Mathematical Programming Problems with Fuzzy Parameters

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

MATHEXATICAL

PROGRAMMING PROBIXMS

WrrH

FUZZY PARAMEnmS

S.A Orlovsb

May 1984 WP-84-38

Working Papers are interim reports on work of t h e 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 i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

An approach t o t h e analysis of mathematical programming problems is presented that is based on a systematic extension of the traditional formulation of the problem t o obtain a formulation applicable for pro- cessing information in t h e form of fuzzy sets. Solutions a r e based on trade-offs among achieving greater possible degree of nondominance and g r e a t e r possible degree of feasibility.

MATHEMATICAL

PROGRAMMING PROBLEMS WITH FUZZY PARAMErERS

S.A Orlovski*

Mathematical programming (MP) problems form a subclass of decision-making problems in which preferences between alternatives a r e described by means of an objective function defined on the s e t of alter- natives in such a way t h a t greater (or smaller) values of this function correspond t o more preferable alternatives. Values of t h e objective func- tion describe effects from choices of one or o t h e r alternative. In economic problems, for an example, these values may reflect profits obtained using various means of production; in water management.prob- lems they may have t h e meaning of electric power production for various water yelds from a reservoir, etc. The set of feasible alternatives in MP problems is described by means of equations and/or inequalities representing relevant relationships between variables. In any case t h e results of the analysis using given formulation of t h e MP problem depend largely upon how adequately various factors of t h e real system or a

*

^{On leave from the} Computing Center of the USSR Academy of Sciences

process are reflected in t h e description of the objective function and of t h e constraints.

Descriptions of the objective function and of t h e constraints in a

MP

problem include parameters. For an example, in problems of rational water, land and other natural resources allocation such parameters can represent crop produc tivities, requirements for irrigation per unit a r e a of land for different crops, economic parameters like prices for various types of production, labour requirements, etc. The n a t u r e of those parameters depends, of course, on t h e detailization accepted for t h e model representation, and their values a r e considered as data which should exogenously be supplied for the analysis.

Clearly, the values of such parameters depend on multiple factors not included into t h e formulation of the problem. For t h e above example these factors may include nutrient contents of the soil, soil t r e a t m e n t practices, solar activity, t h e s t a t e of the external market, and many oth- ers.

If

trying t o make t h e model more representative of the real system we include t h e corresponding complex dependences into it, t h e n t h e model may become cumbersome a n d analytically unacceptable. More- over, i t can happen t h a t such attempts to increase "the precision" of t h e model will be of no practical value due to t h e impossibility t o measure, or to measure t o a sufficient accuracy the values of newly introduced parameters. On the o t h e r hand, t h e model with some fixed values of its parameters may still be too crude, since these values are often chosen in a quite arbitrary way.

An intermediate and flexible approach may be based on t h e intro- duction into the model t h e means of a more adequate representation of experts' understanding of t h e n a t u r e of t h e parameters in t h e form of fuzzy sets of their possible values. The resultant model, although not taking into account many details of the real system in question is a more adequate representation of the reality than that with more o r less arbi- trarily fixed values of t h e parameters. On this way we obtain a new type of

MP

problems containing fuzzy parameters. And treating such problems requires t h e application of fuzzy-set-theoretic tools in a logically con- sistent manner.

MP and related problems with fuzzy information were extensively analyzed and many papers have been published displaying a variety of formulations and approaches to their analysis (see for instance, Dubois a n d Prade, 1978; Negoita and Sulariu, 1976; Z i m m e ~ a n , 1976;

Luhandjula, 1982; Ostaziewich, 1980; Orlovsky, 1977, 1980). Most of t h e approaches to fuzzy MP problems are based on the staightforward use of 1 t h e intersection of fuzzy s e t s representing goals and constraints and on ihe subsequent maximization of the resultant membership function. This approach although mentioned by Bellrnan and Zadeh, 1970 in their underlying paper is in fact some special case of the methodology sug- gested there.

Here we present a different approach based on a systematic exten- sion of t h e traditional formulation of MP problems to obtain a formula- tion applicable for t h e processing information in t h e form of fuzzy sets.

This approach is based on t h e results described in Orlovski. 1978, 1980 a n d is outlined in Orlovsk, 1981, 1983.

