Post on 21-Dec-2015
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
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Outline
• The linear ordering problem
• The Mallows and Plackett-Luce EDAs
• Experimentation
• On the Boltzmann distribution associated to the LOP
• Conclusions and future work
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Combinatorial optimization problems
Permutation optimization problemsDefinition
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Permutation optimization problemsDefinition
Problems whose solutions are naturally represented as permutations
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Permutation optimization problemsGoal
To find the permutation solution that minimizes a fitness function
The search space consists of solutions.
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Permutation optimization problemsExamples
• Travelling salesman problem (TSP)
• Permutation Flowshop Scheduling Problem (PFSP)
• Linear Ordering Problem (LOP)
• Quadratic Assignment Problem (QAP)
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Permutation optimization problemsExamples
• Travelling salesman problem (TSP)
• Permutation Flowshop Scheduling Problem (PFSP)
• Linear Ordering Problem (LOP)
• Quadratic Assignment Problem (QAP)
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The linear ordering problemDefinition
Example extracted from R. Martí and G. Reinelt (2011) The linear ordering problem: exact and heuristic methods in combinatorial optimization.
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The linear ordering problemDefinition
Example extracted from R. Martí and G. Reinelt (2011) The linear ordering problem: exact and heuristic methods in combinatorial optimization.
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The linear ordering problemDefinition
Example extracted from R. Martí and G. Reinelt (2011) The linear ordering problem: exact and heuristic methods in combinatorial optimization.
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The linear ordering problemSome 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
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The linear ordering problem
It is an NP-hard problem(Garey and Johnson 1979)
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Estimation of distribution algorithms Definition
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In previous works
• Implement probability models for permutation domains
– The Mallows model
– The Generalized Mallows model
– The Plackett-Luce model
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In previous works
• Implement probability models for permutation domains
– The Mallows model
– The Generalized Mallows model
– The Plackett-Luce model
Promising performanceon the LOP
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The Mallows modelDefinition
• A distance-based exponential probability model
• Central permutation
• Spread parameter
• A distance on permutations
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The Mallows modelDefinition
• A distance-based exponential probability model
• Central permutation
• Spread parameter
• A distance on permutations
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The Mallows modelDefinition
• A distance-based exponential probability model
• Central permutation
• Spread parameter
• A distance on permutations
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The Ulam distanceDefinition
Calculates the minimum number of insert operations to convert in .
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Distances and neighborhoods
– Two solutions and are neighbors if the Kendall’s-τ distance
between and is
– Two solutions and are neighbors if the Cayley distance
between and is
– Two solutions and are neighbors if the Ulam distance between
and is
Swap neighborhood
Interchange neighborhood
Insert neighborhood
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Distances and neighborhoods
– Two solutions and are neighbors if the Kendall’s-τ distance
between and is
– Two solutions and are neighbors if the Cayley distance
between and is
– Two solutions and are neighbors if the Ulam distance between
and is
Swap neighborhood
Interchange neighborhood
Insert neighborhood
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The Plackett- Luce modelDefinition
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
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The Plackett- Luce modelVase model interpretation
A vase of infinite colored balls
With known proportions of each color
Draw balls from the vase until a permutationof colored balls is obtained
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The Plackett- Luce modelVase model interpretation
Stage 1
We draw a ball.
The probability to extract a red ball atthis stage is:
And it is red.
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The Plackett- Luce modelVase model interpretation
Stage 2
We draw another ball.
The probability to extract a green ball from the remaining balls is:
And it is green.
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The Plackett- Luce modelVase model interpretation
Stage 3
We draw the blue ball.
The probability to extract a blue ball is:
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L-decomposability
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L-decomposability
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• 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
ExperimentsDesign
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ExperimentsResults
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Discussion
Which is the most efficient model to optimize the LOP ?
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Discussion
Theoretically, the Boltzmann distribution associated to the LOP
Boltzmann constant
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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:
Learn from a sample of 106 permutations
Perform a weighted computationof the parameters
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DiscussionProbability concentrates in the fittest solutions
Near uniform distribution
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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.
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Future work
Compare Mallows EDA under the Ulam distance with state-of-the-art algorithms
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Future work
Study the properties of the Boltzmann distribution on the LOP
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