1

Hybrid Genetic Algorithms: Modeling and Application to

the Quadratic Assignment Problem

Patrick Copalu and Pataya Dangprasert

*

Faculty of Business Administration, Assumption University

Bangkok, Thailand

Abstract

Presented in this paper are ‘hybrid genetic algorithms’ (HGA), which are

collaborative performances of ‘genetic algorithms’ (GA) and local searches. Such

algorithms are also called memetic algorithms. Firstly, some of the local searches used

in the hybrid algorithms, such as ‘simulated annealing’ (SA), steepest ascent, and a

modification improvement strategy called ‘tabu search’ (TS) are briefly described.

Then, the hybrid algorithms models, combining the robust TS and CHC algorithms are

conceptually introduced, formulated and mathematically represented. Finally, the

defined HGA, which are practically applied to different instances of the ‘quadratic

assignment problem’ (QAP) by and their results to the best-known QAPLIB problems

from the QAP library are discussed.

The competitiveness of the HGA is demonstrated through experimental results

since 12 of the 16 considered QAP instances produce the best-known solutions. Factors

of that performance and competitiveness are also highlighted in this paper.

Keywords: Tabu search, steepest ascent, simulated annealing, genetic algorithms,

hybrid genetic algorithms, quadratic assignment problem.

Introduction

‘Hybrid genetic algorithms’ (HGA), also

called ‘memetic algorithms’ in some other

literature, are combination performances of

‘genetic algorithms’ (GA) and local search, i.e.

steepest ascent, steepest descent, ‘tabu search’

(TS), and ‘simulated annealing’ (SA).

Traditional GA often explore the

candidate solution encoded in chromosomes

and exploit those with better fitness iterally till

the solution is reached. The local search by

itself explores the solution space using specific

move in its neighborhood. The HGA combine

those two aspects by using the local

(traditionally randomized) of the GA and the

chromosomes that are produced by genetic

operators (N-points search to improve the

initial population crossover operator with N1

*

Department of Computer Information System,

Assumption University, Bangkok, Thailand.

and mutation operator with its associated

mutation rate). The improvement versions of

chromosomes are then carried out by the

traditional genetic algorithms as shown in Fig.1.

The key idea of the HGA is to use

traditional GA to explore in parallel several

regions of the search space and simultaneously

incorporates a good mechanism like SA to

intensify the search around some selected

regions. In addition, the improvement strategy

like TS may be used to keep track of some

states that should be avoided for the second

visit.

In 1995, Whitley described how to model

a basic form of HGA in the context of the

existing models for GA (Whitley 1995). This

model takes into consideration a deterministic

local search called ‘steepest ascent’. With

this basic form of HGA, it is possible to modify

the model by integrating other search

techniques in replacement of steepest ascent.

Even though Whitley (1995) used a deterministic

2

Population

Local

Search

Traditional

GA

Initial

population

Fig. 1. Hybrid genetic algorithm concept

local search, the HGA can be reformulated by

considering the probabilistic moves. For

example, instead of testing all neighbors

orderly and choosing the best improvement, the

neighbors can be tested randomly to choose the

first improvement found. SA is another local

search that drives its moves randomly and can

be considered as a local partner in HGA.

Using QAP problems as test beds, it has

been shown by Vásquez and Whitley (2000)

that the HGA produces better results than any

individual heuristic techniques that proposed to

solve QAP by Cela (1998). Those heuristic

approaches used by Cela included TS, SA, GA,

and ‘greedy randomized search procedure’

(GRASP). In the following section, the

‘steepest ascent’, SA and TS will be briefly

described. The second section will emphasize

on HGA. The third section will discuss the

QAP along with the algorithms implemented

and results obtained will be discussed. The last

part will then offer conclusion.

The Steepest Ascent

The ‘steepest ascent’ (or ‘steepest

descent’) technique uses a fundamental result

from calculus (that the gradient points in the

direction of the steepest increase of a function),

to determine how the initial settings of the

parameters should be changed to yield an

optimal value of the response variable. The

direction of movement is made proportional to

the estimated sensitivity of the performance of

each variable. This is a hill-climber that always

goes for the steepest route up from any point.

