ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2006, Vol. 46, No. 4, pp. 554–562. © MAIK “Nauka /Interperiodica” (Russia), 2006.
Original Russian Text © V.D. Noghin, 2006, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 4, pp. 583–592.
The Edgeworth–Pareto Principle in Terms
of a Fuzzy Choice Function
V. D. Noghin
St. Petersburg State University, Universitetskii pr. 28,
Petrodvorets, St. Petersburg, 198504 Russia
e-mail: [email protected]
Received October 31, 2005
Abstract—The Edgeworth–Pareto principle, the simplest version of which has been known since the
19th century, is stated in general terms of a fuzzy choice function. The application of the principle is
justified for a wide class of fuzzy multicriteria choice problems described by certain axioms of the rational behavior. These results bring the axiomatic substantiation of the Edgeworth–Pareto principle performed previously by the author to the most general form and make it possible to reveal the boundaries
of that class of multicriteria choice problems for which the application of this principle is required.
Based on scalarization methods as applied to multicriteria problems, upper bounds are derived for the
unknown set of selected vectors.
DOI: 10.1134/S096554250604004X
Keywords: Edgeworth–Pareto principle, fuzzy choice functions, axiomatic substantiation, multicriteria
problems
1. INTRODUCTION
According to the widespread (naive) Edgeworth–Pareto principle, an optimal choice in multicriteria
problems must be made within the Pareto set. In [1, 2], this principle was first formulated in the form of an
inclusion and its axiomatic substantiation was performed. It was found that, for a sufficiently wide class of
problems described by certain axioms of the rational behavior of a decision-maker, any set of selected decisions must be contained in the Pareto set. If at least one of the indicated axioms is violated, a selected decision is not necessarily Pareto optimal. In [3], the Edgeworth–Pareto principle was extended to the case when
the values of the criteria are not necessarily numerical.
It is well known that, in applications concerning the best decision making, information received from a
decision-maker and used in decision making is frequently subjective and, hence, fuzzy. In this context, it is
of interest to develop mathematical models that take into account information of this type. Such models are
usually based on the theory of fuzzy sets and relations. For example, a version of the Edgeworth–Pareto
principle was proposed in [4] in the case of a fuzzy preference relation of the decision-maker.
In this paper, all the author’s studies in this direction are summarized and take the most general and complete form. We consider a model of multicriteria choice from a fuzzy set of feasible alternatives. The values
of the criteria involved in the problem are not necessarily numerical. It is only assumed that, in the range of
each criterion, we are given an asymmetric, transitive, and weakly connected relation that is used to compare
alternatives according to that criterion. The result of making a choice from a fuzzy set of feasible alternatives
is generally a nonempty fuzzy subset of this set.
We introduce the concept of a fuzzy choice function, which is used to formulate the rational-choice axioms describing the class of multicriteria fuzzy choice problems for which the Edgeworth–Pareto principle
holds true. According to this principle, any alternative lying outside the Pareto set must have a zero degree
of membership in the selected set. It is shown that this principle can be violated if at least one of the axioms
is discarded. In contrast to the author’s previous substantiation of the Edgeworth–Pareto principle based on
three axioms, only two are used here, namely, the Pareto axiom and an axiom stating that an alternative that
is not selected from a pair cannot be selected from the entire set of feasible alternatives.
Additionally, multicriteria optimization results are used to derive upper bounds for the unknown set of
selected vectors in terms of solutions to various scalar (single-criteria) parametric problems.
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2. MODEL OF FUZZY MULTICRITERIA CHOICE
Suppose that we are given scales (nonempty nonfuzzy abstract sets) Y1, …, Ym (m > 1), which can be
finite or infinite. It is assumed that, on each set Yi, a binary relation i (i = 1, 2, …, m) is defined that is
asymmetric, transitive, and weakly connected. The relation i can be treated as a strict preference relation
on the range of the ith criterion. Recall that the weak connection of i means that, for any two elements s, t ∈
Yi, s ≠ t, we either have s i t or t i s.
Consider the Cartesian product (nonfuzzy set) Ŷ = Y1 × … × Ym. Its elements are referred to as alternatives.
