ISSN 10645624, Doklady Mathematics, 2011, Vol. 83, No. 3, pp. 418–420. © Pleiades Publishing, Ltd., 2011.
Original Russian Text © V.D. Noghin, O.V. Baskov, 2011, published in Doklady Akademii Nauk, 2011, Vol. 438, No. 4, pp. 456–459.
COMPUTER
SCIENCE
Pareto Set Reduction Based on an Arbitrary Finite Collection
of Numerical Information on the Preference Relation
V. D. Noghin and O. V. Baskov
Presented by Academician S.K. Korovin January 21, 2011
Received February 17, 2011
DOI: 10.1134/S1064562411030288
Many applied problems in economics and engi
neering can be formulated as a multicriteria choice
problem with several numerical functions. A specific
feature of multicriteria problems is that, starting a
choice procedure, the decision maker (DM) cannot,
as a rule, precisely express his or her interests and pref
erences, which underlie the choice made. Thus,
beginning the search for a set (in a special case, a sin
gleton) of “best” elements, the DM does not have the
exact definition of this concept. Frequently, these best
elements are detected in the course of decision making
based on available information about the DM’s prefer
ences.
Numerous procedures and methods have been pro
posed for solving multicriteria problems depending on
the type and character of information on the DM’s
preferences [1, 2]. Frequently, these are heuristic pro
cedures that yield substantially different best solutions.
According to the overwhelming majority of research
ers, best solutions have to be sought in the set of Pareto
optimal (effective, tradeoff) alternatives. This circum
stance is expressed in the Edgeworth–Pareto princi
ple, which has relatively recently been axiomatically
substantiated [3]. Thus, the problem of choosing a set
of best alternatives can be reformulated as the problem
of Pareto set reduction.
Consider the model of multicriteria choice [3],
which contains a set X of initial alternatives, a vector
criterion y = f(x) = (f1(x), f2(x), …, fm(x)), and an
asymmetric binary preference relation X defined
on X. Let Y = f(X), and let Y be a binary relation on the
set Y induced by the relation X as follows: x1 X x2 ⇔
f(x1) Y f (x2) for all x1 ∈ x̃ 1 , x2 ∈ x̃ 2 ; x̃ 1 , x̃ 2 ∈ X˜ ,
where X˜ is the collection of equivalence classes gener
ated by the equivalence relation x1 ~ x2 ⇔ f(x1) = f(x2)
St. Petersburg State University, Universitetskii pr. 28,
Petrodvorets, St. Petersburg, 198504 Russia
email: [email protected]
on X. The sets of chosen (best) alternatives and vectors
are denoted by C(X) and C(Y) = f (C(X)), respectively.
These are the sets to be determined in the course of
making a choice. The set of Pareto optimal alternatives
with respect to the vector criterion f on X is denoted
by Pf(X).
The Pareto set Pf(X) can be reduced (i.e., certain
Paretooptimal elements can be eliminated) if we have
additional information on the DM’s preferences. The
most reliable and simple from a practical point of view
is information on the DM’s readiness to make a cer
tain compromise. Frequently, this compromise means
that the DM agrees to lose according to insignificant
criteria while gaining according to significant criteria.
Taking into account information on the DM’s
preference relation, which the DM is usually initially
totally unaware of, underlies an axiomatic approach to
the reduction of the Pareto set. This approach has
been developed for almost three decades [3]. First, it
was established how one should use a single piece of
information (“information quantum”) in the form of
two collections of numbers, of which one indicates the
admissible upper limits of losses for the group of insig
nificant criteria and the other shows the gain values for
the significant criteria that are larger than or equal to
those the DM would agree to receive when making the
compromise. Later, possibilities of using various col
lections of such information (several information
quanta) for Pareto set reduction were studied.
In the geometric language, an information quan
tum about the DM’s preference relation means [3]
that the relation u Y 0 holds for some u ∈ Nm, where
Nm is a set of mdimensional vectors with at least one
strictly positive and one strictly negative component.
The specification of a collection of information
quanta is equivalent to the fulfillment of the relations
ui Y 0, i = 1, 2, …, k.
In this paper, we propose two universal algorithms
(methods) based on taking into account an arbitrary
finite collection of information quanta for Pareto set
reduction. The first (geometric) algorithm was devel
oped by the first author. It involves the solution of the
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PARETO SET REDUCTION BASED ON AN ARBITRARY FINITE COLLECTION
convex analysis problem posed in [3]. The second
(algebraic) algorithm was developed by the second
author. It sequentially takes into account the available
collection of quanta on the basis of the linear operator
used to take into account one quantum in Theorem 3.5
from [3]. In what follows, four reasonable choice axi
oms formulated in [3] are assumed to hold.
