GENETIC ALGORITHMS TO CONSTRAINT SATISFACTION PROBLEMS

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23 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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1

GENETIC ALGORITHMS TO CONSTRAINT SATISFACTION
PROBLEMS

Genetic algorithm is a
population
-
based search

method. Genetic algorithms are
acknowledged as good solvers for tough problems. However, no standard GA takes
constraints into account. This chapter des
cribes how genetic algorithms can be used for
solving constraint satisfaction problems.

I.
WHAT IS A GENETIC AGORITHM

?

The general scheme of a GA can be given as follows:

begin

INITIALIZE population with random candidate solutions;

EVALUATE ea
ch candidate;

repeat

SELECT parents;

RECOMBINE pairs of parents;

MUTATE the resulting children;

EVALUATE children;

SELECT individuals for the next generation

until

TERMINATION
-
CONDITION is satisfied

end

The GA can be represented in form of a diagram:

Figure 1 The general schem
e of Genetic Algorithm

Parents

Children

Population

Initialization

Termination

Recombination

Mutation

Parent selection

Survivor selection

2

It’s clear that this scheme falls in the category of
generate
-
and
-
test

algorithms. The
evaluation function represents a heuristic estimation of solution quality and the search
process is driven by the variation and the selection
operator. GA has a number of
features:

-

GA is population
-
based

-

GA uses recombination to mix information of candidate solutions into a new
one.

-

GA is stochastic.

II
COMPONENTS OF GENETIC ALGORITHMS

The most important components in a GA consist of:

+ re
presentation (definition of individuals)

+ evaluation function (or fitness function)

+ population

+ parent selection mechanism

+ variation operators (crossover and mutation)

+ survivor selection mechanism (replacement)

Representation

Objects fo
rming possible solution within original problem context are called
phenotypes
, their encoding, the individuals within the GA, are called
genotypes
.

The representation step specifies the mapping from the phenotypes onto a set of
genotypes.

Candidate soluti
on
,
phenotype

and
individual

are used to denotes points of the space of
possible solutions. This space is called
phenotype space
.

Chromosome
, and
individual

can be used for points in the
genotye space
.

Elements of a chromosome are called
genes
. A value o
f a gene is called an
allele
.

Variation Operators

The role of variation operators is to create new individuals from old ones. Variation
operators form the implementation of the elementary steps with the search space.

Mutation Operator

A unary variation

operator is called
mutation
. It is applied to one genotype and delivers
a modified
mutant
, the
child

or
offspring

of it.

In general, mutation is supposed to cause a random unbiased change. Mutation has a
theoretical role: it can guarantee that the space
is connected.

Crossover Operator

3

A binary variation operator is called
recombination

or
crossover.

This operator merges
information from two parent genotypes into one or two offspring genotypes.

Similarly to mutation, crossover is a stochastic operato
r: the choice of what parts of
each parent are combined, and the way these parts are combined, depend on random
drawings.

The principle behind crossover is simple: by mating two individuals with different but
desirable features, we can produce an offsprin
g which combines both of those features.

Parent Selection Mechanism

The role of
parent selection

(
mating selection
) is to distinguish among individuals based
on their quality to allow the better individuals to become parents of the next generation.

Pare
nt selection is
probabilistic
. Thus, high quality individuals get a higher chance to
become parents than those with low quality. Nevertheless, low quality individuals are
often given a small, but positive chance, otherwise the whole search could become too

greedy and get stuck in a local optimum.

Survivor Selection Mechanism

The role of survivor selection is to distinguish among individuals based on their quality.
In GA, the population size is (almost always) constant, thus a choice has to be made on
whic
h individuals will be allowed in the next generation. This decision is based on their
fitness values, favoring those with higher quality.

As opposed to parent selection which is stochastic, survivor selection is often
deterministic
, for instance, ranking
the unified multiset of parents and offspring and
selecting the top segment (fitness biased), or selection only from the offspring (age
-
biased).