Let A be an arbitrary nonempty subset. A fuzzy set X in A is specified by the so-called membership function λA defined on A and taking values from the interval [0, 1]. The number λA(x) for x ∈ A is treated as the
degree of membership of x in the fuzzy set X. All the elements x ∈ A for which λA(x) > 0 form the support
of the given fuzzy set, which is denoted by suppX.
Ŷ
Definition 1. A fuzzy choice function is a mapping C defined on the set of all nonempty subsets of 2 \ ∅
that assigns to every A ⊂ Ŷ a certain fuzzy set C(A) with a membership function λA possessing the property
λ A ( y ) ∈ [ 0, 1 ]
∀y ∈ A,
λA( y) = 0
∀y ∈ Ŷ \ A.
In the general case, the equality λA(y) = 0 ∀y ∈ Ŷ is possible for some A. This means that the choice
from A is empty; i.e., C(A) = ∅. In other words, for some A, there is refusal to choose instead of making a
choice from that set.
In the special case where λA takes only two values (0 or 1) for every A ⊂ Ŷ , i.e.,
λ A ( y ) ∈ { 0, 1 }
∀y ∈ A,
the fuzzy choice function becomes the usual choice function (see, e.g., [5, 6]).
Let a fuzzy choice function C be defined. Then, a membership function λA is defined for every nonempty
set A ⊂ Ŷ . In particular, this function is defined on all singleton subsets of A. Define a (nonfuzzy) nonempty
set by the equality
supp Y = { y ∈ Ŷ λ { y } ( y ) > 0 }.
It consists of the alternatives that, when presented as singleton sets, can be chosen with a positive degree of
membership, whereas, for any y ∈ Ŷ \ supp Y, it holds that λ{y}(y) = 0.
Below, a further choice is made within the set suppY, since it is senseless to add an alternative that is not
selected on its own to alternatives selected from a larger set. This assumption can be considered one of the
axioms we accept, although it is likely that there are applications in which this assumption may be violated.
3. RATIONAL-CHOICE AXIOMS
The conditions on the fuzzy choice function under which the Edgeworth–Pareto principle always holds
are formulated as axioms. Thus, the rational-choice axioms single out a sufficiently wide class of fuzzy multicriteria choice problems in which a successful choice is necessarily made within the Pareto set. Outside
this class (i.e., when at least one of the axioms is violated), the best choice is not necessarily Pareto optimal.
In what follows, the fuzzy choice function is again denoted by C and the membership function for a fuzzy
set C(A) is denoted by λA.
Axiom 1 (monotonicity of the membership function). For any nonempty subsets Y', Y'' ⊂ Ŷ such that
Y' ⊂ Y'' and for any alternative y ∈ Y', it holds that λY' (y) ≥ λY'' (y).
According to Axiom 1, an extension of the possibility of choice can perhaps reduce the degree of membership of an arbitrary individual alternative in a selected set. In particular, if an alternative y is not selected
on its own (i.e., from the singleton set {y}), it cannot be selected from any other set containing that alternative. If y is selected on its own (i.e., λ{y}(y) = λ* ∈ (0, 1]), then λ* can be regarded as the degree of membership of that alternative in a fuzzy set of feasible alternatives which is denoted by Y in what follows with
the support suppY in which a choice is made. By convention, λsuppY (·) will be written as λY(·) even if Y is not
a nonfuzzy set and we will talk about a choice from Y rather than from suppY.
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Remark 1. It is usually assumed that some fuzzy choice is made from Y. In other words, there must be
at least one alternative y ∈ suppY for which λY(y) > 0. In this case, one, several, or an infinite number of
alternatives with corresponding positive degrees of membership can be selected.
