Methods of Utility Evaluation in Decision-Making Problems Under Fuzziness and Randomness

Copyright «') IFAC Fuzzy Information Marseille, France, 1983

METHODS OF UTILITY EVALUATION IN DECISION-MAKING PROBLEMS UNDER FUZZINESS AND RANDOMNESS A. N. Borisov and G. V. Merkuryeva Department of Automat Ized Control Systems, Riga Poly technical Institute, 1 Lenin Street, Rlga, Latvian SSR, USSR

Abstract. The probl em of decision- making under fuzziness and randomness is cons idered . The concept of linguistic preference relation is introduc e d allowin g decision- making models to contain the dec ision maker ' s unc ertainty about preferences and hi s opinions on th e s t reng th of these preferences. The L-Iottery is us ed prov iding th e descript ion of undert erm inistic consequences with fuzzy components. The axiomatic definition of util ity function in the decisionmaking problems under fu zz in ess and randomness is given . Methods of linquistic utility eva luatio n are presented . The illustrative exa mpl es are given .

Keywords . Dec ision theory; fuzzy set theory; linguistic lottery; linguistic preference relations; lingui s ti c utility eva luation.

INTRODUCTION App l ication o f th e known models of decis io n rnakin(J llrldpr undetcrministic conseq uences a nd fu zzy desc r iption of cloc ision probl e m e l e m e nt s suc h as c r iter ia l consequ ences estimat cs a nd their probab ilities mcds difficulties in practic e because Ill c thods of linquistic utilit y eva luatioll cion I t e~ i st. These la tter, as a rule, arc assu lll e d to he set up. In th e report SOIllE' methods o f utility eva luation in probI e lll s of decision rnakirt(J ullcler fuzziness and randomn ess are pn~selltcd a nd th e ir a na l ysis is ~J i ven . Some examples of direct, indirec t , deterministic, possibilistic as \Veil as com b ined methods of utility eva lua ti o n are des c rib<:>cI .

The lin g uistic preference relation is defined (Borisov, 198 2a) as a lin g ui s tic variable PREFERENCE =
>.

i'vlethods of th e membership fun c tions con struction for fuz zy sets, which a ll ow to des c ri be lin gu istic preference relat ions numer ica ll y, are cons ider ed in (Borisov, 1982c). Some properties of th e linguistic preference re l ations (such as transitivit y , symmetry, etc . ) are discussed by Borisov (1982a), Efstathiou ( 1980), Zadeh (1977).

BASIC CONCEPTS Linllui s ti c Preference Re lation The dec ision maker I s preference relation is treated as an initial and basic concept, p roviding an opportunity to s tructuriz e and formalize th e dec ision maker I s prefe r e nce sys t em . The fact that fu zzy description of decision problem e l ements l eads to fuzzy preference relations is tak en into account. Lin guistic preference relations are introduced, a llowing decision ma king mod e ls to contain both th e description of th e dec ision maker I S

FIKR-K

uncertainty about preferences and his appro ximate opinions on th e strength of these preferences .

Linguistic Lotterv To describe real - world situations in decision making and construct hypothetical ones under fuzziness and randomness methods of L-Iott eries construction are used.

307

308

A.N. Borisov and G.V. Merkuryeva

Let a random variable X, corresponding to some undeterministic consequence of decision problem assume meanings X. from the 1

set of possible linguistic meanings {X l' •.• , X., .•. , X }, each of them being a fuzzy vadable (X.,r U , X.> and being formalized by 1

X

X.

