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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

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SLIDE 1

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

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SLIDE 2

Permutation-based Problems

2

Travelling Salesman Problem (TSP)

1 2 6 3 5 4 8 7 Combinatorial Optimization Problems Whose solutions are represented as permutations

σ = 12367854 n! 8! = 40320 20! = 2.43 × 1018

The search space consist of solutions

20! = 2.43 × 1018 8! = 40320 n!

σ = 12367854

NP-Hard in most of the cases

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SLIDE 3

3

ß 4920 results!!!

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SLIDE 4
  • Branch & Bound
  • Branch & Cut
  • Linear Programming
  • Genetic Algorithms
  • Variable Neighborhood Search
  • Variable Neighborhood Descent
  • Memetic Algorithm
  • Estimation of Distribution Algorithms
  • Constructive Algorithms
  • Local Search

4

Revised approaches…

  • Ant Colony Optimization
  • Tabu Search
  • Scatter Search
  • Genetic Programming
  • Cutting Plane Algorithms
  • Particle Swarm Optimization
  • Simulated Annealing
  • Cuckoo Search
  • Differential Evolution
  • Artificial Bee Colony Algorithm
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SLIDE 5

5

“…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).

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SLIDE 6

6

Permutation Flowshop Scheduling Problem and Estimation of Distribution Algorithms

Example 1

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SLIDE 7

σ = 13254 σ = 1325 σ = 132 σ = 13

Permutation Flowshop Scheduling Problem

Total flow time (TFT)

f(σ) =

n

X

i=1

cσ(i),m

m1 m2 m3 m4 j4 j1 j3 j2 j5

  • jobs
  • machines
  • processing times

n m pij

5 x 4

7

σ = 1

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SLIDE 8
  • Branch & Bound
  • Branch & Cut
  • Linear Programming
  • Genetic Algorithms
  • Variable Neighborhood Search
  • Variable Neighborhood Descent
  • Memetic Algorithm
  • Estimation of Distribution Algorithms
  • Constructive Algorithms
  • Local Search

8

Revised approaches…

  • Ant Colony Optimization
  • Tabu Search
  • Scatter Search
  • Genetic Programming
  • Cutting Plane Algorithms
  • Particle Swarm Optimization
  • Simulated Annealing
  • Cuckoo Search
  • Differential Evolution
  • Artificial Bee Colony Algorithm

Why?

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SLIDE 9

Estimation of distribution algorithms

9

Generate a set

  • f solutions
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SLIDE 10

10

Generate a set

  • f solutions

Evaluate

Estimation of distribution algorithms

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SLIDE 11

11

Generate a set

  • f solutions

Evaluate Select

Estimation of distribution algorithms

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SLIDE 12

12

P(σ)

Generate a set

  • f solutions

Evaluate Select Learn a probability distribution

Estimation of distribution algorithms

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SLIDE 13

13

P(σ)

Generate a set

  • f solutions

Evaluate Select Sample new solutions Learn a probability distribution

Estimation of distribution algorithms

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SLIDE 14

Estimation of distribution algorithms

14

P(σ)

Generate a set

  • f solutions

Evaluate Select Sample new solutions Evaluate Learn a probability distribution

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SLIDE 15

Estimation of distribution algorithms

15

Generate a set

  • f solutions

Evaluate Select Sample new solutions Evaluate Update the set

  • f solutions

Learn a probability distribution

P(σ)

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SLIDE 16

16

EDAs reported in the literature

Permutation Problems

IDEA-ICE [Bosman, 2001] …

Sn

Combinatorial Problems

UMDA [Mühlenbein, 1998] MIMIC [DeBonet, 1997] FDA [Mühlenbein, 1999] EBNA [Etxeberria, 1999] BOA [Pelikan, 2000] EHBSA [Tsutsui, 2003] NHBSA [Tsutsui, 2006] TREE [Pelikan, 2007] REDA [Romero, 2009]

Ω

Continuous Problems

UMDAc [Larrañaga, 2000] MIMICc [Larrañaga, 2000] EGNA [Larrañaga, 2000] EMNA [Larrañaga, 2001] IDEA [Bosman, 2000]

Rn

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SLIDE 17

17

1 REDAMIMIC NHBSAWT REDAUMDA NHBSAWO EHBSAWT EHBSAWO 2 3 4 5 6

7 EBNABIC UMDA EGNAee UMDAc OmeGA TREE MIMIC 14 13 12 11 10 9 8 ICE

Univariate and bivariate models!!!

