multicriterial optimization using genetic...

Multicriterial Optimization Using Genetic Algorithm

MulticriterialMulticriterial Optimization Optimization Using Genetic Using Genetic AlgorithmAlgorithm

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Best FitnessMean Fitness


Contents

• Optimization, Local and Global Optimization

• Multicriterial Optimization

• Constraints

• Methods of Solution

• Examples

• Task of the Desicion Maker


Global optimization is the process of finding the global extreme value (minimum or maximum) within some search space S.

The single objective global optimization problem can be formally defined as follows:

Global optimization


Global optimization

The maximalization can explain from minimalization with the next fromula:

)x(fmin)x(fmax −=

Then is the global solution(s), f is the objective function, and the set Ω is the feasible region The problem to finding the minimum solution(s) is called the global optimization problem.

).S( ⊂Ω

*x


Optimization Local optimums and the Global optimum

Local optimums

Global optimum

and the others


Altough single-objective optimalization problem may have an unique optimal solution (global optimum).

Multicriterial optimalization

Multiobjective Optimalization Problem (MOPs) as a rule present a possibility of uncountable set of solutions, which when evaluated, produce vectors whose components represent trade-offs of objective space.

A decision maker (DM) then implicitly chooses an acceptable solution (or solutions) by selecting one or more of these vectors.


Multicriterial optimalization

f2

f1

= Best solutions + = ”Normal” solutions

objectivefunctionsF = [ f1, f2 ]


Multicriterial optimalizationThe Multiobjective Optimalization Problem also called multicriteria optimisation or vector optimisation problem can then be determined (in words) as a problem of finding a vector of decision variables which satisfies constraints and optimises a vector function whose elements represent the objective functions.

This functions form a mathematical description of performance criteria which are usually in conflict with each other.

Hence the term ”optimise” means finding such a solution which would give the values of all the objective functions acceptable to the decision maker.


Decision variablesThe decision variables are the numerical quantities for which values are to be chosen in an optimalization problem.


ConstraintsIn most optimalization problem there are always restrictions imposed by the particular characteristics of the environment or resources available (e. g. physical limitations, time restrictions, e.t.c. ).

These restrictions must be satisfied in order to consider that certain solution is acceptable.

All these restrictions in general are called constrains and they describe dependences among decision variables and contants (or parameters) involved in the problem.


ConstraintsThese constrains are expressed in form of mathematical inequalities:

where p < n

and n is the size of decision vector


ConstraintsThe number p of equality constrains, must be less than n, the number of decision variables, because if p >= n the problem is said to be overconstrained, since there are no degrees of freedom left for optimizing (more unknowns than equations).

The number of degrees of freedom is given by (n – p).

Also constrains can be explicit (i.e. given in algebraic form) or implicit in which case the algorithm to compute gi (x) for any given vector x must be known.


Objective FunctionsIn order to know ”how good” a certain solution is, it is nessesary to have some criteria to evaluate it. (For example the profit, the number of employee, etc.)

These criteria are expressed as computable functions of the decision variables, that are called objective functions.

In real word problems, some of them in conflict with others, and some have to be minimized while the others are maximized.

These objective functions may be commensurable (measured in the same unit) or non-commensurable (measured in different units).


Types of Multicriterial Optimization Problem

In multiobjective optimization problems, there are three possible situations:

•Minimize all objective functions

•Maximize all objective functions

•Minimize some and maximize others


Objective Functions

The multiple objectives being optimized almost always conflict, placing a partial, rather than total, ordering on the search space.

In fact finding the global optimum of a general MOP is NP-Complete (Bäck 1996).


Attributes, Critereia, Objectives and Goals

Attributes: are often thought of as differentiating aspects, properties or characteristics or alternatives or consequences.

Criteria: generally denote evaluative measures, dimensions or scales against which alternatives may be gauged in a value or worth sence.

Objectives: are sometimes viewed in the same way, but also denote specific desired levels of attainment or vague ideals.

