Hyper-heuristics Part I · Spaces for Sequencing Problems with Application to Job Shop Scheduling, INFORMS, 38(10), 1495-1509. Fang H.-L., Ross P. and Corne D., 1994. A Promising

Hyper-heuristicsPart I

Ender Özcan

NATCOR – April 2016

Outline Introduction: motivation, relevant concepts

Hyper-heuristics – Definition, Origins andClassification

Selection Hyper-heuristics

Perturbative low level heuristics

Constructive low level heuristics

Hyflex and CHeSC 2011

Case Studies

2

Heuristic Search – revisited

A heuristic is method which seeks good, i.e. near-optimal solutions, at a reasonable cost without beingable to guarantee optimality.

Good for solving ill-structured problems, or complexwell-structured problems (large-scale combinatorialproblems that have many potential solutions toexplore)

A heuristic is a rule of thumb method derivedfrom human intuition.

3

Search Paradigms I

Single point based search vs. Multi-point(population) based search

Constructive

Search on partial candidate solutions

search steps: extend partial solutions, but neverreduce them

Perturbative

Search on complete solutions

4

Need for Search Methodologies (e.g.Heuristics, Metaheuristics) – Example

Travelling salesman problem

N=4, 24

N=5, 120

N=7, 5 040

N=10, 3 628 800

N=81, 5.797 x 10120

Number of particles in the universe is in between 1072 –1087

Tianhe-2: 30.65 petaflops (one thousand million (1015)floating-point operations per second) – ~6 x 1096 years

5

Examples – Heuristics for TSP

The nearest neighbour (NN) algorithm

Constructive

Pairwise exchange (2-opt), or Lin–Kernighan heuristics

Perturbative

6

The nearest neighbour (NN)algorithm

city1

city2

city3city4

10

4

5

7

116

7

city1

city2

city3city4

Select a starting city

<city2>

8


city1

city2

city3city4

choose the nearestunvisited city as the nextmove

<city2, >

4

10

9

6


city1

city2

city3city4


<city2, city1, >

4

511

10


city1

city2

city3city4


<city2, city1, city4, >

4

5

7

11


city1

city2

city3city4

After the choice of the lastcity, algorithm terminates

<city2, city1, city4, city3> : 26

10

4

5

7

12


city1

city2

city3city4

4

7

116

Remove two edges andreplace them with two differentedges, reconnecting thefragments into a new andshorter tour.


Pairwise exchange (2-opt)

13

city1

city2

city3city4

4

7

116



14


city1

city2

city3city4

4

7

116


<city1, city2, city3, city4> : 26 (28+(-2))

15

10

5

(5+10) – (11+6) = -2


city1

city2

city3city4

4

7


10

5

16


Search Paradigms II

Mutational heuristics(diversification/exploration)

vs.

Hill climbing (intensification/exploitation)(objective)

current

17

Mutational Heuristic

MutationalHeuristic

3.0

16.0

22.0

CandidateSolution

16.0

Minimising Fitness /Cost/Penalty/…

e.g., total number of constraintviolations or a weighted sum ofviolations

Processes a given candidate solutionand generates a solution which is notguaranteed to be better than the input

22.0

18

HillClimbing

3.0

16.0

16.0

CandidateSolution

16.0

Minimising Fitness /Cost/Penalty/…

e.g., total number of constraint violations ora weighted sum of violations

Processes a given candidate solutionand generates a better or equal qualitysolution

Hill Climbing Heuristic

19

0-1 Knapsack Problem

Given a set of n items, each item i with aweight wi and a value vi, choose a subset of

those items (where xi =1) yielding themaximum total value without exceeding amaximum weight capacity W

20

Heuristic#1

Sort items by value in decreasing order

Choose the next item if adding its weight to thesum of already chosen items without exceedingthe capacity

Problem Instance

i 1 2 3 4

v [350, 30, 150, 110]

w [ 35, 5, 15, 10]

W=20

Heuristic#1



Problem Instance

i 1 2 3 4 1 3 4 2

v [350, 30, 150, 110] [350, 150, 110, 30]

w [ 35, 5, 15, 10] [ 35, 15, 10, 5]

W=20

Solution: {2,3} (optimal?), Weight: 20, Value: 180

Heuristic#1



Problem Instance

i 1 2 3 4 1 3 4 2

v [350, 30, 150, 190] [350, 150, 190, 30]

w [ 35, 5, 15, 10] [ 35, 15, 10, 5]

