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 3. Interactive Decision Maps technique

• Plan of the lecture
• Few words concerning the psychology of decision
  making
• Why VISUALIZATION is needed?
• Requirements to visualization
• Why decision maps satisfy the requirements?
• Interactive Decision Maps -- main concepts
• Feasible Goals Method
• The first real-life application of the Pareto
  frontier visualization for a large number of
  criteria

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Features of human thinking

• It is well known that a human being cannot
  simultaneously handle very many objects (it has
  been experimentally proven that the number of
  objects should not exceed the magical number
  seven plus or minus two). This statement is true
  in the case of letters, words, sentences,
  paragraphs and even alternatives. Thus, a human
  being cannot think simultaneously about hundreds
  or thousands objective points of the Pareto
  frontier approximation.

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Famous experiment

• There is a sequence of alternatives A1, A2,
  A3,,Ak, which are described by two criteria an
  important and a less important.
• A2 is much better than A1 in the sense of the
  second criterion, but a bit worse in the sense of
  the second criterion. The same for A3 and A2,
  etc. A human being usually prefers A2 to A1, A3
  to A2, etc., i.e.

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Simple methods used by people

1. Instead of several criteria, people often use
   only two of them. By this, they simplify the
   problem.
2. Another simple method. Even having a long list of
   possible alternatives in front, a human being may
   be unable to find the best one. Instead, he/she
   often somehow selects a small number of
   alternatives from the list and compares them.
   Though the most preferred one may be selected
   from this short list, such an approach results in
   missing most of the Pareto optimal solutions one
   of them may be better than the selected one.

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Mental models

• Studies in the field of human psychology have
  resulted in a fairly complicated picture of a
  human decision making process. In particular, the
  concept of a mental model of reality that
  provides the basis of decision making has been
  proposed and experimentally proven.
• The mental models have at least three levels that
  describe the reality in different ways
• Level of logical thinking,
• Level of images, and
• Level of subconscious processes.
• Preferences are connected to processes of all
  three levels. A conflict between the mental
  levels may be one of the reasons of the
  well-known non-transitive behavior of people
  (both in experiments and in real-life
  situations).

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Structure of mental models
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Coordinating the levels

• A large part of human mental activities is
  related to the coordination of the levels. To
  settle the conflict between the levels, time is
  required.
• Psychologists assure that sleeping is used by the
  brain to coordinate the mental levels. (Compare
  with the proverb The morning is wiser than the
  evening'').
• In his famous letter on making a tough decision,
  Benjamin Franklin advised to spend several days
  to make a choice. It is known that group decision
  and brainstorming sessions are more effective if
  they last at least two days.

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Coordinating the levels in MOO problems

• Thus, to settle the conflict between the levels
  of ones mental model in finding a balance
  between different objectives in a multi-objective
  optimization problem, he/she needs to keep
  information on the problem in his/her brains for
  a sufficiently long time.
• Such opportunity is provided by a posteriori
  methods. The absence of method-related time
  pressure is an important advantage of them.
• In contrast, other approaches require fast
  answers to the questions on preferences.

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VISUALIZATION why it is needed?
Visualization is a transformation of symbolic
data into geometric information that must aid in
the formation of mental picture of the symbolic
data.
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Effectiveness of visualization

• As a proverb says A picture is worth a thousand
  words. Another estimate of the role of
  visualization is given by Wierzbicki and Nakamori
  in their book Creative Space, Springer, Berlin,
  2005. To their opinion, a picture is worth a ten
  thousands words.

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Important featureVisualization can influence
all levels of human thinking!
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Visualization in a posteriori methods

• Since visualization can influence all levels of
  thinking, visualization of the Pareto frontier
  can support the mental search for the most
  preferred Pareto optimal solution. Such a search
  may be logically imperfect, but acceptable for
  all levels of human mentality.
• Visualization of the Pareto frontier can be
  repeated as many times as the DM wants to and can
  last as long as needed.
• The question that we consider is how
  visualization can be effectively used in the
  field of multi-objective optimization, namely, in
  a posteriori methods.

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Requirements that must be satisfied by a
visualization technique

• To be effective, a visualization technique must
  satisfy some requirements, which include
• (i) simplicity, that is, visualization must be
  immediately understandable,
• (ii) persistence, that is, the graphs must linger
  in the mind of the beholder, and
• (iii) completeness, that is, all relevant
  information must be depicted by the graphs.

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Visualization of the Pareto frontier in the
bi-objective case satisfies the requirements
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Tradeoff information given by the graph of the
Pareto frontier in a clear form can be accessed
immediately and, if the frontier is not too
complicated, can linger in the mind of the DM for
a relatively long time. In any case, the one can
explore the curve as long as needed. Finally,
the graph provides full information on the
objective values and their mutual dependence
along the tradeoff curve. Thus, visualization of
the Pareto frontier in the bi-objective case
satisfies the requirements.
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Visualization of the Pareto frontier in the case
of three objectives

• The question is what kind of visualization can
  be used in MOO problems in the case of more than
  two criteria?

