Optimization in Fuzzy Environment

Copyright \cl IFAC Fuzzy Information Mar.;eille. France. 1983

OPTIMIZATION IN FUZZY ENVIRONMENT D. Ralescu D epartment of Math ematics, Unive rsity of Cindnnatl~ Cincinnatl~ Ohio 45221 US.A.

Abstrac t . In this paper we study the poss i bility of representing an optimization problem with inexact constraints, as a fuzzy integral . This integral is to be taken with respect to a capacity (or to an outer measure) rather than to a fuzzy measure . We prove a mean- value theorem for the fuzzy integral, which has as consequence the possibility of reducinl! optimizat i on with inexact constraints t o classical optimization . We also give some sufficient conditions such that this reduction holds. 9.

Ke~vords.

1.

Fuzzy constraint; fuzzy measure, fuzzy integral.

IN TRODUC TI ON

The concepts of decision- making in a fuzzy environment were defined by Bellman and Zadeh [ l J . To recall briefl y this approach, let X be a set, P : X ~ [ O,l J a cost function, and u : X ~ [ O,l J a fuzzy constraint . Note that P is a fuz zy set ( fuzzy goal , in [ 1 J ) whi ch can be deri ved fro,.l a pos i ti ve , bOlmded flmction on X, by a simple normali zation . The optimizat ion problem sup P is defined by

= o.

( FMl)

w(0)

(FM2)

AC B = > weAl < weB). ,

( FM3) A C A C ... = > ]J (

1

2

Lf

n=l

An) = li;n W(A ). n+oo n

( FM4) Al':> A2 J ... , W( Al )<=> W(

n An )=limw( An) '

n=l

n+oo

Let f:X ~ [ 0,00) be a measurable function (i . e . {f > a} E A for any a > 0) . The fuzzy integral of f with respect to ]J is

u

s up P u

where fI

=

sup [ p( x ) A u( x )J xE-X

sup ( a fI w{f > o:}) a>O

min .

where {f > a } = {x E Xl f ( x ) > a}. This definition was given in [ ll J-i n a more restrictive context . Pronerties of this i ntegral, esrecially convergence theorems, are studied in [9J.

The dynamic programmi ng approach was used in [ l J t o solve optimizati on prob l ems with inexac t constraints . mainly where X is a finite set.

It i s now clear the similarity between ( 1.2) and ( 1.3) . If, for a cost function P : X ~ [ O,l J , we define the set func t ion ]Jp:P( X) ~ [ O,l J by

In [ 12 J , Tanaka , Okuda and Asai are led to a reformulation of t he problem ( 1.1), under the name fuzzy mathematical programming. Namely, it can be shovm ( see the short proof in [ 5 , p . 160J ) that sup P U

=

sup [0: ;\ sup p( x) J 0:>0 u( x »0:

(1. 3)

wp ( A)

sup p(x)

(1.4)

XEA

(1. 2)

where sup P = 0 , then ( 1 . 2) can be wri t ten

o

Written under this form, it was observed that the right - hand side in ( 1 . 2) is r elated to the fuzzy integral [ ll J .

unde r the form: sup P

To explain this point let A be a a - algeb r a of subsets of X. A fuzzy measure is a positive , extended real- valued set function w: A ~ [ O,ooJ, wi t h the properties:

301

(1. 5 )

D. Rales cu

302

sup p(x) < P(a) , a contradiction. xEA nO

It was pointed out in [ 4, p . 1629 J, [ 6 , p . 665 J , [ 11, p . 12 J that the set function ~P is a fuzzy measure. As we shall see in Section 2 ,

~P

always has

prope r ties (FMl) - (FM)), for any set X and any cost function P. In which proper ty (FM4) is concerned, we show that it only holds for Wp in trivial cases (see also [ 8J ).

(c) It only remains to show that subaddi tive, i.e .

~P

is

00

for any sequence {A } of subsets of X. We study in Section 2 the set functi on ~p; we prove that ~P should be viewed as an outer measur e , rather than as a fuzzy measure . In a topological setting, we show that Wp is a capacity (as it was also mentioned in [ 7J ).

In Section 4 we give sufficient conditions that optimizat ion on fuzzy sets can be r educed to class ical optimization (this will be called the "reduction problem") . Tbese conditions are mo r e gene r al than t hose in [ 12 ]. Rela ted work on fuzzy mathematical programming was re cently reported in [ 3J . THE SET FUNCTION

(2 . ))

and (2 . 2) immediately follows.

