Multi-Objective Programming an overview with a focus on interactive approaches

1
Multi-Objective Programmingan overview with a
focus on interactive approaches

• Carlos Henggeler Antunes
• DEEC University of Coimbra
• INESC Coimbra (www.inescc.pt)
• ch_at_deec.uc.pt

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Introduction

• Real-world problems are multicriteria in nature
• no single measure of what is best exists.
• The complexity of real-world problems in modern
  technologically developed societies is
  characterized by the presence of multiple
  criteria, reflecting economical, social,
  political, physical, engineering, administrative,
  psychological, ethical, aesthetical,...
  evaluation aspects in a given decision context.

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• There is no feasible solution which guarantees
  the best values in all evaluations aspects
• By considering explicitly the different aspects
  of the reality
• mathematical models, and
• the DMs perception of problems
• become more realistic.
• ? broadening the range of solutions under
  analysis.

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• Multicriteria problems
• multiattribute (enumerative definition) the
  potential courses of action, a finite number, are
  explicitly known a-priori, as well as the
  corresponding indexes of merit evaluated for the
  multiple criteria
• multiobjective (analytical definition) the
  potential courses of action form a continuum,
  defined implicitly by a set of constraints
• The decision variable space is mapped into the
  objective function space, in which each
  alternative has a vector-valued representation,
  whose components are the corresponding values for
  each objective function (vector optimization)

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• Decision making in these complex problems cannot
  be reduced to the search for an optimal solution
  of a single objective function (e.g., some type
  of economic indicator).
• Optimal solution (best feasible value for a
  single objective function) ? Nondominated
  solution (there is no other feasible solution
  which improves simultaneously all the objectives
  the improvement in an objective function value
  is obtained by accepting to degrade the value of
  at least one of the other objectives)

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• Since there is no point which optimizes
  simultaneously all the objective functions, the
  simple comparison between nondominated solutions
  does not provide information in the search of a
  nondominated solution which constitutes the final
  solution to the multiobjective problem.
• Two nondominated solutions are incomparable using
  the natural order .
• Some ordering of the set of nondominated
  solutions underlies the intervention of a
• preference relation
• reflecting the DMs preference structure

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• In a single objective optimization problem the
  serach for the optimal solution is purely
  technical
• the best solution is implicit in the
  mathematical model,
• the role of the optimization algorithm is to
  discover it,
• there is no place for decision making.
• In a multiobjective problem it is necessary to
  make intervene in the search process
• technical devices to compute nondominated
  solution
• information on the DMs preferences.

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• The DMs preference structure embodies a set of
  opinions, values, convictions and perspectives of
  the reality, configuring a personal model of the
  reality the DM leans on to evaluate different
  potential courses of action
• It is not possible to classify a solution as good
  or bad just with reference to the mathematical
  model and the resolution techniques
• the quality of a solution is influenced by
  organizational, political, cultural, ...,
  aspects, underlying the decision process.
• Multiobjective problem
• - choice of a solution, among the set of
  nondominated solutions (generally non-countable),
  which constitutes an acceptable compromise
  solution for the DM, having in mind his/her
  preferences, which can evolve throughout the
  decision support process.

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• The decision process is a dynamic entity
  constituted by iterative cycles of
• generation of potential actions,
• evaluation,
• interpretation of information,
• value changes,
• learning, and
• preference adaptation.
• The consideration of multiobjective models
• - reflect better a complex and ill-structured
  reality,
• - enables the exploration of a wider range of
  alternative solutions.
• The criteria heterogeneity arises specific
  problem resulting from
• - conflicts among criteria, since a feasible
  solution that optimizes all the objective
  functions simultaneously does not exist
• - incommensurable criteria, which cannot be
  reduced to a common measuring unit (e.g.,
  monetary)
• - uncertainty, due to the insufficient and/or
  incomplete nature of knowledge and the
  information on the DMs preferences in a
  multidimensional reality.

