Evolutionary approach to optimize the assignment of cells to switches in personal communication...

Evolutionary approach to optimize the assignment of cells to switches

in personal communication networks

Alejandro Quintero*, Samuel Pierre

Mobile Computing and Networking Research Laboratory (LARIM), Department of Computer Engineering, Ecole Polytechnique de Montreal,

C.P. 6079, succ. Centre-Ville, Montreal, Que., Canada H3C 3A7

Received 14 February 2002; revised 17 September 2002; accepted 10 October 2002

Abstract

This paper proposes an evolutionary approach to solve the problem of assigning cells to switches in the planning phase of mobile cellular

networks. Well-known in the literature as an NP-hard combinatorial optimization problem, this problem requires the recourse to heuristic

methods, which can practically lead to good feasible solutions, not necessarily optimal, the objective being rather to reduce the convergence

time toward these solutions. Computational results obtained from extensive tests confirm the effectiveness of this approach to provide good

solutions to problems of a certain size. This approach can be used to solve NP-hard problems, like designing and planning, in the next

generation networking systems.

q 2002 Elsevier Science B.V. All rights reserved.

Keywords: Cellular networks; Cell assignment; Memetic algorithms; Genetic algorithms; Tabu search; Simulated annealing; Migration; Multi-population

algorithm

1. Introduction

A Personal Communication Network (PCN) is a wireless

communication network, which integrates various services

such as voice, video, electronic mail, accessible from a

single mobile terminal and for which the subscriber obtains

a single invoicing. These various services are offered in an

area called cover zone, which is divided, into cells. The cell

is the basic unit of a cellular system (Fig. 1). In each cell is

installed a base station which manages all the communi-

cations within the cell. In the cover zone, cells are connected

to special units called switches, which are located in mobile

switching centers (MSC). When a user in communication

goes from a cell to another, the base station of the new cell

has the responsibility to relay this communication by

allotting a new radio channel to the user. Supporting the

transfer of the communication from a base station to another

is called handoff. This mechanism, which primarily involves

the switches, occurs when the level of signal received by the

user reaches a certain threshold. We distinguish two types of

handoffs. In the case of Fig. 1 for example, when a user

moves from cell B to cell A, it refers to soft handoff because

these two cells are connected to the same switch.The MSC,

which supervises the two cells, remains the same and the

induced cost is weak. On the other hand, when the user

moves from cell B to cell C, there is a complex handoff. The

induced cost is high because both switches 1 and 2 remain

active during the procedure of handoff and the database

containing information on subscribers must be updated.

The total operating cost of a cellular network includes

two components: the cost of the links between the cells

(base station) and the switches to which they are joined, and

the cost generated by the handoffs between cells. It appears

therefore intuitively more discriminating to join cells B and

C to the same switch if the frequency of the handoffs

between them is high. The problem of assigning cells to

switches essentially consists of finding the configuration that

minimizes the total operating cost of the network. Assigning

cells to switches in cellular mobile networks being an NP-

hard problem, enumerative search methods are practically

inappropriate to solve large-sized instances of this problem

[2,18]. Because they exhaustively examine the entire search

space in order to find the optimal solution, they are only

efficient for small search spaces corresponding to small-

sized instances of the problem. For example, for a network

0140-3664/03/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved.

PII: S0 14 0 -3 66 4 (0 2) 00 2 38 -4

Computer Communications 26 (2003) 927–938

www.elsevier.com/locate/comcom

* Corresponding author. Address: Department of Computer Engineering,

Ecole Polytechnique de Montreal, HPO Box 6079, Station, Centre-Ville

Montreal, Canada H3T1J4. Tel.: þ514-340-4711x5077; fax: þ514-340-

3240.

E-mail address: [email protected] (A. Quintero).

http://www.elsevier.com/locate/comcom

with m switches and n cells, m n solutions should be

examined. With 200 cells and 5 switches, that means 5200

possible solutions to examine. Obviously, it is unrealistic to

adopt such a brute-force approach. Thus, heuristic

approaches, like Tabu search, simulated annealing and

genetic algorithms have been developed for this kind of

problem [1,18,32].

Memetic algorithms (MA) are population-based heuristic

search approaches for combinatorial optimization problems

based on cultural evolution [27,28]. MAs are inspired by

Dawkins’ notion of a meme defined as a unit of information

that reproduces itself while people exchange ideas. In

contrast to genes, the people who transmit them before they

are passed on to the next generation typically adapt memes.

While genetic algorithms have been inspired in trying to

emulate biological evolution. Memetic algorithms would try

to mimic cultural evolution. Memetic algorithm is a

marriage between a population based global search and

the heuristic local search made by each of the individuals

[27]. They are in some respects similar to genetic algorithms

(GA), which simulate the process of biological evolution.