Example 1. Let Ŷ = {a, b, c}. Then
Ŷ
2 = { ∅, { a }, { b }, { c }, { a, b }, { a, c }, { b, c }, { a, b, c } }
and a fuzzy choice function satisfying Axiom 1 can be defined, for example, as follows:
λ { a } ( a ) = 1,
λ { b } ( b ) = 0.7,
λ { c } ( c ) = 0,
λ { a, b } ( a ) = 0.9,
λ { a, b } ( b ) = 0.2,
λ { a, c } ( a ) = 1,
λ { a, c }(c) = 0,
λ { b, c }(b) = 0.6,
λ { b, c }(c) = 0,
λ { a, b, c }(a) = 0.9,
λ { a, b, c }(b) = 0.1,
λ { a, b, c }(c) = 0.
Here, suppY = {a, b} and the fuzzy set Y of feasible alternatives consists of the two elements a and b with
their degrees of membership equal to 1 and 0.7, respectively.
Below is a weaker version of Axiom 1 that involves Y and only one- and two-element subsets of alternatives.
Axiom 1'. For any two alternatives y, y' ∈ Ŷ , it is true that λ{y}(y) ≥ λ{y, y'}(y), λ{y, y'}(y) ≥ λY(y), and
λ{y, y'}(y') ≥ λY(y').
The following axiom is less restrictive (which is easy to verify). It is this axiom that will be used below
to prove the Edgeworth–Pareto principle.
Axiom 1". For any pair of alternatives y', y'' ∈ suppY, such that y' ≠ y'' and λ{y', y''}(y'') = 0, it is always
true that λY(y'') = 0.
The equality λ{y', y''}(y'') = 0 means that y" is not selected at all from the pair y', y". In other words, in this
case, y' is perhaps better than y" or y" is no better than y'. It should be noted that, along with λ{y', y''}(y'') = 0,
it is possible that λ{y', y''}(y') = 0 or λ{y', y''}(y') > 0.
Accordingly, Axiom 1" requires that an alternative that is not selected from a pair cannot be selected from
the entire set Y of feasible alternatives. Note that the last axiom refers only to a choice from a fixed subset
of alternatives, namely, from Y. In fact, this means that, when referring to Axiom 1", we have taken into
account information on choices from the singleton sets of alternatives, as discussed at the end of Section 2.
Axiom 2. For any three alternatives y', y'', y''' ∈ Ŷ satisfying λ{y', y''}(y'') = 0 and λ{y', y''}(y''') = 0, it is
always true that λ{y', y'''}(y''') = 0
Axiom 2 establishes a certain sequence (logicality, rationality) in making a choice: if the first alternative
is perhaps better than the second and the second it is perhaps better than the third, then the first is perhaps
better than the third. In this sense, Axiom 2 actually postulates a kind of transitivity of choice from pairs of
alternatives.
It should be noted that, under certain circumstances, the behavior of a decision-maker can be incompatible with this axiom, since humans sometimes behave irrationally. Examples of this kind have been known
to psychologists for a long time.
Axiom 3. For any two alternatives y', y'' ∈ Ŷ such that
y' = ( y '1, …, y 'i – 1, y 'i, y 'i + 1, …, y 'm ),
y'' = ( y '1, …, y 'i – 1, y ''i , y 'i + 1, …, y 'm ),
y 'i i y ''i ,
it is always true that λ{y', y''}(y'') = 0.
According to Axiom 3, an alternative that is less preferable than the other with respect to any one component, all the other components being the same, is never selected from the given pair. The rationality of
Axiom 3 is beyond doubt.
4. GENERALIZED PARETO AXIOM
Before formulating a requirement that can be regarded as a generalization of the Pareto axiom, which is
well known in multicriteria optimization, we give the following definition.
Definition 2 (see [3]). The binary relation defined on the Cartesian product Ŷ by the equivalence
y' y'' ⇔ [ ( y 'i i y ''i or y 'i = y ''i ) for all i = 1, 2, …, m ]
and
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where y' = ( y '1 , …, y 'm ) ∈ Ŷ and y'' = ( y ''1 , …, y ''m ) ∈ Ŷ , is called the Pareto relation.
Pareto axiom. For any two alternatives y', y'' ∈ suppY related by y' y'', it is always true that
λ{y', y''}(y'') = 0.