Initialization

Initialization is kept simple in most GA applications. Whether this step is worth the
extra co
mputational effort or not is very much depending on the application at hand.

Termination Condition

Notice that GA is stochastic and mostly there are no guarantees to reach an optimum.

Commonly used conditions for terminations are the following:

1.

the maxi
mally allowed CPU times elapses

2.

The total number of fitness evaluations reaches a given limit

3.

for a given period of time, the fitness improvement remains under a threshold
value

4.

the population diversity drops under a given threshold.

4

Note
:
Premature conve
rgence

is the well
-
known effect of loosing population diversity
too quickly and getting trapped in a local optimum.

Population

The role of the population is to hold possible solutions. A
population

is a multiset of
genotypes.

In almost all GA applicati
ons, the population size is constant, not changing during the
evolutional search.

III
HOW DO GENETIC ALGORITHMS WORK
?

The details of how Genetic Algorithms work are explained below.

3.1 Initialization

While genetic algorithms are generally stated wit
h an initial population that is
generated
randomly
, some research has been conducted into using
special techniques

to produce a
higher quality initial population. Such an approach is designed to give the GA a good
start and speed up the evolutionary proces
s.

Example
: Some authors propose a GA for exam timetabling problem in which the GA
works only with feasible solutions, implying that the initial population must also be
made up of feasible solution. Then the GA is run to improve the fitness of the initial

population.

Example 3.1
: In a simple exam timetabling problem, we can use a non
-
binary bit string
representation to represent the chromosome because it is easy to understand and
represent. We use six positions representing six exams with each position’s
value as the
time slot assigned to the exam. We can generate the population randomly to assign each
exam a timeslot.

Day

AM

PM

time1 time2 time3 time4

Day1

e1 e3

Day2 e5 e6 e2,e4

If we randomly generate six numbers 3, 8, 4, 8, 6, 7 as six timeslots for
e1
-
e6
, then the
chromosome is
3 8 4 8 6 7
.

If the population size is 5, an initial population can be generated randomly as follows:

5

index

chromosome

fitness

1

3 8 4 8
6 7

0.005

2

7 3 7 6 1 3

0.062

3

5 3 5 5 5 8

0.006

4

7 6 7 7 2 2

0.020

5

1 7 4 5 2 2

0.040

3.2 Reproduction

There are two kinds of reproduction: generational reproduction and steady
-
state
reproduction.

Generational Reproduction

In generationa
l reproduction, the whole of a population is potentially replaced at each
generation. The most often used procedure is to loop N/2 times, where N is the
population size, select
two
chromosomes each time according to the current selection
procedure, produci
ng
two

children from those two parents, finally producing N new
chromosomes.

-
state Reproduction

-
state method selects two chromosomes according to the current selection
procedure, performs crossover on them to obtain one or two children,

perhaps applies
mutation as well, and installs the result back into that population; the least fit of the
population is destroyed.

3.3 Parent Selection mechanism

The effect of selection is to return a probabilistically selected parent. Although this
sel
ection procedure is stochastic, it does not imply GA employ a directionless search.
The chance of each parent being selected is in some way related to its fitness.

Fitness
-
based selection

The standard, original method for parent selection is Roulette Whe
el selection or
fitness
-
based selection. In this kind of parent selection, each chromosome has a chance
of selection that is directly proportional to its fitness. The effect of this depends on the
range of fitness values in the current population.

Example
: if fitness range from 5 to 10, then the fittest chromosome is twice as likely to
be selected as a parent than the least fit.

If we apply fitness
-
based selection on the population given in example 3.1, we select the
second chromosome 7 3 7 6 1 3 as our f
irst parent and 1 7 4 5 2 2 as our second parent.

Rank
-
based selection

6

In the rank
-
based selection method, selection probabilities are based on a chromosome’s
relative rank or position in the population, rather than absolute fitness.

Tournament
-
based se
lection

The original tournament selection is to choose K parents at random and returns the
fittest one of these.