While in nature genes are usually not modified during an

individual’s lifetime, memes are modified [23].

The general idea behind memetic algorithms is to

combine the advantages of evolutionary operators that

determine interesting regions of the search space with local

neighborhood search that quickly finds good solutions in a

small region of the search space. Memetic algorithms have

been applied with success to several other combinatorial

optimization problems [23–25].

The paper is organized as follows. Section 2 presents

background and related work. Section 3 first describes the

evolutionary approach, and then presents some adaptation

and implementations details. Section 4 presents and analyses

results. Finally, Section 5 summarizes the main results.

2. Background and related work

This assignment problem consists of determining a cell

assignment pattern, which minimizes a certain cost

function, while respecting certain constraints, especially

those related to limited switch’s capacity. An assignment

of cells can be carried out according to a single or a

double cell’s homing. A single homing of cells corre-

sponds to the situation where a cell can only be assigned

to a single switch. When a cell is related to two switches

that refers to a double homing. For a double homing, the

call and handoff patterns do not remain the same during

the day. We have, for example, two patterns for the

daytime: one for the morning and another for the

afternoon. The two problems, related to each part of

the day, are not separated and cannot be solved

independently. Optimization should not be performed

separately for each pattern but simultaneously for both

patterns, in a way that minimizes the total cost,

particularly the link cost. In this paper, only single

homing is considered.

Let n be the number of cells to be assigned to m switches.

The location of cells and switches is fixed and known. Let

Hij be the cost per unit of time for a simple handoff between

cells i and j involving only one switch, and H0ij the cost per

time unit for a complex handoff between cells i and j

(I; j ¼ 1; …,n with i – j) involving two different switches.

Hij and H0ij are proportional to the handoff frequency

between cells i and j. Let cik be the amortization cost of the

link between cell i and switch kði ¼ 1;…; n; k ¼ 1;…;mÞ:

Let consider

xik ¼1 if cell i is related to switch k

0 if not

(

The assignment of cells to switches is subject to a

number of constraints. Actually, each cell must be

assigned to only one switch, which can be expressed

by the following formula:

Xmk¼1

xik ¼ 1 for i ¼ 1;…; n: ð1Þ

Let zijk and yij be defined as:

zijk ¼ xikxjk for i; j ¼ 1;…; n and k ¼ 1;…; m; with i – j:

yij ¼Xmk¼1

zijk for i; j ¼ 1;…; n; and i – j:

zijk is equal to 1 if cells i and j, with i – j; are both

connected to the same switch k, otherwise zijk is equal to

0. Thus yij takes the value 1 if cells i and j are both

connected to the same switches and the value 0 if cells i

and j are connected to different switches.

The cost per time unit f of the assignment is expressed as

follows:

f ¼Xn

i¼1

Xmk¼1

cikxikþXn

i¼1

Xn

j¼1;j–i

H 0ijð12yijÞþ

Xn

i¼1

Xn

j¼1;j–1

Hijyij ð2Þ

The first term of the equation represents the link or

cabling cost. The second term takes into account

Fig. 1. Geographic division in a cellular network.

A. Quintero, S. Pierre / Computer Communications 26 (2003) 927–938928

the complex handoffs cost and the third, the cost of

simple handoffs. We should keep in mind that the cost

function is quadratic in xik, because yij is a quadratic

function of xik. Let us mention that an eventual

weighting could be taken into account directly in the

link and handoff costs definitions.

The capacity of a switch k is denoted Mk.If li denotes the

number of calls per unit of time destined to i, the limited

capacity of switches imposes the following constraint:Xn

i¼1

lixik # Mk for k ¼ 1;…;m ð3Þ

accordingtowhich the total loadofall cellswhichareassigned

to the switch k is less than the capacity Mk of the switch.

Finally, the constraints of the problem are completed by:

xik ¼ 0 or 1 for i ¼ 1;…; n and k ¼ 1;…;m: ð4Þ

zijk ¼ xijxik and i; j ¼ 1;…; n and k ¼ 1;…;m: ð5Þ

yij ¼Xmk¼1

zijk for i; j ¼ 1;…; n ð6Þ

Eqs. (1), (3) and (4) are constraints of transport problems. In

fact, each cell i could be assimilated to a factory which

produces a call volumeli. The switches are then considered as

warehouses of capacity Mk where the cells production could

be stored. Therefore, the problem is to minimize Eq. (2) under

Eq. (1), and Eqs. (3)–(6). When the problem is formulated in

this way, it could not be solved with a standard method such as

linear programming because constraint (5) is not linear.