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• Mathematical modeling and optimization techniques
  are adequate for the computation of nondominated
  solutions.
• It is necessary to incorporate into the decision
  process qualitative aspects related with the DMs
  preferences and subjective judgments.
• A good decision
• No other potential action exists that is better
  in some aspects and not worse in all aspects
  under consideration
• A final proposal must be selected within the
  universe of nondominated solutions
• Need to establish balances and compromises among
  the objectives
• Compromise solution
• reach satisfactory goals for the DM
• satisfactory compromise solution.
• maximize a value function,
• the best decision is the one that gives the
  best value

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• Satisfactory solution
• cognitive capabilities limitations of Human
  Beings ? "bounded rationality",
• "satisficing rationality"
• instead of
• "optimizing rationality".
• This is not necessarily a Human fault in decision
  situations that must be corrected,
• it is often a form of intelligence that must
  be refined, and not ignored, by decision aid
  methodologies
• DMs active role
• - the information is not given to the DM
  without requiring him/her any intervention,
• - obliges him/her to obtain the information by
  means of an iterative process of exploration,
  observation, pattern recognition.

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• Multiobjective decision aid methods
• No DMs preference articulation (generating
  methods).
• DMs preference articulation is made
• a-priori (value/utility function methods,)
• progressively (interactive methods)
• Interactive methods
• computation phases,
• dialogue phases
• in which the DM is asked to express his/her
  preferences in face of the solutions which are
  proposed to him/her, until a stop condition is
  fulfilled (depending on the method)
• The DMs intervention is used to guide the
  interactive decision process, reducing the scope
  of the search

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• Interactive methods
• In face of generating methods
• - reduce the computer burden,
• - limit the cognitive effort imposed on the DM.
• In face of value function-based methods
• - avoid the prior preference information
  elicitation to formulate the value function
• - enable learning and preference evolution as a
  more information is being gathered.
• Interactivity
• offer the DM an operational environment
  facilitating exploration, reflection, emergence
  of new intuitions,
• enable the DM to understand more in-depth the
  decision problem at hand,
• contribute to shape and evolve the DMs
  preferences to guiding the search process,
• focus the search process in the regions where
  more interesting decision are located.

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• Learning
• increasing the available knowledge,
• improving the DMs capabilities to make an
  adequate use of that knowledge.
• Trend importance shifting from
• - making decisions (MCDM - "Multiple Criteria
  Decision Making")
• - aiding decisions to be made (MCDA - "Multiple
  Criteria Decision Aid").
• Methods
• - aid the DM by providing him/her better
  information quality,
• - offering the possibility of evolution of
  his/her preference structure by confronting it
  with the proposals generated in the computation
  phases in order to increase his/her understanding
  of the problem

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• Satisfactory compromise solution
• nondominated solution,
• associated with a given compromise between
  the objectives,
• objectives assume satisfactory values for
  the DM
• in such a way that this solution can be accepted
  as the final solution of the decision process.
• What is the meaning of a final solution?
• - the results of the analysis shall be used not
  a definitive prescription but rather as a
  reference or material support to make decisions
  in the sense of finding better actions plans for
  the system.

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Formulation and Definitions

• Linear programming problem with multiple
  objective functions
• max f1(x) c1 x
• max f2(x) c2 x
• ..................
• max fp(x) cp x
• s. to x ? X?x ? ?n x 0, Axb, b ? ?m
• "Max" f (x) C x
• s. to x ? X
• C matrix of objective function coefficients
  the rows are the vectors ck (coefficients of
  objective function fk).
• A matrix of technological coefficients (mxn)
• b RHS vector (available resources or
  requirements imposed)

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• In single objective optimization the decision
  space x ? X is mapped into ?
• In MOP the decision space is mapped into a
  p-dimensional objective function space
• Ff (x) ? ?p x ? X
• Each potential alternative x ? X has as
  representation a vector f(x)(f1(x),f2(x),...,fp(
  x)) the components of which are the values of
  each objective function at that point of the
  feasible region.
• In general, there is no feasible solution x ? X
  that optimizes simultaneously all objective
  functions.
• From an operational perspective, "Max" denotes
  the operation of determining nondominated
  solutions.