When ‘steepest ascent’ is applied to a binary

bit string, as shown in Table 1, the local

neighbors consist of strings with a single bit

changed at a time. The fitness of all neighbors

is evaluated and the search moves toward the

best improvement found. It is a deterministic

algorithm and very easily gets stuck on local

optima. Fig. 2 describes the steepest ascent

local search.

Simulated Annealing

‘Simulated annealing’ (SA) borrows its

basic ideas from statistical mechanics, when a

metal cools down. Those electrons in the metal

align themselves in an optimal pattern to ease

the transfer of energy. In general, left to itself,

a slowly cooling system, eventually finds the

arrangement of atoms that has the lowest

energy. This is the behavior that motivates the

method of optimization by SA. In SA we

construct a model of a system and slowly

decrease the temperature of this theoretical

system until the system assumes a minimal

energy structure.

Table 1. A steepest ascent example

String Fitness

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

1

3

5

10

8

5

2

6

8

12

14

9

5

4

4

4

Choose

0110 (2)

Calculate

neighbors

1110 (4)

0010 (5)

0100 (8)

0111 (6)

Choose

0100

Calculate

neighbors

1100 (5)

0000 (1)

0110 (2)

0101 (5)

No fitness

increase so

choose new

starting point

Current

Hilltop

0110 (2)

0100 (8)

3

C(s): Cost of a solution s

1. Choose a solution x

2. Choose a solutions in the

neighborhood of x such that

C(s) < C(x) (*)

If no neighbor verifies (*), then x is a

local minimum, stop.

3. x ? s

4. Back to step 2

Fig. 2. Steepest ascent algorithm

SA as an optimization technique was first

introduced to solve problems in discrete opti-

mization, mainly combinatorial optimization.

Subsequently, this technique has been

successfully applied to solve optimization

problems over the space of continuous decision

variables. SA is a simulation optimization

technique that allows random ascent moves in

order to escape the local minima, but a price is

paid in terms of a large increase in the

computational time required. It can be proven

that the technique will find an approximated

optimum. The annealing schedule might

require a long time to reach a true optimum.

SA’s major advantage over other methods

is an ability to avoid becoming trapped at local

minima. The algorithm uses a random search,

which not only accepts changes that decrease

objective function f, but also some changes that

increase it.

The latter are accepted with a probability

)exp( Tfp , where f is the increase in f

and T is a control parameter. The parameter T,

by analogy with the original application, is

known as the system ‘temperature’ irrespective

of the objective function involved as shown in

Fig. 3. The Advantages of SA can be therefore

stated in four points. First, it can deal with

arbitrary systems and cost functions. Second, it

statistically guarantees finding an optimal

solution. Third, it is relatively easy to code,

even for complex problems. Finally, it

generally gives a ‘good’ solution (Battiti et

al.1994a). This makes SA an attractive option

for combinatorial optimization problems.

The Tabu Search

‘Tabu search’ (TS) is an iterative procedure

designed for the solution of optimization

Compute a random initial state s

N=0,

*

n

x =s

Repeat I=1,2,…

Repeat j=1,2,….,m

i

Compute a neighbor s’=N(s)

if (f(s’)f(s)) then

s=s’

if (f(s’)< f(

*

n

x )) then

*

n

x =s

n=n+1

endif

else

s=s’ with probability p(T

i

,s,s’)

endif

EndRepeat

EndRepeat

Fig. 3. Simulated annealing pseudocode

problems. TS was invented by Glover (1986)

and has been used to solve a wide range of hard

optimization problems such as job-shop

scheduling, graph coloring problems, the

‘traveling salesman problem’ (TSP) and the

capacitated arc routing problem.

TS can be considered as a ‘metastrategy’

for guiding known heuristics to overcome local

optimality. TS is based on the premise that

problem solving, in order to be qualified as

intelligent, must incorporate adaptive memory

and responsive exploration. The adaptive

memory feature of TS allows the

implementation of procedures that are capable

of searching the solution space economically

and effectively. The role of TS will most often

be to guide and orient the search of another

search procedure.

