Approximation and Visualization of Interactive Decision Maps Short course of lectures

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Approximation and Visualization of Interactive
Decision Maps Short course of lectures

• Alexander V. Lotov
• Dorodnicyn Computing Center of Russian Academy of
  Sciences and
• Lomonosov Moscow State University

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Lecture 2.  Classes of methods for
multi-objective (multi-criteria) optimization. A
posteriori preference methods
Plan of the lecture 1. Old simple-minded
approaches 2. Main types of modern methods for
multi-criteria optimization (MCO) 2a.
No-preference methods 2b. A priori preference
methods 2c. Interactive methods and possible
functions of criteria 2d. A posteriori
preference (generating the Pareto frontier) 3.
Stability of the Pareto frontier and a posteriori
preference methods for constructing a list of
Pareto-optimal points
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Old simple MCO approaches
A priori restrictions DM selects the most
important criterion and specifies restrictions
for others
A priori weights DM specifies weights for all
criteria and solves the problem
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Classification of modern methods according to the
role of the DM
 
MCDA
                 
No-preference methods
A posteriori methods
A priori preference methods
Interactive methods
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No-preference methods

• The opinions of the decision maker are not taken
  into account. The problem is solved by expert
  using some relatively simple method (say, single
  criterion optimization of some function of
  criteria with objectively specified
  parameters). The solution is presented to the
  decision maker who can accept or reject it.

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An example of no-preference methods

• For example, the following optimization problem
  can be solved
• minimize h (y) max (yi - yi)
  i1,2,...,m on Y,
• where 1/(yi - yi), i1,2,...,m,
  and y is the worst possible criterion
  point.
• For y, one can take the criterion point, which
  coordinates are the worst feasible values of
  particular criteria, or the worst values of
  particular criteria over the Pareto frontier, or
  any other bad criterion point.
• Surely, the decision depends to a great extent on
  the worst point. By modifying the worst
  point, one can get any point of the Pareto
  frontier.

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A priori preference methods(constructing the
decision rule)

• Methods based on multi-attribute utility theory
  (MAUT)
• Methods based on direct weighting of criteria
• Methods based on complicated weighting procedures
  (Analytic Hierarchy Process)
• Outranking methods (Electra, etc)
• Methods based on heuristic concepts (say, goal
  identification)

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The simplest method (by Keeny and Raiffa) for
approximation of indifference curves for value
functions based on multi-attribute utility theory

• The case of an additive function v(y1,
  y2)v1(y1)v2(y2)

y2
v2(y2)2
v2(y2)1
v0
v1(y1)1
y1
v1(y1)2
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Methods based on direct weighting of criteria

• DM specifies weights for all criteria and
  maximizes
• Important disadvantage of linear functions
  decrement in the value of one criterion can be
  compensated by the value of another criteria

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Complicated weighting procedures (say, Analytic
Hierarchy Process)

• Procedures consist of weighting and subsequent
  single-criterion optimization.
• The AHP method helps to develop weights. DM has
  to answer m(m-1)/2 questions concerning relative
  importance of criteria. The AHP methods helps to
  study quantitative criteria, too

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ELECTRE method

• French school (led by Prof. Bernard Roi) proposed
  interesting methods for constructing an
  outranking relation these methods are affective
  in the case of a small number of alternatives

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Goal identification - 1

• DM has to identify the goal without information
  on the set Yf(X).

y2
0
y1
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Goal identification - 2

• Then, by using some distance function, the
  closest point of the set Yf(X) is found.

y2
Yf(X)
y0
0
y1
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Goal identification - 3

• The goal programming is the most often used MCDA
  technique, but if the goal is distant from the
  feasible criterion set Yf(X), the solution y0
  depends mainly on the distance function, but not
  on the goal. Qualified experts feel the
  feasibility of criterion values and manage to
  identify the appropriate goals, which are close
  to the feasible criterion set.

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Interactive (iterative) methods

• Methods are based on interaction of the decision
  maker with the computer and consists in a finite
  number of iterations.
• At the first stage of an iteration, the decision
  maker specifies parameters of a function of the
  criteria.
• At the second stage, the computer solves the
  single-criterion optimization problem with the
  criterion function specified by the decision
  maker.

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General scheme of an interactive method

• After the k-th iteration, the vector
  and some other auxiliary information
  must be provided to decision maker.
• Stage 1. DM explores the information obtained at
  the k-th iteration and, may be, previous
  iterations. DM specifies parameters
  of a new optimization problem
  
• while
• Stage 2. Computer solves the problem the new
  decision and criterion vector are computed
• .

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The simplest interactive method

• The method is based on application of the linear
  function h (y) ltc, ygt. An iteration of the
  method consists of two steps
• Computer finds the decision x0 from the set X
  that provides the maximum of the linear
  function h (f(x)) ltc, f(x)gt over X with some
  given vector of parameters c lt0
• DM studies the optimal decision x0 , the
  criterion point y0 f(x0). If DM is not
  satisfied with the decision, he/she changes the
  values of the parameters c and the method goes to
  the next iteration.

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What scalar functions of the criteria can be used?

• Let h(y) be a scalar function of criteria. Let y0
  be the point of maximum of h(y) over Y.
• 1) Does it belong to the Pareto frontier?
• Answer. If results in h(y)
  gt h(y)
• (the scalar function is increasing in respect to
  Pareto domination), then y0 belongs to the
  Pareto frontier.
• 2) But what about the opposite can any point of
  the Pareto frontier can be found by optimization
  of the function h(y) over Y ?

