A Study of Multi-parent Crossover Operators in
a Memetic Algorithm
Yang Wang, Zhipeng L¨u
, and Jin-Kao Hao
LERIA, Universit´e d’Angers, 2 Boulevard Lavoisier, 49045 Angers Cedex 01, France
{yangw,lu,hao}@info.univ-angers.fr
Abstract. Using unconstrained binary quadratic programming problem
as a case study, we investigate the role of multi-parent crossover operators
within the memetic algorithm framework. We evaluate the performance
of four multi-parent crossover operators (called MSX, Diagonal, U-Scan
and OB-Scan) and provide evidences and insights as to why one particu-
lar multi-parent crossover operator leads to better computational results
than another one. For this purpose, we employ several indicators like
population entropy and average solution distance in the population.
Keywords: multi-parent crossover, memetic algorithm, unconstrained
quadratic programming, performance analysis.
1 Introduction
Memetic algorithms (MA) are known to be one of the highly effective metaheuris-
tic approaches for solving a large number of constraint satisfaction and optimiza-
tion problems [1]. One of the most important features of a MA is the crossover
operator for generating offspring solutions. In general, meaningful crossover op-
erators help to create healthy diversification in the population and to avoid a
premature convergence of the population.
In this paper, we provide a case study of multi-parent crossover operators
within memetic algorithms for the unconstrained binary quadratic programming
(UBQP) that can be written
UBQP: Maximize f(x)=x
Qx
x binary
where Q is an n by n matrix of constants and x is an n-vector of binary variables.
The formulation UBQP is notable for its ability to represent a wide range
of important problems [2]. The literature reports a number of evolutionary
and memetic algorithms with two-parent crossover operators for solving the
UBQP problem ([3,4,5,6]). However, one finds no studies concerning multi-parent
crossover operators for UBQP, where multi-parent crossover operators generate
offspring solutions by combining more than two parent solutions. In this work,
we are particularly interested in investigating the role of multi-parent crossover
operators as well as a number of related important questions: why does one par-
ticular multi-parent crossover operator lead to better computational results than
Corresponding author.
R. Schaefer et al. (Eds.): PPSN XI, Part I, LNCS 6238, pp. 556–565, 2010.
c
Springer-Verlag Berlin Heidelberg 2010
A Study of Multi-parent Crossover Operators in a Memetic Algorithm 557
another one? What are the main characteristics of a good multi-parent crossover
operator? To what extent can the crossover operators influence the performance
of the memetic algorithms?
Without claiming to answer all these questions, we present an experimental
analysis of various multi-parent crossover operators within a memetic algorithm.
For this purpose, we use four multi-parent crossover operators, respectively called
MSX, Diagonal, U-Scan and OB-Scan. The last three ones are well known in
the literature while the first one is recently proposed in [7]. The analysis shows
that the computational results are strongly correlated with the characteristics of
the corresponding crossover operators, such as the entropy and average solution
distance of the population, the average solution quality in the population. Fur-
thermore, the analysis sheds light on how a tradeoff between local search and
crossover operator can be achieved when using different crossover operators.
2 Multi-parent Crossover within Memetic Algorithms
2.1 Main Scheme and Initial Population
This study is based on the general memetic framework described in Algorithm 1
that alternates between a combination operator and a local improvement proce-
dure. The combination operator (Section 2.4) is used to generate new offspring
solutions while the local improvement procedure based on tabu search (Section
2.2) aims at optimizing each offspring solution. As soon as an offspring solution
is improved by tabu search, the population is accordingly updated based on two
criteria: the solution quality and the diversity of the population. The individuals
of the initial population are generated randomly (i.e., each variable x
i
of the
n-vector x receives a value of 0 or 1 with equal probability).
