Integration of ACO in a Constraint Programming Language

Size: px

Start display at page:

Download "Integration of ACO in a Constraint Programming Language"

Brooke Newton
10 years ago
Views:

1 Integration of ACO in a Constraint Programming Language Madjid Khichane 12, Patrick Albert 1, and Christine Solnon 2 1 ILOG 2 LIRIS, UMR 5205 CNRS / University of Lyon ANTS 08

2 Motivations Ant Colony Optimization (ACO) Efficient algorithms for solving specific problems...but solving a new problem involves a lot of programming Constraint Programming (CP) High level languages for modelling problems declaratively Branch & Propagate search engine may spend unacceptable time to solve some instances Our goal: use ACO to guide a CP search Describe the problem with the CP language Use the CP solver to propagate and check constraints Use ACO to guide the search First experiments with ILOG solver CP language

declaratively Branch & Propagate search engine may spend unacceptable time to solve some instances Our goal: use ACO to guide a CP

3 Basic principle of Constraint Programming (CP) CP modelling languages Define the problem by means of constraints (X, D, C) Define a set X of variables Define the domain D(x i ) of every variable x i X Define a set C of constraints CP solvers Goal: find an assignment of X which satisfies C Basic idea: Branch & Propagate Branch Construct a search tree Iteratively assign variables Backtrack if a constraint is violated Propagate Use constraints to prune the tree reduce domains by removing inconsistent values

find an assignment of X which satisfies C Basic idea: Branch & Propagate Branch Construct a search tree Iteratively assign

4 Using ACO to solve CSPs Existing work [Solnon 00, Solnon 02, Bridge & Solnon 06] Ants build complete assignments: Repeat Assign a variable to a value chosen w.r.t. ACO Until all variables have been assigned Ants goal = minimize the number of violated constraints very competitive results... but ad hoc algorithms New proposition: CP with ants Ants build partial consistent assignments Repeat Assign a variable to a value chosen w.r.t. ACO Propagate to remove inconsistent values from domains Until propagation detects a failure or all variables assigned Ants goal = maximize the number of assigned variables straightforward integration within a CP language

5 Using ACO to solve CSPs Existing work [Solnon 00, Solnon 02, Bridge & Solnon 06] Ants build complete assignments: Repeat Assign a variable to a value chosen w.r.t. ACO Until all variables have been assigned Ants goal = minimize the number of violated constraints very competitive results... but ad hoc algorithms New proposition: CP with ants Ants build partial consistent assignments Repeat Assign a variable to a value chosen w.r.t. ACO Propagate to remove inconsistent values from domains Until propagation detects a failure or all variables assigned Ants goal = maximize the number of assigned variables straightforward integration within a CP language

6 Table of contents 1 Introduction 2 Description of Ant-CP 3 Application to the Car sequencing 4 Experimental Results 5 Conclusion

7 Ant-CP procedure initialize pheromone trails to τ max repeat 1 each ant builds a partial consistent assignment 2 update pheromone trails until solution found or max cycles reached

8 Ant-CP procedure initialize pheromone trails to τ max repeat 1 each ant builds a partial consistent assignment 2 update pheromone trails until solution found or max cycles reached Pheromone structure associated with a CSP (X, D, C) Pheromone is laid on variable/value couples: For every variable X i, and for every value v i D(X i ), τ Xi,v i = quantity of pheromone associated with X i, v i learnt desirability of assigning v i to X i

associated with a CSP (X, D, C) Pheromone is laid on variable/value couples: For every variable X i, and for

9 Ant-CP procedure initialize pheromone trails to τ max repeat 1 each ant builds a partial consistent assignment 2 update pheromone trails until solution found or max cycles reached Construction of a partial consistent assignment A Iteratively assign variables until all variables assigned or Failure: Choose a non assigned variable X i Choose a value v i D(X i ) with probability p(v i ) = [τ Xi,v i ] α [η(x i,v i )] β v k D(X i ) [τ X i,v k ] α [η(x i,v k )] β where η(x i, v i ) is a problem-dependent heuristic factor Propagate to remove inconsistent values from domains

assigned or Failure: Choose a non assigned variable X i Choose a value v i D(X i ) with probability p(v i ) = [τ Xi,v i ] α [η(x i,v i )] β v k

10 Ant-CP procedure initialize pheromone trails to τ max repeat 1 each ant builds a partial consistent assignment 2 update pheromone trails until solution found or max cycles reached Pheromone updating step Evaporation: multiply pheromone trails by (1 ρ) ρ = evaporation rate (0 ρ 1) Reward the best assignment A of the cycle: X i, v i A, increment τ Xi,v i by 1/(1 + A best A ) where A best is the best assignment found so far

