Multi-Objective Multi-Objective

ProgrammingProgrammingan overview with a focus on interactive an overview with a focus on interactive

approachesapproaches

Carlos Henggeler AntunesCarlos Henggeler AntunesDEEC – University of CoimbraDEEC – University of Coimbra

INESC Coimbra (www.inescc.pt)INESC Coimbra (www.inescc.pt)ch@deec.uc.ptch@deec.uc.pt

IntroductionIntroduction

Real-world problems are multicriteria in Real-world problems are multicriteria in nature …nature …

… … no single measure of what is best no single measure of what is best exists.exists.

The complexity of real-world problems in The complexity of real-world problems in modern technologically developed modern technologically developed societies is characterized by the presence societies is characterized by the presence of multiple criteria, reflecting economical, of multiple criteria, reflecting economical, social, political, physical, engineering, social, political, physical, engineering, administrative, psychological, ethical, administrative, psychological, ethical, aesthetical,... evaluation aspects in a aesthetical,... evaluation aspects in a given decision context.given decision context.

There is no feasible solution which There is no feasible solution which guarantees the best values in all guarantees the best values in all evaluations aspectsevaluations aspects

By considering explicitly the different By considering explicitly the different aspects of the reality aspects of the reality

• • mathematical models, and mathematical models, and

• • the DM’s perception of problems the DM’s perception of problems

become more realistic.become more realistic.

broadening the range of solutions broadening the range of solutions under analysis.under analysis.

Multicriteria problemsMulticriteria problems multiattributemultiattribute (enumerative definition): the potential (enumerative definition): the potential

courses of action, a finite number, are explicitly courses of action, a finite number, are explicitly known a-priori, as well as the corresponding indexes known a-priori, as well as the corresponding indexes of merit evaluated for the multiple criteria; of merit evaluated for the multiple criteria;

multiobjectivemultiobjective (analytical definition): the potential (analytical definition): the potential courses of action form a continuum, defined courses of action form a continuum, defined implicitly by a set of constraints;implicitly by a set of constraints;

The decision variable space is mapped into the The decision variable space is mapped into the objective function space, in which each objective function space, in which each alternative has a vector-valued representation, alternative has a vector-valued representation, whose components are the corresponding whose components are the corresponding values for each objective function (vector values for each objective function (vector optimization)optimization)

Decision making in these complex Decision making in these complex problems cannot be reduced to the search problems cannot be reduced to the search for an for an optimal solutionoptimal solution of a single objective of a single objective function (e.g., some type of economic function (e.g., some type of economic indicator). indicator).

Optimal solutionOptimal solution (best feasible value for a (best feasible value for a single objective function) single objective function) Nondominated Nondominated solution solution (there is no other feasible solution (there is no other feasible solution which improves simultaneously all the which improves simultaneously all the objectives – the improvement in an objectives – the improvement in an objective function value is obtained by objective function value is obtained by accepting to degrade the value of at least accepting to degrade the value of at least one of the other objectives)one of the other objectives)

Since there is no point which optimizes Since there is no point which optimizes simultaneously all the objective functions, simultaneously all the objective functions, the simple comparison between the simple comparison between nondominated solutions does not provide nondominated solutions does not provide information in the search of a information in the search of a nondominated solution which constitutes nondominated solution which constitutes the final solution to the multiobjective the final solution to the multiobjective problem.problem.

Two nondominated solutions are Two nondominated solutions are incomparable using the natural order ≥.incomparable using the natural order ≥.

Some ordering of the set of nondominated Some ordering of the set of nondominated solutions underlies the intervention of asolutions underlies the intervention of a

preference relationpreference relation reflecting the DM’s preference structurereflecting the DM’s preference structure

In a single objective optimization problem the In a single objective optimization problem the serach for the optimal solution is purely serach for the optimal solution is purely technical:technical:

• • the best solution is implicit in the the best solution is implicit in the mathematical model,mathematical model,

• • the role of the optimization algorithm is the role of the optimization algorithm is to discover it, to discover it,

• • there is no place for decision making.there is no place for decision making.

In a multiobjective problem it is necessary to In a multiobjective problem it is necessary to make intervene in the search processmake intervene in the search process

• • technical devices to compute technical devices to compute nondominated solutionnondominated solution

• • information on the DM’s preferences.information on the DM’s preferences.

The DM’s preference structure embodies a set The DM’s preference structure embodies a set of opinions, values, convictions and of opinions, values, convictions and perspectives of the reality, configuring a perspectives of the reality, configuring a personal model of the reality the DM leans on to personal model of the reality the DM leans on to evaluate different potential courses of actionevaluate different potential courses of action

It is not possible to classify a solution as good It is not possible to classify a solution as good or bad just with reference to the mathematical or bad just with reference to the mathematical model and the resolution techniques :model and the resolution techniques :

• • the quality of a solution is influenced by the quality of a solution is influenced by organizational, political, cultural, ..., aspects, organizational, political, cultural, ..., aspects, underlying the decision process.underlying the decision process.

Multiobjective problem:Multiobjective problem:- choice of a solution, among the set of - choice of a solution, among the set of

nondominated solutions (generally non-nondominated solutions (generally non-countable), which constitutes an acceptable countable), which constitutes an acceptable compromise solution for the DM, having in mind compromise solution for the DM, having in mind his/her preferences, which can evolve his/her preferences, which can evolve throughout the decision support process.throughout the decision support process.

The decision process is a dynamic entity constituted by iterative The decision process is a dynamic entity constituted by iterative cycles ofcycles of

• • generation of potential actions, generation of potential actions, • • evaluation,evaluation,• • interpretation of information,interpretation of information,• • value changes,value changes,• • learning, and learning, and • • preference adaptation.preference adaptation.

The consideration of multiobjective modelsThe consideration of multiobjective models- reflect better a complex and ill-structured reality,- reflect better a complex and ill-structured reality,- enables the exploration of a wider range of alternative - enables the exploration of a wider range of alternative

solutions.solutions.

The criteria heterogeneity arises specific problem resulting from :The criteria heterogeneity arises specific problem resulting from :- conflicts among criteria, since a feasible solution that - conflicts among criteria, since a feasible solution that

optimizes all the objective functions simultaneously does not exist;optimizes all the objective functions simultaneously does not exist;- incommensurable criteria, which cannot be reduced to a - incommensurable criteria, which cannot be reduced to a

common measuring unit (e.g., monetary);common measuring unit (e.g., monetary);- uncertainty, due to the insufficient and/or incomplete - uncertainty, due to the insufficient and/or incomplete

nature of knowledge and the information on the DM’s preferences nature of knowledge and the information on the DM’s preferences in a multidimensional reality.in a multidimensional reality.