The Pareto axiom expresses a certain rule of choice from two feasible alternatives related by the Pareto
relation. According to this rule, if one alternative is preferable to the other with respect to one or several
components, all the other components being the same, then the second alternative is never selected from the
given pair. This is quite natural from the point of view of common sense. The Pareto axiom (more exactly,
its simpler nonfuzzy version) is usually unconditionally accepted by most researchers dealing with n-person
games [7] or with social (economic) choice models [8].
Obviously, the Pareto axiom implies Axiom 3, but the converse is not true.
Lemma. The Pareto axiom is a consequence of Axioms 2 and 3.
Proof. Let y', y'' ∈ suppY be two arbitrary alternatives satisfying the relation y' y''. Without loss of generality, we assume that y' y'' means that, for some positive integer 1 l m,
y 'k k y ''k ,
k = 1, 2, …, l,
y 'k = y ''k , k = l + 1, …, m.
Due to Axiom 3, we have
λ { y', ( y1'', y2' , …, ym' ) } ( ( y 1'', y 2' , …, y m' ) ) = 0,
λ { ( y''1, y'2, …, y'm ), ( y''1, y''2, y'3, …, y'm ) } ( ( y ''1 , y ''2 , y '3, …, y 'm ) ) = 0,
………………………………………………………………………
λ { ( y1'', …, y''l – 1, yl', …, ym' ), ( y1'', …, yl'', y'l + 1, …, ym' ) } ( ( y ''1 , …, y ''l , y 'l + 1, …, y 'm ) ) = 0.
Sequentially applying Axiom 2 to these relations yields
λ { y', ( y1'', …, yl'', y'l + 1, …, ym' ) } ( ( y 1'', …, y l'', y 'l + 1, …, y m' ) ) = 0.
(1)
Since y 'k = y ''k for k = l + 1, …, m, equality (1) becomes λ{y', y''}(y'') = 0. The lemma is proved.
5. THE EDGEWORTH–PARETO PRINCIPLE
Below, we need two concepts directly related to the set of feasible alternatives.
Definition 3. The set of Pareto optimal alternatives (Pareto set) is defined as
P(Y) = {y* ∈ suppY| there is no y ∈ suppY such that y y*}.
Definition 4. The set of nondominated alternatives (denoted by Ndom(Y)) is defined as
Ndom(Y) = {y* ∈ suppY| there is no y ∈ suppY, y ≠ y*, such that λ{y, y*}(y*) = 0}.
It should be noted that both introduced sets are nonfuzzy.
Theorem. For any fuzzy choice function satisfying Axiom 1" and the Pareto axiom, it holds that
λY ( y ) = 0
∀y ∈ supp Y \P ( Y ).
(2)
Proof. Fix an arbitrary fuzzy choice function C satisfying Axiom 1" and the Pareto axiom and having a
membership function λY(·) of the fuzzy set C(suppY).
First, we show that
λY ( y ) = 0
∀y ∈ supp Y \Ndom ( Y ).
(3)
Assume the opposite: there is an alternative y ∈ suppY \ Ndom(Y) for which λY(y) > 0. By the definition of
the set of nondominated alternatives, there is y' ∈ suppY such that y' ≠ y and λ{y, y'}(y) = 0. By Axiom 1", the
last equality implies λY(y) = 0, which contradicts the initial assumption λY(y) > 0. Thus, equality (3) is
proved.
Now, we prove equality (2). Assume the opposite: λY(y) > 0 for some y ∈ suppY \ P(Y). In view of y ∉
P(Y) and Definition 3, it follows that there is an alternative y' ∈ suppY for which y' y. By the Pareto axiom,
y' y implies λ{y, y'}(y) = 0, where y ≠ y'. Therefore, y ∉ Ndom(Y). From this, according to (3), we obtain
λY(y) = 0, which contradicts the assumption λY(y) > 0. The theorem is proved.
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This result can be briefly stated as follows: An arbitrary (generally fuzzy) choice from the set of feasible
alternatives satisfying Axiom 1" and the Pareto axiom must not be made outside the Pareto set.
Overall, the conditions imposed on a choice by the indicated axioms can be interpreted as the decisionmaker’s rational behavior in the course of choice making. Therefore, according to the theorem, if the behavior of the decision-maker is rational, the Edgeworth–Pareto principle is always satisfied.
