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 1. General aspects of decision making.
Decision screening. Decision making with multiple
objectives
Plan of the lecture 1. Main phases of decision
making. Decision screening 2. Multi-objective
versus single-objective optimization 3. Main
concepts of multi-objective optimization 4.
Example of Pareto frontier visualization using IDM
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Four main phases of decision making (Herbert
Simon, 1960)

• In the book of Nobel prize winner Herbert Simon
  The New Science of Management Decision', 1960,
  the decision making process is split into four
  main phases
• intelligence, design, choice and review.
• Intelligence concentrates on identification of
  the decision problem and collection of related
  information.
• Design is concentrated on developing a relatively
  small number of decision alternatives that must
  be studied in details.
• Choice is related to selecting a decision
  alternative from the list of alternatives
  prepared at the design phase.
• The final phase, review, is actually the phase of
  implementation of the selected decision and
  obtaining additional experience in this process.
• Thus, the decision making is actually split into
  two stages
• designing a relatively small number of decision
  alternatives, and
• final selecting a decision alternative from a
  short list.

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Main phases of decision makingdecision
screening and final decision making
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Main difference between stages

• Two stages of the decision making have different
  features.
• In the designing stage, selecting a small number
  of the interesting decision alternatives from a
  large (or even infinite) number of possible
  decision alternatives (decision screening) is
  carried out. The procedure can be based on
  relatively rough models that, however, must be
  applied to a very broad set of possible
  decisions.
• The stage of final selecting is devoted to
  choice of the best decision alternative from a
  short list of decision alternatives. The
  procedure must be based on application of the
  most precise adequate models and data for the
  detailed analysis of several alternatives.

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• The course of lectures is devoted to the new
  multi-criteria visualization-based technique
  the Interactive Decision Maps, which is applied
  at the first stage of the decision process,
  namely, for decision screening, i.e. selecting a
  small number of interesting decision
  alternatives, which will be studied during the
  final choice. Thus, relatively rough simplified
  models are applied in our research.

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Often, optimization is considered as a tool for
decision screening.However, one criterion is not
sufficient in various decision problem to
describe different interests related to the
decision. Say, environmental problems are
characterized by at least two criteria cost and
environmental quality. Thus, multi-criteria
(multi-objective) methods must be used.Let us
compare single criterion and multi-criteria
optimization
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Single-criterion optimization

f(x) objective function
optimization criterion
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Optimization problem. Find
Or, find
The problem is denoted as
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Multi-objective (multi-criteria) optimization
However,
f(x) objective vector function, y criterion
vector
What does it mean? It means that less is better
than more for all partial criteria. It is not
sufficient for selecting the unique decision.
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Feasible objective points for water quality
improvement projects cost (F) versus pollution
(Z5)
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Conclusions 1) The frontier of the variety of
possible outcomes is of interest 2) Decision
maker is needed to select the best point of the
frontier 3) Mathematical methods are needed to
construct the frontier.
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Examples of decision problems with multiple
criteria

• Design of environmental projects
• Water management
• National economic development
• Corporate planning
• Machinery design (design of airplanes, cars, etc.
  as well as of their parts)
• Etc.

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Main concepts of multi-criteria
(multi-objective) optimization
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Decision maker

• The decision maker (DM) is a person responsible
  for the decision making. Usually DM is a
  convenient abstraction since many different
  people (advisers, experts, analysts, various
  stakeholders) influence (or try to influence) the
  decision. However, the concept of the DM is used
  in MCDM field.

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Pareto domination (minimization case)
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• Mathematically speaking, in minimization problem,
  the point is better, than the point
  (dominates the point ) means the
  following
• It means that the criterion points dominated by
  y are given by the non-negative cone
  with the vertex in the point y.
• Slater (weak Pareto) domination

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Feasible set in criterion space
Yf(X)
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Non-dominated ( Pareto) frontier
                         
 
P(Y)
f(X)
 
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Ideal point and Edgeworth-Pareto Hull
                         
 
P(Y)
f(X)
y
 
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Objective tradeoffs for two criteria

• Objective tradeoff is a value that helps to
  compare two objective vectors.
• As the objective tradeoff between y1f(x2) (y11,
  y12) and y2f(x2)(y21, y22) (assuming y12 y22
  ?0 ), one understands
• For any Pareto optimal points y1 and y2, the
  value T1,2 (y1, y2) is negative because it
  describes the relation between the improvement of
  one objective and worsening of another. Tradeoff
  information is very important for the DM who
  decides which of these points is more preferable.

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Tradeoff rate in the bi-objective case

• For a Pareto-optimal objective vector yf(x),
  in which the Pareto frontier is smooth, one can
  use the tradeoff rate
• where the derivative is taken along the
  Pareto frontier.
• The value of the tradeoff rate informs the DM
  concerning the exchange between the objective
  points if one moves along the Pareto frontier.

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Tradeoffs can be evaluated visually
y2
y1
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Objective tradeoffs for multi-objective case (mgt2)

• Let us consider two objectives number i and j for
  two criterion points y1 f(x2) and y2f(x2)
  (assuming y1j y2j ?0). The value
• is said to be the partial objective tradeoff
  if other objective values are not taken into
  account. In contrast, it is the total objective
  tradeoff if y1 and y2 satisfy y1k y2k for all
  k?i,j.
• It is clear that the partial tradeoff does not
  give multi-objective information for mgt2. In
  contrast, the total tradeoff has more sense but
  it can only be used for a small part of pairs of
  decisions.