Mathematical modeling and optimization techniques are Mathematical modeling and optimization techniques are adequate for the computation of nondominated adequate for the computation of nondominated solutions.solutions.

It is necessary to incorporate into the decision process It is necessary to incorporate into the decision process qualitative aspects related with the DM’s preferences qualitative aspects related with the DM’s preferences and subjective judgments. and subjective judgments.

A good decision A good decision No other potential action exists that is better in some aspects No other potential action exists that is better in some aspects

and not worse in all aspects under considerationand not worse in all aspects under consideration A final proposal must be selected within the universe of A final proposal must be selected within the universe of

nondominated solutionsnondominated solutions Need to establish balances and compromises among the Need to establish balances and compromises among the

objectives objectives

Compromise solutionCompromise solution• • reach satisfactory goals for the DMreach satisfactory goals for the DM

satisfactory compromise solution.satisfactory compromise solution.• • maximize a value function,maximize a value function,

the best decision is the one that gives the the best decision is the one that gives the best valuebest value

Satisfactory solution:Satisfactory solution: cognitive capabilities limitations of Human Beings cognitive capabilities limitations of Human Beings "bounded rationality", "bounded rationality",

• • "satisficing rationality""satisficing rationality" instead of instead of

• • "optimizing rationality". "optimizing rationality".

This is not necessarily a Human fault in decision This is not necessarily a Human fault in decision situations that must be “corrected”, situations that must be “corrected”,

• • it is often a form of intelligence that must be it is often a form of intelligence that must be refined, and not ignored, by decision aid methodologiesrefined, and not ignored, by decision aid methodologies

DM’s active roleDM’s active role- the information is not given to the DM without - the information is not given to the DM without

requiring him/her any intervention,requiring him/her any intervention,- obliges him/her to obtain the information by - obliges him/her to obtain the information by

means of an iterative process of exploration, means of an iterative process of exploration, observation, pattern recognition. observation, pattern recognition.

Multiobjective decision aid methodsMultiobjective decision aid methods• • No DM’s preference articulation (generating No DM’s preference articulation (generating

methods).methods).• • DM’s preference articulation is made:DM’s preference articulation is made:

* a-priori (value/utility function * a-priori (value/utility function methods,)methods,)

* progressively (interactive methods)* progressively (interactive methods)

Interactive methodsInteractive methods• • computation phases, computation phases, • • dialogue phases:dialogue phases:

in which the DM is asked to express his/her in which the DM is asked to express his/her preferences in face of the solutions which are preferences in face of the solutions which are proposed to him/her, until a stop condition is proposed to him/her, until a stop condition is fulfilled (depending on the method);fulfilled (depending on the method);

The DM’s intervention is used to guide the The DM’s intervention is used to guide the interactive decision process, reducing the scope of interactive decision process, reducing the scope of the searchthe search

Interactive methodsInteractive methods

In face of generating methodsIn face of generating methods- reduce the computer burden, - reduce the computer burden, - limit the cognitive effort imposed on the DM.- limit the cognitive effort imposed on the DM.

In face of value function-based methodsIn face of value function-based methods- avoid the prior preference information elicitation to - avoid the prior preference information elicitation to

formulate the value functionformulate the value function- enable learning and preference evolution as a more - enable learning and preference evolution as a more

information is being gathered.information is being gathered.

InteractivityInteractivity• • offer the DM an operational environment facilitating offer the DM an operational environment facilitating

exploration, reflection, emergence of new intuitions, exploration, reflection, emergence of new intuitions, • • enable the DM to understand more in-depth the enable the DM to understand more in-depth the

decision problem at hand,decision problem at hand,• • contribute to shape and evolve the DM’s preferences contribute to shape and evolve the DM’s preferences

to guiding the search process,to guiding the search process,• • focus the search process in the regions where more focus the search process in the regions where more

interesting decision are located. interesting decision are located.

LearningLearning• • increasing the available knowledge, increasing the available knowledge, • • improving the DM’s capabilities to make an improving the DM’s capabilities to make an

adequate use of that knowledge.adequate use of that knowledge.

Trend: importance shifting fromTrend: importance shifting from- making decisions (MCDM - "Multiple Criteria - making decisions (MCDM - "Multiple Criteria

Decision Making")Decision Making")- aiding decisions to be made (MCDA - "Multiple - aiding decisions to be made (MCDA - "Multiple

Criteria Decision Aid"). Criteria Decision Aid").

MethodsMethods- aid the DM by providing him/her better - aid the DM by providing him/her better

information quality, information quality, - offering the possibility of evolution of his/her - offering the possibility of evolution of his/her

preference structure by confronting it with the preference structure by confronting it with the proposals generated in the computation phases in order proposals generated in the computation phases in order to increase his/her understanding of the problemto increase his/her understanding of the problem

Satisfactory compromise solution Satisfactory compromise solution

• • nondominated solution, nondominated solution,

• • associated with a given associated with a given compromise between the objectives, compromise between the objectives,

• • objectives assume satisfactory objectives assume satisfactory values for the DM values for the DM

in such a way that this solution can be accepted in such a way that this solution can be accepted as the final solution of the decision process.as the final solution of the decision process.

What is the meaning of a final solution?What is the meaning of a final solution?

- the results of the analysis shall be used not a - the results of the analysis shall be used not a definitive prescription but rather as a reference definitive prescription but rather as a reference or material support to make decisions in the or material support to make decisions in the sense of finding better actions plans for the sense of finding better actions plans for the systemsystem. .

Formulation and Formulation and DefinitionsDefinitions

Linear programming problem with multiple objective Linear programming problem with multiple objective functionsfunctions

max fmax f11((xx) = ) = cc11 xx

max fmax f22((xx) = ) = cc22 xx....................................

max fmax fpp((xx) = ) = ccpp xx

s. to s. to xx X X{{xx nn : : xx ≥≥ 00, A, Axx==bb, , bb mm}}

"Max" "Max" ff ( (xx) = C ) = C xxs. to s. to xx X X

C = matrix of objective function coefficients; the rows are the C = matrix of objective function coefficients; the rows are the vectors vectors cckk (coefficients of objective function f (coefficients of objective function fkk). ).