According to this approach two aspects of a fuzzy MP problem a r e of major importance. The first is t h a t whereas in a traditional MP problem t h e objective function represents a complete ordering of alternatives, in a fuzzy MP problem we have only a fuzzy preference relation between alternatives. Due to this fact t h e concept of maximization becomes unde- fined 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 MP problem alternatives can be chosen only on t h e basis of trade-offs among two generally conflicting objectives: achieving greater possible degree of nondominance and g r e a t e r possible degree of feasibility. Both aspects a r e considered in this paper.

2. Problem formulation

We consider t h e following traditional formulation of a mathematical programming

(MP)

problem:

where ( a l , ,..,ap)=E and ( bij

l p x n

=B a r e respectively vector a n d matrix of exogenous parameters.

In the ordinary case when values of parameters

a,

and bij a r e given as numbers the meanings of the inequality signs in the constraints (2) and of the maximization of the performance function (1) a r e well under- stood and are based on our ability of comparing numbers a n d saying for any two of them which is greater o r a t least not smaller t h a n t h e other.

But here we consider the case when values of t h e p a r a m e t e r s a r e described fuzzily by t h e i r respective membership functions which we denote by xi(%), i=1,

....

q , and vij(bij), i=l.

...,

p ; j=l,

...,

n. In this case for any given alternative values of functions

f

and

qj

can also be described only fuzzily a n d the formulation (1)-(2) of o u r problem becomes mathematically meaningless and requires further clarification.

To obtain a mathematically precise formulation of t h e problem in o u r case we should first define how we compare alternatives with e a c h other using fuzzy values of t h e performance function i n t e r m s "greater or equal". More formally, we should extend the natural ordering in the number line onto t h e class of fuzzy numbers or fuzzy subsets of this line.

Second, we should define t h e s e t of those alternatives which in a certain sense "safis,fy" constraints (2) with fuzzy values of parameters bii

2.1.

Mzy

objective function

Let u s consider first the performance function f a n d formulate i t explicitly as a fuzzy performance function p ( z , r ) , z EX, r E R ~ , i.e. a func- tion t h a t gives a fuzzy value for any alternative. To achieve t h a t we can apply what in the fuzzy sets area is traditionally referred t o as t h e exten- sion principle. Let

4,

^i=l. ...,q be some values of t h e parameters; their membership degrees are given by ~ ~ ( 4 ) . i = l , . . . , q . Denote by go the minimum of these values, i.e.

If r 0

=

f (z.a:

....,

a:) is the corresponding value of f for some alternative z € X then it is natural t o accept t h a t this value belongs t o the fuzzy value of t h e performance function for z to a degree not smaller than p O . Using t h i s reasoning we can write the desired fuzzy performance func- tion in the following form:

with

and

x ( E )

=

min xi(ai).

i = l , ....q

For any fixed Z'EX p ( z l , r ) i s t h e membership function of t h e correspond- ing fuzzy evaluation (effectiveness) of alternative z'.

Consider two alternatives z l , z 2 E X and t h e respective fuzzy values of t h e performance function p(z1.r), p(z2.r). Clearly, if p(z1.7) is not worse t h a n ~ ( z ~ T ) to a certain degree t h a n we a r e justified t o consider z l be not worse t h a t z2 to t h e s a m e degree. We define this type of rela- tion between fuzzy values p ( z l , r ) . p(z2,7) (and therefore, between z l , z 2 ) using the extension principle in the following way. A degree r] of

Z r @ z 2 ( ' h o t worse than'' ) i s given by:

Only after having defined this relation (pairwise comparisons between alternatives) we can speak about choosing those alternatives from s e t X* which in some sense if not t h e best a r e not dominated by o t h e r alternatives. Clearly, the fuzziness of relation r](zl,z2) allows us t o speak about alternatives which a r e nondominated only to a certain

*

For the moment we put aside constraints (2) of the original formulation of the problem.

degree. In other words, alternatives can differ in their degrees of non- dominance. Using results from Orlovski, 1976. 1983, we define a degree qND of nondominance of alternative 2 E X as follows:

where q(,;) is t h e relation defined by relationships (3)-(5). Function q N D ( z ) is a description of the fuzzy s e t of nondominated alternatives in s e t

X

with fuzzy binary relation q ( . ..). Using (5) and (6) we obtain the fol- lowing expression for this function: i

Referring now to the original description of the MP problem, we can say t h a t in the fuzzy s e t context (i.e. with given fuzzy values of pararne- t e r s a l ,

...,ap)

we understand t h e "maximization" problem a s that of determining t h e fuzzy s e t of nondominated alternatives

rim.

