METHOD FOR THE BEST SOLUTIONS SEARCH
IN
MULTIOBJECTIVE DECISION PROBLEMS
V . M . O z e r n o i a n d M.G. G a f t
I n s t i t u t e o f C o n t r o l Sciences
b1 P r o f s o y u z n a y a , Moscow
Abstract
p r o b l e m s a c c o m p a n i n g them
development
T h i s paper d e s c r i b e s a method f o r c o n -
very important
structing a decision rule providing the
i s o l a t i o n o f t h e most p r e f e r a b l e
ons
in multiobjective
rules.
for
Criteria
any a b s o l u t e s e n c e ,
decision
o f m e a n i n g - f u l n e s s and
consistency
of received
formulated.
The r e s u l t s a r e g i v e n o f
a p p l y i n g t h e method t o
most
problems
s o l u t i o n of the
search
concerning the mineshaft
layout.
Introduction
D e c i s i o n t h e o r y w h i c h b e l o n g s t o the
area
of
artificial
intelligence
quickly
developing.
closely
related to
problem s o l v i n g ,
i d e n t i c a l processes.
Problem s o l v i n g i m p l i e s the
means t o
is
D e c i s i o n making i s
though these are not
achieve the
search of
clearly visible
g o a l which n e v e r t h e l e s s cannot be a c h i e ved d i r e c t l y
general
[1];
the d e c i s i o n process i n
implies the
l e means o f
search of
achieving the
a l l possib-
Bet o b j e c t i v e ,
p r e f e r e n c e c o m p a r i s o n among t h e m and
choice of the best
The most
one.
important
s p e c i f i c of
decision-making problems
i s t h e presence
o f a d e c i s i o n - m a k e r who h a s t o a c t i n
the
s i t u a t i o n which is
characterized by
l a c k o f i n f o r m a t i o n about
ment,
the
environ-
a b o u t t h e p o s s i b l e outcomes o f t h e
d e c i s i o n s and a b o u t t h e v a l u e s o f some
outcomes.
It
i s not
functions
(optimal
only that
criteria)
objective
i n these
problems are not
stated but
existence
indisputable.
is
not
The n e e d t o
problems
and t h e
also t h e i r
s o l v e complex p r a c t i c a l
interest
to
is
decision theory
of a d e c i s i o n
but
one i n
only f o r a given
d e c i s i o n maker
in connection w i t h the
set o b j e c t i v e .
Thus t h e t w o m a i n p r o b l e m s
of the d e c i s i o n theory are as f o l l o w s :
information are
preferable alternatives
that
It
that could be considered the best
decision problems.
constructing
to q u i c k
decision theory.
postulates nonexistence
soluti-
The m e t h o d i s b a s e d o n t h e i t e r a t i v e
procedure
of
led
scientific
357
1) to s u b s t i t u t e man in r e p e a t i n g
r o u t i n e d e c i s i o n problems (programmable
problems);
2) to h e l p man in complex and
u n c e r t a i n s i t u a t i o n s to choose d e c i s i o n s
c o n s i s t e n t w i t h h i s preference judgments
u s i n g f o r m a l i z e d models and methods
which allow a v o i d i n g mistakes i n long
and complex chains of l o g i c a l arguments
(unprogrammable problems).
Decision t h e o r y must provide t h e
b a s i s f o r development of models and methods whereby a l l the i n f o r m a t i o n on the
problem, i n c l u d i n g the d e c i s i o n maker's
preference judgements i s used i n order
to decide which of the a l t e r n a t i v e
courses of a c t i o n s is the b e s t . When d e c i s i o n problems are f o r m a l i z e d d i f f i c u l t
questions such as a n a l y s i s of complex
and u n c e r t a i n s i t u a t i o n s become concept u a l l y equivalent t o simple problems
t h a t can be solved by "common sense".
Presently i t i s universally
acknowledged t h a t in most important r e a l
l i f e d e c i s i o n problems the s o l u t i o n
q u a l i t y cannot be estimated by one scalar
- v a l u e d performance c r i t e r i o n . Here
a r i s e s the problem of e s t i m a t i n g and
comparing a l t e r n a t i v e s t a k i n g i n t o c o n s i d e r a t i o n a great number of c r i t e r i a
( m u l t i c r i t e r i a o r m u l t i o b j e c t i v e decision
problems[2j. Themain d i f f i c u l t i e s in devel o p i n g f o r m a l i z e d d e c i s i o n methods which
could be applied to a wide range of pract i c a l problems are caused by the fact
that multiobjective decision problems
are ill-structured [3], and in order to
formalize them successfully one must consider a lot of factors such as sociological , psychological, measurement-theoretic,
etc. Disregard of these factors at the
stage of formalizing the process of decision-making shows up as distrust on
the part of the decision-makers of the
results of formalized methods which are
thus downgraded.