fuzzy set

1

=

1

UH _.?X

xe -X

(x)/x. The probabi-

i

1 ity of a variable X to assume linguistic

meaning X. is characterized by fuzzy num1

ber P. or linguistic probability of X.: P.EP ,

U

~

1

1

1

0

P ={},P.= [OlIUn.(P)/P. o 1 1 1 PE , J le"" 1 The list of linguistic probability meanings (p , ••• , p., .•• , P ), corresponding to the 1

r

1

AXIOMATIC DEFINITION OF UTILITY Let's formulate the system of postulates providing numerical decision maker's preferences representation and leading to some rule of risky alternatives ranking in decision maki ng problems under fuzziness and randomness. Let's take as a principle the system of axioms of utility function existence in the nonfuzzy case and nonformal recommendations about choosing these axioms (Keeney, 1976; Luce, 1957; Neumann, 1947). The most meaningful from the practical point of view is t he recommendation, which states that each of postulates has to have an intuitively clear meaning and interpretation.

set of possible meanings (X , ••• , X., .•• , 1

1

X ) of a random variable X, is called linr

guistic probability distribution for this random variable, and the variable X is called a linguistic fuzzy variable. By analogy with the nonfuzzy case linguistic lottery, or L-lottery, is defined (Borisov, 1982b) as a linguistic random variable with known linguistic probability distribution and presented by vector L:

Axiomatic definition of utility function in decision making problems under fuzziness and randomness is based on the justice of the following assumptions. Axiom 1. The completeness of the decision maker I s preference system For any two alternatives Ak and Al the decision maker can always set some preference or equivalence relation R :

L= (p , X ; ... ; P.,X.; ••• P ,X). 1

1

1

1

r

r

Lottery L2 with two consequences is an example of linguistic lottery: L (probability Large, profit increment LOW; 2 .. 5 MALL, profit increment probablhty HIGH) • Methods of L-Iotteries construction, where t heir components X. are described by simple 1

or composite fuzzy variables are discussed by Borisov (1982b). Here basic meanings p in flp. (p) are treated as numerical meanings 1

of a fuzzy event probability, when it is caused by consequence X. : 1

Pi =

{.fx

i

(x) f (x) dx,

where f(x) is a function of probability dens it y on numerical domain of consequence definition U X. Let's also note that linguistic lottery definition, introduced by Dubois and Prade (1982), doesn't assume fuzzy descript ion of its components X. and correponds 1

to the decision-maki ng situations, where only consequences probabilities P l' ••. , P., ••. , P are fuzzy. 1

r

Cet I S define some properties of the preference relation R. Let T(5) = 5 U 5 U{5.} , where o

e

1

5 ,5 correspond to the terms: no preferencg, agproximately equivalent; 5£T(5)n 5 U 5 • 1 0 e Let also R = R uR' , where R is an equivae · ebe- . . I ence re 1atlOn, ut R lS lts complement. e The relation R is symmetri c: e R (A , A ) = 5 ~ R (A , A ) = 5 . ek 1 eel k e The relation R is antisymmetric and antie reflexive:

Re (A , A ) lk

Re(A , AI) = 5.;::;;. k 1

Re (A,

A)

= 5 , 0

= 50 •

Axiom 2. Transitivity of decision maker I s preferences The preference relation is transitive (Zadeh, 1977) : R::>RoR or ~ U lJ k lK kJ

S..

(s. n S .),

where 5 .. denotes fuzzy set, describing corlJ responding linguistic preference relation R(A., A.) = 5 .. , 5 .. E: T(5). 1 J lJ lJ I n particular

309

Methods of Utility Evaluation (A.S .. Aj)" (A,S'kAk)~ 1 IJ J J , (A.S·kA ), S'k> k 1 1 1 IJ Jk (A.S A.)" (A.S A ) ~ (A.S A ). leJ Jek le k

Axiom 5. Equivalence. Let L = (P., X.; P. X.) , 1 1 J' J then if X.S X~, then LS (P . , X ~; P ., X.); 1 e 1 ell J J if X. '" X ~, then L '" ( P ., X ~ ; P., X.). 1 1 1 1 J J

S.. ns·

Axiom 3 . Reduction of composite lotteries A composit e lottery

L = (P , A ; ••• ; P ., 1

1

1

A.; .•• ; P , A ), whose components A. are r

1

r

1

simple lotteries of the form A. = (P. , X; 1

IX

P. , Y), is equivalent to the simpl e lottery Iy L' = (P' , X; P' , Y) with consequences X 1 2 and Y. Probabilities of these consequences are calc ulated by the following expressions :

pi1 ~ peA 1XEt) •••

cIl

A.x (f) ••• 1

$

Ar X),

P'2 = peA 1Y(j) • ••

®

A.1 Y~ ...

