Janos FodorInstitute of Intelligent Engineering Systems

Budapest Tech

Becsi ut 96/b, H-1034 Budapest

Hungary

Email: fodor@bmf.hu

Imre J. RudasInstitute of Intelligent Engineering Systems

Budapest Tech

Becsi ut 96/b, H-1034 Budapest

Hungary

Email: rudas@bmf.hu

Abstract—We investigate the problem of aggregation of fuzzypreference structures in multicriteria decision making. Startingfrom monocriterion weak preferences, there are two ways todefine global strict preference, indifference and incomparability.We prove that, under very mild conditions, the only aggregationfor which those two ways result in the same global preferencestructure is dictatorial.

I. INTRODUCTION

In a multicriteria decision making (MCDM) problem, when

criteria are given in terms of (fuzzy) preference relations on the

set of alternatives, two major steps are distinguished: aggrega-

tion and exploitation. The aggregation phase defines a global

preference relation, and the exploitation phase transforms this

global relation into a ranking of the alternatives.

In the present paper we deal with the aggregation phase

when monocriterion preferences are binary fuzzy relations.

Thanks to keen interest in research in the last decade, theo-

retical aspects of fuzzy preferences have become rather well-

developed and sound. The situation most often considered is

the following. Suppose that A is a set of alternatives, and

R is fuzzy weak preference relation on A (that is, a function

R : A×A → [0, 1] such that R(a, a) = 1 for all a ∈ A), where

the value R(a, b) ∈ [0, 1] expresses the degree to which a is

not worse than b . Three further binary fuzzy relations can be

attached to such an R:

• strict preference: P (a, b) means the degree to which a is

better than b;

• indifference: I(a, b) gives the degree to which a and bare indifferent;

• incomparability: J(a, b) yields the degree to which a and

b cannot be compared.

There is an axiomatic framework of constructing P , I and

J from a given R (see [5]), which we will briefly recall in

the next section. It is easy to prove that the resulted triplet

(P, I, J) is a ϕ-fuzzy preference structure on A. This notion

was introduced in [8].

The natural area of using preference structures is multi-

criteria decision making. In that environment we start from

monocriterion preferences R1, . . . , Rn on A. Concerning any

of the previous three relations, we can define global strict

preference, global indifference and global incomparability in

two different ways. To illustrate this, consider e.g. the strict

preference.

One possibility is to aggregate first R1, . . . , Rn to obtain

global weak preference R := M(R1, . . . , Rn). Then we can

use the above axiomatic construction to define P , I and J .

The other possibility is to define first monocriterion strict

preference, indifference and incomparability, and then to ag-

gregate each type to obtain the related global relations.

Generally speaking, these two ways yield different global

relations. Thus, it is a natural need to search for appropriate

aggregation functions which give the same results in both of

the above approaches.

The main result of the paper says that the only type

of aggregation procedures satisfying these requirements is

dictatorial: it takes into account only one criterion in the global

relations. As a consequence, the global (P, I, J) is a ϕ-fuzzy

preference structure too.

The paper is organized as follows. In Section 2 we recall

some preliminary notions and results that are indispensable in

the sequel. In Section 3 we formulate the aggregation problem

in a mathematical way, by the help of functional equations.

Then we solve the problem for strict preference first. This

results is easily applied for indifference and incomparability.

In some extremely important particular cases the general

dictatorial aggregation is proved to be the identity function.

Finally we present some concluding remarks.

II. PRELIMINARIES

Let (T, S, N) be a continuous De Morgan triplet (i.e., Tis a continuous t-norm, N is a strong negation, and S is

the N -dual t-conorm of T ) representing logic operations in

the fuzzy framework. Consider a set of alternatives A, and

a reflexive binary fuzzy relation R on A. According to the

axiomatic approach in [5], strict preference P , indifference I ,

and incomparability J attached to R is constructed by the use

of some generator function for each of these relations:

P (a, b) := p(R(a, b), R(b, a)),I(a, b) := i(R(a, b), R(b, a)),J(a, b) := j(R(a, b), R(b, a)),

for any a, b ∈ A, where p, i, j are functions from [0, 1]2 to

[0, 1] with some prescribed properties. These functions can be

On Aggregation of Fuzzy Preference Structures

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obtained as solutions of a system of functional equations

S(p(x, y), i(x, y)) = x,

S(p(x, y), j(x, y)) = N(y),

where the t-conorm S and strong negation N are also un-

known. It was proved in [5] that these functional equations

imply the existence of an automorphism ϕ of the unit interval

such that (T, S, N) is of the following form

T (x, y) = ϕ−1(max{ϕ(x) + ϕ(y) − 1, 0}),S(x, y) = ϕ−1(min{ϕ(x) + ϕ(y), 1}),

N(x) = ϕ−1(1 − ϕ(x))

for all x, y ∈ [0, 1]. Then (T, S, N) is called a Łukasiewicz-triplet, and the above functional equations can be written as

ϕ(p(x, y)) + ϕ(i(x, y)) = ϕ(x), (1)

ϕ(p(x, y)) + ϕ(j(x, y)) = 1 − ϕ(y), (2)

and a solution is denoted by < p, i, j >ϕ.

