Numerical Representation of Product Transitive ...

Mathematical and Computer Modelling 53(5-6):617-623 DOI: 10.1016/j.mcm.2010.09.013

Numerical Representation of Product Transitive Complete Fuzzy Orderings ISMAT BEG Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences, 54792- Lahore, Pakistan. E-mail: [email protected] and SAMINA ASHRAF Department of Mathematics, COMSATS Institute of Information Technology, Lahore, Pakistan

Abstract

Let X be a space of alternatives with a preference relation in the

form of product transitive complete fuzzy ordering R. We prove existence of continuous utility functions for R.

2010 Mathematics Subject Classi cation: 46S40; 91B16; 03E72; 68T37; 91B02. Key words and phrases: Fuzzy utility function; product transitivity; fuzzy orderings.

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1

Introduction and Preliminaries

Emerging in 1971, fuzzy relations [14] which express degrees of relationship are proposed as better and softer models for expressing strengths of links between elements of a given universe of discourse. The fuzzy orderings came forth as alternatives of crisp orderings representing degrees of preference. Ponsard [11] studied the in uence of a fuzzy relation of preference on the existence of a fuzzy utility in the case of a countable set of objects . The su ciency proof consists of showing the existence of a utility function to imply the existence of a continuous function in a fuzzy topology. Ponsard's results were based on one hand on the Usawa's theorem and on the other hand on a basic de nition of fuzzy preference from [14]. Later Billot [1] extended the results of [11] to a convex set of objects by giving conditions under which a fuzzy preordering can be represented by a numerical function. He also revealed some astonishing connections between the Debreu's standard model [6] and the fuzzy one they presented. Recently Fono and Andjiga [7] also studied utility function of fuzzy preferences on a countable set under max-*-transitivity. The representation theory deals with existence and construction of real valued order preserving mappings on an ordered universe. The early achievement in the area was made by Cantor [2]. Another milestone on the road was set by Debreu (see [5] and [6]) that added continuity to order preservation. These achievements made Representation Theory a very signi cant area of Mathematics claiming applications in Economics and Social Choice Theory. In this paper, we work on the similar lines but we di er with Billot [1] in his fuzzy model and in particular the de nition of fuzzy transitivity. He constituted his results for the fuzzy relations satisfying the following form of transitivity (we change his notation according to the one used in this work): For all x; y; z 2 X; R(x; y)

R(y; x) and R(y; z)

R(z; y) imply that R(x; z) 2

R(z; x);

here, R(x; y) represents the degree of preference of an element x on an element y. Whereas, we opt to work with product transitive fuzzy orderings which is a much broader class and later yields the above form of transitivity as its crisp counterpart. Another aspect, where this work di er from Billot [1] and all others is the way we de ne complete fuzzy relations. Completeness leads to the situation resembling to Pareto's optimality in which case we want to prefer an alternative a over an alternative b if a0 s degrees of preference over all other alternatives of the universe is greater than those of b . If a complete relation is not guaranteed, it can be converted into a complete relation by a simple scaling procedure. Thus there seems no harm in assuming completeness, while, it helps in obtaining a scenario, where out of every couple of articles one is fully preferable to the other one. Consequently, we obtains an alternative which enjoys 1 degree preference over all the other alternatives. In our fuzzy model, we present a new approach to the study of existence theorems for fuzzy orderings. First we state some preliminaries. A fuzzy set A in a universe X is a mapping from X to [0; 1]. For any x 2 X; the value A(x) is called the degree of membership of x in A. Moreover, F (X) will stand for the set of all fuzzy subsets of X: Given a crisp universe X, a fuzzy binary relation is a fuzzy subset of X

X: Fuzzy binary relations will

be called fuzzy relations throughout this paper. De nition 1.1. [13] Let A; B 2 F (X); A is said to be a subset of B denoted by A

B if and only if A(x)

B(x); for all x 2 X and A; B are said to equal

fuzzy sets denoted by A = B if and only if A(x) = B(x); for all x 2 X: De nition 1.2. [9] Triangular norm (t-norm) T and triangular conorm (tconorm) S are increasing, associative, commutative and [0; 1]2 ! [0; 1] mappings satisfying: T (1; x) = x and S(x; 0) = x; for all x 2 [0; 1]: 3