(f)

Ar Y) -

in accordance with operation rules for fuzzy event probabilities and minimax principle of ext e nsion. Re mark 3.1. If L= (PI' AI; P , A ), then 2 2 t he basic m ea nings of linguistic probability peA 1XEf) A2 X), taking into account the inder e ndc nce of fuzzy events A. and X, are cal-

The axiom of equivalence supposes that alternatives remain equally valuable (or approximately equall y valuable) not only when comparing them alone, but also when substituting t hem into lotteries. Axiom 6. Comparison of fuzzy numbers. The larg est of fuzzy numberstl. and V.

Vmax = {(V., iU CV.))}, 1 v V 1 max

+

1

peA

2

1

).p(X/A ) -

1

membership to this set,

peA A ) p(X/A A )

222112 + P2 Px - P12PxP~ .

12

For the case of lottery L with r consequence we have:

r(A I XG)' •. ®A 1.X (J) ••• IBAr X) i

~

i

j

=

1

1 X

-

i

j

k

where p., p .. , P"k' •.. are interpreted by 1

IJ

1

(v),

is

max interpreted as a degree of relation "V . is 1

greater than V." truth and calcula t ed by the expression: J

Yv

max

(\\)

= sup u. ,u. 1 J

min{yv.(ui),.f'ij(u ), j 1 J

The introduction (Baldwin, 1979) of fuzzy relation R (for e xample, "I s significantly great er") with member ship function fR ( u i ' u} in every particular decision making probl em allows t o take into account and describe the particular decision maker's notions about the essentiality of distinctions u. and u. for as 1

t.... p.p P + L P 'kP P P - •.• IJ X X x " IJ X X i,j, k 1, J r 1 2 r .•. - (-1) P1_? pp ••. p , .•• r x x x

=LP .P

fv

.fR(ui ,u j )} .

+

= rlr~

,..,

in which the degree of fuzzy number's V.

l

X @ A X) = p (A ). p (XI A )

J

1

c ulate d by the following expressions: p (A

J

1

being described by membership functions jl- (u.) and JIv (u.) is a fuzzy set V. 1 . J

IJ

num e rical probabilities of fuzzy events , caus ed by consequences A.,(A.A.),(A.A.A,), ••• 1 1 J 1 J " respectively . Ax iom 4 . Continuity . If XISI.X. and X.S.kX , _ :L.L JJ k SI!E.T(S)nS US , SkE:.T(S)O S U SI' then an J 0 e J 0 L-Iottery L = (P;
j

certaining superiority relation for V. concer1 ning V .• J ~

Axiom 7. Comparison of L-Iotteries. Let 1 I m L = (P., X.; p., X.) and L = (P. , X.; I 1 IJ J m 11 pm, X.), X.S .. X., S.E.T(S)n SUS. J J 1 IJ J IJ 0 e The best is a lottery with the greatest consequence X. probability: ~l

1

.? (Ll 'r Lm)

=

,..m

.f (P i >Pi) •

An effective lottery is described by fuzzy set

L : o

in which

fL o

(L ) is defined as a degree of l

A.N. Borisov and G.V. Merkuryeva

310

-1

-m

t he relation Pi> Pi

truth,

,jlL

-1

(L ) l

=

0

(P . ), and inte rpreted by the d eg -

= jliPi

1

max ree of the decision ma ker I s confidence in preference of Ll concerning Lm • If the assumptions give n above are true, we

come to the following state m e nt. State m e nt 1.

