A distance-based ranking EDA for the

permutation flowshop scheduling problem

Josu Ceberio

Previously…

EDAs for integer domains. EDAs for real value domains.

Few efficient designs for permutation-based problems.

POOR PERFORMANCE

EHBSA and NHBSA (Tsutsui et al.)

Distance-based ranking models

The Mallows model is a distance-based exponential model.

Two parametersConsensus ranking, Spread parameter,

Probability distribution

Distance-based ranking models

Kendall’s tau distance

Decomposition of the distance

Factorization of the probability distribution

1 2 3 4 5 6

2 3 1 6 5 4

2 0 0 2 1

Distance-based ranking EDA

Generalized Mallows EDA is proposed. A generalization of the Mallows model. spread parameters.

Probability distribution

The problem

To check the performance we approach:Permutation Flowshop Scheduling Problem.

Extensively studied.

The Mallows EDA demonstrated good performance.

Permutation Flowshop Scheduling Problem Given a set of n jobs and m machines and processing

times pij.

Find the sequence for scheduling jobs optimally. Optimization criterion: Total Flow Time (TFT).

Codification

1 3 2 5 4

m1m2

m3

m4

j1 j3j2 j5j4

Example

Objective function

Generalized Mallows EDAPreliminary experiments

Spread parameters

Generalized Mallows EDAPreliminary experiments

GM model convergence

Generalized Mallows EDAApproximating spread parameters

Newton-Raphson

An upper bound for the spread parameters is fixed!!

Generalized Mallows EDAApproximating spread parameters

Standart evolutionary shape

Restart mechanism shape

Generalized Mallows EDAPreliminary experiments

Restart mechanism

Improvement !

PFSPstate-of-the-art

LR(n/m)GA

VNSCrossoverVNS

Asynchronus Genetic Algorithm (AGA) – Xu et al. 2009

Local Search (Swap)

Local Search (Insert)

Shake

PFSP state-of-the-art

LR(n/m)Local Search

(Swap)

Local Search (Insert)

Shake

Variable Neighborhood Search 4 (VNS4) – Costa et al. 2012

PFSP state-of-the-art

Fundamentalist approaches rarely achieve optimum solutions.

Hybridization is the path to follow.

High presence of VNS algorithms.

First approach to the PFSP GM-EDA does not succeed. An hybrid approach is considered:

Hybrid Generalized Mallows EDA (HGM-EDA)

Hybrid Generalized Mallows EDA

Generalized Mallows EDA

Local Search (Swap)

Local Search (Insert)

Orbit Shake

VNS

Experimentation

Algorithms: AGA, VNS4, GM-EDA, VNS and HGM-EDA.20 repetitions

Taillard’s PFSP benchmarks: 100 instances• 20 x 05• 20 x 10• 20 x 20• 50 x 05• 50 x 10• 50 x 20

• 100 x 05• 100 x 10• 100 x 20• 200 x 10• 200 x 20• 500 x 20

Experimentation

Spread parameters upper bound.Select the upper-theta that provides the best

solutions for GM-EDA

Stopping criterion: maximum number of evaluations.Evaluations performed by AGA in n x m x 0.4s.

Experimentation

Taillards benchmark

20 x 5 20 x 10 20 x 20

AGA 13932 20003 32911

VNS4 13932 20003 32911

GM-EDA 13934 20009 20003

VNS 13932 20003 32911

HGM-EDA 13932 20003 32911

Experimentation

Taillards benchmark

50 x 5 50 x 10 50 x 20

AGA 66301 85916 121294

VNS466757 86479 121739

GM-EDA 66309 86948 122830

VNS 66309 85980 121386

HGM-EDA 66307 85958 121317

Experimentation

Taillards benchmark

100 x 5 100 x 10 100 x 20

AGA 240102 288988 374974

VNS4242974 292425 378402

GM-EDA 241346 292472 379691

VNS 240162 289438 375410

HGM-EDA 240122 288902 374664

Experimentation

Taillards benchmark

200 x 10 200 x 20 500 x 20

AGA 1039507 1243928 6754943

VNS41048520 1252165 6770472

GM-EDA 1046146 1252545 7225665

VNS 1041846 1246474 6863483

HGM-EDA 1036303 1237959 6861070

Experimentation

Taillard’s benchmark - Summary

20x05 20x10 20x20 50x05 50x10 50x20 100x05 100x10 100x20 200x10 200x20 500x20

AGA ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔VNS4 ✔ ✔ ✔GM-EDA

VNS ✔ ✔ ✔HGM-EDA ✔ ✔ ✔ ✔ ✔ ✔ ✔

Experimentation

Taillard’s benchmark – Results analysisHGM-EDA outperforms state-of-the-art

results in some cases.○ Which is the reason for the performance fall

given in instances of 500x20?

