chapter 1 - emo

8/13/2019 Chapter 1 - EMO

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1EE6701 Evolutionary Computation Chapter 1

Evolutionary Computation

for Multi-objective Optimization

A/Prof Tan Kay Chen

Department of Electrical and Computer Engineering

National University of Singapore

Tel: 6516 2127; Email: [email protected]; Office: E4 08-09
mailto:[email protected]:[email protected]


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Rationale and Motivation

Some real-world problems have several (possibly conflicting)objectives to be optimized

Many of these problems are transformed (or aggregated) into asingle-objective problem (SOP) and are solved using single

objective optimization techniques Transforming multi-objective problem (MOP) into SOP requires

priori knowledge of relative importance of different objectives

Defining the problem in multi-objective framework is more generaland it provides the decision maker different tradeoffs betweendifferent objectives

The multi-objective optimization (also known as multi-criteriaoptimization) aims to find a set of vectors which satisfy the givenconstraints and optimizes a number of objective functions.


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Example:

A two-objective functions,f1 andf2, to be minimized are given as

28

1

812

28

1

811

8

11),...,(

8

11),...,(

i

i

i

i

xexpxxf

xexpxxf

The Pareto optimal front is all

the points on the line defined by

8

1

8

11821

xxxx f1

f2

Trade-off

curve

Unfeasible

region

where -2 xi< 2


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Formal Definitions

A multi-objective optimization problem (MOP) can be written as

min 1() 2() ()

s 1() 2() () 0

1() 2() () 0() , , , is the dimensional decision variable, is the number of objectives , inequality and equality constraints, ()and are respectively the lower and upper bound for each decision

variable

.


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For an MOP, must be greater than unity Based on the constraints, a MOP can be classified into one of the

following classes

() () Type of MOP0 0 Unconstrained MOP

0

0

Bound constrained MOP

> 0 0 x x Inequality constrained MOP0 > 0 x x Equality constrained MOPwhere and are some constants.

For the sake of simplicity, we use denotes the feasible decisionspace. The feasible decision space is the region which satisfies theconstrained in decision space

By using this convention, an MOP can be rewritten as

min 1() 2() ()


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Total-order and Partial-order

Order theory provides a formal framework for describing statementssuch as this is less than that or this precedes that

A relation is a total order on set Sif the following properties hold

Reflexivity: aafor all a S

Anti-symmetry: a b and b a implies a = b

Transitivity: a b and b c implies a c

Comparability: For any a, bS, either a b or b a.

For any partial order set, the comparability property does not hold In single-objective optimization, the feasible set is totally ordered

according to the objective function f

In multi-objective optimization, the feasible set ispartially orderedaccording to the objective function set f.


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Due to the partial ordering in MOP, multitude of trade-off solutionsexist in the objective space

This is one of the major differences between MOP and SOP

Dominance relation

Definition

A solution is said to dominate another solution ( ), if andonly if both of the following conditions are true:

The solution is no worse than in all objectives The solution is strictly better than in at least one objective

Mathematically,

if and only if 1, ,

1, , | <


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Non-dominated Set

Given a set of solutions, we can perform all possible pairwisecomparisons and find the solutions which are not dominated by anysolution in the set

Definition - Non-Dominated Set

Among a set of solutions , the non-dominated set of solutionsare those that are not dominated by any member of the set Definition - Pareto Optimal Set (in decision space)

The set of solutions that are non-dominated in the feasible

objective space

Definition - Pareto Optimal Front (in objective space)

The set of non-dominated solutions in the feasible objectivespace


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Classical Approaches

Generating Pareto optimal solutions plays an important role in multi-objective optimization. The term vector optimization is sometimesused to denote the problem of identifying the Pareto optimal set

Classical approaches often solve the MOP by scalarization.

Scalarization means converting the MOP into a single or a family ofSOP with a real-valued objective function

We are interested in generating Pareto optimal solutions (POS), onlya posteriorimethods are discussed in this module

A Posteriori Methods Generate the Pareto optimal set (or a part of it)

Present it to the decision maker (DM)

Let the DM select one


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Weighted Sum Method

This method scalarizes MOP into SOP by multiplying each objectivewith a predefined weight

min ()=

where is the weight of the -th objective function. The weightcoefficients must be real and positive. It is usual practice that weightare normalized, such that 1=

To use WSM as a posteriori method, a set of weight vectors has to bespecified. The correlation and some nonlinear effects between weights

and MOP are not easy to be understood Weighted sum method is relatively simple and easy to use

Evenly distributed weight vectors do not necessary produce an evenlydistributed representation of the Pareto optimal set. When the MOP is

non-convex, some of the Pareto optimal solution may fail to be found


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Multi-objective Evolutionary Algorithms

(MOEA)

Evolutionary Multi-objective Optimization (EMO) is an application ofevolutionary algorithm to solve multi-objective optimization problem

MOEAs are used to approximate the Pareto-optimal front of multi-objective problems

Many MOEAs have been proposed in the literature

Multi-objective Genetic Algorithm (MOGA)

Non-Dominated Sorting Algorithm II (NSGAII)

Incrementing Multi-objective Evolutionary Algorithm (IMOEA) Multiobjective Evolutionary Algorithm based on Decomposition

(MOEA/D)

These algorithms mainly different in, e.g., Fitness assignmentscheme; Mating selection scheme; Environmental selection scheme;Diversity preservation; Sharing and Elitism


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Common Terms Used in EA


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P2: 4 0 0 3 0 1 6 1

P2: 4 0 03 0 1 6 1

P3: 0 1 6 4 1 8 0 1

P2: 4 0 0 3 0 1 6 1

P2:4 0 13 08 0 1

P3:0 1 6 4 1 1 6 1

f(P1: 1 2 0 9 0 2 1 7)=5%

f(P

2: 4 0 0 3 0 1 6 1)=60%f(P3: 0 1 6 4 1 8 0 1)=35%

f(P2)

f(P3)

f(P1)

Fitness landscape

Decoding

Evaluation

Simulation

designs

Final optimized

designs coded

Initial/random

Selection

Crossover

Mutation

New

generation


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Multi-objective Evolutionary Algorithms

x1, x2, , xN

f(x1), f(x2), , f(xN)