Experiments on

Permutation Flowshop Scheduling Problem

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SLIDE 18

Position 1 2 3 4 5 Item 1

0.2 0.1 0.2 0.1 0.4

2

0.4 0.3 0.2 0.1

3

0.1 0.3 0.3 0.1 0.2

4

0.1 0.2 0.4 0.1 0.2

5

0.2 0.1 0.1 0.5 0.1

  • Node and Edge Histogram-based Sampling Algorithms (EHBSA & NHBSA)

(Tsutsui et al. 2002, Tsutsui et al. 2006)

18

Node Histogram

54123 42351 12354 24351 31452 23415 23451 25431 12543 53124

Population

Experiments on

Permutation Flowshop Scheduling Problem

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SLIDE 19

Item j 1 2 3 4 5 Item i 1

  • 0.4

0.3 0.3 0.4

2

0.4

  • 0.5

0.3 0.3

3

0.3 0.5

  • 0.5

0.4

4

0.3 0.3 0.5

  • 0.6

5

0.4 0.3 0.4 0.6

  • Node and Edge Histogram-based Sampling Algorithms (EHBSA & NHBSA)

(Tsutsui et al. 2002, Tsutsui et al. 2006)

19

Edge Histogram

54123 42351 12354 24351 31452 23415 23451 25431 12543 53124

Population

Experiments on

Permutation Flowshop Scheduling Problem

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SLIDE 20

20

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

n = 3

The group of permutations as a subset

  • f integers group
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SLIDE 21

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

n = 3

The group of permutations as a subset

  • f integers group
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SLIDE 22

Probability Models on Rankings

22

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.

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23

P(σ) = 1 ψ(θ)e−θD(σ,σ0)

Mallows

P(σ) = 1 ψ(θ)e− Pn−1

j=1 θjSj(σ,σ0)

Generalized Mallows

P(σ) =

n−1

Y

i=1

wσ(i) Pn

j=i wσ(j)

Plackett-Luce

P(σ) =

n−1

Y

i=1 n

Y

j=i+1

wσ(i) wσ(i) + wσ(j)

Bradley-Terry Distance-based Order statistics

Probability Models on Rankings

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SLIDE 24
  • A distance-based exponential probability model
  • Central permutation
  • Spread parameter
  • A distance on permutations

σ0 θ

24

P(σ) = e−θD(σ,σ0) ψ(θ)

Probability Models on Rankings

The Mallows Model

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SLIDE 25
  • A distance-based exponential probability model
  • Central permutation
  • Spread parameter
  • A distance on permutations

σ0 θ

25

P(σ) = e−θD(σ,σ0) ψ(θ)

Probability Models on Rankings

The Mallows Model

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SLIDE 26
  • A distance-based exponential probability model
  • Central permutation
  • Spread parameter
  • A distance on permutations

σ0 θ

26

P(σ) = e−θD(σ,σ0) ψ(θ)

Probability Models on Rankings

The Mallows Model

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SLIDE 27

Dτ(σA, σB) = 8

27

  • Kendall’s-τ distance: calculates the number of pairwise disagreements.

1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5

σA = 53412 σB = 12345 Dτ(σA, σB) =?

1 2 3 1 3 ⌃ 1 4 1 4 ⌃ 1 5 1 5 ⌃ 1 3 2 3 ⌃ 2 4 2 5 2 4 ⌃ 2 5 ⌃ 2 3 4 3 4 1 2 5 3 5 ⌃ 3 5 ⌃ 4 5 4 σA σB

Probability Models on Rankings

Kendall’s-τ distance

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SLIDE 28

1 2 3 4 5 6 7 8 6.68 6.7 6.72 6.74 6.76 6.78 6.8 6.82 6.84 x 10

6

Evaluations ( times x max_eval ) Fitness Configuration 500 x 20

AGA HGM−EDA Guided HGM−EDA

28

Probability Models on Rankings

  • 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.