The only t-norm used in this paper is the product t-norm P which is de ned as: P (x; y) = x:y; for all x; y 2 [0; 1]: De nition 1.3. [14] A fuzzy relation R on a universe X is called: (1) re exive if and only if for all x 2 X, R(x; x) = 1; (2) symmetric if and only if for all x; y 2 X; R(x; y) = R(y; x); (3) product transitive if and only if for all x; y; z 2 X; R(x; y):R(y; z)

R(x; z);

(4) complete if and only if for all x; y 2 X; max(R(x; y); R(x; y)) = 1: De nition 1.4. A re exive and product transitive fuzzy relation L on a given universe X, is called a fuzzy preordering. The pair (X; L) will be called a fuzzy preordered space.. For the sake of reference next we state, crisp de nitions of complete ordering and the famous Debreu's lemma [5]. Looking at its historical signi cance, we make no modi cation or adjustment to it's language. De nition 1.5. [5] A complete ordering on a universe X is a binary relation, denoted by 5; satisfying: 1) Given two elements x; y 2 X; x 5 y and/or y 5 x:

2) Given three elements x; y; z 2 X if x 5 y and y 5 z; then x 5 z:

From this relation, can be derived two new ones:' x

y (x is indi erent to y) if x 5 y and y 5 x:

x < y (y is better than x) if x 5 y and not y 5 x: The qoutient set X=

is the set of all indi erence classes of

in X.

Lemma 1.6. [5, Lemma I, p.161] Let X be a completely ordered set whose qoutient set is countable. There exists on X a real order-preserving function, continuous in any topology in which the sets fx 2 X; x

x0 g and fx 2 X; x0 4

xg;

are closed in X.

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Main Results

In crisp settings, a function f from an ordered set (X; ) to an ordered set (Y; ) is said to be order preserving (or a utility function) if and only if for all x; y 2 X; x

y implies f (x)

f (y):

This de nition solely depends upon the crisp relations

and

: The fuzzy

counterparts of such concepts are rather complex in nature. In fuzzy environments, we have a fuzzy relation on a given universe expressing degrees of pairwise preferences. Now, utility function may stand for various meanings e.g., we may want to allocate real numbers to the elements of X, in such a manner that they represent their overall preference in the whole universe for example, Orlovsky's choice functions [12]. While, some authors proved the existence of utility functions that preserve degrees of preference e.g., see [10, 3]. We prove the existence of order preserving mappings on a universe with a fuzzy order in two di erent manners: one which preserves a pair wise comparison between the elements and the other proves the existence of a fuzzy utility which preserves the degrees and converts the given fuzzy ordering into a fuzzy ordering on the real line. In what comes ahead, we shall focus on looking at X and the fuzzy ordering L as a structure and we shall work with the overall preservation of this structure by a continuous mapping. The new construction depends upon the de nition of a fuzzy set Lx on X named a row vector which may be viewed as a way of looking at the fuzzy orderings by its pointwisely character. We start with few results on the behavior of product transitive fuzzy relations. We shall only work with the fuzzy relations for which R(x; y) 6= 0:Throughout 5

this paper R stands for the set of real numbers and N for the set of natural numbers. De nition 2.1. Let R be a fuzzy ordering on a universe X; a row vector of R by an element x of X is a fuzzy set Rx on X de ned as: Rx (y) = R(x; y); for all y 2 X: While, a column vector of R by an element x of X is a fuzzy set Rx on X de ned as: Rx (y) = R(y; x); for all y 2 X: If R is a symmetric fuzzy ordering, then a row vector is called an equivalence class of R (for details see [4]). De nition 2.2. Let (X; R) be a fuzzy preordered space. A function f : X

! R is called a utility function on X if and only if for all x; y 2 X,

R(x; y)

R(y; x) implies that f (x)

f (y):

De nition 2.3. Let (X; R) be a fuzzy preordered space and G : (X; R) ! (R; L). Function G is called a fuzzy utility function if for all x; y 2 X, Rx

Ry implies that G(Rx )

G(Ry ):

Where, L is a product transitive fuzzy ordering on R: Theorem 2.4. Let R be a fuzzy preordering. If for some x; y 2 X; R(x; y) = R(y; x) = a 6= 0; then for all z 2 X; a:R(x; z)

R(y; z)

1 :R(x; z) a

(1)

a:R(y; z)

R(x; z)

1 :R(y; z) a

(2)

and

Proof. Due to product transitivity, following hold for all x; y; z 2 X : R(x; z)