How to select

parent solutions

How to

combine

parent

solutions to

create

offspring

solutions

Initialization

Fitness

Evaluation

Mating

Selection

Recombination/

Crossover

Mutation

Perturb the generated

offspring solution

Fitness

Evaluation

Selection

Termination

Check terminated

condition

Select

individual tosurvive

f(x1), f(x2), , f(xN)


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To achieve good approximation of POF, the output of any MOEAshould fulfill the following characteristics: (1) Sufficient proximity to

the exact Pareto-optimal front (convergence); (2) Well distributedalong the Pareto-optimal front (diversity)

Concept of Sharing And Elitism

To achieve the goals of convergence and diversity,

concept of sharing and elitism are introduced

Sharing:

To avoid the non-dominated solution clustering on someportions of Pareto-optimal front, certain penalty is applied tothe clustering solutions

Example: aand bwill have smaller fitness compared to c

Elitism:

The elites (best solutions or non-dominated solutions) ofthe population should be given the opportunity to be

directly carried over to the next generation

f1

f2

ab

c


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Fitness assignment, preference

Good spread, uniform

distribution, minimum proximity

Many objectives, constraints

Noise, dynamic landscape,

robust optimization

f1

f2

Unfeasible

Region

Pareto front/Non-dominated solutions

Fonseca and Fleming,1995

EMO needs to address several issues

EA is a powerful tool for solving MO optimization problems

Population-based, and capable of searching for the global trade-off

Robust and applicable to a wide range of problems

Capable of handling discontinuous and multi-dimensional problems


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The Pareto rankingassigns the samesmallest costfor all non-dominatedindividuals, while the dominated individualsare ranked according to how

many individuals in the population dominating them

f1

f2

1

1

1

2

24

5

Pareto optimal

ranking

The rank of an individualxin a population can be given by rank(Fx) =1 + qx, where qxis the number of individuals dominating theindividualFxexpressed in the objective domain.


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Goal (MOGA)

Desired value for each objective

Can be used to specify practical

design specification/requirements

Individuals that satisfy the goal

setting have a lower (better) rank

Tan, K. C., Khor, E. F., Lee, T. H. and Sathikannan, R., 'An evolutionaryalgorithm with advanced goal and priority specification for multi-objective optimization', Journal of Artificial Intelligence Research, vol. 18,

pp. 183-215, 2003.

In design optimization, a set of specification or design requirements areoften given in a-priori


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MOEA minimization with a feasible goal setting

Gen = 5 Gen = 70


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Unfeasible goal settingFeasible but extreme goal setting


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(G1G2G3G4) (G1G2G3)

Logical OR and AND connectives among goals


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Priority/Preference

Assign relative importance of each

objective for practical applications

Among strings of equal rank, priority can

be used to determine superiority

Example: Stability in control system design

Soft/hard Objectives

Soft: Always considered in the evolution

Hard: Considered while goal is not satisfied

Example: SS error and actuator saturation

EE6701 E l ti C t ti Ch t 1


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Compared with SO problems, MO optimization often requires alarger population size in order to cover the trade-off surface

Population size can be changed according to the populationdistribution at each generation

Generally, we can start with a small population for initial search.Subsequently increase or decrease the population size according tocurrent Pareto front in the evolution process

The increased individuals can be obtained via local fine-tuning bygenerating additional good individuals to fill up gaps or

discontinuities in the current Pareto front

Tan, K. C., Lee, T. H. and Khor, E. F., 'Evolutionary algorithm withdynamic population size and local exploration for multiobjectiveoptimization', IEEE Transactions on Evolutionary Computation, vol.

5, no. 6, pp. 565-588, 2001.

Dynamic Population



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Local Perturbation



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Population size versus generation Population distribution



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Handling Elements: Provide basic usefulness of finding the non-dominated individuals

Min-Max, Sub-Pop and others received less interests ascompared to Pareto, Weights, Goals and Pref

Weights has attracted significant attentions from 1985-2000.

The popularity of Pareto continues to grow significantly

MO Handling Elements

0

10

20

30

4050

1961

1967

1985

1992

1994

1996

1998

2000

2002

Year

Cumulativenumber

of

Methods

Weights

Min-Max

Pareto

Goals

Pref

Gene

Sub-Pop

Fuzzy

Others



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Supporting Elements

0

10

20

3040

1961

1967

1985

1992

1994

1996

1998

2000

2002

Year

Cumulativenumb

er

ofMethods Dist

Mat

Sub-Reg

Ext-Pop

Elitsm

A-Evo

Supporting Elements: Play an indirect role of supporting the algorithm

to achieve better performance

It was developed more recently than MO handling elements The Dist feature and Elitism/Archive feature have been

incorporated in many algorithms

Other features, such as for noise and many objectives problems

etc., are gaining more attentions recently

28EE6701 E ol tionar Comp tation Chapter 1


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Generational Distance (GD) (Veldhuizen, 1999)

Measures how far the evolved solution set is from

the true Pareto front.

Spacing (S) (Schott, 1995)

Measures how evenly the evolvedsolutions distribute itself.

Maximum Spread (MS)(Zitzler, 1999)

Measures how well the true Pareto front

is covered by the evolved solution set.f

1

f2

Minimization

Minimiza

tion

Non-dominated solution

Pareto Frontier

Non-dominated set

Hyper-Volume Ratio (HVR) (Veldhuizen, 1999)

Calculates the volume covered by the evolved solutions.



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Non-dominance Ratio (NR) (Goh and Tan, 2009)

Compare the quality of the solution set from various algorithms

Measures the ratio of non-dominated solutions contributed by a

particular solution set to the non-dominated solutions provided byall solution sets

Objective Function 1

ObjectiveFunction2

NR

0.8

0.2



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Inverted Generational Distance (IGD) (M. A. Villalobos-Arias, 2005)

Measure the proximity as well as diversity between the obtained

Pareto front and the Pareto Optimal front

The Euclidean distance is measured from each obtained solution tothe optimal solutions set

F2

F1

F2

F1

ParetoOptim

alFront

ParetoOptim

alFront

E

volvedSolutions

EvolvedSolutions

( )=



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TestProblem Features

1 ZDT1 Pareto front is convex.

2 ZDT2 Pareto front is non-convex.

3 ZDT3 Pareto front consists of several noncontiguous convexparts.

4 ZDT4 Pareto front is highly multi-modal where there are 219local Pareto fronts.

5 ZDT6 The Pareto optimal solutions are non-uniformlydistributed along the global Pareto front. The density ofthe solutions is low near the Pareto front and highaway from the front.