The above geometric problem of convex analysis is
stated as follows. Given a finite collection of vectors
a1, a2, …, ak ∈ Rm, k > m, generating an acute convex
mdimensional cone M, the goal is to construct a min
imal (in number) collection of vectors b1, b2, …, bn that
generate the dual {y ∈ Rm | 〈ai, z〉 ≥ 0, i = 1, 2, …, k} of
the cone M. Here, the angle brackets denote the scalar
product of vectors. It should be noted that the sought
vectors b1, b2, …, bn always exist and are determined up
to a positive multiplier.
The following algorithm is proposed for solving this
problem. The collection of vectors a1, a2, …, ak is fed
as input, while the output (in storage) is the desired
collection of vectors.
Algorithm 1
Step 1 (start of a loop over the vectors). Start a loop
m–1
in which all possible (m – 1)
over i from 1 to C k
vector subsets of the collection a1, a2, …, ak are gener
ated.
Step 2 (verification of linear independence). If the
current ith subset ai1, ai2, …, ai(m – 1) chosen from a1, a2,
…, ak is linearly dependent, then increase the number
i by one and go to the beginning of Step 2. If i cannot
m–1
be further increased, i.e., i = C k , go to Step 5. Oth
erwise, i.e., if the subset is linearly independent, go to
Step 3.
Step 3 (construction of an orthogonal vector).
Form a square matrix D of order n from the column
vectors of the subset ai1, ai2, …, ai(m – 1). Preliminarily,
any vector from the set Ii = {a1, a2, …, ak}\{ai1, ai2, …,
ai(m – 1)} that form, together with ai1, ai2, …, ai(m – 1), a
linearly independent system is written to the columns
on the right. Find the last column of the matrix (DT)–1,
where T denotes the transpose. The resulting column
vector (denote by yi) is stored.
Step 4 (verification of whether the vector y i belongs
to the desired set b1, b2, …, bn). Calculate the scalar
products 〈a j, yi〉 for all vectors a j ∈ Ii. If at least one
scalar product is negative, remove y i from the storage.
Increase i by one and go to Step 2 (if this increase is not
possible, go to Step 5).
Remark. To reduce the search and eliminate iden
tical (up to a positive multiplier) desired vectors from
the storage, for each stored vector yi, the correspond
ing collection Yi of all vectors a1, a2, …, ak orthogonal
to y i is stored at Step 4 (i.e., those a j for which
〈aj, yi〉 = 0). At Step 2, if the current subcollection ai1,
ai2, …, ai(m – 1) is a subset of at least one previously
DOKLADY MATHEMATICS
Vol. 83
No. 3
2011
419
formed set Yi, skip this subcollection and increase i
by one.
Step 5 (termination of the algorithm). After exe
cuting a complete cycle over i, the desired column vec
tors b1, b2, …, bn (denoted by yi) are stored.
Theorem 1. The vectors b1, b2, …, bn produced by
finite Algorithm 1 form a minimal collection of vectors
generating a cone that is the dual of the mdimensional
acute convex cone M generated by the collection of vec
tors a1, a2, …, ak ∈Rm, k > m.
Theorem 2. Let four reasonable choice axioms hold
and a consistent collection of information quanta ui Y 0,
i = 1, 2, …, k be given [3].
Then, for any set of selected alternatives C(X),
C ( X ) ⊂ P g ( X ) ⊂ P f ( X ).
(1)
Here, Pg(X) is the set of Pareto optimal alternatives on
the original set X with respect to the new ndimensional
vector criterion g(x) =(〈b1, f(x)〉, 〈b2, f(x)〉, …, 〈bn, f (x)〉),
n ≥ m, where the vectors b1, b2, …, bn are produced by
Algorithm 1 as applied to the collection consisting of vec
tors specifying the information quanta u1, u2, …, uk
together with m unit basis vectors of the space Rm.
According to Theorem 2, when Algorithm 1 is
applied, it is necessary to construct a new vector crite
rion g with respect to which the new Pareto set gives
upper bound (1) for the unknown set of selected alter
natives C(X) in view of the available collection of
information quanta.
The algebraic approach maximally takes into
account the specific features of the given collection
ui Y 0, i = 1, 2, …, k, and is sequential; i.e., informa
tion is taken into account strictly in the order of avail
able indexed quanta.
Given a vector u1 = (u1, u2, …, um), we introduce the
sets A = {i| ui > 0}, B = { j | uj < 0}, and C = {l| ul = 0}. If
the sets A and B are nonempty, the vector u1 generates
an information quantum about the preference relation
if u1 Y 0. To take it into account, Theorem 3.5 in [3]
is used to construct a new vector criterion g whose
components are related to those of the criterion f by
the following relations (the order of the components is
not important):
gi = fi
g ij = u i f j – u j f i
∀i ∈ A ∪ C,
∀( i, j ) ∈ A × B.