In concrete problems, however, the determination of a complete explicit description qND(z) of the fuzzy s e t of

ND

alternatives may be dif- ficult and/or n o t necessary. (This situation is somewhat similar t o that i n multiobjective optimization when t h e determination of a complete explicit description of t h e s e t of Pareto optimal alternatives is often not considered.) More realistic and practically important would be t o have a procedure t h a t allows for the determination of nondominated alterna- tives with some prespecified properties. In our case i t would be useful t o have t h e means of determining alternatives having degrees of nondomi- nance not smaller than some desired level a. Formally, this means the determination of (some) alternatives z satisfying

As is shown in Orlovski, 1981 if the fuzzy values

xi(%)

of parameters

q

a r e such t h a t

xi(%)

² a for some

q ,

i=l. ,...,q then any solution to the following MP problem:

satisfies condition

(a),

i.e. is nondominated to a degree not smaller than a.

I t can further be shown (see Orlovski, 1981) t h a t for continuous functions

xi

( a i ) , i=l.

....

q and f (z ,al,

. . .

^, a q ) problem (9) is equivalent to t h e following

MP

problem:

2.2.

Fuzzy

set of feasible alternatives

Let u s now t u r n our attention to constraints (2) of the original for- mulation of t h e problem. As has been mentioned earlier in this paper the question h e r e is to define how we understand t h e feasibility of alterna- tives with respect t o t h e s e constraints. Clearly, with only fuzzy d e s c r i p tion of values of parameters bij in functions

qj

some alternatives can be more feasible than others. In other words, they can differ in their degree of feasibility and we can only consider a fuzzy s e t of feasible alternatives.

Our purpose here is to obtain an explicit description of this fuzzy set by m e a n s of a membership function (which we shall denote by and we shall reason in t h e following way.

Consider one of the constraints j in (2) a n d l e t b; be some values of t h e respective parameters. Their membership degrees in the respective fuzzy s e t s a r e v i i ( b $ ) . Denote by p; the minirrlum of these degrees, i.e.

If some alternative z

EX

satisfies the inequality q j ( z , b i j

,...,

b;j)

s

0,

then we can naturally accept that this alternative satisfies constraint j t o a degree not smaller t h a n 1;. i.e. we may consider t h a t /1IC(z)z@. For convenience, we introduce the following notations:

Using t h e above reasoning we can write the membership function of t h e fuzzy s e t of alternatives satisfying constraint j in the following form:

p ; ( z ) = - sup v j ( b j ) . b j € P j ( z )

To each alternative z E X this function assigns a degree to which this alternative satisfies constraint j.

We can obtain t h e same function in a more formal way using the extension principle first to extend the definition of function

qj

for fuzzy values of p a r a m e t e r s b i j , a n d then t o extend t h e ordering ( g ) on t h e number line to fuzzy numbers:

For any

E E X

function q j ( z , r ) is the corresponding fuzzy value (fuzzy number) of function

qj

for given fuzzy values v i j ( b y ) of parame- t e r s bij

.

Next we consider n u m b e r 0 a s a fuzzy subset of t h e number line with t h e following membership function:

1, for r

=

0, 0, o t h e r w i s e ,

a n d define for any fixed z E X a degree t o which fuzzy number \ki(z,r) is n o t g r e a t e r t h a n k ( r ) . Using the extension principle we obtain:

pF(z)

=

(degree of

+,

^((rr ) c A o ( r ) j

=

Using t h e above relationship for

*,

^{(z ,T ) we have}

p f b )

= SUP =P

min vij(bi,)

= -

^sup vj(b;) y E ~ l y=$,(z.bU

,...,

b _PI - ) i=1,

...,

p _~,EP,(z)

vso

(5. ^Pi(=),

vj(b;) defined as before) which coincides with equation (11).

Now for a fixed z € X we have degrees p;(z). j=l.

....

^{n to which z}

satisfies t h e respective constraints and it is natural to accept t h a t z simultaneously satisfies all of them t o a degree:

or using (11):

Apparently, here we accept t h a t t h e fuzzy s e t of feasible alternatives i s t h e intersection of t h e fuzzy sets of alternatives satisfying t h e respective constraints.

3.