ZDT1

ZDT2

ZDT3 ZDT4 ZDT6

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1



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Test

Problem

Features

6 FON Pareto front is non-convex.

7 KUR Pareto front consists of several noncontiguous convexparts.

8 POL Pareto front and Pareto optimal solutions consist ofseveral noncontiguous convex parts.

9 TLK Noisy landscape.

10 TLK2 Non-stationary environment.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

FON

-20 -19 -18 -17 -16 -15 -14-15

-10

-5

0

5

KUR

0 5 10 15 200

5

10

15

20

25

POL

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

TLK

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

TLK2


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1. Increasing Dimensionality Tan, K. C., Lee, T. H. and Khor, E. F., 'Evolutionary algorithm with

dynamic population size and local exploration for multiobjectiveoptimization', IEEE Transactions on Evolutionary Computation,vol. 5, no. 6, pp. 565-588, 2001.

Liu, D. S., Tan, K. C., Goh, C. K. and Ho, W. K., 'A multiobjective

memetic algorithm based on particle swarm optimization', IEEETrans on Systems, Man and Cybernetics: Part B (Cybernetics),vol. 37, no. 1, pp. 42-50, 2007.

2. Expensive Function Evaluations

Tan, K. C., Tay, A. and Cai, J., 'Design and implementation of adistributed evolutionary computing software', IEEE Trans onSystems, Man and Cybernetics: Part C, vol. 33, issue 3, pp. 325-338, 2003.

Tan, K. C., Yang, Y. J. and Goh, C. K., 'A distributed cooperativecoevolutionary algorithm for multiobjective optimization', IEEETransactions on Evolutionary Computation, vol. 10, issue 5, pp.527- 549, 2006.



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Decompose a complex problem into smaller problems viaco-evolving subpopulations cooperatively

A divide-and-conquer strategy

Each subpopulation evolves adifferent decision variable

Fitness is dependent on the

collaboration between each

subpopulation

Subpop 1

for

variable 1

Subpop i

for

variable i

Subpop k

for

variable k

Subpop m

for

variable m

Collaborate

Evaluate

Update achive

Rank

Representatives

Representatives

Representatives

Complete solution

Complete solution

and its objective

Achive

Individuals in

subpop i

Assign rank



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Cooperation Methods

C1: The best individual is selected as representative

C2: The best and a random individuals are selected as representatives.

The better solution resulting from the two co-operations is retained

C3: The best two individuals are selected as representatives. The better

solution resulting from the co-operations with the best representative

and either representative is retained

C4: The best two individuals are selected as the representatives.

Cooperation is performed with either one of the representatives

The model of cooperation among subpopulations is animportant issue in cooperative co-evolution The simplest approach is to select the best individual as the representative.

However, this approach tends to perform poorly for problems with high

parameter inter-dependency

Alternatively, a random individual can be selected as the representative.However, this often results in a slower convergence rate

Four different methods have been examined.



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Peer 1

ServerServer

Subpopulations Peers Central server

1

32

5

4

6

1

2

Peer 2

3

4

Peer 3

5

6

Parallelization Strategy

Subpopulations are partitioned

into a number of groups and

assigned to peer computers

Indirect cooperation is

achieved through the

exchange of archive and

representatives between peers

and a central server

Peers are synchronized at

fixed intervals to ensure better

cooperation

A Distributed CCEA



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Developed upon the technology of Java 2

platform of Enterprise Edition (J2EE)

Exploit the inherent parallelism of

evolutionary algorithms

Incorporates the features of robustness,

security, and workload balancing

The DCCEA is designed and embedded in a distributed computingframework named Paladin-DEC.



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Comparative ResultsGenerational Distance

ZDT4

CCEA PAES PESA NSGAII SPEA2 IMOEA

0

0.5

1

1.5

2

Generational Distance


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


ZDT6 FON KUR


0.01

0.02

0.03

0.04

0.05



0

0.01

0.02

0.03

0.04

0.05

0.06



0

0.5

1

1.5

2



0

500

1000

1500

2000


DTLZ2 DTLZ3

CCEA escapes from local optima of

ZDT4 and DTLZ3 CCEA scales well for high-dimensional

MO problems

CCEA performs less well for KUR due tothe high parameter inter-dependency

Observation:



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Comparative ResultsSpacing

ZDT4 ZDT6 FON KUR

DTLZ2 DTLZ3


0.5

1

1.5

2

Spacing


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Spacing


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Spacing


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Spacing

CCEA PAES PESA NSGAII SPEA2 IMOEA0

0.5

1

1.5

2

2.5

3

Spacing


0.5

1

1.5

2

2.5

Spacing

CCEA generally shows good

performance in spacing

CCEA handles the non-uniformdistribution of ZDT6 well

Observation:



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Comparative ResultsMaximum Spread

ZDT4 ZDT6 FON KUR

DTLZ2 DTLZ3


0.7

0.75

0.8

0.85

0.9

0.95

1

Maximum Spread


0.97

0.975

0.98

0.985

0.99

0.995

1

Maximum Spread


0.5

0.6

0.7

0.8

0.9

1

Maximum Spread


0.7

0.75

0.8

0.85

0.9

0.95

1

Maximum Spread

CCEA PAES PESA NSGAII SPEA2 IMOEA0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Maximum Spread


0.5

0.6

0.7

0.8

0.9

1

Maximum Spread

CCEA performs competitively ascompared to other MOEAs

Observation:



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Behaviors of CCEA

Oscillations ofx1help tosample the solutions along thePareto front uniformly

Variablesx2tox10converge to

the optimal location gradually 0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

Generation number

x1

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Generation number

x2~x10

0 50 100 150 200 250 300 350 400-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Generation number

x1~x4

x1

x2

x3

x4

0 50 100 150 200 250 300 350 400-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Generation number

x5~x8

x5

x6

x7

x8

ZDT4

FON

ZDT4

All the variables oscillate and

converge in a similar pace This behavior is related to the

high inter-dependency amongthe parameters

FON



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EE6701 Evolutionary Computation Chapter 1

