Integrating Three Representation Models in Fuzzy Multipurpose

Decision Making Based on Preference Relations

F. Chiclana, F. Herrera, E. Herrera-Viedma

Dept. of Computer Science and Artificial Intelligence

E.T.S. de Ingeniería Informática

University of Granada, 18071- Granada, Spain

E-mail: herrera, viedma@robinson.ugr.es

Abstract

The purpose of this paper is to study a fuzzy multipurpose decision making problem, where

the information about the alternatives provided by the experts can be of a diverse nature.

The information can be represented by means of preference orderings, utility functions and

fuzzy preference relations, and our objective is to establish general models which cover all

possible representations. Firstly, we must make the information uniform, using fuzzy

preference relations. We will present a general procedure to relate preference orderings,

utility values and fuzzy preference relations that generalises the procedures normally used

where there is a combination of different types of information.

We will present generic decision processes based on the concept of fuzzy majority. Fuzzy

majority is represented by a fuzzy quantifier, and applied in the aggregation, by means of

an OWA operator whose weights are calculated by the fuzzy quantifier. We use two

quantifier-guided choice degrees of alternatives. A dominance degree used to quantify the

dominance that one alternative has over all the others, in a fuzzy majority sense, and a non-

dominance degree, that generalises Orlovsky’s non-dominated alternative concept. The

application of the two above choice degrees can be carried out according to two different

selection processes; a sequential choice process and a conjunction choice process.

1. Introduction

The process of decision making, consisting in deriving the best option from a feasible set, is present in just

about every conceivable human task. As a result, the study of decision making is necessary and very

important not only in decision theory but also in areas such as Operations Research, Management Science,

Politics, Social Psychology, etc.. Also, it is obvious that the comparison of different actions according to their

desirability in decision problems, in many cases, cannot be done by using a single criterion or an unique

person. We do not distinguish between “persons” and “criteria”, and interpret the decision process in the

framework of multipurpose decision making (MPDM) [12]. This has led to numerous evaluation schemes,

and has become a major concern of research in decision making [9,13].

The basic model of a decision in a classical normative decision theory has very little in common with real

decision making because it portrays a decision as a clear-cut act of choice, in an environment in which the

goals, constraints, information and consequences of possible actions are supposed to be precisely known.

The only component in which uncertainty is permitted is the occurrence of the different states of nature, for

which probabilistic descriptions are allowed. However, when the uncertainty is of a qualitative nature, the use

of other techniques is necessary. Fuzzy set theory might provide the flexibility needed to represent the

uncertainty resulting from the lack of knowledge. There exist many opportunities to apply fuzzy sets theory in

decision making. Fuzzy theories and methodologies can be used either to translate imprecise and vague

information in the problem specification into fuzzy relationships (fuzzy objectives, fuzzy constraints, fuzzy

preferences,...) or to use fuzzy tools for designing a decision process trying to establish preference orderings

of alternatives. Different fuzzy decision making problems were proposed in [11]. Several authors have

provided interesting results on group decision making or social choice theory and multicriteria decision

making with the help of fuzzy theory [9,7,19,27,28]. In all these decision making problems, procedures have

been established to combine opinions about alternatives related to different points of view. All procedures

are based on pair comparisons, in the sense that processes are linked to some degree of credibility of

preference of any alternative over another, even if the imput data corresponds to an evaluation (utility,

physical or monetary value) related to each alternative considered for each point of view. In this latter case a

pair of alternatives can be compared in a transitive way on the basis of their evaluations.

The classical MPDM procedure follows two steps: aggregation and exploitation. The aggregation phase

defines an outranking relation which indicates the global preference between every ordered pair of

alternatives, taking into consideration the weights of the different points of view. The exploitation phase

transforms the global information about the alternatives into a global ranking of them. This can be done in

different ways, the most common one being the use of a ranking method to obtain a score function.

We will consider fuzzy MPDM problems where, for each purpose (expert or criterion), the information about

the alternatives can be supplied in different ways. Therefore, we will present a general model dealing with

different representations of fuzzy preferences. According to Tanino [21,22], the information about the set of

alternatives supplied by an individual expert can be represented in three different ways:

1. As a preference ordering of the alternatives. In this case the alternatives are ordered from best to worst,

without any other supplementary information.

2. As a fuzzy preference relation. This is the case when an expert supplies a fuzzy binary relation over the set

of alternatives, reflecting the degree to which an alternative is preferred to another.

3. As a utility function. In this case an expert supplies an evaluation (physical or monetary value) for each

alternative, i.e. a function that associates each alternative with a real number indicating the performance of

that alternative according to his point of view.

Our objective in this paper is to establish general models so that we can cover all the possible

representations of the information. With this objective in mind, we will design generic choice processes in

MPDM based on the concept of fuzzy majority [8,16], which is used to represent the concept of a social

opinion, and using the OWA operators [25] as aggregation operators. In order to do this, the paper is set out

as follows:

The problem is presented in section 2. How to make the information uniform is discussed in section 3, where

we present a general model to relate preference orderings, utility values and fuzzy preference relation.

Section 4 presents the decision process, i.e. the classification method of alternatives for multiple preference

relations. Then, and for the sake of illustrating the classification method of alternatives, section 5 is devoted

to developing an example. In section 6 some conclusions are pointed out. Finally, the description of fuzzy

majority concept is presented in an appendix to this paper.

2. Presentation of the Problem

The problem we will deal with is that of choosing the best alternative(s) among a finite set { }X x xn= 1 ,..., .