Improved state-

  • f-the-art !!!
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SLIDE 29

29

Linear Ordering Problem and Neighborhood Topology

Example 2

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SLIDE 30

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

Linear Ordering Problem (LOP)

B = [bk,l]5×5

30

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SLIDE 31

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

5 4 3 2 1 5 4 3 2 1

f(σ) = 138

B = [bk,l]5×5

31

f(σ) =

n−1

X

i=1 n

X

j=i+1

bσ(i),σ(j)

σ = 12345

Linear Ordering Problem (LOP)

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SLIDE 32

9 22 21 26 15 24 25 26 22 14 16 15 7 12 23 13 11 11 28 30

1 2 4 3 5 1 2 4 3 5

B = [bk,l]5×5

32

f(σ) =

n−1

X

i=1 n

X

j=i+1

bσ(i),σ(j)

σ = 53421 f(σ) = 247

Linear Ordering Problem (LOP)

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SLIDE 33

The insert neighborhood

Moving to Landscape Context…

  • Two solutions and are neighbors if is obtained by moving an item
  • f from position to position

σ σ0 σ0 σ i j

5 2 4 1 3

33

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SLIDE 34

5 2 4 1 3

  • Two solutions and are neighbors if is obtained by moving an item
  • f from position to position

σ σ0 σ0 σ i j

34

The insert neighborhood

Moving to Landscape Context…

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SLIDE 35

1 5 2 4 3

  • Two solutions and are neighbors if is obtained by moving an item
  • f from position to position

σ σ0 σ0 σ i j

35

The insert neighborhood

Moving to Landscape Context…

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SLIDE 36
  • Two solutions and are neighbors if is obtained by moving an item
  • f from position to position

σ σ0 σ0 σ i j

1 5 2 4 3

How is the operation translated to the LOP?

36

The insert neighborhood

Moving to Landscape Context…

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SLIDE 37

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

5 4 3 2 1 5 4 3 2 1

B = [bk,l]5×5

37

Linear Ordering Problem (LOP)

An insert operation…

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SLIDE 38

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

5 4 3 2 1 5 4 3 2 1

B = [bk,l]5×5

38

Linear Ordering Problem (LOP)

An insert operation…

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SLIDE 39

15 7 16 11 26 30 22 21 12 13 9 11 14 26 22 24 25 15 23 28

5 4 3 2 1 5 4 3 2 1

B = [bk,l]5×5

39

Linear Ordering Problem (LOP)

An insert operation…

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SLIDE 40

15 7 16 11 26 30 22 21 12 13 9 11 14 26 22 24 25 15 23 28

5 4 3 2 1 5 4 3 2 1

B = [bk,l]5×5

40

Linear Ordering Problem (LOP)

An insert operation…

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SLIDE 41

7 16 15 11 26 30 22 21 12 9 11 22 14 25 24 15 26 23 28 13

5 4 3 2 1 5 4 3 2 1

B = [bk,l]5×5

41

Linear Ordering Problem (LOP)

An insert operation…

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SLIDE 42

f(σ) = 130

7 16 15 11 26 30 22 21 12 9 11 22 14 25 24 15 26 23 28 13

5 4 3 2 1 5 4 3 2 1

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

5 4 3 2 1 5 4 3 2 1

Before After

σ = 12345 σ0 = 14235 f(σ) = 138

42

Linear Ordering Problem (LOP)

An insert operation…

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SLIDE 43

f(σ) = 130

7 16 15 11 26 30 22 21 12 9 11 22 14 25 24 15 26 23 28 13

5 4 3 2 1 5 4 3 2 1

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

5 4 3 2 1 5 4 3 2 1

Before After

σ = 12345 σ0 = 14235 f(σ) = 138

43

Linear Ordering Problem (LOP)

An insert operation…

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SLIDE 44

f(σ) = 130

7 16 15 11 26 30 22 21 12 9 11 22 14 25 24 15 26 23 28 13

5 4 3 2 1 5 4 3 2 1

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

5 4 3 2 1 5 4 3 2 1

Before After

σ = 12345 σ0 = 14235 f(σ) = 138

Two pairs of entries associated to the item 4 exchanged their position.