Effects of the Cooperation

C1 is the greediest method. It performs well for problems with low parameter inter-dependency, but gives poor results for those with high inter-dependency

C4 is the least greedy approach. It performs well for problems with high parameterinter-dependency, but gives poor results for those with low inter-dependency

C2 and C3 provide a balance between greedy search and diversity maintenance,which achieve generally good and robust performances

Problem C1 C2 C3 C4

ZDT1 4.13E-04 4.24E-04 4.04E-04 3.14E-01

ZDT2 4.05E-04 4.81E-04 6.35E-04 5.09E-01

ZDT3 1.45E-03 1.41E-03 1.27E-03 3.64E-01

ZDT4 6.98E-05 7.08E-05 1.48E-04 1.26E+00

ZDT5 1.27E-06 1.27E-06 1.26E-06 2.90E+00

ZDT6 4.89E-07 5.00E-07 4.86E-07 1.43E+00FON 2.35E-02 2.32E-02 2.28E-02 2.13E-02

KUR 3.55E-02 2.31E-02 2.35E-02 1.76E-02

TLK 4.80E-01 5.23E-01 4.90E-01 5.34E-01

DTLZ2 8.57E-04 4.07E-01 2.08E-02 5.44E-01

DTLZ3 1.82E+00 2.24E+00 1.65E+00 5.82E+02

Problem C1 C2 C3 C4

ZDT1 0.1757 0.1771 0.1667 0.8240

ZDT2 0.1613 0.1553 0.1600 0.9328

ZDT3 0.2815 0.2867 0.2733 0.9952

ZDT4 0.1333 0.1436 0.1409 1.9624

ZDT5 0.7114 0.7125 0.7125 1.1676

ZDT6 0.1754 0.1822 0.1881 0.8174FON 0.2687 0.3497 0.5291 0.2676

KUR 0.7919 0.7145 0.7154 0.6817

TLK 1.0905 1.0284 1.0135 1.0772

DTLZ2 0.1245 0.2658 0.1604 0.2929

DTLZ3 0.6638 0.8103 0.7525 0.7201

Median GD for CCEA Median S for CCEA



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y p p

PC Configuration CPU

(MHz)/RAM (MB)

server PIV 1600/512

peer 1 PIII 800/ 512



peer 4 PIII 933/384

peer 5 PIII 933/128

peer 6 PIV 1300/ 128

peer 7 PIV 1300/ 128peer 8 PIII 933/ 512


peer 10 PIII 933/256

11 PCs in LAN

Populations Subpopulation size 20;

archive size 100

Chromosome length 30 bits for each variable

Selection Tournament selection

Crossover operator Uniform crossover

Crossover rate 0.8

Mutation operator Bit-flip mutation

Mutation rate 2/L , where L is the

chromosome length

Number of evaluation 120,000

Exchange interval 5 generations

Sync. interval 10 generations

Configurations



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y p p

Generational Distance Spacing

Maximum Spread Hyper-volume Ratio



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y p p

Observation

Effective in reducing simulation runtime without sacrificing performance

Speedup achievable is more significant for large problems

The increase of communication cost counteracts the reduction of

computation cost with increasing number of peers

Number of peers ZDT1 ZDT2 ZDT3 ZDT4 ZDT6

2 1.52 1.70 1.47 1.23 1.01

3 2.01 1.99 1.88 1.47 1.11

4 2.25 2.21 1.95 1.50 1.14

5 2.48 2.69 2.15 1.56 1.14

6 2.81 3.03 2.83 1.70 1.29

7 2.87 3.32 2.77 1.88 1.25

8 3.38 3.27 2.92 1.82 1.26

9 3.46 3.36 2.96 1.83 1.26

10 3.46 3.18 2.79 1.82 1.25

Speedup

0

50

100

150

200

250

300

1 2 3 4 5 6 7 8 9 10

Number of peers

Runtime(s)

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

Runtime



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y p p

3. Noise and Uncertainty Goh, C. K., and Tan, K. C., 'An investigation on noisy

environments in evolutionary multi-objective optimization', IEEETransactions on Evolutionary Computation, vol. 11, no. 3, pp.354-381, 2007.

Tan, K. C. and Goh, C. K., 'A competitive-cooperativecoevolutionary paradigm for dynamic multi-objective

optimization', IEEE Transactions on Evolutionary Computation,vol. 13, no. 1, pp. 103-127, 2009.

4. Estimation of Distribution Algorithms

Shim, V. A., Tan, K. C., Chia, J.Y. and Mamun, A. Al., 'Multi-

objective optimization with estimation of distribution algorithm innoisy environment', Evolutionary Computation (MIT Press),2012.

Shim, V. A., Tan, K. C. and Cheong, C. Y., 'A hybrid estimationof distribution algorithm with decomposition for solving themultiobjective multiple traveling salesman problem', IEEETransactions on Systems, Man, and Cybernetic: Part C, vol. 42,no. 5, pp. 682-691, Sep 2012.



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y p p

Data encountered in practical applications may often be

influenced by noise

Noise may arise from different sources, e.g., sensor measurementerrors, incomplete simulation of computational models etc

The problem with noise is that it is not simply adding a bias or offsetto the objective function

When a system is presented with noise, each evaluation of the samesolution may result in different objective function values

Noise encountered in EMO optimization is often modeled as arandom perturbation to the objective functions

Performance is greatly influenced by the noise model adopted andthe level of noise intensity encountered



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y p p

1.0

0.8

0.6

0.4

0.2

0.00.2 0.4 0.6 0.8 1.0 1.2 1.4

f1

f2

1.0

0.8

0.6

0.4

0.2

0.00.2 0.4 0.6 0.8 1.0 1.2 1.4

f1

f2

A

BA'B'

Noise may change the way we perceive the solution

E.g., a good solution may be perceivedas a bad solution and vice versa. Theproblem is worse for EMO.

Non-dominated solutions in noisyoptimization may be an inferior orworse a non-feasible solution.

