(Interactive?) Pareto Frontier Visualization (Chapter 7(?) of the Dagstuhl book)

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(Interactive?) Pareto Frontier Visualization(Cha
pter 7(?) of the Dagstuhl book)

• A.V. Lotov
• Dorodnicyn Computing Centre of Russian Academy of
  Sciences, and
• Lomonosov Moscow State University, Russia

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Concerning my research after the Dagstuhl meeting
in November 2004

1. My department has won a tender of Russian Agency
   for Science and Innovations for developing hybrid
   methods for approximating and interactive Pareto
   frontier visualization for the case of 3 to 8
   criteria for non-linear models given as black box
   and characterized by several hundreds inputs.
   Approximation quality measures and stopping rules
   must be provided. Software must be implemented at
   a combination of personal computer and
   multiprocessor systems. The job was completed in
   October 2006.
2. Since 2006, we take part in a project for Web
   support of participatory municipal budget
   planning in Spain supported by the government of
   Madrid. Support is based on Pareto frontier
   visualization in Web.

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Why visualization of the Pareto frontier as a
whole is needed? (Is it an illustration or
practical decision support tool)?
4
Notation
X feasible set in decision space,
Zf(X) feasible set in criterion space
Pareto domination
Non-dominated (efficient, Pareto) set
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Feasible set in criterion space
Zf(X)
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Pareto domination (minimization case)
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Non-dominated (Pareto) frontier
Zf(X)
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Visualization is a transformation of symbolic
data into geometric information. About one half
of human brains neurons is associated with
vision, and this fact provides a solid basis for
successful application of visualization for
transformation data into knowledge. A picture
is worth a thousand words.
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Visualization for illustration of usual goal
programming
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Goal identification - 1

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

z
0
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Goal identification - 2

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

Zf(X)
z0
z
0
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Visualization in decision support

• In MCDA problems, visualization can provide
  geometric information concerning both the
  feasible criterion values and the objective
  tradeoffs
• total objective tradeoff
• local tradeoff rate for a smooth frontier

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Non-dominated (Pareto) frontier and the objective
tradeoff rate
f(x)
f(x1)
f(x2)
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Goal identification at the Pareto frontier

• For a decision maker, criterion tradeoff
  information is important for identification of a
  preferable non-dominated feasible criterion point
  (goal) directly at the non-dominated frontier by
  using the computer mouse.
• Such a goal can be used as a reference point that
  is close to the Pareto frontier

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Pareto frontier and the feasible goal
Zf(X)
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A feasible goal (or its neighborhood) can be used
as the starting information in various
procedures, say, in rules formulation or in
selecting a part of Pareto frontier for
subsequent study
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Can the user identify a goal?

• Real-life decision making proves that people like
  to use the goal approach.
• It means that they are able to identify a
  preferable criterion point.
• Visualization of the Pareto frontier provides an
  opportunity to identify the feasible
  non-dominated goal. Due to the feasibility, the
  traditional problem of specifying the distance
  between the goal and the feasible criterion set
  vanishes.

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One quotation

• In a general bi-criterion case, it has a sense to
  display all efficient decisions by computing and
  depicting the associated criterion points then,
  decision maker can be invited to identify the
  best point at the compromise curve.
• B.Roy
• Decisions avec criteres multiples.
• Metra International, v.11(1), 121-151 (1972)

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Thus, the question is is it possible and is it
profitable to visualize the Pareto frontier in
the case of more than two-three criteria?
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Visualization and psychological aspects of
decision making
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Psychological aspects of thinking
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Important feature of the three-level model of
human mentality

• The three levels have different pictures of the
  reality, and much efforts of the human mental
  activity is related to coordination of the
  levels. The conflict between mental levels may
  result in non-transitive answers concerning their
  preference.

