In the field of evolutionary computation, it is common to use benchmarks of instances of a given problem in order to carry out a performance evaluation of existing and newly developed algorithms. When the final goal is to solve specific real-world problems, real instances are used to perform such comparisons, and, thus, we are not interested in an extensive performance evaluation. However, when the objective of the research is to contribute with methodological developments, then large benchmarks of instances are needed in order to evaluate the efficiency of the proposed algorithm under different scenarios. At this point, it is a usual practice, based on the knowledge of the problem to create new “challenging” instances artificially. In this sense, a recurrent option is to generate instances by sampling their parameters uniformly at random (u.a.r.) in some ranges. Taking this assumption as starting point for the talk, we will present a series of discoveries regarding the generation of random instances for a number of permutation-based combinatorial optimization problems.

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

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
= 12367854
n!
8! = 40320
20! = 2.43 ⇥ 1018
The search space consist of
solutions
20! = 2.43 ⇥ 1018
8! = 40320
n!
= 12367854
NP-Hard in most of the cases

3. 3
ß 4920 results!!!

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

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).

6. 6
Permutation Flowshop Scheduling Problem
and Estimation of Distribution Algorithms
Example 1

7. = 13254= 1325= 132= 13
Permutation Flowshop Scheduling Problem
Total flow time (TFT)
f( ) =
nX
i=1
c (i),m
m1
m2
m3
m4
j4j1 j3j2 j5
• jobs
• machines
• processing times
n
m
pij
5 x 4
7
= 1

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?

9. Estimation of distribution algorithms
9
Generate a set
of solutions

10. 10
Generate a set
of solutions
Evaluate
Estimation of distribution algorithms

11. 11
Generate a set
of solutions
Evaluate Select
Estimation of distribution algorithms

12. 12
P( )
Generate a set
of solutions
Evaluate Select
Learn a probability
distribution
Estimation of distribution algorithms

13. 13
P( )
Generate a set
of solutions
Evaluate Select
Sample new
solutions
Learn a probability
distribution
Estimation of distribution algorithms

14. Estimation of distribution algorithms
14
P( )
Generate a set
of solutions
Evaluate Select
Sample new
solutions
Evaluate
Learn a probability
distribution

15. Estimation of distribution algorithms
15
Generate a set
of solutions
Evaluate Select
Sample new
solutions
Evaluate
Update the set
of solutions
Learn a probability
distribution
P( )

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]
⌦
Continuous Problems
UMDAc [Larrañaga, 2000]
MIMICc [Larrañaga, 2000]
EGNA[Larrañaga, 2000]
EMNA[Larrañaga, 2001]
IDEA[Bosman, 2000]
Rn

17. 17
1
REDAMIMIC
NHBSAWT REDAUMDA
NHBSAWO
EHBSAWT
EHBSAWO
2 3 4 5 6 7
EBNABIC
UMDA EGNAee UMDAc
OmeGA
TREEMIMIC
141312111098
ICE
Univariate and bivariate models!!!
Experiments on
Permutation Flowshop Scheduling Problem

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 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

19. Item j
1 2 3 4 5
Itemi
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

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
of integers group

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
of integers group

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.

23. 23
P( ) =
1
(✓)
e ✓D( , 0)
Mallows
P( ) =
1
(✓)
e
Pn 1
j=1 ✓j Sj ( , 0)
Generalized Mallows
P( ) =
n 1Y
i=1
w (i)
Pn
j=i w (j)
Plackett-Luce
P( ) =
n 1Y
i=1
nY
j=i+1
w (i)
w (i) + w (j)
Bradley-Terry
Distance-based
Order statistics
Probability Models on Rankings

24. • A distance-based exponential probability model
• Central permutation
• Spread parameter
• A distance on permutations
0
✓
24
P( ) =
e ✓D( , 0)
(✓)
Probability Models on Rankings
The Mallows Model

25. • A distance-based exponential probability model
• Central permutation
• Spread parameter
• A distance on permutations
0
✓
25
P( ) =
e ✓D( , 0)
(✓)
Probability Models on Rankings
The Mallows Model

26. • A distance-based exponential probability model
• Central permutation
• Spread parameter
• A distance on permutations
0
✓
26
P( ) =
e ✓D( , 0)
(✓)
Probability Models on Rankings
The Mallows Model

27. D⌧ ( A, B) = 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
A = 53412
B = 12345
D⌧ ( A, B) =?
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 ⌃ 45 4
A B
Probability Models on Rankings
Kendall’s-τ distance

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-
of-the-art !!!