3.4 Crossover Operator

The crossover operator is the most important in GA. Crossover is a process yielding
recombination of bit strings vi
a an exchange of segments between pairs of
chromosomes. There are many kinds of crossover.

One
-
point Crossover

The procedure of one
-
point crossover is to randomly generate a number (less than or
equal to the chromosome length) as the crossover position.
Then, keep the bits before
the number unchanged and swap the bits after the crossover position between the two
parents.

Example: With the two parents selected above, we randomly generate a number 2 as
the crossover position:

Parent1: 7 3 7 6 1 3

Par
ent2: 1 7 4 5 2 2

Then we get two children:

Child 1 : 7 3| 4 5 2 2

Child 2 : 1 7| 7 6 1 3

Two
-
point Cross Over

The procedure of two
-
point crossover is similar to that of one
-
point crossover except
that we must select two positions and only the bits bet
ween the two positions are
swapped. This crossover method can preserve the first and the last parts of a
chromosome and just swap the middle part.

Example: With the two parents selected above, we randomly generate two numbers 2
and 4 as the crossover pos
itions:

Parent1: 7 3 7 6 1 3

Parent2: 1 7 4 5 2 2

Then we get two children:

Child 1 : 7 3| 4 5| 1 3

Child 2 : 1 7| 7 6| 2 2

Uniform Crossover

The procedure of uniform crossover : each gene of the first parent has a 0.5 probability
of swapping with t
he corresponding gene of the second parent.

7

Example: For each position, we randomly generate a number between 0 and 1, for
example, 0.2, 0.7, 0.9, 0.4, 0.6, 0.1. If the number generated for a given position is less
than 0.5, then child1 gets the gene from

parent1, and child2 gets the gene from parent2.
Otherwise, vice versa.

Parent1: 7 *3 *7 6 *1 3

Parent2: 1 *7 *4 5 *2 2

Then we get two children:

Child 1 : 7 7* 4* 6 2* 3

Child 2 : 1 3* 7* 5 1* 2

3.5 Inversion

Inversion operates as a kind of reorde
ring technique. It operates on a single
chromosome and inverts the order of the elements between two randomly chosen points
on the chromosome. While this operator was inspired by a biological process, it requires

Example: Given a chr
omosome 3
8

4 8
6

7. If we randomly choose two positions 2, 5
and apply the inversion operator, then we get the new string: 3
6 8 4 8

7.

3.6 Mutation

Mutation has the effect of ensuring that all possible chromosomes are reachable. With
crossover and ev
en inversion, the search is constrained to alleles which exist in the
initial population. The mutation operator can overcome this by simply randomly
selecting any bit position in a string and changing it. This is useful since crossover and
inversion may no
t be able to produce new alleles if they do not appear in the initial
generation.

Example
: Assume that we have already used crossover to get a new string: 7 3 4 5 1 3.
Assume the mutation rate is 0.001 (usually a small value). Next, for the first bit 7, w
e
generate randomly a number between 0 and 1. If the number is less than the mutation
rate (0.001), then the first bit 7 needs to mutate. We generate another number between 1
and the maximum value 8, and get a number (for example 2). Now the first bit muta
tes
to 2. We repeat the same procedure for the other bits. In our example, if only the first bit
mutates, and the rest of the bits don’t mutate, then we will get a new chromosome as
below:

2 3 4 5 1 3

IV.
CONSTRAINT HANDLING IN GENETIC ALGORITHMS

There

are many ways to handle constraints in a GA. At the high conceptual level we can
distinguish two cases: indirect constraint handling and direct constraint handling.

Indirect constraint handling means that we circumvent the problem of satisfying
constrain
ts by incorporating them in the fitness function
f

such that
f
optimal implies
that the constraints are satisfied, and use the power of GA to find a solution.

8

Direct constraint handling means that we leave the constraints as they are and ‘adapt’
the GA t
o enforce them.