Goals: usually indicate either of the latter notations.A distiction commonly made in Operation Research is to use the term goal to designate potentially attainable levels, and objectives to designate unattainable ideas.


Attributes, Critereia, Objectives and Goals

The convention adopted in this presentation is the same assummed by several researcher Horn (1997), Fishburn (1978) of using the terms objective, criteria, and attribute interchangeably to represent an MOP’s goal or objectives (i.e. distinct mathematical functions) to be achived.

The terms objective space or objective function space are also used to denote the coordinate space within which vectors resulting from evaluating an MOP are plotted.


Objectives Functions


Euclidean space

The set of all n-tuples of real numbers denoted by Rn is called Euclidean n-space.

Two Euclidean spaces are considered:

•The n-dimensional space of decision variables in which each coordinate axis corresponds to a component of vector x.

•The k-dimensional space of objective functions in which each coordinate axis corresponds to a component of vector f(x).


Euclidean space

Every point in the first space (decision variables ) represents a solution and gives a certain point in the second space (objective functions ), which determines a quality of solution in term of the values of the objective functions.


Euclidean space


General Multicriterial Optimization Problem


Convert of Multicriterial Optimization Problem

For simplicity reasons, normally all functions are converted to a maximization or minimization form.

For example, the following identity may be used to convert all functions which are to be maximized into a form which allows their minimalization:

)x(fmax)x(fmin −=


Convert of Multicriterial Optimization Problem

Similarity, of the inequality constrains of the form

can be converted to (1.8) form by multiplying by –1 and changing the sign of the inequality. Thus, the previous equation is equivalent to

i = 1, 2, .... , m 0)x(gi ≤

i = 1, 2, .... , m 0)x(gi ≥−


Multicriterial Optimization ProblemIdeal Solution


Multicriterial Optimization ProblemIdeal Solution

)x(f 2

)x(f 1

)x(f *1

)x(f *2

Singex* solution

vector

Singleoptimal solution

Figure 1.1


Multicriterial Optimization ProblemIdeal Vector


Multicriterial Optimization ProblemConvexity


Multicriterial Optimization ProblemConvex Sets


Multicriterial Optimization ProblemNon-convex Sets


Multicriterial Optimization ProblemPareto Optimality



In words this definition says that is Pareto optimal if there is exists no feasible vector which would decrese some criterion without causing a simultaneous increase in the last one other criterion.

The phrase ”Pareto optimal” is considered to mean which respect to the entire decision variable space unless otherwise specified.



f2

f1

= Pareto Optimal Set + = ”Normal” solutions



Multicriterial Optimization ProblemPareto Front

The minima in the Pareto sence are going to the boundary of the design region, or in the locus of the tangent points of the objective functions.

In the Figure 1.6 a bold curve is used to mark the boundary for a bi-objective problem.The region of the points defined by this bold curve is called the Pareto front.


f2

f1

F

Multicriterial Optimization ProblemPareto Front


Pareto Front


Multicriterial Optimization ProblemGlobal Optimization

Defining an MOP global optimum is not a trivial task as the ”best” compromise solution is really dependent on the specific preferences (or biases) of the (human) decision maker.

Solutions may also have some temporal dependences (e.g. acceptable resource expeditures may vary from month to month).

Thus, there is no universally accepted definition for the MOP global optimization problem.

(But there are implemented more and more individual solutions...)


General Optimization AlgorithmsOverview

Genaral search and optimization techniques are classified into three categories: enumarative, detereministic and stochastic (random). (Figure 1.11 on next page)

As many real-world problems are computationally intensive, some means of limiting the search space must be implemented to find ”acceptable solutions” in ”acceptable time” (Mihalewicz and Fogel 2000)

Deterministic algorithms attempt this by incorporating problem domain knowledge. Many of graph/tree search algorithms are known and applied.


General Optimization AlgorithmsOverview


General Optimization AlgorithmsGenetic Algorithm


1.25 4.67 5.78 98 1.98 3.45…. ….. ….. …. ….. …..…. ….. ….. …. ….. …..…. ….. ….. …. ….. …..…. ….. ….. …. ….. …..

