E L S E V I E R Fuzzy Sets and Systems 77 (1996) 291-297

sets and systems

Fuzziness and randomness in an optimization framework

M . K . L u h a n d j u l a ~

International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy

Received March 1993; revised March 1994

Abstract

A semi-infinite approach for linear programming in the presence of fuzzy random variable coefficients is described. As a byproduct a way for dealing with optimization problems including both fuzzy and random data is obtained. Numerical examples are provided for the sake of illustration.

Keywords: Fuzzy; Random; Optimization

1. Introduction

Several problems which can be cast into a linear programming framework involve both statistical uncertainty and informational or intrinsic impreci- sion. The simplistic way, consisting in replacing arbitrarily uncertain and imprecise data by fixed ones, leads to bad caricatures of the reality. Let me quote Feynman [3] in this connection: "When dealing with a mathematical model careful atten- tion must be paid to imprecision in data". Prob- ability and fuzzy set theories offer ways for constructing metaphors that represent aspects of uncertainty and imprecision. These theories have been extensively applied to solve decision problems in turbulent environments.

Nevertheless, the problem of combining both uncertainty and imprecision in an optimization set-

i On leave of absence from: Department of Mathematics, University of Kinshasa, B.P. 190, Kinshasa XI, Zaire.

ting has not received the attention it deserves. See for instance Czogala [1,2] where the concept of probabilistic set is exploited, Yazenin [15], Roubens and Teghem [10] where comparison be- tween fuzzy and stochastic approaches is discussed without any attempts for integration and Luhan- djula [6] where a laconic discussion on flexible programming with random data is presented.

The purpose of this paper is to describe an ap- proach for solving a linear program with fuzzy random variable coefficients. The main idea behind this approach is to convert the original problem into a stochastic program so that techniques of stochastic optimization can apply. The conversion is made via semi-infinite optimization as described in Luhandjula et al. [7]. An interested reader may also consult [9, 14] where other approaches for incorporating fuzzy random variable coefficients in a linear programming, within the "here and now" and the "wait and see" philosophies, are discussed.

As a byproduct we obtain a means for solving a linear program in the presence of both fuzzy and

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292 M.K. Luhandjula / Fuzzy Sets and Systems 77 (1996) 291 297

random variables. Such a situation arises, for in- stance, when components of the second member of a linear program represent demands which are ran- dom while coefficients of the technological matrix are given by experts who present them as fuzzy numbers in a way to couple their perceptions with hard statistical data.

As a matter of fact by fuzzifying and randomizing the involved random and fuzzy data respectively, the problem can be put in terms of a linear program with fuzzy random variables and the method de- scribed in this paper can be used.

In Section 2 we briefly define a fuzzy random variable. Section 3 is devoted to an approach for solving a linear program with fuzzy random vari- able coefficients. Section 4 is concerned with the problem of combining fuzzy and random variables in a linear programming framework. Numerical examples are included for the sake of illustration.

The paper ends with some concluding remarks together with prospects for the future of fuzzy stochastic optimization.

2. Fuzzy random variable

A fuzzy random variable is a mathematical de- scription of a fuzzy stochastic phenomenon. This concept was first introduced by Kwakernaak [5]. An interested reader may consult [8-11] and [13] where important properties of fuzzy random vari- ables are recorded.

In the following, we restrict to notations used in [13].

Definition. Consider a probability space (0, ~ , P). A fuzzy random variable on this space is a fuzzy- set-valued mapping

~ : f2 ~ Fo(R),

such that for each Borel set B and for every a • (0, 1)

where Fo(R) and 2~, stand for the set of fuzzy numbers with compact supports and the a-level set of the fuzzy set 2,0 respectively.

The following result is crucial for the remaining of this paper.

Theorem 1. ~7 is a fuzzy random variable if and only if given 09 ~t2, ~" X,o is a random interval V~ ~(0, 1].

Proof. Let

X~- (co) = inf{x e R; )~(x) I> a}

and

X~+ (o)) = sup{x e R; 2~(x) ~> a}.

It is clear that

2, = [ x ; (~) , x + (~)].

Let us now show that X~-(to) and X + (m) are ran- dom variables.

For any x e R and any ~ ~ (0, 1) we have

{ ~ , e a l X ; (~o) <. x}

= {co e O I X ; ( ~ ) > x}"

= { ~ e a l X ; ( ~ o ) = ( x + ~ ) } °

= {~o c o l [x;(~o),x,+(o~)] = (x, + oo)} °

--{~o ~o12L = (x, + o0}°~Y.