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• Optimal solution ? efficient / nondominated
  solution
• A feasible solution to a MOP problem is said
  efficient iff no other feasible solution exists
  that improves the value of an objective function
  without worsening the value of, at least, another
  objective function.
• The set of efficient solutions (Pareto optimal)
  is defined by
• XE x ? X ? x' ? X f (x') f (x)
• where f(x') f(x) iff f(x') f(x) and f(x')
  ? f(x) (that is, fk(x') fk(x) for all k and
  fk(x')gtfk(x) for at least one k), and f(x')
  f(x) iff fk(x') fk(x), k1,2,...,p

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• The criterion vector zf (x) is nondominated
  (non-inferior) when x ? XE
• FE zf (x) ? F x ? XE
• Objective function space nondominated solutions
• Decision variable space efficient solutions
• The image of an efficient solution is a
  nondominated solution

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• A feasible solution to a MOP problem is weakly
  efficient iff there is no other feasible solution
  which improves strictly the value of all
  objective functions.
• The set of weakly efficient solutions
• XFE x ? X ? x' ? X f (x') f (x)
• FFE zf (x) ? F x ? XFE

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

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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.
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• Ideal solution (utopia point) z
• would optimize simultaneously all objective
  functions ? its components are the optimum of
  each objective function in the feasible region,
  when optimized individually.
• In general, the ideal solution does not belong to
  the feasible region, but each zk is individually
  reachable.
• The ideal solution is sometimes used as the
  (unreachable) DMs reference point in scalarizing
  functions representing a distance to be minimized
  to determine a compromise efficient solution.
• Although the ideal solution z can always be
  defined in the objective space, its image in the
  decision variable space not always exist ? x may
  not exist such that zf(x)

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• Table of individual optima ("pay-off")
• - organizes the objective values for each
  nondominated solution resulting from the
  individual optimization of each objective
  function in the feasible region X.
• The ideal solution components are obtained in the
  diagonal of the table of individual optima.
• From the table of individual optima the
  anti-ideal solution (nadir point) can be obtained
  selecting in each row the worst value for the
  corresponding objective function ? it intends to
  represent the worst value of objective function
  fk(x) in the efficient region.
• This is just a "convenient" minimum (due to being
  easily determined) which can be different
  (higher) from the actual minimum in the efficient
  region.
• The table of individual optima may not be
  uniquely defined if alternative optimal solutions
  exist for any objective function.
• Also, the anti-ideal solution would not be
  uniquely defined.
• The ideal solution (objective space) is always
  uniquely defined.

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Interactive Methods

• Basic phases
• (a) Initialization automatically establishing
  the initial preference information, stopping
  parameter setting,, etc.
• (b) preparation incorporation of preference
  information into the parameters of the surrogate
  scalar function, which aggregates the multiple
  objectives in a single dimension function to be
  used in the new computation phase
• (c) computation computing one, or more,
  efficient solutions through the optimization of a
  surrogate scalar function, that will be subject
  to the DMs evaluation
• (d) dialogue solution presentation and
  expression of the preference information by the
  DM in face of this solution
• (e) stop according to a given stopping rule,
  which may be simply the DM to consider being
  satisfied with the information gathered so far
  throughout the process

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Initialization
Preparation
Computation
Dialogue
Stop
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• Surrogate scalar functions (scalarizing
  functions)
• aggregate in a single dimension the different
  objectives
• include preference information parameters
• its optimal solution is an efficient solution
  to the MOP
• Interactivity
• offer the DM the central role, leading the
  interactive decision process
• the method shall have active role in this
  mutual convergence
• - dynamic dialogue (adjusted to the different
  stages of the decision process),
• - simple (not demanding too much information
  or information unnecessarily complex)
• - positive (requiring information about what
  the DM wants to improve)
• - divergence mechanisms (enabling to explore
  solutions in a given neighborhood).

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• Two basic (extreme) concepts for the role of
  interactive mechanisms
• Search-oriented.
• - assumes the existence of a pre-existing and
  stable preference structure (e.g. Represented by
  an implicit value function)
• - coherence with this function throughout the
  use of the method, answering in a deterministic
  way to the questions of the interactive protocol
• - reasonable to impose the mathematical
  convergence of the interactive method
• - convergence is guaranteed and controlled by
  the method
• - the aim of the interaction is the search of a
  optimal proposal (which does not depend on the
  evolution of the interactive process) in face of
  the preference structure.
• The preference structure is thus discovered
  during the process.

