Two Approaches to Multiobjective Programming Problems with Fuzzy Parameters

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NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

TWO APPROACHES TO MULTIOaTECTlYE PROGFUUMING PROBLFXS WITH FUZZY

- P

S.k Orlovski

May 1984 WP-84-37

Working Papers a r e interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of t h e Institute or of its National Member Organizations.

INTERNATIONAL INS'l'I'IZTTE FOR APPLIED SYSTEMS ANALYSIS 2381 Laxenburg, Austria

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Two approaches to the analysis of multiobjective programming problems a r e presented based on a systematic extension of t h e traditional formulation of the problem to obtain a formulation applicable for processing information in the form of fuzzy sets. Solutions a r e based on trade-offs among achieving greater possible degree of nondominance and greater possible degree of feasibility.

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TWO APPROACHES

TO

WLTIOBTECTZYE PRDGRAMNING PROBLEMS

WITH FUZZY

-P

S.A. Orlovsk*

1. Introduction

Multiobjective (MO) programming problems with fuzzy information were extensively analyzed and many papers have been published display- ing a variety of formulations a n d approaches to t h e i r analysis (see for instance, Zimmerman, 1978; Yager, 1978; Takeda and Nishida, 1980; Han- nan, 1981; Lohandjula, 1982; Feng, 1983; Backley, 1983; Tong, 1982). Most of t h e approaches t o fuzzy MO problems a r e based on t h e staightforward use of t h e intersection of fuzzy s e t s representing goals a n d constraints and on t h e subsequent maximization of t h e r e s u l t a n t membership function.

Here we p r e s e n t two approaches based on a systematic extension of t h e traditional formulation of MO problems with fuzzy parameters to obtain a formulation applicable for processing information in the form of fuzzy sets. The paper is based on results described in Orlovsk. 1978, 1980, 1981, 1983,1984.

* On leave from the Computing Center of the IISSR Academy of Sciences

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Two aspects of a fuzzy M O problem are of major importance. The first is t h a t whereas in a traditional problem every objective function represents a linear ordering of alternatives, in a fuzzy MO problem we have only fuzzy preference relations between a l t e r n a t i v e s . ' ~ u e to this fact t h e concept of domination requires f u r t h e r definition and we can only speak about determining alternatives with various degrees of nondominance. The second aspect lies in t h a t in a fuzzy MO problem alternatives can be chosen only on t h e basis of trade-offs among two generally conflicting objectives: achieving g r e a t e r possible degree of nondominance and g r e a t e r possible degree of feasibility. Both aspects a r e con- sidered in this paper.

2. Problem formulation

We assume h e r e t h a t alternatives from a given s e t X a r e pairwisely compared with each other using n objective functions Ji(z,Q), i = l . ..., n in such a way t h a t g r e a t e r values of each of these functions a r e con- sidered t o be more preferable. Each of these functions contains a vector of parameters

6

t h e values of which are known fuzzily and described by means of t h e membership function p: Q + [ o , 11. Speaking informally, t h e problem lies in determining feasible alternatives giving g r e a t e r possible values of t h e objective functions.

In a traditional MO problem, when the values of parameters @ a r e specified unambiguously, t h e rational choices a r e Pareto-maximal alternatives which can b e d e t e r m i n e d using well-known computational tech- niques. Here, with fuzzily specified values of parameters

g,

we can have only fuzzy evaluations of t h e corresponding objective functions and, therefore, should additionally more precisely define the meaning of a rational choice.

First, in our reasoning we assume t h a t the s e t feasible alternatives is nonfuzzy and coincides with s e t X, a n d consider t h e fuzzy s e t of feasible alternatives l a t e r for both t h e approaches.

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3. First approach

3.1. Reformulation of the problem

This approach is based on the consideration of levels for t h e values of t h e objective functions which can be achieved t o a h i g h e r possible degree. More formally, we understand our problem i n t h e following way:

Maximization in ( l a ) i s understood, of course, in t h e P a r e t o s e n s e a n d t o indicate t h i s we u s e t h e symbol iiiZE Constraint (2a) r e f l e c t s t h e fact t h a t with fuzzy values of functions

Ji

we can only consider satisfying t h e inequalities Ji(z,Q)2JP t o a certain degree. An essential point in t h i s formulation is t h a t t h e multiobjective choice in t h i s c a s e s h o u l d be based n o t on t h e trade-offs a m o n g t h e values of t h e objective f u n c t i o n s , which a r e fuzzy d u e t o t h e fuzzy n a t u r e of p a r a m e t e r s if, b u t among t h e lower e s t i m a t e s of t h e s e values obtainable t o a certain d e g r e e a. ?"his formulz- tion also implies t h a t when deciding upon t h e trade-offs a m o n g t h e lower e s t i m a t e s of t h e objectives t h e decision-maker should consider t h e possi- bility degree a of t h e s e e s t i m a t e s .