Effect of Noise



50/96

Existing approaches for handling noise in SO and MOproblems can be classified as follows (Jin and Branke, 2005)

Explicit Averaging (Singh, 2003; Bui et al, 2005; Basseur andZitzler, 2006)

Implicit Averaging (Buche et al, 2002)

Selection Modification (Hughes, 2001; Teiche, 2001)

Heuristic Approach (Goh and Tan, 2007)

Design considerations for handling noise in EMO

To minimize error in the selection and elitism process

To improve the efficiency of noise handling mechanism

To handle the final set of solutions in archive for decision-making



51/96

Explicit Averaging

Each solution is evaluated a number of times and is averaged to

compute the expected objective value

By increasing the number of samples, it reduces the degree of

uncertainty in the optimization

f1

f2

Original Solution

Samples



52/96

A large population size is used

There may be many similar solutions and thus influence of noise iscompensated as the search revisits the same region repeatedly

This approach can be computationally expensive

x1

x2

x1

x2

Small population Large population

Implicit Averaging


53/96



54/96

Preliminary InvestigationParameter Settings

1/ chromosome_length

1/ bit_number_per_variable

Basic Algorithm

Employs a fixed-sizepopulation and an archive.

When predetermined archivesize is reached, a recurrent

truncation process based onniche count is used.

Binary tournament selectionscheme is used for thecombined archive andevolving population.

Additive Noise Model2( ) ( ) Normal(0, )f x f x

is represented as a percentage of the maximum of the i-th objective in the

true Pareto front.

2

Chromosome Binary coding; 15 bits per decision variable.

Population Population size 100; Archive (or secondary

population) size 100.

Selection Binary tournament selection

Crossover operator Uniform crossover

Crossover rate 0.8

Mutation operator Bit-flip mutation

Mutation rate for FON and KUR

for ZDT1, ZDT4 and ZDT6

Ranking scheme Pareto ranking

Diversity operator Niche count with radius 0.01 in the normalized

objective space.

Evaluation number 50,000



55/96

0 100 200 300 400 500

-7

-6

-5

-4

-3

-2

-1

0

1

2

Generation

GD

GD

Generation

0 100 200 300 400 500

-1

0

1

2

3

4

5

GD

Generation

0 100 200 300 400 500

-8

-6

-4

-2

0

2

GD

Generation

0 100 200 300 400 500

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

GD

Generation

0 100 200 300 400 500

-7

-6

-5

-4

-3

-2

-1

0

ZDT1 ZDT6ZDT4

KURFON

No noise

0.2%

0.5%

1.0%

5.0%

10.0%

20.0%

Preliminary InvestigationNoise impact on convergence



56/96

Preliminary InvestigationNoise impact on diversity

ZDT1 ZDT6ZDT4

KURFON

0 100 200 300 400 500

0.5

0.6

0.7

0.8

0.9

1

Generation

MS

Generation

MS

0 100 200 300 400 500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Generation

MS

0 100 200 300 400 500

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Generation

MS

0 100 200 300 400 500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Generation

MS

0 100 200 300 400 500

0.4

0.5

0.6

0.7

0.8

0.9

1

No noise

0.2%

0.5%

1.0%

5.0%

10.0%

20.0%



57/96

Preliminary Investigation

ZDT1 ZDT6ZDT4

KURFON

0.0 0.2 0.5 1.0 5.0 10.0 20.00

20

40

60

80

100

Noise Level

ArchiveSize

0.0 0.2 0.5 1.0 5.0 10.0 20.00

20

40

60

80

100

Noise Level

ArchiveSize

0.0 0.2 0.5 1.0 5.0 10.0 20.00

20

40

60

80

100

Noise Level

ArchiveSize

0.0 0.2 0.5 1.0 5.0 10.0 20.00

20

40

60

80

100

Noise Level

ArchiveSize

0.0 0.2 0.5 1.0 5.0 10.0 20.00

20

40

60

80

100

Noise Level

ArchiveSize

Noise impact on archiving

Although MOEA is able to search

for better solutions, the noise-enhanced solutions in the archiveare keeping the newly found non-dominated solutions away, thusresulting in the reduced numberof archived solutions.



58/96


ZDT1 ZDT6ZDT4

KURFON

0 50 100 150 200

0

0.1

0.2

0.3

0.4

0.5

Generation

Decision-errorratio

Generation

Decision-errorratio

0 50 100 150 200

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Generation

Decision-errorratio

0 50 100 150 200

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Generation

Decision-erro

rratio

0 50 100 150 200

0

0.1

0.2

0.3

0.4

0.5

Generation

Decision-erro

rratio

0 50 100 150 200

0

0.05

0.1

0.15

0.2

No noise

0.2%

0.5%

1.0%

5.0%

10.0%

20.0%

Noise impact on decision-making in selection process



59/96


Generation

Search

range

ofvariable,x

0 100 200 300 400 500

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Generation

Search

range

ofvariable,x

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

Generation

S

earch

range

ofvariable,x

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

Generation

S

earch

range

ofvariable,x

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

ZDT1

FON

0% noise level 20% noise level

0% noise level 20% noise level

BMOEA is capable ofnarrowing down the searchrange for better evolutionarysearch optimization in a noise

free environment.

On the other hand, the meanlocation of individuals remainsrelatively the same, and the

range is concentrated near theoptimal region despite thepresence of noise, whichcould be due to the correctdecision-making in theselection process.

Trace of search range for an arbitrary selected parameter



60/96

Better decision-making at early generations: Introduce a momentumterm to accelerate movement in the direction of improvement; whilerestricting movement otherwise

Generation

Decision-errorratio

0 50 100 150 200

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Decision-making error

Experiential Learning Directed Perturbation

( ) ( ) ( 1)j j jp t p t p t

min max( ) ( ), if ( 1)( 1)

Bit-flip, otherwise

j j j

j

p t p t p tp t

Variation increases in magnitudein the same direction of change.