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• Learning-oriented
• - no assumption of a stable and pre-existing
  preference structure with which the DM is always
  consistent
• - the DMs preferences may be partially
  unstable and conflicting
• - the aim of the interaction is the preference
  learning (clarification of what can be a good
  decision according to the DMs perspective)
• - convergence is not guaranteed and it is
  controlled by the DM psychological convergence"
  (meaning the identification of an efficient
  solution as satisfactory based on the information
  gathered so far).
• - the process ends with a satisfactory
  compromise solution according to the available
  information and the DMs will when the emergence
  of new intuitions about new search directions
  seems possible.
• Convergence must be the result of the interaction
  between the DM and the method.

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• Division of the work between the DM and the
  computer
• Differentiation of tasks by the capability of
• - guiding the computer effort for the regions
  where the solutions more in accordance with the
  DMs preference structure are located,
• - accommodating path corrections due to the
  evolution of the preference structure as the
  information gathered makes new hints and
  intuitions to emerge for proceeding the search.
• The future of MOP is in its interactive
  application! (Steuer)

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• Criticisms
• - the DMs preference structure is not
  generally founded on empirical investigation
  (French),
• - too strong assumption the DMs choices are
  always in accordance with an implicit value
  function,
• - anchoring the DM anchors on the first
  proposals, showing some difficulties in
  preferring other solutions,
• - local nature of the preference information
  required.

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• Categorization of interactive methods
• Strategy for reducing the scope of the search
  (explicitly or implicitly)
• - reduction of the feasible region
• - reduction of the weight (parametric) space
• - contraction of the objective function
  gradient cone
• - directional search.
• Type of surrogate scalar function
• - optimization of one of the objective
  functions considering the other functions as
  constraints
• - weighted-sum of the objective functions
• - minimization of a distance to a reference
  point.
• Possibility of the DM/user to request a given
  operation at any time of the interactive process
  or need to comply with a pre-established sequence
  of computation and dialogue phases
• - non-structured
• - structured.

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• These classifications
• - are not mutually exclusive (there are methods
  combining different strategies for reducing the
  scope of the search and techniques for the
  computation of efficient solutions)
• - are not exhaustive there are methods diffcult
  to be classified (e.g., adaptations of feasible
  direction algorithms or cutting planes in MILP or
  MNLP).

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Scalarizing Processes

• Transforming the MOP into an optimization problem
  of a surrogate scalar function
• - the optimal solution of which is an efficient
  solution to the MOP,
• - includes information parameters of the DMs
  preferences.
• Objective properties
• - generate efficient solutions only
• - be able to generate all efficient solutions
• - be independent of non-efficient solutions.
• Subjective properties
• - computer effort involved not too big
• - simple interpretation for the preference
  information parameters.

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• Meaning of the scalarizing functions
• - mere technical device to aggregate
  temporarily the multiple objectives and generate
  efficient solutions to be proposed to the DM
  (with no concern of reflecting a true analytical
  expression of the DMs preferences)
• vs.
• - analytical representation of the DMs
  preferences.

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• MOLP
• max f1(x) c1 x
• max f2(x) c2 x
• ..................
• max fp(x) cp x
• s. to
• x ? X ? x ? ?n x 0, Axb, b ? ?m
• "Max" f (x) C x
• s. to x ? X

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• Optimization of one of the objective functions
  considering the others as constraints
• Surrogate function one of the objective
  functions (the one to which more importance is
  assigned)
• Lower bounds on the other p-1 objectives
  constraints (minimum levels the DM is willing to
  accept!)
• max fi (x) ci x
• s. to fk (x) ck x ek k1,...,p , k?i
• x ? X

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• Theorem If x ? X is a single solution to the
  scalar problem for any i, then x is an efficient
  to the MOLP.
• Optimization of the surrogate function
• efficient solution to the MOLP since
• - the reduced feasible region is not empty
  (which may not happen if the lower bounds ek are
  too severe)
• - no alternative optimal solutions exist for
  the selected objective function (in this case,
  just weakly efficient points are guaranteed).
• Dual variable associated with the constraint
  corresponding to fk(x)
• - local compromise rate between fi and fk in
  the optimal solution to the scalar problem.