In t h e following s u b s e c t i o n we demonstrate t h a t t h e above formulation can be r e d u c e d t o a traditional form of a MO problem.

3.2. Analysis of the problem

For conveniency we shall consider functions J , ( z , ~ ) , i = l ,

...,

n as c o m p o n e n t s of a v e c t o r function

d(z,@)

with values f r o m t h e r e a l vector space

Rn.

If we d e n o t e by t h e vector of levels

( 4 ,

^{. .}

: , G )

t h e n prob- l e m (la)-(2a) c a n be w r i t t e n in the form:

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To formulate constraints (2b) more explicitly, we can directly use the extension principle (Zadeh, 1973) t o obtain:

degreeIT(z,q)%?

=

- sup

~ ( q ) ,

(3) ij: J(Z , q ) l ?

which, in fact, r e p r e s e n t s t h e extention of t h e "greater or equal" relation fmm t h e vector space

Rn

onto the class of fuzzy vectors

-

values of the objective vector-function with fuzzy parameters @.

Finally, using (3) we can formulate problem (1b)-(2b) as follows:

If some tuple ( z s , p S , a S ) is a solution to this problem then the tuple

(y,

.

. .

, q , a s ) is P a r e t o optimal which means t h a t any other alternative z providing for better values of some of t h e components of (?.a) gives worse values of some of the other components of t h i s tuple.

Now i t is a simple excersize to verify t h a t for continuous in ij functions J i ( z , F ) (for any z u ) , i = l ,

...,

n and ~ ( i j ) problem (4) can equivalently be formulated as follows:

and finally, in t h e form:

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4. Second approach

4.1. Reformulation of the problem

This approach is based on t h e extention of t h e n a t u r a l order on t h e r e a l line of values of t h e objective functions onto t h e class of fuzzy sub- s e t s of t h i s line. In t h i s way we obtain preference relations which c a n be u s e d for comparing with e a c h o t h e r fuzzy values of the objective functions for various alternatives. Then, using these relations, we define a fuzzy s t r i c t preference relation on t h e s e t of alternatives and determine t h e corresponding fuzzy s u b s e t of nondorninated alternatives.

As before, we consider n objective functions Ji(z,Q), with Q being a fuzzily-valued v e c t o r of p a r a m e t e r s described by membership function p(q). Using t h e extension principle t h e corresponding fuzzy values of t h e s e functions c a n be obtained in t h e following form:

Now we can obtain t h e following fuzzy nonstrict preference relations i n d u c e d on t h e s e t of alternatives by p i :

The n e x t s t e p i s t o define a way of comparing alternatives with e a c h o t h e r using all t h e s e n preference relations. To do t h a t we define s t r i c t dominance relation on X in t h e following way. Let ~f (z ].z2) be t h e fuzzy s t r i c t preference relation corresponding to qi (z l,zz), and defined a s follows (see Orlovski, 1978):

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Then we say t h a t t h e degree qs (z1,z2) to which a l t e r n a t i v e z l is strictly prefered t o alternative z2 is as follows:

In a nonfuzzy formulation t h i s would m e a n t h a t z l is s t r i c t l y preferable t o z 2 iff it i s strictly b e t t e r t h a n z2 with r e s p e c t t o every objective function. The respective nondominated alternatives a r e c o m m o n l y reffered t o a s serniefficient o r weakly effective.

Having defined q s we c a n describe t h e corresponding fuzzy subset

tlNZ, of n o n d o m i n a t e d alternatives i n t h e form (Orlovski, 1978):

a n d using t h e above formulation of q:, we have:

The value q m ( z ) i s t h e nondominance degree of t h e r e s p e c t i v e a l t e r n a - tive. If 77m(z) 2 a t h e n a l t e r n a t i v e z m a y be s t r i c t l y d o m i n a t e d by s o m e o t h e r a l t e r n a t i v e t o a degree s m a l l e r t h a n 1 -a.