W=20

Solution: {2,3} (optimal?), Weight: 20, Value: 180

Heuristic#2

Sort items by value per unit weight indecreasing order


Problem Instance

i 1 2 3 4

v [350, 30, 150, 190]

w [ 35, 5, 15, 10]

W=20

Heuristic#2



Problem Instance

i 1 2 3 4 4 1 3 2

v [350, 30, 150, 190] [190, 350, 150, 30]

w [ 35, 5, 15, 10] [ 10, 35, 15, 5]

W=20 Solution: {2,4}, Weight: 15, Value: 220

Heuristic#2



Problem Instance

i 1 2 3 4 4 1 3 2

v [350, 30, 150, 190] [190, 350, 200, 30]

w [ 35, 5, 15, 10] [ 10, 35, 20, 5]

W=20 Solution: {2,4}, Weight: 15, Value: 220

What is a Metaheuristic?

A metaheuristic is a high-level problemindependent algorithmic framework thatprovides a set of guidelines or strategies todevelop heuristic optimization algorithms

27

K. Sörensen and F. Glover. Metaheuristics. In S.I. Gass and M. Fu,editors, Encyclopedia of Operations Research and ManagementScience, pp 960–970. Springer, New York, 2013.

Metaheuristics

[Kirkpatrick, 1983] Simulated Annealing (SA)

[Glover, 1986] Tabu Search (TS)

[Voudouris, 1997] Guided Local Search (GLS)

[Stutzle, 1999] Iterated Local Search (ILS)

[Mladenovic, 1999] Variable Neighborhood Search (VNS)

[Holland, 1975] Genetic Algorithm (GA)

[Smith, 1980] Genetic Programming (GP)

[Goldberg, 1989] Genetic and Evolutionary Computation (EC)

[Moscato, 1989] Memetic Algorithm (MA)

[Kennedy and Eberhart, 1995]

Particle Swarm Optimisation (PSO)

[Dorigo, 1992] Ant Colony Optimisation (ACO)

[Resende, 1995] Greedy Randomized Adaptive Search Procedure

(GRASP)

Lo

ca

lS

ea

rch

Po

pu

lati

on

-ba

se

dC

on

str

uc

tive

Main Components ofMetaheuristics

Representation (encoding) ofcandidate solutions

Evaluation function

Initialisation

Search process (guideline)

Neighbourhood relation – moveoperator(s)

Mechanism for escaping fromlocal optima

29Guideline Encoding Initialisation Operator(s)

EscapeMethod

EvaluationFunction

0-1 Knapsack

Parameter Tuning vs Control

30

0-1 Knapsack

Guideline Encoding Initialisation Operator(s)EscapeMethod

EvaluationFunction

Tuning

1

2

3

4

5

Application to Another Domain

31

0-1 Knapsack

Vehicle RoutingDistribution center


EvaluationFunction

Tuning

?

Building a Better Solver

32

Distribution center


EvaluationFunction

Tuning

Vehicle Routing

GuidelineGuideline Encoding InitialisationInitialisation Operator(s)Operator(s)EscapeMethodEscapeMethod

EvaluationFunction

EvaluationFunction

Tuning


EvaluationFunction

TuningEvaluationFunction

State-of-the-art in HeuristicOptimisation

33

Nurse RosteringJohn

Gem

Vehicle RoutingDistribution center

0-1 Knapsack

Automated Design of SearchProcess

Growing area of research motivated by raisingthe level of generality. What are the limits?

Grand Challenge

A CB

Problem Specific Solvers

Doesn’t exist….Significantscope forfutureresearch

34

OBSERVATIONS

Most of the real-world optimization problemsare proven to be NP-hard

The current state of the art in searchmethodologies tend to focus on bespokesystems

In general, these systems are expensive tobuild, but provide successful results

Unfortunately, their application to new problemdomains or even new problem instances from aknown domain still requires expert involvement.

35

OBSERVATIONS (cont.)