44

Linear Ordering Problem (LOP)

An insert operation…

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SLIDE 45

f(σ) = 130

7 16 15 11 26 30 22 21 12 9 11 22 14 25 24 15 26 23 28 13

5 4 3 2 1 5 4 3 2 1

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

5 4 3 2 1 5 4 3 2 1

Before After

σ = 12345 σ0 = 14235 f(σ) = 138

45

The contribution of the item 4 to the objective function varied from 69 to 61.

Linear Ordering Problem (LOP)

An insert operation…

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SLIDE 46

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

5 4 3 2 1 5 4 3 2 1 46

Linear Ordering Problem (LOP)

The contribution of an item to f

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SLIDE 47

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

5 4 3 2 1 5 4 3 2 1 47

Linear Ordering Problem (LOP)

The contribution of an item to f

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SLIDE 48

9 16 14 22 21 15 23 28

2 2 48

Linear Ordering Problem (LOP)

The contribution of an item to f

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SLIDE 49

9 16 14 22 21 15 23 28

2 2

9

  • 5

7 19 (2,2) 9 (2,1) 7 19

  • 5

(2,3)

  • 5

7 19 9 7

  • 5

(2,4) 19 9 7

  • 5

19 (2,5) 9

Contribution: 54 Vector of differences

49

16-21 23-14 22-15 28-9

Linear Ordering Problem (LOP)

The contribution of an item to f

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SLIDE 50

22 28 23 9 15 21 14 16

2 2

9

  • 5

7 19 (2,2) 9 (2,1) 7 19

  • 5

(2,3)

  • 5

7 19 9 7

  • 5

(2,4) 19 9 7

  • 5

19 (2,5) 9

Vector of differences Contribution: 89

50

16-21 23-14 22-15 28-9

Linear Ordering Problem (LOP)

The contribution of an item to f

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SLIDE 51

7 19 (2,4)

  • 5

9

The vector of differences

Local optima

What happens in local optimal solutions? There is no movement that improves the contribution of any item

7 > 0 0 < -5 9 + 7 > 0 19 + 9 + 7 > 0

All the partial sums of differences to the left must be positive

Depends on the overall solution

51

All the partial sums of differences to the right must be negative

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SLIDE 52

But,

  • 13
  • 23

(5,4)

  • 11
  • 19

Negative sums Positive sums

In order to produce local optima, item 5 must be placed in the first position

The vector of differences

Local optima

52

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SLIDE 53

The restrictions matrix

We propose an algorithm to calculate the restricted positions of the items:

9 16 14 22 21 15 23 28

2 2

9

  • 5

7 19 (2,2)

  • 1. Vector of differences.

7 19

  • 5

9

  • 2. Sort differences

(1)

  • 3. Study the most favorable ordering
  • f differences in each positions

53

All the partial sums of differences to the right must be negative

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SLIDE 54

The restrictions matrix

We propose an algorithm to calculate the restricted positions of the items:

9 16 14 22 21 15 23 28

2 2

9

  • 5

7 19 (2,2)

  • 1. Vector of differences.

7 19

  • 5

9

  • 2. Sort differences
  • 3. Study the most favorable ordering
  • f differences in each positions

54

7 (1) 9 19

  • 5

All the partial sums of differences to the right must be negative

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SLIDE 55

The restrictions matrix

We propose an algorithm to calculate the restricted positions of the items:

9 16 14 22 21 15 23 28

2 2

9

  • 5

7 19 (2,2)

  • 1. Vector of differences.