1

seque nces X . exist, such tha t for L-lotte ries 1

I I I Ll = (p 1 ' Xl; ••. ; Pi ' Xi; ••• ; Pr' X r) and

m

=

(p

m m m , X ; ••• ; P . , X .; ••• ; P ,X) 11 11 r r

corr e lation of fu zzy expect ed utilities VI exp ""m d e t erm ines which of the alternatiand V exp ves is p referabl e . The bas i c mear.ings of the utilities are expressed as (1) and membersh ip degrees are cal culated taking into account express ion (1) on the basis of minimax ext e ns io n p rinc ipl e u

exp

=

I . p. U. 1

METHODS OF UTILITY EVALUATION Statement I gives an opportunity to analyse formally alternative decisions unde r risk and fuzzy information. The co mponent of the anal ysis is the process~f dete rmination of cons equenc e s utilitie s V.. The methods of utility 1

1. If preference relat1,ons meet axioms 1 - 7 , t hen fuzz y numbers V., connect e d with con-

L

when determining the superiority relation.

1

1

1: p"lJ u.UJ, + . l ,

-

.'

1,J

r

- •.. - (-1) P 2 1

••• r

1

1,

u1 u

2

J,

p"kU.U,llk lJ 1 J k u • r

e valuation proposed in the given paper are m e thods of empirical utility analysis under fuzziness and known approaches are used (Fishburn, 1967; Keeney , 1976; Kozele t sky, 1979): determination of utilit y on the basis of t he analysis of the c hoices and series of decisions, being made by a s ubj ect in the process of problem solving (indirec t m e thods, axiomatically based) or on the basis of direc t judgements about utilitie s or corr e lations of cons equ e nc es utilities (direc t methods); application in the process o f utility evaluation probabilistic or o nl y deterministic judge m e nts of the decision maker (probabilistic a nd deterministic methods). Let I s give some methods of linguistic utilities e valuation. We shall denote the kind of m e thod by the list T l' T2 ) :

<

T1 - direct (M ) or indirect (M ), 2 1 T2 - probabilistic (P) or deterministic (D).

I1. Comparison of fuzzy estimates of the ex-

v

n ""l of pec t e d utl'1"lUes V~ l I'" , V exp exp , ••• , exp a lternatives L , .•• , L , •.• , L l e ads to 1 n l fuzzy set L o f the most preferabl e alternao ti ve :

Lo = {(LI '.fL (L l )))· o The degree o f the decision maker I s confidence in that alter native Ll is more p r e ferabl e i n compa rison with o ther a lternatives i s calcul a t ed by the foll ow ing formula: .fL (L l ) = sup minf.f"'l(~)"··' f - l (u 2 ),··· o u 1,-)ln V V exp exp

••. , .fvn(un):fR(u1' ••• , u l ' ..• , un)}' exp

Method of Direct Utilit y Evaluatio n < !V1 , D>. 2 We suppose that fr om the set of poss ible con seque nces the decision maker may c hoose three o f the m - the least preferabl e XO, the most preferable X* a nd a n average as to pre-+

f erence, X , which correspo nd to the fol. . " ~o l ow m g utlhty meamngs - the l owest V (about 0), the hig hest V* (say, near 1) and aver age V+ (approx imate l y 0.5) : '""o) - 0 utility (X =V , ,..J

On the basis of the equa tions given above fuzzy restriction R on num e rical meanings g o f utility u and criterial consequences e stimates x is construc t ed : '-0

where .fR(·) is a m embership function for fuz zy r e lation R, describing how es s e ntial are the distinc tions in numerical utilities

""

utility (X·) = V*, .... +) ""+ utility (X = V .

R (x,u) =X g

- 0

xV

-+

-+

+X xV

-'

,.J

+X·xV*

where x and + denote Cartesian produc t and t he union of fuz zy sets.

311

Methods of Utility Evaluation The determination of unknown utility

of a

~ J consequence _ X.J is reduced to the composition

of R and X . : g J

'" V.=XoR J j g

and is made in ac cordance with the following expr es sion:

.fv (u.l j

J

~

,.,

V.