In Sect ion J we give a general mean - value theorem f or the fuzzy integral, and see its relevance to optimization with inexact constraints . One of the main result in [ 12 J is a consequence of this mean- value theo rem.

2.

n

In the same way as in (b) , we can show that

Let us show now that property ( FM4) is not t r ue fo r ~P :

~eneral1y

Example 1 . Take X = JR, p( x) = 1, and the sequences of sets An = ( n, (0). We observe 00

that ~p (

n An) = ~p( n=l

~p(An)

lim n->oo

0) = 0 , while

= 1.

This simple example shows that nroperty (FM4) is much too st r ong to be satisfied by Wp ' In fact, it can be shown that, under fairly ~eneral hypothest s, if ~P satisfies (FM4) , then P is a "trivial" function .

~P

l.et us consider first X to be an arbitrary Sf't, and P : X -> [ 0 ,(0) a positive function . We can prove that the set function ~P as defined by (1 . 4) satis fies proper ties (FMl) (FM3) 0 ; a fu zzy measure : PROPOSITI ON 1.

To be more precise, let us consider X to be a metri c spa ce without isolated points . If we take the cost functi on p(x) = 0 , then ~P satisfies (FM4) . Under the assumption of continui ty , the converse is also true : PROPOSITION 2 . continuous and

~P

If the function P is satisfies (FV4), then

p(x) = 0, for any x E X.

(c)

Wp is an outer - measure .

Proof.

( a)

Proof. ~l/n )

is obvious .

Let us take Xo E X, and let be the ball with center Xo and

radius lino ( b)

Denote by A = U An; we have n=l Wp ( Al ) ~ ~p(A2) ~ ... ~ ~p(A) thus

Since X i s not an isolated

o

point, it follows that An is nonempty, for all n

2

B(xO,l/n) ' {xO} 1.

Also Al :;J A2 ~ ...

00

lim ~p ( An) = sup ~p(An) ex i sts , and n> 1

and

n->oo

nn=l An = 0.

it f ollows that

Since ~P satisfie s ( FM;'), ~p(An) ->

O.

(2 . 1)

Let {x.} . c: X be a sequence, such that J J

Let us suppose strict inequality in (2 . 1); then there is an a E A with sup ~p (A ) < Pta) . n>l n But a E A for some nO ~ 1 . - We get : nO

' X . t xo . Let n > 1 be fixed; O J then Xj E B( xO,l/n) " {xO} for all j -> J. n . Thus : p(x . ) < sup p(x) for j > j . It J xEA - n

x. J

->

X

n

follows that

303

Optimization in a Fuzzy Environment

o~

lim sup p(x . ) ~ j~

J

supp(x) = ~p(An) . x£A

Since

( 1.5) st ill holds, where

this is true for any n > 1 , we conclude that lim sup p(x . ) = 0, thus- lim p(x.) = O. j~

j~

J

~P

is viewed either

as an outer measur e, or as a capacity .

n

THE MEAN- VALUE THEOREM

3.

J

Finally : lim p(y) = O; since P is continuous , y->-xO it follows that P(x O) = O.

In this section we ~i ve a mean- value theorem fo r the fuzzy inte?ral. Its applicability will be in an immediate ~ roof of a basic re sult in fuzzy mathematical p r o~rammin?

We see now that, except for functions which are "almost" zero (p = 0 at any point of continuity), it is not to be expected that ~P is a fuzzy measure .

Let us consider X a set, A a a - algebra of subsets, and ~ :A ->- [0,00) a set function, with the only requirement of beinf nondecreasing:

By a sui table change of ( ~~) we can, however, prove that ~P is a capac i ~y in the sense of Choquet (se e [ 2J ). Suppose that X is a topological space. A capacity is a set function C:P(X) ->- [ O, ooJ with properties : ( cl)

A C B => C( A) < C( B ).

(C2 )

=> C( U A ) n=l n . . . ,Kn compact => lim

n

(2 . 4)

00

I

f

d~

X

a ~ 0,

= sup (a 1\ ]J {f > a}) a>O

( 3.2 )

such that

K n

(3 . 3)

JX f d]J

Proof . Consider the function r-:-[O,oo) ->- [0,00); clearly i t is impossible that F(a) > a for any a > O. From this , and the fact that F is continuous, i t follows that F has a fixed point:

( 3 ) ex

~ 0,

F( Ci.) = ex .