The alternatives will be classified from best to worst, using the information supplied by a set of experts or

criteria (general purpose) (in the following, without lack of generality, we will use the term expert),

{ }E e em= 1 ,..., . As each expert is characterised by their own ideas, attitudes, motivations and personality,

it is quite natural to think that different experts will provide the information in a different way. The information

is presented as preferences over the set of alternatives, and as was pointed out before those preferences will

be of a fuzzy nature. The preferences over the set of alternatives X that the decision maker ek can

provide, may be represented in one of the following three ways:

1. A preference ordering of the alternatives. A classification of the alternatives according to a preference

ordering, i.e. an ordered vector of alternatives. In this case an expert ek provides his opinion on X as an

individual preference ordering { }x xo o n( ) ( ),...,1 , where o( )⋅ is a permutation function over the index set

{ }1,...,n , and o i( ) represents the position of alternative xi in the preference ordering [5,20] .

2. A fuzzy preference relation. This is described by a fuzzy binary relation Rk on X , i.e. a fuzzy set on

the product X X× characterised by a membership function, where ( )µk i jx x, denotes the preference

degree of the alternative xi over the alternative x j for expert ek . If the cardinality of X is small enough,

Rk may be represented by a matrix [ ] ( )[ ]R r x xk ijk

k i j= = µ , , i j n, ,...,= 1 . It is generally assumed that

Rk is reciprocal in that r rijk

jik+ = 1 , and -by definition- rii

k = 0 , for all i j, as given above. Therefore, an

expert ek can supply a fuzzy preference relation Rk over the set of alternatives [7,10,16,17].

3. A fuzzy utility function. This is described as a fuzzy mapping gk , [ ]g Xk : ,→ 0 1 , where ( )g xk i

represents an evaluation of the alternative xi for the point of view of expert ek , i.e. a function that associates

each alternative of X with a real number indicating the performance of that alternative according to the point

of view of ek . In this case an expert ek will give a set of n utility values, one for each alternative, [14,22,20].

The general models that we propose are developed in the following four steps:

1. Making the information uniform. For every preference ordering and set of utility values we derive an

individual preference relation. To do this, a generalisation of methods is used [5,21].

2. Using the concept of fuzzy majority represented by a linguistic quantifier and applied in the aggregation

operations by means of an ordered weighted averaging (OWA) operator [24], a collective preference relation

is obtained from the individual preference relations.

3. Using again the concept of fuzzy majority, two choice degrees of alternatives are used: the quantifier

guided dominance degree and the quantifier guided non-dominance degree [5]. The latter choice degree

generalises Orlovsky’s non-dominated alternative concept [18]. These choice degrees will act over the

collective preference relation supplying a selection set of alternatives.

4. The application of the above choice degrees of alternatives over the collective preference relation may be

carried out according to two different selection processes, called sequential choice process and conjunction

choice process.

It is important to state that we will use the fuzzy majority concept as a basis for our aggregation and

exploitation processes, managing the information under a fuzzy majority represented by fuzzy quantifiers, and

aggregating the information by means of an OWA operator [24], whose weights are calculated by those fuzzy

quantifiers.

Although the impossibility theorems of Arrow are a reality and it is well known the difficulty of the collective-

choice problem, the concept of democratic decision in a group leads us to the impossibility theorems [1].

However, the application of the notion of fuzzy sets in the idea of the fuzzy majority concept is very

appropriate in the multiple criteria (experts) decision making process. The fuzzy majority can provide a

framework with more human-consistency to the aggregation and the choice process in an imprecise

environment [8,10].

3. Making the Information Uniform

In this general framework, where the information provided by a group of experts is supposed to be of a

diverse nature, we need a model which can deal with these three different representations of the information.

To achieve this, as a first step we need to make the information uniform. Due to their apparent merits, the

use of fuzzy preference relations in decision making situations to represent an expert’s opinion about an

alternative set, appears to be a useful tool in modelling decision processes. Furthermore, non-fuzzy

preference relations and utility values are included in the family of fuzzy preference relations [22]; preference

orderings cannot be handled easily and directly to solve the problem but can easily be transformed into fuzzy

preference relations. Finally, the use of fuzzy preference relations appears in a very natural way when we

want to aggregate experts’ preferences into group preferences, that is, in the processes of group decision

making [2,3,8,10,17].

3.1 Utility Values and Preference Relations

In this section we will study the relationship between utility values and preference relations. The relationship

between utility values given on the basis of a difference scale and preference relations will be studied in the

next section. Therefore, this section is devoted to the study of the relationship between the utility values

given on the basis of a positive ratio scale and preference relations.

In such case, it is assumed that each alternative is given a real number indicating the performance of that

alternative according to the expert ek . So, for every set of utility values, ( )U u ukk

nk= 1 ,..., , over the set of

alternatives, we have to obtain a preference relation [ ]R rkijk= . Without loss of generality, we suppose that

the higher the evaluation, the better the alternative satisfies the expert’s criterion. The preference of

alternative xi over alternative x j for expert ek is shown as:

( )r f u uijk

k ik

jk= , ( )1

where f k is a non-decreasing function of the first argument and a non-increasing function of the second

argument. Interpreting u

uik

jk as a ratio of the preference intensity for xi over x j , that is , xi is

u

uik

jk times

as good as x j , and assuming a preference relation being reciprocal, i.e., r rijk

jik+ = 1 , then rij

k may be

defined as:

( )( ) ( )r

u

u

u

u

u

u

u

u uijk

ik

jk

ik

jk

jk

ik

ik

ik

jk

=+

=+

2

2 2 2( )

This formula is equivalent to the one studied by Luce and Suppes (14) and proposed by Tanino (21). In the

latter, if the utility values on the set of alternatives are given on the basis of a positive ratio scale, the following

expression was proposed:

ru

u uijk i

k

ik

jk=

+( )3

Obviously, these are not the only functions we can use to transform utility values given on the basis of a

positive ratio scale into preference relations. As we will show later, both (2) and (3) are particular cases of a

general family of functions.

