Permutation-based Combinatorial Optimization Problems under the Microscope Josu Ceberio Intelligent Systems Group Department of Computer Science and Artificial Intelligence University of the Basque Country (UPV/EHU) 1
Permutation-based Problems Travelling Salesman Problem (TSP) Combinatorial Optimization Problems 8 Whose solutions are represented as permutations 7 6 n ! The search space consist of n ! solutions 8! = 40320 5 8! = 40320 3 20! = 2 . 43 × 10 18 20! = 2 . 43 × 10 18 2 4 NP-Hard in most of the cases 1 σ = 12367854 σ = 12367854 2
ß 4920 results!!! 3
Revised approaches… • Branch & Bound • Ant Colony Optimization • Branch & Cut • Tabu Search • Linear Programming • Scatter Search • Genetic Algorithms • Genetic Programming • Variable Neighborhood Search • Cutting Plane Algorithms • Variable Neighborhood Descent • Particle Swarm Optimization • Memetic Algorithm • Simulated Annealing • Estimation of Distribution Algorithms • Cuckoo Search • Constructive Algorithms • Differential Evolution • Local Search • Artificial Bee Colony Algorithm 4
“ … propose that a theory of heuristic (as opposed to algorithmic or exact) problem-solving should focus on intuition, insight and learning .” “In order to design algorithms practitioners should gain a deep insight into the structure of the problem that is to be solved.” (Sorensen 2012). 5
Permutation Flowshop Scheduling Problem and Estimation of Distribution Algorithms Example 1 6
Permutation Flowshop Scheduling Problem Total flow time (TFT) • jobs n n • machines X m f ( σ ) = c σ ( i ) ,m • processing times p ij i =1 j 2 j 5 j 1 j 3 j 4 5 x 4 m 1 m 2 m 3 m 4 σ = 13254 σ = 1325 σ = 13 σ = 132 σ = 1 7
Revised approaches… • Branch & Bound • Ant Colony Optimization • Branch & Cut • Tabu Search • Linear Programming • Scatter Search • Genetic Algorithms • Genetic Programming • Variable Neighborhood Search • Cutting Plane Algorithms • Variable Neighborhood Descent • Particle Swarm Optimization • Memetic Algorithm • Simulated Annealing • Estimation of Distribution Algorithms • Cuckoo Search • Constructive Algorithms • Differential Evolution • Local Search • Artificial Bee Colony Algorithm Why? 8
Estimation of distribution algorithms Generate a set of solutions 9
Estimation of distribution algorithms Generate a set of solutions Evaluate 10
Estimation of distribution algorithms Generate a set of solutions Evaluate Select 11
Estimation of distribution algorithms Generate a set of solutions Evaluate Select Learn a probability distribution P ( σ ) 12
Estimation of distribution algorithms Generate a set of solutions Evaluate Select Learn a probability distribution P ( σ ) Sample new solutions 13
Estimation of distribution algorithms Generate a set of solutions Evaluate Select Learn a probability distribution P ( σ ) Sample new Evaluate solutions 14
Estimation of distribution algorithms Generate a set of solutions Evaluate Select Learn a probability distribution Update the set P ( σ ) of solutions Sample new Evaluate solutions 15
Combinatorial Problems UMDA [Mühlenbein, 1998] MIMIC [DeBonet, 1997] Continuous Problems FDA [Mühlenbein, 1999] UMDA c [Larrañaga, 2000] EBNA [Etxeberria, 1999] Ω MIMIC c [Larrañaga, 2000] BOA [Pelikan, 2000] EGNA [Larrañaga, 2000] EHBSA [Tsutsui, 2003] EMNA [Larrañaga, 2001] NHBSA [Tsutsui, 2006] IDEA [Bosman, 2000] TREE [Pelikan, 2007] REDA [Romero, 2009] R n S n EDAs reported in the literature Permutation Problems IDEA-ICE [Bosman, 2001] … 16
Experiments on Permutation Flowshop Scheduling Problem 1 2 3 4 5 6 8 9 10 13 7 11 12 14 EHBSA WT UMDA TREE ICE REDA MIMIC EGNA ee UMDA c NHBSA WO MIMIC REDA UMDA OmeGA NHBSA WT EHBSA WO EBNA BIC Univariate and bivariate models!!! 17
Experiments on Permutation Flowshop Scheduling Problem • Node and Edge Histogram-based Sampling Algorithms (EHBSA & NHBSA) (Tsutsui et al. 2002, Tsutsui et al. 2006) Node Histogram Position Population 1 2 3 4 5 54123 0.2 0.1 0.2 0.1 0.4 1 42351 12354 0.4 0.3 0 0.2 0.1 2 24351 31452 Item 0.1 0.3 0.3 0.1 0.2 3 23415 23451 0.1 0.2 0.4 0.1 0.2 4 25431 12543 0.2 0.1 0.1 0.5 0.1 53124 5 18
Experiments on Permutation Flowshop Scheduling Problem • Node and Edge Histogram-based Sampling Algorithms (EHBSA & NHBSA) (Tsutsui et al. 2002, Tsutsui et al. 2006) Edge Histogram Item j Population 1 2 3 4 5 54123 - 0.4 0.3 0.3 0.4 1 42351 12354 0.4 - 0.5 0.3 0.3 2 24351 31452 Item i 0.3 0.5 - 0.5 0.4 3 23415 23451 0.3 0.3 0.5 - 0.6 4 25431 12543 0.4 0.3 0.4 0.6 - 53124 5 19
