1. Josu Ceberio Alexander Mendiburu Jose A. Lozano
A comparison of estimation of distribution algorithms for the linear ordering problem
Josu Ceberio
Alexander Mendiburu
Jose A. Lozano
X Congreso Español de Metaheurísticas, Algoritmos Evolutivos y Bioinspirados - MAEB2015

2. Outline The linear ordering problem The Mallows and Plackett-Luce EDAs
Experimentation
On the Boltzmann distribution associated to the LOP
Conclusions and future work

3. Permutation optimization problems Definition
Combinatorial optimization problems

4. Permutation optimization problems Definition
Problems whose solutions are naturally represented as permutations

5. Permutation optimization problems Goal
To find the permutation solution that minimizes
a fitness function
The search space consists of solutions.

6. Permutation optimization problems Examples
Travelling salesman problem (TSP)
Permutation Flowshop Scheduling Problem (PFSP)
Linear Ordering Problem (LOP)
Quadratic Assignment Problem (QAP)

7. Permutation optimization problems Examples
Travelling salesman problem (TSP)
Permutation Flowshop Scheduling Problem (PFSP)
Linear Ordering Problem (LOP)
Quadratic Assignment Problem (QAP)

8. The linear ordering problem Definition
Example extracted from R. Martí and G. Reinelt (2011) The linear ordering problem: exact and heuristic methods in combinatorial optimization.

9. The linear ordering problem Definition
Example extracted from R. Martí and G. Reinelt (2011) The linear ordering problem: exact and heuristic methods in combinatorial optimization.

10. The linear ordering problem Definition
Example extracted from R. Martí and G. Reinelt (2011) The linear ordering problem: exact and heuristic methods in combinatorial optimization.

11. The linear ordering problem Some applications
Aggregation of individual preferences
Kemeny ranking problem
Triangulation of input-output tables of the branches of an economy
Ranking in sports tournaments
Optimal weighted ancestry relationships

12. The linear ordering problem
It is an NP-hard problem
(Garey and Johnson 1979)

13. Estimation of distribution algorithms Definition

14. In previous works Implement probability models for permutation domains
The Mallows model
The Generalized Mallows model
The Plackett-Luce model

15. Promising performance
In previous works
Implement probability models for permutation domains
The Mallows model
The Generalized Mallows model
The Plackett-Luce model
Promising performance
on the LOP

16. The Mallows model Definition
A distance-based exponential probability model
Central permutation
Spread parameter
A distance on permutations

17. The Mallows model Definition
A distance-based exponential probability model
Central permutation
Spread parameter
A distance on permutations

18. The Mallows model Definition
A distance-based exponential probability model
Central permutation
Spread parameter
A distance on permutations

19. The Ulam distance Definition
Calculates the minimum number of insert operations to convert in

20. Distances and neighborhoods
Swap neighborhood
Two solutions and are neighbors if the Kendall’s-τ distance between and is
Interchange neighborhood
Two solutions and are neighbors if the Cayley distance between and is
Insert neighborhood
Two solutions and are neighbors if the Ulam distance between and is

21. Distances and neighborhoods
Swap neighborhood
Two solutions and are neighbors if the Kendall’s-τ distance between and is
Interchange neighborhood
Two solutions and are neighbors if the Cayley distance between and is
Insert neighborhood
Two solutions and are neighbors if the Ulam distance between and is

22. The Plackett- Luce model Definition
The probability of under the Plackett-Luce model is given by
The vector of scores defines the preference of each item to be ranked in top rank

23. The Plackett- Luce model Vase model interpretation
A vase of infinite colored balls
With known proportions of each color
Draw balls from the vase until a permutation
of colored balls is obtained

24. The Plackett- Luce model Vase model interpretation
Stage 1
We draw a ball.
And it is red.
The probability to extract a red ball at
this stage is:

25. The Plackett- Luce model Vase model interpretation
Stage 2
We draw another ball.
And it is green.
The probability to extract a green ball
from the remaining balls is:

26. The Plackett- Luce model Vase model interpretation
Stage 3
We draw the blue ball.
The probability to extract a blue ball is:

27. L-decomposability

28. L-decomposability

29. Experiments Design Algorithms:
Mallows EDA under the Ulam distance (MaEDA)
Plackett-Luce EDA (PLEDA)
50 instances of sizes: {10, 20, 30, 40, 50, 60, 70, 80, 90, 100}
Average Relative Percentage Deviation (ARPD) of 20 repetitions
Stopping criterion: 100n-1 generations

30. Experiments Results

31. Which is the most efficient model to optimize the LOP ?
Discussion
Which is the most efficient model to optimize the LOP ?

32. Discussion Theoretically,
the Boltzmann distribution associated to the LOP
Boltzmann
constant

33. Discussion
Calculate from the Boltzmann distribution associated to the LOP:
the Mallows model under the Ulam distance
the Plackett-Luce model
4 instances of size n=7
Boltzmann constant c: [0,300]
Kullback-Leibler divergence:
Perform a weighted computation
of the parameters
Learn from a sample of
106 permutations

34. Discussion Probability concentrates in the fittest solutions
Near uniform distribution

35. Conclusions
For small instances, MaEDA and PLEDA obtain similar results.
For large instances, MaEDA is the preferred algorithm.
With respect to the Boltzmann distribution of the LOP:
When the fitness of the solutions is very different, the Mallows model under the Ulam distance is the preferred option.
When the fitness of the solutions is similar, the Plackett-Luce is more accurate.

36. Future work Compare Mallows EDA under the Ulam distance
with state-of-the-art algorithms

37. Study the properties of the Boltzmann distribution on the LOP
Future work
Study the properties of the Boltzmann distribution on the LOP

38. Josu Ceberio Alexander Mendiburu Jose A. Lozano
A comparison of estimation of distribution algorithms for the linear ordering problem
Josu Ceberio
Alexander Mendiburu
Jose A. Lozano
X Congreso Español de Metaheurísticas, Algoritmos Evolutivos y Bioinspirados - MAEB2015