Without loss of generality, we suppose that utility values belong to [ ]0 1, , so that [ ] [ ] [ ]f k : , , ,0 1 0 1 0 1× → has

to verify

1. ( ) [ ]f x y f y x x yk k, ( , ) , , ( )+ = ∀ ∈1 0 1 4

2. ( ) [ ]f x x xk , , , (5)= ∀ ∈12 0 1

3. ( ) ] ]f x xk , , , ( )0 1 0 1 6= ∀ ∈

4. ( )f x y x yk , ( )> ⇔ >12 7

Property (6) means that if an expert has certain knowledge that an alternative does not satisfy his criterion,

then any alternative satisfying his criterion with a positive value should be preferred with the maximum degree

of preference. Property (5) is a consequence of property (4), and indicates the indifference of an expert

between two alternatives verifying his criterion with the same intensity. Property (7) indicates that between

two alternatives, the expert gives a definite preference to the alternative with higher evaluation over the other.

This is a consequence of (5) and the fact that function f k has to be non-decreasing in the first argument and

non-increasing in the second argument.

Without loss of generality, we assume that

( ) ( )f x yh x yk

k

,,

(8)=+

1

1

where [ ] [ ]h Rk : , ,0 1 0 1× → + is a non-increasing function of the first argument and a non-decreasing

function of the second argument. To solve (8) we assume hk is a function able to be written so that

variables are separated multiplicatively, i.e. that hk is a separable function. This assumption is based on the

fact that the set of utility values are given on the basis of a positive ratio scale, and it is necessary and

relevant as we will show later. We then have

( )f x yr x s yk

k k

,( ) ( )

( )=+ ⋅

1

19

where r sk k, are functions with same domain [ ]0 1, , same sign , non-increasing the first and non-

decreasing the second, respectively.

From (2) we have:

[ ]r x s x xk k( ) ( ) , ( )⋅ = ∀ ∈1 0 1 10

Therefore rsk

k

≡1

, and then (9) transforms into

( )f x ys x

s x s ykk

k k

,( )

( ) ( )( )=

+11

being [ ]s Rk : ,0 1 → any non-decreasing function. From (6) we have that sk ( )0 0= , so that

[ ]s Rk : ,0 1 → + . Expression (11) is not defined when ( ) ( )x y, ,= 0 0 , but from property (5) we can define

( )f k 0 0 12, = .

A desirable property to be verified by the defined preference relation should be that if the valuations of a pair

of alternatives change slightly then the preference degree between them should change slightly too, i.e. it

would be desirable for f k to be continuous. This can be achieved if sk is a continuous function.

To support the separable assumption we have made above, let us consider the following. We were trying to

find functions [ ] [ ] [ ]f k : , , ,0 1 0 1 0 1× → verifying

( ) ( ) [ ]f x y f y x x yk k, , , , ( )+ = ∀ ∈1 0 1 4

Without loss of generality, we assume that

( ) ( )[ ] [ ]f x y g x y x yk k, , , , ( )= ∀ ∈2

0 1 12

being [ ]g x yk ( , ) ,∈ 0 1 . Equation (4) transforms into

( )[ ] ( )[ ] [ ]g x y g y x x yk k, , , , ( )2 2

1 0 1 13+ = ∀ ∈

We can represent equation (13) in the parametric form

( ) ( )g x y z g y x zk k, cos , , sin ( )= = 14

where [ ]z ∈ 0 2,π represents the value of the vector angle of a point with Cartesian coordinates ( )x y, or, in

general, ( )t x t yk k( ), ( ) , where [ ]t Rk : ,0 1 → + is any non-decreasing and continuous function, verifying

t k ( )0 0= .

We then have:

[ ] [ ]z

t x

t x t y

k

k k

=+

arccos( )

( ) ( )( )

2 215

and, therefore:

( )[ ] [ ]

g x yt x

t x t yk

k

k k

,( )

( ) ( )( )=

+2 2

16

Finally, writing [ ]s x t xk k( ) ( )=2

, equation (12) becomes

( )f x ys x

s x s ykk

k k

,( )

( ) ( )( )=

+17

which coincides with equation (11).

Summarising, we have the following results:

Proposition 1. For every set of utility values, ( )U u un= 1,..., , over a set of alternatives, { }X x xn= 1 ,..., ,

given on the basis of a positive ratio scale, then the preference of alternative xi over x j , rij , is given by the

following expression:

( ) ( )

( ) ( )r

s u

s u s u if

if

u u

u uij

i

i j i j

i j

=+

≠

=

( )

( ) ( ) , ,

, ,

( )1

2

0 0

0 0

18

where [ ]s R: ,0 1 → + is any non-decreasing and continuous function, verifying s( )0 0= .

Corollary 1.1. When s u u( ) = the (18) reduces to formula (3) proposed by Tanino. When s u u( ) = 2 then

formula (18) reduces to (2).