We also have

{~ e a I X , + (co) ~< x}

= {co c o l x~+ (~) = ( - oo, x ] }

= {~o ~ o l [x;(o~),x,+(o~)] = ( - ~ , x ] }

= {~o ~ O I x , ( c o ) = ( - ~ , x ] } ~ .

3. Linear programming with fuzzy random variable coefficients

Consider the following mathematical program:

(P) maxcx

E(~lij)eJXj ~ (~i)w,, i = 1, m,

x e R " , o9~f2,

where (fiij)o.~bi)~ are fuzzy random variables de- fined on a probability space (C2, F, P), the possibility

M.K. Luhandjula / Fuzzy Sets and Systems 77 (1996) 291-297 293

distributions of which are

H(a0, o, H(T,~)o,

respectively. (P) can merely be written as

max cx

,a,,,x ~< ~,~,,

x~>0, e) • I2,

where/1o, and b,o are m x n and m x 1 matrices, the joint possibility distributions of which are II~,,, and I1~,,, respectively. (P) is an ill-stated problem. A pos- sible interpretat ion can be obtained as follows.

Let T~,,,, T2~ denote suppor t of A,o, support of bo, respectively. Put

T~,, = T1,,, x T2o)

and define

u: T,,, - , [0 1]

by

u(t(~)) = U(tl(Og), tz(eJ))

= min(IIx,,,(tl(~O)), H$,,,(t2(og))).

It is clear that u(t(og)) is the degree of compatibi l i ty of t~(og) • T~o, and t2(o9) • Tz,o with restrictions de- fined by Hx,,, and I-1~,,, respectively.

Consider now a subdivision T~, i = 1 . . . . . p + 1, of T,,, defined as follows:

T,,~, ' = {t(¢o) • To, lu i - , ~ u(t) < ul},

where u~ (i = 0 . . . . . p + 1) are real numbers such that

O = U o < U l < ... < U p < U p + l : 1.

Observe that we have

T,~,k=S'k 'IS'k,

where

SCti ~ i = A,,, xT~, I, x denote ensemblist difference and Cartesian product respectively and ~h ~ is the u-level cut of the fuzzy set ffL

Fur thermore , by Theorem 1, (A,oh and (bo,)i are families of Cartesian product of r andom intervals

and random variables respectively. Consider real numbers ( • i ) i = 1 . . . . . p such that

O = 6 p + 1 < O p < "'" < (~1

chosen to allow some leeways on constraints con- taining t(~o)• To, which are less compatible with restrictions defined by 11~,,, and H~,,,.

Consider now the following mathematical program:

(P') max cx

t , (co)x - t2(a)) <<. a";+ 1;

~p+ 1 (tl(a)),t2(~o)) • T~ ,

tl(a))x - tz(e)) <<. 6~; (t,(oo), tz(O~)) • T~;,

tl(t_O)X -- t2(~ ) ~ 6~; (tl(tO), t2(09)) • T~,',

x~>0, (o • f2,

where 6~ are m vectors, the components of which are 6j. A feasible action for (P') meets feasibility requirements of the original p rogram (P) for most favourable circumstances (t(o)) • ""+' To, ). In addition, strong violations of feasibility requirements are tolerated for less favourable circumstances (t(eo) • To,). In termediary situations are also taken into account by accepting more or less violations of the feasibility according to the value of u(t(e))).

So (P') can be regarded as a stochastic counter- part of the original fuzzy stochastic program.

Theorem 2. The mathematical program (P') is equiv- alent to the following program:

(P") m a x c x

E -Otp+ . - - t l i j ((D)XJ ~p+l ~_t~+'(o)), i 1,m,

~p ~t~ij(eJ)xj - 6r ~< t2i(~o), i = 1,m,

~ t~ i j (~ ) x j 61 ~<t2~(oJ), i = 1,m,

xj~>O, eo e £2,

where t~*o(~o),t~/(~o) are right and left endpoints of random intervals:

(a, jL = {sl H~o, , ( s ) >1 uk}

294 M.K. Luhandjula / Fuzzy Sets and Systems 77 (1996) 291-297

and

(bi)o, = {slllt~,),,,(s) >>- Otk}

respectively.

The proof of this theorem is similar to that of Proposition 6 in [7].

Observe that (P") is a stochastic linear program and according as a provisional or decisional option is chosen, the wait and see or the here and now attitude may be adopted.

Assume now that we have a decisional option in sight and that ~}j(og), k = 1 , . . . ,%÷~, i = 1,m, j = 1, n, are deterministic, then we can apply the chance constrained programming approach. In this case (P") can be written:

(P'") max cx

- o ¢ k ot k P r o b [ ~ t lijxj - dk ~ _t 2i(0))] ) r ,

i = 1,m, k = 1 , . . . , ap+l ,

x~>0, ~ t 2 .