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To settle the conflict between levels, time is
required. In his famous letter, Benjamin
Franklin advised to spend several days to make a
choice. Psychologists assure that sleeping is
used by the brain to coordinate the mental
levels. (Russian proverb The morning is wiser
than the evening).
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Pareto frontier methods and visualization
Standard approach of the Pareto frontier methods
approximating the set P(Y) by a subset of its
points and informing DM concerning such a list of
points. However, selecting from large lists of
multi-objective points is too complicated for a
human being (O. Larichev. Cognitive Validity in
Design of Decision-Aiding Techniques. Journal of
Multi-Criteria Decision Analysis, 1992, 1(3).)
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Visualization of the Pareto frontier can help in
transformation data into knowledge (in formation
of mental picture of the MCDA problems ). Since
visualization can influence all levels of
thinking, it can support the search for a
decision, that is not logically perfect,
acceptable for all levels of human mentality.
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Pareto frontier visualization
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Preliminary remarkStability (robustness) of the
Pareto frontier and correctness of its
approximation problem is not guaranteed
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Example Slater S(Z) and Pareto P(Z) frontiers
for the non-disturbed feasible set in criterion
space
A
Z
B
C
P(Z)
S(Z)
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Stability (robustness) of the Pareto frontier

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

A
Z
B
C
P(Z)
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If some natural requirements hold, the condition
S(Z) P(Z) where Z is the non-disturbed
feasible set in criterion space, is the
necessary andsufficient condition of stability
of P(Z) to the disturbances of parameters.
(Sawaragi Y., Nakayama H., Tanino T., 1985).
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Edgeworth-Pareto Hull
Let
Then
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Stability of the Edgeworth-Pareto Hull

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

A
Zp
Z
B
C
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Stability of the Edgeworth-Pareto Hull

• Disturbed EPH (Zp )

A
Zp
Z
B
C
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Normally, the Edgeworth-Pareto Hull is stable to
the disturbances of parameters of the problem.In
linear case, disturbances can even be estimated.
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Classification of the MCOproblems related
to visualization procedures(in accordance to
the number of decision alternatives and criteria)
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0) bi-criterion case1) finite number of
decision alternatives a) small number of
non-dominated decision alternatives (not greater
than a dozen) b) medium number of
alternatives (not greater than about 1000) c)
large number of alternatives (greater than about
1000).
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2) infinite number of decision alternatives
a) approximation is given by a number of
criterion points (both classical and EMO) b)
tools for a convex case with polyhedral
approximation the case of more then three
criteria.
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Visualization inspired by the bi-criterion case
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The first Pareto frontier method in MCO
generating the Pareto frontier in linear
bi-criterion problem (S.Gass and T.Saaty, 1955).
They used parametric LP method for the linear
system
where changes from 0 to 1.
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Picture was provided!
 

         
The feasible criterion values are provided along
with the objective tradeoffs including local
tradeoff rates as well as the tradeoffs between
any criterion points.
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The main problem in the two-criterion case is
related to the methods for Pareto frontier
approximation, which we do not discuss here.
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Visualization in the case mgt3
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Decision maps

• The technique tries to use the advantages of the
  bi-criterion visualization (including
  visualization of local objective tradeoff rate
  and direct identification of the goal) in the
  case of three or four criteria.
• A series of values of the third (and, may be,
  fourth) criterion is specified (or several
  constraints on their values are imposed).
• Then, a series of bi-criterion graphs for the
  first and the second criteria is constructed and
  displayed in the same graph. Such an approach was
  known even in the 1970s.