Notice that direct and indirect constraint handling can be applied in combination, i.e., in
one application we can handle some constraints directly and others indirectly.

Formally, indirect constraint handling means transforming constra
ints into optimization
objectives.

4.1 Direct constraint handling

Treating constraints directly implies that violating them is not reflected in the fitness
function, thus there is no bias towards chromosomes satisfying them. Therefore, the
population wi
ll not become less and less infeasible w.r.t. these constraints. This means
that we have to create and maintains feasible chromosomes in the population. The basic
problem in this case is that the regular operators are blind to constraints, mutating one or
crossing over two feasible chromosomes can result in infeasible offspring. Typical
approaches to handle constraints directly are the following:

eliminating infeasible candidates

repairing infeasible candidates

preserving feasibility by special opera
tors

decoding, i.e. transforming the search space.

Eliminating infeasible candidates is very inefficient, and therefore hardly applicable.
Repairing infeasible candidates requires a repair procedure that modifies a given
chromosome such that it will not

violate constraints. This technique is thus problem
dependent.

The preserving approach amounts to designing and applying problem
-
specific operators
that do preserve the feasibility of parent chromosomes. Note that the preserving
approach requires the cre
ation of a feasible initial population, which can be NP
-
complete.

Decoding can simplify the problem search space and allow an efficient genetic
algorithm. Formally, decoding can be seen as shifting to a search space that is different
from the Cartesian pr
oduct of the domains of the variables in the original problem
formulation.

4.2 Indirect Constraint Handling

In the case of indirect constraint handling the optimization objectives replacing the
constraints are viewed
penalties

for constraint violation he
nce to be minimized. In
general penalties are given for violated constraints although some GAs allocate
penalties for wrongly instantiated variables or as the distance to a feasible solution.

Advantages of indirect constraint handling are:

-

generality

-

reduction of the problem to ‘simple’ optimization

9

-

possibility of embedding user preferences by means of weights.

Disadvantages of indirect constraint handling are:

-

loss of information by packing everything in a single number

-

does not work well with sparse

problems.

V.
GENETIC ALGORITHM FOR 8
-
QUEENS PROBLEM

5.1 Solution Representation

A chromosome is a permutation of the number 1,…,8 and a given g = < i
1
,…, i
8
>
denotes the board configuration where the
k
-
th column contains exactly one queen
placed on th
e
i
k

th row.

Example: the permutation g = < 1,2,3,4,5,6,7,8> represents a board where the queens
are placed along the main diagonal. The solution space is now the set of all
permutations of 1,…,8.

By using such chromosome we restrict the search to boar
d configurations where
horizontal constraint violation (two queens on the same column) and vertical constraint
violation (two queens on the same row) do not occur. In other words, the representation
guarantees “half” number of the constraints and what rema
ins to be minimized is the
number of diagonal constraint violations.

5.2 Crossover operator

(
cut and crossfill
)

Given two parents, which are two permutations, the following mechanism will create
two child permutations.

1.

select a random position, crossove
r point, i

{1, …, 7}

2.

cut both parents in two segments after this position

3.

copy the first segment of parent 1 into child 1 and the first segment of parent 2
into child 2

4.

scan parent 2 from left to right and fill the second segment of child 1 with values
f
rom parent 2 skipping those that already contained in it.

5.

do the same for parent 1 and child 2.

Example: parent1 1 3 5| 7 6 2 4 8

Parent2 2 1 8| 6 4 3 5 7

Child1: 1 3 5| 2 8 6 4 7

Child2: 2 1

8| 3 5 7 6 4

5.3 Mutation operator

We select two positions in a given chromosome and
swaps

the values standing on those
positions.

10

Note that mutation will cause a small undirected change and crossover creates children
that inherit genetic material from

both parents.

5.4 Parent selection and survivor selection

Parent selection
(
best 2 out of random 5
) choosing 5 individuals randomly from the
population and taking the best two as parents that undergone crossover. This ensures a
bias towards using parent
s with relatively high fitness.