3.2 Preference Orderings and Preference Relations

In this section we will study the relationship between preference orderings and preference relations. For

every preference ordering, Ok , we will assume, without loss of generality, that the higher the position of an

alternative in a preference ordering, the better the alternative satisfies the expert ek . For example, suppose

that an expert supplies his or her opinions about a set of alternatives { }X x x x x= 1 2 3 4, , , by means of the

following preference ordering ( )x x x x2 4 1 3, , , . In this case o o o o( ) , ( ) , ( ) , ( )1 3 2 1 3 4 4 2= = = = . This

means that alternative x2 is the best for that expert, while alternative x3 is the worst. When two or more

preference orderings are given or when preference orderings are presented in a combination with other types

of information (preference relations, utility functions) it is necessary to have a model to help us to decide

when an alternative is better than another. In the second case we must make the information uniform. As

we said before, for every preference ordering, Ok , we will derive a preference ordering relation, which is a

fuzzy preference relation, P k , where pijk reflects the degree (between 0 and 1) to which alternative xi is

declared not worse than alternative x j for expert ek . In our example, alternative x2 is not worse than

alternatives x x x4 1 3, , ; alternative x4 is not worse than alternatives x x1 3, ; and, finally, alternative x1 is not

worse than alternative x3 . When comparing pairs of alternatives from a preference ordering in order to

obtain a credibility of preference of any alternative over any other alternative, it is obvious that the only

difference between two pairs of them is the value of the difference between the alternatives’ position. This

means that, without any other supplementary information except the preference ordering, if p242

3= then

p41 and p13 should be equal to 2/3, but p21 should be greater than or equal to 2/3 and less than or equal to

p23 .

The preference of alternative xi over x j for expert ek depends only on the values o i( ) and o j( ) . This

asserts that there exists a function f k such that

( )p f o i o jijk

k= ( ), ( ) ( )19

where f k is a non-decreasing function of the first argument and a non-increasing function of the second

argument. As the preferences between pairs of alternatives depend on the value o j o i( ) ( )− , then there

exists a function gk such that

( ) ( )p f o i o j g o j o iijk

k k= = −( ), ( ) ( ) ( ) ( )20

where gk is a non-decreasing function.

An example of this type of function has been used in [5]. In that paper the preference ordering relation used

was

pif

if

o i o j

o i o jijk =

<>

1

021

( ) ( )

( ) ( )( )

The simplicity and easy use are the only virtues of this function. Furthermore, this preference relation does

not reflect the case when an expert is not able to distinguish between two alternatives, that is when there is

an indifference between two alternatives, although this can be achieved with an extension of function (21) as

follows

( )p g o j o i

if

if

if

o j o i

o j o i

o j o iijk

k= − =− >− =− <

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

1

0

0

0

0

2212

Functions (21) and (22) do not reflect any kind of intensity of preference between alternatives, that is, they do

not distinguish between the preference of alternative x2 over x4 and the preference of alternative x2 over

x3 . This can be achieved by using another type of function, for example by giving a value of importance or

utility to each alternative. We can assume that the preference of the best alternative over the worst alternative

is the maximum allowed, that is 1. So if, for example, o i( ) = 1 and o j n( ) = , then we assume that

po i o j( ) ( ) = 1 . In this case, the utility value ui associated to alternative xi depends on the value of its position

o i( ) , in such a way that the bigger the value of n o i− ( ) , the bigger the value of ui , that is

( )u s n o ii = − ( ) , being s a non-decreasing function. As an example, we can assign the value

un o i

ni =−

−( )

1 as a degree of importance or utility of the alternative xi . It is clear that the maximum utility

value corresponds to the first alternative and the minimum utility value to the last alternative in the preference

ordering. In this context, we have a normalised set of n utility values, that is, max mini

ii

iu u− ≤ 1 . These

utility values are given on the basis of a difference scale, so that in order to obtain preferences between the

alternatives we can make use of the following expression

( )p u uijk

ik

jk= + −1

2 1 23( )

This expression has been investigated by Dombi [6], and was proposed by Tanino [21]. In [6] Dombi defined

(23) as the universal preference function, and showed that the utility based decision making gives the same

result as the preference based using the universal preference function. In our case (23) becomes

( ) ( )p g o j o iijk

k

o j o in= − = + −

−( ) ( ) ( )( ) ( )1

2 11 24

It is obvious that both (22) and (24) verify:

1. 0 1≤ ≤pijk for any i j n, ,...,= 1 .

2. p pijk

jik+ = 1 for any i j n, ,...,= 1 .

3. When there is an indifference between two alternatives, that is when o i o j( ) ( )= , pijk = 1

2 .

4. p if o i o jijk > <1

2 ( ) ( )

Obviously, these are not the only functions that can be used to transform preference orderings into

preference relations. As we will show later, (22) and (24) are particular cases of a general family of functions

that can be used to transform preference orderings into preference relations.

Suppose o i( ) and o j( ) are the position number of alternatives xi and x j respectively. In what follows,

both o i( ) and o j( ) can be replaced by ( )u s n o ii = − ( ) and ( )u s n o jj = − ( ) , respectively, or in general

by two real numbers u ui j, . We are looking for a general expression of the transformation of preference

orderings into preference relations, that is given a pair of alternatives ( , )x xi j of which we only know their

position numbers in a preference ordering, ( )o i o j( ), ( ) , the preference of xi over x j is given by

( )p f o i o jijk

k= ( ), ( ) ( )19

where f k is a non-decreasing function of the first argument and a non-increasing function in the second

argument, verifying

1. ( ) [ ]f o i o jk ( ), ( ) ,∈ 0 1

2. ( ) ( )f o i o j f o j o ik k( ), ( ) ( ), ( )+ = 1

3. ( )f o i o j if o i o jk ( ), ( ) ( ) ( )> >12

As values o i( ) and o j( ) are given on the basis of a difference scale, then there exists a function gk such

that

( ) ( )p f o i o j g o j o iijk

k k= = −( ), ( ) ( ) ( ) ( )20

where gk is a non-decreasing function verifying

1. ( ) [ ]g ck ∈ 0 1,

2. ( ) ( )g c g ck k+ − = 1

3. ( )g c if ck > >12 0

In case we use utility values given on the basis of a difference scale, then o j o i( ) ( )− can be replaced by

u ui j− . If we use ( )s n o i− ( ) instead of o i( ) , then o j o i( ) ( )− has to be replaced by

( ) ( )s n o i s n o j− − −( ) ( ) .

