# New Graphic Approach to Decision Support for Water Management by d4nR45Y

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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
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
Old simple MCO approaches
A priori restrictions: DM selects the
most important criterion and specifies
restrictions for others: yi  min
*

y  f ( x), x  X , yi  ~i , i  1,2,..., m; i  i*
y
A priori weights: DM specifies weights for
all criteria and solves the problem:
m
 wi yi  max           y  f ( x), x  X
i 1
Classification of modern methods
according to the role of the DM

MCDA

No-preference                 A posteriori
methods                      methods
A priori    Interactive
preference    methods
methods
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.
An example of no-preference methods
For example, the following optimization problem can be
solved:
minimize h (y) = max {  i (yi - yi*) : i=1,2,...,m} on Y,
where  i = 1/(yi** - yi*), i=1,2,...,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
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.
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)
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                                  y  max

v2(y2)=1

v=0            v1(y1)=1          v1(y1)=2     y1
Methods based on direct weighting
of criteria
DM specifies weights for all criteria and
maximizes
m
U ( y )   wi yi  min   y  f ( x), x  X
i 1
decrement in the value of one criterion can be
compensated by the value of another criteria
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
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
Goal identification - 1
• DM has to identify the goal without information
on the set Y=f(X).
y2

0                                     y1
Goal identification - 2
Then, by using some distance function, the closest
point of the set Y=f(X) is found.
y2

y0     Y=f(X)

0                                      y1
Goal identification - 3

The goal programming is the most often used
MCDA technique, but if the goal is distant from
the feasible criterion set Y=f(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.
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.
General scheme of an interactive method
After the k-th iteration, the vectory ( k )  f ( x ( k ) ) 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.
( k 1)
DM specifies parameters                  of a new
optimization problem h( f ( x), (k 1) )  max
while x  X
• Stage 2. Computer solves the problem; the new
decision and criterion vector are computed
.                ( k 1)     ( k 1)
y       f (x    )
The simplest interactive method
The method is based on application of the linear
function h (y) = <c, y>. An iteration of the
method consists of two steps:
a) Computer finds the decision x0 from the set X
that provides the maximum of the linear function
h (f(x)) = <c, f(x)> over X with some given vector
of parameters c <0;
b) 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.
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 y  y results in h(y’’) > 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 ?
Well-known examples of scalar
functions of criteria

1) Linear function h (y) = <c, y>, where c <0
(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 {  i (yi - yi*) : i=1,2,...,m},
where  i : i=1,2,...,m, are some non-negative
coefficients, is a decreasing function in respect to
Slater domination.
Properties of the linear function

1) The maximum over Y of the linear function
h (y) = <c, y>, where c <0, 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) = <c, y> with
some non-positive c only if the EPH is convex.
Example
y2

P(Y)

f(X)

c
y1
Properties of the Tchebycheff distance
2) The point y0 of the minimum over Y of the
Tchebycheff distance
h (y) = max {  i (yi - yi*) : i=1,2,...,m},
where  i : i=1,2,...,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.
Example
y2

P(Y)

f(X)
y*

y1
A posteriori preference methods
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.
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?
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
y2      A

Y

B                  C
P(Y)
S(Y)
y1
Stability of the Pareto frontier-2
P(Y) for the disturbed feasible set in criterion space

A

Y

B

P(Y)                         C
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).
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
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
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.
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.
Parametric LP problem for two criteria

y1  (1   ) y2  min y  Cx, Ax  b
where          changes from 0 to 1.
The problem is solved by using a method
for solving the parametric LP problems
to the list of
y2   objective
points,
picture was
provided!

y1
Different methods
Restrictions-based method
y2

y1
Restrictions-based method: formal
description

yi*  min y  f ( x), x  X ,
yi  li , i  i*, p  0 ,1,...,Pi
p

The result: a large list of points of the Pareto
frontier
Weighted Tchebycheff metric as the
distance from the ideal point
y2

y*                   y1
Formal description
The problems

h( y)  maxi ( yi  yi *) : i  1,2,...,m  min
y  f ( x), x  X
are solved for a large number of parameters
 i  0,    i 1  i  1
m

Result: a large list of Pareto points.
Parametric LP methods for linear
problems with m>2

Direct development of idea by Gass and
Saaty: parametric LP methods for m>2
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).
Evolutionary (including genetic)
multiple criteria optimization
y2

y1

Result: a large list of quasi-Pareto points.
Approximation of the bi-criterion Pareto
frontier by linear segments
NISE (Cohon, 1978)

y2

Picture is
provided
to the DM!                                 y1

The preferred point of an approximation is identified
by the DM.
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