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Well-known examples of scalar functions of
criteria

• 1) Linear function h (y) ltc, ygt, where c lt0
• (note that we consider the minimization
  problem!), is an increasing function in respect
  to Pareto domination
• 2) Tchebycheff distance from the ideal point y
• h (y) max (yi - yi)
  i1,2,...,m,
• where i1,2,...,m, are some
  non-negative coefficients, is a decreasing
  function in respect to Slater domination.

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Properties of the linear function

• The maximum over Y of the linear function
• h (y) ltc, ygt, where c lt0, belongs to the
  Pareto frontier.
• However, the opposite is true only in the case
  of the convex EPH
• any point of the Slater frontier can be the
  maximum over Y of the linear function h (y)
  ltc, ygt with some non-positive c only if the EPH
  is convex.

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Example
                         
 
P(Y)
f(X)
c
 
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Properties of the Tchebycheff distance

• 2) The point y0 of the minimum over Y of the
  Tchebycheff distance
• h (y) max (yi - yi)
  i1,2,...,m,
• where i1,2,...,m, are some
  non-negative coefficients, belongs to the
  Slater frontier.
• Moreover, any point of the Slater frontier can be
  the minimum over Y of the Tchebycheff distance
  with some non-negative coefficients.

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Example
 
P(Y)
f(X)
y
 
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A posteriori preference methods
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A posteriori preference methods are based on
approximating the Pareto frontier and informing
the decision maker concerning it. A posteriori
methods inform the DM about the Pareto optimal
set without asking for his/her preferences. The
DM has to specify a preferred Pareto point, i.e.
non-dominated combination of criterion values,
only after completing the exploration of the
Pareto frontier. Thus, the single-shot
specification of the preferred Pareto optimal
objective point may be separated in time from the
exploration phase.
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Two main problems that must be solved by the a
posteriori preference methods

• How to approximate the Pareto frontier
• How to inform the DM about the Pareto frontier
• In the case of two criteria, information of the
  DM is usually based on graphical display of the
  Pareto frontier. In the case of more, than two
  criteria, a list of objective points is usually
  provided to the DM.
• Question is the problem of approximating the
  Pareto frontier stated correctly?

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Stability of the Pareto frontier-1

• Example Slater (weak Pareto) S(Y) and Pareto
  P(Y) frontiers for the non-disturbed feasible set
  in criterion space Y

A
Y
B
C
P(Y)
S(Y)
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Stability of the Pareto frontier-2

• P(Y) for the disturbed feasible set in criterion
  space

A
Y
B
C
P(Y)
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Stability of the Pareto frontier - 3

• If some natural requirements hold,
• the condition
• S(Y) P(Y)
• where Y is the non-disturbed feasible set
• in criterion space, is the necessary and
• sufficient condition of stability of P(Y)
• to the disturbances of parameters.
• (Sawaragi Y., Nakayama H., Tanino T., 1985).

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Stability of the Edgeworth-Pareto Hull - 1

• Edgeworth-Pareto Hull (EPH) Yp for the
  non-disturbed feasible set in criterion space Y

A
Yp
Y
B
C
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Stability of the Edgeworth-Pareto Hull - 2

• Edgeworth-Pareto Hull (EPH) Yp for the disturbed
  feasible set in criterion space Y

A
Yp
Y
B
C
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Stability of the Edgeworth-Pareto Hull - 3

• If some natural requirements hold, the
  Edgeworth-Pareto Hull is stable to the
  disturbances of parameters of the problem.

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The first a posteriori preference method

• The first a posteriori preference method is the
  method for approximation of the set P(Y) in
  linear bi-criterion problems.
• It was introduced by S.Gass and T.Saaty in 1955
  and is based parametric linear programming.

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Parametric LP problem for two criteria

• where changes from 0 to 1.
• The problem is solved by using a method for
  solving the parametric LP problems

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In addition to the list of objective points,
picture was provided!
 

         
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Different methods
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Restrictions-based method
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Restrictions-based method formal description

• The result a large list of points of the Pareto
  frontier

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Weighted Tchebycheff metric as the distance from
the ideal point
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Formal description

• The problems
• are solved for a large number of parameters

Result a large list of Pareto points.
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Parametric LP methods for linear problems with
mgt2
Direct development of idea by Gass and Saaty
parametric LP methods for mgt2 construct all
Pareto vertices for a linear multi-objective
problem using the movement from a vertex to
another (see R.L.Steuer. Multiple-criteria
optimization. NY John Wiley, 1986). A very large
list of vertices is provided to the DM (sometimes
along with the efficient faces of the set Y).
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Evolutionary (including genetic) multiple
criteria optimization
 
 
Result a large list of quasi-Pareto points.
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Approximation of the bi-criterion Pareto frontier
by linear segmentsNISE (Cohon, 1978)
             
 
Picture is provided to the DM!
The preferred point of an approximation is
identified by the DM.
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Lessons learned from bi-objective problems

• According to Bernard Roy, In a general
  bi-criterion case, it has a sense to display all
  efficient decisions by computing and depicting
  the associated criterion points then, DM can be
  invited to specify the best point at the
  compromise curve.
• It is extremely important that, in bi-objective
  MOO problems, the graphs provide, along with
  Pareto optimal objective points, information
  about the objective tradeoffs.
• Tradeoff information helps to identify the most
  preferred point at the tradeoff curve.
• The question is
• How to apply this experience in the case of mgt2
  ?