Merchant and Sengupta [21,22] replaced it by the following

equivalent set of constraints:

zijk # xik ð7Þ

zijk # xjk ð8Þ

zijk $ xik þ xjk 2 1 ð9Þ

zijk $ 0 ð10Þ

Thus, the problem could be reformulated as follows:

minimizing Eq. (2) under constraints (1), (3) and (4) , and

Eqs. (6)–(10). We can further simplify the problem by

defining:

hij ¼ H 0ij 2 Hij:

hij refers to the reduced cost per time unit of a complex handoff

between cells i and j. Relation Eq. (2) is then re-written as

follows:

f ¼Xn

i¼1

Xmk¼1

cikxik þXn

i¼1

Xn

j¼1; j–1

hijð1 2 yijÞ þXn

i¼1

Xn

j¼1;j–1

Hij|fflfflfflffl{zfflfflfflffl}constant

The assignment problem takes then the following form:Minimize:

f ¼Xn

i¼1

Xmk¼1

cikxik þXn

i¼1

Xn

j¼1; j–1

hijð1 2 yijÞ ð11Þ

subject to: Eqs. (1), (3), (4) and (7)–(10). In this form, the

assignment problem could be solved by usual programming

methods.

The total cost includes two types of cost, namely cost of

handoff between two adjacent cells, and cost of cabling

between cells and switches. The design is to be optimized

subject to the constraint that the call volume of each switch

must not exceed its call handling capacity. This kind of

problem is NP-hard, so enumerative searches are practically

inappropriate for moderate- and large-sized cellular mobile

networks [18,21].

The geographical relationships between cells and

switches are considered in the value of the cost of cabling,

so that the base station of a cell is generally assigned to a

neighbouring switch and not to far switches [37]. In Ref.

[31], an engineering cost model has been proposed to

estimate the cost of providing personal communications

services in a new residential development. The cost model

estimated the costs of building and operating a new PCS

using existing infrastructure such the telephone, cable

television and cellular networks. In Ref. [11], economic

aspects of configuring cellular networks are presented.

Major components of costs and revenues and the major

stakeholders were identified and a model was developed to

determine the system configuration (e.g. cell size, number of

channels, etc.). For example, in a large cellular network, it is

impossible for a cell located in east America to be assigned

to a switch located in west America. In this case, the

variable cost is 1.

3. Evolutionary approach

Evolutionary thought, however, extends beyond the

study of life. Evolution is an optimization process that can

be simulated using a computer or other device and put to

good engineering purpose. The interest in such simu-

lations has increased dramatically in recent years as

applications of this technology have been developed to

supplant conventional technologies in power systems,

pattern recognition, control systems, factory scheduling,

pharmaceutical design, and diverse other areas [8,9].

Evolutionary algorithms have been applied successfully in

various domains of search, optimization, and artificial

intelligence [16,29,33,34,36]. In the field of combinatorial

optimization, it has been shown that combine evolutionary

algorithms with problem-specific heuristics can lead to

highly effective approaches. These hybrid evolutionary

algorithms combine the advantages of efficient heuristics

incorporating domain knowledge and population-based

search approaches.

3.1. Basic principles of classical genetic algorithms

The process used by genetic algorithms to solve

optimization problems is analogous to the natural

A. Quintero, S. Pierre / Computer Communications 26 (2003) 927–938 929

process of evolution by natural selection [17,35]. Genetic

algorithms apply the natural evolutionary processes of

evaluation and selection to string representations of the

arguments of the function being optimized. Structures

(individuals in natural systems) are encoded into one or

more strings (chromosomes). The chromosome represents

individuals or solutions. At each generation, the individuals

reproduced and fit persist, yielding improved results. The

structure is analogous to the phenotype in natural systems

and corresponds to a candidate solution to the optimization

problem. The string encoding is analogous to the genotype

[10].

The initial search space of a GA usually consists of a

population of randomly generated solutions. A diversified

initial population is a prerequisite to good solutions. The

execution of a genetic algorithm can be viewed as a two-

stage process, which starts with a current population to

which selection operations are applied to create an

intermediate population. The selection scheme determines

which individuals are chosen for reproduction. For a

selection, two essential ingredients are required: (1)

inheritance: offspring must retain at least some of the

features that made their parents fitter than average; (2)

variability: at any given time, individuals of varying

fitness must coexist in the population. Then recombina-

tion (crossover) and mutation are applied to the

intermediate population to create a new population

[14]. In genetic algorithms, evolution from generation

to generation is simulated both by preserving the genetic

information contained in the chromosome and by altering

this information by means of random genetic changes.