When stating and solving multiobjective problems it is essential to
consider a great number of meaningful circumstancies and ideas which
cannot be easily given a s t r i c t mathemat i c a l motivation. Another d i f f i c u l t y is
in what c r i t e r i a must be considered in a
specific problem, whether a l l feasible
solutions are considered, etc. These
questions must be solved by a l l means
and a l l of them are outside the mathemat i c a l formulation of the problem. At the
same time working out the methods of
choosing most preferable alternatives is
impossible without using mathematical
methods which are applied to formalized
models rather than to real problems. Thus
in model approach to solving certain
multiobjective problems meaningful considerations which cannot be s t r i c t l y f o r malized have to co-exist with formalized
methods and to be used in combination
with them. It is only in interaction that
they can lead to useful results.
The development and use of a model
it possible to get ordered sets of feasible solutions, provided that the sets are
consistent with the model used. At the
same time it must be stressed that there
is no "right" or "objective" model for
any specific multiobjective problem. In
connection with the fact that in each
problem there is a great number of factors which may be taken into or l e f t out
of consideration, there always exists
358
the possibility to present the same
situation by different models. One can
find whether these models are f i t or not
only using them in real situations.
The process of developing m u l t i c r i t e r i a models in decision problems may be
broken into the following series of
stages: 1) goal formulation and problem
type identification: 2) working out a
feasible solutions set; 3) working out a
c r i t e r i a set; 4) c r i t e r i a scale development; 5) mapping of feasible solutions
set in the set of vector-valued estimates; 6) identification of a decisionmaker's preference judgments; 7) the decision rule construction. The specifics
of the f i r s t six stages of constructing
multicriteria models and means of their
implementation are described in [4]. We
shall discuss in more detail the questions connected with constructing a decision rule.
Construction of Decision Rules
A number of various decision rules
have been proposed, and each of them has
some shortcomings which considerable
l i m i t the sphere of i t s possible application [5-9] . Multicriteria problem
analysis drives one to the conclusion
that the construction of a generally
applicable decision rule appears to be
absolutely impossible. The following
fact may account for i t . Depending on
the decision-makers goals his preference
judgments and the set of assumptions
used in the model, different decision
rules may be constructed that naturally
lead to different ordered sets of feasible solutions [4].. The impossibility of
constructing a generally applicable decision rule requires development of a
method for constructing decision rules
leading to the needed result in every
specific situation.
D i f f i c u l t i e s of finding out preference judgments of the decision-maker
are responsible for the iterative nature
of the decision rule construction procedure involving stage-wise acquisition of
information about decision-maker's preference judgments [8] . Information received
at each stage must be used to construct
an intermediate decision rule on the
basis of which decisions are ordered.
Procedure of constructing the decision
rule leading to the problem solution must
be organized in such a way that the
received sequence of intermediate decision rules has the following characteri s t i c s : the i n i t i a l (weakest) one is
based on the simplest and most evident
assumptions;
the following rules follow
from the previous ones because we make
extra assumptions that do not contradict
the earlier ones and receive extra i n formation. If an intermediate decision
rule leads to the desired result it is
accepted as the f i n a l and t h i s is the
end of the procedure,
Basic Principles
In spite of the fact that decision
rules may be constructed in different
ways depending on the problems at hand,
one may formulate principal requirements
that every iterative procedure of such
a kind must meet:
a) Additional information on preferences, received by the researcher from
the decision-maker at the
i+1-th step
of the procedure must allow establishment of the preference at least between
two vector-valued estimates that cannot
be compared at the i - t h step (meaningfulness criterion of additional information^
b) The preference between any two
vector-valued estimates received at i - t h
step must not change during the subsequent steps (consistency criterion of
adopted assumptions and addittional i n formation) .
When
comparing vector-valued e s t i -
mates, the use of decision rule defines
359
uniquely a binary relation (quasi-order
in the general case) on the set Y of
vector-valued estimates and, thus, on the
set of feasible solutions.
estimates changes over one c r i t e r i o n
Assertion 1 is needed to prove
whether the information received from
the decision-maker in practical s i t u a t i ons is noncontradictory.
Let the expression
denote
the subset of vector-valued estimates
corresponding to feasible solutions.
Definition 2. The subset
scale in comparison with estimates changes over another
criterion scale on the
decisions value in general.