29. 29
Linear Ordering Problem
and Neighborhood Topology
Example 2

30. 22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
Linear Ordering Problem (LOP)
B = [bk,l]5⇥5
30

31. 22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
54321
5
4
3
2
1
f( ) = 138
B = [bk,l]5⇥5
31
f( ) =
n 1X
i=1
nX
j=i+1
b (i), (j)
= 12345
Linear Ordering Problem (LOP)

32. 9
22
21
26
0
15
2425
26
22
14
16157
12 23
13 11 0
0
11
0
0
28 30
12435
1
2
4
3
5
B = [bk,l]5⇥5
32
f( ) =
n 1X
i=1
nX
j=i+1
b (i), (j)
= 53421
f( ) = 247
Linear Ordering Problem (LOP)

33. The insert neighborhood
Moving to Landscape Context…
• Two solutions and are neighbors if is obtained by moving an item
of from position to position
0 0
i j
52 41 3
33

34. 52 41 3
• Two solutions and are neighbors if is obtained by moving an item
of from position to position
0 0
i j
34
The insert neighborhood
Moving to Landscape Context…

35. 1 52 43
• Two solutions and are neighbors if is obtained by moving an item
of from position to position
0 0
i j
35
The insert neighborhood
Moving to Landscape Context…

36. • Two solutions and are neighbors if is obtained by moving an item
of from position to position
0 0
i j
1 524 3
How is the operation translated to the LOP?
36
The insert neighborhood
Moving to Landscape Context…

37. 22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
54321
5
4
3
2
1
B = [bk,l]5⇥5
37
Linear Ordering Problem (LOP)
An insert operation…

38. 22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
54321
5
4
3
2
1
B = [bk,l]5⇥5
38
Linear Ordering Problem (LOP)
An insert operation…

39. 0 15 716 11
26
30
22
21
12
13
9
11
14
26
22
2425
15
23 0
0
28
0
0
54321
5
4
3
2
1
B = [bk,l]5⇥5
39
Linear Ordering Problem (LOP)
An insert operation…

40. 0 15 716 11
26
30
22
21
12
13
9
11
14
26
22
2425
15
23 0
0
28
0
0
54321
5
4
3
2
1
B = [bk,l]5⇥5
40
Linear Ordering Problem (LOP)
An insert operation…

41. 0 71615 11
26
30
22
21
12
9
0 1122
14
2524
15
26 23 0
0
28 0
13
54 321
5
4
3
2
1
B = [bk,l]5⇥5
41
Linear Ordering Problem (LOP)
An insert operation…

42. f( ) = 130
0 71615 11
26
30
22
21
12
9
0 1122
14
2524
15
26 23 0
0
28 0
13
54 321
5
4
3
2
1
22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
54321
5
4
3
2
1
Before After
= 12345 0
= 14235
f( ) = 138
42
Linear Ordering Problem (LOP)
An insert operation…

43. f( ) = 130
0 71615 11
26
30
22
21
12
9
0 1122
14
2524
15
26 23 0
0
28 0
13
54 321
5
4
3
2
1
22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
54321
5
4
3
2
1
Before After
= 12345 0
= 14235
f( ) = 138
43
Linear Ordering Problem (LOP)
An insert operation…

44. f( ) = 130
0 71615 11
26
30
22
21
12
9
0 1122
14
2524
15
26 23 0
0
28 0
13
54 321
5
4
3
2
1
22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
54321
5
4
3
2
1
Before After
= 12345 0
= 14235
f( ) = 138
Two pairs of entries associated to the item 4 exchanged their position.
44
Linear Ordering Problem (LOP)
An insert operation…