(P'") is equivalent to the following program [12]:

max cx

--Ctk - - ~k 2 t tijXj (~k >1 Bio -~) ,

i = l , m , k = l , %+1, xj~>O,

where

N?, _ = ( F ? ) - - B ) ,

F~ '~ denoting the repartition function of the random Nk

variable _tz/(Og) and fl some prescribed level. The resulting problem is then a linear program and existing packages may be used to find a solution of the problem.

For the general case, the two-stage programming approach seems more appropriate. The idea behind this approach [12] is to pay a penalty for any violation of a constraint.

Let g~' be the penalty for a unit of discrepancy between

t2i((D ) and ~ t lij(O,))Xj -- I~ k.

Introduce now the following variables:

y~'(~o) =

{0 2-5` t lq(~o)xj - 6k -- tei(Co) if this quantity is positive,

otherwise.

The resulting problem is then

(PIV) m a x c x - E(~,9~'y~'(og))

- -o tk ~ k - - t2i( o)], y~'(03) : 2 I t lij((.O)Xj (~k

x >>. O, yT'(~o) >>- O, i = 1, m,

k = l , p + l ,

where E denotes the mathematical expectation.

Numerical example 1 Consider the mathematical program

max3xl + 2Xz

x~>0,

where ~ and ~ are 2 x 2 and 2 x l matrices respectively.

Components of these matrices are fuzzy random variables defined on a discrete sample O = (~ol, o)2) as follows:

d,7,) =

where p( to l )= 0.25, p(to2)= 0.75 and ~ denotes a triangular fuzzy number with the following mem- bership function (see Fig. 1):

/ '° x - (m -- 1)

uMx) = l o X + (m + l)

Let 0 = ~ o < 0 . 4 = ~ t < 0 . 6 = ~ 2 < 0 . 8 = ~ 3 < 1 =~4 and 0 = f i 4 < 0 . 0 1 = 6 3 < 0 . 0 2 = 6 2 < 0 . 0 3 = 61. And suppose g~,3 = 1, g~2 = 2, g~' = 3. The

if x ~ < m - - 1,

if x e ]m -- 1,m],

if x e ] m , m + 1],

i f x > m + l .

M.K. Luhandjula / Fuzzy Sets and Systems 77 (1996) 291-297 295

u~ix ) '

1 -

I

m-1 m m÷l

Fig 1.

program (PIV) corresponding to this example is then

min 3x~ + 2 x 2 + Y l 1 3 + Y213+0.75y112

+0.75y212 + 0.5y111 +0.5y211 + 3y~23

+3y223 + 2.25y122+2.25y222+l .5y121

+ 1.5y221

subject to

Y113 = 1.2xl + 1 . 2 x 2 - 2.81,

Y213 = 2.2x~ + 1.2x2 - 3.81,

Y123 = 1.2xl + 3.2x2 - 4.81,

Y223 : 1.2x~ + 2.2x2 - 3.81,

Y112 = 1.4xl + 1.4X 2 -- 2.62,

Y212 = 2.4xl + 1.4x2 - 3.62,

Y~22 = 1.4Xl + 3.4x2 - 4.62,

Y222 = 1.4xl + 2.4x2 - 3.62,

Yl ~ 1 = 1.6xl + 1.6x2 - 2.43,

Y211 = 2.6xl + 1.6x2 - 3.43,

Y121 = 1.6xl + 3.6x2 - 4.43,

Y221 = 1.6xl + 2.6x2 - 3.43,

Xj>/0 , Yitk = yi(wl)~*>>-O;l=l,2, i = 1,2, k = 1,3. The solution of this linear program is xt =1.34,

x 2 = 1, Yl13=0, Y213=0.34, Y112=0.658, Y212 = 1, y111=1.31, Y211 = 1.65, Y 1 2 3 = 0 , Y223=0 , Y122 = 0.658, Y222 = 0.658, Yl21 = 1.316, Y221 =

1.316. Then (1.34,0) may be considered as a solu- tion of the original problem according to desider- ata of the Decider expressed by his choice of 5i and g~k.

4. Linear programming with fuzzy and random data

Consider now a linear program where both fuzzy and random data co occur:

(PFS) min cx

8ijxj <~ T)~, i = 1, m,

xj >/0,

where 5 o are random coefficients and bi fuzzy ones. Random coefficients can be fuzzified as follows:

a , j = ,i j(to).

The membership function of this (degenerate) fuzzy random variable is

ua,j(~,)(t)={~ iftif t =# a°(t°)'a~j(to).

