pressure. Such a synergy of tabu list andaspiration criterion is expected to achieve a harmony indiversity maintenance and selection pressure. Moreover,
the outcome of mating affects the number of parents
participating in multi-parent crossover. Therefore, by the
integration of TS, a harmony in mating and a further
control on the disruptiveness for multi-parent GA are
obtained. However, some corresponding modifications arenecessary for GAs to incorporate the strategy of TS. The
following subsections provide more detailed descriptions
about the components for multi-parent GA to combine thestrategy of TS. The proposed algorithm TMPGA ispresented afterward.
2.1. Representation
Tabu Search (TS) uses a memory structure, called tabu list,
to guide the search in consideration of diversification andintensification [13]. To accommodate such a memory
structure to GA, two components are necessarily appended

to the representation of chromosomes. First, a clan number
is introduced to identify chromosomes. This number isassigned uniquely at the initialization stage. During
reproduction offspring inherit the clan number from one of
their parents. The second component is the tabu list, which
records a set of forbidden clans to mate. Figure 2.1
illustrates a representation in binary-coded GAs. The genes
are encoded in bit strings concerning the solutions of the
optimizing problem. Additionally the clan (8) with the tabu
list (2, 6) carries the information for tabu mating strategy.

11010011……01001011 8 2 6
genes

clan

tabu list

Figure 2.1 Representation of chromosomes

2.2. Mating Strategy
In TMPGA, the mating of multiple parents is not unbridled
but is restricted by the strategy of tabu search. This
restriction further affects the number of parents
participating in a reproduction.
The mating strategy of TMPGA is made up of two
forces in TS, specifically, the tabu restriction and theaspiration criterion. First, the clans indicated in the tabu list
mean that these clans are forbidden for one to mate. A
check is performed on the selected parents to see whether
the mating of these parents is forbidden or not. Such a
restrictive mechanism is helpful to maintain population
diversity through an incest-prevention manner [2, 10, 22].
However, for a moderate restriction and a reduction ofcomputation, TMPGA adopts the concept of polygamy; thetabu-checking procedure is only performed upon the first
selected parent with other selected parents. As Figure 2.2
shows, the tabu restriction occurs when the first parent
detect its clan appears in another parent’s tabu list, and viceversa. Due to the tabu restriction, this mating is judged
invalid unless the following aspiration criterion is satisfied.

Parent 1

110101……001011 8 2 6
Tabu!

Parent 2

011001……011101 3 4Parent 3

100011……101011 4 1 2 8
Figure 2.2 Tabu checking procedures

The aspiration criterion is another measure used tojudge mating. This criterion defines that the tabu restriction
for a mating is overridden if any of offspring reproduced bythis mating is superior to the best chromosome so far. In
other words, a mating is allowed by the aspiration criteria

even though the mating is forbidden by the tabu restriction.
This release from restriction provides moderate
reinforcement in selection pressure. The interaction of thetabu restriction and the aspiration criterion consequently
achieves a harmony in diversity maintenance and selection
pressure.
A mating is classified invalid and is not allowed if thismating incurs a tabu restriction but is unable to meet theaspiration criterion. Only offspring from a valid mating canbe put into subpopulation. Besides, after a valid mating,parents have to update their tabu lists: the first parent adds
all mates’ clans to its tabu list while the rest of the parents
just add the first parent’s clan to their tabu lists. Afterward,
offspring inherit the updated tabu list from one of theirparents. The operation of updating the tabu list works in a
FIFO manner. The oldest clans will be released from thetabu list while some new forbidden clans are added. These
released clans regain the permission for one to mate. Figure
2.3 illustrates the procedures of updating the tabu list. The
first parent adds the clans (3, 4) of the mates to its tabu list;
other parents only add the clan (8) of the first parent to theirrespective tabu list. The released clan (6) means that thefirst parent regains the permission to mate thechromosomes with clan (6).