Genetic operators affect both these goals. The goal

of preserving the genetic information of fit individuals is

achieved through crossover, one of the genetic

operators used to recombine the population’s genetic

material. Fig. 2 shows the flow chart of the genetic

process.

Crossover is the process by which two chosen string

genes arc interchanged. To execute the crossover, strings of

the mating pool are coupled at random. The crossover of a

string pair of length l is performed as follows: a position i is

chosen uniformly between 1 and (1 2 1 ), then two new

strings are created by exchanging all characters between

positions (i þ l) and l of each string of the pair considered.

The new strings can be totally different from their parents.

Mutation is the process by which a randomly chosen

bit in a chromosome is permuted. It is employed to

introduce new information into the population and also to

prevent the population from becoming saturated with

similar chromosomes (premature convergence). Large

mutation rates increase the probability that

good schemata be destroyed, but increase population

diversity.

Diversity is the term used to describe the relative

uniqueness of each individual in the population. However,

crossover and mutation generate new solutions, but with

certain limitations. Crossing nearly identical strings yields

offspring similar to the parent string. Consequently,

crossover cannot reintroduce diversity. Mutation, on the

other hand, can generate the full space, but may take

excessively long time to yield a desirable solution.

3.2. Basic principles of memetic algorithms

In problems characterized by many local optima,

traditional optimization techniques fail to find high-quality

solutions. In these cases, standard genetic algorithm

(SGA) can be considered as an efficient and interesting

option. However, SGA can suffer from excessively slow

convergence before finding an accurate solution because

of their characteristics of using a priori minimal knowl-

edge and failure to exploit local information [6,29,30,34].

This may prevent them from being really of practical

interest for a lot of large-scale constrained applications.

Genetic algorithms promise convergence but not optim-

ality. This implies that the choice of when to stop a

genetic algorithm is not well defined. Since there is no

guarantee of optimality, successive runs of the GA will

provide different chromosomes with varying fitness

measures. If we run an SGA several times, it will

converge each time, possibly at different optimal

chromosomes.

Memetic algorithms (MA) are population-based heuristic

search approaches for combinatorial optimization problems

based on cultural evolution [27,28]. They are inspired by

Dawkins’ notion of a meme defined as a unit of information

that reproduces itself while people exchange ideas. The

person usually modifies a meme before he or she passes it on

to the next generation. Memetic algorithm is a marriage

between a population-based global search and the heuristic

local search made by each of the individuals [27].

A brief description of memetic algorithm could be:

Given a representation of an optimization problem, a certain

number of individuals are created. The state of these

individuals can be randomly chosen or according to a

certain initialization procedure. A heuristic can be chosen to

initialize the population. After that, each individual makes

local search. The mechanism to do local search can be to

reach a local optima or to improve (regarding the objectiveFig. 2. Genetic process flow chart.


cost function) up to a predetermined level. After that, when

the individual has reached a certain development, it interacts

with the other members of the population. The interaction

can be cooperative; it can be similar to the selection

processes of SGA. The cooperative behavior can be

understood as the mechanisms of crossover in SGA or

other types of breeding that result in the creation of a new

individual. More generally, we must understand cooperation

as an interchange of information. The local search and

cooperation (mating, interchange of information) or com-

petition (selection of better individuals) are repeated until a

stopping criterion is satisfied [27]. Fig. 3 shows the flow

chart of the memetic process.

In the context of evolutionary computation, a hybrid

evolutionary algorithm is called memetic if the individuals

representing solutions to a given problem are improved by a

local search or another improvement technique [26]. Kado

et al. [19] compare different implementations of hybrid

genetic algorithms.

In this paper, we propose memetic algorithms with local

refinement strategies to combine the strengths of both by

providing global and local exploitation aspects to the

problem of assigning cells to switches in cellular mobile

networks. The local refinement strategies used with

memetic algorithms are tabu search and simulated

annealing.

A tabu search method is an adaptive technique used in

combinatorial optimization to solve difficult problems [13,

18,21,22]. It is considered as a meta-heuristic method

because it can be applied to different instances of problems

to provide good solutions. The tabu search method is an

improvement to the general descent algorithm, because it

attempts to avoid the trap of local minima. For this purpose,

it is necessary to accept, from time to time, solutions, which

do not improve the objective function, with the hope to

reach better solutions later. However, accepting solutions

that are not necessary, the best introduces a cycle risk,

i.e. return to solutions that had already considered, hence the

idea of keeping a tabu list T of solutions that had already

considered. Thus, during the generation of the set V of

neighbour candidates, the candidates present in the tabu list

are removed.

The simulated annealing (SA) starts from an initial

solution, which they attempt to improve through a series of

greedy moves, while avoiding to be stranded in a local

minimum. The moves used to escape a local minimum

explore only a very limited set of options. They depend on

the initial solution and do not necessarily lead to a good final

solution.