45. f( ) = 130
0 71615 11
26
30
22
21
12
9
0 1122
14
2524
15
26 23 0
0
28 0
13
54 321
5
4
3
2
1
22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
54321
5
4
3
2
1
Before After
= 12345 0
= 14235
f( ) = 138
45
The contribution of the item 4 to the objective function varied from 69 to 61.
Linear Ordering Problem (LOP)
An insert operation…

46. 22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
54321
5
4
3
2
1
46
Linear Ordering Problem (LOP)
The contribution of an item to f

47. 22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
54321
5
4
3
2
1
47
Linear Ordering Problem (LOP)
The contribution of an item to f

48. 9
16
14
22
21 15
23
0
28
2
2
48
Linear Ordering Problem (LOP)
The contribution of an item to f

49. 9
16
14
22
21 15
23
0
28
2
2
9-5 7 19(2,2)
9(2,1) 7 19-5
(2,3)-5 7 199
7-5 (2,4) 199
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

50. 22
28
23
91521 14 0
16
2
2
9-5 7 19(2,2)
9(2,1) 7 19-5
(2,3)-5 7 199
7-5 (2,4) 199
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

51. 719 (2,4) -59
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

52. But,
-13-23 (5,4) -11-19
Negative sumsPositive sums
In order to produce local optima,
item 5 must be placed in the first position
The vector of differences
Local optima
52

53. The restrictions matrix
We propose an algorithm to calculate the restricted positions of the items:
9
16
14
22
21 15
23
0
28
2
2
9-5 7 19(2,2)
1. Vector of differences.
719 -59
2. Sort differences
(1)
3. Study the most favorable ordering
of differences in each positions
53
All the partial sums of
differences to the right
must be negative

54. The restrictions matrix
We propose an algorithm to calculate the restricted positions of the items:
9
16
14
22
21 15
23
0
28
2
2
9-5 7 19(2,2)
1. Vector of differences.
719 -59
2. Sort differences
3. Study the most favorable ordering
of differences in each positions
54
7(1) 9 19-5
All the partial sums of
differences to the right
must be negative

55. The restrictions matrix
We propose an algorithm to calculate the restricted positions of the items:
9
16
14
22
21 15
23
0
28
2
2
9-5 7 19(2,2)
1. Vector of differences.
719 -59
2. Sort differences
3. Study the most favorable ordering
of differences in each positions
-519 7 9(2)
7(1) 9 19-5
(3)9 -5 719
197 (4) -59
9-5 19 (5)7
Non-local
optima
Possible local
optima
55

56. 22
12
13
9
0
11
1116
14
26
22
242530
21 15
26 23 0
0
28
0
0
15 7
54321
5
4
3
2
1
0
0
1
1
0
1
00
0
0
1
001
0 1
1 1 0
0
0
1
0
0 1
54321
5
4
3
2
1
RB
The restrictions matrix
56
Time complexity: O(n3
)

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

58. The restricted insert neighborhood
58
Insert
neighborhood
Restricted Insert
neighborhood

59. The restricted insert neighborhood
59
Insert
neighborhood
Restricted Insert
neighborhood
Evaluations:
Evaluations:
10
5

60. The restricted insert neighborhood
60
Insert
neighborhood
Restricted Insert
neighborhood
Evaluations:
Evaluations:
10
5

61. The restricted insert neighborhood
61
Insert
neighborhood
Restricted Insert
neighborhood
Evaluations:
Evaluations:
10
5

62. The restricted insert neighborhood
62
Insert
neighborhood
Restricted Insert
neighborhood
Evaluations:
Evaluations:
20
11

63. The restricted insert neighborhood
63
Insert
neighborhood
Restricted Insert
neighborhood
Evaluations:
Evaluations:
30
17

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.