Biased instances?- A tabu search algorithm was used for to choose the

hardest instances.

We generate a random benchmark

Experimentation

Random benchmarkNew configurations between 200 and 500.

Total: 100 instances.

• 250 x 10• 250 x 20• 300 x 10• 300 x 20• 350 x 10

• 350 x 20• 400 x 10• 400 x 20• 450 x 10• 450 x 20

Experimentation

Random benchmark - Summary

250x10 250x20 300x10 300x20 350x10 350x20 400x10 400x20 450x10 450x20

AGA ✔ ✔ ✔VNS4

GM-EDA

VNS

HGM-EDA ✔ ✔ ✔ ✔ ✔ ✔ ✔

Experimentation

Random benchmark – Results analysis

Statistical Analysis confirms experimentation.○ Friedman test + Shaffer’s static.

HGM-EDA and AGA are definitely the best algorithms.

VNS4 results do not match with those reported.The performance falls onwards 400x20.

What’s wrong with largest instances?

Analysis – Hybrid approachImprovement ratio EDA vs. VNS

20x0

5

20x1

0

20x2

0

50x0

5

50x1

0

50x2

0

100x

05

100x

10

100x

20

200x

10

200x

20

250x

10

250x

20

300x

10

300x

20

350x

10

350x

20

400x

10

400x

20

450x

10

450x

20

500x

2050%

55%

60%

65%

70%

75%

80%

85%

90%

95%

100%

EDA VNS

Instances

%

Analysis – Generalized Mallows EDAAGA vs. GM-EDA

20x5

20x1

0

20x2

050

x5

50x1

0

50x2

0

100x

5

100x

10

100x

20

200x

10

200x

20

250x

10

250x

20

300x

10

300x

20

350x

10

350x

20

400x

10

400x

20

450x

10

450x

20

500x

201

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

Instances

RP

D (

%)

Analysis – Generalized Mallows EDAThetas convergence

Analysis – Generalized Mallows EDAThetas convergence

Analysis – Generalized Mallows EDAThetas convergence

Analysis – Generalized Mallows EDAThetas convergence

Analysis – Generalized Mallows EDAThetas convergence

Analysis – Generalized Mallows EDAThetas convergence

Stops prematurely!!!

Analysis – HGM-EDA vs. AGAMore evaluations

Max eval. AGA HGM-EDA

x1 6710650 6841042

x2 6708656 6816514

x3 6708162 6769335

x4 6708123 6778298

x5 6708029 6779509

x6 6708029 6775003

x7 6706879

x8 6706879

One instance of 500x20

Analysis – Generalized Mallows EDALR vs. GM-EDA

20x0

5

20x1

0

20x2

0

50x0

5

50x1

0

50x2

0

100x

05

100x

10

100x

20

200x

10

200x

20

250x

10

250x

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300x

10

300x

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350x

10

350x

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400x

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450x

10

450x

20

500x

200.8

0.85

0.9

0.95

1

1.05

1.1

1.15

Instances

%

Analysis – HGM-EDA vs. AGAMore evaluations

Max eval. AGA HGM-EDA

x1 6710650 6841042

x2 6708656 6816514

x3 6708162 6769335

x4 6708123 6778298

x5 6708029 6779509

x6 6708029 6775003

x7 6706879

x8 6706879

One instance of 500x20

Analysis – HGM-EDA vs. AGAMore evaluations

Max eval. AGA HGM-EDA Guided HGM-EDA

x1 6710650 6841042 6743775

x2 6708656 6816514 6721295

x3 6708162 6769335 6732300

x4 6708123 6778298 6707129

x5 6708029 6779509 6716032

x6 6708029 6775003 6712273

x7 6706879

x8 6706879

One instance of 500x20

Analysis – HGM-EDA vs. AGAMore evaluations

One instance of 500x20

1 2 3 4 5 6 7 86600000

6650000

6700000

6750000

6800000

6850000

6900000

AGAHGM-EDA Guided HGM-EDA

Conclusions

Hybrid Generalized Mallows EDA is a efficient algorithm for solving the PFSP.Succeed in 152/220 instances.

The participation of the GM-EDA is essential.

Future Work - PFSP

Test other parameters: evaluations, population size, theta bounds, selection size…

Include information of the instance.

Guided InitializationShake the solution of the LR(n/m) to

build up the population?

Future Work – GM-EDA

Set different upper bounds to the spread parameters

Study other distances. Is suitable Kendall’s-tau distance? Other distances: Cayley, Ulam, Hamming Study the problem.

Other problems: TSP QAP LOP (work in progress)

Eskerrik asko

Josu CeberioEskerrik asko

Josu Ceberio

Distance-based ranking EDA

Mallows EDA Learning and Sampling

0 . . . n - 2

1

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n - 1