7 19

  • 5

9

  • 2. Sort differences
  • 3. Study the most favorable ordering
  • f differences in each positions
  • 5

19 7 9 (2) 7 (1) 9 19

  • 5

(3) 9

  • 5

7 19 19 7 (4)

  • 5

9 9

  • 5

19 (5) 7

Non-local

  • ptima

Possible local

  • ptima

55

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SLIDE 56

22 12 13 9 11 11 16 14 26 22 24 25 30 21 15 26 23 28 15 7

5 4 3 2 1 5 4 3 2 1

1 1 1 1 1 1 1 1 1 1

5 4 3 2 1 5 4 3 2 1

R B

The restrictions matrix

56

Time complexity: O(n3)

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SLIDE 57

57

The restricted insert neighborhood

  • Incorporate the restrictions matrix to the insert neighborhood.
  • Discard the insert operations that move items to the restricted positions.

Theorem Given a non local optima solution σ, for every item σ(i), i = 1, . . . , n, the insert movement that maximises its contribution to the fitness function is not given in a restricted position

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SLIDE 58

The restricted insert neighborhood

58

Insert neighborhood Restricted Insert neighborhood

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SLIDE 59

The restricted insert neighborhood

59

Insert neighborhood Restricted Insert neighborhood

Evaluations: Evaluations: 10 5

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SLIDE 60

The restricted insert neighborhood

60

Insert neighborhood Restricted Insert neighborhood

Evaluations: Evaluations: 10 5

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SLIDE 61

The restricted insert neighborhood

61

Insert neighborhood Restricted Insert neighborhood

Evaluations: Evaluations: 10 5

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SLIDE 62

The restricted insert neighborhood

62

Insert neighborhood Restricted Insert neighborhood

Evaluations: Evaluations: 20 11

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SLIDE 63

The restricted insert neighborhood

63

Insert neighborhood Restricted Insert neighborhood

Evaluations: Evaluations: 30 17

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SLIDE 64

The restricted insert neighborhood

Insert neighborhood Restricted Insert neighborhood Same final solution

Evaluations: Evaluations: 30 17

  • J. Ceberio et al. (2014) The Linear Ordering Problem Revisited. European Journal of Operational Research.
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SLIDE 65

1000n2 evals. 150 250 300 500 750 1000 Total MAr vs MA 35 (4) 31 (8) 39 (11) 43 (7) 41 (9) 37 (13) 226 (52) ILSr vs ILS 37 (2) 37 (2) 49 (1) 48 (2) 50 (0) 50 (0) 271 (7) 5000n2 evals. 150 250 300 500 750 1000 Total MAr vs MA 37 (2) 39 (0) 50 (0) 49 (1) 44 (6) 44 (6) 263 (15) ILSr vs ILS 38 (1) 36 (3) 50 (0) 45 (5) 46 (4) 47 (3) 262 (16) 10000n2 evals. 150 250 300 500 750 1000 Total MAr vs MA 39 (0) 34 (5) 43 (7) 50 (0) 50 (0) 49 (1) 265 (13) ILSr vs ILS 33 (6) 37 (2) 46 (4) 42 (8) 43 (7) 45 (5) 246 (32)

278 instances

Experiments

Two state-of-the-art algorithms

  • Schiavinotto, T., Stützle, T., 2004. The linear ordering problem: instances, search space analysis

and algorithms. Journal of Mathematical Modelling and Algorithms.

65

278 instances

271 7

Iterated Local Search

ILSr Tie 226 52 Memetic Algorithm Mar Tie

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SLIDE 66

66

Quadratic Assignment Problem and Elementary Landscapes

Example 3

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SLIDE 67

Elementary Landscape Decomposition

The quadratic assignment problem (QAP)

Quadratic Assignment Problem (QAP)

67

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

slide-68
SLIDE 68

68

f(σ) =

n

X

i=1 n

X

j=1

di,jhσ(i),σ(j)

D = [di,j]n×n, H = [hk,l]n×n 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

σ = 87625341

Quadratic Assignment Problem (QAP)

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SLIDE 69

Elementary landscapes

Definitions

avg

π∈N(σ)

{f(π)} = f(σ) +

k |N(σ)|( ¯

f − f(σ))

Groover’s wave equation

69

A landscape is

(Sn, f, N)