Algorithm 1. Pseudo-code of the memetic algorithm
1: Input:matrixQ
2: Output: the best solution x
∗
found so far
3: P = {x
1
,...,x
p
}←Population Initialization( )
4: for i = {1,...,p} do
5: x
i
← Tabu Search(x
i
)
6: end for
7: x
∗
= arg max{f (x
i
)|i =1,...,p}
8: repeat
9: randomly choose a subset of individuals E (|E|∈[4, 8])fromP
10: x
0
← Crossover Operator(E)
11: x
0
← Tabu Search(x
0
)
12: if f (x
0
) >f(x
∗
) then
13: x
∗
= x
0
14: end if
15: P ← Pool
Updating(x
0
,P)
16: until a stop criterion is met
558 Y.Wang,Z.L¨uandJ.-K.Hao
2.2 TabuSearchProcedure
In this paper, we employ a simple tabu search algorithm as our local search
procedure. Our tabu search procedure uses a neighborhood defined by the simple
one-flip move, which consists of changing (flipping) the value of a single variable
x
i
to its complementary value 1 −x
i
. The implementation of this neighborhood
uses a fast incremental evaluation technique [9] to calculate the cost (move value)
of transitioning to each neighboring solution.
Tabu search incorporates a tabu list as a “recency-based” memory structure
to assure that solutions visited within a certain span of iterations, called the
tabu tenure, will not be revisited [10]. In our implementation, we elected to set
thetabutenureasTabuTenure(i)=tt + rand(10), where tt is a given constant
(n/100) and rand(10) takes a random value from 1 to 10. Our tabu search method
stops when a given number α of moves are reached, called depth of tabu search.
2.3 Pool Updating
In our memetic algorithm, after an offspring x
0
is obtained by the crossover
operator and improved by tabu search, we decide whether the improved offspring
should be inserted into the population, replacing the existing worst solution. For
this purpose, we define a quality-and-distance goodness score of the offspring x
0
with respect to the population. The main idea is to favor the inclusion of x
0
in the
population if x
0
is “good enough” (in terms of its objective function evaluation)
and is not too similar to any solution currently in the population.
Our aim is not only to maintain a pool of good quality solutions but also to
emphasize the importance of the diversity of the solutions to avoid a premature
convergence of the population. Therefore, if the goodness score of the offspring
solution is good enough, it will have high probability to replace the worst solution
in the population. Interested readers are referred to [7] for more details.
2.4 Combination Operators
In this paper, we use four multi-parent crossover (or combination) operators to
generate offspring solutions, including a “logic” multi-parent combination opera-
tor (MSX), a diagonal multi-parent crossover (Diagonal), a multi-parent uniform
scanning crossover (U-Scan), a multi-parent occurrence based scanning crossover
(OB-Scan). Note that except MSX which is recently proposed for UBQP [7], the
last three ones have been widely used for other combinatorial optimization prob-
lems in the literature [11,12].
All the four combination operators used in our algorithm are applied on a set
E of s (|E| = s) parent solutions randomly selected from the current population,
i.e., E = {x
(1)
,...,x
(s)
},wherex
(i)
=(x
(i)
1
,...,x
(i)
n
). In our implementation,
we set s to be a random number between 4 and 8.
MSX Crossover (MSX): we define a weight w(i) for the solution x
(i)
and a
strength value Strength(j) for variable x
j
as: w(i)=1/
n
j=1
x
(i)
j
and Strength
(j)=
s
i=1
w(i)x
(i)
j
.
A Study of Multi-parent Crossover Operators in a Memetic Algorithm 559
The value Strength(j) gives a relative indication of the tendency of the solu-
tions in E to favor x
j
=1orx
j
= 0. That is, we may say that the larger the
value of Strength(j), the greater is the degree that “E favors x
j
= 1”. Then,
we take an average of the sum(i)valuesoverE to get a value for the num-
ber of x
j
components that should be 1 in an “average” solution, denoted by
Avg =
s
i=1
sum(i)/s.
Thus, the variables with the first Avg largest Strength values receive assign-
ment 1 and other variables receive assignment 0. In practice, it is preferable to
shift Avg slightly in one direction or another to increase the diversity of the
generated offspring solutions [7].
Diagonal Crossover (Diagonal): it is a generalization of the one-point
crossover. For s parent solutions, diagonal crossover divides each parents into
s sections through s − 1 crossover points. Each section has the same length ex-
cept the last section containing the surplus variables when divided unequally.
The offspring is constructed through extracting in a diagonal way respectively
one section from each parent. The formal definition is given as follows.
Given set E with s solutions and s − 1 crossover points {y
1
,y
2
, ..., y
s−1
},
where y
i
= i ∗ n/s and 0 <i<s. Diagonal crossover reproduces the offspring
c =(c
1
,c
2
, ...c
s
)by
c
k
=
⎧
⎪
⎨
⎪
⎩
x
(1)
j
, 1 ≤ j<y
1
k =1;
x
(k)
j
,y
k−1
≤ j<y
k
1 <k<s;
x
(s)
j
,y
s−1
≤ j ≤ nk= s.