In matrix form, these relations can be rewritten as g =
Tf, where T is a rectangular matrix of suitable size.
The criterion g thus constructed generates a new set
of vectors Z = g(X), on which a new relation Z
is induced that is associated with the old as follows:
x' X x'' ⇔ g(x') Z g(x''). In what follows, for brevity,
the index (the symbol of the set on which the consid
ered relation acts) is omitted. If necessary, it can easily
be recovered from the context. It can be proved that
the relation induced on the new set of vectors has all
420
NOGHIN, BASKOV
the properties guaranteeing the fulfillment of four rea
sonable choice axioms.
Now, suppose that we are given a collection of
information quanta ui 0, i = 1, 2, …, k. After taking
into account the first quantum (as described above)
and constructing the linear mapping T, the remaining
vectors ui ∈ Y in the new set of admissible vectors are
assigned their images (vectors) Tui ∈ Z. It can be
shown that ui 0 ⇔ Tui 0, i = 2, 3, …, k. This opens
up the possibility of taking into account each subse
quent information quantum in the image space of T.
For Tui 0, i = 2, 3, …, k, to be treated as relations
specifying information quanta, it is necessary that
each of the vectors Tui have at least one positive and at
least one negative component. If there is a vector with
all nonpositive components, then the original collec
tion of quanta is inconsistent; i.e., the relations ui 0
cannot hold simultaneously within the framework of
the reasonable choice axioms. If among Tui there is a
vector with nonnegative components, then it can be
discarded, since it does not carry additional informa
tion, because, under the indicated axioms, any non
zero vector v with nonnegative components satisfies
the relation v 0.
Thus, the problem of consecutively taking into
account s information quanta on the preference rela
tion is reduced to the problem of taking into account
the preceding s – 1 quanta with the subsequent appli
cation of Theorem 3.5 from [3] to the corresponding
images.
Let us describe Algorithm 2 in consecutive form.
Assume that, by the step indexed by s ≥ 1, s – 1 infor
mation quanta have been taken into account, while the
subsequent quanta Ts – 1…T1ui 0, i = s, s + 1, …, k
remain unaccounted. The lefthand side of the last
relation represents the product of the matrices Ts – 1,
Ts – 2, …, T1 and the vector ui.
Step s.3. To reduce the subsequent computations,
discard the vectors Ts – 1…T1ui ≥ 0.
Step s.4. If among the vectors constructed at
Step s.2, there are ones all of whose components are
nonpositive, then terminates Algorithm 2 with the
message that the original collection of information
quanta is inconsistent.
Step s.5. If there are remaining unaccounted
quanta, go to Step s + 1. Otherwise, terminate Algo
rithm 2 with the output being the matrix TsTs – 1…T1.
Theorem 3. Let four reasonable choice axioms be
satisfied [3]. If the collection of information quanta ui Y 0,
i = 1, 2, …, k, is inconsistent, then Algorithm 2 termi
nates with an inconsistency message. If this collection of
quanta is consistent, then Algorithm 2 terminates, pro
ducing a matrix Q such that, for any set of selected alter
natives C(X), inclusions (1) hold true, where Pg(X) is the
set of Pareto optimal alternatives on the set X with respect
to the vector criterion g(x) = Q f (x). Moreover, the set
Pg(X) is independent of the indexing of information
quanta.
Thus, Algorithm 1 produces a minimal (in the
number of components) vector criterion g for which
inclusions (1) hold. Algorithm 2 generates a new vec
tor criterion g that, generally speaking, differs from
that produced by Algorithm 1, and the dimension of
the last criterion may appear to be noticeably higher
than the minimal one. We intend to continue the study
of the above algorithms, specifically, to modify Algo
rithm 2 so as to eliminate the “redundant” compo
nents of the vector criterion in the course of its opera
tion.
ACKNOWLEDGMENTS
This work was supported by the Russian Founda
tion for Basic Research, project no. 110700449a.
REFERENCES
Algorithm 2
Step s.1. According to Theorem 3.5 in [3], con
struct a linear transformation Ts taking into account
the information quantum Ts – 1Ts – 2…T1us 0.
Step s.2. Construct vectors Ts Ts – 1…T1ui, i = s +
1, …, k.
1. A. B. Petrovskii, Decision Making Theory (Akademiya,
Moscow, 2009) [in Russian].
2. V. D. Noghin, Iskusstv. Intellekt Prinyatie Reshenii,
No. 1, 98–112 (2008).
3. V. D. Noghin, Decision Making in Multicriteria Envi
ronment: A Quantitative Approach (Fizmatlit, Moscow,
2005) [in Russian].
DOKLADY MATHEMATICS
Vol. 83
No. 3
2011