An elementary landscape fulfills

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SLIDE 70

If the neighborhood N is

for all σ, π ∈ Sn, π ∈ N(σ) ⇐ ⇒ σ ∈ N(π)

Symmetric

|N(σ)| = d > 0 for all σ ∈ Sn

Regular then the landscape can be decomposed as a sum of elementary landscapes

70

According to Chicano et al. 2010

Elementary landscape decomposition

… of the QAP

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SLIDE 71

Elementary landscape decomposition

… of the QAP

g(σ) =

n

X

i,j,p,q=1

ψijpqϕ(i,j)(p,q)(σ)

Generalized QAP

71

f(σ) =

n

X

i=1 n

X

j=1

di,jhσ(i),σ(j)

QAP

According to Chicano et al. 2010

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SLIDE 72

g(σ) =

n

X

i, j, p, q = 1 i 6= j p 6= q

ψijpq Ω1

(i,j)(p,q)(σ)

2n + Ω2

(i,j)(p,q)(σ)

2(n − 2) + Ω3

(i,j)(p,q)(σ)

n(n − 2) ! g(σ) =

n

X

i,j,p,q=1

ψijpqϕ(i,j)(p,q)(σ)

Generalized QAP

72

According to Chicano et al. 2010

Landscape 1 Landscape 2 Landscape 3

Under the interchange neighborhood

Elementary landscape decomposition

… of the QAP

slide-73
SLIDE 73

Decomposed QAP

g(σ) =

n

X

i, j, p, q = 1 i 6= j p 6= q

ψijpq Ω1

(i,j)(p,q)(σ)

2n + Ω2

(i,j)(p,q)(σ)

2(n − 2) + Ω3

(i,j)(p,q)(σ)

n(n − 2) !

Landscape 1 Landscape 2 Landscape 3

f(σ) = λ +

n

X

i, j, p, q = 1 i 6= j p 6= q

ψijpq Ω2

(i,j)(p,q)(σ)

2(n − 2) +

n

X

i, j, p, q = 1 i 6= j p 6= q

ψijpq Ω3

(i,j)(p,q)(σ)

n(n − 2)

73

In the classic QAP the matrix is symmetric, as a result

D = [di,j]n×n

Elementary landscape decomposition

… of the QAP

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SLIDE 74

Single-objective Problem

σ∗ = arg max

σ∈Sn

f(σ)

74

Multi-objective Problem

F(σ) = [f1(σ), . . . , fm(σ)] maximize F(σ), σ ∈ Sn where

Decomposed QAP

f(σ) = λ +

n

X

i, j, p, q = 1 i 6= j p 6= q

ψijpq Ω2

(i,j)(p,q)(σ)

2(n − 2) +

n

X

i, j, p, q = 1 i 6= j p 6= q

ψijpq Ω3

(i,j)(p,q)(σ)

n(n − 2)

Multi-objectivization

… of the QAP

slide-75
SLIDE 75

75

Algorithms:

  • SGA
  • NSGA2
  • SPEA2

Movies !!

  • J. Ceberio et al. (2018) Multi-
  • bjectivising Combinatorial

Optimization Problems by means of Elementary Landscape Decomposition. Evolutionary Computation.

slide-76
SLIDE 76

76

Algorithms:

  • SGA
  • NSGA2
  • SPEA2

Movies !!

  • J. Ceberio et al. (2018) Multi-
  • bjectivising Combinatorial

Optimization Problems by means of Elementary Landscape Decomposition. Evolutionary Computation.

slide-77
SLIDE 77

77

Multi-objectivization

… of other problems?

slide-78
SLIDE 78

78

Future Research Possibilities

Other Considerations

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SLIDE 79

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Benchmarking and difficulty:

  • Random generation of

instances

  • Difficulty of instances.
  • Distribution of instances.