Parent 1 110101……00101 8

3 4 2

release6

updateParent 2

011001……01110 3 8 4Parent 3 100011……10101 4

8 1 2

Figure 2.3 Update of tabu list

TMPGA further uses the resulting validity of mating tocontrol the number of parents. If a mating is valid, its
offspring will be put into subpopulation normally.
Otherwise, in case of an invalid mating, TMPGA willremove all the tabu parents to enable this mating valid, andthen re-apply multi-parent crossover with the remaining
parents to reproduce offspring. Provided all mates of thefirst parent are removed, the selection operator will
re-choose another mate from population to keep the mating
has at least two parents. Accordingly, the number ofparents varies with the situation of population. In the
beginning of evolution, the parent number is large because
the mating is less likely to incur a tabu restriction and has ahigher possibility to meet the aspiration criterion. Thislarge number of parents leads to a more diverse searcharound the problem space [8]. Such an enhanced
exploration can contribute to a better solution quality. Next,
the number of parents is on the decrease with convergence.
During evolution the population becomes more and moresimilar and then results in a higher probability to getinvalid mating. Correspondingly, the number of parents

will decrease with the growing number of tabu parents in
an invalid mating. Finally, the number of parents will
decline to two; at that time, multi-parent crossover will
degenerate to traditional two-parent crossover. Thisdeclining procedure keeps the search from violent
perturbation and allows it more exploitation when the
search approaches the promising region of problem space.
On the whole, this adaptive strategy of varying parent
number will take exploration and exploitation into accountin different phases of evolution.
2.3. TMPGA Algorithm
The proposed algorithm TMPGA is described in Figure 2.4.
Most procedures of TMPGA follow the originalarchitecture of GA except the screening process, i.e. steps
12~13, conducted by the tabu strategy. Additionally, a
special rule is designed for the variation of parent number n
in order to reduce the computation cost of the tabu
checking. This rule initially sets the trial number of parents
to a maximal value and then changes the number to thelargest parent number of valid mating in each generation.
As the trial number is reduced to two, the checks of tabu
restriction and aspiration criterion are omitted because at
this time multi-parent crossover degenerates to a
two-parent version. The reason for this rule is that the
population will gradually lose its diversity and yield ahigher and higher probability of invalid mating; that is tosay, it takes increasing computation on the removal of tabu
parents. By the rule, the number of parents is definedaccording to the situation of population in the preceding
generation. Generally the rule causes fewer and fewer
parents participating in a mating. Hence, the number oftrivial tabu checking can be significantly reduced,
especially when the population loses most of its diversity
and the probable number of parents is two.
TMPGA( )1t ←0// t :generation2
initialize P (t )// P (t ) :population
3
evaluate P(t )4n ← MAX_PARENTS// n :parent number
5
while (not terminated) do
6t ← t +1
7
while ( P(t ) is not filled) do
8p1Kn ← select(P(t − 1))// p :parents
9c1Km ← reproduce( p1Kn )// c :offspring
10
evaluate c1Km
11
if ( n > 2 ) then12
if ( Tabu( p1Kn ) ) and//[Tabu](not Aspiration(c1Km ) ) then//[Aspire]13
p1Kn′ ← removeTabu( p1Kn )14
c1Km ← reproduce( p1Kn′ )15P(t ) ← P(t ) ∪ c1Km
16
P(t ) ← survival(P(t − 1), P(t ))17n ← the largest n of valid mating
Figure 2.4 The algorithm TMPGA

3.

Performance Evaluation

In this paper, several experiments are conducted to evaluate
the performance of TMPGA. A test suite consisting of fourcommon test functions [1, 3, 19] is implemented for eachexperiment: De Jong’s second test function (F2), theRastrigin (RAS), the Schwefel (SCH), and the Griewangk(GRI). Table 3.1 presents these test functions and therelated parameters used in our experiments. Besides,
classic two-parent GA and multi-parent GA (MPGA) are
implemented to verify the superiority of TMPGA.
The schemes of GA employed in our experiments are
generational GA, bit-string representation, roulette wheel
selection, flip mutation, and elitism. A population size of
200 chromosomes is used for all problems. Eachexperimental setting includes 50 trials; each trial runs 100
generations. Population is initialized randomly at each run.
For all algorithms the crossover rate Pc is fixed to 1.0 andthe mutation rate Pm is set to (1/chromosome_length).
Besides, traditional 1-point crossover is adopted for GA,
and diagonal crossover is used for MPGA. The algorithm
TMPGA does not prescribe the option of multi-parent
crossover; however, in order to give a fair comparison, thecrossover that TMPGA adopts here follows the same
multi-parent crossover, namely diagonal crossover, as
MPGA in all experiments. The diagonal crossover of
MPGA adopts 15 parents because of its best performance
among our several experiments on the extent of 11 to 15
parents. For TMPGA, the trial parent number is also set to
15 initially, and then it is adjusted adaptively by thestrategy of tabu search as mentioned in Chapter 2. The sizeof tabu list in TMPGA is empirically set to 10 for areasonable restriction.
3.1. Performance Comparison
Figure 3.1 compares the convergence of TMPGA with thatof GA and diagonal-crossover (MPGA) on four test
functions. The results show that TMPGA converges fasterthan both GA and MPGA on all functions except the firsthalf of convergence on function F2. Besides, MPGA