To avoid cycling, solutions that were

recently examined are declared ‘tabu’ (taboo)

for a certain number of iterations. Applying

intensification procedures can accentuate the

search in a promising region of the solution

space. In contrast, diversification can be used

to broaden the search to a less explored region.

Although still in its infancy stages, this meta-

heuristic has been reported in literature

during the last few years as successful

solution approaches for a great variety of

problems.

To prevent the search from endlessly

cycling between the same solutions, the

attribute-based memory of TS is structured at

4

its first level to provide a short- term memory

function, which may be visualized to operate as

follows. Imagine that the attributes of all

explored moves are stored in a list, named a

running list, representing the trajectory of

solutions encountered. Then, related to a sub-

list of the running list a so-called tabu list may

be introduced. Based on certain restrictions the

tabu list implicitly keeps track of moves, or

more precisely salient features of these moves,

by recording attributes complementary to those

of the running list. These attributes will be

forbidden from being embodied in moves

selected in, at least one subsequent iteration,

because their inclusion might lead back to a

previously visited solution.

The goal is to permit ‘good’ moves for

each iteration without re-visiting solutions

already encountered and therefore, the tabu list

restricts the search to a subset of admissible

moves consisting of admissible attributes or

combinations of attributes. A general outline of

TS procedure based on applying such a short-

term memory function is described in Fig. 4

(for solving a minimization problem). Evidently

the key to this procedure lies in the tabu list

management, that is, updating the tabu list and

deciding on how many moves and which ones

have to be set tabu within any iteration of the

search. We may distinguish two main

approaches: static methods and dynamic

methods.

There are in fact several basic ways for

carrying out this management, generally

involving a recently based record that can be

maintained individually for different attributes

or different classes of attributes. In addition,

many applications of TS introduce memory

structures based on frequency (modulated by a

notion of move influence), and the coordination

of these memory elements is made to vary as

the preceding short term memory component

becomes integrated with longer term one.

The purpose of this integration is to

provide a balance between two types of

globally interacting strategies, called

intensification strategies and diversification

strategies. TS is concerned with finding new

and more effective ways of taking advantage of

the mechanisms associated with both adaptive

memory and responsive exploration.

Given a feasible solution x* with objective function

value z*, let x := x* with z(x) = z*.

Iteration:

while stopping criterion is not fulfilled do

begin

(1) select best admissible move that transforms

x into x' with objective function value z(x') and

add its attributes to the running list

(2) perform tabu list management: compute

moves (or attributes) to be set tabu, i.e.,

update the tabu list

(3) perform exchanges: x := x', z(x) = z(x');

if z(x) < z* then z* := z(x), x* := x

endif

endwhile

Result: x* is the best of all determined

solutions, with objective function value z*.

Fig. 4. Tabu search algorithm.

Much remains to be discovered about the

range of problems for which the TS is best

suited. The development of new designs and

strategies mixes makes TS a fertile area for

research and empirical study.

Hybrid Genetic Algorithms

Overview of HGAs

A central goal of the research efforts in

GAs is to find a form of algorithms that is

robust and performs well across a variety of

problems types. GAs exploit only the coding

and the objective function value to determine

plausible future generations. Therefore, it may

be advantageous to incorporate various specific

search techniques, like SA or TS, to form a

hybrid that the global perspective of GAs and

the convergence of problem-specific

techniques.

A promising approach to use domain

knowledge in GAs is the incorporation of local

search or other metaheuristics. The resulting

algorithms, often called ‘hybrid genetic

algorithms’ (HGA), memetic algorithms

(Moscato 1989) or ‘genetic local search

algorithms’ (GLS) (Yamada and Nakano1996),

have proven to be highly effective since they

combine the advantages of an efficient local

search for exploiting the neighborhood of a

5

single solution and population based algorithms

that effectively explore the search space.

The first use of the term ‘memetic

algorithms’ in the computing literature appeared

in 1989 in a paper written by Moscato (1989).