61/96

Gene Adaptation Selection Strategy

Mean location of search range remains relatively invariant: Adjust thesearch range according to the distribution of solutions

It has the advantage of concentrating resources on a smaller region. Byre-evaluating the similar solutions, it gives some form of implicit averaging

j j j

j j j

a lowbd w meanbd

b uppbd w meanbd

Activation of geneadaptation

Generation

Current Search Region

ja

jb

Defined Search Region

jx

jmeanbd

jlowbd

juppbd

Generation

Current Search Region

Defined Search Region

jx

Activation of geneadaptation

jb

ja

jmeanbd

juppbd

jlowbd

j j j

j j j

a meanbd w meanbdb meanbd w meanbd

Convergence model Divergence model



62/96

Possibilistic Archiving

Based on the concept of possibilistic Pareto dominance tostore true non-dominated solutions in the archive

Minimize removal of true non-dominated individuals due to noise

Provide a chance for individuals degraded by noise to survive

2L

22L

1

2x

1x

Lis a reflection of the uncertainty

level present in the system.

The possibilistic archiving removesa solution only if it necessarily

nominates a solution.

The gray region denotes the region

for which this solution dominates.

Because of uncertainty, this solution

can be anywhere denoted in the

box and therefore it is possible that

any solutions in the box may be

non-dominated solutions.



63/96

Adapt gene based on

adopted model

Crossover

ELDP

Add offspring to

population

Combine archive

and evolving

population

Possibilistic

Archiving

Evaluation and

Pareto Ranking

Initialize population

Is stopping criteria met? NoYes

Possibilistic

Archiving

Return Archive

GASSConvergence or

divergence model?

YesNo

BInary Tournament

selection

MOEA-RF

Experiential LearningDirected Perturbation

Gene AdaptationSelection Strategy

Possibilistic Archiving



64/96

0.0 5.0 10.0 20.00

0.5

1

1.5

Noise Level

GD

0.0 5.0 10.0 20.00.5

0.6

0.7

0.8

0.9

1

Noise Level

MS

0.0 5.0 10.0 20.00

0.2

0.4

0.6

0.8

1

Noise Level

HV

R

MOEA-RF

RMOEA

NTSPEA

MOPSEASPEA2

NSGAII

PAES

Performance in Noisy Environment

ZDT1with different noise levels



65/96


FON with 20% noise

0 10000 20000 30000 40000 500000

0.1

0.2

0.3

0.4

0.5

Evaluation

GD

0 10000 20000 30000 40000 500000

0.1

0.2

0.3

0.4

0.5

0.6

Evaluation

MS

MOEA-RF

RMOEA

NTSPEA

MOPSEA

SPEA2

NSGAII

PAES



66/96


MOEA-RF RMOEA NTSPEA MOPSEA SPEA2 NSGAII PAES

1st quartile 28 6 5 7 18 17 4

ZDT1 Median 31 7.5 6 9 21.5 19.5 4.5

3rd quartile 32 10 8 10 27 23 5

1st quartile 29 5 6 9 12 20 5ZDT4 Median 33 7 8 11 26.5 26 6

3rd quartile 41 8 10 13 29 32 9

1st quartile 82 2 4 3 8 8 2

ZDT6 Median 85 3 5 4 9 9 4

3rd quartile 88 5 6 5 11 11 6

1st quartile 9 1 1 1 6 6 1

FON Median 11 2 1.5 2 8.5 8.5 2

3rd quartile 17 2 2 3 12 12 3

1st quartile 25 6 5 8 23 25 7

KUR Median 27 8 5.5 9 25 27 9

3rd quartile 30 9 7 11 28 30 10

Number of non-dominated individuals found for the various problems(20% noise level)



67/96

Baseline MOEA

Generation (a) 0, (b) 10, (c) 60, (d) 200, and (e) 350 for ZDT4 with 219local Pareto fronts.

Baseline MOEA with ELDP

Baseline MOEA with GASS

0 0.5 1

0

20

40

60

80

100

0 0.5 1

0

5

10

15

20

0 0.5 1

0

0.5

1

1.5

2

2.5

0 0.5 1

0

0.5

1

1.5

2

2.5

0 0.5 1

0

0.5

1

1.5

2

2.5

0 0.5 10

20

40

60

80

100

0 0.5 10

1

2

3

4

5

0 0.5 10

0.5

1

1.5

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

20

40

60

80

100

0 0.5 10

5

10

15

20

0 0.5 10

0.5

1

1.5

2

2.5

0 0.5 10

0.5

1

1.5

0 0.5 10

0.5

1

1.5

(a) (b) (c) (d) (e)

(a) (b) (c) (d) (e)

(a) (b) (c) (d) (e)

Effects of the Proposed Features ELDP and GASS have the advantage of overcoming local optimality for ZDT4

The population distribution converges faster when ELDP is incorporated



68/96

ZDT4 with 0% noise

NSGAII

0

0.5

1

1.5


NSGAII-RF SPEA2 SPEA2-RF NSGAII NSGAII-RF SPEA2 SPEA2-RF

4

6

8

x 10-3

Spacing

NSGAII NSGAII-RF SPEA2 SPEA2-RF

0.7

0.8

0.9

1

Maximum Spread


0

0.5

1

Hypervolume Ratio


0

1

2



0.05

0.1

0.15

0.2

0.25

Spacing


0.6

0.7

0.8

0.9

1

Maximum Spread


0

0.2

0.4

0.6

0.8

Hypervolume Ratio

ZDT4 with 20% noise

Effects of ELDP and GASS onNSGA-II and SPEA2



69/96

Performance in Noisy EnvironmentZDT6 with 20% noise

ZDT6 has a non-uniformly distributed and

discontinuous tradeoff

Most algorithms are unable to find the global

tradeoff. MOEA-RF gives a rather consistent

good performance with a relatively small

variance for the performance metrics

MOEA-RFRMOEA NTSPEAMOPSEASPEA2NSGAII PAES


0

1

2

3

4

MOEA-RFRMOEA NTSPEAMOPSEA SPEA2NSGAIIPAES

-0.2

0

0.2

0.4

0.6

Spacing

MOEA-RFRMOEA NTSPEAMOPSEASPEA2NSGAIIPAES

0

0.5

1

Maximum Spread

MOEA-RFRMOEA NTSPEAMOPSEASPEA2NSGAIIPAES

0

0.2

0.4

0.6

0.8

Hypervolume Ratio

70EE6701 Evolutionary Computation

Chapter 1


70/96

Summary The behaviors of EMO in noisy environments are examined

Two heuristics including experiential learning directed

perturbation (ELDP) and gene adaptation selection strategy

(GASS) are applied to exploit MOEA behaviors for better

performance in noisy environment

A possibilistic archiving model to reduce the impact of noise on

the archive

The proposed features also improve performance of general MOEAs,e.g., SPEA2 and NSGA-II, to exhibit competitive or better performancein noisy environments

Three noise-handling features are proposed


Chapter 1


71/96

Hard Disk Drive Servo Specifications The control input should not exceed 2 volts due to physical

constraints on the actual actuator

The overshoots and undershoots of the step response should

be kept less than 5% as the R/W head can start to read or

write within 5% of the target

The 5% settling time in the step response should be less than

2 milliseconds

Excellent steady-state accuracy

Robust in terms of disturbance rejection and uncertainty

Tan, K. C., Sathikannan, R., Tan, W.W. and Loh, A. P., 'Evolutionarydesign and implementation of a hard disk drive servo control system',

Soft Computing, vol. 11, no. 2, pp. 131-139, 2007.