(1)
Uniform Scanning Crossover (U-Scan): this is a generalization of the two-
parent uniform crossover. U-Scan uses a scheme that one of the parents selected
randomly determines the value of the offspring. Thus each parent has the same
probability to be the dominator of the value inherited by the offspring. It breaks
the limitation of traditionary two parents and can extend the number of parents
to an arbitrary number.
Given set E with s solutions, U-Scan generates offspring solution c =(c
1
,c
2
, ...
c
n
) as follows. Value c
j
is obtained by c
j
= x
(i)
j
where x
(i)
j
denotes the jth value
of parent x
(i)
and i is randomly selected from 1 to s with probability 1/s.
Occurrence-Based Scanning Crossover (OB-Scan): OB-Scan relies on
parental occurrence on determining the offspring values. Generally speaking,
each parent votes and the values inherited will be the one favored by the ma-
jority of parents. As for UBQP problem, the value equals either one or zero,
thus we record the frequency of each variable’s value equal to one appearing in
the parents. If this frequency surpasses or is less than the half of the number of
parents, then the value of offspring in this position is assigned to one or zero,
respectively. Otherwise, the variable is assigned to be one or zero randomly. The
following gives a formal definition of OB-Scan.
560 Y.Wang,Z.L¨uandJ.-K.Hao
Given set E, OB-scan reproduces the offspring c =(c
1
,c
2
, ...c
n
)by
c
j
=
⎧
⎪
⎨
⎪
⎩
0,
s
i=1
x
(i)
j
<s/2;
1,
s
i=1
x
(i)
j
>s/2;
rand(0, 1), otherwise.
(2)
where rand(0, 1) ∈{0, 1} is a binary random function.
3 Experimental Results
3.1 Instances and Experimental Protocol
To evaluate the MSX, Diagonal, U-Scan and OB-Scan crossover operators, we
carry out experiments on a set of 15 large random instances with 3000 to 5000
variables from the literature [13]. Our algorithm is programmed in C and com-
piled using GNU GCC on a PC running Windows XP with Pentium 2.66GHz
CPU and 512MB RAM. Given the stochastic nature of the algorithm, problem
instances are independently solved 10 times. The stop condition for a single run
is respectively set to be 5, 10 and 20 minutes on our computer for instances with
3000, 4000 and 5000 variables, respectively. Note that when performing exper-
iments on each crossover, the only difference consists in the crossover operator
and other components of the algorithm are kept unchanged. The parameters
are set as follows: population size p = 30, depth of tabu search α =2n.The
experimental results are summarized in Tables 1 and 2.
3.2 Computational Results
Tables 1 and 2 report the best objective values (in parentheses number of hits
over 10 runs) and the average objective values using the four crossover operators,
Table 1. Computational results on the 15 large random instances with 3000 to 5000
variables: best values (succ rate)
Instance MSX U-Scan OB-Scan Diagonal
p3000.1 3931583(9) 3931583(10) 3931583(8) 3931583(9)
p3000.2
5193073(10) 5193073(10) 5193073(10) 5193073(10)
p3000.3
5111533(7) 5111533(7) 5111533(7) 5111533(6)
p3000.4
5761822(10) 5761822(10) 5761822(9) 5761822(10)
p3000.5
5675625(6) 5675625(1) 5675598(1) 5675625(4)
p4000.1
6181830(10) 6181830(10) 6181830(10) 6181830(10)
p4000.2
7801355(7) 7801355(6) 7801355(4) 7801355(6)
p4000.3
7741685(9) 7741685(9) 7741685(9) 7741685(7)
p4000.4
8711822(10) 8711822(9) 8711822(7) 8711822(9)
p4000.5
8908979(4) 8908979(7) 8908979(3) 8908979(2)
p5000.1
8559015(1) 8559312(1) 8559210(3) 8559210(4)
p5000.2
10835437(1) 10835832(3) 10835437(3) 10835437(5)
p5000.3
10488783(10) 10489137(3) 10489137(1) 10489137(1)
p5000.4
12251211(1) 12251211(2) 12251211(2) 12251211(1)
p5000.5
12731803(10) 12731803(1) 12731803(1) 12731803(1)