converges faster than GA on functions SCH and GRI but
slower than GA on F2 and RAS. In terms of solution
quality, TMPGA yields better solutions than GA does on allfunctions and MPGA does on function RAS. Nevertheless,the differences of the best solutions obtained from MPGA
and TMPGA are not significant on functions F2, SCH, and
GRI.
The results shown in Figure 3.1 are compared withrespect to generations. TMPGA, however, spends
additional computation on tabu checking and invalid
mating in each generation. Thus, the comparison by
generations may not be so fair. Considering this, we further
compare convergence in terms of running time. Figure 3.2
depicts the convergence comparisons regarding the time
taken in each generation. The comparing algorithms arecoded in C language and run on an Intel Pentium III –1.7GHz machine. From Figure 3.2 we conclude that theadditional computation somewhat decreases TMPGA’s
outperformance upon convergence speed if compared withthe results in Figure 3.1. Nevertheless, TMPGA stillachieve a faster convergence than GA does on functions
RAS, SCH, and GRI. TMPGA also converges faster than
MPGA on all test functions even though the convergenceof TMPGA and MPGA are close on functions SCH and
GRI. This superiority supports that the extra computation
cost of TMPGA is worthwhile in terms of convergence
speed.
Table 3.2 quantifies the improvements of MPGA andTMPGA in the convergence speed of GA. The valuest MPGA and tTMPGA respectively indicate the generations(time) taken for MPGA and TMPGA to achieve GA’s best
fitness in our experiments. The results in Table 3.2 showthat TMPGA converges faster than GA by 24 to 66
generations or by 0.881 to 3.352 seconds; in other words,
TMPGA saves 22% to 69% of GA’s convergence time.
MPGA also converges faster than GA except theconvergence on RAS, in which MPGA performs similar to
GA. Altogether, these comparisons further demonstratethat TMPGA is capable of a more efficient convergence
and a more constant outperformance than that of MPGA.

Table 3.1 Test functions

Functions() + (1 − x )
2

2

F2

f = 100 x12 − x 22

RAS

f = 10 N + ∑ xi2 − 10 cos(2πxi )

N

1

(

)

Range of xi

N

Bits of xi

Chrom. length[-2.048, 2.047]

2

12

24

[-5.12, 5.11]

20

10

200

[-512, 511]

10

10

100

[-512, 511]

10

10

100

i =1

N

SCH

f = ∑ − x sin
i =1

(x)
i

N

xi2
x 
− ∏  cos( i ) 
4000
i 
i =1
i =1 
N

GRI

f = 1+ ∑0.035

200

GA
MPGA
TMPGA

GA
MPGA
TMPGA

180

0.03
160
140
Fitness (RAS)

Fitness (F2)

0.025

0.02

0.015

120
100
80
60

0.01

40
0.005
20
0

0

10

20

30

40

50

60

70

80

90

0

100

0

10

20

30

40

Generation

50

60

70

80

90

100

Generation

1800

30

GA
MPGA
TMPGA

1600

GA
MPGA
TMPGA

25
1400
20
Fitness (GRI)

Fitness (SCH)

1200
1000
800
600

15

10

400
5
200
0

0

10

20

30

40

50

60

70

80

90

0

100

0

10

20

30

40

Generation

50

60

70

80

90

100

Generation

Figure 3.1 Comparison of GA, MPGA, and TMPGA in terms of generations
200

0.035

GA
MPGA
TMPGA

GA
MPGA
TMPGA

180

0.03

160
140
Fitness (RAS)

Fitness (F2)

0.025

0.02

0.015

120
100
80
60

0.01

40
0.005

20
0

0

0.5

1

1.5

2

2.5
3
Time (sec)

3.5

4

4.5

5

0

5.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec)