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Decision maps

• A decision map is a collection of bi-criterion
  slices of the Pareto frontier. It is a tool for
  visualization of the Pareto frontier in the case
  of three criteria.

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A traditional decision map
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A question

• The question arises
• Why not to display the three dimensional graphs
  instead of the decision maps?
• Let us consider a dynamic model of long-time
  development of a national economy with such
  objectives as economic growth C, maximal (in
  time) pollution level Z and maximal (in time)
  unemployment U.

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An example of a three dimensional graph

• Objective points 1-10 are depicted in the graph

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An example of the related decision map

• Tradeoff curves (bi-objective slices of the EPH)
  are given in the graph values of U are given
  near the associated tradeoff curves.

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Comparison of the three-dimensional graphs with
the decision maps

• One can find much more information on the
  tradeoff curves than in the three-dimensional
  graph (except the tradeoff curve corresponding to
  U 0 that is actually given in the
  three-dimensional graph since it belongs to the
  slice provided by the plane U 0). In the
  decision map, other slices are seen in a clear
  form, too. The tradeoff rates are visible at any
  point of the tradeoff curves. One can easily
  estimate the zones with qualitatively different
  tradeoff rates.
• One can see the total tradeoff between any two
  points that belong to the same tradeoff curve as
  well as the total tradeoff between any two points
  that have the same value of C or Z.

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Properties of the decision maps

• First of all, let us note that tradeoff curves do
  not intersect in a decision map (though they may
  sometimes coincide). Due to this, they look like
  contour lines of topographic maps. Indeed, a
  value of a third objective (related to a
  particular tradeoff curve) plays the role of the
  height level related to a contour line of a
  topographic map.

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• For example, a tradeoff curve describes such
  combinations of values of the first and the
  second objectives that are feasible for a given
  constraint imposed on the value of the third
  objective (like places lower, than...'' or
  places higher, than...''). Moreover, one can
  easily estimate which values of the third
  objective are feasible for a given combination of
  the first and of the second objectives (like
  height of this particular place is
  between...''). If the distance between tradeoff
  curves is small, this could mean that there is a
  steep ascent or descent in values, that is, a
  small move in the plane of two objectives is
  related to a substantial change in the value of
  the third objective.

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Why the requirements are met?

• Thus, decision maps are fairly similar to
  topographic maps. Thus, one can use topographic
  maps for the evaluation of the effectiveness of
  the visualization given by decision maps.
• Topographic maps have been used for a long time
  and educated people usually understand
  information displayed without difficulties.
  Experience of application of topographic maps
  shows that they are
• simple enough to be immediately understood,
• persistent enough not to be forgotten by people
  after their exploration is over, and
• complete enough to provide information on the
  levels of particular points in the map.
• The analogy between decision maps and topographic
  maps asserts that decision maps satisfy the above
  requirements.

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Old methods for constructing the decision maps

• The simplest approach to constructing and
  displaying a decision map is based on a direct
  conversion of the multi-objective problem to a
  series of bi-objective problems one has to
  select any two objectives fi and fj to be
  minimized. Then, the following bi-objective
  problem is considered
• Here the values of
  must be
  given.
• Various methods for constructing
  bi-objective Pareto frontier can be used for
  solving this problem. As a result, one obtains
  the Pareto frontier for two selected objectives
  for given constraints on other objectives. One
  has to note, however, that this is true only in
  the case if the plane does indeed cut the Pareto
  frontier.

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• To get a full picture of the Pareto frontier in
  this way, one needs to specify a grid in the
  space of m-2 objectives and solve a bi-objective
  problem for any point of the grid. For example,
  in case of five objective functions with 10
  possible constraints for any of m-23 objectives,
  we have to construct the Pareto frontier for 1000
  bi-objective problems, which naturally is a
  tremendous task.
• In addition, one has to visualize these 1000
  bi-objective frontiers somehow. For this reason,
  researchers usually apply this approach only in
  the case of m3 or, sometimes, m4 and restrict
  to a dozen bi-objective problems. As to
  visualization, one usually can find a convenient
  way for the displaying about a dozen tradeoff
  curves.
• One can prove that tradeoff curves obtained in
  this way do not intersect for m3 (though they
  can touch each other).

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Interactive Decision Maps (IDM) technique
The IDM technique provides interactive
display of the decision maps for three to seven
criteria. It is based on visualization of
decision maps by overlapping bi-criterion slices
of the EPH approximated in advance. The
IDM technique is the development of ideas of the
NISE method for the case of more than two
criteria.
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Remind that the EPH is given by
or
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Illustration of the EPH
                         
 
P(Y)
f(X)
 
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The principle differences between the IDM
technique and the NISE method are

• the EPH is approximated instead of the Pareto
  frontier
• slices of the Pareto frontier are visualized as
  frontiers of bi-criterion slices of the EPH.