A = matrix of technological coefficients (mxn) A = matrix of technological coefficients (mxn) bb = RHS vector (available resources or requirements imposed) = RHS vector (available resources or requirements imposed)

In single objective optimization the decision In single objective optimization the decision space space xx X is mapped into X is mapped into

In MOP the decision space is mapped into a p-In MOP the decision space is mapped into a p-dimensional objective function spacedimensional objective function space

F={F={ff ( (xx) ) pp : : xx X} X} Each potential alternative Each potential alternative xx X has as X has as

representation a vector representation a vector ff((xx)=(f)=(f11((xx),f),f22((xx),...,f),...,fpp((xx)) )) the components of which are the values of each the components of which are the values of each objective function at that point of the feasible objective function at that point of the feasible region.region.

In general, there is no feasible solution In general, there is no feasible solution xx X that X that optimizes simultaneously all objective functions.optimizes simultaneously all objective functions.

From an operational perspective, "Max" denotes From an operational perspective, "Max" denotes the operation of determining nondominated the operation of determining nondominated solutions.solutions.

Optimal solution Optimal solution efficient / nondominated efficient / nondominated solutionsolution

A feasible solution to a MOP problem is said A feasible solution to a MOP problem is said efficient iff no other feasible solution exists that efficient iff no other feasible solution exists that improves the value of an objective function improves the value of an objective function without worsening the value of, at least, without worsening the value of, at least, another objective function.another objective function.

The set of efficient solutions (Pareto optimal) is The set of efficient solutions (Pareto optimal) is defined by defined by

XXEE = { = { xx X | X | xx' ' X : X : ff ( (xx') ') ≥≥ ff ( (xx) }) }

where where ff((xx') ') ≥≥ ff((xx) iff ) iff ff((xx') ≥ ') ≥ ff((xx) and ) and ff((xx') ≠ ') ≠ ff((xx) ) (that is, f(that is, fkk((xx') ≥ f') ≥ fkk((xx) for all k and f) for all k and fkk((xx')>f')>fkk((xx) for ) for at least one k), and at least one k), and ff((xx') ≥ ') ≥ ff((xx) iff f) iff fkk((xx') ≥ f') ≥ fkk((xx), ), k=1,2,...,p k=1,2,...,p

The criterion vector The criterion vector zz==ff ( (xx) is ) is nondominated (non-inferior) when nondominated (non-inferior) when xx X XEE::

FFEE = { = {zz==ff ( (xx) ) F : F : xx X XEE } }

Objective function space: nondominated Objective function space: nondominated solutionssolutions

Decision variable space: efficient Decision variable space: efficient solutionssolutions

The image of an efficient solution is a The image of an efficient solution is a nondominated solutionnondominated solution

x1

xj

xn

x'

f1

fk

fp

z'=(f 1(x' ),..., fk(x' ),..., fp(x' ))

A feasible solution to a MOP problem is A feasible solution to a MOP problem is weakly efficient iff there is no other weakly efficient iff there is no other feasible solution which improves strictly feasible solution which improves strictly the value of all objective functions.the value of all objective functions.

The set of weakly efficient solutions The set of weakly efficient solutions

XXFEFE = { = { xx X | X | xx' ' X : X : ff ( (xx') ≥ ') ≥ ff ((xx) }) }

FFFEFE = { = { zz==ff ( (xx) ) F : F : xx X XFEFE } }

Properly efficient solution:Properly efficient solution:

- more restrict notion of efficient solution - more restrict notion of efficient solution to eliminate efficient solutions that to eliminate efficient solutions that present unbounded tradeoffs between present unbounded tradeoffs between the objectives, that is solutions for which the objectives, that is solutions for which the relation improvement / degradation the relation improvement / degradation between the objective function values between the objective function values can be made arbitrarily big. can be made arbitrarily big.

In MOLPIn MOLP X XPEPE X XEE . .

f1

f2

A B

C

D

EG

H

f1

f2A

B

C

(a) (b)

5

f2

f1

1

6

4

3

27

optimal solution to f1

optimal solution to f2

convex dominated by (3,4)

8

dominated by 7 and 4

convex dominated by (4,2)

ideal solution

0

9

dominated by 3

In MILP, nondominated solutions located in the interior of the convex hull (i.e., those which are not vertices) are dominated by a convex combination of vertex solutions (no supporting hyperplane). Generally called convex dominated solutions or unsupported (nondominated) solutions.

Ideal solutionIdeal solution (utopia point) (utopia point) zz* * would optimize simultaneously all objective would optimize simultaneously all objective

functions functions its components are the optimum of its components are the optimum of each objective function in the feasible region, each objective function in the feasible region, when optimized individually. when optimized individually.

In general, the ideal solution does not belong to In general, the ideal solution does not belong to the feasible region, but each zthe feasible region, but each zk* is individually * is individually reachable.reachable.

The ideal solution is sometimes used as the The ideal solution is sometimes used as the (unreachable) DM’s reference point in (unreachable) DM’s reference point in scalarizing scalarizing functionsfunctions representing a distance to be minimized representing a distance to be minimized to determine a compromise efficient solution. to determine a compromise efficient solution.

Although the ideal solution Although the ideal solution zz* can always be * can always be defined in the objective space, its image in the defined in the objective space, its image in the decision variable space not always exist decision variable space not always exist xx* may * may not exist such that not exist such that zz*=*=ff((xx*) *)

Table of individual optimaTable of individual optima ("pay-off") ("pay-off") - organizes the objective values for each nondominated - organizes the objective values for each nondominated

solution resulting from the individual optimization of each solution resulting from the individual optimization of each objective function in the feasible region X. objective function in the feasible region X.

The ideal solution components are obtained in the diagonal of The ideal solution components are obtained in the diagonal of the table of individual optima.the table of individual optima.

From the table of individual optima the anti-ideal solution From the table of individual optima the anti-ideal solution (nadir point) can be obtained selecting in each row the worst (nadir point) can be obtained selecting in each row the worst value for the corresponding objective function value for the corresponding objective function it intends to it intends to represent the worst value of objective function frepresent the worst value of objective function fkk((xx) in the ) in the efficient region.efficient region.

This is just a "convenient" minimum (due to being easily This is just a "convenient" minimum (due to being easily determined) which can be different (higher) from the actual determined) which can be different (higher) from the actual minimum in the efficient region. minimum in the efficient region.

The table of individual optima may not be uniquely defined if The table of individual optima may not be uniquely defined if alternative optimal solutions exist for any objective function. alternative optimal solutions exist for any objective function.

Also, the anti-ideal solution would not be uniquely defined. Also, the anti-ideal solution would not be uniquely defined. The ideal solution (objective space) is always uniquely The ideal solution (objective space) is always uniquely

defined. defined.