Without loss of generality, it can be assumed that:

( )g c h ck k( ) ( )= +12 25

being hk a non-decreasing function verifying

1. ( ) [ ]h ck ∈ −12

12,

2. ( ) ( )h c h ck k+ − = 0

3. ( )h c if ck > >0 0

We then have:

( )p h o j o iijk

k= + −12 26( ) ( ) ( )

Property 2 implies ( ) ( )h c h ck k− = − , that is hk is an odd function. On the other hand, the following result

is well known:

“A function h D R: → with a symmetric domain is an odd function if and only if there exists a function

F D R: → verifying ( )h xF x F x

=− −( ) ( )

2”.

Applying this result to our situation, we have the following results:

Proposition 2. Suppose we have a set of alternatives { }X x xn= 1,..., , and associated with it a preference

ordering { }x xo o n( ) ( ),...,1 , where ( )o ⋅ is a permutation function over the index set {1, …,n}. Then, the

preference degree of alternative xi over x j is given by the following expression

( ) ( )[ ]p F o j o i F o i o jij = + − − −1

21 27( ) ( ) ( ) ( ) ( )

where F is any non-decreasing function.

Corollary 2.1. Suppose that the preference degree of the best alternative over the worst alternative in a

preference ordering is the maximum allowed, that is 1, then:

1. When ( )F x

if

if

if

x

x

x

=>=<

−

12

12

0

0

0

0

, expression (27) reduces to expression (22).

2. When ( )F xa x

=⋅2

, where a R∈ , then an

=−1

1 and (27) reduces to (24).

Proposition 3. Suppose we have a set of alternatives { }X x xn= 1,..., , and a set of n utility values

{ }U u un= 1,..., associated to X . If the utility values are given on the basis of a difference scale then the

preference of xi over x j , pij , is given by the following expression

( ) ( )[ ]p F u u F u uij i j j i= + − − −1

21 28( )

where F is any non-decreasing function.

Corollary 3.1. Expression (28) reduces to Dombi’s universal preference function (23) when F xx

( ) =2

.

Corollary 3.2. Suppose we have a set of alternatives { }X x xn= 1,..., , and associated with it a preference

ordering { }x xo o n( ) ( ),...,1 , where ( )o ⋅ is a permutation function over the index set {1, …,n}. Then, taking

( ) [ ]u s n o i a n o ii = − = ⋅ −( ) ( ) , and F xx

( ) =2

, where a R∈ , then an

=−1

1 (so function s is non-

decreasing), un o i

ni =−

−( )

1 and (28) reduces to (24).

3.3. Preference Orderings, Utility Values and Preference Relations

This section summarises everything we have seen in the previous sections. The result we present includes

all the results we have obtained, and in that sense can be considered as a general theorem to be used when

we wish to make the information uniform.

Proposition 4. Suppose we have a set of alternatives { }X x xn= 1,..., . Suppose that g xi( ) represents an

evaluation of alternative xi , that is a function that associates each alternative xi of the set X with a real

number indicating the performance of that alternative according to a point of view (expert or criterion).

The intensity of preference of alternative xi over alternative x j for that point of view is given by

( ) ( ) ( )[ ]R x x f g x g x f g x g xi j i j j i, ( ), ( ) ( ), ( ) ( )= ⋅ + −1

21 29

where f is a function verifying

1. ( )f a a a R, = ∀ ∈12

2. f is non-decreasing of the first argument and non-increasing of the second argument.

Proof. Without loss of generality, we can assume that the higher the evaluation, the better the alternative

satisfies the expert. The intensity of preference of alternative xi over alternative x j is given by

( ) ( )R x x f g x g xi j i j, ( ), ( )=

where f is a non-decreasing function of the first argument and a non-increasing function of the second

argument. We are assuming a preference relation being reciprocal, i.e., r rijk

jik+ = 1 , then

( ) ( )f g x g x f g x g xi j j i( ), ( ) ( ), ( )+ = 1

We then have

( )f a a a R, = ∀ ∈12

( ) ( )f g x g x f g x g xi j j i( ), ( ) ( ), ( )= −1

Consequently,

( ) ( ) ( )( ) ( )[ ]( ) ( )[ ]

R x x f g x g x f g x g x

f g x g x f g x g x

f g x g x f g x g x

i j i j i j

i j i j

i j j i

, ( ), ( ) ( ), ( )

( ), ( ) ( ), ( )

( ), ( ) ( ), ( )

= = ⋅

= ⋅ +

= ⋅ + −

1

22

1

21

21

Corollary 4.1. (Proposition 2) Suppose we have a set of alternatives { }X x xn= 1,..., , and associated with

it a preference ordering { }x xo o n( ) ( ),...,1 , where ( )o ⋅ is a permutation function over the index set {1, …,n}.

Then, the preference degree of alternative xi over x j is given by expression (29) where

( ) ( )f o i o j F o j o i( ), ( ) ( ) ( )= − , being F any non-decreasing function.

Proof. In this case g x n o ii( ) ( )= − . As values o i( ) and o j( ) are given on the basis of a difference

scale, then

( ) ( ) ( )p f o i o j F g x g x F o j o iij i j= = − = −( ), ( ) ( ) ( ) ( ) ( )

where F is a non decreasing-function. Consequently expression (29) reduces to (27).

Corollary 4.2. (Proposition 3) Suppose we have a set of n utility values { }U u un= 1,..., associated to X .

If the utility values are given on the basis of a difference scale then the preference of xi over x j is given by

expression (29) where ( ) ( )f u u F u ui j i j, = − , being F any non-decreasing function.

Proof. In this case g x ui i( ) = . As the utility values are given on the basis of a difference scale then

( ) ( )f g x g x F g x g xi j i j( ), ( ) ( ) ( )= − , and therefore (29) reduces to (28).