A different heuristic might generategood performance on a differentproblem instance

Balancing the exploration(diversification) and exploitation(intensification) during the search iscrucial

36

Iterated Local Search (ILS)s0= GenerateInitialSolution()

//random or construction heuristic

s* = LocalSearch(s0)

Repeat

s' = Perturbation(s*, memory)

// random move

s*' = LocalSearch(s' )

// hill climbing

s* = AcceptanceCriterion(s*, s*',memory)

// the conditions that the new local optimum

// must satisfy to replace the current solution

Until (termination conditions are satisfied)

Genetic vs Memetic Algorithm

38

Genetic vs Memetic Algorithm

39

Hyper-heuristics

40

A hyper-heuristic is a search method or learningmechanism for selecting or generating heuristics

to solve computationally difficult problems

E. K. Burke, M. Gendreau, M. Hyde, G. Kendall, G. Ochoa, E. Özcan,R. Qu, Hyper-heuristics: A Survey of the State of the Art, Journal of theOperational Research Society, 64 (12) , pp. 1695-1724, 2013.

Potential Solutions

Hyper-heuristic

Operates upon

Low Level Heuristics

Operates upon

41

Different Search Spaces

Potential Solutions

Standard Heuristics

Operates upon

Meta-heuristic

Characteristics ofHyper-heuristics

Operate on a search space of heuristics ratherthan directly on a search space of solutions

Existing (or computer generated) heuristics canbe used within hyper-heuristics

Aim is to take advantage of strengths and avoidweaknesses of heuristics

Easy to implement, practical to deploy (easy,cheap, fast)

? No problem specific information flow fromdomain to hyper-heuristic layer is allowed ?

42

2001

Cowling P.I., Kendall G. and Soubeiga E.,2001. A Hyperheuristic Approach toScheduling a Sales Summit, selectedpapers from PATAT 2000, Springer, LNCS2079, 176-190.1990-95

Storer R. H., Wu S. D. , Vaccari R., 1992. New SearchSpaces for Sequencing Problems with Application to JobShop Scheduling, INFORMS, 38(10), 1495-1509.

Fang H.-L., Ross P. and Corne D., 1994. A PromisingHybrid GA/Heuristic Approach for Open-Shop SchedulingProblems., in'ECAI' , 590-594.

1997

Denzinger J., Fuchs M. and Fuchs M., 1997. Highperformance ATP systems by combining several AImethods. In Proc. of the 15th IJCAI, 102-107.

Hyper-heuristics:Origins

19751961-63

Fisher H. and Thompson G.L., 1963. Probabilistic Learning Combinations ofLocal Job-shop Scheduling Rules. Ch 15,:225-251, Prentice Hall, New Jersey.

Crowston W.B., Glover F., Thompson G.L. and Trawick J.D. Probabilistic andParameter Learning Combinations of Local Job Shop Scheduling Rules. ONRResearch Memorandum, GSIA,CMU, Pittsburgh, (117), 1963

Related Areas Reactive search

Algorithm portfolios

Adaptive operator selection

Meta-learning

Co-evolution/multimeme memeticalgorithms/Memetic computing

Variable Neighbourhood Search

Cooperative (Distributed) Search

Parameter control (e.g., in EAs)

Algorithm configuration44

Classification ofHyper-heuristics

Nature of the heuristic search space

Hyper-heuristics

45

Heuristic generation

constructiveheuristics

perturbativeheuristics

constructiveheuristics

perturbativeheuristics

Heuristic selection

Methodologies to select

Methodologies to generate

Fixed heuristics(mostly humandesigned)

Automaticallygeneratedheuristics fromcomponents

E. K. Burke, M. Hyde, G. Kendall, G. Ochoa, E. Özcan, and J. Woodward (2010). A Classification of Hyper-heuristic Approaches. In Gendreau, M. and Potvin, J.Y. (eds.), Handbook of Metaheuristics, InternationalSeries in Operations Research & Management Science, Volume 146, pp. 449-468. Springer.

Online learning

while solving a probleminstance (adapt)

Examples: reinforcementlearning, meta-heuristics

Offline learning

from a set of training instances(generalise)

Examples: classifier systems,case-based, GP

46

Onlinelearning

Offlinelearning

No-learning

Hyper-heuristics

Feedback

Classification ofHyper-heuristicsE. K. Burke, M. Hyde, G. Kendall, G. Ochoa, E. Özcan, and J. Woodward (2010). A Classification of Hyper-heuristic Approaches. In Gendreau, M. and Potvin, J.Y. (eds.), Handbook of Metaheuristics, InternationalSeries in Operations Research & Management Science, Volume 146, pp. 449-468. Springer.