Interactive MethodsInteractive Methods Basic phases:Basic phases:

(a) Initialization – automatically establishing the initial (a) Initialization – automatically establishing the initial preference information, stopping parameter setting,, etc.; preference information, stopping parameter setting,, etc.;

(b) preparation – incorporation of preference (b) preparation – incorporation of preference information into the parameters of the surrogate scalar information into the parameters of the surrogate scalar function, which aggregates the multiple objectives in a single function, which aggregates the multiple objectives in a single dimension function to be used in the new computation dimension function to be used in the new computation phase;phase;

(c) computation – computing one, or more, efficient (c) computation – computing one, or more, efficient solutions through the optimization of a surrogate scalar solutions through the optimization of a surrogate scalar function, that will be subject to the DM’s evaluation; function, that will be subject to the DM’s evaluation;

(d) dialogue – solution presentation and expression of (d) dialogue – solution presentation and expression of the preference information by the DM in face of this solution; the preference information by the DM in face of this solution;

(e) stop – according to a given stopping rule, which (e) stop – according to a given stopping rule, which may be simply the DM to consider being satisfied with the may be simply the DM to consider being satisfied with the information gathered so far throughout the processinformation gathered so far throughout the process

Initialization

Preparation

Computation

Dialogue

Stop

Surrogate scalar functions (Surrogate scalar functions (scalarizingscalarizing functions) functions)• • aggregate in a single dimension the different aggregate in a single dimension the different

objectivesobjectives• • include preference information parametersinclude preference information parameters• • its optimal solution is an efficient solution to the its optimal solution is an efficient solution to the

MOPMOP

InteractivityInteractivity• • offer the DM the central role, leading the interactive offer the DM the central role, leading the interactive

decision processdecision process• • the method shall have active role in this mutual the method shall have active role in this mutual

convergenceconvergence- - dynamicdynamic dialogue (adjusted to the different dialogue (adjusted to the different

stages of the decision process),stages of the decision process),- - simplesimple (not demanding too much information (not demanding too much information

or information unnecessarily complex)or information unnecessarily complex)- - positivepositive (requiring information about what the (requiring information about what the

DM wants to improve)DM wants to improve)- - divergencedivergence mechanisms (enabling to explore mechanisms (enabling to explore

solutions in a given neighborhood).solutions in a given neighborhood).

Two basic (extreme) concepts for the role of interactive Two basic (extreme) concepts for the role of interactive mechanisms:mechanisms:• • Search-orientedSearch-oriented..

- assumes the existence of a pre-existing and stable - assumes the existence of a pre-existing and stable preference structure (e.g. Represented by an implicit value preference structure (e.g. Represented by an implicit value function)function)

- coherence with this function throughout the use of - coherence with this function throughout the use of the method, answering in a deterministic way to the the method, answering in a deterministic way to the questions of the interactive protocolquestions of the interactive protocol

- reasonable to impose the mathematical - reasonable to impose the mathematical convergence of the interactive methodconvergence of the interactive method

- convergence is guaranteed and controlled by the - convergence is guaranteed and controlled by the methodmethod

- the aim of the interaction is the search of a optimal - the aim of the interaction is the search of a optimal proposal (which does not depend on the evolution of the proposal (which does not depend on the evolution of the interactive process) in face of the preference structure.interactive process) in face of the preference structure.

The preference structure is thus “discovered” during the The preference structure is thus “discovered” during the process.process.

Learning-orientedLearning-oriented- no assumption of a stable and pre-existing - no assumption of a stable and pre-existing

preference structure with which the DM is always preference structure with which the DM is always consistentconsistent

- the DM’s preferences may be partially unstable - the DM’s preferences may be partially unstable and conflictingand conflicting

- the aim of the interaction is the preference - the aim of the interaction is the preference learning (clarification of what can be a good decision learning (clarification of what can be a good decision according to the DM’s perspective)according to the DM’s perspective)

- convergence is not guaranteed and it is controlled - convergence is not guaranteed and it is controlled by the DM: “psychological convergence" (meaning the by the DM: “psychological convergence" (meaning the identification of an efficient solution as satisfactory based identification of an efficient solution as satisfactory based on the information gathered so far). on the information gathered so far).

- the process ends with a satisfactory compromise - the process ends with a satisfactory compromise solution according to the available information and the solution according to the available information and the DM’s will when the emergence of new intuitions about DM’s will when the emergence of new intuitions about new search directions seems possible.new search directions seems possible.

Convergence must be the result of the interaction between Convergence must be the result of the interaction between the DM and the method. the DM and the method.

Division of the work between the DM and the Division of the work between the DM and the computercomputer

Differentiation of tasks by the capability of Differentiation of tasks by the capability of

- guiding the computer effort for the - guiding the computer effort for the regions where the solutions more in regions where the solutions more in accordance with the DM’s preference accordance with the DM’s preference structure are located,structure are located,

- accommodating path corrections due - accommodating path corrections due to the evolution of the preference structure to the evolution of the preference structure as the information gathered makes new as the information gathered makes new hints and intuitions to emerge for hints and intuitions to emerge for proceeding the search.proceeding the search.

The future of MOP is in its interactive The future of MOP is in its interactive application! (Steuer)application! (Steuer)

CriticismsCriticisms

- the DM’s preference structure is not - the DM’s preference structure is not generally founded on empirical generally founded on empirical investigation (French),investigation (French),

- too strong assumption: the DM’s - too strong assumption: the DM’s choices are always in accordance with an choices are always in accordance with an implicit value function,implicit value function,

- anchoring: the DM “anchors” on the - anchoring: the DM “anchors” on the first proposals, showing some difficulties first proposals, showing some difficulties in preferring other solutions,in preferring other solutions,

- local nature of the preference - local nature of the preference information required. information required.

Categorization of interactive methodsCategorization of interactive methods

• • Strategy for reducing the scope of the search (explicitly or Strategy for reducing the scope of the search (explicitly or implicitly):implicitly):

- reduction of the feasible region;- reduction of the feasible region;

- reduction of the weight (parametric) space;- reduction of the weight (parametric) space;

- contraction of the objective function gradient cone;- contraction of the objective function gradient cone;

- directional search.- directional search.

• • Type of surrogate scalar function:Type of surrogate scalar function:

- optimization of one of the objective functions considering - optimization of one of the objective functions considering the other functions as constraints;the other functions as constraints;

- weighted-sum of the objective functions;- weighted-sum of the objective functions;

- minimization of a distance to a reference point.- minimization of a distance to a reference point.