The method is gaining wide acceptance, in

particular in well-known combinatorial

optimization problems where large instances

have been solved to optimality and where other

metaheuristics have failed. An open research

issue is to understand which features of the

representation chosen had lead to

characteristics of the objective functions, which

are efficiently exploited by a hybrid approach.

Mathematical Representation: Models of

HGA can also be represented mathematically

using formal notations (Radcliffe and Surry

1994).

Let (s) be the representation function

that returns the chromosome solution. The

search space will be denoted as S (i.e. the set of

all phenotypes) and the set of chromosomes

will be denoted C (i.e. the set of all genotypes).

The function can therefore be represented as:

CS : (1)

Since all chromosomes are not

necessarily solution of the problem, it is

sufficient for the function to be injective,

formally we get: s S , (s) C (injective)

and c C / c is not solution in C (non

surjective).

The GA is to evaluate a fitness function

for each chromosome and it can be regarded as

a positive real number, the mapping can

therefore be done from the set of genotypes

onto the set of positive real numbers

+

. Let’s

denote this function:

Cf: (2)

to be maximized and the set of global maxima

is denoted

*

C such that CC

*

.

Let Q be a stochastic unary move

operator over C. The stochastic element of

such an operator belongs to a control set

Q

K,

from which a control parameter will be drawn

to determine which of the possible move

actually occur (a binary mask might be used as

the control parameter for mutation of binary

strings). Q can therefore be denoted into

functional form as:

CKSQ

Q

: (3)

With those notations, a chromosome xC

is said to be locally optimal with respect to Q

(or Q-opt) if and only if no chromosome of

higher fitness than x can be generated from it

by a single application of Q, i.e.

k K

Q

, f(Q (x,k)) f(x) (4)

Let’s denote the set of those Q -opt

chromosomes by

C

Q

= {xC / x is Q-opt} (5)

It is also true that:

Q

,

if x C

*

x is Q -opt x C

Q

Thus

Q

CC

*

and therefore, the search for the

genetic algorithm to optimize the fitness

function f over C can be (by transitivity)

formulated over C

Q

instead.

Traditional genetic algorithms combine

crossover (recombination) and mutation to

produce new generations. If the control sets for

crossover and mutation operators are K

X

and

K

M

respectively, the crossover operation can be

functionally denoted as:

CKCC

: (6)

and the recombination operator can be

functionally denoted as:

CKC

: (7)

Therefore the generic reproductive

function

g

R is the composition of the mutation

and reproduction, that is

g

R.

CKKCCR

g

: (8)

defined by:

g

R (x,y,k

M

,

k )=

x,y,

k ),

k ) (9)

The HGA combines the above

functional genetic algorithm with a local

search. Let’s represent that search H with

control set

H

K and define the function

QH

CKCH : (10)

If we now compose

g

R with H, we have the

6

Loc 1

Fac 1

Loc 3

Loc 4

Loc 2

Fac 3

Fac 2

Fac 4

hybrid genetic reproduction function

m

R such

that:

QMHQQm

CKKKCCR

: (11)

that is:

m

R (x,y,k

H

,k

M

,

k )= H ( M (

x,y,k

X

),k

M

),k

H

)

(12)

The Quadratic Assignment Problem

Overview of the QAP

The QAP introduced by Koopmans and

Beckmann (1957) is a combinatorial

optimization problem (COP). The objective of

COP is to minimize the cost function of a

finite, discrete system characterized by a large

number of possible solutions. The term

quadratic comes from the reformulation of the

problem as an optimization problem with a

quadratic objective function.

Cela (1998) pointed out several reasons

why the QAP still gives rise to a lot of

researches. First, a lot of real-world problems

can be modeled by QAP such as the VLSI

module placement, location design of factories,

scheduling problem, statistical data analysis

and process-processor mapping in parallel

computing, etc. Second, many of the

combinatorial optimization problems can be

formulated as QAP, for example the traveling

salesman problem can be formulated as a set of

cities (associated with distances) and a set of

positions (facilities). Third, QAP are difficult

problems from a computational point of view.