4.2. Dete~rminbq alternatives nondominated to a prespecified degree Now we consider t h e problem of determining a l t e r n a t i v e s satisfying:

where a i s t h e desired degree of nondominance.

Let us f o r m u l a t e t h e following nonfuzzy multiobjective problem:

z

EX,

5 E P .

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The following theorem states that under some conditions any solution z to problem (10) satisfies (9).

Theorem.

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f o r any o f t h e f u n c t i o n s p i ( . , . ) , i = l .

...,

n and any ZEX t h e r e e z i s t r i ~ ~ l , i = l .

...,

n s u c h t h a t p i ( z , r , ) l a , then f o r m y ~ b l u t i o n t o prob- lem (20) w e h , m e T m ( z ) l a .

Proof. Let (zO , F O ) be a solution to problem (10). Then, as follows from (8) prove the theorem i t suffices to show that

Assume the contrary, i.e. t h a t y ' u and E>O can be found such t h a t

Using ( 7 ) we can write (11) in the form:

Let us choose z I i , i = l .

...,

n such that ~ ~ ( ~ ' . z Ii)ra for all i = l .

...,

n (the existence of z ' , . i = l .

....

n follows from the assumptions about functions p,). Since ( z O . ? ) is a solution to problem ( l o ) , we have t h a t r t u ' , for a t least one i=io arnong i = l .

...,

n . Thus we have

Therefore, we have

Hence, t h e inequality with index i, in ( l l a ) does not hold, since its first additive t e r m does not exceed 1. This contradiction proves the Theorem.

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Using (6) we can now write problem (11) in the following form:

z

E X ,

or equivalently:

J ( z , i j ) ^-+

s Fs A@)

²a*

z

E X , Q

E

Q.

As can be seen this formulation is quite the s a m e as t h e corresponding

MO

formulation (5) for t h e first approach (see Sect. 3.2) in the case of a fixed a. Therefore, both t h e approaches a r e equivalent t o each o t h e r in the sense t h a t both may lead t o choices of t h e same alternatives.

5.

Puzzy

set of feasible alternatives

Let us n o w additionally assume t h a t the s e t of feasible alternatives is described by t h e following system of inequalities:

with being a given real vector-valued function, and ji being a vector of ^, parameters with t h e m e m b e r s h i p function v:P-r[O,l] describing fuzzily i t s possile values.

To use t h i s type of information we first determine an explicit description of t h e corresponding fuzzy subset of feasible alternatives in the form of a membership function w (z). If we introduce the notation

Then using t h e extension principle we can write this membership function in the form:

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w ( 2 )

=

sup

~ 5 ) .

F E P ( ~ )

The value w ( z ) of this Function is understood as the feasibility degree of the corresponding alternative, and these values should also be taken into account when making choices of alternatives.

Alternatives in this case should be evaluated by two generally conflicting factors: their degree of nondorninance ?*(=) (in the second approach) and their degree of feasibility w ( z ) . Let a be t h e desired degree of nondorninance and

p

be t h e desired level of feasibility. Then an alternative having a degree of nondorninance not smaller than a and feasible to a degree not smaller than

p

should satisfy t h e following inequalities:

Therefore, with the fuzzy set of feasible alternatives formulation (12) will have the following additional constraints:

And i t can easily be verified t h a t with this type of constraints problem (12) can be written a s follows:

By varying the values of a and @ we can determine alternatives with various trade-offs m o n g the degrees of nondominance and feasibility.

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6. Concluding remarks

Two approaches t o

M O

problems with fuzzy p a r a m e t e r s a r e sug- gested in t h i s paper. Both a r e based on t h e systematic use of t h e extension principle as t h e means of processing fuzzy information about parameters. The rationality of choice is based on trade-offs among degrees of feasibility and nondominance. I t is shown t h a t rational alternatives in both t h e approaches can be determined by solving similar MO problems in a traditional form.

The use of fuzzy sets for describing information about real systems i s a relatively new a r e a and much f u r t h e r work is needed in order to find practically effective methods allowing to combine t h e fuzziness of h u m a n judgements with t h e pon~erful logics and tools of mathematical analysis. Successful development in t h i s direction may help overcome one of t h e essential obstacles t o t h e application of t h e m a t h e m a t i c a l modeling to t h e analyses of real systems, namely, t h e existing g a p between t h e language used for m a t h e m a t i c a l models and t h e language u s e d by potential users of those models.

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