Compromise between nondominance and feasibility

Now when we have introduced explicit descriptions of t h e fuzzy rela- tion 7](.;) allowing for t h e comparison of alternatives with each other and of t h e membership function of t h e fuzzy s e t of feasible alternatives pC(z) we can consider rational choices of alternatives on the basis of t h i s information. Apparently, in making these choices we have two gen- erally conflicting objectives: we would like t o choose an alternative hav- ing g r e a t e r possible degree of nondominance a ("maximization"), and a t t h e s a m e time having g r e a t e r possible degree of feasibility

8.

If we fix

some desired levels of a (nondominance) and @ (feasibility) then using the above notation and results we have t h a t any alternative determined as a solution to t h e following problem:

min - sup v j ( q ) s p j =l,. ..,n b j EPj(z)

h a s a degree of nondominance n o t smaller t h a n a a n d is feasible (satis- fies t h e constraints) t o a degree n o t smaller than @. As c a n be verified, if functions v..(b--) and

qj

a r e continuous with respect to bij this problem

31 21

is equivalent t o t h e following:

If problem (13) h a s a solution for a = @ = 1 then s u c h solution is a n alternative t h a t i s unfuzzily ( t o a degree 1) nondominated (i.e. no other alternative is b e t t e r to a positive degree) and a t t h e same time is unfuz- zily ( @ = I ) feasible. If no solution to (13) exists for a=@=l t h e n t h e analyst o r t h e decision-maker

(DM)

should sacrifice either t h e degree of non- dominance or t h e degree of feasibility (or both) and a t t e m p t t o deter- mine less "ideal" alternatives (with smaller a and/or /?) which agree with his tolerances with regard t o a and @. For any alternative z0 EX deter- mined in this way p ( z O . r ) is t h e corresponding fuzzy value of t h e objec- tive function and pC(zO) i s i t s Feasibility degree.

Clearly, "the most rational" solutions would be those corresponding to Pareto optimal pairs ( a m p ) which in this case can be defined as follows.

A pair ( a 0 , $ ) is Pareto optimal for problem (13) if for any other pair (a,@) such t h a t a > a O , P @ O or a > a O ,

@>Po

problem (13) h a s no solution.

In principle. Pareto optimal a,@ can be determined 'by iterating values of t h e s e levels a n d solving problem (13) a t each iteration step.

More realistic, however, would be to solve this problem for some increas- ing values of

a,p

until t h e s e values together with t h e fuzzy value p ( 2 . r ) of the performance function for the solution t o problem (13) satisfy t h e DM.

Remarks:

1. In formulation (12)-(13) of t h e problem it is assumed t h a t all con- straints a r e of equal importance to t h e DM a n d this fact is reflected by assigning the s a m e m i n i m u m desired level of feasibility @ to all of them.

However, if we would like t o take into account differences in t h e impor- tance of constraints we can specify different desired levels of feasibility for different c o n s t r a i n t s j. In t h a t case t h e -respective constraints will be of t h e form (for (12)):

sup v i ( b j ) n @ j . j = l

...

^{n .}

5 ^uj

⁽²⁾

and (for (13)):

This would m e a n t h a t we t r e a t all t h e constraints separately and do not define t h e fuzzy s e t of feasible alternatives p C ( ~ ) as just t h e intersection of t h e respective fuzzy s e t s ^$(t).

2. If fuzzy s e t s xi(%) and vij(bij) in our problem a r e described in a triangular f o r m ( t h a t is extensively discussed in the c u r r e n t literature on fuzzy s e t s ) t h e n they can analytically be described as follows (for

xi

as an example):

with l i l ( a i ) a n d 1?(ai) being two linear functions having slopes of oppo- site signs. In this case, as can easily be seen, condition x i ( % ) 2 a with W a s 1 in formulation (13) is equivalent t o t h e following two linear con- straints:

4. Concluding remark

We outlined h e r e an approach t o processing information in the form of fuzzy s e t s in problems of choice formulated in t h e mathematical pro- gramming form. The use of fuzzy s e t s for describing information about r e a l systems is a relatively new a r e a a n d m u c h f u r t h e r work is needed in order t o find practically sound methods allowing t o combine t h e fuzzi- ness of human judgements with t h e powerful1 logic a n d 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 mathematical modeling t o t h e analyses of r e a l systems, namely, t h e existing g a p between t h e language used for mathematical models and t h e language used by potential users of those models.

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