• • Possibility of the DM/user to request a given operation at any Possibility of the DM/user to request a given operation at any time of the interactive process or need to comply with a pre-time of the interactive process or need to comply with a pre-established sequence of computation and dialogue phases:established sequence of computation and dialogue phases:

- non-structured;- non-structured;

- structured. - structured.

These classificationsThese classifications

- are not mutually exclusive (there are - are not mutually exclusive (there are methods combining different strategies methods combining different strategies for reducing the scope of the search and for reducing the scope of the search and techniques for the computation of techniques for the computation of efficient solutions);efficient solutions);

- are not exhaustive: there are methods - are not exhaustive: there are methods diffcult to be classified (e.g., adaptations diffcult to be classified (e.g., adaptations of feasible direction algorithms or cutting of feasible direction algorithms or cutting planes in MILP or MNLP). planes in MILP or MNLP).

f1

A BC

DF

I

N

f2

ScalarizingScalarizing Processes Processes Transforming the MOP into an optimization problem of a Transforming the MOP into an optimization problem of a

surrogate scalar functionsurrogate scalar function- the optimal solution of which is an efficient - the optimal solution of which is an efficient

solution to the MOP,solution to the MOP,- includes information parameters of the DM’s - includes information parameters of the DM’s

preferences. preferences.

Objective properties:Objective properties:- generate efficient solutions only;- generate efficient solutions only;- be able to generate all efficient solutions;- be able to generate all efficient solutions;- be independent of non-efficient solutions.- be independent of non-efficient solutions.

Subjective properties :Subjective properties :- computer effort involved not too big; - computer effort involved not too big; - simple interpretation for the preference - simple interpretation for the preference

information parameters. information parameters.

Meaning of the scalarizing functions :Meaning of the scalarizing functions :

- mere technical device to aggregate - mere technical device to aggregate temporarily the multiple objectives and temporarily the multiple objectives and generate efficient solutions to be generate efficient solutions to be proposed to the DM (with no concern of proposed to the DM (with no concern of reflecting a true analytical expression of reflecting a true analytical expression of the DM’s preferences);the DM’s preferences);

vs.vs.

- analytical representation of the - analytical representation of the DM’s preferences.DM’s preferences.

MOLPMOLPmax fmax f11((xx) = ) = cc11 xx

max fmax f22((xx) = ) = cc22 xx

....................................

max fmax fpp((xx) = ) = ccpp xx

s. to s. to

xx X X { {xx nn : : xx ≥≥ 00, A, Axx==bb, , bb mm}}

"Max" "Max" ff ( (xx) = C ) = C xx

s. to s. to xx X X

Optimization of one of the objective Optimization of one of the objective functions considering the others as functions considering the others as constraintsconstraints

Surrogate function: one of the objective Surrogate function: one of the objective functions (the one to which more functions (the one to which more importance is assigned)importance is assigned)

Lower bounds on the other p-1 objectives: Lower bounds on the other p-1 objectives: constraints (minimum levels the DM is constraints (minimum levels the DM is willing to accept!)willing to accept!)

max fmax fii ( (xx) = ) = ccii xx

s. to fs. to fkk ( (xx) = ) = cckk xx ≥ ≥ eekkk=1,...,p , k≠ik=1,...,p , k≠i xx X X

x1

x2

A B

C

D

f1

f2

f1 e1

E

TheoremTheorem: If : If xx* * X is a single solution to the X is a single solution to the scalar problem for any i, then scalar problem for any i, then xx* is an efficient to * is an efficient to the MOLP.the MOLP.

Optimization of the surrogate functionOptimization of the surrogate function

• • efficient solution to the MOLP sinceefficient solution to the MOLP since

- the reduced feasible region is not empty - the reduced feasible region is not empty (which may not happen if the lower bounds (which may not happen if the lower bounds eekk are too severe)are too severe)

- no alternative optimal solutions exist for - no alternative optimal solutions exist for the selected objective function (in this case, just the selected objective function (in this case, just weakly efficient points are guaranteed).weakly efficient points are guaranteed).

Dual variable associated with the constraint Dual variable associated with the constraint corresponding to fcorresponding to fkk((xx))

- local compromise rate between f- local compromise rate between fii and f and fkk in in the optimal solution to the scalar problem. the optimal solution to the scalar problem.

Easy to be understood by DMs: capture the attitude of Easy to be understood by DMs: capture the attitude of assigning more importance to one of the objective assigning more importance to one of the objective functions accepting lower bounds for the other functions accepting lower bounds for the other objectives.objectives.

Choice of the function to be optimized may be difficult.Choice of the function to be optimized may be difficult. Setting the function to be optimized during the whole Setting the function to be optimized during the whole

decision aid process makes the method little flexible.decision aid process makes the method little flexible. Results are too dependent on the function selected.Results are too dependent on the function selected. Possible to obtain all points of the nondominated frontier, Possible to obtain all points of the nondominated frontier,

that is vertices and points lying on nondominated faces.that is vertices and points lying on nondominated faces. Preference information:Preference information:

• • inter-criteria informationinter-criteria information

- choice of the objective function to be - choice of the objective function to be optimized;optimized;

• • intra-criteria informationintra-criteria information

- imposing lower bounds on the other - imposing lower bounds on the other objective functions.objective functions.

Weighted-sum of objective functionsWeighted-sum of objective functionsmax { max { 11ff11((xx) + ) + 22ff22((xx) +...+ ) +...+ ppffpp((xx) }) }s. to s. to xx X X

00

Set of feasible weightsSet of feasible weights {{ : : pp, , kk=1, =1, kk ≥ 0, k=1,2,...,p} ≥ 0, k=1,2,...,p} 0 0 { { : : pp, , kk=1, =1, kk > 0, k=1,2,...,p} > 0, k=1,2,...,p}

(interior of (interior of )) Bilinear function defined on X Bilinear function defined on X xx . . By setting a set of weights By setting a set of weights weighted-sum weighted-sum

scalar linear function of the p objective scalar linear function of the p objective functions, to be optimized in X.functions, to be optimized in X.

TheoremTheorem: : xx* * X is a (properly) basic efficient X is a (properly) basic efficient solution to the MOLP iff it is an optimal solution to the MOLP iff it is an optimal solution to the weighted-sum scalar problem for solution to the weighted-sum scalar problem for a set of weights a set of weights * * 00..