SA can be applied to a large variety of technological

domains and in particular to telecommunications. It is a

heuristic optimization method, which consists of a local

search by perturbations. This process gives the possibility of

going away, from time to time, from a local minimum to

allow an extension of the field of research of the best

solution.

According to SA, the current topology considered for

a moment as the better one is constantly compared with

the other topologies, which are very ‘close’ to it, i.e. that

can be obtained by carrying out a small perturbation. If a

perturbation results in a topology better than the current

solution, then this topology is saved as current one.

However, it happens that, further to a disturbance, the

obtained nearby topology is kept as current solution,

even if it is not better than the current one, provided that

it respects a certain probability of acceptance. The fact of

accepting from time to time an empirical solution allows

to avoid being trapped too early into a local optimum.

On the other hand, the probability of acceptance should

be weak enough, so that the algorithm can approach as

good as possible the global optimum. Finally, the

algorithm ends when the stopping criterion is reached.

In this stage, the local search should have ended in a

local minimum or in a global optimum. It follows that

the ideal solution found is, either locally optimal,

considering the high number of local optima, or globally

optimal in the best of the cases.

It is customary, for simulated annealing practitioners, to

use the word temperature to denote the parameter u that

controls the probability of accepting a worsening pertur-

bation over time. At the beginning of the process, u is set to

a relatively high value. Then, it is reduced by a

multiplication by a factor a (0 , a , 1), called the cooling

rate or the annealing factor. This reduction of u takes place

every L iterations or trials.

3.3. Multi-population approach

Classical genetic and memetic algorithms are powerful

and perform well on a broad class of problems. However,

part of the biological and cultural analogies used to motivate

a genetic or memetic algorithm search are inherently

parallels.Fig. 3. Memetic process flow chart.


There exist different ways of exploiting parallelism in

genetic algorithms, and it could be classified into fine-

gained parallel genetic algorithms (also called diffusion or

neighbourhood model) and coarse-gained parallel genetic

algorithms (also called migration model). In the fine-grained

model, one individual resides at each processor. The

individuals have only local interactions and neighbourhood.

In coarse-grained models, the isolated subpopulations help

maintain genetic diversity. Individuals in a subpopulation or

deme (a separately evolving subset of the whole population)

are relatively isolated from individuals on another sub-

population. Therefore, the subpopulation of each island is

exploring a different part of the search space. Every

subpopulation evolves isolate for a few generations before

one or more individuals are exchanged between the sub-

populations. This exchange of individuals is called

migration [12]. Jens [21] indicates that parallel genetic

algorithms in isolated evolving subpopulations with

migrations may offer advantages over sequential

approaches.

The migration model divides the population into multiple

subpopulations, which evolve independently from each

other for a certain number of generations (isolation time).

After the isolation time, some individuals are distributed

between the subpopulations (migration).

By introducing migration, the island model is able to

exploit genetic and cultural differences in the various

subpopulations: this variation in fact represents a source of

genetic and memetic diversities. Each subpopulation is an

island, and there is some designated way in which genetic

and memetic material are moved from one island to another.

The migration algorithm partitions a population of

designs into a set of subpopulations, at specified

intervals, and shares information between these subpopu-

lations. The parameters associated with the migration

algorithm are introduced in Refs. [3–5]: the migration

interval and the migration rate. The migration interval is

the number of generations between each migration, and

the migration rate is the number of individuals selected

for migration.

4. Implementation details

We submitted our memetic algorithms with local

refinement strategies to a series of tests in order to determine

its efficiency and sensitivity to different parameters. Thus,

we will present a few results, which we compare to those

provided by other known heuristics.

4.1. Local search strategies

This section presents the implementations details of the

local refinement strategies used to improve the individuals

representing solutions provided by genetic algorithms: tabu

search and simulated annealing.

Tabu search program was executed by supposing that the

cells are arranged on an hexagonal grid of almost equal

length and width. The antennas are located at the center of

cells and distributed evenly on the grid. However, when two

or several antennas are too close to each other, the antenna

arrangement is rejected and a new arrangement is chosen.