We can also randomize fuzzy quantities b~ by con- sidefing them as a (degenerate) fuzzy random vari- able b~ defined through the function

lt"(~') = ( ~ ifif co = t o o , ~ :~ tOo,

where tOo ~ f2 and p(too) = 1. Replacing ~iij and bi by their corresponding de-

generate fuzzy random variables, (PSF) is a linear program with fuzzy random variable coefficients and the approach described in Section 3 applies.

Numerical example 2

min2xl + x 2

a l l X l + a12x2 ~ h i ,

a21x1 + a22x2 ~ b2,

X 1 ~ 0 ; X 2 ~ 0 ,

296 M.K. Luhandjula / Fuzzy Sets and Systems 77 (1996) 291-297

where

A = and b =

A(o2) (11 2 )

where p ( t o l ) = 0.25, p(co2)= 0.75 and 3,~ fuzzy numbers are defined as in example 1.

After randomizing fuzzy quantities and fuzzify- ing r a n d o m ones, and assuming independence of r andom variables, the resulting p rogram is

m i n 2 x l + X 2 + ZII 3 "3 L Z213 -[- 3Z123 + 3Z223

q- 0.752112 q- 0.75Z212 q- 2.25z122 + 2.25z222

"Jr" 0 . 5 Z l l 1 q -0 .52211 -[- 1.5Z121 "q- 1.5Z221

subject to:

z113 - Xl - x2 = - 4.81,

z213 - 2xl - x2 = - 3.81,

z123 - xl - 3x2 = - 4.81,

z22a - xl - 2x2 = - 3.81,

z112 - x l - x2 = - 4.62,

z122 - x l - 3x2 = - 4.62,

z212 - 2xl - x2 = - 3.62,

z222 - Xl - 2x2 = - 3.62,

z111 - xl - x2 = - 4.43,

z211 - 2xl - x2 = - 3.43,

z121 - Xl - 3x2 = - 4.43,

z221 - xl - 2x2 = - 3.43,

Xj > O, Zil k > O,

where Zitk = Y. aO(Ogl)~Xj - 6k - - b]'~(to0). The solution of this p rog ram is xl = 4.81, x2 = O,

Z l l 3 • O, 7,213 ~--- 5.8, Z123 = O, Z22 a • 1, z112 =

0 .19 ,2212 ----- 6, Z122 = 0-19,2222 = 1.19, z111 = 0.38, z211 = 6.19, z121 = 0.38, z221 = 1.38. Hence (4.81,0)

is the solution of the original p rogram according to desiderata of the Decider expressing by choices of 6 i and ~ g i .

5. Concluding remarks

In this paper we have discussed ways for incor- porat ing fuzzy r andom variables into a linear p rog ramming setting. It has been shown that this problem can be reduced to a stochastic p rogram via semi-infinite optimization. A Bienayme- Chebycheff-like inequality is desirable for fuzzy r andom variables. Under such an inequality the original p rogram can be put into a fuzzy p rogram if supports of variances of involved fuzzy r andom variables are small.

The method described in this paper offers an oppor tun i ty for combining both fuzzy and r andom data in an opt imizat ion context by considering them as degenerate fuzzy r a n d o m variables.

The choice of ~k can be thought of as a way to discretize the interval [0 1]; the more p is large the more we have a fine discretization and the more we have a faithful description of the reality.

As regards 6 j and g~, their choice reflects Deci- der's points of view regarding fuzziness and ran- domness which are in the state of affairs. In this way the approach offers room for working in close rela- t ion with the Decider, a requirement which is un- avoidable in decision making under a turbulent environment.

Nevertheless it is desirable to use techniques of mult iparametr ic p rog ramming in a way to help the Decider by providing him bounds for 6 i . Observe that the size of the resulting problem is bigger than that of the original problem. This is, in our opinion, the price to pay for refusing to caricature badly the original problem by replacing arbitrarily imprecise data by precise ones.

The approach described in this paper focuses on discrete fuzzy r a n d o m variables and on combining fuzzy number with discrete r a n d o m variables. Topics for further inquiries include extension to cont inuous fuzzy r a n d o m variables and extensions to situations where both fuzzy numbers and con- t inuous r a n d o m variables are to be mixed in a lin- ear p rogramming framework.

M.K. Luhandjula / Fuzzy Sets and Systems 77 (1996) 291-297 297

Acknowledgements

The authors would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the Inter- national Centre for Theoretical Physics, Trieste. He would also like to thank the Swedish Agency for Research Cooperation with Developing Countries, SAREC, for financial support during his visit to ICTP under the Associateship scheme. Part of the results described in this paper were obtained while the author was visiting the University of California, Berkeley, as a Fulbright fellow.

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