Initial multiobjective simplex tableau:Initial multiobjective simplex tableau:AA || II || bb

- C- C || 00 || 00 w.r.t. to basis B it is transformed into:w.r.t. to basis B it is transformed into:

BB-1-1 N N || B B-1-1 || B B-1-1 bb

-C-CBB B B-1-1 N N | - C| - CNN C CBB B B-1-1 | C| CBB B B-1-1 bb

A = [ B , N] ; C = [ CA = [ B , N] ; C = [ CBB , C , CNN ]. ]. Reduced cost matrix w.r.t. to basis BReduced cost matrix w.r.t. to basis Baa is is

WWaa= C= CBB B B-1-1 N -C N -CNN.. BBaa is an efficient basis iff the system is an efficient basis iff the system

{{TT W Wa a ≥ ≥ 00, , } is consistent} is consistent. .

The graphical display of the set of weights The graphical display of the set of weights which which lead to a basic efficient solution can be obtained lead to a basic efficient solution can be obtained by decomposing the “weight space” (a p-1 by decomposing the “weight space” (a p-1 dimensional simplex in an Euclidean p dimensional simplex in an Euclidean p dimensional space) dimensional space) . .

Multiobjective simplex tableauMultiobjective simplex tableau

- basic efficient solution to the MOLP- basic efficient solution to the MOLP

- the corresponding - the corresponding is defined by is defined by T T W ≥ W ≥ 00 wwkjkj from the reduced cost matrix W: from the reduced cost matrix W:

- marginal rate of change of objective - marginal rate of change of objective function ffunction fkk((xx) due to the “production” of one unit ) due to the “production” of one unit of the nonbasic variable xof the nonbasic variable xjj. .

W column corresponding to an efficient nonbasic W column corresponding to an efficient nonbasic variable variable unit change trend of the objective unit change trend of the objective functions along the efficient edge.functions along the efficient edge.

Indifference regionIndifference region- set of weights corresponding to a basic - set of weights corresponding to a basic efficient solution (where {efficient solution (where {TTW≥W≥00, , } is } is consistent). consistent).

All the weight combinations in that region lead All the weight combinations in that region lead to the same (basic) efficient solution.to the same (basic) efficient solution.

Parametric (weight) diagram

Projection of the objective function space

Common frontier to 2 indifference regions Common frontier to 2 indifference regions the the respective basic efficient solutions are connected respective basic efficient solutions are connected by an efficient edge, corresponding to making by an efficient edge, corresponding to making basic a nonbasic efficient variable. basic a nonbasic efficient variable.

A point A point belongs to several indifference belongs to several indifference regions regions these regions correspond to efficient these regions correspond to efficient solutions located on the same face (which is solutions located on the same face (which is efficient if efficient if 00).).

Indifference region depend onIndifference region depend on- relative order of magnitude of the - relative order of magnitude of the

objective function values (relative gradient objective function values (relative gradient length),length),

- geometry of the feasible region.- geometry of the feasible region. Area of the indifference regionArea of the indifference region

- robustness measure against weight - robustness measure against weight changeschanges..

Easy to be explained to DMs (apparently!) Easy to be explained to DMs (apparently!) capturing the attitude of expressing the capturing the attitude of expressing the importance assigned to each objective function. importance assigned to each objective function.

Information difficult to be elicited from the DM Information difficult to be elicited from the DM (although apparently simple).(although apparently simple).

There is no guarantee that the solutions There is no guarantee that the solutions computed using a weighted sum scalar function computed using a weighted sum scalar function are in accordance with the preferences are in accordance with the preferences underlying the specification of the weights.underlying the specification of the weights.

It is possible to obtain vertices of the It is possible to obtain vertices of the nondominated frontier only.nondominated frontier only.

Preference information:Preference information:• • inter-criteria informationinter-criteria information

- weights (relative importance - weights (relative importance coefficients?!) for the objective functions. coefficients?!) for the objective functions.

Minimization of a distance to a reference pointMinimization of a distance to a reference point• • Efficient solution closer (according to a given Efficient solution closer (according to a given

metric) to the DM’s aspirations.metric) to the DM’s aspirations.

Using an Lp metric and considering the ideal solution Using an Lp metric and considering the ideal solution as the reference point: as the reference point:

min || min || zz* - * - ff((xx) ||) ||pp

s. to s. to xx X X or, with a weighted Lp metricor, with a weighted Lp metric

min min || || zz* - * - ff((xx) ||) ||pp

s. to s. to xx X X

p=1: all the deviations from the reference point are p=1: all the deviations from the reference point are taken into consideration in the direct proportion of its taken into consideration in the direct proportion of its magnitude.magnitude.

2<p<∞: bigger deviations have more importance as p 2<p<∞: bigger deviations have more importance as p increases,increases,

p=∞: only the biggest deviation is considered.p=∞: only the biggest deviation is considered.

Let Let xx00 be a solution to this problem. By be a solution to this problem. By considering the “worst case” (i.e., biggest considering the “worst case” (i.e., biggest difference between the vector difference between the vector zz* and * and ff((xx00) ) components according to the metric L∞) components according to the metric L∞)

min min maxmax [ z[ zkk* - f* - fkk((xx) ]) ]xx X k=1,...,p X k=1,...,p

has has xx00 as a solution iff as a solution iff xx00 and v and v00 are solutions are solutions

to the linear problem:to the linear problem: minmin vv

s. to s. to v ≥ zv ≥ zkk* - f* - fkk((xx)) k=1,...,pk=1,...,p

xx X X

v ≥ 0 v ≥ 0

(if the reference point (if the reference point zzrr does not satisfy does not satisfy zzrr≥f≥fkk((xx) ) in X, then v in X, then v ).).

Weighted L∞ metricWeighted L∞ metricmin max min max kk [ z [ zkk* - f* - fkk((xx) ]) ]xx X k=1,...,p X k=1,...,p

In MOLP surrogate scalar linear problems results using L1 or L∞ In MOLP surrogate scalar linear problems results using L1 or L∞ metrics.metrics.

To guarantee that the obtained solutions are efficient ones, and To guarantee that the obtained solutions are efficient ones, and not just weakly efficient, the (weighted) augmented Tchebycheff not just weakly efficient, the (weighted) augmented Tchebycheff metric can be used:metric can be used:

min { max min { max kk [z [zkk* - f* - fkk((xx) ] } + ) ] } + [ z [ zkk* - f* - fkk((xx) ]) ]xx X k=1,...,p X k=1,...,p

zzkk* = reference point (arbitrary, may be the ideal solution)* = reference point (arbitrary, may be the ideal solution) is a weighting vector. is a weighting vector.