Domain Barrier

Hyper-heuristic

47

A Hyper-heuristic Framework

48

Heuristic Selection Method Move Acceptance Criteria

Perturbative low level heuristics

Domain Barrier

A Selection Hyper-heuristicFramework – Single Point Search

A Selection Hyper-heuristicFramework – Single Point Search

49

1. generate initial candidate solution p

2. while (termination criteria not satisfied){

3. select a heuristic (or subset of

heuristics) h from {H1, ..., Hn}

4. generate a new solution (or solutions) s

by applying h to p

5. decide whether to accept s or not

6. if (s is accepted) then

7. p=s }

8. return p;

Heuristic Selection

Component name Reference(s)

Simple Random Cowling et al (2000, 2002b)

Random Permutation Cowling et al (2000, 2002b)

Peckish Cowling and Chakhlevitch (2003)

Greedy

Cowling et al (2000, 2002b); Cowling and

Chakhlevitch (2003)

Random Gradient Cowling et al (2000, 2002b)

Random Permutation Gradient Cowling et al (2000, 2002b)

Choice Function

Cowling et al (2000, 2002b); Maashi et al (2015);

Drake et al (2015)

Reinforcement Learning Nareyek (2003); Pisinger and Ropke (2007)

Reinforcement Learning with Tabu Search Burke et al (2003); Dowsland et al (2007)

Quality Index and Tabu based Learning Heuristic Selection Mısır et al (2009, 2012)

Dominance-based Selection Kheiri and Özcan (2011; 2015)

Probability-based Selection Lehrbaum and Musliu (2012)

Adaptive pursuit Walker et al (2012)

Heuristic selection with no learning

Heuristic selection with learning

with no learning with learning

Apply each low level heuristic to the candidatesolution and choose the one that generatesthe best objective value

H1 H2 H3 H4 H5

GR

H6

f1 f2 f3 f4 f5 f6

f3 < f1, f2, f4, f5, f6 at step t

51

Heuristic Selection –Greedy (GR)

A machine learning technique

Inspired by related psychological theory

Reward and punishment

Concerned with how an agent ought to takeactions in an environment to maximize somenotion of long-term reward

Maintains a score for each heuristic

If an improving move then increase (e.g., +1),otherwise decrease (e.g., -1) the score of theheuristic 52

Heuristic Selection –Reinforcement Learning (RL)

The choice function maintains a record of theperformance of each heuristic. Three criteriaare maintained:

1) Its individual performance

2) how well it has performed with otherheuristics

3) the elapsed time since the heuristic has beencalled

53

Heuristic Selection –Choice Function (CF)

''t t t

H1 H2 H3 H4 H5

CF

H6

s1 s2 s3 s4 s5 s6

s2 > s1, s3, s4, s5, s6 at step t

54

Heuristic Selection –Choice Function (CF)

Move Acceptance

Kheiri and Özcan (2015)

Cowling et al (2000, 2002b)

Dowsland et al (2007); Bai et al. (2007a)

Bai and Kendall (2005); Bilgin et al (2006); Pisinger and Ropke (2007);Antunes et al (2009)

Mısır et al (2012)

Özcan et al (2009);Jackson et al. (2013)

Ayob and Kendall (2003)

Kendall and Mohamad (2004b) || Bilgin et al. (2006)

Burke et al (2010); Kheiri and Özcan (2012); Asta and Özcan (2015)

Kendall and Mohamad (2004b)

Mısır et al (2009)

All Moves

Only Improving

Improving&Equal

Late Acceptance

Great Deluge

Threshold Acceptance

Record-to-Record

Naïve Accept.

Exponential Monte Carlo (EMC)

Simulated Annealing (SA)

AM: All Moves Accepted

OI: Only Improving Moves accepted

IE: Improving or Equal moves are accepted.

Naïve Acceptance: Accept all improvingmoves and worsening move with a fixedprobability of p (e.g., 0.5)

56

Move Acceptance –Simple Criteria

Great Deluge

57

0-1 Knapsack

minimise - ݅ ݅ୀଵ

maximise ݅ ݅ୀଵ

minimisingobjective

f(s0)

f(starget)

itermaxiter

0

58

t

c

eU

)1,0(

> 0 inferior solution

- < 0t

t

te

Annealing parameter t, called temperature is slowly decreased:

t is initially high - many inferior moves are acceptedt is decreasing - inferior moves are nearly always rejected

Improving moves are accepted Worsening moves are allowed using Metropolis criterion

= f(s') - f(s) Assume that F has to be minimised

As the temperature decreases, the probability of acceptingworsening moves decreases.