Remark 2. By the lemma, Axioms 2 and 3 imply the Pareto axiom. Therefore, the Edgeworth–Pareto
principle remains valid if the Pareto axiom in the assumptions of the theorem is replaced by Axioms 2 and
3. It is in this form (i.e., under the three axioms indicated) that this principle was earlier stated by the author.
It should be added that the absence of Axiom 2 in the assumptions of the theorem means that the requirement
of the decision-maker’s “transitive” behavior can be violated in the decision making. Thus, the Edgeworth–
Pareto principle turns out to be valid for some (nonfuzzy) multicriteria choice problems with an intransitive
preference relation (for these problems, see [1–3]).
6. MINIMALITY OF THE SET OF AXIOMS
Below are examples showing that the Edgeworth–Pareto principle is violated if at least one of the rational
choice axioms is discarded.
Example 2. Let Y1 = {a1, b1}, Y2 = {a2}, a1 1 b1, Ŷ = {y1, y2}, y1 = (a1, a2), and y2 = (b1, a2).
Here, y1 = (a1, a2) (b1, a2) = y2, since a1 1 b1 . Therefore, P(Y) = {y1}.
Consider a fuzzy choice function such that
1
2
λ { y 1 } ( y ) = 1,
1
λ { y 2 } ( y ) = 1,
2
λ { y 1, y 2 } ( y ) = 0,
λ { y 1, y 2 } ( y ) = 0.9.
For this function, suppY = Y = Ŷ , Ndom(Y) = {y2}, and Axiom 1" is obviously satisfied. In this case, equality
(2) must have the form λ { y 1, y 2 } (y2) = 0. However, it is violated, since the Pareto axiom (and Axiom 3) does
not hold.
Example 3. Let Y1 = {a1, b1, c1}, Y2 = {a2}, a1 1 b1, b1 1 c1, Y = Ŷ = {y1, y2, y3}, y1 = (a1, a2), y2 = (b1,
a2), and y3 = (c1, a2). Here, y1 = (a1, a2) (b1, a2) = y2 and y2 = (b1, a2) (c1, a2) = y3. Therefore, P(Y) = {y1}.
Consider a fuzzy choice function for which
1
2
λ { y 1 } ( y ) = 1,
3
λ { y 2 } ( y ) = 1,
3
2
λ { y 2, y 3 } ( y ) = 0,
1
λ { y 3 } ( y ) = 1,
λ { y 1, y 2 } ( y ) = 0.8,
1
λ { y 2, y 3 } ( y ) = 1,
3
λ { y 1, y 3 } ( y ) = 0.7,
2
2
λ { y 1, y 2 } ( y ) = 0,
λ { y 1, y 3 } ( y ) = 0,
1
λ Y ( y ) = 0,
3
λ Y ( y ) = 1,
λ Y ( y ) = 0.
Here, suppY = Y = Ŷ . It is easy to see that the Pareto axiom holds but equality (2) is violated for the choice
function introduced. The latter is due to the fact that Axiom 1" is not satisfied.
As was indicated in Remark 2, the Edgeworth–Pareto principle can be formulated under Axioms 1", 2,
and 3. Examples 2 and 3 show that the principle can be violated if at least one of the axioms (1" or 3) is not
satisfied. Below is an example confirming that this also occurs if Axiom 2 does not hold.
Example 4. Let Yi = {ai, bi, ci} and ai i bi i ci for i = 1, 2. We have
1
9
1
Ŷ = { y , …, y },
2
y = ( a 1, c 2 ),
6
y = ( c 1, b 2 ),
3
y = ( b 1, b 2 ),
7
y = ( a 1, b 2 ),
4
y = ( c 1, a 2 ),
8
y = ( b 1, a 2 ),
5
y = ( c 1, c 2 ),
y = ( b 1, c 2 ),
9
y = ( a 1, a 2 ).
