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Introduction to Multiobjective

Optimization

Jussi Hakanen

jussi.hakanen@jyu.fi

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Contents

Multiple Criteria Decision Making (MCDM)

Formulation of a multiobjective problem

On solving multiobjective problems

Basic concepts and definitions

Optimality in multiobjective optimization

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What means multiobjective?

Consider several criteria simultaneously

Criteria are conflicting (e.g. usually good quality is not chea) → all the criteria can not be optimized simultaneously

Need for considering compromises between the criteria

Compromise can be better than optimal solution in practice (cf. optimize only costs/profit)

http://en.wikipedia.org/wiki/Multiobjective_optimization

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Multiobjective decision making

process

Need for

optimization

Optimization &

decision making

Optimization problem

formulation Modelling of the

problem (numerical

simulation)

Implementation &

testing of the best

solution found

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Optimization problem formulation

By optimizing only one criterion, the rest are not considered

Objective vs. constraint

Summation of the objectives

– adding apples and oranges

Converting the objectives (e.g. as costs)

– not easy, includes uncertaintes

Multiobjective formulation reveals interdependences between the objectives

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Example 1: Continuous casting

of steel

Optimal control of the secondary cooling of

continuous casting of steel

Long history of research in the Dept. of

Mathematical Information Technology,

Univ. of Jyväskylä

– modelling (1988)

– single objective optimization (1988-1994)

– multiobjective optimization (1994-1998)

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Continuous casting of steel

Liquid steel enters (tundish)

Initial cooling by a water cooled mold → thin crust

Movement supported by rollers

Secondary cooling by water sprays

Rest of the cooling by radiation

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Continuous casting of steel

Measuring temperature in casting difficult →

numerical temperature distribution

Process modelled as a multiphase heat

equation (PDEs, solid & liquid phase) →

temperature distribution

Numerical model by using the finite element

method (FEM)

Dynamic process

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Continuous casting of steel

Secondary cooling significant: intensity of sprays (easy to control) affects significantly to the solidification of steel

Goal: minimize the amount of defects in steel

Quality depends on e.g. the temperature distribution at the surface of steel

– too slow cooling → too long liquid part

– too fast cooling → defects appear

Objective function: keep the surface temperature as close to a given profile as possible

Constraints e.g. for the change of temperature and for the temperature in critical spots

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Continuous casting of steel

Analysis of single objective optimization problem:

– constraints tight (based on metallurgy)

– no feasible solutions

– which constraints to relax?

Convert constraints as objective functions (5 in total)

– enables simultaneous relaxation of different constraints

– information on satisfaction of different constraints and their interdependences

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Example 2: Water allocation

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Water allocation

Papermaking process consumes lots of water (nowadays about 5-10 m3/ton of paper)

Water can be circulated and reused in different parts of the process as long as it remains fresh enough

– dissolved organic material accumulates

Fresh water costs

Process was modelles with the Balas process simulator (http://balas.vtt.fi/)

How to formulate the optimization problem?

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Water allocation

Goal is to minimize the amount of fresh water required for the process

Objective function: minimize the amount of fresh water

Constraints

– the amount of dissolved organic material in the white water of the papermachine

– the amount of dissolved organic material in the pulp entering bleaching

Variables: 5 splitters ja 3 valves

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Water allocation In practice

– set upper bounds for the amounts of organic material

– minimize the amount of fresh water used (one objective function)

How to set the upper bounds?

– based on engineering knowledge and current technology

– what if the bounds could be relaxed a bit?