There is a parametric family of t-norms which plays a key

role in several areas of fuzzy set theory, in particular in fuzzy

preference modelling. Let s ∈ ]0, 1[∪ ]1,∞[, then the Frank

t-norm with parameter s is defined by

T s(x, y) := logs

(1 +

(sx − 1)(sy − 1)s − 1

), x, y ∈ [0, 1].

The limit t-norms are given by the minimum (s → 0), the

algebraic product (s → 1) and the Łukasiewicz t-norm (s →∞). So, we employ the respective notations T 0, T 1 and T∞

for these limit cases. We also identify T1∞ and T 0, as well

as T10 and T∞. The family (T s)s∈[0,∞] is called the Frank

t-norm family [7]. The dual t-conorm is usually denoted by

Ss.

Now we can recall the following result from [5] which is

the most relevant one in studying the aggregation problem.

Theorem 1. Assume that p(x, y) = T1(x,Nϕ(y)) andi(x, y) = T2(x, y), where T1 and T2 are continuous t-norms.Then < p, i, j >ϕ satisfies (1) and (2) if and only if thereexists a number s ∈ [0,∞] such that

T1(x, y) = ϕ−1(T s(ϕ(x), ϕ(y))),T2(x, y) = ϕ−1(T 1/s(ϕ(x), ϕ(y))),

where T s and T 1/s belong to the Frank family.

The notion of fuzzy preference structures has been intro-

duced in [8]. We recall now a minimal definition after [2].

Consider an automorphism ϕ of [0, 1]. A triplet (P, I, J) of

binary fuzzy relations on A is called a ϕ-fuzzy preferencestructure (ϕ-FPS) on A if I is reflexive, I and J are sym-

metric, and we have the following equality for all a, b ∈ A:

ϕ(P (a, b)) + ϕ(P (b, a)) + ϕ(I(a, b)) + ϕ(J(a, b)) = 1. (3)

Because of (3), (P, I, J) can also be called a ϕ-additive FPS

on A.

It is immediate that any (P, I, J) generated by a solution

< p, i, j >ϕ of the functional equation system (1)-(2) is a ϕ-

FPS on A. In [6] the interested reader can find more details

where we presented an overview of fundamentals and recent

advances.

III. FORMULATING THE AGGREGATION PROBLEM

Let A be a set of alternatives and R1, . . . , Rm be reflexive

binary fuzzy relations on A (each weak preference Rk repre-

sents a criterion). Let a, b ∈ A, and denote xk := Rk(a, b),yk := Rk(b, a) for k ∈ {1, . . . ,m}. Then we obtain the global

weak preference as follows:

R(a, b) = MR(R1(a, b), . . . , Rm(a, b))= MR(x1, . . . , xm),

R(b, a) = MR(R1(b, a), . . . , Rm(b, a))= MR(y1, . . . , ym).

Using these global values, finally we can define P, I, J as

follows:

P (a, b) := p(MR(x1, . . . , xm),MR(y1, . . . , ym)),I(a, b) := i(MR(x1, . . . , xm),MR(y1, . . . , ym)),J(a, b) := j(MR(x1, . . . , xm),MR(y1, . . . , ym)).

On the other hand, monocriterion strict preferences, indif-

ferences, and incomparabilities are defined as Pk(a, b) :=p(xk, yk), Ik(a, b) := i(xk, yk), Jk(a, b) := j(xk, yk) for

k ∈ {1, . . . ,m}. Thus, we can introduce P ′, I ′, J ′ as follows:

P ′(a, b) := MP (p(x1, y1), . . . , p(xm, ym)),I ′(a, b) := MI(i(x1, y1), . . . , i(xm, ym)),J ′(a, b) := MJ(j(x1, y1), . . . , j(xm, ym)).

Therefore, equalities P = P ′, I = I ′, J = J ′ can be

formulated as the following functional equations:

p(MR(x1, . . . , xm),MR(y1, . . . , yn))= MP (p(x1, y1), . . . , p(xm, ym)), (4)

i(MR(x1, . . . , xm),MR(y1, . . . , ym))= MI(i(x1, y1), . . . , i(xm, ym)), (5)

j(MR(x1, . . . , xm),MR(y1, . . . , ym))= MJ(j(x1, y1), . . . , j(xm, ym)), (6)

where p, i, j : [0, 1]2 → [0, 1] are given strict preference, in-

difference, and incomparability generator functions (solutions

of (1) and (2)), and we are looking for the unknown functions

MR, MP , MI , and MJ which satisfy the equations (4)–(6)

for all xk, yk ∈ [0, 1] (k ∈ {1, . . . ,m}).

