Better Ants, Better Life? Hybridization of Constraint Programming and Ant Colony Optimization Supervisors: Dr. Bernd Meyer, Dr. Andreas Ernst Martin Held Oct 27th, final presentation -

Hybridizing CP and ACO [2] Introduction Ant Colony Optimization (ACO) nature inspired algorithms to tackle combinatorial optimization problems based on collective behaviour of ants Constraint Programming Programming paradigm that allows to effectively solve hard constraint problems SUDOKU, n-Queens, SEND+MORE = MONEY

Hybridizing CP and ACO [3] Introduction 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

Hybridizing CP and ACO [4] Background Combinatorial Optimization Problems Finding the best solution among a set of discrete solutions interesting COPs are NP-complete solving them exactly needs exponential time near optimal solutions can be obtained by using heuristics Naive way of solving them enumerate all solutions, pick best one bad idea: exponential number of solutions

Hybridizing CP and ACO [5] Background Better: Meta-heuristics stochastic search strategies cleverly sample fraction of solution space given solutions they attempt to generate better ones movement in search space to good solutions mechanisms to prevent getting stuck in local optima Simulated Annealing, Genetic Algorithms, Ant Colony Optimization

Hybridizing CP and ACO [6] Background 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

Hybridizing CP and ACO [7] Background Travelling Salesman Problem (TSP) unconstrained COP Example for a COP A B C D A – C - B - D - A

Hybridizing CP and ACO [8] Background 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 (10, 50) (20, 34) (30, 35) (5, 25)

Hybridizing CP and ACO [9] Background How would ants solve a TSP? based on pheromone trails, ants are able to find shortest paths to food sources translating this into an algorithm (ACO), it can be used to solve COPs A B C D  

Hybridizing CP and ACO [10] Background How would ants solve a TSP? after constructing a tour, corresponding path is rewarded with pheromone amount of pheromone depends on solution quality later ants more likely to regenerate good solutions tours might be infeasible due to constraint violation Reward them? Try again? Prevent construction of infeasible tours? How to solve a TSP-TW?

Hybridizing CP and ACO [11] Background We need smarter ants! (enable them to cope with constraints)

Hybridizing CP and ACO [12] 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 constraints e.g. job scheduling (due date restrictions etc.) X Y Y X=1 Y=1 X=1 Y=2 X=2 Y=1 X=2 Y=2 Background Constraint Programming

Hybridizing CP and ACO [13] Background Ants armed with CP knowledge A B C D (10, 50) (20, 34) (30, 35) (5, 25)

Hybridizing CP and ACO [14] Background Ants armed with CP knowledge look ahead depends on constraint solver strength implemented 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

Hybridizing CP and ACO [15] Project How to increase look ahead? besides CP other options to obtain look ahead bounding: calculation of solution bounds General idea: obtain bound on solution exclude 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

Hybridizing CP and ACO [16] Project How to increase look ahead? relaxation easier to solve than original problem solution of relaxation only lower bound on actual solution Tour Length: 8 (actual TSP optimal 19)

Hybridizing CP and ACO [17] Project How to increase look ahead? Use standard linear programming (lp) to solve relaxation

Hybridizing CP and ACO [18] Project How to increase look ahead? LP solution provides information about lower bound on problem solution cost 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 ants additional pruning Implemented in CPACSlp

Hybridizing CP and ACO [19] Project 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 Ant decision based on pheromone concentration and some bias e.g. prefer shorter edges

Hybridizing CP and ACO [20] Project 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

Hybridizing CP and ACO [21] Results Comparing CPACS vs. (CPACSlp & dynCPACSlp ) all algorithms were applied to a test set of single machine job scheduling problems with various tightness each algorithm was run 10 times on each of the 19 problems Tested different configurations CPACS vs. CPACSlp in absence of pheromone learning CPACS vs. CPACSlp with learning component dynCPACSlp vs. CPACSlp Due to vast amount of computation, experiments had to be carried out on the cluster

Hybridizing CP and ACO [22] Results Comparing CPACS vs. (CPACSlp & dynCPACSlp ) with 10 runs, standard deviation of measurements was to high to draw any conclusions increased the number of runs to 100 analyses commenced 4 weeks ago … in between the cluster exhibited severe stability problems Analysis not finished yet  No final results obtained yet What do we expect? Let’s guess…

Hybridizing CP and ACO [23] Results Possible outcomes and consequences A:significant improvement of search efficiency (labelling steps) B: no measurable improvement

Hybridizing CP and ACO [24] Discussion Possible outcomes and consequences (A) significant improvement of search efficiency (labelling steps) very good, we implemented an algorithm that performs superior to existing ones it needs less labelling steps to find even better solutions than existing ones Further investigations should relate labelling steps to CPU time Less labelling steps not necessarily faster CP and LP solving needs additional time Simpler search heuristic might be able to achieve similar results with same amount of CPU time

Hybridizing CP and ACO [25] Discussion Possible outcomes and consequences (B) no measurable improvement unfortunate obtained bounds might not be tight enough investigation of actual tightness How much pruning is achieved? necessarily improve LP model

Hybridizing CP and ACO [26] Conclusions 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

Hybridizing CP and ACO [27] Future Directions (general) Investigation of labelling steps – CPU time relation Improvement of LP model to get tighter bounds  more search space pruning, better guidance? further investigation of different scalings of dynamic cost values comparison with other algorithms (needs efficient implementations)

Hybridizing CP and ACO [28] Thanks for your attention! Questions?