Corollary 4.3.(Proposition 1) Suppose we have a set of n utility values { }U u un= 1,..., associated to X . If

the utility values are given on the basis of a positive ratio scale then the preference of xi over x j is given by

expression (29) where

( ) ( )

( ) ( )f x y

s x

s x s y if

if

x y

x y

( , )

( )

( ) ( ) , ,

, ,

=+ ≠

=

12

0 0

0 0

being [ ]s R: ,0 1 → + any non-decreasing and continuous function, verifying s( )0 0= .

Proof. Indeed, expression (18) can be rewritten in the following way

rs u

s u s u

s u

s u s u

s u s u s u

s u s u

s u s u

s u s u

s u s u

s u s u

s u

s u s u

iji

i j

i

i j

i j j

i j

i j

i j

i j

i j

i

i j

=+

= ⋅⋅+

= ⋅⋅ + −

+

= ⋅++

+−+

= ⋅ ++

( )

( ) ( )

( )

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( ) (

1

2

2 1

2

2

1

2

1

21

)

( )

( ) ( )−

+

s u

s u s uj

i j

4. The Decision Process

In this section we will deal with choosing the alternative(s) which is(are) considered to be desirable for the

group as a whole. For that reason, and after the information is uniformed into fuzzy preference relations, we

have a set of m individual fuzzy preference relations. The decision process we present here has two steps:

aggregation and exploitation. The aggregation phase defines a collective preference relation, which indicates

the global preference between every ordered pair of alternatives. The exploitation phase transforms the

global information about the alternatives into a global ranking of them.

4.1. Aggregation: The collective preference relation

Once we have made the information uniform, we have a set of m experts, { }E e em= 1 ,..., , and a set of m

individual binary preference relations { }P Pm1 ,..., over the set of n alternatives, { }X x xn= 1 ,..., . From

this set of individual preference relations we derive the collective preference relation, P . Each value,

[ ]pij ∈ 0 1, , represents the preference of alternative xi over alternative x j .

Traditionally, the majority is defined as a threshold number of individuals. Fuzzy majority is a soft majority

concept expressed by a fuzzy linguistic quantifier, which is manipulated via a fuzzy logic based calculus of

linguistically quantified propositions [8]. We will compute pij using an OWA operator , used for aggregating

information. The OWA operator reflects the fuzzy majority calculating its weights by means of the fuzzy

linguistic quantifier. The collective preference relation ( )P pij= is obtained as follows:

( )p p pij Q ij ijm= φ 1 ,...,

where Q is a fuzzy linguistic quantifier used to compute the weight vector of the OWA operator, representing

the fuzzy majority over the individuals.

4.2. Exploitation: Choice Process

At this point, in order to select the alternative(s) “best” acceptable to the group of individuals as a whole, we

will use two choice degrees of alternatives, based on the concept of fuzzy majority: a dominance degree and

a non-dominance degree. Both are based on the use of the OWA operators whose weights are calculated by

means of the quantifiers representing the fuzzy majority. The application of these two choice degrees of

alternatives can be carried out according to two different selection processes: a sequential selection process

and a conjunction selection process [5].

4.2.1. Choice degrees of alternatives

For the alternative x i ni , ,..., = 1 , we compute the quantifier-guided dominance degree, QGDD( )⋅ , used to

quantify the dominance that one alternative has over all the others in a fuzzy majority sense:

( ) ( )QGDD x p j n j ii Q ij= = ≠φ ; ,..., ,1

where φ Q is an OWA operator whose weights are defined using a relative quantifier Q ; and we also

compute the quantifier-guided non-dominance degree, QGNDD( )⋅ ,

( ) ( )QGNDD x p j n j ii Q jid= − = ≠φ 1 1; ,..., ,

where

{ }p max p pijd

ji ij= − ,0

represents the degree in which xi is strictly dominated by x j . In our context, the membership function

QGNDD( )⋅ gives the degree in which each alternative is not dominated by a fuzzy majority of the remaining

alternatives. We note that when the fuzzy quantifier represents the statement “all ”, whose algebraic

aggregation corresponds to the conjunction operator Min, then this non-dominance degree coincides with

Orlovsky’s non-dominated alternative concept.

4.2.2. Selection Process

The application of the above choice degrees of alternatives over the collective preference relation may be

carried out according to two different processes.

• Either selecting and applying one of them according to the preference of the experts, and thus obtaining

a selection set of alternatives. If there is more than one alternative in that selection set, then the other choice

degree may be applied to select the alternative of the above set with the best second choice degree. This

process is called the sequential choice process.

• Or, applying the two choice degrees to X , thus obtaining the selection set of alternatives as the

intersection of the two selection set of alternatives. This process is called the conjunction choice process.

We note that the latter conjunction choice process is more restrictive than the former sequential choice

process because it is possible to obtain an empty selection set. Therefore, a complete selection process

can be applied in three steps .

• Step one. The application of each choice degree of alternatives to the collective preference relation to

obtain the following sets of alternatives:

( ) ( )X x X QGDD x QGDD zQGDD

z X= ∈ =

∈

sup

( ) ( )X x X QGNDD x QGNDD zQGNDD

z X= ∈ =

∈

sup

whose elements are called maximum dominance elements of the fuzzy majority of X quantified by Q and

maximal non-dominated elements by the fuzzy majority of X quantified by Q , respectively.