R(x; y):R(y; z); 6

(3)

R(y; z)

R(y; x):R(x; z);

(4)

R(x; y)

R(x; z):R(z; y);

(5)

R(z; x)

R(z; y):R(y; x);

(6)

R(z; y)

R(z; x):R(x; y);

(7)

R(y; x)

R(y; z):R(z; x):

(8)

If for some x; y 2 X; R(x; y) = R(y; x) = a; then for all z 2 X : R(x; z)

a:R(y; z);

(9)

R(y; z)

a:R(x; z);

(10)

R(x; z):R(z; y);

(11)

R(z; x)

R(z; y):a;

(12)

R(z; y)

R(z; x):a;

(13)

R(y; z):R(z; x):

(14)

a

a

Re-arranging (9), (10) and (11), we get a:R(x; z)

R(y; z)

1 R(x; z) a

a:R(y; z)

R(x; z)

1 R(y; z): a

and

remember that a 6= 0 by assumption. Remark 2.5. Following are immediate consequences of (1) and (2): 1. R(x; z) = 0 if and only if R(y; z) = 0; for any z 2 X. 2. Greater the value of a smaller is the distance between R(x; z) and R(y; z): In fact, increase in the value of a squeezes the region within [0; 1] in which both R(x; z) and R(y; z) lie. 3. If a = 1, then R(x; z) = R(y; z); for all z 2 X: Remark 2.6. In case of a complete fuzzy preordering (see De nition 1.4 (4) ), without loss of generality we can assume that R(y; x) = 1, R(z; y) = 1 and 7

R(z; x) = 1 and thus the inequalities (3 to 8) take the following form for all x; y; z 2 X: R(x; z)

R(x; y):R(y; z)

(15)

R(y; z)

R(x; z)

(16)

R(x; y)

R(x; z)

(17)

R(z; y)

R(x; y)

(18)

R(y; x)

R(y; z)

(19)

i.e., R(y; x) = 1 implies that Rx

Ry : It means that if an element y is prefer-

able to an element x to 1 degree, then its row vector corresponding to x must be a fuzzy subset of the row vector corresponding to y. The phenomenon may be interpreted as: An alternative a is fully preferable to another alternative b if and only if a0 s degrees of preference to all the other alternatives is greater than those of b on the remaining universe. Due to completeness, one of the rows has to be a subset of the other in the sense of De nition 1.2. Remark 2.7. If the given relation is not complete in the sense that max(R(x; y); R(y; x)) = 1; for all x; y 2 X; then we can convert it into a complete relation R0 by de ning R0 (x; y) =

R(x; y) : max(R(x; y); R(y; x))

It is easy to verify that if R is re exive and product transitive, then R0 is a re exive and product transitive fuzzy relation. Remember that this way of making a relation complete is just a scaling procedure and does not correspond to the internal nature of the relation. Next we turn towards de ning a crisp ordering on X with the help of R, a product transitive fuzzy preordering (well known in the literature). Proof is included for completeness sake. Lemma 2.8. Let R be a product transitive complete fuzzy preordering on a universe X. If we de ne a relation on X as: for all x; y 2 X; x 8

y if and only

if R(x; y)

R(y; x); then

is a complete crisp ordering on X (in the sense of

De nition 1.5.). Moreover, the relation

on X de ned as: x

y if and only

if R(x; y) = R(y; x) is a crisp equivalence relation corresponding to Proof. First we prove that

.

is a complete crisp ordering (in the sense of

De nition 1.5.). 1) For all x; y 2 X; either R(x; y) given two elements x and y either x

R(y; x) or R(y; x)

y or y

x:

2) Given three elements x; y; z 2 X such that x R(x; y)

R(y; x) and R(y; z)

R(x; y); thus

y and y

z i.e., that

R(z; y); then due to product transitivity we

have: R(z; y):R(y; x)

R(z; x):

(20)

Due to completeness condition we have: R(z; y) = R(y; x) = 1; this along with R(z; x) ) R(z; x) = 1:

inequality (20) imply that 1 Hence R(x; z)

R(z; x) i.e., x

Next we show that

z:

is the indi erence relation;

Re exivity: R(x; x) = R(x; x) = 1 thus x Symmetry: For any x; y 2 X; if x

x; for all x 2 X:

y; then R(x; y) = R(y; x) = 1:

It further implies that R(y; x) = R(x; y) = 1 i.e., y

x:

Transitivity: For any x; y; z 2 X; if R(x; y) = R(y; x) = 1 and R(y; z) = R(z; y) = 1, then due to product transitivity, we have: R(x; y):R(y; z) It implies that 1 Hence x Looking at

R(x; z) and R(y; x):R(z; y)

R(x; z) and 1

R(z; x):

R(z; x) i.e., R(x; z) = R(z; x) = 1:

z: ; as a complete fuzzy preordering on X, and

; as its correspond-

ing indi erence relation, following is an immediate consequence of Lemma 1.6. due to Debreu [5].