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Bi-objective slices

• To give a geometric interpretation of the total
  tradeoff, it is convenient to consider
  bi-objective slices (cross-sections) of the set Y
  (or the set Yp).
• A bi-objective slice of Y is defined as a set of
  such points in Y , for which all objective values
  except two (i and j, in our case) are fixed. The
  slice is a two-dimensional set containing only
  those pair of criterion points y1 and y2, for
  which it holds y1k y2k for all k?i,j. Thus,
  since only the values of yi and yj change in the
  slice, the slice can be displayed in the ( yi,
  yj)-plane.
• Then, the tradeoff can be evaluated visually
  between any pair of points of the slice. Such a
  comparison is especially informative if both
  objective vectors belong to the Pareto frontier.

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Partial tradeoff rate

• Application of bi-objective slices is even more
  important while studying tradeoff rates between
  objective values.
• If the Pareto frontier is smooth in its point y
  f(x), a tradeoff rate becomes a partial
  tradeoff rate defined as
• where the partial derivative is taken along
  the Pareto frontier. Graphically, it is given by
  the tangent line to the frontier of a
  bi-objective slice. The value of the partial
  tradeoff rate informs the DM about the tradeoff
  rate between values of two objectives under study
  at the point y, while other objectives are fixed
  at some values.

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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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Topic of the course of lectures

• Interactive visualization of decision maps for
  informing the decision makers on the Pareto
  frontier in the case of more than three criteria
  can be carried out by using a special technique
  named Interactive Decision Maps (IDM).
  Description of the IDM technique and its
  applications is the main topic of the course of
  lectures.

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Computer demonstration a simple example of
regional water planning
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The problem

• The problem of economic development of the region
  is studied. If the agricultural (to be precise,
  grain-crops) production would increase, it may
  spoil the environmental situation in the region.
  This is related to the fact that the increment in
  the grain-crops output requires irrigation and
  application of chemical fertilizers. It may
  result in negative environmental consequences,
  namely, a part of the fertilizers may find its
  way into the river and the lake with the
  withdrawal of water. Moreover, shortage of water
  in the lake may occur during the dry season.

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• Two agricultural zones are located in the region.
  Irrigation and fertilizer application in the
  upper zone (located higher than the lake) may
  result in a drop of the level of the lake and in
  the increment in water pollution. Irrigation and
  fertilizer application in the second zone that is
  located lower than the lake may also influence
  the lake. This influence is, however, not direct
  irrigation and fertilizer application in the
  lower zone may require additional water release
  from the lake into the river (the release is
  regulated by a dam) to fulfill the requirements
  of pollution control at the monitoring station
  located in point A .

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The model

• The model consists of three sub-models
• model of agricultural production
• water balances and constraints
• pollution balances and constraints.
• The production in an agricultural zone is
  described by a technological model, which
  includes N agricultural production technologies.
  Let xij, i1,2...,N, j1,2, be the area of the
  j-th zone where the i-th technology is applied.
• The areas xij are non-negative and are
  restricted by the total areas of zones

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• The i-th agricultural production technology in
  the j-th zone is described by the parameters
  aijk, k1,2,3,4,5, given per unit area, where
  aij1 is production, aij2 is water application
  during the dry period, aij3 is fertilizers
  application during the dry period, aij4 is volume
  of the withdrawal (return) flow during the dry
  period, aij5 is amount of fertilizers brought to
  the river with the return flow during the dry
  period. Thus, one can relate the values of
  production, pollution, etc. to the distribution
  of the area among technologies in the zone
• where zj1 is production, zj2 is water
  application during the dry period, zj3 is
  fertilizers application during the dry period,
  zj4 is volume of the withdrawal (return) flow
  during the dry period, zj5 is amount of
  fertilizers brought to the river with the return
  flow during the dry period in the j-th zone.

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• The water balances are fairly simple. They
  include changes in water flows and water volumes
  during the dry period. The deficit of the inflow
  into the lake due to the irrigation equals to z12
  ? z14. The additional water release through the
  dam during the dry period is denoted by d. It is
  supposed that the release d and water
  applications are constant during the dry season.
• Let T be the length of the dry period. The level
  of the lake at the end of the dry period is
  approximately given by
• L(T) L ? (z12 ?
  z14 d)/?,
• where L is the level without irrigation and
  additional release, and ? is a given parameter.
• Flow in the mouth of the river near monitoring
  point A denoted by vA equals to vA0(d ? z22
  z24)/T, where vA0 is the normal flow at point A.
• The constraint is imposed on the value of the
  flow vA ? vA, where the value vA is given.
  Thus, the following constraint is included into
  the model
• vA0(d ? z22 z24)/T ?
  vA.

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• The increment in pollution concentration in the
  lake denoted by wL is approximately equal to
  z15/? , where ? is a given parameter.
• The pollution flow (per day) at the point A
  denoted by wA is given by z25 /T qA0 , where
  qA0 is the normal pollution flow. It means that
  we neglect the influence of fertilizers
  application in the upper zone on pollution
  concentration in the mouth.
• Then, the value of wA equals to (z25 /T qA0 ) /
  vA .
• The constraint wAwA where wA is given is
  transformed into the linear constraint (z25 /T
  qA0 ) wA vA or
• z25 /T qA0 wA( vA0(d ? z22
  z24)/T).

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Decision variables and criteria

• The decision variables are allocations of land
  between different technologies in the
  agricultural zones as well as the additional
  water release through the dam.
• For the criteria, any collection of the variables
  of the model can be used.

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Demonstration of the IDM softwareexploration of
the formulated problem