1800

30

GA
MPGA
TMPGA

1600

GA
MPGA
TMPGA

25
1400
20
Fitness (GRI)

Fitness (SCH)

1200
1000
800
600

15

10

400
5
200
0

0

0.5

1

1.5

2
2.5
Time (sec)

3

3.5

4

4.5

0

0

0.5

1

1.5

2
2.5
Time (sec)Figure 3.2 Comparison of GA, MPGA, and TMPGA in terms of running time

3

3.5

4

4.5

Table 3.2 Generations (time) for MPGA and TMPGA to achieve the best fitness of GA

t GA

t MPGAtTMPGA∆t MPGA−GA (%)∆tTMPGA−GA (%)

F2

96 (4.840)

53 (2.036)

35 (1.488)

44.8 (57.9)

63.5 (69.3)

RAS

100 (4.398)

97 (4.277)

62 (3.041)

3.0 (2.8)

38.0 (30.9)

SCH

100 (4.207)

38 (1.478)

34 (1.470)

62.0 (64.9)

66.0 (65.1)

GRI

100 (4.009)

84 (3.289)

76 (3.128)

16.0 (18.0)

24.0 (22.0)

t X : generations (time) for the algorithm X to achieve GA’s best fitness∆t X −GA (%): relative deviation of generations (time) t X and tGA

3.2. Analysis of Results
Furthermore, we examine the impact of tabu restriction andaspiration criterion upon TMPGA’s performance andnumber of parents. As Figure 3.3 shows, the numbers oftabu events peak around 40 generations on F2 and around
20 to 25 generations on RAS, SCH, and GRI. In
comparison of this circumstance with the convergence of
TMPGA on Figure 3.1, we find that the peak of tabu events
corresponds to the point starting slow convergence. We
attribute such a circumstance to the exhaustion of
population diversity. During evolution the population
gradually loses its diversity and becomes more and moresimilar. When the population is so similar that it fills a lotof chromosomes with the same clans, the probability oftabu events will raise correspondingly. On the other hand, it
reflects that the population is too similar to explore theproblem space effectively. The convergence of TMPGA,
therefore, gets slow in correspondence with the peak oftabu events.
Figure 3.4 depicts the variation in the parent number of
TMPGA on four test functions. Clearly, the number ofparents decreases with evolution until two. This tendency
results from the decreasing population diversity. As

100mentioned in Chapter 2, the validity of mating determines
the number of parents in TMPGA. Initially the population
is diverse and the best solution obtained is far from theglobal optimum; thus it is less likely to incur a tabu
restriction but easier to meet the aspiration criterion. As
shown in Figure 3.3, the beginning has only a few tabu
events and relatively many aspiration events. In the courseof evolution, the probability of valid mating gets lower andlower because of the growing tabu events and the declining
aspiration probability. On the whole, the number of parents
declines with the evolution as Figure 3.4 shows. Hence, thedisruptiveness caused by multi-parent crossover is
progressively restrained, especially when the search
approaches to the promising region. However, the effect oftabu strategy upon the number of parents is not one-way
but interactive. The tabu restriction reduces the number ofparents while this reduced number of parents will further
lower the probability to incur tabu restriction in the next
generation. All in all, the synergy of tabu restriction andaspiration criterion achieves not only a harmonious mating
but an adaptive restraint on disruptiveness; consequently, itleads to the outperformance of TMPGA.

2

F2
RAS
SCH
GRI

90

1.6
Number of aspiration events

Number of tabu events

80
70
60
50
40

1.4
1.2
1
0.8

30

0.6

20

0.4

10

0.2

0

F2
RAS
SCH
GRI

1.8

0

10

20

30

40

50
Generation

60

70

80

90

100

0

0

10

20

30

40

50

60

70

80

90

Generation

Figure 3.3 Variation in number of tabu events (left) and aspiration events (right) for TMPGA on four test functions

100

16

F2
RAS
SCH
GRI

14

Number of parents

12

10

8

6

4

2

0

10

20

30

40

50

60

70

80

90

100

Generation

Figure 3.4 Variation in the number of parents for TMPGA on four test functions

4.