As n grows large it becomes impossible to

enumerate all the possible assignments, even

by fast computers. For example, if n = 25 and a

computer were able to evaluate 10 billion

assignments per second, it would still take

nearly 50 million years to evaluate all

assignments! Last, QAP are considered to be

NP-hard problems and can generally not be

solved using optimization techniques.

In a standard problem in location theory,

we are given a set of n locations and n

facilities, and told to assign each facility to a

location. To measure the cost of each possible

assignment, there are n! of them. We multiply

the prescribed flow between each pair of

facilities by the distance between their assigned

locations, and sum over all the pairs. Our aim

is to find the assignment that minimizes this

cost, and this problem is precisely a quadratic

assignment problem. Fig. 5 shows one possible

solution to a quadratic assignment problem/

facility location problem with four facilities.

The permutation p corresponding to this

graphical solution is (2,1,4,3). This means that:

facility 2 has been assigned to

location 1,

facility 1 assigned to location 2,

facility 4 assigned to location 3,

facility 3 assigned to location 4.

The lines drawn between the facilities

imply that there is a required flow between the

facilities, and the thickness of the line denotes

the value of the required flow. The goal, in

some sense, is to try to get the ‘fat’ lines as

close together as possible. To make the idea

more concrete, let's say that the distances are:

d(1,2) = 22, d(1,3) = 53, d(2,3) = 40 and

d(3,4) = 55. Also, the required flows between

facilities are: f(2,4) = 1, f(1,4) = 2, f(1,2) = 3

and f(3,4) = 4. The assignment cost of the

permutation shown is:

[d(1,2)*f(1,2)]+[d(1,3)*f(2,4)]+d(2,3)*f(1,4)] +

[d(3,4)*f(3,4)]

= 22*3+53*1+40*2+55*4

= 419.

This is not the best possible permutation.

In this case the best answer is 395. As the

problems get larger, it becomes much, much

more difficult to find the optimal solution.

Fig. 5. A quadratic assignment problem

Mathematical Representation

Mathematically, we can formulate the

problem by defining two n by n matrices: a

flow matrix F whose (i,j) element represents

the flow between facilities i and j, and a

7

distance matrix D whose (i,j) element represents

the distance between locations i and j.

We represent an assignment by the vector

p, which is a permutation of the numbers {1, 2,

... ,n}. p(j) is the location to which facility j is

assigned.

With these definitions, the QAP can be

written as:

n

i

n

j

jpipij

p

df

1 1

)()(

min

The most effective algorithms for

optimally solving QAPs are based on branch-

and-bound. The branch-and-bound technique

can be outlined in simple terms. An

enumeration tree of continuous linear programs

is formed, in which each problem has the same

constraints and objective as except for some

additional bounds on certain components of x.

At the root of this tree is the problem with the

requirement x Z

n

removed. The solution to

this root problem will not, in general, have all

integer components.

Heuristic Approaches on QAP

As already stated earlier in this section,

QAP are NP-Hard problems. Therefore,

heuristics or metaheuristics like those

discussed in the first part of this paper like TS,

SA, GAs and HGA, are widely tested and

compared in terms of solution accuracy (are the

best known solutions obtained or even

improved?) and cost of the computation related

to how fast we obtain that solution and how

much memory is required?

In this paper, we review a total of four

algorithms that have been implemented by

Vásquez and Whitley (2000) and compare

them for 16 instances of the QAP. Namely,

those algorithms are:

The first algorithm is a traditional SA

algorithm as described in part I.

The second algorithm is a ‘reactive tabu

search’ (RTS) algorithm, which is an

improvement derived from TS as described by

Battiti and Tecchiolli (1994b). The tabu

scheme based on a fixed list size is not strict

and therefore the possibility of cycles still

remains. The proper choice of the list size is

obviously a critical factor to the ‘good’

convergence of the algorithm. The RTS goes

further in the direction of robustness by

proposing a simple mechanism for adapting the

list size to the properties of the optimization

problem. The configurations visited during the

search and the corresponding iteration numbers

are stored in memory so that, after the last

movement is chosen, one can check for the

repetition of configurations and calculate the

interval between two visits. The basic fast

‘reaction’ mechanism increases the list size

when configurations are repeated. A slower

reduction mechanism is also present so that the

size is reduced in regions of the search space

that do not need large sizes. Compression

techniques can also be implemented if the use

of an excessive memory space to store the

entire configurations is required (hashing

techniques).