Survivor selection
: (
replace worst
) after merging the population and offsprings, then
ranks them according to fitness and deletes the worst two.

5.5 Other issues

Recombination probability

100%

Mutation probability

80%

Population size

100

Initialization

random

Termination

Solution or 10000 evaluations

VI.
A

GENETIC ALGORITHM FOR EXAM TIMETABLING PROBLEM

Burke et al., 1995 [1] proposed a genetic algorithm for solving exam timetabling
problem. This algorithm combin
es
direct representation

and
heuristic crossover
operators

to ensure that the most fundamental constraints are never violated. Heuristic
crossover operators are used to propagate the most desirable features of the timetable to
produce good quality solution
s.

6.1 Solution Representation

When using GA to solve an combinatorial optimization problem, the first important step
is to choose how to represent a solution of the problem as a chromosome. In applying
GA to exam timetabling problem, the most logical ap
proach is to directly encode
solutions with events matched to periods. Figure 5.1 shows such an encoding for n
events where each gene in the chromosome represents which period in which a
particular event is to be scheduled.

Figure 5.1 A direct representation of a timetable

Period Period

1 3

Period Period

2 7

Event 1

Event 2

Event n
-
1

Event n

11

6.2 The Creation of an Initial Population

The following algorithm is used to generate conflict
-
free graph colorings which form
the initial population.

For each population member:

Generate a random
ordering of exams

Take each exam in turn according to that ordering:

Find the first period in which the exam may be placed without conflict and so that

the number of students does not go above a predefined maximum.

Place the exam in

that period.

This algorithm can quickly produce large populations of random feasible exam
timetables.

Note: The method allows the length of the timetable to vary.

The random sequential graph coloring algorithm is used to generate the starting
populat
ion. It uses on average about twice as many periods as the optimal amount.

Then the GA evolves new timetables, possibly reducing the length. This approach
guarantees a feasible timetable and does not create a search space in which no solution
exists.

6.3
Crossover Operators

It is clear that the crossover operator should satisfy the properties of
respect

and
assortment

Respect

is the property that if an exam is timetabled to the same period in both parents
then it will be scheduled to

that period in the child.

Assortment

is the property that the operator can generate a child such that if
Exam1

is
scheduled to
Period 1

in the first parent and
Exam2

is scheduled to
Period 2

in the
second parent then the child may have
Exam 1

in
Period 1

and
Exam 2

in
Period 2

providing that these are compatible.

The crossover operator works for the period
i
as follows:

The operator starts by looking at the first period. It takes exams scheduled in that period
(in both parents) and then uses an algorithm

to select other exams so that none clash
with those already scheduled and the limit on the number of spaces is not violated. Once
this is completed, the crossover looks at period two and so on until all exams are placed.

12

Figu
re 5.2 A Heuristic Crossover Operator

Once an exam is selected, all other exams that clash with it are labeled as
unscheduled

for that period.

The authors construct a number of different crossover operators based on the same
framework but using alternat
ive selection algorithms. The operators are as follows.

Random

Exams are selected at random. This is closest to the standard uniform crossover.

Largest Degree

Exams are selected according to the number of other exams they conflict with.

Most S
imilar Exams

Exams are selected according to how many conflicting exams they have in common

Latest Scheduled in other Parent
.

Exams are selected according to where the same exam is scheduled in the other
pare
nt. Since unplaced exams are passed on the next period, this increases the chances
of shortening the length of the timetable.

Least Conflicting with Previous Period

Exams are selected so as to minimize the number of conflicts with exams in the
previo
usly scheduled period.

6.4 Mutation Operator

Mutation, like crossover, must also ensure that a timetable remains feasible after its
action. It cannot take any exam and shift it to another period at random, since this may
cause a conflict between the move
d exams and ones already scheduled.

We choose to incorporate mutation into the crossover algorithm. This is done by adding
exams to the current search that would otherwise not be considered until a later period.