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

66. 66
Quadratic Assignment Problem
and Elementary Landscapes
Example 3

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

68. 68
f( ) =
nX
i=1
nX
j=1
di,jh (i), (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
= 87625341
Quadratic Assignment Problem (QAP)

69. Elementary landscapes
Definitions
avg
⇡2N( )
{f(⇡)} = f( ) + k
|N( )| ( ¯f f( ))
Groover’s wave equation
69
A landscape is
(Sn, f, N)
An elementary landscape fulfills

70. If the neighborhood N is
for all , ⇡ 2 Sn, ⇡ 2 N( ) () 2 N(⇡)
Symmetric
|N( )| = d > 0 for all 2 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

71. Elementary landscape decomposition
… of the QAP
g( ) =
nX
i,j,p,q=1
ijpq'(i,j)(p,q)( )
Generalized QAP
71
f( ) =
nX
i=1
nX
j=1
di,jh (i), (j)
QAP
According to Chicano et al. 2010

72. g( ) =
nX
i, j, p, q = 1
i 6= j
p 6= q
ijpq
⌦1
(i,j)(p,q)( )
2n
+
⌦2
(i,j)(p,q)( )
2(n 2)
+
⌦3
(i,j)(p,q)( )
n(n 2)
!
g( ) =
nX
i,j,p,q=1
ijpq'(i,j)(p,q)( )
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

73. Decomposed QAP
g( ) =
nX
i, j, p, q = 1
i 6= j
p 6= q
ijpq
⌦1
(i,j)(p,q)( )
2n
+
⌦2
(i,j)(p,q)( )
2(n 2)
+
⌦3
(i,j)(p,q)( )
n(n 2)
!
Landscape 1 Landscape 2 Landscape 3
f( ) = +
nX
i, j, p, q = 1
i 6= j
p 6= q
ijpq
⌦2
(i,j)(p,q)( )
2(n 2)
+
nX
i, j, p, q = 1
i 6= j
p 6= q
ijpq
⌦3
(i,j)(p,q)( )
n(n 2)
73
In the classic QAP the matrix is symmetric, as a resultD = [di,j]n⇥n
Elementary landscape decomposition
… of the QAP

74. Single-objective
Problem
⇤
= arg max
2Sn
f( )
74
Multi-objective
Problem
F( ) = [f1( ), . . . , fm( )]
maximize F( ), 2 Sn
where
Decomposed QAP
f( ) = +
nX
i, j, p, q = 1
i 6= j
p 6= q
ijpq
⌦2
(i,j)(p,q)( )
2(n 2)
+
nX
i, j, p, q = 1
i 6= j
p 6= q
ijpq
⌦3
(i,j)(p,q)( )
n(n 2)
Multi-objectivization
… of the QAP

75. 75
Algorithms:
• SGA
• NSGA2
• SPEA2
Movies !!
J. Ceberio et al. (2018) Multi-
objectivising Combinatorial
Optimization Problems by
means of Elementary
Landscape Decomposition.
Evolutionary Computation.

76. 76
Algorithms:
• SGA
• NSGA2
• SPEA2
Movies !!
J. Ceberio et al. (2018) Multi-
objectivising Combinatorial
Optimization Problems by
means of Elementary
Landscape Decomposition.
Evolutionary Computation.

77. 77
Multi-objectivization
… of other problems?

78. 78
Future Research Possibilities
Other Considerations

79. 79
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

80. 80
Instances as Rankings of Solutions: Linear Ordering Problem
f( ) =
n 1X
i=1
nX
j=i+1
b (i), (j)
2 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?

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

82. Only 48 different
rankings...
105 LOPinstances; n=3 à 720 possible rankings ; bkl sampled from [0,100] u.a.r.
0
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
n! 1 = (41523)
n! = (53241)
1 = (14235)
2 = (32514)
.
.
.
Reverse
✓
n!
2
◆
!
n!/2
X
i=0
✓
n!/2
i
◆

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

84. Experiment
Constraints among consecutive solutions
84
XL Ranking
L Ranking
M Ranking
S Ranking
Defined over all
the parameters
14
0
16
0
26
21
110
23
x12 > x21
x21 > x12
x21 + x23 > x12 + x32
x21 + x23 > x12 + x32

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?

86. 105 instances QAP; n=3 à 720 possible rankings ; parameters sampled from [0,100] u.a.r.
86
0
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?

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
0
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?

88. 88
1. Study the problem and gain insight
2. Algorithm design
3. A lot of research to do yet
To take home…

89. Permutation-based Combinatorial
Optimization Problems under the Microscope
Josu Ceberio
89
Thank you for your attention!