The group of permutations as a subset of integers group n = 3 111 211 311 112 212 312 113 213 313 121 221 321 122 222 322 123 223 323 131 231 331 132 232 332 133 233 333 20
The group of permutations as a subset of integers group n = 3 111 211 311 112 212 312 113 213 313 121 221 321 122 222 322 123 223 323 131 231 331 132 232 332 133 233 333 21
Probability Models on Rankings Bibliography à M. A. Fligner and J. S. Verducci ( 1998 ), Multistage Ranking Models, Journal of the American Statistical Association, vol. 83, no. 403, pp. 892-901. à D. E. Critchlow, M. A. Fligner, and J. S. Verducci ( 1991 ), Probability Models on Rankings, Journal of Mathematical Psychology , vol. 35, no. 3, pp. 294-318. à P. Diaconis ( 1988 ), Group Representations in Probability and Statistics, Institute of Mathematical Statistics. à M. A. Fligner and J. S. Verducci ( 1986 ), Distance based Ranking Models, Journal of Royal Statistical Society, Series B , vol. 48, no. 3, pp. 359-369. à R. L. Plackett ( 1975 ), The Analysis of Permutations, Applied Statistics , vol. 24, no. 10, pp. 193-202. à D. R. Luce ( 1959 ), Individual Choice Behaviour, Wiley. à R. A. Bradley AND M. E. Terry ( 1952 ), Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons, Biometrika , vol. 39, no. 3, pp. 324-345. à L. L. Thurstone ( 1927 ), A law of comparative judgment, Psychological Review , vol 34, no. 4, pp. 273-286. 22
Probability Models on Rankings 1 Distance-based ψ ( θ ) e − θ D ( σ , σ 0 ) P ( σ ) = Mallows 1 ψ ( θ ) e − P n − 1 j =1 θ j S j ( σ , σ 0 ) P ( σ ) = Generalized Mallows n − 1 w σ ( i ) Y P ( σ ) = P n j = i w σ ( j ) i =1 n − 1 n Plackett-Luce w σ ( i ) Y Y P ( σ ) = w σ ( i ) + w σ ( j ) i =1 j = i +1 Order statistics Bradley-Terry 23
Probability Models on Rankings The Mallows Model • A distance-based exponential probability model • Central permutation σ 0 • θ Spread parameter • A distance on permutations P ( σ ) = e − θ D ( σ , σ 0 ) ψ ( θ ) 24
Probability Models on Rankings The Mallows Model • A distance-based exponential probability model • Central permutation σ 0 • θ Spread parameter • A distance on permutations P ( σ ) = e − θ D ( σ , σ 0 ) ψ ( θ ) 25
Probability Models on Rankings The Mallows Model • A distance-based exponential probability model • Central permutation σ 0 • θ Spread parameter • A distance on permutations P ( σ ) = e − θ D ( σ , σ 0 ) ψ ( θ ) 26
Probability Models on Rankings Kendall’s- τ distance Kendall’s- τ distance : calculates the number of pairwise disagreements. • σ A σ B 1-2 1 � 2 1 � 2 1-3 3 ⌃ 1 3 � 1 1-4 4 ⌃ 1 4 � 1 σ A = 53412 1-5 5 ⌃ 1 5 � 1 σ B = 12345 2-3 3 ⌃ 2 3 � 2 2-4 4 � 2 4 ⌃ 2 D τ ( σ A , σ B ) =? D τ ( σ A , σ B ) = 8 2-5 5 � 2 5 ⌃ 2 3-4 3 � 4 3 � 4 3-5 5 � 3 5 ⌃ 3 4-5 5 � 4 5 ⌃ 4 27
Probability Models on Rankings 6 Configuration 500 x 20 x 10 6.84 AGA HGM − EDA Guided HGM − EDA 6.82 6.8 Improved state- of-the-art !!! 6.78 Fitness 6.76 6.74 6.72 6.7 6.68 1 2 3 4 5 6 7 8 Evaluations ( times x max_eval ) J. Ceberio et al. (2013) A Distance-based Ranking Model EDA for the PFSP. IEEE Transactions On Evolutionary Computation , vol 18, No. 2, Pp. 286-300. 28
Linear Ordering Problem and Neighborhood Topology Example 2 29
Linear Ordering Problem (LOP) 0 16 11 15 7 21 0 14 15 9 26 23 0 26 12 22 22 11 0 13 30 28 25 24 0 B = [ b k,l ] 5 × 5 30
Linear Ordering Problem (LOP) 1 2 3 4 5 1 0 16 11 15 7 2 21 0 14 15 9 n − 1 n X X f ( σ ) = b σ ( i ) , σ ( j ) 26 23 0 26 12 3 i =1 j = i +1 22 22 11 0 13 4 5 30 28 25 24 0 B = [ b k,l ] 5 × 5 σ = 12345 f ( σ ) = 138 31
Linear Ordering Problem (LOP) 5 3 4 2 1 0 25 24 28 30 5 12 0 26 23 26 3 n − 1 n X X f ( σ ) = b σ ( i ) , σ ( j ) 4 13 11 0 22 22 i =1 j = i +1 9 14 15 0 21 2 7 11 15 16 0 1 B = [ b k,l ] 5 × 5 σ = 53421 f ( σ ) = 247 32
The insert neighborhood Moving to Landscape Context… σ 0 σ 0 • Two solutions and are neighbors if is obtained by moving an item σ of from position to position i j σ 1 2 3 4 5 33
The insert neighborhood Moving to Landscape Context… σ 0 σ 0 • Two solutions and are neighbors if is obtained by moving an item σ of from position to position i j σ 1 2 3 4 5 34
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