6.5 Fitness Calculation and Selection

The

evaluation function can be made up of any timetabling related factors. For example,
we may focus on two particular common requirements:

-

The length of the timetable

-

The number of conflicts between exams in adjacent periods.

Given a space P of candidate so
lutions to a problem, fitness function f(p) for p

P
measures the quality of a solution p.

Period
i

of child Timetable:

-

Take those exams schedules in period
i

in both parents 1 and 2.

-

se
lect extra exams from the exams scheduled in period
i

in either parent 1 or
parent 2 or left over from period
i
-
1
.

(Any unscheduled exams are passed onto period
i+1
).

13

Notice that the quality of a solution p may not vary smoothly as the genes comprising p
vary since the genetic operators such as crossover and mutation do not vary

the gene
values smoothly.

It seems reasonable to distinguish between timetables in terms of fitness based on the
numbers and kinds of different constraints violated. For instance, if V(p) is the number
of violated soft constraints in candidate
p
, one cou
ld choose:

f(p) = 1/(1 + V(p))

so that the range of f(p) is from 0 to 1.

If we have
n

kinds of soft constraints, the penalty associated with constraint
-
type
i

is
w
i
,
p

is a timetable, and
c
i
(p)

is the number of violations of constraints of type
i
in
p
, then
the fitness function becomes:

n

f(p) = 1/(1 +

w
i
c
i
(p))

i =1

6.6 Other Issues

The Genetic algorithm for exam timetabling problem uses:

-

Generational Reproduction

-

Popula
tion size = 200

-

Rank
-
based selection mechanism

6.7. Remarks

Burke et al. [1] combines traditional CSP solving heuristics with genetic algorithms.
heuristics (which can le
-
ends) and the blindness of the stochastic genetic
search. The heuristics can be incorporated into the genetic operators mutation and
crossover.

VII
CONCLUSIONS

Constraint handling is not straightforward in a GA because the search operators su
ch as
mutation and recombination are ‘blind’ to constraints. There is no guarantee that if the
parents satisfy some constraints, the offspring will satisfy them as well.

However, genetic algorithms can be effective constraint solvers when knowledge about

the constraints is incorporated either into the genetic operators, in the fitness function,
or in repair mechanisms.

References

[1] Burke, E. K., Elliman, D.G., Weave, R. F., A Hybrid Genetic Algorithm for Highly
Constrained Timetabling Problems,
Proc. o
f 6
th

International Conference on the

14

Practice and Theory of Automated Timetabling
, Napier University, Edinburgh, UK,
1995.

[2] Craenen, B. C. W., Eiben, A. E. and Marchiori, E., How to Handle Constraint with
Evolutionary Algorithms. In L. Chambers (Ed.),
The Practical Handbook of Genetic
Algorithms: Applications, 2
nd

Edition
, volume 1, Chapman & Hall/CRC, 2001, pp. 341

361.

APPENDIX: Randomly Sequential Graph Coloring Algorithm

A
coloring

of a graph is an assignment of a color to each vertex of the gr
aph so that no
two vertices connected by an edge have the same color.

One strategy for graph coloring is the following “greedy” algorithm.

Initially we try to color as many as vertices as possible with the first color, then as many
as possible of the un
colored with the new color, then as many as possible of the un
-
colored vertices with the second color, and so on.

To color vertices with a new color, we perform the following steps.

1.

Select some uncolored vertex and color it with the new color.

2.

Scan the
list of uncolored vertices. For each uncolored vertex, determine
whether it has an edge to any vertex already colored with the new color. If there
is no such edge, color the present vertex with the new color.

Example
: In figure A1 having colored vertex 1

red, we can color vertices 3 and 4 red
also.

Figure A1. Coloring vertices of a graph by randomly sequential graph coloring
algorithm

There are some heuristics on the order of selecting one vertex from the list of uncolored

vertices.

5

1

4

3

2

red

red

red