Chromosome- (Floating point Coding)G

ener

áció

Individual

Objective Values

Fitn

ess V

alue

s…. ….. ….. …. ….. …..

…. ….. ….. …. ….. …..

Selection

General Optimization Algorithms Genetic Algorithm


1.00 4.12 5.10 12.5 1.99 4.15

Crossover

1.25 4.67 5.78 98 1.98 3.45

0.15 3.32 1.83 7.54 2.00 6.12Parents

CROSSOVER

0.78 3.65 2.61 34.5 1.98 5.12Children

Mutation



Mutation

Crossover

Parents

Variable_1

Var

iabl

e_2

Children


Decision Variable Space


Objective 1

Obj

ectiv

e 2

Rank = 1Rank = 2Rank = 3

Local Pareto Front

Pareto Front

MOGA Optimization Algorithms Genetic Algorithm

Objective Function Space


Nonlinear Fitness Assignment

Rank

(Dum

my)

Fitn

ess

RankMAX



Obj

ectiv

e_2

MOGA operation (theoretically)

Objective_1Pareto Front



Genetic Drift (real operation of MOGA)

Objective_1

Obj

ectiv

e_2

Pareto Front



Objective_1

Obj

ectiv

e_2

Genetic Drift Break with Fitness Correction

Normalization

[0,1]x[0,1]

∆2

∆1



1

1

0

distance

1Niche count

σshare

Σ(Niche Count) = 1/Wi



Calculation of Fitness Correction Factors



Genetic Drift Break with Fitness Correction

Rank Values

Fitn

ess

0 0 01 12 2 23 3 3



Examples:

( )55

21

21 10x10 where

2x)x(f

x)x(f≤≤−

−=

=

( ) ( )

1x,y0

where x8sin

y101x

y101x1y101)y,x(f

x)y,x(f2

2

1

≤≤

+

−

+

−+=

=

π

MOP-1

MOP-2

MOGA Optimization Examples

MOP-1normal

MOP-1 with Drift Break

MOP-2normal MOP-2 with Drift Break


( )( )

conditions

y,x)ycos(5.0)ysin(2)xcos()xsin(5.1B)ycos(5.1)ysin()xcos(2)xsin(5.0B

)2cos(5.0)2sin(2)1cos()1sin(5.1A)2cos(5.1)2sin()1cos(2)1sin(5.0A

where)1y()3x(1)y,x(f

)BA()BA(1)y,x(f

2

1

2

1

222

222

2111

≤≤−−+−=−++=−+−=−++=

++++−=

−+−+−=

ππ

MOP 3




4x4

and 2 1,i where

n

1xexp1)x(f

n1xexp1)x(f

i

n

1i

2

ii2

n

1i

2

ii1

≤≤−

=

+−−=

−−−=

∑

∑

=

=

MOP 4




Decision Maker

Mathematically, every Pareto optimal point is an equally acceptable solution of the multiobjective optimalization problem. However, it is generally desirable to obtain one point as a solution. Selecting one of the set of Pareto optimal solutions call for information that is not contained in the objective function. That is why – compared to single objective optimalization - a new element is added in multiobjective optimalization.


Decision Maker

We need a decision maker to make the selection. The decision maker is a person (or a group of persons) who is supposed to have better insight into the problem and who can express preference repations between different solutions. Usually, the decision maker is responsible for the final solution.


Decision Maker

Solving a multiobjective optimalization problem calls for the co-operation of the decision maker and an analyst. By an analyst we have mean a person or a computer program responsibile for mathematical side of the solution process. The analyst generates information for the decisition maker to consider and the solution is selected according to the preferences of the decision maker.


Decision Maker

It is assummedin the following that we have a single decision maker or an unanimous group of decision makers. Generally, group decision making is a world of its own. It calls for negotiations and specific methods when searching for compromises between different interest groups.


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multicriterial optimization using genetic...

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