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Theorem 2.9. If X=~ is countable, then there exists a real valued function f on X, which is continuous in any topology in which the sets fx 2 X : R(x; y) R(y; x)g; for all y 2 X and fx 2 X : R(y; x)

R(x; y)g; for all y 2 X; are

closed. For the sake of reference, we shall call f , the Debreu's utility function in remaining part of this paper. Remark 2.10. Let F = fRx : x 2 Xg be the class of all row vectors of a product transitive complete fuzzy preordering R. Then it is easy to observe from the on going discussion that x where,

y , Rx

Ry ;

is inclusion of fuzzy sets (De nition 1.2) and x

y , Rx = Ry ;

where = is in the sense of equality of fuzzy sets (De nition 1.2). Moreover, we observe that

is a complete crisp ordering on F and = is its corresponding

indi erence relation. In next theorems following two notations will be used : Range(f ) = ff (x); for all x 2 Xg and Range(R) = fR(x; y); for all x; y 2 Xg: Theorem 2.11. If the qoutient set X=~ is countable, then there exists a mapping G : F ! F 0 ; where F 0 is the set of row vectors of a fuzzy preordering on Range(f ) such that: for all Rx ; Ry 2 F; Rx

Ry implies that G(Rx )

G(Ry ). Moreover, G is continuous in a topology de ned by the sup norm on mappings de ned as: d1 (Rx ; Ry ) = sup jRx (t) t2X

Ry (t)j ; for all Rx ; Ry 2 F:

Proof. If Range(R) is countable, then there exists a mapping g : Range(R) ! [0; 1]; de ned as: 1 g(R(x; y)) = R(x; y); for all x; y 2 X: 2 10

Mapping g is continuous in the usual topology on R. It is easy to observe that g so de ned, preserves the usual less than or equal to ordering

and is

continuous in usual topology. De ne G with the help of g and Debreu's utility function f as follows: For any Rx 2 F; G(Rx ) = Lf (x) ; where, f is the Debreu's utility function on the completely preordered space (X; ) and Lf (x) is a fuzzy set de ned on Range(f ) as follows: Lf (x) (f (y)) = g(R(x; y)); for all f (y) 2 Range(f ): Let F 0 = fG(Rx ); for all Rx 2 F g: Mapping G preserves ordering on F as follows: for all Rx ; Ry 2 F; Rx

Ry implies that G(Rx )

This is so because Rx Thus g(R(x; t))

G(Ry ):

Ry implies that R(x; t)

R(y; t); for all t 2 X:

g(R(y; t)): It further implies that Lf (x) (t)

t 2 X. Hence Lf (x)

Lf (y) i.e., G(Rx )

Lf (y) (t); for all

G(Ry ):

G is continuous in the topology de ned by sup norm: This is so because g is continuous in the usual topology which implies that: Given any

> 0; there exists an > 0 such that:

jg(R(x; t))

g(R(y; t))j

whenever jR(x; t)

R(y; t)j

G(Ry )(t)j

whenever jRx (t)

Ry (t)j

:

It implies that, jG(Rx )(t)

:

It further implies that, sup jG(Rx )(t)

G(Ry )(t)j

t2X

whenever sup jRx (t)

Ry (t)j

whenever d1 (Rx ; Ry )

:

t2X

So, d1 (G(Rx ); G(Ry ))

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:

Therefore G is continuous and order preserving. What remains to prove is that F 0 is the set of row vectors of a product transitive complete fuzzy relation. L is product transitive: For all f (x); f (y); f (z) 2 Range(f ); L(f (x); f (y)):L(f (y); f (z)) = g(R(x; y)):g(R(y; z)) 1 1 = R(x; y): R(y; z) 2 2 1 = R(x; y):R(y; z) 4 1 R(x; y):R(y; z) 4 1 R(x; z) = g(R(x; z)) 2 = L(f (x); f (z)): Thus, L so de ned is product transitive. Finally, we work with the preservation of the overall structure of fuzzy preordering as a multivalued mapping on X by a mapping which converts it to a multivalued mapping on the set of real numbers. De ne an ordering X

on

F as: (x; Rx )

Then

(y; Ry ) , x

y and Rx

Ry :

is a complete ordering in the sense of De nition 1.5.

