Ant Colony Optimization (ACO): Applications to Scheduling Franco Villongco IEOR 4405 4/28/09

Definition Metaheuristic: similar to genetic algorithms, simulated annealing etc. Flexible enough to be applied to combinatorial optimization problems.

Inspiration Foraging behavior of real ants Blind ants communicate through stigmergy Leave pheromone trails to make a certain path more likely to be traversed by other ants

Two-bridge Experiment NEST FOOD

Two-bridge Experiment NEST FOOD

Problem Representation (S, f, Ω) S: set of candidate solution f: objective function of s є S Ω: set of constraints Set C ={ c1, c2… cN} where N is the number of components Problem states are defined as x = ( ci, cj… ch) We call χ the set of all states

Problem Representation Nonempty set S* of optimal solutions GC = (C,L) whose nodes are the components. Artificial ants then build solutions by performing walks on the complete graph Like in the two-bridge experiment, arcs (trails) that have more pheromone will have a higher probability of being chosen.

Scheduling Applications Jm||Cmax We use Ant System algorithm GC = (C,L) consists of all the operations and two additional nodes for a source and sink node. Our constraints Ω are simply the precedence constraints.

Scheduling Applications Pheromone trail τij on the arc (i,j) indicates the desirability of choosing operation j directly after operation i. heuristic information associated with that operation ηj

Scheduling Applications At each iteration of the construction procedure, m ants concurrently build solutions After each iteration, pheromone evaporation will be applied on all arcs: Where the parameter ρ є (0,1)

Scheduling Applications The better Cmax is for the solution constructed by a particular ant k, the more pheromone there will be to the arcs corresponding to that solution:

Scheduling Applications Any ant at node i will choose node j with probability Where Nk is the set of feasible operations nj is the heuristic value proportional to the amount of work remaining corresponding to the job of the operation considered

Scheduling Applications 1||ΣTjwj We use the Ant Colony System algorithm Same as AS but with differences in pheromone updates and ant decision rule For our construction graph, we have for our node the n positions and n jobs Pheromone trail τij indicate the desirability of scheduling job j to position i heuristic information ηj inversely proportional to job j’s deadline

Scheduling Applications Main differences Pheromone update (global): Only the best-so-far solution increases in pheromone For all (i,j) in sbs (best-so-far solution) and where

Scheduling Applications Pheromone update (local): applied during the iteration to the arcs (i,j) that were traversed

Scheduling Applications Now, in choosing the next job j to schedule the probability of choosing job j is Where J is the random variable that will equal j with probability

Thank You!