Conclusions

This work presents a tabu multi-parent genetic algorithm(TMPGA), which integrates the strategy of tabu search into
the mating of parents for the multi-parent crossover. First,
an additional memory structure consisting of a clan number
and a tabu list is appended to the representation ofchromosomes. This memory structure records the
trajectory of evolution and further acts as the basis ofmating strategy. The tabu restriction contributes tomaintain diversity in an incest-prevention manner byforbidding parents to mate with the chromosomes on thetabu list. The aspiration criterion, on the other hand,
supplies moderate selection pressure by overriding the tabu
restriction for those superior solutions. TMPGA utilizes thesynergy of tabu restriction and aspiration criterion tocontrol the mating of multiple parents for a good balance ofdiversity maintenance and selection pressure. The outcomeof mating further adaptively adjusts the number of parent
for the multi-parent crossover. Therefore, thedisruptiveness caused by multiple parents is restrained
progressively in the course of evolution.
The effectiveness of TMPGA is verified by theexperiments on four common test functions. The resultspresent the influence of tabu restriction and aspiration
criterion upon the number of parents; nevertheless, theparent number affects tabu restriction as well. TMPGA’s
well performance reflects the advantages of such aninteraction. The performance comparisons indicate that
TMPGA achieves faster convergence and better solution
quality than classic two-parent GA and multi-parent GA
with diagonal crossover on the test functions. Altogether,
these favorable results demonstrate the superiority of
TMPGA due to the integration of tabu search into themating strategy of multi-parent genetic algorithms.
However, more tests and analyses are needed to confirm
the general outperformance of TMPGA. This paper onlyuses the diagonal crossover as the multi-parent crossover in
TMPGA. More multi-parent crossovers such as scanningcrossover or multi-cut crossover are needed to test forvalidating TMPGA’s effectiveness on multi-parent GAs.
On the performance evaluation, we compare theapproaches in terms of running time to present the general
influence of additional computation in tabu and aspiration
checking upon the convergence. Nevertheless, othercriterion, such as the number of function evaluations,should be further considered. Besides, in this paper,
performance is only evaluated experimentally withfunctional optimization problems. Currently we areconducting a series of comprehensive experiments on thecomponents of TMPGA. Theoretical analysis and further
improvements in TMPGA are also underway.

Acknowledgements
The authors would like to thank the anonymous reviewers
for their valuable comments. This work was supported by
the International Graduate School of Dynamic Intelligent
Systems, University Paderborn, Germany.

References[1] T. Bäck and Z. Michalewicz, “Test landscapes,”
Handbook of Evolutionary Computation, pp.
B2.7:14-B2.7:20, 1997.
[2] R. Craighurst and W. Martin, “Enhancing GA
performance through crossover prohibitions based onancestry,” Proceeding of the 6th International
Conference on Genetic Algorithms, pp. 130-135, 1996.
[3] K.A. DeJong, “An analysis of the behavior of a class ofgenetic adaptive Systems,” PhD thesis, University ofMichigan[4] A.E. Eiben, “Multiparent recombination,” Evolutionary
Computation 1: Basic Algorithms and Operators, pp.
289-307, Institute of Physics Publishing, 2000.

