better ants, better life? hybridization of constraint programming and ant colony optimization...

Better Ants, Better Life?Better Ants, Better Life?

Hybridization of Constraint Programming and Ant Colony Optimization

Supervisors: Dr. Bernd Meyer, Dr. Andreas Ernst

Martin Held

Oct 27th, 2005

- final presentation -

Hybridizing CP and ACO [2]

IntroductionIntroduction

Ant Colony Optimization (ACO)

nature inspired algorithms to tackle combinatorial optimization problemsbased on collective behaviour of ants

Constraint Programming

Programming paradigm that allows to effectively solve hard constraint problemsSUDOKU, n-Queens, SEND+MORE = MONEY


IntroductionIntroduction

Project Aims

Analysis of different constraint handling techniques in Ant Colony Optimization algorithms (ACO)Improvement of an existing hybrid algorithm (CPACS)Combination of hard and soft constraint handling algorithms


BackgroundBackground

Combinatorial Optimization Problems

Finding the best solution among a set of discrete solutionsinteresting COPs are NP-completesolving them exactly needs exponential timenear optimal solutions can be obtained by using heuristics

Naive way of solving them

enumerate all solutions, pick best onebad idea: exponential number of solutions



Better: Meta-heuristics

stochastic search strategiescleverly sample fraction of solution spacegiven solutions they attempt to generatebetter onesmovement in search space to good solutionsmechanisms to prevent getting stuckin local optimaSimulated Annealing, Genetic Algorithms,Ant Colony Optimization



Constrained COPs

most real world COPs are constrained

practical relevance (timetabling, rostering,

job scheduling, …)

constraints impose difficulties for search procedure

how to generate feasible solutions?

should infeasible solutions be used?

…

no ultimate way of handling constraints in MH



Travelling Salesman Problem (TSP)

unconstrained COP

Example for a COP

A

B

C

D

2

4

5 23

3

7

A – C - B - D - A



TSP with Time windows (TSP-TW)Each city has a time windows in which is has to be visited

single machine job scheduling problem

Example for a constrained COP

A

B

C

D

8

5

30 3

15

12

(10, 50)(20, 34)

(30, 35)

(5, 25)



How would ants solve a TSP?

based on pheromone trails, ants are able to find shortest paths to food sourcestranslating this into an algorithm (ACO), it can be used to solve COPs

A

B

C

D

2

4

5 23

3

7



How would ants solve a TSP?

after constructing a tour, corresponding path is rewarded with pheromoneamount of pheromone depends on solution qualitylater ants more likely to regenerate good solutions

tours might be infeasible due to constraint violationReward them? Try again? Prevent construction of infeasible tours?

How to solve a TSP-TW?



We need smarter ants!(enable them to cope with constraints)


Prunes search space, provides look ahead

Find a feasible instantiation for X and Y X ≥ Y, X+Y < 4, X Є {1,2}, Y Є {1,2}

Can be used to express more complex constraintse.g. job scheduling (due date restrictions etc.)

X

Y Y

1 2

1 2 1 2

X=1Y=1

X=1Y=2

X=2Y=1

X=2Y=2


Constraint Programming



Ants armed with CP knowledge

A

B

C

D

8

5

30 3

15

12

(10, 50)(20, 34)

(30, 35)

(5, 25)



Ants armed with CP knowledge

look ahead depends on constraint solver strengthimplemented in CPACS

A

8

5(10, 50) (20, 34)

(30, 35)

(5, 25)

B

C

D

30

C

D

B

D

B

C

D

C

D

B

C

B

15

12

15

3

12

3

15

15

12

12

3

3


ProjectProject

How to increase look ahead?

besides CP other options to obtain look ahead bounding: calculation of solution boundsGeneral idea:

obtain bound on solutionexclude choices that lead to worse solutions than best found so far

To obtain bounds and information about costs for potential variable assignments we solve a relaxation of the actual TSP-TW


ProjectProject


relaxation easier to solve than original problemsolution of relaxation only lower bound on actual solution

A

B

C

D

A

B

C

D

A B

D C

5

2

23

34 7

Tour Length: 8 (actual TSP optimal 19)


ProjectProject


Use standard linear programming (lp) to solve relaxation


ProjectProject


LP solution provides information aboutlower bound on problem solutioncost for each edge, by which amount would lower bound increase if edge would be in solution

Allows to exclude edges from consideration:

LB + cost[i][j] > best solution found so far

Edges (i,j) were this is true do not have to be considered by antsadditional pruning

Implemented in CPACSlp


ProjectProject

Using LP information for dynamic guidance

A

8

5

B

C

D

30

C

D

B

D

B

C

D

C

D

B

C

B

15

12

15

3

12

3

15

15

12

12

3

3

Ant decision based on pheromone concentration and some bias

e.g. prefer shorter edges


ProjectProject

Using LP information for dynamic guidance

bias is static, dependent on fixed problem properties

Lp calculated edge costs can be used as bias instead

change dynamically while tour construction

represent ‘current’ desirability of edges

might have better search guiding abilities

Implemented in dynamicCPACSlp


ResultsResults

Comparing CPACS vs. (CPACSlp & dynCPACSlp )

all algorithms were applied to a test set of single machine job scheduling problems with various tightnesseach algorithm was run 10 times on each of the 19 problemsTested different configurations

CPACS vs. CPACSlp in absence of pheromone learningCPACS vs. CPACSlp with learning componentdynCPACSlp vs. CPACSlp

Due to vast amount of computation, experiments had to be carried out on the cluster


ResultsResults

Comparing CPACS vs. (CPACSlp & dynCPACSlp )

with 10 runs, standard deviation of measurements was to high to draw any conclusionsincreased the number of runs to 100analyses commenced 4 weeks ago …in between the cluster exhibited severe stability problemsAnalysis not finished yet No final results obtained yetWhat do we expect?

Let’s guess…


ResultsResults

Possible outcomes and consequences

A:significant improvement of search efficiency (labelling steps)

B: no measurable improvement


DiscussionDiscussion

Possible outcomes and consequences (A)

significant improvement of search efficiency (labelling steps)

very good, we implemented an algorithm that performs superior to existing onesit needs less labelling steps to find even better solutions than existing onesFurther investigations should relate labelling steps to CPU timeLess labelling steps not necessarily fasterCP and LP solving needs additional timeSimpler search heuristic might be able to achieve similar results with same amount of CPU time


DiscussionDiscussion

Possible outcomes and consequences (B)

no measurable improvement

unfortunateobtained bounds might not be tight enoughinvestigation of actual tightnessHow much pruning is achieved?necessarily improve LP model


ConclusionsConclusions

implemented new algorithmic approaches to handle constraints in ACO

results will allow to asses the relevance of these new approaches

proved justification of using Stochastic Ranking as constraint handling technique in ACO


Future Directions (general)Future Directions (general)

Investigation of labelling steps – CPU time relationImprovement of LP model to get tighter bounds more search space pruning, better guidance?further investigation of different scalings of dynamic cost valuescomparison with other algorithms (needs efficient implementations)


Thanks for your attention!

Questions?

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