3) Information on the effect of
estimates changes over the scales of
c r i t e r i a belonging to one group in comparison with the estimates changes over
the scales of c r i t e r i a belonging to
another group on decisions value in gener a l ; at the same time at least one of the
compared groups must contain more than
one scale.
Information of each type is c a l s s i f i e d with due regard to the degree of
i t s effect on elimination of vectorvalued estimates incomparability. The sef
of a l l possible combinations of the r e sultant ordered information classes may
be presented as a flowchart which determines the sequence of obtaining d i f f e rent types information on decisionmaker's preference judgments. The c l a s s i f i c a t i o n suggested in [8] allows finding
a decision rule for each type of i n f o r mation presented by the flowchart.
Application
Information on Decision Maker's Preferences
The suggested procedure of constructing the decision rule was applied to
When organizing a decision rule
generating procedure it is essential to
define and substantiate a sequence in
which different kinds of information on
the decision-maker's preferences are
received. Thus it becomes necessary to
make a classification of such informat i o n . Reference [8] classifies the i n formation in terms of i t s relation with
one or several c r i t e r i a . This approach
makes it possible to distinguish three
types of information:
1) Information on the effect of
estimates changes over one criterion
scale on the decisions value ( u t i l i t y )
in general.
2) Information on the effect of
developing the method of isolating the
needed amount of the most preferable
versions of a mineshaft layout
[10].
When
designing coal mines the number of
feasible layout versions may be as high
as tens of thousands. At the same time
the mineshaft layout is such a complex
plant that the number of i t s versions
cannot be estimated by one scalar-valued
performance c r i t e r i o n . On the basis of
the detailed design examination which
consumes much time no more than 20-30
versions may be studied. Thus the selection of the best design version through
a detailed design examination is to a
great extent determined by the versions
under study.
360
The developed method f o r most p r e f e r a b l e s o l u t i o n s search makes i t p o s s i b l e a t the e a r l y stages o f d e s i g n i n g :
- to work out a l l f e a s i b l e l a y o u t
v e r s i o n s f o r a wide range of m i n e r a l g e o l o g i c a l f a c t o r s on the b a s i s of morphological analysis;
- t o estimate a l l versions w i t h
respect to 25 n o n a n a l y t i c a l t e c h n o l o g i c a l
and economic c r i t e r i a ;
- to compare a l l f e a s i b l e v e r s i o n s
and i s o l a t e from them the desired
number of most p r e f e r a b l e ones.
When developing the method the d e c i s i o n r u l e was c o n s t r u c t e d i n t h r e e s t a ges. A t the f i r s t stage only i n f o r m a t i o n
on d i f f e r e n t estimate preferences over
each c r i t e r i o n scale was used; at the
second stage e x t r a i n f o r m a t i o n was
received in terms of some c r i t e r i a on
the preference of estimates change over
the scale of one of them from the best
estimate to some other in comparison
w i t h s i m i l a r estimates change over the
scale o f another c r i t e r i o n ; a t the t h i r d
stage i n f o r m a t i o n on u t i l i t y changes
r e l a t i o n s corresponding t o estimates
changes both over one c r i t e r i o n scale
and over some d i f f e r e n t c r i t e r i a scales
was r e c e i v e d .
The method of search f o r the most
p r e f e r a b l e d e c i s i o n s was repeatedly
a p p l i e d t o d e s i g n i n g mines f o r c e r t a i n
c o a l d e p o s i t s . I n one problem the i n i t i a l set
contained 7704 f e a s i b l e v e r sions of the l a y o u t ; from these 21 v e r sions were chosen as most p r e f e r a b l e .
This subset contained l a y o u t s t h a t had
been disregarded e a r l i e r as w e l l as
those t h a t had been u s u a l l y included in
the number of the best ones.
At the same time some v e r s i o n s
which were t r a d i t i o n a l l y used in d e s i g n i n g were not i n c l u d e d in the number of
most p r e f e r a b l e ones. The subsequent
research and design examination f u l l y
confirmed the wisdom of t h a t s e l e c t i o n .
361
A n a l y s i s of the r e s u l t s of the methodology a p p l i c a t i o n to c e r t a i n problems
s o l u t i o n shows t h a t i t allows c o n s i d e r a b le saving of time and cost of design
work and improvement of the q u a l i t y of
t h e d e c i s i o n s made. The methodology
allows s u b s t i t u t i o n of man in choosing
the most p r e f e r a b l e system v e r s i o n s at
e a r l y stages of d e s i g n .
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Business and I n d u s t r i a l Problem
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