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• Easy to be understood by DMs capture the
  attitude of assigning more importance to one of
  the objective functions accepting lower bounds
  for the other objectives.
• Choice of the function to be optimized may be
  difficult.
• Setting the function to be optimized during the
  whole decision aid process makes the method
  little flexible.
• Results are too dependent on the function
  selected.
• Possible to obtain all points of the nondominated
  frontier, that is vertices and points lying on
  nondominated faces.
• Preference information
• inter-criteria information
• - choice of the objective function to be
  optimized
• intra-criteria information
• - imposing lower bounds on the other objective
  functions.

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• Weighted-sum of objective functions
• max ?1f1(x) ?2f2(x) ... ?pfp(x)
• s. to x ? X
• ? ? ?0
• Set of feasible weights
• ? ? ? ? ? ?p, ? ?k1, ?k 0,
  k1,2,...,p
• ?0 ? ? ? ? ?p, ? ?k1, ?k gt 0,
  k1,2,...,p (interior of ?)
• Bilinear function defined on X x ?.
• By setting a set of weights ? weighted-sum scalar
  linear function of the p objective functions, to
  be optimized in X.
• Theorem x ? X is a (properly) basic efficient
  solution to the MOLP iff it is an optimal
  solution to the weighted-sum scalar problem for a
  set of weights ? ? ?0.

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• Initial multiobjective simplex tableau
• A I b
• - C 0 0
• w.r.t. to basis B it is transformed into
• B-1 N B-1 B-1 b
• -CB B-1 N - CN CB B-1 CB B-1 b
• A B , N C CB , CN .
• Reduced cost matrix w.r.t. to basis Ba is
• Wa CB B-1 N -CN.
• Ba is an efficient basis iff the system
• ?T Wa 0, ? ? ? is consistent.

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• The graphical display of the set of weights ?
  which lead to a basic efficient solution can be
  obtained by decomposing the weight space (a p-1
  dimensional simplex in an Euclidean p dimensional
  space) ?.
• Multiobjective simplex tableau
• - basic efficient solution to the MOLP
• - the corresponding ? is defined by ?T W 0
• wkj from the reduced cost matrix W
• - marginal rate of change of objective function
  fk(x) due to the production of one unit of the
  nonbasic variable xj.
• W column corresponding to an efficient nonbasic
  variable ? unit change trend of the objective
  functions along the efficient edge.

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• Indifference region
• - set of weights corresponding to a basic
  efficient solution (where ?TW0, ? ? ? is
  consistent).
• All the weight combinations in that region lead
  to the same (basic) efficient solution.

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Parametric (weight) diagram
Projection of the objective function space
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• Common frontier to 2 indifference regions ? the
  respective basic efficient solutions are
  connected by an efficient edge, corresponding to
  making basic a nonbasic efficient variable.
• A point ? ? ? belongs to several indifference
  regions ? these regions correspond to efficient
  solutions located on the same face (which is
  efficient if ? ? ?0).
• Indifference region depend on
• - relative order of magnitude of the objective
  function values (relative gradient length),
• - geometry of the feasible region.
• Area of the indifference region
• - robustness measure against weight changes.

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• Easy to be explained to DMs (apparently!)
  capturing the attitude of expressing the
  importance assigned to each objective function.
• Information difficult to be elicited from the DM
  (although apparently simple).
• There is no guarantee that the solutions computed
  using a weighted sum scalar function are in
  accordance with the preferences underlying the
  specification of the weights.
• It is possible to obtain vertices of the
  nondominated frontier only.
• Preference information
• inter-criteria information
• - weights (relative importance coefficients?!)
  for the objective functions.

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• Minimization of a distance to a reference point
• Efficient solution closer (according to a
  given metric) to the DMs aspirations.
• Using an Lp metric and considering the
  ideal solution as the reference point
• min z - f(x) p
• s. to x ? X or, with a weighted Lp metric
• min ? z - f(x) p
• s. to x ? X
• ? ? ?
• p1 all the deviations from the reference point
  are taken into consideration in the direct
  proportion of its magnitude.
• 2ltplt8 bigger deviations have more importance as
  p increases,
• p8 only the biggest deviation is considered.