Consider the class of fuzzy choice functions such that
i
λ { y i } ( y ) = 1,
i = 1, 2, 3,
3
λ { y 2, y 3 } ( y ) = 0,
j
λ { y j } ( y ) = 0,
2
j = 4, …, 9,
λ { y 2, y 3 } ( y ) = 1,
1
1
λ { y 1, y 2 } ( y ) = 0.8,
λ { y 1, y 3 } ( y ) = 0,
2
λ { y 1, y 2 } ( y ) = 0,
3
λ { y 1, y 3 } ( y ) = 0.9.
For each choice function of this class, suppY = P(Y) = {y1, y2, y3} and Axiom 2 is violated, since
λ { y 1, y 2 } (y2) = 0 and λ { y 2, y 3 } (y3) = 0, but λ { y 1, y 3 } (y3) > 0. The conditions imposed on the choice function with
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membership functions λA(·) are supplemented with another 36 equalities of the form λ { y 1, y 4 } (y1) = 1,
λ { y 1, y 4 } (y4) = 0, λ { y 4, y 5 } (y4) = 0, λ { y 4, y 5 } (y5) = 0, λ { y 1, y 7 } (y1) = 0, λ { y 1, y 7 } (y7) = 0, etc., so that Axiom 3 is
satisfied.
If the membership functions satisfying the above conditions additionally satisfy Axiom 1", then, in view
of λ { y 2, y 3 } (y2) = 0, λ { y 2, y 3 } (y3) = 0, and λ { y 1, y 3 } (y1) = 0, each such function satisfies λY(y i ) = 0 (i = 1, 2, 3),
which contradicts the requirement that a choice from the set of feasible alternatives be nonempty (see
Remark 1 in Section 3). Thus, the rejection of Axiom 2 in Example 4 leads to the violation of the Edgeworth–Pareto principle in the sense indicated in Remark 2.
7. UPPER BOUNDS FOR THE UNKNOWN SET OF SELECTED ALTERNATIVES
The set of alternatives selected from Y is a solution to the choice problem and has to be found in applications. Equality (2) gives an upper bound for the unknown set of selected alternatives. Specifically, under
the rational choice axioms, the best (generally fuzzy) choice must be made only within the Pareto set. If
certain additional conditions are imposed on Y, then (as shown below) other useful (in applications) upper
bounds for the unknown set of selected alternatives can be formulated in terms of solutions to various scalar
parametric optimization problems.
In what follows, we assume that
Y i ⊂ ,
i = 1, 2, …, m,
m
m
Ŷ ⊂ ,
supp Y ⊂ ,
and that each relation i coincides with the restriction of the “strictly greater” relation > to the number set
Yi. Thus, each feasible alternative y ∈ suppY is an m-dimensional number vector (point). It is easy to see
that, by Definition 3, we have P(Y) = P(suppY).
Corollary 1. Let the set suppY be convex. For any fuzzy choice function satisfying Axiom 1" and the
Pareto axiom, it holds that
λY ( y ) = 0
∀y ∈ supp Y \Y ≥ ,
(4)
where Y≥ is the union, over all the vectors µ = (µ1, …, µm) of the sets of maximizers of the linear function
m
µ y on the set suppY; moreover,
i=1 i i
∑
m
µ 1, …, µ m ≥ 0,
∑µ
i
(5)
= 1.
i=1
Proof. According to L. Hurwicz’s theorem (see, e.g., Theorem 2.2.2 in [9]), which states that, if the set
of feasible vectors is convex, then any Pareto optimal vector can be obtained by maximizing the linear funcm
µ y over the indicated set for some µ of form (5), we have the inclusion P(Y) ⊂ Y≥. Then
tion
i=1 i i
{suppY \Y≥} ⊂ {suppY \P(Y)} and (2) immediately implies (4). Corollary 1 is proved.
∑
Since lnyi is convex (for positive yi) and
m
∑
i=1
m
µ i ln y *i ∑
m
µ i ln y i ⇔
i=1
∏
µi
( y *i ) i=1
m
∏y
µi
i ,
i=1
Corollary 1 implies the following result.