We show tha t

Ix f d~ = a = F(a) . Suppose a < ex; then F(a) > F(ex) = ex > a . Thus a 1\ F(a) = a < a . If a > ex, then F(a) < F(ex) = ex < a . Then a K F(a)=F(a) O. The refore

Suppose that strict

inequa lity holds in (2 .4 ). Then : inf l1 p ( K ) > c > ~p( K), for some c > O. Thus n>l n W~(K ) = sup P > c > ~p(K) , for any n ~ 1 .

I

f dw

X

But

ex

sup [ a 1\ F(a) ] < a a>O

ex J\ F("&) <

Ix

(3 . L)

f d]J, which ends the

proof . ( 3 ) xn

p( xn ) > c > ~p( K ).

£

Kn' such that

Since {xn}

there is a subsequence x

n.

->-

C

Xo

K , l £ K.

Applied to a cost function P : X ->- [ O, l J and to the set function Up, Theorem 1 yie~ds Fr om

J

p( x

(J . l)

If f : X ->- [0,00) is a measurable function, the fuzzy integral can st ill be defined by :

r

Pr oper t ie s (Cl) and (C2) were already proved in Pr oposition 1. It only remains to prove (C3) ; let Kl -::J K2 ~ ... , Kn compact . We always have

I t follows :

~(B)

THEOREHl. If the function F(a) = w{r>a} is continuous , then there exists an -

K ) n=l n

Proof. Since P is upper semi - continuous , then sup p(x) < 00 for any compact set K; x£K therefore the last assertion follows.

n

<

c( fl

for any compact subset K of X.

~l Kn .

~ ( A)

The next result i s a counterpart of the mean - value theorem in classical measure theory:

Proposition 3. If P is upper semi- cont inuous, then ~P is a capacity . Moreover ~p(K) < 00 ,

where K

A, A C B =>

£

lim C( A ). n n->-oo

C( K ).

n~

A,B

) > c > Wp( K) and the uppe r semi n. J

continuity of P, we conclude that P(x O) ~ c > ~p ( K), a contradiction. It is clear that the integral repre sentat ion

the f ollowing COROLLARY 1. F(a) - sUP

u( x-)~a

( [12J ) . If the function p(x) is continuous, then

there exists an

ex

£

[ O,l J, such that

D. Ralescu

304

sup P

s up p( x) . u( x » a

sup P u

u

The meaning of this r esult is the f ollowing : if F is continuous , then the f uzzy mat hemat i cal programmi ng problem sup P can be r educed u

to the ordi nar y opt i miL"ation pr oblem p( x ), f or some Lt E (O , l J ( if P -! 0) . sup u(x»a In other wo r ds , for a given cost function P and fuzzy constraint u , only par t of the info r mation contained in u is needed . We shall call the possibility e f wri t ing sup P = sup P(x) , the r eduction pr oblem . u u( x »a In the next section we discuss some sufficient conditions for this reduct i on problem.

s up p( x) . u( x »a

Pr oo f. We only have to che ck the ri /!ht continuity F(a) = suo p( x ). Take u(x»a a l .:. a 2 :. .. . , an ~ a O' a O -! 1 : then 00

{u~al} C {u:.a) c

...

and ~l {u:.an } = {u>a OL

By Pr opos i tion 1 , then F( a n )

~

suo p(x) . u( X'»Ci O p( x)= sup p( x )=F( Ci )'

If we show that

sup O U(X»Ci O U(X)~CiO Suppose that stri ct i nequali ty sup p( x) holds . Then ( -3 )

we are done . sup p( x) < u(x»a u( x )~.a0 O a E: {u ~ a } wi th sup p(x) O u( x »a

<

Pea) .

In fact

O

SUFFICIENT CONDIT I ONS FOR THE REDUCTION PROBLEM

~ .

The hypothesis of continuity for F is compl icated from the pr actical point of view . This was "Iso observed i n [ 12 J, whe r e a sufficient condition is p,iven in te r ms of fuzzy convexity . We shall given here a ne w proof o f this result , unde r more general assumpt.ions . Our conditions will also include , as a particular case , classical optimization (i . e . u = X , the characteristic f unction of A a compact set A) . Let us consider X a topological space . The ru ~zy constraint u : X ~ [ O,l J is supposed to be upper semi - continuous (u . s . c . ) , with a compa ct support supp u. The cost f unction P : X ~ [ O, l J is also assumed to be u . s . c . LEMMA 1 .