The 2nd term is a perturbation, with The 2nd term is a perturbation, with >0 sufficiently small, to >0 sufficiently small, to ensure the efficiency of the solution.ensure the efficiency of the solution.

minmin v + v + [ z [ zkk* - f* - fkk((xx) ]) ] s. to v ≥ zs. to v ≥ zkk* - f* - fkk((xx) ) k=1,...,pk=1,...,p

xx X X

f1z1*

z2*z*

zz1z2

45º

f2

TheoremTheorem: If : If xx* * X is a solution to the augmented X is a solution to the augmented weighted Tchebycheff problem for any reference weighted Tchebycheff problem for any reference point, then point, then xx* is an efficient solution to the MOLP. * is an efficient solution to the MOLP.

Minimization of the “discomfort" of getting a Minimization of the “discomfort" of getting a compromise nondominated solution compromise nondominated solution zz00 rather rather than the ideal point (or other reference point) than the ideal point (or other reference point) zz*.*.

It is possible to obtain all points on the It is possible to obtain all points on the nondominated frontier (feasible region vertices nondominated frontier (feasible region vertices or faces).or faces).

Preference information:Preference information:• • intra-criteria informationintra-criteria information

- setting the reference point(s);- setting the reference point(s);• • inter-criteria information:inter-criteria information:

- weighting coefficients for the L∞ metric.- weighting coefficients for the L∞ metric.

Step Method (STEM)Step Method (STEM) Feasible region reduction. Feasible region reduction. Dialogue phase: quantities that the DM is willing to sacrifice Dialogue phase: quantities that the DM is willing to sacrifice

in the functions for which the objective values are already in the functions for which the objective values are already satisfactory in order to improve the remaining functions. satisfactory in order to improve the remaining functions.

Computation phase: minimizing a weighted Tchebycheff Computation phase: minimizing a weighted Tchebycheff distance to the ideal solution.distance to the ideal solution.

The compromise solution computed in each iteration by The compromise solution computed in each iteration by minimizing the weighted Tchebycheff distance to the ideal minimizing the weighted Tchebycheff distance to the ideal solution is presented to the DM.solution is presented to the DM.

If the objective function values are considered as satisfactory If the objective function values are considered as satisfactory the process ends.the process ends.

Otherwise, the DM is asked to specify the objective he/she is Otherwise, the DM is asked to specify the objective he/she is willing to relax, and in what amount, in order to improve the willing to relax, and in what amount, in order to improve the remaining objective functions. remaining objective functions.

Step 1Step 1

Individual optimization of each Individual optimization of each objective functionobjective function

Building the table of individual optima Building the table of individual optima (pay-off).(pay-off).

Step 2Step 2

““Calibration” of the weights to be used Calibration” of the weights to be used in the computation phase of iteration hin the computation phase of iteration h Higher weight for objectives with larger relative Higher weight for objectives with larger relative

variations.variations. Normalization factor (using L2 norm) of the Normalization factor (using L2 norm) of the

objective function gradientsobjective function gradients..

Set S:Set S:

- indices of the objective functions that will be - indices of the objective functions that will be relaxed in the next iteration (according to the relaxed in the next iteration (according to the DM’s indications, by considering their values as DM’s indications, by considering their values as satisfactory).satisfactory).

At the beginning S=ø and X(1)At the beginning S=ø and X(1)X.X.

Step 3Step 3

- Weights which define the weighted - Weights which define the weighted metric L∞metric L∞

- The weights of the objective functions - The weights of the objective functions whose values are considered satisfactory (that whose values are considered satisfactory (that will be permitted to be relaxed) are zero. will be permitted to be relaxed) are zero.

Step 4Step 4Computation phase: solving the LP minimization of the Computation phase: solving the LP minimization of the weighted Tchebycheff distance to the ideal solution:weighted Tchebycheff distance to the ideal solution:

minmin vvs. tos. to v ≥ av ≥ a 1 ≤ k ≤ p1 ≤ k ≤ p

xx X X(h)(h)

v ≥ 0v ≥ 0

Dialogue phase: the solution Dialogue phase: the solution zz(h) = (h) = ff ( (xx(h)(h)) resulting from ) resulting from solving the problem in iteration h is presented to the DM:solving the problem in iteration h is presented to the DM:

xx(h)(h) is the point of the reduced feasible region X is the point of the reduced feasible region X(h)(h) closer to closer to zz* according to the weighted Tchebycheff metric.* according to the weighted Tchebycheff metric.

Step 5Step 5If the DM considers this solution as satisfactory then the If the DM considers this solution as satisfactory then the process ends with process ends with xx(h)(h) as the final solution. as the final solution.

Otherwise, the DM is asked to indicateOtherwise, the DM is asked to indicate- what are the objective functions f- what are the objective functions fkk ( (xx) he/she is ) he/she is

willing to sacrifice (S:=S willing to sacrifice (S:=S {k}), {k}), - what is the maximum amount - what is the maximum amount kk to be sacrificed in to be sacrificed in

each one. each one.

StepStep 6 6Preparation of a new computation phase: building the Preparation of a new computation phase: building the new reduced feasible region by introducing constraints new reduced feasible region by introducing constraints on the objective function values. on the objective function values.

The feasible region for the iteration (h+1) will include The feasible region for the iteration (h+1) will include the additional constraintsthe additional constraints

cckk xx ≥ f ≥ fkk ( (xx(h)(h)) - ) - kk k k S S

cckk xx ≥ f ≥ fkk ( (xx(h)(h)) ) k k S S

Returns to step 3.Returns to step 3.

In STEM’s original versionIn STEM’s original version- each function can be relaxed just once, - each function can be relaxed just once, - in each iteration a single function can be - in each iteration a single function can be

relaxed.relaxed.These limitations can be overridden in order to make These limitations can be overridden in order to make the method more flexible.the method more flexible.

TRIMAP MethodTRIMAP Method Free search: Free search:

- progressive and selective learning of the - progressive and selective learning of the nondominated solution setnondominated solution set

Preference informationPreference information

- lower bounds for the objective function - lower bounds for the objective function values,values,

- constraints on the weights.- constraints on the weights.

Computation phaseComputation phase

- optimizing a weighted-sum of the - optimizing a weighted-sum of the objective functions.objective functions.

Parametric (weight) space: coherent means for gathering and presenting Parametric (weight) space: coherent means for gathering and presenting the information to the DM (+ analyst).the information to the DM (+ analyst).

Devoted to three-objective LP problemsDevoted to three-objective LP problems

- limitation, but- limitation, but- enables the use of graphical means adequate for the dialogue - enables the use of graphical means adequate for the dialogue

with the DM.with the DM. Enables a progressive and selective filling of the parametric space:Enables a progressive and selective filling of the parametric space:

- information about the shape of the nondominated frontier,- information about the shape of the nondominated frontier,- avoid an exhaustive study of regions in which the objective - avoid an exhaustive study of regions in which the objective

functions are similar. functions are similar.