Evaporation: multiply pheromone trails by (1 ρ) ρ = evaporation rate (0 ρ 1) Reward the best assignment A

11 Pheromone strategies Default pheromone strategy pheromone is laid on variable/value couples Specific pheromone strategies The user has to define a set of pheromone trails a function τ pheromone factors a function comp rewarded components

strategies The user has to define a set of pheromone trails

12 Table of contents 1 Introduction 2 Description of Ant-CP 3 Application to the Car sequencing 4 Experimental Results 5 Conclusion

13 The car sequencing problem Goal: Sequence cars along an assembly line Each car requires a set of options Space cars requiring a same option Example Set of cars to be sequenced: Sequencing contraints: Solution: 1/2 ; 2/5 ; 1/5 ; 1/3

cars requiring a same option Example Set of cars to be

14 CP model for the car sequencing problem First model of the User s manual of ILOG Solver Variables For each position i in the sequence, car i = class of the ith car For each position i in the sequence and each option j, opt ij = 1 if car i requires option j and opt ij = 0 otherwise Constraints Constraints on the number of cars to be produced: car class c, #{car i = c} = nb of cars of class c to be produced IloDistribute Constraint between car and opt variables: car i and option j, opt ij = 1 iff car i requires option j IloBoolAbstraction Capacity constraints: option j, subseq. s of q j cars, i s opt ij p j

Constraints on the number of cars to be produced: car class c, #{car i = c} = nb of cars of class c to be produced IloDistribute Constraint between car

15 Pheromone strategies Comparison of 4 pheromone strategies Default pheromone is laid on couples (position/car class) Classes [Gravel et al 04] pheromone is laid on couples of consecutive car classes Cars [Solnon 00] pheromone is laid on couples of consecutive cars Empty pheromone is not used

04] pheromone is laid on couples of consecutive car classes Cars [Solnon

16 Heuristics Recall: choice of a value for a variable p(v i ) = [τ Xi,v i ] α [η(x i,v i )] β P v k D(X i ) [τ X i,v k ] α [η(x i,v k )] β where η(x i, v i ) is a problem-dependent heuristic factor Utilisation rate UR(o i ) of an option o i [Smith 97] UR(o i ) = number of required slots / number of available slots UR(o i ) > 1 no solution Comparison of 2 heuristics DSU = sum of utilization rates of required options favor cars that require options with high utilisation rates DSU+P = sum of utilization rates of required options + failure when UR(o i ) > 1 + filter domains when UR(o i ) = 1

of available slots UR(o i ) > 1 no solution Comparison of 2 heuristics DSU = sum of utilization rates of required options favor cars that require

17 Table of contents 1 Introduction 2 Description of Ant-CP 3 Application to the Car sequencing 4 Experimental Results 5 Conclusion

18 Test suites Instances of [Lee et al. 95] available in CSP lib 70 satisfiable instances with 200 cars All solved by Ant-CP in a few cycles and less than one second (whatever the pheromone strategy and the heuristic factor) these instances are too easy to evaluate Ant-CP Instances of [Perron & Shaw 04] 82 instances from 100 to 500 cars; 8 options; 20 car classes all satisfiable nearly half of these instances are difficult

less than one second (whatever the pheromone strategy and the heuristic factor) these instances are too

19 Comparison of the 4 pheromone structures with the DSU heuristic Cars Success rate (for 10 runs per instance) Default Classes no pheromone Number of cycles

per instance) 75 70 65 60 55 50 Default Classes no

20 Comparison of the 2 heuristics Success rate (for 10 runs per instance) Cars Default Classes no pheromone DSU+P DSU DSU+P DSU DSU+P DSU DSU+P DSU Number of cycles

21 Table of contents 1 Introduction 2 Description of Ant-CP 3 Application to the Car sequencing 4 Experimental Results 5 Conclusion

22 Using ACO to guide a CP search Complementarity of CP and ACO Use CP for modelling the problem and for propagating and checking constraints Use ACO for guiding the search as a generic value ordering heuristic First results on the car sequencing problem Ant-CP outperforms complete approaches: Complete approaches still have difficulties to solve the 70 instances of Lee (see CP 06, CP 07)...whereas these instances are all quickly solved by Ant-CP. Ant-CP is an order slower than heuristic approaches dedicated to the car sequencing problem...but the programming effort is also much smaller

23 Further work Validate our approach on other CSPs Adaptive parameter tuning towards reactive ACO use resampling and similarity ratio to dynamically tune parameters Use filtering procedures dedicated to a non backtracking search

Ant Colony Optimization and Constraint Programming

Ant Colony Optimization and Constraint Programming Christine Solnon Series Editor Narendra Jussien WILEY Table of Contents Foreword Acknowledgements xi xiii Chapter 1. Introduction 1 1.1. Overview of the