Possible Future Lines

Permutation Problems

Non-Standard Permutation Problems:

  • Partially Permutation Problems
  • Quasi Permutation Problems
  • Multi Permutation Problems

Possible codifications:

  • Vector of integers
  • Vector of continuous values
  • Matrices
  • Cycles

Problem Types:

  • Problems with constrains
  • Multi-objective
  • Deceptive problems
  • Dynamic Problems
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SLIDE 80

80

Instances as Rankings of Solutions: Linear Ordering Problem f(σ) =

n−1

X

i=1 n

X

j=i+1

bσ(i),σ(j)

σ ∈ Sn

123 213 132 231 312 321

n = 3

How many rankings can be generated?

(|Sn|)! = (n!)! (3!)! = 720

Are We Generating Instances Uniformly At Random?

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SLIDE 81

Only 48 different rankings... 105 LOP instances; n=3 à 720 possible rankings ; bkl sampled from [0,100] u.a.r.

1000 2000 3000 4000 1 23 45 67 89 111 133 155 177 199 221 243 265 287 309 331 353 375 397 419 441 463 485 507 529 551 573 595 617 639 661 683 705 Count Ranking ID

81

Are We Generating Instances Uniformly At Random?

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SLIDE 82

Only 48 different rankings... 105 LOP instances; n=3 à 720 possible rankings ; bkl sampled from [0,100] u.a.r.

1000 2000 3000 4000 1 23 45 67 89 111 133 155 177 199 221 243 265 287 309 331 353 375 397 419 441 463 485 507 529 551 573 595 617 639 661 683 705 Count Ranking ID

82

Are We Generating Instances Uniformly At Random?

Linear Ordering Problem

σn!−1 = (41523) σn! = (53241) σ1 = (14235) σ2 = (32514)

. . .

Reverse

✓n! 2 ◆ !

n!/2

X

i=0

✓n!/2 i ◆

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SLIDE 83

Only 48 different rankings... 105 LOP instances; n=3 à 720 possible rankings ; bkl sampled from [0,100] u.a.r.

1000 2000 3000 4000 1 23 45 67 89 111 133 155 177 199 221 243 265 287 309 331 353 375 397 419 441 463 485 507 529 551 573 595 617 639 661 683 705 Count Ranking ID

83

Are We Generating Instances Uniformly At Random?

The rankings were not uniformly sampled: XL: 3560 ± 84 L: 2531 ± 62 M: 1268 ± 63 S: 752 ± 25

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SLIDE 84

Experiment

Constraints among consecutive solutions

84

XL Ranking L Ranking M Ranking S Ranking Defined over all the parameters

14 16 26 21 11 23

x12 > x21 x21 > x12 x21 + x23 > x12 + x32 x21 + x23 > x12 + x32

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SLIDE 85

Analysis of Local Optima

85

132 312 321 231 213 123 XL ranking 231 312 321 132 213 123 L ranking 213 321 132 312 123 231 M ranking 231 312 123 132 213 321 S ranking

Within each group equal number of local optima were found! And at the same ranks!

Are We Generating Instances Uniformly At Random?

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SLIDE 86

105 instances QAP; n=3 à 720 possible rankings ; parameters sampled from [0,100] u.a.r.

86

50 100 150 200 250 300 350 1 25 49 73 97 121 145 169 193 217 241 265 289 313 337 361 385 409 433 457 481 505 529 553 577 601 625 649 673 697 Count Ranking ID

All the possible rankings were created (720) Symmetry can be observed

Are We Generating Instances Uniformly At Random?

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SLIDE 87

105 instances PFSP; n=3 x m=10 à 720 possible rankings ; parameters sampled from [0,100] u.a.r.

87

All the possible rankings were created (720) Symmetry can be observed Large variance

200 400 600 800 1000 1200 1400 1600 1 26 51 76 101 126 151 176 201 226 251 276 301 326 351 376 401 426 451 476 501 526 551 576 601 626 651 676 701 Count Ranking ID

Are We Generating Instances Uniformly At Random?

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SLIDE 88

88

  • 1. Study the problem and gain insight
  • 2. Algorithm design
  • 3. A lot of research to do yet

To take home…

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SLIDE 89

Permutation-based Combinatorial Optimization Problems under the Microscope

Josu Ceberio

89

Thank you for your attention!