Index of /site/fileadmin/seminaria/2005

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Systematic construction of recombination operators. . . The Capacitated Vehicle Routing Problem (CVRP) The crossover operators Comparison experiments Conclusions and further research

Comparison of Crossover Operators

for the Capacitated Vehicle Routing Problem

Marek Kubiak

Faculty of Computer Science and Management

Poznań University of Technology

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Outline

1 Systematic construction of recombination operators for problems

of combinatorial optimisation

2 The Capacitated Vehicle Routing Problem (CVRP)

3 The crossover operators

4 Comparison experiments

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Systematic construction of operators: motivation

metaheuristic algorithms are not general tools of optimisation

(the No Free Lunch Theorem)

the way a given metaheuristic is adapted to a problem

may have crucial influence on the algorithm’s performance

domain knowledge has to be introduced in algorithms

for the sake of efficiency

schemes of systematic and justified adaptation

of metaheuristic algorithms to problems are welcome

(4)

Systematic construction of operators: the procedure

make hypotheses about features of solutions which may

influence the goal function

define/implement distance functions which exploit the features

test the distance functions against global convexity

global convexity is revealed ⇒ implement recombination

operators which preserve the distances (DPX)

Global convexity examination is made

a tool of searching for important features of solutions

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The Capacitated Vehicle Routing Problem (CVRP)

Input:

one depot

set of geographically

distributed customers with

certain demands

travel cost between

customers and the depot

set of identical vehicles with

limited capacity

Goal: find a set of routes which

satisfies customers’ demands and

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Systematic construction of operators for the CVRP

two features of solutions examined:

clusters of customers: sim

cpn

(s

1 , s

2 )

edges: sim

ce

(s

1 , s

2 )

high absolute values of correlation in sets of 1000 local optima

for most of instances found

3 DPX operators implemented:

CPX: preserves clusters of customers

CEPX: preserves common edges

CECPX: preserves both

a mutation operator CPM implemented which preserves

clusters of customers

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Systematic construction of operators for the CVRP

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Systematic construction of operators for the CVRP

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Systematic construction of operators for the CVRP

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RBX SplitX CPX2 CEPX CECPX2

Crossover operators for the CVRP

route-based crossover (

RBX

), sequence-based crossover (SBX)

Potvin J.-Y., Bengio S., The Vehicle Routing Problem with Time Windows - Part II: Genetic Search,

INFORMS Journal of Computing, 8, 2, 1996.

Jozefowiez N. et al., Parallel and Hybrid Models for Multi-Objective Optimization: Application to the Vehicle Routing Problem, in: Parallel Problem Solving from Nature - PPSN VII, Springer-Verlag, 2002

generic crossover (GCX), specific crossover (SCX)

Tavares J. et al., Crossover and Diversity: a Study about GVR, GECCO 2003.

split crossover (

SplitX

)

Prins C., A Simple and Effective Evolutionary Algorithm for the Vehicle Routing Problem, 4th

Metaheuristics International Conference, July 2001.

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RBX: route-based crossover

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SplitX: Prins’ representation

a permutation only

optimal division into

routes

arbitrary merge points

makes no difference for

local optima

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SplitX: split crossover

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SplitX: split crossover

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CPX2: clusters preserving crossover

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Evolvability Speed Preservation of distances Optimisation efficiency

Aspects of comparison

evolvability

speed

preservation of distances

optimisation efficiency

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Evolvability Speed

Preservation of distances Optimisation efficiency

Motivation for measuring evolvability

is an operator able to introduce something new to a

population?

what is the probability (for the given operator) to produce an

offspring better than its parents?

Altenberg L., The Schema Theorem and Price’s Theorem. In: Whitley D., Vose M. (eds.), Foundations of

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The evolvability experiment

1000 trials of each crossover on the same set of local optima

after each crossover perform local search with 3 merged

neighbourhoods:

merge 2 routes

exchange 2 edges

swap 2 customers

examine the offspring after local search:

is it different from parents?

is it better than the worse parent?

(an approximation of an acceptance criterion)

is it better than the better parent?

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Evolvability: results

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Evolvability: results

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Evolvability: results

Op1>Op2

RBX

SplitX

CPX2

CEPX

CECPX2

Sum

RBX

0

4 -3

-20

-19

-38

SplitX

-4

0 -5

-20

-19

-48

CPX2

3

5

0 -21

-21

-34

CEPX

20

21

0

10

71 CECPX2

19

21 -10

0

49

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The speed experiment

1000 trials of each crossover on the same set of local optima

after each crossover perform local search with 3 merged

neighbourhoods:

merge 2 routes

exchange 2 edges

swap 2 customers

examine the number of iterations of local search

compute the average time of one crossover (

to be done

)

investigate the relationship between the number of iterations

and quality of the offspring (

to be done

)

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Speed: results

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Motivation for measuring the preservation of distances

preservation of certain features in a VRP

Jaszkiewicz A., Kominek P., Genetic local search with distance preserving recombination operator for a vehicle routing problem, European Journal of Operational Research, 151, 2003, 352-364

preservation of certain features in the ROADEF 2003

Challenge problem

Jaszkiewicz A., Adaptation of the genetic local search algorithm to the management of Earth observation satellites, in: 7th National Conference on Evolutionary Computation and Global Optimization, Kazimierz

Dolny, Poland, 2004

important features for the CVRP were identified

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A distance metric for sets

A normalized distance metric:

dists

(S

1 , S

2 ) =

|S

1 ∪ S

2 | − |S

1 ∩ S

2 |

|S

1 ∪ S

2 |

=

|S

1 | + |S

2 | − 2 · |S

1 ∩ S

2 |

|S

1 | + |S

2 | − |S

1 ∩ S

2 |

=

18 + 18 − 2 · 9

18 + 18 − 9

=

2

3

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A distance metric for clusterings

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A distance metric for clusterings

distc

(D

1 , D

2 )

=

1/2 · max

C

1i

∈D

1

{min

C

2j

∈D

2

{dist

s

(C

1i

, C

2j

)}} +

+1/2 · max

C

2j

∈D

2

{min

C

1i

∈D

1

{dist

s

(C

1i

, C

2j

)}}

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A distance metric for clusterings

dist

s

C

11 C

12 C

13 min

C

21

0.29

1

0.29 C

22

0.86

0.93

0.69

0.69 C

23

1

0.44

0.85

0.44 min

0.29

0.44

0.69 distc

=0.69

Table:

Computation of dist

c

for the exemplary clusterings.

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The distance preservation experiment

1000 trials of each crossover on the same set of local optima

after each crossover perform local search with 3 merged

neighbourhoods:

merge 2 routes

exchange 2 edges

swap 2 customers

examine the distances:

between parents

between the offspring and its parents

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Distance preservation: initial results

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Distance preservation: initial results

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The optimisation efficiency experiment

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Conclusions

high evolution potential in systematically constructed

operators

the fastest operators are not necessarily the best

forbidding LS moves should be beneficial (preserve distance,

speed up computation)

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Further research

process results of distance preservation experiments

perpare optimisation efficiency experiment

implement and compare to other operators

evolvability:

how does the probability change with increasing quality of

solutions?

does the probability correlate with strength of global convexity?

implement the forbidding of moves in local search

exploit distance metrices in convergence analysis of the

memetic algorithm

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