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Recent example (from a paper of A.Mattson and
A.Messac)
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Such graphs are known as the decision maps. They
show local objective tradeoff rates for two
criteria and total tradeoffs for criterion points
in different bi-criterion graphs. Thus, decision
maps provide graphic information on tradeoffs
between all three (or even four) criteria.
However, such decision maps cannot be displayed
interactively since they require substantial time
to be computed especially if the models of the
decision situation are complicated.
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Visualization in the case of a finite number of
alternatives
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Small number of non-dominated alternatives (not
greater than a dozen)
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Bar chart (histogram) 4 alternatives, 6 criteria
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Value paths of 8 alternatives for 22 criteria
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Value paths of 48 alternatives
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Radar diagrams for 4 alternatives and 20 criteria
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Medium number of alternatives (not greater than
about 1000)
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Value paths graph for thousands of alternatives

• It is a usual value paths graph, while all paths
  are given with the same color.
• User can selects a part of the paths to be given
  in a contrasting color and explore them. If a
  small number of alternatives is actually studied,
  one can analyze them using other alternatives as
  a background, which provides the feasibility
  properties for the whole set of alternatives.
• (Example from the paper of R.M.Cooke and J.M. van
  Noortwijk follows).

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Method by Sanaz Mostaghim (several hundreds of
Pareto points)
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Scatterplot Matrix

• It is an attempt to display criterion points for
  the case of three, four and more criteria by
  their projections at all possible two-criterion
  planes.

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Example of scatterplot matrix
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Overlay of three scatterplots
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Disadvantage of Scatterplot Matrix

• It is important that such an approach may be
  misleading.
• Consider a simple example with three criteria (to
  be maximized) and 7 non-dominated criterion
  points. The points were selected in such a way,
  that all projections at two-criterion planes look
  just the same.

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Consider a short list of non-dominated points
point Criterion 1 Criterion 2 Criterion 3
1 0 1 1
2 1 0 1
3 1 1 0
4 0.2 0.2 0.8
5 0.2 0.8 0.2
6 0.8 0.2 0.2
7 0.4 0.4 0.4
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A two-criterion plane
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The point 7 is non-dominated and balanced, it
may happen to be the preferred one. However, it
is deep inside all the projections, and so the
user may not recognize its merits. Thus, the
simple-minded projecting at the criteria planes
may result in misunderstanding of the Pareto
frontier.
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Interactive visualization by bi-criterion slices
of the EPH

• Let us consider N criterion points (N is about
  several thousands)
• z1, z2,, zN
• Let the number of criteria be not greater,
  than 8. Instead of points, their Edgeworth-Pareto
  Hulls (EPH), that is, cones
• can be studied.

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Bi-criterion slices (but not projections!) of the
collection of cones can be displayed. They are
computed and displayed sufficiently fast.
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Overlayed slices for the EPH of the seven
criterion points
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This picture is a decision map, too. However,
interactive exploration of the decision maps can
be provided in the case of more than three
criteria. It means, dozens or even hundreds of
different decision maps can be displayed in
seconds.
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Another example 990 alternative designs of gear
transmission
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Another possible display
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One can see the values and the tradeoffs for
three criteria, the influence of the forth
criterion (fifth, etc., if exist) can be studied
by moving sliders of the scroll-bars.User can
select a different allocation of criteria among
scrollbars and axes.User can hit any point of
the EPH and get the related non-dominated vector.
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Large number of alternatives (examples with 1012
alternatives were studied)
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Interactive visualization of the EPH of the
convex hull of criterion points

• Once again, we consider N criterion points, which
  can be given explicitly or implicitly (mlt8-9)
• For example, they can be given as criterion
  values of integer decisions that belong to the
  feasible set.
• The EPH of the convex hull (envelope) of these
  points is approximated. Then, several
  two-criterion slices (not projections!) of such
  approximation can be computed and displayed
  fairly fast. It results in interactive
  visualization of the Pareto frontier of the
  envelope.

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Let us consider the same example of 990
alternative designs of gear transmission

• One can see 36 two-criterion slices of the EPH of
  the envelope.

