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

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

3. 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

4. Two-bridge Experiment
NEST
FOOD

5. Two-bridge Experiment
NEST
FOOD

6. 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

7. 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.

8. 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.

9. 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

10. 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)

11. 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:

12. 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

13. 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

14. 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

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

16. 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

17. Thank You!