= max min{.rx(xi)'YR (xi,u.ll. x €~ 1 g J

The method of direc t eva luation assumes that m e mbe rship fun c tions for fuz zy sets, describ ing the semantics o f cr iter ial consequences estimates , are known . Method of Linguistic Re lations Evaluation . 2

1 0 1

,w

-.j

X. SX· are determined and the corresponding 1

membership functions for utilities are calcu1 ated taking into account expression (4). For the other consequences X., utilities V. J J are reconstructed according to expression (2) and (3). The joint application of the mentioned methods f or determination of linguistic consequences utilities allows to vary the size of information being r e quired from the d ec ision maker. At t he same time the inte rpolation of utility V. o n the basis of expression (2) and (3) is J probably more correc t in the sense that it is based on a greater amount o f information in co mparison with the algorithms constructe d on the method of dir ec t utility evaluation only .

,.,

Let th e utility of a cons eque n ce X., equa l to 1

Vi with m e mbers hip fun c tion

.f'V. (u)

and

1

lingu istic re l ation S . . between X. and cons e ,., IJ 1 quence X. with unknown utilit y be known: J S .. €. T(S) and is forma lized by fuz zy set with 1J mcmbershir function JL s . . (s) • Ta king into account the int-6rpre tation of basic variable s, th e ratio between p~ssi0e num e rical utilities of conseque nces X., X. is ex 1 J pressed as foll ows ; u. = s · U . ' The n the reI J construc tion of membership funstion of the unknow n utilit y o f conseque nce X. is made in J acc ordance with the fo ll ow in g expr e ssion: fys (s), .fv,.,J. (u.l=maxmin J s .cU " S IJ ~

fUV,",( S .u»). J

J.

1

(4) If we tr ea t

{Us (s), where s = u/u. as a .. J IJ no n ev ident representation of fuzzy relation R (u. , u.l, the n the la tte r express ion is re g 1 J duced t o the form: J

tU

J

v- . (u J ) J

,.,

= V

0

i

S . .• 1J

The Me thod of Choice Analysis
P>.

This m e thod is an ext e nsio n o f the method of s t a ndard game (Fishburn, 196 7) for the case o f fuz zy d e cision analysis . The r e are two modifi cations of the m e thod, depending on the t ype o f information, be ing r equired from the decis io n make r: 1) The d ec ision maker is asked to set suc h a m eaning of linguistic probability pO, which J crea t e s for him approximately equivalent d e cis ion-making situations and the corresponding . s, caused by lottery L. = (0 ga m P., X 0 ; p·,X* ) -J J J J a nd cons e que nc e X . with unknown utilit y; J 2) Th e decision m ake r is as ked to set a pref erence r e lation, S between lotte ry L. with ~o __ J sgn~quences X , X· an9.-fixed probabilities p . , P . * and consequence X . with unknown utilitJ. J I n both cases the d e t e rminatio n of unknown utilit y '1 . of consequence X. includes: J J 1) d e t e rmination of the expec t ed linguistic utility of lottery L . on the basis of State ment I; J ..J 2) r econstruction of utilit y V . taking into consideration utilit y of lotte r y LJ and prefe r e nJ

The method of linguistic r e lations eval uation ass um e s the linguistic preference variable to b e given . Combined JVl e thod of the Kind

N

X . linguistic relations S such as X S X. or

.

The decision make r fix es up two charact e r istic consequence s - the l~ast preferable )(0 and the most p r e ferable X *, corresponding to t he utility meanings Vo and '1*. For some

ce r e lation S on the basis of e xpression (4). The method of choice analysis (as well as other indirect methods of utility evaluation) is based on the accepted axiomatic concerning the decision maker I s preference system and in t hat sense is more grounded. At the same time from the computational point of view it turns out more complicated and more difficult in co mparison with other above mentioned methods.

