Fixed Channel Assignment in Cellular Radio Networks Using a

Modified Genetic Algorithm

Speaker : Li, SHAN-CHENG

Date:2004/06/09

1.each diagonal element represents the CSC , i.e.,the minimum separation distance between any two channels at cell i

2.each nondiagonal element represents the minimum separation distance in frequency between any two frequencies assigned to cells i and j ,respectively.

In this matrix, CCC is represented by ,ACC is represented by ,and cells that are free to use the same channels are represented by . In all cases,

Problem Formulation(1/2)

iic

ijc

1ijc

Using an appropriate mapping, channels can be represented by consecutive positive integers. Therefore, the RMC constraints can be described by an nXn symmetric matrix called the compatibility matrix C

1iic0ijc2ijc

Problem Formulation(2/2)

By analyzing the traffic at each cell, the traffic demand requirement can be obtained. This can be represented by an n-element demand vector denoted as d . In this vector,each element represents the number of channels to be assigned to cell i.

id

Graph-Theoretic Formulation

ikki

sm,

max

Mathematically, it means that if sik is the channel number of the kth call assigned to cell I, then the problem can be expressed as

Minimize

Subject to ijjlik css

for i,j=1,2,…,n

and k=1,2,…,di,l=1,2,…,dj

and (i,k)≠(j,l)

Formulation

In particular, we represent the solution space F as an nXm binary matrix,where n is the number of radio cells and m is the total number of available channels. Each element fjk in the matrix is either ona or zero such that

jcelltodnotassigne

assignediskchanneliff jk _____

0

1

Cost Function(1/4)

01

m

qiiq df

If the assignment to cell i violates the demand constraint , then

If the assignment of channel p to cell i violates CSC , then

1

1

1

0ii

ii

cp

mq

pq

cpqiqf

Cost Function(2/4)

If the assignment of channel p to cell i violates CCC and/or ACC , then

0

0

1

1

1

1

n

c

ij

j

cp

mq

cpqjq

ij

ij

ij

f

Therefore, a generic choice of cost function can be expressed as

n

i

m

qiiq

n

i

m

pip

cp

mq

pq

cpqiq

n

i

m

pip

n

c

ij

j

cp

mq

cpqjp

df

ff

ffFC

ii

ii

ij

ij

ij

1 1

1 1

1

1

1

1 1

0

1

1

1

1

Cost Function(3/4)

Cost Function(4/4)

Crossover in Genetic Fix(1/2)

Crossover in Genetic Fix(2/2)

Crossover

Let bi be the ith bit position of an individual . To mutate bi ,we need to find a random bj such that .Then , we swap bi with bj . In the case of binary array representation,both bi with bj must be in the same row.

Mutation

1 ji bb

Minimum-Separation Encoding Scheme(1/2)

There exists some combinatorial optimization problems,which may require a minimum separation between consecutive elements in the solution. By encoding each element properly, the solution space can be greatly reduced.

In the sequel, we will present an encoding technique,called the minimum-separation encoding scheme ,which can accomplish this objective

P=10 q=3 dmin=3

Minimum-Separation Encoding Scheme(2/2)

However , a problem still remains is a ‘one’ is at a position within (dmin-1) from the end of the string . To cope with this shortcoming,we need to first augment the original individual with (dmin-1) “zero” before performing the encoding scheme.

Impelmentation Issues(1/4)Therefore, using the minimum-separation encoding scheme , we can eliminate the CSC requirement from the cost function and further reduce the search space.As a result, row i of the solution matrix F will consist of di with a total length of s'1

~

bitscdm iii 11

Hence, using the genetic-fix algorithm and the minimum-separation encoding scheme,the cost function of the channel-assignment problem can be simplified to

ip

n

i

m

p

n

c

ij

j

cp

mq

cpqjq ffFC

ij

ij

ij

1 1

0

1

1

1

1

Impelmentation Issues(2/4)

Impelmentation Issues(3/4)

By exploiting the symmetry of the compatibility matrix C,the cost function can be further simplified to

m

cp

p

cpq

m

pipjpipjq

n

i

n

c

ij

c

p

p

qipjq

ij ij

ij

ij

ffff

ffFC

1

1 1

1

10

1

1

1

1

1

2

1

Impelmentation Issues(4/4)

Local-Search Routine

The routine starts with a given binary array solution F and an empty penalty vector with size np and proceeds as follows.

Results(1/4)

Results(2/4)

Results(3/4)

Results(4/4)