The third algorithm is the CHC

algorithm. The CHC is a generational genetic

search algorithm that uses truncation selection.

The choice for the CHC is made because it

produces the best results for GA. The CHC

generational genetic search is similar to a

traditional GA without regular mutation.

Instead, when a convergence is detected, the

search restarts with the mutation of the current

best individual (for example 35% mutation

rate) to regenerate the entire population.

In addition to this mutational difference,

‘uniform crossover’ (HUX) in which all pairs

of parents are allowed to produce offspring is

changed into ‘distance preserving crossover’

(DPX). DPX can be viewed as a threshold

crossover equipped with an ‘incest prevention

strategy’, i.e. the parents are mated randomly, a

threshold is set and only pairs exceeding that

threshold are allowed to reproduce. Since this

operator relies only on the notion of a distance

between solutions, it can be used for the QAP

problem. In general, the DPX is aimed at

producing an offspring, which has the same

distance to each of his parents, and this

distance is equal to the distance between the

parents themselves. The alleles that are

identical for the same genes in both parents

will be copied to the offspring. The alleles for

all other genes change. Thus, although the

crossover is highly disruptive, the local search

procedure subsequently applied is forced to

8

explore a region of the search space that is

defined by the genes with different alleles in

the parents, which is the region where better

solutions are most likely to be found. The DPX

operator for the QAP works as follows.

Suppose that two parents A and B (as shown in

Fig. 6) are given. First, all assignments that are

contained in both parents are copied to the

offspring C. The remaining positions of the

genome are then randomly filed with the yet

unassigned facilities, taking care that no

assignment that can be found in just one parent

is inserted into the child. After that, we end up

with a child C, for which the condition d(C,A)

= d(C,B) = d(A,B) holds. In the example, d=5.

A

2

4

7

1

8

9

3

5

0

B

7

4

5

3

8

9

2

1

0

C

4

8

9

0

C

5

4

1

2

8

9

7

3

0

Fig. 6. The DPX crossover operator

The fourth and last algorithm is the HGA

(CHC+TS). Three different types of crossover

operators are implemented through the GA.

The first one is the OBC operator (order-based

crossover) in which a number of elements are

selected from one parent and copied to the

offspring. The missing elements are taken from

the other parent in order. The second one is the

HUX operator, that is to say, the traditional

uniform crossover operator. The third and last

one is the DPX operator described previously.

For comparative reasons, two different versions

of GA are implemented. Namely a generational

GA for which mating individuals are selected

in a fitness biased fashion, the operators are

applied and the best individuals are selected

(form the old population and new offspring),

and Steady-State GA for which two parent are

selected randomly and two new offspring are

created. The best offspring replaces the worst

individual of the population. As for the

parameters involved, the CHC with no

hybridization restarts the population five times.

The hybrid version restarts the population only

three times and then run the robust TS for n

iterations, n being the problem size.

Heuristic vs Random Initialization

When performing search techniques in

general like simulated annealing steepest ascent

or genetic techniques (GT), the question of

how to generate the initial solution often arises

as a primordial matter of concern.

Should initialization of the solution be

based on a heuristic rule or on a randomly

generated one? Theoretically, it should not

matter, but in practice this may depend on the

problem. In some cases, a pure random

solution systematically produces better final

results. On the other hand, a good initial

solution may lead to lower overall run times.

This can be important, for example, in cases

where each iteration takes a relatively long

time; therefore, one has to use some clever

termination rule, as simulation time is a crucial

bottleneck in an optimization process.

In many cases, a simulation is run several

times with different initial solutions. Such a

technique is most robust, but it requires the

maximum number of replications compared

with all other techniques. The pattern search

technique applied to small problems with no

constraints or qualitative input parameters

requires fewer replications than the GT.