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• Let x0 be a solution to this problem. By
  considering the worst case (i.e., biggest
  difference between the vector z and f(x0)
  components according to the metric L8)
• min max zk - fk(x)
• x ? X k1,...,p
• has x0 as a solution iff x0 and v0 are
  solutions to the linear problem
• min v
• s. to v zk - fk(x) k1,...,p
• x ? X
• v 0
• (if the reference point zr does not satisfy
  zrfk(x) in X, then v ? ?).

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• Weighted L8 metric
• min max ?k zk - fk(x)
• x ? X k1,...,p
• In MOLP surrogate scalar linear problems results
  using L1 or L8 metrics.
• To guarantee that the obtained solutions are
  efficient ones, and not just weakly efficient,
  the (weighted) augmented Tchebycheff metric can
  be used
• min max ?k zk - fk(x) ? zk -
  fk(x)
• x ? X k1,...,p
• zk reference point (arbitrary, may be the
  ideal solution)
• ? ? ? is a weighting vector.
• The 2nd term is a perturbation, with ? gt0
  sufficiently small, to ensure the efficiency of
  the solution.
• min v ? zk - fk(x)
• s. to v zk - fk(x) k1,...,p
• x ? X

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• Theorem If x ? X is a solution to the augmented
  weighted Tchebycheff problem for any reference
  point, then x is an efficient solution to the
  MOLP.
• Minimization of the discomfort" of getting a
  compromise nondominated solution z0 rather than
  the ideal point (or other reference point) z.
• It is possible to obtain all points on the
  nondominated frontier (feasible region vertices
  or faces).
• Preference information
• intra-criteria information
• - setting the reference point(s)
• inter-criteria information
• - weighting coefficients for the L8 metric.

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Step Method (STEM)

• Feasible region reduction.
• Dialogue phase quantities that the DM is willing
  to sacrifice in the functions for which the
  objective values are already satisfactory in
  order to improve the remaining functions.
• Computation phase minimizing a weighted
  Tchebycheff distance to the ideal solution.
• The compromise solution computed in each
  iteration by minimizing the weighted Tchebycheff
  distance to the ideal solution is presented to
  the DM.
• If the objective function values are considered
  as satisfactory the process ends.
• Otherwise, the DM is asked to specify the
  objective he/she is willing to relax, and in what
  amount, in order to improve the remaining
  objective functions.

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• Step 1
• Individual optimization of each objective
  function
• Building the table of individual optima
  (pay-off).
• Step 2
• Calibration of the weights to be used in the
  computation phase of iteration h
• Higher weight for objectives with larger relative
  variations.
• Normalization factor (using L2 norm) of the
  objective function gradients.

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• Set S
• - indices of the objective functions that will
  be relaxed in the next iteration (according to
  the DMs indications, by considering their values
  as satisfactory).
• At the beginning Sø and X(1)?X.
• Step 3
• - Weights which define the weighted metric L8
• - The weights of the objective functions whose
  values are considered satisfactory (that will be
  permitted to be relaxed) are zero.

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• Step 4
• Computation phase solving the LP minimization
  of the weighted Tchebycheff distance to the ideal
  solution
• min v
• s. to v a 1 k p
• x ? X(h)
• v 0
• Dialogue phase the solution z(h) f (x(h))
  resulting from solving the problem in iteration h
  is presented to the DM
• x(h) is the point of the reduced feasible
  region X(h) closer to z according to the
  weighted Tchebycheff metric.
• Step 5
• If the DM considers this solution as
  satisfactory then the process ends with x(h) as
  the final solution.
• Otherwise, the DM is asked to indicate
• - what are the objective functions fk (x)
  he/she is willing to sacrifice (SS ? k),
• - what is the maximum amount ?k to be
  sacrificed in each one.

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• Step 6
• Preparation of a new computation phase
  building the new reduced feasible region by
  introducing constraints on the objective function
  values.
• The feasible region for the iteration (h1) will
  include the additional constraints
• ck x fk (x(h)) - ?k k ? S
• ck x fk (x(h)) k ? S
• Returns to step 3.
• In STEMs original version
• - each function can be relaxed just once,
• - in each iteration a single function can be
  relaxed.
• These limitations can be overridden in order to
  make the method more flexible.

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TRIMAP Method

• Free search
• - progressive and selective learning of the
  nondominated solution set
• Preference information
• - lower bounds for the objective function
  values,
• - constraints on the weights.
• Computation phase
• - optimizing a weighted-sum of the objective
  functions.