Chapter 1


72/96

103

104

-40

-20

0

20

40

60

frequency: rad/sec

ma

gnitude:dB

:: Measured

: Identified

103

104

-400

-300

-200

-100

0

frequency: rad/sec

phase:de

gree

Gv(s) 4.4 1010s4.87 1015

s2 (s2 1.45 103s 1.1 108 )

x(k 1) 1 1.664

0 1

x(k)

1.384

1.664

u

HDD Model

Sampling frequency of 4 kHz

KpKfz ff1z ff2

Controllers in Discrete Form

Ks Kbz fb1z fb2


Chapter 1


73/96

Time domain and frequency domain design specificationsCustomer specifications Objective Goal

1. Stability ( Closed-loop poles) Nr([eig(A_clp)] > 0) 0

Frequency

Domain

2. Closed-loop sensitivity or

Disturbance rejection [S( j )]1

3. Plant uncertainty [T( j )] 1

4. Actuator saturation Max(u) 2 volts

Time 5. Rise time Trise 2 milli sec

Domain 6. Overshoots Oshoot 0.05

7. Settling time 5%Tsettling 2 milli sec

8. Steady-state error SSerror 0

SumReference

Input Position Output

y(n)=Cx(n)+Du(n)

x(n+1)=Ax(n)+Bu(n)

Position Feedback Controller

y(n)=Cx(n)+Du(n)

x(n+1)=Ax(n)+Bu(n)

Plant

y(n)=Cx(n)+Du(n)

x(n+1)=Ax(n)+Bu(n)

FeedForwardController

Controller Output

Kp (0.038597) z0.63841

z0.39488

Ks

(0.212) z0.783

z0.014001


Chapter 1


74/96

Layout of MOEA Toolbox

Population Handling

Simulation Objectives

Remote Control

Graphical Results

Quick Setup

Master Window


Chapter 1


75/96

Design Trade-off Graph Progress Ratio

Tan, K. C., Lee, T. H., Khoo, D. and Khor, E. F., 'A multi-objectiveevolutionary algorithm toolbox for computer-aided multi-objectiveoptimization', IEEE Transactions on Systems, Man and Cybernetics:Part B (Cybernetics), vol. 31, no. 4, pp. 537-556, 2001.


Chapter 1


76/96

Servo Output Response Disturbance Rejection


77/96


Chapter 1


78/96

Real-time Implementation


Chapter 1


79/96

Vehicle routing problem (VRP) involves

the routing of a set of identical vehicles

with limited capacity from a central depot

to a set of geographically dispersedcustomers to satisfy their demands

For VRPSD, customers demands are

stochastic and other parameters such as

vehicle capacity, customers and depotsare known a-priori

Tan, K. C., Cheong, C. Y. and Goh, C. K., 'Solving multiobjective vehiclerouting problem with stochastic demand via evolutionary computation',

European Journal of Operational Research, vol. 177, pp. 813-839, 2007.


Chapter 1


80/96

Capacity constraint Each customer has a demand. Vehicle capacity cannot be exceeded

Treated as a hard constraint, e.g., a route failure occurs when this

constraint is violated

Time constraint

Service and travel time for each route should not exceed the length of

a drivers workday, e.g., 8 hours

Treated as a soft constraint in the form of remuneration for drivers. $10 for each of first 8 hours of work and $20 for each additional hour

subsequently

This is to penalize exceedingly long routes which may not be feasible

to implement in practice

Constraints in VRPSD


Chapter 1


81/96

Stochastic demand

Demand of each customer is a random variable whose

distribution is known

A normal distribution is usually used

Actual demand is not known but is revealed only when the

vehicle arrives at the customers location

Examples of VRPSD

Beer and soft drinks distribution

Provision of ATMs with cash

Trash collection

Stochastic Demands


Chapter 1


82/96

Route failure A vehicle may find that it is not able to satisfy a customers

demand upon arrival due to the capacity constraint

Recourse policy

The vehicle returns to the depot to restock and then continues

delivery according to the originally planned route

Main obstacle

The actual cost of a solution cannot be known before it is

actually implemented, e.g., the main obstacle for solving

VRPSD is to find a suitable objective function

Difficulty in Solving VRPSD


Chapter 1


83/96

Route Simulation Method (RSM)

RSM is applied to evaluate the cost of routes for a particular

realization of the customers demands

Generate Nsets of demands based on the known

distributions of the customers demands

Averaging technique is used to obtain the expected cost of

the solution

Depot

1

2

3

4

5

6

Customer GeneratedDemand

1 5

2 6

3 2

4 13

5 9

6 5

Example: Vehicle capacity: 15


Chapter 1


84/96

Multiple objectives

Travel distance

Driver remuneration

Number of vehicles

Variable-length chromosome

Encodes the number of

routes/vehicles and the order of

customers served by these vehicles

Every chromosome can have a

different number of routes

0

2

5

7

0

0

1

3

4

8

9

0

0

6

10

0

Chromosome

0

2

5

7

0

0

1

3

4

8

9

0

0

6

10

0

Chromosome

Each route contains a

sequence of customers

A chromosome encodes a

complete routing solution


Chapter 1


85/96

Route-exchange Crossover

Allows good sequence of routes to be shared

Infeasibility after the change can be eradicated easily

A random shuffling operator is applied to increase the diversity

of chromosomes to explore the large VRPSD search space


Chapter 1


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Multi-mode Mutation

Partial swap Merge shortest route

Split longest route

Split LongestRoute

Random Shuffle

Partial Swaprand[0,1) < elastic rate

Chromosomeselected for

mutation

No

Yes

rand[0,1) < squeeze rate

No

Merge ShortestRoute

Yes


Chapter 1


87/96

Shortest Path (SP) search

Exploit the structure that route

failures are more likely to occur

at the end of a route

Which Directional (WD) search

Exploit the structure that the cost

of a route is different for both

directions

Rebuild the route in the oppositedirection

For both methods, the new route

is compared with the original one

and the better route is retained Flowchart

Build customer

database

Population

initialization

Route Simulation

Method and

Pareto ranking

Tournamentselection

Route-exchange

crossover

Multi-mode

mutation

Elitism

Local search

exploitation

Stopping

criterion

met?