The maximum cell size for the service area is related to the

desired availability level. Principle factors affecting cell

radius and availability include the rain region, the antenna

and its height, foliage loss, modulation, Tx power, Rx

sensitivity, and sectorization. These effects are generally

related to the service area, such as dense urban, suburban,

and low-density. The cost of cabling between a cell and a

switch is proportional to the distance separating both. We

took a proportionality coefficient equal to the unit. The call

rate gi of a cell i follows a gamma law of average and

variance equal to the unit. The call duration inside the cells

are distributed according to an exponential law of parameter

equal to 1. This is justified by the fact that we are

considering a simple model. For more realistic call duration

distributions and models, the reader is referred to Ref. [7]. If

a cell j has k neighbours, the [0,1] interval is divided into

k þ 1 sub-intervals by choosing k random numbers

distributed evenly between 0 and 1. At the end of the

service period in cell j, the call could be either transferred to

the ith neighbour (i ¼ 1;…; k) with a handoff probability rij

equal to the length of ith interval, or ended with a

probability equal to the length of the k þ 1th interval. To

find the call volumes and the rates of coherent handoff, the

cells are considered as M/M/1 queues forming a Jackson

network [20]. The incoming rates ai in cells are obtained by

solving the following system:

ai 2Xn

j¼1

ajrji ¼ gi avec i ¼ 1;…; n

If the incoming rate ai is greater than the service rate, the

distribution is rejected and chosen again. The handoff rate

hij is defined by:

hij ¼ li·rij

All the switches have the same capacity M calculated as

follows:

M ¼1

mð1 þ

K

100ÞXn

i¼1

li

where K is uniformly chosen between 10 and 50, which

insures a global excess of 10–50% of the switches’ capacity

compared to the cells’ volume of call.

In simulated annealing, the parameters are the tempera-

ture u, the annealing factor að0 , a , 1Þ; the current

solution S1; the solution after perturbations S2; and the

stopping criterion. The steps in simulated annealing are:

Create the initial solution S1 (at random, each cell being

allocated to a switch in an unpredictable way, we create a

solution free from capacity constraints on the switches, but


respecting the constraint of unique assignment of cells to

switches), u is set to a relatively high value (the best solution

cost); Select a new solution S2 that can be obtained by

carrying out a small perturbation over S1;If the solution S2 is

better than S1, then this topology is saved as current one.

However, it happens that, further to a disturbance, the

obtained nearby topology is kept as current solution, even if

it is not better than the current one, provided that it respects

a certain probability of acceptance (the fact of accepting a

loss of quality or fitness allows to avoid being trapped too

early into a local optimum); u is reduced by a multiplication

by an annealing factor aðukþ1 ¼ uk*aÞ; where a ¼ 0:95; If

the stopping criterion will not have been reached, go to step

to select a new solution S2.

4.2. Memetic algorithm implementation

It is desirable that the encoding makes the representation

as robust as possible. This means that even if a piece of the

representation is randomly changed, it will still represent a

viable individual. We have introduced a simple notation to

represent cells and switches, and for encoding chromosomes

and genes. We opted for a non-binary representation of the

chromosomes [15]. In this representation, the genes

(squares) represent the cells, and the integers they contain

represent the switch to which the cell of row i (gene of the

ith position) is assigned. Our chromosomes have therefore a

length equal to the number of cells in the network n, and the

maximal value that a gene can take is equal to the maximal

number of switches m. A chromosome represents the set of

cells in the cellular mobile network, and the length is the

number of cells. A particular value of the string is called a

gene and the possible values are called alleles, taken from

the alphabet V ¼ {1; 2;…;m}: For example, the position 3

in the chromosome represents the third cell and it is

assigned to switch in (Fig. 4).

The first element of the initial population is the one

obtained when all cells are assigned to the nearest switch.

This first chromosome is created therefore in a deterministic

way. The creation of other chromosomes of the population

is probabilistic and follows the strategy of population

without doubles. This strategy permits to ensure the

diversity of the population and a good cover of the search

space. All chromosomes of the population verify the unique

assignment constraint, but not necessarily the one of the

switches’ capacity. The maximum size of this initial

population must be lower than or equal to m n to respect

the principle of the population without doubles.

The crossover operator crosses, under a certain prob-

ability, two randomly chosen chromosomes in the population.

The two chosen chromosomes (parents) are directly inserted

into the new population. Then, the two obtained chromo-

somes (children) are then inserted if the generated probability

of crossover is lower than or equal to the prefixed one.

The mutation operator transfers, under a certain prob-

ability, the elements of the population. It is necessary to

bring back the genetic material that would have been

forgotten during the generations. The size of the population

obtained after applying the operators is then the double of

the one of the initial population.

The choice of the candidates is based on the evaluation

function. In our adaptation, every chromosome is evaluated

according to the criterion of cost in a first time. The sort by

ascending order of the objective value of those chromo-

somes permits to have the best potential chromosomes as

the first elements of population. The second stage of

evaluation consists of verifying the chromosomes in relation

to the capacity constraint on the switches and to determine

the best chromosome that verifies this constraint.