Multiobjective formulation where the constraints would also be objectives (3)

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Example 3: Chemical separation

process Consider a chemical separation process based

on chromatography

Applied to many important separationsin sugar,

petrochemical,and pharmaceutical industries

* http://www.pharmaceutical-technology.com

Utilizes the difference

in the migration speeds

of different chemical

components in liquid

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5. Recover 2nd product 4. Recover 1st product 2. Feed

Desorbent Feed (Mixture of

two components)

1. Initial state

Column is filled with desorbent

3. Elution

Chromatography (Single Column)

Chromatographic Column (Vessel packed with adsorbent particles)

Pump

Adapted from Y. Kawajiri, Carnegie Mellon University

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November 11, 2009 Bergische Universität Wuppertal

Process simulation

Cycle

Step

Liquid Flow

FeedDesorbent

Extract Raffinate

1

Liquid Flow

FeedDesorbent

Extract Raffinate

2

Liquid Flow

FeedDesorbent

Extract Raffinate

3

Liquid Flow

FeedDesorbent

Extract Raffinate

4

Liquid Flow

FeedDesorbent

Extract Raffinate

5

Liquid Flow

FeedDesorbent

Extract Raffinate

6

Liquid Flow

FeedDesorbent

ExtractRaffinate

7

Liquid Flow

FeedDesorbent

ExtractRaffinate

8

Liquid Flow

FeedDesorbent

ExtractRaffinate

9

Liquid Flow

FeedDesorbent

ExtractRaffinate

10

Liquid Flow

Feed Desorbent

ExtractRaffinate

11

Liquid Flow

Feed Desorbent

ExtractRaffinate

12

Liquid Flow

Feed Desorbent

ExtractRaffinate

13

Liquid Flow

Feed Desorbent

ExtractRaffinate

14

Liquid Flow

Feed Desorbent

Extract Raffinate

15

Liquid Flow

Feed Desorbent

Extract Raffinate

16

Adapted from Y. Kawajiri, Carnegie Mellon University

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Chemical separation process

Two inlet and two outlet streams are switched in the direction of the liquid flow at a regular interval (steptime)

Operating variables

– Switching interval (Step Time)

– Liquid velocities

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Chemical separation process

Typically a profit function is optimized

Formulation of a profit is not easy

Multiobjective formulation

– max throughput

– min desorbent consumption

– max purity of the product

– max recovery of the product

Enables more flexible consideration and reveals how different objectives affect the solution

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Multiobjective optimization

(MOO) problem

Multiple objective functions, number denoted by k ( k > 1)

– special case: two objectives

– Objective vectors can be visualized when k = 2, 3

Variables: values change the solution

Constraints: same as in single objective problems

Feasible region S: consists of all the points satisfying the constraints

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Mathematical formulation

• Vector valued objective function

• Objective vector

• Image of the feasible region

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Optimality: objective space

f1

min

f2

min Z = f(S)

Best values are located down and left

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Optimality for multiple

objectives

When objectives to be optimized are

conflicting → no single optimal solution

– cf. single objective optimization

Compromise

There are potentially infinitely many optimal

solutions

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Optimality

Which solutions can be ’’optimal’’?

How to find such solutions?

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Optimality: objective space

f1

min

f2

min Z = f(S)

Best values are located down and left

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Pareto optimality (PO)

Mathematical definition:

In other words: a vector is PO if no objective can be improved without impairing some other one

Note: PO solutions can not be compared mathematically without some additional information

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Pareto optimal set

Also known as the Pareto front

Consists of all the PO solutions

Usually presented in the objective space

– e.g. k=2 → PO set is a subset of two

dimensional space

f1

f2

f1

f2

f1

f2

discrete linear nonlinear

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What means solving a problem?

Find all PO solutions

– theoretical approach, not feasible in practice

Find an approximation for PO front

– approximation with good diversity (representatives in all parts of the PO set) and good spread (no similar solutions)

– can also be an approximation of some specific part of the front

Find a best compromise (PO solution)

– requires preferences from a DM

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How to choose ’’a best’’

compromise?