• Step two. The application of the conjunction choice process, obtaining X X XQGCP QGDD QGNDD= ∩ . If

X QGCP ≠ ∅ , then End,

otherwise

• Step three. The application of the two sequential choice processes, according to either a dominance or

non-dominance criterion, i.e.,

.Dominance based sequential selection process. If ( )# X QGDD = 1 , then End, otherwise

obtaining:

( ) ( )X x X QGNDD x QGNDD zQG DD NDD QGDD

x X QGDD

− −

∈= ∈ =

sup

or

.Non-dominance based sequential selection process. If ( )# X QGNDD = 1 , then End, otherwise

obtaining

( ) ( )X x X QGDD x QGDD zQG NDD DD QGNDD

z X QGNDD

− −

∈= ∈ =

sup

5. Example

Consider the following illustrative example of the classification method of alternatives studied in this paper.

Suppose that we have a set of six experts, { }E e e e e e e= 1 2 3 4 5 6, , , , , , and a set of four alternatives,

{ }X x x x x= 1 2 3 4, , , . Suppose that experts e e1 2, supply their opinions in terms of ordering preferences of

the alternatives, experts e e3 4, in terms of utility values associated to each alternative, and experts e e5 6, in

terms of preferences between pairs of alternatives. Suppose the information is the following:

O x x x x

O x x x x

U

U

P

P

1 2 4 1 3

2 3 2 1 4

3

4

5

6

05 0 7 1 01

0 7 0 9 0 6 0 3

01 0 6 0 7

0 9 0 8 0 4

0 4 0 2 0 9

0 3 0 6 01

05 0 7 1

0 5 08 0 6

0 3 0 2

=

=

=

=

=

−−

−−

=

−−

−

( , , , )

( , , , )

( . , . , , . )

( . , . , . , . )

. . .

. . .

. . .

. . .

. .

. . .

. . 08

0 0 4 0 2

.

. . −

Using expressions (2) and (24) to make the information uniform, we have:

P

P

P

1

2

3

1 6 2 3 1 2

5 6 1 2 3

1 3 0 1 6

1 2 1 3 5 6

1 3 1 6 2 3

2 3 1 3 1 6

5 6 2 3 1

1 3 5 6 0

25 74 0 2 25 26

49 74 49 149 0 98

08 100 149 100 101

1 26 0 02 1 101

=

−−

−−

=

−−

−−

=

−−

−−

/ / /

/ /

/ /

/ / /

/ / /

/ / /

/ /

/ /

/ . /

/ / .

. / /

/ . /

=

−−

−−

P4

49 130 49 85 49 58

81 130 81 117 0 9

36 85 36 117 0 8

9 58 01 0 2

/ / /

/ / .

/ / .

/ . .

Using the fuzzy majority criterion with the linguistic quantifier “At least half ” with the pair ( , . )0 0 5 and the

corresponding OWA operator with the weight vector W = ( / , / , / , , , )1 31 31 3 0 0 0 , the collective preference

relation is:

P =

−−

−−

0 40492 0 65556 0 93546

08 0 86667 0 84889

0 68562 05485 0 96337

0 37778 0 61111 0 41111

. . .

. . .

. . .

. . .

We apply the choice process with the fuzzy quantifier “Most ” with the pair ( . , . )0 3 08 and the corresponding

OWA operator with the weight vector W = ( / , / , / )1 1510 15 4 15 . The choice degrees of alternatives acting

over the collective preference supply the following values:

x x x x

QGDD

QGNDD

1 2 3 4

0 60738 083703 0 66757 0 41556

0 87461 1 0 91515 0 46726

. . . .

. . .

These values represent the dominance that one alternative has over the “most” alternatives according to “at

least half” of the experts, and the nondominance degree to which the alternative is not dominated by “most”

alternatives according to “at least half” of the experts, respectively.

Clearly the maximal sets are

X x and X xQGDD QGNDD= ={ } { }2 2

therefore the selection set of alternatives for all selection procedures is the singleton { }x2 .

If we use the linguistic quantifier “As many as possible” instead of “At least half ”, with the pair ( . , )0 51 and the

corresponding OWA operator with the weight vector W = ( , , , / , / , / )0 0 0 1 3 1 3 1 3 , then the collective

preference relation is:

P =

−−

−−

0 2 0 31438 0 62222

059508 0 4515 0 38889

0 34444 013333 0 58889

0 06454 015111 0 03663

. . .

. . .

. . .

. . .

Applying the choice process with the fuzzy quantifier “Most ” with the pair ( . , . )0 3 08 and the corresponding

OWA operator with the weight vector W = ( / , / , / )1 1510 15 4 15 , then the choice degrees of alternatives

acting over the collective preference supply the following values:

x x x x

QGDD

QGNDD

1 2 3 4

0 3044 0 44437 0 30444 0 06288

087461 1 0 91515 0 46726

. . . .

. . .

The maximal sets are

X x and X xQGDD QGNDD= ={ } { }2 2

and therefore the selection set of alternatives for all selection procedures is the singleton { }x2 . As we see,

the solution to our example is the same using different linguistic quantifiers in the aggregation of the

individual preference relations.

6. Conclusions

In this paper we have presented a general model for a multi-person decision making problem, where the

information supplied by the group of experts can be of a diverse nature, based on the concept of fuzzy

majority for the aggregation and exploitation of the information in decision making. As a first step, it was

necessary to make the information uniform, for which we used fuzzy preference relations according to their

apparent merits and also because utility values and non-fuzzy preference relations are included in the family

of fuzzy preference relations. We have presented a general method to relate preference orderings, utility

values and fuzzy preference relations. As we have seen, this general method generalises the procedures

normally used, in particular those suggested by Tanino, Chiclana et. al., and Dombi´s universal preference

function.

We have used two quantifier-guided choice degrees of alternatives; a dominance degree used to quantify the

dominance that one alternative has over all the others in a fuzzy majority sense, and a non-dominance

degree that generalises Orlovsky’s non dominated alternative concept. We have shown that the above choice

degrees can be carried out according to two proposed selection processes.