The generality of having possibly different aggregation

functions is apparent, as we can see by the following lemma.

3rd International Symposium on Computational Intelligence and Intelligent Informatics – ISCIII 2007 • Agadir, Morocco • March 28-30, 2007

86

Lemma 1. (1) If MR and MP are solutions of (4) withMR(0, . . . , 0) = 0 then MR = MP . Moreover, MR is self-dual with respect to N .

(2) If MR and MI are solutions of (5) with MR(1, . . . , 1) =1 then MR = MI .

(3) If MR and MJ are solutions of (6) with MR(0, . . . , 0) =0 then MR = MJ .

Thus, we have to deal only with one unknown aggregation

function, denoted by M(= MR = MP = MI = MJ), in

all cases. Moreover, M is self-dual with respect to the strong

negation N ; that is, we have for all x1, . . . , xm ∈ Rm:

M(x1, . . . , xn) = N(M(N(x1), . . . , N(xm))).

In addition, we require the following properties too (see

conditions in Lemma 1).

M(0, . . . , 0) = 0, (7)

M(1, . . . , 1) = 1, (8)

M is nondecreasing and continuous (9)

IV. ONLY DICTATORIAL AGGREGATION EXISTS

In this section we show a general result on the form of the

aggregation function M . First we investigate strict preferences,

because the corresponding functional equation (4) is very

restrictive in itself.

A. Aggregating Strict Preferences: Solution of (4)

The following theorem was already stated in [4].

Theorem 2. The general solution M of (4) is of the followingform:

M(x1, . . . , xm) = gk(xk), (10)

where gk : [0, 1] → [0, 1] is a nondecreasing function withgk(0) = 0, gk(1) = 1, satisfying

gk(p(x, y)) = p(gk(x), gk(y)) (11)

for all x, y ∈ [0, 1].

Therefore, the only possible aggregation takes into account

one criterion, and thus it is dictatorial, even if we consider only

strict preferences. The obtained class of aggregation functions

will be narrowed further if we consider indifferences and

incomparabilities.

B. Indifferences and Incomparabilities

Since MI = MJ = MP = MR = M , now we must use

the same formula (10) for defining global indifference and

incomparability. In addition to (11), the function gk has to

satisfy the following functional equations too:

gk(i(x, y)) = i(gk(x), gk(y)) (x, y ∈ [0, 1]), (12)

gk(j(x, y)) = j(gk(x), gk(y)) (x, y ∈ [0, 1]). (13)

These are simple consequences of the respective equations (5)

and (6).

C. The General Form of the Aggregation

We can summarize the previous facts in the following

statement.

Theorem 3. The general solution M of equations (4), (5), (6)is of the following form:

M(x1, . . . , xm) = gk(xk),

where gk : [0, 1] → [0, 1] is a nondecreasing function withgk(0) = 0, gk(1) = 1, satisfying the following three functionalequations:

gk(p(x, y)) = p(gk(x), gk(y)),gk(i(x, y)) = i(gk(x), gk(y)),gk(j(x, y)) = j(gk(x), gk(y))

for all x, y ∈ [0, 1]. In particular, gk must be self-dual withrespect to the strong negation N :

gk(N(x)) = N(gk(x)) (x ∈ [0, 1]).

We can be certain that there is at least one function gk which

satisfies all the previous conditions. Indeed, gk(x) := x for all

x ∈ [0, 1] is a continuous solution. Generally speaking, we do

not know if there are other continuous solutions. The study in

the next section might suggest a negative answer.

V. PARTICULAR CLASSES

Now we determine continuous solutions of equations (11),

(12), and (13) in the case indicated by Theorem 1. That

is, generator functions of strict preference, indifference and

incomparability are of the following form now:

ps,ϕ(x, y) := T sϕ(x, Nϕ(y)), (14)

is,ϕ(x, y) := T 1/sϕ (x, y), (15)

js,ϕ(x, y) := T 1/sϕ (Nϕ(x), Nϕ(y)) (16)

for all x, y ∈ [0, 1], where T sϕ is a ϕ-transform of the t-norm

T s belonging to the Frank family with parameter s ∈ [0,∞],and 1/∞ := 0, 1/0 := ∞. Fortunately, we are able to use

some of our previous results related to decomposable measures

[3].

A. Case 1: s = 0If s = 0 then i0,ϕ(x, y) = ϕ−1(max(ϕ(x) + ϕ(y) − 1, 0)),

the ϕ-transform of the Łukasiewicz t-norm. It is continuous,

Archimedean, with zero divisors (i.e., it is nilpotent).