Start

End

No

Yes

Update archive

Perform

local

search?

Yes

No

Generation loop

Build customer

database

Population

initialization

Route Simulation

Method and

Pareto ranking

Tournamentselection

Route-exchange

crossover

Multi-mode

mutation

Elitism

Local search

exploitation

Stopping

criterion

met?

Start

End

No

Yes

Update archive

Perform

local

search?

Yes

No

Generation loop

While evolutionary operators focus on global exploration, local

search contributes to the intensification of optimization results


Chapter 1


88/96

Performance of Local Search Operators

All Solutions Non-dominated Solutions

0 100 200 300 4001000

1500

2000

2500

3000

Generation

Averagetraveldistance

WD/SP

SP/WD

SP

WD

RAN

NLS

0 100 200 300 4001000

1500

2000

2500

3000

Generation

Averagetraveldistance

WD/SP

SP/WD

SP

WD

RAN

NLS

0 100 200 300 400

1000

1200

1400

1600

1800

2000

2200

Generation

Averagedriverremun

eration

WD/SP

SP/WD

SP

WD

RANNLS

0 100 200 300 400

800

1000

1200

1400

1600

1800

2000

Generation

Averagedriverremun

eration

WD/SP

SP/WD

SP

WD

RANNLS


Chapter 1


89/96

Comparison of Different Optimization Criteria

Multiobjective Performance

0.00E+00

1.00E+06

2.00E+06

3.00E+06

4.00E+06

5.00E+06

6.00E+06

7.00E+06

All solutions Non-dominated solutions

Traveld

istanceXDriverremu

neration

MO

DORV

DODV

DODR

SOD

SOR

SOV


Chapter 1


90/96

Search Space of MO


Chapter 1


91/96

Performance for different value of N (e.g., expected cost of a solutionas compared to the actual cost of implementing the solution).

Travel distance

Driver remuneration

Robust VRPSD solutions

1 3 5 10 30 50 70

-100

0

100

200

300

Test Demand Set 1

Increaseintraveldistance

N1 3 5 10 30 50 70

-100

0

100

200

300

Test Demand Set 2


N1 3 5 10 30 50 70

-100

0

100

200

300

Test Demand Set 3


N1 3 5 10 30 50 70

-100

0

100

200

300

Test Demand Set 4


N

1 3 5 10 30 50 70

-100

0

100

200

300

Test Demand Set 1

Increasein

driverremuneration

N1 3 5 10 30 50 70

-100

0

100

200

300

Test Demand Set 2

Increasein

driverremuneration

N1 3 5 10 30 50 70

-100

0

100

200

300

Test Demand Set 3

Increasein

driverremuneration

N1 3 5 10 30 50 70

-100

0

100

200

300

Test Demand Set 4

Increasein

driverremuneration

N


Chapter 1


92/96

Robust VRPSD solutions (Travel Distance)

GEG GEM AEM DET

-100

0

100

200

300

400

Test Demand Set 1

In

creaseintraveldistance

GEG GEM AEM DET

-100

0

100

200

300

400

Test Demand Set 2

In

creaseintraveldistance

GEG GEM AEM DET

-100

0

100

200

300

400

Test Demand Set 3


GEG GEM AEM DET

-100

0

100

200

300

400

Test Demand Set 4


(Dror and Trudeau)


Chapter 1


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Robust VRPSD solutions (Driver Remuneration)

GEG GEM AEM DET

-100

0

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400

Test Demand Set 1

Incre

aseindriverremuneratio

n

GEG GEM AEM DET

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Test Demand Set 2

Incre

aseindriverremuneratio

n

GEG GEM AEM DET

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Test Demand Set 3

Increaseindriverremu

neration

GEG GEM AEM DET

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Test Demand Set 4

Increaseindriverremu

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Robust VRPSD solutions

ExpectedTravel

Distance

Increasein TravelDistance

ActualTravel

Distance

ExpectedDriver

Remuneration

Increase inDriver

Remuneration

Actual DriverRemuneration

MultiplicativeAggregate

(X106)

GEG 1086.59 33.85 1120.44 985.95 32.87 1018.82 1.142

GEM 1120.52 16.54 1137.06 990.33 18.64 1008.97 1.147

AEM 1002.08 122.35 1124.43 947.04 125.04 1072.08 1.205

DET 970.17 213.40 1183.57 909.59 217.35 1126.94 1.334

Averaged over the non-dominated solutions at the termination of the simulation.

Test demand set 1


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Significance of RSM

The RSM was implemented by using demand sets randomly

generated based on the customers demand distributions

In actual implementation, there may not be a need to randomly

generate the demand sets if the company keeps past demand

records of their customers

Past records are useful, and can be used to provide the demand sets

for the RSM to operate on

Important if the customers demand distributions are not known

Multiple objectives were considered in solving the VRPSD problem

New way of assessing quality of solutions (RSM)

Solutions of the MOEA are robust to stochastic nature of the problem

Summary


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Goh, C. K. and Tan, K. C., Evolutionary Multi-

objective Optimization in Uncertain Environments:

Issues and Algorithms, Springer-Verlag, 2009

Tan, K. C., Khor, E. F. and Lee, T. H. Multiobjective

Evolutionary Algorithms and Applications, Springer-

Verlag, United Kingdom, 2005
http://localhost/var/www/apps/conversion/tmp/scratch_6/noise.htm

chapter 1 - emo

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