Corollary 2. Let suppY be convex and all of its vectors have positive components. Then, for any fuzzy
choice function satisfying Axiom 1" and the Pareto axiom, it holds that
λY ( y ) = 0
∀y ∈ supp Y \Y è ,
where Yè is the union over all the vectors µ satisfying (5) of the sets of maximizers of
suppY.
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Corollary 3. Let suppY be a convex and closed set. For any fuzzy choice function satisfying Axiom 1"
and the Pareto axiom, it holds that
λY ( y ) = 0
∀y ∈ supp Y \Y > ,
(6)
where Y > denotes the closure of the union over all the vectors µ = (µ1, …, µm) of the sets of maximizers of
the linear function
∑
m
µy
i=1 i i
on suppY; moreover,
m
µ 1, …, µ m > 0,
∑µ
i
(7)
= 1.
i=1
Proof. We apply Theorem 3.1.8 from [9], in which X is the convex closed set suppY and the vector function f is replaced by the linear vector function (y1, …, ym). All the assumptions of the theorem are satisfied.
Therefore, P(Y) ⊂ Y > and, hence, {suppY \ Y > } ⊂ {suppY \ P(Y)}. Then (6) immediately follows from (2).
n
Corollary 4. Suppose that X ⊂ is polyhedral; i.e., it is the solution set of a finite set of linear inequaln
ities. Moreover, let all the components f1, …, fm of the vector function f be polyhedral concave on . Then,
for any fuzzy choice function satisfying Axiom 1" and the Pareto axiom, it holds that
λY ( y ) = 0
∀y ∉ Y > ,
where Y> is the union over all the vectors µ form (7) of the sets of maximizers of the linear function
m
µ y on the set Y = f(X).
i=1 i i
∑
Proof. According to Theorem 2.2.8 in [9], under the conditions of Corollary 4, the set of Pareto optimal
vectors P(Y) coincides with the set of properly effective vectors G(Y) (see [9]). Therefore, by using
A.M. Geoffrion’s theorem (see [9, Theorem 2.2.4]), we can write P(Y) = G(Y) = Y>. Then, the required result
follows from (2).
Corollary 5. Suppose that suppY consists of positive vectors. Then, for any fuzzy choice function satisfying Axiom 1" and the Pareto axiom, it holds that
λY ( y ) = 0
∀y ∉ Y maxmin ,
where Ymaxmin is the union over all the vectors µ satisfying (5) of the sets of maximizers of
min
µ i y i on
i = 1, 2, … , m
the set suppY.
Proof is similar to that of Corollary 1, with the only difference being that Germeier’s theorem (see [9,
Theorem 2.2.1]) is used instead of Gurvich’s theorem.
The following corollary makes use of the scalarization method, which was first described in [10, 11].
Corollary 6. Suppose that the components of all the vectors of suppY are bounded above; i.e., there is a
n
vector p ∈ such that y < p for all y ∈ suppY. Then, for any fuzzy choice function satisfying Axiom 1"
and the Pareto axiom, it holds that
λY ( y ) = 0
∀y ∉ Y minmax ,
where Yminmax is the union over all the vectors µ of form (7) of the sets of minimizers of
max µ i (pi – yi )
i = 1, 2, …, m
on the set suppY.
Proof. This result is proved by applying the Edgeworth–Pareto principle and Germeier’s theorem (for
the minimization problem) to the multicriteria problem with a vector function having the components pi –
yi , i = 1, 2, …, m.
Corollary 7. Let α be an arbitrary given number. For any fuzzy choice function satisfying Axiom 1" and
the Pareto axiom, it holds that
λY ( y ) = 0
∀y ∉ Ŷ minmax ,
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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THE EDGEWORTH–PARETO PRINCIPLE IN TERMS
where Ŷ minmax is the union over all the vectors p = (p1, …, pm) with
max { p i – yi}
i = 1, 2, …, m
m
sup y i .
i=1
y ∈ supp Y
∑
m
i=1
561
p i = α of the minimizers of
on the set suppY. In particular, if all sup y i (i = 1, 2, …, m) are finite, we can set α =
y ∈ supp Y
∑
Proof. Let y* be an arbitrary weakly effective (Slater optimal, see [9]) vector of the set suppY. We introduce a vector p with the components
⎞
1⎛
p i = y *i – ---- ⎜ y *i + α⎟ ,
m⎝
⎠
i=1
m
∑
i = 1, 2, …, m.