In this case equation (12) is given as follows (x, y ∈ [0, 1]):

gk(ϕ−1(max(ϕ(x) + ϕ(y) − 1, 0)))= ϕ−1(max(ϕ(gk(x)) + ϕ(gk(y)) − 1, 0)). (17)

Lemma 2. The general continuous solution of (17) is of thefollowing form (x ∈ [0, 1]):

gk(x) = ϕ−1(max{1 − αk + αkϕ(x), 0}), (18)

where αk ≥ 1 is a real parameter.

J. Fodor, I. J. Rudas • On Aggregation of Fuzzy Preference Structures

87

Since gk must be self-dual with respect to the strong

negation N (see Lemma 1 (1)), it is immediate that this can

hold if and only if αk = 1, i.e., if and only if gk(x) = x for

all x ∈ [0, 1] .

B. Case 2: s = ∞If s = ∞ then i∞,ϕ(x, y) = min(x, y), the minimum

t-norm (independently of ϕ). It is continuous and non-

Archimedean. Moreover, p∞,ϕ(x, y) = ϕ−1(max(ϕ(x) −ϕ(y), 0)).

Therefore in the present particular case equation (12) is

automatically satisfied by any nondecreasing function gk.

However, equation (11) is as follows (x, y ∈ [0, 1]):

gk(ϕ−1(max(ϕ(x) − ϕ(y), 0)))= ϕ−1(max(ϕ(gk(x)) − ϕ(gk(y)), 0)). (19)

After a suitable substitution, this equation can be transformed

into (17), thus the general solution of (19) is given by (18),

and among those functions there is only one which is self-dual

with respect to N : this is again gk(x) := x for all x ∈ [0, 1].

C. Case 3: s ∈ ]0,∞[

If s ∈ ]0,∞[ then is,ϕ(x, y) = T1/sϕ (x, y) is a continuous,

Archimedean t-norm without zero divisors (i.e., it is strict).

In this case is,ϕ can be obtained by an appropriate ϕs-

transformation of the product t-norm:

T 1/sϕ (u, v) = ϕ−1

s (ϕs(u) · ϕs(v)) (u, v ∈ [0, 1]).

More explicitly, we have for all x ∈ [0, 1]:

ϕs(x) =

⎧⎪⎪⎨⎪⎪⎩

(1/s)ϕ(x) − 1(1/s) − 1

if s ∈]0,∞[, s �= 1,

ϕ(x) if s = 1

, (20)

and thus

ϕ−1s (x) =

⎧⎨⎩

ϕ−1(x) if s = 1

ϕ−1(log(1/s)(1 + ((1/s) − 1)x)) otherwise,.

From the solution of Cauchy’s famous functional equations

(see e.g. [1]) we can prove the following result (see [3]).

Lemma 3. For s ∈]0,∞[, the general continuous solution of(12) is of the following form:

gk(x) = ϕ−1s (ϕλk

s (x)) (x ∈ [0, 1]), (21)

where ϕs is the automorphism of the unit interval given in(20), and λk is a positive parameter.

As in the previous two particular cases, it is easy to see that

the only self-dual function (with respect to N ) among (21) is

gk(x) := x for all x ∈ [0, 1].We can summarize the results of this section in the following

theorem.

Theorem 4. Let (T, S, N) be a given Łukasiewicz triplet. Sup-pose generator functions of strict preference, indifference and

incomparability are given by (14), (15), and (16), respectively.Then the only continuous aggregation function which satisfiesalso equations (11), (12), (13) is given as follows:

M(x1, . . . , xm) = xk

for some k ∈ {1, . . . ,m} (x1, . . . , xm ∈ [0, 1]).

Therefore, if we have a profile of ϕ-fuzzy preference

structures

〈(P1, I1, J1), . . . , (Pk, Ik, Jk), . . . , (Pm, Im, Jm)〉then the aggregated triplet is (Pk, Ik, Jk). Consequently, the

aggregated triplet is a ϕ-FPS.

VI. CONCLUDING REMARKS

As we have shown, the only continuous aggregation of

ϕ-fuzzy preference structures is dictatorial. That is, it is

impossible to assign weights other than 0 or 1 to each criterion

in order to have the same global preferences in both types of

aggregation procedures mentioned in the Introduction.

Note, however, that there exist non-dictatorial continuous

aggregations of particular preference structures (for instance,

when monocriterion weak preferences are complete and nega-

tively transitive fuzzy relations). More results on such partic-

ular cases can be found in [5].

ACKNOWLEDGMENT

The authors would like to thank supports from OTKA

T046762 and from SZVT (Scientific Society for Management

and Organization, Hungary).

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