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• The requirements to visualization are met by the
  IDM since it displays the decision maps, which
  satisfy such requirements. In particular, the
  IDM-produced decision maps are complete since
  they can display information on Pareto frontiers
  with any desired precision.
• On-line calculation of decision maps in the IDM
  technique provides additional options. One can
• change objectives located on axes,
• change the number of tradeoff curves on decision
  maps,
• zoom the picture, and
• change graphic features of the display such as
  the color of the background, colors of the
  slices, etc.

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• Since the IDM technique satisfies the above
  requirements, the decision maker can consider the
  decision maps mentally as long as needed and
  select the most preferred objective point from
  the whole set of Pareto optimal solutions. If
  some features of the graphs are forgotten, he/she
  can look at the frontier again and again.

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Feasible Goals Method (FGM)
A preferred point of the Pareto frontier can
be specified by the user directly on computer
display. It is considered as a goal. Since the
goal is feasible, a Pareto-optimal decision can
be found that results in the goal. It can be
found by solving a special single-criterion
optimization problem.
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Single-criterion optimization used in the FGM
Let us consider the multi-objective minimization
problem y f (x)?min, x ?
X. Let y be the goal point specified by the
user. Since the EPH was approximated, the graph
of the Pareto frontier is an approximation, too.
It means that the goal point specified by the
user is only approximately feasible or
Pareto-optimal.
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• For this reason, we consider the goal y as a
  reference point, which is used in a special
  goal-related optimization problem for computing
  the related Pareto-optimal decision
• where y f (x), x ? X, the values are
  small positive parameters. Since the goal y is
  close to the Pareto frontier, the Pareto-optimal
  feasible goal y f (x) is close to the goal
  specified by the user. Such a procedure was
  proposed by A.Wierzbicki in 1981.

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Feasible Goals Method/Interactive Decision Maps
technique

• Combination of such idea with the IDM results in
  the FGM/IDM technique.

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The first real-life application of Pareto
frontier visualization specification of
national goals in the USSR
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• In the 1980s, the State Planning Agency of the
  Soviet Union has started the design of a
  computer-based decision support system for a
  long-term national economy planning. The DSS was
  based on application of the hierarchical system
  of dynamic input-output models that described the
  development of the USSR economy with different
  levels of aggregation.

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The model

• The most aggregated (upper level) model described
  the possible long-time development of the USSR
  production system. It was a dynamic input-output
  model, in the framework of which 17 production
  industries were considered. Yearly outputs of
  production industries were defined to be equal to
  the sum of investments, exports minus imports,
  final consumption, and raw materials consumption
  of all other industries (balance economic model
  of the type proposed by the Nobel prize winner
  W.Leontiev).

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• Balance of the product of the i-th industry,
  where xi is the product, parameters aij are
  coefficients of direct production consumption, is
  yi the final consumption.
• Decision variables xi and invi are non-negative.
  Coefficients ?i describe labor requirements per
  unit of production, values ximax are constraints
  imposed by capital limitations.
• The parameters of the model depend on time.

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• Decision variables, which are time-dependent, are
  related to production of industries (it resulted
  in distribution of the labor force among
  industries), allocation of investments between
  industries, etc.
• Feasible labor statistics were projected, and the
  capacities of production industries depended upon
  investment. The delay between investment and the
  resulting capacity growth was given, its value
  depended on the industry.

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The criteria

• The upper level model was used for identification
  of long-term national social-economic goals for a
  time period of 15 years.
• For the particular goals, values of several
  performance indicators of the national economic
  system were used. They included consumption of
  several population groups, development of health
  care and educational systems, etc. Seven criteria
  were considered in total

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Experience 1

• In the early variant of the DSS, officials of the
  State Planning Agency had to identify the
  particular goals on the basis of their
  experience, without any computer support.
• As a rule, the goals identified by them were not
  feasible. Then, some optimization software was
  used to compute a feasible criterion vector as
  close as possible to the identified goal.
  Usually, such feasible criterion vectors were
  distant from the goals identified by the
  officials, and so it turned out that the
  identified goals had nothing to do with the
  reality.

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• The officials were disappointed with such
  results. After several attempts, they refused to
  use the DSS.
• It seems that the officials regarded such results
  as undermining their prestige since it might be
  attributed to their incompetence.
• It was clear for the DSS developers that an
  additional decision support tool was needed to
  help officials to identify goals that are close
  to feasible performance vectors. They decided to
  use the FGM and the related software.

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Experience 2

• At that time (at the very beginning of the 1980s)
  the State Planning Agency was unable to get
  personal computers, and so the authors had to
  approximate the feasible goals set by the
  mainframe computer and to print out a large
  number of graphs that contained collections of
  bi-criterion tradeoff curves.
• The officials of the State Planning Agency
  studied the album of feasible social-economic
  goals by themselves with his help. Indeed, it
  turned out that the album of graphs worked
  sufficiently well without any support from the
  authors. In total, the officials used the album
  for more than three years.

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• Perestrojka in the USSR destroyed the planning
  system, and application of the DSS was halted.