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The local tradeoff rates are seen more clearly,
then in approximation by cones. However, it is
an enveloping tradeoff the convex hull is
displayed, which includes additional infeasible
criterion points that simplify the graph. Thus,
the goal is only reasonable (the word
introduced by Stewart, Zionts a.o.)
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Because of it, several feasible alternatives,
which are close to the goal, are provided. One
needs to apply additional efforts to select one
of them. In the following graph, nine feasible
points are selected.
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Another example the graph for 390 625 decision
alternatives
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Convex EPH for 1012 and 5 criteria were
approximated and interactive decision maps were
displayed, too.
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Visualization in the case of infinite number of
alternatives
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Visualization of approximations given by a finite
number of points

• In EMO and in the most of the classical Pareto
  frontier approximation methods, a finite number
  (N) of criterion points approximates the Pareto
  frontier P(Z).
• Thus, one can use the same technique as for
  visualization of a finite number of criterion
  points.
• Say, if Nlt12, value paths can be used.

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If the N is not greater than several thousands
and mlt9, overlapping of two-criterion slices
(interactive decision maps) of the EPH is
possible.Let us consider an example of a
non-linear model that featured 325 input
(control) variables and four criteria. The
Pareto frontier was approximated by 2879 points.
The decision map looked as follows.
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It is important to realize that any finite set of
approximating criterion points (not greater than
several thousands points) can be visualized by
displaying its EPH (using interactive decision
maps). The approximation method plays no role.
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For this reason, a special software for Pareto
frontier visualization was developed Pareto
Front Viewer (PFV).

• To apply PFV, it is sufficient to prepare the
  criterion points in a simple text format.

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Once again, it is important that visualization is
separated from approximation. Thus, visualization
can be carried out on-line.
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Visualization of Pareto frontier approximations
given by a very large finite number of points

• The number N is much greater, than several
  thousands and mlt9, one can apply the interactive
  decision maps for visualization of the Pareto
  frontier of the CEPH, that is, the EPH of the
  convex hull of points. Though the Pareto frontier
  of the envelope is displayed, it can help to
  study the frontier and select some interesting
  points.

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Visualization in the convex case
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Various methods have been proposed for Pareto
frontier approximation in the convex case for
two-three criteria. Though they result in graphs,
we do not consider them here since visualization
in such a case is trivial.
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If mgt3, even in convex case most of the methods
apply approximation of the Pareto frontier by
criterion points, visualization of which has
already been discussed. As an alternative, one
can visualize bi-criterion approximations
(non-interactive decision maps) discussed
earlier. Another approach consists in
visualization based on the polyhedral
approximation.
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The interaction option of the interactive
decision maps makes it possible to apply them in
the case of larger number of criteria, till 7-8.
The following figure shows the decision map for 5
criteria.
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In contrast to the finite number of points, such
decision maps show real local criterion tradeoff
rates. To obtain the interactive decision maps,
however, one needs to have a method for
approximating the convex EPH. Such methods for
mlt7-8 have already been developed in the
beginning of 1980s and are intensively used since
then.
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Interactive Pareto frontier visualization in
interactive MCO procedures
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Pareto Race with Interactive Decision Maps
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Pareto Step Method

• Stage 1 classification

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Stage 2 changing the goal
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Interactive Pareto frontier visualization in Web
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Scheme of the Web server
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Example of the RGDB applet display
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User architecture in commerse
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Application in e-democracy
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Conclusion

• Software tools for Pareto frontier visualization
  for three to eight criteria do exist and can be
  used for constructing efficient decisions by
  identification of feasible (or reasonable) goals.

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Many thanks for your attention!Please send your
comments to Lotov06_at_ccas.ru(in 2006) or to
Lotov07_at_ccas.ru (in 2007)
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Pareto frontier visualization based on polyhedral
approximation in the case of three criteria
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Two main approaches

• Displaying an approximation of the Pareto
  frontier (or of the whole EPH) in the form of
  three-dimensional picture.
• Displaying an approximation of the Pareto
  frontier (or of the whole EPH) in the form of
  the decision map.

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Example of a three-dimensional picture (Pareto
frontier is dashed)
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Example of the related decision map
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To our opinion, decision maps provide more
precise information on feasible criterion values
and especially on criterion tradeoffs.