To select the elements of the new generation, we used the

method of the casino caster. As the problem that we have to

solve is a minimization problem, we applied the caster to the

inverses of the objective values of the chromosomes of

the population whose the size is the double of the

initial population one. We recover then in the new selected

population either chromosomes that verify the constraint on

Fig. 4. Representation of individuals.

Fig. 5. MA convergence results.


the capacity of the switches or those that rape it. The number

of generations is fixed at the beginning of the execution. We

inserted into our adaptation the concept of cycle that permits

to run several successive genetic processes. At every cycle,

a new initial population is created.

Our multi-population memetic algorithm (MA) is con-

trolled by many parameters that affect their efficiency and

accuracy. Among other things, one must decide the number

and the size of the populations, the rate of the migration, and

the destination of the migrants.

For the migration algorithm used in this paper,

subpopulations are arranged in fully-meshed topology.

Here, individuals may migrate from any subpopulation to

another. For each subpopulation, a pool of potential

emigrants is constructed from the other subpopulations.

The migration interval is incorporated into the parallel

algorithm as a probability Pm; and the migration rate is

incorporated as a maximum value Sm: For each subpopu-

lation in the parallel algorithm, migration is achieved as

follows. At the end of a generation, a uniformly distributed

random number x is generated. If x , Pm then migration is

initialized. During migration, a uniform random number

determines the number of individual ns between 1 and Sm to

send. The selection of the individuals for migration is

a fitness-based process. The best ns individuals

in the subpopulation are sent to the other subpopulations.

Whether or not emigrants are sent to the other

subpopulations, each subpopulation then checks to see if

emigrants are arriving from its neighbour. If so, a uniform

random number pr determines the number of accepted

individuals, then the best pr individuals are received into the

subpopulation and replace the pr least fit individuals.

4.3. Some computational experiments

In the first step, we generate an initial population of size

200 chromosomes, then we choose at random the values,

which compose each chromosome: it is the first generation

of chromosomes. In the second step, we estimate each

chromosome by the objective function, what allows to

deduct its value of capacity. Finally, in the last step, the

cycle of generations of the populations begins then, each

new population replacing the previous one. The number of

400 generations is defined at first. In each generation, we

will apply the various genetic operators and the local search

to 200 chromosomes. After each generation, 200 newly

created chromosomes replace the previous generation. After

the ith generation, chromosomes will have evolved in such a

way that this last generation contains chromosomes better

than those in previous generations (i 2 1)th.

To determine the number of subpopulations in parallel,

MA was executed over a set of 600 test cases with 3 instances

of problem in series of 20 tests for each assignment pattern,

with a number of populations varying between 1 and 10. This

experience shows that MA converges to good solutions, close

the lower bound, with a number of populations varying

between 7 and 10, as shown in Fig. 5. An intuitive lower

bound for the problem is the link cost of the solution obtained

by assigning each cell ito the nearest switch k. This lower

bound does not take into account handoff cost. In fact, we

suppose that capacity constraint is being relaxed and that all

cells could be assigned to a single switch.

To define the population size, MA was executed over a set

of600 test caseswith3mobilenetworks inseriesof20 tests for

each assignment pattern with 8 populations. This experience

shows that MA converges to provide good solutions with a

population size varying between 80 and 140 (Fig. 6).

Fig. 6. Population size in MA.

Table 1

CPU Time for MA execution

Element CPU Time percentage (%)

Generate an initial population 1

Evaluation of the chromosomes 42

Crossover, mutation, local-search 20

Immigration procedure 1

Selection procedure 25

Emigration procedure 1

Others 7


The values used by MA and SGA are: the number of

generations is 200; the population size is 100; the number of

populations is 8 for MA and 1 for SGA; the crossover

probability is 0.9; the mutation probability is 0.08; the

migration interval (Pm) is0.1; themigration rate (Sm) is0.4and

the emigrants accepted (Pr) is 0.2. The number of exchanged

individuals (migration rate) determines the importance

of genetic and memetic diversities in the subpopulations.

Theexperiences showthat anumber ofexchanged individuals

between 40 and 80% of the subpopulation size provide good

solutions. In terms of evaluation fitness, a migration rate of

40% or 80% provides similar results.

Whereas the migration algorithm seeks to improve the

normalized cost and reliability of the SGA, it is important

also to ensure that unacceptable time overhead is not

introduced by migration (Table 1). In order to analyze the

performance of the memetic algorithm independent of

the migration algorithm, migration is turned off ðpm ¼ 0:0Þ:

Table 2

MA and SGA parameters

Number

of generations

Population

size

Number

of cycles

Number

of populations

Crossover

probability

Mutation

probability

SGA 100 100 40 1 0.9 0.08

MA1 (SMA-Simulated

annealing)

100 100 1 1 0.9 0.08

MA2 (SMA-tabu

search)

100 100 1 1 0.9 0.08

Fig. 7. Comparison between MA and SGA.