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• Parametric (weight) space coherent means for
  gathering and presenting the information to the
  DM ( analyst).
• Devoted to three-objective LP problems
• - limitation, but
• - enables the use of graphical means adequate
  for the dialogue with the DM.
• Enables a progressive and selective filling of
  the parametric space
• - information about the shape of the
  nondominated frontier,
• - avoid an exhaustive study of regions in which
  the objective functions are similar.
• Reducing the scope of the search
• - impose bounds on the objective function
  values (type of information that does not require
  a great effort from the DM),
• - translated onto the parametric (weight)
  space.
• By making a comparative analysis of the
  parametric (weight) space and the objective space
  displays, the DM is able to make a progressive
  and selective covering of the diagram, assessing
  in each interaction the interest to search for
  solution in areas not yet exploited.

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• TRIMAP combines 3 main procedures
• - decomposition of the parametric (weight)
  diagram,
• - introduction of constraints into the
  objective space,
• - introduction of constraints on the weights.
• The constraints introduced into the objective
  function values are translated into the
  parametric (weight) diagram.
• Computation of the efficient solutions that
  optimize each objective (provides a first
  overview about the range of values for each
  objective function).
• Auxiliary information computation of the
  efficient solution that minimizes a weighted
  Tchebycheff distance to the ideal solution
  (similar to STEM).

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• Weight selection for the computation of
  nondominated solutions
• - selecting a set of weights in a region of the
  parametric diagram display (triangle) not yet
  filled, which seems important to proceed the
  search
• - build a weighted function whose gradient is
  normal to the plane passing through 3
  nondominated solutions already computed selected
  by the user.
• Introduction of additional bounds on the
  objective function values
• - graphically translated onto the parametric
  diagram,
• - enables the dialogue with the Dm to be
  carried out in terms of the objective function
  values, accumulating the resulting information in
  the parametric (weight) diagram display.
• Imposing the additional bound
• fk (x) Lk (Lk ? ?, k ? 1,2,3)

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• ? building the auxiliary problem
• max fk(x)
• s. to x ? Xa
• Xa? x ? X fk (x) Lk
• By maximizing fk (x) in Xa (basic) alternative
  optimal solutions are obtained.
• The vertices of the feasible polyhedron Xa that
  optimize the auxiliary problem are selected. The
  sub-regions of the parametric (weight) diagram
  corresponding to each one of these points are
  computed and displayed graphically.
• These are the indifference regions defined by ?T
  W0, w.r.t each alternative efficient basis.
• The union of all these indifference regions
  determines the sub-region of the parametric
  diagram where the additional bound on the
  objective function value is satisfied.

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• If the DM is interested only in solutions
  satisfying fk (x) Lk, then it is sufficient
  from now on to restrict the search to this
  sub-region.
• If the DM wants to impose more than one bound
  then the auxiliary problem is solved for each one
  of them and the corresponding sub-regions in the
  parametric diagram are filled with different
  patterns, thus enabling to visualize clearly the
  zones where intersection exists.
• Imposing direct limitations on the weights
• ?k uij, i,j ? 1,2,3, i?j, uij ? ?
• 0 lt uL ?k uH lt 1, with k ? 1,2,3

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• Two main graphs
• - parametric (weight) diagram displaying the
  indifference regions corresponding to the (basic)
  efficient solutions already computed,
• - projection of the objective function space
  displaying the solutions already known.
• Complementary indicators for each solution
• - distances L1, L2 and L8 to the ideal
  solution,
• - area of the indifference region ( occupied
  of the total triangle area).

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• Other interactive methods
• ICW criterion cone contraction
• Pareto Race line search
• Zionts-Wallenius weight space reduction
• Nimbus for nondifferentiable functions
• Methods for MILP
• GDF
• SPOT
• ....

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• Dealing with uncertainty
• Sensitivity analysis
• Stochastic programming
• Interval programming
• Fuzzy programming
• Robustness analysis (min-max, min-max regret)

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New trends

• MOP meta-heuristics, particularly based on
  solution populations (AG/EP, PSO)
• Genetic Algorithms / Evolutionary Programming for
  combinatorial MOPs