An inferior solution s' (yielding > 0)is accepted with a probability of te

Simulated Annealing

Simulated Annealing

59

Hyper-heuristic Tools

HYFLEX

G. Ochoa, M. Hyde, T. Curtois, J. A. Vazquez-Rodriguez, J. Walker, M.

Gendreau, G. Kendall, B. McCollum, A. J. Parkes, S. Petrovic, E. K. Burke(2012). European Conference on Evolutionary Computation inCombinatorial Optimisation (EvoCOP 2012), J.-K. Hao and M. Middendorf(Eds.), LNCS 7245, pp. 136-147. Springer, Heidelberg

HYPERION

J. Swan, E. Özcan, G. Kendall, Hyperion - A Recursive Hyper-heuristic

Framework, The Learning and Intelligent OptimizatioN Conference (LION5),Lecture Notes in Computer Science 6683, pp. 616-630, 2011.

60

Selection Hyper-heuristic –revisited

61

HyFlexHyper-heuristics Flexible Interface

62

Defines behaviours of components andarranges the interaction between them

Separation between theproblem-specific and thegeneral-purpose parts, bothof which are reusable andinterchangeable throughthe HyFlex interface

http://www.hyflex.org/

HyFlex v1.0 JavaImplementation

63

Currently there are 6 problem domain implementations

heuristic types: mutational, ruin-recreate, local search, crossover

parameters: intensity, depth of search

BinPacking

Flow Shop

PersonnelScheduling

TSP

MAX-SAT

VRP http://www.hyflex.org/

CHeSC 2011 benchmark based on HyFlex v1.0

Organising Partners:

Sponsor:

BinPacking

Flow Shop

PersonnelScheduling

TSP

MAX-SAT

VRP

• 10 public training instances• 5 test instances(3 training + 2 hidden/all hidden)

• Set problem instance• Set time limit (10 min.)• Perform 31 runs• Report median

Hidden


Ranking: Formula 1scoring system

64

65


And the winner is...

AdapHH – M. MısırK. VerbeeckP. De CausmaeckerG. Vanden Berghe

AdapHH – Overview

66

Case Study: An IteratedMulti-stage SelectionHyper-heuristic

A. Kheiri and E. Özcan, An Iterated Multi-stageSelection Hyper-heuristic, European Journal ofOperational Research, (250)1:77–90, 2016

A Multi-stage Hyper-heuristic

68

Stage 1

Select a low level heuristic i with probability

Apply the chosen heuristic

Accept/reject based on an adaptive thresholdacceptance method

A Multi-stage Hyper-heuristic

69

LLH1=2, LLH2=1, LLH3=150% 25% 25%

6 LLHs 3 LLHs

Reduce theNumber of LLHs

(N n)+

Assign Scores

Stage 2

LLH3LLH1 + LLH1

LLH4LLH2 + LLH2

LLH5LLH1 + LLH2

LLH6LLH2 + LLH1

Given N LLHs, e.g., LLH1, LLH2

Pair up all and increase the number of LLHs to N+N2

Relay Hybridisation

PS TSP

70

SAT

71

BP

72

PS

73

Performance Comparison

74

Topwith aCHeSC2011score of163.60

75

Performance Comparison

Case Study: A Hybrid Approach tothe Multi-mode Resource-

constrained Multi-projectScheduling Problem – Winner of the

MISTA 2013 Challenge School ofComputer Science

ASAP Team:ID#3Shahriar Asta, Daniel Karapetyan,Ahmed Kheiri, Ender Özcan andAndrew J. Parkes

S. Asta, D. Karapetyan, A. Kheiri, E. Özcan, and A.J. Parkes,Combining Monte-Carlo and Hyper-heuristic methods for theMulti-mode Resource-constrained Multi-project SchedulingProblem, in review.

Problem Description

Resource-ConstrainedProject Scheduling

Schedule given jobs

Limited resources

Precedence relations

Minimise makespan

Multi-modeResource-constrainedMulti-project Scheduling

Multiple modes for each job

Multiple projects

Local and global resources

Minimise the sum ofmakespans

MISTA 2013 Challenge

Aim: Develop an algorithm that produces thebest possible solution to any given problemin 5 minutes.