[5] A.E. Eiben, “Multiparent recombination in
evolutionary computing,” Advances in Evolutionary
Computing, Natural Computing Series, Springer, 2002.
[6] A.E. Eiben, R. Hinterding, and Z. Michalewicz,“Parameter control in evolutionary algorithms,” IEEE
Transaction on Evolutionary Computation, vol. 3, no. 2,
pp. 124-141, 1999.
[7] A.E. Eiben, P-E. Raué, and Zs. Ruttkay, “Genetic
algorithms
withmulti-parent
recombination,”
Proceedings of the 3rd Conference on Parallel Problem
Solving from Nature, pp. 78-87, 1994.
[8] A.E. Eiben, C.H.M. van Kemenade, and J.N. Kolk,“Orgy in the computer: multi-parent reproduction in
genetic algorithms,” Proceedings of the 3rd European
Conference on Artificial Life, pp. 934-945, 1995.
[9] A.E. Eiben and C.H.M. van Kemenade, “Diagonalcrossover in genetic algorithms for numerical
optimization,” Journal of Control and Cybernetics, vol.
26(3), pp.447-465, 1997.
[10] L.J. Eshelman and J.D. Schaffer, “Preventing
premature convergence in genetic algorithms by
preventing incest,” Proceeding of the 4th International
Conference on Genetic Algorithms, pp. 115-122, 1991.
[11] S.C. Esquivel, H.A. Leiva, R.H. Gallard, “Multiplicity
in genetic algorithms to face multicriteria
optimization,” Proceedings of the 1999 Congress on
Evolutionary Computation, 1999, vol. 1, pp. 85-90,
1999.
[12] S.C. Esquivel, H.A. Leiva, R.H. Gallard, “Multiple
crossovers between multiple parents to improve searchin evolutionary algorithms,” Proceedings of the 1999
Congress on Evolutionary Computation, vol. 2, pp.
1589-1594, 1999.
[13] F. Glover and M. Laguna, “Tabu search,” Kluwer
academic publishers, 1997.
[14] R. Hinterding, Z. Michalewicz, and A.E. Eiben,“Adaptation in evolutionary computation: a survey,”
IEEE International Conference on Evolutionary
Computation, pp. 65-69, 1997.
[15] J. H. Holland, “Adaptation in natural and artificial
systems,” University of Michigan Press, 1975.
[16] H. Kita, I. Ono, and S. Kobayashi, “Multi-parental
extension of the unimodal normal distribution
crossover for real-coded genetic algorithms,”
Proceedings of the 1999 Congress on Evolutionary
Computation, vol. 2 , pp. 1581-1588, 1999.
[17] J. Lis and A.E. Eiben, “A multi-sexual genetic
algorithm
formultiobjective
optimization,”
Proceedings of the 4th IEEE Conference on
Evolutionary Computation, pp. 59-64, 1997.
[18] H.A. Leiva, S.C. Esquivel, R.H. Gallard, “Multiplicity
and local search in evolutionary algorithms to build thePareto front,” Proceedings. XX International
Conference of the Chilean Computer Science Society,pp. 7-13, 2000.
[19] H. Mühlenbein, M. Schomisch, and J. Born, “Theparallel genetic algorithm as function optimizer,”
Parallel Computing, vol. 17, pp.619-632, 1991.

[20] H. Mühlenbein and H.-M. Voigt, “Gene pool
recombination in genetic algorithms,” Proceedings of
6th International Conference on Genetic Algorithms, pp.
104-113, 1995.
[21] C.A. Schippers, “Multi-parent scanning crossover andgenetic drift,” Theoretical Aspects of Evolutionary
Computing, pp. 307-330, Springer, 1999.
[22] H. Shimodaira, “DCGA: a diversity control oriented
genetic algorithm,” IEEE International Conference on
Tools with Artificial Intelligence, pp. 367-374, 1997.
[23] M. Srinivas and L.M. Patnaik, “Adaptive probabilities
of crossover and mutation in genetic algorithms,” IEEE
Transaction on Systems, Man, and Cybernetics, vol. 24,
no. 4, pp. 656-667, 1994.
[24] C.K. Ting, S.T. Li, and C.N. Lee, “TGA: a new
integrated approach to evolutionary algorithms,”
Proceeding of the 2001 IEEE Congress on
Evolutionary Computation, pp. 917-924, 2001.
[25] C.K. Ting, S.T. Li, and C.N. Lee, “On the harmonious
mating strategy through tabu search,” Information
Sciences, 2003.
[26] S. Tsutsui, “Multi-parent recombination in genetic
algorithms with search space boundary extension by
mirroring, ” Proceedings of the 5th International
Conference on Parallel Problem Solving from Nature,
pp. 428-437, 1998.
[27] S. Tsutsui and A. Ghosh, “A study on the effect ofmulti-parent recombination in real coded genetic
algorithms, ” Proceedings of the 1998 IEEE
International Conference on Evolutionary Computation,
pp. 828-833, 1998.
[28] S. Tsutsui and L.C. Jain, “On the effect ofmulti-parents recombination in binary coded genetic
algorithms,” Proceedings of the 2nd International
Conference on Knowledge-based Intelligent Electronic
Systems, pp. 155-160, 1998.
[29] S. Tsutsui, M. Yamamura, and T. Higuchi,“Multi-parent recombination with simplex crossoverin real coded genetic algorithms,”Proceedings of the1999 Genetic and Evolutionary Computation
Conference, pp. 657-664, 1999.
[30] H.-M. Voigt and H. Mühlenbein, “Gene pool
recombination and utilization of covariances for the
breeder genetic algorithm,” Proceedings of the 2nd IEEE
International Conference on Evolutionary Computation,
pp.172-177, 1995.