The function F(a)

suo

p( x)

i s left continuous . n n

:J {U':'Cl 2 } ~ ... and

Then

n

{u.:.aJ:l} = {u.:.aOL n =l Also {u>a 1 are compact , since u is U. S. c . -n and supp u is compact . Then , by Proposition J, we p,et F( a O ) = lim F( an ). n->OO {11:'0: }

1

Suppose now X a topolop: i cal vector space. A fuzzy set u : X ~ LO, l J is called st r ictly convex if u( h

+ (l - A)y) > u( x) ,\ u( y )

a O' ob vi ously .

Let us fix an x

E:

{u>ao

(4 . 1 )

O

By the cont i nuity of P we vet pea) < S,,-\, p(x) < Pea) , a contradiction - U(X»Ci O which ends the proof . Other sufficient conditions are given in the next theorem . Recall that a function u :X is called concave, if :

for any x,y

E

X, A

C

+

4

[ O,l J

(l - A)u\y)

[ 0 ,1 ).

~HEOREM J . If P i s continuous , u is u . s . c ., concave , and supp U is compact , then there ex i sts an Ci E (O , :!. J such that

sup P = SUD U

u(~ »a

p~x) .

The proof is carried out in much Proof . the same way as in the previous theo r em . Instead of (4 . 2) , we have : u( ( l - l/n)a +( l/r.)x ) > (l- l / n )u\ a H I/n )u( x »a (4 .4 )

for any X, y E X, x -! y , and A E (0,1) . an d the r est i s identical to the proo f above . THEOREM 2 ( [ 12 J ). If P i s continuous , u is u. S. c ., strictly convex , and supp u is compact , the r e exists an a E (O ,l J such that

};

cle arly x -! a and , by the stri ct convexity of u , we can wr i te : u( (l - l/n)a + (l/n)x ) > u(a) 11 u(x) = a O (4 .2) Thus (l - l/n)a + (l/n)x £ {u>a } and O (l - l/n)a+(l/n)x ~ a . Therefore : p( (l - l/n)a + (l/n)x < sup p(x) < Pea) . - U,X ( ) >Ci

Observe that u concave implies u quasi concave (i. e . u( h+(l - A)y) > u( x) . u( y), see [ lO J ) , but , generally, ~ concave does not imply u stric tly fU3zy convex .

Consider {a } C [ O, l J,

\(1 ::. \(2 < ... , Cl n ' ('(0 '

=

uOx + (l - A)y).:. b(x)

u( x»a

Proof .

u( a )

Both Theo r em 2 and Theor em J give suffic i ent condit i ons fo r the reduct i on pr oblem , when X

Optimization in a Fuzzy Environment

is assumed to have a linear structure . typical example in applications is X

The

= mn .

If X is not assumed to be a vector space , the following topological sufficient conditions can be useful: THEOREM 4 . If P is continuous, u is u . S. c ., supp u is compact, and if, for any Xo € X, u(x)O I 1, there exists a sequence {xn }n , xn ~ x ' u( x ) > u(x O) , then the reduction o n problem holds. Is obtained by examining the proof Pro of . of Theorem 2 . Informally , the last condition in Theorem 4 can be stated in the following way: any point Xo € X with membership degree I 1, can be approximated arbitrarily close by points x € X with stri ctly greater membership degrees. REFERENCES 1.

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2.

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3.

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4.

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5.

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6.

H. T. Nguyen, On fuzziness and li:1guis t ic probabilities, .J. Math . Anal. Appl. 61 ( 1977), 658- 671-:-

7.

H. T. Nguyen , Some mathematical tools for linguistic probab i lities , Fuzzy Sets and Syst . 2 (1979), 53- 65 .

8.

M. L. Puri and D. Ralescu, A poss i bility measure is no t a fuzzy measure , Fuzzy Sets and Systems 2 (1982) , 3ll - 3~

9.

D. Ralescu and G. Adams, The fuzzy integral , Journal of Math. Analysis and Applic. 75 (1980), 562 - 570.

10 . A. W. Robe r ts and D.E. Varberg, Convex Functions, Academic Pr ess , New York, 1973 . 11. M. Sugeno, Theory of fuzzy integrals and its applications, Ph . D. Dissertation , Tokyo Inst. of Technology , 1974 .

12 .

305

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