Problem instances are not known in advance.

21 teams registered, 16 teams qualified afterthe first round, 9 teams qualified after thefinal round.

We designed a memetic algorithm –construct and improve

78

Memetic Algorithm

79

Schedule generator: scheduleseach job to the earliestavailable time in the given order

Monte Carlo Search Treebased initialisation

Decide on good initial sequenceof projects

Sequence basedrepresentation

# 1 2 3 4 5 6job 3 6 1 5 2 4

mode 1 1 3 2 2 1

Memetic Algorithm

80

Hyper-heuristic

Memetic Algorithm

81

Hyper-heuristic

Memetic Algorithm

82

Core1

Core2

Corek

Hyper-heuristic

Hyper-heuristic

Hyper-heuristic

83

• swap jobs• change mode of a job

3

• reshuffle several jobs• change mode of several jobs

10

• swap projects• move a project

4

Low LevelHeuristics/Operators

Iterated Multi-stage Hyper-heuristic

Results

84

MISTA 2013 Challenge – Result

We produced the bestsolutions for 17 out ofthe 20 instances

On the 12th secondour algorithmbecomes the winner

85

Case Study: A Tensor-based Selection

Hyper-heuristic for Cross-domain Heuristic Search

School ofComputer Science

S. Asta and E. Özcan, A Tensor-based SelectionHyper-heuristic for Cross-domain Heuristic Search,Information Sciences, vol. 299, pp. 412-432, 2015.

Two Simple Hyper-heuristicsMixing Heuristics(Stochastic Local Search)

Simple Random Heuristic Selection –Improving and Equal Move Acceptance (IE)

Reject any worsening move

Simple Random Heuristic Selection – NaïveMove Acceptance (NA)

Accept a worsening move with a fixed probabilityof p (0.5 in this study)

87

Proposed Approach – Ideas

The balance between diversification andintensification is crucial

Mix move acceptance methods

Use machine learning to partition the low levelheuristics associated with each method

ts 2ts 3ts? ?IE NAhIE hNA = h (hIE hNA = )

h: set of low level heuristics(MU+RC+LS)

(e.g. ILS)

Intensify Diversify Intensify Diversify Intensify

time

88

Tensors

Many real-world data are multidimensional

Very high-dimensional (big) with a large amountof redundancy

Multi-dimensional arrays representing suchdata describe a tensor

Many applications insignal processing,psychometrics, andmore

SOURCE:http://en.wikipedia.org/wiki/File:Video_represented_as_a_third-order_tensor.jpg

89

Tensor Factorisation

There are different decomposition methods,we use Canonical Polyadic (CP) factorisation

This gives a projection of 3D data onto 1Dvectors

Helps to discoverlatent structures indata, quantifying therelationship betweenpairs of differentcomponents

SOURCE: B. Krausz, C. Bauckhage, Actionrecognition in videos using nonnegative tensorfactorization., in: ICPR, IEEE, 2010, pp. 1763–1766.

90

Proposed Approach –TeBHA-HH

91

-NA

Noise Elimination

(Exclude Poor

Performing

Heuristic Group)

Construct Tensor

Tensor

Factorization (CP

Decomposition)

Analysis: Extract

two subgroups of

and

Switch the subgroup

and move

acceptance, XX

XX←NA XX←IE

Apply SR-XX

using ܆܆

ି

Tmax

reached

?

No

YesReturn Solution

(Stop)

Tmax

tp tp

ts

Perform Search

Basic

FrameUse SR-NA

TeBHA-HH:Tensor Construction Phase

-NA

Noise Elimination

(Exclude Poor

Performing

Heuristic Group)

Construct Tensor

Tensor

Factorization (CP

Decomposition)

Analysis: Extract

two subgroups of

and

Switch the subgroup

and move

acceptance, XX

XX←NA XX←IE

Apply SR-XX

using ܆܆

ି

Tmax

reached

?

