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

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

Permutation optimization problems Definition Combinatorial optimization problems

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

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

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

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

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.

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.

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.

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

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

Estimation of distribution algorithms Definition

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

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

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

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

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

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

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

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

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

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

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:

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:

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

L-decomposability

L-decomposability

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

Experiments Results

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

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

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

Discussion Probability concentrates in the fittest solutions Near uniform distribution

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.

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

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

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