A.N. Borisov and G.V. Merkuryeva

312

Application of the method of choice analysis is worth while in the cases, when in practice a number of decisions are known - alternative choice, made by an individual in the given class of problems.

EXAMPLE Let I s give an example, which illustrates construction of utility function with the help of applied program package, METHOD, which is written in FORTRAN and realizes the above ment ioned methods of utility evaluation in decision making problems under fuzziness. Let I s show a fragment of the dialogue between t he decision maker (DM) and a computer (C) when determining utility by the method of linguistic relations evaluation: C. Tell a consequence, the least preferable for you DM.PROFIT INCREMENT. WW C. Tell a consequence, the most preferable for you DM.PROFIT INCREMENT. HIGH C. Tell a consequence, which utility it IS necessary to determine DM. PROFIT INCREMENT. LARGE C. Tell the preference of the given consequence concerning the least preferable consequence DM.PREFERENCE. CONSIDERABLY PREFERABLE C. Tell the preference of the most preferable consequence concerning the given consequence DM. PREFERENCE. SLIGHTLY PREFERABLE C. Give the next meaning of the consequence which utility it I S necessary to determine DM.PROFIT INCREMENT. AVERAGE C. Tell the preference of the given consequence concerning the least preferable consequence DM.PREFERENCE. SLIGHTLY PREFERABLE The membership functions of linguistic utilities f or the consequences LARGE PROFIT INCREMENT and AVERAGE PROFIT INCREMENT, obt ained with the help of the method of linguistic relations evaluation, are approximated by the terms - AVERAGE and SMALL.

CONCLUSION The methods of linguistic utility evaluation described in this paper give an opportunity to determine utility function also in the decision making problems under fuzziness and randomness with multidimensional consequences.

REFERENCES Baldwin, J.F., and N.C. Guild (1979). Comparison of fuzzy sets on the same decision space. Fuzzy Sets & Systems, ~, 213-231. Borisov, A.N. (Ed.) (1982a). Decision Making Models Based on Linguistic Variable. Zinatne, Riga. 256 pp. Borisov, A.N., and G.V. Merkuryeva (1982b). Linguistic Lotteries - Construction and Properties. Busefal, II, 39-46. Borisov, A.N., and G.V. Merkuryeva (1982c). Linguistic Preference Relations Modeling in the Decision Making Problems. In R. Trapple, N. P., Findler, and W. Korn (Eds.), Progress in Cybernetics and Systems Research, vol. XI. Hemisphere Publishing Corporation, Washington. pp. 269-274. Dubois, D. , and H. Prade (1982). The use of fuzzy numbers in decision analysis. In M.M. Gupta, and E. Sanchez (Eds.), Fuzzy Information and Decision Processes. North-Holland. Efstathiou, J., and R. Tong (1980). Ranking Fuzzy Sets Using Linguistic Preference Relations. Proc. 10th Int. Symp. Multiple-Valued Logic. Evanston, 1980, N.Y. pp. 137-142. Fishburn, P.C. (1967). Methods of estimating additive utilities. Management Science, ..!l, 435-453. Keeney, R. 1., and H. Raiffa (1976). Decis ions with Multiple Objectives: Preferences and Value Tradeoffs. Wiley & Sons, New York. 569 pp. Kozeletsky, Y. (1979). Psychological Decision Theory. Progress, Moscow. 504 pp. Kuzmin, V. B. (1982). Construction of Group Decisions in the Spaces of Nonfuzzy and Fuzzy Binary Relations. Nauka, Moscow. 168 pp. Luce, R.D., and H. Raiffa (1957). Games and Decisions. Wiley, New York. Neumann J. , and O. Morgenstern (1947). Theory of Games and Economical Behavi~ 2 nd. ed. Princeton University Press, Princeton. Zadeh, L.A. (1977). Linguistic Characterization of Preference Relations as a Basis of Choice in Social Systems. Mem. No. UCB/ERL M77 /24. University of California, Berkeley. 37 pp.