PO solutions can not be compared

mathematically without some additional

information

– cf. ordering vectors in a plane

There typically exist infinitely many PO

solutions for continuous problems

Additional information related to the

problem considered is needed

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Decision maker (DM)

Person(s) who is an expert in the

application area

Is able to express preferences related to

objectives

– e.g. is able to compare Pareto optimal

solutions

No need for expertize in optimization

Helps in finding a best compromise

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Weak Pareto optimality

Some objective can be

improved without worsening

others

PO solution is also weakly PO

PO solutions are better but more

difficult to compute than weakly

PO ones

Weakly PO solutions

f1

f2

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Ranges for the PO set

Ranges for the objective function values in the PO set provide information about achievable solutions

Ideal objective vector (best values)

– how good values can be obtained

Nadir objective vector (worst values)

– how bad values possibly have to be accepted

Usefull information in decision making

Are utilized also in some MOO methods

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Ideal objective vector

Consists of best values for each objective

when optimized independently

– smallest values for minimization

– k objective functions → optimization of k

single objective problems

Ideal objective vector is not feasible!

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Ideal objective vector

f1

min

f2

min Z = f(S)

z2*

z1*

z*

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Nadir objective vector

Consists of worst values for each objective

in the PO set

– largest values for minimization

Generally difficult to compute, need for

approximation

– Note: when k=2 → easier to compute

– e.g. by using a pay-off table

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Pay-off table

Is obtained by evaluating all the objective functions in the points where ideal values were obtained

– i:th row the values of objectives in a point where fi has it’s optimal value

– diagonal has z*

– zinad = the worst value of the i:th column

Can give either optimistic or pessimistic approximation for the Nadir objective vector (depends on the problem considered)

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Nadir objective vector

f1

min

f2

min Z = f(S)

z2nad

z1nad

znad

z2*

z1*

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Reference point

Reference point = a vector in the objective

space that contains desirable values for the

objectives

The components of a reference point are

called aspiration levels

One way for the DM to express preferences

(intuitive)

Utilized also in some MOO methods

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Special case: 2 objectives

PO set can be visualized (if available)

f1

min

f2

min

If one of the objectives is

improved, the other one

will impair

A desired solution can

be chosen from the

”curve”

Very common in

practical applications

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Scalarizing the problem

Often the idea of MOO methods is to some way convert the problem into single objective one

– methods of single objective optimization can be utilized

This is called scalarization

Can be done in a good way or in a bad way

– examples of scalarization will come in later lectures

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Properties of a good MOO

method

Methods based on scalarization produce

usually one solution at a time

Good method should have the following

properties

– produce (weakly) PO solutions

– is able to find any (weakly) PO solution

(by using suitable parameters of the

method)

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Approaches

Plenty of methods developed for MOO

MOO methods can de categorized based on the role of the DM

– No-preference methods (no DM)

– Aposteriori methods

– Apriori methods

– Interactive methods

The approaches are discussed more in the coming lectures – just an appetizer!

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Examples of MOO literature

V. Changkong & Y. Haimes, Multiobjective Decision Making: Theory and Methodology, 1983

Y. Sawaragi, H. Nakayama & T. Tanino, Theory of Multiobjective Optimization, 1985

R.E. Steuer, Multiple Criteria Optimization: Theory, Computation and Applications, 1986

K. Miettinen, Nonlinear Multiobjective Optimization, 1999

K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms, 2001

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Examples of MOO literature

M. Ehrgott, Multicriteria Optimization, 2005

J. Branke, K. Deb, K. Miettinen & R.

Slowinski (eds): Multiobjective Optimization:

Interactive and Evolutionary Approaches,

2008

G.P. Rangaiah (editor), Multi-Objective

Optimization: Techniques and Applications

in Chemical Engineering, 2009

E. Talbi, Metaheuristics: from Design to

Implementation, 2009

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http://www.mcdmsociety.org/

Newsletter

22nd International Conference on Multiple Criteria Decision Making

17-21 June 2013, Málaga (Spain)

11th MCDA/M Summer School 2013

Helmut-Schmidt-Universität, Hamburg, Germany, July 22nd– August 2nd, 2013

Membership does not cost you anything!

January 23-27, 2012 Dagstuhl Seminar on Learning in

Multiobjective Optimization