Reducing the scope of the searchReducing the scope of the search- impose bounds on the objective function values (type of - impose bounds on the objective function values (type of

information that does not require a great effort from the DM),information that does not require a great effort from the DM),- translated onto the parametric (weight) space.- translated onto the parametric (weight) space.

By making a comparative analysis of the parametric (weight) space and By making a comparative analysis of the parametric (weight) space and the objective space displays, the DM is able to make a progressive and the objective space displays, the DM is able to make a progressive and selective covering of the diagram, assessing in each interaction the selective covering of the diagram, assessing in each interaction the interest to search for solution in areas not yet exploited.interest to search for solution in areas not yet exploited.

TRIMAP combines 3 main procedures:TRIMAP combines 3 main procedures:- decomposition of the parametric (weight) - decomposition of the parametric (weight)

diagram,diagram,- introduction of constraints into the objective - introduction of constraints into the objective

space,space,- introduction of constraints on the weights. - introduction of constraints on the weights.

The constraints introduced into the objective function The constraints introduced into the objective function values are translated into the parametric (weight) values are translated into the parametric (weight) diagram. diagram.

Computation of the efficient solutions that optimize each Computation of the efficient solutions that optimize each objective (provides a first overview about the range of objective (provides a first overview about the range of values for each objective function).values for each objective function).

Auxiliary information: computation of the efficient Auxiliary information: computation of the efficient solution that minimizes a weighted Tchebycheff distance solution that minimizes a weighted Tchebycheff distance to the ideal solution (similar to STEM).to the ideal solution (similar to STEM).

Weight selectionWeight selection for the computation of nondominated for the computation of nondominated solutions: solutions:

- selecting a set of weights in a region of the - selecting a set of weights in a region of the parametric diagram display (triangle) not yet filled, parametric diagram display (triangle) not yet filled, which seems important to proceed the search; which seems important to proceed the search;

- build a weighted function whose gradient is - build a weighted function whose gradient is normal to the plane passing through 3 nondominated normal to the plane passing through 3 nondominated solutions already computed selected by the user. solutions already computed selected by the user.

Introduction of additional boundsIntroduction of additional bounds on the objective on the objective function valuesfunction values

- graphically translated onto the parametric - graphically translated onto the parametric diagram, diagram,

- enables the dialogue with the Dm to be carried - enables the dialogue with the Dm to be carried out in terms of the objective function values, out in terms of the objective function values, accumulating the resulting information in the accumulating the resulting information in the parametric (weight) diagram display. parametric (weight) diagram display.

Imposing the additional boundImposing the additional boundffkk ( (xx) ≥ L) ≥ Lkk (L(Lkk , k , k {1,2,3}) {1,2,3})

building the auxiliary problembuilding the auxiliary problem max fmax fkk((xx))

s. to s. to xx X Xaa

XXaa { { xx X : f X : fkk ( (xx) ≤ L) ≤ Lkk } }

By maximizing fBy maximizing fkk ( (xx) in X) in Xaa (basic) alternative optimal (basic) alternative optimal solutions are obtained. solutions are obtained.

The vertices of the feasible polyhedron XThe vertices of the feasible polyhedron Xaa that optimize that optimize the auxiliary problem are selected. The sub-regions of the auxiliary problem are selected. The sub-regions of the parametric (weight) diagram corresponding to each the parametric (weight) diagram corresponding to each one of these points are computed and displayed one of these points are computed and displayed graphically. graphically.

These are the indifference regions defined by These are the indifference regions defined by TT W≥0, W≥0, w.r.t each alternative efficient basis. w.r.t each alternative efficient basis.

The union of all these indifference regions determines The union of all these indifference regions determines the sub-region of the parametric diagram where the the sub-region of the parametric diagram where the additional bound on the objective function value is additional bound on the objective function value is satisfied.satisfied.

If the DM is interested only in solutions If the DM is interested only in solutions satisfying fsatisfying fkk ( (xx) ≥ L) ≥ Lkk, then it is sufficient from , then it is sufficient from now on to restrict the search to this sub-region. now on to restrict the search to this sub-region.

If the DM wants to impose more than one bound If the DM wants to impose more than one bound then the auxiliary problem is solved for each then the auxiliary problem is solved for each one of them and the corresponding sub-regions one of them and the corresponding sub-regions in the parametric diagram are filled with in the parametric diagram are filled with different patterns, thus enabling to visualize different patterns, thus enabling to visualize clearly the zones where intersection exists. clearly the zones where intersection exists.

Imposing direct limitations on the weightsImposing direct limitations on the weights

kk ≥ u ≥ uijij, i,j , i,j {1,2,3}, i≠j, u {1,2,3}, i≠j, uijij ++

0 < u0 < uLL ≤ ≤ kk ≤ u ≤ uHH < 1, with k < 1, with k {1,2,3} {1,2,3}

Two main graphsTwo main graphs

- parametric (weight) diagram - parametric (weight) diagram displaying the indifference regions displaying the indifference regions corresponding to the (basic) efficient corresponding to the (basic) efficient solutions already computed,solutions already computed,

- projection of the objective function - projection of the objective function space displaying the solutions already space displaying the solutions already known. known.

Complementary indicators for each solution:Complementary indicators for each solution:

- distances L1, L2 and L∞ to the ideal - distances L1, L2 and L∞ to the ideal solution,solution,

- area of the indifference region (% - area of the indifference region (% occupied of the total triangle area).occupied of the total triangle area).

Other interactive methodsOther interactive methods

ICW – criterion cone contractionICW – criterion cone contraction Pareto Race – line searchPareto Race – line search Zionts-Wallenius – weight space reductionZionts-Wallenius – weight space reduction Nimbus – for nondifferentiable functionsNimbus – for nondifferentiable functions Methods for MILPMethods for MILP GDFGDF SPOTSPOT ........

Dealing with uncertaintyDealing with uncertainty

Sensitivity analysisSensitivity analysis Stochastic programmingStochastic programming Interval programmingInterval programming Fuzzy programmingFuzzy programming Robustness analysis (min-max, min-max Robustness analysis (min-max, min-max

regret)regret)

New trendsNew trends

MOP meta-heuristics, MOP meta-heuristics, particularly based on particularly based on solution populations solution populations (AG/EP, PSO)(AG/EP, PSO)

Genetic Algorithms / Genetic Algorithms / Evolutionary Evolutionary Programming for Programming for combinatorial MOPscombinatorial MOPs