Obviously,
m
∑p
i
= α.
i=1
Since any effective (Pareto optimal) vector is weakly effective, it is sufficient to show that the point y*
can be obtained by minimizing the function max { p i – yi} on suppY for the indicated vector p.
i = 1, 2, … , m
Assume the opposite: there is a point y ∈ suppY for which
max { p i – y i } <
i = 1, 2, …, m
max { p i – y *i }.
i = 1, 2, … , m
Then, for any i ∈ {1, 2, …, m}, we obtain the inequality
m
pi – yi <
max { p i – y *i } =
i = 1, 2, …, m
m
⎧
⎫
1
α
1
α
max ⎨ y *i – ---- y *i – ---- – y *i ⎬ = – ---- y *i + ---- ,
m
m
m
m
i = 1, 2, …, m
⎩
⎭
i=1
i=1
∑
∑
which implies
1
y *i – ---m
m
∑
i=1
1
α
y *i + ---- – y i < – ---m
m
m
α
∑ y* + ---m- .
i
i=1
Hence, y *i < yi , which contradicts the weak effectiveness of y*. Thus, the proof is completed.
The vectors p satisfying the condition
m
∑
i=1
m
pi = α
for
α =
∑
i=1
sup y i
y ∈ supp Y
form a hyperplane in the criteria space, which in a sense can be called an ideal unattainable set.1 Then, the
minimization of max { p i – yi} on the set suppY in Corollary 7 means that we search (in this set) for the
i = 1, 2, …, m
vector nearest to the point p of this hyperplane provided that the uniform (Chebyshev) metric is used.
ACKNOWLEDGMENTS
This work was supported by the Russian Foundation for Basic Research (project no. 05-01-00310).
REFERENCES
1. V. D. Noghin, “A Logical Justification of the Edgeworth–Pareto Principle,” Zh. Vychisl. Mat. Mat. Fiz. 42, 951–
957 (2002) [Comput. Math. Math. Phys. 42, 915–920 (2002)].
2. V. D. Noghin, Decision Making in Multicriteria Environment: A Quantitative Approach, 2nd ed. (Fizmatlit, Moscow, 2002) [in Russian].
3. V. D. Noghin, “Generalized Edgeworth–Pareto Principle and Its Range of Applicability,” Ekon. Mat. Metody 41
(3), 128–131 (2005).
1 This
hyperplane can be attainable in the degenerate case when suppY contains the largest vector with respect to .
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NOGHIN
4. V. D. Noghin, “The Edgeworth-Pareto Principle and the Relative Importance of Criteria in the Case of a Fuzzy
Preference Relation,” Zh. Vychisl. Mat. Mat. Fiz. 43, 1666–1676 (2003) [Comput. Math. Math. Phys. 43, 1604–
1612 (2003)].
5. M. A. Aizerman and F. T. Aleskerov, Choice of Variants: Theoretical Foundations (Nauka, Moscow, 1990) [in
Russian].
6. I. M. Makarov, T. M. Vinogradskaya, A. A. Rubchinsky, and V. B. Sokolov, The Theory of Choice and Decision
Making (Nauka, Moscow, 1982; Mir, Moscow, 1987).
7. H. Moulin, Axioms of Cooperative Decision Making (Cambridge Univ. Press, Cambridge, MA, 1988; Mir, Moscow, 1991).
8. A. Sen, On Ethics and Economics (Blackwell, London, 1987; Nauka, Moscow, 1996).
9. V. V. Podinovskii and V. D. Noghin, Pareto Optimal Solutions of Multicriteria Problems (Nauka, Moscow, 1982)
[in Russian].
10. J. Jahn, “Scalarization in Vector Optimization,” Math. Program. 29, 203–218 (1984).
11. J. Jahn, “Some Characterization of the Optimal Solutions of a Vector Optimization Problem,” Operat. Res. Spektrum 7, 7–17 (1985).
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