An Analysis of

Edge Assembly Crossover for the Traveling Salesman Problem

Yuichi Nagata and Shigenobu Kobayashi

IEEE, Conference on Evolutionary Computation, 1999

Genetic AlgorithmHolland, 1975 -

Imitation of Evolution of life form in the Natureindividuals - members of species - in the nature

model of evolution processes where the basic operations are natural selection, crossover and mutationSchema Theorem - analysis of reproduction model

Nothing to do with the real genetic organism

Problem SolvingPolynomial time function

y = ax2 + b• Given constant a, b, know x, calculate y• Find x that gives Maximum y in a given range of x

No Calculation Search for the Solution

Search Space• For small space, use classical exhaustive techniques • For larger space, need special techniques• Analysis of space• Global Optima vs Local Optima

Stochastic SearchLocal Search TechniquesAdaptive Search Techniques

Random searchAnt ColonySimulated AnnealingNeural Network

search space

Standard Genetic Algorithm

Procedure GAbegin

t := 0 ;initialize P(t) ;evaluate P(t) ;while (not termination-condition) dobegin

t := t + 1 ;select P(t) from P(t-1) ;alter P(t) ;evaluate P(t) ;

end

end

Step 0: InitializationStep 0: Initialization

Step 1: SelectionStep 1: Selection

Step 2 : CrossoverStep 2 : Crossover

Step 3 : MutationStep 3 : Mutation

Step 5 : Termination TestStep 5 : Termination Test

Step6: EndStep6: End

Step 4: EvaluationStep 4: Evaluation

GAs: Terminology

– Representation : gene, chromosome, Population– Evaluation : objective function, fitness function– Selection – Operator : crossover, mutation– Replacement : new Generation– Termination : Generation count, Convergence

Step 0: InitializationStep 0: Initialization

Step 1: SelectionStep 1: Selection

Step 2 : CrossoverStep 2 : Crossover

Step 3 : MutationStep 3 : Mutation

Step 5 : Termination TestStep 5 : Termination Test

Step6: EndStep6: End

Step 4: EvaluationStep 4: Evaluation

Representation

• Very crucial step• representation should satisfy the presumption that the whole

chromosome is decomposable to building blocks • String of genes of given alphabet:

– Binary – Float– Integer

• More complex representation– matrices– rules– trees

Initialization of the Starting Population

• Aspects affecting a performance of GA– schemata sampled in the initial population

• Initialization mechanisms – random – informed - uses prior knowledge of the desired solution shape

• Pre-processing– runs several short pre-processing runs– samples the promising areas of the search space identified

during the foregone pre-processing runs

Selection

• Models nature’s survival-of-the-fittest principle• Selection strategies:

– Roulette wheel (proportionate)– Ranking– Tournament

• Selection process:

– determination of Expected values: EVi = fitnessi / fitnessavg

– sampling algorithm - conversion of EVi to the actual number of individuals

Roulette Wheel Selection

Crossover

• Provides random information exchange - works on couples of individuals

• Simple 1-point crossover

Mutation

• Mutation - preserves population diversity– works on single individual

Replacement Strategy• Replacement strategy defines:

– how big portion of the old population will be replaced in each generation of the new population, and

– the rule that determines which individuals from the old population will be replaced and which individuals will be placed in the new population

• Generational - the old population is entirely rebuilt in each generation (short-lived species)

• Steady-state - just a few individuals are replaced in each generation (longer-lived species)

Premature Convergence

• The ratio of the best-fit individual’s reproduction rate to the average reproduction rate is too high

• selection kills ‘worse’ individuals too early

Theory

of GAs

Schema Theorem

• Schema = Pattern• Schema Theorem

– Short, low-order, above-average schemata – receive exponentially increasing trials in subsequent generations

of a genetic algorithm

• Building Block Hypothesis – GA seeks near-optimal performance through short, low-order,

high-performance schemata

mc pso

m

SptFtSevaltStS )(

1

)(1)(/),(),()1,(

• Schema In binary representation - 2L strings, 3L schemata

L = 7, S = (**0*1*1) - covers 24 strings– {0,1, *}– S = {*1*01***, 1*0*11*0, 10111011, *******1, ****0*** }

• Fitness of a schema - average fitness computed over all covered strings

• Schema property– order

• the length of string minus the number of *• defining the specialty of a schema• 8 bits : 11010011, schema and building block 1*010*1*

– defining length • the distance between the first and the last fixed string positions• defining the compactness of information contained in a schema (*11**1*0) = 6, (1******1) = 7

)(So

)(S

Selection• eval(S,t) is the average fitness of all strings in the

population matched by the schema S at time t ;

• Expecting to have strings matched by schema S

– the average fitness of the population

– becomes ;

p

j j pvevaltSeval1

/)(),(

)(/),(|)(|),()1,( tFtSevaltptStS

|)(|/)()( tptFtF

)(/),(),()1,( tFtSevaltStS

)1,( tS

Crossover

)10****************************111(

*)*************************111***(*

1

0

S

S

)011101011111110010101000001010000(

,)110000000100010001000111110111110(

b

a

v

v

)110000000100010010101000001010000(

,)011101011111110001000111110111110(

b

a

v

v

– the string is matched by these two schemata

survives

destroyed

– the probability of destruction of a schema S :

– the probability of survival of a schema S :

1

)()(

m

SSpd

1

)(1)(

m

SSps

(S) = 7, m = 8 ?

mc pso

m

SptFtSevaltStS )(

1

)(1)(/),(),()1,(

• Selective probability of crossover

• The combined effect of selection and crossover

• a new schema growth equation :

cp

1

)(1)(

m

SpSp cs

1

)(1)(/),(),()1,(

m

SptFtSevaltStS c

• All of the fixed positions of a schema must remain unchanged to survive mutation

• mutate at least one of these bits would destroy the schema

• the probability of destruction of a schema S :

• the probability of survival of a schema S :

)110001111101101110000101110111011(av

mpso )(

mpso1 )(

mc pso

m

SptFtSevaltStS )(

1

)(1)(/),(),()1,(

Mutation

TSP with GA

Path representation

(5 1 7 8 9 4 6 2 3) 5-1-7-8-9-4-6-2-3

• Crossover operators Node Orientation vs Edge Orientation

• Mutation operators–insertion 5-2-1-7-8-9-4-6-2-3–Reciprocal Exchange 5-9-7-8-1-4-6-2-3–Inversion 5-9-8-7-1-4-6-2-3

Information of the parents transferred to offsprings

Node crossover = simple but information discarded

Edge crossover = tough but information enclosed

TSP: Edge-Recombination Operator

b-c-e-a-d

b-d-e-c-a

a-b-c-d-e

Edge Assembly Crossover

Edge Assembly Crossover

Previous work

Exx crossover

Ex crossover

Test Library

EXXCrossover

EXCrossover

EXXCrossover

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