No

YesReturn Solution

(Stop)

Tmax

tp tp

ts

Perform Search

Basic

FrameUse SR-NA

Represent the searchhistory of SR-NA usingremaining low levelheuristics andconstruct a 3rd ordertensor in time tp

Proposed Approach –Tensor Construction

93

No. of active entries =

=

Pre

vious

heurist

icin

dex

Current heuristic index

01234567

0 1 2 3 4 5 6 7


94


=

Pre

vious

heurist

icin

dex


01234567

0 1 2 3 4 5 6 7


95


=

Pre

vious

heurist

icin

dex


01234567

0 1 2 3 4 5 6 7


96


=

Pre

vious

heurist

icin

dex


01234567

0 1 2 3 4 5 6 7


97

• The frame is now put in the empty tensor .• The label of the frame is the change in the objective

value ( ) resulted by applying the active elementsof the frame collectively.

Pre

vious

heurist

icin

dex


Frame 1 of tensor ञ


98

Pre

vious

heurist

icin

dex


01234567

0 1 2 3 4 5 6 7


=


99

Pre

vious

heurist

icin

dex


01234567

0 1 2 3 4 5 6 7


=


100

Pre

vious

heurist

icin

dex


01234567

0 1 2 3 4 5 6 7


=


101

Pre

vious

heurist

icin

dex


01234567

0 1 2 3 4 5 6 7


=


102

Pre

vious

heurist

icin

dex



01234567

0 1 2 3 4 5 6 7• The frame is now appended to the tensor .• The label of the frame is the change in the objective

value ( ) resulted by applying the active elementsof the frame collectively.


103

Frame 2 of tensor ञFrame 2 of tensor ञ





Continuing thisprocess results inan initial tensor.


104






Frames withconsecutivepositive labels( ) are selectedand put into thefinal tensor .


105



Frames withconsecutivepositive labels( ) are selectedand put into thefinal tensor .

Emphasis on inter-frame correlations

TeBHA-HH:Tensor Factorisation

-NA

Noise Elimination

(Exclude Poor

Performing

Heuristic Group)

Construct Tensor

Tensor

Factorization (CP

Decomposition)

Analysis: Extract

two subgroups of

and

Switch the subgroup

and move

acceptance, XX

XX←NA XX←IE

Apply SR-XX

using ܆܆

ି

Tmax

reached

?

No

YesReturn Solution

(Stop)

Tmax

tp tp

ts

Perform Search

Basic

FrameUse SR-NA

Decompose the tensorusing CP (AlternatingLeast Squarealgorithm)

: model fitness

Produce a basic frame

Basic Frame

TeBHA-HH:Tensor Analysis

-NA

Noise Elimination

(Exclude Poor

Performing

Heuristic Group)

Construct Tensor

Tensor

Factorization (CP

Decomposition)

Analysis: Extract

two subgroups of

and

Switch the subgroup

and move

acceptance, XX

XX←NA XX←IE

Apply SR-XX

using ܆܆

ି

Tmax

reached

?

No

YesReturn Solution

(Stop)

Tmax

tp tp

ts

Perform Search

Basic

FrameUse SR-NA

Locate the pair withmax score: LS0,LS1

Top half goes to hNA, the rest to hIE

Sort all entries on the column:

(LS0,LS1,MU3,MU2,MU5,MU4,MU1,MU0)

TeBHA-HH:Final Phase: Perform Search

-NA

Noise Elimination

(Exclude Poor

Performing

Heuristic Group)

Construct Tensor

Tensor

Factorization (CP

Decomposition)

Analysis: Extract

two subgroups of

and

Switch the subgroup

and move

acceptance, XX

XX←NA XX←IE

Apply SR-XX

using ܆܆

ି

Tmax

reached

?

No

YesReturn Solution

(Stop)

Tmax

tp tp

ts

Perform Search

Basic

FrameUse SR-NA

Run the cyclic multi-stage hyper-heuristicSR−IE with ® SR-NA withalternating at every time period ts

Results–CHeSC2011

MAX-SAT

VRP

2nd in BP4th in TSP4th in PSWorst in FS

109

Conclusion

Tensors can be used to represent the trail ofheuristic invocations in a concise manner undera selection hyper-heuristic framework

They can be further used to extract the latentrelationship between the low level heuristicswhich emerges during the search process

Tensor analysis can help improve theperformance of a multi-stage hyper-heuristic(allowing hybridisation of move acceptance)yielding counter-intuitive (basis) results

110

Documents

Hyper-heuristics Part I · Spaces for Sequencing Problems with Application to Job Shop Scheduling, INFORMS, 38(10), 1495-1509. Fang H.-L., Ross P. and Corne D., 1994. A Promising