Turning off migration for analysis of the memetic algorithm

ensures that timing measurements indicate the effects of

parallel algorithm, rather than the effects of migration.

5. Performance evaluation and numerical results

In order to compare the performance of memetic

algorithm (MA) and genetic algorithm (SGA) with that of

the other heuristics, two types of experiments were

performed: a set of experiments to evaluate the quality of

the solutions in terms of their costs, and another set to

evaluate the performance of MA and SGA in terms of CPU

times. All the test runs described in this section were

performed in a networked workstation environment operat-

ing at 100 Mbps with 10 PCs (Pentium 500 MHz).

5.1. Comparison with standard genetic algorithm approach

In this section, we present the comparative results,

solution costs and CPU times, between memetic

algorithm (MA) and standard genetic algorithm (SGA).

Fig. 8. Comparison between MA, tabu search and simulated annealing.


For the experiments, MA and SGA are executed over a set

of 600 test cases with 3 topologies in series of 50 tests for

each assignment pattern. Table 2 shows the parameter

values used by MA and SGA.

We have tested two memetic algorithms (standard

genetic algorithm in combination with simulated annealing

and tabu search). For these experiments, SMA and MA are

executed over a set of 600 test cases with 3 instances of

problem in series of 50 tests for each assignment pattern.

MA were executed with 8 parallel populations. The results

of this experiment show that the simulated annealing and

tabu search improved the individuals representing solutions

provided by sequential genetic algorithm and multi-

population algorithm. In the case of simulated annealing,

the average improvement rate for sequential algorithm is

39.69% and the average improvement rate for multi-

population algorithm is 22.34%. In the case of tabu search,

the average improvement rate for sequential algorithm is

40.24% and the average improvement rate for multi-

population algorithm is 16.5%.

A comparison between MA and SGA is undertaken in

order to measure the evolution cost and the execution

time. In the first set of experiments, the results obtained

by SGA are compared directly with those provided by

MA. MA and SGA always find the feasible solutions. In

each of the three considered series of tests, MA yields an

improvement in terms of both CPU times and evaluation

cost, in comparison with SGA. In terms of CPU times,

MA2 is between 10 and 20 faster than MA1 and SGA to

find a feasible solution (Fig. 7a). In terms of costs, MA

provide always feasible solutions with a very similar

cost, with better results than sequential memetic

algorithm and standard genetic algorithm. Fig. 7b

shows the comparison between MA and SGA in which

each simulation represents the average over 50 tests of

each algorithm. In conclusion, experimental results from

solving different instances of assignment problem show

that MA approach provides better results than a standard

genetic algorithm.

5.2. Comparison with other heuristics

In this section, we compare tabu search method,

simulated annealing and memetic algorithm (MA) against.

For the experiments, memetic algorithm and the other

heuristics are executed over a set of 600 test cases with a

number of cells varying between 100 and 200, and a number

of switches varying between 5 and 7, that means the search

space size is between 5100 and 7200. We did not consider

problems with small search spaces, because in these cases it

is possible to use enumerative searches to solve them.

The three heuristics always find feasible solutions. In

each of the all considered series of tests, MA yields an

improvement in the cost function in comparison with the

other two heuristics. In terms of costs, MA provides better

results than tabu search and simulated annealing and

the average improvement rates are 1.6 and 4.5% (Fig. 8).

However, in terms of CPU MA is slower than simulated

annealing and tabu search.

6. Conclusions

In this paper, we proposed an evolutionary approach

(memetic algorithms) to solve the problem of assigning cells

to switches in cellular mobile networks. Experiments have

been conducted to measure the quality of solutions provided

by these algorithms. Experimental results from solving

different instances of assignment problem show that our

approach provides better results than a standard genetic

algorithm (SGA). In each of the series of tests considered,

memetic algorithms (MA) yield an improvement in CPU

times and in costs in comparison with SGA.

Finally, the results obtained have been compared with

the results provided by tabu search and simulated

annealing. MA, tabu search and simulated annealing

provide feasible solutions with a very similar cost. In

general, they results obtained with MA are better than

those generated by tabu search and simulated annealing.

The MA were tested with data supplied by a local

network operator and the obtained results were

also satisfying. However the best guarantee of the

efficiency of the algorithm is the large range of data

used in our tests. In summary, the numerical results have

shown that MA can adequately solve this NP-hard

problem and they provide good solutions for moderate-

and large-sized cellular networks (from 5 switches and

100 cells).

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