Theorem 2.12. If the qoutient set X=~ is a countable, then there exists a continuous (in the product topology such that (x; Rx )

) mapping H : X

(y; Ry ) implies that H(x; Rx )

F !R

F0 ;

H(y; Ry ):

Proof. Since X=~ is countable, consequently = divides F into countable number of fuzzy sets which are completely ordered by the inclusion ordering. Hence there exist two functions f (Theorem 2.9) and G (2.11). Now de ne a mapping: H : X

F !R

F0 ;

H(x; Rx ) = (f (x); Lf (x) ) 12

where, f is the utility function de ned in Theorem 2.9 and Lf (x) is de ned as in Theorem 2.11. H is continuous in ( on X and

) where,

is the natural topology

is the topology on F de ned by sup norm on mappings.

Next we propose an other formula to de ne a preference relation in the form of product transitive complete fuzzy preordering just with the help of Debreu's utility function. Since we have already assumed R(x; y) 6= 0; for all x; y 2 X; so, there seems no harm in assuming that the Debreu's utility function takes only non zero values. Theorem 2.13. Let (X; R) be a complete fuzzy preordered space and let f be the utility function obtained via Theorem 2.9. De ne a fuzzy relation L on Range(f ) as follows: 8 < 1 if f (y) f (x); L(f (x); f (y)) = : f (x) otherwise. f (y)

(21)

Then L is a product transitive complete fuzzy preordering on Range(f ) such that: If R(x; y)

R(y; x); then L(f (x); f (y))

L(f (y); f (x)) and R(x; y) = 1

imply that L(f (x); f (y)) = 1: Proof. L is product transitive: For all f (x); f (y); f (z) 2 Range(f ); a). Let f (x) < f (y) < f (z); so, L(f (x); f (y)):L(f (y); f (z)) = b). f (x) < f (y) and f (y)

f (x) f (y) : f (y) f (z)

=

f (x) f (z)

= L(f (x); f (z)):

f (z):

Hence L(f (x); f (y)):L(f (y); f (z)) =

f (x) :1 f (y)

=

f (x) f (y)

f (x) ; f (z)

by assumption.

Similarly, in all other cases product transitivity holds. Next if R(x; y) = 1, then R(y; x)

R(x; y) which implies that y

is a utility function on (X; ); it implies that f (y)

x. Since f

f (x), then using (21)

we get, L(f (x); f (y)) = 1 (it also proves the completeness of the preordering L) and also L(f (y); f (x))

L(f (x); f (y)):

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3

Conclusion

In this paper, we have presented two approaches to use Debreu's utility function in order to obtain continuous order preservations of product transitive complete fuzzy preordering by mappings that also yield product transitive complete fuzzy orderings on the set of real numbers. Acknowledgement: The authors are thankful to referee for precise remarks to improve the presentation of the paper.

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[7] L. A. Fono and N. G. Andjiga, Utility function of fuzzy preferences on a countable set under max-*-transitivity, J. Social Choice and Welfare 28(4)(2007), 667-683 [8] K. Menger, Probabilistic theories of relations, Proc. of the Nat. Acad. of Sci., 37 (1951) 178-180. [9] H. T. Nguyen and E. A. Walker, Fuzzy Logic, (Third Edition) CRC Press 2006. [10] S. V. Ovchinnikov, Numerical representations of transitive fuzzy relations, Fuzzy Sets and Systems, 126 (2) (2002) 225-232. [11] C. Ponsard, Fuzzy mathematical models in economics, Fuzzy Sets and Systems, 28 (1987) 273-283. [12] K. Sengupta, Fuzzy preference and Orlovsky choice procedure, Fuzzy Sets and Systems, 93 (1998) 231{234. [13] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338-353. [14] L. A. Zadeh, Similarity relations and fuzzy orderings, Inf. Sci., 3 (2)(1971) 177-200.

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