However, GT can easily handle

constraints that have lower computational

complexity. Finally, SA can be embedded

within the ‘tabu’ search to construct a probabilistic

technique for global optimization. The next

section will present results and discuss those

results obtained by the implemented HGA.

Results and Discussion

The results obtained by Vazquez and

Whitley (2000) when implementing the HGA

(CHC + TS) are compared with existing results

found from literature for SA, TS (both robust

and reactive), CHC results and ‘genetic

hybrids’. These results are summarized in

Table 2. A total of 16 problems have been

selected for testing, ten problems are Taillard’s

QAP instances and six problems are Skorin-

9

Kapov’s QAP instances. Twenty different trials

have been performed for each algorithm and

compared to the best-known solutions from

literature.

During those experiments, TS and CHC

produced a good near-optimal solution for the

QAP. As for SA, OB and GLS, the results are

not satisfying and tuning parameters need to be

implemented to improve the performance. It is

also true to say that the Taillard instances for

the QAP are harder than the Skorin-Kapov

instances (TS can produce results within 0.07%

above the best-known solution).

In order to improve those results, two

types of tuning were implemented: the first

tuning consisted of adding a backtracking

mechanism to the RTS. In that situation, all

the Skorin-Kapov instances are consistently

solved and near optimal solutions within

0.73% above the best-known solution are

obtained for Taillard instances. The second

tuning is to Combine TS and CHC (hybrid

solution). In that case, 12 of the 16 problems

can be solved and obtain the best-known

solution.

According to the experimental results

obtained and shown in Table 2, it is reasonable

to say that the HGA proposed (TS+CHC) is

very competitive and produces good near-

optimal solutions. It was beaten in only

three cases and found the best-known

solution for 12 of the 16 chosen benchmark

problems.

Conclusion

In this paper, a HGA for solving several

instances of the QAP was presented. The

ingredients of the ‘memetic algorithm’,

evolutionary operators and local neighborhood

search, were described. The performance of

the HGA was investigated on a set of QAP

instances and compared to the best-known

results known from the literature and to the

performance of some very good heuristic

approaches to the QAP like RTS, SA and

CHC.

Table 2. Percentage above the best-known

solutions

(Vásquez et al. 2000)

Tai: Taillard Problems

Sko: Skorin-Kapov Problems

‘*’: best-known solution has been reached.

The hybrid algorithm combining the RTS

and CHC algorithm was able to outperform

these alternative heuristics on all QAP

instances of practical interest. Furthermore, the

approach proves to be very robust, since the

best-known solutions can be found for 12 of

the 16 instances considered. This derives form

the properties of both RTS and CHC. The CHC

explores in parallel several regions of the

search space and the RTS intensifies the search

around some selected regions.

There are at least two avenues for future

research. First, the algorithm should be applied

to larger instances of the QAP to investigate its

scalability. Second, the algorithm performance

with other parameter settings for population

size, operator rates and running times should be

investigated.

Problem RTS SA CHC CHC+TS

Tai10a 0.000 (*) 0.000 (*) 0.000 (*) 0.000 (*)

Tai20a 0.000 (*) 0.000 (*) 0.000 (*) 0.000 (*)

Tai30a 0.000 (*) 1.735 0.641 0.000 (*)

Tai40a 0.344 2.335 0.936 0.000 (*)

Tai50a 0.825 2.016 1.279 0.219

Tai60a 0.785 2.398 1.313 0.253

Tai80a 0.387 2.170 1.014 0.239

Tai100a 0.730 1.771 1.491 0.434

Tai150b 0.788 0.752 0.444 0.000 (*)

Tai256c 0.299 0.122 0.076 0.000 (*)

Sko100a 0.052 1.452 0.247 0.000 (*)

Sko100b 0.027 0.241 0.115 0.000 (*)

Sko100c 0.021 1.136 0.715 0.000 (*)

Sko100d 0.062 1.450 0.347 0.000 (*)

Sko100e 0.021 0.855 0.744 0.000 (*)

Sko100f

0.068

1.142

0.224

0.000 (*)

10

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