APPENDIX A. Fuzzy Majority

As we said before, the majority is traditionally defined as a threshold number of individuals. Fuzzy majority is

a soft majority concept expressed by a fuzzy linguistic quantifier, which is manipulated via a fuzzy logic based

calculus of linguistically quantified propositions.

In this appendix we present the fuzzy linguistic quantifiers, used for representing the fuzzy majority, and the

OWA operators, used for aggregating information. The OWA operator reflects the fuzzy majority calculating

its weights by means of the fuzzy quantifiers.

A.1. Fuzzy Linguistic Quantifiers

Quantifiers can be used to represent the amount of items satisfying a given statement. Classical logic is

restricted to the use of the two quantifiers, there exists and for all, that are closely related respectively with the

or and and connectives. Human discourse is much richer and more diverse in its quantifiers, e.g. about 5,

almost all, a few, many, most, as many as possible, nearly half, at least half. In an attempt to bridge the gap

between formal systems and natural discourse and, in turn, provide a more flexible knowledge representation

tool, Zadeh introduced the concept of linguistic quantifiers [29].

Zadeh suggested that the semantic of a linguistic quantifier can be captured by using fuzzy subsets for its

representation. He distinguished between two types of linguistic quantifiers, absolute and proportional or

relative. Absolute quantifiers are used to represent amounts that are absolute in nature such as about 2 or

more than 5. These absolute linguistic quantifiers are closely related to the concept of counting or number of

elements. He defined these quantifiers as fuzzy subsets of the non-negative real numbers, R + . In this

approach, an absolute quantifier can be represented by a fuzzy subset Q , such that for any r R∈ + the

membership degree of r in Q , Q r( ) indicates the degree in which the amount r is compatible with the

quantifier represented by Q ,. Proportional quantifiers, such as most, at least half, can be represented by

fuzzy subsets of the unit interval [0,1]. For [ ]r ∈ 0 1, , Q r( ) indicates the degree in which the proportion r is

compatible with the meaning of the quantifier it represents. Any quantifier of natural language can be

represented as a proportional quantifier or, given the cardinality of the elements considered, as an absolute

quantifier. Functionally, linguistic quantifiers are usually one of three types, increasing, decreasing, or

unimodal. An increasing type quantifier is characterised by the relationship

( ) ( )Q r Q r if r r1 2 1 2≥ >

These quantifiers are characterised by values such as most, at least half. A decreasing type is characterised

by the relationship

( ) ( )Q r Q r if r r1 2 1 2≤ >

These quantifiers characterise terms such as a few, at most k. Unimodal type quantifiers have the property

that

( ) ( ) ( ) ( )Q a Q b Q c Q d≤ ≤ = ≥1

for some a b c d≤ ≤ ≤ . These are useful for representing terms like about q.

An absolute quantifier, [ ]Q R: ,+ → 0 1 , satisfies:

( ) ( )Q and k such that Q k0 0 1= ∃ =,

A relative quantifier, [ ] [ ]Q: , ,0 1 0 1→ , satisfies:

( ) [ ] ( )Q and r such that Q r0 0 0 1 1= ∃ ∈ =, ,

A non-decreasing quantifier satisfies:

( ) ( )∀ > ≥a b if a b then Q a Q b,

The membership function of a non decreasing relative quantifier can be represented as

Q rr a

b a

if

if

if

r a

a r b

r b

( ) =−−

<≤ ≤

>

0

1

with [ ]a b r, , ,∈ 0 1 .

Yager in [25] describes a formalism for evaluating the truth of linguistically quantified propositions based upon

a logical interpretation that uses a generalisation of the and and or operations via OWA operators. In this

way the OWA operators reflect the fuzzy majority calculating their weights by means of the fuzzy quantifiers.

A.2. The Ordered Weighted Averaging Operator

The OWA operators were proposed by Yager in [24] and more recently characterised in [25], and provide a

family of aggregation operators which have the and operator at one extreme and the or operator at the other

extreme.

An OWA operator of dimension n is a functionφ , [ ] [ ]φ : , ,0 1 0 1n → , that has associated a set of weights.

Let { }a an1 ,..., be a list of values to aggregate, then the OWA operator φ is defined as

( )φ a a W B w bnT

ii

n

i11

,..., = ⋅ = ⋅=∑

where { }W w wn= 1 ,..., is a weighting vector verifying [ ]wi ∈ 0 1, and wii

n

==∑ 1

1

; and B is the associated

ordered value vector. Each element b Bi ∈ is the i th− largest value in the collection a an1 ,..., .

The OWA operators fill the gap between the operators Min and Max. It can be immediately verified that the

OWA operators are commutative, increasing and idempotent, but in general non-associative.

A natural question in the definition of the OWA operator is how to obtain the associated weighting vector. In

[24,25], Yager proposed two ways to obtain it. The first one is to use some kind of learning mechanism by

means of some sample data; and the second one is by trying to give some semantics or meaning to the

weights. The latter has allowed multiple applications in areas of fuzzy and multi-valued logic, evidence

theory, designing of fuzzy controllers, and quantifier-guided aggregations.

We are interested in the area of quantifier-guided aggregations, because our idea is to calculate weights

using linguistic quantifiers to represent the concept of fuzzy majority in the aggregation operations that are

made in our decision process. Therefore, in the aggregations of OWA operator the concept of fuzzy majority

is underlying by means of the weights. In [24,25], Yager suggested an interesting way to compute the

weights of the OWA aggregation operator using linguistic quantifiers, which, in the case of a non-decreasing

proportional quantifier Q , is given by this equation:

( ) ( )w Q in Q i

n i ni = − − =1 1, ,...,

When a fuzzy linguistic quantifier Q is used to compute the weights of the OWA operatorφ , this is

symbolised byφ Q .

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